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tomekkorbak

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[ "It’s that time again…Time for giving to those we care about. ", "Gifts that say “I love you”, “Thank You” and “Wish you well”. ", "How many of really need a new pair slippers or a picture frame? ", "A new blender or some kitchen gadget?", "\nWhat we really need is relaxation…Yes, time to ourselves to refresh and renew. ", "And the perfect place is here at Pret-a-Porter Salon & Spa..Give a gift certificate this year for a full body Deep Tissue massage, a European Facial, or a Spa Manicure and Pedicure.", "\nThis year give the gift of relaxation..It will be one that they won’t forget!", "\n\nContact details\n\nFind us\n\nAbout Prêt-á-Porter Salon & Spa\n\nPrêt-á-Porter Salon & Spa opened in 1998. ", "Owned by Roz and Dan Oserin, it is located in the downtown area of La Mesa, just 20 minutes from downtown San Diego. ", "Prêt-á-Porter Salon & Spa, are proud to partner up with Aveda. ", "Aveda is a leader in the beauty industry as well as being one of the top 10 green companies in the world." ]

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[ "/**\n * oscP5oscArgument by andreas schlegel\n * example shows how to parse incoming osc messages \"by hand\".", "\n * it is recommended to take a look at oscP5plug for an alternative way to parse messages.", "\n * oscP5 website at http://www.sojamo.de/oscP5\n */\n\nimport oscP5.*;", "\nimport netP5.*;", "\n\nOscP5 oscP5;\nNetAddress myRemoteLocation;\n\nint a;\nfloat b;\n\nvoid setup() {\n background(0);\n\n oscP5 = new OscP5(this,12000);\n \n \n// myRemoteLocation = new NetAddress(\"192.168.0.177\",10000);\n myRemoteLocation = new NetAddress(\"224.0.0.0\",10000); //multicast address\n \n \n a=0;\n b=0.0;\n \n}\n\nvoid draw() {\n \n sender();\n \n delay(1000);\n \n}\n\nvoid sender(){\n a+=1;\n b+=0.1;\n if(a>999) a=0;\n if(b>999.0) b=0.0;\n \n OscMessage myMessage = new OscMessage(\"/ard/aaa\");\n \n myMessage.add(a); \n myMessage.add(b); \n myMessage.add(\"some text\");\n oscP5.send(myMessage, myRemoteLocation);\n}\n\n\n\nvoid oscEvent(OscMessage theOscMessage) {\n print(\"### received an osc message.\");", "\n \n theOscMessage.print();\n}\n" ]

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[ "Michael Keane is wank, and has been since about the 3rd game of the season, 6'4 and cant win a header and has the pace of a slug. ", "Williams gets the most shit off our fans but he's been as bad, Jagielka is one of the worst leaders in recent memory too. ", "Crap." ]

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[ "Georgia couple sentenced to prison for racial threats\n\nA Georgia couple will be spending years behind bars after yelling racial slurs while driving with Confederate flags at an African-American child's birthday party." ]

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[ "Stream order\n\nThe stream order or waterbody order is a positive whole number used in geomorphology and hydrology to indicate the level of branching in a river system.", "\n\nThere are various approaches to the topological ordering of rivers or sections of rivers based on their distance from the source (\"top down\") or from the confluence (the point where two rivers merge) or river mouth (\"bottom up\"), and their hierarchical position within the river system. ", "As terminology, the words \"stream\" and \"branch\" tend to be used rather than \"river\".", "\n\nClassic stream order \n\nThe classic stream order, also called Hack's stream order or Gravelius' stream order, is a \"bottom up\" hierarchy that allocates the number \"1\" to the river with its mouth at the sea (the main stem). ", "Its tributaries are given a number one greater than that of the river or stream into which they discharge. ", "So, for example, all immediate tributaries of the main stem are given the number \"2\". ", "Tributaries emptying into a \"2\" are given the number \"3\" and so on.", "\n\nThis type of stream order indicates the river's place in the network. ", "It is suitable for general cartographic purposes, but can pose problems because at each confluence, a decision must be made about which of the two branches is a continuation of the main channel, and whether the main channel has its source at the confluence of two other smaller streams. ", "The first order stream is the one which, at each confluence, has the greatest volumetric flow, usually reflecting the long-standing naming of rivers. ", "Associated with this stream order system was the quest by geographers of the 19th century to find the \"true\" source of a river. ", "In the course of this work, other criteria were discussed to enable the main stream to be defined. ", "In addition to measuring the length of rivers (the distance between the farthest source and the mouth) and the size of the various catchments, geographers searched for the stream which deviated least at the actual confluence, as well as taking into account the successive names of rivers and their tributaries, such as the Rhine and the Aare or the Elbe and the Vltava.", "\n\nStrahler stream order\n\nAccording to the \"top down\" system devised by Strahler, rivers of the first order are the outermost tributaries. ", "If two streams of the same order merge, the resulting stream is given a number that is one higher. ", "If two rivers with different stream orders merge, the resulting stream is given the higher of the two numbers.", "\n\nThe Strahler order is designed to reflect the morphology of a catchment and forms the basis of important hydrographical indicators of its structure, such as its bifurcation ratio, drainage density and frequency. ", "Its basis is the watershed line of the catchment. ", "It is, however, scale-dependent. ", "The larger the map scale, the more orders of stream may be revealed. ", "A general lower boundary for the definition of a \"stream\" may be set by defining its width at the mouth or, referencing a map, by limiting its extent. ", "The system itself is also applicable for other small-scale structures outside of hydrology.", "\n\nShreve stream order \n\nThe Shreve system also gives the outermost tributaries the number \"1\". ", "Unlike the Strahler method, at a confluence the two numbers are added together.", "\n\nShreve stream order is preferred in hydrodynamics: it sums the number of sources in each catchment above a stream gauge or outflow, and correlates roughly to the discharge volumes and pollution levels. ", "Like the Strahler method, it is dependent on the precision of the sources included, but less dependent on map scale. ", "It can be made relatively scale-independent by using suitable normalization and is then largely independent of an exact knowledge of the upper and lower courses of an area.", "\n\nHorton Stream Order and Other systems \nOther systems include the Horton stream order, an early top down system devised by Robert E. Horton, and the topological stream order system, which is \"a bottom up\" system, and where the stream order number increases by one at every confluence.", "\n\nClassic Stream Order vs Horton & Strahler Methods \nClassical or Topological ordering systems are assigned a dimensionless numerical order of \"one\" starting at the mouth of a stream, which is its lowest elevation point. ", "The vector order then increases as it traces upstream and converges with other smaller streams, resulting in a correlation of higher order numbers to more highly elevated headwaters. ", "Horton proposed to establish a reversal of this order. ", "Horton's 1947 research report established a stream ordering method based on vector geometry. ", "Five years later, in 1952, Arthur Strahler proposed a modification to Horton's method. ", "Both Horton's and Strahler's methods established the assignment of the lowest order, number 1, starting at the river's headwater, which is the highest elevation point. ", "Whereas classical order number assignment correlates to height and elevation and traces upstream, Horton and Strahler's stream ordering methods correlate to gravity flow and trace downstream.", "\n\nBoth Horton's and Strahler's stream ordering methods rely on principles of vector point-line geometry. ", "Horton's and Strahler's rules form the basis of programming algorithms which interpret map data as queried by Geographic Information Systems.", "\n\nUsage \nThe classic use of stream order is in general hydrological cartography. ", "Stream order systems are also important for the systematic mapping of a river system, enabling the clear labelling and ordering of streams.", "\n\nThe Strahler and Shreve methods are particularly valuable for the modelling and morphometric analysis of river systems, because they define each section of a river. ", "That allows the network to be separated at each gauge or outflow into upstream and downstream regimes, and for these points to be classified. ", "These systems are also used as a basis for modelling the water budget using storage models or time-related, precipitation-outflow models and the like.", "\n\nIn the GIS-based earth sciences these two models are used because they show the graphical extent of a river object.", "\n\nResearch activity following Strahler's 1952 report has focused on solving some challenges when converting two-dimensional maps into three-dimensional vector models. ", "One challenge has been to convert rasterized pixel images of streams into vector format. ", "Another problem has been that map scaling adjustments when using GIS may alter the stream classification by a factor or one or two orders. ", "Depending on the scale of the GIS map, some fine detail of the tree structure of a river system can be lost.", "\n\nResearch efforts by private industry, universities and federal government agencies such as the EPA and USGS have combined resources and aligned focus to study these and other challenges. ", "The principal intent is to standardize software and programming rules so GIS data is consistently reliable at any map scale. ", "To this end, both the EPA and USGS have spearheaded standardization efforts, culminating in the creation of The National Map. ", "Both federal agencies, as well as leading private industry software companies have adopted Horton's and Strahler's stream order vector principles as the basis for coding logic rules built into the standardized National Map software.", "\n\nSee also \n Waterbody number – an overview\n GIS and Hydrology -\n The National Map\n\nReferences\n\nLiterature \n Drwal, J. Wykształcenie i organizacja sieci hydrograficznej jako podstawa oceny struktury odpływu na terenach młodoglacjalnych, Rozprawy i monografie, Gdansk, 1982, 130 pp (in Polish)\n Hack, J. Studies of longitudinal stream profiles in Virginia and Maryland, U.S. Geological Survey Professional Paper, 1957, 294-B \n Horton, R. E., Erosional development of streams and their drainage basins: hydro-physical approach to quantitative morphology, Geological Society of America Bulletin 56 (3): 275-370, 1945\n Koschitzki, Thomas. ", "GIS-basierte, automatische Erfassung natürlicher Fließgewässerhierarchien und ihre Abbildung in Datenbanken, beispielhaft dargestellt am Einzugsgebiet der Salza. ", "Dissertation, Martin-Luther-Universität Halle-Wittenberg, Halle (Saale), 2004, URN (NBN) urn:nbn:de:gbv:3-000007179 (Weblink, archive server DNB)\n Scheidegger A. E., (1966), Statistical Description of River Networks. ", "Water Resour. ", "Res., ", "2(4): 785-790 \n Shreve, R., (1966), Statistical Law of Stream Numbers, J. Geol., ", "74, 17-37. ", "\n Strahler, A.N. Dynamic basis of geomorphology. ", "In: Geological Society of America Bulletin 63/1952, pp.", " 923–938.", "\n Strahler, A.N. Quantitative analysis of watershed geomorphology. ", "Transactions of the American Geophysical Union. ", "1957; 38(6), pp. ", "913-920.", "\n Strahler, A.N. \"Quantitative geomorphology of drainage basins and channel networks.\" ", "Chow, V.T., Editor. ", "Handbook of Applied Hydrology. ", "New York: McGraw-Hill; 1964; pp. ", "4-39, 4-76.", "\n Shreve, R. Statistical law of stream numbers. ", "In: Journal of Geology 74/1966. ", "\n Weishar, Lee L. Development of Marsh Hydrogeomorphology and Marsh Vegetation with a Salt Hay Farm Wetland Restoration Site. ", "Retrieved 16 Apr 2017.", "\n Woldenberg, M. J., (1967), Geography and properties of surfaces, Harvard Papers in Theoretical Geography, 1: 95-189. ", "\n Rivertool'' – Extension für ArcView, usages documentation, Office of Applied Hydrology (Weblink, gis-tools.de)\n\nExternal links \n International Glossary of Hydrology (pdf file; 1.24 MB)\n \n\nCategory:Hydrology\nCategory:Limnology" ]

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[ "Kazakhstan is bracing to revive its tungsten mining industry with a helping hand from China.", "\n\nAlmas Aydarov, deputy governor of the Karaganda region, on January 17 told business news website Atameken Business that Xiamen Tungsten, which is based in China’s southeastern Fujian province, will spend $750 million on building infrastructure to mine and process the rare metal.", "\n\nThe start of operations is slated for 2023 and the main buyers of the product will be in China, which currently accounts for 60 percent of the global tungsten consumption. ", "Other potential markets include the United States, Germany and France.", "\n\nTungsten is commonly used as an alloy with other metals to give them additional strength. ", "Other applications are as components in light bulbs and other equipment in which its high heat-resistant properties are useful.", "\n\nA preliminary agreement on the project was reached in November, when Prime Minister Bakytzhan Sagintayev was visiting China.", "\n\nAydarov said Xiamen Tungsten intends to buy out Tau-Ken Samruk, the company that holds the tungsten concessions, from the Samruk-Kazyna sovereign fund.", "\n\nA technical survey carried out by Xiamen Tungsten found that one of the sites that will be developed — Upper Kayrakty — is the second-largest deposit of tungsten in the world. ", "The field was first unearthed in Soviet times, but development was halted in the wake of the collapse of the Soviet Union.", "\n\nAccording to Aydarov, this undertaking will put Kazakhstan in a position to produce up to 12,500 tons of tungsten annually and provide work for 2,000 people." ]

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[ "In recent years, various cigarettes with a desired property imparted are known. ", "The wrapping paper of such cigarettes has a band for imparting a property, and the band is formed by applying one of materials, which vary in the property to be imparted, onto the wrapping paper.", "\nFor example, a cigarette disclosed in PCT Application Published Japanese Translation No. ", "2001-509366 has a low ignition propensity, and this low ignition propensity is imparted by a plurality of bands. ", "The bands are formed on the wrapping paper, at predetermined intervals along the axial direction of the cigarette.", "\nA cigarette disclosed in the pamphlet of International Publication No. ", "01/84969 has a property of improving burning by suppressing the production of an undesired substance or aldehyde at the time of ignition. ", "This burning-improvement property is imparted by a single band. ", "Specifically, the single band is formed on the wrapping paper to be located at the distal end (ignition end) of the cigarette.", "\nFurther, a cigarette may have a band for improving the taste and flavor thereof, and such band is also formed on the wrapping paper.", "\nIn order for a cigarette as mentioned above to fully show its property, it is desirable that a band be formed on the wrapping paper accurately. ", "Particularly in the case of the cigarette disclosed in the above-mentioned pamphlet, unless the band is accurately located at the distal end of the cigarette, the band cannot impart the desired burning-improvement property.", "\nHowever, in making cigarettes of this type, it is very difficult to locate the band accurately at the distal end of a cigarette. ", "Specifically, the common cigarette making machine comprises a garniture tape for making a web of wrapping paper travel at a fixed speed, a wrapping section, and a cutting section. ", "While passing through the wrapping section, cut tobacco on the traveling web is continuously wrapped in the web, and the wrapping section continuously delivers a resultant tobacco rod to the cutting section. ", "Then, while passing through the cutting section, the tobacco rod is cut to a predetermined length, so that individual cigarettes are obtained.", "\nIf the bands are formed on the web at predetermined intervals in advance, and the garniture tape and the web are made to travel in an integrated manner, the common cigarette making machine can make cigarettes having an above-mention band, accurately.", "\nWhile the tobacco rod is being formed, however, if a slip occurs between the web and the garniture tape, no matter how slight the slip is, the slip causes a band for a cigarette to displace from its desired position, so that the band cannot be accurately located at the distal end of a cigarette.", "\nIn order to obviate this problem, it is thinkable, in the cigarette making machine, to relatively change the traveling speed of the garniture tape with respect to the timing of cutting the tobacco rod, or in other words, change the advance phase of each band in traveling relative to the timing of cutting the tobacco rod.", "\nHowever, in the cigarette making machine, high-speed and stable manufacturing of cigarettes is ensured by keeping the traveling speed of the garniture tape and the timing of cutting the tobacco rod constant. ", "Therefore, it is not realistic to relatively change the traveling speed of the garniture tape or with respect to the timing of cutting the tobacco rod as mentioned above." ]

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[ "fxaudio\n\nFxaudio - Royalty Free SFX\n\nEnvato marketplace authors –\nIf you would like to use any of my watermarked sounds for any of your projects please do not hesitate in doing so. ", "Please contact me to tell me where you have used my work:)" ]

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[ "The authors confirm that all data underlying the findings are fully available without restriction. ", "All relevant data are within the paper and its Supporting Information files.", "\n\nIntroduction {#s1}\n============\n\n*Paulownia*, one of the fastest growing trees in the world, is native to China. ", "In recent years, it has been introduced in many other countries [@pone.0112533-Jrgensen1]. ", "Due to the biological and wood properties of this tree, it is ideal for use in house construction, pulp and paper, furniture, farm implements and handicrafts [@pone.0112533-Lpez1], even as an intercropping species in modern agroforestry and environmental protection [@pone.0112533-Hiruki1], [@pone.0112533-Stankovi1].", "\n\nPaulownia witches\\' broom (PaWB), the most destructive infectious disease of *Paulownia*, is caused by phytoplasma of Aster Yellows group '*Candidatus* Phytoplasma asteri' [@pone.0112533-Lee1]. ", "Paulownia trees with this disease show numerous morphological symptoms, including axillary bud germination, shorter internodes and smaller leaf etiolation, resulting in significant decline in vigor and growth of the tree, even premature death of the tree [@pone.0112533-Hiruki1]. ", "Since the 1970s, abundant of researches about the way and route of pathogen transmission [@pone.0112533-Jin1], the prevention and control of insect vectors [@pone.0112533-Cao1], [@pone.0112533-Fan1], and physiological and biochemical variation of Paulownia during the occurrence of PaWB were carried out, and several metabolic pathways related to PaWB were reported by Liu et al. [", "@pone.0112533-Liu1] and Mou et al. [", "@pone.0112533-Mou1]. ", "Although these researches are helpful to understand the interaction of Paulownia and phytoplasma, the molecular mechanism of PaWB occurrence is still remain elusive.", "\n\nDNA methylation is a widespread epigenetic modification, mainly occurs at gene promoter, or transcribed regions [@pone.0112533-Zilberman1]--[@pone.0112533-Chekanova1], playing an important role in regulation of gene expression mediating variation of plant morphology [@pone.0112533-Mason1]. ", "For example, flower abnormalities caused by the tomato stolbur phytoplasma were correlated with gene-specific demethylation [@pone.0112533-Pracros1]. ", "In Paulownia, previous studies have shown that the global DNA methylation level of PaWB seedlings (PS) was lower than that of healthy seedlings (HS) [@pone.0112533-Li1]. ", "MMS, a DNA methylating agent, can modify guanine to 7-methylguanine and adenine to 3-methyladenine, and increase the methylation level of 5-cytosine [@pone.0112533-Yao1]. ", "Our previous results showed that the PS could recover a healthy morphology by treatment with suitable concentration of MMS, in which the phytoplasma could be removed [@pone.0112533-Fan1], [@pone.0112533-Zhai1], [@pone.0112533-Cao2]. ", "However, the relationship between morphological changes of PaWB seedlings and DNA methylation has not been studied.", "\n\nAmplified fragment length polymorphism (AFLP) and methylation-sensitive amplification polymorphism (MSAP) are efficient and reliable methods to detect DNA polymorphism and DNA methylation, respectively [@pone.0112533-Wu1], [@pone.0112533-Dong1]. ", "Here, with these two approaches, the variations of DNA polymorphisms and DNA methylation in PS, the ones treated with MMS and HS were investigated in order to reveal the genes associated with PaWB. ", "The results will provide new insights for further studies into the mechanism of PaWB.", "\n\nMaterials and Methods {#s2}\n=====================\n\nPlant materials {#s2a}\n---------------\n\nHealthy *Paulownia fortunei* and PaWB tissue cultured seedlings were obtained from the Institute of Paulownia, Henan Agricultural University, Zhengzhou, China. ", "The two types of tissue cultured seedlings were first cultivated on 1/2 Murashige - Skoog (MS) medium [@pone.0112533-Murashige1] for 30 days, before uniform terminal buds of about 1.5 cm in length from the PS were transferred into 100 mL flasks containing 1/2 MS medium (40 mL) including 25 mg·L^−1^ sucrose and 8 mg·L^−1^ agar (Sangon, Shanghai, China) with 0 mg·L^−1^MMS, 20 mg·L^−1^MMS (PS-20), 60 mg·L^−1^MMS (PS-60), 100 mg·L^−1^MMS (PS-100). ", "The terminal buds of the HS were transferred into 1/2 MS medium without MMS. ", "For each treatment, 60 terminal buds were planted into 20 flasks, each treatment was performed in triplicate. ", "All samples were cultured initially at 20°C in the dark for 5 days. ", "Thereafter, they were transplanted at 25±2°C and a light intensity of 130 µmol·m^−2^·s^−1^ with a 14∶10 h (light/dark) photoperiod. ", "The method of morphological observation was performed according to Fan et al. [", "@pone.0112533-Fan1]. ", "Thirty days after the beginning of transplantation, terminal buds of 1.5 cm in length, growing in consistent condition, were sheared from the different seedlings, then immediately frozen in liquid nitrogen and stored at −80°C.", "\n\nNucleic acid extraction {#s2b}\n-----------------------\n\nTotal DNAs were extracted from the terminal buds of different samples according to the cetyl trimethylammonium bromide (Beijing Chemical Co., Beijing, China) method, as described by Zhang et al. [", "@pone.0112533-Zhang1], RNase (Invitrogen, Carlsbad, CA, USA) was used to render the DNA free of genomic RNA contamination. ", "Total RNAs were extracted following by an Aidlab total RNA extraction kit (Aidlab, Beijing, China). ", "RNase free - DNase I (Invitrogen) was used to render the RNA free of genomic DNA contamination. ", "The DNAs and RNAs were assessed with NanoDrop 2000 (Thermo Scientific, Wilmington, DE, USA).", "\n\nPaWB phytoplasma detection {#s2c}\n--------------------------\n\nPaWB phytoplasma was detected by nested-PCR as described by Lee et al. [", "@pone.0112533-Lee2]. ", "The PCR procedure and the method of agarose gel electrophoresis were performed by Fan et al. [", "@pone.0112533-Fan1].", "\n\nAFLP and MSAP analysis {#s2d}\n----------------------\n\nThe AFLP digestion reaction comprised 3 U *Pst*I and 3 U *Mse*I (Li-COR, Co., Lincoln, NE, USA), the pre-amplification and selective amplification reaction conditions and the method of electrophoresis were adapted from Cao et al. [", "@pone.0112533-Cao3]. ", "The AFLP adapter and selective amplification primer sequences are listed in [Table S1](#pone.0112533.s002){ref-type=\"supplementary-material\"}.", "\n\nThe MSAP experiment comprised two digestion reactions, the first digestion reaction included 16 U of *Eco*RI (TaKaRa, Dalian, China) plus 10 U of *Msp*I (TaKaRa), the second digestion reaction was the same as the first digestion except the *Hpa*II (TaKaRa) instead of *Msp*I. The MSAP pre-amplification and selective amplification reaction conditions and the method of electrophoresis were followed by Cao et al. [", "@pone.0112533-Cao4]. ", "The MSAP adapter and selective amplification primer sequences are listed in [Table S2](#pone.0112533.s003){ref-type=\"supplementary-material\"}.", "\n\nData analysis {#s2e}\n-------------\n\nAfter silver staining, only clear and reproducible bands were scored, where the presence of a band was scored as \"1\" and the absence was scored as \"0\". ", "For the MSAP analysis, the bands were scored according to the presence or absence of the bands in the products of *Eco*RI/*Hpa*II (H) and *Eco*RI/*Msp*I (M) digestions in different samples, according to these bands on the electrophoresis gels, the DNA methylation could be divided into three classes: class I indicated no methylation (the bands were present in both H and M), class II presented DNA hemi-methylation (the bands were present in H but absent in M), class III showed the DNA fully methylation (the bands were absent in H but present in M). ", "Compared with the bands of PS, the DNA methylation patterns of PS treated with MMS or HS were classified into DNA methylation polymorphism and monomorphism. ", "The DNA methylation polymorphisms included type A (DNA methylation), type B (DNA demethylation) and type C (uncertain DNA methylation). ", "Among them, A~1~ and A~2~ were regarded as DNA de novo methylation (the bands were present in both H and M in PS, but only in H or M in the ones treated with MMS or HS), A~3~ and A~4~ were regarded as DNA hypermethylation (the bands were present only in H or M in PS, but absent in both H and M in the ones treated with MMS or HS). ", "Type B (B~l~, B~2~, B~3~ and B~4~) showed DNA demethylation, the bands were the opposite to type A. Type C represented uncertain DNA methylation (the DNA methylation bands could not be determined between PS and the ones treated with MMS or HS). ", "Type D (D~1~, D~2~ and D~3~) represented monomorphism (the status of the bands in M and H were the same in PS and the ones treated with MMS or HS). ", "The statistical formulates used to score the bands were as follows: total DNA methylation level (%)  =  \\[(class II+class III)/(class I+class II+class III)\\]×100; DNA methylation polymorphism (%)  =  \\[(A+B+C)/(A+B+C+D)\\]×100; DNA methylation monomorphism (%)  =  \\[D/(A+B+C+D)\\]×100.", "\n\nDNA methylation patterns in different seedlings {#s2f}\n-----------------------------------------------\n\nIn order to identify the DNA methylation patterns related to PaWB, we compared the DNA methylation patterns in different seedlings (the seedlings in different morphology and the seedlings in the same morphology) ([Figure 1](#pone-0112533-g001){ref-type=\"fig\"}). ", "First, we compared the DNA methylation patterns in different morphological seedlings, in the PS vs. HS comparison, the factors for different DNA methylation patterns referred to PaWB and plant development difference (PDD); In the PS-60 vs. PS-20 comparison, the factors referred to PaWB, PDD and MMS treatments difference (MMST). ", "In order to depart these factors from PaWB, we further picked out the same DNA methylation patterns in these two comparisons results. ", "Obviously, the DNA methylation patterns involved in MMST were ruled out, and the DNA methylation patterns involved in PaWB and PDD were reserved. ", "Second, we compared the DNA methylation patterns in same morphological seedlings, in the PS vs. PS-20 comparisons, the factors for same DNA methylation patterns involved in PaWB and PDD, in the PS-60 vs. HS comparisons, the factor only involved in PDD. ", "In order to rule out the interference of PDD from the DNA methylation related to PaWB, we further reserved the different DNA methylation pattern in these two comparisons results, it was clear that the DNA methylation patterns involved in PDD were ruled out, and PaWB was reserved. ", "At last, the same DNA methylations patterns related to PaWB were obtained from the first and second comparisons.", "\n\n{#pone-0112533-g001}\n\nIsolation and sequencing of polymorphic methylated fragments {#s2g}\n------------------------------------------------------------\n\nClear and reproducible bands from DNA methylation patterns related to PaWB were cut carefully with a clean blade and recovered using a UNIQ-10 Column DNA Gel Extraction Kit (Sangon, Shanghai, China), following the manufacturer\\'s instructions. ", "The DNA fragments were reamplified with the same selective primer combinations. ", "The purified DNA was ligated into the pMD18-T easy vector (TaKaRa) and sequenced. ", "The sequences obtained were analyzed using the Blastx programs at the NCBI website (<http://www.ncbi.nlm.nih.gov/>).", "\n\nQuantitative Real Time-PCR (qRT-PCR) analysis {#s2h}\n---------------------------------------------\n\nThe expressions of candidate genes were determined using qRT-PCR. ", "First-strand cDNA was synthesized using an iScript cDNA Synthesis Kit (Bio-Rad, Hercules, CA, USA), according to the manufacturer\\'s instructions. ", "The PCR reaction contained SsoFast Supermix 10 µL (Bio-Rad), forward primer 0.4 µM (Sangon), reverse primer 0.4 µM (Sangon) and cDNA 1 µL, in a total volume of 20 µL. The qRT-PCR reaction mixture was run on a CFX96TM Real-Time PCR Detection System (Bio-Rad), starting with 95.0°C, 1 min; Then 40 cycles of 95.0°C for 10 s and 55.0°C for 15 s. 18SrRNA served as the internal reference gene. ", "The results were analyzed using the 2^−ΔΔCt^ method [@pone.0112533-Livak1], each qRT-PCR analysis was performed in triplicate. ", "The primers used for the quantification of gene expression are listed in [Table S3](#pone.0112533.s004){ref-type=\"supplementary-material\"}. ", "Statistical analysis was performed using SPASS 19.0 (SPASS, Inc., Chicago, IL, USA).", "\n\nResults {#s3}\n=======\n\nMorphological changes of different seedlings of the *P. fortunei* plants {#s3a}\n------------------------------------------------------------------------\n\nMorphological changes of PS showed that the seedlings infected phytoplasma could recover a healthy morphology after MMS treatment ([Figure 2](#pone-0112533-g002){ref-type=\"fig\"}). ", "The small, light yellow leaves without seta turned into green leaves with seta, and the short internodes changed into normal internodes. ", "Among these morphologic changes, only very tiny axillary buds were discovered in the PS-20, which disappeared in the PS-60 and PS-100 ([Table 1](#pone-0112533-t001){ref-type=\"table\"}). ", "In addition, rooting rates increased significantly with the extension of incubation time at the same MMS concentration, but declined with MMS concentration increasing (*p*\\<0.05), simultaneously, the time of the first root of PS was also delayed with MMS concentration increasing. ", "These results indicated that the PS could recover a healthy morphology after treatment with a suitable concentration of MMS.", "\n\n{#pone-0112533-g002}\n\n10.1371/journal.pone.0112533.t001\n\n###### Development of MMS treated seedlings.", "\n\n![](", "pone.0112533.t001){#pone-0112533-t001-1}\n\n MMS concentrations Rooting ratio/% Rooting Axillary Leaves color and internodes Terminal crown \n -------------------- ----------------- --------- ---------- ----------------------------- ---------------- ------------------------------------------------------------ --------\n 0 90.0a 100a 100a 6 Yes Small, light yellow leaf without seta and short internodes Expand\n 20 75.0b 90a 100a 7 Yes Green leaf with seta and normal internodes Normal\n 60 30.0c 75.0b 86.7b 12 None Green leaf with seta and normal internodes Normal\n 100 0d 11.1c 31.7c 17 None Green leaf with seta and normal internodes Normal\n HS 100a 100a 100a 5 None Green leaf with seta and normal internodes Normal\n\nHS: Healthy seedlings. ", "The different letters within a column indicate significant difference, while the same letters within a column indicate no significant difference (*p*\\<0.05).", "\n\nPaWB phytoplasma detection {#s3b}\n--------------------------\n\nTo detect PaWB phytoplasma in PS and the ones treated with MMS, nested-PCR was performed to detect 16SrDNA of phytoplasma by universal phytoplasma primers. ", "The result showed that the specific 1.2 kb fragments of phytoplasma were only detected in the PS and PS-20 ([Figure 3](#pone-0112533-g003){ref-type=\"fig\"}), but the specific band was not detected in the healthy morphology seedlings, such as PS-60, PS-100, and HS. ", "These results illustrated that the key reason for the recovery was because MMS removed the PaWB phytoplasma.", "\n\n{#pone-0112533-g003}\n\nPaulownia DNA sequence polymorphism {#s3c}\n-----------------------------------\n\nTo study the effect of the phytoplasma infection on DNA sequence of Paulownia seedlings, Paulownia DNA polymorphisms of PS, the ones treated with MMS and HS were detected with 96 pairs of AFLP primer combinations ([Figure S1](#pone.0112533.s001){ref-type=\"supplementary-material\"}). ", "The results showed that no polymorphic DNA fragments were amplified by the same primer combinations, and the sizes of the fragments were identical for each seedling, showing that phytoplasma infection did not change the DNA sequence at AFLP level.", "\n\nDNA methylation variations {#s3d}\n--------------------------\n\n### Variations of DNA methylation levels {#s3d1}\n\nDNA methylation level of PS significantly increased with MMS concentration increasing (*p*\\<0.05) ([Table 2](#pone-0112533-t002){ref-type=\"table\"}). ", "The DNA methylation levels of PS, PS-20, PS-60 and PS-100 were 26.01%, 29.33%, 32.29% and 33.59%, respectively, and the DNA methylation level of HS was 35.97%. ", "This finding indicated that the DNA methylation levels of PS treated with MMS were higher than that of PS, but lower than that of HP. ", "This implied that, to some extent, variations of DNA methylation levels of the PS and the ones treated with MMS were associated with the morphological changes.", "\n\n10.1371/journal.pone.0112533.t002\n\n###### Changes of DNA methylation levels in PaWB seedlings with MMS treatment.", "\n\n![](", "pone.0112533.t002){#pone-0112533-t002-2}\n\n MMS concentration/(mg·L^−1^) Total amplified bands[a](#nt102){ref-type=\"table-fn\"} Band of class I Band of class II Band of class III Total methylated bands[b](#nt103){ref-type=\"table-fn\"} Methylation level/%[c](#nt104){ref-type=\"table-fn\"}\n ------------------------------ ------------------------------------------------------- ----------------- ------------------ ------------------- -------------------------------------------------------- -----------------------------------------------------\n 0 2691 1991 247 453 700 26.01a\n 20 2713 1919 257 539 796 29.33b\n 60 2589 1753 283 553 836 32.29c\n 100 2477 1645 295 537 832 33.59d\n HS 2357 1509 283 565 848 35.97e\n\n: Total amplified bands  =  band of class I+ band of class II+ band of class III;\n\n: total methylated bands  =  band of class II+ band of class III;\n\n: methylation level (%)  =  (total methylated bands)/(total amplified bands)×100; HS: Healthy seedlings. ", "The different letters within a column indicate significant difference, while the same letters within a column indicate no significant difference (*p*\\<0.05).", "\n\n### Variations of DNA methylation patterns {#s3d2}\n\nAbundant DNA methylation patterns were detected by 96 pairs of MSAP primer combinations in the PS treated with MMS or HS ([Table 3](#pone-0112533-t003){ref-type=\"table\"}). ", "The DNA methylation and demethylation polymorphisms increased with MMS concentration increasing ([Figure 4](#pone-0112533-g004){ref-type=\"fig\"}) ([Table 4](#pone-0112533-t004){ref-type=\"table\"}), respectively. ", "DNA methylation polymorphisms of PS-20, PS-60 and PS-100 were 16.75%, 17.37% and 17.76%, respectively, simultaneously, DNA demethylation polymorphisms were 7.14%, 7.66% and 11.11%, respectively. ", "These results demonstrated that there exist DNA methylation and DNA demethylation events in the process of morphological changes, and more DNA methylation events than DNA demethylation events occurred. ", "A similar trend was detected in the HS. ", "These observations suggested that changes of DNA methylation patterns were closely related to morphological changes of Paulownia.", "\n\n{#pone-0112533-g004}\n\n10.1371/journal.pone.0112533.t003\n\n###### Changes of DNA methylation patterns in PaWB seedlings with MMS treatment.", "\n\n![](", "pone.0112533.t003){#pone-0112533-t003-3}\n\n .Digestion[a](#nt105){ref-type=\"table-fn\"} Changes of methylation patterns Number of differences bands[b](#nt106){ref-type=\"table-fn\"} Types of methylation pattern \n -------------------------------------------- --------------------------------- ------------------------------------------------------------- ------------------------------ ------------------------- ------------------------- ------ ------ ------ ----- ------\n 1 1 0 1 CCGG C[C]{.ul}GG 103 103 119 69 A~1~\n GGCC GG[C]{.ul}C \n 1 1 1 0 CCGG [CC]{.ul}GG [CC]{.ul}GG 89 115 117 67 A~2~\n GGCC GGC[C]{.ul} GGCC \n 0 1 0 0 C[C]{.ul}GG [CC]{.ul}GG 85 127 123 155 A~3~\n GG[C]{.ul}C GG[CC]{.ul} \n 1 0 0 0 [CC]{.ul}GG [CC]{.ul}GG [CC]{.ul}GG 145 117 111 149 A~4~\n GGC[C]{.ul} GGCC GG[CC]{.ul} \n 0 1 1 1 C[C]{.ul}GG CCGG 33 29 79 53 B~1~\n GG[C]{.ul}C GGCC \n 1 0 1 1 C[C]{.ul}GG [CC]{.ul}GG [CC]{.ul}GG 25 11 49 51 B~2~\n GG[C]{.ul}C GGC[C]{.ul} GGCC \n 0 0 0 1 [CC]{.ul}GG C[C]{.ul}GG 65 85 71 153 B~3~\n GG[CC]{.ul} GG[C]{.ul}C \n 0 0 1 1 [CC]{.ul}GG CCGG 57 81 95 157 B~4~\n GG[CC]{.ul} GGCC \n 0 1 1 0 C[C]{.ul}GG [CC]{.ul}GG [CC]{.ul}GG 5 12 16 15 C\n GG[C]{.ul}C GGC[C]{.ul} GGCC \n 1 1 1 1 CCGG CCGG 1505 1589 1563 997 D~1~\n GGCC GGCC \n 1 0 1 0 [CC]{.ul}GG [CC]{.ul}GG [CC]{.ul}GG [CC]{.ul}GG 67 63 33 89 D~2~\n GGC[C]{.ul} GGCC GGC[C]{.ul} GGCC \n 0 1 0 1 C[C]{.ul}GG C[C]{.ul}GG 341 327 271 381 D~3~\n GG[C]{.ul}C GG[C]{.ul}C \n\n: H~1~ and M~1~: bands digested by *Eco*RI/*Hpa*II (H) and *Eco*RI/*Msp*I (M) in PS; H~x~ and M~x~: bands digested by H and M in MMS treated seedlings or HS; [C]{.ul} and [CC]{.ul}: cytosine methylation;\n\n: 0--20: the number of DNA methylation patterns of PS-20 relative to PS; 0--60: the number of DNA methylation patterns of PS-60 relative to PS; 0--100: the number of DNA methylation patterns of PS-100 relative to PS; 0--HS: the number of DNA methylation patterns of HS relative to PS.", "\n\n10.1371/journal.pone.0112533.t004\n\n###### DNA methylation status in PaWB seedlings with MMS treatment.", "\n\n![](", "pone.0112533.t004){#pone-0112533-t004-4}\n\n Combination Total methylated bands[a](#nt107){ref-type=\"table-fn\"} Type A[b](#nt108){ref-type=\"table-fn\"} Type B[c](#nt109){ref-type=\"table-fn\"} Type C[d](#nt110){ref-type=\"table-fn\"} Type D[e](#nt111){ref-type=\"table-fn\"} \n ------------- -------------------------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---- ------ ------ -------\n 0--20 2520 422 16.75 180 7.14 5 0.20 1913 75.91\n 0--60 2659 462 17.37 206 7.75 12 0.45 1979 74.43\n 0--100 2647 470 17.76 294 11.11 16 0.60 1867 70.53\n 0--HS 2336 440 18.84 414 17.72 15 0.64 1467 62.80\n\n: Total methylation bands  =  band of type A+ band of type B+ band of type C+ band of type D;\n\n: type A, DNA methylation type, type A (%)  =  (band of type A)/(total methylation bands)×100;\n\n: type B, DNA demethylation type, type B (%)  =  (band of type B)/(total methylation bands)×100;\n\n: type C, uncertain DNA methylation type, type C (%)  =  (band of type C)/(total methylation bands)×100;\n\n: type D, DNA methylation monomorphism, type D (%)  =  (band of type D)/(total methylation bands)×100.", "\n\nAnalysis of polymorphic fragment sequences {#s3e}\n------------------------------------------\n\nEighty-one clear and reproducible methylated fragments related to PaWB through the comparison of DNA methylation patterns in different seedlings were sequenced, of which 36 (44.44%) represented unannotated sequences, forty-five (55.56%) fragments were homologous with annotated sequences ([Table S4](#pone.0112533.s005){ref-type=\"supplementary-material\"}), these genes encoded proteins with a variety of functions, including substance metabolism, transcription, pathogen defense and signal transduction.", "\n\nExpression analysis of polymorphic fragments {#s3f}\n--------------------------------------------\n\nThe expressions of six methylated genes were analyzed using qRT-PCR. ", "The results showed that two genes encoding proteins (chitin-inducible gibberellin-responsive protein and uncharacterized protein LOC100796964) ([Figure 5A and E](#pone-0112533-g005){ref-type=\"fig\"}) were up-regulated and four genes encoding proteins (leucyl aminopeptidase, cytochrome P450 76B6, ring finger protein and beta-hydroxyacyl-ACP dehydrase 1) ([Figure 5B, C, D and F](#pone-0112533-g005){ref-type=\"fig\"}) were down-regulated with MMS concentration increasing (*p*\\<0.05), indicating that the expressions of all six genes were consistent with the changes of DNA methylation patterns.", "\n\n{#pone-0112533-g005}\n\nDiscussion {#s4}\n==========\n\nThe changes of DNA polymorphism and DNA methylation in the PS, the ones treated with MMS and HS, based on AFLP and MSAP approaches in this paper showed that phytoplasma infection did not change the DNA sequence of seedlings at AFLP level, but changed the DNA methylation levels and patterns, and the observation from AFLP was consistent with the previous results [@pone.0112533-Cao1], [@pone.0112533-Zhai1], [@pone.0112533-Cao2], but differed from the TMV-infected tobacco [@pone.0112533-Kathiria1] in which the host exhibited an increase in the homologous recombination frequency (HRF), implying that the morphological changes of the seedlings treated with MMS may be related to the epigenetic modifications. ", "DNA methylation is one of the main epigenetic modifications, plays a vital role in plant growth and development, changes of DNA methylation levels has closely related to plant phenotypic changes, the DNA methylation pattern, such as DNA hypermethylation or hypomethylation can result in their morphological abnormalities [@pone.0112533-Richards1], [@pone.0112533-Finnegan1]. ", "The MSAP analysis showed that DNA methylation level of PS was lower than those treated with MMS and HS. ", "This finding is in agreement with our previous HPLC result [@pone.0112533-Li1], demonstrating that the decrease of host DNA methylation levels was related to PaWB. ", "This coincides with the results in *Arabidopsis* [@pone.0112533-Finnegan2], [@pone.0112533-Finnegan3], 5-azacytidine treated brassica-oleracea [@pone.0112533-King1] and '*Candidatus* Phytoplasma asteris'-infected periwinkles [@pone.0112533-LeljakLevanic1]. ", "Moreover, variations of DNA methylation patterns were also discovered when the PS acquired a healthy morphology. ", "The DNA methylation or demethylation polymorphisms increased with MMS concentration increasing. ", "Although DNA demethylation polymorphisms also occurred at a high frequency, the DNA methylation polymorphisms were generally the most frequent, these results were consistent with the previous observations [@pone.0112533-Cao2], suggesting that the variations of DNA methylation patterns are a dynamic process. ", "These results indicated that the occurrence of PaWB is a complex process, morphological changes of Paulownia after phytoplasma infection has relation with DNA methylation.", "\n\nPlant responds to pathogen attack by modifying gene expression. ", "DNA methylation is one of the mechanisms in regulating gene expressions [@pone.0112533-Finnegan1], similar results were also reported in Tomato Yellow Leaf Curl Sardinia Virus-Tomato [@pone.0112533-Mason1] and *Mycosphaerella fijiensis* toxins-Musa [@pone.0112533-Gimenez1] systems. ", "In the present study, we identified several DNA methylation genes related to PaWB, and qRT-PCR showed that their expressions significantly changed in the process of the PS morphological changes. ", "Among these genes, their functions not only involved in phytoplasma virulence, but also implicated in symptom formation and disease defense of PaWB. ", "The genes encoded leucyl aminopeptidase implicated in phytoplasma vitamine metabolic pathways [@pone.0112533-TranNguyen1], so the higher expression of leucyl aminopeptidase was related to the phytoplasma virulence; The gene encoded beta-hydroxyacyl-ACP dehydrase 1 involved in fatty acid biosynthesis pathway [@pone.0112533-Mou1], [@pone.0112533-RismaniYazdi1], which is necessary for phytoplasma [@pone.0112533-Bertaccini1], so the higher gene expression of these genes in the PS may be related to phytoplasma overgrowth, this means that over-expression of these genes further disturbed the normal growth of Paulownia, and resulted in the changes of set of genes expressions. ", "For example, the gene encoding chitin-inducible gibberellin-responsive protein (CIGR) was down-regulation in the PS, it was reported that CIGR belongs to the GRAS family, mainly contributes to the stem elongation of plant [@pone.0112533-Kovi1], so the lower expression of this gene in the PS might correlate with the dwarf of the Paulownia; The gene encoding cytochrome P450 76B6 was up-regulation in the PS, which consistent with the global transcriptome result [@pone.0112533-Liu1], [@pone.0112533-Mou1]. ", "Previous research have evidenced that cytochrome P450 76B6 played an important role in biosynthesis of flavonoids [@pone.0112533-Sung1], which had closely related to decrease of the leaf cell death after phytoplasma infection [@pone.0112533-Himeno1], numerous studies have revealed that flavonoids were induced in response to pathogen infection [@pone.0112533-Treutter1]--[@pone.0112533-Margaria1], so the higher expression of this gene in the phytoplasna infection seedlings might implicate in plant defense; Another protein involved in plant defense was ring finger protein [@pone.0112533-Yu1], the trend of this gene expression in the PS and the ones treated with MMS was similar with that reported by Mou et al. [", "@pone.0112533-Liu1]. ", "Overall, morphological changes of Paulownia seedlings after phytoplasma infection resulted in various changes of the DNA methylation patterns related to PaWB, which further induced the changes of corresponding gene expressions in the PS and the ones treated with MMS, interesting, several genes were detected by both DNA methylation and transcriptome analysis. ", "Besides these genes, we also found 36 unannotated genes whose functions were not clear and worthy of further investigation in future studies.", "\n\nThe occurrence of PaWB involved in many factors. ", "Even though several genes involved in metabolic pathways of Paulownia had been identified, but a few of these genes might only be associated with the growth and development of *P. fortunei* itself. ", "The similar result was got in *Paulownia tomentosa × Paulownia fortunei* [@pone.0112533-Cao2]. ", "In order to identify some genes more closely related to the occurrence of PaWB, the genes associated with the growth and development had to be discarded. ", "Through comparing the sequences and the sizes of the methylation genes generated in *P. fortunei* and *P. tomentosa × P. fortunei*, we found that three genes appeared simultaneously in two species of Paulownia seedlings with PaWB, including chase 2 sensor protein, cation proton exchanger, and transcription factor HB29. ", "The roles of these three genes in the process of occurrence of PaWB retain unknown, we will put emphasis on them in our next research.", "\n\nIn conclusion, phytoplasma infection resulted in the *P. fortunei* morphological changes of the seedlings, but these changes could be recovered by more than 60 mg·L^−1^MMS treatment. ", "DNA polymorphisms analysis showed that Paulownia DNA sequence was not changed in the process of morphological changes at AFLP level, conversely, these variations regulated by changes of DNA methylation levels and patterns, providing further clues to clarify the molecular mechanism of PaWB.", "\n\nSupporting Information {#s5}\n======================\n\n###### \n\n**AFLP gels electrophoresis of PaWB seedlings with MMS treatment.** ", "a: bands amplification obtained from PS; b: bands amplification obtained from PS-20; c: bands amplification obtained from PS-60; d: bands amplification obtained from PS-100; e: bands amplification obtained from HS; M: DNA Marker; P~1~/M~21~ -- P~1~/M~x~: primer combinations.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**AFLP adapters and primers used in this study.** ", "P~1~/M~1~--P~64~/M~64~ are the selective-amplification primer combinations.", "\n\n(DOCX)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**MSAP adapters and primers used in this study.** ", "^\\*^Selective-amplification primer combinations comprised each *Eco*RI primer combined with each *Hpa*II/*Msp*I primer.", "\n\n(DOCX)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**Primers used for qRT - PCR analysis.**", "\n\n(DOCX)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**List of MSAP fragment with different methylation profiles in PaWB seedlings with MMS treatment.** ", "^a^: BB1--BB52: MSAP polymorphic fragments during PS, MMS treated PS and HS; ^b^: the sequence information obtained from the GenBank database.", "\n\n(DOCX)\n\n###### \n\nClick here for additional data file.", "\n\nWe thank J.Y.C for his interest and invaluable assistance during this work, and thank the anonymous reviewers for their valuable suggestions for improving the manuscript.", "\n\n[^1]: **Competing Interests:**The authors have declared that no competing interests exist.", "\n\n[^2]: Conceived and designed the experiments: GQF XBC ZLZ MJD YPD. ", "Performed the experiments: GQF XBC ZLZ MJD YPD. ", "Analyzed the data: GQF XBC ZLZ MJD YPD. ", "Contributed reagents/materials/analysis tools: GQF XBC ZLZ MJD YPD. ", "Wrote the paper: XBC GQF.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "Q:\n\nIterating over XML with simplexml_load_string\n\nI'm trying to retrieve data from an XML document that looks like:\n<scores sport=\"soccer\" updated=\"08.05.2018 09:39:42\">\n<category name=\"Argentina: Superliga\" gid=\"1081\" id=\"1081\" file_group=\"argentina\" iscup=\"False\">\n<matches date=\"May 08\" formatted_date=\"08.05.2018\">\n<match status=\"FT\" timer=\"\" date=\"May 08\" formatted_date=\"08.05.2018\" time=\"00:15\" commentary_available=\"argentina\" venue=\"Estadio Brigadier General Estanislao López\" v=\"60\" static_id=\"2288469\" fix_id=\"2172208\" id=\"2442089\">\n<localteam name=\"Colon Santa FE\" goals=\"0\" id=\"5947\"/>\n<visitorteam name=\"River Plate\" goals=\"0\" id=\"6042\"/>\n<events>\n<event type=\"yellowcard\" minute=\"39\" extra_min=\"\" team=\"visitorteam\" player=\"C. Mayada\" result=\"\" playerId=\"91623\" assist=\"\" assistid=\"\" eventid=\"24420891\"/>\n<event type=\"yellowcard\" minute=\"40\" extra_min=\"\" team=\"localteam\" player=\"M. Fritzler\" result=\"\" playerId=\"14256\" assist=\"\" assistid=\"\" eventid=\"24420892\"/>\n<event type=\"subst\" minute=\"46\" extra_min=\"\" team=\"visitorteam\" player=\"J. Quintero\" result=\"\" playerId=\"74510\" assist=\"G. Martinez\" assistid=\"51346\" eventid=\"24420893\"/>\n<event type=\"subst\" minute=\"63\" extra_min=\"\" team=\"visitorteam\" player=\"L. Pratto\" result=\"\" playerId=\"17657\" assist=\"R. Mora\" assistid=\"55877\" eventid=\"24420894\"/>\n<event type=\"subst\" minute=\"63\" extra_min=\"\" team=\"localteam\" player=\"N. Leguizamon\" result=\"\" playerId=\"443149\" assist=\"C. Bernardi\" assistid=\"186181\" eventid=\"24420895\"/>\n<event type=\"subst\" minute=\"70\" extra_min=\"\" team=\"localteam\" player=\"P. Ledesma\" result=\"\" playerId=\"13875\" assist=\"A. Bastia\" assistid=\"15082\" eventid=\"24420896\"/>\n<event type=\"subst\" minute=\"73\" extra_min=\"\" team=\"visitorteam\" player=\"R. Borre\" result=\"\" playerId=\"320836\" assist=\"I. Scocco\" assistid=\"15172\" eventid=\"24420897\"/>\n<event type=\"subst\" minute=\"82\" extra_min=\"\" team=\"localteam\" player=\"L. Heredia\" result=\"\" playerId=\"337735\" assist=\"A. Ruiz\" assistid=\"189511\" eventid=\"24420898\"/>\n<event type=\"yellowcard\" minute=\"90\" extra_min=\"1\" team=\"visitorteam\" player=\"R. Borre\" result=\"\" playerId=\"320836\" assist=\"\" assistid=\"\" eventid=\"24420899\"/>\n</events>\n<ht score=\"[0-0]\"/>\n<ft score=\"[0-0]\"/>\n</match>\n</matches>\n</category>\n<category name=\"Argentina: Primera B Nacional - Promotion - Play Offs\" gid=\"3899\" id=\"1078\" file_group=\"argentina\" iscup=\"False\">\n<matches date=\"May 08\" formatted_date=\"08.05.2018\">\n<match status=\"23:05\" timer=\"\" date=\"May 08\" formatted_date=\"08.05.2018\" time=\"23:05\" commentary_available=\"\" venue=\"\" v=\"0\" static_id=\"2369960\" fix_id=\"2253925\" id=\"2443571\">\n<localteam name=\"Almagro\" goals=\"?\" ", "id=\"5900\"/>\n<visitorteam name=\"Agropecuario\" goals=\"?\" ", "id=\"21385\"/>\n<events/>\n<ht score=\"\"/>\n</match>\n</matches>\n</category>\n</scores>\n\nWhat I have tried so far:\n$xml=simplexml_load_string($content);\n//print_r($xml);\nforeach ($xml->scores->category->matches->match as $value) {\n echo (string)$value;\n}\n\nMessage I'm getting:\nNotice: Trying to get property of non-object in C:\\xampp\\htdocs\\proj\\inc\\get.php on line 23\n\nNotice: Trying to get property of non-object in C:\\xampp\\htdocs\\proj\\inc\\get.php on line 23\n\nWarning: Invalid argument supplied for foreach() in C:\\xampp\\htdocs\\proj\\inc\\get.php on line 23\n\nTo be specific, I'm trying to extract what's inside the attributes like Date, team name, goals, etc.", "\nThank you in advance.", "\n\nA:\n\nWhen you load an XML file using SimpleXML, the root element is the starting point for any access. ", " So you don't need scores as this is $xml.", "\nforeach ($xml->category->matches->match as $value) {\n echo (string)$value['status'];\n}\n\nThis outputs the status element (FT) to show it's actually the <match> element.", "\nUpdate: As Syscall mentioned, the structure of this XML would need something like...\nforeach ($xml->category as $value1) {\n foreach ( $value1->matches->match as $value) {\n echo (string)$value['status'];\n }\n}\n\nto fetch all of the <match> elements.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Headlines\n\nGreat American Beer Festival Winners Announced\n\nThe 2012 Great American Beer Festival (GABF) competition awarded 254 medals to some of the best commercial breweries in the United States. ", "Presented by the Brewers Association, GABF is the largest commercial beer competition in the world." ]

{ "pile_set_name": "Pile-CC" }

[ 0.005050505050505051, 0.020202020202020204 ]

[ "Introduction {#Sec1}\n============\n\nIn eukaryotic cells, functional mRNA expression is a multi-step process in which the DNA-encoded message is transcribed into a pre-mRNA molecule that undergoes numerous modifications such as 5′-end capping, splicing, 3′-end cleavage and polyadenylation, together with the assembly of diverse factors required for the formation of a messenger ribonucleoprotein particle (mRNP)^[@CR1],[@CR2]^. The adequately processed mRNPs are then competent for their export to the cytoplasm where they will be translated^[@CR3]^. All these processes are intimately linked: 5′-end capping, splicing and 3′-end maturation occur co-transcriptionally due to the important role played by the carboxy-terminal domain (CTD) of RNA polymerase II's (RNAP-II) largest subunit^[@CR4],[@CR5]^. However, mRNA processing is error-prone and improperly matured mRNPs have to be degraded in order to avoid the synthesis of nonfunctional proteins. ", "While the synthesis of the mRNPs progresses, surveillance mechanisms that detect malformed mRNPs are also operating. ", "Aberrant mRNPs^[@CR6]^ that fail to pass the quality control steps are retained in the nucleus and degraded by different ribonucleases. ", "In human cells, two major degradation pathways are responsible for mRNA decay of defective transcripts in the nucleus: (i) the 5′-3′ exoribonuclease XRN2, together with the decapping factor DCP2, and (ii) the RNA exosome^[@CR7],[@CR8]^.\n\nThe RNA exosome complex, first described in yeast, is conserved in all eukaryotic cells. ", "In human cells, it is composed of a core of nine subunits which serves as a binding platform for two active ribonucleases - hRRP6 and hDIS3/hRRP44 - that have 3′-5′ RNA exonuclease and endonuclease activities^[@CR9],[@CR10]^. This complex recognizes and degrades improperly-formed RNAs in the nucleus^[@CR11]^. To exert its function, the nuclear RNA exosome uses cofactors that directly stimulate its enzymatic activity and serve as adaptors for its many substrates^[@CR12]^. Several proteins or complexes have recently been identified for their capacity to recruit the nuclear RNA exosome onto its target RNAs. ", "In the yeast system for which several exosome-associated adaptors have been characterized, it appears that the nuclear RNA exosome depends largely on the activities of the TRAMP (Trf4p/5p-Air1p/2p-Mtr4p polyadenylation) complex^[@CR13]--[@CR19]^. However, in human, at least three distinct RNA exosome adaptors have recently been identified. ", "All critically depends on the RNA helicase hMTR4: the hTRAMP complex, which is homologous to the yeast complex and localizes in the nucleolus^[@CR20]--[@CR22]^, the PAXT (poly(A) tail exosome targeting) complex formed by hMTR4-ZFC3H1 and the NEXT (nuclear exosome targeting) complex which is not conserved in yeast and localizes in the nucleoplasm^[@CR21],[@CR23]--[@CR25]^. The RBM7 protein, a putative pre-mRNA splicing factor, and the ZCCHC8 (zinc finger CCHC domain-containing protein 8) protein form the NEXT complex. ", "Interestingly, ZCCHC8 has also been shown to interact with the cap-binding complex (CBC) and several members of the SR protein family^[@CR21]^, and one study has reported that, *in vitro*, RBM7 interacts with the splicing factor 3b subunit 2 (SAP145) and the SRSF3 protein^[@CR26]^. The interaction between SRSF3 and RBM7 was also found by a proteomic approach^[@CR27]^. This suggests that the cap-binding complex and the spliceosome are involved in the recruitment of the RNA exosome to its target RNAs via their interaction with the NEXT complex. ", "Moreover, it was recently demonstrated that the recruitment of the RNA exosome by CBC promotes the degradation of promoter upstream transcripts (PROMPTs)^[@CR27]^. The role of the spliceosome in mRNA decay is also suspected because an interaction has been found between hMTR4 and hRRP6 and several components of the spliceosomal U4/U6.U5 tri-snRNP complex^[@CR28]^. However, a direct functional link between the splicing reaction and the mRNA decay machinery is lacking.", "\n\nEpstein-Barr virus (EBV), a ubiquitous human herpes virus, is the causative agent of infectious mononucleosis and is also associated with various cancers observed in immuno-competent individuals (Burkitt's lymphoma, Hodgkin's disease, nasopharyngeal carcinoma, T/NK lymphoma, and a subtype of gastric carcinoma) as well as in immuno-deficient patients (lymphoma and post-transplant lympho-proliferative diseases)^[@CR29]^. EBV persists in an infected host for life by establishing a latent infection in memory B cells^[@CR30]^. Following cell differentiation, viral reactivation into the productive cycle leads to the production of progeny viruses that spread to new cells and hosts^[@CR31]^. For these two phases of the viral life cycle different viral gene expression programs are used. ", "During latency, a small subset of EBV genes is expressed that allows maintenance of the viral genome and induction of cell proliferation and differentiation in the absence of viral production. ", "Genes expressed during this phase have the same structure as cellular genes. ", "By contrast, during the productive cycle, the structure of a majority of the viral genes expressed during this phase differs from that of cellular genes because most of the viral genes are intronless. ", "This characteristic may represent an handicap for the viral mRNAs in terms of cytoplasmic accumulation and translation efficiency since it is now well documented that pre-mRNA splicing efficiently increases both mRNA export and translation^[@CR3],[@CR32],[@CR33]^. However, EBV - like all herpesviruses - encodes a protein called EB2 (also referred to as BMLF1 or SM) whose role is to increase the cytoplasmic accumulation and translation of viral mRNAs expressed from intronless genes^[@CR34]--[@CR38]^. EB2 is an RNA-binding protein (RBP)^[@CR39]^ essential for virus replication^[@CR40]^ that shuttles between the nucleus and the cytoplasm^[@CR41],[@CR42]^. It has been reported to facilitate the nuclear export of viral mRNAs expressed from intronless genes by interacting with the cellular ALYREF and TAP-NXF1/p15 proteins that promote cellular mRNA export^[@CR41],[@CR42]^. Thus, EB2 may serve as a bridge between viral RNAs and the cellular export complex (TREX), which is recruited to the 5´-end of cellular RNAs as a result of splicing^[@CR43]^. In this model, EB2, like homologous proteins from other herpesviruses, compensates for the absence of splicing of the viral mRNAs expressed from intronless genes, by recruiting the TREX complex. ", "Accordingly, EB2 is dispensable for mRNAs expressed from intron-containing genes. ", "In addition, EB2 is able to inhibit the splicing reaction^[@CR35],[@CR44]^ and modify the splicing pattern of specific cellular mRNAs^[@CR45]^. The interaction observed between EB2 and SR proteins like SRSF1 (ASF/SF2), SRSF3 (SRp20) and SRSF7 (9G8) is probably involved both in the modifications of specific cellular genes splicing and in the increase in nuclear mRNA export of viral mRNA expressed from intronless genes^[@CR46]--[@CR48]^. Interestingly, EB2, like its homologous protein ORF57 from Kaposi's sarcoma-associated herpesvirus, has also been suggested to be involved in increasing mRNA stability. ", "However, the mechanisms involved are not yet understood^[@CR49],[@CR50]^.\n\nIn the present study, we show that in addition to its well-documented role in mRNA export and translation, the viral RNA-binding protein EB2 is also involved in mRNA stability control. ", "In the absence of EB2, its target mRNAs - produced from intronless genes - are rapidly degraded by the nuclear RNA exosome. ", "We have identified the splicing factor SRSF3 as being responsible for the destabilization of these viral mRNAs. ", "Our data reveal, for the first time, that SRSF3 associates with both the NEXT complex and the RNA exosome to induce degradation of mRNAs in the nucleus. ", "The viral protein EB2 appears to interfere with this processus, thus rescuing viral mRNAs from degradation. ", "Our findings reveal an important novel function of the SR protein SRSF3, in the recognition and degradation of some mRNAs.", "\n\nResults {#Sec2}\n=======\n\nThe EB2 protein is not only an mRNA export factor {#Sec3}\n-------------------------------------------------\n\nIt has been extensively shown that the viral protein EB2, as well as its homologs from other herpesviruses - ICP27, UL69 and ORF57 - are involved in the cytoplasmic accumulation of viral mRNAs expressed from intronless genes. ", "That EB2 also has a function in nuclear stabilization of its target mRNAs was suggested by the observation that the EB2's target mRNAs are not well accumulated in the cell nucleus in the absence of EB2^[@CR49]^. To explore this further, cytoplasmic and nuclear RNAs were purified from HEK293~EBV∆BMLF1~ cells^[@CR40]^ which contain a recombinant virus defective for the expression of EB2. ", "The relative amounts of several viral mRNAs whose cytoplasmic accumulation was known to be dependent on EB2 (BDLF1, BdRF1 and BFRF3) or not (BMRF1) was then measured, by RT-PCR (Fig.", " [1a](#Fig1){ref-type=\"fig\"}). ", "U6 snRNA and ß-actin mRNA were used as controls for the purification efficiency of the nuclear and cytoplasmic mRNAs, respectively. ", "As expected, in the absence of viral reactivation, the viral early (BMRF1) and late (BDLF1, BdRF1 and BFRF3) mRNAs were not expressed. ", "When the viral productive cycle was activated by ectopic expression of the viral transcription factor EB1, the late mRNAs known to be targeted by EB2 were barely detected in the cell cytoplasm but accumulated in the cell nucleus. ", "This nuclear accumulation argues in favor of a role for EB2 in viral mRNA export as has already been reported^[@CR34]^. However, when EB2 was expressed in the cells, EB2-dependent viral mRNAs not only accumulated in the cytoplasm as a result of EB2-mediated mRNA export, but their expression was also clearly enhanced in the nucleus. ", "These results suggest that EB2 could play a role in nuclear mRNA stability.", "Figure 1EB2 stabilizes late viral mRNAs in the nucleus. (**", "a**) *mRNA accumulation:* Cytoplasmic and nuclear RNAs from HEK293~EBV∆BMLF1~ cells transiently transfected as indicated at the top of the figure were submitted to RT-PCR analysis using specific primers to detect cellular U6 snRNA and ß-actin mRNA, or EBV-encoding mRNAs (BDLF1, BdRF1, BFRF3 and BMRF1). ", "The PCR products were loaded on a 2% agarose gel and visualized by ethidium bromide staining. ", "The RT-PCR results were in the linear range of the PCR reaction. ", "Expression of EB2, EB1 and Tubulin proteins expressed in HEK293~EBV∆BMLF1~ cells that have been transfected, or not (lane 1), with an EB1 expression plasmid (lane 2), or cotransfected with expression plasmids for both EB1 and EB2 (lane 3) were controlled by western-blotting. \\* ", "Indicates an unspecific band recognized by the anti-EB2 serum. (**", "b**) Schematic representation of the pTRE2-BDLF1 construct which contains the viral gene BDLF1 under the control of the Tet-responsive promoter and the pCMV-RLuc construct which contains the Renilla Luciferase gene, RLuc, under the control of the CMV promoter. (**", "c**) (**e**) (**g**) and (**i**) *Nuclear and cytoplasmic BDLF1and luciferase mRNAs accumulation:* Quantification by RT-qPCR of the nuclear (**c** and **g**) and cytoplasmic (**e** and **i**) BDLF1 and luciferase mRNA expressed from HeLa cells co-transfected with the pTRE2-BDLF1 construct or the pCMV-RLuc construct without or with an EB2 expression vector (time 0 of the kinetic). ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). \\*\\* ", "Indicates *P*-value of \\<0.01 and \\*\\*\\* indicates *P*-value of \\<0.001. (**", "d**) (**f**) (**h**) and (**j**) *Nuclear and cytoplasmic BDLF1 or luciferase mRNA stability:* Relative nuclear (**d** and **h**) and cytoplasmic (**f** and **j**) BDLF1 or luciferase mRNA level after transcription inhibition (expressed as a percentage of the amount at time point 0) was determined by RT-qPCR from HeLa cells co-transfected with the pTRE2-BDLF1 construct or the pCMV-RLuc construct without or with an EB2 expression vector. ", "Numbers above the panel refer to hours after transcriptional shutoff of the BDLF1 or luciferase intronless gene. ", "Half-lives (*t*~1/2~) and s.d were calculated from three independent experiments (*n* = 3).", "\n\nTo test this hypothesis, the nuclear stability of an EB2 target mRNA, BDLF1, and a Luciferase reporter mRNA, were compared, in the presence or absence of the EB2 protein. ", "For this, HeLa cells were transfected with the pTRE2-BDLF1 reporter plasmid in which BDLF1's transcription is under the control of doxycycline or the pCMV-Renilla Luciferase reporter plasmid (Fig.", " [1b](#Fig1){ref-type=\"fig\"}), together, or not, with an EB2 expression plasmid. ", "24 h after transfection, the cells were treated either with doxycycline in order to stop transcription of the BDLF1 reporter gene, or with actinomycin D to stop transcription of the Luciferase gene, and both nuclear and cytoplasmic mRNA, purified at different time points, were quantified by RT-qPCR. ", "As expected, at time zero, the BDLF1 and the Luciferase mRNA levels were more important in both the nucleus and the cytoplasm of cells transfected with an EB2 expression vector than in the control cells (Fig.", " [1c,e,g,i](#Fig1){ref-type=\"fig\"}). ", "Quantification of BDLF1 and Luciferase mRNAs during the time-course following inhibition of their transcription, revealed that they were much more stable in the nucleus of cells expressing the EB2 protein (Fig.", " [1d,h](#Fig1){ref-type=\"fig\"}). ", "In the presence of EB2, the half-life of the reporter mRNAs was increased over four-fold. ", "Interestingly, the half-life of cytoplasmic reporter mRNAs was very similar to that observed for the nuclear mRNA, both in the absence or presence of EB2 (Fig.", " [1f,j](#Fig1){ref-type=\"fig\"}). ", "This observation suggests that nuclear export is not likely to be responsible for the reporter mRNA half-life changes observed in the different conditions. ", "Taken together, these results clearly demonstrate that EB2 enhances nuclear mRNA stability of specific transcripts.", "\n\nThe nuclear RNA exosome is involved in the decay of EB2's target mRNAs {#Sec4}\n----------------------------------------------------------------------\n\nEukaryotic mRNAs contain two features important for their stability: the 5′ 7-methylguanosine cap and the 3′ poly(A) tail. ", "In the cell nucleus, these determinants interact respectively with the cap-binding complex (CBC) and the nuclear poly(A)-binding protein (PABPN1) to protect the transcripts from exoribonucleases and to promote the export of the mRNAs. ", "To initiate mRNA decay, either one of these two structures must be compromised or the mRNA must be cleaved by endonucleolytic attack. ", "Two main pathways have been described for the degradation of nuclear mRNAs: either the 5′cap structure is removed by the decapping enzyme DCP2 thus allowing the mRNA body to be degraded in the 5′-3′ direction by the XRN2 exoribonuclease or the 3′ poly(A) tail is attacked by a large exonuclease complex known as the RNA exosome^[@CR51]^. In order to determine which of these pathways is responsible for the nuclear decay of intronless mRNAs, expression of specific components of these different machineries was down-regulated using siRNAs. ", "Depletion efficiency of the proteins was determined by western blotting (Fig.", " [2a,c](#Fig2){ref-type=\"fig\"}). ", "Interestingly, down-regulation of hRRP40, a component of the core RNA exosome complex, induced a net stabilization of the Renilla luciferase reporter mRNA (Fig.", " [2b,d](#Fig2){ref-type=\"fig\"}). ", "Down-regulation of either of the two ribonucleases associated with the RNA exosome core complex resulted in only a weak effect (hDIS3/hRRP44) or the absence (hRRP6) of an effect on the RLuc reporter mRNA stability. ", "However, simultaneous down-regulation of the two ribonucleases enhanced the stability of the target mRNA to a level similar to what was seen when hRRP40, the core component of the RNA exosome, was down-regulated (Fig.", " [2b](#Fig2){ref-type=\"fig\"}). ", "Taken together, these results indicate first, that the RNA exosome pathway is important for decay of EB2's target mRNAs, and second, that both ribonucleases associated with the RNA exosome core complex are involved in the degradation pathway.", "Figure 2EB2's target mRNAs are mainly degraded by the nuclear RNA exosome. (**", "a**) and (**c**) *Immunoblot:* Western blotting analysis of cell extracts showing protein depletion upon the indicated siRNA treatments. ", "Hela cells were treated with specific or control (Firefly) siRNAs. ", "Membranes were probed with the indicated antibodies (anti-hRRP40, -hDIS3, -hRRP6, -DCP2, -XRN2 and -Tubulin). ", "Anti-Tubulin antibody was used as loading control. \\* ", "Indicates non-specific bands. (**", "b**) and (**d**) *Nuclear mRNA accumulation:* RT-qPCR analysis of nuclear RNAs expressed from HeLa cells transfected with the indicated siRNAs and the Renilla luciferase intronless gene. ", "Data are displayed as mean values normalized to the control siRNA and GAPDH mRNA as an internal control. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). \\* ", "Indicates a *P*-value of \\<0.05 and \\*\\* indicates a *P*-value of \\<0.01.", "\n\nBy contrast, when the 5′-3′ decay pathway was down-regulated by siRNAs directed against either DCP2 or XRN2 proteins or both, the stability of the RLuc reporter mRNA was not significally altered compared to what was observed in the case of down-regulation of the RNA exosome pathway (Fig.", " [2d](#Fig2){ref-type=\"fig\"}). ", "Thus, nuclear decay of EB2's target mRNAs appears to depend mainly on the action of the nuclear RNA exosome.", "\n\nThe nuclear RNA exosome impacts EBV viral gene expression and virus production {#Sec5}\n------------------------------------------------------------------------------\n\nSince RNA exosome depletion has an important effect on the nuclear stability of RLuc intronless mRNA, we asked whether depletion of hRRP40 (one of the core RNA exosome subunits) could also affect the EB2-dependent nuclear accumulation of specific viral mRNAs as well as viral production. ", "For this, expression of the nuclear RNA exosome was down-regulated in the HEK293~EBV∆BMLF1~ cell line, by transfection of a siRNA specifically directed against hRRP40 (Fig.", " [3a](#Fig3){ref-type=\"fig\"}). ", "48 h after activation of the viral productive cycle, the viral nuclear and cytoplasmic mRNAs were quantified by RT-qPCR, and viruses produced in the cell culture supernatant were titrated. ", "Interestingly, in the absence of EB2, depletion of the RNA exosome allowed a net nuclear accumulation of the late viral mRNAs BDLF1, BdRF1 and BFRF3, whereas the amount of the BMRF1 early mRNA was not affected (Fig.", " [3b](#Fig3){ref-type=\"fig\"}). ", "By contrast, in the presence of EB2, depletion of the RNA exosome had only a weak but reproducible effect on late viral nuclear mRNA accumulation. ", "It is interesting to note that these results are very similar to what was observed for the cytoplasmic accumulation of viral mRNAs in cells from which SRSF3 had been depleted (Fig.", " [3c](#Fig3){ref-type=\"fig\"}) as already reported^[@CR47]^. As expected from the cumulative effect on the stabilization of different viral and cellular mRNAs following RNA exosome depletion, virus production was reproducibly higher from cells trans-complemented with EB1 and EB2, when hRRP40 was down-regulated (Fig.", " [3d](#Fig3){ref-type=\"fig\"}). ", "However, in the absence of EB2, hRRP40 depletion was clearly not sufficient to allow virus production even though this depletion enhances the nuclear and cytoplasmic accumuation of the late viral mRNAs as shown above in Fig.", " [3b and c](#Fig3){ref-type=\"fig\"}. ", "This absence of virus production could be easily explained by the fact that EB2 is also involved in mRNA export and translation^[@CR36],[@CR52]^. Taken together, these results suggest that in the absence of the viral protein EB2, the nuclear RNA exosome induces mRNA decay of some viral mRNAs. ", "However, the presence of EB2 counteracts this negative effect of the RNA exosome on these intronless mRNAs.", "Figure 3RNA exosome depletion increases both nuclear accumulation of viral intronless mRNAs and virion production. (**", "a**) *Immunoblot:* Western blotting analysis of HEK293~EBV∆BMLF1~ cells extracts showing protein depletion upon hRRP40 siRNA treatments. ", "HEK293~EBV∆BMLF1~ cells were treated with specific (hRRP40) or control (Firefly) siRNAs and transfected with a control vector (lanes 1 and 4) or an EB1 expression vector in order to induce the viral productive cycle (lanes 2, 3, 4 and 5) and transcomplemented with an EB2 expression vector (lanes 3 and 5). ", "Membranes were probed with the indicated antibodies (anti-hRRP40, -EB1, -EB2, -gp350 and -Tubulin). ", "Anti-Tubulin antibody was used as a loading control. (**", "b**) and (**c**) *Nuclear and cytoplasmic mRNA accumulation:* Quantification by RT-qPCR of nuclear (**b**) and cytoplasmic (**c**) EBV-encoding early (E) mRNA (BMRF1) or late (L) mRNAs (BDLF1, BdRF1 and BFRF3) in HEK293~EBV∆BMLF1~ cells transfected as described in (**a**). ", "Data are displayed as mean values normalized to the control siRNA and GAPDH mRNA as an internal control. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). \\* ", "Indicates a *P*-value of \\<0.05 and \\*\\* indicates a *P*-value of \\<0.01. (**", "d**) *Virus production:* HEK293~EBV∆BMLF1~ cells supernatant were collected 48 h after the induction of the productive cycle, filtered and used to infect Raji cells. ", "The number of EBV-infected Raji cells was evaluated by quantification of GFP-expressing cells 72 h later by FACS analysis. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). \\*\\* ", "Indicates a *P*-value of \\<0.01.", "\n\nSRSF3 is involved in viral mRNA stability {#Sec6}\n-----------------------------------------\n\nInteraction between EB2 and the splicing machinery has been previously documented^[@CR46],[@CR47]^ and interestingly, we recently found that SRSF3 (SRp20) depletion results in an increase in the cytoplasmic accumulation of EB2's target mRNAs^[@CR47]^. This suggests that EB2 functions by antagonizing SRSF3 and that the splicing factor SRSF3 might affect mRNA stability. ", "To investigate this possibility, the pTRE2-BDLF1 reporter plasmid was transfected into HeLa cells previously depleted of SRSF3 by siRNA treatment (Fig.", " [4a](#Fig4){ref-type=\"fig\"}). ", "The amount of nuclear and cytoplasmic BDLF1 mRNA expressed was then quantified by RT-qPCR (Fig.", " [4b,c](#Fig4){ref-type=\"fig\"}) and its nuclear stability assessed after addition of doxycycline (Fig.", " [4d](#Fig4){ref-type=\"fig\"}). ", "As shown in Fig.", " [4b,c](#Fig4){ref-type=\"fig\"}, EB2 increased the nuclear and cytoplasmic accumulation of BDLF1 mRNA. ", "Down-regulation of SRSF3 also led to an accumulation of nuclear and cytoplasmic BDLF1 mRNA and this accumulation was slightly affected by co-expression of EB2. ", "However, the cytoplasmic accumulation was less important than the nuclear accumulation because as we have previously shown^[@CR47]^, SRSF3 is required for the cytoplasmic export of this intronless mRNA. ", "This combined effect of EB2 and SRSF3 could be the consequence of either the presence of residual SRSF3 in cells treated with the siRNA directed against SRSF3 or a slight increase in the expression of EB2 as detected by western blot (Fig.", " [4a](#Fig4){ref-type=\"fig\"}, lane 4) or both. ", "During the time-course study, it appeared that down-regulation of SRSF3 stabilized the BDLF1 mRNA in the nucleus to a level similar to that seen when EB2 was expressed in the cells (Fig.", " [4d](#Fig4){ref-type=\"fig\"}). ", "These results suggest that EB2 antagonizes the destabilizing effect of SRSF3 on nuclear BDLF1 mRNA. ", "SRSF3 plays an important role in the metabolism of cellular mRNA, especially intronless mRNA. ", "It has been shown that SRSF3 has to be recruited onto the target mRNA to have an effect^[@CR47],[@CR48]^. In order to demonstrate that SRSF3 is recruited onto its target mRNA, HeLa cells were transfected with a HA-SRSF3 expression vector together with the BDLF1 or Luciferase reporter plasmids. ", "48 h later, the SRSF3 protein was immunoprecipitated and the presence of the BDLF1 or Luciferase mRNA associated with SRSF3 was analysed by RT-qPCR (Fig.", " [4e](#Fig4){ref-type=\"fig\"}). ", "The results show that both the BDLF1 and the Luciferase mRNA specifically interact with SRSF3. ", "Taken together, these data suggest that SRSF3, when bound onto its target mRNAs, recruits a complex involved in mRNA degradation.", "Figure 4Depletion of the splicing factor SRSF3 stabilizes intronless mRNAs in the nucleus. (**", "a**) *Immunoblot:* Western blotting analysis of HeLa cells transfected with the pTRE2-BDLF1 construct without or with an EB2 expression plasmid as indicated. ", "Cells were previously transfected with either a control siRNA (lanes 1 and 2) or a siRNA specific for SRSF3 (lanes 3 and 4). ", "The western blots were probed with either an anti-Flag antibody to detect Flag-tagged EB2, an anti-SRSF3 or an anti-Tubulin antibody as a loading control. (**", "b**) and (**c**) *Nuclear and cytoplasmic BDLF1 mRNA accumulation:* Quantification by RT-qPCR of nuclear (**b**) and cytoplasmic (**c**) BDLF1 mRNA from HeLa cells transfected as described in (**a**). ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). \\* ", "Indicates a *P*-value of \\<0.05 and \\*\\* indicates a *P*-value of \\<0.01. (**", "d**) *Nuclear BDLF1 mRNA stability:* Relative nuclear BDLF1 mRNA level after transcription inhibition (as a percentage of the amount at time point 0) was determined by RT-qPCR from HeLa cells transfected as described in (**a**). ", "Numbers above the panel refer to hours after doxycycline-mediated transcriptional shutoff of the BDLF1 intronless gene. ", "Half-lives (*t*~1/2~) and s.d were calculated from three independent experiments (*n* = 3). (**", "e**) *RNA immunoprecipitation:* HeLa cells were transfected with the pCMV-BDLF1 or the pCMV-RLuc reporter constructs together with a HA-SRSF3 expression plasmid as indicated. ", "SRSF3 containing complexes were immunoprecipitated using an anti-HA antibody. ", "RNA was then purified and the amount of specific BDLF1 and RLuc transcripts co-immunoprecipitated was evaluated by RT-qPCR. ", "The amount of SRSF3 expressed and immunoprecipitated in the samples was evaluated by western blot analysis. ", "Error bars represent the s.d. ", "from two independent experiments (*n* = 2). \\* ", "Indicates a *P*-value of \\<0.05 and \\*\\* indicates a *P*-value of \\<0.01.", "\n\nThe splicing factor SRSF3 and the RNA exosome complex are linked in the nuclear mRNA decay of unspliced mRNA {#Sec7}\n------------------------------------------------------------------------------------------------------------\n\nDown-regulation of SRSF3 expression in the cells leads to the stabilization of nuclear mRNAs whose expression are dependent on EB2. ", "Furthermore, we have previously shown that the nuclear RNA exosome pathway is the major contributor to the nuclear mRNA decay observed for EB2's target mRNAs. ", "In order to determine whether SRSF3 and the nuclear RNA exosome function (or not) within the same pathway, the Renilla luciferase and BDLF1 reporter plasmids were transfected into HeLa cells from which the RNA exosome (hRRP40 or hRRP46 core components) or/and SRSF3 were depleted (Fig.", " [5a](#Fig5){ref-type=\"fig\"}). ", "The nuclear accumulation of the reporter mRNA was then measured 24 h post-transfection (Fig.", " [5b](#Fig5){ref-type=\"fig\"}). ", "As expected, depletion of either the nuclear RNA exosome or SRSF3, alone, resulted in an increased accumulation of the reporter mRNAs. ", "However, co-depletion of either SRSF3 and hRRP40 or SRSF3 and hRRP46, did not further increase the amount of accumulated nuclear mRNA, suggesting that there was no additive effect for the co-depletion of both complexes. ", "This result argues in favour of the hypothesis that SRSF3 and the RNA exosome complex are in the same pathway and suggests that SRSF3 could aid in the recruitment of the nuclear RNA exosome onto the target mRNAs.", "Figure 5Intronless mRNAs do not hyperaccumulate in the nucleus upon co-depletion of the RNA exosome and the SRSF3 protein. (**", "a**) *Immunoblot:* Western blotting analysis showing protein depletion upon the indicated siRNA treatments. ", "Anti-Tubulin antibody was used as a loading control. \\* ", "indicates non-specific bands. (**", "b**) *Nuclear mRNA accumulation:* RT-qPCR analysis of nuclear RNAs from HeLa cells transfected with the indicated siRNAs together with the Renilla luciferase and BDLF1 intronless constructs. ", "Data are displayed as mean values normalized to the control siRNA and GAPDH mRNA as an internal control. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3).", "\n\nTo assay for a possible association between SRSF3 and the RNA exosome *in vivo*, each Flag-tagged subunit of the RNA exosome expressed in HeLa cells was immunoprecipitated, and co-purification of SRSF3 was assessed by western blotting. ", "hRRP6, one of the ribonucleases associated with the core RNA exosome, was used as a positive control. ", "These analyses revealed that SRSF3 was co-immunoprecipitated with each subunit of the RNA exosome (Fig.", " [6a](#Fig6){ref-type=\"fig\"}), suggesting an association of SRSF3 with the RNA exosome. ", "This interaction did not require the presence of RNA since SRSF3 still co-immunoprecipitated with the hMTR3 subunit of the RNA exosome following RNase treatment (Fig.", " [6b,c](#Fig6){ref-type=\"fig\"}). ", "In order to be certain that SRSF3 was not unspecifically immuno-precipitated, a control IP experiment using a non-specific IgG antibody was performed. ", "The result, presented in Fig.", " [S1](#MOESM1){ref-type=\"media\"}, shows that neither the Flag-tagged transfected proteins nor SRSF3 were immuno-precipitated by the control antibody.", "Figure 6The RNA exosome and the NEXT complex interact with the splicing factor SRSF3. (**", "a**) HA-SRSF3 was co-expressed with individual Flag epitope-tagged RNA exosome subunits in HeLa cells as indicated. ", "The top panels show western blotting of 1/10 of total extract ('Input'), and the bottom panels show western blotting following immunoprecipitation with an anti-Flag antibody (IP: Flag). ", "Anti-hRRP6 antibody was used as a positive control. (**", "b**) and (**c**) HA-SRSF3 and Flag-hMTR3 were expressed by transient transfection in HeLa cells as indicated. ", "Cellular extracts were immunoprecipitated with an anti-Flag antibody (**b**) or an anti-HA antibody (**c**) in the presence or absence of RNase A. The immunoprecipitated complexes were then analyzed by western blotting using an anti-Flag polyclonal antibody (**c**) or an anti-HA antibody (**b**). (**", "d**) and (**e**) HA-SRSF3 was coexpressed with individual Flag epitope-tagged hTRAMP (**d**) or NEXT (**e**) complex subunits in HeLa cells. ", "Cellular extracts were immunoprecipitated with an anti-Flag antibody and then the immunoprecipitated complexes were analyzed by western blotting using an anti-Flag polyclonal antibody to visualize the hTRAMP and NEXT complex subunits (top panel), or an anti-HA antibody to detect SRSF3 (bottom panel). (**", "f**) and (**g**) HA-SRSF3 and RBM7 Flag-tagged NEXT subunit were expressed by transient transfection in HeLa cells as indicated. ", "Cellular extracts were immunoprecipitated with an anti-Flag antibody (**f**) or with an anti-HA antibody (**g**) in the presence or absence of RNase A. The immunoprecipitated complexes were then analyzed by western blotting using an anti-Flag polyclonal antibody (**g**) or an anti-HA antibody (**f**). (**", "h**) HA-SRSF3 was expressed by transient transfection in HeLa cells previously treated with control or specific siRNAs for the NEXT complex, either alone or together with the hMTR3 Flag-tagged RNA exosome subunit. ", "Cellular extracts were immunoprecipitated with an anti-Flag antibody in the presence or absence of RNase A. The immunoprecipitated complexes were then analyzed by western blotting using an anti-Flag polyclonal antibody to visualize hMTR3 (top panel), anti-RBM7 or anti-ZCCHC8 antibodies (middle panels), or an anti-HA antibody to detect SRSF3 (bottom panel). \\* ", "indicates non-specific bands.", "\n\nTaken together, our findings indicate that SRSF3 associates, either directly or indirectly, with the RNA exosome complex in cells. ", "The RNA exosome consists of a catalytically inactive nine-subunit core that acquires its ribonucleolytic activity through its association with the two subunits, hRRP6 and hDIS3/hRRP44. ", "Although active *in vitro*, the RNA exosome complex requires appropriate adaptors to be fully functional *in vivo*. ", "Different subtypes of adaptors like the hTRAMP and NEXT complexes have been described in the nucleus of mammalian cells. ", "These adaptors direct the RNA exosome to its target substrates, are specific for discrete processing/degradation pathways and can even recognize specific RNA features. ", "We thus asked whether the RNA exosome could be recruited to SRSF3 by one of these adaptors. ", "Interestingly, when Flag-tagged versions of each hTRAMP subunit were individually expressed along with HA-SRSF3 in HeLa cells, none of these hTRAMP components was able to pull down SRSF3 (Fig.", " [6d](#Fig6){ref-type=\"fig\"}). ", "However, the same experiment made with individual subunits of the NEXT complex showed that RBM7 co-precipitated with SRSF3 (Fig.", " [6e](#Fig6){ref-type=\"fig\"}). ", "This association was resistant to RNase treatment (Fig.", " [6f](#Fig6){ref-type=\"fig\"}) and was also found in the reverse experiment in which HA-SRSF3 was immunoprecipitated (Fig.", " [6g](#Fig6){ref-type=\"fig\"}). ", "Taken together, these results show that SRSF3 interacts with both the RNA exosome and the NEXT complex and suggest that the nuclear RNA exosome can be recruited onto specific target mRNAs by SRSF3 via its association with the NEXT complex. ", "In order to confirm this hypothesis, the NEXT subunits, RBM7 and ZCCHC8, were depleted from cells using specific siRNAs, and the exosome subunit hMTR3 immunoprecipitated to determine whether SRSF3 was still associated with the RNA exosome (Fig.", " [6h](#Fig6){ref-type=\"fig\"}). ", "As expected, immunoprecipitation of the RNA exosome complex (through its hMTR3 subunit) allowed co-immunoprecipitation of SRSF3 together with the NEXT components, RBM7 and ZCCHC8. ", "However, when expression of RBM7 and ZCCHC8 was downregulated, co-immunoprecipitation of SRSF3 with the RNA exosome complex (again through its hMTR3 subunit) was greatly impaired. ", "It has to be noted that downregulation of the NEXT subunits (RBM7 and ZCCHC8) induces a decrease in the level of SRSF3, further suggesting that NEXT and the SRSF3 protein are linked. ", "Taken together, these results argue strongly in favor of a direct interaction between SRSF3 and the NEXT subunit RBM7 and suggest that the interaction between SRSF3 and the RNA exosome is mediated by RBM7.", "\n\nFunctional implications of the SRSF3 and RNA exosome interaction {#Sec8}\n----------------------------------------------------------------\n\nIn order to study the functional relevance of the interactions we had found between the SR protein SRSF3 and the RNA exosome, we then assessed the effect of depletion of various components of these complexes on the steady-state levels of our reporter mRNAs. ", "For this purpose, HeLa cells were treated with siRNAs targeting either (i) the RNA exosome core subunit hRRP40, (ii) the SRSF3 protein, (iii) the NEXT complex subunits RBM7 and ZCCHC8, or (iv) the RNA helicase hMTR4.", "\n\nKnockdown efficiencies were controlled by western blotting analyses (Fig.", " [7a](#Fig7){ref-type=\"fig\"}) and by quantification of a specific ncRNA known to be degraded by the RNA exosome and the NEXT complex, the PROMPT proEXT1 (Fig.", " [7b](#Fig7){ref-type=\"fig\"})^[@CR27]^. As shown previously, RT-qPCR analyses revealed that the level of RLuc and BDLF1 mRNAs increased upon depletion of hRRP40 and SRSF3 (Fig.", " [7b](#Fig7){ref-type=\"fig\"}). ", "Depletion of hMTR4 also led to an increase in mRNA accumulation similar to that observed with the depletion of SRSF3 or the RNA exosome. ", "RBM7 or ZCCHC8 down-regulation also led to an increased accumulation of the reporter mRNA, but their impact was not as efficient as the down-regulation of hMTR4 or SRSF3. ", "It is important to note that the downregulation of RBM7 and ZCCHC8 was not as efficient as the downregulation of SRSF3. ", "However, down-regulation of hMTR4 and SRSF3 also decreased the expression level of both RBM7 and ZCCHC8 (Fig.", " [7a](#Fig7){ref-type=\"fig\"}), which might explain why the depletion of these two factors had a stronger impact. ", "In summary, the data collectively suggest that SRSF3, the NEXT complex and the RNA exosome are linked in the process of nuclear mRNA decay.", "Figure 7The SRSF3 protein and the NEXT complex associated with the RNA exosome are functionally connected to degrade intronless mRNAs. (**", "a**) Western blot analysis showing protein depletion upon the indicated siRNA treatments. ", "Anti-Tubulin antibody was used as a loading control. \\* ", "Indicates non-specific bands. (**", "b**) *mRNA stabilization by exosome depletion:* RT-qPCR analysis of nuclear RNAs from HeLa cells transfected with the indicated siRNAs together with the Renilla luciferase and BDLF1 intronless constructs. ", "proEXT1 PROMPT served as a positive control. ", "Data are displayed as mean values normalized to the control siRNA and GAPDH mRNA as an internal control. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). (**", "c**) Western blotting analysis showing the overexpression of the Flag tagged-hMTR4 protein and depletion of the SRSF3 protein after siRNA treatment. ", "Anti-Tubulin antibody was used as a loading control. (**", "d**) RT-qPCR analysis of nuclear RNAs from HeLa cells transfected with the indicated siRNAs together with the Renilla luciferase and BDLF1 intronless constructs. ", "Data are displayed as mean values normalized to the control siRNA and GAPDH mRNA as an internal control. ", "Error bars represent the s.d. ", "from three independent experiments (*n* = 3). (**", "e**) Western blotting analysis showing expression of the NEXT components RBM7, ZCCHC8 and hMTR4 in HeLa cells transfected with either a control siRNA (lane 1), or a siRNA specific for SRSF1 (lane 2), SRSF3 (lane 3) or SRSF7 (lane 4). ", "Anti-Tubulin antibody was used as a loading control. ", "\\*Indicates non-specific bands.", "\n\nThe observation that SRSF3's depletion impacts on RBM7, ZCCHC8 and hMTR4 levels (Fig.", " [7a](#Fig7){ref-type=\"fig\"}) suggests that a strong link exists between the NEXT complex and the SR protein, SRSF3. ", "In order to demonstrate that these results were not due to an off-target effect of the specific siRNA used, we repeated the experiments with two other independent siRNAs targeting SRSF3 and obtained similar results (Fig.", " [S2](#MOESM1){ref-type=\"media\"}). ", "In order to prove that the effect seen at the protein level was not the result of an effect of SRSF3 on hMTR4 stability or splicing, both the amount and the integrity of the hMTR4 mRNA were controlled by RT-PCR in cells treated with the siRNA directed against SRSF3. ", "From this analysis we concluded that downregulation of SRSF3 does not significatively affect the expression level or splicing of hMTR4 mRNA (Fig.", " [S2](#MOESM1){ref-type=\"media\"}). ", "This argues in favor of a direct impact of SRSF3 on the stability of the hMTR4 protein. ", "This observation could suggest that SRSF3 exerts its effect on the stability of intronless mRNAs via a downregulation of hMTR4 expression. ", "To test this hypothesis, we analyzed, by RT-qPCR, the stability of the BDLF1 and Luciferase reporter genes in cells overexpressing hMTR4 - and in which SRSF3 was first downregulated (Fig.", " [7d](#Fig7){ref-type=\"fig\"}). ", "As previously found, downregulation of SRSF3 impacted hMTR4, RBM7 and ZCCHC8 expression levels. ", "However, in these conditions, overexpression of hMTR4 did not rescue the expression level of RBM7 and ZCCHC8 (Fig.", " [7c](#Fig7){ref-type=\"fig\"}). ", "Moreover, the RLuc and BDLF1 mRNAs accumulated in the cell nucleus at similar levels as in the absence of hMTR4 overexpression (Fig.", " [7d](#Fig7){ref-type=\"fig\"}). ", "This demonstrates that SRSF3's effect on the stability of the intronless mRNAs is not due to hMTR4 downregulation alone.", "\n\nFinally, in order to determine whether the link between the SR protein SRSF3 and the RNA exosome is specific for SRSF3, the expression of two other SR proteins, SRSF1 (ASF/SF2) and SRSF7 (9G8), was down-regulated and the level of the NEXT complex components assessed by western blot (Fig.", " [7e](#Fig7){ref-type=\"fig\"}). ", "SRSF3 was the only protein of the three whose down-regulation was able to disturb the expression of NEXT complex components. ", "This suggests that SRSF3 makes a specific contribution to mRNA decay that other SR proteins do not.", "\n\nDiscussion {#Sec9}\n==========\n\nIn this study, we demonstrate that some mRNAs synthesized from intronless genes - which is the case for most herpesviruses mRNAs produced during the viral productive cycle - are unstable and are degraded in the nucleus. ", "We also show that the splicing factor SRSF3 is involved in the nuclear destabilization of these intronless mRNAs by the RNA exosome. ", "The RNA exosome complex is recruited onto the target mRNAs by the NEXT complex associated with SRSF3 via its RBM7 component. ", "In addition, SRSF3 appears to play an important role in the stabilization of the NEXT complex. ", "The interaction between the NEXT complex, the RNA exosome and the SRSF3 protein suggests that the mRNA degradation machinery is co-transcriptionally recruited onto the mRNA.", "\n\nHerpesvirus mRNAs synthesized from intronless genes, inefficiently accumulate in the cytoplasm in the absence of a specific viral protein^[@CR40],[@CR53],[@CR54]^. In the case of EBV, viral mRNAs synthesized from intronless genes, not only accumulate in the cytoplasm but are also stabilized in the cell nucleus. ", "However, it is surprising to observe that the BDLF1 mRNA accumulates in the cytoplasm, although its nuclear half-life is over 7 hours. ", "In effect, it should not be expected to stay in the nucleus for such a long period if it is efficiently exported by EB2. ", "This suggests that EB2 has only a minor role in mRNA export and that its major function is to stabilize its target mRNAs. ", "It has been shown that some RNA molecules can spend hours in the nucleus before being exported to the cytoplasm. ", "mRNA nuclear retention can effectively buffer the burst of mRNA transcription which could induce temporal fluctuations in the level of cytoplasmic mRNA^[@CR55]^. Since, in our assay, the BDLF1 gene is under the control of a very strong promoter, we can hypothesize that the high level of BDLF1 mRNA produced may be retained in the nucleus and protected from nuclear degradation when EB2 is expressed.", "\n\nBased on our findings, we propose a novel model for the coupling between pre-mRNA splicing and nuclear RNA quality control revealed by our study on the function of the EBV protein EB2 (Fig.", " [8](#Fig8){ref-type=\"fig\"}). ", "In the absence of EB2 expression, the SR protein SRSF3 loaded onto the viral mRNAs expressed from intronless genes, recruits the splicing machinery and the NEXT complex together with the nuclear RNA exosome, leading to the destabilization of the mRNAs and their decay. ", "By contrast, when EB2 is expressed, it interacts with SRSF3^[@CR46],[@CR47]^, hence preventing the nuclear RNA exosome's access to the mRNAs that are then protected from the degradation machinery. ", "However, overexpression of EB2 does not appear to inhibit the interaction between SRSF3 and the NEXT complex suggesting that EB2 and RBM7 do not interact with the same domain of SRSF3. ", "Another way in which EB2 could interfere with RNA exosome recruitment, could be by changing the ability of the SRSF3/NEXT complex loaded onto the mRNA, to interact with the RNA exosome. ", "Concomitantly, the interaction between EB2 and the TAP-NXF1/p15 complex favors mRNAs export from the nucleus^[@CR42]^ and their translation in the cytoplasm^[@CR36],[@CR52]^.Figure 8Model. ", "During their transcription, intronless nascent mRNAs associate with numerous RNA-binding proteins like the cap-binding complex (CBC), the nuclear poly(A)-binding protein (PABPN1), and some spliceosome components such as the SRSF3 protein. ", "In the absence of EB2, the association of the splicing factor SRSF3 to newly synthesized mRNAs led to the recruitment of the RNA exosome via the NEXT complex, and consequently to their degradation in the nucleus. ", "In the presence of EB2, EB2 stabilizes these intronless mRNAs in the nucleus and then allow their export to, and translation, in the cytoplasm.", "\n\nInterestingly, our present work demonstrating the implication of specific components of the spliceosome (i. e. SRSF3) and the RNA exosome in the nuclear degradation of intronless RNAs extends a previous work from Bresson *et al*. ", "showing that the nuclear poly(A)-binding protein and the exosome are also implicated in the nuclear degradation of intronless RNAs^[@CR56]^.\n\nAs discussed above, our work provides evidence of an interaction between SRSF3 and the NEXT complex. ", "In this context, it is noteworthy that depletion of SRSF3 induces a concomitant depletion of all three components of the NEXT complex - RBM7, ZCCHC8 and hMTR4. ", "Similarly, depletion of MTR4 correlates with a depletion of both RBM7 and ZCCHC8. ", "Taken together, these observations suggest that SRSF3 contributes to the stabilization of the NEXT complex, hence also increasing the stability of its components. ", "However, from our results, it would appear that the depletion of hMTR4 is complete when SRSF3 is downregulated. ", "This is surprising since only a fraction of the hMTR4 protein pool is expected to be associated with the NEXT complex. ", "In effect, in human cells, the hMTR4 RNA helicase which is required for all activities of the nuclear RNA exosome^[@CR22]^ is associated with the NEXT, PAXT and hTRAMP complexes. ", "However, the antibody used in our assays to detect hMTR4 is not very efficient and only detects hMTR4 expressed at a high level. ", "The level of remaining hMTR4, following SRSF3 depletion, is probably insufficient to be detected by the antibody. ", "Nevertheless, the modification of the protein level of the different components of the NEXT complex observed when SRSF3 is depleted, argues in favor of the formation of a stable complex between SRSF3 and NEXT and comforts the hypothesis that SRSF3 plays an important role in RNA exosome activity. ", "In addition, the recently published results of independent proteomics analysis^[@CR21],[@CR22],[@CR27]^ demonstrating that RBM7, ZCCHC8, hMTR4 and SRSF3 co-purify, completely support our conclusion that SRSF3 interacts with the NEXT complex and not only with RBM7, outside of the NEXT context.", "\n\nIn the cell nucleus, several types of coding and non-coding RNAs are synthesized and processed. ", "To ensure the survival of cells, only functional competent RNAs must be produced. ", "All the aberrant and inproperly processed transcripts have to be recognized and degraded by RNA quality control mechanisms^[@CR57],[@CR58]^. mRNA expression is the result of a coupling between transcription by RNA polymerase II and pre-mRNA processing reactions such as 5′-end capping, splicing, 3′-end maturation and mRNP formation. ", "All these events can affect the localization, translation and degradation of the mRNA. ", "Recent data have shown that the 5′cap of the mRNA associated with the cap-binding complex (CBC) is a type of hub that mediates key events in the life of the mRNA^[@CR59],[@CR60]^. Indeed, as recently demonstrated, the CBC is very important both for the recruitment of the mRNA export machinery and the recruitment of the RNA exosome through the NEXT complex^[@CR27],[@CR61]^, suggesting a direct link between mRNA 5′-end capping and nuclear RNA surveillance. ", "Interestingly, it has also been shown that the CBC is necessary, although not sufficient, for co-transcriptional spliceosome assembly in yeast^[@CR62]^. This demonstration of an essential link between the CBC and spliceosome assembly indicates that 5′-end capping couples transcription to pre-mRNA splicing and probably also to mRNA decay via recruitment of the RNA exosome by specific splicing factors. ", "Our results support a direct coupling between the splicing and the mRNA degradation machineries. ", "Thus, in addition to the recruitment of NEXT by the CBC complex, there appears to be an alternative mode of recruitment of the nuclear RNA exosome involving specific spliceosome components.", "\n\nMaterials and Methods {#Sec10}\n=====================\n\nPlasmids {#Sec11}\n--------\n\npCMV-Renilla Luciferase, pCMV-HA-SRSF3, pCMV-EB1, pCI-Flag-EB2 and pCMV-BDLF1 have been described elsewhere^[@CR34],[@CR36],[@CR40],[@CR47]^. For pTRE2-BDLF1, the EBV BDLF1 CDS was amplified by PCR and cloned into the BamHI and XbaI sites of the pTRE2 expression plasmid (Clontech). ", "The pTet-On vector was supplied by Clontech (Tet-Off and Tet-On gene expression systems). ", "Individual Flag-tagged RNA exosome components, hTRAMP and NEXT subunits were a generous gift from Drs Frederick W. Alt and Torben Heick Jensen^[@CR21],[@CR63]^. All plasmid sequences are available upon request.", "\n\nCell culture and transfections {#Sec12}\n------------------------------\n\nHeLa cells were grown at 37 °C in DMEM supplemented with 10% Foetal Bovine Serum (FBS) and penicillin-streptomycin. ", "HEK293~EBV∆BMLF1~ cells^[@CR40]^ were maintained in DMEM supplemented with 10% FBS, penicillin-streptomycin and hygromycin B (100 µg/ml). ", "Raji cells were maintained in RPMI supplemented with 10% FBS and penicillin-streptomycin. ", "48 h prior to DNA transfection, siRNAs were transfected using INTERFERin transfecting reagent (Polyplus-transfection) at 20 nM final concentration, following the manufacturer's instructions (see Table [S1](#MOESM1){ref-type=\"media\"} of Supplementary Data for the list of siRNAs used). ", "Plasmid transfections were performed using cationic polymers (jetPEI from Polyplus-transfection) as specified by the manufacturer and 24 h or 48 h after DNA transfection, cells were collected for further analysis.", "\n\nVirus production and titration {#Sec13}\n------------------------------\n\nHEK293~EBV∆BMLF1~ cells were transfected with pCMV-EB1 in order to activate the viral productive cycle. ", "Cell supernatants were harvested 48 h post-transfection and filtered through a 0.45 -µm pore size filter. ", "Raji cells (1 × 10^5^) were incubated with 1 ml of cell supernatant for 6--8 h at 37 °C in a 24-well plate. ", "Cells were then washed, re-suspended in 1 ml of RPMI supplemented with 10% FBS and penicillin-streptomycin and incubated for an additional 72 h at 37 °C. ", "GFP-expressing Raji cells were quantified by FACS analysis.", "\n\nPulse-chase mRNA decay assays {#Sec14}\n-----------------------------\n\nHeLa cells were transfected in 60 mm dishes with the pTet-On vector and the pTRE2-BDLF1 plasmid. ", "24 h after transfection, 1 μg/ml of doxycycline (Sigma-Aldrich) was added to repress transcription. ", "Cells were then harvested at the time points indicated in individual experiments. ", "Actinomycin D treatment of HeLa cells was performed using 5 μg/ml of actinomycin D (Sigma-Aldrich).", "\n\nRNA extraction and RT-qPCR analysis {#Sec15}\n-----------------------------------\n\nNuclear and cytoplasmic RNAs extraction were performed as previously described^[@CR36]^ and reverse-transcribed using the qScript cDNA Synthesis Kit (Quanta BioSciences). ", "Standard PCRs were performed using GoTaq DNA polymerase (Promega) and the PCR-amplified fragments analyzed on 2% agarose gels. ", "All PCR reactions were made using several dilutions of the RT products in order to insure that the RT-PCR results were in the linear range of the PCR reaction. ", "We used ß-actin mRNA and U6 snRNA as internal controls. ", "qPCRs were performed using FastStart Universal SYBR Green Master (Rox) (Roche) on an Applied Biosystems 7000 thermocycler. ", "Cycling conditions were 5 min at 95 °C and 45 cycles of 15 s at 95 °C, 30 s at 60 °C on a 96-well thermoblock. ", "This program was followed by melting curve analysis in order to verify the specificity of the PCR products. ", "PCR results were normalized with the parallel amplification of GAPDH mRNA. ", "The various set of primers used in the study are listed in Table [S2](#MOESM1){ref-type=\"media\"} of Supplementary Data.", "\n\nCo-immunoprecipitation assays {#Sec16}\n-----------------------------\n\n48 h after transfection, cells from a 100 mm plate were washed in PBS (Life Technologies) and collected in 0.5 ml of RSB100 buffer (10 mM Tris-HCl pH 7.4, 100 mM NaCl, 2.5 mM MgCl~2~), containing 0.5% Triton X-100 and protease inhibitors (Roche). ", "Cells were gently lysed by sonication and centrifuged (4000 g, 15 min) at 4 °C. ", "Supernatants were incubated with 2 μg of a mouse anti‐Flag M2 antibody (Sigma-Aldrich, F3165) or 2 μg of a mouse anti‐HA antibody (Sigma-Aldrich, H3663) overnight at 4 °C, 10 r.p.m. ", "rotation. ", "Protein-antibody complexes were recovered with 40 μl of Protein G Sepharose 4 Fast Flow beads (GE Healthcare Life Sciences) during 2 h at 4 °C, 10 r.p.m. ", "and recovered complexes were washed extensively with RSB100/0.5% Triton X-100 buffer. ", "RNase treatment was performed by adding 2 μl of RNase A (10 mg/ml) to the incubation mixture where indicated. ", "Immunopurified proteins were then resuspended in SDS‐loading buffer and analysed by western blotting.", "\n\nRNA immunoprecipitation {#Sec17}\n-----------------------\n\nRIPs were carried out from uncrosslinked extracts. ", "Briefly, 10^8^ cells were washed in PBS and then resuspended in 1 ml of RSB100 buffer (10 mM Tris-HCl pH 7.4, 100 mM NaCl, 2.5 mM MgCl~2~), containing 0.5% Triton X-100, protease inhibitors (Roche) and RNasin (Promega). ", "Extracts obtained after centrifugation were incubated with 2 μg of a mouse anti‐HA antibody (Sigma-Aldrich, H3663) overnight at 4 °C. ", "PureProteome ProteinA/G Mix Magnetic Beads (Merck Millipore) were mixed and coated with PBS + 10% BSA, supplemented with tRNA and RNasin overnight. ", "After re-equilibration with RSB100 buffer, the beads were added to the lysate for 2 h at 4 °C before extensive washing in RSB100 buffer containing 0.5% Triton X-100 and 0.05% NP-40. ", "Half of the immunoprecipitates were analyzed by Western blotting. ", "The other half were treated 15 min at 55 °C with proteinase K (0.5 mg/ml[)]{.ul} and SDS (0.1%), and the RNAs were recovered by phenol/chloroforme extraction and ethanol precipitation.", "\n\nWestern blot analysis {#Sec18}\n---------------------\n\nCell lysates were prepared in RSB100/0.5% Triton X-100 and protease inhibitors (Roche) buffer and quantified by Bradford assay. ", "Equivalent amounts of each sample were resolved by SDS-PAGE and analysed by western blotting using the following antibodies: mouse anti-Tubulin (Sigma-Aldrich, clone B-5-1-2; 1:1000), mouse monoclonal anti-EB1 (Z125 antibody; 1:1000)^[@CR64]^, rabbit polyclonal anti-EB2 (1:500)^[@CR40]^, mouse monoclonal anti-gp350 (1:500)^[@CR65]^, rabbit polyclonal anti-Flag (Sigma-Aldrich, F7425; 1:2500), rabbit polyclonal anti-HA (Sigma-Aldrich, H6908; 1:1000), mouse monoclonal anti-SRSF1 (Life Technologies, clone 7B4; 1:1000), mouse monoclonal anti-SRSF3 (Life Technologies, clone 96; 1:1000), rabbit polyclonal anti-SRSF7 (Sigma-Aldrich, HPA043850; 1:1000), rabbit polyclonal anti-XRN2 (a gift from N J Proudfoot; 1:1000), rabbit polyclonal anti-DCP2 (a gift from M Kiledjian; 1:1000), mouse polyclonal anti-hDIS3 (Abnova, H00022894-A01; 1:1000), rabbit polyclonal anti-hRRP40, anti-hRRP46 and anti-hRRP6 (a gift from Ger J M Pruijn; 1:1000), human anti-hMTR4 (a gift from Ger J M Pruijn; 1:1000), rabbit polyclonal anti-ZCCHC8 (Sigma-Aldrich, HPA037484; 1:1000) or rabbit polyclonal anti-RBM7 (Sigma-Aldrich, HPA013993; 1:1000).", "\n\nStatistical analysis {#Sec19}\n--------------------\n\nAll results are expressed as means ± standard deviations of experiments independently repeated at least three times. ", "Unpaired Student's *t*-test was used to evaluate the statistical difference between samples and significance was evaluated with *P*-values as follows: \\* *P* \\< 0.05; \\*\\* *P* \\< 0.01; \\*\\*\\* *P* \\< 0.001.", "\n\nElectronic supplementary material\n=================================\n\n {#Sec20}\n\nSupplementary Information\n\n**Publisher\\'s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.", "\n\nElectronic supplementary material\n=================================\n\n**Supplementary information** accompanies this paper at 10.1038/s41598-018-31078-1.", "\n\nWe are grateful to Prof. Torben Heick Jensen and Dr. Michal Lubas (Aarhus University, Denmark), Prof. Frederick W. Alt (Howard Hughes Medical Institute, Boston, USA), Prof. Ger J. M. Pruijn (Radboud University, Netherlands), Prof. Nick J. Proudfoot (University of Oxford, UK) and Prof. Megerditch Kiledjian (Rutgers University, USA) for providing plasmids and antibodies. ", "We are thankful to Dr. R. Buckland and Dr. E.P. Ricci for critical reading of the manuscript. ", "We also acknowledge the contribution of the 'AniRA - Analyse génétique' platform of the SFR BioSciences (UMS3444/US8). ", "We finally thanks our funders: La Ligue Contre le Cancer (LNCC) Rhône, the Fondation ARC, the 'Institut National de la Santé et de la Recherche Médicale' (INSERM) and the 'Agence Nationale pour la Recherche' (ANR n° RPV09056CSA). ", "HG, EM and FM are part of the French Laboratory of Excellence project ECOFECT (ANR-11-LABX-0048).", "\n\nF.M. and H.G. designed the research. ", "F.M. and H.G. performed the experiments. ", "F.M. and H.G. analyzed the data. ", "E.M., A.C., N.H.B. and E.B. provided materials and resources. ", "H.G. wrote the paper, with input from the other co-authors.", "\n\nCompeting Interests {#FPar1}\n===================\n\nThe authors declare no competing interests.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "Nutritional aspects of cancer-related fatigue.", "\nFatigue, which may well be the most common experience of patients with cancer, remains underappreciated by health care professionals. ", "Perhaps one reason is that because of its complexity and many components, fatigue is not completely understood. ", "Knowledge of fatigue models, such as the integrated Fatigue Model of Piper, can help dietitians identify potential causes of fatigue such as activity-rest patterns, and identification can lead dietitians to early intervention. ", "Understanding cancer treatment factors, such as nausea and decreased participation in activities of daily living, that are believed to play a part in fatigue form another level on which dietitians can provide intervention. ", "Through intervention, dietitians, working with patients and other members of the multidisciplinary team, may increase the understanding and appreciation of fatigue as well as provide relief from it. ", "Efforts to maintain nutritional status can decrease or prevent some of the fatigue associated with cancer and its treatment. ", "Therefore, the goal of clinical dietitians who work with a fatigued patient with cancer is to use nutrition management to minimize therapeutic side effects and maximize the patient's nutritional parameters." ]

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[ "A backlight is a form of illumination used in liquid crystal displays (LCDs). ", "Because an LCD does not produce its own light, another light source, the “backlight,” illuminates the LCD so that a visible image is produced. ", "LCDs with backlights are used in many electronic user devices, such as flat panel displays, LCD televisions, mobile devices such as cell phones, etc.", "\nSome LCDs use a backlight that gives off a uniform light over its surface, such as an electroluminescent panel (ELP). ", "Other LCDs use multiple light sources to enable localized dimming, such as light emitting diodes (LEDs), or cold or hot cathode fluorescent lamps (CCFLs or HCFLs). ", "Compared to LCDs with a single uniform backlight, well-designed LCDs with local dimming backlights can exhibit higher energy/power efficiencies, better image qualities, and mura-free (defect free) display qualities." ]

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[ "Q:\n\nIs $\\Bbb{Q}[\\sqrt[2]{2},\\sqrt[3]{2}]$ isomorphic to $\\Bbb{Q}[x,y]/(x^2-y^3)$?", "\n\nSay we have the field extension $\\Bbb{Q}[\\sqrt[2]{2},\\sqrt[3]{2}]$. Is this field isomorphic to $\\Bbb{Q}[x,y]/(x^2-y^3)$?", "\nI made some preliminary investigation, and this doesn't seem to be true. ", "Is $\\Bbb{Q}[\\sqrt[2]{2},\\sqrt[3]{2}]$ isomorphic to $\\Bbb{Q}[x,y]/(x^2-y^3,x^2-2,y^3-2)$?", "\nHow should we deal with such situations?", "\n\nA:\n\nHere's one way to show it (there are many others!). ", " The strategy is to use the universal property of quotient rings to construct homomorphisms that can differentiate the two rings. ", " In particular, there is a homomorphism $f:\\mathbb{Q}[x,y]\\to\\mathbb{Q}$ which sends $x$ and $y$ to $0$. Since $f(x^2-y^3)=0$, this factors through a homomorphism $g:\\mathbb{Q}[x,y]/(x^2-y^3)\\to\\mathbb{Q}$. On the other hand, there does not exist any homomorphism $h:\\mathbb{Q}[\\sqrt{2},\\sqrt[3]{2}]\\to\\mathbb{Q}$, since $h(\\sqrt{2})$ would need to be some element of $\\mathbb{Q}$ whose square is $2$ and no such element exists.", "\n\n" ]

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[ "The present invention relates generally to an antenna assembly, and more particularly to a loop antenna assembly for a hand-held radio frequency transceiver, such as a cellular telephone or PCS device operating in the 800-900 or 1850-1990 MHz. ", "frequency ranges, respectively." ]

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[ "Lindsie Chrisley is speaking out on the heels of her reality television star dad and his wife being indicted on tax evasion and other federal charges this week.", "\n\n“Lindsie would like to thank all of her fans and supporters. ", "The circumstances Todd and Julie find themselves in, is quite unfortunate,\" her Atlanta-based attorney Musa Ghanayem released in a statement on her behalf. “", "It was reported that Lindsie was the source of the information that led to her father’s arrest. ", "That is untrue.\"", "\n\n“Chrisley Knows Best” stars Todd and Julie Chrisley pleaded not guilty Wednesday to tax evasion and other fraud charges that their lawyers said stemmed from false allegations made by a former employee.", "\n\nThe allegations:'Chrisley Knows Best' star Todd Chrisley pleads not guilty, says ex-employee framed him\n\nThe couple, who live in Nashville, were indicted just one day earlier and entered the pleas during their initial court appearance before a federal magistrate judge in Atlanta.", "\n\n“Lindsie has been a constant target of lies, harassment and threats from her family and as a result, has been distancing herself from the Chrisley family since 2017,\" the statement continues. “", "Lindsie is currently processing the events that have unfolded. ", "We will continue to cooperate with law enforcement and pray for a just resolution.”", "\n\nBefore the couple's indictment was handed up, Todd Chrisley had already aired his grievances on social media, blaming a former employee for the federal tax evasion allegations.", "\n\nThe 12-count indictment issued against the couple also includes charges of wire fraud, conspiracy to commit bank fraud and conspiracy to defraud the United States, U.S. Attorney Byung J. \"BJay\" Pak said at a news conference.", "\n\nOn Thursday Todd and Julie Chrisley remained free on $100,000 bond. ", "Under conditions of their release they were required to surrender their passports and are not allowed to travel outside of parts of Georgia and Tennessee without letting their probation officers know.", "\n\nThe bond conditions were based on requests from the couple's lawyers for travel permission so they can continue filming the reality television show." ]

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[ "Gravity flow systems, such as sewer systems, regularly require maintenance or repair and it is well known to block, or isolate, a sewer pipe from fluid flowing therethrough, to allow the necessary work to be carried out downstream from the blockage. ", "The plugs in common usage for creating the necessary blockage are expandable or inflatable plugs, an example of which is the Muni Ball® manufactured by Cherne Industries, Inc. of Minnesota U.S.A. This inflatable plug includes an inflatable central portion capped at each end by steel plates. ", "The central portion is inflated using pneumatic pressure to grip the interior surface of the pipe, thereby preventing fluid flow therethrough.", "\nThere are considerable problems associated with such inflatable plugs. ", "Firstly, there are problems associated with the under-pressure of the plug i.e. when insufficient pneumatic pressure is supplied, for example in the event of failure of the pneumatic pump. ", "In this case, the sealing grip of the inflatable portion against the interior surface of the pipe will be lost leading initially to undesirable leaks and then ultimately to the potentially high speed projection of the plug within the pipe as a result of water pressure which may have built up behind the plug. ", "The projectile plug can then travel, with considerable force and energy, towards the workers in the downstream section of the pipe, which is obviously highly undesirable. ", "To avoid such problems, it is intended that an inflatable plug is used in conjunction with a restraint but such restraints may be readily omitted by a user. ", "Secondly, there are problems associated with over-pressurisation of the plug. ", "If too much pressure is applied by the pneumatic pump, the inflatable portion may over-inflate to such an extent that it can burst, or otherwise fail. ", "This will lead to a flood of water towards any workers present down-stream of the plug, together with high-speed projection of the remains of the plug, as a result of both the build-up of water pressure and the bursting action of the plug. ", "Several workers have lost their lives as a result of an inflatable plug bursting or failing in such a manner.", "\nIn attempt to protect workers in such conditions, it is recommended that an exclusion area is created around the plug. ", "It is also recommended that the pressure supplied to the plug be continuously monitored to ensure that under- or over-pressure does not occur. ", "Adhering to both of these recommendations can be inconvenient for workers, particularly when space requirements are at a premium to make exclusion areas difficult to maintain and when personnel is limited to allow for constant monitoring of pressure supply levels.", "\nFinally, the requirement for the provision of a compressor to provide compressed air to inflate the plug can be inconvenient as it is necessary to transport the compressor to the location of an access shaft of the gravity flow system and access shafts may be located in rough or steep terrain.", "\nThe present invention is directed towards providing a device for isolating a pipe, or sections of a pipe, which substantially ameliorates the problems associated with the known inflatable plugs.", "\nAny discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is solely for the purpose of providing a context for the present invention. ", "It is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present invention as it existed before the priority date of each claim of this application." ]

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[ "On Tuesday, the European Union (EU) announced a new online speech code to be enforced by four major tech companies, including Facebook and YouTube and on Wednesday, Facebook deleted the account of Ingrid Carlqvist, Gatestone’s Swedish expert who has been reporting on the Muslim crime and rapes against Swedish women in Sweden.", "\n\nThe Gatestone Institute, a non-partisan, not-for-profit international policy council and think tank reported through email and stated that her account was deleted because Ingrid had posted their latest video to her Facebook feed called, called “Sweden’s Migrant Rape Epidemic.”", "\n\nThe Gatestone Institute reported that it is a video version of this research paper that they had published last year and that the video has gone viral and has had more than 80,000 views in its first two days after production.", "\n\nThe Gatestone Institute said, “However, the EU is quite candid as it is applying a political lens to their censorship, and it now has teams of political informants with the Orwellian title of “trusted reporters” to report any cases of “xenophobia” or “hate speech” to Facebook for immediate deletion.”", "\n\nThe video profiles of how Sweden was a low-crime and a peaceful nation who allowed Muslim refugees into their nation as a sign of good will only to watch its nation become the second highest number of rapes, only after Lesotho.", "\n\n“It’s political censorship. ", "It’s outrageous… and it’s contrary to our western values of free speech, political freedom and the separation of mosque and state, “the Institute said. “ ", "But in another way, it’s a tremendous compliment as the world’s censors think that Gatestone Institute’s work is important enough and persuasive enough that it needs to be silenced.”", "\n\nThe Gatestone Institute continued and said, “Well, not if we have anything to say about it. ", "We raised such a ruckus about this attack that the Swedish media started reporting on Facebook’s heavy-handed censorship. ", "It backfired, and Facebook went into damage-control mode. ", "They put Ingrid’s account back up — without any explanation or apology. ", "Ironically, their censorship only gave Ingrid’s video more attention.”", "\n\n“Facebook and the EU have backed down… for today. ", "But they’re deadly serious about stopping ideas they don’t like. ", "They’ll be back.”", "\n\n[Editor’s note: Be watching for a new book by Joe Newby and Adina Kutnicki that details how the social media giant enables militant Islamic jihad. ", "That book is set to be published this fall. ", "More on that here.]", "\n\nRelated News:\n\nTrending: Two Campaign Workers For Ilhan Omar’s GOP Opponent Shot, One Fatally\n\nIf you haven’t checked out and liked our Facebook page, please go here and do so." ]

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[ "Most people know that MSG makes their food taste better, but they don't have a clue what it does to their body.", "\n\nAs we've been learning more about MSG's impact -- on obesity, the brain, the heart and even cancer -- consumers are wondering what to do.", "\n\nWhat do you choose at the grocery or restaurant -- or better yet, what do you avoid?", "\n\nKeeping monosodium glutamate off the table is no easy task. ", "Your waiter or chef rarely knows anything about hidden MSG or what food ingredients contain it.", "\n\nIn the grocery, there are dozens of hidden names for MSG.", "\n\nDr. Russell Blaylock said, \"The consumer needs to tell these manufacturers, 'We're not going to buy your product until that's taken out -- and don't disguise the names because we'll find out what they are.'\"", "\n\nAnd Blaylock says that science is finding more and more problems with the sneaky additive -- even infertility.", "\n\nHe said, \"Because the male sperm cannot get to the egg, and MSG decreases the number of sperm …we know the ovary is filled with glutamate receptors.\"", "\n\nMSG stimulates those receptors so that a woman's eggs become damaged. ", "And with basic life issues like that, is the Food and Drug Administration sounding any alarms?", "\n\n\"The FDA has never walked into a flavoring house or into a food producer, and said, 'Give me that flavoring -- I want to test it for free glutamic acid.' ", "So it's never been tested,\" said Blaylock.", "\n\nJack Samuels learned that MSG caused his heart spasms years ago.", "\n\n\"On two occasions they thought I had died because they could not get a pulse,\" he said.", "\n\nSamuels decided to take up the MSG fight by publishing its hidden names on his Web site, truthinlabeling.org.", "\n\nAnd for his own safety, he only shops at a natural foods store, with a specific strategy.", "\n\nHere are some of his secrets:\n\n-- Pick organic vegetables without cosmetic wax.", "\n\nSamuels: And to fill in the crannies that are normal… CBN News: …And that could be a glutamate-type? ", "Samuels: …they sometimes put a hydrolyzed protein into that mixture. ", "In so doing, it breaks down some of the protein, and an individual like me will get sick.", "\n\n-- Buy milk that is pasteurized, but not homogenized -- but not one that's ultra-pasteurized.", "\n\nTerry Strouss blames MSG for the migraines he suffered for almost 20 years. ", "He looks for those secret names at a regular grocery store.", "\n\n\"We stay away from bologna and hot dogs, and all of the cheaper processed meats,\" Strouss said. \"", "This (he shows us regular meat) is all really safe for our family. ", "The ranch dressing -- it might be fat-free, but it has MSG right in the ingredient list.\"", "\n\nSo there are definite ways to keep healthy while grocery shopping. ", "And our careful consumers say that grocery stores shouldn't worry about losing business.", "\n\nStrouss says that his family actually spends a little more -- yet eats much healthier. ", "So everybody wins.", "\n\nBut what about those restaurants? ", "Some chains, like Pizza Hut, list their ingredients online.", "\n\nPizza Hut even has five food items they officially warn as having MSG, such as their wings and salad dressings.", "\n\nUnfortunately, CBN News found at least nine other food items from Pizza Hut with hidden forms of MSG.", "\n\nBut Pizza Hut is not alone. ", "All of the major chains we investigated had items with hidden MSG.", "\n\nWhen deciding what to order, be aware that salad dressing, sauces, meat products and toppings are common offenders.", "\n\nClassier restaurants are generally safer because they often cook from scratch and add real spices instead of flavor enhancers. ", "But to be sure, you may have to talk to the chef to find out what he or she uses.", "\n\nSo, what should you do in general? ", "Blaylock says, go for the fruits and vegetables.", "\n\nHe has a DVD on nutrition so families will know the good things that counteract MSG's effects. ", "He also warns about aspartame -- the diet sweetener -- which is a related compound.", "\n\nBlaylock especially recommends dark green vegetables because they are high in antioxidants to defuse MSG damage. ", "They're also high in magnesium, which can actually block MSG from those glutamate receptors and damage.", "\n\n\"Name something that it doesn't affect,\" he said, \"because there are receptors in the spleen, they're in the liver, they're in the heart, in the GI tract, in the immune system -- they're in the brain, in peripheral nerves, in the vagus nerves, in the heart, and all the arteries.\"", "\n\nFor centuries, the Japanese have used a flavor enhancer that was seaweed. ", "It had a little glutamate, but it didn't seem to cause any problems.", "\n\nThen, 100 years ago, they started manufacturing it -- laying the foundation for today's problems: massive worldwide use of MSG, improper labeling, and a federal agency that refuses to label MSG so consumers can make intelligent decisions." ]

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[ "Yes, there are fish living in and around Davis. ", "If you go fishing, you can catch some. ", "Some are considered pets while others are classified as wildlife. ", "You can even find them (along with other kinds of seafood) in some restaurants." ]

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[ "Alterations in nonspecific cross-reacting antigen localization during cell culture. ", "An immunoelectron microscopic study using a human lung adenocarcinoma cell line.", "\nNonspecific cross-reacting antigen (NCA), a constituent of the carcinoembryonic antigen family, was localized ultrastructurally in a human lung adenocarcinoma cell line, PC-9. ", "NCA was distributed predominantly on the plasma membrane in the early phases of cell culture. ", "Deletion of fetal bovine serum (FBS) from the culture medium suppressed cell division without significantly altering cell viability, and induced a dramatic but reversible change in NCA localization. ", "Under these conditions, NCA was localized to membrane degradation products within cytoplasmic vesicles and vacuoles. ", "Acid phosphatase activity was also present in some of these intracellular structures. ", "Similar changes in NCA localization were seen in cells cultured with FBS at day 6 when the cells reached a plateau stage of growth. ", "These findings strongly suggest that plasma membrane degradation is accelerated by the cessation of cell growth. ", "Cytoplasmic reactivity for NCA in cancer cells may therefore reflect degradation of plasma membrane-associated NCA and may not necessarily be correlated with increased systhesis of this glycoprotein." ]

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[ "\"911.\" \"", "What's your emergency?\" \"", "I've got an officer down!\" \"", "Could you repeat that?\" \"", "This is Detective Vic Mackey!\" \"", "An officer's down!\" \"", "You have an officer down.\" \"", "The goddamn police radio in the van isn't working.\" \"", "What's the officer's condition?\" \"", "Condition?\" \"", "He's dying!\" \"", "We're in a...\" \"I have the address.\" \"", "Where are we?\" \"", "An ambulance is en route.\" \"", "I'm coming!\" \"", "Detective?\" \"", "Detective?\" \"", "He was about 10, 15 feet in front of me and Shane when I hear this gunshot, and I see Terry go down.\" \"", "Jesus.\" \"", "Shane and I turn the corner, and we see the guy in this back bathroom.\" \"", "He gets off a few rounds.\" \"", "We hit back.\" \"", "At least you killed the prick.\" \"", "Hey.\" \"", "Terry's a tough son of a bitch.\" \"", "He's gonna be okay.\" \"", "What happened?\" \" ", "Terry's down?\" \" ", "Yeah.\" \"", "How is he?\" \"", "Took a head shot.\" \"", "Yeah.\" \"", "I didn't clear the room.\" \"", "I was point.\" \"", "The first thing you do is clear the room.\" \"", "We both missed him.\" \"", "At 1:29 a.m., Detective Terry Crowley was admitted for a single gunshot wound to the face.\" \"", "Surgeons made every attempt to resuscitate him.\" \"", "Shit.\" \"", "Ultimately, their attempts were unsuccessful.\" \"", "Detective Crowley was pronounced dead at 2:13 this morning.\" \"", "No!\" \"", "Damn.\" \"", "I'm responsible.\" \"", "I'm responsible.\" \"", "Responsible how?\" \"", "It was my operation, and I called the shots.\" \"", "What could you have done differently?\" \"", "Terry asked me to move him up.\" \" ", "He was part of your team.\" \" ", "He wasn't ready for this.\" \"", "Eight years as a cop, a stint in robbery, two commendations.\" \"", "Sounds to me like he was ready.\" \"", "Strike Team's different.\" \"", "Really?\" \"", "We knock down the doors other cops don't want to.\" \"", "My guys accept the risk because they know I will take care of them.\" \"", "We go out on a mission, I expect every single one of them to come home.\" \"", "Anything else is unacceptable.\" \"", "Why so long?\" \"", "They're asking questions.\" \"", "Yeah, how many questions can they have?\" \"", "How would it look if I.A.D. only took five minutes to decide?\" \"", "They have to be thorough.\" \"", "Aceveda's gonna use this to shut us down.\" \"", "Aceveda's a water boy.\" \"", "He can't do dick.\" \"", "Why is it taking him so long?\" \"", "This is the last time.\" \"", "I answered your question.\" \"", "Detective Vendrell.\" \"", "Yeah, that's me.\" \"", "Just tell them what happened.\" \"", "Yeah.\" \"", "What are you doing back here?\" \"", "Not our crime scene anymore.\" \"", "I.A.D. gave us a friendly shove.\" \"", "You didn't shove back?\" \"", "Assistant Chief Gilroy gave the order.\" \"", "Gilroy?\" \"", "He's there?\" \"", "He was.\" \"", "I don't like spending five hours getting evidence for someone else.\" \"", "Are we on this case, or aren't we?\" \"", "David.\" \"", "Sir.\" \"", "Your office?\" \"", "Thanks for your good work this morning, Claudette.\" \"", "It hasn't been wasted.\" \"", "No, just our time.\" \"", "Sir.\" \"", "Where the hell were you last night?\" \"", "What do you mean?\" \"", "Detective Crowley got shot.\" \"", "I left you a message.\" \"", "Yeah, I got it.\" \"", "So?\" \"", "You didn't say I had to come down.\" \"", "I shouldn't have to.\" \"", "A fellow officer was dying in that hospital.\" \"", "I said a prayer for him.\" \"", "A prayer?\" \"", "Yeah.\" \"", "What?\" \"", "I've been on the job for three weeks, I didn't even know that guy.\" \"", "Rule number one, he was a cop.\" \"", "That means you knew him.\" \"", "Two deaths, two investigations.\" \"", "David, please.\" \"", "You can't have your cops investigating your cops.\" \"", "If you clear the Strike Team, that's gonna smack of favoritism.\" \"", "If you actually find something, you'll have a civil war on your hands.\" \"", "Let Internal Affairs do their job.\" \"", "Has anyone sent Department sympathies to Crowley's family?\" \"", "I spoke to his mother an hour ago.\" \"", "Someone from our side should speak at the funeral.\" \"", "Perhaps you...\" \"No.\" \"", "He was your man.\" \"", "I'm sure you'll represent us well.\" \"", "You're the first Mackey brought aboard when he assembled the Strike Team.\" \" ", "We worked things together.\" \" ", "So there was a comfort level.\" \"", "Yeah.\" \"", "Same with the others?\" \"", "Vic knew them, too.\" \"", "Detective Crowley?\" \"", "No, he was recommended by someone.\" \"", "I don't recall who.\" \"", "So Terry was an outsider?\" \"", "No, he was a new guy, but once Vic puts you on the team then you're on the team.\" \"", "Hey.\" \"", "Hi.\" \"", "Strike Team's getting the third degree.\" \"", "Yeah, looks like it.\" \"", "You're assigned the crime scene, right?\" \"", "We're off it now.\" \"", "The guys are curious.\" \"", "I said I'd find out what's going on.\" \"", "Any idea?\" \"", "Looks to me like Vic and his cowboys went in undermanned with a half-ass plan and managed to get a guy killed.\" \"", "Then again, I'm off the case.\" \"", "Later, okay?\" \"", "Tell him this is L.A., not the Wild West.\" \" ", "I'd know those fingernails anywhere.\" \" ", "How you holding up?\" \"", "I lost a guy.\" \"", "I don't like it.\" \" ", "I saw you at the hospital last night.\" \" ", "I saw you, too.\" \" ", "You had a crowd.\" \"", "I didn't want...\" \" It's all right.\" \" ", "Look, I don't want you to think that...\" \" I don't.\" \"", "If you need a break from all this and want to talk, come find me.\" \"", "That's good to know.\" \"", "Anytime someone kills a cop scumbags get it into their heads that it's open season.\" \" ", "Police rule number...\" \" Three.\" \"", "Cop goes down, we re-establish our dominance on the street.\" \"", "Someone mouths off, they hit the pavement.\" \"", "Someone resists, we make sure they make a pit stop at the emergency room.\" \"", "We gotta make sure these assholes know who's in charge.\" \"", "Right?\" \"", "Loud and clear.\" \"", "What happened to your hand?\" \"", "I put it through a window.\" \"", "Last night during the raid?\" \"", "No.\" \"", "After, when I found out about Terry.\" \"", "Is it all right?\" \"", "Yeah.\" \"", "Cut up pretty bad, though.\" \"", "Any tendon damage?\" \"", "Can we just get this the hell over with, please?\" \"", "Mackey sent Terry to get killed.\" \"", "You don't know that.\" \"", "Please.\" \"", "Cops get killed in the line of duty.\" \"", "Did you tell anyone Terry was trying to take down Mackey?\" \"", "Did you?\" \"", "David...\" \"I didn't, and Terry's dead.\" \"", "I followed protocol.\" \"", "Protocol.\" \"", "Things that Terry wanted, a job, a car, I'm not authorized to sign off on.\" \"", "So you broke the circle.\" \"", "I sent his requests up the ladder.\" \"", "This is the Justice Department.\" \"", "So they can be trusted?\" \"", "I've got to go to I.A.D. Why?\" \"", "To let them know exactly why Mackey wanted Terry dead.\" \"", "You can't do that.\" \"", "They'll let him walk, Moses.\" \"", "If they have motive, they'll dig deeper.\" \"", "You initiated an off-the-books investigation into one of your own men.\" \"", "You come forward, your career's dead.\" \"", "We sent Terry to his own death.\" \"", "What am I supposed to do?\" \"", "What do you hope to accomplish?\" \"", "It won't prove that Vic set Terry up.\" \"", "You'll lose the power you've worked your whole life to acquire.\" \"", "I'm not gonna let him get away with it.\" \"", "There is an epitaph for an unknown soldier that goes:\" \"\"Do not stand on my grave and weep\" \"\"For I am not there, I do not sleep.\" \"\"", "With my brothers in the field, I kneel\" \"\"Their lives, my merit\" \"\"My death, their zeal.\"\" \"", "I believe these words do express the sentiments of Detective Terry Crowley.\" \"", "A good man and an exemplary police officer who served his community and his department with dignity, loyalty, and courage.\" \"", "In the brief time that I was fortunate enough to work with him I came to know him as a man who embraced life adored his friends and family and loved being a cop.\" \"\"", "I am the resurrection and the light.\" \"\"", "He that believeth in me\" \"\"Even though he were dead\" \"\"Yet, shall he live.\"\" \"", "Amen.\" \"", "Yes, Lord.\" \"\"", "The Lord is my shepherd, I shall not want.\" \"\"", "He maketh me to lie down in green pastures\" \"\"He leadeth me beside the still waters.\"\" \"", "We will miss his service and his good nature.\" \"", "And I know I speak on behalf of the entire force when I say Terry, our lives are your merit and your memory gives us zeal.\" \"\"", "He leadeth me beside the still waters.\" \"\"", "He restoreth my soul.\" \"\"", "Yea, though I walk through the valley of the shadow of death\" \"\"I will fear no evil\" \"\"For thou art with me\" \"\"Thy rod and thy staff, they comfort me.\"\" \"", "Fire!\" \"", "Fire!\" \"", "Baby, I'm sorry!\" \"", "Mama's sorry!\" \"", "It's all right, now.\" \"", "Mama's sorry, baby!\" \"", "It's gonna be all right.\" \"", "What are you doing?\" \" ", "What?\" \" ", "Save it for the car ride, Shane.\" \"", "What's the matter with you?\" \" ", "I didn't expect to see his mother here...\" \" He had a mom.\" \"", "So what?\" \"", "So what did we do?\" \"", "What we had to.\" \" ", "Christ, come on, man.\" \" ", "Enough.\" \"", "Come on.\" \"", "I'm cold, man.\" \"", "Look, we made him one of us, and what did he do?\" \" ", "What did he do?\" \" ", "He turned rat.\" \"", "He tried to stick us both in a cage with the same goddamn animals we fight every day.\" \"", "Now, did I enjoy it?\" \"", "No.\" \"", "Did it need to be done?\" \"", "You're goddamn right it did.\" \"", "Now, listen to me we're gonna ride this out.\" \"", "Right?\" \"", "I'm so sorry for your loss.\" \"", "Sir, step back.\" \"", "It's cool.\" \"", "I called you.\" \" ", "Step away from the body.\" \" ", "I need the money he owes me.\" \"", "Don't make me warn you again.\" \"", "He's got my cash.\" \"", "I'm not gonna warn you again.\" \"", "I need the money he owes me.\" \"", "What'd she say?\" \"", "Let's go!\" \"", "What?\" \"", "I'm a good Samaritan here!\" \"", "What were you doing to the body?\" \"", "I heard he had the $100 he owed me so I come here and see somebody shot him!\" \"", "So I went down the block and called for 911 and come back here to get my money.\" \"", "Jesus, Julien, this guy's still alive.\" \"", "This is 1-Tango-13.\" \"", "I need a R.A. to respond.\" \"", "Keep fighting, man!\" \" ", "Keep fighting!\" \"", "I'll collect from you later!\" \" ", "Let's go.\" \"", "Shane barely touched his pork chops.\" \"", "I don't think he's doing well.\" \"", "He's gonna be fine.\" \"", "He don't seem fine.\" \"", "It hasn't even been a week yet.\" \"", "I'm coming, sweetie.\" \"", "It's like he blames himself for the whole thing.\" \"", "It's not normal.\" \"", "I mean, it's not like he killed Terry.\" \"", "Ace of clubs.\" \"", "I'm sitting poolside with this guy named Frank...\" \" Transvestite?\" \" ", "He's buying drinks.\" \"", "That ugly one?\" \"", "Yeah, the guy that I always hang out with.\" \"", "50 years old!\" \"", "You gotta be joking me!\" \"", "You holding out on me, Shane?\" \"", "I got the fresh meat.\" \"", "This here's Terry.\" \"", "He comes to us from Robbery.\" \"", "Robbery?\" \"", "You mean, like, the fag squad, man?\" \"", "They ran out of fresh asses over there so I figured I'd try you prom queens on for size.\" \"", "If you're looking for fresh asses, I'd cross Shane off your list.\" \"", "Stop it, mister.\" \"", "Stop it!\" \"", "It looks like it'll be a blast to run with you guys.\" \"", "Just make sure you keep up.\" \"", "I'll try.\" \"", "Don't let these pussies haze you too bad either.\" \"", "We love you pretty boys from Robbery.\" \"", "Know how to play pitch?\" \"", "No.\" \"", "You bring your wallet?\" \"", "Don't let them stick you with Shane.\" \"", "He always loses.\" \"", "'Cause you're cheaters.\" \"", "And he's a crybaby.\" \"", "You're all...\" \"Shut up!\" \"", "Don't trust one of these guys.\" \"", "You know what?\" \"", "He's the worst player in here.\" \"", "To infinity and beyond!\" \"", "Okay, Captain Infinity let's get you ready for school.\" \"", "Come on.\" \"", "Body was over there.\" \"", "We found one shell casing a few feet away.\" \"", "How's the victim?\" \"", "Paramedic says he's swirling down the drain.\" \" ", "Who said that?\" \"", "Randy?\" \" ", "Yeah.\" \"", "Sounds like Randy.\" \"", "Cynical bastard.\" \"", "Does the rookie ever talk?\" \"", "Yeah, he talks.\" \"", "What does this tell you?\" \"", "Ma'am?\" \"", "You're our first eyes and ears on a crime scene.\" \"", "What does this one tell you?\" \"", "Overturned carts, footprints, clear sign of a struggle.\" \"", "Actually, the struggle occurred after we arrived.\" \"", "A guy was going through the victim's pockets.\" \"", "After repeated requests to stop, I pulled him off the body.\" \"", "Knocking over the churro cart and getting sugar spread all over?\" \"", "Yes, ma'am.\" \"", "Actually, it was my fault.\" \"", "I should've...\" \"I hope this isn't the way you teach a trainee how to preserve a crime scene.\" \"", "This is how murderers go free, son.\" \"", "Don't take it hard.\" \"", "She's in a pissy mood 'cause it's her birthday.\" \"", "Did we...\" \"Is this scheduled?\" \"", "No.\" \"", "Frances came by to give me her preliminary report.\" \"", "I wanted you to get a sneak peek at it, too.\" \"", "We wrapped up the interviews.\" \"", "I'll fast-track ballistics and get you a final copy in the next couple of weeks.\" \"", "Bottom line, it wasn't a model operation, but it wasn't a crime, either.\" \"", "Vic and his boys look okay on this one.\" \"", "Really?\" \"", "The drug-dealer's girlfriend confirms the team's account.\" \"", "She puts the gun that killed Crowley in the dealer's hands seconds before the shooting.\" \"", "Thanks for the fast work, Frances.\" \"", "She must be good.\" \"", "Two men killed, a dozen rounds fired no ballistics, and she's already making her call.\" \"", "It's a preliminary report, David.\" \"", "I just don't want us accused of whitewashing anything.\" \"", "The last thing we need is another chapter in the Rodney King-O.J. book.\" \"", "I don't wanna be a chapter in that book, either.\" \"", "I'll make sure she doesn't get lazy on this one, okay?\" \"", "It was a great eulogy, by the way.\" \"", "It was very nice.\" \"", "If my dad dies, it's all my fault.\" \"", "Why's that?\" \"", "Dad wanted me to go to college around here.\" \"", "Live at home.\" \"", "He's a vendor.\" \"", "He doesn't make much money.\" \"", "But I wanted Stanford.\" \"", "I had gotten a scholarship, but it didn't cover everything.\" \"", "There was still books, housing.\" \"", "When he found out, he said if that was my dream he'd make it happen.\" \"", "That's when he stopped paying.\" \" ", "Stopped paying what?\" \" ", "Protection money.\" \"", "This bully in the neighborhood made all the vendors pay.\" \" ", "My father wasn't licensed, so...\" \" So he didn't go to the police.\" \"", "I told him he couldn't stop paying the guy but he said it was the only way I could go to Stanford.\" \"", "He was so proud when I had gotten in.\" \"", "I begged him to pay the money, but...\" \"The thug that did this to my dad can you guys really get him?\" \"", "We can try.\" \"", "Who is he?\" \"", "His name's Marlon Demrall.\" \"", "Marlon Demrall?\" \"", "Yeah.\" \"", "So?\" \"", "You need to come with us.\" \"", "Thanks, but I'll pass.\" \"", "You can go peacefully or not.\" \"", "Let's go.\" \"", "How 'bout I don't go at all?\" \"", "What's the emergency, brother?\" \"", "Questioning in a shooting.\" \"", "When?\" \"", "Today.\" \"", "I didn't shoot nobody today.\" \"", "What're you doin' to him?\" \"", "Come on, man.\" \"", "Get the camcorder!\" \"", "They're taking Marlon away!\" \"", "Get the camcorder!\" \"", "They're taking Marlon away!\" \"", "What is wrong with you?\" \"", "Stay inside.\" \"", "Let him go.\" \"", "He didn't do it.\" \"", "He's my son's father!\" \"", "I didn't shoot nobody.\" \"", "Right, yeah.\" \"", "You didn't do anything.\" \"", "You a brother?\" \"", "Or did you become one of them?\" \"", "Look at what you're doing to a brother!\" \"", "He's not my brother.\" \"", "Let go!\" \"", "Mama!\" \"", "Get the camcorder!\" \"", "They're taking Marlon!\" \"", "You got nothin' to do today?\" \"", "He did not...\" \"You ain't nothin' but a Uncle Tom!\" \"", "Uncle Tom?\" \"", "That's right.\" \"", "If the Oreo wrapper fits, wear it, brother.\" \"", "Shut up.\" \"", "Don't go making it black and white, Marlon.\" \"", "Take pictures!\" \"", "Smile for the camera!\" \"", "Goddamn cops!\" \"", "But I'm like, where am I supposed to find a dental dam at 2:00 in the morning?\" \"", "What's a dental dam?\" \"", "It's a rubber for your mouth when you go down on a chick.\" \"", "Are you serious?\" \"", "If you need a dental dam, I gotta question what your face is doing down there in the first place.\" \"", "She had made a specific request.\" \"", "Sounds like a petri dish with legs to me.\" \"", "I'd invest in a couple of syringes of penicillin.\" \"", "So I find this all-night Vietnamese store that carries them.\" \"", "Leave it to Charlie to put the dental dams with the toothpaste.\" \"", "Are you serious?\" \"", "I rush back.\" \"", "I slap this thing on, and I am goin' to town.\" \"", "I swear, I'm like Gene Simmons on that.\" \"", "I was like the tao of poon, I swear to God.\" \"", "And she's like, \"I don't feel anything.\"\" \"", "And you're like, \"Really?\" \"", "'Cause that's my dick.\"\" \"", "Have you ever actually tried to use one of these things?\" \"", "I guess you and me dated a different breed of woman.\" \"", "Down and dirty.\" \"", "No pun intended.\" \"", "Don't make me hurt you.\" \"", "What time is lunch here?\" \"", "Shut up.\" \"", "You worked me up an appetite.\" \"", "Tom here could use a bite, too.\" \"", "Sir to you.\" \"", "Yes, sir.\" \"", "Uncle Tom is looking very hungry.\" \"", "Detectives, meet Marlon Demrall.\" \" ", "Pleasure.\" \" ", "Shut up.\" \"", "Is that the only word you ladies know?\" \"", "Enjoy.\" \"", "Happy birthday, by the way.\" \" ", "You told her?\" \" ", "No.\" \"", "Yes.\" \"", "By accident.\" \"", "Come with me, Marlon.\" \"", "Whatever you say, birthday girl.\" \"", "Hey, Lowe.\" \"", "You gonna grab a beer with us tonight?\" \"", "Can't.\" \"", "I've got other plans.\" \"", "What's with your partner?\" \"", "Can't spend quality time with his fellow boys in blue?\" \"", "I'm just going to church tonight.\" \"", "God can wait.\" \"", "Tonight, my friend, you get your first b-and-b.\" \"What's a b-and-b?\" \"", "It's a tradition we have here.\" \"", "Nothing like your very first one.\" \"", "Be at the motor pool at the end of the shift.\" \" ", "You skip out on us, it's your ass.\" \" ", "Let's get out of here.\" \"", "What's a b-and-b?\" \"", "It's stupid and juvenile and the sooner you get it over with the sooner you'll be accepted by Jackson and the rest.\" \"", "Why should I care what they think?\" \"", "Rule number seven those guys might be overgrown frat boys, but might someday be the only thing that comes between you and a bullet.\" \"", "What y'all got to eat around here?\" \"", "How 'bout a nice churro?\" \"", "You like churros, Marlon?\" \"", "Yeah, that sugary shit is all right.\" \"", "But what I'm really jonesing for is a cheeseburger.\" \"", "Let me see if I can accommodate you.\" \"", "My man.\" \"", "Come on, sweet thing.\" \"", "Now I know you got a smile for me underneath all that.\" \"", "Dutch?\" \" ", "Dutch?\" \" ", "Captain.\" \"", "I could use your help with an interrogation.\" \"", "I'm in the middle of a case.\" \"", "Claudette's got it under control.\" \"", "All right.\" \"", "That's what all this is about?\" \"", "Look, you could have saved me a trip.\" \"", "Read my lips.\" \"", "I did not kill Poncho.\" \"", "Okay?\" \"", "Nivan Gonzalez.\" \"", "He's still alive at the moment.\" \"", "Whatever.\" \"", "I didn't shoot nobody this morning, but I'll turn you on to who did, though.\" \"", "Will you?\" \"", "Yeah.\" \"", "But I want something first.\" \"", "And what is that?\" \"", "A cheeseburger!\" \"", "I'm hungry as hell!\" \"", "Talk first, then you eat.\" \"", "Come on.\" \"", "All right.\" \"", "I had a protection thing going on with those churro chumps.\" \"", "It was a good gig, too.\" \"", "Till last month.\" \"", "This new Mexican gang comes in, packing some serious heat tells me they movin' in on my business.\" \"", "And what gang was this?\" \"", "Call themselves the Los Magnificos.\" \"", "See, that's what's wrong with this country.\" \"", "I'm prime example numero uno.\" \"", "I'm trying to make a serious buck, and then comes some wetback that's taking away my job.\" \"", "Your job?\" \"", "My job.\" \"", "Your job was illegally shaking down churro vendors.\" \"", "Look, a job is a job, lady.\" \"", "Now, I like mine well done.\" \"", "Just like me.\" \"", "With everything.\" \"", "No onions.\" \"", "They give me a rash.\" \"", "I don't find you funny.\" \"", "You lookin' for me?\" \"", "We'd like to ask you some questions.\" \"", "About what?\" \"", "About how Terry died.\" \"", "You I.A.D. now?\" \" ", "Seems right up your alley.\" \" ", "I don't feel they checked every angle.\" \"", "So now you got questions.\" \"", "I've got a few.\" \"", "What's stopping me from having my P.B.A. rep tell you to blow me?\" \"", "Nothing.\" \"", "If Vic Mackey wants to hide behind a lawyer then that's the way it'll be.\" \"", "You got questions?\" \"", "Fine.\" \"", "Ask your goddamn questions.\" \" ", "Why'd you go in without more backup?\" \" ", "A battalion of cops makes noise.\" \"", "It's called the element of surprise.\" \"", "Imagine my surprise when I heard your guy died.\" \"", "Those of us who venture out past our desks sometimes get shot at.\" \"", "More firepower wouldn't have tipped him off.\" \"", "Who should I have had watching our backs?\" \"", "You?\" \"", "I'd hate to have to change your diaper afterwards.\" \"", "Better than burying one of your own.\" \"", "So you fired at the dealer, the dealer returned fire, hitting Terry.\" \"", "No.\" \"", "Like I told I.A.D., Two-Time shot Terry then Shane and I returned fire.\" \"", "Sounds textbook to me.\" \"", "That's all we got.\" \"", "Thanks for your time.\" \"", "That's all we've got?\" \"", "Was that supposed to make him spontaneously confess?\" \"", "No.\" \"", "It's supposed to make him feel safe.\" \"", "Now we go after the weak link in the chain.\" \"", "Who's the weak link?\" \"", "What was that all about?\" \"", "Aceveda is spinning his wheels, as usual.\" \"", "Enter.\" \"", "What can you tell me about a new gang called Los Magnificos?\" \"", "They're not so new they been clawing up the food chain for the last six months.\" \"", "Someone from the gang pulled the trigger on my churro guy.\" \"", "Any bushes you can beat?\" \"", "Looks like we've been invited to the party, ladies.\" \"", "Let's go.\" \"", "I hear it's your birthday.\" \"", "Happy birthday.\" \"", "I need to ask Shane a few questions.\" \"", "Me?\" \"", "I need 10 minutes.\" \"", "I'll send him to catch up with you.\" \"", "I'm cool.\" \"", "I'm gonna grab a Coke first.\" \"", "Do you need anything?\" \"", "No.\" \"", "How's the paint brush, Van Bro?\" \"", "Don't be coming around here unless you lookin' to buy.\" \"", "And no more freebies.\" \"", "You want info, you have to buy something this time.\" \"", "All right.\" \" ", "Lem, pick the poison.\" \" ", "What do I know about art, man?\" \"", "Ronnie?\" \"", "I like the one with the clouds.\" \"", "You got a place for it?\" \"", "Yeah, I guess.\" \"", "All right.\" \"", "How much?\" \"", "Gimme $100.\" \"", "And don't be hangin' it in no bathroom, neither.\" \"", "Looking for a Los Mag.\" \"", "Shot up a churro vendor.\" \"", "The people say amigo didn't want to pay his freight.\" \"", "He ignored a couple of warnings.\" \"", "Los Mags sent in a baby banger to do the dirty deed.\" \"", "Earned him his stripes.\" \"", "Initiation shooting?\" \"", "Things they got these kids doing.\" \"", "Did you have to bake a batch of brownies back in your day, Van Bro?\" \"", "Getting into life was a lot more fun than getting out.\" \"", "I heard you lost a man a few days back.\" \" ", "Yeah.\" \" ", "Tough times, Vic.\" \"", "I don't know.\" \"", "Maybe someday somebody even win this war.\" \"", "I intend to.\" \"", "VAN See you around, baby.\" \"", "All right, man.\" \"", "This little vato ready for the barrio?\" \"", "The wannabe getting initiated, he's our shooter.\" \"", "I'll take him.\" \"", "Anybody else gets caught in our net, it's gravy.\" \" ", "Are we goin' in?\" \" ", "No.\" \"", "His punk ass wants a beating.\" \"", "I say we let him have it.\" \"", "What's he saying?\" \"\"", "Welcome to the life.\"\" \"", "You were going to be his backup.\" \"", "We were his backup.\" \"", "Why is Terry dead, then?\" \"", "Wrong place, wrong time.\" \"", "Aren't you and Vic supposed to be these kick-ass top cops?\" \"", "How does somebody you're backing up get blown away by some pissant drug dealer?\" \"...", "get blown away by some pissant drug dealer?\" \"", "I.A.D. cleared us.\" \"", "I didn't clear you.\" \"", "It just went bad.\" \"", "That's all.\" \"", "You lie.\" \"", "He squints when he lies.\" \"", "You do.\" \"", "No, I don't.\" \"", "You're squinting now.\" \"", "All right.\" \"", "Play time's over.\" \"", "Last one down the hill gets to drink the backwash.\" \"", "All right.\" \"", "We're a go in three, two, one, go!\" \"", "Go.\" \"", "Police!\" \"", "Stop right there!\" \"", "Freeze!\" \"", "Bienvenido a la vida.\" \"", "You let Terry get killed.\" \"", "Why would we do that?\" \"", "You know why.\" \"", "I.A.D. cleared us.\" \"", "You know a drug dealer named Rondell Robinson?\" \"", "No.\" \"", "You're squinting.\" \"", "You don't know him?\" \"", "I know who he is.\" \"", "I meant I don't know him personally.\" \"", "So if I had a witness that saw you and Vic acting real friendly with Rondell Robinson that would be a lie.\" \"", "That's right.\" \"", "So he didn't see the three of you at a car wash?\" \"", "Vic was just puttin' a scare into him.\" \"", "Lettin' him know that we were watching him.\" \"", "So you lied.\" \"", "No, I didn't.\" \"", "I meant something else.\" \"", "Terry knew you were in bed with Robinson, so he had to die.\" \"", "Terry was in our team.\" \"", "He was your errand boy.\" \"", "Bullshit.\" \"", "You led Terry there to die.\" \"", "Not true.\" \"", "The one guy who disapproved is the only one who didn't come out of that house alive.\" \"", "That's a coincidence?\" \"", "Believe what you want.\" \"", "Terry was gonna turn you in, so Vic wanted him dead.\" \" ", "Vic needed him dead!\" \" ", "No!\" \"", "God.\" \"", "The shot.\" \" ", "That's what the last shot was.\" \" ", "The last shot?\" \"", "Two-Time's girlfriend said that she heard the last shot 10, 15 seconds after the others.\" \"", "When we approached Two-Time, he moved.\" \"", "Vic couldn't leave it to chance.\" \"", "Vic fired once more.\" \"", "That's the shot that killed Terry.\" \" ", "No way.\" \" ", "Vic fired that shot, didn't he?\" \" ", "No.\" \" ", "Stop squinting!\" \"", "I know.\" \"", "I know.\" \"", "You're wrong.\" \"", "I want Mackey.\" \" ", "You don't have to go down with him.\" \" ", "That is not how it happened.\" \"", "Stop lying to me.\" \"", "I know what happened.\" \"", "I have a witness, Shane.\" \"", "He connects you and Vic with Rondell Robinson.\" \"", "I have a witness.\" \"", "Give me Vic and your problems end.\" \"", "Just give me Mackey.\" \"", "What the hell's going on here?\" \"", "You've had him here for two hours?\" \"", "He's making up sick shit about us.\" \"", "Wait for me downstairs.\" \"", "This whole thing is a joke.\" \"", "Grab a sandwich, Dutch-boy.\" \"", "I know what you did.\" \"", "Up until now, I've tolerated you.\" \"", "You come near my men again, I'm gonna lose patience.\" \"", "You listening?\" \"", "I know.\" \"", "The churro guy you shot's in surgery.\" \"", "They don't know if he'll live.\" \"", "Tell her you didn't shoot this man.\" \"", "If you tell me who had you do this, I might get you into juvie court.\" \"", "If you wait and he dies, you'll be tried for murder as an adult.\" \"", "Please tell her you didn't do this.\" \"", "Of course I didn't.\" \"", "You see?\" \"", "I shoot someone, he don't almost end up dead.\" \"", "Bitch ends up all dead.\" \"", "That's your alibi?\" \"", "If you'd have shot him, he'd be dead?\" \"", "Hell, yeah.\" \"", "That's my alibi.\" \"", "Guess what?\" \"", "Churro guy's actually gonna make it.\" \"", "Good.\" \"", "You're not gonna believe what went down between Aceveda and Mackey.\" \" ", "I don't want to know.\" \" ", "Yeah, I know, but this is...\" \"Anyone gets between those two will end up with a hose full of piss running down his leg.\" \"", "Good news and bad news.\" \"", "Bad news is the churro vendor just died.\" \"", "Good news for your son, though, someone else is taking credit for the killing.\" \"", "Thank you, Jesus.\" \"", "Some guy named Marlon walked in off the street said your gang was trying to muscle in on his business so he decided to show all the vendors your gang couldn't protect them.\" \"", "Marlon didn't do no shooting.\" \"", "He did.\" \"", "Everyone's talking about how he showed your boys up.\" \"", "That old man stopped paying us.\" \"", "That's my shooting!\" \" ", "He's just looking to piggyback a rep.\" \" ", "Olman, no.\" \"", "You're trying to take credit for it now?\" \" ", "Stop...\" \" I told you if I shot him, he'd be dead, right?\" \"", "People need to know.\" \"", "Los Mags own these streets.\" \" ", "Who asked you to do the shooting?\" \" ", "I ain't never telling you that.\" \"", "These are my brothers now.\" \"", "Do you know what this means?\" \"", "Means I earned my way in.\" \"", "No.\" \"", "It means you're going to jail.\" \"", "At least I'll have family there.\" \"", "Tell me you didn't reopen an investigation that wasn't even yours.\" \"", "There were unanswered questions.\" \"", "You accused Mackey of working with a drug dealer.\" \" ", "Who's your witness?\" \" ", "Anonymous tip.\" \"", "Anyone who can corroborate it?\" \"", "Is there something I don't know that explains why you're still pursuing this?\" \"", "Well?\" \"", "No.\" \"", "Then it ends here.\" \"", "Christ, David.\" \"", "You're doing a good job.\" \"", "Chief said so the other day.\" \"", "Don't lose focus now.\" \"", "I won't.\" \"", "Good.\" \"", "Thanks for getting Captain Ass-lnvader off my back.\" \"", "It wouldn't kill you to show him more respect.\" \"", "Come on.\" \"", "Vic, the men here take their cues from you.\" \"", "I'm just saying dial it down.\" \"", "I don't do my best work with him nipping at my heels.\" \" ", "Which is why I'm watching out for you.\" \" ", "Good.\" \"", "Question is, are you watching out for me?\" \"", "Meaning?\" \"", "My guys tipped you off that Terry was looking to take you down.\" \" ", "Ben, the bust just went bad.\" \" ", "My source at Justice is all over my ass.\" \"", "He wants to know if you made us all accomplices to a murder.\" \"", "I told him you'd never do that to me.\" \"", "Would you?\" \"", "Terry got overeager.\" \"", "He got himself killed.\" \"", "End of story?\" \"", "End of story.\" \"", "It better be.\" \"", "Is this from you?\" \"", "No.\" \"", "I know how you feel about birthdays.\" \"", "Denture cream.\" \"", "Thank you.\" \"", "'Cause I'm getting old.\" \"", "I don't see a card.\" \"", "Anyone want to take credit for this?\" \"", "Not me.\" \"", "I didn't think so.\" \" ", "How's the birthday girl doing?\" \" ", "Don't say that word.\" \" ", "I came to take you dancing.\" \" ", "I don't want to go dancing.\" \"", "Now we gotta do something, honey.\" \"", "It's your birthday.\" \"", "So?\" \"", "So, it's ladies night at the Crimson.\" \"", "I have it on very good authority that Franklin Green's gonna be there.\" \" ", "His divorce final?\" \" ", "Two weeks ago.\" \"", "I can't let Franklin Green see me looking like this.\" \"", "Which is why we'll spruce up first, then go to the Crimson.\" \" ", "We too old to be chasing men around.\" \" ", "I'm not.\" \"", "Neither am I. Come on.\" \"", "Drink!\" \"", "Drink!\" \"", "Drink!\" \"", "Drink!\" \"", "Who said that Jesus freak couldn't drink?\" \"", "Your b-and-b's almost complete, buddy boy.\" \"", "You had the beer.\" \"", "Now it's time for the blow.\" \" ", "You're kidding.\" \" ", "I kid you not.\" \"", "She's a badge sucker.\" \"", "What's a badge sucker?\" \"", "A badge sucker, man.\" \"", "She digs cops, okay?\" \"", "She's like the neighborhood welcoming committee.\" \"", "Listen.\" \"", "A better hummer you're not gonna find.\" \" ", "Trust me.\" \" ", "No...\" \"I cannot...\" \"No, stay, listen!\" \"", "Sit down!\" \"", "This isn't torture, okay?\" \"", "We've all gone through this.\" \"", "Julien, say hi to Betty.\" \"", "So this is the new guy?\" \"", "Go!\" \"", "Welcome to the force.\" \"", "Handsome.\" \"", "And he's lucky, too.\" \"", "Wanna go for a walk?\" \"", "Fellas, I cannot.\" \"", "No.\" \"", "Come on, now.\" \"", "I can't!\" \"", "Go!\" \"", "Give it up, man.\" \"", "Wait.\" \"", "Come on.\" \"", "Come on.\" \"", "All right.\" \"", "Let's get you over here.\" \"", "Relax, Officer.\" \"", "This is a lot more fun than roll call.\" \"", "You don't have to do this.\" \"", "Yeah.\" \"", "She love him long time.\" \"", "Go!\" \"", "Thanks a lot.\" \"", "You don't put me in an interrogation room without giving me all the facts.\" \"", "I wanted them to think you had nothing.\" \"", "Then I could take my shot.\" \"", "You didn't think they'd respect me enough to fear me?\" \"", "No.\" \"", "I didn't.\" \"", "That's pretty shitty.\" \"", "Sorry.\" \"", "You really think Vic had something to do with Terry dying?\"" ]

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[ "Active exploration of gas deposits in the Eastern Mediterranean has boosted the region’s importance for the local powers. ", "Most European states depend on imports of energy resources, which means that taking hold of new gas sources is an important element for strengthening their energy security and diversifying their sources of hydrocarbon supplies.", "\n\nCurrently, Greece, Cyprus, France, and Italy are among the main players that have divided up the known and future gas deposits in the Mediterranean among themselves. ", "All these states are EU members. ", "We should add that other EU states also indirectly benefit from new resources, even if they do not have immediate access to gas deposits. ", "They will, however, gain an opportunity to diversify their gas imports and distribute their hydrocarbon dependency among a greater number of suppliers.", "\n\nThe discovery of a new treasure trove of hydrocarbons often produces not only profits, but also additional problems since natural resources frequently turn into a source of conflict. ", "The case of the Eastern Mediterranean is no exception, as another power has staked its claim to a share of the region’s resources, a power that had officially received no piece of the gas “pie” that the European states had divided up among themselves. ", "This power is Turkey, which has decided to actively explore the gas deposits in the Eastern Mediterranean and has also visibly increased its military presence in the region. ", "Over the last few months, Turkish and Greek warships have been involved in several dangerous incidents, with both parties declaring their readiness to open fire at a pinch. ", "Ankara has also warned that it would “not back down” in a potential confrontation. ", "Like Greece, Turkey has already held military manoeuvres in the region.", "\n\nTurkey’s Motives\n\nWhy does Turkey need the gas deposits of the Mediterranean? ", "Today, Ankara is forced to import most of the gas it needs. ", "According to 2016 data, imported gas accounts for 99 per cent of Turkey’s total gas consumption. ", "Most of this gas (over 50 per cent) is purchased from Russia, with Iran, Azerbaijan, Algeria, and Nigeria being among Turkey’s other important suppliers. ", "Multibillion natural resource purchases are a heavy burden on Turkey’s struggling economy. ", "Its GDP has been stagnating since 2017, with a growth of just 0.877 per cent in 2019, compared to over 7 per cent two years ago . ", "These negative trends have been exacerbated by the coronavirus pandemic. ", "It has been a particularly painful time for Turkey, as the country has had to deal with the consequences of the lockdown, the partial suspension of economic activities and a sharp drop in tourist flows, which have always been an important source of revenues for Ankara. ", "The timing of the shortened 2020 holiday season could not have been worse for Turkey. ", "According to official data from the Turkish government, by June 2020, Turkey’s GDP had dropped by 9.9 per cent compared with the previous quarter.", "\n\nIt is extremely important under such circumstances that Turkey finds new energy sources: the gas deposits in the Mediterranean will lift the overwhelming burden on the country’s budget and give its weakened economy room to breathe. ", "In such a situation, decreasing dependence on gas imports could be posited as the short-term goal. ", "In the long term, Turkey intends to become a net gas exporter, which will require huge gas deposits, including those outside the Mediterranean.", "\n\nFighting for resources fits well into Recep Erdogan’s “neo-Ottoman” foreign policy concept that envisions a Turkey that is more willing to engage in confrontation with Western powers. ", "Additionally, the “neo-Ottoman doctrine” entails bolstering Turkey’s regional influence—and gaining new resources in the Mediterranean fits well within this task.", "\n\nInternational Legal Conflicts within the Dispute\n\nAnkara’s problem is that the formal provisions of the law of the sea do not allow Turkey to explore and develop potential and known gas deposits in the Eastern Mediterranean. ", "The situation, however, is complicated by the fact that the law of the sea, like any other international legal norms, has understandable problems in terms of compliance. ", "Additionally, the provisions of the law of the sea are very complex, and different states frequently interpret them differently, which is true for both Turkey and Greece. ", "For instance, Turkey is actively exploring gas deposits in the Aegean Sea, although legally it does not have the right to do this: under the law of the sea, virtually all of the Aegean Sea belongs to Greece’s exclusive economic zone due to a chain of Greek islands that are closer to Turkey’s coasts than to continental Greece itself. ", "Ankara, however, insists that the islands should not be taken into account when determining exclusive economic zones, which has created the first international legal conflict in the dispute.", "\n\nThe second conflict pertains to another stretch of the Mediterranean between Italy and Libya. ", "Turkey has staked its claim to this stretch, citing its agreement with Libya’s Government of National Accord. ", "The problem is that the GNA does not control all of Libya’s territory, which could put a question mark over the government’s legitimacy. ", "On the other hand, the GNA enjoys international recognition, a fact that Turkey repeatedly stresses.", "\n\nAnother case is connected with gas deposits closer to the coasts of Cyprus. ", "Turkey does not recognize Cyprus; it only recognizes the Turkish Republic of Northern Cyprus (it is the only country to do so). ", "Consequently, Ankara views exploring and developing gas deposits in the Exclusive Economic Zone of Cyprus as a violation of Turkey’s rights. ", "In the meantime, the colossal Calypso gas deposit that was discovered off the coast of Cyprus in 2018 is one of the main bones of contention in the present energy dispute.", "\n\nThe Role of the European Union and Individual European Stakeholders\n\nFrom the very outset, Brussels supported Greece and condemned Ankara’s aggressive actions. ", "However, the European Union is not entirely homogeneous in its attitude to the dispute. ", "Firstly, some of its members are locked in a confrontation with Turkey, such as Greece and Cyprus, and their stance in unequivocal. ", "There are stakeholder states, such as France and Italy, two European Mediterranean powers that also have an interest in the region’s gas deposits. ", "Their oil and gas companies, France’s Total, and Italy’s Eni, have already bought shares in the discovered Mediterranean gas reserves and made relevant arrangements with Athens and Nicosia. ", "In the standoff between Greece and Turkey, Paris and Rome are solidly behind Greece. ", "Moreover, France has not limited itself to rhetoric, and has sent warships to the Eastern Mediterranean, thus demonstrating its willingness to support the Hellenic Navy in a critical situation. ", "This is a particularly important step, since it entails a radical shift in the military balance of power within the dispute.", "\n\nOut of all the EU member states, particular mention should be made of Germany, which has a special connection with Turkey and currently holds the presidency of the Council of the European Union. ", "Tellingly, Berlin also sided with Greece, although, unlike France, it has been far more restrained in its conduct. ", "Germany did not send its Navy to the region. ", "Berlin’s principal message is the need for dialogue between the opposing parties and a détente in the conflict. ", "This is Germany’s typical foreign policy stance since it prefers to avoid exerting pressure by force. ", "Additionally, Germany has no additional incentives within the dispute since it stakes no claim to the resources of the Mediterranean.", "\n\nAs for the European Union in general, the overall support for Greece is easy to explain. ", "Brussels proceeds from the official provisions of the law of the sea and, unlike Turkey, it recognizes Cyprus and, consequently, the right of Athens and Nicosia to the gas deposits. ", "In the long term, this new source of gas could help stabilize the European Union and serve as a safety net in the event of a crisis. ", "It was not that long ago that the global financial crisis and the subsequent Eurozone troubles, which hit Greece especially hard, almost resulted in Athens defaulting and withdrawing from the European Union—a fact that could have set a very dangerous precedent and entailed a chain reaction in other Eurozone states with major financial woes (such as Italy). ", "With this is mind, European politicians may very well count on the fact that the revenues from developing the gas fields will help keep the Greek economy on an even keel and insure both Athens and Brussels against possible new economic shocks. ", "We should keep in mind here that the European Union had to establish a financial aid programme and spend significant funds to save Greece from bankruptcy.", "\n\nAdditionally, as we have already mentioned, the new source of gas will allow many EU countries to diversify their energy suppliers and thus to boost their energy security.", "\n\nHow Likely is the Dispute to Turn into a “Hot” Conflict?", "\n\nDespite several critical incidents, an open conflict over the gas deposits in the Eastern Mediterranean is not particularly likely, mostly due to the forces being unequal. ", "Turkey has found itself almost completely isolated, and the only agreement Ankara can rely on has been achieved with Libya’s unstable Government of National Accord. ", "On the other side, there is an entire coalition of states, with Greece and France having already held joint military exercises.", "\n\nFrance’s military intervention radically changes the balance of power. ", "Turkey’s Navy is larger and stronger than Greece’s (149 warships vs. 116, according to the Global Firepower Index), but significantly smaller than that of France (180 warships). ", "However, it is not only a matter of how many warships each side has. ", "What is important here is their quality: for instance, France has four aircraft carriers, while Turkey has none.", "\n\nThe European Union’s general support for Greece is also important. ", "The idea of imposing sanctions against Turkey was evoked at the most recent EU Foreign Ministers Meeting. ", "Financial penalties could have a major effect on Turkey, given that the European Union is Ankara’s principal trade partner, accounting for 42.4 per cent of its exports and 32.3 per cent of its imports. ", "In such a situation, trade sanctions may prove very painful for Turkey, especially given its stagnating economy and the significant losses it has suffered as a result of the coronavirus pandemic.", "\n\nAdditionally, the scope of the European Union’s non-military leverage against Turkey is not confined to economic sanctions. ", "In the event of an open conflict between Athens and Ankara, Brussels can strip Turkey of its current benefits in trading with European states. ", "In particular, the question of excluding Turkey from the EU Customs Union may appear on Brussels’ agenda. ", "Additionally, the European Union could take Turkey’s potential EU membership off the table forever and strike Ankara from the list of candidates.", "\n\nStill, we should not discount the serious obstacles in the way of Brussels imposing sanctions against Turkey and using other measures to apply pressure on Ankara. ", "One such obstacle is Ankara’s geopolitical significance for Washington. ", "Despite all the recent complications in their relations, Turkey remains one of the key U.S. allies in the region and a NATO stronghold in the Middle East.", "\n\nAs for Turkey itself, a “hot” conflict could prove detrimental to the country in several ways at once. ", "First, given the unequal military power, it is extremely unlikely that Turkey would emerge victorious from such a conflict. ", "Second, a war will undermine Turkey’s global standing and its membership in international organizations. ", "Third, Turkey cannot afford in its current economic state to either actively build up its military power (even though its authorities claim the opposite and have announced significant increases in the naval budget, with the construction on aircraft carriers being top of the spending list) or bear the burden of possible sanctions which, given the country’s many connections with the European Union, could prove very painful.", "\n\nThe rhetoric of the Turkish leadership is highly belligerent rhetoric, yet Ankara is very well aware of the real consequences of breaking up with Europe and starting an open conflict with a country that is a member of both the European Union and NATO. ", "It is possible that, instead of instigating a “hot” conflict, Turkey could attempt to use its own instruments of applying non-military pressure, such as the huge number of refugees present on Turkish territory. ", "Since 2016, Brussels and Ankara have had a refugee agreement in place. ", "However, Recep Erdogan has already demonstrated in the past that he is capable of suspending this agreement and “cracking open” the door to Europe for migrants, which would set new crises in motion at the borders to the European Union.", "\n\nDoes the Gas Dispute in the Mediterranean Affect Russia?", "\n\nSpecial attention should be paid here to the possible prospects for Russia in the ongoing dispute. ", "Naturally, Russia has a very tangential relation to the confrontation in the Mediterranean, although the outcome of this confrontation may be important for Moscow.", "\n\nOn the one hand, Russia can hardly profit from Turkey gaining its own major sources of gas. ", "Currently, Moscow is the main supplier of gas to the Turkish market. ", "Undoubtedly, Russia is interested in preserving this status quo. ", "The recent launch of the Turkish Stream confirms that Moscow intends to maintain its dominant standing in the Turkish energy resources market.", "\n\nOn the other hand, a new source of gas for European countries could shake Russia’s position in the even more important European market. ", "It is no secret that the EU countries are attempting to diversify their resource suppliers for greater energy security. ", "However, abandoning Russian gas is very difficult since a gas pipeline infrastructure has already been created in Europe, making Russian gas relatively inexpensive. ", "Much will depend on whether Greece, Cyprus, and Israel will succeed in jointly building the EastMed gas pipeline meant to deliver gas from the Eastern Mediterranean to Greece. ", "Theoretically, EastMed could be extended to other European states. ", "It currently has a design capacity of 10 billion cubic metres, which may be increased by tapping the currently unexplored resources of the Eastern Mediterranean. ", "This is a very ambitious and expensive project, but if it does materialize, it could change the situation in the European gas market, since pricewise, it could compete with cheap Russian gas. ", "If there is no pipeline running from the Mediterranean, Mediterranean gas will have a hard time pushing Russia aside in the European market: without the gas pipeline, gas will be shipped as liquefied natural gas (LNG), which will significantly increase its price and make it far less attractive to European countries.", "\n\nFrom our partner RIAC" ]

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[ "Selective use of preoperative chemoradiotherapy for T3 rectal cancer can be justified: analysis of local recurrence.", "\nPreoperative chemoradiotherapy (CXRT) is performed to reduce local recurrence and improve the rate of radical resection for rectal cancer. ", "Currently, it is recommended for all suspected T3 rectal cancers. ", "However, the survival benefit of this procedure is controversial, and complications of RT are often overlooked. ", "The present study was designed to assess the validity of our institutional policies regarding selective use of preoperative CXRT for T3 rectal cancer. ", "From September 2006 to May 2010, we retrospectively analyzed data for patients with pathologic T3 rectal cancer, especially those with follow-up for more than 18 months. ", "Patients who underwent RT pre-/postoperatively were excluded. ", "Patients who fulfilled these criteria but did not undergo RT or had incurable distant metastasis were also excluded. ", "Of the 319 patients who underwent surgery for rectal cancer, 183 were diagnosed with T3 cancer. ", "After excluding 58 patients, the final sample included 125 subjects with a mean follow-up of 26.0 months. ", "In all, 94 patients had a follow-up of more than 18 months. ", "Four patients had local recurrences, and the cumulative local recurrence rate was 3.4 % at 18 months and 5.6 % after 2 years. ", "All four had low rectal cancer (mean distance 4.5 ± 1.9 cm). ", "Three were diagnosed as N0 and one as N1. ", "The mean disease-free period was 15.0 months. ", "The distance between the tumor and the anal verge was the only significant risk factor for local recurrence. ", "In this study, the oncologic outcome for T3 rectal cancer without preoperative CXRT was acceptable in terms of local recurrence. ", "Selective use of preoperative CXRT for T3 rectal cancer needs to be further evaluated prospectively." ]

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[ "). ", "Solve -9 = -4*y + u for y.\n4\nLet a(p) = -6*p + 13*p**2 + p**3 - 14*p + 3*p**2 + 4*p + 12. ", "Let b be a(-17). ", "Let c(s) = s**2 + 4*s - 1. ", "Let q be c(b). ", "Solve 2*w + 2 + q = 0 for w.\n-3\nSuppose -5*n = -0*n - 20. ", "Let r(v) = -v + 2. ", "Let k be r(-5). ", "Let u(l) = 13*l**3 - 3*l**2 + 7*l - 5. ", "Let i be u(1). ", "Solve k*a - n*a = i for a.\n4\nLet p be ((-30)/(-20))/((-2)/(-116)) - -3. ", "Suppose -2*c = -p + 74. ", "Solve -15 = 3*o - c*o for o.\n3\nLet b be -1 + 7 + (-18)/6. ", "Let i be (1/b)/(5/30). ", "Solve i*c = 5*c for c.\n0\nLet m(w) = 10*w - 215. ", "Let y be m(32). ", "Solve 0 = 103*q - y*q - 8 for q.\n-4\nLet m be 12/4 - 1825/(-25). ", "Solve -m*w + 10 = -74*w for w.\n5\nLet n be (-360)/(-105) + 3/(-7). ", "Solve -5*o = 2 + n for o.\n-1\nLet l(f) = 4*f**2 - 1. ", "Let w be l(-1). ", "Let n(x) = -x + 2. ", "Let k be n(-2). ", "Let t = k - 0. ", "Solve t*m = w - 7 for m.\n-1\nSuppose 0*x = 4*x. ", "Let r be (x - 1)/(3/(-33)). ", "Suppose -5*n + 5*v - 2 = v, 4*n + v = r. Solve -2*f - n = 8 for f.\n-5\nSuppose 0 = -4*i - 5*q + 4*q - 216, -2*i + 2*q - 98 = 0. ", "Let t = i + 57. ", "Solve -4*k = t + 8 for k.\n-3\nLet j be -6*(1 - (-52)/12). ", "Let i = -27 - j. Solve -i*y + 0*y + 15 = 0 for y.\n3\nLet o(n) = n**3 - 15*n**2 - 14*n - 28. ", "Suppose 0 = 4*l, 0*l + 32 = 2*x + l. Let w be o(x). ", "Solve w*z + 21 = 5 for z.\n-4\nLet s = 212 - 209. ", "Solve -5*d - 10 = -s*d for d.\n-5\nSuppose b = 8*b. ", "Suppose -8*u + 6*u = b. Solve u = p - 2 - 1 for p.\n3\nSuppose -5*b + 0*x - x = 2, 8 = 5*b - 4*x. ", "Suppose b = -2*n - 0 + 4. ", "Solve -5*c - 3 - n = 0 for c.\n-1\nLet l be (2/6)/((-3)/(-36)). ", "Suppose 8 - 28 = -l*m. ", "Let q(b) = -4*b**2 + 35*b - 17. ", "Let g be q(8). ", "Solve -2*h = -g*h + m for h.\n1\nLet y(m) = m**2 + 2*m + 11. ", "Let h be y(3). ", "Solve 20*c = h*c - 30 for c.\n5\nSuppose -1 - 8 = 3*v. ", "Let t be 1 + 52/4 - v. Let x = t - 13. ", "Solve 0 = 5*a - 14 + x for a.\n2\nSuppose -19*s = -20*s - 1, -5*s + 55 = 3*t. ", "Solve -19*p - 3 = -t*p for p.\n3\nSuppose -3*s - 6*v + v = -21, 0 = -5*s + v + 7. ", "Solve s*q + q - 3 = 0 for q.\n1\nSuppose -c + 3 - 5 = 0, -3*q + c = -11. ", "Solve 2*h = q*h - 1 for h.\n1\nLet j(w) = w**3 + 22*w**2 + 39*w - 20. ", "Let u be j(-20). ", "Solve u = 6*i - 0*i - 7*i for i.\n0\nSuppose 6*z = 180 - 168. ", "Solve 8*g + z = 6*g for g.\n-1\nSuppose 0 = m - 2 - 4. ", "Let i be 11 - 6/(9/3). ", "Let v be m/i*3/2. ", "Solve -7 = -3*w - v for w.\n2\nLet a(l) = 3*l**2 - 142*l - 415. ", "Let n be a(-3). ", "Solve -4*i - 30 = -n for i.\n2\nLet c(r) = -r - 14. ", "Let v be c(-5). ", "Let g(j) = -j**2 - 8*j + 12. ", "Let h be g(v). ", "Suppose f - 4*k = -f - 12, 0 = 5*k - 20. ", "Solve -f*b + h*b + 4 = 0 for b.\n-4\nLet t(x) = x**3 - 5*x**2 + 6*x - 3. ", "Let q be t(4). ", "Let l be ((-3)/6)/(2/8). ", "Let s = q - l. Solve -3*c = -s*c + 8 for c.\n2\nLet z = 252 + -244. ", "Solve z*u - 5 = -13 for u.\n-1\nLet l be -1 - ((1 + -3 - -1) + -88). ", "Let o = -83 + l. Solve -o*g = -4 - 1 for g.\n1\nLet q be -1 + -2 + 40/8. ", "Let u be (-1)/(q/(-26)) + 2. ", "Suppose -3*b - 3 = -u. ", "Solve 0*a = 4*a - b for a.\n1\nLet s(q) = -4 - q**3 - 13*q**2 - 4*q**2 - 17*q - 5. ", "Let m be s(-16). ", "Solve m - 1 = -3*t for t.\n-2\nSuppose 6 - 22 = 16*g. ", "Let d(h) be the second derivative of -h**5/20 + h**4/6 + h**3/3 + h**2/2 + 3*h. ", "Let f be d(g). ", "Solve f = 4*m - 18 for m.\n5\nLet z(o) = o**2 - 4*o - 6. ", "Let t be z(6). ", "Suppose 2*u - t = -0*u. ", "Solve 3*f = 6*f - u for f.\n1\nLet o(u) = u**3 - 5*u**2 + 10*u - 21. ", "Let g be 2 + (-12)/2 - (-19 - -11). ", "Let h be o(g). ", "Solve 0 = -5*d + h + 7 for d.\n2\nLet i be (((-1)/3)/(-1))/((-2)/(-12)). ", "Let v be 4 + 1*(i - 4) + 0. ", "Solve 6 = -d - v*d for d.\n-2\nSuppose -1 = -7*l + 8*l. ", "Let y(c) = -5*c**3 + 2*c + 2. ", "Let b be y(l). ", "Solve 0*p + p = -b for p.\n-5\nLet u be -4 + (-115)/(-30) - 93/(-18). ", "Solve 3*s = u*s + 8 for s.\n-4\nLet w be (12 - -6 - 13)*(-12)/(-10). ", "Solve w*y + 52 = -7*y for y.\n-4\nLet a(d) = -d**2 - 9*d - 12. ", "Let f be a(-6). ", "Suppose 4*j - 40 = -4*m, -2*j = f*m - 2*m - 44. ", "Solve 0 = 3*b + b + m for b.\n-3\nLet l(z) = z**3 + 9*z**2 + 7. ", "Let o be l(-9). ", "Let u = 9 + -8. ", "Let d be -1 + 4 + u + -2. ", "Solve 10 = o*a - d*a for a.\n2\nLet t(b) = b**2 + 11*b - 18. ", "Let z be t(-14). ", "Let q = z + -23. ", "Let s(k) = -k**2 + 7*k - 5. ", "Let g be s(6). ", "Solve -v = -q - g for v.\n2\nSuppose -9*t + 25*t = 96. ", "Solve 1 = v + t for v.\n-5\nSuppose 10*g + 51 = 61. ", "Solve p - g = -5 for p.\n-4\nLet c be -46*2*(-3)/4. ", "Suppose -c*d + 66*d + 15 = 0. ", "Solve -5*b + d = -4*b for b.\n5\nLet l(v) = -v**2 + 21*v - 44. ", "Let s(t) = 3*t - 6. ", "Let u be s(8). ", "Let y be l(u). ", "Solve -g - g = -y for g.\n5\nLet h(y) be the second derivative of -y**5/20 - 13*y**4/12 - 11*y**3/6 + 17*y**2/2 - 22*y. ", "Let t be h(-12). ", "Solve t*u - 6 = 4 for u.\n2\nLet p be -13 + 34 + 1*-1. ", "Suppose p = -2*m + 26. ", "Solve b = m*b for b.\n0\nLet a be ((-4)/(-10))/(17/340). ", "Suppose a*f = 12*f - 8. ", "Solve 1 - 11 = -f*d for d.\n5\nLet f(b) = b**3 + 3*b**2 + 1. ", "Let i(m) = -3*m + 6. ", "Let q be i(3). ", "Let n be f(q). ", "Solve -2*v - n = -v for v.\n-1\nSuppose 0*h - 3*h = -12. ", "Suppose -3 = -5*w - h*p, -8*p + 4*p = 12. ", "Suppose 0*f - 4*f = -8, 3*a - 3*f = 0. ", "Solve -w*c = 4 + a for c.\n-2\nLet n(j) = j + 6. ", "Let o be n(-6). ", "Suppose 3*g + 2 = 2*k, 0 = -5*k - g + 6*g + 10. ", "Solve k*s + s = o for s.\n0\nSuppose -x + 3*d + 2 = 0, -x - 2*d = -0*d + 8. ", "Let h be 3*(6 + x)/2. ", "Suppose h*s = -1 + 7. ", "Solve b - s = 2*b for b.\n-2\nSuppose 0 = 5*m + 5*w + 5, 5*m - 3*w = 4*m + 3. ", "Solve 4*z + 1 + 11 = m for z.\n-3\nLet k(g) = 14*g - 6*g - 7*g. ", "Let f be k(5). ", "Solve 0 = f*a - 10 for a.\n2\nLet g(b) = b + 3*b**3 - 7*b**3 + b**3. ", "Let m be g(-1). ", "Suppose 4*c = 2 + 30. ", "Solve -2*q = m*q - c for q.\n2\nLet s be (-1 - 0)/((-2)/6). ", "Suppose -1 = -s*g + 5. ", "Solve -3*x + 20 = g*x for x.\n4\nLet r(f) = -f**3 - 13*f**2 - 5*f - 5*f + 5*f**2 - 15 + 0*f. ", "Let h be r(-7). ", "Solve -7*p - h = -4*p for p.\n-2\nLet c be (2/(-2) - -43)*2/4. ", "Let x(o) = -o**3 + 20*o**2 + 21*o + 6. ", "Let r be x(c). ", "Solve 2 = -4*w - r for w.\n-2\nLet x be ((-50)/(-250))/((-2)/(-20)). ", "Solve -x*u - 4 = -2 for u.\n-1\nSuppose 0 = -3*x - 15, -8*x + 4*x - 20 = -2*r. ", "Solve 4*l + r*l + 8 = 0 for l.\n-2\nLet n(s) = -3*s**2 + 78*s + 44. ", "Let o be n(26). ", "Solve p + 10*p - o = 0 for p.\n4\nSuppose -3*a + 6 = -3*b - 45, -11 = -a - 2*b. ", "Solve -3*u = 18 - a for u.\n-1\nSuppose 264 - 260 = x. Solve x*a = -2*a for a.\n0\nSuppose 4*h = 3*h - 3, 3*j - 5*h - 18 = 0. ", "Let v be -3*((-6)/(-2) + -4). ", "Solve -r = v - j for r.\n-2\nLet a(u) = u**2 - 33*u - 229. ", "Let q be a(39). ", "Suppose 0 = -2*c + 36 + 54. ", "Suppose 3*i = 3*f - c, i - 7 = 2*f - 3*f. ", "Solve q*k + f = -4 for k.\n-3\nLet a be 10/(-8)*(-9 - -5). ", "Suppose 4*o + 0 = 2*d - 6, -26 = -a*d - o. Solve -d*x + 3 = -2 for x.\n1\nSuppose 0 = 93*f - 87*f - 24. ", "Let y be 1*(47 - -3)/(f/2). ", "Solve 0 = 4*i + i - y for i.\n5\nLet r(d) = d**3 - 5*d**2 + 4*d + 5. ", "Let v be r(4). ", "Suppose 3*p + v = 4*p, -115 = -4*j + 5*p. ", "Let b = j - 23. ", "Solve -4*h = -0*h - b for h.\n3\nLet m be (-2)/(-5) + 92/20. ", "Suppose -5*a = -m*k - 10, 10 = 6*a - a + 4*k. ", "Suppose -a*j = -5*f + 18, 0 = 2*f - 6*f - 3*j - 4. ", "Solve -u = -f*u + 1 for u.\n1\nLet i(a) = a**3 - 12*a**2 - 14*a + 18. ", "Let m be i(13). ", "Suppose m*g = -3*q + 5, g - 3*q - 23 = -g. ", "Solve g*o + 5 = -15 for o.\n-5\nLet l(k) = -2*k + 26. ", "Suppose 0*h + 33 = 3*h. ", "Let i be l(h). ", "Solve -q = i - 0 for q.\n-4\nSuppose 0 = 733*p - 742*p + 1710. ", "Solve p = -2*s + 180 for s.\n-5\nLet x be 12 - (-3 - 3/(15/(-25))). ", "Solve x*i + 4 = -36 for i.\n-4\nSuppose -w - 5*r = -12, 2*w - 3*r - 5 = 6. ", "Let p be -1 - (-2 - 1) - (9 - w). ", "Solve 4*k - 7 + 23 = p for k.\n-4\nLet n(m) = m**3 + 4*m**2 + 2*m + 10. ", "Let q be n(-4). ", "Suppose q*x - 48 = -x. ", "Solve 5*i - x = i for i.\n4\nLet w be 3*8/192 + (-6)/48. ", "Solve -8*l - 14 - 26 = w for l.\n-5\nSuppose -2*w = 4*y + 2, -2*w + 0*y - 3*y - 3 = 0. ", "Let h be (-12)/(3 - (w + 7)). ", "Suppose f = 2*o - 6*o + 39, -2*f = -2*o + h. Solve -15 = -o*a + 4*a for a.\n3\nLet v be -1 - (-1 - (-3 - 27))/1. ", "Let k be (8/(-20))/(6/v). ", "Solve -8 = -k*r - 2*r for r.\n2\nLet g be (152/(-16))/((-1)/2). ", "Let q = 23 - g. Solve -2*w - 2 = -q*w for w.\n1\nSuppose v = 3*t - 2, 3*v - 1 + 2 = 4*t. ", "Let a be (v - 1 - 3)*-1. ", "Suppose 4*o = a*d + 57, 0*d = d - 1. ", "Solve -b = -4*b - o for b.\n-5\nLet h = -202 - -207. ", "Solve 6*q = h + 1 for q.\n1\nLet x(g) = -g**3 + 9*g**2 - 20*g + 12. ", "Let f be x(6). ", "Solve 2*k = -f - 6 for k.\n-3\nLet t = -300 + 332. ", "Solve m = 35 - t for m.\n3\nLet m be 1/1*(-10)/(-30)*12. ", "Solve -3 + 23 = m*x for x.\n5\nSuppose -4*o = -5*o. ", "Suppose -5*g + 1 = -19. ", "Suppose g = 2*l - 0. ", "Solve 8 + o = l*x for x.\n4\nSuppose -a - 36 = 5*k, a = -4*k + 3*a - 26. ", "Let r be -8*((-21)/(-4))/k. ", "Let w(z) = 5*z**2 - 3*z + " ]

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[ "Article rings false... lots of subtle inconsistencies such as, \"The alien in the alleged interview indicated that they are not Extraterrestrial because they descended from the dinosaurs on Earth. ", "The reason for this destruction was not a natural disaster -- an asteroid impact as your scientists believe falsely -- but a war between two enemy groups that took place mainly in the orbit and high atmosphere of your planet.\"", "\n\nOkay, so if you're not an ET, then why are you saying \"your planet\" instead of \"our planet\"?", "\n\nFTFA: The interview further suggested that a sentient reptilian-like entity evolved from the remaining decimated dinosaur population, and that allegedly still lives on Earth, and that it operates some UFOs.", "\n\nCould the extraterrestrials, please, kill off, all those extra commas this guy keeps sticking in there, so that I can read past, possibly beyond, the first two paragraphs, without getting annoyed? ", "Not that I really cared about reading any further, anyway.", "\n\nWhat does this even mean? ", "I see it once in just about every other thread. ", "Are we just supposed to stop talking? ", "What exactly is \"done\"?", "\n\nYou should know I imagine you wearing a backwards baseball cap and calling people \"brah\".", "\n\nAnd we picture you as a dumbass young earth creationist and subby.", "\n\nIt takes a unique brand of retardation to make even the most down syndrome child go, \"holy crap that's some stupid logic.\"", "\n\nI have no idea why you would think I'm a YEC. ", "Is it because I asked for a reason as to why a person would bother to log in only to say something as stupid as \"Done in one LOL\"? ", "Because I'm not seeing the connection.", "\n\nI can't wait for the first person to come through the museum citing this. ", "After all, I just got done with another local that claims she has real life modern \"carnosaurs\" that live in her backyard, up here in the rockies.", "\n\nThat must be impossible...because that guy who made that huge boat and put two of every species in it to survive a flood and repopulate the planet would have certainly written about alien-like reptilians. ", "And all Christians KNOW that this man existed and that really happened.", "It's written in a book, right?It must be true.", "\n\nThere can be NO OTHER possibilities in regards to human and Earth history other than what we're told RIGHT NOW.", "\n\nRight?", "\n\n(I'm SO glad I don't let the authoritarianism of this \"society\" stop me from asking questions about what I HAVEN\"T EXPERIENCED FOR MYSELF).", "\n\ndpzum1:That must be impossible...because that guy who made that huge boat and put two of every species in it to survive a flood and repopulate the planet would have certainly written about alien-like reptilians. ", "And all Christians KNOW that this man existed and that really happened.", "It's written in a book, right?It must be true.", "\n\nThere can be NO OTHER possibilities in regards to human and Earth history other than what we're told RIGHT NOW.", "\n\nRight?", "\n\n(I'm SO glad I don't let the authoritarianism of this \"society\" stop me from asking questions about what I HAVEN\"T EXPERIENCED FOR MYSELF).", "\n\nWhat does this even mean? ", "I see it once in just about every other thread. ", "Are we just supposed to stop talking? ", "What exactly is \"done\"?", "\n\nYou should know I imagine you wearing a backwards baseball cap and calling people \"brah\".", "\n\nThe \"thread is over\" meme here on Fark is a compliment to someone who posts a particularly funny, witty, surprising, ironic etc. ", "response to the thread topic especially if said post is among the very Weenerss. ", "A different way to say, \"That was a funny post. ", "I salute you .\"", "\n\nIt is not to be taken literally.", "\n\nYou may continue to participate in the thread as you see fit - in your case by insulting the author of a post you completely did not understand.", "\n\nMy sister is one of those fundamentalist baptists that literally believe the Noah story. ", "Try to make any sort of comment about how ridiculous it is and she will (no joke) cover her ears and hum church songs until you stop. ", "Lucky for me she just moved out of state. ", "WAY out of state.", "\n\nSo yeah, people believe weird shiat and this is NOT surprising. ", "Knew a guy in the Army that insisted that everything in News of the World was true because otherwise, the Government wouldn't let them print it." ]

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[ "Media playback is unsupported on your device Media caption The singer wins BBC privacy case at High Court but how did it get to this\n\nSir Cliff Richard has won his privacy case against the BBC over its coverage of a police raid on his home.", "\n\nHigh Court judge Mr Justice Mann awarded an initial £210,000 in damages.", "\n\nThe singer claimed the BBC's reporting of the 2014 raid, which was part of an investigation into historical child sex allegations, was a \"serious invasion\" of privacy. ", "He was never arrested or charged.", "\n\nThe BBC said journalists acted in good faith and it is considering an appeal.", "\n\nSpeaking outside the High Court in London, the BBC's director of news and current affairs Fran Unsworth apologised to Sir Cliff and said: \"In retrospect, there are things we would have done differently.\"", "\n\nBut, she said, the case marked a \"significant shift\" against press freedom and an \"important principle\" around the public's right to know was at stake.", "\n\nMedia playback is unsupported on your device Media caption The BBC's director of news says the ruling is a \"significant shift against press freedom\"\n\nIn his judgement, Mr Justice Mann said the BBC had infringed Sir Cliff's privacy rights in a \"serious\" and \"sensationalist\" way.", "\n\nHe rejected the BBC's case that its reporting, which included footage filmed from a helicopter, was justified under rights of freedom of expression and of the press.", "\n\nMr Justice Mann said a suspect in a police investigation \"has a reasonable expectation of privacy\" and while Sir Cliff being investigated \"might be of interest to the gossip-monger\", there was not a \"genuine public interest\" case.", "\n\nHe also said while the case could have a \"significant impact on press reporting\", it did not mean the law was changing or he was setting a precedent - as the Human Rights Act already covers the issues at stake, namely the right to privacy versus right to freedom of expression.", "\n\n'Choked up'\n\nHe awarded Sir Cliff £190,000 damages and an extra £20,000 in aggravated damages after the BBC submitted its coverage of the raid for an award.", "\n\nThe BBC must pay 65% of the £190,000 and South Yorkshire Police, which carried out the raid, 35%.", "\n\nSouth Yorkshire Police had earlier agreed to pay Sir Cliff £400,000 after settling a claim he brought against the force.", "\n\nAt court, Sir Cliff told the BBC he was \"choked up\" at the judgement, adding: \"It's wonderful news.\"", "\n\nFans supporting him sang the singer's hit Congratulations as the judgement was announced.", "\n\nMedia playback is unsupported on your device Media caption Fans of Sir Cliff Richard cheered and sang his 1968 song Congratulations outside court\n\nThe 77-year-old singer took legal action against the BBC over broadcasts of a South Yorkshire Police raid on his home in Sunningdale, Berkshire, in August 2014.", "\n\nOfficers were investigating an allegation made by a man who claimed he was sexually assaulted by Sir Cliff at an event at Sheffield United's Bramall Lane in 1985 when he was a child.", "\n\nAnalysis: 'Dark day for news reporting'\n\nBy BBC legal correspondent Clive Coleman\n\nToday's judgement is very significant.", "\n\nThe judge found it was not merely the BBC's use of helicopter pictures which breached Sir Cliff's right to privacy. ", "The simple naming of Sir Cliff as a suspect in the police investigation amounted to a breach of his privacy.", "\n\nIt means, going forward, people who are suspects in police investigations, save in exceptional circumstances, are entitled to reasonably expect the matter is kept private and not covered by the media.", "\n\nThat is why the BBC is broadening this out and saying, in effect, this is a dark day for news reporting.", "\n\nLooking at some of the police investigations covered in the past, the BBC points out that naming the suspects has sometimes resulted in additional complainants coming forward.", "\n\nStanding alongside Sir Cliff outside court, his solicitor Gideon Benaim said the singer's motivation was \"not for personal gain\" but to \"right a wrong\".", "\n\nMr Benaim said Sir Cliff, who has sold 250 million records since he began performing in 1958, had \"never expected his privacy and reputation would be tarnished in this way\".", "\n\nHe said his client had offered to settle earlier with the BBC for \"reasonable\" damages and an apology, but the BBC had been \"defiant\".", "\n\nMedia playback is unsupported on your device Media caption Sir Cliff Richard's lawyer said the singer \"aimed to right a wrong\"\n\nThe case raised \"serious questions\", he said, about the way BBC management scrutinises the work of its journalists and how the BBC focused on preserving an exclusive story rather than respecting Sir Cliff's rights.", "\n\nSir Cliff told reporters he would not comment further at that time, adding: \"It's going to take a while to get over the whole emotional factor.\"", "\n\nLater, he told ITV News the judgement was \"just the most wonderful relief\".", "\n\nHe said he \"couldn't speak\" and his \"eyes were watering\" in the court room.", "\n\n'Creative limbo'\n\nResponding to the judgement, South Yorkshire Police chief constable Stephen Watson said he accepted the findings and the force accepted and apologised for its mistakes at a \"very early stage\".", "\n\n\"I would like to take this opportunity to again offer our sincere apologies for the distress Sir Cliff Richard has suffered,\" he added.", "\n\nThe judge said he would hold another hearing to determine further damages after the singer said his plans for \"professional work\" were \"seriously disrupted\" in the wake of the coverage.", "\n\nHe said that in the years leading up to August 2014, he had worked regularly, released a new album every 18 months or so and usually played a number of concerts.", "\n\nBut he said he had been left \"in effect in creative limbo\" for two years until prosecutors said he would not face any charges.", "\n\nSir Cliff claimed his right to privacy under the Human Rights Act had been violated while the BBC argued that the same act protects freedom of expression.", "\n\nImage copyright REUTERS/Toby Melville Image caption Presenter Gloria Hunniford returned to the High Court to support her friend Sir Cliff\n\nTV and radio presenter Gloria Hunniford, who told the trial her friend Sir Cliff felt \"violated and betrayed\" by the BBC's report, returned to the High Court for the judgement.", "\n\nShe later told the BBC that Sir Cliff spent £4m on his case.", "\n\nMs Hunniford said it was a \"landmark case\" and thought it would \"eventually change the law\", meaning suspects could have anonymity until they are charged with an offence.", "\n\nShe said Sir Cliff had been distraught and was so upset he had only returned to the flat for five minutes after the raid before he went on to sell it.", "\n\nWhat was a \"total invasion of privacy\" for Sir Cliff, she said, was footage from a camera zooming in on police searching inside his home.", "\n\n'Recognise responsibilities'\n\nConservative MP Anna Soubry also called for a law change at Prime Minister's Questions.", "\n\nCiting what she dubbed \"Cliff's' law\", Ms Soubry said: \"Suspects should not be named by the media until such time as they are charged.\"", "\n\nIn response, Theresa May said it was a very \"difficult issue\" and needed \"careful judgement\".", "\n\n\"There may well be cases where the publication of a name enables other victims to come forward and therefore strengthen a case,\" the prime minister said.", "\n\nShe also said both the police and media \"have to recognise their responsibilities\".", "\n\nImage copyright Victoria Jones/PA Wire Image caption Fans of Sir Cliff were outside the court\n\nBBC director of editorial policy and standards David Jordan told BBC Radio 4's World At One the judgement could have a \"chilling effect\" on the ability of the media to report when somebody is the subject of a police investigation.", "\n\nHe said the BBC will look in depth at the 200-page judgement before deciding on whether or not to appeal.", "\n\nThe judge was also critical of various BBC witnesses who gave evidence.", "\n\nGary Smith, the BBC's UK news editor at the time and now head of news at BBC Scotland, was \"obsessed\" with scooping rivals which \"probably affected some of his judgement\" at the time of the coverage, he said.", "\n\nMs Unsworth, who gave final approval for the story, was also, but to a lesser extent, \"affected by the desire to protect the scoop\" at the time while Dan Johnson, the reporter who broke the story, was \"not dishonest\" but let his \"enthusiasm get the better of him\"." ]

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[ "3 Historic Entrepreneur Moms To Celebrate This Mother's Day\n\nWhile Jason Porath was working at DreamWorks Animation, he got really tired of seeing women depicted in the same narrow roles over and over again. ", "So in 2014 he quit his job and dedicated himself to telling the stories of amazing women in all of their varied and fantastic ways. ", "You’ve probably read his celebrated first book, Rejected Princesses: Tales of History’s Boldest Heroines, Hellions, and Heretics, which details all the princesses Disney couldn't handle.", "\n\nNow, just in time for Mother’s Day, feast your eyes on Porath’s second book, Tough Mothers: Amazing Stories of History’s Mightiest Matriarchs. ", "In it you’ll find 50 stories about fascinating matriarchs throughout history, from all corners of the globe. ", "In both books, Porath puts his animation background to good use with gorgeous color illustrations of these women and their worlds.", "\n\nIn honor of mothers everywhere, we’re highlighting three women featured in the book who were entrepreneurs and businesspeople long before Oprah, Arianna Huffington, or Sheryl Sandberg.", "\n\nAmerica’s First Female CEO\n\nRebecca LukensJason Porath/Dey Street Books\n\nWhen Rebecca Lukens’ husband died in 1825, she found herself in a precarious situation: $15,000 in debt, suddenly in charge of the family iron-working plant, still mourning the loss of her father and two of her children (who had died the year before), and pregnant. “", "With her sixth child,” Porath writes. ", "Yikes.", "\n\nTwenty-two years of hard work later, “she’d accumulated over $100,00 in personal property” by running a successful business. ", "After her death the company was even renamed in her honor: Lukens Steel. ", "For 125 years after her death her descendants ran the Fortune 500 company, and “in 1994, Fortune magazine recognized Rebecca as America’s first female industrialist leader, and inducted her into the American National Business Hall of Fame.”", "\n\nKorea’s First Female Businessperson\n\nMan-deok KimJason Porath/Dey Street Books\n\nIn early 1800s, Man-deok Kim was a celebrated, successful businesswoman with a government post and a residence in Seoul. ", "But Kim didn’t start life out with a silver spoon. ", "Orphaned at a young age she soon became a gisaeng, “an entertainer/concubine/slave of the government.” ", "After she learned that her father had been a nobleman, she petitioned for her freedom and won it. ", "She then opened a hotel for merchants, which she parlayed into a mercantile empire, trading clothing, jewelry, cosmetics, and more and making a tidy bundle. ", "She’s become part of Korean culture at large, with an award named after her, TV show based on her life, and even her own Korean proverb (\"work like a dog, spend like Man-deok\").", "\n\nSo how did she get that government post, a house in Seoul, and even an official title of Uinyeo Bansu (\"medicine woman\")? ", "Through her generosity: During an unfathomable famine, she donated hundreds of sacks of rice to her Jeju Island countrymen. ", "The plentiful food (which even the government couldn't provide) got the attention of the king, who granted her more fortune than she could have wished for.", "\n\nOne of America’s Wealthiest Self-Made Women\n\nMadam C.J. WalkerJason Porath/Dey Street Books\n\nThe child of ex-slaves who was married by age 14 and a mother by 18, Sarah Breedlove “didn’t exactly start life with a ton of options,” as Porath writes, noting that she first worked as a laundress in the late 1800s. ", "Yet a few decades later, she was the wealthiest black woman in America. ", "She wore the best clothes in the country, drove new automobiles, lived in an enormous mansion, and was known by a new name: Madam C.J. Walker.", "\n\nWalker made her fortune selling hair-care products aimed at the African-American community, and “improving the lot of African Americans became part of her company’s mission.” ", "When she died at age 61, “newspapers celebrated her as America’s first female self-made millionaire.”", "\n\nMegan Giller is the author of the best-selling book Bean-to-Bar Chocolate: America’s Craft Chocolate Revolution. ", "Her blog Chocolate Noise was a 2016 Saveur Blog Awards finalist, and her work has been published in The New York Times, Slate, Zagat, Food & Wine, and Modern ..." ]

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[ "2017 Alumni Summer Soirée\n\nIt’s our 20th birthday this year! ", "In 1997, Maynooth officially became an independent university, but we know full well that the memories and milestones began long before that. ", "Come celebrate this historic chapter—and the start of summer! —", "with the annual Maynooth Alumni Summer Soirée on 8th June.", "\n\nIrish film is the theme for this special year. ", "Join Prof Maria Pramaggiore from the Department of Media Studies as she gives a special commemorative talk on 'Cinema and the Three Ages of Maynooth: 1795, 1910 and 1997,' with clips and context about the state of Irish cinema at each of these Maynooth milestones. ", "Following will be guest alumni Denis Clohessy (Class of '95), an award-winning composer for film and theatre, and international soprano Rachel Croash (Class of '09).", "\n\nDon’t miss this chance to celebrate our 20th anniversary with a special Summer Soirée focused on Irish Film and the music behind it. ", "Beginning with a wine reception at 7pm, invite a friend and get your summer off to a great start with the very best of Maynooth culture. ", "Register now for tickets." ]

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[ "SHENZHEN, China, March 15, 2012 /PRNewswire-Asia-FirstCall/ -- China Information Technology, Inc. (Nasdaq: CNIT), a leading provider of information technologies and display technologies based in China, today announced that its Board of Directors has approved the termination of its stock purchase plan. ", "At the same time, the Company's chairman and chief executive officer, Mr. Jiang Huai Lin, entered into a new $2 million purchase plan. ", "Mr. Lin also agreed to purchase 1,084,895 shares in a private transaction outside the purchase plan at a purchase price per share of $1.20.", "\n\nThe Company has approximately 27.0 million shares outstanding, approximately 41.0% of which are currently held by Mr. Lin, not including the pending acquisition of approximately 1.1 million shares as described above.", "\n\nMr. Lin commented, \"To reemphasize my commitment to the Company and my belief in its potentially strong future, I have decided to continue the increase in my equity holdings in the Company by reinitiating my purchases under our purchase plan and making a separate private purchase. ", "At the same time, the termination of the Company's share repurchase plan will enable it to devote more resources to other priorities such as marketing and research and development.\"", "\n\nThe purchases under the new purchase plan will be made from time to time on the open market or in privately negotiated transactions at the discretion of Mr. Lin's broker, subject to market conditions and other factors, including black-out periods during which Mr. Lin and others are prohibited from trading in the Company's shares.", "\n\nWhile black-out periods typically occur near the end of a fiscal quarter in anticipation of the public release of quarterly earnings, the Company may impose a black-out period at any time without advance public notice. ", "The purchase plan is expected to continue until September 2012 unless extended or shortened by the parties. ", "Mr. Lin intends to effect the share purchase plan in compliance with the conditions of Rule 10b-18 and Rule 10b5-1 under the Securities Exchange Act of 1934, as amended.", "\n\nAbout China Information Technology, Inc.\n\nChina Information Technology, Inc., through its subsidiaries and other consolidated entities, specializes in geographic information systems (GIS), digital public security technology (DPST), and hospital information systems (HIS), as well as high-end digital display products and solutions in China. ", "Headquartered in Shenzhen, China, the Company's integrated solutions include specialized software, hardware, systems integration, and related services to help its customers improve efficiency in information management. ", "To learn more about the Company, please visit its corporate website at http://www.chinacnit.com.", "\n\nSafe Harbor Statement\n\nThis press release may contain certain \"forward-looking statements\" relating to the business of China Information Technology, Inc., and its subsidiary companies. ", "All statements, other than statements of historical fact included herein are \"forward-looking statements\" including statements regarding: the amount and timing of any purchases of the Company's common stock by the Company's chairman and chief executive officer; and any other statements of non-historical information. ", "These forward-looking statements are often identified by the use of forward-looking terminology such as \"believes,\" \"expects\" or similar expressions, involve known and unknown risks and uncertainties. ", "Although the Company believes that the expectations reflected in these forward-looking statements are reasonable, they do involve assumptions, risks and uncertainties, and these expectations may prove to be incorrect. ", "Investors should not place undue reliance on these forward-looking statements, which speak only as of the date of this press release. ", "The Company's actual results could differ materially from those anticipated in these forward-looking statements as a result of a variety of factors, including those discussed in the Company's periodic reports that are filed with the Securities and Exchange Commission and available on its website (http://www.sec.gov). ", "All forward-looking statements attributable to the Company or persons acting on its behalf are expressly qualified in their entirety by these factors. ", "Other than as required under the securities laws, the Company does not assume a duty to update these forward-looking statements." ]

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[ "Q:\n\nSpring STS: Moving classes to other folders causes not found error when running url\n\nWhen I place ALL my classes, that is Controllers, Models, etc under the MAIN umbrella folder, my app works as expected:\nsrc\\main\\java\n|- myapp package\n |-all classes here\n\nBut if I try to place classes under appropriate folders to keep a cleaner hierarchy, then the app will throw an object not found error. ", "For example:\nsrc\\main\\java\n|- myapp package\n |-myapp.java\n|- controller package\n |- controllerclass.java\n|- model package\n |- modelclass.java\n\nThis causes object not found when url invoked even though the app builds.", "\nI'm new to Spring, so not sure if I missed something in the setup somewhere. ", "Any advice appreciated.", "\nThanks\n\nA:\n\nI am assuming you are using spring boot. ", "In order to make your classes eligible for auto wire candidate, they should be in same package or you have to specify the packages through @ComponentScan annotation in your MyApp class\n@ComponentScan({\"controller\", \"model\"})\npublic class MyApp {\n ...\n}\n\n" ]

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[ "Metabolism of ropivacaine in humans is mediated by CYP1A2 and to a minor extent by CYP3A4: an interaction study with fluvoxamine and ketoconazole as in vivo inhibitors.", "\nPotential drug-drug interactions can be identified in vitro by exploring the importance of specific cytochrome P450 (CYP) isozymes for drug metabolism. ", "The metabolism of the local anesthetic ropivacaine to 3-hydroxyropivacaine and (S)-2',6'-pipecoloxylidide was shown in vitro to be dependent on CYP1A2 and 3A4, respectively. ", "In this in vivo model study we quantitated the role of these 2 isozymes for the metabolism of ropivacaine. ", "In a randomized, 3-way crossover study, 12 healthy subjects received a single dose of 40 mg ropivacaine intravenously alone or combined either with 25 mg fluvoxamine as a CYP1A2 inhibitor or with 100 mg ketoconazole as a CYP3A4 inhibitor twice daily for 2 days. ", "Venous plasma and urine samples were collected over 10 hours and 24 hours, respectively. ", "The samples were analyzed for ropivacaine base, 3-hydroxyropivacaine, and (S)-2',6'-pipecoloxylidide. ", "Coadministration with fluvoxamine decreased the mean total plasma clearance of ropivacaine from 354 to 112 mL/min (68%), whereas ketoconazole decreased plasma clearance to 302 mL/min (15%). ", "The relative changes in unbound plasma clearance were similar to the changes in total plasma clearance. ", "The ropivacaine half-life (t1/2) of 1.9 hours was almost doubled during fluvoxamine administration and the plasma concentration at the end of infusion increased slightly, whereas the corresponding parameters after ketoconazole administration remained unchanged. ", "Coadministration with ketoconazole almost abolished the (S)-2',6'-pipecoloxylidide concentrations in plasma, whereas fluvoxamine administration increased the (S)-2',6'-pipecoloxylidide levels. ", "The fraction of dose excreted as 3-hydroxyropivacaine in urine decreased during fluvoxamine administration from 39% to 13%. ", "CYP1A2 is the most important isozyme for the metabolism of ropivacaine. ", "Drug-drug interactions with strong inhibitors of this isozyme could be of clinical relevance during repeated administration. ", "A potent inhibitor of CYP3A4 causes a minor decrease in clearance, which should be of no clinical relevance." ]

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[ "Planning instruction in gerontology.", "\nThe gerontology course taught in the Physical and Occupational Therapy Programme, McGill University, was scrutinized using a) knowledge identified by researchers as necessary for professionals tending the aged and b) the results of a questionnaire administered to graduates working with the aged. ", "Researchers revealed that professionals should be able to provide support for the aged and their families, frequently evaluate their problems, encourage their independence and avoid stereotyping them. ", "Results of the questionnaire showed that graduates prioritized the amount of time they spent on various tasks in this order; listening, talking to clients, consulting with team members and lastly performing physiotherapeutic tasks. ", "An approach to Multidisciplinary Services is described which would ensure quality care for the aged and enhance the accessibility of these services." ]

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[ "Listeria monocytogenes meningitis at King Edward VIII Hospital, Durban. ", "A 10-year experience, 1981-1990.", "\nNine cases, 3 adults and 6 children, with Listeria monocytogenes meningitis were seen over a 10-year period at King Edward VIII Hospital, Durban. ", "These cases accounted for 0.8% (3/374) and 0.6% (6/1,210) of all culture-positive cases of acute bacterial meningitis in adults and children, respectively, and represented 2.9% (4/136) of all culture-positive cases in the neonatal age group and 5.7% (3/53) of culture-positive cases in adults 50 years and older. ", "The patients had positive blood and cerebrospinal fluid (CSF) cultures. ", "All isolates were sensitive to ampicillin, chloramphenicol, sulphamethoxazole-trimethoprim combination and gentamicin. ", "One isolate in an 11-month-old child was resistant to penicillin and 2 isolates in the adult patients displayed intermediate sensitivity to this antibiotic. ", "The adults were over 50 years of age and presented with an abrupt onset of a pyrexial illness, meningitis and focal neurological signs; only 1 survived. ", "Only 1 8-week-old infant of the paediatric cases survived. ", "A polymorphonuclear leucocytosis, low serum glucose and elevated protein values were common findings in the CSF and the features in some patients mimicked tuberculous or viral meningitis. ", "The fulminant course of the disease and the fact that penicillin and not ampicillin is the first-line antibiotic makes it essential to consider listeriosis as a possible diagnosis, particularly in the very ill patient." ]

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[ "Introduction {#s1}\n============\n\n*Pseudomonas syringae* is a widespread bacterial pathogen that causes disease on a broad range of economically important plant species. ", "In order to infect, *P. syringae* produces a number of toxins and uses a type III secretion system (TTSS) to deliver effector proteins into eukaryotic cells [@pbio.1001792-OBrien1],[@pbio.1001792-Xin1]. ", "This mechanism is essential for successful infection by both plant- and animal-associated bacteria as bacterial mutants deficient in the TTSS are no longer pathogenic [@pbio.1001792-Nomura1]. ", "Effectors contribute collectively to pathogenesis inside the host cell by targeting host molecules and defeating plant defenses, which are based on two tiers of recognition by the innate immune system [@pbio.1001792-Jones1]. ", "The first branch is triggered by the recognition of highly conserved microbe-associated molecular patterns (MAMPs) by host cell transmembrane proteins that function as pattern recognition receptors (PRRs), which in turn, activate MAMP-triggered immunity (MTI) [@pbio.1001792-Jones1]. ", "The second branch recognizes type III effectors inside the plant cell via nucleotide-binding site-leucine-rich repeat (NB-LRR) resistance (R) proteins [@pbio.1001792-Jones1]. ", "This leads to activation of effector-triggered immunity (ETI), and is characteristically associated with programmed cell death known as the hypersensitive response (HR). ", "Accumulating evidence suggests that a primary function of microbial effectors is suppression of both MTI and ETI to avoid pathogen recognition during the infection process [@pbio.1001792-Lindeberg1]. ", "However, despite the fact that elucidating effector action is essential to understanding bacterial pathogenesis, the molecular function and host targets of the vast majority of effectors remain largely unknown.", "\n\nPlant immunity relies on a complex network of small-molecule hormone signaling pathways [@pbio.1001792-RobertSeilaniantz1]. ", "Classically, salicylic acid (SA) signaling mediates resistance against biotrophic and hemi-biotrophic microbes such as *P. syringae*, whereas a combination of jasmonic acid (JA) and ethylene (ET) pathways activates resistance against necrotrophs such as the fungal pathogen *Botrytis cinerea* [@pbio.1001792-RobertSeilaniantz1]. ", "SA and JA/ET defense pathways generally antagonize each other and thus, elevated resistance against biotrophs is often correlated with increased susceptibility to necrotrophs, and vice versa [@pbio.1001792-Grant1]. ", "The collective contribution of these two hormones during plant-pathogen interactions is crucial to the success of the interaction. ", "Remarkably, some *P. syringae* strains have evolved a sophisticated strategy for manipulating hormonal homeostasis by producing coronatine (COR), a mimic of the bioactive jasmonate hormone, JA-isoleucine (JA-Ile) [@pbio.1001792-Fonseca1]. ", "COR contributes to disease symptomatology by inducing chlorotic lesions [@pbio.1001792-Uppalapati1]--[@pbio.1001792-Brooks1], facilitates entry of the bacteria into the plant host by stimulating the opening of stomata [@pbio.1001792-Melotto1],[@pbio.1001792-Melotto2], and promotes bacterial growth by inhibiting SA-dependent defenses required for *P. syringae* resistance, because of its activation of the antagonistic JA pathway [@pbio.1001792-LaurieBerry1],[@pbio.1001792-Cui1]. ", "COR, as the JA-Ile phytohormone, is perceived through a receptor complex formed by the F-box protein CORONATINE-INSENSITIVE 1 (COI1) and JASMONATE ZIM DOMAIN (JAZ) proteins [@pbio.1001792-Sheard1]--[@pbio.1001792-Thines1]. ", "COI1 is the F-box component of an SCF-(Skip-cullin-F-box)-type E3 ubiquitin ligase required for all JA-dependent responses tested so far [@pbio.1001792-Fonseca1],[@pbio.1001792-Feys1]--[@pbio.1001792-Katsir1]. ", "JAZ co-receptors are COI1 substrates that negatively regulate the JA-signaling pathway by directly interacting with and repressing transcription factors (TFs) that control JA-regulated genes [@pbio.1001792-Sheard1]--[@pbio.1001792-Thines1],[@pbio.1001792-Pauwels1],[@pbio.1001792-FernandezCalvo1]. ", "Repression of TFs by JAZ is mediated by a general co-repressor machinery involving TOPLESS (TPL) and TPL-related proteins that interact with JAZ repressors through the adaptor protein NINJA [@pbio.1001792-Pauwels2]. ", "The JAZ family of JA-repressors consists of 12 members in *Arabidopsis* that have emerged as central modulators of JA signaling [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Yan1]. ", "Under stress conditions, COR or JA-Ile promotes the formation of JAZ-COI1 complexes, triggering JAZ degradation via the 26S proteasome [@pbio.1001792-Sheard1]--[@pbio.1001792-Thines1]. ", "This leads to de-repression of the TFs that initiate the transcription of JA-dependent genes, and repression of SA-dependent defenses against the bacteria. ", "Thus, COR acts as a potent virulence factor in plants by triggering the degradation of JAZs. ", "Acquisition of COR by bacterial pathogens has been of tremendous adaptive importance during host-pathogen evolution because it has allowed bacteria to manipulate the host hormonal network to promote susceptibility.", "\n\nCOR is produced by several bacterial strains distributed throughout the *P. syringae* phylogeny, but is particularly common among *P. syringae* pv. *", "tomato* strains such as DC3000 (*Pto* DC3000) [@pbio.1001792-Bender1],[@pbio.1001792-Hwang1]. ", "Interestingly, strains such as *P. syringae* pv. *", "tabac*i (*Pta*) 11528 that do not produce COR can still open stomata, suggesting that other virulence factors, probably type-III effectors, are used to activate the JA pathway instead of COR [@pbio.1001792-Melotto1],[@pbio.1001792-Lee1],[@pbio.1001792-Studholme1]. ", "This is supported by several studies suggesting that COR and effectors act synergistically to induce JA responses [@pbio.1001792-Thilmony1]--[@pbio.1001792-Cui2]. ", "Indeed, several effectors have been shown to modulate the expression of JA-inducible genes [@pbio.1001792-He1]. ", "Moreover, gene expression profiles indicate that several JA-regulated genes are still induced in *coi1* mutants upon *P. syringae* infection, indicating that JA responses are activated downstream or independently of COI1 [@pbio.1001792-Chen1]. ", "Therefore, bacterial effectors should target other components of the JA pathway downstream of COI1 to activate JA responses, the best candidates being JAZ repressors. ", "Recently, Mukhtar and colleagues developed a large-scale map of physical interactions between proteins from the reference plant *Arabidopsis thaliana* and effector proteins from *P. syringae* and the obligate biotrophic oomycete *Hyaloperonospora arabidopsidis* (*Hpa*) [@pbio.1001792-Mukhtar1]. ", "The experiment yielded a map of 6,200 interactions, and showed that pathogens from different kingdoms deploy independently evolved virulence effectors that interact with a limited set of highly connected cellular hubs to facilitate their diverse life-cycle strategies. ", "Strikingly, two out of the five most significantly targeted plant hub proteins by effectors (namely RESPONSE TO LOW SULPHUR \\[LSU1, AT3G49580\\] and an unknown kinesin light-chain related protein \\[AT3G27960\\]), physically interact with several JAZ proteins [@pbio.1001792-Mukhtar1]. ", "Thus, it is plausible that pathogens may attempt to manipulate the JA pathway directly through JAZ proteins.", "\n\nIn previous studies, we generated a draft genome sequence of *Pta* 11528 and used a functional screen to infer its repertoire of T3SS effectors [@pbio.1001792-Studholme1]. ", "This led to the identification of *Pta* 11528 proteins with homology to previously described effectors. ", "Since *Pta* 11528 does not produce COR, we hypothesized that it might have followed an alternative evolutionary strategy to activate the JA pathway by developing effector proteins that target JA signaling components. ", "As noted above, we hypothesized that JAZ repressors would be the best candidate targets. ", "Therefore, to study whether any such effector proteins target JAZ repressors, we developed a screen to analyze the stability of JAZ proteins in the presence of each *Pta* 11528 effector. ", "We found that the *Pta* 11528 effector HopX1 encodes a cysteine protease that interacts with and degrades JAZ proteins in a COI1-independent manner. ", "Ectopic expression of HopX1 in *Arabidopsis* induced the expression of JA-dependent genes, compromised the induction of SA-marker gene *PR1* upon SA treatment, and complemented the growth of a COR-deficient *Pto* DC3000 strain during natural bacterial infections. ", "Moreover, *Pto* DC3000 *COR−* growth increased by about one log (colony forming units \\[cfu\\]/cm^2^) when naturally expressing *hopX1*, but not its catalytic mutant version, indicating that HopX1 can effectively promote susceptibility when delivered by the natural TTSS. ", "This increase in susceptibility was similar to the effect of supplementing bacteria with 2 µM of COR and independent of COI1. ", "Altogether, the results indicate that similar to COR, HopX1 acts within the plant cell to promote activation of jasmonate-induced defenses and bacterial disease bypassing the need of JA-Ile perception.", "\n\nResults {#s2}\n=======\n\nHopX1 Compromises the Accumulation of JAZ Proteins {#s2a}\n--------------------------------------------------\n\nIt has been suggested that *P. syringae* pv. *", "tabaci* activates JA responses such as stomatal aperture without producing COR [@pbio.1001792-Melotto1],[@pbio.1001792-Lee1],[@pbio.1001792-Studholme1], which suggests that effectors from this strain could provide the same function by targeting components of the JA pathway. ", "Whereas stomatal closure is part of a plant innate immune response triggered upon pathogen perception to restrict bacterial invasion, plant pathogenic bacteria have evolved specific virulence factors such as COR to promote stomata opening in order to circumvent such innate immune responses [@pbio.1001792-Melotto1],[@pbio.1001792-Melotto2]. ", "To investigate whether live *Pta* 11528 bacteria can re-open plant stomata without producing COR, we first incubated host *Nicotiana benthamiana* leaves with virulent *Pta* 11528 bacteria. ", "In this experiment, *Pto* DC3000 was used as a positive control, and a non-pathogenic TTSS defective strain *Pta* 11528 *hrpV−* unable to secrete effector proteins into the plant cell served as a negative control [@pbio.1001792-Oh1]. ", "Incubation of *N. benthamiana* leaves with *Pta* 11528 *hrpV−* for 5 hours induced stomatal closure as described previously ([Figure S1](#pbio.1001792.s001){ref-type=\"supplementary-material\"}) [@pbio.1001792-Melotto1]. ", "In contrast, virulent *Pta* 11528 or *Pto* DC3000 bacteria maintained stomatal apertures similar to mock treatments. ", "This indicates that effector proteins from *Pta* 11528 might play a role in re-opening of stomata as COR does, probably through action on the JA-signaling pathway.", "\n\nTo identify *Pta* 11528 effector proteins that could target JAZs, we analyzed a *Pta* 11528 effector library constructed in a binary vector within the T-DNA region, in which individual genes are expressed from the *35S* promoter as genetic fusions to three C-terminal hemagglutinin (HA) epitope tags ([Table S1](#pbio.1001792.s016){ref-type=\"supplementary-material\"}). ", "We transiently co-expressed *JAZ5-HA* with individual effector genes from this library or an empty vector (*EV*) control in *N. benthamiana* by agroinfiltration. ", "Using this approach, we identified HopX1 as a *Pta* 11528 effector capable of compromising JAZ5 accumulation ([Figure S2](#pbio.1001792.s002){ref-type=\"supplementary-material\"}). ", "To confirm this result and to exclude a potential effect of the HA tag, we developed an independent form of HopX1 with an N-terminal green fluorescent protein (GFP) fusion. ", "We then transiently co-expressed an *EV* construct or *GFP-hopX1* under the control of the *35S* promoter with *35S:JAZ5-HA* in *N. benthamiana* leaf tissue. ", "Western blot analysis showed that GFP-HopX1 accumulated in *N. benthamiana* ([Figure 1A](#pbio-1001792-g001){ref-type=\"fig\"}). ", "Similar to previous results, JAZ5-HA protein was detectable when co-expressed with the EV control but not (or only weakly) with GFP-HopX1, despite the fact that *GFP-hopX1* co-expression did not affect *JAZ5-HA* mRNA expression levels ([Figure 1B](#pbio-1001792-g001){ref-type=\"fig\"}). ", "These results indicate that HopX1 compromised the accumulation of JAZ5 protein when transiently co-expressed in *N. benthamiana* without affecting gene expression levels.", "\n\n{#pbio-1001792-g001}\n\nThe *Arabidopsis* JAZ family contains 12 members grouped into four major phylogenetic clades ([Figure S3](#pbio.1001792.s003){ref-type=\"supplementary-material\"}) [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Yan1]. ", "To test if HopX1 could compromise the accumulation of JAZ proteins other than JAZ5, we transiently co-expressed the 12 *JAZ*-HA genes individually with *GFP-hopX1* or an *EV* control in *N. benthamiana*, and analyzed JAZ accumulation by Western blotting. ", "We successfully detected eight out of 12 JAZ proteins when co-infiltrated with the EV control ([Figure 1C](#pbio-1001792-g001){ref-type=\"fig\"}). ", "Interestingly, HopX1 compromised the accumulation of all eight JAZ proteins detected, indicating that HopX1 activity is not restricted to JAZ5 but targets the whole JAZ family ([Figure 1C](#pbio-1001792-g001){ref-type=\"fig\"}). ", "We also analyzed the effect of HopX1 on the stability of additional JA related proteins such as the JA receptor COI1 and the downstream TF MYC2, which is a direct target of JAZ repressors [@pbio.1001792-Chini1]. ", "HopX1 did not alter COI1 or MYC2 protein levels compared to the EV control ([Figure 1D and 1E](#pbio-1001792-g001){ref-type=\"fig\"}). ", "Altogether, these results indicate that HopX1 compromises the accumulation of the JAZ family of JA-repressors in a specific manner.", "\n\n*HopX*1 Encodes a Cysteine Protease and This Activity Is Required for HopX1-Mediated JAZs Degradation {#s2b}\n-----------------------------------------------------------------------------------------------------\n\nHopX1 family members from different *P. syringae* strains are modular proteins that contain a putative cysteine-based catalytic triad and a novel conserved N-terminal domain [@pbio.1001792-Nimchuk1]. ", "The putative catalytic triad is required for effector function and consists of cysteine (C), histidine (H), and aspartic acid (D) residues conserved with cysteine proteases [@pbio.1001792-Nimchuk1]. ", "In order to determine whether these domains are conserved in HopX1 from *Pta* 11528, we performed a PSI-BLAST search with the HopX1*~Pta~* ~11528~ protein sequence. ", "BLAST analysis revealed a near identical match with HopX1 from *P. syringae* pv. *", "phaseolicola* bacterial strains such as race 4 (99% identity), but showed significantly less homology with the respective *Pto* DC3000 homolog (72% identity) ([Figure S4A](#pbio.1001792.s004){ref-type=\"supplementary-material\"}). ", "The putative catalytic triad residues were strictly conserved in HopX1*~Pta~* ~11528~, as was the novel N-terminal domain of unknown function typical of this family of effectors ([Figure S4A](#pbio.1001792.s004){ref-type=\"supplementary-material\"}). ", "This conservation suggests that the domains may play an important role in the activity of this effector inside the plant cell, and that HopX1*~Pta~* ~11528~ might have cysteine protease activity.", "\n\nTo determine if HopX1 has cysteine protease activity *in vitro*, we used a kit designed for the detection of protease activity (serine, aspartic, cysteine, and metalloproteinases) using fluorometry based on the hydrolysis of a labeled casein general substrate [@pbio.1001792-Twining1]. ", "As previously described by Nimchuck and colleagues [@pbio.1001792-Nimchuk1], we did not detect any protease activity when purified recombinant HopX1 protein fused to maltose binding protein (MBP) was incubated with the casein-labeled substrate *in vitro*, indicating that this recombinant protein may be inactive or that it might lack a co-factor ([Figure S4B](#pbio.1001792.s004){ref-type=\"supplementary-material\"}). ", "However, we detected significant protease activity when the casein substrate was incubated with HopX1-HA immunopurified directly from stable transgenic *Arabidopsis* plants expressing the *hopX1* gene from a *dexamethasone*-inducible promoter (*DEX*) ([Figure 2A](#pbio-1001792-g002){ref-type=\"fig\"}), suggesting that HopX1 expressed *in planta* has protease activity. ", "To test whether this activity of HopX1 required its conserved cysteine protease catalytic triad, we substituted the conserved Cys-179 residue within this domain for an alanine residue to generate the HopX1^C179A^ mutant. ", "HopX1^C179A^-HA inmunoprecipitated from transgenic *Arabidopsis* plants did not show any proteolytic activity compared to negative controls. ", "The trypsin enzyme used as a positive control in these experiments showed much higher activity on the casein substrate than HopX1 ([Figure 2A](#pbio-1001792-g002){ref-type=\"fig\"}). ", "These data indicate that HopX1 has protease activity, but seems to operate suboptimally on a general substrate *in vitro*. ", "To test whether HopX1 may have evolved specific substrate selectivity, we incubated inmunoprecipitated HopX1 and HopX1^C179A^ from *Arabidopsis* with recombinant MBP-JAZ5 expressed and purified from *Escherichia coli* cells with or without protease inhibitors. ", "The amount of MBP-JAZ5 diminished significantly when incubated with HopX1 but not with HopX1^C179A^ or buffer in the absence of protease inhibitors, but not in its presence ([Figure 2B](#pbio-1001792-g002){ref-type=\"fig\"}). ", "Thus, HopX1, but not HopX1^C179A^, is capable of inducing JAZ5 degradation *in vitro* suggesting that the effector indeed acts as a protease on the JAZ5 substrate.", "\n\n{#pbio-1001792-g002}\n\nTo confirm that degradation of JAZ proteins by HopX1 requires its putative cysteine protease activity *in vivo*, we next tested the effect of the C179A mutation on JAZ5-HA accumulation when the proteins were co-expressed transiently in *N. benthamiana* leaves ([Figure 2C](#pbio-1001792-g002){ref-type=\"fig\"}). ", "JAZ5-HA was detectable in the presence of HopX1^C179A^, but not HopX1, indicating that this catalytic residue is indeed critical for the effect of HopX1 on JAZ5 levels ([Figure 2C](#pbio-1001792-g002){ref-type=\"fig\"}). ", "Similarly, only wild-type HopX1, but not HopX1^C179A^, compromised the accumulation of additional JAZs such as JAZ1, JAZ2, JAZ9, and JAZ10 when transiently co-expressed in *N. benthamiana* leaves. ", "However, HopX1 did not alter MYC2 protein levels ([Figure S4C](#pbio.1001792.s004){ref-type=\"supplementary-material\"}). ", "To exclude that JAZ5 degradation is a general property of cysteine proteases, we analyzed the effects of previously described cysteine proteases such as HopC1 [@pbio.1001792-Dowen1] and HopN1 [@pbio.1001792-LopezSolanilla1] or an independent HopAD1 effector on the stability of JAZ5 when coexpressed transiently in *N. benthamiana*. ", "As expected, JAZ5-HA accumulation was compromised in the presence of HopX1, but not when co-expressed with the EV control or with HopC1 or HopN1 proteins ([Figure S5](#pbio.1001792.s005){ref-type=\"supplementary-material\"}). ", "The HopAD1 effector could not be detected in this assay. ", "Taken together, these data suggest that degradation of JAZ proteins by HopX1 is specific and depends on a cysteine protease enzymatic activity that requires a conserved cysteine within the proposed catalytic triad.", "\n\nHopX1 Associates with and Degrades JAZs in a COI1-Independent Manner {#s2c}\n--------------------------------------------------------------------\n\nIncreased JA-Ile levels promote binding of JAZs to SCF^COI1^ and subsequent degradation of JAZ repressors via the ubiquitin/26S proteasome pathway [@pbio.1001792-Sheard1]--[@pbio.1001792-Thines1]. ", "Therefore, degradation of JAZs by HopX1 might be direct (through its protease activity) or an indirect effect mediated by JA-Ile synthesis and COI1. ", "To investigate if JAZ degradation by HopX1 is direct or indirect, we first analyzed whether HopX1-induced degradation of JAZ5 was dependent on the 26S proteasome pathway by using the proteasomal inhibitor MG132. ", "In HopX1 coexpression experiments in *N. benthamiana*, JAZ5 was not detected in the presence of MG132, indicating that degradation does not require the proteasome ([Figure S6](#pbio.1001792.s006){ref-type=\"supplementary-material\"}). ", "We next checked whether HopX1-induced degradation of JAZ proteins occurred in a COI1-dependent or independent manner. ", "To test this, we first used a stable transgenic *N. tabacum* line silenced for expression of the *NtCOI1* gene [@pbio.1001792-Shoji1]. ", "Reverse transcription (RT)-PCR analysis confirmed that control *N. tabacum* plants (Line VC, transformed with *EV*) accumulated *NtCOI1* mRNA, whereas *NtCOI1* transcripts were undetectable in *N. tabacum* plants silenced for the *NtCOI1* gene (Line L18) ([Figure S7A](#pbio.1001792.s007){ref-type=\"supplementary-material\"}). ", "Interestingly, *NtCOI1*-silenced *N. tabacum* plants produced few seeds, a phenotype reminiscent of infertile *Arabidopsis coi1-1* plants ([Figure S7B](#pbio.1001792.s007){ref-type=\"supplementary-material\"}) [@pbio.1001792-Feys1]. ", "We next analyzed the ability of HopX1 to trigger JAZ5 degradation in both *EV*- and *NtCOI1*-silenced plants when transiently co-expressed in *N. tabacum*, a species that also allows facile transient gene expression assays [@pbio.1001792-Sparkes1]. ", "Strikingly, GFP-HopX1 compromised the accumulation of JAZ5 in both *EV*- and *NtCOI1*-silenced plants to the same extent ([Figure 3A](#pbio-1001792-g003){ref-type=\"fig\"}). ", "This suggests that HopX1 triggers the degradation of JAZ proteins in a COI1-independent manner.", "\n\n![", "HopX1 interacts with and degrades JAZ proteins in a COI1-independent manner.\\\n(A) HopX1 compromises the accumulation of JAZ5 in *N. tabacum* plants silenced for the *NtCOI1* gene. ", "Immunoblots showing JAZ5-HA accumulation in the presence of GFP-HopX1 or GFP alone, when co-expressed transiently in *N. tabacum* plants silenced for the *NtCOI1* gene (line 18) or EV-transformed (line VC). ", "CBB, Coomassie brilliant blue staining. ", "This experiment was repeated three times with similar results. (", "B) HopX1 triggers the degradation of JAZΔJas proteins in a *COI1*-independent manner. *", "N. benthamiana* plants were transiently co-transformed with *GFP-hopX1* or *GFP* alone, and the dominant-negative JAZ variants *JAZ1*Δ*Jas-HA*, *JAZ2*Δ*Jas-HA*, or *JAZ7*Δ*Jas-HA* proteins as indicated. ", "Protein stability was analyzed by immunoblot. ", "This experiment was repeated twice times with similar results. (", "C) HopX1 interacts with JAZ repressors in PD assays. ", "Immunoblots with anti-HA antibody of HopX1-HA or HopX1^C179A^-HA recovered after PD experiments using crude protein extracts from *DEX*:*hopX1-HA* (X1), *DEX*:*hopX1* ^C179A^-*HA* (CA), or Col-0 (C) *Arabidopsis* plants, and resin-bound recombinant MBP or MBP-fused JAZ proteins (top). ", "Input lanes show the level of expression of recombinant HopX1 proteins in transgenic and control plants. ", "CBB staining shows the amount of recombinant JAZ-MBP or MBP proteins used in the resin (bottom). ", "The results are representative of five independent experiments. (", "D) Schematic representation of the JAZ5 protein and its conserved domains. ", "The NT, the ZIM, and the Jas domains are depicted and the corresponding JAZ5 fragments are represented. (", "E) HopX1 interacts with JAZ proteins through their conserved ZIM domains in PD assays. ", "Immunoblot (anti-HA antibody) of HopX1-HA and HopX1^C179A^-HA recovered from PD reactions (using extracts of *DEX:hopX1-HA* \\[X1\\], *DEX:hopX1* ^C179A^-*HA* \\[CA\\], or Col-0 \\[C\\] *Arabidopsis* plants) using MBP or MBP-fused JAZ5, JAZ5~1--91~ (JAZ5 NT), JAZ5~92--163~ (JAZ5 ZIM), or JAZ5~164--274~ (JAZ5 Jas) derivatives (top). ", "The lower panels show the CBB staining of the input quantity of recombinant MBP proteins used on the column. ", "The results are representative of three independent experiments. (", "F) Subcellular localization of HopX1 in plant cells. ", "Confocal microscopy localization of transiently expressed GFP-HopX1 or GFP alone in *N. benthamiana* leaves 48 hours post-infiltration (green). ", "Nuclei were stained with DAPI (blue). ", "This experiment was repeated three times with similar results.](pbio.1001792.g003){#pbio-1001792-g003}\n\nJAZ proteins are characterized by three sequence motifs, namely a relatively conserved N-terminal (NT) motif and the two highly conserved ZIM (central) and Jas (C-terminal) domains ([Figure 3D](#pbio-1001792-g003){ref-type=\"fig\"}) [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Pauwels1],[@pbio.1001792-Yan1],[@pbio.1001792-Fonseca2]. ", "The ZIM domain mediates homo- and heteromeric interactions between *Arabidopsis* JAZ proteins [@pbio.1001792-Chini2],[@pbio.1001792-Chung1] and interacts with the general adaptor protein NINJA [@pbio.1001792-Pauwels2], whereas the C-terminal Jas domain is responsible for the interaction with COI1 and the TFs [@pbio.1001792-Chini2],[@pbio.1001792-FernandezCalvo1],[@pbio.1001792-Melotto3]. ", "Dominant-negative JAZ variants lack part of the C-terminal Jas domain [@pbio.1001792-Sheard1],[@pbio.1001792-Chini1],[@pbio.1001792-Katsir1]. ", "Consistently, these truncated JAZ forms (JAZΔJas) are resistant to COI1-dependent degradation after jasmonate treatment, and plants overexpressing them are JA-insensitive [@pbio.1001792-Chini1],[@pbio.1001792-Thines1]. ", "To confirm whether degradation of JAZ proteins by HopX1 is COI1-independent, we analyzed the effect of HopX1 on three JAZΔJas proteins, namely JAZ1ΔJas, JAZ2ΔJas, and JAZ7ΔJas. ", "We detected *JAZ1*Δ*Jas*, *JAZ2*Δ*Jas*, and *JAZ7*Δ*Jas* proteins when co-expressed individually with the *EV* control in *N. benthamiana* leaf tissue ([Figure 3B](#pbio-1001792-g003){ref-type=\"fig\"}). ", "However, none of these JAZΔJas protein forms accumulated when co-expressed with GFP-HopX1 ([Figure 3B](#pbio-1001792-g003){ref-type=\"fig\"}), confirming that HopX1 triggers the degradation of JAZ proteins in a COI1-independent manner.", "\n\nAll of our previous results imply that HopX1 targets JAZ proteins directly. ", "Therefore, we next examined the ability of HopX1 and HopX1^C179A^ to interact with all 12 *Arabidopsis* JAZ proteins in pull-down (PD) experiments. ", "To do this, we used recombinant JAZ proteins fused to MBP and cell extracts of either wild-type plants (as negative controls), or transgenic plants expressing HopX1-HA or HopX1^C179A^-HA from the *DEX* inducible promoter. ", "These experiments included protease inhibitors so that degradation of JAZs would not diminish a potential interaction with HopX1. ", "As shown in [Figure 3C](#pbio-1001792-g003){ref-type=\"fig\"}, all full-length MBP--JAZ proteins tested interacted with wild-type HopX1-HA, but not with the mutant HopX1^C179A^-HA form. ", "Similarly, MBP-HopX1 purified from *E. coli* cells co-immunoprecipitated with JAZ5-GFP when the recombinant effector protein was incubated with *N. benthamiana* plants extracts transiently expressing the JAZ5 transgene ([Figure S8](#pbio.1001792.s008){ref-type=\"supplementary-material\"}). ", "We could also detect weak interaction with MBP-HopX1^C179A^ but to a lesser extent than MBP-HopX1. ", "Overall, this indicates that HopX1 interacts with JAZ proteins.", "\n\nTo further determine the requirements for HopX1 interaction with JAZ proteins, we expressed the JAZ deletion mutants JAZ5~1--91~ (NT domain), JAZ5~92--163~ (ZIM domain), and JAZ5~164--274~ (Jas domain) fused to MBP in *E. coli* and purified these fragments for PD analysis with cell extracts of transgenic plants expressing HopX1-HA ([Figure 3D](#pbio-1001792-g003){ref-type=\"fig\"}). ", "As shown in [Figure 3E](#pbio-1001792-g003){ref-type=\"fig\"} the N-terminal fragment alone, or the C-terminus containing the Jas domain, did not interact with HopX1. ", "However, the JAZ5~92--163~ derivative containing the ZIM domain was sufficient for interaction with HopX1. ", "Notably, the mutant version HopX1^C179A^ could not be pulled down with any fragment of JAZ5, as we observed previously with full-length JAZ proteins ([Figure 3E](#pbio-1001792-g003){ref-type=\"fig\"}). ", "The results indicate that HopX1 interacts directly with JAZ proteins though the central ZIM domain, whereas the other conserved domains seem to be dispensable. ", "The data support our previous results showing that HopX1 triggers the degradation of all JAZ proteins, as the ZIM domain is present in all members of the JAZ family of repressors.", "\n\nJAZ proteins are translated in the cytoplasm and localized predominantly in the nucleus [@pbio.1001792-Chini1],[@pbio.1001792-Thines1]. ", "We next examined in which subcellular compartment the degradation of JAZs proteins by HopX1 takes place. ", "To test this, we expressed GFP or GFP-HopX1 in *N. benthamiana* leaves and analyzed its subcellular localization using confocal microscopy. ", "As reported previously [@pbio.1001792-Nimchuk1], the fluorescent signal corresponding to GFP-HopX1 accumulated mainly in the cytoplasm 48 hours post inoculation (HPI) ([Figure 3F](#pbio-1001792-g003){ref-type=\"fig\"}). ", "We could also detect GFP-HopX1 fusion protein within the nucleus in cross-middle nuclear sections using confocal microscopy analysis ([Figures 3F](#pbio-1001792-g003){ref-type=\"fig\"} and [S9A](#pbio.1001792.s009){ref-type=\"supplementary-material\"}), indicating that a pool of the effector enters the nucleus. ", "To confirm this result, we further performed separation of nuclear extracts from the cytoplasmic fraction of *N. benthamiana* leaves transiently expressing GFP alone or GFP-HopX1 fusion protein. ", "Crude fractions enriched for nuclei contained detectable levels of full length GFP-HopX1 although to a lesser extent than cytoplasmic fractions, confirming the microscopy results ([Figure S9B](#pbio.1001792.s009){ref-type=\"supplementary-material\"}). ", "These results indicate that HopX1 could potentially degrade JAZ proteins both in the cytoplasm and nucleus.", "\n\nHopX1 Activates JA-Dependent Gene Expression in *Arabidopsis* {#s2d}\n-------------------------------------------------------------\n\nThe results described above suggest that HopX1 could be responsible for, or at least participate in, the induction of JA-dependent responses by *Pta* 11528. ", "To test if HopX1 is sufficient for this activation, we analyzed JA marker gene expression in stable transgenic *Arabidopsis* lines expressing *hopX1* or *hopX1* ^C179A^ genes from a *dexamethasone*-inducible promoter (*DEX*). ", "These transgenic lines were constructed in *Arabidopsis* accession Aa--0, an ecotype that does not trigger cell death in response to certain *HopX1* alleles [@pbio.1001792-Nimchuk1]. ", "As JA markers, we chose genes induced early after JA treatment such as *JAZ5*, *JAZ10*, and *JAZ12*. ", "As shown in [Figures 4A](#pbio-1001792-g004){ref-type=\"fig\"} and [S10](#pbio.1001792.s010){ref-type=\"supplementary-material\"}, induction of *hopX1* expression by DEX treatment for 36 hours strongly up-regulated all three *JAZ* marker genes in transgenic *hopX1 Arabidopsis*, whereas transcript levels remained low in transgenic plants expressing *hopX1* ^C179A^ or an EV control. ", "Moreover, transgenic plants expressing *hopX1* developed chlorotic symptoms after DEX treatment, in contrast with *hopX1* ^C179A^ plants ([Figure 4B and 4C](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Development of chlorosis was correlated with losses in chlorophyll content ([Figure S11](#pbio.1001792.s011){ref-type=\"supplementary-material\"}), a hallmark response of the JA pathway [@pbio.1001792-Creelman1]. ", "Similarly, transient expression of *hopX1* by DEX treatment in *N. benthamiana* for 3 days, but not of *hopX1* ^C179A^ or an *EV* control, also induced strong chlorotic symptoms in the infiltrated area ([Figure S12](#pbio.1001792.s012){ref-type=\"supplementary-material\"}). ", "Therefore, HopX1 triggers the activation of JA-dependent gene expression and JA-related phenotypes in a cysteine catalytic triad-dependent manner when ectopically expressed in *Arabidopsis*.", "\n\n{#pbio-1001792-g004}\n\nCrosstalk between JA and SA signaling pathways plays an important role in the regulation and fine-tuning of induced defenses against pathogens. ", "The SA and JA/ET defense pathways generally antagonize each other, and induction of the JA pathway by *P. syringae* strains counteracts SA-dependent defenses [@pbio.1001792-Grant1]. ", "Thus, we next determined whether activation of the JA-pathway by HopX1 could interfere with SA-dependent gene expression. ", "We analyzed the expression of the SA marker gene *PR1* in *DEX*-inducible transgenic *Arabidopsis* Aa--0 plants expressing *hopX1 or hopX1* ^C179A^ pre-treated with 1 mM of SA or a mock solution for 24 hours. ", "Treatment with SA strongly induced *PR1* expression in non-induced *hopX1* or *hopX1* ^C179A^ transgenic plants ([Figure 4D](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Strikingly, DEX induction of the *hopX1* effector gene for 24 hours strongly reduced subsequent SA induction of *PR-1* expression ([Figure 4D](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Pre-induction of the *hopX1* ^C179A^ gene interfered weakly with *PR-1* expression, but to a much lesser extent than wild-type HopX1. ", "Overall, these results suggest that HopX1 activates the JA pathway to suppress SA-dependent defense responses to induce plant susceptibility.", "\n\nOn the basis of these results, we hypothesized that HopX1 could contribute to bacterial pathogenicity by mimicking COR-induced susceptibility. ", "The jasmonate mimic COR is a bacterial toxin that contributes to bacterial invasion of the apoplast by *Pto* DC3000 [@pbio.1001792-Melotto1]. ", "Consistently, coronatine-deficient (*COR−*) *Pto* DC3000 mutants are less virulent on *Arabidopsis* plants when surface-inoculated [@pbio.1001792-Melotto1]. ", "To test if HopX1 contributes to bacterial pathogenicity during natural infections when delivered by the bacterial TTSS we compared bacterial replication of a *Pto* DC3000 *COR−* strain expressing *hopX1* ~Pta11528~ or the *hopX1* ^C179A^ gene on *Arabidopsis* Col-0 plants infected by spray inoculation. ", "This type of inoculation mimics natural infection conditions and is one of the most sensitive techniques to assess plant susceptibility to bacterial pathogens [@pbio.1001792-Zipfel1]. *", "Pto* DC3000 *COR−* growth was increased by about one log (cfu/cm^2^) when expressing *hopX1* compared to the same strain containing *hopX1* ^C179A^ or an empty vector construct ([Figure 4E](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Strikingly, differences in bacterial growth promoted by HopX1 were abolished when DC3000 *COR−* strains expressing *EV* or the *hopX1* or *hopX1* ^C179A^ genes were supplemented with 2 µM of COR ([Figure 4E](#pbio-1001792-g004){ref-type=\"fig\"}). ", "This result supports the idea that HopX1 and COR act redundantly. ", "Similar results were obtained when we compared bacterial replication on transgenic *Arabidopsis* Aa--0 plants expressing *hopX1* or *hopX1* ^C179A^ infected with *Pto* DC3000 or the isogenic *Pto* DC3000 *COR−* strain by spray inoculation ([Figure S13](#pbio.1001792.s013){ref-type=\"supplementary-material\"}). ", "These results indicate that HopX1 can complement the deficiency in COR production of *Pto* DC3000 *COR−*, and supports a key role for the effector catalytic triad in the activation of the jasmonate pathway.", "\n\nWe further studied whether *Pto* DC3000 *COR−* growth promotion by HopX1 was COI1-dependent by performing similar experiments in *Arabidopsis* plants lacking the *COI1* receptor gene (*coi1-30*). ", "As described previously, external application of COR did not restore *Pto* DC3000 *COR−* growth on *coi1-30* plants ([Figure 4F](#pbio-1001792-g004){ref-type=\"fig\"}) [@pbio.1001792-Kloek1],[@pbio.1001792-Brooks1]. ", "In contrast, expression of *hopX1*, but not *hopX1* ^C179A^ or *EV*, enhanced the growth of *Pto* DC3000 *COR−* in *coi1-30* plants, similar to previous results in Col-0 plants ([Figure 4F](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Moreover, we examined whether JAZ proteins were degraded by HopX1 *in vivo* after bacterial infections on transgenic *Arabidopsis* Col-0 plants expressing the dominant negative J*AZ1ΔJas-HA* variant from the *35S* promoter. ", "As described above, JAZ proteins lacking the C-terminal Jas domain do not interact with COI1 and cannot be ubiquitinated and degraded by the proteasome, thus promoting JA-insensitivity [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Chini2],[@pbio.1001792-FernandezCalvo1],[@pbio.1001792-Melotto3]. ", "J*AZ1ΔJas-HA* levels were strongly reduced in total plant extracts 24 hours after infiltration of *Pto* DC3000 *COR−* bacteria expressing *hopX1*, whereas infiltration of buffer or *Pto* DC3000 *COR−* expressing either *hopX1* ^C179A^ or *EV* did not cause any significant changes ([Figure 4G](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Consistent with the fact that J*AZ1ΔJas-HA* is resistant to COI1-dependent degradation by bacterial COR, challenge with *Pto* DC3000 did not alter J*AZ1ΔJas-HA* levels ([Figure 4G](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Finally, we investigated whether *Pto* DC3000 *COR−* bacteria expressing *hopX1* could re-open plant stomata. ", "To do this, we incubated *N. benthamiana* leaves with *Pto* DC3000 as a positive control or *Pto* DC3000 *COR−* bacteria expressing *hopX1*, *hopX1* ^C179A^, or an *EV* control. ", "Incubation of *N. benthamiana* leaves with *Pto* DC3000 *COR−* expressing an *EV* for 5 hours induced stomatal closure whereas stomata of *N. benthamiana* leaves incubated with *Pto* DC3000 remained open similar to mock treated control leaves as it was described previously ([Figure S14](#pbio.1001792.s014){ref-type=\"supplementary-material\"}) [@pbio.1001792-Melotto1]. ", "Strikingly, only *Pto* DC3000 *COR−* bacteria expressing *hopX1*, but not *hopX1* ^C179A^, could re-open plant stomata of *N. benthamiana* leaves. ", "This indicates that HopX1 from *Pta* 11528 plays a role in re-opening of stomata as COR does. ", "These data are consistent with functional redundancy between HopX1 and the phytotoxin COR. ", "Taken together, our data indicate that HopX1 acts inside the plant cell to promote activation of the JA pathway and induce susceptibility in *Arabidopsis*.", "\n\nDiscussion {#s3}\n==========\n\nLevels of resistance in whole plants are influenced by systemic signals mediated, in many cases, by plant hormones. ", "The importance of the role of hormones in biotic interactions is underlined by the increasing number of pathogenic microbes that are known to produce phytohormones or phytohormone mimics to perturb hormonal homeostasis and promote disease [@pbio.1001792-RobertSeilaniantz1]. ", "To date, production of cytokinins (CKs) [@pbio.1001792-Kakimoto1], abscisic acid (ABA) [@pbio.1001792-Siewers1], auxin [@pbio.1001792-Spaepen1], JA [@pbio.1001792-Mittal1], and ET [@pbio.1001792-Weingart1] has been reported in various bacterial or fungal species [@pbio.1001792-RobertSeilaniantz1]. ", "Remarkably, some pathogens produce hormone mimics. ", "This is the case of some strains of *P. syringae* that produce COR, a mimic of the bioactive jasmonate JA-Ile, but synthesized via an unrelated biosynthetic pathway involving ligation of coronamic acid (cma) to the polyketide coronafacic acid (cfa) [@pbio.1001792-Xin1],[@pbio.1001792-RobertSeilaniantz1]. ", "Like JA-Ile, COR functions as an SA antagonist to promote virulence via suppression of host defenses. ", "Notably, COR is produced by only a few *P. syringae* pathovars, whereas all *Pseudomonas* inject an array of effector proteins into the host cell that collectively promote disease by targeting and altering host cellular activities. ", "The identification of their cellular targets is crucial to understand virulence. ", "Recent data indicate that some effectors might impinge the JA pathway. ", "For instance, Cui and colleagues showed that the effector AvrB indirectly perturbs JA signaling by interfering with the *Arabidopsis* mitogen-activated protein kinase MAP kinase 4 (MPK4) [@pbio.1001792-Cui2]. ", "Moreover, several studies suggest that even in strains producing COR, effectors act synergistically with COR to induce the JA pathway [@pbio.1001792-Thilmony1]--[@pbio.1001792-Chen1]. ", "Thus, it seems plausible that *Pseudomonas* manipulates the plant hormonal network through bacterial effectors to induce susceptibility. ", "Here we have uncovered a novel molecular mechanism by which a bacterial strain (*Pta* 11528), which does not produce COR, activates the JA pathway to promote susceptibility through the degradation of JAZ repressors by the effector HopX1. ", "In this work, we analyzed a *Pta* 11528 effector library containing ten identified effectors whereas the full *Pta* 11528 effector repertoire is predicted to secrete a suite of about 30 virulence effector proteins into the host cytoplasm [@pbio.1001792-Studholme1]. ", "Thus, it is plausible that additional effectors among the *Pta* 11528 effector repertoire not tested in this work may also destabilize JAZ proteins in a redundant manner with HopX1 to ensure activation of JA signaling and bacterial pathogenesis.", "\n\nExpression of *hopX1*, but not *hopX1* ^C179A^, induced expression of early JA-responsive genes while reducing SA-mediated induction of the SA-marker gene *PR1*, a hallmark response of the JA pathway. ", "These phenotypes are evidently associated with the virulence function of the effector, as the presence of HopX1 promotes *Pto* DC3000 growth both in stable transgenic *Arabidopsis* plants expressing the effector under the control of a inducible promoter, or when the effector is delivered naturally by the TTSS of the bacteria. ", "These results highlight a novel bacterial strategy to subvert the antagonistic relationship between host SA and JA signaling pathways. ", "JA-Ile activates plant responses by promoting physical interaction between the E3 ligase COI1 and JAZ repressors. ", "This interaction leads to JAZ polyubiquitination (poly-Ub) and subsequent degradation by the 26S proteasome, releasing TFs from repression [@pbio.1001792-Sheard1]--[@pbio.1001792-Thines1]. ", "Remarkably, HopX1 compromises the accumulation of JAZ repressors in *NtCOI1*-silenced plants, and triggers the degradation of dominant-negative JAZ variants lacking the C-terminal Jas domain, which is required for the interaction with COI1 [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Yan1]. ", "Therefore, HopX1-triggered degradation of JAZ proteins occurs in a COI1-independent manner, thus allowing activation of the pathway in the absence of the hormone. ", "The ability of HopX1 to promote degradation of JAZ proteins and activate JA responses suggests that this is a strategy to promote disease. ", "PD and co-immunoprecipitation experiments demonstrated that HopX1 interacts with JAZ proteins through its central ZIM domain, conserved in all 12 *Arabidopsis* JAZ repressors and their JAZΔJas variants. ", "The interaction with the ZIM domain is consistent with HopX1-triggered degradation of all full-length JAZs and truncated derivatives tested in our assays, and indicates that the effect of HopX1 on JAZs is direct or at least through the same protein complex. ", "JAZ proteins are translated in the cytoplasm and then localized mainly, but not exclusively, in the nucleus [@pbio.1001792-Chini1],[@pbio.1001792-Thines1],[@pbio.1001792-Withers1]. ", "For example, JAZ1 can be detected in both cytosol and nucleus [@pbio.1001792-Jiang1]. ", "Moreover, Withers and colleagues recently reported a MYC2-dependent mechanism for nuclear import of cognate JAZs transcriptional repressors in *Arabidopsis* [@pbio.1001792-Withers1], which implied an equilibrium of JAZ proteins between the nucleus and the cytoplasm. ", "Thus, it is conceivable that HopX1 could potentially degrade JAZ proteins both in the cytoplasm and nucleus.", "\n\nInteraction and degradation of JAZs by HopX1 suggest that this effector has proteolytic activity. ", "HopX1 family members are modular proteins that contain a putative cysteine-based catalytic triad and a novel conserved N-terminal domain [@pbio.1001792-Nimchuk1]. ", "The putative catalytic triad consists of cysteine (C), histidine (H), and aspartic acid (D) and is similar to that utilized by diverse enzyme families such as cysteine proteases and peptide N-glycanases (PNGases), which are all members of the transglutaminase (TGase) protein superfamily [@pbio.1001792-Nimchuk1]. ", "While previous reports failed to detect protease activity *in vitro* using general substrates, genetic approaches demonstrated the functional relevance of the conserved catalytic triad of HopX1 family members [@pbio.1001792-Nimchuk1]. ", "For example, mutation of this putative catalytic triad of various HopX family members abolished avirulence activity on *R2*-expressing bean cultivars and also prevented initiation of cell death in *Arabidopsis* following transient expression assays [@pbio.1001792-Nimchuk1]. ", "We found that HopX1 has protease activity on both general and specific substrates when the effector was directly immunopurified from plants and that this activity required Cys-179 within the conserved catalytic triad. ", "Previously, Coaker and colleagues reported that the cysteine protease effector AvrRpt2 requires host general folding catalyst cyclophilins like *Arabidopsis* ROC1 to activate AvrRpt2 cysteine protease activity once the effector is translocated into the plant cell [@pbio.1001792-Coaker1],[@pbio.1001792-Coaker2]. ", "Activated AvrRpt2 undergoes autoprocessing of its N-terminus to yield the mature protease that cleaves the host target RIN4 *in planta* [@pbio.1001792-Chisholm1]. ", "Thus, it is possible that HopX1 also requires a host eukaryotic chaperone into the plant cell for the activation of the protease activity in a similar fashion as AvrRpt2. ", "These results, together with the ability of HopX1 to interact with JAZs in PD and co-immunoprecipitation assays, suggest a model in which HopX1 acts on JAZs directly or indirectly through a third plant protein in the same complex. ", "Noteworthy, HopX1^C179A^ shows markedly reduced affinity for JAZ proteins in PD and co-immunoprecipitation assays compared to wild-type HopX1 ([Figures 3C, 3D](#pbio-1001792-g003){ref-type=\"fig\"}, and [S8](#pbio.1001792.s008){ref-type=\"supplementary-material\"}). ", "Thus, it is possible that the lack of activity of HopX1^C179A^ on JAZs could be due not exclusively to a missing enzymatic activity but also to a compromised ability to interact with its host targets, or a combination of both simultaneously. ", "In this regard, we can also not completely exclude that HopX1 activity on JAZs could be due to an additional plant protease that co-purifies between the same protein complex. ", "Despite previous reports showing that the cysteine protease effector AvrPphB cleaved host targets into smaller fragments [@pbio.1001792-Chisholm1],[@pbio.1001792-Shao1], we did not see smaller bands in our assays including free MBP. ", "We speculate that this could be due to the small size of JAZ proteins (25--35 KDa). ", "However, it is also possible that HopX1 degrades JAZ proteins at multiple sites and triggers target degradation in a similar fashion to the cysteine protease effector AvrRpt2 on its 30-kDa host substrate protein RIN4 [@pbio.1001792-Chisholm1],[@pbio.1001792-Shao1]. ", "On the other hand, it is possible that the immunoprecipitation of HopX1-HA or C179A-HA from transgenic plants contains additional proteases that could affect MBP stability. ", "This would explain why in [Figure 2B](#pbio-1001792-g002){ref-type=\"fig\"} MBP-JAZ5 protein purified from *E. coli* is more stable in the presence of general cocktail protease inhibitors. ", "Despite this, unspecific degradation would occur in all samples to the same extent and therefore, the effect we observed in MBP-JAZ5 in the presence of HopX1 is likely to be specific. ", "Moreover, we found that substitution of the conserved Cys-179 amino acid of HopX1 to alanine also compromised HopX1-mediated JAZ5 degradation *in vivo*. ", "Similarly, activation of JA-dependent gene expression and JA-related phenotypes occurred in a Cys-179 dependent manner when expressed ectopically in *Arabidopsis*. ", "Furthermore, in contrast to wild-type HopX1, HopX1^C179A^ delivered by the TTSS was unable promote growth of *Pto* DC3000 *COR−* bacteria. ", "Overall, these data suggest that HopX1 family members are proteolytic enzymes of the cysteine protease family, and that this activity is required for function.", "\n\nSeveral JAZ pre-mRNAs are subject to alternative splicing, which, at least for JAZ10, results in truncated JAZΔJas proteins that are resistant to proteasomal degradation because they are unable to interact with COI1, thereby causing dominant JA-insensitive phenotypes [@pbio.1001792-Yan1],[@pbio.1001792-Chung1],[@pbio.1001792-Chung2]. ", "The alternative splicing of *JAZ* genes provides a feedback mechanism to efficiently repress the JA signal output and reduce the fitness costs associated with over-stimulation of the signaling pathway [@pbio.1001792-Chung1],[@pbio.1001792-Chung2]. ", "Degradation of JAZΔJas variants by HopX1 suggests that this effector should promote a sustained activation of the JA pathway, favoring susceptibility. ", "In agreement with this hypothesis, *Pto* DC3000 COR− expressing *hopX1* promoted bacterial replication in *Arabidopsis* plants ([Figure 4E and 4F](#pbio-1001792-g004){ref-type=\"fig\"}). ", "Moreover, *Pto* DC3000 COR−, which does not produce COR, grew better in the presence of the effector (DEX in [Figure S13](#pbio.1001792.s013){ref-type=\"supplementary-material\"}) than the wild-type strain producing COR in the absence of the effector (mock in [Figure S13](#pbio.1001792.s013){ref-type=\"supplementary-material\"}). ", "Therefore, our results suggest that evolution has shaped this effector--host interaction to maximize the activation of the JA response pathway through direct targeting of most forms of JAZs. ", "Besides the 12 JAZ repressors, six additional *Arabidopsis* proteins contain a ZIM domain. ", "Collectively, these 18 proteins are known as the TIFY proteins [@pbio.1001792-Pauwels1],[@pbio.1001792-Vanholme1]. ", "It remains to be determined whether HopX1 targets any of these ZIM domain containing proteins, and if they also play a role in plant defense against bacteria.", "\n\nNotably, HopX homologs are found in diverse phytopathogenic bacteria including *Pto* DC3000. ", "BLAST analysis revealed a near identical match of HopX1*~Pta~* ~11528~ with HopX1 from *P. syringae* pv. *", "phaseolicola* bacterial strains such as race 4 (*Pph* race 4) (99% identity) but significantly less homology with the respective *Pto* DC3000 homolog (72% identity) ([Figure S4A](#pbio.1001792.s004){ref-type=\"supplementary-material\"}). ", "Several lines of evidence suggest that the *Pto* DC3000 and *Pta* 11528 alleles of HopX1 play different roles in the plant cell. ", "Firstly, HopX1*~Pto~* ~DC3000~ triggers cell death in several *Arabidopsis* ecotypes including Col-0 and Ws-0 whereas HopX1*~Pph~* ~race\\ 4~ (which is almost identical to the *Pta* 11528 allele) does not [@pbio.1001792-Nimchuk1]. ", "Secondly, HopX1*~Pto~* ~DC3000~ did not compromise the accumulation of JAZ5 when co-expressed transiently in *N. benthamiana* ([Figure S15](#pbio.1001792.s015){ref-type=\"supplementary-material\"}). ", "Consistently, transgenic *Arabidopsis* Col-0 plants expressing the dominant negative J*AZ1ΔJas* variant infected with *Pto* DC3000 did not alter J*AZ1ΔJas* levels ([Figure 4G](#pbio-1001792-g004){ref-type=\"fig\"}). ", "This indicates that these plants are resistant to both HopX1*~Pto~* ~DC3000~ and COI1-dependent degradation by bacterial COR. ", "The ability of HopX1*~Pta~* ~11528~ to trigger JAZ degradation and activation of JA responses may reflect specialized functions among the broad divergence across the HopX family members. ", "It is tempting to speculate that redundancy between COR and HopX1 has allowed strains that produce the phytotoxin to evolve HopX1 for other functions and in doing so, lose its JA-inducing activity. ", "Overall, HopX1 exemplifies a bacterial strategy for pathogenicity in which the JA pathway is targeted through direct degradation of JAZ repressors to promote pathogenesis.", "\n\nFinally, emerging data suggest that bacterial pathogens have evolved effectors to manipulate important plant hormones that regulate defense response such as SA and JA [@pbio.1001792-RobertSeilaniantz1]. ", "For, example, the *P. syringae* effector HopI1 directly targets Hsp70 in choloroplasts to suppress SA accumulation [@pbio.1001792-Jelenska1] whereas the effector AvrB promotes JA signaling likely through an indirect mechanism via the activation of MPK4 [@pbio.1001792-Cui2]. ", "Very recently, the *P. syringae* effector HopZ1a was reported to interact and acetylate several JAZ proteins through a putative acetyltransferase activity [@pbio.1001792-Jiang1]. ", "HopZ1a-mediated acetylation induces JAZ1 degradation through an undefined mechanism that is dependent on COI1. ", "This leads to activation of JA-dependent gene expression and plant susceptibility [@pbio.1001792-Jiang1]. ", "It is possible that post-translational modifications of JAZ1 induces or represses 26S proteasome degradation triggered by COI1. ", "In this study, we found that *hopX1* encodes a cysteine protease, activity that is required for degradation of JAZs by HopX1. ", "HopX1 associates with JAZ proteins through its central ZIM domain and degradation occurs in a proteasome- and COI1-independent manner, highlighting the different strategies used by bacterial effectors to target similar host components. ", "Importantly, ectopic expression of HopX1 in *Arabidopsis* induces the expression of JA-dependent genes and represses SA-induced markers, whereas delivery of HopX1 by the natural TTSS of *Pseudomonas* partially re-opens stomata during the infection process and promotes susceptibility to a similar extent as the addition of COR. ", "These results highlight a novel molecular mechanism by which bacterial effectors directly manipulates core regulators of hormone signaling to facilitate infection. ", "Indeed, oomycete pathogens were also found to produce effectors that interact with JAZ3 [@pbio.1001792-Mukhtar1]. ", "The recent findings highlight the JA receptor complex as a major common and critical hub of suppression for diverse pathogens during the arms race between plants and pathogens.", "\n\nMaterials and Methods {#s4}\n=====================\n\nPlant Materials and Growth Conditions {#s4a}\n-------------------------------------\n\n*Arabidopsis* Aa--0 plants were transformed as described previously [@pbio.1001792-Bechtold1] using the binary vector pTA7002 [@pbio.1001792-Aoyama1] containing the *hopX1* or the *hopX1* ^C179A^ gene fused to a C-terminal HA under the control of a *dexamethasome* (*DEX*)-inducible promoter. ", "Homozygous single-insertion transformants were selected in the T3 generation and HopX1 or HopX1^C179A^ accumulation was confirmed by Western blots after induction with 30 µM DEX with 0.01% Silwet L-77. ", "To generate transgenic plants expressing JAZ1ΔJas-HA in Col-0 background, JAZ1ΔJas was amplified with Expand High Fidelity polymerase (Roche) using Gateway-compatible primers (*AtJAZ1 ΔJas*, 5′-ggggacaagtttgtacaaaaaagcaggcttcATGTCGAGTTCTATGGAATGTTCTG-3′ and 5′-ggggaccactttgtacaagaaagctgggtcCTTGGCTAAGCTATTAGCGGT-3′). ", "PCR products were cloned into pDONR207 with a Gateway BP II kit (Invitrogen) and sequence verified. ", "This plasmid, a Gateway LR II kit (Invitrogen), and the pGWB14 [@pbio.1001792-Mita1] destination vector was used to generate *35S:JAZ1ΔJas-HA*. ", "These constructs were transferred to *Agrobacterium tumefaciens* strain C58C1 by freeze thawing and then transformed in Col-0 plants by floral dipping method [@pbio.1001792-Clough1]. ", "Hygromycin-resistant plants were selected and their T2 progenies propagated for subsequent analysis. ", "The knockout line *coi1-30* [@pbio.1001792-Yang1] and stable *N. tabacum* lines silenced for an EV (line VC) or the *NtCOI1* gene (line18) [@pbio.1001792-Shoji1] were previously described.", "\n\n*N. benthamiana* and *N. tabacum* plants were grown in controlled environment chambers at an average temperature of 24°C (range 18°C--26°C), with 45%--65% relative humidity under long day conditions (16 h light). ", "In contrast, *A. thaliana* plants were grown in controlled environment chambers at an average temperature of 22°C (range 16°C--24°C), with 45%--65% relative humidity under short day conditions (10 h light). *", "N. benthamiana* and *N. tabacum* plants were grown for three to four weeks prior to *A. tumefaciens*-mediated transformation. *", "Arabidopsis* plants of four to six weeks were usually analyzed in bacterial growth assays.", "\n\nBacterial Strains {#s4b}\n-----------------\n\nThe *E. coli* strain DH5α was used for cloning, and small- and large-scale plasmid isolation. *", "A. tumefaciens* C58C1 was used for transient expression in *N. benthamiana. ", "Pseudomonas* strains used in this study were *Pseudomonas syringae* pv. *", "tomato* (*Pto*) DC3000, *Pseudomonas syringae* pv. *", "tabaci* (*Pta*) 11528, a non-pathogenic TTSS defective *Pta* 11528 *hrpV−* bacteria unable to secrete effector proteins into the plant cell [@pbio.1001792-Oh1] and the coronatine-deficient *Pto* DC3000 strain (*Pto* DC3000 *COR−*) which is a *Pto* DC3000 AK87 mutant that carries mutations in cmaA (coronamic acid A) and cfa6 (coronafacic acid 6) [@pbio.1001792-Brooks1]. ", "HopX1 or HopX1^C179A^ were cloned into pCPP5040, a derivative of the broad-host-range vector pML123 [@pbio.1001792-Labes1], which expresses insert genes from the *nptII* promoter, and generates protein products for expression in *Pseudomonas* species [@pbio.1001792-LopezSolanilla1]. ", "This vector was a kind gift from Emilia Lopez Solanilla and Pablo Rodriguez Palenzuela. ", "These plasmids were introduced into *Pto* DC3000 *COR−* by triparental mating in which pRK600 was used as a helper plasmid.", "\n\nStomatal Aperture Measurements {#s4c}\n------------------------------\n\nLeaf discs from 4- to 5-week-old *N. benthamiana* young leaves were exposed to white light for 1 hour while submerged in a solution containing 50 mM KCl, 10 µm CaCl~2~, and 10 mm MES-KOH, (pH = 6.1) to induce stomatal aperture. ", "Subsequently, leaf discs were immersed in buffer or bacterial suspension at 5×10^8^ cfu/ml (optical density at 600 nm \\[OD~600~\\] = 1) in 50 mM KCl, 10 µm CaCl~2~, and 10 mm MES-KOH (pH = 6.1). ", "The samples were incubated under the same conditions for 5 hours. ", "Abaxial leaf surfaces were observed with a microscope (Leica DMR), and stomatal aperture was measured using ImageJ software.", "\n\nStatistical Methods {#s4d}\n-------------------\n\nStatistical significance based on *t* test analysis was developed by GraphPad Prism program. ", "Seven independent samples were used to analyze the significance of bacterial growth results.", "\n\nSeventeen independent stomata were used to analyze stomatal aperture in each condition.", "\n\nTransient Expression Assays and Inhibitory Assays {#s4e}\n-------------------------------------------------\n\nGrowth and transient expression conditions were as described previously [@pbio.1001792-Wu1] using the *A. tumefaciens* strain C58C1. ", "For transient gene expression, *A. tumefaciens* was syringe infiltrated in *N. benthamiana* or *N. tabacum* leaves at OD~600~ = 0.4, whereas for transient gene co-expression assays both constructs were infiltrated at OD~600~ = 0.3. ", "Samples were collected after two days. ", "The genes expressed in this paper were *35S:hopX1-HA, DEX:hopX1-HA*, *DEX:hopX1* ^C179A^-*HA*, *35S:GFP, 35S:GFP-hopX1*, *35S:GFP-hopX1* ^C179A^, *35S:JAZ-HA* (*Arabidopsis* cDNA coding sequences cloned in constructs for all 12 JAZs), *35S:COI1-GFP, 35S:MYC2-HA, 35S:JAZ1ΔJas-HA, 35S:JAZ2ΔJas-HA*, and *35S:JAZ7ΔJas-HA*. ", "We also used the *Pto* DC3000 effector genes *35S:hopX1~Pto~* ~DC3000~-*HA*, *35S:HopC1~Pto~* ~DC3000~-*HA*, *35S:HopAD1~Pto~* ~DC3000~-*HA*, and *35S:HopN1~Pto~* ~DC3000~-*HA*. ", "For inhibitory assays *in planta*, MG132 (Sigma) inhibitor was co-infiltrated with *A. tumefaciens* at a concentration of 100 µM.\n\nProtein Extraction and Immunoblotting {#s4f}\n-------------------------------------\n\nTotal proteins were extracted from leaf tissue by homogenization in extraction buffer (100 mM Tris-HCl \\[pH 7.5\\], 150 mM NaCl, 5 mM EDTA, 5% glycerol, 10 mM DTT, 2% PVPP, 1 mM PMSF, protease inhibitors cocktail \\[Roche\\], and 0.5% Triton X-100). ", "In all experiments protein samples were equilibrated to equivalent concentrations of total proteins. ", "Extracted proteins were fractioned by 8%--10% SDS-PAGE, transferred onto HybondTM-P membranes (Amersham) and incubated with anti-HA-horseradish peroxidase (Roche) or anti-GFP-horseradish peroxidase antibody (Milteny Biotec). ", "Immunodetection was performed with ECL chemiluminiscence reagent (GE Healthcare) or Supersignal West Femto (Thermo Scientific).", "\n\nPull-Down Assays {#s4g}\n----------------\n\nMBP-JAZ fusion proteins were generated as previously described [@pbio.1001792-Chini2]. ", "Ten-day-old *Arabidopsis* wild-type Col-0 seedlings and lines expressing *DEX:hopX1-HA* or *DEX:hopX1* ^C179A^-*HA* were induced with 30 µM DEX plus 0.01% Silwet L-77 or a mock solution for 24 hours. ", "Seedlings were ground in liquid nitrogen and homogenized in extraction buffer containing 50 mM Tris-HCl, (pH 7.4), 150 mM NaCl, 10% glycerol, 0.2% NP-40, 1 mM DTT, 1 mM phenylmethylsulphonyl fluoride, 5 mM MgC~l2~, 50 mM MG132 (Sigma-Aldrich), and complete protease inhibitor (Roche). ", "After centrifugation (16,000 *g* at 4°C), the supernatant was collected. ", "For in vivo PD experiments, 6 µg of resin-bound MBP fusion protein was added to 250 µg of pre-equilibrated total protein extract and incubated for 1 h at 4°C with rotation. ", "After washing, samples were denaturalized, loaded on 8% SDS-PAGE gels, transferred to nitrocellulose membranes, and incubated with anti-HA-horseradish peroxidase (Roche). ", "A 5 µl aliquot of MBP-fused protein of each sample was run into SDS-PAGE gels and stained with Coomassie brilliant blue to confirm equal protein loading.", "\n\nProtease Activity Assays {#s4h}\n------------------------\n\nImmunoprecipitation of HopX1-HA and HopX1^C179A^-HA from stable transgenic *Arabidopsis* plants using the anti-HA affinity matrix (Roche) was performed according to the manufacturer\\'s instructions with some modifications (Anti-HA Affinity Matrix, catalogue number 11 815 016 001, Roche). ", "Briefly, transgenic *Arabidopsis* seedlings expressing *hopX1-HA* and *hopX1* ^C179A^-HA or wild-type Col-0 used as a negative control were induced by spraying with 30 µM DEX plus 0.01% Silwet L-77 for 24 hours. ", "The material was ground in liquid nitrogen and homogenized in extraction buffer containing 100 mM Tris-HCl (pH 7.5), 150 mM NaCl, 5 mM EDTA, 5% glycerol, 10 mM DTT, 2% PVPP, 1 mM PMSF, protease inhibitors cocktail (Roche), and 0.5% Triton X-100. ", "The samples were centrifuged twice at 16,000 *g* at 4°C. ", "The supernatant was incubated for 5 hours (4°C, with rotation) with the anti-HA affinity matrix (Roche), and then washed once with 1 ml of extraction buffer and three times with 1 ml of cool TBS (50 mM Tris-HCl \\[pH 7.5\\], 150 mM NaCl, \\[pH = 7.4\\]) to remove all protease inhibitors from the anti-HA affinity matrix. ", "Elution of HA proteins from the anti-HA affinity matrix was performed by incubating with 1 mg/ml of HA peptide (Roche) in TBS buffer for 30 minutes at 37°C with strong shaking. ", "Supernatant was recovered by centrifugation and transferred to a fresh tube for protease activity assays. ", "Immunoprecipitated HopX1-HA and HopX1^C179A^-HA was pre-equilibrated for similar amounts of protein and maintained at 4°C until used.", "\n\nIn vitro Protease Fluorescent Detection kit to determine protease activity was performed according to the manufacturer\\'s instructions (PF0100-1KT, Sigma). ", "Briefly, 10 µl of immunoprecipitated HopX1-HA or HopX1^C179A^-HA (or MBP-HopX1 for [Figure S4B](#pbio.1001792.s004){ref-type=\"supplementary-material\"}) was incubated with 20 µl of Incubation buffer and 20 µl of fluorescein isothiocyanate (FITC)-casein substrate overnight in the dark at 37°C with moderate shaking. ", "The reaction was stopped by treatment with 150 µl of 10% trichloroacetic acid (TCA) for 30 min at 37°C in the dark. ", "Samples were then centrifuged, and supernatant was recorded for fluorescence intensity with excitation at 485 nm and monitored for the emission wavelength of 535 nm (485/535 nm).", "\n\nFor *in vitro* experiments regarding protease activity of HopX1 on JAZs, we incubated 20 µl of immunoprecipitated HopX1-HA or HopX1^C179A^-HA with 5 µl of recombinant JAZ5 fused to MBP expressed and purified from *E. coli* cells in a total volume of 50 µl TBS (with or without general protease inhibitors, Roche). ", "The reaction was incubated overnight at 37°C with moderate shaking. ", "To stop the reactions all samples were denaturalized, and then loaded on 10% SDS-PAGE gel, transferred to a nitrocellulose membrane, and incubated with anti-MBP (AbCAM). ", "Immunodetection was performed with ECL chemiluminiscence reagent (GE Healthcare)\n\nConfocal Microscopy {#s4i}\n-------------------\n\nHopX1 was placed into pENTR/D-TOPO (Invitrogen) using the 5′-ggggacaagtttgtacaaaaaagcaggcttcATGAGAATTCACAGTGCTGGTCA-3′ and 5′-ggggaccactttgtacaagaaagctgggtcTTATCTTCGTGGAGGCATGCCTTTAGACG-3′ primers, and then recombined into the gateway binary vector pGWB6 to create constructs that express N-terminal GFP fusion proteins. ", "This was then transiently expressed in *N. benthamiana* leaves by *Agrobacterium*-mediated transformation. ", "Localization of GFP fusion was visualized with sequential laser scanning confocal microscopy, using a Leica Confocal SP5 with sequential imaging at 488 nm excitation and 505--525 nm emission (green/GFP) and 633 nm excitation and 660 nm emission (red/chlorophyll).", "\n\nPlant Cell Fractionation {#s4j}\n------------------------\n\nPlant cell fractionation was performed by using the CelLytic PN Isolation/Extraction kit for plant leaves (CELLYTPN1-1KT, Sigma) according to the manufacturer\\'s instructions.", "\n\nMeasure of Chlorophyll Content {#s4k}\n------------------------------\n\nThe chlorophyll content of leaves was measured by acetone extraction according to Arnon [@pbio.1001792-Arnon1]. ", "Briefly, 100--150 mg tissue per sample was extracted for two hours in 400 µl of acetone 80% (v/v) at 4°C in the dark with constant shacking. ", "The homogenate was centrifuged at 3,000 rpm for 2 minutes. ", "The supernatant was saved (V~1~) and 0.2 ml of the supernatants were diluted in a known volume of acetone 80% until the absorbance of the extract at 663 nm and 645 nm was read between 0.2--0.8 (V~2~) using a SpectraMax Absorbance Microplate Reader. ", "The concentration of chlorophyll a (C~a~), b (C~b~), and total chlorophyll (C~t~) was calculated using Arnon\\'s equations: C~a~ = 12.7 \\* A~663~−2.63 \\* A~645~, C~b~ = 22.9 \\* A~645~−4.68 \\* A~663~, and C~t~ = C~a~+C~b~ \\* \\[V1 \\* (V2+0.2)/(0.2 \\* P\\]. ", "Ca, Cb are expressed in µg \\* ml^−1^, P in grams of fresh leave tissue, V in ml, and C~t~ in µg \\* ml^−1^ of fresh weight.", "\n\nQuantitative RT-PCR {#s4l}\n-------------------\n\nQuantitative RT-PCR for JA-dependent gene expression experiments were performed with RNA extracted from 10-day-old seedlings grown on liquid MS media that were treated with 5 µM DEX for 36 h or a mock solution. ", "For quantitative RT-PCR analysis of *PR1* expression in stable transgenic *Arabidopsis* lines expressing the *hopX1* or *hopX1* ^C179A^ genes, 10-day-old seedlings grown on liquid MS media were DEX-induced for 24 hours and then treated with 1 mM SA or a mock solution for additional 24 hours. ", "For each experiment, three biological replicates, consisting of tissue pooled from 15 to 20 plants were taken. ", "For semi-quantitative RT-PCR of AtJAZ5 expression levels when co-expressed with HopX1 in *N. benthamiana*, discs from six independent leaves were collected and frozen in liquid nitrogen two days after infiltration. ", "For semi-quantitative RT-PCR analysis of stable *NtCOI1*-silenced *N. tabacum* plants, discs from six independent leaves of four weeks old plants were collected and frozen in liquid.", "\n\nRNA extraction and cleanup was done using Trizol reagent (Invitrogen) followed by RNeasy mini kit (Qiagen) and DNase digestion to remove genomic DNA contamination. ", "cDNA was synthesized from 0.5 to 1 µg of total RNA with the high-capacity cDNA reverse transcription kit (Applied Biosystems). ", "Two microliters from one-tenth diluted cDNA was used to amplify selected genes. ", "For quantitative PCR analysis, Power SYBR Green was used for gene amplification (Applied Biosystems). ", "Quantitative PCR was performed in 96-well optical plates in a 7300 Real Time PCR system (Applied Biosystems). ", "Data analysis shown was done using three technical replicates from one biological sample; similar results were obtained with two additional independent biological replicates.", "\n\nPrimers for genes used here are as follows: *AtJAZ10*, 5′-GAGAAGCGCAAGGAGAGATTAG-3′ and 5′-CTTAGTAGGTAACGTAATCTCC-3′; *AtJAZ5*, 5′-AAAGATGTTGCTGACCTCAGTG -3′ and 5′-CCCTCCGAAGAATATGGTCA-3′; *AtJAZ*12, 5′-CATCTAATGTGGCATCACCAG-3′ and 5′-TGCCTCCTTGCAATAGGTAGA-3′; *AtPR1*, 5′-AAGTCAGTGAGACTCGCATGTGC-3′ and 5′-GGCTTCTCGTTCACATAATTCCC-3′; *AtACTIN8*, 5′-CCAGTGGTCGTACAACCGGTA -3′ and 5′- TAGTTCTTTTCGATGGAGGAGCTG-3′; *HopX1*, 5′-TAGCAAGCTTCGCTTACG-3′ and 5′-GTTTCACGCGTAACCTTG-3′; *NtCOI1*, 5′-GAAGATCTTGAATTGATGGC-3′ and 5′-CCCAGAAGCATCCATCTCAC-3′; and *NtαTubulin*, 5′-AGTTGGAGGAGGTGATGATG-3′ and 5′-TATGTGGGTCGCTCAATGTC-3′.\n\nBacterial Growth Curves {#s4m}\n-----------------------\n\nTransgenic *Arabidopsis* plants expressing *hopX1* and *hopX1* ^C179A^ were induced by spraying with 30 µM DEX plus 0.01% Silwet L-77 or a mock solution for 24 hours, and then infected with selected bacteria. ", "All bacterial growth assays in *Arabidopsis* were performed by spray inoculation as described [@pbio.1001792-Zipfel1]. ", "Briefly, overnight bacterial cultures were pelleted and resuspended in sterile 10 mM MgCl~2~. Plants were sprayed with a bacterial suspension containing 10^8^ (cfu)/ml bacteria (OD~600~ = 0.2) with 0.04% Silwet L-77. ", "Leaf discs were harvested after two days and ground in 10 mM MgCl~2~. Serial dilutions of leaf extracts were plated on LB agar with appropriate antibiotics. ", "Each data point represents the average of seven replicates, each containing two leaf discs from different plants. ", "Error bars indicate standard errors of the mean (SEM). ", "These experiments were repeated at least three times with similar results, and representative results are shown.", "\n\nImmunoblots showing JAZ1ΔJas degradation in *Pto* DC3000 COR− expressing *hopX1* infected Col-0 *Arabidopsis* plants were performed after syringae infiltration of the indicated bacterial suspension at 10^8^ (cfu)/ml (OD~600~ = 0.2) into the leaves. ", "Samples were collected after 24 hours for protein analysis.", "\n\nSupporting Information {#s5}\n======================\n\n###### \n\n**Live** ***Pta*** **11528 bacteria manipulate stomatal aperture in a TTSS-dependent manner.** ", "Stomatal aperture in wild-type *N. benthamiana* leaves measured after 5 hours of incubation with mock or bacterial strains *Pto* DC3000, *Pta* 11528, or *Pta* 11528 *hrpV−*. ", "Error bars indicate SEM (*n* = 17). ", "Asterisks indicate significant differences compared with mock-treated samples at *\\*\\*p*\\<0.01. ", "The results are representative of three independent experiments.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1 compromises the accumulation of JAZ5.** ", "Immunoblots showing JAZ5-HA accumulation in the presence of HopX1-HA when co-expressed transiently in *N. benthamiana* for two days. ", "Proteins were detected with anti-HA antisera. ", "CBB, Coomassie brilliant blue staining. ", "The results are representative of three independent experiments.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**Phylogenetic tree of the** ***Arabidopsis*** **JAZ proteins.** ", "Phenogram representation of the neighbor-joining phylogenetic tree of the 12 full-length JAZ proteins. ", "The sequence alignment was generated using DiAlign (Genomatix) and the tree was created by Phylodendron (University of Indiana). ", "Branch lengths are proportional to the estimated evolutionary distance. ", "Bootstrap values are included. ", "JAZ proteins can be tentatively grouped into four clades.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX family members contain a consensus cysteine-based catalytic triad and a conserved N-terminal domain.** (", "A) HopX family members contain a consensus cysteine-based catalytic triad and a conserved N-terminal domain. ", "Conservation of the putative catalytic residues (red dots) and the N-terminal domain (red line) between *P. syringae* HopX1 alleles from *Pta* 11528, *Pto* DC3000, and *Pph* race 4. ", "Sequences were downloaded from the National Center for Biotechnology Information and were aligned using DiAlign (Genomatix). (", "B) MBP-HopX1 has no protease activity *in vitro* on the general substrate casein. ", "Protease activity of recombinant MBP-HopX1 or MBP-HopX1^C179A^ incubated with fluorescein isothiocyanate (FITC)-labeled casein at 37°C overnight with moderate shaking. ", "Trypsin was used as a positive control. ", "Fluorescence units record the fluorescence intensity with excitation at 485 nm and monitor the emission wavelength of 535 nm (485/535 nm). ", "An immunoblot showing MBP-HopX1 and MBP-HopX1^C179A^-HA effector inputs is also shown. ", "This experiment was repeated three times with similar results. (", "C) Degradation of JAZ1, JAZ2, JAZ9 and JAZ10 by HopX1 requires the cysteine-based catalytic triad of a putative protease *in vivo*. ", "The immunoblots show JAZ1-HA, JAZ2-HA, JAZ9-HA, JAZ10-HA and MYC2-HA accumulation in the presence of GFP-HopX1 or GFP-HopX1^C179A^ when co-expressed transiently in *N. benthamiana*. ", "This experiment was repeated twice with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**The cysteine proteases HopC1 and HopN1, or the unrelated effector HopAD1-HA, do not compromise JAZ accumulation.** (", "A) Immunoblots showing JAZ5-HA accumulation in the presence of HopX1-HA, the cysteine protease HopC1-HA or an empty vector control when co-expressed transiently in *N. benthamiana* for 2 days. ", "Proteins were detected with anti-HA antisera. ", "CBB, Coomassie brilliant blue staining. ", "This experiment was repeated twice with similar results. (", "B) Immunoblots showing JAZ5-HA accumulation in the presence of HopX1-HA, the cysteine protease HopN1-HA, the unrelated effector HopAD1-HA, or an empty vector control when co-expressed transiently in *N. benthamiana* for 2 days. ", "Proteins were detected with anti-HA antisera. ", "CBB is shown. ", "This experiment was repeated three times with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**JAZ5 degradation by HopX1 is independent of the 26S proteasome.** ", "Immunoblots showing JAZ5-HA accumulation in the presence of GFP-HopX1 or an empty vector control when co-expressed transiently in *N. benthamiana* for 2 days after treatment with 100 µM of MG132. ", "This experiment was repeated twice with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**Molecular and phenotypic analysis of** ***NtCOI1*** **-silenced** ***N. tabacum*** **plants.** (", "A) RT-PCR of *NtCOI1* expression in leaves of *N. tabacum*-silenced EV plants (Line VC) or *N. tabacum* plants silenced for the *NtCOI*1 gene (Line L18). *", "N. tabacum α-Tubulin* (*NtTUB*) was used as an amplification control. ", "This supporting figure is related to [Figure 3A](#pbio-1001792-g003){ref-type=\"fig\"}. (", "B) Capsule and seeds production obtained from transgenic *N. tabacum* plants silenced with an EV construct (Line VC) or *NtCOI1* (Line 18).", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**MBP-HopX1 co-immunoprecipitates with JAZ5-GFP from** ***N. benthamiana*** **plant extracts.** ", "MBP-HopX1 and MBP- HopX1^C179A^ proteins purified from *E. coli* cells were incubated 2 hours with *N. benthamiana* plants extracts transiently expressing the JAZ5-GFP transgene and then subjected to co-immunoprecipitation analysis using GFP agarose beads. ", "CBB, Coomassie brilliant blue staining. ", "This experiment was repeated twice with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1 subcellular localization in plant cells.** (", "A) Confocal microscopy localization of GFP- HopX1 or GFP alone in transiently transformed *N. benthamiana* leaves 48 hours post-infiltration (green). ", "The nuclear GFP signal of a middle cross section is shown. ", "This supporting figure is related to [Figure 3F](#pbio-1001792-g003){ref-type=\"fig\"}. ", "Note that the pictures show confocal sections of the nuclei and, therefore, the signal is nuclear and not derived from GFP protein surrounding the nuclear membrane. (", "B) Crude subcellular fractionation of GFP-HopX1 in *N. benthamiana*. ", "Immunoblots showing GFP-HopX1 localization after subcellular fractionation when GFP-HopX1 or GFP alone was expressed transiently in *N. benthamiana* leaves for 48 hours. ", "Total, cytoplasmic and nuclear fractions are shown. ", "Actin, cytoplasmic intrinsic protein for control of the purity of the nuclear fraction. ", "RNA polymerase II, nuclear intrinsic protein for control of enrichment of the nuclear fraction. ", "Equivalent fraction volumes were loaded. ", "This experiment was repeated twice with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1 triggers the activation of JA-dependent gene expression in** ***Arabidopsis*** **.** ", "Quantitative RT-PCR analysis of *JAZ5* and *JAZ12* expression on Col-0 (EV) and stable transgenic *Arabidopsis* Aa--0 lines expressing the *hopX1* or *hopX1* ^C179A^ genes 36 hours after treatment with DEX or a mock solution. ", "The measurements (three technical replicates) represent the relative expression levels between mock (control) and DEX-treated plants in each *Arabidopsis* background. ", "All samples were normalized against the housekeeping gene *AtACT8*. ", "Error bars represent standard deviation (SD). ", "The results are representative of three independent experiments.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**Ectopic expression of** ***hopX1*** **but not** ***hopX1*** **^C179A^** **in stable transgenic** ***Arabidopsis*** **Aa--0 correlates with loss in chlorophyll content.** (", "A) Contents of leaf chlorophyll in stable transgenic *Arabidopsis* lines expressing the *hopX1* or *hopX1* ^C179A^ genes two, five, and nine days after DEX treatment. ", "Error bars represent standard deviation (SD). ", "L12 and L13 are two independent stable transgenic *Arabidopsis* lines expressing the *hopX1* gene. ", "L15 and L16 are two independent stable transgenic *Arabidopsis* lines expressing the *hopX1* ^C179A^ gene. ", "The results are representative of three independent experiments. (", "B) Immunoblots showing HopX1-HA or HopX^1C179A^-HA accumulation in stable *Arabidopsis* lines expressing the *hopX1* or *hopX1* ^C179A^ genes. ", "Plants were induced with DEX or a mock solution for five hours. ", "Line designations are as for [Figure S11A](#pbio.1001792.s011){ref-type=\"supplementary-material\"}.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1 induces chlorosis in** ***N. benthamiana*** **leaves when transiently expressed.** ", "Chlorotic symptoms in transiently expressed *EV*, *hopX1*, or *hopX1* ^C179A^ *N. benthamiana* leaf tissue after DEX treatment for 48 hours.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1 complements the growth of a COR-deficient** ***Pto*** **DC3000 strain when expressed ectopically in** ***Arabidopsis*** **Aa--0 plants.** ", "Growth of *Pto* DC3000 and the isogenic *Pto* DC3000 *COR−* strain on stable transgenic *Arabidopsis* lines expressing the *hopX1* or *hopX1* ^C179A^ genes two days after spray inoculation with bacteria at 10^8^ cfu/ml^−1^. Plants were pre-treated with DEX or a mock solution 24 hours prior to bacterial infection. ", "Pto DC3000 growth increased by about one log (cfu/cm^2^) in transgenic HopX1 *Arabidopsis* pre-treated with DEX compared to mock treated control plants whereas ectopic expression of *hopX1* ^C179A^ could only promote Pto DC3000 growth to about half a log (cfu/cm^2^). ", "The data indicate that HopX1 contributes to pathogenicity over 2-fold compared to HopX1C179A when effectors are ectopically overexpressed in *Arabidopsis* plants. ", "Furthermore, in mock treated transgenic HopX1 *Arabidopsis* plants, Pto DC3000 COR− growth was restricted by one log (cfu/cm^2^) compared to Pto DC3000. ", "However, both strains grew to similar levels when expressing *hopX1* in response to DEX treatment. ", "In contrast, Pto DC3000 COR− growth returned to wild-type levels when HopX1 carried the C179A mutation. ", "Error bars indicate standard error of the mean (SEM). ", "Red asterisks indicate statistically significant differences between DEX and mock-treated plants in each *Arabidopsis* line/bacterial strain (Student\\'s *t* test, *\\*p*\\<0.05 and *\\*\\*p*\\<0.01). ", "Black asterisks indicate statistically significant differences between *Pto* DC3000 and *Pto* DC3000 *COR−* in each condition (Student\\'s *t* test, *\\*p*\\<0.01). ", "The results are representative of three independent experiments.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n***Pto*** **DC3000** ***COR−*** **bacteria expressing** ***hopX1*** **manipulate stomatal aperture.** ", "Stomatal aperture in wild-type *N. benthamiana* leaves measured after 5 hours of incubation with mock, *Pto* DC3000, or *Pto* DC3000 *COR−* bacteria expressing *hopX1*, *hopX1* ^C179A^, or an *EV* control. ", "Error bars indicate SEM (*n* = 20). ", "Asterisks indicate significant differences compared with *Pto* DC3000 *COR−* bacteria expressing an *EV* control at *\\*\\*p*\\<0.01. ", "The results are representative of two independent experiments.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**HopX1** ***~Pto~*** ~**DC3000**~ **does not compromise the accumulation of JAZs when transiently co-expressed in** ***N. benthamiana*** **.** ", "Immunoblots showing JAZ5-HA and JAZ10-HA accumulation in the presence of HopX1*~Pta~* ~11528~, HopX1*~Pto~* ~DC3000~, or an empty vector control when co-expressed transiently in *N. benthamiana*. ", "Proteins were detected with anti-HA. ", "CBB, Coomassie brilliant blue staining. ", "A non-specific band is shown as an additional loading control in the blot of JAZ10. ", "This experiment was repeated twice with similar results.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\n###### \n\n**List of** ***Pta*** **11528 secreted effector proteins included in this study.** ", "Putative activities of type III effectors are shown. ", "The *Pta* 11528 effector identity was determined by BLAST against all known effector proteins.", "\n\n(TIF)\n\n###### \n\nClick here for additional data file.", "\n\nWe thank Jeff H. Chang for the library of *Pta* 11528 effectors. ", "We also thank Tsubasa Shoji and Takashi Hashimoto for materials provided (*COI1*-silenced tobacco plants), Carmen Castresana for critical review of the manuscript, and Matthew Smoker for *Arabidopsis* transformations. ", "We are finally very grateful to Emilia Lopez Solanilla and Pablo Rodriguez Palenzuela for providing the pCPP5040 for *Pseudomonas* expression.", "\n\nCOR\n\n: coronatine\n\ncfu\n\n: colony forming units\n\nDEX\n\n: dexamethasome\n\nET\n\n: ethylene\n\nEV\n\n: empty vector\n\nGFP\n\n: green fluorescent protein\n\nHA\n\n: hemagglutinin\n\nJA\n\n: jasmonic acid\n\nJA-Ile\n\n: jasmonate-Isoleucine\n\nMBP\n\n: maltose binding protein\n\nPD\n\n: pull down\n\n*Pta*\n\n: *Pseudomonas syringae* pv. *", "tabaci*\n\n*Pto* DC3000\n\n: *P. syringae* pv. ", "tomato stain DC3000\n\nSA\n\n: salicylic acid\n\nTF\n\n: transcription factor\n\nTTSS\n\n: type III secretion system\n\n[^1]: The authors have declared that no competing interests exist.", "\n\n[^2]: The author(s) have made the following declarations about their contributions: Conceived and designed the experiments: SG-I RS. ", "Performed the experiments: SG-I MB GF-B. Analyzed the data: SG-I MB GF-B AC JPR RS. ", "Contributed reagents/materials/analysis tools: AC. ", "Wrote the paper: SG-I JPR RS.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "Dongria Saree\n\nRegular price\nRs. ", "3,799.00\nSale price\nRs. ", "0.00\n\nCheck Pincode Serviceability\n\nQuantity\n\nThe Dongria Kondh identity of tribal identity, In a particular village in India Adivasis are involved in producing Dongria textile , the producers are group of weavers , to make it more real they changed the technique from embroidery to weaving- shift in technology. ", "It was translated in such a way that it appears like an embroidered kapdagonda in similar colors and exact pattern, takes around 4-5 days minimum to 8 days to produce a single Saree." ]

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[ "Answers\n1\n\nAfter the shipwreck, Viola is ridden with grief, something that briefly shadows her optimistic and gentle nature. ", "Though she fears that she has lost her brother forever, yet she hopes that he is still alive, and tries her best not to succumb to her grief. ", "Her words are plain and straightforward, denoting grief but also her sensibility. ", "Although she does not know the Captain, she presumes that he has a \"fair and outward character\" from their limited interaction, and his offers to help her (l. 48); she assumes the best of him, rather than the worst, though she admits even while she makes her judgment, that appearances can be deceiving." ]

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[ "To be able to access the service on your browser, you'll need to make sure you're using Safari version 11.1 (released in 2018 and available for some of the latest versions of Mac OS) or later. ", "That's pretty much all you need, though -- the advantage of watching in a browser, after all, is that you don't need to download and run a separate app anymore." ]

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[ "HobbitShop.com has all of your favorite The Hobbit: The Battle of the Five Armies and The Lord of the Rings official merchandise. ", "Celebrate The Hobbit: The Battle of the Five Armies with a Smaug action figure, t-shirt, or phone case! ", "We also carry a great selection of The Lord of the Rings merchandise, including One Ring jewelry, a high-end chess set, and more! ", "While one does not simply walk into Mordor, we make it easy to bring home all of your favorite Middle-earth merchandise at HobbitShop.com!" ]

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[ "Implementation of analytical gradients and of a mixed real and momentum space DVR method for excess electron systems described by a self-consistent polarization model.", "\nThis work presents two extensions of our self-consistent polarization model for treating non-valence excess electron systems. ", "The first extension is the implementation of analytical gradients, and the second extension is the implementation of a mixed real space plus momentum space approach combined with fast Fourier transforms to reduce the computational time compared to a purely real space discrete variable representation approach. ", "The performance of the new algorithms is assessed in calculations of the excess electron states of various size water clusters and of the non-valence correlation-bound anion of the C240 fullerene." ]

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[ "Q:\n\nPython: Why does %r not accurately represent the raw data on \\\" versus \\'\n\nI've been going through Zed's LPTHW and I have been messing around with escape characters after doing lesson 10. ", " While fooling around with %r I came across this, and I have no idea why it's happening (I'm so new to any form of programming/coding it hurts):\ntest = \"10'5\\\"\"\ntest_2 = '10\\'5\"'\n\nprint \"%r\" % test\nprint \"%r\" % test_2 \n\nWhen I run this, I get:\n'10\\'5\"'\n'10\\'5\"'\n\nI'm confused. ", " I had assumed that I would get output in the following:\n\"10'5\\\"'\n'10\\'5\"'\n\nIt was my understanding that %r would return the string identical to how it is written, yet it seems to convert it to test_2 by moving the \\ to the left.", "\nAm I missing something here?", "\nThanks.", "\n\nA:\n\nIt was my understanding that %r would return the string identical to how it is written\n\nYour understanding is incorrect. ", "Python does not \"remember\" how a string was written in the source code; all that matters to the interpreter is that it contains the characters:\n10'5\"\n\nPrinting a repr of that string will use whichever type of quotation marks Python feels is most appropriate for its contents. ", "Since both strings contain the same characters, they are printed identically by repr (and, hence, by the %r format string).", "\n\n" ]

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[ "All relevant data are within the paper and its Supporting Information files.", "\n\nIntroduction {#sec001}\n============\n\nIn an increasingly human-dominated world, it is necessary to understand the effects of human disturbances in order to achieve effective management and conservation of wildlife. ", "Human activities such as motorized vehicle use \\[[@pone.0184018.ref001]\\], tourism \\[[@pone.0184018.ref002]\\], and legal and illegal hunting \\[[@pone.0184018.ref003], [@pone.0184018.ref004]\\] have a variety of direct and indirect effects on animals. ", "Wildlife managers need to be able to identify and quantify these effects for a successful management of populations. ", "Anthropogenic effects can be estimated through different methodologies whose use depends on several conditions, for example animal species, ease of use, and economic costs \\[[@pone.0184018.ref005]\\], among others.", "\n\nBehavioral ecological methods are frequently used to assess anthropogenic stressors because ultimately a species' behavioral response to humans will influence its ability to coexist with them. ", "Flight-initiation distance (FID) is among behavioral responses, and is the distance between an animal and an observer at the moment the animal begins to flee \\[[@pone.0184018.ref006]\\]. ", "FID is frequently used by managers because it is relatively simple to evaluate, non-invasive, and inexpensive. ", "This measure is used to quantify shyness or sensitivity to human disturbances because flight is a common animal behavioral response to human presence \\[[@pone.0184018.ref007], [@pone.0184018.ref006]\\]. ", "While managers acknowledge the variability in FID, they nevertheless use estimates of a species' FID to attempt to minimize human impact. ", "This measure is best used by comparing disturbed and undisturbed populations, where differences should be attributable to differential habituation \\[[@pone.0184018.ref008], [@pone.0184018.ref009]\\].", "\n\nFID has been used in several ungulate species: guanacos (*Lama guanicoe*) \\[[@pone.0184018.ref003], [@pone.0184018.ref002]\\], impala (*Aepyceros melampus*) and greater kudu (*Tragelaphus strepsiceros*) \\[[@pone.0184018.ref010]\\], mule deer (*Odocoileus hemionus*) \\[[@pone.0184018.ref011]\\], Thomson's gazelle (*Eudorcas thomsonii*) \\[[@pone.0184018.ref012]\\], among others. ", "The general pattern is that ungulates exhibit longer FID when they are under hunting pressure \\[[@pone.0184018.ref003]\\], however this is not clear in guanacos, the largest ungulate inhabiting arid and semiarid ecosystems in South America. ", "Donadio and Buskirk \\[[@pone.0184018.ref003]\\] in San Guillermo National Park, San Juan Province, found no statistical differences in FID between areas with and without poaching. ", "Nor was poaching found to affect FID in La Payunia Provincial Reserve in Mendoza Province either \\[[@pone.0184018.ref013]\\]. ", "Nevertheless, Malo and collaborators \\[[@pone.0184018.ref002]\\] found that the probability of neutral human encounters generated a decrease in FID in this species.", "\n\nThe effect of legal and illegal hunting pressure can be highly heterogeneous, which makes it important to analyze complementary biological and behavioral aspects, including FID, if we want to know the effect of a certain disturbance upon any animal population \\[[@pone.0184018.ref006]\\]. ", "Group living is another behavioral strategy exhibited by different ungulate species in areas with high predation risk \\[[@pone.0184018.ref014], [@pone.0184018.ref015], [@pone.0184018.ref016]\\]. ", "Increasing group size improves the likelihood of detecting predators due to the collective detection effect \\[[@pone.0184018.ref017]\\]. ", "In a variety of species, a positive relation has been found between group size and FID, which indicates that animals in larger groups initiate flight at greater distances \\[[@pone.0184018.ref018]\\]. ", "For this reason, group formation is important to ungulate females with offspring that are vulnerable to predation \\[[@pone.0184018.ref019]\\], and that must choose areas with low predation risk, such as inside a protected area.", "\n\nThe effect of human activities, like poaching, on animal behavior could be influenced by habitat structure, which can be characterized through biophysical predictors. ", "Measures of these predictors are difficult to identify on the basis of field studies across large areas \\[[@pone.0184018.ref020]; [@pone.0184018.ref021]\\]. ", "Topography and vegetation structure \\[[@pone.0184018.ref022], [@pone.0184018.ref013]\\] can be assessed with remotely sensed data \\[[@pone.0184018.ref023], [@pone.0184018.ref024]\\] through a multi-scale analysis in broad areas \\[[@pone.0184018.ref025], [@pone.0184018.ref026]\\]. ", "However, this information has been scarcely used for explaining ungulate behavior. ", "Taraborelli and collaborators \\[[@pone.0184018.ref013]\\] found shorter alert distances of guanacos on steeper slopes. ", "Estes and collaborators \\[[@pone.0184018.ref027]\\] used texture analysis of SPOT, ASTER and MODIS remote sensing images, and found them to be a good tool for understanding habitat selection by different ungulates.", "\n\nThe guanaco is a resource-defense polygynous species, with family groups comprising a territorial male with multiple females and their offspring \\[[@pone.0184018.ref028], [@pone.0184018.ref029]\\]. ", "Towards the end of the 19th century, guanaco populations were present in nearly all Argentine biomes, occupying open woodlands and scrub-dominated areas. ", "Currently, guanacos are abundant only in Patagonian steppes and Andean foothills \\[[@pone.0184018.ref002]\\]. ", "There are several causes for this decline: legal and illegal hunting, competition with domestic livestock and/or exotic herbivores, and habitat loss or fragmentation by agricultural development \\[[@pone.0184018.ref030], [@pone.0184018.ref031]\\]. ", "As a consequence of this, the guanaco was listed in Appendix II of the CITES (Convention on International Trade in Endangered Species) in 1978, which only permits the use of fiber from live animals \\[[@pone.0184018.ref032]\\].", "\n\nHunting of guanacos is controlled in Argentina by Law No. ", "20961/75, and is particularly prohibited in the province of San Juan (Law No. ", "6911), notwithstanding, they are poached in almost the entire Province. ", "Based on poaching pressure and habitat destruction, in San Juan province the guanaco has been deemed a \"vulnerable\" species by the Secretary of State for the Environment and Sustainable Development (Resolution No. ", "0656-SEAyDS-11).", "\n\nAt a time when biodiversity loss is a major conservation concern, protected areas are the only sites where native species population conservation is expected to occur. ", "In these areas it becomes necessary to find assessment tools and assess the factors affecting these tools. ", "FID is a commonly used method to evaluate the poaching pressure on ungulates and other animals. ", "In guanacos, no strong evidence has been found that FID is affected by poaching in a similar way. ", "Due to this, our objectives were: 1- to evaluate whether poaching affects the FID and group structure (group size and kind of group) of a guanaco population in a high cold desert in San Juan province; 2- to assess whether habitat structure influences FID and group structure in this population.", "\n\nMaterials and methods {#sec002}\n=====================\n\nEthics statement {#sec003}\n----------------\n\nThis study was conducted with wild free-ranging animals and was completely observational. ", "Research was conducted in accordance with guidelines of the Bioethics Commission, Departamento de Biología, Facultad de Ciencias Exactas, Físicas y Naturales, Universidad Nacional de San Juan, Argentina and the provincial authorities where the work occurred. ", "Approval and permission to conduct research was granted by the Mr. Arturo Curatola owner of Don Carmelo Reserve and the Dirección de Conservación y Áreas Protegidas dependent of Secretaría de Estado, Ambiente y Desarrollo Sustentable of San Juan Province.", "\n\nStudy area {#sec004}\n----------\n\nThe study was conducted in La Invernada Valley, Ullúm department (31° 10\\' S, 69° 46\\' W), San Juan province, Argentina \\[[@pone.0184018.ref033]\\] ([Fig 1](#pone.0184018.g001){ref-type=\"fig\"}). ", "The study area lies at between 2000 and 3100 m a.s.l., ", "and comprises sectors of the Puna Desert and Monte Desert which correspond to the Puna and Monte of hills and closed basins (Monte de Sierras y Bolsones) ecoregions respectively. ", "The vegetation in the Puna is dominated by *Lycium chanar*, *Artemisia mendozana*, *Maihueniopsis glomerata*, *Adesmia* aff. *", "horrida* and *Stipa* spp., ", "with *Lilaeopsis macloviana*, *Taraxacum officinale*, *Nastanthus agglomeratus*, *Azorella* spp. ", "and *Adesmia pinnifolia* being less abundant \\[[@pone.0184018.ref033]\\]. ", "The Monte is characterized by *Larrea divaricata*, *Monttea aphylla*, *Gochnatia glutinosa* and *Trycicla spinosa* \\[[@pone.0184018.ref033]\\]. ", "In general, the area is a shrub steppe dominated by a gently undulating relief and flatlands with short xerophytic vegetation.", "\n\n{#pone.0184018.g001}\n\nLa Invernada Valley covers about 58000 ha and includes a protected area, Don Carmelo Reserve (Provincial decree 1220/1993), of about 40000 ha (hereafter called DCR). ", "The remaining unprotected area (hereafter called UA, 18000 ha) is vulnerable to poaching events because there is no law enforcement. ", "This UA stretches between the Reserve and Provincial Route No. ", "436. ", "The animals can move freely into and out of DCR because this protected area is delimited by a single wire that is 15 cm above the ground. ", "Density of guanacos in La Invernada Valley is one of the highest in the Province (4--6 ind/km^2^), and livestock are not present \\[[@pone.0184018.ref034]\\].", "\n\nField sampling {#sec005}\n--------------\n\nThe study was conducted in May 2010, March 2011 and February 2012. ", "The study area was divided into the area where poaching occurs, Unprotected area (UA), and does not occur, Don Carmelo Reserve (DCR, [Fig 1](#pone.0184018.g001){ref-type=\"fig\"}). ", "We monitored the FID of guanaco groups in both areas. ", "We used different dirt roads that run across the study area; these were travelled by four observers in a pickup truck (20--35 km/h). ", "We used this methodology because poachers travel in pickup trucks along dirt roads seeking to shoot the animals (Secretary of State for the Environment and Sustainable Development, pers. ", "comm.). ", "The low vegetation and gentle slope in the study area allow a suitable visibility for monitoring the behavior of guanacos.", "\n\nWe identified individuals or groups based on the cohesive behavior among members \\[[@pone.0184018.ref003]\\]; thus, individuals were considered in the same group when they were at a maximum distance of 50 m from each other. ", "We measured FID with a laser rangefinder (7x26 Bushnell ELITE 1500) from the center of the group when the first individual fled. ", "To analyze group structure, we recorded size and age composition for each group. ", "Categorizing family and bachelor groups at long distances is difficult without marked animals \\[[@pone.0184018.ref030]\\]; due to this, we considered three group composition categories: 1-groups with offspring; 2- groups without offspring, and 3- solitary adult. ", "If we saw a group with the same structure (size and composition) and position as a group recorded on the previous day, we did not record this data. ", "We treated each group as an independent sample because we travelled the dirt roads only once a day.", "\n\nHabitat structure {#sec006}\n-----------------\n\nFor habitat structure, we considered slope and heterogeneity of vegetation from remote sensing data. ", "References used for images were: WGS 84 datum, UTM (Universal Transverse Mercator) projection; zone: Argentina 19 South. ", "Slope angle (in degrees) was modelled on the basis of a MDE-Ar (Modelo Digital de Elevaciones de la República Argentina 30 m-resolution). ", "This model is based on SRTM (Shuttle Radar Topography Mission, see <http://www.ign.gob.ar/NuestrasActividades/Geodesia/ModeloDigitalElevaciones/Introduccion>). ", "For the analysis of vegetation, Landsat 8 OLI scenes (30-m resolution) of the study area (path 232 and row 81--82 for La Invernada Valley) acquired on 27 February 2011 (<http://earthexplorer.usgs.gov/>) were used. ", "This date was selected because the images had no cloud cover. ", "Images were rescaled to the Top Of Atmosphere (TOA) reflectance with a correction for the sun angle using coefficients provided in the product metadata file (MTL file). ", "The SATVI green index (Soil Adjusted Total Vegetation Index) \\[[@pone.0184018.ref035]\\] was the basis for the image texture analyses to assess the heterogeneity of vegetation \\[[@pone.0184018.ref026], [@pone.0184018.ref036]\\]. ", "Several window sizes were evaluated, so the different scales were represented by the extent of the moving window of an image texture measure, i.e. with 3 x 3 (0.81 ha), 5 x 5 (2.25 ha), 7 x 7 (4.41 ha) and 9 x 9 (7.29 ha) 30-m pixel moving windows.", "\n\nAll remote sensing variables were finally stored as separate layers in the GIS and were extracted for each group recorded. ", "For image analysis we used Quantum GIS (Version 2.14 Essen, <http://qgis.osgeo.org/>) and ENVI GIS (ENVI 2004, Research Systems, Boulder, Colorado, USA).", "\n\nStatistical analyses {#sec007}\n--------------------\n\nAll analyses were performed using Generalized Linear Mixed Models (GLMMs). ", "To assess the effect of poaching on FID, we fitted a model with FID as response variable, poaching situation as explanatory variable with two levels: with (UA) and without poaching (DCR), and with group composition categories (group with and without offspring, and solitary adult) and year (2010, 2011 and 2012) as random variables. ", "Because the error structure was not Gaussian, in spite of the log transformation, FID was modelled as a Gamma distribution.", "\n\nTo assess the poaching effect on group size (response variable) we fitted a Negative Binomial distribution because the model showed overdispersion \\[[@pone.0184018.ref037]\\] (*ĉ*\\>1). ", "We did not use solitary individuals for this analysis. ", "To assess whether poaching affected the proportion of group composition categories, we fitted models for each group category with a Binomial distribution (used for proportion data). ", "Thus, for each fitted model we included poaching situation as explanatory variable and year as random variable.", "\n\nTo evaluate how FID (log transformed) and group size respond to habitat structure at different spatial scales, GLMMs with a Gaussian and Negative Binomial error structure respectively \\[[@pone.0184018.ref038]\\] were fitted slope (MDE-Ar) plus each scale considered (i.e. 3 x 3, 5 x 5, 7 x 7 and 9 x 9 30-m) for SATVI mean texture measure. ", "We fitted four models, one for each scale considered. ", "We used poaching situation (UA and DCR) as random factor. ", "The habitat structure measures (slope and texture measure) that were uncorrelated were included as explanatory variables (Spearman rank correlation: r \\< \\|0.7\\|) \\[[@pone.0184018.ref039]\\]. ", "A backward elimination procedure was performed to remove insignificant terms without losing important information. ", "Backward elimination started with all of the predictors in a full model. ", "The least significant variable, i.e. the one with the largest *P* value, was removed and the model was refitted. ", "Each subsequent step removed the least significant variable in the model until all remaining variables had individual *P* values less than 0.05. ", "The sign of parameters having significant effects was used to interpret the results \\[[@pone.0184018.ref040], [@pone.0184018.ref041]\\].", "\n\nAll statistical analyses were carried out using R version 3.2.2 \\[[@pone.0184018.ref042]\\]. ", "We used nlme package \\[[@pone.0184018.ref043]\\] for Gamma and Binomial distribution and glmmADMB package \\[[@pone.0184018.ref044]\\] for Negative Binomial distribution.", "\n\nResults {#sec008}\n=======\n\nWe recorded a total of 862 guanaco observations. ", "In the UA, we observed 30 groups (with offspring = 12, without offspring = 15, solitary = 3) and 70 groups in DCR (with offspring = 27, without offspring = 30, solitary = 13). ", "FID was significantly higher in the UA ($\\overline{x} = 515.93\\text{~m}$, SD = ±257.72 m) than in DCR ($\\overline{x} = 354.18\\text{~m}$, SD = ±260.38 m; GLMM~Gamma~, t = 2.71, P = 0.007, [Fig 2](#pone.0184018.g002){ref-type=\"fig\"}). ", "Poaching also affected group size, all groups being larger (without solitary individuals) in the UA than in DCR (GLMM~Neg,Binom~, Z = -3.52, P\\<0.001). ", "Notwithstanding, the number of groups for each category in the UA was less than half that in DCR, the proportion of group categories was similar in both areas, and we found no significant differences in either case (P\\>0.3 in all cases, [Table 1](#pone.0184018.t001){ref-type=\"table\"}).", "\n\n10.1371/journal.pone.0184018.t001\n\n###### Mean group size ($\\overline{x}$) with Standard Error (±SE), number of groups recorded (n) and proportion of each group category (Prop) according to the poaching situation.", "\n\nSignificance factor for proportion of group category was \\<0.05.", "\n\n![](", "pone.0184018.t001){#pone.0184018.t001g}\n\n With poaching (UA) Without poaching (DCR) *P* \n ------------------- -------------------- ------------------------ ----- ------------ ---- ------ ------\n With offspring 13.16 ±10.49 12 0.4 9.46 ±5.36 27 0.38 0.85\n Without offspring 15.4 ±17.83 15 0.5 6.62 ±4.19 30 0.43 0.52\n Solitary 1 3 0.1 1 13 0.18 0.31\n\n{#pone.0184018.g002}\n\nFID and group size were not affected by slope ($\\overline{x} = 6.90$, SD±6.05) but SATVI mean texture measure affected FID at all scales (P = 0.03, [Table 2](#pone.0184018.t002){ref-type=\"table\"}). ", "FID decreased when SATVI mean texture measure increased, i.e. greater vegetation cover. ", "SATVI mean texture measure did not affect group size (P\\>0.1) on any of the scales examined.", "\n\n10.1371/journal.pone.0184018.t002\n\n###### Generalized linear mixed model for Flight-initiation distance related to image texture values from SATVI at different scales (number of pixels).", "\n\nPoaching situation was considered a random effect.", "\n\n![](", "pone.0184018.t002){#pone.0184018.t002g}\n\n Fixed effect Estimate SE t P\n -------------------------------- ---------- --------- ------- ------\n 3 x 3 moving windows (0.81 ha) \n  Intercept 2531.50 919.15 2.75 0.00\n  Mean of SATVI -58.45 25.46 -2.30 0.02\n 5 x 5 moving windows (2.25 ha) \n  Intercept 2838.61 983.00 2.89 0.00\n  Mean of SATVI -66.97 27.23 -2.46 0.01\n 7 x 7 moving windows (4.41 ha) \n  Intercept 2968.36 1033.10 2.87 0.00\n  Mean of SATVI -70.57 28.61 -2.46 0.01\n 9 x 9 moving windows (7.29 ha) \n  Intercept 3094 1084.13 2.85 0.00\n  Mean of SATVI -74.10 30.05 -2.46 0.01\n\nDiscussion {#sec009}\n==========\n\nWe found that guanacos exhibited longer flight distances and larger group size in the UA, where they were subjected to poaching, than in DCR. ", "Regarding the proportion of group composition categories, we found that each category was equally represented in both areas. ", "However, we found less than half the number of groups in the UA than in DCR.", "\n\nMany studies use behavioral changes as a tool for measuring disturbance; FID is frequently used to assess the effect of different types of disturbances on wildlife \\[[@pone.0184018.ref007], [@pone.0184018.ref002]\\]. ", "Unlike Donadio and Buskirk \\[[@pone.0184018.ref003]\\] and Taraborelli and collaborators \\[[@pone.0184018.ref013]\\], who did not find any poaching effect on FID, we found that guanacos in DCR showed shorter FID than in the UA. ", "This is consistent with findings for other ungulates \\[[@pone.0184018.ref007], [@pone.0184018.ref010]\\] which show greater FID when hunting pressure increases. ", "Poachers used the full extent of La Invernada Valley until the Reserve was created (1993), thereafter they habitually poach only in the outside area, which probably explains our results on FID. ", "Marino and Johnson \\[[@pone.0184018.ref001]\\] suggest that guanacos can change their tolerance to humans if stimuli shift from negative (poaching) to neutral (inside the Reserve, without poaching).", "\n\nOur results show, similarly to other studies, that ungulates can reduce predation risk by forming groups \\[[@pone.0184018.ref045]\\], we found larger groups in the UA than in DCR. ", "Probably, as proposed by Taraborelli and collaborators \\[[@pone.0184018.ref017]\\], the benefit from living in groups for the guanaco in our study area is due to cooperative vigilance. ", "Moreover, unlike Stephens and Peterson \\[[@pone.0184018.ref019]\\], who found that in moose the groups with offspring are more vulnerable than groups without them \\[[@pone.0184018.ref019]\\], we did not find a differential effect of poaching on each group category. ", "We found a similar proportion for each group category in both areas.", "\n\nEnvironmental factors also have a significant influence on ungulate behavior \\[[@pone.0184018.ref010], [@pone.0184018.ref013]\\]. ", "In relation to habitat structure, similarly to Taraborelli and collaborators \\[[@pone.0184018.ref013]\\], we found that neither FID nor group size were affected by slope, probably due to the great ability of guanacos to climb. ", "On the other hand, vegetation cover was not important for group size but it was an important factor in modulating the risk assumed by guanacos, since our results show a low FID (high risk) in areas with high vegetation cover. ", "For the same population, it was found that guanacos use vegas (sites with high vegetation quality) even if they entail high predation risk \\[[@pone.0184018.ref015]\\]. ", "Probably, due to the energy benefits gained in areas with high vegetation cover, the guanaco could stay in areas with greater food availability despite a high predation risk. ", "Different aspects of foraging behavior (i.e. food items selected, selection of sites with high vegetation quality) deserve further research in order to assess this hypothesis.", "\n\nAccording to our results, the guanaco population in this cold desert responds to poaching pressure as do other species of ungulates. ", "Most of the groups occurred in the protected area and showed higher FID and larger group size compared to the unprotected area. ", "For this population, FID can be a useful tool for evaluating the effect of poaching on guanacos, and how they respond to this human disturbance. ", "In addition, managers must take habitat structure into account, since the guanaco appears to assume a higher risk (lower FID) in areas with high vegetation cover.", "\n\nSupporting information {#sec010}\n======================\n\n###### \n\n(XLSX)\n\n###### \n\nClick here for additional data file.", "\n\nThe authors wish to thank Arturo Curatola and his family for allowing us to work on their property, also Silvina Bongiovanni, Andrea Astié and Claudia Campos for their critical reading, and Nelida Horak for assisting us with the English version. ", "This article is dedicated to the memory of Andrés Calderon.", "\n\n[^1]: **Competing Interests:**The authors have declared that no competing interests exist.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "Q:\n\nHow to find coordinates $(x,y) $ from distance rule\n\nI have 3 coordinates of parallelogram $A(3,\\,2)\\,, B(4,\\,-5),\\, C(0,\\,-3)$ and $D(x,\\,y)$\nit's possible to get $x$ and $y$ (coordinate of $D$) from distance rule\n\nA:\n\n1. ", "way\nSince the diagonal halves each other, the midpoint of each coincide:\n$${A+C\\over 2} = {B+D\\over 2} \\Longrightarrow D = A-B+C =(-1,4)$$\n2. ", "way\n$$AB = CD\\;\\;\\;{\\rm and}\\;\\;\\; AD = BC$$\n....\nBut this is like taking something from the left pocket with right hand.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "define(function(){\n\n\treturn {\n\t\tload_ipython_extension: function(){\n\t\t\tconsole.debug('krisk loaded')\n\n\t\t\trequire.config({\n\t\t\t\tmap: {\n\t\t\t \"*\" : {\n\t\t\t \"echarts\": \"nbextensions/krisk/echarts.min\",\n\t\t\t \"dark\": \"nbextensions/krisk/dark\",\n\t\t\t \"infographic\": \"nbextensions/krisk/infographic\",\n\t\t\t \"roma\": \"nbextensions/krisk/roma\",\n\t\t\t \"vintage\": \"nbextensions/krisk/vintage\",\n\t\t\t \"macarons\": \"nbextensions/krisk/macarons\",\n\t\t\t \"shine\": \"nbextensions/krisk/shine\",\n\n\t\t\t }\n\t\t\t }\n\t\t\t // Solution above is not clean and preferable below.", "\n\t\t\t // Forced to use it because of Jupyter issue. ", "See\n\t\t\t // https://github.com/jupyter/notebook/issues/626#issuecomment-160593027\n\t\t\t \t \n // baseUrl : \"nbextensions/krisk\",\n // paths: {\n // echarts: \"echarts.min\"\n\n // }\n });\n\t\t}\n\t}\n\t\n})" ]

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[ "RAF's unmanned Reaper aircraft had been operated from Creech airforce base in Nevada, but missions from Lincolnshire began this week\n\nRemotely controlled armed drones used to target insurgents in Afghanistan have been operated from the UK for the first time, the Ministry of Defence said on Thursday.", "\n\nMissions of the missile-carrying Reaper aircraft began from a newly built headquarters at RAF Waddington in Lincolnshire earlier this week – five years after the MoD bought the unmanned aerial vehicles (UAVs) to monitor and attack the Taliban.", "\n\nSince then the UK has been controlling the RAF's five Reaper aircraft from Creech airforce base in Nevada because the British military did not have the capability to fly them from here.", "\n\nHowever, the MoD made building a new UAV hub at Waddington a priority following the 2010 strategic defence and security review, and the centre \"stood up\" at the end of last year.", "\n\nWaddington has become the home of XIII squadron, and defence officials said pilots from the unit have now started to take command of Reapers, working in tandem with the team in America.", "\n\nThere are three operating terminals at the base in Lincolnshire, and they had to go through extensive technical trials before they were deemed ready for use.", "\n\n\"We aren't flying any more operations than we were before, but with the time differences between the US, Afghanistan and the UK, it is now possible for pilots at Waddington to work in relay with the those in the US,\" said a source.", "\n\nThere are no current plans to disband the squadron in the US, which is expected to continue operating until the end of next year, when all Nato combat operations in Afghanistan will finally come to an end.", "\n\nThe RAF has bought five more Reaper aircraft, which are expected to be deployed in Afghanistan over the summer, bringing the total to 10. ", "British UAVs have flown 45,000 hours in Afghanistan, and fired 350 weapons, including Hellfire missiles.", "\n\nThough the MoD insists it operates with aircraft only in support of British troops, and only in Helmand province, the use of UAVs has been dominated by the CIA's controversial programme to target insurgent leaders in Pakistan.", "\n\nThese strikes have sometimes caused civilian casualties, and have raised questions over the legality and morality of using remotely piloted systems in areas that are not conflict zones.", "\n\nThe disclosure comes at a sensitive time for the MoD – just two days before a protest outside RAF Waddington organised by CND, the Drone Campaign Network, Stop the War and War on Want.", "\n\nThe coalition has warned that switching control of drones to Waddington from US bases marks an unwelcome expansion in the UK's UAV programme.", "\n\n\"Drones, controlled far away from conflict zones, ease politicians' decisions to launch military strikes and order extra-judicial assassinations, without democratic oversight or accountability to the public,\" said Rafeef Ziadah, from War on Want.", "\n\nChris Nineham, vice-chair of the Stop the War Coalition, added: \"Drones are being used to continue the deeply unpopular War on Terror, with no public scrutiny. ", "They're using them to fight wars behind our backs. ", "These remote-controlled killing machines should be banned.\"" ]

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[ "Q:\n\nhow to remove sticky (permanent) notifications from notification bar?", "\n\nI have a buggy app installed in my phone, which sometimes creates permanent download progress notification with animated icon but fails to remove it from notification bar by its own.", "\nI tried to remove the notification manually (swipe it off) but it is sticky and can't be removed. ", "I tried \"clear\" option in notification bar, but again no success and it is still there! ", "I even stopped that app which created the notification, but no change.", "\nThe only option that I have to remove such sticky and buggy notifications is to restart the device, which is very unpleasant.", "\nWhich app, hack, command, gesture, etc... I can use to remove that sticky notification?", "\n*My device is Sony Xperia S running ICS 4.0.4 stock ROM (rooted)\n\nA:\n\nIf you're using Jelly Bean, then in manage apps, you can turn off notifications for the specific app. ", "Keep in mind, this will turn off all notifications for the app.", "\n\nA:\n\nI have the same problem... You should go to settings, apps, all, find there download manager and clear all data there. ", "It helps in my case. ", "Good luck :-)\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Borehole fluid telemetry systems, often referred to as mud pulse systems, use borehole fluid, such as so-called drilling mud, as a medium to transmit information from the bottom of a borehole to the surface. ", "Such information is useful during operations for the exploration and/or discovery of hydrocarbons such as oil and gas. ", "Virtually any type of data that may be collected downhole can be communicated to the surface using borehole fluid telemetry systems, including information about the drilling operation or conditions, as well as logging data relating to the formations surrounding the well. ", "Information about the drilling operation thus transmitted may include, for example, pressure, temperature, direction and/or deviation of the wellbore, as well as drill bit condition. ", "Formation data may include, by way of an incomplete list of examples, sonic density, porosity, induction, and pressure gradients of the formation. ", "The transmission of this information is important for control and monitoring of drilling operations, as well as for diagnostic purposes.", "\nBorehole fluid telemetry systems produce fluid pulse telemetry signals comprising transient borehole fluid pressures variations. ", "The fluid pulse telemetry signals often comprise data pulses produced by a valve arrangement (e.g. a rotary shear valve or a poppet valve). ", "The rate of data pulse production, and therefore of transmission bandwidth, may be limited by the mechanics of the particular apparatus used in generating fluid pulses downhole." ]

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[ "Monday, June 30, 2014\n\nLast week I received the great news that pictures from my personal project \"Scuffy- A Scarecrow's Tale\" have been accepted into the 3X3 illustration Annual. ", "I've been trying to get the story published ever since I started the book as part of my MFA project back in 2009.", "\n\nFinally, a picture book annual that had an unpublished category has come along and I am gratified to finally see at least part of the book in print. ", "There was a snag in the process when I read in the submission guidelines that even unpublished entries needed to be submitted as \"published\" spreads and include the \"cover\".", "\n\nI enlisted the help of friend and fellow artist Val Paul Taylor to give my spread mock ups and cover a nice type treatment. ", "I am super excited with the wonderful type treatment he came up with for the \"book\". ", "Scuffy is currently being shop by my literary agent, so hopefully I will actually get to finish the book at some point. ", "Fingers crossed!", "\n\nMonday, June 23, 2014\n\nI recently finished another piece of art for Pioneer Theater. ", "This one for a new play that will have it's world premier this fall called Alabama Story written by Kenneth Jones. ", "The plot revolves around a little known incident (at least I had not been aware of ) that happened in Alabama during the racial tension of the late 1950's.", "\n\nIn 1958, Garth Williams, famously the illustrator for Charlotte's Web and the Little House on the Prairie books, wrote and illustrated a charming little tale of two rabbits who frolic together and fall in love. ", "At the end of the book the two rabbits wed in a ceremony attended by their fellow woodland creatures. ", "Innocent enough, right?", "\n\nWell in the racially charged South, this book became the center of a firestorm because, in the book, one rabbit was white and the other was black. ", "The book was censored and a lawsuit raised to remove the book from the state funded library system. ", "The story revolves around a white librarian, her childhood friend who is a black man and various other lawyers and State legislators who are immersed in sorting out this murky topic.", "\n\nAs I began sketching the art, and admittedly before I read the script, I thought it would be fun to use the rabbits as symbols for the racial tension. ", "Early sketches revolved around the idea of the black and white rabbits and the book that precipitated the argument. ", "Even after I changed the setting to the library shelves, the client felt that the furry rabbit angle gave too much of an impression that this was a children's play so that idea was scrapped.", "\n\nI still wanted to evoke a feeling racial tension and imply the idea that the story somehow revolved around books, so we switched to the idea of the black character and the white librarian being on opposite sides of the the divide, in this case, a stack of books. ", "The light filtering from the left side illuminates her as she reads and the the other side is symbolically more in shadow.", "\n\nAfter this idea was approved, I set to work. ", "I took photos and proceeded to the final drawing and rendering. ", "I think it turned out pretty well considering my deadline was cut about three days short due to a planned excursion with my son. ", "I had to scramble to deliver the art before I left town which included a 2:00 am bedtime one evening. ", "I don't enjoy the late nights and avoid them as much as possible, but sometimes the deadline just has to be met, regardless of the lack of sleep.", "\n\nMonday, June 9, 2014\n\nSome of you may be familiar with the Peter and the Starcatchers series of books by Dave Barry and Ridley Pearson. ", "Well, the story has been turned into a play that recently closed it's run on Broadway and is now being staged in other parts of the country. ", "I had the luxury of picking a play to illustrate from several of the plays on the Pioneer Theater Company schedule for next season and this was among those on the list. ", "I guess being old (or at least having a long track record working with the theater) has it's perks. ", "It was a tough choice between this and The Crucible but Peter ultimately won out.", "\n\nFor this one, I went through an unusually high number of comps before we settled on the winning design. ", "At times this can be frustrating, because personally, I felt that they were all visually compelling and any one of the designs could work. ", "But, I have learned over the years that the client is the one that has to be happy and I am hired to make that happen. ", "Here's a run down of the idea sketches:\n\nThe theater wanted to play on the pirate idea, so the first two were pretty straight forward pirate cliche scenes with a nod to N.C. Wyeth.", "\n\nIdea three hinged on the \"magic flying dust\" that is being transported via old fashioned steamer trunk. ", "I thought it would be fun to focus on that aspect of the story since everyone in the play wants to get their hands on it. ", "That idea was liked except the creative director wanted more of a focus on the pirate ship.", "\n\nSo the next idea I had was to do a split screen of sorts with the top of the ship above the water and the hull being shown below the water. ", "The twist was to show the hull as a pirate skull. ", "Great idea right? ", "I love this one and will probably finish a version like this for fun, but the theater thought is was a little much. \"", "Just a regular hull\" was the directive, so back to the drawing board.", "\n\nAfter decapitating the skull from the ship, it was obvious that there was something lacking in that big empty space. ", "It needed a little magic, so I decided to let the sky full of stars filter down underneath the water. ", "I intended to leave the area open for the title treatment, when I thought to myself, \"why not do your own hand lettering?\" ", "Having done a few projects over the years with my own lettering, I decided to give it a shot.", "\n\nI came up with a nice \"piratey\" style that feels like it could have washed right up on treasure map. ", "The theater loved it and agreed to bump up my fee to cover the hand done type. ", "Moral of the story, if you happen to get an idea or two shot down, keep after it and find a way to make it your own. ", "If I had not decided to tackle the lettering myself, I doubt I would be feeling so good about the final result. ", "Who knows, maybe I will get more chances to do hand lettering after this.", "\n\nWednesday, June 4, 2014\n\nI just finished up the illustration for the Summerbelle fruit carton. ", "It's always fun to work with John Ball at BDG. ", "I really enjoyed the process and it turned out to be one of my favorite jobs of the year so far with virtually no changes. ", "I have to reinforce the fact that I followed all the steps that I have taught to students in the past which basically follows the basic recipe for illustration success. ", "I have outlined it before, but it bears repeating:\n\n1-Think. ", "Get the idea right in your mind before you start, so you have a target you are shooting for. ", "Think of possible alternate solutions and list them. ", "On this job, it was pretty clear what the intention was, but I brought the idea of the umbrella to the solution which ultimately was what worked best.", "\n\n2- Thumbnail Sketches. ", "This is the most basic form of ideation on paper. ", "Don't just chicken scratch out something vague and indecipherable. ", "Take some care to get proportions, shape relationships, angles, POV, and value patterns defined at this stage. ", "No details yet, but the \"skeleton\" is established. ", "I made sure the angles of the pose were as dynamic as possible given the parameters of the space.", "\n\nFinal Summerbelle Art by Greg Newbold with graphic design by John Ball\n\n3- Reference Gathering. ", "In this case, I went to the trouble of renting a real costume and props from a professional theater resource. ", "It cost a little more, but this was a good paying project and it was worth the added expense. ", "I posed my model based on my thumbnail ideas and took lots of photos. ", "I gave myself 2-3 alternate poses as well in case the client did not like what I proposed. ", "I submitted two versions for consideration and the umbrella pose won out. ", "Having great photo reference saved endless headaches and guessing.", "\n\n4- Final Drawing. ", "I draw my final drawings by hand despite the fact that this project and most of my illustration work is painted digitally. ", "This gives me a chance to make adjustments and edits that deviate from the reference material, to stylize and to generally put my personal fingerprints on a piece.", "\n\n5- Value Study. ", "I made sure that my final drawing had the values and shadows established in a monochromatic fashion. ", "My actual physical drawing has a certain amount of value in it and I reinforced it with multiply layer glazes of gray before stepping to color.", "\n\nSummerbelle carton looking sweet with art on three sides\n\n6- Color Study. ", "I work out the color plan before diving in. ", "When I don't, disasters happen. ", "I've been doing this for a long time, so I admit that sometimes this is a mental color study for me. ", "I thought about how I wanted the colors to pop based on the input from the client. ", "The overall desire was that it wanted to fell light and summery. ", "I did a quick Photoshop paint over on top of my sketch at a small size just to reinforce my instincts and then dove into the final art.", "\n\n7- Final Art. ", "At this point, most of the questions had been asked and answered. ", "It was just a matter of getting all the details painted and then making sure the nuances were appealing. ", "For instance, I went back into the shadows of the skin tones to bring some cool light into the up facing planes and warming up the down facing planes or adding reflected color as in the underside of the arm picking up the orange glow of the citrus fruit.", "\n\nThe Story So Far\n\nI have been making art for as long as I can remember. ", "These days I work in acrylic paint or Photoshop when creating illustrations for magazines, children's book publishers and advertising clients. ", "I have even been given some cool awards for it. ", "Lately, I am loving oil painting out of doors and for galleries.", "\n\nPermissions\n\nAll images and content are copyright 2010-2015 Greg Newbold and/or their respective owners. ", "For those wishing to use text or images in any traditional print media or for commercial licensing rights, please email me regarding permissions.", "\n\nIf you want to quote this site for use on a non-commercial blog, website, or Facebook page, please feel free as long as you give a credit and link back. ", "Students may also quote text or reference images in their school reports." ]

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[ "Photo: Spencer Platt/Getty\n\nThere are many dirty jobs in America, and not all of them can be handled by television personality Mike Rowe. ", "Some jobs are so toxic and soul-corroding that a shift minding the shit-chewing machine sounds like a golden opportunity by comparison. ", "United Airlines—one of 2017’s top trending brands—is now advertising just such a position as a “brand public relations manager.”", "\n\n\nAs everyone who hasn’t recently suffered head trauma on a United Airlines flight will remember, the company is in a pickle following its brutal treatment of a passenger, its heartless non-apology for his “re-accommodation,” and its later attempt to pull the human mask back over its wire face with a re-apology. ", "For the many who watched at home thinking, “Hell, I could run an airline better than these clowns,” here’s your chance!", "\n\nIf joining us in building a forward-thinking, path-blazing, world class communications team sounds intriguing—let’s talk. ", "Being part of big, transformative moments like this are rare—so, if you have the courage, the creativity, the conviction, the agility, the leadership and most importantly the passion to help United reclaim its reputation as one of the world’s great and resilient companies we hope you’ll consider embarking on this journey with us.", "\n\n\nYes, it could be you helping United blaze a path toward a future where everybody forgets what they did to that one guy and how their role in a great, resilient oligopoly will make flying suck forever. ", "This is a big, transformative moment: It’s your chance to help United reclaim its reputation as a company that is merely “Bill Lumbergh from Office Space evil” and not “Omni Consumer Products evil.”", "\n\n\n\nAnyway, if you’re a rockstar brand manager who eats communication plans for breakfast and shits them into the mouths of media contacts after lunch, check this job out. ", "Just make sure you’ve got the qualifications they’re looking for:\n\n* Bachelor’s degree in communications, marketing, or journalism * 5-7 years of public relations experience * Willingness to hose remains of predecessor off work station * Access to the Men In Black devices that make people forget videos they saw * Ideas to replace our old slogan, “Move Fast And Break People” * Proficiency with Excel and InDesign\n\n[Note: Fusion, like The A.V. Club, is owned by Univision Communications.]", "\n\n\n\n[via Live And Let’s Fly]" ]

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[ "Breast Lift\n\nA breast lift, or mastopexy, can reverse the effects of pregnancy, gravity, aging and weight change, all of which can take their toll on the shape and position of a woman’s breasts. ", "Over time, the skin loses its elasticity while breast volume may decrease, causing breasts to sag and take on a droopy shape. ", "A breast lift surgically removes excess skin and repositions the breasts for a firmer and more youthful appearance.", "\n\nA board-certified plastic surgeon, Dr. Sugene Kim is recognized as one of the top female cosmetic surgeons in the greater Houston area and a breast lift specialist. ", "Dr. Kim and the team at SGK Plastic Surgery have helped people from Spring to Conroe and all the way up to College Station bring out the best in their body with both surgical and nonsurgical cosmetic procedures.", "\n\nContact us online or call our practice today at 281-363-4546 to schedule your breast lift consultation at SGK Plastic Surgery, conveniently located in The Woodlands near Grand Central Park.", "\n\nWho Should Consider a Breast Lift?", "\n\nWomen who may benefit from a mastopexy procedure usually have some degree of ptosis, or breast sagging, characterized by nipples that point downward. ", "Many candidates for breast lift surgery have loose breast skin or breasts that appear droopy. ", "While women with naturaly moderate to large breasts tend to experience the greatest degree of sagging, even women with small breasts can experience ptosis.", "\n\nSome of the most common candidates for a breast lift are women who have been pregnant, nursed, lost weight or simply have noticed the effects of gravity on their breasts. ", "Many women choose to combine a breast lift with breast augmentation to restore the position of their breasts while also increasing volume. ", "Many women also include a breast lift in their mommy makeover.", "\n\nBreast Lift Surgery Techniques\n\nTechniques used in breast lift procedures vary depending on breast size and the severity of drooping. ", "There are three common incision types used during mastopexy:\n\nThe anchor incision is the most common breast lift incision. ", "This incision circles the areola, travels vertically down to the crease at the bottom of the breast and follows the curve of the breast on either side. ", "While this technique is the most involved, it also provides the most lift and can be highly effective for women with large breasts and a considerable degree of ptosis.", "\n\nThe lollipop incision circles the areola and travels vertically down to the crease at the bottom of the breast, ending there.", "\n\nThe circumareolar incision circles around the areola and does not scar the rest of the breast. ", "While this technique can leave the breasts virtually scarless, it is usually only effective for women with a mild degree of sagging.", "\n\nIn addition to tightening the breast skin and excising any extra, Dr. Kim also repositions the nipples to a natural central location and can resize the areolas as needed to provide the look you desire. ", "During your consultation with Dr. Kim, she can explain the incision types used for breast lift surgery in greater detail, as well as give you an idea of the best approach for your needs. ", "If you wish to resolve asymmetry of your breasts, she may recommend a customized breast augmentation or breast reduction in combination with the lift in order to create the most natural-looking results.", "\n\nWhy Choose Dr. Kim for Your Breast Lift\n\nDr. Sugene Kim is double board-certified and experienced with breast lift surgery. ", "She will listen to your expectations and evaluate your physical needs to determine the surgery techniques that are most appropriate for you. ", "She can also make recommendations for other procedures or treatments that can complement your breast lift results and help you meet your broader appearance goals.", "\n\nDuring your initial one-on-one consultation with Dr. Kim, she can determine whether you might be a good candidate for breast lift surgery. ", "She can also help you better understand the results you can expect after your procedure.", "\n\nTo learn more about breast lifts at SGK Plastic Surgery, contact us online or call our practice in The Woodlands today at 281-363-4546 to schedule your consultation." ]

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[ "The burning keeps rising\n\nMy first chart shows the amount of global energy produced by burning fossil fuels each year. ", "As you can clearly see, humanity just keeps burning more and more.", "\n\nIf this seems surprising to you, it could be because the BP report doesn't have any charts or tables showing total fossil fuel use. ", "To create this and the other charts for this article, I had to download and combine separate spreadsheets for oil, gas and coal.", "\n\nBack in 1992, the world gathered at the Rio Summit promising to prevent fossil fuel pollution from unleashing a full-blown climate crisis onto future generations. ", "As the chart shows, humans burned 7.1 billion \"tonnes of oil equivalent\" in fossil fuels per year back then.", "\n\nNote: A \"tonne of oil equivalent\" (toe) is an energy metric equal to ~42 gigajoules that BP and others use to compare different sources of energy. ", "An \"Mtoe\" is a million \"toe\" and a \"Gtoe\" is a billion \"toe.\"", "\n\nSince that original global climate conference, there have been 24 more, as shown on the chart. ", "And yet, as the chart also makes clear, the amount of fossil fuels burned each year hasn't gone down. ", "It hasn't levelled off. ", "It has only surged upward.", "\n\nLast year, the fossil fuel pyre jumped to another record — 11.7 Gtoe. ", "Both oil and gas burning also set new records. ", "Even coal burning, which fell slightly for a few years, has started rising again. ", "Its now just two per cent below its all-time high. ", "To paraphrase Mark Twain: rumours of its death have been greatly exaggerated.", "\n\nIn total, humanity now burns two-thirds more fossil fuels than in 1990.", "\n\nDo you see any hint of \"climate progress\" in that data? ", "I sure don't. ", "I see the very opposite.", "\n\nAs Greta Thunberg, the teenage Swedish climate activist leading the school strikes, recently said: \"We are facing a disaster of unspoken sufferings for enormous amounts of people... We have to acknowledge that the older generations have failed... it is now time for civil disobedience. ", "It is time to rebel.\"", "\n\nHalf since 1990\n\nTo appreciate just how extreme the burning has been in recent decades, it helps to see it in the greater context. ", "My next chart shows just that. ", "It shows cumulative global fossil fuel burning since humans started doing it in earnest with the industrial revolution.", "\n\nThe solid black part of the line shows that half of all fossil fuels burned in human history have been burned since 1990. ", "This has occurred during the \"climate crisis\" years in which adults understood the threat it posed, and repeatedly promised to do something about it.", "\n\nIn fact, more than 80 per cent of all fossil fuel burning has occurred during the lifetime of the \"baby boomer\" generation. ", "That's my generation. ", "We got to enjoy the amazing benefits a stable climate provides.", "\n\nInstead of safeguarding this heritage to pass along to future generations, we've set it on fire. ", "And each year, we make the pyre even bigger.", "\n\nWhat we are passing down to younger generations is a turbocharged climate Hydra which is starting to lash out at humans and the ecosystems we depend on.", "\n\nIn the Greek myth, Hercules fought the Hydra. ", "In our modern reality, we are feeding ours ever-greater amounts each year. ", "The serpents of drought, heat waves, floods, rising seas, super storms, wildfires, acidifying oceans and eco-collapse keep growing more powerful and aggressive.", "\n\nRenewables, wherefore art thou?", "\n\nWhat about renewable energy?", "\n\nOne of the great climate hopes has been that rising amounts of renewables and nuclear energy would cause fossil fuel burning to start falling.", "\n\nMy next chart lets you see what happened instead. ", "The green dotted line shows how much climate-safe energy was used each year. ", "This line includes the BP data for renewables (like hydro, solar and wind) as well as nuclear energy.", "\n\nClimate-damaging energy use is not just rising. ", "It is pulling away from the climate-safe alternatives.", "\n\nFossil fuel's lead has widened from 6.1 Gtoe in 1990 to 9.6 Gtoe today.", "\n\nIn practical terms, this means that each year, the amount of clean energy that must be added to replace fossil burning keeps getting larger and larger — not smaller.", "\n\nIn a Greek myth symbolizing futility, Sisyphus kept moving a boulder to the top of a hill only to have it roll down again, forcing him to do it over and over.", "\n\nHumanity is stuck in the same endless cycle, trying to build enough new clean energy to replace dirty energy. ", "Except in our modern version, the size of the clean-energy boulder that needs to be shouldered keeps getting larger each time.", "\n\nDespite the task growing ever-larger, Bloomberg just reported that worldwide investments in clean-energy projects have fallen yet again, to a six-year low. ", "Clean-energy investments declined in all three major markets — the U.S., Europe and China.", "\n\nAt this point, it is just another form of climate denial to expect clean energy to force fossil fuel burning to fall — let alone fall all the way down to zero as required to avoid a climate crisis.", "\n\nYear by year\n\nMy next chart shows the annual changes in global energy use.", "\n\nAs the black bars show, fossil fuel burning has increased in 27 of the past 28 years.", "\n\nThe only outlier year was 2009, and that was caused by a sudden global recession. ", "The very next year, however, fossil burn rebounded with the biggest increase in human history.", "\n\nFor a few years after this, the annual increases in fossil burning were getting smaller. ", "Hope flickered briefly that the fossil burn might stop increasing. ", "Adding to this hope was the global Paris Climate Agreement in late 2015, in which the entire world promised to increase climate ambition.", "\n\nUnfortunately, as the red arrow on the chart makes clear, as soon as the ink dried on the agreement, the world reversed course and started increasing the increases in fossil fuel burning.", "\n\nAnd last year, humanity's sprint away from climate safety accelerated yet again as the global fossil burn jumped to fill two-thirds of new energy demand.", "\n\nRenewables, in contrast, stumbled last year. ", "According to the BP report summary, in 2018, \"all fuels grew faster than their 10-year averages, apart from renewables.\"", "\n\nWhen down really means up\n\nThe orange line shows fossil fuel's share of global energy use. ", "It fell three percentage points since 1990. ", "However, the black line shows that the actual amount of fossil fuels burned rose dramatically. ", "The mathematical reason why these both could be true is that global energy use expanded faster than fossil did. ", "As a result, the larger amount of fossil took up a slightly smaller share of the even larger energy pie.", "\n\nUnfortunately, the climate doesn't care what share of the energy pie fossil fuels have. ", "The climate only reacts to the total amount of fossil fuels we burn — that's the black line on the chart. ", "And as long as that black line is rising, we are accelerating away from climate safety.", "\n\nUnfortunately, we are still headed in the opposite direction. ", "The dotted lines on the chart show where we will end up if we continue with the global energy trends since 2010. ", "Fossil burning will continue rocketing upward, even as its share keeps falling.", "\n\nWho cut back? ", "Who burned more?", "\n\nSo far, we've looked only at data for the world as a whole. ", "Next, I'll drill down into the BP data to show what Canada and some other countries have done.", "\n\nThere are 200 countries, so to keep the charts manageable, I'll focus on just the G20 countries, which includes the European Union (EU). ", "Collectively, these countries account for roughly 90 per cent of global GDP and fossil burn.", "\n\nAmount burned per capita\n\nMy first G20 chart shows fossil fuels burned per capita last year.", "\n\nIndia burns the least per person, at less than half the global average.", "\n\nThe Chinese and Europeans are in the middle of the pack. ", "They all burn more than the global average.", "\n\nCanadians and Americans are at the extreme end — burning four times the global average. ", "Only the Saudis, in dead last, burn more than we do.", "\n\nBack in 1990, Canadians burned much less than Americans, per person. ", "Now, we burn more than they do.", "\n\nChanges per country\n\nMy second G20 chart ranks countries by the amount they have changed their fossil burning since 1990.", "\n\nAt the top, in green, are countries that reduced their fossil fuel use. ", "These are the major European economies and Russia.", "\n\nCanada is in the middle of the pack, in red. ", "We increased our annual burn by 57 Mtoe.", "\n\nAt the bottom of this chart, in orange, are the world's three most populous countries. ", "Combined, these three accounted for two-thirds of the global increase in fossil fuel burning. ", "China alone accounted for around half the global increase.", "\n\nChange per person\n\nMy next chart shows the same country changes — but this time divided by their current population.", "\n\nThis lets you compare what has happened on a person-to-person basis.", "\n\nAs you would expect, the biggest shifts in the rankings are for the world's most populous countries. ", "China moves off the bottom, but still has an increase far above the global average. ", "The U.S. moves to the middle. ", "And India moves even higher.", "\n\nCanada falls toward the bottom.", "\n\nCanada's much larger increase than the U.S. since 1990 explains why we've now passed them up in the amount we burn per capita.", "\n\nOh, Canada\n\nLet's drill down one more level into the BP data to get a more detailed view of Canada's fossil fuel burning history.", "\n\nThe resulting chart on the right shows the amount of oil, gas and coal we've burned each year since 1965.", "\n\nThe first thing that jumps out is that Canada just keeps burning more and more as the years go by.", "\n\nLast year, we set an all-time record by burning 219 Mtoe worth of fossil fuels.", "\n\nCanada's coal burning has fallen to less than half of what it was at its peak in 2002. ", "But all that climate progress has been more than wiped out by surging oil and gas burning. ", "Fossil oil burn has grown by a third since 1990 and fossil gas burning has nearly doubled. ", "Both are still rising, driving total fossil burn to record heights.", "\n\nWhat now?", "\n\nWhere could Canadians look for ideas to reduce our fossil burn?", "\n\nMy final chart compares Canada's fossil burn to those from three of our closest allies. ", "It shows changes since 1990.", "\n\nThe U.S. and the EU are the world's two biggest economies. ", "Fossil fuel burning in both appears to have peaked more than a decade ago.", "\n\nIf they can stop increasing fossil fuel burning, surely Canada can as well.", "\n\nThe Europeans, in particular, have made significant cuts in fossil fuel use since peaking. ", "We could adopt some of the climate policies that are working for them.", "\n\nCanadian politicians like to say that we are world leaders in the climate fight.", "\n\nThe energy data in this BP report shows the opposite. ", "It shows that Canadians burn more fossil fuels per capita than almost any other country. ", "And it shows that instead of cutting back as required to prevent a full-blown climate crisis, we just keep burning more and more.", "\n\nIf Canadians want to pass along a safe and sane climate system to our future generations, we will need to quickly change course. ", "Many of our peer countries peaked fossil burning years ago and have policies we could adopt.", "\n\nHomo sapiens claim to be an intelligent species. ", "Maybe so, but, we are certainly not very smart. ", "We are the main driver behind our own likely extinction as a species. ", "Greed continues to dominate common sense as we focus on quarterly income statements, balance sheets, corporate bonuses and consumerism as mainstays of our daily activities.", "\n\nThere is a price to be paid for climate change. ", "Who will pay it? ", "Our grandchildren? ", "Our children? ", "Us? ", "or all three?", "\n\nNot us. ", "Our children & grandchildren.", "\nIt is up to us to rally. ", "Pressure politicians.", "\nChange our driving and transportation habits, quit pointing fingers and talk about it to everyone to the extent the government makes a war against the climate crisis.", "\n\nWe've been paying for it for years now ... it's just that absent living where the climate extremes have triggered significant property damage, it's been fairly easy for people to not notice it.", "\nA/C, auto-pay utilities bills, sufficient income for necessities, etc. ", "make all those incremental changes over the years harder to notice.", "\n\nAs has been pointed out numerous times (mostly by Elizabeth May), most of Canada's carbon footprint is generated by a handful of heavy producers in the oil/gas, cement, and, and transportation industries, and as such, a concentrated legislative focus on these heavy hitters could easily make major inroads in our carbon output as a country. ", "Obviously, approving and paying for new pipelines such as TransMountain, which only expand Canadian oil/gas activity, is a hugely regressive policy.", "\nClearly though, most of the fossil fuel consumption increase is coming from the growing economies of India, and especially China, and that has to stop. ", "Foot-dragging by the US, being the world's largest economy, also has to be a major area of focus.", "\nWhile Canada has historically had a big effect on world affairs as a progressive example, we certainly haven't been that on climate change, and that needs to change. ", "A good place to start making that change is with the next federal election.", "\n\nCould you break down that horrific figure for Canadian fossil fuel burning? ", "How much comes from: Heating? ", "Transportation? ", "Industry? ", "Tar sands extraction? ", "And does it include emissions from the customers for tar sands bitumen?", "\n\nSelling my car is not an option for me or most Canadians but a small but important step should be incentives to switch to electric transportation - personal and commercial use - cars, trucks, busses, aircraft etc.", "\n\nI've heard lots of people say they absolutely need a car, with no elaboration. ", "And yet, millions of Canadians live without one, in both urban and rural areas.", "\nAnd many also say they *must* have a recent-year vehicle ... not because they plan on keeping it for a long time, or because of better gas mileage or lower manufacturing outputs, but usually, when pressed, it comes down to they feel it is an entitlement of the economic status they've attained.", "\nWhen push comes to shove, almost all of the increase since 2005 comes from the tar sands.", "\nCheck out the whole series of articles by Barry Saxifrage. ", "He has produced some incredibly enlightening diagrams!", "\n\nLooks like Canadians don't get it, why we are not sure, the last time they made sacrifices was WW 2, rationing, growing gardens no overspending on personal items, is that what is needed to get people out of SUVs, get the politicians out of jet planes, elect someone who has a grip on reality. ", "Pretty sad.", "\n\nThere is little indication that the Liberals or Conservatives will address the carbon footprint issue, head on. ", "Could/should the ‘ lesser’ political parties ,with a creative climate action plan, gain sufficient electoral support to upset our political apple cart ?", "\n\nA somewhat similar climate incident occurred in the 1980’s, when acid rain was created by electric power generating plants that burned high sulphur coal… after much debate scientists were able to convince political leaders to switch to ‘cleaner’ natural gas. ", "Remember call in earlier days, when coal was the major heating energy source…. ", "Guess who saved us then…a cousin of the fossil fuel industry…the petroleum industry ! ", "I remember how happy we were when the old wood/coal furnace was retired and new natural gas furnace was installed ! ", "Overnight the air became cleaner! ", "Nature has a way of adjusting to interference…but unfortunately we have reached the limit now because of the overloading the atmosphere carbon dioxide…\nThe petroleum industry, along with much research, has left us where you cannot eat, wear, or see anything that is not at least partly created with petroleum derivatives….! ", "Now let the industry work with governments to help fix the carbon footprint …! ", "New ‘energy ‘concepts are appearing …what about Small Modular Reactors(SMR), modern nuclear power ,ITER ,wind ,solar .geothermal, retrofitting…Opportunities galore for everyone ! ", "Is ‘oil’ up for the challenge or have they chosen the route of the long forgotten buggy whip industry ?", "\nWhat’s required is a challenge to the many technical groups that reside in universities, Government and industry.", "\nWith much luck, planning, sacrifice ,hard work ,we managed to win WW2…This Climate War seeks another FDR, Winston to lead us …now !", "\n\nWhat you say is true. ", "However for us to curtail our oil exports from this country does nothing for anyone! ", "Quite the contrary. ", "If we do not fill the market with our product, someone else will. ", "We have a market opportunity on the Trans Mountain west improvement giving us access to China, Indonesia and even Australia. ", "Shipping from the east coast adds too much expense we wouldn't be competitive. ", "If we do not ship; someone else will!", "\n\nTo quote my mother, when her kids were teenagers:\n\"I see. ", "And if everyone else jumps in the lake, will you do it too?\"", "\nThe truth of the matter is that doing the right thing does not depend upon anyone else!", "\nAnd if someone else sells the oil, that's not on us, now, is it?", "\nThe truth of the matter is that most of the population has been making changes (aka \"sacrifices\") for decades now, and the petro industry wipes out all the gains we've made. ", "Why should the 85% of us in non-tar-sands provinces, and the significant numbers of the 15% in the \"have tar\" provinces put up with the bogus reasoning that it's worth burning the entire planet, to the extent that we are justified in producing/burning 4X the per capita share of the rest of the world ... and all so that a few people draw unreasonable pay in between when their towns burn down, so that shareholders get mighty dividends, and executives get huge paycheques and bonuses ... and oh yeah, so that a few politicians get big cheques for their re-election warchests?", "\nNo!!!", "\nJust ... *NO* Enough is enough, and too much is too much! (", "That's also quoting my late mother, in the same era) ... and we're already far past too much.", "\nPlease quit spouting bogus and irrelevent \"logic\" that a 10 year old in my day would realize was faulty!!!", "\nWe're better than that, and smarter than that. ", "And we care more about our kids and grandkids than that, and the people in the rest of the world. ", "Not to mention the rapidly reducing numbers of species.", "\nI'm not going crazy. ", "I'm getting really Pee-Oh'ed.", "\n\nIsn't that a bit like saying if we don't rape the girls in Mexico, someone else will?", "\nGet a grip, please!!!", "\nThere are people who sell all their stuff and give away all their money, and take up life as a mendicant.", "\nAny takers for that one?", "\n\nJohn Clark - whoever you are and whatever you represent - your comment now ranks with those of the criminally egregiously ignorant and blinkered apologists for the poison that is killing the earth. ", "If we don't do it, someone else will? ", "What a piece of schoolyard \"logic\". ", "Even China and India are thinking about abandoning fossil fuels. ", "Their respective pollution indices are through the roof and their people are increasingly crippled by it. ", "Except for the monsoon season India is blessed/cursed? ", "with abundant solar power. ", "China's land mass can support an untold amount of solar and wind energy. ", "Either we change course, or we exterminate ouselves and most other life on earth. ", "Do we really have a choice?", "\n\nAllow me to name two elephants in the room. ", "First, all economic value creation requires energy, so the current economic model's requirement for continual exponential growth is a big part of the problem (and the reason the only drop in 28 years was the global recession). ", "Second, there are simply too many humans for the planet's ecosystem to support but no one will grapple with this issue because of reproductive rights and fewer humans would consume less (and therefore hurt economic growth). ", "So our leaders are trying to figure out how to reduce the main drivers of economic growth using an economic model that demands exponential economic growth -- time to change the economic model to one that serves humanity better (read Donut Economics).", "\n\nA final observation: renewable energy fights climate change if and only if it results in fossil fuels staying in the ground. ", "That is why the responsible thing to do is to stop all extraction from the tar sands as one of the world's dirtiest sources of fossil fuels.", "\n\nThe last paragraph is utterly true. ", "The two elephants are fallacies.", "\nThere *is* such a thing as sustainability ... and the world's population has been levelling out in burps, fits, starts and bumps since WW2.", "\nGoogle Hans Rosling's great presentations on the matter on TouTube.", "\nThe only problem with a sustainable economy is that it doesn't provide massive profits for some, and not enough to live on for the rest. ", "That is the real elephant.", "\n\nWe're taking too much, mining all of the Earth's resources, living and non-living. ", "The bioproductivity of the planet is less than 3% per year, yet successful industries and investments demand double-digit annual returns. ", "This \"free lunch\", aka bubble, means using, for example, much more fresh water than Nature can provide and clean up for us each year. ", "We pumping acquifers dry, using up the rapidly melting glaciers, and getting buckets of rain from the heated, saturated atmosphere that runs off over-paved cities and takes away agricultural soil that's the wind is already depleting alarmingly every year. ", "And that's just one for-instance.", "\n\nWhich is to say, the crisis isn't, at its roots, about carbon -- i.e. if we lower carbon use to reduce Earth's fever, then we can carry on taking too much with renewable energy. ", "Fossil fuels just grease the wheels of our greed and trashing. ", "The question is: which is stupider, to outstrip Nature's ability to provide and replenish using dirty fuels, or keep taking way too much with clean, green energy?", "\n\nThe beauty of the dirty, climate-changing stuff is that it creates the fever, which gives us a powerful clue that some underlying thing we're doing is wrong, like the fever that accompanies a serious disease or injury, telling us that bodily systems are under attack. ", "We don't just lower the overheating person or animal and consider the infection or injury solved.", "\n\nIt's strangely deluded to think that addressing the planet's fever will fix, in any meaningful way, what's causing the problem. ", "We can't continue to operate on a business model that takes profits by digging into core capital.", "\n\nWe have to figure out what's enough and live within Earth's means. ", "We need to factor into every decision, individually and connectively, the cost to the Earth. ", "We need guidance in this, which research, governments, and educators can provide. ", "The McKinlay family motto & crest that I grew up with, \"Not Too Much\", has all the salvation in it we need, and it's a beautiful thing, full of satisfaction with enough.", "\n\nNot likely? ", "Well, then, get used to what General Dick Little of Frank [satire] Magazine used to say: \"It was us against the environment, and we won.\"" ]

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[ "Q:\n\nHow to make radio buttons with same value reusable?", "\n\nI am a beginner to Vue, and learning something by doing. ", "I was able to make a checkbox reusable, but getting some weird result for radio buttons.", "\nI have the data in an array format in ProgramDesign.vue:\ndata() {\n return {\n strategies: [\n \"Not Important\",\n \"Slightly Important\",\n \"Moderately Important\",\n \"Very Important\",\n \"Extremely Important\",\n ],\n };\n},\n\nThese are the options that get repeated on every question.", "\nI made a separate component for the radio like this:\n<template>\n <div>\n <span v-for=\"strategy in groups\" :key=\"strategy\">\n <input :id=\"strategy\" class=\"radio-style\" name=\"strategy\" type=\"radio\" />\n <label :for=\"strategy\" class=\"radio-style-3-label\">{{strategy}}</label>\n </span>\n </div>\n</template>\n<script>\nexport default {\n props: {\n groups: Array,\n },\n};\n</script>\n\nThis is how it's used in ProgramDesign.vue:\n<p>first question goes here ?", "</p>\n<RadioButton :groups=\"strategies\" />\n<div class=\"line\"></div>\n<p>second question goes here ?", "</p>\n<RadioButton :groups=\"strategies\" />\n\nI was able to get the reusable output, but when I click on the radio button for the second question, the buttons for the first question get selected. ", "How can I fix this?", "\n\nA:\n\nThe problem is the input IDs and names are not unique between component instances, as can be seen in the rendering of your two RadioButton components (simplified for brevity):\n<!-- ", "RadioButton 1 -->\n<div>\n <span>\n <input id=\"Not Important\" name=\"strategy\" type=\"radio\">\n <label for=\"Not Important\">Not Important</label>\n </span>\n</div>\n\n<!-- ", "RadioButton 2 -->\n<div>\n <span>\n <input id=\"Not Important\"❌ name=\"strategy\"❌ type=\"radio\">\n <label for=\"Not Important\">Not Important</label>\n </span>\n</div>\n\nEach label is linked to an input by matching the for and id attributes, such that clicking the label causes the linked radio input to change values. ", "When there are multiple inputs with the same identifier, the browser links the label to the first matching input, causing the behavior you observed.", "\nThe name must also be unique between groups (RadioButton instances), since the browser creates radio groups of inputs that have matching names.", "\nSolution\nAlternatively, a label and input can be linked by putting the input inside the label, resolving the id/for duplication (and improving readability):\n<label>\n <input name=\"strategy\" type=\"radio\">\n Not Important\n</label>\n\nAnd one way to resolve the duplicate names is to base the name on a counter incremented per instance:\n<template>\n <div>\n <label v-for=\"strategy in groups\" :key=\"strategy\">\n <input :name=\"'strategy' + groupId\" type=\"radio\">\n {{strategy}}\n </label>\n </div>\n</template>\n\n<script>\nlet groupId = 0\n\nexport default {\n props: {\n groups: Array\n },\n data() {\n return {\n groupId: groupId++\n }\n }\n}\n</script>\n\n" ]

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[ "[workspace]\nmembers = [\n # Main storage database\n \"utils/alexandria\",\n\n # main qaul.net service library\n \"libqaul\",\n\n # libqaul additions\n \"libqaul/http\",\n \"libqaul/ipc\",\n \"libqaul/rpc\",\n #\"libqaul/ws\",\n\n # qaul.net main services\n \"libqaul/service/chat\",\n \"libqaul/service/files\",\n \"libqaul/service/voice\",\n\n # decentralised routing protocol\n \"ratman\",\n \"ratman/harness\",\n \"ratman/identity\",\n \"ratman/netmod\",\n \"ratman/configure\",\n\n # various utility crates\n \"utils/async-notify\",\n \"utils/clockctrl\",\n \"utils/permute\",\n \"utils/visn\",\n \n # available netmod drivers\n \"netmods/netmod-mem\",\n \"netmods/netmod-tcp\",\n \"netmods/netmod-udp\",\n \"netmods/netmod-wd\",\n\n # android build support\n \"utils/android-support\",\n\n # android logging support\n \"utils/android-tracing\",\n\n # test binaries\n \"clients/multinode-test\",\n \"clients/linux-http-test\",\n \"clients/linux-voice-test\",\n \"clients/mini-tcp\",\n\n # Actual target binaries\n \"clients/linux-cli\",\n \"clients/hubd\",\n]\n" ]

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[ "Do you know the worst Transformers game of all time? ", "Arguably it’s Transformers: Mystery of Convoy. (", "The direct translation of the title is: Fight! ", "Super Robot Lifeform Transformers: Mystery of Convoy.) ", "This 1986 cartridge game was created exclusively for the Japanese Nintendo Famicom system. ", "It’s a game so bad they misspelled “Convoy”.", "\n\nSo why does this game exist? ", "Because the 1986 the animated Transformers movie didn’t get released in Japan until 1990. ", "Fans watching The Transformers cartoon went from season 2 to season 3 without a movie in between. ", "This left a question with Japaneses Transformers fans. ", "What the hell happened to Optimus Prime?", "\n\nTo capitalize on this, Transformers: Mystery of Convoy, was made to answer that question. ", "The game’s title “Convoy” being Optimus Prime and mystery of who killed him.", "\n\nMystery of Convoy is a 2D platformer game consisting of 10 notoriously difficult levels. ", "You play as Ultra Magnus, firing your short range gun at waves of enemies.", "\n\nI’ve read about people who’ve played this game. ", "All express their frustration over how difficult it is. ", "Sluggish controls, poor hit mechanics, and an insane number of enemies are just some of the reasons this game sucks.", "\n\nThe game has a few secrets and four possible endings. ", "Bumblebee is in the game to help you skip levels, but only if you kill special red enemy jets that appear at random. ", "If you collect hidden letters to spell “RODIMUS” you get one of two good ending screens. ", "The first telling you that Rodimus Prime will continue the investigation. ", "Now you can play the whole game again as Rodimus Prime. ", "If you beat the game as Rodimus Prime, you are shown the second good ending.", "\n\nThe funny thing is, I can’t find anything about this game that actually answers the question of what happened to Opimtus Prime. ", "Do you know? ", "Leave a comment below.", "\n\nGallery\n\nNintendo Famicom System Nintendo Famicom System Flyer: Front Flyer: Back Stage 1: Wilderness Zone Stage 1: Cybertron Moon Base Stage 2: Temple Ruins Stage 2: Two Cybertron Moon Bases Stage 3: Time Tunnel Stage 3: Decepticon Insignia A Stage 4: Subspace City Stage 4: Decepticon War Ship Nemesis Stage 5: Floating City Stage 5: Gestalt Menasor Stage 6: Time Tunnel 2 Stage 6: Decepticon Insignia B Stage 7: Glacial Zone Stage 7: Gestalt Bruticus Stage 8: Time Tunnel 3 Stage 8: Decepticon Insignia C The Secret Battle Secret Battle Hint Stage 9: Underground Machine City Stage 9: Gigantic Megatron Stage 10: Time Tunnel 4 Stage 10: Trypticon 1st Good Ending Screen 2nd Good Ending screen: Rodimus Prime Rodimus Prime Rodimus Prime Sprites. ", "Bumblebee on the bottom right?", "\n\nUseful Links" ]

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[ "#\n# Copyright 2020 Red Hat, Inc. and/or its affiliates\n# and other contributors as indicated by the @author tags.", "\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.", "\n# You may obtain a copy of the License at\n#\n# http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.", "\n# See the License for the specific language governing permissions and\n# limitations under the License.", "\n#\n\norg.keycloak.connections.jpa.", "QuarkusJpaConnectionProviderFactory\n" ]

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[ "Assessing pre-clinical ventricular dysfunction in obese children and adolescents: the value of speckle tracking imaging.", "\nObesity has become a major health problem worldwide. ", "Cardiovascular abnormalities have been described not only in obese adults but also in obese children and adolescents. ", "The aim of the present study was to investigate left and right, systolic and diastolic ventricular dysfunction in obese paediatric patients without comorbidities using 2D speckle tracking longitudinal strain. ", "Doppler echocardiogram was performed on 50 obese children and adolescents with body mass index (BMI) above the 95th percentile (OG) and 46 non-obese sex- and age-matched controls (CG). ", "Systolic and diastolic functions of both ventricles were investigated through conventional Doppler echocardiography. ", "Tissue Doppler imaging (TDI), colour Doppler myocardial imaging (CDMI), and two-dimensional (2D) speckle tracking were also used to analyse ventricular performance in both groups. ", "Left-ventricular (LV) ejection fraction was similar between groups (68.2 ± 6.2 vs. 68.3 ± 5.3, P = 0.931). ", "Left-ventricular diastolic parameters did not differ between groups, except for a lower mitral A wave (61.6 ± 13.0 vs. 51.9 ± 10.0 cm/s, P > 0.001) and higher E/A ratio (1.8 ± 0.5 vs. 2.1 ± 0.4, P = 0.007) in the controls. ", "Left-ventricular global strain was lower in the OG by both methods (CDMI: 22.0 ± 2.8 vs. 24.6 ± 2.7%, P = 0.020; 2D speckle tracking: 18.4 ± 1.6 vs. 20.4 ± 1.7%, P < 0.001). ", "In multivariate analysis, 2D longitudinal global strain correlated negatively with BMI, r = -0.433, p = 0.002. ", "Although EF was not different between the two groups, LV 2D speckle tracking longitudinal strain was lower in the obese group, even in the absence of other comorbidities, indicating that obesity effects on LV function is an early finding in obesity." ]

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[ "Portland timbers reserves Video\n\nHighlights - Timbers Reserves vs. LA Galaxy Reserves This page was last edited on 17 June , at The Timbers staff operates the games as full first-team ones, however, meaning concessions are open. ", "Mexico v Italy Bo5. ", "League Asia Challenge J. Sports teams based in Oregon. ", "Football Change Sport American Football Aussie Rules Baseball Basketball Boxing Cricket Current Affairs Cycling Darts ESports Football Formula 1 Gaelic Games Golf Greyhound Racing Hockey Horse Racing Ice Hockey Motor Sport Politics Rugby League Rugby Union Snooker Speedway Tennis TV and Film Specials UFC and MMA Virtual Sports Winter Sports. ", "Williams, Augustine Augustine Williams. ", "Projekt und Interface von RahU. Startseite Jetzt Live Teamvergleich Gratis Wetten Kontakt Werbung. ", "In T2's first season, the team finished in 8th place in the Western Conference of the league table with 11 wins, 15 losses, and 2 ties. ", "DNQ Website Club website Current season Portland Timbers 2 is an American professional soccer team based in Portland, Oregon , and play in the United Soccer League , in the second tier of the United States Soccer Pyramid. ", "Articles provided by OMNISPORT. ", "Column 1 Stand Together Online Auctions." ]

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[ "Memories of environmental change and local adaptations among molapo farming communities in the Okavango Delta, Botswana-A gender perspective.", "\nThis paper focuses on ways in which three riparian communities (Xobe, Shorobe and Tubu) practising flood recession (molapo) farming along the fringes of the Okavango Delta in Ngamiland District in north-western Botswana, present memories of experiential impacts of and adaptation to key environmental and anthropogenic change events. ", "Participatory methodological tools were used to capture local knowledge of people who had resided in the Okavango wetlands for many years. ", "Findings indicate that key environmental change events were characterized by intergenerational experiences of severe and frequent droughts, floods, and recurrent outbreaks of human and animal disease. ", "These events had impacted livelihoods and well-being of communities. ", "Community adaptation strategies were embedded in local institutions of governance, especially chieftainship and the Kgotla, as legitimate platforms for community re-organization against unpredictable environmental change. ", "We concluded that policy/program formulation processes need to take cognisance of local communities' historical knowledge of environmental change and adaptation. ", "In particular it emerged that men and women, and people of different ages have differentiated memories of historical events which are complementary and necessary in developing a comprehensive adaptation strategy." ]

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[ "Target Phasing Out Amazon’s Kindle\n\nNEW YORK (AP) - Target Corp. is phasing out Amazon.com Inc.'s e-reader Kindle at its more than 1,700 stores and its website.", "\n\nTarget spokeswoman Molly Snyder said Wednesday that the decision to stop selling Kindles this spring came after an \"ongoing review\" of Target's merchandise that evaluates quality and prices of the chain's offerings.", "\n\nThe move coincides with the discounter's plan to create mini shops of Apple Inc. products in 25 of its stores this year.", "\n\nDespite competition from cheaper tablet computers such as Amazon's Kindle Fire, Apple's iPad remains the most popular tablet. ", "Apple has sold more than 55 million iPads since the tablet's debut in 2010.", "\n\nTarget, which is based in Minneapolis, started selling Kindles two years ago. ", "Target announced in late November that Amazon's Kindle was the best-selling tablet in its stores on the day after Thanksgiving, typically the busiest day of the year.", "\n\nTarget will also stop selling Kindle accessories like covers and chargers.", "\n\nSnyder declined to comment further about its partnership with Apple, only saying, \"We will continue to offer our guests a full assortment of e-readers and supporting accessories.\"", "\n\nA spokeswoman for Amazon didn't immediately return calls.", "\n\nTarget's decision to phase out the Kindle is also occurring as the retailer, along with other major merchants, are trying to fight a growing practice called \"showrooming.\" ", "That's when shoppers, armed with smartphones, browse products in physical stores and then shop online for a better price. ", "Earlier this year, Target sent out a letter to vendors asking for help in developing exclusive merchandise and matching rivals' online prices.", "\n\nCopyright Associated Press. ", "All rights reserved. ", "This material may not be published, broadcast, rewritten, or redistributed.", "\n\nPopular Stories in Technology\n\nApple's iPhones and iPads have gotten free software updates, including battery improvements and a smarter virtual assistant. ", "The new features and capabilities in the update, iOS 9, are primarily refinements rather than anything transformative." ]

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[ "package org.gearvrf.widgetlibviewer;\nimport android.graphics.", "Color;\nimport android.graphics.", "PointF;\nimport android.graphics.drawable.", "Icon;\n\nimport org.gearvrf.", "GVRContext;\nimport org.gearvrf.", "GVRMesh;\nimport org.gearvrf.", "GVRRenderData;\nimport org.gearvrf.widgetlib.adapter.", "BaseAdapter;\nimport org.gearvrf.widgetlib.log.", "Log;\nimport org.gearvrf.widgetlib.main.", "GVRBitmapTexture;\nimport org.gearvrf.widgetlib.main.", "WidgetLib;\nimport org.gearvrf.widgetlib.widget.", "GroupWidget;\nimport org.gearvrf.widgetlib.widget.", "ListWidget;\nimport org.gearvrf.widgetlib.widget.", "Widget;\nimport org.gearvrf.widgetlib.widget.basic.", "LightTextWidget;\nimport org.gearvrf.widgetlib.widget.basic.", "TextParams;\nimport org.gearvrf.widgetlib.widget.layout.", "Layout;\nimport org.gearvrf.widgetlib.widget.layout.", "OrientedLayout;\nimport org.gearvrf.widgetlib.widget.layout.basic.", "LinearLayout;\nimport org.gearvrf.widgetlib.widget.properties.", "JSONHelpers;\nimport org.gearvrf.widgetlibviewer.", "GVRWidgetViewer.", "R;\nimport org.json.", "JSONObject;\n\nimport static org.gearvrf.utility.Log.tag;\nimport static org.gearvrf.widgetlib.widget.properties.", "JSONHelpers.getFloat;\nimport static org.gearvrf.widgetlib.widget.properties.", "JSONHelpers.getJSONObject;\nimport static org.gearvrf.widgetlib.widget.properties.", "JSONHelpers.optPointF;\npublic class NotificationsContentScene extends BaseContentScene {\n NotificationsContentScene(GVRContext gvrContext, Widget.", "OnTouchListener homeListener) {\n super(gvrContext);\n mControlBar.addControlListener(\"Home\", homeListener);\n }\n\n private enum Properties {\n list, panel, padding\n }\n\n @Override\n protected Widget createContent() {\n JSONObject properties = WidgetLib.getPropertyManager().getInstanceProperties(getClass(), TAG);\n JSONObject panelProperties = getJSONObject(properties, Properties.panel);\n\n GroupWidget notificationPanel = new GroupWidget(mGvrContext, panelProperties);\n notificationPanel.setTexture(WidgetLib.getTextureHelper().getSolidColorTexture(Color.", "RED));\n // main screen layout\n final float padding = JSONHelpers.getFloat(properties, Properties.padding);\n\n LinearLayout notificationPanelLayout = new LinearLayout();\n notificationPanelLayout.setOrientation(OrientedLayout.", "Orientation.", "VERTICAL);\n notificationPanelLayout.setGravity(LinearLayout.", "Gravity.", "TOP);\n notificationPanelLayout.enableOuterPadding(true);\n notificationPanelLayout.setDividerPadding(padding, Layout.", "Axis.", "Y);\n notificationPanelLayout.setOffset(-0.01f, Layout.", "Axis.", "Z);\n\n //for loading notification_panel_background.fbx, its model is actually the child of the root node.", "\n //Because of this hierarchy thing, we have to group label and list as children of notificationPanel.", "\n //Then place notificationPanel as a sibling of the model\n notificationPanel.applyLayout(notificationPanelLayout);\n\n LinearLayout listLayout = new LinearLayout();\n listLayout.setOrientation(OrientedLayout.", "Orientation.", "VERTICAL);\n listLayout.setGravity(LinearLayout.", "Gravity.", "TOP);\n listLayout.setDividerPadding(padding, Layout.", "Axis.", "Y);\n listLayout.setOffset(-0.01f, Layout.", "Axis.", "Z);\n listLayout.enableClipping(true);\n listLayout.enableOuterPadding(true);\n\n //create List\n JSONObject listProperties = getJSONObject(properties, Properties.list);\n ListWidget list = new ListWidget(mGvrContext, listProperties);\n list.setAdapter(new NotificationAdapter(mGvrContext));\n list.applyLayout(listLayout);\n list.enableClipRegion();\n list.enableTransitionAnimation(true);\n\n notificationPanel.addChild(list);\n\n return notificationPanel;\n }\n\n static class NotificationContentWidget extends GroupWidget {\n private enum Properties {content_with_big_icon}\n private enum ContentProperties {nameNtime, title, text}\n private ResizableLightTextWidget mTitle;\n private ResizableLightTextWidget mText;\n\n private final DisplayParams mBigIconParams;\n\n NotificationContentWidget(GVRContext context) {\n super(context);\n\n JSONObject widgetProperties = getObjectMetadata();\n\n LinearLayout layout = new LinearLayout();\n layout.setOrientation(OrientedLayout.", "Orientation.", "VERTICAL);\n layout.setOffset(-0.01f, Layout.", "Axis.", "Z);\n //layout.setGravity(LinearLayout.", "Gravity.", "CENTER);\n applyLayout(layout);\n\n JSONObject bigIconContentProperties = getJSONObject(widgetProperties, Properties.content_with_big_icon);\n\n mBigIconParams = new DisplayParams(context, bigIconContentProperties);\n\n //mAppNameNTime = new NotificationAppNameNTimeWidget(context);\n mTitle = new ResizableLightTextWidget(context, 1.0f, 1.0f);\n //mText = new ResizableLightTextWidget(context, 1.0f, 1.0f);\n\n\n //mAppNameNTime.setTouchable(false);\n mTitle.setTouchable(false);\n //mText.setTouchable(false);\n //mAppNameNTime.setFocusEnabled(false);\n mTitle.setFocusEnabled(false);\n //mText.setFocusEnabled(false);\n\n //addChild(mAppNameNTime);\n addChild(mTitle);\n //addChild(mText);\n }\n\n public void loadNotificationInfo() {\n\n meshUpdate(mBigIconParams);\n //mAppNameNTime.loadNotificationInfo(notificationInfo);\n //mText.setText(notificationInfo.text);\n\n //Svetlana +++++\n //set mTitle rendering order to TRANSPARENT\n mTitle.setText(\"NotificationTitle\");\n mTitle.setRenderingOrder(GVRRenderData.", "GVRRenderingOrder.", "TRANSPARENT);\n //Svetlana -----\n // mText.setRenderingOrder(GVRRenderData.", "GVRRenderingOrder.", "TRANSPARENT);\n }\n\n private void meshUpdate(DisplayParams displayParams) {\n mTitle.updateMesh(displayParams.title.mesh);\n// mText.updateMesh(displayParams.text.mesh);\n TextParams params = mTitle.getTextParams();\n params.setFromJSON(getGVRContext().getActivity(), displayParams.title.properties);\n mTitle.setTextParams(params);\n\n// params = mText.getTextParams();\n// params.setFromJSON(getGVRContext().getActivity(), displayParams.text.properties);\n // mText.setTextParams(params);\n }\n\n class ResizableLightTextWidget extends LightTextWidget {\n public ResizableLightTextWidget(GVRContext context, float width, float height) {\n super(context, width, height);\n }\n protected void updateMesh(GVRMesh mesh) {\n setMesh(mesh);\n }\n }\n\n\n private static class DisplayParams {\n static class Item {\n final GVRMesh mesh;\n final JSONObject properties;\n final String name;\n\n Item(GVRContext context, JSONObject properties, String name) {\n this.properties = properties;\n this.name = name;\n\n PointF size = optPointF(properties, Widget.Properties.size);\n Log.d(TAG, name + \": (w,h) = \" + size.x + \",\" + size.y);\n mesh = context.createQuad(size.x, size.y);\n }\n }\n\n final Item title;\n final Item text;\n\n DisplayParams(GVRContext context, JSONObject properties) {\n JSONObject titleProperties = getJSONObject(properties, ContentProperties.title);\n JSONObject textProperties = getJSONObject(properties, ContentProperties.text);\n\n title = new Item(context, titleProperties, \"title\");\n text = new Item(context, textProperties, \"text\");\n }\n }\n }\n\n public class NotificationAdapter extends BaseAdapter {\n private GVRContext mGvrContext;\n\n public NotificationAdapter(GVRContext gvrContext) {\n mGvrContext = gvrContext;\n }\n\n @Override\n public int getCount() {\n return 1;\n }\n\n @Override\n public String getItem(int position) {\n return \"NotificationItem\";\n }\n\n public Icon getLargeIcon() {\n return null;\n }\n\n @Override\n public long getItemId(int position) {\n return position;\n }\n\n @Override\n public Widget getView(int position, Widget convertView, GroupWidget parent) {\n NotificationItemGroupWidget itemGroupWidget= new\n NotificationItemGroupWidget(mGvrContext, getLargeIcon());\n\n return itemGroupWidget;\n }\n }\n\n static class NotificationItemWidget extends GroupWidget {\n private static final String TAG = NotificationItemWidget.class.getSimpleName();\n private NotificationContentWidget mContentWidget;\n private Widget mLargeIcon;\n Icon largeIcon;\n\n private enum Properties {large_icon}\n\n public NotificationItemWidget(GVRContext context, Icon icon) {\n super(context);\n largeIcon = icon;\n //NotificationItemWidget has two children, mLargeIcon (in Geometry rendering order), and mContentWidget\n JSONObject properties = getObjectMetadata();\n Log.d(TAG, \"NotificationItemWidget(): properties: %s\", properties);\n LinearLayout layout = new LinearLayout();\n layout.setGravity(LinearLayout.", "Gravity.", "CENTER);\n layout.setOrientation(OrientedLayout.", "Orientation.", "HORIZONTAL);\n layout.setOffset(-0.01f, Layout.", "Axis.", "Z);\n applyLayout(layout);\n\n mContentWidget = new NotificationContentWidget(context);\n mContentWidget.setTouchable(false);\n mContentWidget.setFocusEnabled(false);\n\n JSONObject largeIconProperties = getJSONObject(properties, Properties.large_icon);\n mLargeIcon = new Widget(getGVRContext(), largeIconProperties);\n mLargeIcon.setTouchable(false);\n mLargeIcon.setFocusEnabled(false);\n mLargeIcon.setRenderingOrder(GVRRenderData.", "GVRRenderingOrder.", "GEOMETRY);\n\n addChild(mContentWidget);\n addChild(mLargeIcon);\n\n setTexture(WidgetLib.getTextureHelper().getBitmapTexture(R.drawable.notification_item));\n setRenderingOrder(GVRRenderData.", "GVRRenderingOrder.", "GEOMETRY);\n loadNotificationInfo();\n }\n\n public void loadNotificationInfo() {\n mContentWidget.loadNotificationInfo();\n if (largeIcon !", "= null) {\n GVRBitmapTexture texture = WidgetLib.getTextureHelper().getBitmapTexture(R.drawable.vr_app);\n mLargeIcon.setTexture(texture);\n mLargeIcon.setVisibility(Visibility.", "VISIBLE);\n } else {\n mLargeIcon.setVisibility(Visibility.", "PLACEHOLDER);\n }\n }\n }\n\n static class NotificationItemGroupWidget extends GroupWidget {\n private final NotificationItemWidget mItemWidget;\n private final CancelButton mCancelButton;\n\n private enum Properties {\n cancel\n }\n public NotificationItemGroupWidget(GVRContext context, Icon icon) {\n super(context);\n setName(\"NotificationItemGroupWidget\");\n\n JSONObject properties = getObjectMetadata();\n\n JSONObject cancelProperties = getJSONObject(properties, Properties.cancel);\n Log.d(TAG, \"cancel button: Properties: %s\" + cancelProperties);\n\n LinearLayout layout = new LinearLayout();\n layout.setGravity(LinearLayout.", "Gravity.", "TOP);\n layout.setOrientation(OrientedLayout.", "Orientation.", "HORIZONTAL);\n layout.setDividerPadding(0.25f, Layout.", "Axis.", "X);\n layout.setOffset(-0.01f, Layout.", "Axis.", "Z);\n applyLayout(layout);\n\n mItemWidget = new NotificationItemWidget(context, icon);\n\n mCancelButton = new CancelButton(context, cancelProperties);\n mCancelButton.setTouchable(true);\n\n addChild(mItemWidget);\n addChild(mCancelButton);\n }\n\n private class CancelButton extends Widget {\n String key;\n\n public CancelButton(GVRContext context, float width, float height) {\n super(context, width, height);\n key = \"Cancel\";\n }\n\n public CancelButton(GVRContext context, JSONObject properties) {\n super(context, properties);\n }\n\n public String getkey() {\n return key;\n }\n }\n }\n\n @Override\n public String getName() {\n return TAG;\n }\n\n @Override\n public void onProximityChange(boolean onProximity) {\n if (onProximity) {\n }\n }\n\n private static final String TAG = tag(NotificationsContentScene.class);\n\n}\n" ]

{ "pile_set_name": "Github" }

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[ "[Antibiotic concentration in kidney cell fractions].", "\nA two-compartment model of kidney cells is designed. ", "The first compartment is the cytoplasm, the other is a large organelle fraction consisting of lysosomes and mitochondria. ", "Drugs easily accumulated in the second compartment are considered to be harmful to the kidney. ", "Although small molecular antibiotics move across membranes, they reach a certain equilibrium within 30 minutes. ", "Therefore reproducible values of antibiotic concentrations in both fractions can be obtained. ", "The two-compartment model provides a new approach to pharmacokinetic study at a subcellular level." ]

{ "pile_set_name": "PubMed Abstracts" }

[ 0, 0, 0, 0, 0, 0, 0 ]

[ "Treatment of liver metastases from uveal melanoma.", "\nUveal melanoma patients with liver metastases have a poor prognosis. ", "The effect of screening and multimodality treatment (including surgery) should be evaluated. ", "A total of 602 patients treated for uveal melanoma during a 14-year period had abdominal ultrasonography screening every 6 months. ", "Sixty-three developed liver metastases as the first extraocular metastatic site. ", "When possible, liver surgery and intra-arterial catheter implantation were performed. ", "The influence on survival of demographics, uveal tumor characteristics, liver metastasis presentation, and treatment was studied. ", "The median time to liver metastasis was 29 months. ", "Twenty-eight patients (44%) were operated on: 14 (22%) had R0 liver surgery, and 14 with diffuse liver involvement had R2 liver surgery (there were no significant surgical complications). ", "Thirty-five patients with diffuse liver involvement received systemic chemotherapy or best supportive care only. ", "The median overall survival was 15 months (range, 3-110 months): 25 months for the 14 patients with R0 surgery, 16 months for the 14 with R2 surgery, and 11 months for the 35 with chemotherapy or supportive care. ", "By univariate analysis, age (< or =70 years), number of metastases (< or =10), and quality of operation (R0) were predictive of a better prognosis. ", "In the case of liver metastases from uveal melanoma, aggressive treatment permitting tumor eradication seems to offer a chance of long-term survival to selected patients. ", "Nevertheless, neither ultrasound screening nor quality of operation had an effect on the outcome of most patients (78%). ", "Better screening tests and more effective multimodality treatments are required to improve survival in uveal melanoma patients with hepatic metastases." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Image designed for 20th Century Fox social media channels, promoting the film, \"Exodus Gods and Kings,\" alongside the major college basketbal rivalry between Ohio State University & University of Michigan:\n\nCharacter poster designed for the Facebook page of the film, \"Calvary,\" leading up to its Digital HD release:\n\nGraphics created for 20th Century Fox Halloween social media campaign featuring its catalog of horror films:" ]

{ "pile_set_name": "Pile-CC" }

[ 0.009389671361502348 ]

[ "If you’re anything like me, then you love Speed Racer with all of your heart. ", "Eviscerated upon release, it has come to be seen as the Wachowskis’ true masterpiece by a growing segment of those who are referred to in hushed tones as “film people.” ", "They’ll tell you it’s actually beautiful and earnest, a pure expression of the potential of cinema without a cynical bone in its body. ", "I am one of those film people, and I am here to tell you that it’s happening again.", "\n\nJupiter Ascending is not the quite cinematic revelation that Speed Racer was, buts its more conventional aesthetic choices are balanced with its nostalgic commitment to genre and a greater thematic richness. ", "A space opera in the most literal of senses, it is a melodramatic love story, a wondrous tour through decadent costume and set design, and a pointed takedown of the underlying amorality of capitalism.", "\n\nSummarising Jupiter Ascending is more than a little difficult, the plot initially laying the groundwork for a chosen-one teen drama, before instead shifting into the action-packed proceedings of intergalactic corporate legalese. ", "Warring members of one of the universe’s largest family businesses fight over the deeds to the Earth, and somehow at the heart of all this is Mila Kunis’ Jupiter Jones, a poor girl still cleaning toilets every day. ", "She is the film’s emotional heart, swept up into the drama through nothing but chance, shepherded from plot point to plot point, a cog in a machine that cares not one iota for her agency or personhood. ", "The convoluted story and Jupiter’s passive nature are reminiscent of recurring complaints levied at your Twilights, your Divergents etc., ", "but here the film elevates them from narrative flaws to integral thematic components. ", "Jupiter Ascending doesn’t inherit the problems of its genre, it confronts them.", "\n\nAll that makes Jupiter Ascending seem like a dry affair, but the reality couldn’t be further than the truth. ", "It’s dripping in camp, from Eddie Redmayne’s villainous drawl to the time it decides to just turn into Brazil for about five minutes. ", "The film’s true strength is the lost art of sincerity, it embraces the inherent stupidity of its space opera universe and still commits to every single beat. ", "Much like Lucy last year, it is smart and stupid in equal measure, celebrating its pulpy nature and never undercutting either it or its thematic ideas in order to bolster the other. ", "I likeGuardians of the Galaxy as much as the next guy, but if the only way we’re going to get space bombast in the future is to couch it in a self-effacing layer of snarky detachment, then we’re living in a sad world indeed.", "\n\nUltimately, these are not the words I truly want to write about Jupiter Ascending. ", "Those words would be full of spoilers, a parsing of the films specific themes and ambitions, a celebration of every campy line read and overwrought piece of set design. ", "It is a film that demands its audience to meet it half way, and if you do, there is so much worth talking about on the side. ", "For a film that also features Bees genetically engineered to recognise space royalty, I cannot think of a greater compliment.", "\n\nBe on the right side of history, this time. ", "Go see Jupiter Ascending, then we’ll have the right conversation." ]

{ "pile_set_name": "Pile-CC" }

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[ "This application is the national stage of PCT FR 09/03255 filed Dec. 22, 1999, now WO 00/040526.", "\nA subject matter of the present invention is a process for the preparation of xcex2-phosphorated nitroxide radicals of formula: \nin which R1, R2, R3, R4 and R5 will be defined subsequently. ", "These compounds are used in particular as regulators of radical polymerization.", "\nThese compounds can be obtained in particular by the oxidation of N-alkylaminophosphonates of formula: \nThe N-alkylaminophosphonates (II) can be obtained in a known way by reacting a carbonyl compound R1R2C(O), a primary amine R3NH2 and a phosphorus compound HP(O)R4R5 having a mobile hydrogen according to a Mannich-type reaction: \nA process is disclosed in international patent application WO 96/24620 which consists in reacting, in a first stage, a carbonyl compound R1R2C(O) with a primary amine R3NH2 according to a carbonyl compound/primary amine molar ratio substantially equal to 1 and then, in a second stage, in adding, to the compound obtained in the first stage, a phosphorus compound HP(O)R4R5 according to a phosphorus compound/product obtained in the first stage molar ratio ranging from 1.5 to 2.5, indeed even more. ", "There are several disadvantages to this way of proceeding.", "\nThus, the water formed in the first stage during the reaction of the carbonyl compound with the primary amine which results in the formation of an imine according to the scheme:\n greater than Cxe2x95x90O+xe2x80x94NH2xe2x86x92 greater than Cxe2x95x90Nxe2x80x94+H2O\nis not removed before the addition of the phosphorus compound, which may be of such a nature as to result in a possible hydrolysis of said phosphorus compound, in particular when the latter is a phosphite.", "\nIn addition, the use of a very large excess of phosphorus compound (150% to 250%, indeed even more) with respect to the compound obtained in the first stage (imine) is prohibitory for an industrial process.", "\nFurthermore, this excess, in addition to the fact of the difficulties in removing it, is of such a nature as to generate numerous impurities by reaction with, in particular, the unconverted carbonyl compound to result in hydroxyphosphonates ( greater than C(OH)xe2x80x94P(O) less than ) in the case where the phosphorus compound used is a phosphite. ", "This excess of phosphorus compound can also result in the formation of heavy products resulting from the reaction between (II), the carbonyl compound and the excess phosphorus compound HP(O)R4R5.", "\nAll these impurities liable to be present in the crude product (II) make the purification of the N-alkylaminophosphonate (II) difficult and consequently render its subsequent use difficult.", "\nThe xcex2-phosphorated nitroxide (I) can be obtained by oxidation of the N-alkylaminophosphonate (II) by replacing the hydrogen of greater than Nxe2x80x94H with an oxygen atom.", "\nThis oxidation can be carried out according to various techniques known to a person skilled in the art. ", "A few appropriate techniques are listed below in a nonexhaustive fashion:\nreaction of a secondary amine with aqueous hydrogen peroxide solution in the presence of a solid catalyst (titanium silicalite), the principle of which is disclosed in patent U.S. Pat. ", "No. ", "5,218,116;\nreaction of a secondary amine with a dioxirane or its carbonyl precursors in combination with oxone(copyright), the technique of which is disclosed in patent U.S. Pat. ", "No. ", "5,087,752,\nreaction of a secondary amine with metachloroperbenzoic acid, according to a protocol described in the Journal of the American Chemical Society, 1967, 89 (12), pages 3055-3056.", "\nAccording to international application WO 96/24620, diethyl 2,2-dimethyl-1-(1,1-dimethylethylamino)propylphosphonate could only be efficiently oxidized by means of meta-chloroperbenzoic acid (mCPBA) according to a protocol which consists in introducing the mCPBA, in solution in CH2Cl2, into a CH2Cl2 solution of the abovementioned aminophosphonate. ", "The xcex2-phosphorated nitroxide obtained is purified by passing through a column of silica gel, which involves the use of a large volume of elution solvent.", "\nThis way of operating can only be applied to the preparation of small amounts of (I). ", "In addition, the analysis of this product by HPLC has shown that this technique results in nitroxides with a purity of less than 80%.", "\nIn order to improve the process for the oxidation of the aminophosphonate (II) by mCPBA, the assignee of the present application has carried out the preliminary drying of the technical mCPBA, on the one hand, and optimized the mCPBA/aminophosphonate molar ratio, on the other hand.", "\nAlthough a satisfactory yield of xcex2-phosphorated nitroxide is obtained, this way of operating produces amounts of effluents (metachlorobenzoic acid salt) which are difficult to remove.", "\nFurthermore, even the optimized use of mCPBA does not make possible viable translocation to the industrial scale.", "\nThe applicant has found that, by using nonhalogenated organic peracids in a water/organic solvent two-phase medium with an aqueous phase buffered at a pH ranging from 5 to 12, it can oxidize aminophosphonates (II) to xcex2-phosphorated nitroxides (I) while significantly reducing the manufacturing costs and while generating effluents which can be easily removed.", "\nA subject matter of the present invention is thus a process for the manufacture of compounds of general formula: \nin which R1 and R2, which are identical or different, represent a hydrogen atom, a linear, branched or cyclic alkyl radical having a number of carbon atoms ranging from 1 to 10, an aryl radical, or an aralkyl radical having a number of carbon atoms ranging from 1 to 10, or else R1 and R2 are connected to one another so as to form a ring which includes the carbon atom carrying said R1 and R2 said ring having a number of carbon atoms, including the carbon carrying the R1 and R2 radicals, ranging from 3 to 8; R3 represents a linear or branched and saturated or unsaturated hydrocarbonaceous radical which can comprise at least one ring, said radical having a number of carbon atoms ranging from 1 to 30; and R4 and R5, which are identical or different, represent a linear or branched alkyl radical having a number of carbon atoms ranging from 1 to 20 or a cycloalkyl, aryl, alkoxyl, aryloxyl, aralkyloxyl, perfluoroalkyl, aralkyl, dialkyl- or diarylamino, alkylarylamino or thioalkyl radical, or else R4 and R5 are connected to one another so as to form a ring which includes the phosphorus atom, said heterocycle having a number of carbon atoms ranging from 2 to 4 and being able in addition to comprise one or more oxygen, sulfur or nitrogen atoms;\nsaid process consisting in oxidizing an aminophosphonate of formula \nobtained by reaction of a carbonyl compound R1R2C(O), of a primary amine R3NH2 and of a phosphorus compound HP(O)R4R5, R1, R2, R3, R4 and R5 having the meanings given above; said process being characterized in that it consists first in preparing and optionally in isolating the compound II and then subsequently in oxidizing it by carrying out the following (successive) stages:\na) a carbonyl compound R1R2C(O) is reacted with a primary amine R3NH2 according to an R1R2C(O)/R3NH2 molar ratio of between 0.8 and 1.5 and preferably between 0.9 and 1.1 at a temperature of between 0xc2x0 C. and 120xc2x0 C. and a pressure ranging from 1 to 10 bar; then the water formed is removed from the reaction medium;\nb) the compound obtained in a) is reacted with a phosphorus compound HP(O)R4R5 used according to an HP(O)R4R5 /compound a) molar ratio at most equal to 1.5 and preferably of between 1 and 1.5 at a temperature of between 0xc2x0 C. and 120xc2x0 C.;\nc) an acidic treatment of the reaction medium obtained in b) is carried out, an organic solvent is subsequently added, separation by settling is carried out, the aqueous phase is recovered and then a basic treatment of said aqueous phase is carried out;\nd) the aminophosphonate is extracted by means of an organic solvent identical to that used previously in c);\ne) said solvent is then completely removed and said aminophosphorate (II) is isolated and is oxidized to xcex2-phosphorated nitroxide according to the following stages:\na1) the aminophosphonate (II) obtained in e) is dissolved in a water-immiscible organic solvent, then;\nb1) subsequently, an amount of non-halogenated organic peracid, according to a peracid/aminophosphonate (II) molar ratio ranging from 1.5 to 2.5, and a sufficient amount of a basic aqueous solution of an alkali metal carbonate or hydrogencarbonate or of an alkaline earth metal carbonate or hydrogencarbonate or alternatively of an ammonia solution to produce a pH ranging from 5 to 12 and preferably ranging from 6 to 9 are simultaneously added with vigorous stirring, at a temperature of between xe2x88x9210xc2x0 C. and +40xc2x0 C. and preferably of between xe2x88x925xc2x0 C. and +30xc2x0 C., to the medium thus obtained, until the aminophosphonate (II) has been completely converted;\nc1) the organic phase is then recovered by simple separation by settling and the xcex2-phosphorated nitroxide is isolated by evaporation of the organic solvent under reduced pressure.", "\nThe purity of the xcex2-phosphorated nitroxide obtained can be improved by flash distillation under reduced pressure or by low temperature crystallization.", "\nAccording to an alternative form of the process of the present invention, it is possible not to isolate the aminophosphonate (II) and to carry out the oxidation on the aqueous phase obtained in c), which comprises the aminophosphonate hydrochloride.", "\nShould that arise, the procedure is thus carried out in the following way:\nafter having carried out the acidic treatment of the reaction medium obtained in b) and then added a water-immiscible organic solvent, the aqueous phase is separated by settling, an organic solvent is added to the aqueous phase and then stages b1) and c1) as described above are carried out.", "\nMention will be made, by way of illustration of carbonyl compounds R1R2C(O) which can be used according to the present invention, of trimethylacetaldehyde (pivalaldehyde), isobutyraldehyde, cyclohexanecarboxyaldehyde, diethyl ketone, dibutyl ketone, methyl ethyl ketone, cyclohexanone, 4-tert-butylcyclohexanone or xcex1-tetralone.", "\nMention will be made, by way of illustration of primary amines R3xe2x80x94NH2 which can be used according to the present invention, of methylamine, ethylamine, propylamine, isopropylamine, tert-butylamine, diphenylmethylamine, triphenylmethylamine, aniline, xcex1-naphthylamine, benzylamine, 1-phenylethylamine, cyclohexylamine or cyclopentylamine.", "\nUse will preferably be made of tert-butylamine, isopropylamine, diphenylmethylamine, 1-phenylethylamine or cyclohexylamine.", "\nThe reaction between the carbonyl compound R1R2C(O) and the amine R3NH2 (stage a) is carried out with vigorous stirring at a temperature of between 0xc2x0 C. and 120xc2x0 C. and preferably at a temperature of between 0xc2x0 C. and 60xc2x0 C. The reaction is generally carried out at a pressure of between 1 bar and 10 bar, preferably at atmospheric pressure, and under an inert gas atmosphere, such as nitrogen or argon. ", "The reaction time can vary within wide limits. ", "It depends on the reactivity of the amine employed. ", "The complete conversion of the carbonyl compound R1R2C(O) can be confirmed by chromatographic (GC) analysis.", "\nOn completion of the reaction, stirring is halted and the reaction medium is allowed to separate by settling. ", "Separation by settling is generally rapid. ", "The aqueous phase, consisting virtually entirely of the water formed during the reaction between the carbonyl compound and the primary amine which results in the imine (III), according to the scheme: \nis subsequently removed.", "\nThe removal of the water can be completed by the addition of a dehydrating agent, such as a molecular sieve, to the phase which has been separated by settling or alternatively azeotropic distillation can be carried out.", "\nThe imine (III) obtained in stage a) is reacted with a compound HP(O)R4R5 (stage b)).", "\nMention will be made, by way of illustration of phosphorus compounds HP(O)R4R5 which can be used according to the present invention, of dimethyl phosphite, diethyl phosphite, n-propyl phosphite, dibenzyl phosphite, diisopropyl phosphite, di(n-dodecyl)phosphite; diphenylphosphine oxide or dibenzylphosphine oxide.", "\nThe reaction between the phosphorus compound HP(O)R4R5 and the imine (III) obtained in stage a) is carried out with vigorous stirring at a temperature of between 0xc2x0 C. and 120xc2x0 C. and preferably at a temperature of between 10xc2x0 C. and 80xc2x0 C. As in stage a), the reaction is carried out under an inert gas atmosphere and preferably at atmospheric pressure.", "\nUse is made of a molar excess of phosphorus compound HP(O)R4R5 with respect to the imine obtained in stage a) at most equal to 50% and preferably of an excess which is as low as possible, indeed even zero phosphorus compound/imine (III) molar ratio equal to 1).", "\nThe reaction medium is kept stirred for a time which can vary within wide limits; this time is preferably at most equal to 25 hours.", "\nThe reaction medium is subsequently acidified (stage c)) with an aqueous hydrochloric acid solution.", "\nThe concentration by weight of HCl is at most equal to 20% and preferably between 5% and 15%. ", "This acidification is preferably carried out at a temperature of between 0xc2x0 C. and 20xc2x0 C. and preferably at a temperature in the region of 10xc2x0 C.\nWhen the pH of the reaction medium is less than 3, a first extraction is carried out which consists in removing the organic impurities and unconverted reactants with an organic solvent which is a good solvent of the impurities. ", "Methylene chloride (CH2Cl2) will advantageously be used.", "\nSubsequently, the acidic aqueous phase (comprising the aminophosphonate hydrochloride) is separated by settling and is subjected to a basic treatment.", "\nThis acidic aqueous phase is preferably treated with an aqueous solution of an alkali metal carbonate or hydrogencarbonate, such as K2CO3, NaHCO3, KHCO3 or Na2CO3, or with an ammonia solution.", "\nA second extraction (stage d)) of the released aminophosphonate (II) is subsequently carried out with the same solvent used in stage c).", "\nThe organic phase is advantageously washed with pure water and then concentrated under reduced pressure.", "\nA virtually pure aminophosphonate (II) is obtained.", "\nThe oxidation of the aminophosphonate (II) to the nitroxide (I) is carried out according to stages a1) to c1) as described above.", "\nMention will be made, by way of illustration of organic solvents which can be used according to the present invention in stages a1) to c1), of aliphatic hydrocarbons, such as pentane, heptane or cyclohexane;\nchlorinated solvents, such as CH2Cl2; esters of aliphatic acids, such as ethyl acetate or ethyl propionate, or a mixture of at least two of the abovementioned solvents.", "\nMention will be made, by way of illustration of nonhalogenated organic peracids which can be used according to the present invention, of peracetic acid, perpropionic acid or perbutanoic acid.", "\nThe compounds (I) can be identified by elemental analysis, HPLC, IR and EPR.", "\nThe compounds (II) obtained according to the invention can be identified by proton, 13C and 31P NMR, by IR and by elemental analysis.", "\nThe compounds obtained according to the process of the present invention have a sufficient purity to be used as regulators of radical polymerizations.", "\nThe process according to the present invention exhibits the advantage of resulting in high yields of xcex2-phosphorated nitroxide radicals." ]

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[ "Preservation of bacteria in natural polymers.", "\nA new inexpensive and simple method for preserving microorganisms has been developed. ", "Natural polymers of acacia gum and pullulan were used to preserve model bacteria Escherichia coli and Bacillus subtilis via immobilization and storage under various conditions. ", "Formulation of E. coli and B. subtilis in acacia gum significantly increased the viability of both cultures during desiccation at 40 degrees C as well as during the storage at various temperatures and relative humidity. ", "In the ranges of temperatures and humidity used in experiments, the high humidity affected the viability of E. coli more than high temperature. ", "Thermodynamic parameters for E. coli thermal degradation were used for quantification of results and characterization of the preservation process. ", "Viability of B. subtilis in acacia gum polymer was not significantly changed during the storage in the temperature and humidity experiments. ", "The number of viable B. subtilis recovered after storage in pullulan, and in PBS under various humidity conditions was 1-2 logs less in comparison with the number of cells before storage. ", "It was found that acacia gum provides better protection than pullulan for both bacteria during the preservation process." ]

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[ "A resin containing phenolic hydroxyl groups is used in an adhesive, a molding material, paint, a photoresist material, an epoxy resin raw material, a curing agent for an epoxy resin, and the like. ", "Since the heat resistance and moisture resistance of the cured product of the resin containing phenolic hydroxyl groups are excellent, the resin is also widely used in the electrical and electronic field such as a semiconductor sealing material or an insulating material for a printed wiring board, as a curable composition including the resin containing phenolic hydroxyl groups itself as a main agent or as a curing agent of an epoxy resin or the like.", "\nAmong these, in the field of a photoresist, a wide variety of resist pattern forming methods subdivided according to the use or the function have been developed one after another. ", "Accordingly, performances required for a resin material for a resist have become more sophisticated and diversified. ", "For example, high developability is required in order to accurately form a fine pattern on a highly-integrated semiconductor with high production efficiency, a cured product is required to have properties of being flexible and hard to be broken in the case of forming a thick film, and dry etching resistance and heat resistance are required in the case of a resist underlayer film. ", "Further, toughness such as substrate followability, as well as the high heat resistance, is required in the case of a resist permanent film." ]

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[ "Q:\n\nUnique random based on year\n\nIm developing a app which needs will display one random element from a list every day (list of 366 items).", "\nBut I would like this app to display the same item for all users. ", "Also I'd like this calculation to display a different (random) order the next year.", "\ni.e \nuser A: Jan 26th 2017 : item 32\nuser B: Jan 26th 2017 : item 32\nuser A: Jan 26th 2017 : item 27\nuser B: Jan 26th 2017 : item 27\nAlso I would like the calulation to not show the same item more than once in a given year.", "\nIm not particulary good at maths but I am a programmer, being new to this forum apologies if this question is really basic! ", "\nOrdinarly this could be done quite easily with a central repository firing out the item to use with a internet connection but I'd like to make the app work offline.", "\n\nA:\n\nTake the current year as a seed ot a pseudo random number generator and then use the prng to compute a random permutation of $\\{1,\\ldots,366\\}$. Display the item corresponding to the current day of year as index.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Carlos Pascual, U.S. Ambassador to Mexico, has resigned after a political fallout that occurred when WikiLeaks published a cable in December 2010 in which the U.S. Embassy to Mexico was critical of the Mexican government’s efforts in the ongoing war on drugs. ", "The cable said Mexican President Filipe Calderon “has struggled with an unwieldy and uncoordinated interagency and spiraling rates of violence that have made him vulnerable to criticism that his anti-crime strategy has failed.“", "\n\nAccording to Reuters, Calderon was critical of the ambassador in an interview published February 22 in Mexican newspaper El Universal. ", "Diplomatic tensions rose and Pascual resigned.", "\n\nJulian Assange would see this as a victory; removing “conspirators” in control helps to dismantle the “conspiracy” of government. ", "In many ways, he is right to see it that way. ", "Instead of taking a cooperative approach (“Hey, we clearly aren’t doing this as well as you think, so lets work on solutions.”), ", "President Calderon was standoffish, and the honesty of the embassy ended up hurting the relationship between the two nations.", "\n\nWhere honest dialogue is unwelcome, progress is impossible; as he sits on house arrest in the U.K., Assange is grinning to himself." ]

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[ "<!", "DOCTYPE html>\n<html>\n <head>\n <title>JavaScript Project checklist</title>\n <link href='http://fonts.googleapis.com/css?family=Lato:400,700' rel='stylesheet' type='text/css'>\n <link href=\"//maxcdn.bootstrapcdn.com/bootstrap/3.3.1/css/bootstrap.min.css\" rel=\"stylesheet\">\n <link href=\"resources/styles.css\" rel=\"stylesheet\">\n </head>\n <body>\n <script src=\"//ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js\"></script>\n <script src=\"//canjs.com/release/2.2.4/can.jquery.js\"></script>\n <script src=\"resources/drawDoughnut.js\"></script>\n <script type=\"text/javascript\" src=\"//rawgit.com/daffl/jquery.dform/master/dist/jquery.dform-1.1.0.min.js\"></script>\n <script src=\"resources/form-params.js\"></script>\n <script src=\"resources/converter.js\"></script>\n\n <div class=\"container\">\n <div class=\"row\">\n\n <div class=\"col-md-8 col-md-offset-2\">\n <button class=\"btn btn-link pull-right btn-small open-paste-tool-js\">Paste in previous results...</button>\n <div class=\"panel panel-default paste-tool\">\n <div class=\"panel-heading\">\n <h3 class=\"panel-title\">Paste in your JSON data...</h3>\n </div>\n <div class=\"panel-body\">\n <div class=\"form-group\">\n <label class=\"control-label\" for=\"paste-bucket\">Paste JSON data</label>\n <textarea class=\"form-control paste-bucket\" id=\"paste-bucket\" name=\"paste-bucket\"></textarea>\n </div>\n <button class=\"btn btn-default\">Apply</button>\n </div>\n </div>\n </div>\n <div class=\"col-md-8 col-md-offset-2\">\n <div class=\"jumbotron\">\n <h1>JavaScript Project Checklist</h1>\n <p class=\"text-center\">\n <a href=\"https://github.com/bitovi/checklist\">The Github Repository</a> | <a href=\"http://blog.bitovi.com/why-checklist/\">The Original Article</a></p>\n </div>\n \n <div class=\"links panel panel-default\">\n <div class=\"panel-heading\">\n <h3 class=\"panel-title\">Jump to section...</h3>\n </div>\n <div class=\"panel-body\">\n <ul>\n <li><a href='#management'>Management</a></li>\n <li><a href='#design'>Design & UX</a></li>\n <li><a href='#dev'>Development</a></li>\n </ul>\n </div>\n </div>\n\n <form id=\"checklist\" role=\"form\"></form>\n </div>\n </div>\n </div>\n </body>\n</html>\n" ]

{ "pile_set_name": "Github" }

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[ "All relevant data are available via Dryad (<http://dx.doi.org/10.5061/dryad.bn987>).", "\n\nIntroduction {#sec001}\n============\n\nPerturbation-based, gait assessment paradigms can aid in identifying dysfunction often associated with age \\[[@pone.0142083.ref001]--[@pone.0142083.ref003]\\], disease \\[[@pone.0142083.ref004], [@pone.0142083.ref005]\\], or injury \\[[@pone.0142083.ref006], [@pone.0142083.ref007]\\]. ", "Perturbations can be cognitive \\[[@pone.0142083.ref001], [@pone.0142083.ref008]--[@pone.0142083.ref010]\\], physical \\[[@pone.0142083.ref006], [@pone.0142083.ref011]--[@pone.0142083.ref013]\\], or visual \\[[@pone.0142083.ref012], [@pone.0142083.ref014]--[@pone.0142083.ref016]\\] in nature and are typically selected specific to deficits suspected in a population. ", "In contrast, the temporal-spatial \\[[@pone.0142083.ref003], [@pone.0142083.ref017]--[@pone.0142083.ref019]\\], kinematic variability \\[[@pone.0142083.ref013], [@pone.0142083.ref015], [@pone.0142083.ref020]\\], and dynamic stability \\[[@pone.0142083.ref012], [@pone.0142083.ref020], [@pone.0142083.ref021]\\] measures collected during these assessments are often used interchangeably to quantify and characterize gait stability responses to perturbations. ", "Unfortunately, the psychometric properties for many of these measures are not established for perturbation-based gait assessments, thus preventing their widespread utilization and the interpretation of published findings. ", "Ideally, these measures would exhibit low minimal detectable change (MDC) values and strong between-session reliability to effectively identify deficits and track changes over time. ", "Additionally, these measures should demonstrate sensitivity to expected gait responses across a range of commonly used perturbations.", "\n\nDuring unperturbed gait, temporal-spatial measures like step width (SW), step length (SL), and stride time (ST) were reported to exhibit excellent between-session reliability (interclass correlation coefficient; ICC \\> 0.87) in healthy adults \\[[@pone.0142083.ref022], [@pone.0142083.ref023]\\]. ", "Similarly, the between-session reliability of SW, SL, and ST was reported as excellent (ICC ≥ 0.75) when young adults walking at fast speeds (\\~1.60 m/s) were exposed to underfoot physical perturbations \\[[@pone.0142083.ref024]\\]. ", "While between-session reliability was excellent, these results may not generalize to different types of perturbations. ", "For example, between-session reliability for stride velocity variability decreased from moderate (ICC = 0.656) during unperturbed gait to poor (ICC = 0.226) during cognitively perturbed gait in older adults \\[[@pone.0142083.ref025]\\]. ", "In contrast, ICC values for velocity and cadence measures remained excellent during both unperturbed and cognitively perturbed gait \\[[@pone.0142083.ref025]\\]. ", "The authors suggested that measures of variability such as stride velocity variability might be inherently less reliable due to high between and within subject variance and may require hundreds of strides to increase between-session reliability during cognitively perturbed gait. ", "To our knowledge, no group has reported the between-session reliability of temporal-spatial measures during visually perturbed gait.", "\n\nAs an alternative to temporal-spatial measures, multiple groups \\[[@pone.0142083.ref026]--[@pone.0142083.ref032]\\] have used local and orbital non-linear measures to assess dynamic gait stability, particularly in circumstances \\[[@pone.0142083.ref012], [@pone.0142083.ref021], [@pone.0142083.ref033]\\] and/or with populations \\[[@pone.0142083.ref026], [@pone.0142083.ref027], [@pone.0142083.ref029], [@pone.0142083.ref032]\\] where instability is prevalent. ", "Local and orbital stability measures quantify how quickly responses to perturbations grow or decay over time. ", "They have shown sensitivity and directional specificity in identifying gait responses to physical and visual perturbations \\[[@pone.0142083.ref012], [@pone.0142083.ref033]\\]. ", "However, their utility in determining stability deficits in response to cognitive perturbations has proved inconclusive \\[[@pone.0142083.ref020]\\] and may require a large number of perturbed strides to detect deficits \\[[@pone.0142083.ref034]\\].", "\n\nRecently, two groups have reported the within-session and between-session reliability of local dynamic stability measures (i.e. local divergence exponents) in healthy adults during unperturbed over ground \\[[@pone.0142083.ref030], [@pone.0142083.ref035]\\] and treadmill walking \\[[@pone.0142083.ref036]\\]. ", "Short-term local dynamic stability measures were found to have good within-session reliability (ICC ≥ 0.70) and poor to fair between-session reliability (ICC ≤ 0.63) during over ground walking \\[[@pone.0142083.ref035]\\]. ", "ICC and MDC values were strongly influenced by the state space reconstruction method employed. ", "Similarly, short-term local dynamic stability measures were found to have greater within-session reliability (ICC ≥ 0.77) compared to between-session reliability (ICC \\~ 0.60) during treadmill walking \\[[@pone.0142083.ref036]\\]. ", "Further, short-term local dynamic stability measures demonstrated greater reliability than long-term local dynamic stability measures with both showing increased reliability as more strides were analyzed \\[[@pone.0142083.ref036]\\]. ", "While these studies do provide some indication as to the reliability of local dynamic stability measures during unperturbed gait, to date, no study has reported both the between-session reliability and MDC of local and orbital stability measures during perturbed gait.", "\n\nThe general lack of reliability and MDC values for temporal-spatial, kinematic variability, and dynamic stability measures during perturbed gait limits their usability for quantifying, identifying, and tracking dysfunction. ", "Further, the interpretation of published perturbation-based assessments findings is made more difficult without reliability and MDC reference values to help distinguish between differences associated with dysfunction and those due to measurement error. ", "The purpose of this study was to systematically determine both the between-session reliability and MDC values of reported temporal-spatial, kinematic variability, and dynamic stability measures during perturbed and unperturbed gait. ", "As part of this systematic investigation, three commonly used perturbations; color-interference Stroop (cognitive) \\[[@pone.0142083.ref006], [@pone.0142083.ref019], [@pone.0142083.ref020], [@pone.0142083.ref034]\\], walking surface oscillations (physical)\\[[@pone.0142083.ref012], [@pone.0142083.ref015], [@pone.0142083.ref033], [@pone.0142083.ref037]\\] and visual field oscillations (visual) \\[[@pone.0142083.ref012], [@pone.0142083.ref015], [@pone.0142083.ref033], [@pone.0142083.ref037]--[@pone.0142083.ref041]\\] were used to study the reliability and sensitivity of measured responses. ", "Previous investigations indicate that the magnitude and type of responses may vary between selected measures and perturbation type \\[[@pone.0142083.ref012], [@pone.0142083.ref015], [@pone.0142083.ref020]\\], thus these factors are likely to affect reliability. ", "The information from the present study could help facilitate the interpretation of other study results and improve the clinical utility of these measures during perturbation-based assessments.", "\n\nMethods {#sec002}\n=======\n\nParticipants {#sec003}\n------------\n\nTwenty participants (5 females, age: 26.1 ± 6.8 years, height: 1.75 ± 0.10 m, and mass: 78.3 ± 9.7 kg) completed two identical testing sessions. ", "The experimental protocol was reviewed and approved by the Institutional Review Board at Brooke Army Medical Center. ", "Written consent was obtained from each participant prior to enrollment. ", "Exclusion criteria included neurologic and orthopedic injuries or disorders that would alter normal gait. ", "Participants also had to pass a visual acuity and color discrimination screen. ", "Glasses or contact lens were worn by participants who required them for corrected vision.", "\n\nExperimental apparatus {#sec004}\n----------------------\n\nAll tasks were performed in a virtual reality environment (Computer Assisted Rehabilitation Environment; Motekforce Link, Amsterdam, Netherlands; [Fig 1A](#pone.0142083.g001){ref-type=\"fig\"}) comprised of a 7 m diameter dome with 270-degrees of horizontal visual field projection and a 6-degrees of freedom motion platform \\[[@pone.0142083.ref038]\\]. ", "Participants stood or walked in the center of a 1.8 x 2.8 m (width x length) instrumented treadmill wearing a safety harness tethered to a metal frame mounted outside their field of view. ", "Full body 3-dimensional kinematic data were collected at 60 Hz using a 24-camera infrared motion capture system (Vicon Motion Systems, Oxford, UK) to track 57 reflective markers \\[[@pone.0142083.ref033]\\] place on hand, arm, head, trunk, pelvis, thigh, leg and foot segments.", "\n\n{#pone.0142083.g001}\n\nExperimental protocol {#sec005}\n---------------------\n\nA wooded country scene with a centered walking path was displayed in the virtual reality environment during all conditions ([Fig 1B](#pone.0142083.g001){ref-type=\"fig\"}). ", "Participants were asked to walk down the path while maintaining a forward head orientation. ", "In addition to unperturbed walking (NOP), participants walked while being perturbed (1) cognitively using a color-interference Stroop task (COG), (2) physically through translations of the walking surface (PLAT), or (3) visually with translations of the visual field (VIS). ", "Three 3 minute trials of each gait condition were performed in random order and participants were allowed to rest in between trials. ", "For all trials, each participant walked on the motorized treadmill in the virtual reality environment at the same constant speed scaled to their leg length: $$speed = \\sqrt{0.16 \\cdot g \\cdot l}$$ where *g* = 9.81 m/s^2^ and *l* = leg length in meters \\[[@pone.0142083.ref042]\\]. ", "Optic flow in the virtual reality environment was scaled to match the speed of the treadmill. ", "Demonstrations, instructions, and practice time were provided before each condition to minimize learning effects. ", "Participants returned for a second identical testing session within 14 ± 2 days.", "\n\nThe color-interference Stroop task consisted of words of common colors (i.e. red, blue, green, and yellow) displayed in an incongruent colored font ([Fig 1B](#pone.0142083.g001){ref-type=\"fig\"}). ", "Participants were required to avoid reading the word and instead report only the color of the font. ", "Words were randomly displayed in front of the participant at a rate of one per second. ", "The PLAT and VIS perturbation conditions consisted of continuous pseudo-random medial-lateral oscillations of either the visual scene or treadmill surface \\[[@pone.0142083.ref012], [@pone.0142083.ref015]\\]. ", "Perturbations were applied as a pseudo-random sum of sines with 4 incommensurate frequencies (0.16, 0.21, 0.24 and 0.49Hz) in the following equation: $$D\\left( t \\right) = A\\begin{bmatrix}\n{1.0\\,\\text{sin}\\left( {0.16 \\cdot 2\\pi t} \\right) + 0.8\\,\\text{sin}\\left( {0.21 \\cdot 2\\pi t} \\right)} \\\\\n{+ 1.4\\,\\text{sin}\\left( {0.24 \\cdot 2\\pi t} \\right) + 0.5\\,\\text{sin}\\left( {0.49 \\cdot 2\\pi t} \\right)} \\\\\n\\end{bmatrix}$$ where D(t) is the translation distance (m), A is a scaling factor, and t is time (sec) \\[[@pone.0142083.ref012], [@pone.0142083.ref015]\\]. ", "PLAT and VIS perturbations were scaled with A = 0.05 and A = 0.5 such that the maximum displacement was approximately 0.33m and 3.32m, respectively.", "\n\nInitial data processing {#sec006}\n-----------------------\n\nAll data reduction and analyses were performed using Vicon Nexus 1.7 (Vicon Motion Systems, Oxford, United Kingdom), Visual3D (C-Motion Inc., Rockville, MD), and MatLab (The Mathworks, Natick, MA). ", "Raw marker data were filtered with a zero-lag Butterworth filter at a low-pass cutoff frequency of 8 Hz. ", "Heel strike and toe off events were determined using a velocity-based algorithm for comparing the anterior-posterior foot velocity relative to the pelvis \\[[@pone.0142083.ref043]\\].", "\n\nTemporal-spatial measures {#sec007}\n-------------------------\n\nStep length (SL) was defined as the distance between the foot centers in the anterior-posterior direction at heel strike. ", "Step width (SW) was calculated as the medial-lateral, heel-to-heel distance between the two heel markers in double-limb stance. ", "Stride time (ST) was quantified as the duration between consecutive heel strikes of the same foot. ", "Means and standard deviations (SD) of SL, SW, and ST were calculated for each 3-minute walking trial.", "\n\nState Spaces {#sec008}\n------------\n\nDelay embedded state spaces \\[[@pone.0142083.ref044]\\] were constructed using raw C7 vertebral marker velocity data and time delayed copies \\[[@pone.0142083.ref045]\\], such that: $$S(t) = \\lbrack v(t),v(t + \\tau),\\ldots,v(t + (d_{E} - 1)\\tau)\\rbrack$$ where *S*(*t*) was the *d* ~*E*~-dimensional state vector, *v*(*t*) was the original data, *τ* was the time delay and *d* ~*E*~ was the embedding dimension. ", "Time delays of 15, 10, and 30 samples, for anterior-posterior (AP), vertical (VT), and medial-lateral (ML) directions, respectively, were used as determined from assessments of the first minima of Average Mutual Information functions \\[[@pone.0142083.ref046]\\]. ", "An embedding dimension of *d* ~*E*~ = 5 \\[[@pone.0142083.ref032]\\] was used for all trials. ", "To calculate orbital and local stability, state spaces were constructed using 124 continuous strides from each trial. ", "For local stability analyses, these 124 continuous strides were first re-sampled to 12,400 total data points yielding an average of 100 data points per stride \\[[@pone.0142083.ref034], [@pone.0142083.ref047]\\] used during the delay embedding process.", "\n\nOrbital stability {#sec009}\n-----------------\n\nOrbital stability was quantified by calculating the magnitude of the maximum Floquet multipliers (MaxFM), which quantify the rate of convergence or divergence from a limit cycle due to small perturbations, using established procedures \\[[@pone.0142083.ref012], [@pone.0142083.ref048], [@pone.0142083.ref049]\\]. ", "If MaxFM \\> 1, a system is orbitally unstable as small perturbations would grow by the next cycle. ", "Consequently, if MaxFM \\< 1, the system is orbitally stable indicating small perturbations diminish by the next cycle. ", "Each delay embedded state space was divided into individual strides and each stride was time normalized to 101 samples, corresponding to 0--100% of the gait cycle. ", "Poincare maps were defined at each percent of the stride as: $$S_{k + 1} = F(S_{\\ k})$$ where *S* was the state of the system at stride *k* at each given Poincare section. ", "The average trajectory for all strides in a given trial was used to define the fixed points as: $$S^{*} = F(S^{*})$$\n\nThe orbital stability of the system was then quantified by estimating the Floquet multipliers by linearizing [Eq 3](#pone.0142083.e003){ref-type=\"disp-formula\"} about these fixed points as: $$\\lbrack S_{k + 1} - S^{*}\\rbrack \\cong J(S^{*})\\lbrack S_{k} - S^{*}\\rbrack$$ where *J*(*S\\**) is the Jacobian matrix for the system at each Poincare section. ", "The eigenvalues of each of the 101 *J*(*S\\**) are the Floquet multipliers. ", "Orbital stability was then defined as the maximum value from the magnitudes of the largest eigenvalues (i.e., MaxFM).", "\n\nLocal stability {#sec010}\n---------------\n\nLocal stability was quantified by calculating the local divergence exponents. ", "Nearest neighbor points in the reconstructed state space represent the effects of small local perturbations to the system. ", "The local divergence exponents quantify the response of a system to small local perturbations \\[[@pone.0142083.ref012], [@pone.0142083.ref020], [@pone.0142083.ref050], [@pone.0142083.ref051]\\]. ", "The local divergence exponents (LDE) were estimated using the slopes of linear fits to the mean log divergence curve: $$y(i) = \\frac{1}{\\Delta t}{\\langle{\\text{ln}\\lbrack d_{j}(i)\\rbrack}\\rangle} = \\lbrack\\lambda^{*}\\rbrack i + c$$ where *d* ~*j*~(*i*) was the Euclidean distance between the *j* ^*th*^ pair of initially nearest neighbors after *i* discrete time steps (i.e., *i*Δ*t* seconds) and 〈∙〉 denotes the average over all values of *j*. ", "Short-term (*λ*\\*~*S*~) and long-term (*λ*\\*~*L*~) LDE were calculated as the slopes of the linear fits of the divergence curve between 0 and 1 stride and between 4 and 10 strides \\[[@pone.0142083.ref052]\\], respectively. ", "Positive LDE indicate local instability.", "\n\nTrunk kinematic variability {#sec011}\n---------------------------\n\nTrunk kinematic variability during walking conditions was characterized using C7 marker velocities in the AP, VT, and ML directions \\[[@pone.0142083.ref015], [@pone.0142083.ref020]\\]. ", "Data for each individual stride were time normalized to 101 samples, corresponding to 0--100% of the gait cycle. ", "Standard deviations were calculated across all strides at each time normalized point within a single trial. ", "Standard deviations were then averaged over the normalized stride to yield the *MeanSD* for each trial using: $$MeanSD(V_{x}) = {\\langle{SD_{n}\\lbrack V_{x}\\rbrack}\\rangle}$$ where *V* ~*x*~ is the velocity in each direction (i.e., *x* ∈{*AP*, *ML*, *VT*}), *n* indicates each time normalized point of the gait cycle (0%, ..., 100%), and 〈∙〉 indicates the average over all *n* \\[[@pone.0142083.ref020], [@pone.0142083.ref045]\\].", "\n\nStatistical analyses {#sec012}\n--------------------\n\nTemporal-spatial, kinematic variability, and dynamic stability means and standard deviations were used to describe group response magnitudes during the unperturbed and perturbed conditions. ", "For each measure, within session differences between the unperturbed and each of the three perturbed conditions were evaluated using three paired t-tests. ", "A Bonferroni-Holm correction was performed to correct for these multiple comparisons. ", "The Bonferroni--Holm method uses a step-down approach to account for multiple comparisons by arranging p-values from the smallest to the largest and comparing them to sequential significance cutoffs \\[[@pone.0142083.ref053]\\]. ", "A correction factor accounting for the three comparisons was applied with the smallest p-value cutoff set to 0.05/3 = 0.0167.", "\n\nBetween-session differences (i.e. session 1 vs. session 2) in temporal-spatial, kinematic variability, and dynamic stability measures were determined using paired t-tests. ", "Effect size was determined for each comparison using the Cohen d statistic using the following equation: $$d = t_{s} \\cdot \\sqrt{2 \\cdot \\left( {1 - r} \\right)/N}$$ where *t* ~*s*~ is the effect size t-value from the comparisons, r is the correlation between the comparisons, and N is the number of participants \\[[@pone.0142083.ref054]\\]. ", "This equation uses the correlation coefficient to limit overestimation of the effect magnitude. ", "An effect size of 0.2--0.3 signifies a \"small\" effect, 0.5--0.7 a \"medium\" effect, and ≥ 0.8 a \"large\" effect \\[[@pone.0142083.ref055]\\].", "\n\nThe between-session (i.e. session 1 vs. session 2) reliability (ICC) of each temporal-spatial, kinematic variability, and dynamic stability measure was calculated using a two-way random model (2, k) for consistency \\[[@pone.0142083.ref056]\\]. ", "ICC values ≥ 0.75 were considered \"excellent\", 0.40--0.74 \"fair to good\", and \\< 0.40 \"poor\" \\[[@pone.0142083.ref057]\\]. ", "In order to calculate MDC values, the standard error of the measurement (SEM) was first determined using the equation $$SEM = SD \\times \\sqrt{(1 - ICC)}$$ where *SD* is the standard deviation from the first testing session. *", "MDC* values were calculated using the equation: $$MDC = SEM \\times 1.96 \\times \\sqrt{2}$$ \\[[@pone.0142083.ref058]\\]. ", "SEM and MDC values were determined for each temporal-spatial, kinematic variability, and dynamic stability measure using Microsoft Excel 2007 (Microsoft Corp., Redmond, WA). ", "All t-tests and *ICC* calculations were performed using SPSS Statistics 19.0 (IBM, [www.spss.com](http://www.spss.com/)).", "\n\nResults {#sec013}\n=======\n\nTemporal-spatial measures {#sec014}\n-------------------------\n\nParticipants walked at an average speed of 1.20 ± 0.04 m/s across all walking conditions. ", "In response to all perturbation conditions, participants tended to walk with increased SW, SW variability, SL variability, and ST variability and decreased SL and ST ([Fig 2](#pone.0142083.g002){ref-type=\"fig\"} and [Table 1](#pone.0142083.t001){ref-type=\"table\"}). ", "Reponses exhibited during COG were smaller than PLAT and VIS. ", "Significant differences (p \\< 0.05) were seen during COG in SW mean, SL variability, and ST variability compared to NOP. ", "During both sessions, the effect sizes for SW mean and SL variability were small to medium, while the ST variability differences exhibited large effect sizes ([Table 1](#pone.0142083.t001){ref-type=\"table\"}). ", "PLAT and VIS conditions elicited the largest responses with mean and variability values for SW, SL, and ST demonstrating significant differences (p \\< 0.05) compared to NOP. ", "In general, the effect sizes for these differences were large during both sessions ([Table 1](#pone.0142083.t001){ref-type=\"table\"}).", "\n\n{#pone.0142083.g002}\n\n10.1371/journal.pone.0142083.t001\n\n###### Temporal-spatial p-values and effect sizes (Cohen's d) for paired unperturbed to perturbed condition comparisons and paired between-session comparisons.", "\n\n![](", "pone.0142083.t001){#pone.0142083.t001g}\n\n p-value \\[Cohen's d\\] NOP1-COG1 NOP2-COG2 NOP1-PLAT1 NOP2-PLAT2 NOP1-VIS1 NOP2-VIS2 NOP1-NOP2 COG1-COG2 PLAT1-PLAT2 VIS1-VIS2\n ------------------------------ -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- ---------------- ---------------- -------------------- --------------------\n Step Width Mean (cm) **0.001 \\[0.35\\]** **0.027 \\[0.17\\]** **0.001 \\[0.85\\]** **0.001 \\[0.72\\]** **0.001 \\[0.82\\]** **0.001 \\[0.56\\]** 0.634 \\[0.04\\] 0.164 \\[0.13\\] 0.082 \\[0.18\\] **0.027 \\[0.30\\]**\n Step Width Variability (cm) 0.466 \\[0.12\\] 0.079 \\[0.21\\] **0.001 \\[5.09\\]** **0.001 \\[3.72\\]** **0.001 \\[1.89\\]** **0.001 \\[1.65\\]** 0.270 \\[0.19\\] 0.302 \\[0.18\\] 0.651 \\[0.04\\] **0.001 \\[0.78\\]**\n Step Length Mean (cm) 0.673 \\[0.04\\] 0.269 \\[0.20\\] **0.001 \\[0.86\\]** **0.003 \\[0.67\\]** **0.001 \\[0.82\\]** 0.049 \\[0.31\\] 0.160 \\[0.23\\] 0.761 \\[0.03\\] 0.585 \\[0.12\\] **0.001 \\[0.45\\]**\n Step Length Variability (cm) **0.049 \\[0.63\\]** **0.048 \\[0.48\\]** **0.001 \\[2.26\\]** **0.001 \\[1.97\\]** **0.001 \\[1.77\\]** **0.001 \\[1.65\\]** 0.163 \\[0.32\\] 0.096 \\[0.40\\] **0.004 \\[0.46\\]** **0.001 \\[0.92\\]**\n Stride Time Mean (s) 0.708 \\[0.04\\] 0.918 \\[0.01\\] **0.001 \\[1.14\\]** **0.001 \\[0.84\\]** **0.00 \\[1.14\\]** **0.001 \\[0.55\\]** 0.772 \\[0.03\\] 0.913 \\[0.01\\] **0.007 \\[0.38\\]** **0.001 \\[0.68\\]**\n Stride Time Variability (s) **0.006 \\[1.03\\]** **0.001 \\[1.04\\]** **0.001 \\[1.27\\]** **0.001 \\[1.37\\]** **0.001 \\[1.49\\]** **0.001 \\[1.12\\]** 0.419 \\[0.24\\] 0.359 \\[0.18\\] **0.039 \\[0.60\\]** **0.001 \\[0.88\\]**\n\nWalking conditions are labeled as NOP--no perturbation, COG--cognitive perturbation, PLAT--physical perturbation, and VIS--visual perturbation. ", "The numbers 1 and 2 indicate conditions performed during session one and two, respectively. ", "Bold values depict significant differences.", "\n\nSignificant between sessions differences (p \\< 0.05) were observed for the VIS condition with a change in all temporal-spatial measures toward NOP values ([Fig 2](#pone.0142083.g002){ref-type=\"fig\"} and [Table 1](#pone.0142083.t001){ref-type=\"table\"}). ", "The effect sizes for these differences ranged from small (d = 0.30) to large (d = 0.92, [Table 1](#pone.0142083.t001){ref-type=\"table\"}). ", "These between-session differences had a negative effect on reliability with VIS demonstrating lower between-session reliability for SW, SW variability, and ST variability compared to the other conditions ([Table 2](#pone.0142083.t002){ref-type=\"table\"}). ", "Overall, temporal-spatial measures during PLAT demonstrated the best reliability with 5 of the 6 measures exhibiting ICC values in the excellent range (0.83--0.95). ", "Further, SW, SW variability, SL, and ST measures demonstrated excellent reliability across all unperturbed and perturbed conditions.", "\n\n10.1371/journal.pone.0142083.t002\n\n###### Temporal-spatial ICC and MDC values during unperturbed and perturbed walking.", "\n\n![](", "pone.0142083.t002){#pone.0142083.t002g}\n\n ICC \\[MDC\\] NOP COG PLAT VIS\n ------------------------------ -------------------- -------------------- -------------------- --------------------\n Step Width Mean (cm) **0.96** \\[0.58\\] **0.96** \\[0.63\\] **0.95** \\[0.81\\] **0.91** \\[1.22\\]\n Step Width Variability (cm) **0.80** \\[0.58\\] **0.83** \\[0.39\\] **0.94** \\[0.50\\] **0.79** \\[1.80\\]\n Step Length Mean (cm) **0.87** \\[1.90\\] **0.97** \\[0.93\\] **0.83** \\[1.88\\] **0.94** \\[1.54\\]\n Step Length Variability (cm) 0.49 \\[0.43\\] 0.54 \\[0.59\\] **0.89** \\[0.42\\] 0.61 \\[1.39\\]\n Stride Time Mean (s) **0.98** \\[0.010\\] **0.96** \\[0.013\\] **0.94** \\[0.031\\] **0.91** \\[0.026\\]\n Stride Time Variability (s) 0.34 \\[0.007\\] 0.65 \\[0.009\\] 0.38 \\[0.013\\] 0.58 \\[0.015\\]\n\nWalking conditions are labeled as NOP--no perturbation, COG--cognitive perturbation, PLAT--physical perturbation, and VIS--visual perturbation. ", "Bold values exhibited excellent between-session reliability (ICC ≥ 0.75).", "\n\nThe significant differences identified between the unperturbed and perturbed conditions were all above the MDCs calculated for each temporal-spatial measure ([Table 2](#pone.0142083.t002){ref-type=\"table\"}). ", "In general, mean temporal-spatial gait measures exhibited lower MDC values compared to variability measures when normalized to group means ([Fig 3](#pone.0142083.g003){ref-type=\"fig\"}). ", "Furthermore, all temporal-spatial gait measures for the PLAT condition consistently demonstrated MDC (percent of mean) values comparable to or less than the NOP condition ([Fig 3](#pone.0142083.g003){ref-type=\"fig\"}).", "\n\n{#pone.0142083.g003}\n\nTrunk variability and stability measures {#sec015}\n----------------------------------------\n\nIn general, participants exhibited similar trunk kinematic variability (i.e. velocity MeanSD) and dynamic stability (i.e. MaxFM and LDE) during the COG and NOP conditions ([Fig 4](#pone.0142083.g004){ref-type=\"fig\"} and [Table 3](#pone.0142083.t003){ref-type=\"table\"}). ", "In contrast, trunk movement variability and instability significantly increased (p \\< 0.003) with PLAT and VIS compared to NOP with medium to large effects sizes ([Fig 4](#pone.0142083.g004){ref-type=\"fig\"} and [Table 3](#pone.0142083.t003){ref-type=\"table\"}). ", "These differences were greatest in the ML direction compared to the AP. ", "Further, significant increases in MaxFM occurred primarily in the oscillation direction (i.e. ML) during PLAT and VIS conditions. ", "In contrast, significant differences in LDE were seen in all 3 principle directions during PLAT and VIS conditions.", "\n\n{#pone.0142083.g004}\n\n10.1371/journal.pone.0142083.t003\n\n###### Trunk movement variability and dynamic stability p-values and effect sizes (Cohen's d) for paired unperturbed to perturbed condition within-session comparisons and paired between-session comparisons.", "\n\n![](", "pone.0142083.t003){#pone.0142083.t003g}\n\n p-value \\[Cohen's d\\] NOP1-COG1 NOP2-COG2 NOP1-PLAT1 NOP2-PLAT2 NOP1-VIS1 NOP2-VIS2 NOP1-NOP2 COG1-COG2 PLAT1-PLAT2 VIS1-VIS2\n ------------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- ---------------- -------------------- -------------------- --------------------\n AP \n Velocity MeanSD (cm/s) 0.281 \\[0.29\\] **0.028 \\[0.29\\]** **0.001 \\[1.84\\]** **0.001 \\[1.22\\]** **0.001 \\[1.76\\]** **0.001 \\[0.79\\]** 0.237 \\[0.41\\] **0.027 \\[0.56\\]** **0.003 \\[1.01\\]** **0.001 \\[1.40\\]**\n MaxFM 0.909 \\[0.03\\] 0.342 \\[0.28\\] 0.700 \\[0.09\\] 0.091 \\[0.46\\] 0.466 \\[0.17\\] 0.668 \\[0.09\\] 0.659 \\[0.09\\] 0.062 \\[0.31\\] **0.021 \\[0.57\\]** 0.101 \\[0.44\\]\n Short-term LDE (λ\\*~S~) 0.076 \\[0.37\\] **0.003 \\[0.41\\]** **0.001 \\[1.04\\]** **0.001 \\[0.67\\]** **0.001 \\[0.98\\]** **0.001 \\[0.73\\]** 0.193 \\[0.30\\] **0.018 \\[0.40\\]** **0.002 \\[0.60\\]** **0.011 \\[0.60\\]**\n Long-term LDE (λ\\*~L~) 0.714 \\[0.08\\] 0.780 \\[0.06\\] **0.001 \\[1.41\\]** **0.001 \\[0.86\\]** **0.001 \\[1.54\\]** **0.001 \\[0.73\\]** 0.741 \\[0.05\\] 0.809 \\[0.04\\] **0.003 \\[0.44\\]** **0.004 \\[0.63\\]**\n ML \n Velocity MeanSD (cm/s) 0.881 \\[0.03\\] 0.818 \\[0.02\\] **0.001 \\[3.00\\]** **0.001 \\[1.58\\]** **0.001 \\[1.93\\]** **0.001 \\[1.93\\]** 0.545 \\[0.19\\] 0.575 \\[0.14\\] 0.055 \\[0.27\\] **0.001 \\[1.02\\]**\n MaxFM **0.002 \\[0.58\\]** 0.835 \\[0.05\\] **0.001 \\[4.09\\]** **0.001 \\[1.70\\]** **0.001 \\[3.34\\]** **0.001 \\[2.27\\]** 0.096 \\[0.55\\] 0.746 \\[0.10\\] **0.006 \\[0.56\\]** **0.001 \\[1.19\\]**\n Short-term LDE (λ\\*~S~) 0.583 \\[0.09\\] 0.431 \\[0.09\\] **0.001 \\[3.41\\]** **0.001 \\[2.40\\]** **0.001 \\[2.96\\]** **0.001 \\[1.94\\]** 0.717 \\[0.09\\] 0.310 \\[0.24\\] 0.164 \\[0.29\\] **0.001 \\[1.12\\]**\n Long-term LDE (λ\\*~L~) 0.860 \\[0.04\\] 0.444 \\[0.21\\] **0.001 \\[1.03\\]** **0.001 \\[1.30\\]** **0.014 \\[0.73\\]** **0.003 \\[0.65\\]** 0.792 \\[0.05\\] 0.180 \\[0.20\\] 0.407 \\[0.20\\] 0.875 \\[0.04\\]\n VT \n Velocity MeanSD (cm/s) 0.206 \\[0.30\\] **0.001 \\[0.14\\]** **0.001 \\[2.91\\]** **0.001 \\[0.72\\]** **0.001 \\[1.85\\]** **0.001 \\[0.35\\]** 0.754 \\[0.10\\] 0.899 \\[0.04\\] 0.501 \\[0.21\\] **0.017 \\[0.80\\]**\n MaxFM 0.134 \\[0.34\\] **0.011 \\[0.61\\]** **0.001 \\[1.06\\]** **0.001 \\[1.00\\]** **0.001 \\[0.69\\]** 0.065 \\[0.34\\] 0.575 \\[0.12\\] **0.005 \\[0.42\\]** 0.507 \\[0.13\\] 0.163 \\[0.28\\]\n Short-term LDE (λ\\*~S~) 0.577 \\[0.08\\] 0.090 \\[0.23\\] **0.001 \\[2.27\\]** **0.001 \\[1.29\\]** **0.001 \\[2.08\\]** **0.005 \\[0.67\\]** 0.840 \\[0.03\\] 0.421 \\[0.16\\] **0.018 \\[0.35\\]** **0.001 \\[1.01\\]**\n Long-term LDE (λ\\*~L~) **0.010 \\[0.80\\]** 0.984 \\[0.01\\] **0.003 \\[1.10\\]** **0.002 \\[1.08\\]** **0.002 \\[1.09\\]** 0.089 \\[0.50\\] 0.229 \\[0.37\\] 0.271 \\[0.30\\] 0.078 \\[0.60\\] **0.001 \\[0.86\\]**\n\nWalking conditions are labeled as NOP--no perturbation, COG--cognitive perturbation, PLAT--physical perturbation, and VIS--visual perturbation. ", "The numbers 1 and 2 indicate conditions performed during session one and two, respectively. ", "Values shown in each direction of motion: anterior-posterior (AP), medial-lateral (ML), and vertical (VT). ", "Bold values depict significant differences.", "\n\nSignificant between sessions differences (p \\< 0.05) were observed for the PLAT and VIS conditions with a change in most variability and stability measures toward NOP values ([Fig 4](#pone.0142083.g004){ref-type=\"fig\"} and [Table 3](#pone.0142083.t003){ref-type=\"table\"}). ", "This change was greatest in the ML direction during Visual perturbations with large effect sizes (d \\> 1.02, [Table 3](#pone.0142083.t003){ref-type=\"table\"}). ", "However, the between-session reliability ([Table 4](#pone.0142083.t004){ref-type=\"table\"}) for trunk velocity MeanSD, MaxFM, and short-term LDE in the ML direction were greater during VIS (ICC \\> 0.70) compared to the NOP condition (ICC \\< 0.59). ", "Overall, between-session reliability for all trunk variability and stability measures improved during the perturbed conditions compared to the NOP.", "\n\n10.1371/journal.pone.0142083.t004\n\n###### Trunk movement variability and dynamic stability ICC and MDC values during unperturbed and perturbed walking.", "\n\n![](", "pone.0142083.t004){#pone.0142083.t004g}\n\n ICC \\[MDC\\] NOP COG PLAT VIS \n ------------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- ------------------- -------------------- -------------------- ---------------- ----------------\n Velocity MeanSD (cm/s) -0.31 \\[1.27\\] 0.04 \\[1.45\\] 0.03 \\[0.49\\] 0.60 \\[0.45\\] 0.51 \\[0.80\\] -0.09 \\[0.38\\] 0.25 \\[1.02\\] **0.87** \\[1.28\\] 0.09 \\[0.65\\] -0.08 \\[1.86\\] 0.70 \\[4.19\\] 0.13 \\[1.06\\]\n MaxFM 0.74 \\[0.051\\] 0.03 \\[0.095\\] 0.73 \\[0.063\\] **0.86** \\[0.038\\] 0.19 \\[0.067\\] **0.90** \\[0.028\\] 0.65 \\[0.033\\] 0.61 \\[0.045\\] **0.76** \\[0.033\\] 0.53 \\[0.047\\] 0.71 \\[0.063\\] 0.70 \\[0.040\\]\n Short-term LDE (λ\\*~S~) 0.66 \\[0.070\\] 0.59 \\[0.048\\] **0.84** \\[0.053\\] **0.86** \\[0.043\\] 0.60 \\[0.049\\] 0.73 \\[0.080\\] **0.83** \\[0.053\\] 0.72 \\[0.040\\] **0.88** \\[0.054\\] 0.70 \\[0.090\\] 0.70 \\[0.067\\] 0.68 \\[0.101\\]\n Long-term LDE (λ\\*~L~) **0.86** \\[0.005\\] **0.80** \\[0.008\\] 0.18 \\[0.012\\] **0.81** \\[0.006\\] **0.88** \\[0.005\\] 0.42 \\[0.011\\] **0.91** \\[0.004\\] 0.60 \\[0.007\\] -0.09 \\[0.017\\] **0.77** \\[0.007\\] 0.49 \\[0.009\\] 0.60 \\[0.017\\]\n\nWalking conditions are labeled as NOP--no perturbation, COG--cognitive perturbation, PLAT--physical perturbation, and VIS--visual perturbation. ", "Values shown in each direction of motion: anterior-posterior (AP), medial-lateral (ML), and vertical (VT). ", "Bold values exhibited excellent between-session reliability (ICC ≥ 0.75).", "\n\nThe significant differences identified between the unperturbed and perturbed conditions were all above the MDCs calculated for each variability and stability measure ([Table 4](#pone.0142083.t004){ref-type=\"table\"}). ", "In general, MaxFM and short-term LDE measure exhibited lower MDC values compared velocity MeanSD and long-term LDE measures when normalized to group means ([Fig 5](#pone.0142083.g005){ref-type=\"fig\"}). ", "Trunk velocity MeanSD, MaxFM and short-term LDE measures for the PLAT condition consistently demonstrated MDC (percent of mean) values comparable to or less than the NOP condition ([Fig 5](#pone.0142083.g005){ref-type=\"fig\"}).", "\n\n{#pone.0142083.g005}\n\nDiscussion {#sec016}\n==========\n\nGait performance during perturbations can be used to assess an individual's ability to effectively respond to challenges. ", "Gait responses identified will often depend on population specific deficits, but can also be strongly affected by the psychometric properties of the gait measure selected. ", "However, it remains unclear which measures most consistently detect corrective responses. ", "Therefore, we systematically determined the between-session reliability and MDC values of commonly reported temporal-spatial, kinematic variability, and dynamic stability measures during three different types of perturbed walking. ", "Interclass correlation coefficient (ICC) and minimum detectable change (MDC) values in healthy young adults were calculated to provide an objective assessment of reliability and facilitate the selection of gait measures during perturbation-based assessments.", "\n\nTemporal-spatial measures {#sec017}\n-------------------------\n\nThe temporal-spatial differences observed between perturbation and NOP conditions were all above calculated MDCs ([Fig 2](#pone.0142083.g002){ref-type=\"fig\"} and [Table 2](#pone.0142083.t002){ref-type=\"table\"}) signifying true gait changes in response to perturbations that were not due to chance. ", "In general, these differences indicated that participants walked with shorter, wider, and quicker steps that were also more variable when exposed to perturbations ([Fig 2](#pone.0142083.g002){ref-type=\"fig\"}). ", "These findings closely match previous reports which used similar visual and physical perturbations in virtual reality environments \\[[@pone.0142083.ref011], [@pone.0142083.ref015]\\]. ", "In contrast, observed responses to COG did not match the results of Grabiner and Troy \\[[@pone.0142083.ref019]\\] who used an identical Stroop task to perturb gait. ", "Specifically, they reported a non-significant 4% decrease (p = 0.10) in mean SW and a significant 16% decrease (p = 0.029) in SW variability while performing the Stroop task during gait. ", "Our participants during session 1 exhibited a significant 8% increase (p = 0.001, d = 0.35) in mean SW compared to NOP with no significant change in SW variability. ", "The addition of optic flow to our Stoop task may have assisted participants in maintaining their gait heading \\[[@pone.0142083.ref059]\\], thus mitigating responses to the Stroop. ", "Also, we postulate that the larger area of our treadmill allowed participants to adopt a larger step width without needing to make corrective steps at the sides of the treadmill. ", "On a narrower treadmill belt with rails (as described in \\[[@pone.0142083.ref010]\\]), participants might potentially need to more tightly control step width and possibly make more corrections to avoid walking off the sides \\[[@pone.0142083.ref019]\\] or touching the hand rails. ", "This illustrates how the physical environment, regardless of the perturbation properties, can influence gait responses and possibly the reliability of their measurement.", "\n\nThe type of perturbation modality used can also affect gait response over time and may change the between-session reliability of temporal-spatial measures. ", "For example, participants walking in a virtual reality environment were reported \\[[@pone.0142083.ref037]\\] to adapt to visual field oscillations in as little as one exposure. ", "In our study, temporal-spatial measures from the second VIS session normalized toward values observed in the NOP condition suggesting a habituation to a repeated exposure of visual field oscillations. ", "The between-session differences observed during the VIS condition contributed to small reductions in ICC and %MDC values. ", "However, all temporal-spatial measures remained significantly different from NOP values indicating continued gait alteration in response to the visual perturbations that were above our calculated MDCs. ", "Despite an expected habituation to visual perturbations, the temporal-spatial measures demonstrated excellent between-session reliability (ICC ≥ 0.75, [Table 2](#pone.0142083.t002){ref-type=\"table\"}) with MDC values sufficient for tracking treatment effects to visual perturbations. ", "Besides their use during assessments, visual perturbations have been used to address balance impairments associated with visuo-spatial deficits in conditions like traumatic brain injury \\[[@pone.0142083.ref060]--[@pone.0142083.ref062]\\]. ", "Our reported MDCs are important as they give clinical researchers a means to objectively interpret differences seen between treatment and assessment sessions.", "\n\nPerturbation-based assessment paradigms and temporal-spatial measures are often used to identify deficits during ambulation often associated with instability \\[[@pone.0142083.ref003], [@pone.0142083.ref017]--[@pone.0142083.ref019]\\]. ", "However, the relationship between gait instability and changes in temporal-spatial mean and variability measures is still unclear. ", "Temporal-spatial mean differences may indicate the presence of compensatory strategies in response to perturbations while changes in variability may represent positive adaptations to destabilizing conditions. ", "Of the temporal-spatial measures reported here, step width mean \\[[@pone.0142083.ref003]\\] and variability \\[[@pone.0142083.ref063]\\] are said to be greater discriminators of instability compared to step length and stride time measures. ", "Older adults who fell, and thus consider unstable, were found to have increased mean step widths compared to younger adults suggesting a compensatory strategy to increase balance by widening base of support \\[[@pone.0142083.ref064]\\]. ", "Further, older adults who fell displayed decreased step width variability compared to older adults with no history of falls (i.e. increased stability) \\[[@pone.0142083.ref064]\\] suggesting a diminished ability to vary responses to destabilizing conditions. ", "In this study, none of our young participants display instability responses which may have led to a fall suggesting that they were capable of producing appropriate compensations and adaptations. ", "Mean measures did exhibited greater ICC and lower %MDC values compared to variability measures across all perturbation types ([Fig 3](#pone.0142083.g003){ref-type=\"fig\"}) which may indicate consistency in the compensatory strategy employed during the perturbations. ", "Specifically, participants continued to walk with shorter, wider, and quicker steps during the second session but variability decreased suggesting that participants did not need vary compensatory strategies as much during the second session. ", "While this may point to learning or adaptation to the perturbation over time, it is important to note that the significant difference between perturbed and NOP conditions above MDC values were still observed in the second session.", "\n\nThe temporal-spatial measures studied here demonstrated excellent between-session reliability (ICC ≥ 0.75) with the exception of SL and ST variability. ", "In the second session compared to the first, SL and ST variability decreased across all conditions resulting in slightly lower ICC values compared to the other measures. ", "The reduction in SL and ST variability may indicate that subjects were able to improve gait speed control to better match treadmill speed. ", "When walking on a treadmill, gait control is thought to be organized around a goal function that produces a gait velocity equal to the fixed treadmill speed \\[[@pone.0142083.ref065], [@pone.0142083.ref066]\\]. ", "A \"Goal Equivalent Manifold\" (GEM) can be defined as all combinations of SL and ST which yield the treadmill speed. ", "Humans have been shown to minimize errors relative to this GEM by making small consistent stride-to-stride changes in SL and ST during non-perturbed \\[[@pone.0142083.ref066]\\] gait in order to remain on the treadmill's surface. ", "In contrast, when young adults walking on a treadmill were challenged with cognitive perturbations, variability in the non-goal-equivalent direction of the GEM increased \\[[@pone.0142083.ref065]\\]. ", "This meant that more combinations of SL and ST that produced velocities not equivalent to the treadmill speed were observed during perturbations. ", "Similarly, we measured a significant increase in SL and ST variability during perturbations compared to the non-perturbed condition signifying gait velocity inconsistencies. ", "During the perturbation conditions, participants were observed to drift backwards on the treadmill belt suggesting a preference to slow their walking velocity. ", "However, they had to speed up once they drifted too far back in order to stay on the treadmill belt. ", "The ability to drift on the treadmill was likely due to its large surface area that allowed for gait speeds over several strides that were slower than the treadmill. ", "The decrease in SL and ST variability during the second session most likely reflected improved gait speed control even during perturbed gait. ", "This improvement may also reflect a learning effect following repeated exposures which enforced speed control in order to maintain a positioning on the treadmill.", "\n\nTrunk variability and dynamic stability measures {#sec018}\n------------------------------------------------\n\nCompared to the temporal-spatial measures, between-session reliability for trunk kinematic variability and dynamic stability measures varied over a greater range of ICC values and were, generally, not as high. ", "Of these measures, short-term (*λ*\\*~*S*~) and long-term (*λ*\\*~*L*~) LDE are the only measures to have reliability and MDC values reported in the literature \\[[@pone.0142083.ref030], [@pone.0142083.ref035], [@pone.0142083.ref036]\\]. ", "Between-session ICC for *λ*\\*~*S*~ were poor to moderate (≤ 0.63) and MDCs ranged from 17% to 46% of their mean with both strongly influenced by the state space reconstruction method utilized \\[[@pone.0142083.ref035]\\]. ", "The best between-session reliability and MDC values achieved were with state space reconstructions using fixed time delays (6, 24 samples), embedding dimensions (7, 9), and 200 strides \\[[@pone.0142083.ref035]\\]. ", "We used a similar state space reconstruction method, albeit with slightly different time delays (15, 10, 30 samples), embedding dimensions (5), and number of strides (124 per trial). ", "With our reconstruction method, we observed between-session ICC values for *λ*\\*~*S*~ that were fair to excellent (≥ 0.59) with MDCs ranging from 3.9% to 7.8% ([Fig 5](#pone.0142083.g005){ref-type=\"fig\"}) across all non-perturbed and perturbed conditions. ", "In addition, we observed between-session ICC values for *λ*\\*~*L*~ that were fair to excellent (0.49--0.91) with %MDCs ranging from 16.6% to 49.1% in ML and AP directions across all conditions. ", "In contrast, a previous report found between-session reliability (ICC: 0.47--0.67) and %MDCs (67--107%) for *λ*\\*~*L*~ to be considerably poorer than *λ*\\*~*S*~ \\[[@pone.0142083.ref036]\\]. ", "We report substantially greater between-session reliability and lower %MDCs for *λ*\\*~*S*~ and *λ*\\*~*L*~ than previously described which may be the result of using velocity versus acceleration \\[[@pone.0142083.ref030], [@pone.0142083.ref035], [@pone.0142083.ref036]\\] data for reconstruction of the state spaces. ", "The characteristics (e.g. variability and noise) are likely different across these kinematic signals which may have improved *λ*\\*~*S*~ and *λ*\\*~*L*~ values reliability \\[[@pone.0142083.ref035]\\]. ", "Between-session reliability and MDCs may have been further improved because we chose to average *λ*\\*~*S*~ and *λ*\\*~*L*~ values across the three trials in each session. ", "Reported ICC values indicate good (≥ 0.67, \\[[@pone.0142083.ref030], [@pone.0142083.ref035]\\]) to excellent (≥ 0.84, \\[[@pone.0142083.ref036]\\]) *λ*\\*~*S*~ within-session reliability which improved as more strides were analyzed \\[[@pone.0142083.ref036]\\] supporting the averaging of *λ*\\*~*S*~ and *λ*\\*~*L*~ during a single session to better estimate local dynamic stability.", "\n\nTrunk kinematic variability (i.e. velocity MeanSD) and orbital stability (i.e. MaxFM) between-session reliability and MDC values have not previously been reported in the literature. ", "Compared to *λ*\\*~*S*~ and *λ*\\*~*L*~, velocity MeanSD and MaxFM between-session reliability and MDCs values were less consistent across non-perturbed and perturbed conditions. ", "In general, velocity MeanSD and MaxFM reliability and MDCs values improved during PLAT and VIS conditions where responses demonstrated directional specificity to walking surface and visual field ML perturbations. ", "In contrast, local stability (i.e. LDE) measures exhibited less directional specificity with similar responses in all directions. ", "These results are in agreement with reports of Floquet multipliers demonstrating greater specificity in their responses to perturbation direction \\[[@pone.0142083.ref012]\\] compared to LDE. ", "Floquet multipliers were also stated to have less sensitivity during walking surface and visual field perturbations \\[[@pone.0142083.ref012]\\] compared to LDE. ", "However, if %MDC is considered a measure of sensitivity to change, MaxFM demonstrated %MDCs of 5.4% and 7.5% in the ML direction during PLAT and VIS ([Fig 5](#pone.0142083.g005){ref-type=\"fig\"}). ", "These values are near equivalent to the %MDCs of *λ*\\*~*S*~ (4.8%, 7.8%) and much smaller than *λ*\\*~*L*~ (25.5%, 33.0%) in the ML direction during PLAT and VIS. ", "Thus, MaxFM demonstrated specificity to perturbation direction and sensitivity that can be used to detect and track changes in response to physical and visual perturbations. ", "Between-session reliability for velocity MeanSD was the highest (ICC = 0.87, 0.70) in the ML direction during PLAT and VIS conditions. ", "However, MDCs were only low during the PLAT condition. ", "Similar to the temporal-spatial measures during the VIS condition, velocity MeanSD MDCs increased possibly due to habituation to the visual oscillations over the two sessions.", "\n\nSimilar to other reports \\[[@pone.0142083.ref020], [@pone.0142083.ref034]\\], trunk kinematic variability and dynamic stability measures demonstrated little sensitivity to cognitive perturbations. ", "We found during COG of the first session that MaxFM in the ML direction was significantly different than NOP. ", "Also, during the second session COG condition, we observed significant decreases in velocity MeanSD and *λ*\\*~*S*~ values. ", "These differences were below our calculated MDCs for each measure suggesting that these differences maybe be the result of biological variation and measurement error. ", "In a similar study using the Stroop as a cognitive perturbation, changes in dynamic stability could only be detected using a substantial number of strides (\\>150) \\[[@pone.0142083.ref034]\\].", "\n\nWhile trunk kinematic variability and dynamic stability measures during the COG condition compared to the NOP did not appear to differ much, between-session reliability and MDC values did improve. ", "The presence of the Stoop task may have offered subjects a more specific point to focus on visually that facilitated their ability to maintain gait heading. ", "This may help explain the higher level of consistency across the two sessions \\[[@pone.0142083.ref059]\\]. ", "Further, young healthy participants have also been shown to prioritize gait at the expense of cognitive performance \\[[@pone.0142083.ref009], [@pone.0142083.ref010], [@pone.0142083.ref019], [@pone.0142083.ref067]\\] when presented with a cognitive challenge during gait. ", "Thus, the low level challenge of the Stroop task may have contributed to the small effects observed \\[[@pone.0142083.ref020]\\] and the improved between-session reliability and MDC values. ", "For future studies, cognitive tasks with internal interfering factors like mental tracking or arithmetic problems may elicit greater effects on gait than those with external interfering factors such as the Stroop task \\[[@pone.0142083.ref068]\\].", "\n\nConclusions {#sec019}\n===========\n\nIn this present study, temporal-spatial, MaxFM, and LDE measures were the most reliable measures with the lowest MDC values across all perturbation types, supporting their use for tracking changes over multiple testing sessions. ", "Of the temporal-spatial measures, SW mean and variability measures demonstrated the greatest response to perturbations with excellent between-session reliability and low MDCs. ", "MaxFM demonstrated specificity to perturbation direction and sensitivity with excellent between-session reliability and low MDC values supporting their use in detecting and tracking changes in response to physical and visual perturbations. ", "We report substantially greater between-session reliability and lower %MDCs for *λ*\\*~*S*~ and *λ*\\*~*L*~ than previously described which may be the result of using velocity versus acceleration \\[[@pone.0142083.ref030], [@pone.0142083.ref035], [@pone.0142083.ref036]\\] data for reconstruction of the state spaces and averaging across trials within a session.", "\n\nThe three perturbation conditions described in this study were chosen because of their frequent use with temporal-spatial, trunk kinematic, and dynamic stability measures in identifying deficits during perturbation-based gait assessments \\[[@pone.0142083.ref011], [@pone.0142083.ref012], [@pone.0142083.ref015], [@pone.0142083.ref018], [@pone.0142083.ref019], [@pone.0142083.ref034]\\]. ", "They represent three types of perturbations (i.e. cognitive, physical, and visual) often described during dual-task studies \\[[@pone.0142083.ref059], [@pone.0142083.ref068]--[@pone.0142083.ref072]\\], but do not encompass all perturbations reported in the literature. ", "Cognitive, physical, and visual perturbation-based assessments each provide unique information for evaluating gait performance, as response magnitude and measure reliability vary by perturbation type.", "\n\nIn this study, we established between-session reliability and MDC values not previously reported for temporal-spatial, trunk kinematic, and dynamic stability measures during perturbed gait. ", "These measures demonstrated fair-excellent reliability across three types of perturbations. ", "Often, between-session reliability and MDC values are specific to the application, presentation environment, instructions given, and subject populations as they are affected by biological variability and methodological error. ", "While investigators will likely want to determine between-session reliability and MDC values for each unique application and population, the values reported here provide normative (i.e. young healthy) reference data to assist in the interpreting of changes observed during perturbed walking in populations (i.e. elderly, non-healthy) with histories of gait instability. ", "Further study would be necessary to determine what specific effects that factors like learning, adaptation, perturbation type/direction, and state space reconstruction methods may have on the reliability of the temporal-spatial, kinematic variability, and dynamic stability measures.", "\n\nThe authors would like to our virtual reality specialist, Michael Vernon, for developing the Computer Assisted Rehabilitation Environment applications used in this study; Mustafa Shinta, Newton Chan, and Emily Sinitksi for assisting with data collection and processing.", "\n\n[^1]: **Competing Interests:**The authors have declared that no competing interests exist. ", "The view(s) expressed herein are those of the author(s) and do not reflect the official policy or position of Brooke Army Medical Center, the U.S. Army Medical Department, the U.S. Army Office of the Surgeon General, the Department of the Army and Department of Defense or the U.S. Government.", "\n\n[^2]: Conceived and designed the experiments: CAR JBD JMW. ", "Performed the experiments: CAR. ", "Analyzed the data: CAR. ", "Contributed reagents/materials/analysis tools: CAR JBD JMW. ", "Wrote the paper: CAR JBD JMW.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "---\nabstract: |\n The Temperley-Lieb (TL) family of algebras is well known for its role in building integrable lattice models. ", "Even though a proof is still missing, it is agreed that these models should go to conformal field theories in the thermodynamic limit and that the limiting vector space should carry a representation of the Virasoro algebra. ", "The fusion rules are a notable feature of the Virasoro algebra. ", "One would hope that there is an analogous construction for the TL family. ", "Such a construction was proposed by Read and Saleur \\[Nucl. ", "Phys. ", "B 777, 316 (2007)\\] and partially computed by Gainutdinov and Vasseur \\[Nucl. ", "Phys. ", "B 868, 223-270 (2013)\\] using the bimodule structure over the Temperley-Lieb algebras and the quantum group Uq(sl2).", "\n\n We use their definition for the dilute Temperley-Lieb (dTL) family, a generalisation of the original TL family. ", "We develop a new way of computing fusion by using induction and show its power by obtaining fusion rules for both dTL and TL. ", "We recover those computed by Gainutdivov and Vasseur and new ones that were beyond their scope. ", "In particular, we identify a set of irreducible TL- or dTL-representations whose behavior under fusion is that of some irreducibles of the CFT minimal models.", "\n\n **Keywords** dilute Temperley-Lieb algebra$\\cdot$Temperley-Lieb algebra$\\cdot$fusion rules$\\cdot$dilute loop models$\\cdot$ Virasoro algebra\naddress: |\n CRM and Département de physique\\\n Université de Montréal\\\n Montréal, QC, Canada, H3C 3J7\nauthor:\n- Jonathan Belletête\ntitle: |\n Fusion rules for the Temperley-Lieb algebra\\\n and its dilute generalisation\n---\n\n[Introduction]{} The Temperley-Lieb family of algebras $\\lbrace {\\mathsf{TL}_{n}}(q) \\rbrace_{n\\in \\mathbb{Z}_{>0}}$, introduced in [@TemperleyLieb], is well-known for its use in building integrable lattice models that correspond to a large variety of different physical systems [@Martin; @Nienhuis], particularly to quantum spin chains. ", "Many properties of these physical models can be interpreted in terms of the algebraic properties of the family, which can be obtained by studying the representation theory of these algebras. ", "As such, it has received a lot of attention over the years. ", "Since its introduction, many generalizations have been proposed: the periodic Temperley-Lieb algebras [@GrahamLehrer; @GreenFan; @Greenseul; @ErdmannGreen], the boundary or blob algebras [@GrimmMartin], the multi-colored Temperley-Lieb algebras [@GrimmPearce], etc... One such generalization which is of particular interest is the dilute Temperley-Lieb family $\\lbrace {\\mathsf{dTL}_{n}}(q) \\rbrace_{n\\in \\mathbb{Z}_{>0}}$ [@Grimm; @BelSY], which has been introduced to build dilute lattice models, i.e., ones where lattice sites can be empty.", "\n\nIt has been conjectured that the lattice models built from ${\\mathsf{TL}_{n}}(q)$ should correspond, in the continuum limit, to conformal field theories [@KooSaleur; @PasSaleur; @PeRasZu]. ", "A consequence to these conjectures is that the Temperley-Lieb family should bear a structure of Virasoro-module when $n$ goes to infinity. ", "In order to study these conjectures, or at least to give them credibility, there has been a lot of interest towards identifying similar algebraic structures between ${\\mathsf{TL}_{n}}$ and the Virasoro algebra, like module structure [@GoodmanWenzl; @Westbury; @RiSY; @BelRiSy] and fusion rules [@ReadSaleur1; @ReadSaleur2; @GainVass; @PeaRasm].", "\n\nFusion rules, from a physical point of view, describe how fields interact at short distance. ", "From a mathematical point of view however, it is a way of defining a product between modules over the algebras underlying the theory. ", "For chiral algebras in CFT’s, these rules have been widely studied, and while defining these rules in terms of functors is relatively simple, computing them explicitly as proven to be very challenging. ", "The recursive algorithm described by Nahm [@Nahm] and developed by Gaberdiel and Kausch [@GabKa] remains the leading tool.", "\n\nOn the Temperley-Lieb family, there has been two main suggestions on how to define and then compute such functors. ", "The first, suggested by Pearce, Rasmussen [@PeaRasm], is built in terms of the lattice models and rely on properties of their transfer matrices instead of relying directly on the algebras. ", "The second, proposed by Read and Saleur [@ReadSaleur1; @ReadSaleur2] and later studied by Gainutnidov and Vasseur [@GainVass], is built around the following description. ", "To compute the fusion product between two spin chains, one joins them together at one of their extremities and then one lets them evolve. ", "While heuristic, they used this idea to build a purely categorical description of the fusion rules which, while motivated from spin chain analysis, rely entirely on algebraic properties of the algebras. ", "This paper will focus on the latter definition.", "\n\nInstead of computing these fusion rules directly, Gainutnidov and Vasseur opted to follow a route closer to how these rules are defined in the Virasoro algebra. ", "There, fusion is defined by first pushing modules to modules over a quantum group, using the co-multiplication on Virasoro, and are then pulled back to modules over the Lie algebra. ", "However, there is no co-multiplication on ${\\mathsf{TL}_{n}}$, so they instead used the quantum Schur-Weyl duality between ${\\mathsf{TL}_{n}}$ and the quantum group $U_{q}(sl_{2})$ [@Jimbo; @SWMartin; @SWCorfWeston]. ", "Modules over ${\\mathsf{TL}_{n}}$ are first pushed to modules over this quantum group, where the co-multiplication naturally defines a fusion product, and the result is then pulled back to ${\\mathsf{TL}_{n}}$. They then argued that the resulting construction was equivalent to Read and Saleur’s original one. ", "Using this argument, they were able to compute fusion rules for most of the main classes of Temperley-Lieb modules[@GainVass].", "\n\nWe are interested in generalizing this construction for the other, more exotic Temperley-Lieb algebras, in particular, the dilute ${\\mathsf{dTL}_{n}}$. While generalizing Read and Saleur’s construction is simple enough, generalizing Gainutnidov and Vasseur’s argument is not, mainly because the duality between ${\\mathsf{dTL}_{n}}$ and $U_{q}(sl_{2})$ is not so clear. ", "Our goal is thus to compute directly this fusion product, without using this duality. ", "We instead rely purely on category theory and the representation theory of ${\\mathsf{TL}_{n}}$ and ${\\mathsf{dTL}_{n}}$.\n\nThe outline of the paper is as follows. ", "In section \\[sec:rappel\\_tl\\], we present a quick overview of the the representation theory of the regular ${\\mathsf{TL}_{n}}$ and the dilute ${\\mathsf{dTL}_{n}}$ families. ", "None of these results are proved here; the reader can consult [@RiSY; @BelSY; @BelRiSy] for their proofs. ", "In section \\[sec:deffusion\\], we present the generalization of Read and Saleur’s construction for general family of algebras and then for dilute and regular cases. ", "A natural consequence of this construction is the existence of a dual product, the *fusion quotient*. ", "Studying this new operation is beyond the scope of this paper but some results are nevertheless presented in appendix \\[sec:fusion\\_quotient\\]. ", "The fusion of projective modules is studied in section \\[sec:fusionproj\\]. ", "These turn out to admit a representation in terms of Chebyshev polynomials of the second kind. ", "In section \\[sec:fusion.standard\\], we study the fusion of standard modules, first with projective modules and then with other standard modules. ", "Fusion rules for irreducible modules are first studied in sections \\[sec:fusionIvsP\\] and \\[sec:fusionIvsS1\\]. ", "These show the appearance of two other classes of modules, the $B$s and the $T$s. ", "The fusion rules for those are studied in sections \\[sec:fusionBvsP\\], \\[sec:fusionBvsS\\] and \\[sec:fusionTvsP\\], \\[sec:fusionTvsS\\], respectively. ", "Fusion rules for pairs of irreducible modules are finally computed in section \\[sec:fusionIvsIfinal\\]. ", "In particular, a subset of irreducible modules is shown to behave under fusion like primary fields in a minimal model of the Virasoro algebra.", "\n\n[Temperley-Lieb algebras]{}\\[sec:rappel\\_tl\\] *The results of this section first appeared in [@GoodmanWenzl; @Martin; @Westbury]. ", "The definitions and results presented here are based on [@RiSY; @BelSY; @BelRiSy].*", "\n\nThe Temperley-Lieb algebras can be defined in terms of generators or in terms of diagrams. ", "The later is presented here and will be used throughout the paper as it gives a more intuitive description of the fusion product. ", "After introducing this definition, the classes of indecomposable modules are introduced in terms of extensions. ", "Loewy diagrams are given and can be used as a quick way of assessing the various properties of these modules. ", "Finally, the algebra’s families are described in terms of the induction and restriction functors. ", "The basic objects, $n$-diagrams, are first introduced. ", "Draw two vertical lines, each with $n$ points on it, $n$ being a positive integer. ", "Choose first $2m$ points, $0\\leq m \\leq n$ an integer, and put a $\\circ$ on each of them. ", "A point with a $\\circ$ will be called a *vacancy*. ", "Now connect the remaining points, pairwise, with non-intersecting strings. ", "The resulting object is called a [*$n$-diagram*]{}. ", "If the diagram contains no vacancy, it is said to be *dense*, and is called *dilute* otherwise. ", "If the number of vacancies on the left side of a $n$-diagram is odd (even), it is called *odd*, (*even*). ", "For example,\n\n$$\\underbrace{\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,5/2);\n \\draw[very thick] (3,-1/2) -- (3,5/2);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\draw (0,2) .. controls (1,2) and (2,0) .. (3,0);\n \\draw (3,1) .. controls (2,1) and (2,2) .. (3,2);\n \\end{tikzpicture}}_{\\text{dense, even $3$-diagram}}, \\qquad\n \\underbrace{\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,7/2);\n \\draw[very thick] (3,-1/2) -- (3,7/2);\n \\draw (0,0) .. controls (1,0) and (2,2) .. (3,2);\n \\draw (3,0) .. controls (2,0) and (2,1) .. (3,1);\n \\draw (0,1) .. controls (1,1) and (1,3) .. (0,3);\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (3,3) circle (1/5);\n \\fill[white] (3,3) circle (1/5)};\n \\end{tikzpicture}}_{\\text{dilute, odd $4$-diagram}}, \\qquad\n \\underbrace{\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,9/2);\n \\draw[very thick] (3,-1/2) -- (3,9/2);\n \\draw (0,0) .. controls (1,0) and (2,2) .. (3,2);\n \\draw (0,1) .. controls (1,1) and (1,4) .. (0,4);\n \\draw (3,3) .. controls (2,3) and (2,4) .. (3,4);\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (0,3) circle (1/5);\n \\fill[white] (0,3) circle (1/5)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\end{tikzpicture}}_{\\text{dilute, even $5$-diagram}}$$\n\nOn the set of formal linear combinations of all $n$-diagrams a product is defined by extending linearly the product of two $n$-diagrams obtained as follows. ", "The two diagrams are put side by side, the inner borders and the points on them are identified, then removed. ", "A string which no longer ties two points is called a *floating string*. ", "A floating string that closes on itself is called a *closed loop*. ", "If all floating strings are closed loops, the result of the product of the two $n$-diagrams is then the diagram obtained by reading the vacancies on the left and right vertical lines and the strings between them multiplied by a factor of $\\beta = q+q^{-1}$, $q$ a non-zero complex number, for each closed loop. ", "Otherwise, the product is the zero element of the algebra. ", "For example, $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,5/2);\n \\draw[very thick] (3,-1/2) -- (3,5/2);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\draw (0,2) .. controls (1,2) and (2,0) .. (3,0);\n \\draw (3,1) .. controls (2,1) and (2,2) .. (3,2);\n \\end{tikzpicture} \\times\n \\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,5/2);\n \\draw[very thick] (3,-1/2) -- (3,5/2);\n \\draw (0,0) .. controls (1,0) and (2,2) .. (3,2);\n \\draw (0,2) .. controls (1,2) and (1,1) .. (0,1);\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\end{tikzpicture} = \\beta\n \\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,5/2);\n \\draw[very thick] (3,-1/2) -- (3,5/2);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\draw (0,2) -- (3,2);\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\end{tikzpicture}$$ If $q$ is a root of unity, the integer $\\ell $ is defined as the smallest strictly positive integer such that $q^{2\\ell}=1$. If $q$ is not a root of unity, $\\ell $ is said to be infinite. ", "A dashed string represents the formal sum of two diagrams: one where the points are linked by a regular string, and one where the points are both vacancies. ", "For example, $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,3)};\n {\\draw[very thick] (3,-1/2) --++ (0,3)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw[dashed] (0,2) --++ (3,0);}\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\node at (4,1) {$=$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,3)};\n {\\draw[very thick] (3,-1/2) --++ (0,3)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (0,2) --++ (3,0)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\node at (4,1) {$+$};\n {\\draw[very thick] (5,-1/2) --++ (0,3)};\n {\\draw[very thick] (8,-1/2) --++ (0,3)};\n {\\draw (5,0) .. controls (6,0) and (6,1) .. (5,1)};\n {\\draw (5,2) circle (1/5);\n \\fill[white] (5,2) circle (1/5)};\n {\\draw (8,2) circle (1/5);\n \\fill[white] (8,2) circle (1/5)};\n {\\draw (8,0) circle (1/5);\n \\fill[white] (8,0) circle (1/5)};\n {\\draw (8,1) circle (1/5);\n \\fill[white] (8,1) circle (1/5)};\n\\end{tikzpicture}\\ \\ .$$ Note that the diagram where each point is linked by a dashed line to the corresponding point on the opposite side acts as the identity on all $n$-diagrams and is a sum of $2^n$ $n$-diagrams.", "\n\nNote finally that the product is clearly associative: the reading of how the left and right sides are connected in a product of three diagrams is blind to the order of glueing, and so is the number of closed loops. ", "The set of $n$-diagrams with the formal sum with complex number coefficients and the product just introduced is the dilute Temperley-Lieb algebra ${\\mathsf{dTL}_{n}}={\\mathsf{dTL}_{n}}(\\beta)$. The subset spanned by only even (odd) diagrams is closed under the product and this subalgebra will be called the even (odd) dilute Temperley-Lieb subalgebra, denoted by ${\\mathsf{edTL}_{n}}$ (${\\mathsf{odTL}_{n}}$). ", "Clearly any dilute $n$-diagram is either even or odd. ", "Since the product of two diagrams of distinct parities is zero, it is clear that the even and odd subalgebras are two-sided ideals of ${\\mathsf{dTL}_{n}}$ and $${\\mathsf{dTL}_{n}}={\\mathsf{edTL}_{n}}\\oplus {\\mathsf{odTL}_{n}}.$$ A module on which every odd (even) diagram acts as zero is called even (odd). ", "It follows that every module can be split into a direct sum of an even, and an odd modules.", "\n\nThe regular Temperley-Lieb algebra ${\\mathsf{TL}_{n}} = {\\mathsf{TL}_{n}}(\\beta)$ is obtained by considering only dense diagrams, that is, those containing no vacancies. ", "As such, every non-zero ${\\mathsf{TL}_{n}}$-module is even. ", "In the case $\\beta =0 $ ($\\ell = 2$), the structure of ${\\mathsf{TL}_{n}}$ will be slightly more complicated than for the other cases. ", "It will thus be treated separately in many calculations and definitions.", "\n\n[The indecomposable modules]{}\\[sec:rappel.indec\\] Since the Temperley-Lieb algebras are finite dimensional associative algebras over the complex numbers, they have finitely many non-isomor phic, irreducible modules. ", "In both algebras, these can be indexed by a single integer $0 \\leq k \\leq n$, which must be of the same parity as $n$ in ${\\mathsf{TL}_{n}}$, and are written ${\\mathsf I_{n,k}}$. The only exception is when $\\ell = 2$ in ${\\mathsf{TL}_{n}}$, where ${\\mathsf I_{n,0}} \\equiv 0$.\n\nThese integers $k$ are first classified in orbits. ", "If $\\ell$ is a finite number, an integer $k \\geq 0$ is said to be *critical*, and is written ${k_c}$ if $k+1 \\equiv 0 \\mod \\ell$. If $\\ell$ is not a finite number, every integer is said to be critical; this is also the case if $\\ell = 1$. For a non-critical integer $k$, define $k_{1}$ to be the smallest non-critical integer strictly bigger than $k$ such that $(k_{1} + k )/2 +1 \\equiv 0 \\mod \\ell$. Similarly, define $k_{-1}$ to be the biggest non-critical integer strictly smaller than $k$ such that $(k_{-1} + k)/2 +1 \\equiv 0 \\mod \\ell$. Define inductively $(k_{i})_{j}= k_{i+j}$, so that for instance $(k_{1})_{-1} = k_{0} = k$. Two integers $r,k$ are then said to be in the same orbit if there exist $i \\in \\mathbb{Z}$ such that $r=k_{i}$; the modules ${\\mathsf I_{n,k}},{\\mathsf I_{n,r}}$ are also said to be on the same orbit. ", "The irreducible modules ${\\mathsf I_{n,{k_c}}}$ are each alone on their orbit. ", "For instance, when $\\ell = 3$, figure \\[fig:orbits\\] shows the orbits between $-3$ and $16$.\n\n; in [-3,...,]{} [ at (/2,0) ; ]{} in [2,8,14]{} [ (/2 -1/2,0) .. controls (/2 -1/2,-1/2) and (/2 +1/2,-1/2) .. (/2+1/2,0); (/2 -1,1/2) .. controls (/2 -1,1) and (/2 +1,1) .. (/2+1,1/2); ]{} in [-1, 5, 11]{} [ (/2 -2/2,0) .. controls (/2 -2/2,-1/2) and (/2 +2/2,-1/2) .. (/2+2/2,0); (/2 -1/2,1/2) .. controls (/2 -1/2,1) and (/2 +1/2,1) .. (/2+1/2,1/2); ]{} in [-1,2,5,8,11,14]{} [ (/2,1/4) circle (8 pt); ]{}\n\n\\[prop:extensions\\] For $0 \\leq r,k \\leq n$, $$\\text{\\rm{Ext}}\\left({\\mathsf I_{n,r}},{\\mathsf I_{n,k}}\\right) \\simeq \\mathbb{C}\\delta_{r,k_{\\pm 1}}.$$\n\nThere is then a unique indecomposable module ${\\mathsf S_{n,k}}$, up to isomorphism[^1], satisfying the short exact sequence $$0 \\longrightarrow {\\mathsf I_{n,k_{1}}} \\longrightarrow {\\mathsf S_{n,k}} \\longrightarrow {\\mathsf I_{n,k}} \\longrightarrow 0.$$ This defines the *standard module* ${\\mathsf S_{n,k}}$. In ${\\mathsf{TL}_{n}}$, when $\\ell =2$, ${\\mathsf I_{n,0}}=0$, so that ${\\mathsf S_{n,0}}\\equiv {\\mathsf I_{n,2}}$. Note also that if $k_{1}>n$, the module ${\\mathsf I_{n,k_{1}}}$ is simply not defined, in which case ${\\mathsf S_{n,k}} \\simeq {\\mathsf I_{n,k}}$. It is generally consistent to set undefined irreducible modules to the zero module; we shall use this convention unless otherwise noted.", "\n\nThere is also a unique indecomposable module ${U_{n,k}}$, satisfying the short exact sequence $$0 \\longrightarrow {\\mathsf I_{n,k}} \\longrightarrow {U_{n,k}} \\longrightarrow {\\mathsf I_{n,k_{1}}} \\longrightarrow 0.$$ This defines the *dual standard module* ${U_{n,k}}$.\n\nLet ${\\mathsf{T}^{1}_{n,k}}= {\\mathsf S_{n,k}}$, then ${\\mathsf{T}^{2i}_{n,k}}$ is defined as the unique indecomposable extension of ${\\mathsf I_{n,k_{2i}}}$ by ${\\mathsf{T}^{2i-1}_{n,k}}$ and ${\\mathsf{T}^{2i+1}_{n,k}}$ as the unique indecomposable extension of ${\\mathsf{T}^{2i}_{n,k}}$ by ${\\mathsf I_{n,k_{2i+1}}}$. Figure \\[fig:LoewyofT\\] shows the Loewy diagrams of the smallest $\\mathsf{T}$ modules.", "\n\n(0,0) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k}}$]{} – (3,-3) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{1}}}$]{}; (0,0) – (1.5,-1.5); at (1.5,-5) [${\\mathsf{T}^{1}_{n,k}}$]{};\n\n(0,0) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k}}$]{} – (3,-3) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{1}}}$]{} – (6,0) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{2}}}$]{}; (0,0) – (1.5,-1.5); (6,0) – (4.5,-1.5); at (3,-5) [${\\mathsf{T}^{2}_{n,k}}$]{};\n\n(0,0) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k}}$]{} – (3,-3) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{1}}}$]{} – (6,0) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{2}}}$]{} – (9,-3) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{3}}}$]{}; (0,0) – (1.5,-1.5); (6,0) – (4.5,-1.5); (6,0) – (7.5,-1.5); at (4.5,-5) [${\\mathsf{T}^{3}_{n,k}}$]{};\n\nSimilarly, let ${\\mathsf{B}^{1}_{n,k}} = {U_{n,k}}$ and define ${\\mathsf{B}^{2i}_{n,k}}$ as the unique indecomposable extension of ${\\mathsf{B}^{2i-1}_{n,k}}$ by ${\\mathsf I_{n,k_{2i}}}$, and ${\\mathsf{B}^{2i+1}_{n,k}}$ as the unique indecomposable extension of ${\\mathsf I_{n,k_{2i+1}}}$ by ${\\mathsf{B}^{2i}_{n,k}}$. Figure \\[fig:LoewyofB\\] shows the Loewy diagrams of a few $\\mathsf{B}$ modules.", "\n\n(0,0) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k}}$]{} – (3,3) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{1}}}$]{}; (3,3) – (1.5,1.5); at (1.5,-2) [${\\mathsf{B}^{1}_{n,k}}$]{};\n\n(0,0) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k}}$]{} – (3,3) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{1}}}$]{} – (6,0) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{2}}}$]{}; (3,3) – (1.5,1.5); (3,3) – (4.5,1.5); at (1.5,-2) [${\\mathsf{B}^{2}_{n,k}}$]{};\n\n(0,0) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k}}$]{} – (3,3) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{1}}}$]{} – (6,0) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{2}}}$]{} – (9,3) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{3}}}$]{}; (3,3) – (1.5,1.5); (3,3) – (4.5,1.5); (9,3) – (7.5,1.5); at (1.5,-2) [${\\mathsf{B}^{3}_{n,k}}$]{};\n\nThe $\\mathsf{P}$ modules are defined a bit differently. ", "In the case ${\\ell}=2$ of ${\\mathsf{TL}_{n}}$, ${\\mathsf P_{0}}$ is the zero module. ", "For all other cases, when $k $ is critical or smaller than $\\ell-1$, ${\\mathsf P_{n,k}}={\\mathsf S_{n,k}}$; otherwise, ${\\mathsf P_{n,k}}$ is the unique indecomposable extension of ${\\mathsf S_{n,k}}$ by ${\\mathsf S_{n,k_{-1}}}$. Figure \\[fig:LoewyofP\\] shows the Loewy diagrams of the $\\mathsf{P}$ modules.", "\n\n(0,0) circle (6pt) node\\[anchor = east\\] [${\\mathsf I_{n,i}}$]{} – (3,-3) circle (6pt) node\\[anchor = west\\] [${\\mathsf I_{n,2\\ell -2 -i}}$]{}; (0,0) – (1.5,-1.5); at (1.5,-4.5) [${\\mathsf P_{n,i}}$]{}; at (1.5,-6) [with $0\\leq i<\\ell -1$]{}; (-9,-1.5) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,{k_c}}}$]{}; at (-9,-4.5) [${\\mathsf P_{n,{k_c}}}$]{}; at (-9,-6) [where ${k_c}+1\\equiv 0 \\mod \\ell$]{}; (9,0) circle (6pt) node\\[anchor = east\\] [${\\mathsf I_{n,{k_c}-j}}$]{} – (12,-3) circle (6pt) node\\[anchor = north\\] [${\\mathsf I_{n,{k_c}+j}}$]{} – (15,0) circle (6pt) node\\[anchor = west\\] [${\\mathsf I_{n,{k_c}+2\\ell -j}}$]{} – (12,3) circle (6pt) node\\[anchor = south\\] [${\\mathsf I_{n,{k_c}+j}}$]{} – (9,0); (12,3) – (10.5,1.5); (12,3) – (13.5,1.5); (9,0) – (10.5,-1.5); (15,0) – (13.5,-1.5); at (12,-4.5) [${\\mathsf P_{n,{k_c}+ j}}$]{}; at (12,-6) [with $0 <j\\leq\\ell-1 $]{};\n\nThese modules satisfy several exact sequences which can all be read from their Loewy diagrams. ", "For example, the short exact sequence $$0 \\longrightarrow {\\mathsf I_{n,k}} \\longrightarrow {\\mathsf{B}^{3}_{n,k}} \\longrightarrow {\\mathsf{T}^{2}_{n,k_{1}}} \\longrightarrow 0,$$ can be seen by noticing that in the Loewy diagram of ${\\mathsf{B}^{3}_{n,k}}$, the part circled in a dashed line is precisely the Loewy diagram of ${\\mathsf{T}^{2}_{n,k_{1}}}$: $$\\begin{tikzpicture}[scale=1/3]\n \\filldraw\n (0,0) circle (6pt) node[anchor = north] {${\\mathsf I_{n,k}}$}\n -- (3,3) circle (6pt) node[anchor = south] {${\\mathsf I_{n,k_{1}}}$}\n -- (6,0) circle (6pt) node[anchor = north] {${\\mathsf I_{n,k_{2}}}$}\n -- (9,3) circle (6pt) node[anchor = south] {${\\mathsf I_{n,k_{3}}}$};\n \\draw[->] (3,3) -- (1.5,1.5);\n \\draw[->] (3,3) -- (4.5,1.5);\n \\draw[->] (9,3) -- (7.5,1.5);\n \\draw[dashed, blue] (1.5,5) -- (10.5,5) -- (10.5,-2) -- (1.5,-2) -- (1.5,5);\n\\end{tikzpicture}$$ The Hom spaces can similarly be read off their diagrams.", "\n\n\\[prop:indec.prop\\]\n\n- The $P$ modules are all projective; they form a complete set of non-isomorphic indecomposable projective modules.", "\n\n- ${\\mathsf P_{n,k}}$ is the projective cover of ${\\mathsf I_{n,k}}$.\n\n- If $f:{\\mathsf{B}^{2i}_{n,k}} \\to \\overset{i}{\\underset{j=0}{\\bigoplus}} {\\mathsf P_{n,k_{2j}}} $ is injective, then $\\operatorname{Coker}f \\simeq {\\mathsf{B}^{2(i+1)}_{n,k_{-1}}}$. If $k_{-1}<0$, there are no such morphism.", "\n\n- If $g:{\\mathsf{T}^{2i+1}_{n,k_{-1}}} \\to \\overset{i}{\\underset{j=0}{\\bigoplus}} {\\mathsf P_{n,k_{2j}}}$ is injective, then $\\operatorname{Coker}g \\simeq {\\mathsf{T}^{2i+1}_{n,k}}$.\n\n- The modules ${\\mathsf P_{n,k}}$ are injective for all $k\\geq \\ell-1$, and the modules ${\\mathsf{B}^{1}_{n,k}}$ are also for all $k<\\ell -1$, except if $\\ell =2$ in ${\\mathsf{TL}_{n}}$ in which case ${\\mathsf{B}^{1}_{n,0}}$ is not injective. ", "They form a complete set of non-isomorphic indecomposable injective modules.", "\n\n- The injective hull of ${\\mathsf I_{n,k}}$ is ${\\mathsf{B}^{1}_{n,k}}$ if $k<\\ell-1$ and ${\\mathsf P_{n,k}}$ otherwise.", "\n\n[A basis of ${\\mathsf S_{n,k}}$]{}\\[sec:rappel.base.standard\\] Our computations will almost all be based on the short-exact sequences satisfied by the various modules and on their homological properties, they will therefore be completely independent of a choice of basis. ", "However, a basis of the standard module ${\\mathsf S_{n,k}}$ will be needed. ", "The bases we present here are the usual ones used in the Temperley-Lieb algebras so the reader should feel free to skip this section if they are already familiar with them.", "\n\nStart by defining the basic objects, the $n$-link diagrams, which are built in the following way. ", "First, take a dilute $n$-diagram and remove its right (left) side as well as the points that were on it. ", "An object, whether it is a string or a vacancy that no longer touches any point, is simply removed. ", "The other floating strings are straightened out and called *defects*. ", "For example, $$\\begin{aligned}\n{3}\n\\begin{tikzpicture}[scale = 1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw[very thick] (3,-1/2) --++ (0,4)}; \n {\\draw (0,3) .. controls (1,3) and (2,1) .. (3,1)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (3,2) .. controls (2,2) and (2,3) .. (3,3)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n \\node at (5,1.5) {$\\rightarrow$ };\n\\end{tikzpicture}&\\quad\n\\begin{tikzpicture}[scale=1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw (0,3) .. controls (1,3) and (2,1) .. (3,1)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (3,2) .. controls (2,2) and (2,3) .. (3,3)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n \\node at (5,1.5) {$\\rightarrow$ };\n\\end{tikzpicture}& \\quad\n\\begin{tikzpicture}[scale=1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw (0,3) --++ (3,0)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n \\node at (4,1.5) {\\ };\n\\end{tikzpicture}\\end{aligned}$$ The resulting diagram is called a [*left $n$-link*]{} ([*right $n$-link*]{}). ", "It is seen that a dilute $n$-diagram induces a unique pair of one left and one right $n$-link diagrams and that, given such a pair, there can be at most one $n$-diagram, if any, that could have induced them. ", "It will thus be useful to denote an $n$-diagram by its induced $n$-links, $b= |l r|$, where $l$ ($r$) is the left (right) link diagram induced from $b$. This notation can also be used for linear combinations of $n$-diagrams as in $b=|(l+j)r| + |uv|$ where $l, j, u$ are left $n$-links and $r, v$ right ones. ", "If $u$ is a left link, then $\\bar u$ will denote its (right) mirror image.", "\n\nA natural action can be defined of $n$-diagrams on left (and right) $n$-link diagrams. ", "We start with the left action. ", "Draw the $n$-diagram on the left side of the left $n$-link, identify the points on its right side with those on the link and remove them. ", "Each floating string that is not connected to the remaining side is removed and yields a factor $\\beta$ if it is closed and zero if it opened, or touches a vacancy. ", "If a floating string starting on the remaining side is connected to a defect in the $n$-link diagram, it becomes a defect. ", "Finally, remove any remaining vacancies on the right side. ", "The remaining drawing is the resulting $n$-link diagram, weighted by factors of $\\beta$, one for each closed floating strings. ", "For example $$\\begin{aligned}\n\\begin{tikzpicture}[scale = 1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw[very thick] (3,-1/2) --++ (0,4)};\n {\\draw (0,3) .. controls (1,3) and (2,1) .. (3,1)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (3,2) .. controls (2,2) and (2,3) .. (3,3)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw[very thick] (4,-1/2) --++ (0,4)};\n {\\draw (4,3) --++ (3,0)};\n {\\draw (4,2) .. controls (5,2) and (5,1) .. (4,1)};\n {\\draw (4,0) circle (1/5);\n \\fill[white] (4,0) circle (1/5)};\n \\node at (8,2) {$=$ };\n\\end{tikzpicture}&\n\\begin{tikzpicture}[scale = 1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw (0,3) .. controls (1,3) and (2,1) .. (3,1)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (3,2) .. controls (2,2) and (2,3) .. (3,3)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,3) --++ (3,0)};\n {\\draw (3,2) .. controls (4,2) and (4,1) .. (3,1)};\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n \\node at (7,2) {$=$ };\n\\end{tikzpicture}\n\\begin{tikzpicture}[scale = 1/3]\n {\\draw[very thick] (0,-1/2) --++ (0,4)};\n {\\draw (0,3) --++ (3,0)};\n {\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1)};\n {\\draw (0,2) circle (1/5);\n \\fill[white] (0,2) circle (1/5)};\n\\end{tikzpicture}\\end{aligned}$$\n\nOver ${\\mathsf{dTL}_{n}}$, the formal sums of all $n$-link diagrams having exactly $k$-defects, with the action defined above, defines a basis of ${\\mathsf S_{n,k}}$.\n\nOver ${\\mathsf{TL}_{n}}$, the formal sums of all $n$-link diagrams having exactly $k$-defects and no vacancies, with the action defined above, defines a basis of ${\\mathsf S_{n,k}}$.\n\n[The Temperley-Lieb families]{}\\[sec:rappel.induction\\] There is a natural inclusion of the symmetric group $\\mathsf{S}_{n}$ into $\\mathsf{S}_{n+1}$. There are similar inclusion for the Temperley-Lieb algebras. ", "Consider the following transformation: take a $n$-diagram and add a dashed line at its bottom. ", "The result is an element of ${\\mathsf{dTL}_{n+1}}$. Similarly, taking a dense $n$-diagram and adding a straight line at its bottom yields a dense $(n+1)$-diagram which is an element of ${\\mathsf{TL}_{n+1}}$. Extending the first transformation linearly gives a subalgebra of ${\\mathsf{dTL}_{n+1}}$ isomorphic to ${\\mathsf{dTL}_{n}}$, while doing the same thing to the second yields a subalgebra of ${\\mathsf{TL}_{n+1}}$ isomorphic to ${\\mathsf{TL}_{n}}$. There are thus two ascending families of algebras $${\\mathsf{dTL}_{1}} \\subset {\\mathsf{dTL}_{2}} \\subset {\\mathsf{dTL}_{3}} \\subset \\hdots, \\qquad \\text{and }{\\mathsf{TL}_{1}} \\subset {\\mathsf{TL}_{2}} \\subset {\\mathsf{TL}_{3}} \\subset \\hdots$$ The functor ${-{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}^{n+1}_{n}$ is the induction functor from ${\\mathsf{dTL}_{n}}$ to ${\\mathsf{dTL}_{n+1}}$, or from ${\\mathsf{TL}_{n}}$ to ${\\mathsf{TL}_{n+1}}$. While this really defines multiple functors, they will have similar properties so we write them all ${-{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}$, unless it is not clear which one we are talking about from the context. ", "The induction functor from a subalgebra $B$ to an algebra $A$ is always a right-exact linear functor defined on all $B$-module $U$ by $${U{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} = A \\otimes_{B} U,$$ where $A$ is seen as a left $A$-module and a right $B$-module, and the index $B$ next to the tensor product sign means that elements of $B$ can pass freely through it.", "\n\nAs the induction functors “moves up\" along the families, its adjoint, the restriction functor ${-{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em}$ “moves down\", taking ${\\mathsf{dTL}_{n+1}}$-modules to ${\\mathsf{dTL}_{n}}$-modules or ${\\mathsf{TL}_{n+1}}$-modules to ${\\mathsf{TL}_{n}}$-modules. ", "The restriction functor from an algebra $A$ to a subalgebra $B$ is always an exact, linear functor defined on an $A$-module $V$ by $${V{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} = \\text{Hom}_{A} \\left( A , V \\right),$$ where $A$ is seen as a left $A$-module and a right $B$-module.", "\n\nThese functors have been computed before for all indecomposable modules over either family of Temperley-Lieb algebras in [@RiSY; @BelSY; @BelRiSy]. ", "These results will be very important for computing the fusion rules and they will be stated where they are needed.", "\n\n[The Fusion ring]{}\\[sec:deffusion\\] Fusion is first defined for left modules over a general family of algebras. ", "This definition is a straightforward generalization of the definition in [@ReadSaleur1; @ReadSaleur2; @GainVass], which works for the regular Temperley-Lieb family. ", "Some general results are then proven before studying fusion in the Temperley-Lieb families.", "\n\n[The fusion product]{} Consider $(A_i)_{i \\in \\mathbb{N}}$ a family of associative algebras over $\\mathbb{C}$ such that for all positive integers $i,j$ the tensor algebra $A_{i} \\otimes_{\\mathbb{C}} A_{j}$ is isomorphic to a subalgebra of $A_{i+j}$. The tensor algebra $A_{i} \\otimes_{\\mathbb{C}} A_{j}$ is defined such that $(a \\otimes b)(c\\otimes d) = ac \\otimes bd$ for all $a,c \\in A_{i}$ and all $b,d \\in A_{j}$. Given $U$ a $A_{i}$-module and $V$ a $A_{j}$-module, the *fusion* of $U$ and $V$ is defined as $$U {\\times\\hspace{-0.27em}_f}V = A_{i+j} \\otimes_{A_{i}\\otimes_{\\mathbb{C}} A_{j}}\\left( U \\otimes_{\\mathbb{C}} V \\right)\\text{.}$$ Note that $U \\otimes_{\\mathbb{C}} V$ is naturally a $A_{i}\\otimes_{\\mathbb{C}} A_{j}$-module. ", "The fusion can thus be seen as a simple induction from $A_{i}\\otimes_{\\mathbb{C}} A_{j}$ to $A_{i+j}$ and, hence, $U {\\times\\hspace{-0.27em}_f}V$ is an $A_{i+j}$-module. ", "Note that to each induction functor corresponds an adjoint restriction functor. ", "As such, there exists a construction adjoint to the fusion product which is called the *fusion quotient*. ", "This construction will only be used while computing the fusion product of irreducible modules, and the argument to obtain the needed fusion quotients are slightly different from those used to compute fusion products. ", "These results will therefore be presented in appendix \\[sec:fusion\\_quotient\\].", "\n\nThe following propositions follow readily from the properties of tensor products.", "\n\n\\[prop:fusion\\_resproj\\] For $U$,$V$ two $A_{i}$-modules and $W$,$Z$ two $A_{j}$-modules, $$(U\\oplus V){\\times\\hspace{-0.27em}_f}(W\\oplus Z) \\simeq (U{\\times\\hspace{-0.27em}_f}W) \\oplus (U {\\times\\hspace{-0.27em}_f}Z) \\oplus (V{\\times\\hspace{-0.27em}_f}W) \\oplus (V{\\times\\hspace{-0.27em}_f}Z) \\text{.}$$Furthermore, if $U$ and $W$ are projective then so is $U{\\times\\hspace{-0.27em}_f}W$.\n\nThe first result follows readily from the linearity of tensor products.", "\n\nSuppose now that $U$ and $W$ are two projective $A_{i}$- and $A_{j}$-modules respectively. ", "By definition, this means that there are two sets $\\Lambda$ and $\\Sigma$ and two projective modules $P$ and $Q$ such that $A^{\\Lambda}_{i} \\simeq U \\oplus P$ and $A^{\\Sigma}_{j}\\simeq W \\oplus Q$. Here $A^{\\Lambda}_{i}$ is a direct sum of copies of $A_{i}$ indexed by the elements of $\\Lambda$ and similarly for $A^{\\Sigma}_{j}$. Using the first result, $$A^{\\Lambda}_{i} {\\times\\hspace{-0.27em}_f}A^{\\Sigma}_{j} \\simeq (U {\\times\\hspace{-0.27em}_f}W) \\oplus (U{\\times\\hspace{-0.27em}_f}Q) \\oplus (P{\\times\\hspace{-0.27em}_f}W) \\oplus (P{\\times\\hspace{-0.27em}_f}Q) \\simeq A^{\\Gamma}_{i+j},$$ were $\\Gamma$ is a set whose elements are the pairs $(\\lambda, \\sigma)$ with $\\lambda \\in \\Lambda$,$\\sigma \\in \\Sigma$. The second equality is obtained by noting that the induction to an algebra $A$ of a subalgebra $B$ is always isomorphic to $A$. ", "Since $A^{\\Gamma}_{i+j} $ is a free module by definition, $U{\\times\\hspace{-0.27em}_f}W$ is projective.", "\n\n\\[prop:fusionexact\\] If the sequence $$0 \\longrightarrow U \\overset{f}{\\longrightarrow} V \\overset{g}{\\longrightarrow} W \\longrightarrow 0$$ of $A_{i}$-modules is exact, the sequence of $A_{i+j}$-modules $$U{\\times\\hspace{-0.27em}_f}S \\overset{\\bar{f}}{\\longrightarrow} V{\\times\\hspace{-0.27em}_f}S \\overset{\\bar{g}}{\\longrightarrow} W{\\times\\hspace{-0.27em}_f}S \\longrightarrow 0$$ is also exact for all $A_{j}$-modules $S$.\n\nNote that $\\mathbb{C}$ is semi-simple, so that all $\\mathbb{C}$-modules are flat. ", "The sequence of $A_{i}\\otimes_{\\mathbb{C}} A_{j}$-modules $$0 \\longrightarrow U \\otimes_{\\mathbb{C}} S \\overset{f \\otimes_{\\mathbb{C}} \\text{id}_{A_{j}}}{\\longrightarrow} V\\otimes_{\\mathbb{C}} S \\overset{g\\otimes_{\\mathbb{C}} \\text{id}_{A_{j}}}{\\longrightarrow} W\\otimes_{\\mathbb{C}} S \\longrightarrow 0 ,$$ is therefore exact. ", "The conclusion is obtained by using the fact that induction is right-exact.", "\n\nIt should also be noted that for any $A_{i}$-module $U$, $$U {\\times\\hspace{-0.27em}_f}A_{j} = A_{i+j} \\otimes_{A_{i}\\otimes_{\\mathbb{C}} A_{j}} U \\otimes_{\\mathbb{C}} A_{j} \\simeq A_{i+j} \\otimes_{A_{i}} U \\text{,}$$ which is simply the induction functor from $A_{i}$ to $A_{i+j}$. Note also that just like the induction functor, it will depend on the actual embedding $A_{i} \\to A_{i+j}$.\n\n[Fusion on the dilute Temperley-Lieb family]{} Of the many ways of including ${\\mathsf{dTL}_{n}}$ as a subalgebra of ${\\mathsf{dTL}_{n+p}}$, we focus on two. ", "The first is to insert $p$ dashed lines at the bottom of every diagram in ${\\mathsf{dTL}_{n}}$ and the other is to add them at the top. ", "The simplest way to define the inclusion of ${\\mathsf{dTL}_{n}}\\otimes{\\mathsf{dTL}_{p}}$ in ${\\mathsf{dTL}_{n+p}}$ is thus to draw the diagram $a\\in {\\mathsf{dTL}_{n}}$ on top of $b\\in{\\mathsf{dTL}_{p}}$. For example, $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n\\draw[very thick] (0,-3/2) -- (0,3/2);\n\\draw[very thick] (3,-3/2) -- (3,3/2);\n \\draw (0,-1) .. controls (1,-1) and (2,1) .. (3,1);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n {\\draw (3,0) circle (1/5);\n \\fill[white] (3,0) circle (1/5)};\n {\\draw (3,-1) circle (1/5);\n \\fill[white] (3,-1) circle (1/5)};\n\\node at (4,0) {$\\otimes$};\n\\draw[very thick] (5,-2) -- (5,2);\n\\draw[very thick] (8,-2) -- (8,2);\n \\draw (5,-3/2) .. controls (6,-3/2) and (7,-1/2) .. (8,-1/2);\n \\draw (5,-1/2) .. controls (6,-1/2) and (6,3/2) .. (5,3/2);\n {\\draw (5,1/2) circle (1/5);\n \\fill[white] (5,1/2) circle (1/5)};\n {\\draw (8,-3/2) circle (1/5);\n \\fill[white] (8,-3/2) circle (1/5)};\n \\draw (8,1/2) .. controls (7,1/2) and (7,3/2) .. (8,3/2);\n\\end{tikzpicture}\n\\longrightarrow\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n\\draw[very thick] (0,-7/2) -- (0,7/2);\n\\draw[very thick] (3,-7/2) -- (3,7/2);\n \\draw (0,1) .. controls (1,1) and (2,3) .. (3,3);\n \\draw (0,2) .. controls (1,2) and (1,3) .. (0,3);\n {\\draw (3,2) circle (1/5);\n \\fill[white] (3,2) circle (1/5)};\n {\\draw (3,1) circle (1/5);\n \\fill[white] (3,1) circle (1/5)};\n \\draw (0,-3) .. controls (1,-3) and (2,-2) .. (3,-2);\n \\draw (0,-2) .. controls (1,-2) and (1, 0) .. (0,0);\n {\\draw (0,-1) circle (1/5);\n \\fill[white] (0,-1) circle (1/5)};\n {\\draw (3,-3) circle (1/5);\n \\fill[white] (3,-3) circle (1/5)};\n \\draw (3,-1) .. controls (2,-1) and (2,0) .. (3,0);\n\\end{tikzpicture}.$$ Notice that we could have defined it the other way around, drawing $b$ on top of $a$. It can be shown that the two inclusions yield isomorphic bi-module structures on ${\\mathsf{dTL}_{n+m}}$. It follows that fusion is commutative on the dilute Temperley-Lieb family.", "\n\nFor $U,V$, modules over ${\\mathsf{dTL}_{n}}$ and ${\\mathsf{dTL}_{p}}$, respectively, $$U{\\times\\hspace{-0.27em}_f}V \\simeq V{\\times\\hspace{-0.27em}_f}U.$$\n\nNote that the inclusion used is compatible with the parity of diagrams. ", "Take $a,b$ two diagrams with well-defined parity in ${\\mathsf{dTL}_{n}}$ and ${\\mathsf{dTL}_{m}}$, respectively. ", "If $a$ is odd but $b$ is even, $a \\otimes b$ is odd while if they are both odd or even, $a\\otimes b$ is even. ", "It follows that fusing two modules with the same parity yields an even module while if their parities are different it yields an odd one. ", "Note also that fusing a module with ${\\mathsf{dTL}_{1}}$ gives the induction of this module as defined in \\[sec:rappel.induction\\]. ", "Since ${\\mathsf{dTL}_{1}}\\simeq {\\mathsf P_{1,1}} \\oplus {\\mathsf P_{1,0}}$ the following proposition is obtained.", "\n\n\\[prop:fusion\\_parity\\] For a ${\\mathsf{dTL}_{n}}$-module $V$ with a well-defined parity, $$V{\\times\\hspace{-0.27em}_f}{\\mathsf{dTL}_{1}} \\simeq {V{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq V{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\oplus V {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}},$$ $V{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}$ has the same parity as $V$, while $V{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}}$ has a different parity.", "\n\nFurthermore, tensor products are associative and it is easy to verify that the chosen inclusion process is also. ", "It thus follows that the fusion algebra of the dilute Temperley-Lieb family is associative.", "\n\nFor $U$ a ${\\mathsf{dTL}_{n}}$-module, $V$ a ${\\mathsf{dTL}_{m}}$-module and $W$ a ${\\mathsf{dTL}_{p}}$-module, $$\\left(U {\\times\\hspace{-0.27em}_f}V\\right){\\times\\hspace{-0.27em}_f}W \\simeq U{\\times\\hspace{-0.27em}_f}\\left(V {\\times\\hspace{-0.27em}_f}W \\right).$$\n\n[Fusion on the regular Temperley-Lieb algebra]{} Fusion for the regular Temperley-Lieb family is very similar to that on the dilute family. ", "Again inclusion of ${\\mathsf{TL}_{n}}$ in ${\\mathsf{TL}_{n+p}}$ can be obtained by adding straight lines below or above an $n$-diagram and inclusion of ${\\mathsf{TL}_{n}}\\otimes{\\mathsf{TL}_{p}}$ in ${\\mathsf{TL}_{n+p}}$ by drawing $n$-diagrams atop $p$-diagrams. ", "For example, $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n\\draw[very thick] (0,-3/2) -- (0,3/2);\n\\draw[very thick] (3,-3/2) -- (3,3/2);\n \\draw (0,-1) .. controls (1,-1) and (2,1) .. (3,1);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\draw (3,-1) .. controls (2,-1) and (2,0) .. (3,0);\n\\node at (4,0) {$\\otimes$};\n\\draw[very thick] (5,-1) -- (5,1);\n\\draw[very thick] (8,-1) -- (8,1);\n \\draw (5,-1/2) .. controls (6,-1/2) and (6,1/2) .. (5,1/2);\n \\draw (8,-1/2) .. controls (7,-1/2) and (7,1/2) .. (8,1/2);\n\\end{tikzpicture}\n\\to\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n\\draw[very thick] (0,-5/2) -- (0,5/2);\n\\draw[very thick] (3,-5/2) -- (3,5/2);\n \\draw (0, -2) .. controls (1,-2) and (1,-1) .. (0,-1);\n \\draw (0,0) .. controls (1,0) and (2,2) .. (3,2);\n \\draw (0,1) .. controls (1,1) and (1,2) .. (0,2);\n \\draw (3,-2) .. controls (2,-2) and (2,-1) .. (3,-1);\n \\draw (3,0) .. controls (2,0) and (2,1) .. (3,1);\n\\end{tikzpicture}$$ The definition mimics very closely that on the dilute family and the proofs of the various results will be nearly identical. ", "In particular, the same arguments yields the following proposition.", "\n\n\\[prop:fusion\\_regular\\] For $U$ a ${\\mathsf{TL}_{n}}$-module, $V$ a ${\\mathsf{TL}_{m}}$-module and $W$ a ${\\mathsf{TL}_{k}}$-module, $$U {\\times\\hspace{-0.27em}_f}{\\mathsf{TL}_{1}} \\simeq U {\\times\\hspace{-0.27em}_f}{\\mathsf S_{1,1}} \\simeq {U{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em},$$ $$U {\\times\\hspace{-0.27em}_f}V \\simeq V{\\times\\hspace{-0.27em}_f}U,$$ $$\\left(U {\\times\\hspace{-0.27em}_f}V\\right){\\times\\hspace{-0.27em}_f}W \\simeq U{\\times\\hspace{-0.27em}_f}\\left(V {\\times\\hspace{-0.27em}_f}W \\right).$$\n\n[Fusion of projective modules]{}\\[sec:fusionproj\\] It was proved in proposition \\[prop:fusion\\_resproj\\] that the fusion of two projective modules always yields a projective module. ", "Since the projective modules of the Temperley-Lieb algebras are all known, it is natural to start by computing their fusion rules. ", "The projective indecomposables of ${\\mathsf{dTL}_{n}}$ and ${\\mathsf{TL}_{n}}$ falls in three different classes (see section \\[sec:rappel.indec\\]), the standard modules ${\\mathsf S_{n,k}}={\\mathsf P_{n,k}}$ with $k<{\\ell}-1$, which we will often call the *small projectives*, the standard modules ${\\mathsf S_{n,{k_c}}}={\\mathsf P_{n,{k_c}}}$ where ${k_c}$ is critical and the projective indecomposable ${\\mathsf P_{n,{k_c}+i}}$ for $0<i<{\\ell}$. We use the same notation for the two families, but recall that in ${\\mathsf{dTL}_{n}}$, modules such as ${\\mathsf P_{n,k}}$ are defined for all integer $k \\in [0,n]$, while in ${\\mathsf{TL}_{n}}$, they are only defined when $k \\equiv n \\mod 2$. Propositions \\[prop:fusion\\_parity\\] and \\[prop:fusion\\_regular\\] show that fusion is closely related to the process of induction; the following proposition gives the induction of projective modules [@BelSY; @BelRiSy].", "\n\nFor all critical ${k_c}$, $0<i<{\\ell}$, and $n-1\\geq {k_c}+i$, $${{\\mathsf P_{n-1,i-1}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\begin{cases}\n{\\mathsf P_{n,i-2}} \\oplus {\\mathsf P_{n,i-1}} \\oplus {\\mathsf P_{n,i}} \\text{,} & \\text{ on } {\\mathsf{dTL}_{n}}\\\\\n{\\mathsf P_{n,i-2}} \\oplus {\\mathsf P_{n,i}} , & \\text{ on } {\\mathsf{TL}_{n}}\n\\end{cases},$$ $${{\\mathsf P_{n-1,{k_c}}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\begin{cases}\n {\\mathsf P_{n,{k_c}}} \\oplus {\\mathsf P_{n,{k_c}+1}} \\text{,} & \\text{ on } {\\mathsf{dTL}_{n}}\\\\\n{\\mathsf P_{n,{k_c}+1}} , & \\text{ on } {\\mathsf{TL}_{n}}\n \\end{cases},$$ $$\\begin{gathered}\n{{\\mathsf P_{n-1,{k_c}+i}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\left. ", "\\begin{cases}\n{\\mathsf P_{n,{k_c}+i}},& \\text{ on } {\\mathsf{dTL}_{n}}\\\\\n0, & \\text{ on } {\\mathsf{TL}_{n}}\n\\end{cases} \\right\\}\n \\oplus \\left.\\begin{cases}\n{\\mathsf P_{n,{k_c}}} \\oplus {\\mathsf P_{n,{k_c}}}, & \\text{ if } i=1\\\\\n{\\mathsf P_{n,{k_c}+i-1}}, & \\text{ otherwise}\n\\end{cases}\\right\\} \\\\\n\\oplus \\left.\\begin{cases}\n{\\mathsf P_{n,{k_c}-{\\ell}}} \\oplus {\\mathsf P_{n,{k_c}+ {\\ell}}}, & \\text{ if } i={\\ell}-1\\\\\n {\\mathsf P_{n,{k_c}+i+1}}, & \\text{ otherwise}\n \\end{cases}\\right\\},\\end{gathered}$$ where it is understood that ${\\mathsf P_{n,j}} \\simeq 0$ if $j<0$.\n\nProposition \\[prop:fusion\\_parity\\] described how fusion behaves regarding parity of modules: the fusion of two odd or even modules yields an even module while the fusion of an odd and an even module yields an odd one. ", "A projective module ${\\mathsf P_{n,k}}$ is odd (even) if $n-k$ is odd (even); the following proposition is thus easily proven.", "\n\n\\[prop\\_fusion\\_pvs01\\] For all critical ${k_c}$, $0<i<{\\ell}$, and $n-1\\geq {k_c}+i$, on the dilute family $${\\mathsf P_{n-1,i-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf P_{n,i-1}} \\text{,} \\qquad {\\mathsf P_{n-1,{k_c}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf P_{n,{k_c}}},$$ $${\\mathsf P_{n-1,{k_c}+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf P_{n,{k_c}+i}} \\text{,}$$ and in both families $${\\mathsf P_{n-1,i-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf P_{n,i-2}} \\oplus {\\mathsf P_{n,i}}\\text{,}\\qquad {\\mathsf P_{n-1,{k_c}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf P_{n,{k_c}+1}} \\text{,}$$ $${\\mathsf P_{n-1,{k_c}+i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq \\begin{cases}\n{\\mathsf P_{n,{k_c}}} \\oplus {\\mathsf P_{n,{k_c}}} & \\text{ if } i=1\\\\\n{\\mathsf P_{n,{k_c}+i-1}} & \\text{ otherwise}\n\\end{cases} \\oplus \\begin{cases}\n{\\mathsf P_{n,{k_c}-{\\ell}}} \\oplus {\\mathsf P_{n,{k_c}+ {\\ell}}} & \\text{ if } i={\\ell}-1\\\\\n {\\mathsf P_{n,{k_c}+i+1}} & \\text{ otherwise}\n \\end{cases}.$$\n\nIt follows from the previous proposition together with the linearity of fusion, the breakdown according to parity and the fact that ${\\mathsf{dTL}_{1}} \\simeq {\\mathsf P_{1,0}} \\oplus {\\mathsf P_{1,1}}$ and ${\\mathsf{TL}_{1}} \\simeq {\\mathsf P_{1,1}}$.\n\nFor all projective modules in the dilute family, fusion of projectives with ${\\mathsf P_{1,0}}$ simply increases the parameter $n$ by one. ", "Since fusion is associative, fusions can be computed using the smallest $n$ for which the modules make sense, and fuse the result with the appropriate number of ${\\mathsf P_{1,0}}$ needed to reach the required $n$. For instance $${\\mathsf P_{10,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{6,4}} \\simeq {\\mathsf P_{1,0}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{9,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{6,4}}) \\simeq {\\mathsf P_{2,0}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{8,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{6,4}}) \\simeq \\hdots \\simeq {\\mathsf P_{9,0}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{3,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4,4}}) .$$ In the regular family, this role is played by ${\\mathsf P_{2,0}}$, when ${\\ell}\\neq 2$. Then $${\\mathsf P_{2,0}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{n,p}} \\simeq {\\mathsf P_{n+2,p}},$$ for all $p$. The proof is much more involved and based on diagrammatic arguments; it is presented in appendix \\[sec:magicp2\\]. ", "When ${\\ell}=2$, it will be proved as a corollary of proposition \\[prop:fusionIvsP\\] that this role is played by ${\\mathsf I_{4,2}} \\simeq {\\mathsf S_{4,0}}$. The results could therefore depend on the parity of $n/2$. Nevertheless, most of our proofs will be independent of $n$, so we will simply write ${\\mathsf P_{p}} = {\\mathsf P_{n,p}}$ and assume that $n$ is big enough for the module to exist. ", "Proofs where $n$ is important will be dealt with separately.", "\n\n[The fusion matrix]{}\n\nFor a projective module $\\mathsf{P}$, define the *fusion matrix* $\\text{F}(\\mathsf{P})$ by $$\\mathsf{P} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq \\bigoplus_{k} (\\text{F}(\\mathsf{P}))_{j}^{k} {\\mathsf P_{k}}$$ where it is understood that a non-negative integer multiple of a module stands for that many copies of this module. ", "To simplify the notation, $k$ is allowed to run over all non-negative integers, but it is assumed that ${\\mathsf P_{n,k}}\\simeq 0$ when $k >n$, or when $k \\not\\equiv n \\mod 2$ in the regular family. ", "Define also $X = \\text{F}({\\mathsf P_{1,1}})$, ${F_{i}} = \\text{F} ({\\mathsf P_{i}})$ and write $({F_{i}})^{k}_{j} = {F_{i,j}}^{k}$. This definition will reduce the computation of fusion rules to simple products of matrices. ", "Note that since fusion is commutative, ${F_{i,j}}^{k} = {F_{j,i}}^{k}$.\n\nProposition \\[prop\\_fusion\\_pvs01\\] already gives the fusion matrices of ${\\mathsf P_{0}}$ and ${\\mathsf P_{1}}$: $${F_{0,i}}^{j} = \\delta_{i,j},$$ $${F_{1,i}}^{j} = \\left. ", "\\begin{cases}\n\\delta_{j,i+1}, & \\textrm{if } i=0 \\text{ or } i+1 \\equiv 0 \\mod {\\ell}\\\\\n2\\delta_{j,i-1} + \\delta_{j,i+1}, & \\textrm{if } i \\equiv 0\\mod {\\ell}\\text{ and } {\\ell}\\neq 2\\\\\n\\delta_{j,i-1} + \\delta_{j,i+1} + \\delta_{j, i-2{\\ell}+1}, & \\textrm{if } i>{\\ell}-1 \\text{ and } i+2 \\equiv 0\\mod {\\ell}\\text{ and } {\\ell}\\neq 2\\\\\n2\\delta_{j,i-1} + \\delta_{j,i+1} + \\delta_{j, i-2{\\ell}+1}, & \\textrm{if } i>{\\ell}-1 \\text{ and } i+2 \\equiv 0\\mod {\\ell}\\text{ and } {\\ell}=2 \\\\\n\\delta_{j,i-1} + \\delta_{j,i+1}, & \\textrm{otherwise}\n\\end{cases}\\right\\}$$ where $\\delta_{i,j}$ is the Kronecker delta.", "\n\nThe following proposition shows that a finite projective module is uniquely determined by its fusion matrix.", "\n\n\\[prop:fmatrixunique\\] For $P$,$Q$ two finite projective ${\\mathsf{dTL}_{n}}$- or ${\\mathsf{TL}_{n}}$-modules, $$\\text{F}(P) = \\text{F}(Q) \\iff P \\simeq Q.$$\n\nEvery finite projective module is isomorphic to a direct sum of principal indecomposable modules. ", "For a projective module $T$, define the set $\\alpha(T)$ as the set of integers such that $$T \\simeq \\oplus_{i \\in \\alpha(T)} {\\mathsf P_{i}},$$ where each integer can occur more than once. ", "Define $i(T)$ as the maximum of $\\alpha(T)$, and $\\# i(T)$ as the number of times this maximum appears. ", "From proposition \\[prop\\_fusion\\_pvs01\\], it is clear that $ i(T{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}) = i(T) + 1$ and $ \\# i(T{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}) = \\# i(T)$.\n\nNow, for $P,Q$ two projective ${\\mathsf{dTL}_{n}}$- or ${\\mathsf{TL}_{n}}$-modules, if $\\text{F}(P) = \\text{F}(Q)$, then in particular $ P {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq Q{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}$. Thus $i(P{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}) = i(Q{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}})$ and $\\# i(P{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}) = \\# i(Q{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}})$. Therefore $i(P)= i(Q)$, $\\# i(P)=\\# i(Q)$, and $$P \\simeq P' \\oplus \\# i(P){\\mathsf P_{i(P)}}, \\qquad Q \\simeq Q' \\oplus \\# i(P) {\\mathsf P_{i(P)}},$$ where $i(P') < i(P)$ and $i(Q')<i(Q)$. Since fusion is linear, $ P' {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq Q'{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}$. Proceeding by recursion on the cardinality of $i(P)$, the result is obtained.", "\n\n[Fusion matrices of small projectives]{}\\[sec:fmatrixsmall\\] By using the formulas in proposition \\[prop\\_fusion\\_pvs01\\], for $0 \\leq i<{\\ell}-1$ and all $j$, $${\\mathsf P_{1}} {\\times\\hspace{-0.27em}_f}( {\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}}) \\simeq ({\\mathsf P_{i-1}} \\oplus {\\mathsf P_{i+1}}) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\text{.}$$ In terms of the fusion matrices, this is simply $$\\sum_{m}{F_{1,m}}^{p} {F_{i,j}}^{m} = {F_{i-1,j}}^{p} + {F_{i+1,j}}^{p}$$ and this gives the recurrence relation $$X{F_{i}} = {F_{i-1}} + {F_{i+1}} \\text{,}\\qquad {F_{0}} = \\text{id} \\text{, } {F_{1}} = X ,$$ where $$(X{F_{i}})^{p}_{j} = \\sum_{m=0}^{n} X_{m}^{p}({F_{i}})^{m}_{j}=\\sum_{m=0}^{n} {F_{1,m}}^{p} {F_{i,j}}^{m}.$$ One should recognize the recurrence relation[^2] of Chebyshev polynomials of the second kind ${U_{i}}\\left(\\frac{X}{2}\\right)$ and thus find $${F_{i}} = {U_{i}}\\left(\\frac{X}{2}\\right) \\text{,}\\qquad 0\\leq i \\leq l-1 \\text{.}$$ Since the matrix $X$ is known, this can be used in principle to compute the fusion matrix of all small projectives. ", "Note that this proof fails when ${\\ell}=2$ on the regular family because in this case, there are no small projectives.", "\n\n[Fusion matrices for the indecomposable projective ${\\mathsf P_{{k_c}+i}}$]{} Using again proposition \\[prop\\_fusion\\_pvs01\\], for $0\\leq i \\leq {\\ell}-1$ $${\\mathsf P_{1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{k_c}+i}} \\simeq \n\\left. ", "\\begin{cases}\n0, & \\text{ if } i=0 \\\\\n{\\mathsf P_{{k_c}}} \\oplus {\\mathsf P_{{k_c}}} & \\text{ if } i=1\\\\\n{\\mathsf P_{{k_c}+i-1}} & \\text{ otherwise}\n\\end{cases}\\right\\} \\oplus \\left. ", "\\begin{cases}\n{\\mathsf P_{{k_c}-{\\ell}}} \\oplus {\\mathsf P_{{k_c}+ {\\ell}}} & \\text{ if } i={\\ell}-1\\\\\n {\\mathsf P_{{k_c}+i+1}} & \\text{ otherwise}\n \\end{cases}\\right\\} \\text{.}$$ Expressing this in terms of fusion matrices gives the following recurrence relation $$X {F_{{k_c}}} = {F_{{k_c}+1}},$$ $$\\label{eq:fusionmatrix1}\nX {F_{{k_c}+1}} = 2 {F_{{k_c}}} + {F_{{k_c}+2}},$$ $$X {F_{{k_c}+i}} = {F_{{k_c}+i-1}} + {F_{{k_c}+i +1}} , \\qquad \\text{if } {k_c}+ i\\pm 1 \\text{ are not critical}$$ $$\\label{eq:fusionmatrix2}\nX {F_{{k_c}+ {\\ell}-1}} = {F_{{k_c}+ {\\ell}-2}} + {F_{{k_c}+ {\\ell}}} + {F_{{k_c}- {\\ell}}},$$ where it was implicitly assumed that ${\\ell}\\neq 2$. When ${\\ell}=2$, equations and becomes $$X {F_{{k_c}+ {\\ell}-1}} = X {F_{{k_c}+1}} = 2{F_{{k_c}}} + {F_{{k_c}+ 2}} + {F_{{k_c}- 2}}.$$ Using the fact that ${F_{{\\ell}-1}} = {U_{{\\ell}-1}}\\left(\\frac{X}{2}\\right)$, it can be checked directly that the solution to this system is\n\nFor $0 \\leq i \\leq \\ell -1$, $${F_{{k_c}+ i}} = \\begin{cases}\n {U_{{k_c}}}\\left(\\frac{X}{2}\\right), & \\text{ if } i =0\\\\\n {U_{{k_c}-i }}\\left(\\frac{X}{2}\\right) + {U_{{k_c}+ i}}\\left(\\frac{X}{2}\\right), & \\text{ otherwise }\n \\end{cases}.$$\n\n[A closed expression for ${\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}}$]{} Using fusion matrices, computing fusion rules is reduced to evaluating a Chebyshev polynomial at a matrix $X $, but since this matrix is not diagonal, computing this polynomial may be far from trivial. ", "However, since the projective indecomposable modules are all finite dimensional, proposition \\[prop:fmatrixunique\\] implies that if $$F_{i}F_{j} = F(G),$$ where $F(G)$ is the fusion matrix of some finite-dimensional projective module $G$, then $${\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq G.$$ Computing fusion rules thus reduces to expressing a product of Chebyshev polynomials as a linear combination of other Chebyshev polynomials. ", "Using this fact will greatly simplify the proof of the following explicit formulas. ", "These are written in a particular way to express the fact that they are identical to those obtained by Gainutnidov and Vasseur [@GainVass].", "\n\n\\[prop:fusionproj\\] If $k,r \\geq 1$, $0<i,j<{\\ell}$, $${\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq \\overset{i+j-{\\ell}+1}{\\underset{\\sigma = (i+j+{\\ell}+1)\\bmod{2}}{\\bigoplus}}{\\mathsf P_{{\\ell}-1 +\\sigma}} \\oplus \\overset{\\text{\\rm{Min}}(i+j,2{\\ell}- (i+j) -4)}{\\underset{\\sigma = |i-j|}{\\bigoplus}}{\\mathsf P_{\\sigma}},$$ $${\\mathsf P_{i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{k {\\ell}-1}} \\simeq \\overset{i}{\\underset{\\sigma = i\\bmod{2}}{\\bigoplus}}{\\mathsf P_{k {\\ell}-1 + \\sigma}},$$ $$\\begin{aligned}\n{\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1}} & \\simeq \\overset{k+r-1}{\\underset{\\rho = |k-r| +1}{\\bigoplus}}\\overset{{\\ell}-1}{\\underset{\\sigma = ({\\ell}+1)\\bmod{2}}{\\bigoplus}}{\\mathsf P_{\\rho {\\ell}-1 +\\sigma}}\\\\\n & \\simeq \\overset{k+r-1}{\\underset{\\rho = |k-r| +1}{\\bigoplus}} \\left({\\mathsf P_{\\rho {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right),\\end{aligned}$$ $$\\begin{aligned}\n{\\mathsf P_{j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{k {\\ell}-1+i}} &\\simeq \\overset{i+j-{\\ell}}{\\underset{\\sigma = (i+j+{\\ell})\\bmod{2}}{\\bigoplus}}\\left({\\mathsf P_{(k-1) {\\ell}-1 + \\sigma}} + {\\mathsf P_{(k+1){\\ell}-1 +\\sigma}}\\right) \\oplus 2 \\overset{j-i}{\\underset{\\sigma = |i-j|\\bmod{2}}{\\bigoplus}}{\\mathsf P_{k {\\ell}-1 +\\sigma}} \\notag \\\\\n& \\oplus \\overset{\\text{\\rm{Min}}(i+j, 2{\\ell}- (i+j)-2)}{\\underset{\\sigma = \\text{\\rm{Max}}(i-j,j-i+2)}{\\bigoplus}}{\\mathsf P_{k {\\ell}-1 +\\sigma}},\\end{aligned}$$ $$\\begin{aligned}\n{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{k {\\ell}-1+i}} &\\simeq \\overset{i-1}{\\underset{\\sigma = (i+1)\\bmod{2}}{\\bigoplus}}\\left({\\mathsf P_{|k-r| {\\ell}-1 + \\sigma}} + {\\mathsf P_{(k+r){\\ell}-1 +\\sigma}}\\right)\\notag \\\\\n&\\oplus 2\\overset{k+r -1}{\\underset{\\rho = |k-r|+1}{\\bigoplus}}\\overset{{\\ell}-i-1}{\\underset{\\sigma = (i+{\\ell}+1)\\bmod{2}}{\\bigoplus}}{\\mathsf P_{\\rho {\\ell}-1 +\\sigma}} \\oplus 2\\overset{k+r -2}{\\underset{\\rho = |k-r|+2}{\\bigoplus}}\\overset{i-1}{\\underset{\\sigma = (i-1)\\bmod{2}}{\\bigoplus}}{\\mathsf P_{\\rho {\\ell}-1 +\\sigma}}\\\\\n&\\simeq \\left({\\mathsf P_{|k-r| {\\ell}-1}} + {\\mathsf P_{(k+r){\\ell}-1 }}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\notag \\\\\n& \\oplus 2\\overset{k+r -1}{\\underset{\\rho = |k-r|+1}{\\bigoplus}}\\left({\\mathsf P_{\\rho {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}- i -1}}\\right) \\oplus 2\\overset{k+r -2}{\\underset{\\rho = |k-r|+2}{\\bigoplus}}\\left({\\mathsf P_{\\rho {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\right)\\end{aligned}$$ $$\\begin{gathered}\n{\\mathsf P_{k {\\ell}-1+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}- 1+j }} \\simeq \\overset{k+r+1}{\\underset{\\rho = |k-r|-1}{\\bigoplus} }\\left( 4\\phi_{\\rho} \\overset{{\\ell}- (l+i+j+1)\\bmod{2}}{\\underset{\\sigma = 2{\\ell}- (i+j)+1}{\\bigoplus} } {\\mathsf P_{(\\rho+1){\\ell}-1-\\sigma}}\\right)\\\\\n\\oplus 2 \\overset{k+r-1}{\\underset{\\rho = |k-r|+1}{\\bigoplus} }\\left( \\overset{\\text{\\rm{Min}}(i+j-1,2{\\ell}- i - j -1)}{\\underset{\\sigma = |i-j| +1}{\\bigoplus} }{\\mathsf P_{(\\rho + 1){\\ell}-1 -\\sigma}} \\oplus 2\\overset{{\\ell}- (l+i+j+1)\\bmod{2}}{\\underset{\\sigma = i+j +1}{\\bigoplus} }{\\mathsf P_{(\\rho + 1){\\ell}-1 -\\sigma}} \\right)\\\\\n\\oplus 2 \\overset{k+r}{\\underset{\\rho = |k-r|}{\\bigoplus} }\\psi_{\\rho}\\left( \\overset{\\text{\\rm{Min}}({\\ell}-i+j-1,{\\ell}+ i - j -1)}{\\underset{\\sigma = |{\\ell}-i-j| +1}{\\bigoplus} }{\\mathsf P_{(\\rho + 1){\\ell}-1 -\\sigma}} \\oplus2\\overset{{\\ell}- \\gamma_1}{\\underset{\\sigma = \\text{\\rm{Min}}({\\ell}- i +j +1, {\\ell}+i-j+1)}{\\bigoplus} }{\\mathsf P_{(\\rho + 1){\\ell}-1 -\\sigma}} \\right)\\text{,}\\end{gathered}$$\n\nwhere $\\gamma_1 = (i+j+1)\\bmod{2}$,$\\gamma_2 = (i+j+q+{\\ell})\\bmod{2}$,$\\phi_{\\rho} = 1 - \\frac{3}{4}\\delta_{\\rho,|k-r|-1} - \\frac{1}{4}\\delta_{\\rho,|k-r|+1} - \\frac{1}{4}\\delta_{\\rho,k+r-1} - \\frac{3}{4}\\delta_{\\rho,k+r+1}$, $\\psi_{\\rho} = 1- \\frac{1}{2} \\delta_{\\rho,|k-r|} - \\frac{1}{2}\\delta_{\\rho,k+r}$ and all sums have “step=2\".", "\n\nThe proof of all these are done using the same argument. ", "Start by using the identity $${U_{i}}(y){U_{j}}(y) = \\sum_{\\underset{step =2}{k=|i-j|}}^{i+j} {U_{k}}(y),$$ to write the product of the fusion matrices as a sum of Chebyshev polynomials then gather them in appropriate combinations to obtain a linear combination of fusion matrices. ", "Using the fact that a fusion matrix uniquely determines a projective module and that fusion of two projective modules always yields a projective modules, the conclusion is obtained. ", "Here are a few examples on how this is done. ", "Since the argument of the polynomials involved will always be $\\frac{X}{2}$, we will simply omit them and write ${U_{i}}$ instead of ${U_{i}}\\left(\\frac{X}{2}\\right)$.\n\nFor ${\\ell}=5$, here are some fusion of small projectives. ", "$${F_{3}}{F_{2}}= {U_{3}}{U_{2}} = {U_{1}} + {U_{3}} + {U_{5}} = {F_{1}} + {F_{5}} \\text{,}$$ $${F_{3}}{F_{4}} = {U_{1}} + {U_{3}} +{U_{5}} + {U_{7}} = ({U_{1}} + {U_{7}}) + ({U_{3}} + {U_{5}}) = {F_{5}} + {F_{7}} \\text{,}$$ $${F_{2}}{F_{4}} = {U_{2}} + {U_{4}} + {U_{6}} = {U_{4}} + ({U_{2}} + {U_{6}}) = {F_{4}} + {F_{6}}\\text{.}$$ For the fusion of a small projective and a projective indecomposable, $$\\begin{tabular}{r l}\n${F_{4}}{F_{8}} $&$= {U_{4}}({U_{0}} + {U_{8}}) = 2 {U_{4}} + {U_{6}} + {U_{8}} + {U_{10}} + {U_{12}}$\\\\\n&$ = 2{U_{4}} + ({U_{8}} + {U_{10}}) + ({U_{6}} + {U_{12}}) = 2{F_{4}} + {F_{10}} + {F_{12}}$\n\\end{tabular}$$ giving the fusion rule $${\\mathsf P_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq 2{\\mathsf P_{4}} \\oplus {\\mathsf P_{10}} \\oplus {\\mathsf P_{12}} \\text{.}$$ The fusion matrix of ${\\mathsf P_{11}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{28}}$ is $$\\begin{aligned}\n{F_{11}}{F_{28}} & = ({U_{7}} + {U_{11}})({U_{20}} + {U_{28}}) = \\sum_{\\underset{step =2}{i = 13}}^{27}({U_{i}}) + \\sum_{\\underset{step =2}{i = 9}}^{31}({U_{i}})+\\sum_{\\underset{step =2}{i = 21}}^{35}({U_{i}})+\\sum_{\\underset{step =2}{i = 17}}^{39}({U_{i}})\\notag \\\\\n& = {U_{9}} + ({U_{11}} + {U_{17}}) + 2({U_{13}} + {U_{15}}) + 3 {U_{19}} + 2 ({U_{17}} + {U_{21}}) + 4({U_{23}} + {U_{25}})\\notag\\\\\n& + 2 ({U_{21}}+{U_{27}}) + 3{U_{29}} + 2({U_{27}}+{U_{31}}) + 2({U_{33}}+{U_{35}}) + ({U_{31}} + {U_{37}}) + {U_{39}}\\notag\\\\\n& = {F_{9}} + {F_{17}} + 2 {F_{15}} + 3{F_{19}}+ 2 {F_{21}} + 4 {F_{25}} + 2{F_{27}} + 3{F_{29}}\\notag\\\\\n& +2{F_{31}}+2{F_{35}} + {F_{37}} + {F_{39}}\\end{aligned}$$ giving the fusion rule $$\\begin{aligned}\n{\\mathsf P_{11}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{28}} & \\simeq {\\mathsf P_{9}}\\oplus 2{\\mathsf P_{15}}\\oplus {\\mathsf P_{17}} \\oplus 3{\\mathsf P_{19}} \\oplus 2{\\mathsf P_{21}} \\oplus 4{\\mathsf P_{25}} \\oplus 2{\\mathsf P_{27}} \\oplus 3{\\mathsf P_{29}} \\notag\\\\\n& \\oplus 2 {\\mathsf P_{31}} \\oplus 2 {\\mathsf P_{35}} \\oplus {\\mathsf P_{37}} \\oplus {\\mathsf P_{39}}.\\end{aligned}$$ Note that we used the same notation in this proposition than in [@GainVass], where they compute the fusion rules in ${\\mathsf{TL}_{n}}$. This makes it obvious that the two fusion rules are identical.", "\n\n[The semi-simple case]{} When $q$ is not a root of unity different from $\\pm 1$, the algebras ${\\mathsf{TL}_{n}}$ and ${\\mathsf{dTL}_{n}}$ are semi-simple and the standard modules ${\\mathsf S_{n,i}}$ are all irreducible and projective. ", "They satisfy the induction rules $${{\\mathsf S_{n,i}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq {\\mathsf S_{n+1,i-1}}\\oplus {\\mathsf S_{n+1,i}} \\oplus {\\mathsf S_{n+1,i+1}},$$ where it is understood that ${\\mathsf S_{n,i+1}} = 0$ if $n \\neq i+1 \\mod 2$ in the regular family. ", "Using arguments identical to those in section \\[sec:fmatrixsmall\\] yields $${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{1,1}} \\simeq {\\mathsf S_{n+1,i-1}} \\oplus {\\mathsf S_{n+1,i+1}}, \\qquad {\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{1,0}} \\simeq {\\mathsf S_{n+1,i}},$$ where the second rule is replaced by $${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2,0}} \\simeq {\\mathsf S_{n+2,i}},$$ in the regular family. ", "This gives the following recurrence relation for the fusion matrices $$X{F_{i}} = {F_{i+1}} + {F_{i-1}}, \\qquad {F_{0}} = \\text{id}, \\qquad {F_{1}} = X,$$ where $X$ is simply $(X)_{i}^{j} = \\delta_{i}^{j+1} + \\delta_{i}^{j-1}.$ Using the same argument as in section \\[sec:fmatrixsmall\\] then gives the following fusion rules.", "\n\nIf $q$ is not a root of unity different from $\\pm 1$, then for $0\\leq i \\leq n$, $0 \\leq j \\leq m$, $${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,j}} \\simeq \\bigoplus_{\\underset{step=2}{k = |i-j|}}^{i+j} {\\mathsf S_{n+m,k}}.$$\n\n[Fusion of standard modules]{}\\[sec:fusion.standard\\] It was noted in section \\[sec:deffusion\\] that fusion is closely related with induction, we thus start by giving the behaviour of the non-projective standard modules under the induction functor [@RiSY; @BelSY; @BelRiSy].", "\n\nIf $i$ with $0\\leq i \\leq n-1$ is not critical, $${{\\mathsf S_{n-1,i}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\begin{cases}\n{\\mathsf S_{n,i-1}} \\oplus {\\mathsf S_{n,i}} \\oplus {\\mathsf S_{n,i+1}}, & \\text{ in the dilute family},\\\\\n{\\mathsf S_{n,i-1}} \\oplus {\\mathsf S_{n,i+1}}, & \\text{ in the regular family},\n\\end{cases}$$ where it is understood that ${\\mathsf S_{n,-1}} = 0$.\n\nUsing the same arguments as in proposition \\[prop\\_fusion\\_pvs01\\], this gives the following fusion rules.", "\n\n\\[prop\\_fusion\\_svs01\\] If $i$ with $0\\leq i \\leq n-1$ is not critical, in the dilute family $${\\mathsf S_{n-1,i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf S_{n,i}} \\text{,}$$ while in both families $$\\label{eq:fusion_svs1}\n{\\mathsf S_{n-1,i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf S_{n,i-1}} \\oplus {\\mathsf S_{n,i+1}}\\text{,}$$ where ${\\mathsf S_{n,j}}\\simeq{\\mathsf P_{n,j}}$ if $j$ is critical.", "\n\nUsing the same argument as in the projective case with the first fusion rule, and proposition \\[prop:magickSP2\\] in the regular case, $${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,j}} \\simeq {\\mathsf P_{n-i + m-j,0}} {\\times\\hspace{-0.27em}_f}({\\mathsf S_{i,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{j,j}}) \\text{.}$$ We will therefore always omit the parameter $n$, writing ${\\mathsf S_{n,i}} = {\\mathsf S_{i}}$, and assume that $n$ is big enough and of the right parity, in the regular case, for the module to exists. ", "Note that in the regular case when ${\\ell}=2$, the module ${\\mathsf S_{n,0}}$ is very particular because ${\\mathsf S_{n,0}} \\simeq {\\mathsf I_{n,2}}$. This module will therefore be treated in section \\[sec:tl\\_l2\\].", "\n\nOnce a formula for the fusion of ${\\mathsf S_{k \\ell}}$, $k \\in \\mathbb{N}$, with some module $M$ is obtained, the second fusion rules will be used to obtain a formula for the fusion of $M$ with the other standard modules by simple induction. ", "We start by studying the fusion of a standard module with a projective module then consider the fusion of two standard modules. ", "Finally, we give a simple rule that can be used to quickly compute the fusion of standard modules.", "\n\n[Fusion of a standard and a projective module]{} The general formula that will be obtained is quite complex and the inductive proof is very technical. ", "The argument is thus split in four propositions that will be simpler to prove. ", "Each one will be preceded by an example with ${\\ell}=5$ before moving to the general case. ", "The proof for general ${\\ell}$ is very straightforward once these examples are understood so we highly suggest that the reader works them out carefully.", "\n\nConsider the case ${\\ell}=5$ and the standard module ${\\mathsf S_{n,25}} = {\\mathsf S_{25}}$ which is not projective. ", "Proposition \\[prop\\_fusion\\_svs01\\] then gives $$\\label{eq_s25vsp1}\n{\\mathsf S_{25}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{24}} \\oplus {\\mathsf S_{26}} \\text{.}$$ Note that ${\\mathsf S_{24}}\\simeq{\\mathsf P_{24}}$ is projective. ", "Fusing the left side of this isomorphism with ${\\mathsf P_{1}}$ and using the associativity of fusion with proposition \\[prop\\_fusion\\_svs01\\] and \\[prop\\_fusion\\_pvs01\\] gives $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\simeq {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{0}} \\oplus {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq {\\mathsf S_{25}} \\oplus {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}},$$ while fusing its right side with ${\\mathsf P_{1}}$ and using the same propositions gives $$\\left({\\mathsf P_{24}} \\oplus {\\mathsf S_{26}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{24}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf P_{26}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{25}} \\oplus {\\mathsf S_{25}} \\oplus {\\mathsf S_{27}}.$$ Comparing the two results yields $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq {\\mathsf P_{25}} \\oplus {\\mathsf S_{27}} \\text{.}$$ Repeating the same arguments gives the fusion rules $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\simeq {\\mathsf P_{24}} \\oplus {\\mathsf P_{26}} \\oplus {\\mathsf S_{28}},$$ $$\\label{eq_s25vsp4}\n{\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\simeq {\\mathsf P_{25}} \\oplus {\\mathsf P_{27}} \\oplus {\\mathsf S_{29}}$$ where ${\\mathsf S_{29}}={\\mathsf P_{29}}$ is projective. ", "A pattern can be identified here: for all $i<5$, $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq {\\mathsf P_{24}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\oplus {\\mathsf S_{25+i}} \\text{.}$$\n\n\\[prop\\_fusion\\_sklvspi\\] For $i < {\\ell}$, ${\\ell}>2$ in ${\\mathsf{TL}_{n}}$, and $k \\in \\mathbb{N}$, $$\\label{eq_sklvspi}\n{\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq {\\mathsf P_{k {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\oplus {\\mathsf S_{ k {\\ell}+i}} \\text{.}$$\n\nThe proof proceeds by induction on $i$. Proposition \\[prop\\_fusion\\_svs01\\] already gives the case $i=0$ and $i=1$. Suppose therefore the result for $i<{\\ell}-1$ and $i-1$. Applying propositions \\[prop\\_fusion\\_svs01\\] and \\[prop\\_fusion\\_pvs01\\] on the left side of equation gives $${\\mathsf S_{ k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf S_{ k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\oplus {\\mathsf S_{ k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i+1}}.$$ Using the same proposition on the right side of yields $${\\mathsf P_{i-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf S_{ k {\\ell}+i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i-2}} \\oplus {\\mathsf P_{i}}\\right) \\oplus {\\mathsf S_{k {\\ell}+i-1}} \\oplus {\\mathsf S_{k {\\ell}+i+1}}.$$ Comparing the two results and using the induction hypothesis for $i-1$ gives the conclusion. ", "Note that we implicitly assumed that ${\\ell}\\neq 2$. In this case, there is only $i=0$ and $i=1$, which are both covered by proposition \\[prop\\_fusion\\_svs01\\].", "\n\nLet us return to the preceding ${\\ell}=5$ example. ", "Using again the associativity and commutativity of fusion with proposition \\[prop\\_fusion\\_pvs01\\] if equation is fused with ${\\mathsf P_{1}}$, the left side gives $${\\mathsf S_{25}}{\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\simeq {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5}},$$ while the right one becomes $$\\left({\\mathsf P_{24}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus \\underbrace{{\\mathsf S_{29}}}_{\\simeq {\\mathsf P_{29}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{25}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{30}}.$$ Comparing the two gives $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5}} \\simeq {\\mathsf P_{25}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{30}}.$$ Repeating this operation yields $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{6}} \\simeq {\\mathsf P_{26}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{31}}\\text{,}$$ $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{7}} \\simeq {\\mathsf P_{27}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{32}}\\text{,}$$ $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{33}}\\text{.}$$ Fusing ${\\mathsf P_{1}}$ again on the last rule[^3], the left side becomes $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{8}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\simeq {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{7}} \\oplus {\\mathsf P_{9}}\\right),$$ while the right one becomes $$\\begin{aligned}\n \\left({\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{33}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} &\\simeq \\left({\\mathsf P_{19}} \\oplus {\\mathsf P_{27}} \\oplus {\\mathsf P_{29}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{24}} \\oplus {\\mathsf P_{32}} \\oplus {\\mathsf P_{34}}\\\\\n & \\simeq \\underbrace{\\left({\\mathsf P_{27}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{32}} \\right)}_{\\simeq {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{7}}} \\oplus \\left({\\mathsf P_{19}}\\oplus {\\mathsf P_{29}}\\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{24}} \\oplus {\\mathsf P_{34}}.\\end{aligned}$$ This is simply $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{9}} \\simeq ({\\mathsf P_{19}} \\oplus {\\mathsf P_{29}}) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{24}} \\oplus {\\mathsf P_{34}} \\text{.}$$ We can then proceed with the general case.", "\n\n\\[prop\\_fusion\\_sqlvspkl\\] For $0\\leq i<{\\ell}$, $k,s \\in \\mathbb{Z}_{>0}$, $$\\label{eq_sqlvspkl}\n{\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+i}} \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1+i}} \\text{.}$$ In the case ${\\ell}=2$ in ${\\mathsf{TL}_{n}}$, the fusion with ${\\mathsf P_{{\\ell}-2}}$ must be removed.", "\n\nThe proof proceeds by induction on $s$ and $i$. Let us start by proving that for a given $k$, if the result stands for $i=0$ then it will also stands for all $i \\leq {\\ell}$. Note that the case $i=1$ follows directly from the case $i=0$ since for all $p\\geq 1$, ${\\mathsf P_{p {\\ell}}} \\simeq {\\mathsf P_{1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{p {\\ell}-1}}$. Suppose therefore that the result stands for $i-1, i<{\\ell}$. Fusing with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_pvs01\\] with the associativity and commutativity of fusion then gives, on the left side $${\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{s {\\ell}-1 +i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\simeq (1+\\delta_{i,1}){\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-2 +i}} \\oplus {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{s {\\ell}+i}} \\oplus \\delta_{i,{\\ell}-1} {\\mathsf P_{(s-1){\\ell}-1}}\\right),$$ and on the right side $$\\begin{aligned}\n \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}&\\left( \\left({\\mathsf P_{r {\\ell}-1+i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\\\\n & \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( \\left((1+\\delta_{i,1}){\\mathsf P_{r {\\ell}-2+i}} \\oplus {\\mathsf P_{r {\\ell}+ i}} \\oplus \\delta_{i,{\\ell}-1}{\\mathsf P_{(r-1){\\ell}-1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right)\\\\\n &\\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}}\\left((1+\\delta_{i,1}) {\\mathsf P_{r {\\ell}-2+i}} \\oplus {\\mathsf P_{r{\\ell}+i}} \\oplus \\delta_{i,{\\ell}-1} {\\mathsf P_{(r-1){\\ell}-1}}\\right)\\\\\n & \\simeq (1+\\delta_{i,1})\\left(\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-2+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}}{\\mathsf P_{r {\\ell}-2+i}}\\right)\\\\\n &\\oplus \\delta_{i,{\\ell}-1} \\left(\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left({\\mathsf P_{(r-1){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} {\\mathsf P_{(r-1){\\ell}-1}} \\right)\\\\\n & \\oplus \\left(\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1+i+1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}}{\\mathsf P_{r {\\ell}-1+i+1}}\\right).\\end{aligned}$$ If $i \\neq {\\ell}-1 $, collecting the relevant terms, comparing the two sides and applying the induction hypothesis then gives the result for $i+1$. If $i ={\\ell}-1 $, there is a slight subtlety involved. ", "In the preceding expression, collect the terms being fused with ${\\mathsf P_{{\\ell}-2}}$ and note that $$\\begin{aligned}\n \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left({\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{r {\\ell}-1+{\\ell}}} \\right)& \\simeq \\overset{k+s-2}{\\underset{\\underset{step =2}{r' = |k-s|}}{\\bigoplus}} {\\mathsf P_{r'{\\ell}-1}} \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r' = |k-s|+2}}{\\bigoplus}} {\\mathsf P_{r'{\\ell}-1}}\\\\\n & \\simeq \\overset{k+(s-1)-1}{\\underset{\\underset{step =2}{r = |k-(s-1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}} \\oplus \\overset{k+s+1-1}{\\underset{\\underset{step =2}{r = |k-(s+1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}},\\end{aligned}$$ where we rearranged the terms between the two sums and used the fact that ${\\mathsf P_{-1}} \\equiv 0$. The exact rearranging required depends on the value of $k-s$. Doing the same rearranging on the other terms gives $$\\begin{aligned}\n {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{(s+1) {\\ell}-1}} \\oplus {\\mathsf P_{(s-1){\\ell}-1}}\\right) & \\simeq \\overset{k+(s-1)-1}{\\underset{\\underset{step =2}{r = |k-(s-1)| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |s-1-(k+1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}} \\\\\n & \\oplus \\overset{k+s+1-1}{\\underset{\\underset{step =2}{r = |k-(s+1)| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+s+1}{\\underset{\\underset{step =2}{r = |s+1-(k+1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}}. ", " \\end{aligned}$$ It follows that if the statement holds for $(s-1,i=0)$, $(s,i=0)$, it will also stand for $(s,i)$ for all $i \\leq \\ell -1$ and $(s+1,i =0)$. The only remaining step is to prove that the result stands for $k=1$, $i=0$. This is precisely proposition \\[prop\\_fusion\\_sklvspi\\]. ", "In the case ${\\ell}=2$ of ${\\mathsf{TL}_{n}}$, the result and its proof are slightly different, because then $1 = {\\ell}-1$, so that $${\\mathsf P_{s {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq 2{\\mathsf P_{s {\\ell}-1}} \\oplus {\\mathsf P_{(s+1){\\ell}-1}}\\oplus {\\mathsf P_{(s-1){\\ell}-1}}.$$ However, the same arguments can be used to induce on $s$ and on $i$.\n\nNow that the expression for the fusion of ${\\mathsf S_{k {\\ell}}}$ is known, proposition \\[prop\\_fusion\\_svs01\\] can be used to compute the fusion of the other standard modules with the projective. ", "We return to the ${\\ell}=5$ example. ", "It was previously found that $${\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq {\\mathsf P_{28}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{33}}.$$ Fusing the left side with ${\\mathsf P_{1}}$ gives, $$\\label{eq:exampleS26P8.1}\n {\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq \\left({\\mathsf S_{25}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq \\left( {\\mathsf P_{24}} \\oplus {\\mathsf S_{26}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}},$$ while fusing the right side with ${\\mathsf P_{1}}$ yields $$\\begin{aligned}\n\\label{eq:exampleS26P8.2}\n {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{3}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\oplus {\\mathsf P_{33}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{2}} \\oplus {\\mathsf P_{4}} \\right) \\oplus {\\mathsf P_{33}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\notag \\\\\n & \\simeq \\left( {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\oplus {\\mathsf P_{33}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\oplus {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}}.\\end{aligned}$$ Using proposition \\[prop:fusionproj\\], notice that $$\\begin{aligned}\n {\\mathsf P_{24}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq \\left({\\mathsf P_{19}} \\oplus {\\mathsf P_{29}}\\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus 2 {\\mathsf P_{24}}\\simeq {\\mathsf P_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{28}}.\\end{aligned}$$ Comparing with then gives the fusion rule $${\\mathsf S_{26}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\oplus {\\mathsf P_{33}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}.$$ Repeating the same steps gives $${\\mathsf S_{27}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf P_{33}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}},$$ $${\\mathsf S_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8}} \\simeq {\\mathsf P_{28}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{0}} \\oplus {\\mathsf P_{33}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}}.$$\n\n\\[prop:fusion.standvsproj\\] For $0<i<{\\ell}$, $0 \\leq j <{\\ell}$ and $k,s\\in\\mathbb{Z}_{>0}$, $$\\label{eq:fusionSP}\n{\\mathsf S_{k {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+j}} \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1-i}}\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\right)\\text{.}$$\n\nIn this case the proof is a simple induction on $i$. The case $i = 1$ is covered by proposition \\[prop\\_fusion\\_sqlvspkl\\]. ", "Fusing the left side of ($i=1$) with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_svs01\\] gives $$\\label{eq:fusionSP.G}\n \\left( {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1 +j}} \\simeq \\left({\\mathsf P_{k {\\ell}-1}} \\oplus {\\mathsf S_{k {\\ell}-1 + 2}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1 +j}},$$ while fusing the right side of the same equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_pvs01\\] yields $$\\begin{aligned}\n\\label{eq:fusionSP.D}\n\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}&\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\notag\\\\\n& \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{{\\ell}-3}} \\right)\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\notag\\\\\n& \\simeq \\Big( \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}} \\right)\\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\Big) \\notag\\\\\n&\\oplus \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right).\\end{aligned}$$ However, proposition \\[prop:fusionproj\\] gives $$\\begin{aligned}\n{\\mathsf P_{k{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+j}} & \\simeq \\left({\\mathsf P_{|s-k| {\\ell}-1}} + {\\mathsf P_{(s+k){\\ell}-1 }}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\notag \\\\\n& \\oplus 2\\overset{k+s -1}{\\underset{\\rho = |k-s|+1}{\\bigoplus}}\\left({\\mathsf P_{\\rho {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}- j -1}}\\right) \\oplus 2\\overset{k+s -2}{\\underset{\\rho = |k-s|+2}{\\bigoplus}}\\left({\\mathsf P_{\\rho {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\right),\\end{aligned}$$ and $$\\begin{aligned}\n\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right) &\\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( \\left({\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(r+1){\\ell}-1}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\oplus 2 {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}- (j+1)}}\\right) \\notag\\\\\n& \\simeq 2 \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}- (j+1)}}\\right) \\oplus \\overset{k+s-2}{\\underset{\\underset{step =2}{r' = |k-s|}}{\\bigoplus}}\\left({\\mathsf P_{r'{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\right) \\notag\\\\\n& \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r' = |k-s| + 2}}{\\bigoplus}}\\left({\\mathsf P_{r'{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\right).\\end{aligned}$$ Collecting identical terms in the last two sums then gives the identity $${\\mathsf P_{k{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+j}} \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right).$$ Comparing equations , and and using this identity then gives the result for $i=2$.\n\nSuppose now that the result stands for $i-1,i$ with $1<i<{\\ell}$. Fusing the left side of equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_svs01\\] gives $$\\left( {\\mathsf S_{k {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+j}} \\simeq \\left( {\\mathsf S_{k {\\ell}-1 +(i-1)}} \\oplus {\\mathsf S_{k {\\ell}-1 +(i+1)}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{s {\\ell}-1+j}},$$ while fusing the right side of equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_pvs01\\] yields $$\\begin{aligned}\n\\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}&\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-1-i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right)\\right) \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\right) \\notag \\\\\n& \\simeq \\overset{k+s-1}{\\underset{\\underset{step =2}{r = |k-s| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-1-(i-1)}}\\oplus {\\mathsf P_{{\\ell}-1-(i+1)}}\\right)\\right) \\notag \\\\\n& \\oplus \\overset{k+s}{\\underset{\\underset{step =2}{r = |s-(k+1)| + 1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i-2}} \\oplus {\\mathsf P_{i}}\\right) \\right).\\end{aligned}$$ Comparing these two results and using the induction hypothesis then gives the result for $i+1$.\n\nNote that it was implicitly assumed that ${\\ell}\\neq 2$, because this case is covered by proposition \\[prop\\_fusion\\_sqlvspkl\\].", "\n\n[Fusion of two standard modules]{} The action of ${\\mathsf P_{1}}$ has played a central role so far in the proofs. ", "Projective modules can all be expressed as “polynomials\" in ${\\mathsf P_{1}}$ and even the standard modules ${\\mathsf S_{k {\\ell}+i}}$ could be obtained by fusing ${\\mathsf S_{k l}}$ with it. ", "However, fusing ${\\mathsf S_{k {\\ell}}}$ repeatedly with ${\\mathsf P_{1}}$ produced a sum of projective module, so that ${\\mathsf S_{(k+1){\\ell}}}$ cannot be obtained from ${\\mathsf S_{k {\\ell}}}$. Another argument will thus be needed to “cross\" the critical lines without obtaining projective modules. ", "It will eventually be proved that this is done by fusing with ${\\mathsf S_{{\\ell}}}$. The proofs are identical for the dilute and the regular family, except when ${\\ell}=2$. The proof of proposition \\[prop:fusionstandvssl\\] below is then very different. ", "The result still stands in this case, but the proof will be presented in section \\[sec:tl\\_l2\\].", "\n\nThe first step is to compute the dimension of $ {\\mathsf S_{k, k}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r,r}}$ as it will make the proof of proposition \\[prop:fusionstandvssl\\] much easier. ", "Note that the parameter $n$ in ${\\mathsf S_{n,k}}$ is now important as the dimension of the modules depends on it. ", "The general case is very simple but somewhat long. ", "We compute the dimension of $ {\\mathsf S_{3,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{3,3}}$. Define (see section \\[sec:rappel.base.standard\\]) $$z = \\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3/2) -- (0,3/2);\n \\draw (0,-1) -- (3,-1);\n \\draw (0,0) -- (3,0);\n \\draw (0,1) -- (3,1);\n\\end{tikzpicture},$$ which is such that ${\\mathsf S_{3,3}} = A_{3}z$, where $A_n = {\\mathsf{TL}_{n}}$ or ${\\mathsf{dTL}_{n}}$. Then $${\\mathsf S_{3,3}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{3,3}} \\simeq A_{6} \\left(\\operatorname{id}_{A_6} \\otimes_{A_{3} \\otimes A_{3}} ( z \\otimes_{\\mathbb{C}} z)\\right) .$$ Furthermore, notice that the only diagram in $A_{3}$ which does not act as zero on $z$ is the identity. ", "It follows that the only diagrams of $A_{6}$ which do not act as zero on $\\operatorname{id}_{A_6} \\otimes_{A_{3} \\otimes A_{3}} ( z \\otimes_{\\mathbb{C}} z)$ are those of the following form $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3) -- (0,3);\n \\draw[very thick] (3,-3) -- (3,3);\n \\draw (3,-5/2) -- (0,-5/2);\n \\draw (3,-3/2) -- (0,-3/2);\n \\draw (3,-1/2) -- (0,-1/2);\n \\draw (3,1/2) -- (0,1/2);\n \\draw (3,3/2) -- (0,3/2);\n \\draw (3,5/2) -- (0,5/2);\n\\end{tikzpicture},\\qquad\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3) -- (0,3);\n \\draw[very thick] (3,-3) -- (3,3);\n \\draw (3,-5/2) -- (2,-5/2);\n \\draw (3,-3/2) -- (2,-3/2);\n \\draw (3,-1/2) .. controls (2,-1/2) and (2,1/2) .. (3,1/2);\n \\draw (3,3/2) -- (2,3/2);\n \\draw (3,5/2) -- (2,5/2);\n \\node at (1,0) {$x_1$};\n\\end{tikzpicture}, \\qquad\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3) -- (0,3);\n \\draw[very thick] (3,-3) -- (3,3);\n \\draw (3,-5/2) -- (2,-5/2);\n \\draw (3,-3/2) .. controls (3/2,-3/2) and (3/2,3/2) .. (3,3/2);\n \\draw (3,-1/2) .. controls (2,-1/2) and (2,1/2) .. (3,1/2);\n \\draw (3,5/2) -- (2,5/2);\n \\node at (1,0) {$x_2$};\n\\end{tikzpicture}, \\qquad\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3) -- (0,3);\n \\draw[very thick] (3,-3) -- (3,3);\n \\draw (3,-5/2) .. controls (3/2,-5/2) and (3/2,5/2) .. (3,5/2);\n \\draw (3,-3/2) .. controls (2,-3/2) and (2,3/2) .. (3,3/2);\n \\draw (3,-1/2) .. controls (5/2,-1/2) and (5/2,1/2) .. (3,1/2);\n \\node at (1,0) {$x_3$};\n\\end{tikzpicture},$$ where $x_{i}$ is a link diagram in ${\\mathsf S_{6,6-2i}}$. It also follows that for $b \\in A_{3} \\otimes A_{3}$, $ b ( z \\otimes_{\\mathbb{C}} z) = 0$, unless $b$ can be expressed as $b = (\\operatorname{id}\\otimes \\operatorname{id}) + c,$ for some $c\\in{\\mathsf{dTL}_{3}}\\otimes_{\\mathbb{C}}{\\mathsf{dTL}_{3}}$. We thus conclude that these diagrams form a basis of ${\\mathsf S_{3,3}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{3,3}}$ and thus that $$\\dim {\\mathsf S_{3,3}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{3,3}} = \\dim {\\mathsf S_{6,6}} + \\dim {\\mathsf S_{6,4}} + \\dim {\\mathsf S_{6,2}} + \\dim {\\mathsf S_{6,0}}.$$ The general case is obtained by a straightforward generalisation of this argument.", "\n\n\\[lem:magickdim\\] For all $k,r \\in \\mathbb{N}$, $$\\dim \\left( {\\mathsf S_{k,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r,r}} \\right) = \\sum_{i=0}^{\\min\\left(k,r \\right)} \\dim {\\mathsf S_{k+r, k+r-2i}}.$$\n\nThe proof of the general case ${\\mathsf S_{k {\\ell}+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r {\\ell}+j}}$ will be done by induction on $k$,$r$,$i$ and $j$. Fusion with ${\\mathsf P_{1}}$ will be used to induce from $i$ to $i+1$, and from $j$ to $j+1$, while fusion with ${\\mathsf S_{{\\ell}}}$ will be used to induce from $k$ to $k+1$,and $r$ to $r+1$. The inductive proof is split into numerous lemmas so that the various steps are clearer. ", "Each lemma will be accompanied by an example to illustrate the result.", "\n\nUse again the particular case ${\\ell}=5$, and recall (see section \\[sec:rappel.indec\\]) that the projective module ${\\mathsf P_{5,5}}$ satisfies the short exact sequence $$0 \\longrightarrow {\\mathsf P_{5,3}} \\longrightarrow {\\mathsf P_{5,5}} \\longrightarrow {\\mathsf S_{5,5}} \\longrightarrow 0 \\text{,}$$ and using the right-exactness of fusion, proposition \\[prop:fusionexact\\], this implies the exact sequence $${\\mathsf P_{5,3}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\overset{f}{\\longrightarrow} {\\mathsf P_{5,5}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\longrightarrow {\\mathsf S_{5,5}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\longrightarrow 0\\text{.}$$ Using the previously obtained fusion rules, note that $$({\\mathsf P_{5, 5}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5, 5}})/({\\mathsf P_{5, 3}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5, 5}}) \\simeq \\frac{2 {\\mathsf P_{10,4}} \\oplus 2{\\mathsf P_{10,6}} \\oplus {\\mathsf P_{10,8}} \\oplus {\\mathsf P_{10,10}}}{{\\mathsf P_{10,4}} \\oplus {\\mathsf P_{10,6}} \\oplus {\\mathsf S_{10,8}}} \\simeq {\\mathsf P_{10,4}} \\oplus {\\mathsf P_{10,6}} \\oplus {\\mathsf P_{10,8}} \\oplus {\\mathsf S_{10,10}}$$ by using the fact that ${\\mathsf P_{10,10}}/{\\mathsf S_{10,8}} \\simeq {\\mathsf S_{10,10}}$. However, lemma \\[lem:magickdim\\] gives $$\\dim{{\\mathsf S_{5,5}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}}} = \\dim{{\\mathsf P_{10,4}} \\oplus {\\mathsf P_{10,6}} \\oplus {\\mathsf P_{10,8}} \\oplus {\\mathsf S_{10,10}}},$$ so it follows that $f$ must be injective and thus $${\\mathsf S_{5,5}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\simeq {\\mathsf P_{6,4}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{4,4}} \\oplus {\\mathsf S_{10,10}}.$$ Fusing the left side of this result with ${\\mathsf P_{1,1}}$ and using proposition \\[prop\\_fusion\\_svs01\\] gives $$\\label{eq:exampleSvsS.1}\n {\\mathsf S_{5,5}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{5,5}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\right) \\simeq {\\mathsf S_{5,5}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{6,4}} \\oplus {\\mathsf S_{6,6}}\\right),$$ while fusing its right side with ${\\mathsf P_{1,1}} $ and using propositions \\[prop\\_fusion\\_svs01\\], and \\[prop\\_fusion\\_pvs01\\] yields $$\\label{eq:exampleSvsS.2}\n {\\mathsf P_{6,4}}{\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{4,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}\\right) \\oplus \\left({\\mathsf S_{10,10}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\right) \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5,5}} \\oplus {\\mathsf P_{11,9}} \\oplus {\\mathsf S_{11,11}}.$$ Using proposition \\[prop:fusionproj\\], and \\[prop\\_fusion\\_sqlvspkl\\], note that $${\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5,5}} \\simeq 2 {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5,3}} \\oplus {\\mathsf P_{11,9}} \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5,3}} \\oplus {\\mathsf S_{5,5}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{6,4}}.$$ Comparing equations , and and using this observation give the fusion rule $${\\mathsf S_{6,6}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{5,3}} \\oplus {\\mathsf P_{11,9}} \\oplus {\\mathsf S_{11,11}}$$ Repeating these arguments yields $${\\mathsf S_{7,7}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{6,2}} \\oplus {\\mathsf P_{11,9}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\oplus {\\mathsf S_{12,12}}\\text{,}$$ $${\\mathsf S_{8,8}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{7,1}} \\oplus {\\mathsf P_{11,9}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,2}} \\oplus {\\mathsf S_{13,13}}\\text{,}$$ $${\\mathsf S_{9,9}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5,5}} \\simeq {\\mathsf P_{6,4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{8,0}} \\oplus {\\mathsf P_{11,9}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{3,3}} \\oplus {\\mathsf S_{14,14}}\\text{.}$$ Note that since ${\\mathsf S_{9,9}} \\simeq {\\mathsf P_{9,9}}$ is projective, the last one could be obtained from proposition \\[prop\\_fusion\\_sqlvspkl\\].", "\n\n\\[prop:fusionstandvssl\\] For $0 < i < {\\ell}$, $k \\in \\mathbb{Z}_{>0}$ and in the regular family ${\\ell}> 2$, $$\\label{eq:fusionstandvssl}\n{\\mathsf S_{k {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\left({\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-i}}\\right) \\oplus \\left({\\mathsf P_{(k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i - 2}}\\right) \\oplus {\\mathsf S_{(k+1){\\ell}-1+i}}.$$\n\nThe proof proceeds by induction on $k$ and $i$. Let us start by proving that for a given $k$, if the result stands for $i=1$, it will also stand for all $i \\leq {\\ell}-1 $.", "\n\nSuppose that the result stands for $i=1$. Fusing the left side of equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_svs01\\] gives $$\\label{eq:fusionSvsS.1}\n \\left({\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\left( {\\mathsf P_{k {\\ell}-1}} \\oplus {\\mathsf S_{k {\\ell}+1}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$$ while fusing its right side and using propositions \\[prop\\_fusion\\_svs01\\], and \\[prop\\_fusion\\_pvs01\\] yields $$\\label{eq:fusionSvsS.2}\n \\left({\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\right) \\oplus {\\mathsf S_{(k+1){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq \\left( {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\right) \\oplus {\\mathsf P_{(k+1){\\ell}-1}} \\oplus {\\mathsf S_{(k+1){\\ell}-1+2}}.$$ However, proposition \\[prop:fusionproj\\] gives $${\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq 2{\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k-1) {\\ell}-1}} \\oplus {\\mathsf P_{(k+1){\\ell}-1}},$$ and proposition \\[prop:fusion.standvsproj\\], $${\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k-1) {\\ell}-1}} \\oplus {\\mathsf P_{(k+1){\\ell}-1}}.$$ Comparing equations , and , and using these two results gives the result for $i =2$. Suppose then that the result stands for $i-1, i$, with $2 \\leq i< {\\ell}-1$. Fusing the left side of with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_svs01\\] gives $$\\left({\\mathsf S_{k {\\ell}-1 +i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\left({\\mathsf S_{k {\\ell}-1 + i-1}} \\oplus {\\mathsf S_{k {\\ell}-1 + i +1}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}},$$ while fusing its right side with ${\\mathsf P_{1}}$ and using propositions \\[prop\\_fusion\\_svs01\\], and \\[prop\\_fusion\\_pvs01\\] yields $$\\begin{aligned}\n \\left({\\mathsf P_{k {\\ell}-1}} \\right. & ", "\\left. {", "\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right)\\right) \\oplus \\left({\\mathsf P_{(k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i - 2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right)\\right) \\oplus {\\mathsf S_{(k+1){\\ell}-1+i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\notag\\\\\n & \\simeq \\left({\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-(i-1)}} \\oplus {\\mathsf P_{{\\ell}- (i+1)}}\\right)\\right) \\oplus \\left({\\mathsf P_{(k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i - 3}} \\oplus {\\mathsf P_{i-1}}\\right)\\right) \\notag\\\\\n & \\qquad \\oplus {\\mathsf S_{(k+1){\\ell}-1+i-1}}\\oplus {\\mathsf S_{(k+1){\\ell}-1 + i+1}}\\notag\\\\\n & \\simeq \\left({\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-(i-1)}}\\right) \\oplus \\left({\\mathsf P_{(k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{(i-1) - 2}}\\right) \\oplus {\\mathsf S_{(k+1){\\ell}-1+(i-1)}} \\notag\\\\\n & \\qquad \\oplus \\left({\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-(i+1)}}\\right) \\oplus \\left({\\mathsf P_{(k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{(i+1) - 2}}\\right) \\oplus {\\mathsf S_{(k+1){\\ell}-1+(i+1)}}\\end{aligned}$$ Comparing the two and using the induction hypothesis yields the result for $i+1$.\n\nWe must now do the induction on $k$. Note that when $k=0$, ${\\mathsf S_{0 {\\ell}-1+i}} \\simeq {\\mathsf P_{i-1}}$ and is thus projective. ", "Proposition \\[prop\\_fusion\\_sklvspi\\] then gives the result when $k=0$. Suppose now that the result holds for $k$ and $i={\\ell}-1$. There is a short-exact sequence $$0 \\longrightarrow {\\mathsf S_{(k+1) {\\ell}, k {\\ell}-1 + ({\\ell}-1)}} \\longrightarrow {\\mathsf P_{(k+1 ){\\ell},(k+1) {\\ell}}} \\longrightarrow {\\mathsf S_{(k+1) {\\ell}, (k+1) {\\ell}}} \\longrightarrow 0.$$ Note that the $n = (k+1) {\\ell}$ is important in this case so it is written explicitly. ", "Fusing this sequence with ${\\mathsf S_{{\\ell},{\\ell}}}$ gives the exact sequence $${\\mathsf S_{(k+1) {\\ell}, k {\\ell}-1 + ({\\ell}-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} \\overset{f}{\\longrightarrow} {\\mathsf P_{(k+1){\\ell}, (k+1) {\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} \\longrightarrow {\\mathsf S_{(k+1) {\\ell}, (k+1) {\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} \\longrightarrow 0.$$ We thus have the following inequality $$\\begin{tabular}{r l}\n$\\dim{{\\mathsf S_{(k+1) {\\ell},(k+1) {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}}} $&$\\leq \\dim{{\\mathsf P_{(k+1){\\ell},(k+1){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}}} - \\dim{{\\mathsf S_{(k+1) {\\ell}, k {\\ell}-1 + ({\\ell}-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} }$\\\\\n& $= \\dim{{\\mathsf P_{(k+1){\\ell}-1, (k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1,{\\ell}-1}} \\oplus {\\mathsf S_{(k+2){\\ell},(k+2){\\ell}}}}$, \n\\end{tabular}$$ where equality stands if and only if $\\ker f = 0$, and the second line is obtained by using proposition \\[prop\\_fusion\\_sqlvspkl\\] and the induction hypothesis with the structure of the projective modules (see section \\[sec:rappel.indec\\]). ", "However, lemma \\[lem:magickdim\\] gives $$\\dim {\\mathsf S_{(k+1){\\ell}, (k+1) {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} = \\dim{{\\mathsf P_{(k+1){\\ell}+1, (k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1, {\\ell}-1}} \\oplus {\\mathsf S_{(k+2){\\ell},(k+2){\\ell}}}}.$$ It follows that $\\ker f =0$, and thus that $$\\begin{tabular}{r l}\n${\\mathsf S_{(k+1) {\\ell},(k+1) {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} $&$\\simeq \\left({\\mathsf P_{(k+1){\\ell},(k+1){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}}\\right)/ \\left({\\mathsf S_{(k+1) {\\ell}, k {\\ell}-1 + ({\\ell}-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell},{\\ell}}} \\right)$\\\\\n& $\\simeq {\\mathsf P_{(k+1){\\ell}-1, (k+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1, {\\ell}-1}} \\oplus {\\mathsf S_{(k+2){\\ell},(k+2){\\ell}}}$, \n\\end{tabular}$$ where the second equality is obtained by using proposition \\[prop\\_fusion\\_sqlvspkl\\] and the induction hypothesis with the structure of the projective modules. ", "Note that once the result stands for $n= (k+1){\\ell}$, fusing it repeatedly with ${\\mathsf P_{2,0}}$ will give the result for all $n\\geq (k+1){\\ell}$. It follows that if the result stands for $k$ and $i={\\ell}-1$, it stands for $k+1$ and $i=1$. Using the first part of the proof, the conclusion is obtained.", "\n\nFusion with ${\\mathsf S_{{\\ell}}}$ can thus be used to “cross\" the critical lines. ", "The following continuation of the ${\\ell}=5$ example illustrate how the argument works. ", "Proposition \\[prop:fusionstandvssl\\] gives $${\\mathsf S_{10}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5}} \\simeq {\\mathsf P_{9}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{15}}\\text{.}$$ Fusing the left side of this equation with ${\\mathsf S_{5}}$ and using propositions \\[prop:fusionstandvssl\\], and \\[prop:fusion.standvsproj\\] produces $$\\begin{aligned}\n {\\mathsf S_{10}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{5}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{5}}\\right) &\\simeq {\\mathsf S_{10}} {\\times\\hspace{-0.27em}_f}\\left(\\left({\\mathsf P_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}}\\right) \\oplus {\\mathsf S_{10}} \\right) \\notag\\\\\n & \\simeq \\left({\\mathsf P_{9}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{14}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{10}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{10}},\\end{aligned}$$ while fusing its right side with ${\\mathsf S_{5}}$ and using the same propositions gives $$\\begin{aligned}\n \\left({\\mathsf P_{9}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{15}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{5}} &\\simeq \\left( {\\mathsf P_{9}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}}\n\\oplus {\\mathsf P_{4}} \\oplus {\\mathsf P_{14}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus \\left({\\mathsf P_{14}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{20}} \\right) \\notag\\\\\n & \\simeq \\left( {\\mathsf P_{9}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf P_{14}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus \\left({\\mathsf P_{4}} \\oplus {\\mathsf P_{14}} \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{20}}.\\end{aligned}$$ Comparing the two gives the fusion rule $${\\mathsf S_{10}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{10}} \\simeq ({\\mathsf P_{4}} \\oplus {\\mathsf P_{14}}){\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\oplus {\\mathsf S_{20}}.$$\n\n\\[prop:fusionSqlvsSkl\\] For $q$,$k \\in \\mathbb{Z}_{>0}$, $$\\label{eq:propSqlvsSkl}\n{\\mathsf S_{q {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{ k {\\ell}}} \\simeq \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right) \\oplus {\\mathsf S_{(q+k){\\ell}}}\\text{.}$$\n\nSince fusion is commutative, suppose without loss of generality that $k \\leq q$. The proof then proceeds by induction on $k$. For $k=1$, proposition \\[prop:fusionstandvssl\\] gives the result for all $q$. Suppose then that the result holds for some $k < q$. Fusing the left side of equation with ${\\mathsf S_{{\\ell}}}$ and using propositions \\[prop:fusionstandvssl\\], and \\[prop\\_fusion\\_sqlvspkl\\] gives $$\\begin{aligned}\n{\\mathsf S_{q {\\ell}}} & {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{ k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right) \\simeq {\\mathsf S_{q {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(k+1) {\\ell}}}\\right)\\notag\\\\\n& \\simeq \\Big( \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k| + 1}}{\\bigoplus}}\\left( {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\oplus \\overset{k+q}{\\underset{\\underset{step =2}{r = |k-(q+1)| + 1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}} \\Big) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{q {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{(k+1){\\ell}}}.\\end{aligned}$$ Fusing its right side with ${\\mathsf S_{{\\ell}}}$ and using the same propositions yields $$\\begin{aligned}\n\\overset{q+k-1}{\\underset{\\underset{step =2}{r = q-k+1}}{\\bigoplus}}&\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right) \\right) \\oplus {\\mathsf S_{(q+k){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\notag\\\\\n&\\simeq \\overset{q+k-1}{\\underset{\\underset{step =2}{r = q-k+1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{2 {\\ell}-1}} \\right) \\right) \\oplus {\\mathsf P_{(q+k){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(q+k+1){\\ell}-1}} \\notag\\\\\n& \\simeq \\overset{q+k-1}{\\underset{\\underset{step =2}{r = q-k+1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus \\left({\\mathsf P_{(r-1 ){\\ell}-1}}\\oplus{\\mathsf P_{(r+1){\\ell}-1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right) \\notag\\\\\n&\\qquad \\oplus {\\mathsf P_{(q+k){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(q+k+1){\\ell}-1}}.\\end{aligned}$$ Comparing these two equations gives $$\\begin{aligned}\n\\overset{k+q}{\\underset{\\underset{step =2}{r = q+2-k}}{\\bigoplus}} &{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{q {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{(k+1){\\ell}}} \\\\\n&\\simeq \\overset{q+k-1}{\\underset{\\underset{step =2}{r = q-k+1}}{\\bigoplus}}\\left({\\mathsf P_{(r-1 ){\\ell}-1}}\\oplus{\\mathsf P_{(r+1){\\ell}-1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{(q+k){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(q+k+1){\\ell}-1}} \\notag\\\\\n& \\simeq \\overset{q+k}{\\underset{\\underset{step =2}{r = q-k+2}}{\\bigoplus}}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus \\overset{q+k}{\\underset{\\underset{step =2}{r = q-(k+1) +1 }}{\\bigoplus}}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(q+k+1){\\ell}-1}},\\end{aligned}$$ where the second equality is obtained by rearranging the terms in the sum. ", "Comparing the two sides of this equation gives the conclusion.", "\n\nKnowing the fusion ${\\mathsf S_{q {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{k {\\ell}}}$, the fusion ${\\mathsf S_{q{\\ell}+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{k {\\ell}+j}}$ can be computed by using the fusion of ${\\mathsf S_{k {\\ell}+j'}}$ with ${\\mathsf P_{1}}$.\n\nFor $q,k\\in \\mathbb{Z}_{>0}$, $0 \\leq i,j<{\\ell}$, $$\\begin{gathered}\n\\label{eq:fusionStand}\n{\\mathsf S_{q {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{k {\\ell}-1 +j}} \\simeq \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-|i-j|-1}} \\right) \\oplus \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j)| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{|i-j|-1}} \\right) \\\\\n\\oplus \\overset{{\\ell}-|{\\ell}-(i+j)|-1}{\\underset{\\underset{step =2}{s = |i-j|+1}}{\\bigoplus}}\\left({\\mathsf S_{(q+k) {\\ell}-1 + s}} \\right) \\oplus {\\mathsf P_{(k+q+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i+j - {\\ell}-1}}.\\end{gathered}$$\n\nThe proof proceeds by induction on $i, j$ and involves many different particular cases: $i<j$, $i=j$ or $i>j$ with $i+j < {\\ell}$ or $i+j>{\\ell}$.\n\nWithout loss of generality, suppose $q \\geq k$. Proposition \\[prop:fusionSqlvsSkl\\] gives the case $i=j=1$, proposition \\[prop:fusion.standvsproj\\] gives the case $j=0$ for all $i>0$, and $i=0$ for all $j>0$, while proposition \\[prop:fusionproj\\] gives the case $i=j=0$. Suppose now $j \\geq 1$, fusing the left side of equation with ${\\mathsf P_{1}}$ and using propositions \\[prop\\_fusion\\_svs01\\] and \\[prop:fusion.standvsproj\\] gives $$\\begin{aligned}\n\\label{eq:fusionSvsS.3}\n {\\mathsf S_{q {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{k {\\ell}-1 +j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\simeq {\\mathsf S_{q {\\ell}-1+i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{k {\\ell}-1 +j-1}} \\oplus {\\mathsf S_{q {\\ell}-1 +i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{k {\\ell}-1 +(j+1)}},\\end{aligned}$$ while fusing the right side of this equation with ${\\mathsf P_{1}}$ yields $$\\begin{aligned}\n \\overbrace{\\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-|i-j|-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\right)}^{a} \\oplus \\overbrace{\\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j)| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{|i-j|-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\right)}^{b} \\notag \\\\\n\\oplus \\underbrace{\\overset{{\\ell}-|{\\ell}-(i+j)|-1}{\\underset{\\underset{step =2}{s = |i-j|+1}}{\\bigoplus}}\\left({\\mathsf S_{(q+k) {\\ell}-1 + s}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)}_{c} \\oplus \\underbrace{{\\mathsf P_{(k+q+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{i+j - {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)}_{d}.\\end{aligned}$$\n\nThe terms in $a$ can be written $$\\begin{aligned}\n\\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}}&\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-|i-j|-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right)\\right) \\notag\\\\\n& \\overset{1}{\\simeq} \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}} {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left. ", "\\begin{cases}\n {\\mathsf P_{{\\ell}}}, & \\text{ if } i=j \\\\\n{\\mathsf P_{{\\ell}-|i-j|-1 -1}}\\oplus {\\mathsf P_{{\\ell}-|i-j|-1 +1}}, & \\text{otherwise}\n\\end{cases} \\right\\rbrace \\notag\\\\\n& \\overset{2}{\\simeq} \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}} \\left. ", "\\begin{cases}\n2{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(r+1){\\ell}-1}}, & \\text{ if } i=j,\\\\\n{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-|i-j-1| -1}}\\oplus {\\mathsf P_{{\\ell}-|i-j+1| - 1}} \\right) , & \\text{otherwise}\n\\end{cases}\\right\\rbrace \\notag\\\\\n& \\overset{3}{\\simeq} \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-|i-j-1| -1}}\\oplus {\\mathsf P_{{\\ell}-|i-j+1| - 1}} \\right) \\right) \\notag\\\\\n& \\qquad \\oplus \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}} \\left. ", "\\begin{cases}\n{\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(r+1){\\ell}-1}}, & \\text{ if } i=j,\\\\\n0, & \\text{otherwise}\n\\end{cases}\\right\\rbrace \\notag\\\\\n& \\overset{4}{\\simeq} \\overset{q+k-1}{\\underset{\\underset{step =2}{r = |q-k|+1}}{\\bigoplus}} \\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-|i-j-1| -1}}\\oplus {\\mathsf P_{{\\ell}-|i-j+1| - 1}} \\right) \\right) \\notag\\\\\n & \\qquad \\oplus \\left. ", "\n \\begin{cases}\n\\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-(j-1))| + 1}}{\\bigoplus}}\\Big({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{\\underbrace{|i-(j-1)|-1}_{= 0}}}\\Big) & \\\\\n\\oplus \\overset{q+k-2}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-(j+1))| + 1}}{\\bigoplus}}\\Big({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{\\underbrace{|i-(j+1)|-1}_{= 0}}}\\Big) & \\text{ if } i=j,\\\\\n0 & \\text{otherwise}.", "\n\\end{cases}\\right\\rbrace\\end{aligned}$$ The first equality is obtained by using proposition \\[prop\\_fusion\\_pvs01\\], the second by using proposition \\[prop:fusionproj\\] and the third and fourth are obtained by noting that $|i-j+1| = |i-j-1| = 1$ when $i=j$ and rearranging the terms in the sums, respectively.", "\n\nThe terms in $b$ can be written $$\\begin{aligned}\n \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j)| + 1}}{\\bigoplus}}&\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{|i-j|-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\right) \\notag\\\\\n &\\simeq \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j)| + 1}}{\\bigoplus}}\\left.", "\n \\begin{cases} \n 0, & \\text{ if } i=j,\\\\\n {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{|i-j|-2}} \\oplus {\\mathsf P_{|i-j|}}\\right), & \\text{ otherwise}\n \\end{cases}\\right\\rbrace \\notag \\\\\n & \\simeq \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j)| + 1}}{\\bigoplus}}\\left.", "\n \\begin{cases} \n 0, & \\text{ if } i=j,\\\\\n {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{|i-j-1|-1}} \\oplus {\\mathsf P_{|i-j+1|-1}}\\right), & \\text{ otherwise}\n \\end{cases}\\right\\rbrace \\notag\\\\\n & \\simeq \\left. ", "\\begin{cases}\n 0, & \\text{ if } i=j,\\\\\n \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j-1)| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{|i-j-1|-1}} \\right) &\\\\\n \\oplus \\overset{q+k}{\\underset{\\underset{step =2}{r = |k-q-\\text{sign}(i-j+1)| + 1}}{\\bigoplus}}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{|i-j+1|-1}} \\right),& \\text{ otherwise}\n \\end{cases}\\right\\rbrace .\\end{aligned}$$ The first equality is obtained by using proposition \\[prop\\_fusion\\_pvs01\\] and the fact that ${\\mathsf P_{-1}} \\equiv 0$, while the second one is obtained by noting that if $i>j$, $|i-j| -2 = |i - (j+1)| -1$, $|i-j| = |i-(j-1)|-1$ while if $i<j$, $|i-j|-2 = |i-(j-1)|-1$, $|i-j| = |i - (j+1)|-1$. The third one is obtained by noting that if $i \\neq j$ and $\\text{sign}(i-j) \\neq \\text{sign}(i-j \\pm 1)$, then $|i-j \\pm 1| -1 < 0$, and thus ${\\mathsf P_{|i-j \\pm 1| -1}} \\equiv 0$.\n\nThe terms in $c$ can be written $$\\begin{aligned}\n \\overset{{\\ell}-|{\\ell}-(i+j)|-1}{\\underset{\\underset{step =2}{s = |i-j|+1}}{\\bigoplus}}&\\left({\\mathsf S_{(q+k) {\\ell}-1 + s}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\notag\\\\\n & \\simeq \\overset{{\\ell}-|{\\ell}-(i+j)|-1}{\\underset{\\underset{step =2}{s = |i-j|+1}}{\\bigoplus}}\\left(\n {\\mathsf S_{(q+k){\\ell}-1 + s -1}} \\oplus {\\mathsf S_{(q+k){\\ell}-1 + s+1}}\\right) \\notag\\\\\n & \\simeq \\overset{{\\ell}-|{\\ell}-(i+j)|-2}{\\underset{\\underset{step =2}{s = |i-j|}}{\\bigoplus}}{\\mathsf S_{(q+k) {\\ell}-1 + s}} \\oplus \\overset{{\\ell}-|{\\ell}-(i+j)|}{\\underset{\\underset{step =2}{s = |i-j|+2}}{\\bigoplus}}{\\mathsf S_{(q+k) {\\ell}-1 + s}} \\notag\\\\\n & \\simeq \\overset{{\\ell}-|{\\ell}-(i+j+1)|-1}{\\underset{\\underset{step =2}{s = |i-j-1|+1}}{\\bigoplus}}{\\mathsf S_{(q+k) {\\ell}-1 + s}} \\oplus \\overset{{\\ell}-|{\\ell}-(i+j-1)|-1}{\\underset{\\underset{step =2}{s = |i-j+1|+1}}{\\bigoplus}}{\\mathsf S_{(q+k) {\\ell}-1 + s}} \\notag\\\\\n & \\oplus \\delta_{0,|i-j|} {\\mathsf P_{(k+q){\\ell}-1}} \\oplus \\delta_{i+j, {\\ell}} {\\mathsf P_{(k+q+1){\\ell}-1}}.\\end{aligned}$$ The first equality is obtained by applying proposition \\[prop\\_fusion\\_svs01\\], the second by splitting the sum in two and renaming the indices while the third is obtained by considering the different possibilities for the absolute values and rearranging the two sums accordingly.", "\n\nThe terms in $d$ can be written $$\\begin{aligned}\n {\\mathsf P_{(k+q+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}& \\left({\\mathsf P_{i+j - {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\notag\\\\\n & \\simeq {\\mathsf P_{(k+q+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left. ", "\\begin{cases}\n 0,& \\text{ if } i+j < {\\ell}+1,\\\\\n ({\\mathsf P_{i+j-1 - {\\ell}-1}} \\oplus {\\mathsf P_{i+j+1 - {\\ell}-1}} ), & \\text{ otherwise}\n \\end{cases}\\right\\rbrace\\end{aligned}$$ by simply using proposition \\[prop\\_fusion\\_pvs01\\] and the fact that ${\\mathsf P_{t}} \\equiv 0$ when $t <0$.\n\nPutting all of these together, grouping the terms in the appropriate manner and comparing the result with equation yields the conclusion for $i, j+1$, provided that it stands for $i,j$, $i,j-1$. The induction to $i+1,j$ from $i-1,j$ is done using the same arguments, except that in equation , ${\\mathsf P_{1}}$ is fused with ${\\mathsf S_{q {\\ell}-1 +i}}$ instead of ${\\mathsf S_{k {\\ell}-1 +j}}$, and the rearranging used to reorder the sums in the different terms is slightly different.", "\n\n[The case ${\\ell}=2$ in ${\\mathsf{TL}_{n}}$]{}\\[sec:tl\\_l2\\] We treat here the regular Temperley-Lieb family when ${\\ell}=2$. Recall that in this case the module ${\\mathsf P_{0}}$ which was used to remove the dependence on $n$ is trivial, so the proof of proposition \\[prop:fusionstandvssl\\] does not work. ", "The method used here is more tedious than that of the previous section but it will ultimately give the same results.", "\n\n\\[prop:fusion\\_ds2vsds2\\] When ${\\ell}=2$, in the regular family, $${\\mathsf S_{n,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} \\simeq {\\mathsf P_{n+m,2}} \\oplus {\\mathsf S_{n+m,4}}.$$ If $n \\geq 4$, $${\\mathsf I_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} \\simeq {\\mathsf S_{n+m,2}}.$$\n\nThe case $n=m=2$ is particular and it must be computed by hand. ", "Using the same arguments as in lemma \\[lem:magickdim\\] the following set is a basis of ${\\mathsf S_{2,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2,2}}$ : $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-2) -- (0,2);\n \\draw[very thick] (3,-2) -- (3,2);\n \\draw (3,-3/2) -- (0,-3/2);\n \\draw (3,-1/2) -- (0,-1/2);\n \\draw (3,1/2) -- (0,1/2);\n \\draw (3,3/2) -- (0,3/2);\n\\end{tikzpicture}\n\\otimes \\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-5/2) -- (0,-1/2);\n \\draw[very thick] (0,1/2) -- (0,5/2);\n \\draw (0,-2) -- (2,-2);\n \\draw (0,-1) -- (2,-1);\n \\draw (0,1) -- (2,1);\n \\draw (0,2) -- (2,2);\n\\end{tikzpicture} ,\\qquad\\underbrace{\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-2) -- (0,2);\n \\draw[very thick] (3,-2) -- (3,2);\n \\draw (3,-3/2) -- (2,-3/2);\n \\draw (3,-1/2) .. controls (2,-1/2) and (2,1/2) .. (3,1/2);\n \\draw (3,3/2) -- (2,3/2);\n \\node at (1,0) {$x_1$};\n\\end{tikzpicture}\\otimes \\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-5/2) -- (0,-1/2);\n \\draw[very thick] (0,1/2) -- (0,5/2);\n \\draw (0,-2) -- (2,-2);\n \\draw (0,-1) -- (2,-1);\n \\draw (0,1) -- (2,1);\n \\draw (0,2) -- (2,2);\n\\end{tikzpicture}, \\qquad\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-2) -- (0,2);\n \\draw[very thick] (3,-2) -- (3,2);\n \\draw (3,-3/2) .. controls (3/2,-3/2) and (3/2,3/2) .. (3,3/2);\n \\draw (3,-1/2) .. controls (2,-1/2) and (2,1/2) .. (3,1/2);\n \\node at (1,0) {$x_2$};\n\\end{tikzpicture}\\otimes \\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-5/2) -- (0,-1/2);\n \\draw[very thick] (0,1/2) -- (0,5/2);\n \\draw (0,-2) -- (2,-2);\n \\draw (0,-1) -- (2,-1);\n \\draw (0,1) -- (2,1);\n \\draw (0,2) -- (2,2);\n\\end{tikzpicture}}_{V},$$ where $x_{i}$ are the link diagrams in ${\\mathsf S_{4,4-2i}}$ and it can be seen directly that the elements of $V$ spans a submodule of ${\\mathsf S_{2,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2,2}} $. ", "However when ${\\ell}=2$, ${\\mathsf P_{4,2}}\\simeq{\\mathsf P_{3,1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} $ is spanned by $$\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-2) -- (0,2);\n \\draw[very thick] (3,-2) -- (3,2);\n \\node at (1,0) {$x_1$};\n \\draw (2,-3/2) -- (3,-3/2);\n \\draw (2,-1/2) -- (3,-1/2);\n \\draw (3,1/2) .. controls (2,1/2) and (2,3/2) .. (3,3/2);\n\\end{tikzpicture} \\otimes\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-1) -- (0,2);\n \\draw (0,3/2) -- (3,3/2);\n \\draw (0,-1/2) .. controls (1,-1/2) and (1,1/2) .. (0,1/2);\n \\draw[very thick] (0,-5/2) -- (0,-3/2);\n \\draw (0,-2) -- (3,-2);\n\\end{tikzpicture} , \\qquad\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-2) -- (0,2);\n \\draw[very thick] (3,-2) -- (3,2);\n \\node at (1,0) {$x_2$};\n \\draw (3,-3/2) ..controls (2,-3/2) and (2,-1/2) .. (3,-1/2);\n \\draw (3,1/2) .. controls (2,1/2) and (2,3/2) .. (3,3/2);\n\\end{tikzpicture} \\otimes\n\\begin{tikzpicture}[baseline={(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-1) -- (0,2);\n \\draw (0,3/2) -- (3,3/2);\n \\draw (0,-1/2) .. controls (1,-1/2) and (1,1/2) .. (0,1/2);\n \\draw[very thick] (0,-5/2) -- (0,-3/2);\n \\draw (0,-2) -- (3,-2);\n\\end{tikzpicture},$$ where $x_{i}$ are the link diagrams in ${\\mathsf S_{4,4-2i}}$. A simple verification shows that $\\text{span}\\{V\\}\\simeq {\\mathsf P_{4,2}}$, and that $({\\mathsf S_{2,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2,2}})/\\text{Span}\\{V\\} \\simeq {\\mathsf S_{4,4}}$. Using the fact that ${\\mathsf P_{4,2}}$ is injective (see section \\[sec:rappel.indec\\]) yields the conclusion.", "\n\nSuppose that $n\\geq m$, $n \\geq 4$ and start with the exact sequence $$0 \\longrightarrow {\\mathsf I_{n,2}} \\longrightarrow {\\mathsf P_{n,2}} \\longrightarrow {\\mathsf S_{n,2}} \\longrightarrow 0,$$ which becomes $${\\mathsf I_{n,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} \\overset{f}{\\longrightarrow} {\\mathsf P_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}} \\longrightarrow {\\mathsf S_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} \\longrightarrow 0,$$ by using the right-exactness of fusion with the fusion rules \\[prop\\_fusion\\_sqlvspkl\\]. ", "To find ${\\mathsf I_{n,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}}$, fuse the sequence $${\\mathsf P_{m,2}} \\longrightarrow {\\mathsf P_{m,2}} \\longrightarrow {\\mathsf S_{m,2}} \\longrightarrow 0,$$ with ${\\mathsf I_{n,2}}$ to obtain,, $${\\mathsf P_{n+m,2}} \\longrightarrow {\\mathsf P_{n+m,2}} \\longrightarrow {\\mathsf S_{m,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{n,2}} \\longrightarrow 0,$$ where proposition \\[prop:fusionIvsP\\] was used. ", "Note that the proof of this proposition is independent of this one so it can safely be used. ", "It follows that there are three possibilities $${\\mathsf S_{m,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{n,2}} \\simeq \\begin{cases}\n{\\mathsf P_{n+m,2}}, & \\\\\n{\\mathsf S_{n+m,2}}, & \\\\\n0\n\\end{cases}.$$ But, proposition \\[prop:fusionIvsP\\] gives ${\\mathsf S_{m,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf S_{m,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{n+1,1}} \\simeq {\\mathsf P_{n+m+1,1}} \\oplus {\\mathsf P_{n+m+1,3}}$. Since ${\\mathsf P_{n+m,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq 2{\\mathsf P_{n+m+1,1}} \\oplus {\\mathsf P_{n+m+1,3}}$, it follows that $${\\mathsf S_{m,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{n,2}} \\simeq {\\mathsf S_{n+m,2}}.$$ Now, the morphisms from ${\\mathsf S_{n+m,2}}$ to $ {\\mathsf P_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}}$ are known (see their Loewy diagrams) and the cokernel of $f$ must be one of the following modules $${\\mathsf P_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}}, \\qquad {\\mathsf S_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}}, \\qquad {\\mathsf P_{n+m,2}} \\oplus {\\mathsf S_{n+m,4}}.$$ Using propositions \\[prop\\_fusion\\_pvs01\\] and \\[prop\\_fusion\\_svs01\\], $$\\left( {\\mathsf P_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}}\\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq 3{\\mathsf P_{n+m+1,1}} \\oplus 3{\\mathsf P_{n+m+1,3}} \\oplus {\\mathsf P_{n+m+1,5}},$$ $$({\\mathsf S_{n+m,2}} \\oplus {\\mathsf P_{n+m,4}}){\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq 2{\\mathsf P_{n+m+1,1}} \\oplus 3{\\mathsf P_{n+m+1,3}} \\oplus {\\mathsf P_{n+m+1,5}},$$ $$({\\mathsf P_{n+m,2}} \\oplus {\\mathsf S_{n+m,4}}){\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq 2{\\mathsf P_{n+m+1,1}} \\oplus 2{\\mathsf P_{n+m+1,3}} \\oplus {\\mathsf P_{n+m+1,5}},$$ while $${\\mathsf S_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf S_{n,2}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{m+1,1}} \\oplus {\\mathsf P_{m+1,3}}) \\simeq 2{\\mathsf P_{n+m+1,1}} \\oplus 2{\\mathsf P_{n+m+1,3}} \\oplus {\\mathsf P_{n+m+1,5}}.$$ It thus follows that ${\\mathsf S_{n,2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2}} \\simeq {\\mathsf P_{n+m,2}} \\oplus {\\mathsf S_{n+m,4}}$, as long as one of $n$ or $m$ is bigger or equal to $4$.\n\nNow that the fusion of ${\\mathsf S_{n,2}}$ with itself is known, it can be used to compute the fusion of the other standard modules. ", "Note that the fusion of ${\\mathsf S_{2,2}}$ with standard modules other than ${\\mathsf S_{2,2}}$ can be obtained by the same arguments as in proposition \\[prop:fusionstandvssl\\], so we will only give the proof for ${\\mathsf S_{n,2}} = {\\mathsf S_{2}}$ with $n\\geq 4$. We present a few examples before proving the general case. ", "There is an exact sequence $$0 \\longrightarrow {\\mathsf S_{2}} \\longrightarrow {\\mathsf P_{4}} \\longrightarrow {\\mathsf S_{4}} \\longrightarrow 0,$$ which becomes $${\\mathsf S_{2}} \\overset{f}{\\longrightarrow} {\\mathsf P_{4}} \\longrightarrow {\\mathsf I_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\longrightarrow 0,$$ by fusing it with ${\\mathsf I_{2}}$ and using the preceding proposition with proposition \\[prop:fusionIvsP\\]. ", "Note that ${\\mathsf I_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf S_{4}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{3}} \\oplus {\\mathsf P_{5}}$. Since the cokernel of $f$ is either ${\\mathsf P_{4}}$ or ${\\mathsf S_{4}}$, it follows that ${\\mathsf I_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\simeq {\\mathsf S_{4}}$. Now, the exact sequence $$0 \\longrightarrow {\\mathsf I_{2}} \\longrightarrow {\\mathsf P_{2}} \\longrightarrow {\\mathsf S_{2}} \\longrightarrow 0,$$ when fused with ${\\mathsf S_{4}}$, yield the exact sequence $${\\mathsf S_{4}} \\longrightarrow {\\mathsf P_{4}} \\oplus {\\mathsf P_{6}} \\longrightarrow {\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\longrightarrow 0,$$ by using proposition \\[prop:fusion.standvsproj\\]. ", "There are thus three possibilities $${\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\simeq {\\mathsf P_{4}} \\oplus {\\mathsf P_{6}}, \\qquad {\\mathsf{B}^{2}_{2}} \\oplus {\\mathsf P_{6}} \\qquad \\text{ or } {\\mathsf P_{4}} \\oplus {\\mathsf S_{6}}.$$ But, using proposition \\[prop:fusion.standvsproj\\], we get $${\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq ({\\mathsf P_{1}} \\oplus {\\mathsf P_{3}}){\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\simeq {\\mathsf P_{1}} \\oplus 2{\\mathsf P_{3}} \\oplus 2{\\mathsf P_{5}} \\oplus {\\mathsf P_{7}}.$$ Then, we verify which of the three possibilities satisfies this rule: $$({\\mathsf P_{4}} \\oplus {\\mathsf P_{6}}) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{1}} \\oplus 3{\\mathsf P_{3}} \\oplus 3{\\mathsf P_{5}} \\oplus {\\mathsf P_{7}},$$ $$({\\mathsf{B}^{2}_{2}} \\oplus {\\mathsf P_{6}}) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{1}} \\oplus 2{\\mathsf P_{3}} \\oplus 3 {\\mathsf P_{5}} \\oplus {\\mathsf P_{7}},$$ $$({\\mathsf P_{4}} \\oplus {\\mathsf S_{6}}) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{1}} \\oplus 2{\\mathsf P_{3}} \\oplus 2 {\\mathsf P_{5}} \\oplus {\\mathsf P_{7}},$$ where propositions \\[prop\\_fusion\\_pvs01\\], \\[prop\\_fusion\\_svs01\\], and \\[prop:fusionBvsP\\] were used. ", "We are allowed to do so because the proofs of these propositions are independent of the fusion rules for standard modules. ", "Comparing these fusion with ${\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} $, it follows that $${\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{4}} \\simeq {\\mathsf P_{4}} \\oplus {\\mathsf S_{6}}.$$ The proof of the general result that follows is obtained by induction and repeats the preceding arguments.", "\n\nFor $n \\geq 4$, $m/2 \\geq k \\geq 1$, $${\\mathsf S_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2k}} \\simeq {\\mathsf P_{n+m,2k}} \\oplus {\\mathsf S_{n+m, 2(k+1)}},$$ and $${\\mathsf I_{n,2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{m,2k}} \\simeq {\\mathsf S_{n+m,2k}}.$$\n\nNote that a simple corollary of this proposition is that ${\\mathsf I_{n,2}}\\simeq {\\mathsf S_{n,0}}$ when ${\\ell}=2 $, plays the role of ${\\mathsf P_{n,0}}$ when ${\\ell}\\neq 2$, except that in this case $n\\geq 4$ instead of $n\\geq 2$.\n\n[A simple rule for fusion]{} The fusion rules for standard modules and projective modules can be hard to apply in practice because of the numerous direct sum and fusions involved; we thus present a simple “rule of thumb\" to quickly compute fusion of standard modules.", "\n\nTo a standard module ${\\mathsf S_{i}}$ ($i$ can be critical), associate the Chebyshev polynomial of the second kind $U_{i}(\\frac{x}{2})$ where $x$ is a formal parameter. ", "To a projective module ${\\mathsf P_{{k_c}+j}}$, ${\\ell}>j>0$, associate the sum of Chebyshev polynomials $U_{{k_c}-j}(\\frac{x}{2}) + U_{{k_c}+j}(\\frac{x}{2})$. Call this association the *polynomial representation* of the modules. ", "Furthermore, since the polynomials all have the same argument, it will simply be omitted. ", "To take the fusion of two modules, multiply their polynomial representations and split the result by using the product rule $$U_{i}U_{j} = \\overset{i+j}{\\underset{step =2}{\\underset{k=|i-j|}{\\Sigma}}} U_{k}.$$ Collect the terms in this sum to form the polynomial representation of projective modules, starting with the smallest $k$. Remaining terms are then identified with the corresponding standard modules.", "\n\nIt is straightforward but tedious to prove that all fusion rules obtained so far respect this simple rule.", "\n\n[Fusion of quotients]{}\\[sec:fusionquotients\\] We are now trying to compute the fusion of two irreducible modules. ", "We begin by explaining the general idea which we will use to compute them. ", "Suppose there are two modules $U,V$ and two resolutions $$U_{2} \\longrightarrow U_{1} \\longrightarrow U \\longrightarrow 0, \\qquad V_{2} \\longrightarrow V_{1} \\longrightarrow V \\longrightarrow 0,$$ by modules $U_{i},V_{i}$. It is a standard exercise in diagram chasing to obtain the exact sequence $$U_{2}{\\times\\hspace{-0.27em}_f}V_{1} \\oplus U_{1}{\\times\\hspace{-0.27em}_f}V_{2} \\longrightarrow U_{1} {\\times\\hspace{-0.27em}_f}V_{1} \\overset{\\phi}{\\longrightarrow} U {\\times\\hspace{-0.27em}_f}V \\longrightarrow 0.$$ If $\\phi$ can be computed somehow, the knowledge of the fusion rules for $U_{1} {\\times\\hspace{-0.27em}_f}V_{1}$, $U_{2} {\\times\\hspace{-0.27em}_f}V_{1}$ and $U_{1} {\\times\\hspace{-0.27em}_f}V_{2}$ will give $U {\\times\\hspace{-0.27em}_f}V$. If $U_{1}$, $V_{1}$ are “close\" to $U$ and $V$, the kernel of $\\phi$ will be small, and its image will be much easier to compute. ", "The idea is therefore to find the $U_{1}$, $V_{1}$ that are the “closest\" to $U$ and $V$ but such that their fusion can be computed. ", "Of course the “closest\" module to an irreducible ${\\mathsf I_{n,i}}$ is ${\\mathsf I_{n,i}}$ itself, the second closest would be the standard module ${\\mathsf S_{n,i}}$ and the third would be the projective module ${\\mathsf P_{n,i}}$. The goal is thus to find the fusion of irreducible modules with projective ones, which will then be used to compute the fusion of irreducible modules with standard modules. ", "This is where the modules ${\\mathsf{B}^{2i}_{k}}$s appear. ", "We will then compute the fusion rules for these modules, introducing yet another class of modules, the ${\\mathsf{T}^{2i+1}_{k}}$s. ", "Computing the fusion of these modules with projective and standard modules will be the last step before arriving at the fusion of two irreducibles. ", "Note that the same arguments will be used over and over again so we will not detail the proofs as much as in the preceding sections.", "\n\n[Fusion of irreducible and projective modules]{}\\[sec:fusionIvsP\\] We start by giving the rules for the induction of ${\\mathsf I_{n,k}}$ [@BelSY].", "\n\nIf $n\\geq k {\\ell}- 1+i$, $0<i<{\\ell}$, $$\\begin{gathered}\n{{\\mathsf I_{n,k {\\ell}-1 -i}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\left. ", "\\begin{cases}\n{\\mathsf I_{n+1,k {\\ell}-1 -i}}& \\text{ in } {\\mathsf{dTL}_{n}} \\\\\n0 & \\text{ in } {\\mathsf{TL}_{n}}\n\\end{cases}\\right\\} \\oplus \\begin{cases}\n{\\mathsf P_{n+1, (k-1) {\\ell}-1}} & \\text{if } i ={\\ell}-1\\\\\n{\\mathsf I_{n+1,k{\\ell}-1 -i-1}} & \\text{otherwise}\n\\end{cases}\\\\\n\\oplus \\begin{cases}\n0 & \\text{if } i=1\\\\\n{\\mathsf I_{n+1,k {\\ell}-1 - i +1}} & \\text{ otherwise}\\\\\n\\end{cases}.\\end{gathered}$$\n\nThe condition on $n$ ensures that the module under study is not a standard module. ", "Using proposition \\[prop:fusion\\_parity\\] with the parity of the irreducibles gives the following fusion rules.", "\n\n\\[prop:fusionIvs01\\] If $n\\geq k {\\ell}- 1+i$, $0 < i < {\\ell}$,then in the dilute Temperley-Lieb family $${\\mathsf I_{n, k {\\ell}-1 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf I_{n+1, k {\\ell}-1 -i}},$$ while in both families $${\\mathsf I_{n, k {\\ell}-1 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq \\begin{cases}\n{\\mathsf P_{n+1, (k-1) {\\ell}-1}} & \\text{if } i ={\\ell}-1\\\\\n{\\mathsf I_{n+1,k{\\ell}-1 -i-1}} & \\text{otherwise}\n\\end{cases}\n\\oplus \\begin{cases}\n0 & \\text{if } i=1\\\\\n{\\mathsf I_{n+1,k {\\ell}-1 - i +1}} & \\text{ otherwise}\\\\\n\\end{cases}.$$\n\nIn the standard family, when ${\\ell}\\neq 2$, $${\\mathsf I_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf I_{n+2,i}},$$ which is proven in proposition \\[prop:magickOthersP\\]. ", "The proofs in this section will be independent of $n$ as long as it is big enough for the irreducible modules to be distinct from the standard modules; we will therefore simply omit the $n$. Note now that $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf I_{k {\\ell}-3}}.$$ Fusing the left side of this equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_pvs01\\] gives $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\simeq {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{0}} \\oplus {\\mathsf P_{2}}\\right),$$ while fusing its right side with ${\\mathsf P_{1}}$ and using proposition \\[prop:fusionIvs01\\] gives $${\\mathsf I_{k {\\ell}-3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf I_{k{\\ell}-2}} \\oplus {\\mathsf I_{k {\\ell}-4}}.$$ Comparing the two results then yields the fusion rule $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq {\\mathsf I_{k {\\ell}-4}}.$$ The following proposition is then obtained by simply repeating these arguments.", "\n\n\\[prop:fusionIvssmallP\\] For all $0\\leq i<{\\ell}-1$, $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq {\\mathsf I_{k {\\ell}-2-i}}.$$\n\nOnce the fusion rules for ${\\mathsf I_{k{\\ell}-2}}$ are known, this proposition will be used to quickly compute the fusion of the other irreducible modules, since for all $0 < i < {\\ell}$ and any module $M$, $${\\mathsf I_{k {\\ell}-1 -i}} {\\times\\hspace{-0.27em}_f}M \\simeq \\left({\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}M \\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}}.$$\n\nFor $k>1$, $i={\\ell}-1$, the same arguments give $${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{(k-1){\\ell}-1}}.$$ Fusing this repeatedly with ${\\mathsf P_{1}}$ then yields $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq {\\mathsf P_{(k-1){\\ell}}},$$ $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}} \\simeq {\\mathsf P_{(k-1){\\ell}+1}},$$ $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+2}} \\simeq {\\mathsf P_{(k-1){\\ell}+2}}.$$ Continuing in this manner eventually yields $${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+ {\\ell}-1}} \\simeq {\\mathsf P_{(k-2){\\ell}-1}} \\oplus {\\mathsf P_{k {\\ell}-1}}.$$ Note that if $k=2$, ${\\mathsf P_{(k-2){\\ell}-1}} \\simeq {\\mathsf P_{ -1}}\\simeq 0$. The following proposition gives the general formula.", "\n\n\\[prop:fusionIvsP\\] For all $k>1$,$r\\geq 1 $, $0\\leq i < {\\ell}-1$, $0\\leq j<{\\ell}$, $$\\label{eq:fusionIvsP}\n{\\mathsf I_{k {\\ell}-2- i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1 +j}} \\simeq \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}} {\\mathsf P_{p {\\ell}-1 +j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}.$$\n\nThe proof proceeds by induction on $r$ and $j$. The cases $r=1$ (for all $j$) and $r =2$, $j=0$ were proved in the preceding discussion, so suppose that the result stands for some $r$ and $j =0$. Fusing the left side of equation with ${\\mathsf P_{1}}$ and using proposition \\[prop\\_fusion\\_pvs01\\] then gives $${\\mathsf I_{k {\\ell}-2- i}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\simeq {\\mathsf I_{k {\\ell}-2 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}}} ,$$ while fusing its right side with ${\\mathsf P_{1}}$ and using the same proposition yields $$\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 }} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}} {\\mathsf P_{p {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}.$$ The case $j=1$ is then obtained by simply comparing the two results. ", "Now, assume that the result stands for this $q$ and $j-1,j$, $1 \\leq j < {\\ell}-1$. Fusing the left side of equation and using proposition \\[prop\\_fusion\\_pvs01\\] gives $${\\mathsf I_{k {\\ell}-2- i}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r {\\ell}-1 +j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\simeq {\\mathsf I_{k {\\ell}-2 -i}} {\\times\\hspace{-0.27em}_f}\\left((1+\\delta_{j,1}){\\mathsf P_{r {\\ell}-1 + (j-1)}} \\oplus {\\mathsf P_{r {\\ell}-1 +j+1}} \\right),$$ while fusing its right side and using the same proposition yields $$\\begin{aligned}\n &\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\notag\\\\\n &\\qquad \\simeq \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( (1 + \\delta_{j,1}){\\mathsf P_{p {\\ell}-1 +j-1}} \\oplus {\\mathsf P_{p {\\ell}-1 +j+1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\notag\\\\\n & \\qquad \\simeq (1 + \\delta_{j,1})\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +j-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +j+1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) .\\end{aligned}$$ Comparing these two results and using the induction hypothesis then yields the conclusion for $j+1$. Note that doing the same thing for the case $j ={\\ell}-1$ gives, for the left side $${\\mathsf I_{k {\\ell}-2- i}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{(r+1) {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\right) \\simeq {\\mathsf I_{k {\\ell}-2 -i}} {\\times\\hspace{-0.27em}_f}\\left((1+\\delta_{{\\ell}, 2}){\\mathsf P_{r {\\ell}-1 + ({\\ell}-2)}} \\oplus {\\mathsf P_{(r+1) {\\ell}-1 }} \\oplus {\\mathsf P_{(r-1){\\ell}-1}} \\right),$$ and for the right side $$\\begin{aligned}\n &\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +{\\ell}- 1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\\\\n &\\qquad \\simeq (1 + \\delta_{\\ell,2})\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +\\ell -2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left\\{k-r, r-k +2 \\right\\}}}{\\bigoplus}}\\left( {\\mathsf P_{(p+1) {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) \\\\\n & \\qquad \\qquad \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left\\{k-r, r-k +2 \\right\\}}}{\\bigoplus}}\\left( {\\mathsf P_{(p-1) {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right)\\\\\n & \\qquad \\simeq (1 + \\delta_{\\ell ,2})\\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-r, r-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1 +\\ell -2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) \\oplus \\overset{k+r+1-2}{\\underset{\\underset{step=2}{p = \\max\\left(k-(r+1), r+1-k +2 \\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right) \\\\\n & \\qquad \\qquad \\oplus \\overset{k+r-1-2}{\\underset{\\underset{step=2}{p = \\max\\left\\{k-(r-1), r-1-k +2 \\right\\}}}{\\bigoplus}}\\left( {\\mathsf P_{p {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\right),\\end{aligned}$$ where the last equality is obtained by rearranging the terms between the sums and considering the different values of $r-k$. Comparing the two sides, it follows that if the conclusion holds for $r-1, j=0$, $r,j={\\ell}-1, {\\ell}-2 $, it will also hold for $r+1,j=0$.\n\nNote that if $k=1$, repeating the arguments leading to proposition \\[prop:fusionIvssmallP\\] gives $${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}} \\simeq {\\mathsf I_{1}},$$ $${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\simeq {\\mathsf I_{0}},$$ $${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq 0.$$ This implies of course that ${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}\\simeq 0$ for all $i\\geq {\\ell}-1$.\n\n\\[prop:fusionIWeird\\] For all $i\\geq {\\ell}-1$, $j < \\ell -1$, $${\\mathsf I_{j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq 0.$$\n\nNote also that since fusion is right-exact, fusing ${\\mathsf I_{{\\ell}-2}}$ with any quotient of ${\\mathsf P_{k}}$ will always yield $0$. This include the standard non-projective modules as well as all irreducibles ${\\mathsf I_{k}}$ with $k>{\\ell}-1$.\n\n[Fusion of irreducible and standard modules, first part]{}\\[sec:fusionIvsS1\\] Proposition \\[prop:fusionstandvssl\\] can be used to obtain the non-projective standard modules by repeatedly fusing ${\\mathsf S_{n,{\\ell}}}$ with itself and small projectives. ", "The first step to obtain the fusion of irreducible modules with standard modules is thus to compute ${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$, for $k>1$. There is a short exact sequence $$0 \\longrightarrow {\\mathsf P_{{\\ell}-2}} \\longrightarrow {\\mathsf P_{{\\ell}}} \\longrightarrow {\\mathsf S_{{\\ell}}} \\longrightarrow 0.$$ Using the right-exactness of fusion together with known fusion rules, this yields the exact sequence $${\\mathsf I_{(k-1){\\ell}}} \\overset{f}{\\longrightarrow} {\\mathsf P_{(k-1){\\ell}}} \\longrightarrow {\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow 0.$$ Since ${\\mathsf I_{(k-1){\\ell}}}$ is irreducible, $f$ is either zero or injective. ", "If it is injective, then ${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{\\ell}} \\simeq {\\mathsf{B}^{2}_{(k-1){\\ell}-2}}$ by proposition \\[prop:indec.prop\\] while if $f=0$, ${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf P_{(k-1){\\ell}}}$. However, note that by propositions \\[prop:fusionIvsP\\] and \\[prop\\_fusion\\_sklvspi\\] $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{(k-1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}},$$ while by proposition \\[prop:fusionproj\\] $${\\mathsf P_{(k-1){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{(k-1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\oplus {\\mathsf P_{(k-1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}.$$ It follows that $f$ cannot be zero, and thus that ${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{l}} \\simeq {\\mathsf{B}^{2}_{(k-1){\\ell}-2}}$.\n\nNote that the case ${\\ell}=2$ in the regular family cannot be obtained from this discussion, since in this case the exact sequence satisfied by ${\\mathsf P_{2}}$ is instead $$\\label{eq:sesP2l2}\n 0 \\longrightarrow {\\mathsf I_{2}} \\longrightarrow {\\mathsf P_{2}} \\longrightarrow {\\mathsf S_{2}} \\longrightarrow 0.$$ In this case, proposition \\[prop:fusion\\_ds2vsds2\\] gives $${\\mathsf I_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2k}} \\simeq {\\mathsf S_{2 k}}.$$ Instead, use the exact sequence $${\\mathsf S_{2(k+1)}} \\longrightarrow {\\mathsf S_{2k}} \\longrightarrow {\\mathsf I_{2k}} \\longrightarrow 0,$$ which becomes $${\\mathsf S_{2(k+1)}} \\longrightarrow {\\mathsf S_{2k}} \\longrightarrow {\\mathsf I_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{2}} \\longrightarrow 0,$$ by fusing it with ${\\mathsf I_{2}}$. Since ${\\mathsf I_{2k}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf I_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{2k-1}}, $ it follows that $${\\mathsf I_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{2}} \\simeq {\\mathsf I_{2k}}.$$ Using this fact with the exact sequence gives $${\\mathsf I_{2k}} \\longrightarrow {\\mathsf P_{2k}} \\longrightarrow {\\mathsf I_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2}} \\longrightarrow 0.$$ Now, since ${\\mathsf I_{2k}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{2k -1 }} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2}} \\simeq {\\mathsf P_{2(k-1) -1}} \\oplus {\\mathsf P_{2(k +1)-1}} $, while ${\\mathsf P_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{2k-1}} \\oplus {\\mathsf P_{2(k+1) -1}} \\oplus 2 {\\mathsf P_{2(k-1)-1}}$, it follows that $${\\mathsf I_{2k}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2}} \\simeq {\\mathsf{B}^{2}_{2(k-1)}}.$$\n\n\\[prop:fusionIvsSl\\] For all $k>1$, and $\\ell \\geq 2$ $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf{B}^{2}_{(k-1){\\ell}-2}}.$$\n\nTo proceed and compute the fusion of the irreducibles with the other standard modules, we therefore need the fusion of ${\\mathsf{B}^{2}_{(k-1){\\ell}-2}}$ with ${\\mathsf S_{{\\ell}}}$, which requires the fusion of ${\\mathsf{B}^{2}_{(k-1){\\ell}-2}}$ with projective modules. ", "This is our next step.", "\n\n[Fusion of ${\\mathsf{B}^{2i}_{n,k}}$ and projective modules]{}\\[sec:fusionBvsP\\]\n\n(0,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{n,k}}$]{} – (3,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{1}}}$]{} – (6,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{2}}}$]{} – (9,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{3}}}$]{}; (9,3) – (10.5,1.5); (10.5,1.5) – (12,0); at (12,1.5) [$\\hdots$]{}; (13,3) – (14.5,1.5); (14.5,1.5) – (16,0); (16,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{2(i-1)}}}$]{} – (19,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{n,k_{2i -1}}}$]{} – (22,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{n,k_{2i}}}$]{}; (3,3) – (1.5,1.5); (3,3) – (4.5,1.5); (9,3) – (7.5,1.5); (9,3) – (10.5,1.5); (19,3) – (17.5,1.5); (19,3) – (20.5,1.5);\n\nThe rules for the induction of these modules are [@BelRiSy] $$\\begin{gathered}\n {{\\mathsf{B}^{2i}_{n,k}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq \\left. ", "\\begin{cases}\n {\\mathsf{B}^{2i}_{n+1,k}}, & \\text{ in } {\\mathsf{dTL}_{n}}\\\\\n 0, & \\text{ in } {\\mathsf{TL}_{n}}\n \\end{cases}\\right\\} \\oplus \\begin{cases}\n \\overset{i}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{n+1,k+2p{\\ell}-1}} & \\text{ if } k = 0 \\mod {\\ell}\\\\\n {\\mathsf{B}^{2i}_{n+1,k-1}} & \\text{otherwise}\n \\end{cases}\\\\\n \\oplus \\begin{cases}\n \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{n+1,k+2p{\\ell}+1}} & \\text{ if } k+2 = 0 \\mod {\\ell}\\\\\n {\\mathsf{B}^{2i}_{n+1,k+1}} & \\text{otherwise}\n \\end{cases}.\\end{gathered}$$ The usual argument on the parity of the modules gives the following fusion rules.", "\n\n\\[prop:fusionBvs01\\] In the dilute family $${\\mathsf{B}^{2i}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf{B}^{2i}_{n+1,k}},$$ while in both families $$\\begin{gathered}\n{\\mathsf{B}^{2i}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq \\begin{cases}\n \\overset{i}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{n+1,k+2p{\\ell}-1}} & \\text{ if } k = 0 \\mod {\\ell}\\\\\n {\\mathsf{B}^{2i}_{n+1,k-1}} & \\text{otherwise}\n \\end{cases}\\\\\n \\oplus \\begin{cases}\n \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{n+1,k+2p{\\ell}+1}} & \\text{ if } k+2 = 0 \\mod {\\ell}\\\\\n {\\mathsf{B}^{2i}_{n+1,k+1}} & \\text{otherwise}\n \\end{cases}.\\end{gathered}$$\n\nThe first formula shows that the parameter $n$ can be adjusted by simply fusing the module with ${\\mathsf P_{1,0}}$. In the regular family, proposition \\[prop:magickOthersP\\] gives $${\\mathsf{B}^{2i}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf{B}^{2i}_{n+2,k}}.$$ Like for the standard modules, we therefore omit this parameter and simply assume $n$ to be big enough for the modules to exist.", "\n\nWe start by studying the fusion of ${\\mathsf{B}^{2i}_{23}}$ in the ${\\ell}=5$ case. ", "The preceding proposition gives $${\\mathsf{B}^{2i}_{23}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{0}} \\simeq {\\mathsf{B}^{2i}_{23}},$$ $${\\mathsf{B}^{2i}_{23}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(5+2p) 5 - 1}} \\oplus {\\mathsf{B}^{2i}_{22}}.$$ Fusing the last equation with ${\\mathsf P_{1}}$ yields $${\\mathsf{B}^{2i}_{23}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{0}} \\oplus {\\mathsf P_{2}}) \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(5+2p)5-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf{B}^{2i}_{21}} \\oplus {\\mathsf{B}^{2i}_{23}}.$$ Comparing this result with proposition \\[prop:fusionBvs01\\], it follows that $${\\mathsf{B}^{2i}_{23}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(5+2p)5-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf{B}^{2i}_{21}}.$$ Repeating the argument gives $${\\mathsf{B}^{2i}_{23}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(5+2p)5-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\oplus {\\mathsf{B}^{2i}_{20}}.$$\n\n\\[prop:fusionBvssmallP\\] For all $0 \\leq j <{\\ell}-1$, ${\\ell}\\geq 2$, $k > 1$ $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\oplus {\\mathsf{B}^{2i}_{k{\\ell}-2-j}},$$ $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}-1}}.$$\n\nIf $j=1,0$, proposition \\[prop:fusionBvs01\\] already gives the conclusion. ", "Suppose that the result stands for $j-1,j$ with $ j <{\\ell}-2$. Then $$\\begin{aligned}\n{\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq {\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{j-1}} \\oplus {\\mathsf P_{j+1}} \\right)\\\\\n& \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}({\\mathsf P_{j}} \\oplus {\\mathsf P_{j-2}})\\right) \\oplus {\\mathsf{B}^{2i}_{k{\\ell}-2-j-1}} \\oplus {\\mathsf{B}^{2i}_{k {\\ell}-2 - j+1}}.\\end{aligned}$$ Comparing the two lines and using the induction hypothesis yields the conclusion for $j+1$. In particular, this yields $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}} \\oplus {\\mathsf{B}^{2i}_{(k-1){\\ell}}}.$$ Fusing this result with ${\\mathsf P_{1}}$ gives $$\\begin{aligned}\n{\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-1}}\\oplus {\\mathsf P_{{\\ell}-3}}\\right) \\simeq & \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-4}}\\oplus {\\mathsf P_{{\\ell}-2}}\\right)\\right) \\oplus {\\mathsf{B}^{2i}_{(k-1){\\ell}+1}}\\\\ \n& \\oplus \\overset{i}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k-1+2p){\\ell}-1}}.\\end{aligned}$$ Comparing the two sides, using the result of the first part and rearranging the terms gives $$\\begin{aligned}\n{\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} & \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k+2p +1){\\ell}-1}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}-1}}\\\\\n&\\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}-1}}\\end{aligned}$$ where the second line follows from proposition \\[prop\\_fusion\\_sklvspi\\].", "\n\nThe last formula can be used to quickly obtain the fusion rules with the bigger projectives. ", "Thus, $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf{B}^{2i}_{k{\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}}},$$ $${\\mathsf{B}^{2i}_{kl -2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}+1}},$$ $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+2}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+2}}\\right) \\oplus {\\mathsf P_{(k-1){\\ell}+2}}.$$ Continuing in this manner eventually gives $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2{\\ell}-1}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2{\\ell}-1}}\\right) \\oplus {\\mathsf P_{(k-2){\\ell}-1}} \\oplus {\\mathsf P_{k {\\ell}-1}}.$$ Note that if $k=2$, ${\\mathsf P_{(k-2){\\ell}-1}}\\simeq 0$. Repeating these arguments, the proof for the general formula is straightforward.", "\n\n\\[prop:fusionBvsP\\] For all $k>1,r>0$, $i>0$, $0\\leq j<{\\ell}$, $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1 +j}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1+j}}\\right) \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left( k-r,r-k+2\\right)}}{\\bigoplus}} {\\mathsf P_{p{\\ell}-1+j}}.$$\n\nThe same method can be used to obtain the formulas for the fusion of ${\\mathsf{B}^{2i}_{k{\\ell}-t}}$. Fusing the formula in the preceding proposition with ${\\mathsf P_{1}}$ yields $$\\begin{aligned}\n{\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1 +j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1 +j}}\\right) \\oplus {\\mathsf{B}^{2i}_{k{\\ell}-3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1 -j}} \\\\\n& \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left(\\left({\\mathsf S_{(k+2p){\\ell}+1}} \\oplus {\\mathsf P_{(k+2p){\\ell}-1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1+j}}\\right)\\\\\n& \\qquad \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left( k-r,r-k+2\\right)}}{\\bigoplus}} {\\mathsf P_{p{\\ell}-1+j}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}.\\end{aligned}$$ Comparing the two lines yields $${\\mathsf{B}^{2i}_{k{\\ell}-3}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1 +j}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}+1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1+j}}\\right) \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left( k-r,r-k+2\\right)}}{\\bigoplus}} \\left({\\mathsf P_{p{\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right).$$ Once again, this operation can be repeated and gives the following general formula.", "\n\nFor all $k>1,r,i>0$, $0<t<{\\ell}$, $0\\leq j <{\\ell}$, $${\\mathsf{B}^{2i}_{k{\\ell}-1-t}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1 +j}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}-1+t}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1+j}}\\right) \\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = \\max\\left( k-r,r-k+2\\right)}}{\\bigoplus}}\\left( {\\mathsf P_{p{\\ell}-1+j}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{t-1}}\\right).$$\n\nNote that, in this section, the case ${\\mathsf{B}^{2i}_{{\\ell}-2}}$ has been avoided. ", "In this case, there is a short-exact sequence $$0 \\longrightarrow {\\mathsf I_{{\\ell}-2}} \\longrightarrow {\\mathsf{B}^{2i}_{{\\ell}-2}} \\longrightarrow {\\mathsf{T}^{2i-1}_{{\\ell}}} \\longrightarrow 0,$$ and since ${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq 0 $, for all $j\\geq {\\ell}-1 $, $${\\mathsf{B}^{2i}_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq {\\mathsf{T}^{2i-1}_{{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}}.$$ This case will be treated in section \\[sec:fusionTvsP\\]\n\n[Fusion of ${\\mathsf{B}^{2i}_{k}}$ and standard modules]{}\\[sec:fusionBvsS\\]\n\nWe now want to compute the fusion of ${\\mathsf{B}^{2i}_{k}}$ with the standard modules ${\\mathsf S_{q}}$ that are not projective. ", "The first step is to find a formula for ${\\mathsf{B}^{2i}_{k{\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$.\n\nUsing the projective cover of ${\\mathsf S_{{\\ell}}}$ (see section \\[sec:rappel.indec\\]) and the right-exactness of fusion, one can obtain the exact sequence $$\\label{eq:fusionBvsS.1}\n {\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\overset{f}{\\longrightarrow} {\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\longrightarrow {\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow 0.$$ Using propositions \\[prop:fusionBvssmallP\\], \\[prop:fusionBvsP\\], and \\[prop\\_fusion\\_sqlvspkl\\] gives $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}} \\oplus {\\mathsf{B}^{2i}_{(k-1){\\ell}}},$$ and $${\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-3}} \\oplus {\\mathsf P_{{\\ell}-1}}\\right)\\right) \\oplus \\overset{i}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k-1+2p){\\ell}}}.$$ Therefore $$\\left({\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right)/ \\left( {\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right) \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+1)}_{(k-1){\\ell}-2}}$$ where we used proposition \\[prop:indec.prop\\]. ", "If it can be proved that $f$ is injective, this will give a formula for ${\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$. To do this, we will prove that the dimension of ${\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$ is that of $\\left({\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right)/ \\left( {\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right)$, and this will be done by induction on $i$.\n\nNote first that by proposition \\[prop:fusionIvsSl\\] $${\\mathsf{B}^{2 \\times 0}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf{B}^{2}_{(k-1){\\ell}-2}} \\simeq \\left({\\mathsf{B}^{0}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right)/ \\left( {\\mathsf{B}^{0}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right).$$ This gives the case $i=0$ for all $k>1$. Assume now that $${\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+1)}_{(k-1){\\ell}-2}}$$ for a certain $i$ and all $k>1$. To proceed with the induction, we will use the short exact sequence $$0 \\longrightarrow {\\mathsf{B}^{2i}_{k {\\ell}-2}} \\longrightarrow {\\mathsf{B}^{2(i+1)}_{k{\\ell}-2}} \\longrightarrow {\\mathsf S_{(k+2i){\\ell}}}\\longrightarrow 0.$$ It can be seen by inspecting the Loewy diagram of the $B$ modules and proved using techniques developed in [@BelRiSy]. ", "$$\\begin{tikzpicture}[scale=1/3]\n\\filldraw \n (0,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{k {\\ell}-2}}$}\n -- (3,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{k {\\ell}}}$}\n -- (6,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{(k+2){\\ell}-2}}$}\n -- (9,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{(k+2){\\ell}}}$};\n\\draw (9,3) -- (10.5,1.5);\n\\draw[dotted] (10.5,1.5) -- (12,0);\n\\node at (12,1.5) {$\\hdots$};\n\\draw[dotted] (13,3) -- (14.5,1.5);\n\\draw (14.5,1.5) -- (16,0);\n\\filldraw\n (16,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{(k+2i){\\ell}-2}}$}\n -- (19,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{(k+2i){\\ell}}}$}\n -- (22,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{ (k+2i+2){\\ell}-2}}$};\n\\draw[->] (3,3) -- (1.5,1.5);\n\\draw[->] (3,3) -- (4.5,1.5);\n\\draw[->] (9,3) -- (7.5,1.5);\n\\draw[->] (9,3) -- (10.5,1.5);\n\\draw[->] (19,3) -- (17.5,1.5);\n\\draw[->] (19,3) -- (20.5,1.5);\n\\draw[dashed, blue] (-1.5,-2) -- (-1.5,5) -- (17,5) -- (17,-2) -- (-1.5,-2);\n\\node[anchor=east] at (-1.5,1.5) {${\\mathsf{B}^{2i}_{k{\\ell}-2}}$};\n\\draw[dotted, red] (17.5,-2) -- (17.5,5) -- (25,5) -- (25,-2) -- (17.5,-2);\n\\node[anchor =west] at (25,1.5) {${\\mathsf S_{(k+2i){\\ell}}}$};\n\\end{tikzpicture}$$ The right-exactness of fusion yields the exact sequence $${\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\overset{g}{\\longrightarrow} {\\mathsf{B}^{2(i+1)}_{k {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow {\\mathsf S_{(k+2i){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\longrightarrow 0.$$ It follows that $$\\begin{aligned}\n \\dim \\left( {\\mathsf{B}^{2(i+1)}_{k {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right) & \\leq \\dim\\left({\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right) + \\dim\\left( {\\mathsf S_{(k+2i){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right) \\\\\n & = \\dim \\left( \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+1)}_{(k-1){\\ell}-2}}\\right) \\\\\n & \\qquad + \\dim \\left( {\\mathsf P_{(k+2i){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(k+2i +1){\\ell}}}\\right) \\\\\n & = \\dim \\left(\\overset{i+1-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+2)}_{(k-1){\\ell}-2}} \\right)\\end{aligned}$$\n\nwhere the equality occurs if and only if $g$ is injective. ", "The exact sequence gives $$\\begin{aligned}\n \\dim \\left({\\mathsf{B}^{2(i+1)}_{k {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right)& = \\dim \\left( {\\mathsf{B}^{2(i+1)}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\right) - \\dim \\operatorname{im}f \\\\\n & \\geq \\dim \\left( {\\mathsf{B}^{2(i+1)}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\right) - \\dim \\left( {\\mathsf{B}^{2i}_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\right) \\\\\n & = \\dim \\left( \\overset{i+1-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+2)}_{(k-1){\\ell}-2}}\\right).\\\\\\end{aligned}$$ It follows that $\\ker f \\simeq 0$, and the following result is thus proved.", "\n\n\\[prop:fusionBvsSl\\] For all $i\\geq 0$, $k>1$, $${\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+1)}_{(k-1){\\ell}-2}}.$$\n\nFusion rules for bigger standard modules will not be needed to compute the fusion of irreducible modules but we include them for the sake of completeness.", "\n\nFor all $0<r <k$, $$\\label{eq:BvsSql}\n{\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r {\\ell}}} \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} \\oplus {\\mathsf{B}^{2(i+r)}_{(k-r){\\ell}-2}}.$$\n\nWe proceed by induction on $r$, the case $r=1$ being given by the previous proposition. ", "Assume the result for some $r<k-1$. Using propositions \\[prop:fusionBvsP\\], \\[prop\\_fusion\\_sqlvspkl\\], and \\[prop:fusionproj\\], we start by noting that $$\\begin{aligned}\n {\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} & \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right)\\oplus \\overset{k+r-2}{\\underset{\\underset{step=2}{p = k-r}}{\\bigoplus}}\\left({\\mathsf P_{p {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right) \\notag\\\\\n & \\simeq \\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k+2p+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1}} \\right) \\notag\\\\\n & \\qquad \\oplus \\overset{r-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k-r+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\right),\\end{aligned}$$ and $${\\mathsf P_{(k+2p+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1}} \\simeq {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(k+r+2p){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}.$$ Next, we fuse the left side of equation with ${\\mathsf S_{{\\ell}}}$ and use propositions \\[prop:fusionBvsSl\\], \\[prop\\_fusion\\_sqlvspkl\\], and \\[prop:fusionstandvssl\\] to obtain $$\\begin{aligned}\n {\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{r {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right)& \\simeq {\\mathsf{B}^{2i}_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(r+1){\\ell}}}\\right), \\end{aligned}$$ while fusing the right side of equation with ${\\mathsf S_{{\\ell}}}$ gives $$\\begin{aligned}\n\\overset{i-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p){\\ell}-1}} & {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\oplus {\\mathsf{B}^{2(i+r)}_{(k-r){\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\notag\\\\\n& \\simeq\\overset{i-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(r+1){\\ell}-1}}\\right)\\right) \\notag\\\\\n& \\oplus \\overset{i+r-1}{\\underset{p = 0}{\\bigoplus}}\\left({\\mathsf P_{(k-r+2p){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{B}^{2(i+1+r)}_{(k-r-1){\\ell}-2}}.\\end{aligned}$$ Comparing the two results and using the previous observations gives the conclusion for $r+1$.\n\nNote that in all of these calculations, we carefully avoided the case $k=1$ (and $r=k$). ", "There is a short exact sequence $$0 \\longrightarrow {\\mathsf I_{{\\ell}-2}} \\longrightarrow {\\mathsf{B}^{2i}_{{\\ell}-2}} \\longrightarrow {\\mathsf{T}^{2i -1}_{{\\ell}}} \\longrightarrow 0,$$ which can be seen by inspecting the Loewy diagram of ${\\mathsf{B}^{2i}_{{\\ell}-2}}$ and proved using techniques developed in [@BelRiSy]. ", "$$\\begin{tikzpicture}[scale=1/3]\n\\filldraw \n (0,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{{\\ell}-2}}$}\n -- (3,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{{\\ell}}}$}\n -- (6,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{2{\\ell}-2}}$}\n -- (9,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{2{\\ell}}}$};\n\\draw (9,3) -- (10.5,1.5);\n\\draw[dotted] (10.5,1.5) -- (12,0);\n\\node at (12,1.5) {$\\hdots$};\n\\draw[dotted] (13,3) -- (14.5,1.5);\n\\draw (14.5,1.5) -- (16,0);\n\\filldraw\n (16,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{(2i-1){\\ell}-2}}$}\n -- (19,3) circle (6 pt) node[anchor = south] {${\\mathsf I_{(2i-1){\\ell}}}$}\n -- (22,0) circle (6 pt) node[anchor = north] {${\\mathsf I_{2i{\\ell}-2}}$};\n\\draw[->] (3,3) -- (1.5,1.5);\n\\draw[->] (3,3) -- (4.5,1.5);\n\\draw[->] (9,3) -- (7.5,1.5);\n\\draw[->] (9,3) -- (10.5,1.5);\n\\draw[->] (19,3) -- (17.5,1.5);\n\\draw[->] (19,3) -- (20.5,1.5);\n\\draw[dashed,blue]\n (2,6) -- (24,6)\n -- (24,-3) -- (2,-3)\n -- (1.5,6);\n\\node[anchor = west] at (24,1.5) {${\\mathsf{T}^{2i-1}_{{\\ell}}}$};\n\\end{tikzpicture}$$ Since it was already noted in proposition \\[prop:fusionIWeird\\] that ${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{q{\\ell}}} \\simeq 0$ for all $q\\geq 1$, $$\\label{eq:fusionBSWeird}\n{\\mathsf{B}^{2i}_{{\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{q{\\ell}}} \\simeq {\\mathsf{T}^{2i -1}_{{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{q{\\ell}}}.$$ Therefore, to compute ${\\mathsf I_{2{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{2{\\ell}}}$, we will have to compute $$\\left({\\mathsf I_{2{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf{B}^{2}_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq {\\mathsf{T}^{1}_{{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}.$$ The fusion rules for ${\\mathsf{T}^{2i-1}_{{\\ell}}}$ will thus be needed to compute the fusion rules of the irreducible modules.", "\n\n[Fusion of ${\\mathsf{T}^{2i+1}_{k}}$ and projective modules]{}\\[sec:fusionTvsP\\] The formulas for the induction of ${\\mathsf{T}^{2i+1}_{k}}$ are [@BelRiSy]: $${{\\mathsf{T}^{2i+1}_{n,k}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq {\\mathsf{T}^{2i+1}_{n+1,k-1}} \\oplus {\\mathsf{T}^{2i+1}_{n+1,k+1}}\\oplus \\left. ", "\\begin{cases}\n {\\mathsf{T}^{2i+1}_{n+1,k}}, & \\text{ in } {\\mathsf{dTL}_{n}}\\\\\n 0, & \\text{ in } {\\mathsf{TL}_{n}}\n \\end{cases}\\right\\},$$ where $${\\mathsf{T}^{2i+1}_{n,k}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}} {\\mathsf P_{n,k+2p{\\ell}}}$$ if $k$ is critical and $${\\mathsf{T}^{2i+1}_{n,-1}} \\simeq \\overset{i}{\\underset{p=1}{\\bigoplus}} {\\mathsf P_{n,2p{\\ell}-1}}.$$ Using the parity of the relevant modules gives the following fusion rules.", "\n\n\\[prop:fusionTvs01\\] For all $k$,$i$, in the dilute family $${\\mathsf{T}^{2i+1}_{n,k}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf{T}^{2i+1}_{n+1,k}},$$ while in both families $${\\mathsf{T}^{2i+1}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf{T}^{2i+1}_{n+1,k-1}} \\oplus {\\mathsf{T}^{2i+1}_{n+1,k+1}}.$$\n\nOnce again, fusing these modules with ${\\mathsf P_{1,0}}$ simply increases the parameter $n$. In the regular case, proposition \\[prop:magickOthersP\\] gives $${\\mathsf{T}^{2i+1}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf{T}^{2i+1}_{n+2,k}},$$ as long as ${\\ell}\\neq 2$. As before, the proofs will be independent of $n$ so we simply omit this parameter and assume $n$ to be big enough for the modules to exist.", "\n\nWe start by studying the modules ${\\mathsf{T}^{2i+1}_{k{\\ell}}}$. Note that $${\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}} {\\mathsf P_{(k+2p){\\ell}-1}} \\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}+1}}.$$ Fusing this expression with ${\\mathsf P_{1}}$ yields $$\\begin{aligned}\n{\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} &\\simeq {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{0}} \\oplus {\\mathsf P_{2}} \\right)\\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}} {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}}} \\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}+2}}\\end{aligned}$$ Comparing the first and second lines and using proposition \\[prop:fusionTvs01\\] give the fusion rule $${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}+2}}.$$ It is a simple exercise to repeat this argument and obtain the fusion rules for the other small projectives.", "\n\n\\[prop:fusionTvssmallP\\] For all $i,k \\geq 0$, $0\\leq j<{\\ell}-1$, $${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j-1}} \\right)\\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}+j}}.$$\n\nIn particular, $${\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}-2}}.$$ Fusing this expression with ${\\mathsf P_{1}}$ gives $$\\begin{aligned}\n{\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq {\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-3}} \\oplus {\\mathsf P_{{\\ell}-1}}\\right) \\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-4}} \\oplus {\\mathsf P_{{\\ell}-2}} \\right)\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}-3}} \\\\\n & \\qquad \\oplus \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p+1){\\ell}-1}}\\right).\\end{aligned}$$ Comparing the first and second lines gives $$\\begin{aligned}\n\\label{eq:fusionTvsP.1}\n {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k+1+2p){\\ell}-1}}\\right) \\notag \\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) ,\\end{aligned}$$ where the known fusion rules for standard modules (proposition \\[prop\\_fusion\\_sklvspi\\]) were used in the second line. ", "Fusing this expression with ${\\mathsf P_{1}}$ gives $${\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right) .$$ Fusing the latter expression again with ${\\mathsf P_{1}}$ gives $$\\begin{aligned}\n {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}\\left(2{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{{\\ell}+1}}\\right)\\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}\\left( 2{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{{\\ell}+1}}\\right)\\right).\\end{aligned}$$ Comparing the two lines yields the fusion rule $${\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}}\\right).$$ The same arguments prove the following proposition.", "\n\n\\[prop:fusionTvsP\\] For all $i, k \\geq 0$, $r\\geq {\\ell}-1$, $$\\label{eq:fusionTklvsP}\n {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}}\\right),$$\n\nThe fusion rules for ${\\mathsf{T}^{2i+1}_{k{\\ell}+i}}$ can be obtained from these formulas. ", "We start by fusing with ${\\mathsf P_{1}}$. $$\\begin{aligned}\n {\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} & \\simeq \\left(\\overset{i}{\\underset{p=0}{\\bigoplus}} {\\mathsf P_{(k+2p){\\ell}-1}} \\oplus {\\mathsf{T}^{2i+1}_{k{\\ell}+1}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left(\\left( {\\mathsf S_{(k+2p){\\ell}+1}} \\oplus {\\mathsf P_{(k+2p){\\ell}-1}}\\right) {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}}\\right).\\end{aligned}$$ Comparing the two lines yields $${\\mathsf{T}^{2i+1}_{k{\\ell}+1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}+1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}}\\right).$$ This argument can be repeated to obtain the following proposition.", "\n\nFor all $i,k \\geq 0 $, $0<j<{\\ell}$, $r\\geq {\\ell}-1$, $${\\mathsf{T}^{2i+1}_{k{\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}-1+j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}}\\right).$$\n\n[Fusion of ${\\mathsf{T}^{2i+1}_{k}}$ and standard modules]{}\\[sec:fusionTvsS\\]\n\nWe want to compute fusions of ${\\mathsf{T}^{2i+1}_{k}}$ with non-projective standard modules. ", "Proceeding as in the previous sections, we start by computing ${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}$, where $k\\neq 0$.\n\nThere is a short-exact sequence $$0 \\longrightarrow {\\mathsf P_{{\\ell}-2}} \\longrightarrow {\\mathsf P_{{\\ell}}} \\longrightarrow {\\mathsf S_{{\\ell}}} \\longrightarrow 0,$$ which gives the exact sequence $$\\label{eq:sesTvsS.1}\n{\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\overset{f}{\\longrightarrow} {\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\longrightarrow {\\mathsf{T}^{2i+1}_{kl}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow 0$$ by using the right-exactness of fusion. ", "Propositions \\[prop:fusionTvssmallP\\] and equation give $${\\mathsf{T}^{2i+1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-3}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}-2}},$$ $${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}}{\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{{\\ell}-3}}\\oplus {\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf P_{(k+1+2p){\\ell}}}\\right).$$ Therefore $$\\left({\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}\\right) / \\left( {\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right) \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}}},$$ where proposition \\[prop:indec.prop\\], which gives $ \\Big(\\overset{i}{\\underset{p=0}{\\bigoplus}} {\\mathsf P_{(k+1+2p){\\ell}}} \\Big)/ {\\mathsf{T}^{2i+1}_{(k+1){\\ell}-2}} \\simeq {\\mathsf{T}^{2i+1}_{(k+1){\\ell}}}$, was used. ", "The goal is now to prove that $$\\label{eq:fusionTvsSl}\n{\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{l}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}}},$$ which is equivalent to $f$ being injective. ", "Note that for $i=0$, this is just the fusion of two standard modules, and proposition \\[prop:fusionstandvssl\\], or \\[prop:fusion\\_ds2vsds2\\] if $k=1$, ${\\ell}=2$ in the regular family, agrees with . ", "We thus proceed by induction on $i$. Assume that stands for $i$ and use the short exact sequence $$0 \\longrightarrow {\\mathsf{T}^{2i+1}_{k{\\ell}}} \\longrightarrow {\\mathsf{T}^{2i+3}_{k{\\ell}}} \\longrightarrow {\\mathsf S_{(k+2i+2){\\ell}}} \\longrightarrow 0,$$ to obtain the exact sequence $${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow {\\mathsf{T}^{2i+3}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow {\\mathsf S_{(k+2i+2){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\longrightarrow 0.$$ It gives the inequality $$\\begin{aligned}\n \\dim \\left({\\mathsf{T}^{2i+3}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right) & \\leq \\dim \\left({\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right) + \\dim \\left( {\\mathsf S_{(k+2i+2){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\right)\\\\\n &= \\dim \\left( \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}}}\\right)\\\\\n & \\qquad + \\dim \\left(\\left( {\\mathsf P_{(k+2i+2){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf S_{(k+2i + 3){\\ell}}} \\right)\\\\\n & = \\dim \\left(\\overset{i+1}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+3}_{(k+1){\\ell}}} \\right)\\end{aligned}$$\n\nHowever, the exact sequence also give the inequality $$\\begin{aligned}\n\\dim \\left( {\\mathsf{T}^{2i+3}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right) & = \\dim \\left({\\mathsf{T}^{2i+3}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\right) - \\dim \\operatorname{im}f \\\\\n& \\geq \\dim \\left( \\overset{i+1}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+3}_{(k+1){\\ell}}} \\right)\\end{aligned}$$\n\nComparing the two bounds shows that $\\dim\\left(\\text{\\rm im}f\\right) = \\dim\\left({\\mathsf{T}^{2i+3}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}}\\right)$ and thus that $f$ is injective. ", "Formula must therefore stand for $i+1$, proving the following proposition.", "\n\n\\[prop:fusionTvsSl\\] For $i\\geq 0$, $k>0$, $${\\mathsf{T}^{2i+1}_{k{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+1){\\ell}}}.$$\n\nFusions with the bigger standard modules and the other ${\\mathsf{T}^{2i+1}_{k{\\ell}+i}}$ will not be needed but are presented for the sake of completeness.", "\n\nFor all $i\\geq0$, $k,r>0$, $$\\label{eq:fusionTvsSql}\n{\\mathsf{T}^{2i+1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r {\\ell}}} \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+r){\\ell}}}.$$\n\nWe proceed by induction on $r$. The case $r=1$ being contained in proposition \\[prop:fusionTvsSl\\] , suppose that the result holds for a certain $r>1$. Then, we start by noticing that by propositions \\[prop:fusionTvsP\\], and \\[prop\\_fusion\\_sklvspi\\], $$\\begin{aligned}\n {\\mathsf{T}^{2i+1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} &\\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf S_{(k+2p){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right)\\\\\n & \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{l-2}} \\oplus {\\mathsf P_{(k+2p+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} \\right),\\end{aligned}$$ and by proposition \\[prop:fusionproj\\], $${\\mathsf P_{(k+2p+1){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1}} \\simeq {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{(r-1){\\ell}-1}} \\oplus {\\mathsf P_{(k+2p +r){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}.$$ Then, fuse the left side of with ${\\mathsf S_{{\\ell}}}$ and use propositions \\[prop:fusionTvsP\\], and \\[prop\\_fusion\\_sklvspi\\], with equation to obtain $${\\mathsf{T}^{2i+1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{r{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right) \\simeq {\\mathsf{T}^{2i+1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(r+1){\\ell}}} \\right),$$ while fusing its right side with ${\\mathsf S_{{\\ell}}}$ and using propositions \\[prop\\_fusion\\_sqlvspkl\\], and \\[prop:fusionTvsSl\\] gives $$\\begin{aligned}\n\\overset{i}{\\underset{p=0}{\\bigoplus}}& \\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r{\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}}\\right)\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+r){\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\notag \\\\\n & \\qquad \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(r+1){\\ell}-1}} \\oplus {\\mathsf P_{(r-1){\\ell}-1}}\\right)\\right)\\notag\\\\\n & \\qquad \\simeq \\overset{i}{\\underset{p=0}{\\bigoplus}}\\left({\\mathsf P_{(k+2p+r){\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2i+1}_{(k+r+1){\\ell}}}.\\end{aligned}$$ Comparing the two sides and using the preceding observations gives the conclusion for $i+1$.\n\n[The fusion of irreducible and standard modules, second part]{} We now have the tools needed to compute the fusion of an irreducible module and a non-projective standard module.", "\n\n\\[prop:fusionIlvsSql\\] For $k>1$,$r>0$, and in the regular family if ${\\ell}\\neq 2$, $$\\label{eq:fusionIvsS.1}\n{\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r{\\ell}}} \\simeq \\begin{cases}\n{\\mathsf{B}^{2r}_{(k-r){\\ell}-2}} & \\text{ if } k>q\\\\\n{\\mathsf{T}^{2k-3}_{(2+r-k){\\ell}}} & \\text{ if } k\\leq q\n\\end{cases}.$$ In the regular family, if ${\\ell}= 2$, $${\\mathsf I_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r{\\ell}}} \\simeq \\begin{cases}\n{\\mathsf{B}^{2r}_{(k-r){\\ell}}} & \\text{ if } k>r\\\\\n{\\mathsf{T}^{2k+1}_{(1+r-k){\\ell}}} & \\text{ if } k\\leq r\n\\end{cases}.$$\n\nWe proceed by induction on $r$. Proposition \\[prop:fusionIvsSl\\] already gives the case $r=1$, so suppose that the result holds for some $1 \\leq r<k-1$. Fuse the left side of equation with ${\\mathsf S_{{\\ell}}}$ and use propositions \\[prop:fusionstandvssl\\], and \\[prop:fusionIvsP\\] to obtain $$\\begin{aligned}\n {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf S_{r {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{ {\\ell}}}\\right) &\\simeq {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(r+1){\\ell}}} \\right)\\notag\\\\\n & \\simeq \\overset{r-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p-r){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{(r+1){\\ell}}}.\\end{aligned}$$ Then, fuse the right side of equation with ${\\mathsf S_{{\\ell}}}$ and use propositions \\[prop:fusionBvsSl\\], and \\[prop:fusionTvsSl\\] to obtain $${\\mathsf{B}^{2q}_{(k-r){\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\overset{r-1}{\\underset{p = 0}{\\bigoplus}}{\\mathsf P_{(k+2p-r){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus \\left. ", "\\begin{cases}\n{\\mathsf{B}^{2(r+1)}_{(k-1-r){\\ell}-2}}, & \\text{ if } r< k-1\\\\\n{\\mathsf{T}^{2(k-2)+1}_{2{\\ell}}}, & \\text{ if } r=k-1\n\\end{cases}\\right\\rbrace .$$ Comparing the two results gives the conclusion for $r+1$. In particular, this gives the conclusion for all $r \\leq k$.\n\nSuppose now that the result holds for some $r\\geq k$. Fuse the left side of equation with ${\\mathsf S_{{\\ell}}}$ and use propositions \\[prop:fusionstandvssl\\], and \\[prop:fusionIvsP\\] to obtain $$\\begin{aligned}\n {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf S_{r {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{ {\\ell}}} \\right)& \\simeq {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{r{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf S_{(r+1){\\ell}}} \\right) \\notag \\\\\n & \\simeq \\overset{k-2}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(2+r-k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{(r+1){\\ell}}}.", "\n \\end{aligned}$$ Fusing the right side of equation with ${\\mathsf S_{{\\ell}}}$ and using propositions \\[prop:fusionTvsSl\\] instead gives $${\\mathsf{T}^{2k-3}_{(2+r-k){\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{{\\ell}}} \\simeq \\overset{k-2}{\\underset{p=0}{\\bigoplus}}\\left( {\\mathsf P_{(2+r-k+2p){\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}\\right) \\oplus {\\mathsf{T}^{2k-3}_{(2-k+r+1){\\ell}}}.$$ Comparing the two results then give the conclusion for $r+1$.\n\nIn the regular family, the case where ${\\ell}=2$ is slightly different because then ${\\mathsf{B}^{2i}_{0}} \\simeq {\\mathsf{T}^{2i-1}_{2}}$. Nevertheless, the arguments are nearly identical.", "\n\n\\[prop:fusionIvsstand\\] For $k>1$, $r\\geq 1$, $0<i,j<{\\ell}$, $$\\label{eq:fusionIvsstand}\n{\\mathsf I_{k {\\ell}-1 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf S_{r{\\ell}-1+j}} \\simeq \\begin{cases}\n{\\mathsf{B}^{2r}_{(k-r){\\ell}-1-j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} & \\text{ if } k>r\\\\\n{\\mathsf{T}^{2k-3}_{(2+r-k){\\ell}+(j-1)}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} & \\text{ if } k\\leq r\n\\end{cases}.$$\n\nThe proof mimics those of previous sections so we will only give a rough outline. ", "Proceed by induction on $i,j$, using proposition \\[prop:fusionIlvsSql\\] for the case $i=j=1$. To induce on $i$, fuse both sides of equation with ${\\mathsf P_{1}}$, use propositions \\[prop:fusionIvssmallP\\], \\[prop:fusionBvssmallP\\], and \\[prop:fusionTvssmallP\\] and compare the two results. ", "Then, induce on $j$ by doing the same thing but with propositions \\[prop\\_fusion\\_sklvspi\\], \\[prop:fusionBvssmallP\\], and \\[prop:fusionTvssmallP\\], instead.", "\n\n[Fusion of two irreducible modules, first part]{}\\[sec:fusionIvsIp1\\] Now that the fusion of standard modules with irreducible ones are know, the fusion of two irreducible modules can be directly computed.", "\n\n\\[prop:fusion.", "IvsI1\\] For $k\\geq r >1$, and in the regular family, ${\\ell}\\neq 2$, $${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r {\\ell}-2}} \\simeq \\begin{cases}\n\\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r+2}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} & \\text{ if } r < k \\\\\n{\\mathsf{B}^{1}_{0}} \\oplus \\overset{2k-2}{\\underset{\\underset{step=2}{p=4}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} & \\text{ if } r=k\n\\end{cases}.$$\n\nStart with the exact sequence $${\\mathsf S_{r {\\ell}}} \\longrightarrow {\\mathsf S_{r {\\ell}-2}} \\longrightarrow {\\mathsf I_{r {\\ell}-2}} \\longrightarrow 0,$$ which becomes $$\\left.\\begin{cases}\n{\\mathsf{B}^{2 r}_{(k-r){\\ell}-2}} & \\text{ if } r<k\\\\\n{\\mathsf{T}^{2k-3}_{2{\\ell}}} & \\text{ if } r=k\n\\end{cases}\\right\\} \\overset{g}{\\longrightarrow} {\\mathsf{B}^{2(r-1)}_{(k-r){\\ell}}} \\longrightarrow {\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}} \\longrightarrow 0$$ by using the right-exactness of fusion together with proposition \\[prop:fusionIvsstand\\]. ", "Then, build the following exact commuting diagram: $$\\begin{tikzpicture}[scale=1/3]\n \\node[anchor = west] at (3,0) {$\\left.\\begin{cases}\n{\\mathsf{B}^{2 r}_{(k-r){\\ell}-2}} & \\text{ if } r<k\\\\\n{\\mathsf{T}^{2k-3}_{2{\\ell}}} & \\text{ if } r=k\n\\end{cases}\\right\\}$};\n \\draw[->] (14,0) -- (17,0);\n \\node[anchor = south] at (15.5,0) {$g$};\n \\node[anchor=west] at (17,0) {${\\mathsf{B}^{2(r-1)}_{(k-r){\\ell}}}$};\n \\draw[->] (21,0) -- (24,0);\n \\node[anchor=south] at (22.5,0) {$\\bar{g}$};\n \\node[anchor=west] at (24,0) {${\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}} \\longrightarrow 0$};\n \\node at (0,-5) {$0$};\n \\draw[->] (1,-5) -- (7,-5);\n \\node[anchor=west] at (7,-5) {$\\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r}}{\\bigoplus}}{\\mathsf I_{p {\\ell}}}$};\n \\draw[->] (11,-5) -- (17,-5);\n \\node[anchor=north] at (14,-5) {$\\alpha$};\n \\node[anchor=west] at (17,-5) {${\\mathsf{B}^{2(r-1)}_{(k-r){\\ell}}}$};\n \\draw[->] (21,-5) -- (24,-5);\n \\node[anchor=north] at (22.5,-5) {$\\bar{\\alpha}$};\n \\node[anchor=west] at (24,-5) {$\\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r+2}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} \\longrightarrow 0$};\n \\draw[->] (9,-1.5)--(9,-3);\n \\node[anchor=east] at (9,-2) {$\\gamma $};\n \\draw[->] (19,-1) -- (19,-4);\n \\node[anchor=west] at (19,-2.5) {$\\operatorname{id}$};\n \\draw[->] (27,-1)--(27,-3);\n \\node[anchor=west]at (27,-2) {$f $};\n \\draw[->] (19,3) -- (19,1);\n \\node[anchor=south] at (19,3) {$0$};\n \\draw[->] (27,3) -- (27,1);\n \\node[anchor=south] at (27,3) {$\\ker f $};\n \\draw[->] (19,-6) -- (19,-8);\n \\node[anchor=north] at (19,-8) {$0$};\n\\end{tikzpicture}.$$ Here, $f$ exists by universality of the Cokernel of $g$ because $${\\text{\\rm{Hom}}\\big( {\\mathsf{B}^{2 r}_{(k-r){\\ell}-2}},\\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r+2}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} \\big)} \\simeq {\\text{\\rm{Hom}}\\big( {\\mathsf{T}^{2k -3}_{2{\\ell}}},\\overset{2k-2}{\\underset{\\underset{step=2}{p=2}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} \\big)} \\simeq 0,$$ and thus $\\bar{\\alpha}g = 0$, which also give the existence of $\\gamma$ by universality of $\\ker \\bar\\alpha$. The snake lemma then gives $\\operatorname{Coker}{f} \\simeq 0$ and $\\ker f \\simeq \\operatorname{Coker}{\\gamma}$. Our goal is now to prove that $$\\label{eq:IvsI.Goal}\n {\\text{\\rm{Hom}}\\big( \\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r}}{\\bigoplus}}{\\mathsf I_{p {\\ell}}},{\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}} \\big)} \\simeq 0,$$ because that would imply that $\\ker f = 0 $, and thus that $f$ is an isomorphism. ", "But, if there is a non-zero morphism from some ${\\mathsf I_{p{\\ell}}}$ to ${\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}}$, it has to be injective since ${\\mathsf I_{p{\\ell}}}$ is irreducible, and there must thus be a morphism from ${\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}}$ to ${\\mathsf P_{p{\\ell}}}$, the injective hull of ${\\mathsf I_{p{\\ell}}}$ (when $p\\neq 0$). ", "We are therefore trying to compute $${\\text{\\rm{Hom}}\\big( {\\mathsf I_{r {\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}}, {\\mathsf P_{p {\\ell}}} \\big)},$$ for $k-r \\leq p \\leq k+r-2$.\n\nNow, recall that ${\\mathsf I_{r {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{(r-1){\\ell}-1}}$ which implies that $$\\label{eq:fusionII}\n{\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{(r-1){\\ell}-1}} \\simeq\\overset{k+r-2}{\\underset{\\underset{step=2}{s=k-r+2}}{\\bigoplus}}{\\mathsf I_{s{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}}.$$ Using this observation with the definition of the fusion quotient (see section \\[sec:fusion\\_quotient\\]) and proposition \\[prop:fusionq.proj\\] give $$\\begin{aligned}\n {\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}},{\\mathsf P_{p{\\ell}-({\\ell}-1)}} \\big)} & \\simeq {\\text{\\rm{Hom}}\\big( \\overset{k +r -2}{\\underset{\\underset{step=2}{s=k-r+2}}{\\bigoplus}}{\\mathsf I_{s {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}},{\\mathsf P_{p{\\ell}- ({\\ell}-1)}} \\big)} \\notag\\\\\n & \\simeq {\\text{\\rm{Hom}}\\big( \\overset{k +r -2}{\\underset{\\underset{step=2}{s=k-r+2}}{\\bigoplus}}{\\mathsf I_{s {\\ell}-2}} ,{\\mathsf P_{p{\\ell}- ({\\ell}-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\big)}\\notag\\\\\n & \\simeq 0,\\label{eq:fusionIvsI.1}\\end{aligned}$$ where the last line is obtained in the following way. ", "Start by using proposition \\[prop:fusionproj\\] to obtain $${\\mathsf P_{p{\\ell}- ({\\ell}-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{p{\\ell}}} \\oplus {\\mathsf P_{(p-2){\\ell}}} \\oplus \\overset{{\\ell}-3}{\\underset{\\underset{step=2}{\\sigma = ({\\ell}-1) \\mod 2}}{\\bigoplus}} {\\mathsf P_{(p-1){\\ell}-1 + \\sigma}}$$ if ${\\ell}\\neq 2$, and $${\\mathsf P_{p \\times 2 - (2-1)}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2 -1}} \\simeq {\\mathsf P_{p\\times 2}},$$ when ${\\ell}=2$. Then, notice that the projective modules ${\\mathsf P_{s {\\ell}-2}}$, the only projective module containing ${\\mathsf I_{s{\\ell}-2}}$ as a submodule never appears in these fusions for any $p \\in [k-r, k-r+2, \\hdots, k+r-2 ] $.", "\n\nHowever, using the definition of the fusion quotient (see section \\[sec:fusion\\_quotient\\]) and proposition \\[prop:fusionq.proj\\] also give $$\\begin{aligned}\n {\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}},{\\mathsf P_{p{\\ell}-({\\ell}-1)}} \\big)} & \\simeq {\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}}, {\\mathsf P_{p{\\ell}- ({\\ell}-1)}} \\div_{f} {\\mathsf P_{{\\ell}-1}} \\big)} \\notag\\\\\n & \\simeq {\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}}, {\\mathsf P_{p{\\ell}- ({\\ell}-1)}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\big)}. ", "\\label{eq:fusionIvsI.2}\\end{aligned}$$ It follows that $${\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}}, {\\mathsf P_{p{\\ell}- ({\\ell}-1)}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\big)} \\simeq 0,$$ and in particular $${\\text{\\rm{Hom}}\\big( {\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r{\\ell}-2}}, {\\mathsf P_{p{\\ell}}} \\big)}\\simeq 0$$ for all $p \\in [k-r, k-r+2, \\hdots, k+r-2 ] $. ", "Equation is thus proved, and the conclusion when $r \\neq k$ is obtained.", "\n\nWhen $r=k$, the proof above does not work because then the injective hull of ${\\mathsf I_{(k-r){\\ell}}}\\simeq{\\mathsf I_{0}}$ is ${\\mathsf{B}^{1}_{0}} $ instead of ${\\mathsf P_{0}}$. But the Loewy diagram of ${\\mathsf{T}^{2k-3}_{2 {\\ell}}}$, figure \\[fig:LdiagT2lc\\], shows that ${\\mathsf I_{0}}$ is not one of its quotient, and thus ${\\mathsf I_{0}} \\subset \\ker f$.\n\n(3,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{2 {\\ell}}}$]{} – (6,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{4{\\ell}-2}}$]{} – (9,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{4{\\ell}}}$]{}; (9,3) – (10.5,1.5); (10.5,1.5) – (12,0); at (12,1.5) [$\\hdots$]{}; (13,3) – (14.5,1.5); (14.5,1.5) – (16,0); (16,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{2(k-1){\\ell}-2}}$]{} – (19,3) circle (6 pt) node\\[anchor = south\\] [${\\mathsf I_{2(k-1){\\ell}}}$]{} – (22,0) circle (6 pt) node\\[anchor = north\\] [${\\mathsf I_{2k{\\ell}-2 }}$]{}; (3,3) – (4.5,1.5); (9,3) – (7.5,1.5); (9,3) – (10.5,1.5); (19,3) – (17.5,1.5); (19,3) – (20.5,1.5);\n\nThe same argument as for the case $k\\neq r$ can then be used to rule out the appearance of the other irreducible modules, and it follows that ${\\mathsf I_{0}} \\simeq \\ker f$. However, proposition \\[prop:extensions\\] shows that the only irreducible module which can be extended by ${\\mathsf I_{0}}$ is ${\\mathsf I_{2{\\ell}-2}}$, giving $${\\mathsf I_{k {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2}} \\simeq M \\oplus \\overset{2k-2}{\\underset{\\underset{step=2}{p=4}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}},$$ where $M$ satisfy the short exact sequence $$0 \\longrightarrow {\\mathsf I_{0}} \\longrightarrow M \\longrightarrow {\\mathsf I_{2 {\\ell}-2}} \\longrightarrow 0.$$ Note that this sequence cannot split since ${\\text{\\rm{Hom}}\\big( {\\mathsf{B}^{2(k-1)}_{0}},{\\mathsf I_{0}} \\big)} \\simeq 0$. Comparing this sequence with the definition of the $B$ modules then gives $$M \\simeq {\\mathsf{B}^{1}_{0}}.$$\n\nUsing proposition \\[prop:fusionIvssmallP\\], this can be used to compute the fusion of the other irreducibles. ", "However to do so requires the fusion of ${\\mathsf{B}^{1}_{0}}$ with projective modules.", "\n\n[Fusion of ${\\mathsf{B}^{1}_{n,k}}$ and projective modules]{} We start by giving the behaviour of ${\\mathsf{B}^{1}_{n,k}}$ under induction [@BelRiSy].", "\n\nFor all $n \\geq k^{1}$, and $k$ not critical, $${{\\mathsf{B}^{1}_{n,k}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em} \\simeq {\\mathsf{B}^{1}_{n+1,k-1}} \\oplus {\\mathsf{B}^{1}_{n+1,k+1}} \\oplus \\left. ", "\\begin{cases}\n{\\mathsf{B}^{1}_{n+1,k}}, & \\text{ in } {\\mathsf{dTL}_{n}}\\\\\n0, & \\text{ in } {\\mathsf{TL}_{n}}\n\\end{cases}\\right\\}.$$ where ${\\mathsf{B}^{1}_{n,k \\pm 1}} \\simeq {\\mathsf P_{n,k \\pm 1}}$ when $k$ is critical.", "\n\nUsing proposition \\[prop:fusion\\_parity\\] together with the parity of the modules yields the following fusion rules.", "\n\nFor all non-critical $k$, in the dilute Temperley-Lieb family $${\\mathsf{B}^{1}_{n,k}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,0}} \\simeq {\\mathsf{B}^{1}_{n+1,k}},$$ while in both families $${\\mathsf{B}^{1}_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\simeq {\\mathsf{B}^{1}_{n+1,k-1}} \\oplus {\\mathsf{B}^{1}_{n+1,k+1}}.$$\n\nAs usual, fusing ${\\mathsf{B}^{1}_{n,k}}$ with ${\\mathsf P_{1,0}}$ simply increases the parameter $n$. We will thus omit this parameter and always assume that it is big enough for the modules to exist.", "\n\nWe now compute the fusion rules for ${\\mathsf{B}^{1}_{k l}}$, $k\\geq 0$. The preceding proposition gives $${\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf P_{k {\\ell}-1}} \\oplus {\\mathsf{B}^{1}_{k {\\ell}+1}},$$ where it is understood that ${\\mathsf P_{k {\\ell}-1}} \\simeq 0$ if $k=0$. Fusing this result with ${\\mathsf P_{1}}$ yields $$\\begin{tabular}{r l}\n${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} $&$\\simeq {\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{0}} \\oplus {\\mathsf P_{2}} \\right)$\\\\\n&$\\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf{B}^{1}_{k {\\ell}}} \\oplus {\\mathsf{B}^{1}_{k {\\ell}+2}}$.\n\\end{tabular}$$ Comparing the two lines gives the fusion rule $${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\simeq {\\mathsf P_{k {\\ell}-1}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\oplus {\\mathsf{B}^{1}_{k {\\ell}+2}}.$$ Repeating the argument yields $${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2}} \\oplus {\\mathsf{B}^{1}_{k {\\ell}+3}},$$ $${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{4}} \\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{3}} \\oplus {\\mathsf{B}^{1}_{k{\\ell}+4}}.$$ This arguments can be repeated as needed to obtain the following fusion rules.", "\n\n\\[prop:fusion.", "B1vsSmallP\\] For $0< i \\leq {\\ell}-1$, $k\\geq 0$, $${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq {\\mathsf P_{k {\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i-1}} \\oplus {\\mathsf{B}^{1}_{k{\\ell}+i}},$$ where it is understood that ${\\mathsf P_{-1}} \\simeq 0$.\n\nIf $k\\neq 0$, this proposition gives $${\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} \\simeq {\\mathsf P_{k{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-2}} \\oplus {\\mathsf P_{(k+1){\\ell}-1}} \\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}},$$ where we used proposition \\[prop\\_fusion\\_sklvspi\\]. ", "Fusing this expression with ${\\mathsf P_{1}}$ gives $${\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} \\simeq {\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}} \\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}.$$ Fusing this with ${\\mathsf P_{1}}$ again gives $$\\begin{tabular}{r l}\n${\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{1}} $&$\\simeq {\\mathsf{B}^{1}_{k{\\ell}}} {\\times\\hspace{-0.27em}_f}\\left(2{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{{\\ell}+1}}\\right)$\\\\\n& $\\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}\\left(2{\\mathsf P_{{\\ell}-1}} \\oplus {\\mathsf P_{{\\ell}+1}} \\right)$.\n\\end{tabular}$$ Comparing the two lines gives the fusion rule $${\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}} \\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{{\\ell}+1}}.$$ It is simple enough to repeat this argument and obtain the general formula.", "\n\n\\[prop:fusion.", "B1vsP\\] For $k>0$,$r\\geq {\\ell}-1$, $${\\mathsf{B}^{1}_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\simeq {\\mathsf S_{k {\\ell}}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}}.$$\n\nFor $k=0$, recall the short exact sequence $$0 \\longrightarrow {\\mathsf I_{0}} \\longrightarrow {\\mathsf{B}^{1}_{0}} \\longrightarrow {\\mathsf I_{2{\\ell}-2}} \\longrightarrow 0.$$ Since ${\\mathsf I_{0}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r}} \\simeq 0$ for all $r\\geq {\\ell}-1$ (see propositions \\[prop:fusionIWeird\\] and \\[prop:fusionIvssmallP\\]), the following result is obtained.", "\n\nFor all $r\\geq 1$, $0\\leq j< {\\ell}$, $${\\mathsf{B}^{1}_{0}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r{\\ell}-1 +j}} \\simeq {\\mathsf I_{2{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{r {\\ell}-1 +j}} \\simeq {\\mathsf P_{r {\\ell}-1 +j}}.$$\n\nMore general results could be easily obtained but we will stop here since we have all we need to finish the computation of fusions of irreducible modules.", "\n\n[Fusion of two irreducible modules, second part]{}\\[sec:fusionIvsIfinal\\] Proposition \\[prop:fusion.", "IvsI1\\] gives $${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r {\\ell}-2}} \\simeq \\begin{cases}\n\\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r+2}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} & \\text{ if } r < k \\\\\n{\\mathsf{B}^{1}_{0}} \\oplus \\overset{2k-2}{\\underset{\\underset{step=2}{p=4}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2}} & \\text{ if } r=k\n\\end{cases},$$ and proposition \\[prop:fusionIvssmallP\\] gives $${\\mathsf I_{k{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} \\simeq {\\mathsf I_{k{\\ell}-2-i}},$$ for all $0\\leq i <{\\ell}-1$. To obtain ${\\mathsf I_{k {\\ell}-2-i}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r {\\ell}-2 -j}}$ we must therefore compute ${\\mathsf I_{p {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}}$ and ${\\mathsf{B}^{1}_{0}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}}$. Using propositions \\[prop:fusionproj\\], and \\[prop:fusionIvsP\\], $$\\begin{aligned}\n{\\mathsf I_{p {\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} & \\simeq {\\mathsf I_{p {\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\Big(\\overset{\\min (i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf P_{\\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+ 1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{{\\ell}-1 +\\sigma}} \\Big)\\\\\n & \\simeq \\overset{\\min(i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2 - \\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{(p-1){\\ell}-1 +\\sigma}}.\\end{aligned}$$\n\nSimilarly, using proposition \\[prop:fusionproj\\] with propositions \\[prop:fusion.", "B1vsSmallP\\] and \\[prop:fusion.", "B1vsP\\] gives $${\\mathsf{B}^{1}_{0}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\simeq \\overset{\\min(i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf{B}^{1}_{\\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{{\\ell}-1 +\\sigma}}.$$ These give the final result.", "\n\nFor $1<r< k$, $0<i,j<{\\ell}-1$, $${\\mathsf I_{k{\\ell}-2-i}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r {\\ell}-2-j}} \\simeq \\overset{k+r-2}{\\underset{\\underset{step=2}{p=k-r+2}}{\\bigoplus}}\\Big( \\overset{\\min(i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2 - \\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{(p-1){\\ell}-1 +\\sigma}} \\Big),$$ $$\\begin{gathered}\n{\\mathsf I_{k {\\ell}-2 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{k {\\ell}-2 -j}} \\simeq \\overset{\\min(i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf{B}^{1}_{\\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{{\\ell}-1 +\\sigma}} \\\\\n\\oplus \\overset{2k-2}{\\underset{\\underset{step=2}{p=4}}{\\bigoplus}}\\Big( \\overset{\\min(i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf I_{p {\\ell}-2 - \\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{(p-1){\\ell}-1 +\\sigma}} \\Big).\\end{gathered}$$\n\nWe still need to compute the fusion rules for the irreducibles with $k=1$. Recall that it was established in proposition \\[prop:fusionIWeird\\] that $${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{r}} \\simeq 0$$ for all $r\\geq {\\ell}$. Using the short-exact sequence $$0 \\longrightarrow {\\mathsf I_{{\\ell}}} \\longrightarrow {\\mathsf P_{{\\ell}-2}} \\longrightarrow {\\mathsf I_{{\\ell}-2}} \\longrightarrow 0,$$ with the right exactness of fusion, it follows that $${\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{{\\ell}-2}} \\simeq {\\mathsf P_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{{\\ell}-2}} \\simeq {\\mathsf I_{0}},$$ and thus that $$\\begin{aligned}\n{\\mathsf I_{{\\ell}-2 -i}} {\\times\\hspace{-0.27em}_f}{\\mathsf I_{{\\ell}-2-j}} & \\simeq {\\mathsf I_{{\\ell}-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{{\\ell}-2}} {\\times\\hspace{-0.27em}_f}\\left( {\\mathsf P_{i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{j}} \\right)\\notag \\\\\n & \\simeq {\\mathsf I_{0}}{\\times\\hspace{-0.27em}_f}\\left(\\overset{\\min (i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf P_{\\sigma}} \\oplus \\overset{i+j -{\\ell}+1}{\\underset{\\underset{step=2}{\\sigma =(i+j+{\\ell}+1)\\text{mod} 2 }}{\\bigoplus}}{\\mathsf P_{{\\ell}-1 +\\sigma}} \\right)\\notag\\\\\n & \\simeq \\overset{\\min (i+j,2{\\ell}-(i+j)-4)}{\\underset{\\underset{step=2}{\\sigma = |i-j|}}{\\bigoplus}}{\\mathsf I_{\\sigma}},\\end{aligned}$$ where the last line is obtained by proposition \\[prop:fusionIvssmallP\\]. ", "The following theorem is then obtained by changing the indices.", "\n\nFor all $0\\leq i,j <{\\ell}-1$, $$\\label{eq:fusionI.Kacs}\n{\\mathsf I_{i}}{\\times\\hspace{-0.27em}_f}{\\mathsf I_{j}} \\simeq \\overset{\\min\\left\\{ i+j, 2{\\ell}-(i+j)-4\\right\\}}{\\underset{\\underset{step=2}{p=|i-j|}}{\\bigoplus}}{\\mathsf I_{p}}.$$\n\nIt should be noted that for a minimal model $M(p',p)$ of the Virasoro algebra, the fusion rule between two primary fields is $$\\phi_{1,1+s} {\\times\\hspace{-0.27em}_f}\\phi_{1,1+r} = \\sum_{\\underset{step=2}{l=|r-s|}}^{\\text{Min}\\left(r+s, 2p -(s+r) - 4) \\right)} \\phi_{1,1+l},$$ which is identical to under the correspondence $\\ell \\to p$, ${\\mathsf I_{i}} \\to \\phi_{1,1+i}$.\n\n[Conclusion]{} The main results of the paper are now reviewed. ", "A definition of a fusion product on the Temperley-Lieb family as been proposed by Read and Saleur [@ReadSaleur1; @ReadSaleur2]; in section \\[sec:deffusion\\], we generalize their definition to more general families of associative algebras, including the dilute Temperley-Lieb algebra. ", "A straightforward consequence of this definition is that the fusion of pairs of projective modules are also projective. ", "In the Temperley-Lieb algebras, when $q$ is not a root of unity, the projective modules ${\\mathsf P_{n,k}}$ behaves under fusion like irreducible $su(2)$ representations under tensor product: $${\\mathsf P_{n,k}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{m,r}} \\simeq \\overset{k+r}{\\underset{\\underset{step =2}{p = |k-r|}}{\\bigoplus}} {\\mathsf P_{n+m,p}}.$$ When $q$ is a root of unity, they behave like a polynomial ring, with a basis of Chebyshev polynomials of the second kind: $${\\mathsf P_{n,i}} \\to U_{i}(x), \\qquad {\\mathsf P_{n,{k_c}}} \\to U_{{k_c}}(x), \\qquad {\\mathsf P_{n,{k_c}+i}} \\to U_{{k_c}-i}(x) + U_{{k_c}+i}(x).$$\n\nIn section \\[sec:fusion.standard\\], we use this information to compute fusion products of standard modules ${\\mathsf S_{n,k}}$ with projective modules and other standard modules. ", "It is shown that these can once again be interpreted as a polynomial ring with a basis of Chebyshev polynomials, albeit with a different product. ", "The correspondence is $${\\mathsf S_{n,k}} \\to U_{k}(x),$$ and when taking a product, the result must first be re-written in terms of the polynomials representing projective modules ${\\mathsf P_{n,p}}$, starting with the smallest $p$; the remaining polynomials are then identified with the standard modules.", "\n\nIn section \\[sec:fusionquotients\\], it is shown how to use fusion rules obtained previously to construct more complex ones. ", "In particular, we compute the fusion product of an irreducible modules, and a standard modules. ", "This shows the appearance of two other classes of indecomposable modules, the $B$’s and the $T$’s. ", "After computing their fusion rules in section \\[sec:fusionBvsP\\],\\[sec:fusionBvsS\\],\\[sec:fusionTvsP\\] and \\[sec:fusionTvsS\\], the fusion product of pairs of irreducible modules is computed in section \\[sec:fusionIvsIp1\\]. ", "Here, we use the adjoint of the fusion product, the fusion quotient, which simplifies the proofs greatly. ", "Finally, in section \\[sec:fusionIvsIfinal\\] we find a general formula for the fusion product of pairs of irreducible modules lying on the left of the first critical line.", "\n\nIt is then recognized that the irreducible modules ${\\mathsf I_{n,i}}$, with $i\\leq \\ell -2$, behave under fusion like primary fields in the first line of the Kacs table of a Virasoro minimal model $M(p',p)$, with $p = \\ell$.\n\nThere are still many fusion rules between indecomposable modules which we have yet to compute. ", "We chose to limit ourselves to the projective, standard and irreducible modules because they are very important in the representation theory of the Temperley-Lieb algebras, but it would be interesting to find out how the other, more exotic, modules behave under this fusion product, as they do appear in physical problems CIT. ", "We believe that the arguments used here could be extended to obtain these fusions.", "\n\nThe appearance of the fusion quotient is a simple consequence of the definition of the fusion product. ", "However, while it is conjectured that the fusion product corresponds to the fusion product on the Virasoro algebra in the limit, the meaning of this fusion quotient is unclear. ", "Is there a corresponding functor on the Virasoro algebra?", "\n\n[Fusion of ${\\mathsf P_{2,0}}$ in ${\\mathsf{TL}_{n}}$]{}\\[sec:magicp2\\] We investigate here the fusion of ${\\mathsf P_{2,0}}$ with various modules in the regular family ${\\mathsf{TL}_{n}}$ when ${\\ell}\\neq 2$.\n\n\\[prop:magickSP2\\] If $\\beta \\neq 0$, then ${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf S_{n+2,i}}.$\n\nPick $$z = \\begin{tikzpicture}[baseline={(current bounding box.center)}, scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,15/2);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\node[anchor = south] at (1/2,1) {$\\vdots$};\n \\draw (0,3) .. controls (1,3) and (1,4) .. (0,4);\n \\draw (0,5) -- (3,5);\n \\node[anchor = south] at (1/2,5) {$\\vdots$};\n \\draw (0,7) -- (3,7);\n \\node[anchor = east] at (0,7) {\\small{$1$}};\n \\node[anchor = east] at (0,5) {\\small{$i$}};\n \\node[anchor = east] at (0,4) {\\small{$i+1$}};\n \\node[anchor = east] at (0,0) {\\small{$n$}};\n\\end{tikzpicture}, \\qquad x = \\begin{tikzpicture}[baseline= {(current bounding box.center)},scale = 1/3]\n \\draw[very thick] (0,-3/4) -- (0,3/4);\n \\draw (0,-1/2) .. controls (1,-1/2) and (1,1/2) .. (0,1/2);\n\\end{tikzpicture},$$ where $z$ is a generator of ${\\mathsf S_{n,i}}$ because $<z,z> = \\beta^{\\frac{n-i}{2}} \\neq 0$, and $x$ is a generator of ${\\mathsf P_{2,0}} = \\mathbb{C}x$. Note that $$\\begin{tabular}{r l}\n ${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}$&$ \\simeq {\\mathsf{TL}_{n+2}} \\otimes \\left({\\mathsf S_{n,i}} \\otimes {\\mathsf P_{2,0}}\\right)$\\\\\n &$ \\simeq {\\mathsf{TL}_{n+2}} \\otimes \\left({\\mathsf{TL}_{n}}\\otimes {\\mathsf{TL}_{2}} \\right)(z \\otimes x )$\\\\\n & $\\simeq {\\mathsf{TL}_{n+2}} \\otimes \\left(z \\otimes x \\right)$.\n\\end{tabular}$$ With the usual generator, $$e_{j}=\n \\begin{tikzpicture}[baseline={(current bounding box.center)}, scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,15/2);\n \\draw[very thick] (3,-1/2) -- (3,15/2);\n \\draw (0,0) -- (3,0);\n \\node[anchor = east] at (0,0) {\\small{$n$}};\n \\node[anchor = south] at (3/2,0) {$\\vdots$};\n \\draw (0,2) -- (3,2);\n \\node[anchor = east] at (0,2) {\\small{$j+2$}};\n \\draw (0,3) .. controls (1,3) and (1,4) .. (0,4);\n \\node[anchor = east] at (0,3) {\\small{$j+1$}};\n \\node[anchor = east] at (0,4) {\\small{$j$}};\n \\draw (3,3) .. controls (2,3) and (2,4) .. (3,4); \n \\draw (0,5) -- (3,5);\n \\node[anchor = east] at (0,5) {\\small{$j-1$}};\n \\node[anchor = south] at (3/2,5) {$\\vdots$};\n \\draw (0,7) -- (3,7);\n \\node[anchor = east] at (0,7) {\\small{$1$}};\n \\end{tikzpicture},$$ we note that $$e_{i+1}e_{i+3}\\hdots e_{n-1} \\otimes (z \\otimes x) = \\beta^{\\frac{n-i}{2} +1} (z\\otimes x).$$ It thus follows that $$\\begin{tabular}{r l}\n ${\\mathsf S_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}$&$ \\simeq {\\mathsf{TL}_{n+2}} \\otimes \\left(e_{i+1}e_{i+3}\\hdots e_{n-1} \\otimes e_{1}\\right) (z \\otimes x)$\\\\\n & $\\simeq {\\mathsf{TL}_{n+2}} \\left(e_{i+1}e_{i+3}\\hdots e_{n-1} e_{n+1} \\right)\\otimes (z \\otimes x)$\\\\\n & $\\simeq \\text{Span}_{\\mathbb{C}} \\left\\{ \\;\n \\begin{tikzpicture}[baseline={(current bounding box.center)}, scale=1/3]\n \\draw[very thick] (0,-1/2) -- (0,15/2);\n \\draw[very thick] (3,-1/2) -- (3,15/2);\n \\node[anchor = west] at (0,4) {$u$};\n \\draw (3,0) .. controls (2,0) and (2,1) .. (3,1);\n \\node[anchor=west] at (3,0) {\\small{$n+2$}};\n \\node[anchor=west] at (3,1) {\\small{$n+1$}};\n \\node[anchor=south] at (5/2,1) {$\\vdots$};\n \\draw (3,3) .. controls (2,3) and (2,4) .. (3,4);\n \\node[anchor=west] at (3,3) {\\small{$i+2$}};\n \\node[anchor=west] at (3,4) {\\small{$i+1$}};\n \\draw (3,5) -- (2,5);\n \\node[anchor=west] at (3,5) {\\small{$i$}};\n \\node[anchor=south] at (5/2,5) {$\\vdots$};\n \\draw (3,7) -- (2,7);\n \\node[anchor=west] at (3,7) {\\small{$1$}};\n \\end{tikzpicture} \\otimes (z \\otimes x)| \\text{ where } u\\in {\\mathsf S_{n+2,i}} \\right\\}$\\\\\n & $\\simeq {\\mathsf S_{n+2,i}}$,\n\\end{tabular}$$ where the last two lines are obtained by straightforward calculations.", "\n\n\\[prop:magickPP2\\] If $\\beta \\neq 0$, then ${\\mathsf P_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf P_{n+2,i}}.$\n\nIf $i_{-}<0$(see definition of $i_{\\pm}$ in section \\[sec:rappel.indec\\]), then ${\\mathsf P_{n,i}} \\simeq {\\mathsf S_{n,i}}$ and the result is given by proposition \\[prop:magickSP2\\]. ", "If $i_{-} \\geq 0$, there is a short exact sequence $$0 \\longrightarrow {\\mathsf S_{n,i_{-}}} \\longrightarrow {\\mathsf P_{n,i}} \\longrightarrow {\\mathsf S_{n,i}} \\longrightarrow 0,$$ which becomes $${\\mathsf S_{n+2,i_{-}}} \\longrightarrow {\\mathsf P_{n,i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\longrightarrow {\\mathsf S_{n+2,i}} \\longrightarrow 0,$$ by fusing it with ${\\mathsf P_{2,0}}$ and using proposition \\[prop:magickSP2\\]. ", "However, since the fusion of two projective modules is projective, it follows that ${\\mathsf P_{n,i}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} $ is a projective module having ${\\mathsf S_{n+2,i}}$ as a quotient, and whose dimension is at most $\\dim {\\mathsf S_{n+2,i_{-}}} + \\dim {\\mathsf S_{n+2,i}} = \\dim {\\mathsf P_{n+2,i}}$. Since ${\\mathsf P_{n+2,i}}$ is the projective cover of ${\\mathsf S_{n+2,i}}$, the conclusion follows.", "\n\nSimilar arguments can be used to compute the action of ${\\mathsf P_{2,0}}$ on other modules. ", "We simply state the result.", "\n\n\\[prop:magickOthersP\\] If $n\\geq i_{+}$, $${\\mathsf I_{n,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf I_{n+2,i}},$$ and if $n\\geq k$, $${\\mathsf{B}^{j}_{n,k}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf{B}^{2j}_{n+2,k}},$$ $${\\mathsf{T}^{j}_{n,k}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}} \\simeq {\\mathsf{T}^{j}_{n+2,k}}.$$\n\n[Fusion quotient]{}\\[sec:fusion\\_quotient\\] We present here a brief study of the operator adjoint to the fusion product, the fusion quotient. ", "We begin with the definition then present the basic properties that follows from it. ", "Finally, we give the fusion quotients of a few Temperley-Lieb modules to show that the two operations, while giving similar results, are not equivalent.", "\n\n[Definition of the fusion quotient]{}\n\n\\[f\\_frob\\] Consider a family of algebras $(A_i)_{i \\in \\mathbb{N}}$ on which fusion is defined (see the beginning of section \\[sec:deffusion\\]), $U$ a $A_i$-module, $V$ a $A_{j}$-module and $W$ a $A_{i+j}$-module. ", "There is an isomorphism of vector spaces $$\\text{\\rm Hom}_{A_{i+j}}\\left(U {\\times\\hspace{-0.27em}_f}V, W\\right) \\simeq \\text{\\rm Hom}_{A_{i}}\\left(U,\\text{Hom}_{A_{i+j}}\\left( A_{i}{\\times\\hspace{-0.27em}_f}V, W\\right)\\right)$$ where $A_{i}{\\times\\hspace{-0.27em}_f}V$ is seen as a left $A_{i+j}$-module and a right $A_{i}$-module.", "\n\nFor $U$ a $A_{i}$-module and $V$ a $A_{i+j}$-module. ", "The *fusion quotient* of $V$ by $U$, denoted by $V \\div_{f} U$, is the $A_{j}$-module $$V \\div_{f} U = \\text{\\rm Hom}_{A_{i+j}}\\left(A_{j}{\\times\\hspace{-0.27em}_f}U, V\\right)$$ where the module structure is given by $$(a g): b \\otimes_{A_{j}\\otimes A_{i}}\\left(c \\otimes_{\\mathbb{C}} x\\right) \\mapsto g\\left(b \\otimes_{A_{j}\\otimes A_{i}}\\left(c a\\otimes_{\\mathbb{C}} x\\right) \\right),$$ where $a,c \\in A_{j}$, $b \\in A_{i+j}$, $x \\in U$,$g \\in \\text{Hom}_{A_{i+j}}\\left(A_{j} {\\times\\hspace{-0.27em}_f}U,V \\right)$.\n\nIf the fusion product has additional properties, like linearity, associativity and commutativity the fusion quotient will inherit some of those.", "\n\n\\[prop:fusionq.prop\\] Let $Q$ and $\\bar{Q}$ be a pair of $A_{i+j+k}$-modules, $U, \\bar{U}$ two $A_{j}$-modules and $V$ a $A_{k}$-module, $$\\label{fq_linear}\n\\left(Q \\oplus \\bar{Q} \\right)\\div_{f} \\left(U \\oplus \\bar{U}\\right) \\simeq \\left(Q \\div_{f} U \\right) \\oplus \\left(\\bar{Q} \\div_{f} U \\right) \\oplus\\left(Q \\div_{f} \\bar{U} \\right) \\oplus\\left(\\bar{Q} \\div_{f} \\bar{U} \\right).$$ If the fusion product on the family $\\lbrace A_{i} \\rbrace$ is associative, then $$\\label{fq_assoc}\n\\left(Q \\div_{f} U\\right) \\div_{f} V \\simeq Q \\div_{f} \\left(V {\\times\\hspace{-0.27em}_f}U\\right).$$ If the fusion product is also commutative, then $$\\label{fq_comm}\n\\left(Q \\div_{f} U\\right) \\div_{f} V \\simeq \\left(Q \\div_{f} V\\right) \\div_{f} U.$$\n\nThe proof of follows from the linearity of the fusion product and of the $\\text{Hom}$ functor. ", "If the fusion product on the family $\\lbrace A_{i} \\rbrace$ is associative, then $$\\begin{aligned}\n\\left(Q \\div_{f} U\\right) \\div_{f} V & = \\text{Hom}_{A_{i+k}}\\left(A_{i}{\\times\\hspace{-0.27em}_f}V, Q \\div_{f}U\\right) \\\\\n& \\simeq \\text{Hom}_{A_{i+k+j}}\\left(\\left(A_{i}{\\times\\hspace{-0.27em}_f}V\\right){\\times\\hspace{-0.27em}_f}U, Q \\right)\\\\\n& \\simeq \\text{Hom}_{A_{i+j+k}}\\left(A_{i}{\\times\\hspace{-0.27em}_f}\\left(V{\\times\\hspace{-0.27em}_f}U \\right),Q\\right)\\\\\n& = Q \\div_{f} \\left(V {\\times\\hspace{-0.27em}_f}U \\right).\\end{aligned}$$ The first and last lines are simply the definition of the fusion quotient, while the second is proposition \\[f\\_frob\\] and the third is obtained by using the associativity of the fusion product. ", "If the product is also commutative, it is clear that $$\\left(Q \\div_{f} U\\right) \\div_{f} V \\simeq Q \\div_{f} \\left(V {\\times\\hspace{-0.27em}_f}U\\right) \\simeq Q \\div_{f} \\left(U {\\times\\hspace{-0.27em}_f}V\\right) \\simeq \\left(Q \\div_{f} V\\right) \\div_{f} U$$ by using .", "\n\nThe following proposition gives the behaviour of short exact sequences under the fusion quotient.", "\n\n\\[prop:fusionq.functor\\] Let $$0 \\longrightarrow U \\longrightarrow V \\longrightarrow W \\longrightarrow 0$$ be a short exact sequence of $A_{i}$-modules and $Q$ be a $A_{j}$-module. ", "If $i>j$, the sequence of $A_{i-j}$-modules $$0 \\longrightarrow U \\div_{f} Q \\longrightarrow V\\div_{f} Q \\overset{f}{\\longrightarrow} W \\div_{f} Q$$ is exact. ", "If $Q$ is projective, then $f$ is surjective. ", "If $j>i$, the sequence of $A_{j-i}$-module $$0 \\longrightarrow Q\\div_{f} W \\longrightarrow Q\\div_{f} V \\longrightarrow Q \\div_{f} U$$ is exact.", "\n\nFor the case $i>j$ simply use the fact that ${\\text{\\rm{Hom}}\\big( P, - \\big)}$ is always left-exact for all module $P$. If moreover $Q$ is projective, proposition \\[prop:fusion\\_resproj\\] shows that $A_{i-j} {\\times\\hspace{-0.27em}_f}Q$ is projective so that $\\text{Hom}_{A_{i+j}}\\left\\{A_{i-j} {\\times\\hspace{-0.27em}_f}Q, - \\right\\}$ is also right-exact. ", "For the other case, the right-exactness of the fusion product is used to obtain the exact sequence $$A_{j-i} {\\times\\hspace{-0.27em}_f}U \\longrightarrow A_{j-i} {\\times\\hspace{-0.27em}_f}V \\longrightarrow A_{j-i} {\\times\\hspace{-0.27em}_f}W \\longrightarrow 0.$$ The final result is obtained by using the fact that ${\\text{\\rm{Hom}}\\big( - ,P \\big)}$ is always left-exact and contravariant.", "\n\nNote also that the fusion quotient of an $A_{i+j}$-module $U$ by $A_{j}$ has the structure of a $A_{i}\\otimes_{\\mathbb{C}}A_{j}$-module. ", "It can be seen that this quotient is in fact isomorphic to the restriction of $U$ to the subalgebra $A_{i}\\otimes_{\\mathbb{C}}A_{j}$. The following proposition relates this structure to the quotient of $U$ by a $A_{j}$-module $V$.\n\n\\[prop:fusionq.hom\\] For $U$ a $A_{i+j}$-module and $V$ a $A_{j}$-module, $$U\\div_{f} V \\simeq \\text{\\rm Hom}_{A_{j}}\\left(V, U \\div_{f} A_{i}\\right)$$ where the action of $A_{i}$ on $\\text{\\rm Hom}_{A_{j}}\\left(V, U \\div_{f} A_{i}\\right)$ is given by $$(a_i g): v_{j} \\mapsto \\left(b_{i+j} \\otimes_{A_{j}\\otimes A_{i}} \\left(c_{j}\\otimes_{\\mathbb{C}} d_i\\right) \\mapsto g(v_{j})\\left(b_{i+j} \\otimes_{A_{j}\\otimes A_{i}} \\left(c_{j} \\otimes_{\\mathbb{C}} d_{i} a_{i}\\right)\\right) \\right),$$ where the indices on $a_i,b_{i+j},\\hdots$ refers to which of $A_{i},A_{i+j},\\hdots$ they belong.", "\n\nThe proof proceeds by construction. ", "Define the vector space homomorphism $\\phi:U\\div_f V \\to \\text{Hom}_{A_{j}}\\left(V, U \\div_{f} A_{i}\\right)$ by $$\\phi(g)= \\left(v_j \\mapsto\\left(b_{i+j} \\otimes_{A_{j}\\otimes A_{i}} \\left(c_{j} \\otimes_{\\mathbb{C}} d_{i}\\right) \\mapsto g(b_{i+j} \\otimes_{A_{j}\\otimes A_{i}} \\left(c_{j}v_{j} \\otimes_{\\mathbb{C}} d_{i}\\right)) \\right)\\right)$$ and another homomorphism $\\psi: \\text{Hom}_{A_{j}}\\left(V, U \\div_{f} A_{i}\\right) \\to U\\div_f V $ by $$\\psi(g) = \\left(b_{i+j}\\otimes_{A_{j}\\otimes A_{i}} (v_{j}\\otimes_{\\mathbb{C}} d_{i}) \\mapsto g(v_{j})(b_{i+j}\\otimes_{A_{j}\\otimes A_{i}} (\\operatorname{id}_{A_{j}}\\otimes_{\\mathbb{C}}d_{i}) ) \\right).$$ It is straightforward to verify that these two morphisms are inverse of each other, and that the action of $A_{i}$ defined in the proposition makes them into $A_{i}$-module homomorphisms.", "\n\n[Fusion quotient in the Temperley-Lieb families]{} We present here the fusion quotient of some modules in the ${\\mathsf{TL}_{n}}$ and ${\\mathsf{dTL}_{n}}$ families.", "\n\nLet $A_n$ be ${\\mathsf{TL}_{n}}$ or ${\\mathsf{dTL}_{n}}$. For any $A_{n+1}$-module $U$, $$U \\div_f A_{1} \\simeq {U{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em},$$ where the restriction functor is ${-{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} = _{A_{n}}\\left(A_{n+1}\\right)_{A_{n+1}} \\otimes_{A_{n+1}} -$.\n\nThe functor ${-{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em}$ is the adjoint of the functor ${-{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}$ defined in section \\[sec:rappel.induction\\]. ", "Since $ - {\\times\\hspace{-0.27em}_f}A_{1}$ is equivalent to ${-{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}$, their adjoints must also be equivalent.", "\n\nThis restriction functor as also been computed [@RiSY; @BelSY; @BelRiSy].", "\n\nFor $0 \\leq i <\\ell$, $0 < j<\\ell $ such that $n+1 \\geq {k_c}+i $, $$\\begin{aligned}\n{{\\mathsf P_{n+1,{k_c}+i}}{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} \\simeq & \\left. ", "\\begin{cases}\n2 {\\mathsf P_{n,{k_c}}},& \\text{ if } i=1\\\\\n0, & \\text{ if } i=0 \\\\\n{\\mathsf P_{n,{k_c}+i-1}}, & \\text{otherwise}\n\\end{cases}\\right\\} \\oplus \\left. ", "\\begin{cases}\n{\\mathsf P_{n,{k_c}+i}}, &\\text{ in } {\\mathsf{dTL}_{n}} \\text{ and } n\\geq {k_c}+i\\\\\n{\\mathsf S_{n,{k_c}-i}}, &\\text{ in } {\\mathsf{dTL}_{n}} \\text{ and } n < {k_c}+i\\\\\n0, &\\text{ in } {\\mathsf{TL}_{n}}\n\\end{cases}\\right\\} \\notag\\\\\n& \\oplus \\left. ", "\\begin{cases}\n{\\mathsf P_{n,{k_c}-\\ell}} \\oplus {\\mathsf P_{n,{k_c}+\\ell}}, & \\text{ if } i = \\ell -1 \\text{ and } n \\geq {k_c}+ \\ell\\\\\n{\\mathsf P_{n,{k_c}-\\ell}}, & \\text{ if } i = \\ell -1 \\text{ and } n < {k_c}+ \\ell\\\\\n{\\mathsf S_{n,{k_c}-( i+1)}}, & \\text{ if } i \\neq \\ell -1 \\text{ and } n < {k_c}+ i + 1\\\\\n{\\mathsf P_{n,{k_c}+ i+1}}, & \\text{ if } i \\neq \\ell -1 \\text{ and } n \\geq {k_c}+ i + 1\n\\end{cases}\\right\\},\\end{aligned}$$ $$\\begin{aligned}\n{{\\mathsf S_{n+1, {k_c}+i}}{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} \\simeq & \\left. ", "\\begin{cases}\n {\\mathsf P_{n,{k_c}}}, & \\text{ if } i=1\\\\\n {\\mathsf S_{n,{k_c}+ i-1}}, & \\text{ otherwise }\n\\end{cases}\\right\\rbrace \\oplus \n\\left. ", "\\begin{cases}\n{\\mathsf S_{n, {k_c}+ i}}, &\\text{ in } {\\mathsf{dTL}_{n}} \\text{ and } n\\geq {k_c}+i\\\\\n0, &\\text{ in } {\\mathsf{dTL}_{n}} \\text{ and } n < {k_c}+i\\\\\n0, & \\text{ in } {\\mathsf{TL}_{n}}\n\\end{cases}\\right\\rbrace \\notag\\\\\n& \\oplus \\left. ", "\\begin{cases}\n {\\mathsf P_{n,{k_c}+ \\ell}}, & \\text{ if } i = {\\ell}-1 \\text{ and } n \\geq {k_c}+ \\ell \\\\\n {\\mathsf S_{n,{k_c}+ i +1}}, & \\text{ if } i \\neq {\\ell}- 1 \\text{ and } n \\geq {k_c}+ i+1 \\\\\n 0, & \\text{ if } n < {k_c}+ i +1\n\\end{cases} \\right\\rbrace.\\end{aligned}$$\n\n\\[coro:res.ind.proj\\] If $k \\leq n$, $${{\\mathsf P_{n+2,k}}{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} \\simeq {{\\mathsf P_{n,k}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}, \\qquad {{\\mathsf S_{n+2,k}}{}\\hspace{-0.18em} \\downarrow{}\\hskip -0.18em} \\simeq {{\\mathsf S_{n,k}}{}\\hspace{-0.18em} \\uparrow{}\\hskip -0.18em}.$$\n\nAs for the fusion product, we now need to compute the fusion quotient of a standard module by ${\\mathsf P_{2,0}}$ in ${\\mathsf{TL}_{n}}$.\n\nIn the regular ${\\mathsf{TL}_{n}}$ family, if $\\ell \\neq 2$, and $n- 2m \\geq q $, then $${\\mathsf S_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\simeq {\\mathsf S_{n-2m,q}}.$$ If $n-2m < q$, then ${\\mathsf S_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\simeq 0$.\n\nStart with the case $m=1$. The first step is to prove that the two modules have the same dimension. ", "For this, note that $$\\begin{tabular}{r l}\n ${\\mathsf S_{n,q}}\\div_{f} {\\mathsf P_{2,0}}$ &$ \\overset{1}{\\simeq} {\\text{\\rm{Hom}}\\big( {\\mathsf{TL}_{n-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}, {\\mathsf S_{n,q}} \\big)}$\\\\\n &$ \\overset{2}{\\simeq} \\overset{n-2}{\\underset{\\underset{step=2}{i=n \\mod 2}}{\\bigoplus}}\\left( \\left(\\dim{\\mathsf I_{n-2,i}}\\right) {\\text{\\rm{Hom}}\\big( {\\mathsf P_{n-2,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}, {\\mathsf S_{n,q}} \\big)}\\right)$\\\\\n &$ \\overset{3}{\\simeq} \\overset{n-2}{\\underset{\\underset{step=2}{i=n \\mod 2}}{\\bigoplus}}\\left( \\left(\\dim{\\mathsf I_{n-2,i}}\\right) {\\text{\\rm{Hom}}\\big( {\\mathsf P_{n,i}}, {\\mathsf S_{n,q}} \\big)}\\right)$\\\\\n &$ \\overset{4}{\\simeq} \\mathbb{C} \\left(\\dim {\\mathsf I_{n-2,q}} + \\dim {\\mathsf I_{n-2,q_{1}}}\\right),$\n \\end{tabular}$$ where the isomorphism are morphism of vector spaces. ", "Here, $1$ is simply the definition of the fusion quotient while $2$ is Wedderburn’s theorem with linearity of Hom. ", "The morphism $3$ is obtained by using proposition \\[prop:fusionproj\\] while $4$ is obtained by inspecting the Loewy diagrams of the projective modules to find the morphism from the $\\mathsf{P}$s to ${\\mathsf S_{n,q}}$. It follows that $$\\dim ({\\mathsf S_{n,q}} \\div_{f}{\\mathsf P_{2,0}}) = \\dim {\\mathsf I_{n-2,q}} + \\dim {\\mathsf I_{n-2,q_{1}}} = \\dim {\\mathsf S_{n-2,q}}.$$ Note that one or both of these irreducible modules may not be defined, in which case we simply set their dimension to zero. ", "In particular if both $q,q_{1} > n-2$, then ${\\mathsf S_{n,q}} \\div_{f}{\\mathsf P_{2,0}} \\simeq 0$.\n\nTo identify the action of ${\\mathsf{TL}_{n-2}}$ on ${\\mathsf S_{n,q}} \\div_{f} {\\mathsf P_{2,0}}$, we proceed as follows. ", "Note that ${\\mathsf{TL}_{n-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}$ is isomorphic as a left ${\\mathsf{TL}_{n}}$-module and as a right ${\\mathsf{TL}_{n-2}}$-module to $J$, the left ideal of ${\\mathsf{TL}_{n}}$ spanned by diagrams where the bottom two nodes on their right side are linked together, i.e. those of the form $$\\begin{tikzpicture}[scale=1/3]\n \\draw (0,-1/2) -- (0,9/2);\n \\draw (3,-1/2) -- (3,9/2);\n \\draw (3,0) .. controls (2,0) and (2,1) .. (3,1);\n \\node[anchor = west] at (0,2) {$u_i$};\n \\node[anchor = east] at (3,3) {$v_i$};\n \\node[anchor= west] at (3,2) {,};\n \\end{tikzpicture}$$ where $u_i \\in {\\mathsf S_{n,i}}$, $v_i \\in {\\mathsf S_{n-2,i}}$ for some $0 \\leq i \\leq n-2$, and where the action of ${\\mathsf{TL}_{n-2}}$ on $J$ is obtained by adding two straight lines at the bottom of every diagram. ", "To see this, verify that $\\phi: a \\mapsto az$, defines a bi-module isomorphism between the two, where $$z= \\operatorname{id}_{{\\mathsf{TL}_{n}}} \\otimes_{{\\mathsf{TL}_{n-2}}{\\times\\hspace{-0.27em}_f}{\\mathsf{TL}_{2}}} \\left(\\operatorname{id}_{{\\mathsf{TL}_{n-2}}} \\otimes \\:\\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1/3]\n\\draw (0,-1/2) -- (0,3/2);\n\\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n\\end{tikzpicture} \\right).$$ Next, notice that $g$ is an homomorphism from ${\\mathsf{TL}_{n}}$ to ${\\mathsf S_{n,q}}$ if and only if there exists a unique $x$ in ${\\mathsf S_{n,q}}$ such that $g \\equiv g_{x} :a \\mapsto a x$. Furthermore, since $J$ is isomorphic to ${\\mathsf{TL}_{n-2}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}$, it is a direct summand of ${\\mathsf{TL}_{n}}$, and thus every morphism from $J$ to ${\\mathsf S_{n,q}}$ must be of the form $g_{x} \\circ i$ for some $x$, where $i$ is the canonical injection.", "\n\nNow, consider the diagram $$\\begin{tikzpicture}[scale=1/3]\n \\draw (0,-1/2) -- (0,9/2);\n \\draw (3,-1/2) -- (3,9/2);\n \\draw (0,0) .. controls (1,0) and (1,1) .. (0,1);\n \\draw (3,0) .. controls (2,0) and (2,1) .. (3,1);\n \\draw (0,2) -- (3,2);\n \\node[anchor = south] at (3/2,2) {$\\vdots$};\n \\draw (0,4) -- (3,4);\n \\node[anchor = east] at (0,2) {$e=$};\n \\end{tikzpicture}$$ in ${\\mathsf{TL}_{n}}$ and notice that for any $a \\in J$, $a e \\frac{1}{\\beta} = a$. It follows that $${\\text{\\rm{Hom}}\\big( J, {\\mathsf S_{n,q}} \\big)} \\simeq \\left\\lbrace (a\\mapsto a x)| x \\in {\\mathsf S_{n,q}} \\text{ such that } e x = \\beta x \\right\\rbrace.$$ Note now that any link diagram in ${\\mathsf S_{n,q}}$ where the two bottom nodes are linked together will define such a morphism. ", "These span a vector space of dimension $\\dim {\\mathsf S_{n-2,q}}$.\n\nUsing the action of ${\\mathsf{TL}_{n-2}}$ defined on ${\\mathsf S_{n,q}}$ by adding two straight lines at the bottom of every diagram, it can be directly verified that for any $b \\in {\\mathsf{TL}_{n-2}}$, $$(bg_{x} \\circ i): a \\mapsto abx = g_{bx} \\circ i,$$ and thus, that ${\\text{\\rm{Hom}}\\big( J,{\\mathsf S_{n,q}} \\big)}$ is isomorphic, as a left ${\\mathsf{TL}_{n-2}}$-module, to the submodule of ${\\mathsf S_{n,q}}$ spanned by diagrams where the two bottom nodes are linked together. ", "Comparing these link diagrams with a basis of ${\\mathsf S_{n-2,q}}$ gives the conclusion.", "\n\nThe proof then proceed by induction on $m$. The case $m=1$ is proved so assume that the result stands for some $m$. Then $$\\left({\\mathsf S_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\right)\\div_{f} {\\mathsf P_{2,0}} \\simeq {\\mathsf S_{n,q}} \\div_{f}\\left({\\mathsf P_{2m,0}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{2,0}}\\right) \\simeq {\\mathsf S_{n,q}}\\div_{f} {\\mathsf P_{2(m+1),0}} \\simeq {\\mathsf S_{n-2(m+1),q}},$$ where we simply used propositions \\[prop:fusionproj\\] and \\[prop:fusionq.prop\\].", "\n\n\\[coro:fusionq.proj.reg\\] In ${\\mathsf{TL}_{n}}$, if $\\ell \\neq 2$ and $n-2m \\geq q$, $${\\mathsf P_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\simeq {\\mathsf P_{n-2m,q}}.$$\n\nIf $q<\\ell-1$ or if $q$ is critical, this is trivial. ", "If $q >\\ell-1$ is not critical, there is a short-exact sequence $$0 \\longrightarrow {\\mathsf S_{n,q_{-1}}} \\longrightarrow {\\mathsf P_{n,q}} \\longrightarrow {\\mathsf S_{n,q}} \\longrightarrow 0,$$ which gives the short-exact sequence of ${\\mathsf{TL}_{n-m}}$-modules $$0 \\longrightarrow {\\mathsf S_{n-2m,q_{-1}}} \\longrightarrow {\\mathsf P_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\longrightarrow {\\mathsf S_{n-2m,q}} \\longrightarrow 0,$$ by using proposition \\[prop:fusionq.functor\\]. ", "Since $${\\text{\\rm{Hom}}\\big( {\\mathsf S_{n-2m,q}}, {\\mathsf P_{n,q}}\\div_{f}{\\mathsf P_{2m,0}} \\big)} \\simeq {\\text{\\rm{Hom}}\\big( {\\mathsf S_{n,q}},{\\mathsf P_{n,q}} \\big)} \\simeq \\mathbb{C},$$ the only morphism from ${\\mathsf S_{n-2m,q}}$ to ${\\mathsf P_{n,q}}\\div_{f} {\\mathsf P_{2m,0}}$ must be the one which goes through ${\\mathsf S_{n-2m,q_{-1}}}$, and thus this sequence does not split. ", "Comparing this sequence with the definition of ${\\mathsf P_{n-2m,q}}$ gives the conclusion.", "\n\nNote that a consequence of this is that ${\\mathsf P_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\simeq {\\mathsf S_{n-2m,q_{-1}}}$ if $q_{-1}\\leq n < q$, and ${\\mathsf P_{n,q}}\\div_{f} {\\mathsf P_{2m,0}} \\simeq 0$ if $q_{-1}>n$.\n\nIf $U$ is a ${\\mathsf{dTL}_{n+m}}$-module, $V$ a ${\\mathsf{TL}_{m}}$-module, both with well-defined parity, then $U\\div_{f} V$ is even if they are both of the same parity and odd otherwise.", "\n\nIt was argued in a comment preceding proposition \\[prop:fusion\\_parity\\] that for $W,V$, two modules with well-defined parities, $W{\\times\\hspace{-0.27em}_f}V$ is even if they are both of the same parity and odd otherwise. ", "In particular, take $W = {\\mathsf{edTL}_{n}}$ , the even ideal of ${\\mathsf{dTL}_{n}}$. Then $W {\\times\\hspace{-0.27em}_f}V$ is even (odd), if and only if $V$ is even (odd). ", "But, by definition, $${\\text{\\rm{Hom}}\\big( W{\\times\\hspace{-0.27em}_f}V, U \\big)} \\simeq {\\text{\\rm{Hom}}\\big( W,U \\div_{f} V \\big)}.$$ The right side of this equality is non-zero if and only if $U \\div_{f} V$ is even, while the left side vanishes unless $U$ is of the same parity as $W {\\times\\hspace{-0.27em}_f}V$. It follows that $U \\div_{f} V$ is even if and only if $U$ is of the same parity as $V$.\n\n\\[prop:fusionq.proj\\] Unless $\\ell =2$ in the regular family, for all $0\\leq i \\leq n$ and $0\\leq j \\leq m$, $$\\label{eq:fusionq.proj}\n{\\mathsf P_{m+2n,j}} \\div_{f} {\\mathsf P_{n,i}} \\simeq {\\mathsf P_{m,j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{n,i}},$$ $${\\mathsf S_{m+2n,j}} \\div_{f} {\\mathsf P_{n,i}} \\simeq {\\mathsf S_{m,j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{n,i}}.$$ If $\\ell =2$ in the regular family, the statement is still true for $i=n=1$.\n\nWe do the proof for the first equality as that of the second is identical. ", "Using the restriction of ${\\mathsf P_{m+2n,j}}$ with the preceding proposition gives the conclusion for $i=1$ in both families, and $i=0$ in the dilute family. ", "The case $i=0$ in the regular family is contained in corollary \\[coro:fusionq.proj.reg\\]. ", "We thus proceed by induction on $i$. If the result stands for $i$, then $$\\begin{aligned}\n \\left({\\mathsf P_{m+2n,j}}\\div_{f} {\\mathsf P_{n-1,i}}\\right) \\div_{f} {\\mathsf P_{1,1}} & \\overset{1}{\\simeq} {\\mathsf P_{m+2n,j}} \\div_{f} \\left({\\mathsf P_{n-1,i}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}} \\right)\\\\\n & \\overset{2}{\\simeq} {\\mathsf P_{m+2n,j}} \\div_{f} \\left({\\mathsf P_{n,i-1}} \\oplus {\\mathsf P_{n,i+1}} \\right),\\end{aligned}$$ and $$\\begin{aligned}\n \\left({\\mathsf P_{m+2n,j}}\\div_{f} {\\mathsf P_{n-1,i}}\\right) \\div_{f} {\\mathsf P_{1,1}} & \\overset{3}{\\simeq} \\left({\\mathsf P_{m+2,j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{n-1,i}} \\right)\\div_{f} {\\mathsf P_{1,1}}\\\\\n &\\overset{4}{\\simeq} \\underset{\\lambda \\in \\Lambda}{\\bigoplus} {\\mathsf P_{m+n+1,\\lambda}} \\div_{f} {\\mathsf P_{1,1}}\\\\\n &\\overset{5}{\\simeq} \\underset{\\lambda \\in \\Lambda}{\\bigoplus} {\\mathsf P_{m+n-1,\\lambda}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}\\\\\n &\\overset{6}{\\simeq} \\left({\\mathsf P_{m,j}}{\\times\\hspace{-0.27em}_f}{\\mathsf P_{n-1,i}} \\right){\\times\\hspace{-0.27em}_f}{\\mathsf P_{1,1}}\\\\\n &\\overset{7}{\\simeq} {\\mathsf P_{m,j}} {\\times\\hspace{-0.27em}_f}\\left({\\mathsf P_{n,i-1}} \\oplus {\\mathsf P_{n,i+1}}\\right),\\end{aligned}$$ where we assumed, for simplicity, that $i, i\\pm 1$ were not critical. ", "These cases are simple generalizations of the same arguments. ", "The isomorphism $1$ is simply proposition \\[prop:fusionq.prop\\], while $2$ is proposition \\[prop:fusionproj\\]. ", "The isomorphism $3$ is obtained by applying $- \\div_{f} {\\mathsf P_{1,1}}$ on the right side of , and $4$ is obtained by applying proposition \\[prop:fusionproj\\], where all the index appearing in the projective modules were grouped in the family $\\Lambda$. Noting that $\\lambda \\leq j+i \\leq m+n$, for all $\\lambda \\in \\Lambda$, proposition \\[prop:fusionq.proj\\] with $i=1$ can be used, obtaining $5$. Finally, use again proposition \\[prop:fusionproj\\] to obtain $6$, and use the associativity of the fusion product with, again proposition \\[prop:fusionproj\\], to obtain $7$. Comparing $2$ and $7$ and using the induction hypothesis gives the conclusion.", "\n\nWhat happens when we take a quotient of the form ${\\mathsf P_{n+2m,j}} \\div_{f} {\\mathsf P_{m,i}}$, but $j>n$? ", "It can be seen that $$\\begin{gathered}\n{\\mathsf P_{n+2m,j}}\\div_{f} {\\mathsf P_{m,i}} \\simeq \\left({\\mathsf P_{n+2m+(j-n),j}} \\div_{f} {\\mathsf P_{j-n,0}} \\right)\\div_{f} {\\mathsf P_{m,i}} \\\\\n\\simeq \\left( {\\mathsf P_{j+2m,j}} \\div_{f} {\\mathsf P_{m,i}}\\right) \\div_{f} {\\mathsf P_{(j-n),0}} \\simeq \\left({\\mathsf P_{j,j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{m,i}} \\right)\\div_{f} {\\mathsf P_{j-n,0}},\\end{gathered}$$ where we simply used propositions \\[prop:fusionq.prop\\] and \\[prop:fusionq.proj\\]. ", "There is thus the following “recipe\": start by computing ${\\mathsf P_{n,j}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{m,i}}$, by applying proposition \\[prop:fusionproj\\], ignoring the fact that ${\\mathsf P_{n,j}}$ is not well-defined. ", "Then, use the fact that, by definition $${\\mathsf P_{n,k}} \\equiv \\begin{cases}\n{\\mathsf S_{n,k_{-}}}, & \\text{when } k_{-}\\leq n< j\\\\\n0, & \\text{when } k_{-}>n\n\\end{cases}.$$ For instance, in $\\ell =5$, $${\\mathsf P_{10,9}} \\div_{f} {\\mathsf P_{4,4}} \\simeq \\underbrace{{\\mathsf P_{6,9}}}_{0} \\oplus \\underbrace{{\\mathsf P_{6,11}}}_{0} \\oplus \\underbrace{{\\mathsf P_{6,13}}}_{{\\mathsf S_{6,5}}} \\simeq {\\mathsf S_{6,5}}.$$\n\nMore complex fusion quotients could be computed by using arguments similar to those we used to compute fusion products. ", "However, the focus of this paper is on the fusion product, we only give one fairly simple case to show that the two operations are distinct.", "\n\nFor $n\\geq \\ell$, $${\\mathsf P_{m+2n, j}} \\div_{f} {\\mathsf S_{n,\\ell}} \\simeq \\begin{cases}\n{\\mathsf P_{m,j-1}} {\\times\\hspace{-0.27em}_f}{\\mathsf P_{n ,\\ell -1}} \\oplus {\\mathsf{T}^{2}_{m+n, \\ell -2- j}}, & 0\\leq j < \\ell -1\\\\\n{\\mathsf P_{m,j}}{\\times\\hspace{-0.27em}_f}{\\mathsf S_{n,\\ell}}, & \\text{otherwise}\n\\end{cases}.$$\n\n[10]{}\n\nH.N.V. Temperley, E.H. Lieb, [*Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem*]{}, Proc.", " Roy.", " Soc.", " Lond.", " A [**322**]{} (1971) 251–280.", "\n\nF. Goodman, H, Wenzl, [*The Temperley-Lieb Algebra at Roots of Unity*]{}, Pac.", " J. Math.", " [**161**]{} (1993) 307–334.", "\n\nP. Martin, [*Potts Models and Related Problems in Statistical Mechanics*]{}, volume 5 of [*Advances in Statistical Mechanics*]{} (1991) World Scientific, Singapore.", "\n\nB. Nienhuis, [*Critical and multicritical $\\mathcal{O}(n)$ models*]{}, Physica [**A163**]{} (1990) 152–157.", "\n\nJ.J. Graham, G.I. Lehrer, [*The representation theory of affine Temperley-Lieb algebras*]{}, Enseign. 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[**", "A40**]{} 8845-8872 (2007)\n\nA. Morin-Duchesne, J. Rasmussen, P. Ruelle, *Dimer representations of the Temperley-Lieb algebra*, N. Physics B, [**890**]{} 262-387 (2015)\n\n[^1]: Whenever we say that a module is unique, we will always mean “up to isomorphism\", but it will not always be mentioned.", "\n\n[^2]: Note that the Chebyshev solution to this recurrence relation is valid on $\\mathbb{C}[ X ]$ even when $X$ is a matrix.", "\n\n[^3]: Note that $9,14,19,24,29$ and $34$ are critical.", "\n" ]

{ "pile_set_name": "ArXiv" }

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[ "Blocking cerebral lymphatic drainage deteriorates cerebral oxidative injury in rats with subarachnoid hemorrhage.", "\nSubstances and fluid in the brain and subarachnoid spaces may be drained into extracranial lymphatics. ", "This study aimed to investigate the possible role of cerebral lymphatic drainage in the process of cerebral injury following subarachnoid hemorrhage (SAH). ", "Wistar rats were divided into non-SAH, SAH, and SAH plus cervical lymphatic blockage (SAH + CLB) groups. ", "Autologous arterial hemolysate was injected into rats' cisterna magna to induce SAH. ", "At time of 24 and 72 h after SAH, the rats were sacrificed for serum lactate dehydrogenase (LDH) activity, brain tissue superoxide dismutase (SOD) activity, and brain tissue malonaldehyde (MDA) content detection. ", "It was found that serum LDH activity increased in rats of SAH group comparing with non-SAH group. ", "SAH also resulted in decreased brain tissue SOD activity and increased brain tissue MDA content. ", "In rats of SAH + CLB group, the increase of serum LDH activity was to a lager extent. ", "Meanwhile, brain tissue SOD activity decreased and MDA content increased to a lager extent, as compared with SAH group. ", "It was concluded that blockage of cerebral lymphatic drainage deteriorates cerebral oxidative injury after SAH, indicating cerebral lymphatic drainage may exert intrinsic protective effects against cerebral injury following SAH." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Senator Hatch Says Global Fund Advocating For Generic Drugs To Solve Healthcare Crises Is Abusing Funds\n\nfrom the also-he's-lying dept\n\nRecipients must procure their products in accordance with national and international laws. ", "The Global Fund encourages recipients to apply the flexibilities provided within national laws and in the World Trade Organization's Agreement on Trade-related Aspects of Intellectual Property rights (TRIPS)...\n\nBy advocating for developing countries to disregard the Agreement on Trade-Related Aspects of Intellectual Property Rights (TRIPS) through issuing compulsory licenses to gain access to Global Fund grants, we are abusing the system. ", "Access to these products is vital to our success in fighting the war on HIV/AIDS and actions inconsistent with patent law such as these will only hinder our ability to work in partnership with the companies that have provided the intellectual property rights to develop generic versions of their products.", "\n\nThank you for reading this Techdirt post. ", "With so many things competing for everyone’s attention these days, we really appreciate you giving us your time. ", "We work hard every day to put quality content out there for our community. ", "Techdirt is one of the few remaining truly independent media outlets. ", "We do not have a giant corporation behind us, and we rely heavily on our community to support us, in an age when advertisers are increasingly uninterested in sponsoring small, independent sites — especially a site like ours that is unwilling to pull punches in its reporting and analysis. ", "While other websites have resorted to paywalls, registration requirements, and increasingly annoying/intrusive advertising, we have always kept Techdirt open and available to anyone. ", "But in order to continue doing so, we need your support. ", "We offer a variety of ways for our readers to support us, from direct donations to special subscriptions and cool merchandise — and every little bit helps. ", "Thank you.", "\n\n–The Techdirt Team\n\nWe've written a few times about governments around the globe seeking to make use of lower-priced (and often more widely available) generic medicines for responding to serious diseases, rather than sticking with the more expensive name brand. ", "Current international agreements, like TRIPs, allow countries to effectively \"opt-out\" of promises to respect other country's patent laws for the sake of supplying such medicine and, you know, saving lives. ", "It's pretty clear that breaking the patent on these drugs is quite effective , driving costs down on a massive scale, making this part of healthcare much more cost-effective.", "Of course, the pharma companies don't like this at all. ", "So it should hardly be that big of a surprise that Senator Orrin Hatch directly tried to put pressure on Hillary Clinton and the Obama administration to stop efforts to promote more widespread use of generics in the developing world using these methods. ", "The link above highlights that Hatch sent then Secretary of State Hillary Clinton a letter that suggested he's quite upset that the US-funded \"Global Fund to Fight AIDS, Tuberculosis and Malaria\" was training people in different countries to explore compulsory licenses of drugs or other ways to make use of generics.", "Of course, if you just read Hatch's letter (embedded below), you'd be forgiven for being confused. ", "In it, he claims to be outraged that the Global Fund is encouraging various countries to buyof \"cheaper\" brand name drugs. ", "Huh? ", "In what environment do generic drugs cost more than the brand name? ", "Also, he appears to just be wrong. ", "Part of the data that he based his complaint on comes from a presentation (embedded with the letter) from the Global Fund, in which they have a few graphs showing that (as you would expect if you were anyone but Orrin Hatch) the generic versions are, in fact,, which is obviously why they were recommending making use of the option.", "So, even as Hatch's letter is full of bluster about misspending money, the details actually seem to suggest that they saved money.", "In the letter, Hatch also claims that the presentation directly calls on countries to \"disregard\" the TRIPS Agreement. ", "But, again, the attached presentation appears to tell the opposite story, with a specific call to seek solutions that comply with TRIPs.", "In other words, directly contrasting Hatch's claims, the Global Fund appeared to tell countries to comply with TRIPs, but (correctly) noted that there were some exceptions built within the law that allowed them to seek compulsory licensing and the use of much cheaper generics. ", "Still, there may have been some behind-the-scenes maneuvering as well. ", "As the IP-Watch report notes, soon after all of this, the executive director of the Global Fund, Michel Kazatchkine, resigned. ", "The report suggests this was due to US pressure over Global Fund's daring effort to tell developing countries. ", "Also: the replacement, Mark Dybul, is a US official, likely to make sure that the Global Fund works in the same manner as Hatch would like -- artificially keeping the big drug makers happily without competition and with artificially high profits, even as healthcare in the developing world might suffer.", "Incredibly, Hatch actually seems to be arguing that by suggesting developing countries explore generics, it's putting people at risk because... well, there's a gap in the explanation, but it appears to be due to big pharma companies feeling they need such subsidies to make the drugs in the first place. ", "From the letter:Got it? ", "If you're a developing nation seeking to stem the AIDS epidemic by making AIDS drugs actually available to those with the disease, you are are actually decreasing the success of the war on AIDS because [something, something, something] big pharma will take their ball and go home. ", "Of course, there's no actual evidence to support this, and tons of evidence that shows that developing countries who actually set a compulsory license on drugs related to critical diseases, actually do help deal with serious problems. ", "There is no evidence that this suddenly scares companies into not making the drugs at all. ", "They still make plenty of money in the developed world off of those drugs.", "Just to summarize, though, an actual US Senator, Orrin Hatch argued that, by having the Global Fund advocate for better, cheaper generic medicines, as clearly allowed under international agreements, that the Global Fund is somehow paying more for medicines (they're not) and convincing big pharma to fail in the war against these diseases (they're not). ", "It's no secret that politicians aren't always in sync with the truth, but this is significantly more blatant a misrepresentation than is normal.", "\n\nFiled Under: developing nations, drugs, generic drugs, global fund, healthcare, hillary clinton, orrin hatch, patents, trips" ]

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[ "// Copyright 2009 The Go Authors. ", "All rights reserved.", "\n// Use of this source code is governed by a BSD-style\n// license that can be found in the LICENSE file.", "\n\n// +build !", "gccgo\n\n#include \"textflag.h\"\n\n//\n// System calls for arm, Linux\n//\n\n// Just jump to package syscall's implementation for all these functions.", "\n// The runtime may know about them.", "\n\nTEXT ·Syscall(SB),NOSPLIT,$0-28\n\tB\tsyscall·Syscall(SB)\n\nTEXT ·Syscall6(SB),NOSPLIT,$0-40\n\tB\tsyscall·Syscall6(SB)\n\nTEXT ·SyscallNoError(SB),NOSPLIT,$0-24\n\tBL\truntime·entersyscall(SB)\n\tMOVW\ttrap+0(FP), R7\n\tMOVW\ta1+4(FP), R0\n\tMOVW\ta2+8(FP), R1\n\tMOVW\ta3+12(FP), R2\n\tMOVW\t$0, R3\n\tMOVW\t$0, R4\n\tMOVW\t$0, R5\n\tSWI\t$0\n\tMOVW\tR0, r1+16(FP)\n\tMOVW\t$0, R0\n\tMOVW\tR0, r2+20(FP)\n\tBL\truntime·exitsyscall(SB)\n\tRET\n\nTEXT ·RawSyscall(SB),NOSPLIT,$0-28\n\tB\tsyscall·RawSyscall(SB)\n\nTEXT ·RawSyscall6(SB),NOSPLIT,$0-40\n\tB\tsyscall·RawSyscall6(SB)\n\nTEXT ·RawSyscallNoError(SB),NOSPLIT,$0-24\n\tMOVW\ttrap+0(FP), R7\t// syscall entry\n\tMOVW\ta1+4(FP), R0\n\tMOVW\ta2+8(FP), R1\n\tMOVW\ta3+12(FP), R2\n\tSWI\t$0\n\tMOVW\tR0, r1+16(FP)\n\tMOVW\t$0, R0\n\tMOVW\tR0, r2+20(FP)\n\tRET\n\nTEXT ·seek(SB),NOSPLIT,$0-28\n\tB\tsyscall·seek(SB)\n" ]

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[ "Possible Italian F1 team to join the grid in 2019\n\nZoran Stefanovic has revealed plans for a second attempt to enter a team in Formula 1 after meeting Ross Brawn in Austria, having previously to enter as Stefan GP.", "\n\nStefanovic had previously owned the Stefan GP team that fell to the wayside after the FIA denied him an entry, despite the team owner claiming the team had completed its cars and signing Kazuki Nakajima to drive.", "\n\nThe project was then canned but with recent rumours that a number of teams, including the much discussed China F1 Racing Team, were looking to enter F1, Stefanovic has publically stated that he is launching a second attempt to enter.", "\n\nHe has set up facilities in Parma and hired former Williams and Ferrari man Enrique Scalabroni to start putting together the technical aspects of the outfit, already reaching an agreement to use a wind tunnel facility.", "\n\nSpeaking exclusively to Autosport about his plans Stefanovic said: \"I am here to meet Ross Brawn because I am setting up an F1 team based in Italy.", "\n\n\"Before coming to Austria I've put together some very important agreements: I defined that it will be based in Parma, which is located a reasonable distance from the wind tunnel.", "\n\n\"I've also signed a contract with a specialist in aerodynamics to help work on the project.\"", "\n\nStefeanovic plans to build the team over the next 12 months before setting a target of securing an entry for 2019, but any grid addition would be dependent on the FIA deciding to open the tender for a new slot.", "\n\nFIA president Jean Todt has made it clear that the entries will only be opened up if he feels there are serious candidates ready to fill the places.", "\n\n\"When we feel it is time we will be able to make a tender,\" Todt said last month.", "\n\n\"At the moment we have ten teams and the idea is to have up to 12 teams. ", "So we have an opportunity, if we have one or two strong newcomers it could be possible.\"", "\n\nStefan Grand Prix, was founded by Serbian-born engineer and businessman Zoran Stefanović, with hopes of becoming Serbia's first Formula One team. ", "Stefanović and Stefan Grand Prix had attempted to enter Formula One on two previous occasions; once in 1996 and again one year later with an attempt to purchase the remains of Lola's abortive 1997 entry.", "\n\nDisputed entry\nIn 2009, Formula One's governing body, the FIA, opened applications for new teams to join the 2010 grid. ", "Applications from fifteen new teams were received, with three (Campos Meta (later to be renamed Hispania Racing), Manor Grand Prix (later to be renamed Virgin Racing) and Team USF1) being selected in July 2009 to join the existing ten teams for 2010. ", "Stefan GP was one of the unsuccessful applicants.", "\n\nThe Formula One community was first made aware of Stefanović's ambitions after he filed a complaint with the European Commission, of a similar nature to one submitted by N.Technology. ", "Stefanović claimed that the entry selection process had been biased in favour of teams who had nominated to run the Cosworth engines that were being introduced for 2010. ", "He claimed that the FIA had not simply shown bias against non-Cosworth teams, but also against teams which would be classified as manufacturers, citing that new teams Virgin Racing, Campos and USF1 had planned to outsource the design of their chassis to other firms and design studios, whereas Stefan Grand Prix had the support of Serbian company AMCO, and thus could build its own cars independently. ", "This bias, he claimed, was supported by Prodrive's failure to make the grid; like Stefan, Prodrive would have the facilities to build its own chassis. ", "Stefanović also had the support of Mike Coughlan, the disgraced McLaren engineer. ", "The team has released photographs showing Coughlan in its offices.", "\n\nIn November 2009, Toyota officially backed out of the championship, following a manufacturer exodus led by Honda and BMW. ", "Despite initial plans not to sell the team on, Stefanović acquired the rights to use the team's chassis, gearbox and the 2010 spec engine. ", "It also employed several former Toyota team members, then out of work after being made redundant by Toyota. ", "With the vacancy left by Toyota, the FIA moved to fill the thirteenth and final grid position, which was contested between Stefanović and Peter Sauber, after BMW sold the team back to him. ", "Sauber was eventually accepted to the grid. ", "Stefanović was later reported as pursuing a 2010 grid position despite the thirteenth grid position having been awarded to Sauber.", "\n\nThe more teams the better but they have to be compitive I don't want anymore caterhams and hrts I doubt his entry will have the funding to enter Stefano gp was a disaster in 2010 I don't see it changing in 2019\n\nAny word on this guy as far as what the locals think of him...if they know him? ", "Is the guy legit or does he have a \"shady\" persona with regards to who he affliates himself with?", "\n\nFrom what I know him, when I meet him couple of times, and speak with him, I cannot say he is shady. ", "Very smart. ", "Educated. ", "Private business. ", "For what I know he is into mechanical engineering. ", "Who know... Maybe he can pull it...\n\nFrom what I know him, when I meet him couple of times, and speak with him, I cannot say he is shady. ", "Very smart. ", "Educated. ", "Private business. ", "For what I know he is into mechanical engineering. ", "Who know... Maybe he can pull it...\n\nFrom what I know him, when I meet him couple of times, and speak with him, I cannot say he is shady. ", "Very smart. ", "Educated. ", "Private business. ", "For what I know he is into mechanical engineering. ", "Who know... Maybe he can pull it...\n\nHaving gone through a failed attempt it's reasonable to assume he knows why he failed and what he needs to add to this attempt. ", "So perhaps there is some chance of it happening. ", "It's probably far more likely than any VAG entry happening" ]

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[ "Q:\n\nAdBlock into ReactJs\n\nI'm struggling to find a solution without any progress until now . ", "I am trying to catch if the user has an AdBlocker and if it does i want to display a message advising him to turn it off. ", "But , until now no success .", "\ni import my component into my main container like :\n<DetectAdBlock pathname={window.location.pathname} />\n\nand then this is my adblocker.js\nimport React from 'react';\nimport PropTypes from 'prop-types'\n\nclass DetectAdBlock extends React.", "Component {\n\n static propTypes = {\n pathname: PropTypes.string.isRequired\n};\n\n constructor(props) {\n super(props);\n this.state = {\n adBlockDetected: false\n }\n this.detectAdBlocker = this.detectAdBlocker.bind(this);\n }\n componentDidMount() {\n this.detectAdBlocker();\n }\n componentWillUpdate(nextProps, nextState) {\n if (this.props.pathname !", "== nextProps.pathname) {\n this.detectAdBlocker();\n }\n }\n detectAdBlocker() {\n const head = document.getElementsByTagName('head')[0];\n\n const noAdBlockDetected = () => {\n this.setState({\n adBlockDetected: false\n });\n }\n const adBlockDetected = () => {\n this.setState({\n adBlockDetected: true\n });\n }\n // clean up stale bait\n const oldScript = \n document.getElementById('adblock-detection');\n if (oldScript) {\n head.removeChild(oldScript);\n }\n // we will dynamically generate some 'bait'.", "\n const script = document.createElement('script');\n script.id = 'adblock-detection';\n script.type = 'text/javascript';\n script.src = '/ads.js';\n script.onload = noAdBlockDetected;\n script.onerror = adBlockDetected;\n head.appendChild(script);\n }\n noticeContentJSX() {\n return (\n <div id=\"adblock-notice\">\n <div className=\"message\">\n <h3>Hey, you!</h3>\n <p>Your adblocker is on again.</p>\n <button \n onClick={this.detectAdBlocker}\n >\n Check for Adblocker again\n </button>\n </div>\n </div>\n );\n }\n render() {\n return (\n <div id=\"adblock-wrapper\">\n { this.state.adBlockDetected \n ? ", "this.noticeContentJSX()\n : null\n }\n </div>\n )\n }\n}\n// DetectAdBlock.propTypes = {\n// pathname: PropTypes.string.isRequired\n// };\n\nDetectAdBlock.defaultProps = {\n pathname: ''\n}\nexport default DetectAdBlock;\n\nThe problem is that there is nothing to show either i have my AdBlock enabled .", "\n\nA:\n\nI think it should be easier than that. ", "I can't actually test this as I'm at work with adblock off but something like this should work:\nclass AdblockDetect extends Component {\n state = {\n usingAdblock: false,\n }\n\n componentDidMount() {\n this.setState({ usingAdblock: this.fakeAdBanner.offsetHeight === 0 });\n }\n\n render() {\n if (this.state.usingAdblock === true) {\n return this.props.children;\n }\n\n return (\n <div\n ref={r => (this.fakeAdBanner = r)}\n style={{ height: '1px', width: '1px', visiblity: 'none', pointerEvents: 'none' }}\n className=\"adBanner\"\n />\n );\n }\n}\n\nclass App extends Component {\n render() {\n return (\n <div className=\"App\">\n <AdblockDetect>You are using adblock</AdblockDetect>\n </div>\n );\n }\n}\n\n" ]

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[ "Attensity Q makes it easier for businesses to drink from the Twitter firehose\n\nIn the NLP (natural language processing) business for a while, Attensity sees an opportunity to get new customers with Q, a visualization tool it says can help non-technical users like marketers find insights in oceans of social media data.", "\n\n\"Any type of end user can pick up and start listening,\" said Martin Onofrio, Attensity's chief revenue officer. \"", "You spin it up and start listening. ", "It's that high-level observation tool.\"", "\n\nAttensity's underlying platform provides real-time access to Facebook, more than 150 million blogs and forums, and the Twitter firehose.", "\n\nOf course, within those data sets and particularly Twitter, \"there's a massive amount of garbage\" that Attensity's technology can clean out through spam filters, heuristics, taxonomies, domains and other techniques, said Sean Timmins, director of enterprise solutions.", "\n\nHowever, with Q \"the goal is not to target individual people,\" Timmins said. \"", "We're not an ad engine.\"", "\n\nQ users can simply put in topic phrases and the system will start harvesting conversations that seem to be related, and presenting the data in a visual format.", "\n\nIf a Q user spots something interesting -- say, a sudden burst of social media messages that are critical of a company's product -- that data set can be moved into another Attensity tool, Analyze, for deeper, root-cause analysis. ", "Attensity has created a workflow that makes the data migration process simple, according to Timmins.", "\n\nThere are at least a dozen components to Attensity's NLP engine that examine various parts and types of speech, such as entities, voices, topics and intent, Timmins said. ", "The system can also be tweaked to properly handle entities with multiple meanings, such as by separating the McDonald's restaurant chain from a local farm called McDonald's, he added.", "\n\nQ will be available as an on-demand service later this month.", "\n\nThere are many vendors in the NLP market, all of which are scrambling to develop the best algorithms, filters and presentation tools to make sense of the Web's ever-larger pool of social media data.", "\n\nRecently, NLP vendors have made progress in doing the same things as before, but for additional languages, said analyst Curt Monash of Monash Research. ", "They've also gotten better at parsing \"shorter messages with more questionable spelling and grammar,\" such as Twitter posts, he added.", "\n\nHowever, NLP vendors haven't made much progress in deepening their engines' understanding of language, according to Monash.", "\n\nWhat Q does should also be taken into context, he said.", "\n\n\"Analyzing the Twitter firehose is a bit of a fad,\" Monash said. \"", "The largest consumer companies certainly gain value from it; other cases are less clear.\"", "\n\n\"Simple stand-alone social media monitoring, like what Attensity is introducing, is worth having,\" he added. \"", "But it's like any other [business intelligence]. ", "If you don't get the occasional insight from it, you're doing something wrong.\"", "\n\nCopyright 2018 IDG Communications. ", "ABN 14 001 592 650. ", "All rights reserved. ", "Reproduction in whole or in part in any form or medium without express written permission of IDG Communications is prohibited." ]

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[ "Synthesis and biological evaluation of inhibitors of thymidine monophosphate kinase from Bacillus anthracis.", "\nNineteen lipophilic thymidine phosphate-mimicking compounds were designed and synthesized as potential inhibitors of thymidine monophosphate kinase of Bacillus anthracis, a Gram-positive bacterium that causes anthrax. ", "These thymidine analogues were substituted at the 5'-postion with sulfonamide-, amide-, (thio)urea-, or triazole groups, which served as lipophilic surrogates for phosphate. ", "Three of the tested compounds produced inhibition of B. anthracis Sterne growth and/or thymidine monophosphate activity. ", "Additional studies will be necessary to elucidate the potential of this type of B. anthracis thymidine monophosphate inhibitors as novel antibiotics in the treatment of anthrax." ]

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[ "Selfie-viewing and facial dissatisfaction among Chinese adolescents: A moderated mediation model of general attractiveness internalization and body appreciation.", "\nThe present study examined whether selfie-viewing (i.e., viewing other people's selfie posts and related likes/comments) on social networking sites (SNSs) was positively associated with adolescents' facial dissatisfaction and whether this association was mediated by general attractiveness internalization. ", "We also tested whether the indirect link between selfie-viewing and facial dissatisfaction by general attractiveness internalization would be moderated by body appreciation. ", "Furthermore, potential gender differences were explored. ", "The model was tested with 1496 Chinese adolescent boys and girls who completed questionnaires regarding selfie-viewing, general attractiveness internalization, body appreciation, and facial dissatisfaction. ", "Results indicated that selfie-viewing was positively related to facial dissatisfaction and this relationship was mediated by general attractiveness internalization. ", "Furthermore, body appreciation moderated the indirect relation between selfie-viewing and facial dissatisfaction via general attractiveness internalization. ", "Gender did not moderate the mediation model. ", "Findings from the current study provide new insights into the relation between SNS use and body image and highlight the potential value of promoting body appreciation in the prevention of body image concerns." ]

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[ "Q:\n\nHow to use a method in a class to compare two objects of the class\n\nHi guys I am totally new in oop in python. ", "I am trying to write a class of students that gets the number of students, their ages, their heights and their weights and store these information as 3 separate lists. ", "\nI aim to compute the average of ages and weights as well as heights. ", "I don't have any problem so far. ", "\nIn the next step I want to compare the average age of two instances of the class plus the average of the weights. ", "\nAs it's an exercise in oop I should do it by a method of the class. ", "\nBut, I don't know that is it possible to do it using a method in the original class (class School) or I should create a subclass to compare the attributes of two instances of the School class.", "\nAny help is really appreciated. ", " \nHere is my code:\nclass School:\n\n avg_age = 0\n avg_heigt = 0\n avg_weight = 0\n\n def __init__(self):\n\n self.n =int(input())\n\n self.list_ages = [float(x) for x in input().split(\" \")]\n\n self.list_higt = [float(x) for x in input().split(\" \")]\n\n self.list_weight = [float(x) for x in input().split(\" \")] \n\n def get_av(self):\n School.avg_age = sum(self.list_ages) / len(self.list_ages)\n\n School.avg_heigt = sum(self.list_higt)/len(self.list_higt)\n\n Scoohl.avg_weight = sum(self.list_weight)/len(self.list_weight)\n return(\"{},{},{}\".format(School.avg_age,School.avg_heigt,School.avg_weight)) \n\nA:\n\nYou mix class attributes and instance attributes. ", "Class attributes are shared between instances:\nclass Banks:\n total = 0\n def __init__(self,name,money):\n self.name=name\n self.money=money\n Banks.total += money\n\nb1 = Banks(\"one\",100)\nb2 = Banks(\"two\",5000000)\n\n# prints 5000100 - all the money of all banks, \n# does not help you comparing avgs ov instances at all\nprint(b1.total) \n\nOutput:\n5000100\n\nYou need seperated averages per instance and a function that compares one instance (self) agiant another instance:\nclass School: \n def __init__(self): \n # self.n =int(input()) # not needed - use len(one of your lists) \n list_ages = [float(x) for x in input(\"Gimme ages, space seperated: \").split()] \n list_hight = [float(x) for x in input(\"Gimme hights, space seperated: \").split()] \n list_weight = [float(x) for x in input(\"Gimme weights, space seperated: \").split()] \n\n # shortest list downsizes all other lists \n self.list_ages, self.list_hight, self.list_weight = zip(\n *zip( list_ages,list_hight,list_weight ))\n\n def get_av(self):\n self.avg_age = sum(self.list_ages) / len(self.list_ages)\n print(self.avg_age)\n self.avg_height = sum(self.list_hight) / len(self.list_hight)\n print(self.avg_height)\n self.avg_weight = sum(self.list_weight) / len(self.list_weight)\n print(self.avg_weight) \n return self.avg_age, self.avg_height, self.avg_weight\n\n def compare(self,other):\n self.get_av()\n other.get_av()\n print(\"Our pupils are younger: \", self.avg_age < other.avg_age)\n print(\"Our pupils are smaller: \", self.avg_height < other.avg_height)\n print(\"Our pupils are lighter: \", self.avg_weight < other.avg_weight)\n\nc = School() # 4 5 6 22 44 66 88 99 20.2 20.2 20.2 20.2 20.2 20.2\nd = School() # 100 100 100\n\nc.compare(d)\n\nOutput (formatted with newlines in between):\nGimme ages, space seperated: 4 5 6\nGimme hights, space seperated: 22 44 66 88 99 \nGimme weights, space seperated: 20.2 20.2 20.2 20.2 20.2 20.2\n\nGimme ages, space seperated: 100\nGimme hights, space seperated: 100 \nGimme weights, space seperated: 100\n\n5.0\n44.0\n20.2\n\n100.0\n100.0\n100.0\n\nOur pupils are younger: True\nOur pupils are smaller: True\nOur pupils are lighter: True\n\nMore info:\n\nWhat is the difference between class and instance attributes?", "\npython class documentation\n\n" ]

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[ "Q:\n\nlinks inside svg not working on iOS\n\nI have an inline SVG inside which I have links to other divs. ", " \nOn desktop and android mobile devices everything works fine whereas on iPad and iPhone it doesn't. ", " \nWhen I tap on the link it flashes as if it has recognised that it is a link but no action takes place.", "\nMy SVG is a map with 15 icons on it and is very very long so there's a greatly simplified\njsfiddle here\nOr CSS and HTML below\n\n #link {\r\n margin-top: 1000px;\r\n height: 100px;\r\n width: 100px;\r\n background: red;\r\n }\n<svg id=\"mapImage\" version=\"1.1\" fill=\"none\" stroke=\"none\" stroke-linecap=\"square\" stroke-miterlimit=\"10\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\r\n <clipPath id=\"p.0\">\r\n <path d=\"m0 0l960.0 0l0 720.0l-960.0 0l0 -720.0z\" clip-rule=\"nonzero\"></path>\r\n </clipPath>\r\n <g clip-path=\"url(#p.0)\">\r\n <path fill=\"#cfe2f3\" d=\"m56.047245 81.91601l143.3386 0l0 129.35434l-143.3386 0z\" fill-rule=\"nonzero\"></path>\r\n <a xlink:href=\"#link\">\r\n <path fill=\"#0000ff\" d=\"m102.396324 123.95276l45.25985 0l0 36.629913l-45.25985 0z\" fill-rule=\"nonzero\"></path>\r\n </a>\r\n </g>\r\n</svg>\r\n\r\n<div id=\"link\">Linked div</div>\n\nLinks to external websites embedded in the SVG work fine on iOS as do normal HTML a href links to both external websites and other divs.", "\nCan I edit the inline SVG so that the links work on iOS? ", " If not would javascript be the solution?", "\n\nA:\n\nThis may or not be the right way to do things but it works.", "\nI removed xlink:href=\"#\" from the anchor tag, gave each one an id then used really simple jquery to open the hidden div.", "\nI have a feeling this breaks rules somewhere but it's the best I can come up with at the moment. ", " Other advice and opinions are welcome.", "\nHTML, CSS, jQuery below:\n\n $('#linkTo').click(function() {\r\n $(\"#hiddenDiv\").toggle();\r\n });\n #hiddenDiv {\r\n display: none;\r\n height: 100px;\r\n width: 100px;\r\n background: red;\r\n }\r\n .link {\r\n cursor: pointer;\r\n }\n<script src=\"https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js\"></script>\r\n<svg id=\"mapImage\" version=\"1.1\" fill=\"none\" stroke=\"none\" stroke-linecap=\"square\" stroke-miterlimit=\"10\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\r\n <clipPath id=\"p.0\">\r\n <path d=\"m0 0l960.0 0l0 720.0l-960.0 0l0 -720.0z\" clip-rule=\"nonzero\"></path>\r\n </clipPath>\r\n <g clip-path=\"url(#p.0)\">\r\n <path fill=\"#cfe2f3\" d=\"m56.047245 81.91601l143.3386 0l0 129.35434l-143.3386 0z\" fill-rule=\"nonzero\"></path>\r\n <a id=\"linkTo\" class=\"link\">\r\n <path fill=\"#0000ff\" d=\"m102.396324 123.95276l45.25985 0l0 36.629913l-45.25985 0z\" fill-rule=\"nonzero\"></path>\r\n </a>\r\n </g>\r\n</svg>\r\n\r\n<div id=\"hiddenDiv\">\r\n</div>\n\njsfiddle here\n\n" ]

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[ "Visa policy of Kenya\n\nVisitors to Kenya require a visa unless they come from one of the visa exempt countries. ", "All visitors must hold a passport valid for 6 months.", "\n\nVisa policy map\n\nVisa exemption\nCitizens of the following 43 countries can visit Kenya with no visa for stays up to 90 days (unless otherwise noted):\n\nVisa is not required for holders of diplomatic or service category passports of Brazil, China, Israel, Iran, and Turkey, nor for holders of diplomatic passports of India.", "\n\nCitizens of Uganda, Rwanda, and Kenya can travel freely between their respective nations using national identification cards.", "\n\nVisa waiver agreement for ordinary passports was signed with in November 2018 and it is yet to come into force.", "\n\nA visa waiver protocol for diplomatic passports was signed with , but it is yet to come into force.", "\n\nAn agreement on visa exemption for diplomatic and service passports was signed with in May 2018, but it is yet to come into force.", "\n\nE-Visa\nKenya introduced an e-Visa system from 2 July 2015. ", "According to the government an application should be done at least 7 days in advance and visitors can stay for up to 90 days with e-visa.", "\n\nVisa on arrival\nVisa on arrival is still available and visitors can stay for up to 90 days with this visa.", "\n\nConsular visa\n\nCitizens of the following countries and territories require a visa that cannot be obtained online but instead the application is referred to the Director of Immigration services for processing and approval:\n\nChildren under the age of 16 are not required to pay a visa fee.", "\n\nEast African Tourist Visa\nFrom February 2014 Kenya, Rwanda and Uganda began issuing an East African Tourist Visa. ", "The visa costs 100 USD and has no restrictions on citizens of any country. ", "The visa is a non-extendable multiple-entry 90-day visa that has to be first used to enter the country that issued the visa.", "\n\nSee also\n\nVisa requirements for Kenyan citizens\n\nReferences\n\nExternal links\nWho Requires a Visa to enter The Republic of Kenya?, ", "Official Website\neVisa for Kenya, Official Website\n\nKenya\nCategory:Foreign relations of Kenya" ]

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[ "Update on platelet glycoprotein IIb/IIIa inhibitors: recommendations for clinical practice.", "\nAntiplatelet therapy is the cornerstone of treatment for patients with acute coronary syndrome (ACS) and undergoing percutaneous coronary intervention (PCI). ", "Glycoprotein IIb/IIIa receptors mediate platelet aggregation, representing the final common pathway of platelet-mediated thrombosis. ", "Therefore, agents blocking this pathway may be desirable for the treatment of patients with ACS and PCI. ", "Glycoprotein IIb/IIIa receptor inhibitors have been widely investigated and have been key to the pharmacological advancements in the field. ", "However, although GPIs have been important to reduce ischemic complications, their elevated risk of bleeding complications remains a major limitation. ", "The poor prognostic implications, including increased mortality, associated with bleeding complication underscores the need for alternative treatment options. ", "Over the past years there have been several advancements in antithrombotic pharmacology which have led to changes in recommendations for GPI usage in clinical practice. ", "This is an overview of the most recent clinical trial data on GPIs, and provides practical insight on their modern day use in ACS therapy." ]

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[ "---\nabstract: 'It is shown that membership in rational subsets of wreath products $H \\wr V$ with $H$ a finite group and $V$ a virtually free group is decidable. ", "On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath product $\\mathbb{Z}\\wr\\mathbb{Z}$ with an undecidable membership problem.'", "\nauthor:\n- Markus Lohrey\n- Benjamin Steinberg\n- Georg Zetzsche\ntitle: 'Rational subsets and submonoids of wreath products [^1]'\n---\n\nIntroduction\n============\n\nThe study of algorithmic problems in group theory has a long tradition. ", "Dehn, in his seminal paper from 1911 [@Dehn11], introduced the word problem (Does a given word over the generators represent the identity?), ", "the conjugacy problem (Are two given group elements conjugate?) ", "and the isomorphism problem (Are two given finitely presented groups isomorphic?), ", "see [@LySch77] for general references in combinatorial group theory. ", "Starting with the work of Novikov and Boone from the 1950’s, all three problems were shown to be undecidable for finitely presented groups in general. ", "A generalization of the word problem is the [*subgroup membership problem*]{} (also known as the [*generalized word problem*]{}) for finitely generated groups: Given group elements $g, g_1, \\ldots, g_n$, does $g$ belong to the subgroup generated by $g_1, \\ldots, g_n$? ", "Explicitly, this problem was introduced by Mihailova in 1958, although Nielsen had already presented an algorithm for the subgroup membership problem for free groups in his paper from 1921 [@Nie21].", "\n\nMotivated partly by automata theory, the subgroup membership problem was further generalized to [*the rational subset membership problem*]{}. ", "Assume that the group $G$ is finitely generated by the set $X$ (where $a \\in X$ if and only if $a^{-1} \\in X$). ", "A finite automaton $A$ with transitions labeled by elements of $X$ defines a subset $L(A) \\subseteq G$ in the natural way; such subsets are the rational subsets of $G$. The rational subset membership problem asks whether a given group element belongs to $L(A)$ for a given finite automaton (in fact, this problem makes sense for any finitely generated monoid). ", "The notion of a rational subset in a monoid can be traced back to the work of Eilenberg and Schützenberger from 1969 [@EiSchu69]. ", "Other early references are [@Ani71; @Gilman96]. ", "Rational subsets of groups also found applications for the solution of word equations (here, quite often the term rational constraint is used) [@DiMu06; @LohSen06icalp]. ", "In automata theory, rational subsets are tightly related to valence automata: For any group $G$, the emptiness problem for valence automata over $G$ (which are also known as $G$-automata) is decidable if and only if $G$ has a decidable rational subset membership problem. ", "See [@FeSt02; @Kam09; @KaSiSt06] for details on valence automata and $G$-automata.", "\n\nFor free groups, Benois [@Benois69] proved that the rational subset membership problem is decidable using a classical automaton saturation procedure (which yields a polynomial time algorithm). ", "For commutative groups, the rational subset membership can be solved using integer programming. ", "Further (un)decidability results on the rational subset membership problem can be found in [@LohSte08] for right-angled Artin groups, in [@Rom99] for nilpotent groups, and in [@LohSt09tocs] for metabelian groups. ", "In general, groups with a decidable rational subset membership problem seem to be rare. ", "In [@LohSt10] it was shown that if the group $G$ has at least two ends, then the rational subset membership problem for $G$ is decidable if and only if the submonoid membership problem for $G$ (Does a given element of $G$ belong to a given finitely generated submonoid of $G$?) ", "is decidable.", "\n\nIn this paper, we investigate the rational subset membership problem for wreath products. ", "The wreath product is a fundamental operation in group theory. ", "To define the wreath product $H \\wr G$ of two groups $G$ and $H$, one first takes the direct sum $K = \\bigoplus_{g \\in G} H$ of copies of $H$, one for each element of $G$. An element $g \\in G$ acts on $K$ by permuting the $G$-copies of $H$ according to the left action of $g$ on $G$. The corresponding semidirect product is the wreath product $H \\wr G$.\n\nIn contrast to the word problem, decidability of the rational subset membership problem is not preserved under wreath products. ", "For instance, in [@LohSt09tocs] it was shown that for every nontrivial group $H$, the rational subset membership problem for $H \\wr (\\mathbb{Z} \\times \\mathbb{Z})$ is undecidable. ", "The proof uses an encoding of a tiling problem, which uses the grid structure of the Cayley graph of $\\mathbb{Z} \\times \\mathbb{Z}$.\n\nIn this paper, we prove the following two new results concerning the rational subset membership problem and the submonoid membership problem for wreath products:\n\n1. ", " The submonoid membership problem is undecidable for $\\mathbb{Z} \\wr \\mathbb{Z}$. The wreath product $\\mathbb{Z} \\wr \\mathbb{Z}$ is one of the simplest examples of a finitely generated group that is not finitely presented, see [@DaOl11; @Clea06] for further results showing the importance of $\\mathbb{Z} \\wr \\mathbb{Z}$.\n\n2. ", " For every finite group $H$ and every virtually free group[^2] $V$, the group $H \\wr V$ has a decidable rational subset membership problem; this includes for instance the famous lamplighter group $\\mathbb{Z}_2 \\wr \\mathbb{Z}$.\n\nFor the proof of (i) we encode the acceptance problem for a 2-counter machine (Minsky machine [@Min67]) into the submonoid membership problem for $\\mathbb{Z} \\wr \\mathbb{Z}$. One should remark that $\\mathbb{Z} \\wr\n\\mathbb{Z}$ is a finitely generated metabelian group and hence has a decidable subgroup membership problem [@Rom74; @Rom80]. ", "For the proof of (ii), an automaton saturation procedure is used. ", "The termination of the process is guaranteed by a well-quasi-order (wqo) which refines the classical subsequence wqo considered by Higman [@Hig52]. ", "Wqo theory has also been applied successfully for the verification of infinite state systems. ", "This research led to the notion of well-structured transition systems [@FinkelS01]. ", "An application in formal language theory is the decidability of the membership problem for leftist grammars [@MotwaniPSV00]. ", "Usually, a disadvantage of using wqo theory is that it does not yield algorithms with good complexity bounds. ", "In the context of well-structured transition systems, several natural reachability problems (e.g. for lossy channel systems) were shown to be not primitive recursive [@ChambartS07; @Schnoebelen02]. ", "Also the membership problem for leftist grammars was shown be not primitive recursive [@Jurdzinski08]. ", "The complexity status for the rational subset membership problem for wreath products $H \\wr V$ ($H$ finite, $V$ virtually free) remains open. ", "Actually, we do not even know whether the rational subset membership problem for the lamplighter group $\\mathbb{Z}_2 \\wr\n\\mathbb{Z}$ is primitive recursive.", "\n\nAs mentioned earlier, the rational subset membership problem is undecidable for every wreath product $H \\wr (\\mathbb{Z} \\times \\mathbb{Z})$, where $H$ is a nontrivial group. ", "We conjecture that this can be generalized to the following result: For every nontrivial group $H$ and every non-virtually free group $G$, the rational subset membership problem for $H \\wr G$ is undecidable. ", "The reason is that the undecidability proof for $H \\wr (\\mathbb{Z} \\times \\mathbb{Z})$  [@LohSt09tocs] only uses the grid-like structure of the Cayley graph of $\\mathbb{Z} \\times \\mathbb{Z}$. In [@KuLo04annals] it was shown that the Cayley graph of a group $G$ has bounded tree width if and only if the group is virtually free. ", "Hence, if $G$ is not virtually free, then the Cayley-graph of $G$ has unbounded tree width, which means that finite grids of arbitrary size appear as minors in the Cayley-graph of $G$. One might therefore hope to reduce again a tiling problem into the rational subset membership problem for $H \\wr G$ (for $H$ non-trivial and $G$ not virtually free).", "\n\nOur decidability result for the rational subset membership problem for wreath products $H \\wr V$ with $H$ finite and $V$ virtually free can be also interpreted in terms of tree automata with additional data values. ", "Consider a tree walking automaton operating on infinite rooted trees. ", "Every tree node contains an additional data value from a finite group such that all but finitely many nodes contain the group identity. ", "Besides navigating in the tree, the tree automaton can multiply (on the right) the group element from the current tree node with another group element (specified by the transition). ", "The automaton cannot read the group element from the current node. ", "Our decidability result basically says that reachability for this automaton model is decidable.", "\n\nRational subsets of groups\n==========================\n\nLet $G$ be a finitely generated group and $X$ a finite symmetric generating set for $G$ (symmetric means that $X$ is closed under taking inverses). ", "For a subset $B \\subseteq G$ we denote with $B^*$ the [*submonoid*]{} of $G$ generated by $B$. The subgroup generated by $B$ is $\\langle B \\rangle$. The set of [*rational subsets*]{} of $G$ is the smallest set that (i) contains all finite subsets of $G$ and (ii) that is closed under union, product, and $^*$. Alternatively, rational subsets can be represented by finite automata. ", "Let $A = (Q,G,E,q_0,Q_F)$ be a finite automaton, where transitions are labeled with elements of $G$: $Q$ is the finite set of states, $q_0 \\in Q$ is the initial state, $Q_F \\subseteq Q$ is the set of final states, and $E \\subseteq Q \\times G \\times Q$ is a finite set of transitions. ", "Every transition label $g \\in G$ can be represented by a finite word over the generating set $X$. In this way, $A$ becomes a finite object. ", "The subset $L(A) \\subseteq G$ accepted by $A$ consists of all group elements $g_1 g_2 g_3 \\cdots g_n$ such that there exists a sequence of transitions $(q_0, g_1, q_1), (q_1, g_2, q_2), (q_2, g_3, q_3), \\ldots, (q_{n-1}, g_n,q_n) \\in E$ with $q_n \\in Q_F$. The [*rational subset membership problem*]{} for $G$ is the following decision problem:\n\nINPUT: A finite automaton $A$ as above and an element $g \\in G$.\n\nQUESTION: Does $g \\in L(A)$ hold?", "\n\nSince $g \\in L(A)$ if and only if $1_G \\in L(A)g^{-1}$, and $L(A)g^{-1}$ is rational, too, the rational subset membership problem for $G$ is equivalent to the question of deciding whether a given automaton accepts the group identity.", "\n\nThe [*submonoid membership problem*]{} for $G$ is the following decision problem:\n\nINPUT: Elements $g, g_1, \\ldots, g_n \\in G$.\n\nQUESTION: Does $g \\in \\{g_1, \\ldots, g_n\\}^*$ hold?", "\n\nClearly, decidability of the rational subset membership problem for $G$ implies decidability of the submonoid membership problem for $G$. Moreover, the latter generalizes the classical subgroup membership problem for $G$ (also known as the generalized word problem), where the input is the same as for the submonoid membership problem for $G$ but it is asked whether $g \\in \\langle g_1, \\ldots, g_n\\rangle$ holds.", "\n\nIn our undecidability results in Section \\[sec-undec\\], we will actually consider the non-uniform variant of the submonoid membership problem, where the submonoid is fixed, i.e., not part of the input.", "\n\nWreath products\n===============\n\nLet $G$ and $H$ be groups. ", "Consider the direct sum $$K = \\bigoplus_{g \\in G} H_g,$$ where $H_g$ is a copy of $H$. We view $K$ as the set $H^{(G)}=\\{ f \\in H^G \\mid f^{-1}(H \\setminus\\{1_H\\})\\ \\text{is finite}\\}$ of all mappings from $G$ to $H$ with finite support together with pointwise multiplication as the group operation. ", "The group $G$ has a natural left action on $H^{(G)}$ given by $$gf(a) = f(g^{-1}a)$$ where $f \\in H^{(G)}$ and $g, a \\in G$. The corresponding semidirect product $H^{(G)} \\rtimes G$ is the wreath product $H \\wr G$. In other words:\n\n- Elements of $H \\wr G$ are pairs $(f,g)$, where $g \\in G$ and $f \\in H^{(G)}$.\n\n- The multiplication in $H \\wr G$ is defined as follows: Let $(f_1,g_1), (f_2,g_2) \\in H \\wr G$. Then $(f_1,g_1)(f_2,g_2) = (f, g_1g_2)$, where $f(a) = f_1(a)f_2(g_1^{-1}a)$.\n\nThe following intuition might be helpful: An element $(f,g) \\in H\\wr G$ can be thought of as a finite multiset of elements of $H \\setminus\\{1_H\\}$ that are sitting at certain elements of $G$ (the mapping $f$) together with the distinguished element $g \\in G$, which can be thought of as a cursor moving in $G$. If we want to compute the product $(f_1,g_1) (f_2,g_2)$, we do this as follows: First, we shift the finite collection of $H$-elements that corresponds to the mapping $f_2$ by $g_1$: If the element $h \\in H\\setminus\\{1_H\\}$ is sitting at $a \\in G$ (i.e., $f_2(a)=h$), then we remove $h$ from $a$ and put it to the new location $g_1a \\in H$. This new collection corresponds to the mapping $f'_2 \\colon a \\mapsto f_2(g_1^{-1}a)$. After this shift, we multiply the two collections of $H$-elements pointwise: If in $a \\in G$ the elements $h_1$ and $h_2$ are sitting (i.e., $f_1(a)=h_1$ and $f'_2(a)=h_2$), then we put the product $h_1h_2$ into the location $a$. Finally, the new distinguished $G$-element (the new cursor position) becomes $g_1 g_2$.\n\n\\[finiteindex\\] Let $K$ be a subgroup of $G$ of finite index $m$ and let $H$ be a group. ", "Then $H^m\\wr K$ is isomorphic to a subgroup of index $m$ in $H\\wr G$.\n\nLet $T$ be a set of right coset representatives for $G/K$; it has $m$ elements. ", "The action of $G$ on $H^{(G)}$ restricts to an action of $K$ on $H^{(G)}$ and so $H^{(G)}\\rtimes K$ is a subgroup of $H\\wr G$. There is a $K$-equivariant[^3] group isomorphism $\\alpha\\colon H^{(G)}\\to (H^T)^{(K)}$ given by $[\\alpha(f)(k)](t) = f(kt)$, where $f \\in H^{(G)}$, $k \\in K$, and $t \\in T$. This $\\alpha$ is indeed bijective; the inverse $\\alpha^{-1}$ is given by $[\\alpha^{-1}(f)](kt) =[f(k)](t)$ for $f \\in (H^T)^{(K)}$, $k\\in K$, and $t\\in T$ (which has finite support because $T$ is finite and $f$ has finite support). ", "That $\\alpha$ is $K$-equivariant follows from $$[k\\alpha(f)(k')](t)=[\\alpha(f)(k^{-1}k')](t)=f(k^{-1}k't) = [kf](k't) = [\\alpha(kf)(k')](t).$$ It follows that $H^m\\wr K\\cong (H^T)^{(K)}\\rtimes K\\cong H^{(G)}\\rtimes K$.\n\nIt thus remains to prove that $H^{(G)}\\rtimes K$ has index $m$ in $H\\wr G$. Indeed, let $e \\in H^{(G)}$ be the map sending all of $G$ to the identity of $H$. Then the elements of the form $(e,t)$ with $t\\in T$ form a set of right coset representatives of $H^{(G)}\\rtimes K$ in $H\\wr G$. Indeed, it is easy to see that these elements are in distinct cosets. ", "If $g=kt$ with $k\\in K$ and $t\\in T$, then $(f,g) = (f,k)(e,t)$, which is in the coset of $(e,t)$.\n\nDecidability {#sec-dec}\n============\n\nWe show that the rational subset membership problem is decidable for groups $G=H\\wr V$, where $H$ is finite and $V$ is virtually free. ", "First, we will show that the rational subset membership problem for $G=H\\wr F_2$, where $F_2$ is the free group generated by $a$ and $b$, is decidable. ", "For this we make use of a particular well-quasi-order.", "\n\nA well-quasi-order {#sec-wqo}\n------------------\n\nRecall that a [*well-quasi-order*]{} on a set $A$ is a reflexive and transitive relation $\\preceq$ such that for every infinite sequence $a_1, a_2, a_3,\\ldots$ with $a_i \\in A$ there exist $i < j$ such that $a_i \\preceq a_j$. In this paper, $\\preceq$ will be always antisymmetric as well; so $\\preceq$ will be a well partial order.", "\n\nFor a finite alphabet $X$ and two words $u,v \\in X^*$, we write $u\\preceq v$ if there exist $v_0,\\ldots,v_n\\in X^*$, $u_1,\\ldots,u_n\\in X$ such that $v=v_0u_1v_1\\cdots u_nv_n$ and $u=u_1\\cdots u_n$. The following theorem was shown by Higman [@Hig52] (and independently Haines [@Hai69]).", "\n\nThe order $\\preceq$ on $X^*$ is a well-quasi-order.", "\n\nLet $G$ be a group. ", "For a monoid morphism $\\alpha\\colon X^*\\to G$ and $u,v \\in X^*$ let $u\\preceq_\\alpha v$ if there is a factorization $v=v_0u_1v_1\\cdots u_nv_n$ with $v_0,\\ldots,v_n\\in X^*$, $u_1,\\ldots,u_n\\in X$, $u=u_1\\cdots u_n$, and $\\alpha(v_i)=1$ for $0\\le i\\le n$. It is easy to see that $\\preceq_\\alpha$ is indeed a partial order on $X^*$. Furthermore, let $\\preceq_G$ be the partial order on $X^*$ with $u\\preceq_G v$ if $v=v_0u_1v_1\\cdots u_nv_n$ for some $v_0,\\ldots,v_n\\in X^*$, $u_1,\\ldots,u_n\\in\nX$, and $u=u_1\\cdots u_n$ such that $\\alpha(v_i)=1$ for every morphism $\\alpha\\colon\nX^*\\to G$ and $0\\le i\\le n$. Note that if $G$ is finite, there are only finitely many morphisms $\\alpha\\colon X^*\\to G$. This means that for given $X$ and $G$, we can construct a finite automaton for the upward closure $U\\subseteq X^*$ of $\\{{\\varepsilon}\\}$ with respect to $\\preceq_G$. Since for $w=w_1\\cdots w_n$, $w_1,\\ldots,w_n\\in X$, the upward closure of $w$ equals $Uw_1\\cdots Uw_nU$, we can also construct a finite automaton for the upward closure of any given singleton provided that $G$ is finite. ", "In the latter case, we can also show that $\\preceq_G$ is a well-quasi-order:\n\nLet $G$ be a group. ", "Then the following statements are equivalent:\n\n1. ", " \\[st:wqo\\] $(X^*, \\preceq_G)$ is a well-quasi-order for each finite alphabet $X$.\n\n2. ", " \\[st:bounded\\] For every $n\\in{\\mathbb{N}}$, there is a $k\\in{\\mathbb{N}}$ with $|\\langle g_1,\\ldots,g_n\\rangle|\\le k$ for all $g_1,\\ldots,g_n\\in G$.\n\nSuppose \\[st:bounded\\] does not hold. ", "Then there is a finite alphabet $X$ and a sequence of morphisms $\\alpha_1,\\alpha_2,...\\colon X^*\\to G$ such that $|\\alpha_i(X^*)|\\ge i$ for each $i\\ge 1$. We inductively define a sequence of words $w_1,w_2,\\ldots\\in X^*$. Choose $w_1={\\varepsilon}$ and suppose $w_1,\\ldots,w_i$ have been defined. ", "Since $|\\alpha_{i+1}(X^*)|\\ge i+1$, we can choose $w_{i+1}\\in X^*$ to be a word such that $\\alpha_{i+1}(w_{i+1})$ is outside of $\\{\\alpha_{i+1}(w_1),\\ldots,\\alpha_{i+1}(w_i)\\}$. We claim that the words $w_1,w_2,\\ldots$ are pairwise incomparable with respect to $\\preceq_G$. Observe that $u\\preceq_G v$ implies $\\alpha(u)=\\alpha(v)$ for any morphism $\\alpha\\colon X^*\\to G$. Since for any $i,j\\in{\\mathbb{N}}$, $i<j$, the construction guarantees $\\alpha_j(w_j)\\ne\\alpha_j(w_i)$, the words are pairwise incomparable.", "\n\nSuppose \\[st:bounded\\] does hold and let $X$ be a finite alphabet. ", "First, we claim that there is a finite group $H$ such that $\\preceq_G$ coincides with $\\preceq_H$. By \\[st:bounded\\] there are only finitely many non-isomorphic groups that appear as $\\alpha(X^*)$ for morphisms $\\alpha\\colon X^*\\to G$, say $H_1,\\ldots,H_m$, and each of them is finite. ", "For $H=H_1\\times\\cdots\\times H_m$, we have $$\\bigcap_{\\alpha:X^*\\to G} \\ker(\\alpha) = \\bigcap_{\\alpha:X^*\\to H}\\ker(\\alpha).$$ Hence, $\\preceq_G$ coincides with $\\preceq_H$. There are only finitely many morphisms $\\alpha\\colon X^*\\to H$, say $\\alpha_1, \\ldots, \\alpha_\\ell$. If $\\beta:\nX^*\\to H^\\ell$ is the morphism with $\\beta(w)=(\\alpha_1(w),\\ldots,\\alpha_\\ell(w))$, then $$\\bigcap_{\\alpha:X^*\\to H}\\ker(\\alpha)=\\ker(\\beta).$$ Thus, $\\preceq_H$ coincides with $\\preceq_\\beta$. Therefore, it suffices to show that $\\preceq_\\beta$ is a well-quasi-order.", "\n\nLet $w_1,w_2,\\ldots\\in X^*$ be an infinite sequence of words. ", "Since $H^\\ell$ is finite, we can assume that all the $w_i$ have the same image under $\\beta$; otherwise, choose an infinite subsequence on which $\\beta$ is constant. ", "Consider the alphabet $Y=X\\times H^\\ell$. For every $w\\in X^*$, $w=a_1\\cdots a_r$, let $\\bar{w}\\in Y^*$ be the word $$\\label{eq:embed}\\bar{w}=(a_1, \\beta(a_1))(a_2, \\beta(a_1a_2))\\cdots(a_r,\\beta(a_1\\cdots a_r)).$$ Applying Higman’s Lemma to the sequence $\\bar{w}_1,\\bar{w}_2,\\ldots$ yields indices $i<j$ such that $\\bar{w}_i\\preceq \\bar{w}_j$. This means $\\bar{w}_i=u'_1\\cdots u'_r$, $\\bar{w}_j=v'_0u'_1v'_1\\cdots u'_rv'_r$ for some $u'_1,\\ldots, u'_r\\in Y$, $v'_0,\\ldots,v'_r\\in Y^*$. By definition of $\\bar{w}_i$ and $\\bar{w}_j$, we have $u'_s=(u_s, h_s)$ for $1\\le s\\le r$, where $h_s=\\beta(u_1\\cdots u_s)$ and $w_i=u_1\\cdots u_r$. Let $\\pi_1\\colon Y^*\\to X^*$ be the morphism extending the projection onto the first component, and let $v_s=\\pi_1(v'_s)$ for $0\\le s\\le r$. Then clearly $w_j=v_0u_1v_1\\cdots u_rv_r$. We claim that $\\beta(v_s)=1$ for $0\\le s\\le r$, from which $w_i\\preceq_\\beta w_j$ and hence the lemma follows. ", "Since $\\bar{w}_j$ is also obtained according to , we have $$\\beta(u_1\\cdots u_{s+1}) = h_{s+1} = \\beta(v_0 u_1v_1 \\cdots u_sv_s u_{s+1})$$ for $0\\le s\\le r-1$. By induction on $s$, this allows us to deduce $\\beta(v_s)=1$ for $0\\le s\\le r-1$. Finally, $\\beta(w_i)=\\beta(w_j)$ entails $$\\beta(u_1\\cdots u_r)=\\beta(w_i)=\\beta(w_j)=\\beta(v_0u_1v_1\\cdots u_rv_r)=\\beta(u_1\\cdots u_r v_r),$$ implying $\\beta(v_r)=1$.\n\nLoops\n-----\n\nLet $G=H\\wr F_2$ and fix free generators $a,b\\in F_2$. Recall that elements of $G$ are pairs $(k, f)$, where $k\\in\nK=\\bigoplus_{g\\in F_2} H$ and $f\\in F_2$. In the following, we simply write $kf$ for the pair $(k,f)$. Fix an automaton $A=(Q,G,E,q_0,Q_F)$ with labels from $G$ for the rest of Section \\[sec-dec\\]. ", "We want to check whether $1 \\in L(A)$. Since $G$ is generated as a monoid by $H\\cup\n\\{a,a^{-1},b,b^{-1}\\}$, we can assume that $E\\subseteq Q\\times (H\\cup\n\\{a,a^{-1},b,b^{-1}\\})\\times Q$.\n\nA *configuration* is an element of $Q\\times G$. For configurations $(p,g_1)$, $(q,g_2)$, we write $(p,g_1){\\to_{A}}(q,g_2)$ if there is a $(p,g,q)\\in E$ such that $g_2=g_1g$. For elements $f,g\\in F_2$, we write $f\\le g$ ($f<g$) if the reduced word representing $f$ is a (proper) prefix of the reduced word representing $g$. We say that an element $f\\in F_2\\setminus\\{1\\}$ is *of type* $x \\in \\{a,a^{-1},b,b^{-1}\\}$ if the reduced word representing $f$ ends with $x$. Furthermore, $1\\in F_2$ is of type $1$. Hence, the set of *types* is $T=\\{1,a,a^{-1},b,b^{-1}\\}$. When regarding the Cayley graph of $F_2$ as a tree with root $1$, the children of a node of type $t$ are of the types $C(t)=\\{a,a^{-1},b,b^{-1}\\}\\setminus \\{t^{-1}\\}$. Clearly, two nodes have the same type if and only if their induced subtrees of the Cayley graph are isomorphic. ", "The elements of $D=\\{a,a^{-1},b,b^{-1}\\}$ will also be called *directions*.", "\n\nLet $p,q\\in Q$ and $t\\in T$. A sequence of configurations $$(q_1, k_1f_1) {\\to_{A}} (q_2, k_2f_2) {\\to_{A}} \\cdots {\\to_{A}} (q_n, k_nf_n)\\label{comp}$$ is called a *well-nested $(p,q)$-computation in $t$* if\n\n1. ", " $q_1=p$ and $q_n=q$,\n\n2. ", " $f_1=f_n$ is of type $t$, and\n\n3. ", " \\[cond:subtree\\] $f_i \\ge f_1$ for $1< i < n$.\n\nWe define the *effect* of the computation to be $f_1^{-1}k_1^{-1}k_nf_1\\in\nK$. Hence, the effect describes the change imposed by applying the corresponding sequence of transitions, independently of the configuration in which it starts. ", "For $f\\in F_2$, let $|f|$ be the length of the reduced word representing $f$. The *depth* of the computation is the maximum value of $|f_1^{-1}f_i|$ for $1\\le i\\le n$. Of course, if $f_1=f_n=1$, condition \\[cond:subtree\\] is satisfied automatically. ", "Hence, we have $1\\in L(A)$ if and only if for some $q\\in Q_F$, there is a well-nested $(q_0,q)$-computation in $1$ with effect $1$.\n\nFor $d \\in C(t)$, a well-nested $(p,q)$-computation in $t$ is called a *$(p,d,q)$-loop in $t$* if in addition $f_1d \\leq f_i$ for $1<i<n$. Note that there is a $(p,d,q)$-loop in $t$ that starts in $(p,kf)$ (where $f$ has type $t$) with effect $e$ and depth $m$ if and only if there exists a $(p,d,q)$-loop in $t$ with effect $e$ and depth $m$ that starts in $(p, t)$.\n\nGiven $p,q\\in Q$, $t\\in T$, $d\\in C(t)$, it is decidable whether there is a $(p,d,q)$-loop in $t$: This amounts to checking whether a given automaton with input alphabet $\\{a,a^{-1},b,b^{-1}\\}$ accepts a word representing the identity of $F_2$ such that no proper prefix represents the identity of $F_2$. Since this can be accomplished using pushdown automata, we can compute the set $$X_t = \\{(p,d,q)\\in Q\\times C(t)\\times Q \\mid \\text{there is a\n$(p,d,q)$-loop in $t$}\\}.$$\n\nLoop patterns\n-------------\n\nGiven a word $w=(p_1,d_1,q_1)\\cdots (p_n,d_n,q_n)\\in X_t^*$, a *loop assignment for $w$* is a choice of a $(p_i,d_i,q_i)$-loop in $t$ for each position $i$, $1\\le i\\le n$. The *effect* of a loop assignment is $e_1\\cdots e_n\\in K$, where $e_i\\in K$ is the effect of the loop assigned to position $i$. The *depth* of a loop assignment is the maximum depth of an appearing loop. ", "A *loop pattern for $t$* is a word $w\\in X_t^*$ that has a loop assignment with effect $1$. The depth of the loop pattern is the minimum depth of a loop assignment with effect $1$. Note that applying the loops for the symbols in a loop pattern $(p_1,d_1,q_1)\\cdots (p_n,d_n,q_n)$ does not have to be a computation: We do not require $q_i=p_{i+1}$. Instead, the loop patterns describe the possible ways in which a well-nested computation can enter (and leave) subtrees of the Cayley graph of $F_2$ in order to have effect $1$. The sets $$P_t = \\{w\\in X_t^* \\mid \\text{$w$ is a loop pattern for $t$} \\}$$ for $t\\in T$ will therefore play a central role in the decision procedure.", "\n\nRecall the definition of the well-quasi-order $\\preceq_H$ from Section \\[sec-wqo\\].", "\n\n\\[lemma:upwardclosed\\] For each $t\\in T$, the set $P_t$ is an upward closed subset of $X_t^*$ with respect to $\\preceq_H$.\n\nSince $K$ is a direct sum of copies of $H$, the orders $\\preceq_H$ and $\\preceq_K$ coincide. ", "It therefore suffices to show that $P_t$ is upward closed with respect to $\\preceq_K$. Let $u\\in P_t$ and $u \\preceq_K v$, $v\\in X_t^*$, meaning $v=v_0u_1v_1\\cdots u_nv_n$ with $u=u_1\\cdots u_n$ and $\\alpha(v_i)=1$, $0\\le\ni\\le n$, for every morphism $\\alpha\\colon X_t^*\\to K$. Since $u\\in P_t$, there is a loop assignment for each $u_i$, $1\\le i\\le n$, with effect $e_i$ such that $e_1\\cdots e_n=1$. By construction of $X_t$, for each $(p,d,q)\\in X_t$, there is a $(p,d,q)$-loop, say $\\ell_{p,d,q}$, in $t$. Let $\\varphi\\colon X_t^*\\to K$ be the morphism such that for each $(p,d,q)\\in X_t$, $\\varphi((p,d,q))$ is the effect of $\\ell_{p,d,q}$. Choosing $\\ell_{p,d,q}$ for each occurrence of $(p,d,q)$ in a subword $v_i$ and reusing the loop assignments for the $u_i$ defines a loop assignment for $v$. Since $\\varphi(v_i)=1$ for $0\\le i\\le n$, the effect of this loop assignment is $\\varphi(v_0)e_1\\varphi(v_1)\\cdots e_n\\varphi(v_n)=e_1\\cdots e_n=1$. Hence, $v\\in P_t$.\n\nSince $\\preceq_H$ is a well-quasi-order, the previous lemma already implies that each $P_t$ is a regular language. ", "On the one hand, this follows from the fact that the upward closure of each singleton is regular. ", "On the other hand, this can be deduced by observing that $\\preceq_H$ is a monotone order in the sense of [@EhHaRo83]. ", "Therein, Ehrenfeucht, Haussler, and Rozenberg show that languages that are upward closed with respect to monotone well-quasi-orders are regular. ", "Our next step is a characterization of the sets $P_t$ that allows us to compute finite automata for them. ", "In order to state this characterization, we need the following definitions.", "\n\nLet $X,Y$ be alphabets. ", "A *regular substitution* is a map $\\sigma\\colon X\\to\n{2^{Y^*}}$ such that $\\sigma(x)$ is a regular language for every $x\\in X$. For $w\\in X^*$, $w=w_1\\cdots w_n$, $w_i\\in X$, let $\\sigma(w)=R_1\\cdots R_n$, where $\\sigma(w_i)=R_i$ for $1\\le i\\le n$. Given a set $R\\subseteq Y^*$ and a regular substitution $\\sigma\\colon X\\to {2^{Y^*}}$, let $\\sigma^{-1}(R)=\\{w\\in\nX^*\\mid \\sigma(w)\\cap R\\ne\\emptyset\\}$. Note that if $R$ is regular, then $\\sigma^{-1}(R)$ is regular as well [@Sak09 Proposition 2.16], and an automaton for $\\sigma^{-1}(R)$ can be constructed effectively from an automaton for $R$ and automata for the $\\sigma(x)$.[^4]\n\nThe alphabet $Y_t$ is given by $$Y_t = X_t \\cup ( (Q\\times H\\times Q) \\cap E).$$ The morphism $\\pi_t\\colon Y_t^*\\to X_t^*$ is the projection onto $X_t^*$, meaning $$\\pi_t(y)= \\begin{cases}\ny & \\text{ for } y\\in X_t \\\\\n{\\varepsilon}& \\text{ for } y\\in Y_t\\setminus X_t .", "\n\\end{cases}$$ The morphism $\\nu_t\\colon Y_t^*\\to H$ is defined by $$\\begin{aligned}\n\\nu_t((p,d,q)) & = & 1 \\text{ for } (p,d,q)\\in X_t \\\\\n\\nu_t((p,h,q)) & = & h \\text{ for } (p,h,q)\\in Y_t\\setminus X_t.\\end{aligned}$$ For $p,q\\in Q$ and $t\\in T$, define the regular set $$R_{p,q}^t = \\{ (p_0,g_1,p_1) ( p_1, g_2, p_2) \\cdots (p_{n-1}, g_n,p_n) \\in Y_t^* \\mid p_0 = p, p_n = q \\}.$$ Given $t\\in T$ and $d\\in C(t)$, the regular substitution $\\sigma_{t,d}\\colon X_t\\to{2^{Y_d^*}}$ is defined by $$\\begin{aligned}\n\\sigma_{t,d}((p,d,q)) &=& \\bigcup \\{ R^d_{p',q'} \\mid (p,d,p'), (q',d^{-1},q)\\in E\\} \\\\\n\\sigma_{t,d}((p,u,q)) &=& \\{\\varepsilon\\} \\ \\text{ for } u\\in C(t) \\setminus\\{d\\}.\\end{aligned}$$ Given two tuples, $(U_t)_{t\\in T}$ and $(V_t)_{t\\in T}$ with $U_t,V_t\\subseteq\nX_t^*$, we write $(U_t)_{t\\in T}\\le(V_t)_{t\\in T}$ if $U_t\\subseteq V_t$ for each $t\\in T$.\n\n\\[lemma:smallest\\] $(P_t)_{t\\in T}$ is the smallest tuple such that for every $t\\in T$ we have ${\\varepsilon}\\in P_t$ and $$\\bigcap_{d\\in C(t)} \\sigma_{t,d}^{-1}\\left(\\pi_d^{-1}(P_d) \\cap \\nu_d^{-1}(1)\\right) \\subseteq P_t. ", "\\label{eq:substclosure}$$\n\nFor each $i\\in{\\mathbb{N}}$, let $P^{(i)}_t\\subseteq X_t^*$ be the set of loop patterns for $t$ whose depth is at most $i$. Then $P^{(0)}_t=\\{{\\varepsilon}\\}$ and $$P^{(i+1)}_t = P^{(i)}_t \\cup \\bigcap_{d\\in C(t)} \\sigma_{t,d}^{-1}\\left(\\pi_d^{-1}(P^{(i)}_d) \\cap \\nu_d^{-1}(1)\\right).$$ The lemma follows since $P_t=\\bigcup_{i\\ge 0} P_t^{(i)}$.\n\nGiven a language $L\\subseteq X_t^*$, let ${{L}{\\uparrow_{t}}}=\\{v\\in X_t^* \\mid \\text{$u\\preceq_H v$ for some $u\\in L$}\\}$.\n\n\\[maindecidablethm\\] The rational subset membership problem is decidable for every group $G=H\\wr F$, where $H$ is finite and $F$ is a finitely generated free group.", "\n\nSince $H\\wr F$ is a subgroup of $H\\wr F_2$ (since $F$ is a subgroup of $F_2$), it suffices to show decidability for $G=H\\wr F_2$. First, we compute finite automata for the languages $P_t$. We do this by initializing $U_t^{(0)}:={{\\{{\\varepsilon}\\}}{\\uparrow_{t}}}$ for each $t\\in T$ and then successively extending the sets $U^{(i)}_t$, which are represented by finite automata, until they equal $P_t$: If there is a $t\\in T$ and a word $$w\\in \\bigcap_{d\\in C(t)} \\sigma_{t,d}^{-1} \\left(\\pi_d^{-1}(U_d^{(i)})\\cap\\nu_d^{-1}(1)\\right) \\setminus U^{(i)}_t,$$ we set $U_t^{(i+1)}:=U^{(i)}_t \\cup {{\\{w\\}}{\\uparrow_{t}}}$ and $U^{(i+1)}_{u}:=U^{(i)}_{u}$ for $u\\in T\\setminus \\{t\\}$. Otherwise we stop. ", "By induction on $i$, it follows from Lemma \\[lemma:upwardclosed\\] and Lemma \\[lemma:smallest\\] that $U^{(i)}_t\\subseteq P_t$.\n\nIn each step, we obtain $U^{(i+1)}_t$ by adding new words to $U^{(i)}_t$. Since the sets $U^{(i)}_t$ are upward closed by construction and there is no infinite (strictly) ascending chain of upward closed sets in a wqo, the algorithm above has to terminate with some tuple $(U^{(k)}_t)_{t\\in T}$. This, however, means that for every $t\\in T$ $$\\bigcap_{d\\in C(t)} \\sigma_{t,d}^{-1}\\left(\\pi_d^{-1}(U^{(k)}_d)\\cap \\nu_d^{-1}(1)\\right)\\subseteq U^{(k)}_t.$$ Since on the other hand ${\\varepsilon}\\in U^{(k)}_t$ and $U^{(k)}_t\\subseteq P_t$, Lemma \\[lemma:smallest\\] yields $U^{(k)}_t=P_t$.\n\nNow we have $1\\in L(A)$ if and only if $\\pi_1^{-1}(P_1)\\cap\\nu_1^{-1}(1)\\cap\nR^1_{q_0,q}\\ne\\emptyset$ for some $q\\in Q_F$, which can again be checked by constructing and analyzing finite automata.", "\n\nThe rational subset membership problem is decidable for every group $H\\wr V$ with $H$ finite and $V$ virtually free.", "\n\nThis is immediate from Theorem \\[maindecidablethm\\] and Proposition \\[finiteindex\\], because if $F$ is a free subgroup of index $m$ in $V$, then $H^m\\wr F$ is isomorphic to a subgroup of index $m$ in $H\\wr V$ and decidability of rational subset membership is preserved by finite extensions [@Gru99; @KaSiSt06].", "\n\nUndecidability {#sec-undec}\n==============\n\nIn this section, we will prove the second main result of this paper: The wreath product $\\mathbb{Z}\\wr\\mathbb{Z}$ contains a fixed submonoid with an undecidable membership problem. ", "Our proof is based on the halting problem for 2-counter machines (also known as Minsky machines), which is a classical undecidable problem.", "\n\n2-counter machines\n------------------\n\nA 2-counter machine (also known as Minsky machine) is a tuple $C = (Q, q_0, q_f, \\delta)$, where\n\n- $Q$ is a finite set of states,\n\n- $q_0 \\in Q$ is the initial state,\n\n- $q_f \\in Q$ is the final state, and\n\n- $\\delta \\subseteq (Q \\setminus \\{q_f\\}) \\times \\{c_0,c_1\\} \\times \\{+1,-1,=0\\} \\times\n Q$ is the set of transitions.", "\n\nThe set of configurations is $Q \\times \\mathbb{N} \\times \\mathbb{N}$. On this set we define a binary relation $\\to_C$ as follows: $(p, m_0, m_1) \\to_C (q, n_0, n_1)$ if and only if one of the following three cases holds:\n\n- There exist $i \\in \\{0,1\\}$ and a transition $(p, c_i, +1, q) \\in \\delta$ such that $n_i = m_i+1$ and $n_{1-i}=m_{1-i}$.\n\n- There exist $i \\in \\{0,1\\}$ and a transition $(p, c_i, -1, q) \\in \\delta$ such that $n_i = m_i-1$ (in particular, we must have $m_i > 0$) and $n_{1-i}=m_{1-i}$.\n\n- There exist $i \\in \\{0,1\\}$ and a transition $(p, c_i, =0, q) \\in \\delta$ such that $n_i=m_i=0$ and $n_{1-i}=m_{1-i}$.\n\nIt is well known that every Turing-machine can be simulated by a 2-counter machine, see e.g. [@Min67]. ", "In particular, we have:\n\n\\[thm-2-counter\\] There exists a fixed 2-counter machine $C= (Q, q_0, q_f, \\delta)$ such that the following problem is undecidable:\n\nINPUT: Numbers $m,n \\in \\mathbb{N}$.\n\nQUESTION: Does $(q_0,m,n) \\to^*_C (q_f,0,0)$ hold?", "\n\nSubmonoids of $\\mathbb{Z} \\wr \\mathbb{Z}$\n-----------------------------------------\n\nIn this section, we will only consider wreath products of the form $H\\wr\\mathbb{Z}$. An element $(f,m) \\in H\\wr\\mathbb{Z}$ such that the support of $f$ is contained in the interval $[a,b]$ (with $a,b \\in \\mathbb{Z}$) and $0,m \\in [a,b]$ will also be written as a list $[f(a),\\ldots, f(b)]$, where in addition the element $f(0)$ is labeled by an incoming (downward) arrow and the element $f(m)$ is labeled by an outgoing (upward) arrow.", "\n\nIn this section, we will construct a fixed finitely generated submonoid of the wreath product $\\mathbb{Z} \\wr \\mathbb{Z}$ with an undecidable membership problem.", "\n\nLet $C= (Q, q_0, q_f, \\delta)$ be the 2-counter machine from Theorem \\[thm-2-counter\\]. ", "Without loss of generality we can assume that there exists a partition $Q = Q_0 \\cup Q_1$ such that $q_0 \\in Q_0$ and $$\\delta \\subseteq (Q_0 \\times \\{c_0\\} \\times \\{+1,-1,=0\\} \\times Q_1) \\cup\n(Q_1 \\times \\{c_1\\} \\times \\{+1,-1,=0\\} \\times Q_0).$$ In other words, $C$ alternates between the two counters. ", "Hence, a transition $(q,c_i, x, p)$ can be just written as $(q,x,p)$. Let $$\\Sigma = Q \\cup\\{c,\\#\\}.$$ Let $\\mathbb{Z}^{\\Sigma}$ be the free abelian group generated by $\\Sigma$. First, we will prove that there is a fixed finitely generated submonoid $M$ of the wreath product $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ with an undecidable membership problem. ", "Let $$K = \\bigoplus_{m \\in \\mathbb{Z}} \\mathbb{Z}^{\\Sigma}.$$ An infinite presentation for $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ is $$\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z} = \\langle \\Sigma,a \\mid [a^n s\na^{-n}, a^m t a^{-m}] = 1\\; (n,m \\in \\mathbb{Z}, s,t \\in \\Sigma)\\rangle.$$ Here, $a$ generates the right $\\mathbb{Z}$-factor of the wreath product $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$. In the following, we will freely switch between the description of elements of $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ by words over $(\\Sigma \\cup \\{a\\})^{\\pm 1}$ and by pairs from $K \\rtimes \\mathbb{Z}$. For a finite-support mapping $f \\in K$, $m \\in \\mathbb{Z}$, and $x \\in \\Sigma$, we also write $f(m,x)$ for the integer $f(m)(x)$.\n\nOur finitely generated submonoid $M$ of $\\mathbb{Z}^{\\Sigma} \\wr\n\\mathbb{Z}$ is generated by the following elements. ", "The right column shows the generators in list notation, where elements of the free abelian group $\\mathbb{Z}^{\\Sigma}$ are written additively, i.e., as $\\mathbb{Z}$-linear combinations of elements of $\\Sigma$: $$\\begin{aligned}\n{2}\n & p^{-1} a \\# a^2 \\# a q \\ \\text{for} \\ (p,=0,q) \\in \\delta && [\\overset{\\shpos}{-p}, \\#, 0, \\#, \\overset{\\shneg}{q}] \\label{generator=0} \\\\\n & p^{-1} a \\# a c a^2 q a^{-2} \\ \\text{for} \\ (p,+1,q) \\in \\delta && [\\overset{\\shpos}{-p}, \\#, \\overset{\\shneg}{c}, 0, q] \\label{generator+1} \\\\\n & p^{-1} a \\# a^3 q a^6 c^{-1} a^{-8} \\ \\text{for} \\ (p,-1,q) \\in \\delta \\qquad && [\\overset{\\shpos}{-p}, \\#, \\overset{\\shneg}{0}, 0,q,0,0,0,0,0,-c] \\label{generator-1} \\\\\n & c^{-1} a^8 c a^{-8} && [\\overset{\\shpos\\shneg}{-c}, 0, 0, 0, 0, 0, 0, 0, c] \\label{generator-copy1} \\\\\n & c^{-1} a \\# a^7 c a^{-6} && [\\overset{\\shpos}{-c}, \\#, \\overset{\\shneg}{0}, 0, 0, 0, 0, 0, c] \\label{generator-copy2} \\\\\n & q_f^{-1} a^{-1} && [\\overset{\\shneg}{0},\\overset{\\shpos}{-q_f}] \\label{generator-terminate} \\\\\n & \\#^{-1} a^{-2} && [\\overset{\\shneg}{0},0,\\overset{\\shpos}{-\\#}]\\label{generator-go-left}\\end{aligned}$$ For initial counter values $m,n \\in \\mathbb{N}$ let $$I(m,n) = a q_0 a^2 c^m a^4 c^n a^{-6}.$$ The list notation for $I(m,n)$ is $$\\label{list-initial}\n[\\overset{\\shpos}{0}, \\overset{\\shneg}{q_0}, 0, m \\cdot c, 0, 0,0,n \\cdot c] .$$ Here is some intuition: The group element $I(m,n)$ represents the initial configuration $(q_0,m,n)$ of the 2-counter machine $C$. ", "Lemma \\[lemma-main-undec\\] below states that $(q_0,m,n) \\to^*_C (q_f,0,0)$ is equivalent to the existence of $Y \\in M$ with $I(m,n) Y = 1$, i.e., $I(m,n)^{-1} \\in M$. Generators of type – simulate the 2-counter machine $C$. ", "States of $C$ will be stored at cursor positions $4k+1$. The values of the first (resp., ", "second) counter will be stored at cursor positions $8k+3$ (resp., ", "$8k+7$). ", "Note that $I(m,n)$ puts a single copy of the symbol $q_0 \\in \\Sigma$ at position $1$, $m$ copies of symbol $c$ (which represents counter values) at position 3, and $n$ copies of symbol $c$ at position 7. ", "Hence, indeed, $I(m,n)$ sets up the initial configuration $(q_0,m,n)$ for $C$. ", "Even cursor positions will carry the special symbol $\\#$. Note that generator is the only generator which changes the cursor position from even to odd or vice versa. ", "It will turn out that if $I(m,n) Y = 1$ ($Y \\in M$), then generator has to occur exactly once in $Y$; it terminates the simulation of the 2-counter machine $C$. ", "Hence, $Y$ can be written as $Y = U (q_f^{-1} a^{-1}) V$ with $U,V \\in M$. Moreover, it turns out that $U \\in M$ is a product of generators –, which simulate $C$. ", "Thereby, even cursor positions will be marked with a single occurrence of the special symbol $\\#$. In a second phase, that corresponds to $V \\in M$, these special symbols $\\#$ will be removed again and the cursor will be moved left to position $0$. This is accomplished with generator . ", "In fact, our construction enforces that $V$ is a power of .", "\n\nDuring the simulation phase (corresponding to $U \\in M$), generators of type implement zero tests, whereas generators of type (resp., ) ", "increment (resp., ", "decrement) a counter. ", "Finally, and copy the counter value to the next cursor position that is reserved for the counter (that is copied). ", "During such a copy phase, is first applied $\\geq 0$ many times. ", "Finally, is applied exactly once.", "\n\n\\[lemma-main-undec\\] For all $m,n \\in \\mathbb{N}$ the following are equivalent:\n\n- $(q_0,m,n) \\to^*_C (q_f,0,0)$\n\n- There exists $Y \\in M$ such that $I(m,n) Y = 1$.\n\nAssume first that $I(m,n) Y = 1$ for some $Y \\in M$. We have to show that $(q_0,m,n) \\to^*_C (q_f,0,0)$; this is the more difficult direction. ", "Let $$Y = y_1 \\cdots y_k,$$ where each $y_i$ is one of the generators of $M$. For $0 \\leq i \\leq k$ let $$Y_i = y_1 \\cdots y_i$$ (thus, $Y_0 = 1$) and assume that $$I(m,n) Y_i = (f_i,m_i) \\in K \\rtimes \\mathbb{Z}.$$ Hence, $f_k = 0$ is the zero-mapping and $m_k=0$. Moreover $(f_0,m_0) = I(m,n)$.\n\n[*Claim 1.*]{} ", "For all $0 \\leq i \\leq k$, $q \\in Q$, and $\\ell \\in \\mathbb{Z}$ we have $f_i(2\\ell,q) = 0$.\n\n[*Proof of Claim 1.*]{} ", "Assume that $f_i(2\\ell,q) \\neq 0$ for some $0 \\leq i \\leq k$, $q \\in Q$, and $\\ell \\in \\mathbb{Z}$. Choose $0 \\leq i \\leq k$ minimal such that there exist $q \\in Q$ and $\\ell \\in \\mathbb{Z}$ with $f_i(2\\ell,q) \\neq 0$. Since $f_0(2\\ell,q)=0$ for all $q \\in Q$ and $\\ell \\in \\mathbb{Z}$ (the list notation for $(f_0,m_0)$ is ), we must have $i \\geq 1$. Hence, $f_{i-1}(2\\ell,q) = 0$ for all $q \\in Q$ and $\\ell \\in \\mathbb{Z}$. An inspection of the generators shows that if $m_{i-1}$ were odd, we would also have $f_i(2\\ell,q) = 0$ for all $q \\in Q$ and $\\ell \\in \\mathbb{Z}$. Therefore, $m_{i-1}$ must be even. ", "An inspection of the generators of $M$ shows that there exist $j \\in \\mathbb{Z}$ and $p \\in Q$ such that $$f_i(2j,p) < 0 \\text{ and } f_i(2j',p') = 0 \\text{ for all $j' < j$ and $p' \\in Q$.}$$ But then, for all $i \\leq i' \\leq k$ there exist $j \\in \\mathbb{Z}$ and $p \\in Q$ such that $$f_{i'}(2j,p) < 0 \\text{ and } f_{i'}(2j',p') = 0 \\text{ for all $j' < j$ and $p' \\in Q$.}$$ For $i'=k$ we obtain a contradiction, since $f_k = 0$.\n\nClaim 1 implies that for all $1 \\leq i \\leq k$ with $m_{i-1}$ even, the generator $y_i$ cannot be of type , , , or .", "\n\n[*Claim 2.*]{} ", "For all $0 \\leq i \\leq k$ and $\\ell \\in \\mathbb{Z}$ we have $f_i(2\\ell,c) = 0$.\n\n[*Proof of Claim 2.*]{} ", "Assume that $f_i(2\\ell,c) \\neq 0$ for some $0 \\leq i \\leq k$ and $\\ell \\in \\mathbb{Z}$. Choose $0 \\leq i \\leq k$ minimal such that there exists $\\ell \\in \\mathbb{Z}$ with $f_i(2\\ell,c) \\neq 0$. Since $f_0(2\\ell,c)=0$ for all $\\ell \\in \\mathbb{Z}$, we must have $i \\geq 1$. Hence, $f_{i-1}(2\\ell,c) = 0$ for all $\\ell \\in \\mathbb{Z}$. An inspection of the generators shows that if $m_{i-1}$ were odd, we would also have $f_i(2\\ell,c) = 0$ for all $\\ell \\in \\mathbb{Z}$. Therefore, $m_{i-1}$ must be even. ", "The generator $y_i$ must be of one of the types , , , or . ", "But the types and are excluded by the remark before Claim 2. ", "Therefore, $y_i$ must be either or . ", "Thus, there exists $j \\in \\mathbb{Z}$ such that $$f_i(2j,c) < 0 \\text{ and } f_i(2j',c) = 0 \\text{ for all $j' < j$.}$$ Note that for all $i < i' \\leq k$ with $m_{i'-1}$ even, the generator $y_{i'}$ is not of type (again by the remark before Claim 2). ", "This implies that for all $i \\leq i' \\leq k$ there exists $j \\in \\mathbb{Z}$ such that $$f_{i'}(2j,c) < 0 \\text{ and } f_{i'}(2j',c) = 0 \\text{ for all $j' < j$.}$$ For $i'=k$ we obtain a contradiction, since $f_k = 0$.\n\nClaim 1 and 2 imply that for all $1 \\leq i \\leq k$ with $m_{i-1}$ even, the generator $y_i$ is .", "\n\n[*Claim 3.*]{} ", "For all $0 \\leq i \\leq k$ and $\\ell \\in \\mathbb{Z}$ we have $f_i(2\\ell+1,\\#) = 0$.\n\n[*Proof of Claim 3.*]{} ", "Assume that $f_i(2\\ell+1,\\#) \\neq 0$ for some $0 \\leq i \\leq k$ and $\\ell \\in \\mathbb{Z}$. Choose $0 \\leq i \\leq k$ minimal such that there exists $\\ell \\in \\mathbb{Z}$ with $f_i(2\\ell+1,\\#) \\neq 0$. Since $f_0(\\ell,\\#)=0$ for all $\\ell \\in \\mathbb{Z}$, we must have $i \\geq 1$. Hence, $f_{i-1}(2\\ell+1,\\#) = 0$ for all $\\ell \\in \\mathbb{Z}$. There are two possible cases:\n\n1. ", " $m_{i-1}$ is odd and $y_i$ is the generator .", "\n\n2. ", " $m_{i-1}$ is even and $y_i$ is a generator of type – or .", "\n\nBut the second case is not possible by the remark before Claim 3. ", "Hence, $m_{i-1}$ is odd and $y_i$ is the generator . ", "Thus, there exists $j \\in \\mathbb{Z}$ with $f_i(2j+1,\\#) < 0$. Since for every $i \\leq i' \\leq k$ with $m_{i'-1}$ even, the generator $y_{i'}$ can only be of type (again by the remark before Claim 3), it follows that for every $i \\leq i' \\leq k$ we have $f_{i'}(2j+1,\\#) < 0$. For $i'=k$ we obtain a contradiction, since $f_k = 0$.\n\n[*Claim 4.*]{} ", "There is exactly one $1 \\leq i \\leq k$ such that $y_i$ is the generator .", "\n\n[*Proof of Claim 4.*]{} ", "For $g = (f,m) \\in \\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ and $b \\in \\{0,1\\}$ we define $$\\sigma_Q(g,b) = \\sum_{k \\in \\mathbb{Z}} \\sum_{q \\in Q} f(2k+b,q) .$$ An inspection of all generators of $M$ shows that for every $g \\in \\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ and every generator $z$ of $M$ we have:\n\n- If $z$ is not the generator , then $\\sigma_Q(gz,b) = \\sigma_Q(g,b)$ for both $b=0$ and $b=1$.\n\n- If $z$ is the generator , then there is $b \\in \\{0,1\\}$ such that $\\sigma_Q(gz,b) = \\sigma_Q(g,b)-1$ and $\\sigma_Q(gz,1-b) = \\sigma_Q(g,1-b)$.\n\nThe claim follows, since $\\sigma_Q(I(m,n),0)= \\sigma_Q(I(m,n)Y,0)=\n\\sigma_Q(I(m,n)Y,1) = 0$ and $\\sigma_Q(I(m,n),1)=1$.\n\nBy Claim 1–4, there exists a unique $1 \\leq i \\leq k$ such that the following three properties hold:\n\n- For every $1 \\leq j < i$, $y_j$ is a generator of type –.", "\n\n- $y_i$ is the generator .", "\n\n- For every $i < j \\leq k$, $y_j$ is the generator .", "\n\nHence, $I(m,n) Y_{i-1}$ must be of the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#, \\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q_f}],$$ since only such an element can be reduced to 1 by right-multiplication with generator followed by a positive power of generator . ", "We show that this implies $(q_0,m,n) \\to^*_C (q_f,0,0)$. Note that every generator of type – (those generators that occur in $Y_{i-1}$) moves the cursor $2d$ (for some $d \\geq 0$) to the right along the $\\mathbb{Z}$-line. ", "This means that for every $0 \\leq j \\leq i-1$, $m_j$ is odd and moreover, for every odd $m < m_j$, the group element $f_j(m) \\in \\mathbb{Z}^\\Sigma$ is zero.", "\n\n[*Claim 5.*]{} ", "Let $0 \\leq j < i-1$ and assume that $I(m,n)Y_j$ is of the form $$\\label{config-8k+1}\n[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{p},0,a \\cdot c,0,0,0,b \\cdot c],$$ where $p \\in Q_0$, $a,b \\in \\mathbb{N}$, and $\\overset{\\shneg}{p}$ occurs at position $\\ell = 8k+1$ for some $k \\geq 0$ (hence, represents the configuration $(p,a,b)$). ", "Then there exists $j' > j$ and a valid $C$-transition $(p,a,b) \\to_C (q,a',b')$ such that $I(m,n)Y_{j'}$ is of the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q},0,b' \\cdot c,0,0,0,a' \\cdot c].$$ Here $\\overset{\\shneg}{q}$ occurs at position $\\ell+4$.\n\n[*Proof of Claim 5.*]{} ", "Generator $y_{j+1}$ has to be of the form , , or , because otherwise we leave at position $\\ell$ a negative copy of $c$, which cannot be compensated later. ", "Let us first assume that $y_{j+1}$ has the form , i.e., $(p,=0,q) \\in \\delta$. Then $I(m,n)Y_{j+1}$ is of the form $$\\label{claim-5-zero-test}\n[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,a \\cdot c,\\#,\\overset{\\shneg}{q},0,b \\cdot c,0,0,0,0],$$ where $\\overset{\\shneg}{q}$ occurs at position $\\ell+4$. If $a > 0$, then the $a$ many $c$’s at position $\\ell+2$ cannot be removed in the future. ", "Hence, we must have $a=0$. Setting $a'=0$ and $b'=b$ shows that has the form required in the conclusion of Claim 5.", "\n\nNext, assume that $y_{j+1}$ has the form . ", "Hence $(p,+1,q) \\in \\delta$ and $I(m,n)Y_{j+1}$ is of the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,\\overset{\\shneg}{(a+1) \\cdot c},0,q,0,b \\cdot c,0,0,0,0],$$ where $\\overset{\\shneg}{(a+1) \\cdot c}$ occurs at position $\\ell+2$. So we have to remove $a+1$ many copies of $c$ from position $\\ell+2$. Hence, the only way to continue is to apply $a$ many times generator followed by a single application of generator . ", "Hence, $I(m,n)Y_{j+a+2}$ must be of the form $$\\label{claim-5-+1}\n[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q},0,b \\cdot c,0,0,0,(a+1)\\cdot c],$$ where $\\overset{\\shneg}{q}$ occurs at position $\\ell+4$. Setting $b'= b$ and $a'=a+1$ shows that has the form required in the conclusion of Claim 5.", "\n\nFinally, assume that $y_{j+1}$ has the form , hence $(p,-1,q) \\in \\delta$ and $I(m,n)Y_{j+1}$ is of the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,\\overset{\\shneg}{a \\cdot c},0,q,0,b \\cdot c,0,0,0,-c],$$ where $\\overset{\\shneg}{a \\cdot c}$ occurs at position $\\ell+2$. First, assume that $a=0$. Then there is no way to move the cursor to the right without leaving a negative copy of a symbol from $Q \\cup \\{c\\}$ at position $\\ell+2$, and this negative copy cannot be eliminated later. ", "Hence, we must have $a > 0$. Now, the only way to continue is to apply $a-1$ many times generator followed by a single application of generator . ", "Hence, $I(m,n)Y_{j+a+1}$ must be of the form $$\\label{claim-5--1}\n[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q},0,b \\cdot c,0,0,0,(a-1)\\cdot c],$$ where $\\overset{\\shneg}{q}$ occurs at position $\\ell+4$. Setting $b'= b$ and $a'=a-1$ shows that has the form required in the conclusion of Claim 5.", "\n\nThis concludes the proof of Claim 5. ", "Completely analogously to Claim 5, one can show:\n\n[*Claim 6.*]{} ", "Let $0 \\leq j < i-1$ and assume that $I(m,n)Y_j$ is of the form $$\\label{config-8k+5}\n[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{p},0,a \\cdot c,0,0,0,b \\cdot c],$$ where $p \\in Q_1$, $a,b \\in \\mathbb{N}$, $\\overset{\\shneg}{p}$ occurs at position $\\ell = 8k+5$ for some $k \\geq 0$ (hence, represents the configuration $(p,b,a)$). ", "Then there exists $j' > j$ and a valid $C$-transition $(p,b,a) \\to_C (q,b',a')$ such that $I(m,n)Y_{j'}$ is of the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#,\n\\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q},0,b' \\cdot c,0,0,0,a' \\cdot c].$$ Here $\\overset{\\shneg}{q}$ occurs at position $\\ell+4$.\n\nUsing Claim 5 and 6 we can now easily conclude that $(q_0,m,n) \\to^*_C\n(q_f,0,0)$ holds.", "\n\nThe other direction (if $(q_0,m,n) \\to^*_C (q_f,0,0)$ then there exists $Y \\in M$ with $I(m,n) Y = 1$) is easier. ", "A computation $$(q_0,m,n) \\to_C (q_1,m_1,n_1) \\to_C \\cdots \\to_C (q_{\\ell-1},m_{\\ell-1},n_{\\ell-1}) \\to_C (q_f,0,0)$$ can be directly translated into a sequence of $M$-generators $y_1 y_2 \\cdots y_k$ such that the group element $I(m,n) y_1 y_2 \\cdots y_k$ has the form $$[\\overset{\\shpos}{0},0,\\#,0,\\#,0,\\#, \\ldots,0,\\#,0,\\#,\\overset{\\shneg}{q_f}],$$ Multiplying this element with generator followed by a positive power of generator yields the group identity.", "\n\nThe following result is an immediate consequence of Theorem \\[thm-2-counter\\] and Lemma \\[lemma-main-undec\\].", "\n\n\\[thm:undec1\\] There is a fixed finitely generated submonoid $M$ of the wreath product $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ with an undecidable membership problem.", "\n\nFinally, we can establish the main result of this section.", "\n\nThere is a fixed finitely generated submonoid $M$ of the wreath product $\\mathbb{Z} \\wr \\mathbb{Z}$ with an undecidable membership problem.", "\n\nBy Theorem \\[thm:undec1\\] it suffices to reduce the submonoid membership problem of $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$ to the submonoid membership problem of $\\mathbb{Z} \\wr \\mathbb{Z}$. If $m=|\\Sigma|$, then Proposition \\[finiteindex\\] shows that $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}\\cong \\mathbb Z^m\\wr m\\mathbb Z$ is isomorphic to a subgroup of index $m$ in $\\mathbb Z\\wr \\mathbb Z$. So if $\\mathbb Z\\wr \\mathbb Z$ had decidable submonoid membership for each finitely generated submonoid, then the same would be true of $\\mathbb{Z}^{\\Sigma} \\wr \\mathbb{Z}$.\n\nWe remark that, together with the undecidability of the rational subset membership problem for groups $H\\wr\n(\\mathbb{Z}\\times\\mathbb{Z})$ for non-trivial $H$ [@LohSt09tocs], our results imply the following: For finitely generated non-trivial abelian groups $G$ and $H$, the wreath product $H\\wr G$ has a decidable rational subset membership problem if and only if (i) $G$ is finite[^5] or (ii) ($G$ has rank 1 and $H$ is finite). ", "Furthermore, for virtually free groups $G$ and $H$, the rational subset membership problem is decidable for $H\\wr G$ if and only if (i) $G$ is trivial or (ii) $H$ is finite, or (iii) ($G$ is finite and $H$ is virtually abelian).", "\n\nBy [@Clea06], the wreath product $\\mathbb{Z} \\wr \\mathbb{Z}$ is a subgroup of Thompson’s group $F$ as well as of Baumslag’s finitely presented metabelian group $\\langle a, s, t \\mid [s,t]=[a^t, a]=1, a^s = aa^t \\rangle$. Hence, we get:\n\nThompson’s group $F$ as well as Baumslag’s finitely presented metabelian group both contain finitely generated submonoids with an undecidable membership problem.", "\n\nOpen problems\n=============\n\nAs already mentioned in the introduction, we conjecture that the rational subset membership problem for a wreath product $H \\wr G$ with $H$ non-trivial and $G$ not virtually free is undecidable. 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", "In [*Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008)*]{}, volume 5126 of [*Lecture Notes in Computer Science*]{}, pages 51–62. ", "Springer, 2008.", "\n\nM. Kambites. ", "Formal languages and groups as memory. , ", "37(1):193–208, 2009.", "\n\nM. Kambites, P. V. Silva, and B. Steinberg. ", "On the rational subset problem for groups. , ", "309(2):622–639, 2007.", "\n\nD. Kuske and M. Lohrey. ", "Logical aspects of [Cayley]{}-graphs: the group case. , ", "131(1–3):263–286, 2005.", "\n\nM. Lohrey and G. S[é]{}nizergues. ", "Theories of [HNN]{}-extensions and amalgamated products. ", "In [ *Proceedings of the 33st International Colloquium on Automata, Languages and Programming (ICALP 2006)*]{}, number 4052 in Lecture Notes in Computer Science, pages 681–692. ", "Springer, 2006.", "\n\nM. Lohrey and B. Steinberg. ", "The submonoid and rational subset membership problems for graph groups. , ", "320(2):728–755, 2008.", "\n\nM. Lohrey and B. Steinberg. ", "Submonoids and rational subsets of groups with infinitely many ends. , ", "324(4):970–983, 2010.", "\n\nM. Lohrey and B. Steinberg. ", "Tilings and submonoids of metabelian groups. , ", "48(2):411–427, 2011.", "\n\nR. C. Lyndon and P. E. Schupp. . ", "Springer, 1977.", "\n\nM. L. Minsky. . ", "Prentice-Hall International, Englewood Cliffs, 1967.", "\n\nR. Motwani, R. Panigrahy, V. A. Saraswat, and S. Venkatasubramanian. ", "On the decidability of accessibility problems (extended abstract). ", "In [*Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000)*]{}, pages 306–315. ", "ACM, 2000.", "\n\nJ. Nielsen. ", "Om regning med ikke kommutative faktoren og dens anvendelse i gruppeteorien. , ", "pages 77–94, 1921. ", "In Danish.", "\n\nV. Roman’kov. ", "On the occurence problem for rational subsets of a group. ", "In V. Roman’kov, editor, [*International Conference on Combinatorial and Computational Methods in Mathematics*]{}, pages 76–81, 1999.", "\n\nN. S. Romanovski[ĭ]{}. ", "Some algorithmic problems for solvable groups. , ", "13(1):26–34, 1974.", "\n\nN. S. Romanovski[ĭ]{}. ", "The occurrence problem for extensions of abelian groups by nilpotent groups. , ", "21:170–174, 1980.", "\n\nJ. Sakarovitch. . ", "Cambridge University Press, 2009.", "\n\nP. Schnoebelen. ", "Verifying lossy channel systems has nonprimitive recursive complexity. , ", "83(5):251–261, 2002.", "\n\n[^1]: This work was supported by the DAAD research project RatGroup. ", "The second author was partially supported by a grant from the Simons Foundation (\\#245268 to Benjamin Steinberg).", "\n\n[^2]: Recall that a group is virtually free if it has a free subgroup of finite index.", "\n\n[^3]: A $K$-equivariant group isomorphism $\\alpha\\colon H^{(G)}\\to (H^T)^{(K)}$ is an isomorphism that commutes with the action of $K$: $k\\alpha(f) = \\alpha(kf)$.\n\n[^4]: In [@Sak09], it is shown that the class of regular languages is closed under arbitrary inverse substituitions. ", "Moreover, the construction is effective if $\\sigma$ is a regular substitution.", "\n\n[^5]: If $G$ has size $m$, then by Proposition \\[finiteindex\\], $H^m \\cong H^m \\wr 1$ is isomorphic to a subgroup of index $m$ in $H \\wr G$. Since $H^m$ is finitely generated abelian, decidability of the rational subset membership problem of $H \\wr G$ follows from the fact that decidability is preserved by finite extensions [@Gru99; @KaSiSt06].", "\n" ]

{ "pile_set_name": "ArXiv" }

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[ "January 13, 2008\n\nIs Vitamin D an Autism Treatment? …", "it has now been suggested that autism may be caused by insufficient vitamin D during pregnancy and/or early years. ", "Autistic children have difficulty in forming relationships, but they also tend to have larger heads, changes similar to those found in baby rats that are bred with insufficient vitamin D. (The Telegraph, Why is Vitamin D So Vital?).", "\n\nExtracted from a periodic newsletter from the Vitamin D Council by John Cannell, MD\n\nThe following is Dr. Cannell’s reply to one of a number of questions sent to him regarding vitamin D and autism.", "\n\nDear Dr. Cannell:\n\nWho are you to write an article on autism? ", "You didn't even publish it in a medical journal. ", "You are not with a university. ", "You have not published very much. ", "You have no expertise on autism. ", "No autism experts support your theory. ", "There is no evidence to support the theory. ", "Shouldn't you leave this to experts before you give parents more false hopes?", "\n\nMary, Trenton, New Jersey.", "\n\nDear Mary:\n\nYou are right, I am a nobody; just ask my ex-wife. ", "In the Toronto Globe, I explained why I have not yet submitted the paper. ", "As far as giving false hopes, I've thought about that charge. ", "Right now, regardless of what advocacy groups say, autism is rather hopeless. ", "That is, no treatment, including vitamin D, has been shown to materially affect the clinical course of autism. ", "As a psychiatrist, my observation is that people would rather live with a false hope than with no hope.", "\n\nFurthermore, if autistic children began taking vitamin D, the worse that can happen is that a period of false hope will followed by dashed hopes and then parents will be back to hopelessness.", "In the meantime, they will have made their child vitamin D sufficient.", "Vitamin D deficiency is a serious problem in childhood. (", "Postgrad Med J. 2007 Apr;83(978):230-5).", "\n\nAs far as the theory having no support from experts, Dr. Richard Mills, research director of the National Autistic Society in England, was quoted in the Telegraph article on the autism/vitamin D theory: \"There has been speculation in the past about autism being more common in high-latitude countries that get less sunlight and a tie-up with rickets has been suggested - observations which support the theory.\"", "\n\nFinally, you said there is no evidence to support the theory.", "I assume you meant there is no proof.", "The first statement is absolutely false, the second absolutely true.", "As I detailed in my paper, there is a lot of evidence to support the theory.", "In fact, if anyone can come up with an autism fact, that the theory cannot explain, I'd like to know about it.", "\n\nEven the announcement of a link between television viewing and autism supports the theory.", "Furthermore, the TV/autism link is actually evidence of a treatment effect.", "That is, if autistic children who play outside in the sunshine more - watching less TV - have less severe illness, it may be due to the Sun-God, who bestows her precious gift of calcitriol (natural vitamin D) into the brains of children playing outside in her sunlight but not into the brains of children watching TV inside in the darkness. (", "Natl Bur Econ Res Bull Aging Health. ", "2007 Winter;(18):2-3).", "\n\nAs far as proof the theory is true, there is, of course, none.", "In medicine, proof means randomized controlled human trials, the gold standard for proof.", "However, proof is the last step, not the first.", "First comes evidence, then comes a theory, then comes researchers disproving those theories.", "It works that way.", "Sometimes we never get to the last step, proof.", "For example, please point me to a single randomized controlled human trial proving cigarette smoking is dangerous?Instead, the convincing evidence of smoking's dangerousness lies in epidemiological (population) studies, not randomized controlled trials.", "\n\nProof, or disproof, of the autism vitamin D theory will take years, years during which young autistic brains will continue to suffer irreparable damage.", "Perhaps vitamin D' powerful anti-inflammatory actions will help prevent that damage, perhaps not.", "\n\nIt's something of a Pascal's wager, betting on vitamin D instead of the existence of God, risking your child's brain instead of eternal damnation.", "\"If you believe vitamin D helps autism and turn out to be incorrect, you have lost nothing - but if you don't believe in vitamin D and turn out to be incorrect, your child will suffer irreparable brain damage.\"", "\n\nDisclaimer: The information contained in this article is presented for information purposes only and is in no way intended to replace professional medical care or attention by a qualified practitioner. ", "It cannot and should not be used as a basis for diagnosis or choice of treatment.", "\n\nEnter your search termsSubmit search form\n\nWeb\n\nwww.kflatthealthnews.com\n\nCopyright 2007 Kevin Flatt. ", "Reproduction of any information on other websites is PROHIBITED.", "\n\nDisclaimer: The information and opinions on this website is for information purposes only and is believed to be accurate and sound, based on the best judgment available to the author. ", "Readers should consult appropriate health professionals on any matter relating to their health and well-being. ", "Readers who fail to consult appropriate health authorities assume the risk of any injuries.", "\n\nNatural Remedies Recent Articles\n\nSubscribe to RSS headline updates from: Powered by FeedBurner\n\nNatural Remedies Google Analytics\n\nCopyright 2007 Kevin Flatt. ", "Reproduction of any information on other websites is PROHIBITED.", "\n\nDisclaimer: The information and opinions on this website is for information purposes only and is believed to be accurate and sound, based on the best judgment available to the author. ", "Readers should consult appropriate health professionals on any matter relating to their health and well-being. ", "Readers who fail to consult appropriate health authorities assume the risk of any injuries." ]

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[ "Q:\n\nCatchable fatal error: Argument 1 passed to DOMDocument::saveXML() must be an instance of DOMNode, string\n\nI'm having a problem when saving the data into XML when there's no existing XML like orders.xml. ", "But when there is existing file it saves. ", "It shows \n\nCatchable fatal error: Argument 1 passed to DOMDocument::saveXML() must be an instance of DOMNode, string given in C:\\xampp\\htdocs\\CWP413\\MidtermExam\\buy.php on line 75\"...\n\nThe first line is for loading the form from buying the product from another form.", "\nHere is my code:\n// script by candy21 \n$products =simplexml_load_file(\"product.xml\") or die(\"ERROR\");\n\nforeach ($products->product as $product){\n if($product['id']==$_GET['id']){\n $id = $product['id'];\n $name =$product->name;\n $price = $product->price;\n break;\n }\n\n }\n\nif(isset($_POST['submitSave'])){\n $quan = $_POST['quan'];\n\n $total = $price * $quan;\n\n $xml = new DOMDocument(\"1.0\",\"UTF-8\");\n $xml -> preserveWhiteSpace = TRUE;\n $xml -> formatOutput = true;\n\n if(file_exists(\"orders.xml\")){\n $xml = simplexml_load_file('orders.xml');\n $order = $xml->addChild('order');\n $order->addChild('id', $id);\n $order->addChild('name', $name);\n $order->addChild('price', $price);\n $order->addChild('quan', $quan);\n $order->addChild('total', $total);\n\n //echo \"<xmp>\" .$xml -> saveXML(). \"", "</xmp>\";\n $xml->saveXML(\"orders.xml\");\n $xml=simplexml_load_file(\"orders.xml\") or die(\"ERROR\");\n $a=1;\n foreach($xml->children() as $order)\n {\n echo\"<div class = 'orderlist' \";\n echo\"<b>Order No.:\". ", "$a++.\"</b><br>\";\n echo\"Product Id: \".$order->id.", "\"<br>\";\n echo\"Product Name: \".$order->name.", "\"<br>\";\n echo\"Price: \".$order->price.", "\"<br>\";\n echo\"Quantity Ordered: \".$order->quan.", "\"<br>\";\n echo\"Total Price: \".$order->total.", "\"<br>\";\n echo\"</div><br>\";\n }\n\n }\n else{\n $orders = $xml -> createElement(\"orders\");\n $xml -> appendChild($orders);\n\n $order = $xml -> createElement(\"order\");\n $orders -> appendChild($order);\n\n $id = $xml -> createElement(\"id\", $id);\n $order -> appendChild($id);\n\n $name = $xml -> createElement(\"name\", $name);\n $order -> appendChild($name);\n\n $price = $xml -> createElement(\"price\", $price);\n $order -> appendChild($price);\n\n $quan = $xml -> createElement(\"quan\", $quan);\n $order -> appendChild($quan);\n\n $total = $xml -> createElement(\"total\", $total);\n $order -> appendChild($total);\n\n $xml->saveXML(\"orders.xml\");\n $xml->saveXML($order->item(0));\n\n $xml=simplexml_load_file(\"orders.xml\") or die(\"ERROR\");\n $a=1;\n foreach($xml->children() as $order)\n {\n echo\"<div class = 'orderlist' \";\n echo\"<b>Order No.:\".$a++.\"</b><br>\";\n echo\"Product Id: \".$order->id.", "\"<br>\";\n echo\"Product Name: \".$order->name.", "\"<br>\";\n echo\"Price: \".$order->price.", "\"<br>\";\n echo\"Quantity Ordered: \".$order->quan.", "\"<br>\";\n echo\"Total Price: \".$order->total.", "\"<br>\";\n echo\"</div><br>\";\n }\n }\n\nA:\n\nDOMDocument::saveXML() doesn't take a filename argument. ", "It returns the result as a string, you need to write that to the file yourself. ", "So change:\n$xml->saveXML(\"orders.xml\");\n\nto:\nfile_put_contents(\"orders.xml\", $xml->saveXML());\n\n" ]

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[ "//-------------------------------------------------------------------------\n/*\nCopyright (C) 1996, 2003 - 3D Realms Entertainment\n\nThis file is part of Duke Nukem 3D version 1.5 - Atomic Edition\n\nDuke Nukem 3D is free software; you can redistribute it and/or\nmodify it under the terms of the GNU General Public License\nas published by the Free Software Foundation; either version 2\nof the License, or (at your option) any later version.", "\n\nThis program is distributed in the hope that it will be useful,\nbut WITHOUT ANY WARRANTY; without even the implied warranty of\nMERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. ", " \n\nSee the GNU General Public License for more details.", "\n\nYou should have received a copy of the GNU General Public License\nalong with this program; if not, write to the Free Software\nFoundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.", "\n\nOriginal Source: 1996 - Todd Replogle\nPrepared for public release: 03/21/2003 - Charlie Wiederhold, 3D Realms\n*/\n//-------------------------------------------------------------------------\n\n#include \"duke3d.h\"\n\nextern char everyothertime;\nshort which_palookup = 9;\n\n\ntloadtile(short tilenume)\n{\n gotpic[tilenume>>3] |= (1<<(tilenume&7));\n}\n\nvoid cachespritenum(short i)\n{\n char maxc;\n short j;\n\n if(ud.monsters_off && badguy(&sprite[i])) return;\n\n maxc = 1;\n\n switch(PN)\n {\n case HYDRENT:\n tloadtile(BROKEFIREHYDRENT);\n for(j = TOILETWATER; j < (TOILETWATER+4); j++)\n if(waloff[j] == 0) tloadtile(j);\n break;\n case TOILET:\n tloadtile(TOILETBROKE);\n for(j = TOILETWATER; j < (TOILETWATER+4); j++)\n if(waloff[j] == 0) tloadtile(j);\n break;\n case STALL:\n tloadtile(STALLBROKE);\n for(j = TOILETWATER; j < (TOILETWATER+4); j++)\n if(waloff[j] == 0) tloadtile(j);\n break;\n case RUBBERCAN:\n maxc = 2;\n break;\n case TOILETWATER:\n maxc = 4;\n break;\n case FEMPIC1:\n maxc = 44;\n break;\n case LIZTROOP:\n case LIZTROOPRUNNING:\n case LIZTROOPSHOOT:\n case LIZTROOPJETPACK:\n case LIZTROOPONTOILET:\n case LIZTROOPDUCKING:\n for(j = LIZTROOP; j < (LIZTROOP+72); j++)\n if(waloff[j] == 0)\n tloadtile(j);\n for(j=HEADJIB1;j<LEGJIB1+3;j++)\n if(waloff[j] == 0)\n tloadtile(j);\n maxc = 0;\n break;\n case WOODENHORSE:\n maxc = 5;\n for(j = HORSEONSIDE; j < (HORSEONSIDE+4); j++)\n if(waloff[j] == 0)\n tloadtile(j);\n break;\n case NEWBEAST:\n case NEWBEASTSTAYPUT:\n maxc = 90;\n break;\n case BOSS1:\n case BOSS2:\n case BOSS3:\n maxc = 30;\n break;\n case OCTABRAIN:\n case OCTABRAINSTAYPUT:\n case COMMANDER:\n case COMMANDERSTAYPUT:\n maxc = 38;\n break;\n case RECON:\n maxc = 13;\n break;\n case PIGCOP:\n case PIGCOPDIVE:\n maxc = 61;\n break;\n case SHARK:\n maxc = 30;\n break;\n case LIZMAN:\n case LIZMANSPITTING:\n case LIZMANFEEDING:\n case LIZMANJUMP:\n for(j=LIZMANHEAD1;j<LIZMANLEG1+3;j++)\n if(waloff[j] == 0)\n tloadtile(j);\n maxc = 80;\n break;\n case APLAYER:\n maxc = 0;\n if(ud.multimode > 1)\n {\n maxc = 5;\n for(j = 1420;j < 1420+106; j++)\n if(waloff[j] == -1)\n tloadtile(j);\n }\n break;\n case ATOMICHEALTH:\n maxc = 14;\n break;\n case DRONE:\n maxc = 10;\n break;\n case EXPLODINGBARREL:\n case SEENINE:\n case OOZFILTER:\n maxc = 3;\n break;\n case NUKEBARREL:\n case CAMERA1:\n maxc = 5;\n break;\n }\n\n for(j = PN; j < (PN+maxc); j++)\n if(waloff[j] == 0)\n tloadtile(j);\n}\n\nvoid cachegoodsprites(void)\n{\n short i;\n\n if(ud.screen_size >= 8)\n {\n if(waloff[BOTTOMSTATUSBAR] == 0)\n tloadtile(BOTTOMSTATUSBAR);\n if( ud.multimode > 1)\n {\n if(waloff[FRAGBAR] == 0)\n tloadtile(FRAGBAR);\n for(i=MINIFONT;i<MINIFONT+63;i++)\n if(waloff[i] == 0)\n tloadtile(i);\n }\n }\n\n tloadtile(VIEWSCREEN);\n\n for(i=STARTALPHANUM;i<ENDALPHANUM+1;i++)\n if (waloff[i] == 0)\n tloadtile(i);\n\n for(i=FOOTPRINTS;i<FOOTPRINTS+3;i++)\n if (waloff[i] == 0)\n tloadtile(i);\n\n for( i = BIGALPHANUM; i < BIGALPHANUM+82; i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = BURNING; i < BURNING+14; i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = BURNING2; i < BURNING2+14; i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = CRACKKNUCKLES; i < CRACKKNUCKLES+4; i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = FIRSTGUN; i < FIRSTGUN+3 ; i++ )\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = EXPLOSION2; i < EXPLOSION2+21 ; i++ )\n if(waloff[i] == 0)\n tloadtile(i);\n\n tloadtile(BULLETHOLE);\n\n for( i = FIRSTGUNRELOAD; i < FIRSTGUNRELOAD+8 ; i++ )\n if(waloff[i] == 0)\n tloadtile(i);\n\n tloadtile(FOOTPRINTS);\n\n for( i = JIBS1; i < (JIBS5+5); i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = SCRAP1; i < (SCRAP1+19); i++)\n if(waloff[i] == 0)\n tloadtile(i);\n\n for( i = SMALLSMOKE; i < (SMALLSMOKE+4); i++)\n if(waloff[i] == 0)\n tloadtile(i);\n}\n\nchar getsound(unsigned short num)\n{\n short fp;\n long l;\n\n if(num >= NUM_SOUNDS || SoundToggle == 0) return 0;\n if (FXDevice == NumSoundCards) return 0;\n\n fp = kopen4load(sounds[num],loadfromgrouponly);\n if(fp == -1) return 0;\n\n l = kfilelength( fp );\n soundsiz[num] = l;\n\n if( (ud.level_number == 0 && ud.volume_number == 0 && (num == 189 || num == 232 || num == 99 || num == 233 || num == 17 ) ) ||\n ( l < 12288 ) )\n {\n Sound[num].lock = 2;\n allocache((long *)&Sound[num].ptr,l,&Sound[num].lock);\n if(Sound[num].ptr !", "= NULL)\n kread( fp, Sound[num].ptr , l);\n }\n kclose( fp );\n return 1;\n}\n\nvoid precachenecessarysounds(void)\n{\n short i, j;\n\n if (FXDevice == NumSoundCards) return;\n j = 0;\n\n for(i=0;i<NUM_SOUNDS;i++)\n if(Sound[i].ptr == 0)\n {\n j++;\n if( (j&7) == 0 )\n getpackets();\n getsound(i);\n }\n}\n\n\nvoid cacheit(void)\n{\n short i,j;\n\n precachenecessarysounds();\n\n cachegoodsprites();\n\n for(i=0;i<numwalls;i++)\n if( waloff[wall[i].picnum] == 0 )\n {\n if(waloff[wall[i].picnum] == 0)\n tloadtile(wall[i].picnum);\n if(wall[i].overpicnum >= 0 && waloff[wall[i].overpicnum] == 0 )\n tloadtile(wall[i].overpicnum);\n }\n\n for(i=0;i<numsectors;i++)\n {\n if( waloff[sector[i].floorpicnum] == 0 )\n tloadtile( sector[i].floorpicnum );\n if( waloff[sector[i].ceilingpicnum] == 0 )\n {\n tloadtile( sector[i].ceilingpicnum );\n if( waloff[sector[i].ceilingpicnum] == LA)\n {\n tloadtile(LA+1);\n tloadtile(LA+2);\n }\n }\n\n j = headspritesect[i];\n while(j >= 0)\n {\n if(sprite[j].xrepeat !", "= 0 && sprite[j].yrepeat !", "= 0 && (sprite[j].cstat&32768) == 0)\n if(waloff[sprite[j].picnum] == 0)\n cachespritenum(j);\n j = nextspritesect[j];\n }\n }\n\n}\n\nvoid docacheit(void)\n{\n long i,j;\n\n j = 0;\n\n for(i=0;i<MAXTILES;i++)\n if( (gotpic[i>>3]&(1<<(i&7))) && waloff[i] == 0)\n {\n loadtile((short)i);\n j++;\n if((j&7) == 0) getpackets();\n }\n\n clearbufbyte(gotpic,sizeof(gotpic),0L);\n\n}\n\n\n\nvoid xyzmirror(short i,short wn)\n{\n if (waloff[wn] == 0) loadtile(wn);\n\tsetviewtotile(wn,tilesizy[wn],tilesizx[wn]);\n\n\tdrawrooms(SX,SY,SZ,SA,100+sprite[i].shade,SECT);\n\tdisplay_mirror = 1; animatesprites(SX,SY,SA,65536L); display_mirror = 0;\n\tdrawmasks();\n\n\tsetviewback();\n\tsquarerotatetile(wn);\n}\n\nvoid vscrn(void)\n{\n long i, j, ss, x1, x2, y1, y2;\n\n\t if(ud.screen_size < 0) ud.screen_size = 0;\n\t else if(ud.screen_size > 63) ud.screen_size = 64;\n\n if(ud.screen_size == 0) flushperms();\n\n\t ss = max(ud.screen_size-8,0);\n\n\t x1 = scale(ss,xdim,160);\n\t x2 = xdim-x1;\n\n\t y1 = ss; y2 = 200;\n if ( ud.screen_size > 0 && ud.coop !", "= 1 && ud.multimode > 1)\n\t {\n j = 0;\n for(i=connecthead;i>=0;i=connectpoint2[i])\n if(i > j) j = i;\n\n if (j >= 1) y1 += 8;\n if (j >= 4) y1 += 8;\n if (j >= 8) y1 += 8;\n if (j >= 12) y1 += 8;\n\t }\n\n\t if (ud.screen_size >= 8) y2 -= (ss+34);\n\n\t y1 = scale(y1,ydim,200);\n\t y2 = scale(y2,ydim,200);\n\n\t setview(x1,y1,x2-1,y2-1);\n\n pub = NUMPAGES;\n pus = NUMPAGES;\n}\n\nvoid pickrandomspot(short snum)\n{\n struct player_struct *p;\n short i;\n\n p = &ps[snum];\n\n if( ud.multimode > 1 && ud.coop !", "= 1)\n i = TRAND%numplayersprites;\n else i = snum;\n\n p->bobposx = p->oposx = p->posx = po[i].ox;\n p->bobposy = p->oposy = p->posy = po[i].oy;\n p->oposz = p->posz = po[i].oz;\n p->ang = po[i].oa;\n p->cursectnum = po[i].os;\n}\n\nvoid resetplayerstats(short snum)\n{\n struct player_struct *p;\n short i;\n\n p = &ps[snum];\n\n ud.show_help = 0;\n ud.showallmap = 0;\n p->dead_flag = 0;\n p->wackedbyactor = -1;\n p->falling_counter = 0;\n p->quick_kick = 0;\n p->subweapon = 0;\n p->last_full_weapon = 0;\n p->ftq = 0;\n p->fta = 0;\n p->tipincs = 0;\n p->buttonpalette = 0;\n p->actorsqu =-1;\n p->invdisptime = 0;\n p->refresh_inventory= 0;\n p->last_pissed_time = 0;\n p->holster_weapon = 0;\n p->pycount = 0;\n p->pyoff = 0;\n p->opyoff = 0;\n p->loogcnt = 0;\n p->angvel = 0;\n p->weapon_sway = 0;\n// p->select_dir = 0;\n p->extra_extra8 = 0;\n p->show_empty_weapon= 0;\n p->dummyplayersprite=-1;\n p->crack_time = 0;\n p->hbomb_hold_delay = 0;\n p->transporter_hold = 0;\n p->wantweaponfire = -1;\n p->hurt_delay = 0;\n p->footprintcount = 0;\n p->footprintpal = 0;\n p->footprintshade = 0;\n p->jumping_toggle = 0;\n p->ohoriz = p->horiz= 140;\n p->horizoff = 0;\n p->bobcounter = 0;\n p->on_ground = 0;\n p->player_par = 0;\n p->return_to_center = 9;\n p->airleft = 15*26;\n p->rapid_fire_hold = 0;\n p->toggle_key_flag = 0;\n p->access_spritenum = -1;\n if(ud.multimode > 1 && ud.coop !", "= 1 )\n p->got_access = 7;\n else p->got_access = 0;\n p->random_club_frame= 0;\n pus = 1;\n p->on_warping_sector = 0;\n p->spritebridge = 0;\n p->palette = (char *) &palette[0];\n\n if(p->steroids_amount < 400 )\n {\n p->steroids_amount = 0;\n p->inven_icon = 0;\n }\n p->heat_on = 0;\n p->jetpack_on = 0;\n p->holoduke_on = -1;\n\n p->look_ang = 512 - ((ud.level_number&1)<<10);\n\n p->rotscrnang = 0;\n p->newowner =-1;\n p->jumping_counter = 0;\n p->hard_landing = 0;\n p->posxv = 0;\n p->posyv = 0;\n p->poszv = 0;\n fricxv = 0;\n fricyv = 0;\n p->somethingonplayer =-1;\n p->one_eighty_count = 0;\n p->cheat_phase = 0;\n\n p->on_crane = -1;\n\n if(p->curr_weapon == PISTOL_WEAPON)\n p->kickback_pic = 5;\n else p->kickback_pic = 0;\n\n p->weapon_pos = 6;\n p->walking_snd_toggle= 0;\n p->weapon_ang = 0;\n\n p->knuckle_incs = 1;\n p->fist_incs = 0;\n p->knee_incs = 0;\n p->jetpack_on = 0;\n setpal(p);\n}\n\n\n\nvoid resetweapons(short snum)\n{\n short weapon;\n struct player_struct *p;\n\n p = &ps[snum];\n\n for ( weapon = PISTOL_WEAPON; weapon < MAX_WEAPONS; weapon++ )\n p->gotweapon[weapon] = 0;\n for ( weapon = PISTOL_WEAPON; weapon < MAX_WEAPONS; weapon++ )\n p->ammo_amount[weapon] = 0;\n\n p->weapon_pos = 6;\n p->kickback_pic = 5;\n p->curr_weapon = PISTOL_WEAPON;\n p->gotweapon[PISTOL_WEAPON] = 1;\n p->gotweapon[KNEE_WEAPON] = 1;\n p->ammo_amount[PISTOL_WEAPON] = 48;\n p->gotweapon[HANDREMOTE_WEAPON] = 1;\n p->last_weapon = -1;\n\n p->show_empty_weapon= 0;\n p->last_pissed_time = 0;\n p->holster_weapon = 0;\n}\n\nvoid resetinventory(short snum)\n{\n struct player_struct *p;\n short i;\n\n p = &ps[snum];\n\n p->inven_icon = 0;\n p->boot_amount = 0;\n p->scuba_on = 0;p->scuba_amount = 0;\n p->heat_amount = 0;p->heat_on = 0;\n p->jetpack_on = 0;p->jetpack_amount = 0;\n p->shield_amount = max_armour_amount;\n p->holoduke_on = -1;\n p->holoduke_amount = 0;\n p->firstaid_amount = 0;\n p->steroids_amount = 0;\n p->inven_icon = 0;\n}\n\n\nvoid resetprestat(short snum,char g)\n{\n struct player_struct *p;\n short i;\n\n p = &ps[snum];\n\n spriteqloc = 0;\n for(i=0;i<spriteqamount;i++) spriteq[i] = -1;\n\n p->hbomb_on = 0;\n p->cheat_phase = 0;\n p->pals_time = 0;\n p->toggle_key_flag = 0;\n p->secret_rooms = 0;\n p->max_secret_rooms = 0;\n p->actors_killed = 0;\n p->max_actors_killed = 0;\n p->lastrandomspot = 0;\n p->weapon_pos = 6;\n p->kickback_pic = 5;\n p->last_weapon = -1;\n p->weapreccnt = 0;\n p->show_empty_weapon= 0;\n p->holster_weapon = 0;\n p->last_pissed_time = 0;\n\n p->one_parallax_sectnum = -1;\n p->visibility = ud.const_visibility;\n\n screenpeek = myconnectindex;\n numanimwalls = 0;\n numcyclers = 0;\n animatecnt = 0;\n parallaxtype = 0;\n randomseed = 17L;\n ud.pause_on = 0;\n ud.camerasprite =-1;\n ud.eog = 0;\n tempwallptr = 0;\n camsprite =-1;\n earthquaketime = 0;\n\n numinterpolations = 0;\n startofdynamicinterpolations = 0;\n\n if( ( (g&MODE_EOL) !", "= MODE_EOL && numplayers < 2) || (ud.coop !", "= 1 && numplayers > 1) )\n {\n resetweapons(snum);\n resetinventory(snum);\n }\n else if(p->curr_weapon == HANDREMOTE_WEAPON)\n {\n p->ammo_amount[HANDBOMB_WEAPON]++;\n p->curr_weapon = HANDBOMB_WEAPON;\n }\n\n p->timebeforeexit = 0;\n p->customexitsound = 0;\n\n}\n\nvoid setupbackdrop(short sky)\n{\n short i;\n\n for(i=0;i<MAXPSKYTILES;i++) pskyoff[i]=0;\n\n if(parallaxyscale !", "= 65536L)\n parallaxyscale = 32768;\n\n switch(sky)\n {\n case CLOUDYOCEAN:\n parallaxyscale = 65536L;\n break;\n case MOONSKY1 :\n pskyoff[6]=1; pskyoff[1]=2; pskyoff[4]=2; pskyoff[2]=3;\n break;\n case BIGORBIT1: // orbit\n pskyoff[5]=1; pskyoff[6]=2; pskyoff[7]=3; pskyoff[2]=4;\n break;\n case LA:\n parallaxyscale = 16384+1024;\n pskyoff[0]=1; pskyoff[1]=2; pskyoff[2]=1; pskyoff[3]=3;\n pskyoff[4]=4; pskyoff[5]=0; pskyoff[6]=2; pskyoff[7]=3;\n break;\n }\n\n pskybits=3;\n}\n\nvoid prelevel(char g)\n{\n short i, nexti, j, startwall, endwall, lotaglist;\n short lotags[65];\n\n\n clearbufbyte(show2dsector,sizeof(show2dsector),0L);\n clearbufbyte(show2dwall,sizeof(show2dwall),0L);\n clearbufbyte(show2dsprite,sizeof(show2dsprite),0L);\n\n resetprestat(0,g);\n numclouds = 0;\n\n for(i=0;i<numsectors;i++)\n {\n sector[i].extra = 256;\n\n switch(sector[i].lotag)\n {\n case 20:\n case 22:\n if( sector[i].floorz > sector[i].ceilingz)\n sector[i].lotag |= 32768;\n continue;\n }\n\n if(sector[i].ceilingstat&1)\n {\n if(waloff[sector[i].ceilingpicnum] == 0)\n {\n if(sector[i].ceilingpicnum == LA)\n for(j=0;j<5;j++)\n if(waloff[sector[i].ceilingpicnum+j] == 0)\n tloadtile(sector[i].ceilingpicnum+j);\n }\n setupbackdrop(sector[i].ceilingpicnum);\n\n if(sector[i].ceilingpicnum == CLOUDYSKIES && numclouds < 127)\n clouds[numclouds++] = i;\n\n if(ps[0].one_parallax_sectnum == -1)\n ps[0].one_parallax_sectnum = i;\n }\n\n if(sector[i].lotag == 32767) //Found a secret room\n {\n ps[0].max_secret_rooms++;\n continue;\n }\n\n if(sector[i].lotag == -1)\n {\n ps[0].exitx = wall[sector[i].wallptr].x;\n ps[0].exity = wall[sector[i].wallptr].y;\n continue;\n }\n }\n\n i = headspritestat[0];\n while(i >= 0)\n {\n nexti = nextspritestat[i];\n\n if(sprite[i].lotag == -1 && (sprite[i].cstat&16) )\n {\n ps[0].exitx = SX;\n ps[0].exity = SY;\n }\n else switch(PN)\n {\n case GPSPEED:\n sector[SECT].extra = SLT;\n deletesprite(i);\n break;\n\n case CYCLER:\n if(numcyclers >= MAXCYCLERS)\n gameexit(\"\\nToo many cycling sectors.\");", "\n cyclers[numcyclers][0] = SECT;\n cyclers[numcyclers][1] = SLT;\n cyclers[numcyclers][2] = SS;\n cyclers[numcyclers][3] = sector[SECT].floorshade;\n cyclers[numcyclers][4] = SHT;\n cyclers[numcyclers][5] = (SA == 1536);\n numcyclers++;\n deletesprite(i);\n break;\n }\n i = nexti;\n }\n\n for(i=0;i < MAXSPRITES;i++)\n {\n if(sprite[i].statnum < MAXSTATUS)\n {\n if(PN == SECTOREFFECTOR && SLT == 14)\n continue;\n spawn(-1,i);\n }\n }\n\n for(i=0;i < MAXSPRITES;i++)\n if(sprite[i].statnum < MAXSTATUS)\n {\n if( PN == SECTOREFFECTOR && SLT == 14 )\n spawn(-1,i);\n }\n\n lotaglist = 0;\n\n i = headspritestat[0];\n while(i >= 0)\n {\n switch(PN)\n {\n case DIPSWITCH:\n case DIPSWITCH2:\n case ACCESSSWITCH:\n case PULLSWITCH:\n case HANDSWITCH:\n case SLOTDOOR:\n case LIGHTSWITCH:\n case SPACELIGHTSWITCH:\n case SPACEDOORSWITCH:\n case FRANKENSTINESWITCH:\n case LIGHTSWITCH2:\n case POWERSWITCH1:\n case LOCKSWITCH1:\n case POWERSWITCH2:\n break;\n case DIPSWITCH+1:\n case DIPSWITCH2+1:\n case PULLSWITCH+1:\n case HANDSWITCH+1:\n case SLOTDOOR+1:\n case LIGHTSWITCH+1:\n case SPACELIGHTSWITCH+1:\n case SPACEDOORSWITCH+1:\n case FRANKENSTINESWITCH+1:\n case LIGHTSWITCH2+1:\n case POWERSWITCH1+1:\n case LOCKSWITCH1+1:\n case POWERSWITCH2+1:\n for(j=0;j<lotaglist;j++)\n if( SLT == lotags[j] )\n break;\n\n if( j == lotaglist )\n {\n lotags[lotaglist] = SLT;\n lotaglist++;\n if(lotaglist > 64)\n gameexit(\"\\nToo many switches (64 max).\");", "\n\n j = headspritestat[3];\n while(j >= 0)\n {\n if(sprite[j].lotag == 12 && sprite[j].hitag == SLT)\n hittype[j].temp_data[0] = 1;\n j = nextspritestat[j];\n }\n }\n break;\n }\n i = nextspritestat[i];\n }\n\n mirrorcnt = 0;\n\n for( i = 0; i < numwalls; i++ )\n {\n walltype *wal;\n wal = &wall[i];\n\n if(wal->overpicnum == MIRROR && (wal->cstat&32) !", "= 0)\n {\n j = wal->nextsector;\n\n if(mirrorcnt > 63)\n gameexit(\"\\nToo many mirrors (64 max.)\");", "\n if ( (j >= 0) && sector[j].ceilingpicnum !", "= MIRROR )\n {\n sector[j].ceilingpicnum = MIRROR;\n sector[j].floorpicnum = MIRROR;\n mirrorwall[mirrorcnt] = i;\n mirrorsector[mirrorcnt] = j;\n mirrorcnt++;\n continue;\n }\n }\n\n if(numanimwalls >= MAXANIMWALLS)\n gameexit(\"\\nToo many 'anim' walls (max 512.)\");", "\n\n animwall[numanimwalls].tag = 0;\n animwall[numanimwalls].wallnum = 0;\n\n switch(wal->overpicnum)\n {\n case FANSHADOW:\n case FANSPRITE:\n wall->cstat |= 65;\n animwall[numanimwalls].wallnum = i;\n numanimwalls++;\n break;\n\n case W_FORCEFIELD:\n if(waloff[W_FORCEFIELD] == 0)\n for(j=0;j<3;j++)\n tloadtile(W_FORCEFIELD+j);\n case W_FORCEFIELD+1:\n case W_FORCEFIELD+2:\n if(wal->shade > 31)\n wal->cstat = 0;\n else wal->cstat |= 85+256;\n\n\n if(wal->lotag && wal->nextwall >= 0)\n wall[wal->nextwall].lotag =\n wal->lotag;\n\n case BIGFORCE:\n\n animwall[numanimwalls].wallnum = i;\n numanimwalls++;\n\n continue;\n }\n\n wal->extra = -1;\n\n switch(wal->picnum)\n {\n case WATERTILE2:\n for(j=0;j<3;j++)\n if(waloff[wal->picnum+j] == 0)\n tloadtile(wal->picnum+j);\n break;\n\n case TECHLIGHT2:\n case TECHLIGHT4:\n if(waloff[wal->picnum] == 0)\n tloadtile(wal->picnum);\n break;\n case W_TECHWALL1:\n case W_TECHWALL2:\n case W_TECHWALL3:\n case W_TECHWALL4:\n animwall[numanimwalls].wallnum = i;\n// animwall[numanimwalls].tag = -1;\n numanimwalls++;\n break;\n case SCREENBREAK6:\n case SCREENBREAK7:\n case SCREENBREAK8:\n if(waloff[SCREENBREAK6] == 0)\n for(j=SCREENBREAK6;j<SCREENBREAK9;j++)\n tloadtile(j);\n animwall[numanimwalls].wallnum = i;\n animwall[numanimwalls].tag = -1;\n numanimwalls++;\n break;\n\n case FEMPIC1:\n case FEMPIC2:\n case FEMPIC3:\n\n wal->extra = wal->picnum;\n animwall[numanimwalls].tag = -1;\n if(ud.lockout)\n {\n if(wal->picnum == FEMPIC1)\n wal->picnum = BLANKSCREEN;\n else wal->picnum = SCREENBREAK6;\n }\n\n animwall[numanimwalls].wallnum = i;\n animwall[numanimwalls].tag = wal->picnum;\n numanimwalls++;\n break;\n\n case SCREENBREAK1:\n case SCREENBREAK2:\n case SCREENBREAK3:\n case SCREENBREAK4:\n case SCREENBREAK5:\n\n case SCREENBREAK9:\n case SCREENBREAK10:\n case SCREENBREAK11:\n case SCREENBREAK12:\n case SCREENBREAK13:\n case SCREENBREAK14:\n case SCREENBREAK15:\n case SCREENBREAK16:\n case SCREENBREAK17:\n case SCREENBREAK18:\n case SCREENBREAK19:\n\n animwall[numanimwalls].wallnum = i;\n animwall[numanimwalls].tag = wal->picnum;\n numanimwalls++;\n break;\n }\n }\n\n //Invalidate textures in sector behind mirror\n for(i=0;i<mirrorcnt;i++)\n {\n startwall = sector[mirrorsector[i]].wallptr;\n endwall = startwall + sector[mirrorsector[i]].wallnum;\n for(j=startwall;j<endwall;j++)\n {\n wall[j].picnum = MIRROR;\n wall[j].overpicnum = MIRROR;\n }\n }\n}\n\n\nvoid newgame(char vn,char ln,char sk)\n{\n struct player_struct *p = &ps[0];\n short i;\n\n if(globalskillsound >= 0)\n while(Sound[globalskillsound].lock>=200);\n globalskillsound = -1;\n\n waitforeverybody();\n ready2send = 0;\n\n if( ud.m_recstat !", "= 2 && ud.last_level >= 0 && ud.multimode > 1 && ud.coop !", "= 1)\n dobonus(1);\n\n if( ln == 0 && vn == 3 && ud.multimode < 2 && ud.lockout == 0)\n {\n playmusic(&env_music_fn[1][0]);\n\n flushperms();\n setview(0,0,xdim-1,ydim-1);\n clearview(0L);\n nextpage();\n\n playanm(\"vol41a.anm\",6);\n clearview(0L);\n nextpage();\n playanm(\"vol42a.anm\",7);\n// clearview(0L);\n // nextpage();\n playanm(\"vol43a.anm\",9);\n clearview(0L);\n nextpage();\n\n FX_StopAllSounds();\n }\n\n show_shareware = 26*34;\n\n ud.level_number = ln;\n ud.volume_number = vn;\n ud.player_skill = sk;\n ud.secretlevel = 0;\n ud.from_bonus = 0;\n parallaxyscale = 0;\n\n ud.last_level = -1;\n lastsavedpos = -1;\n p->zoom = 768;\n p->gm = 0;\n\n if(ud.m_coop !", "= 1)\n {\n p->curr_weapon = PISTOL_WEAPON;\n p->gotweapon[PISTOL_WEAPON] = 1;\n p->gotweapon[KNEE_WEAPON] = 1;\n p->ammo_amount[PISTOL_WEAPON] = 48;\n p->gotweapon[HANDREMOTE_WEAPON] = 1;\n p->last_weapon = -1;\n }\n\n display_mirror = 0;\n\n if(ud.multimode > 1 )\n {\n if(numplayers < 2)\n {\n connecthead = 0;\n for(i=0;i<MAXPLAYERS;i++) connectpoint2[i] = i+1;\n connectpoint2[ud.multimode-1] = -1;\n }\n }\n else\n {\n connecthead = 0;\n connectpoint2[0] = -1;\n }\n}\n\n\nvoid resetpspritevars(char g)\n{\n short i, j, nexti,circ;\n long firstx,firsty;\n spritetype *s;\n char aimmode[MAXPLAYERS];\n STATUSBARTYPE tsbar[MAXPLAYERS];\n\n EGS(ps[0].cursectnum,ps[0].posx,ps[0].posy,ps[0].posz,\n APLAYER,0,0,0,ps[0].ang,0,0,0,10);\n\n if(ud.recstat !", "= 2) for(i=0;i<MAXPLAYERS;i++)\n {\n aimmode[i] = ps[i].aim_mode;\n if(ud.multimode > 1 && ud.coop == 1 && ud.last_level >= 0)\n {\n for(j=0;j<MAX_WEAPONS;j++)\n {\n tsbar[i].ammo_amount[j] = ps[i].ammo_amount[j];\n tsbar[i].gotweapon[j] = ps[i].gotweapon[j];\n }\n\n tsbar[i].shield_amount = ps[i].shield_amount;\n tsbar[i].curr_weapon = ps[i].curr_weapon;\n tsbar[i].inven_icon = ps[i].inven_icon;\n\n tsbar[i].firstaid_amount = ps[i].firstaid_amount;\n tsbar[i].steroids_amount = ps[i].steroids_amount;\n tsbar[i].holoduke_amount = ps[i].holoduke_amount;\n tsbar[i].jetpack_amount = ps[i].jetpack_amount;\n tsbar[i].heat_amount = ps[i].heat_amount;\n tsbar[i].scuba_amount = ps[i].scuba_amount;\n tsbar[i].boot_amount = ps[i].boot_amount;\n }\n }\n\n resetplayerstats(0);\n\n for(i=1;i<MAXPLAYERS;i++)\n memcpy(&ps[i],&ps[0],sizeof(ps[0]));\n\n if(ud.recstat !", "= 2) for(i=0;i<MAXPLAYERS;i++)\n {\n ps[i].aim_mode = aimmode[i];\n if(ud.multimode > 1 && ud.coop == 1 && ud.last_level >= 0)\n {\n for(j=0;j<MAX_WEAPONS;j++)\n {\n ps[i].ammo_amount[j] = tsbar[i].ammo_amount[j];\n ps[i].gotweapon[j] = tsbar[i].gotweapon[j];\n }\n ps[i].shield_amount = tsbar[i].shield_amount;\n ps[i].curr_weapon = tsbar[i].curr_weapon;\n ps[i].inven_icon = tsbar[i].inven_icon;\n\n ps[i].firstaid_amount = tsbar[i].firstaid_amount;\n ps[i].steroids_amount= tsbar[i].steroids_amount;\n ps[i].holoduke_amount = tsbar[i].holoduke_amount;\n ps[i].jetpack_amount = tsbar[i].jetpack_amount;\n ps[i].heat_amount = tsbar[i].heat_amount;\n ps[i].scuba_amount= tsbar[i].scuba_amount;\n ps[i].boot_amount = tsbar[i].boot_amount;\n }\n }\n\n numplayersprites = 0;\n circ = 2048/ud.multimode;\n\n which_palookup = 9;\n j = connecthead;\n i = headspritestat[10];\n while(i >= 0)\n {\n nexti = nextspritestat[i];\n s = &sprite[i];\n\n if( numplayersprites == MAXPLAYERS)\n gameexit(\"\\nToo many player sprites (max 16.)\");", "\n\n if(numplayersprites == 0)\n {\n firstx = ps[0].posx;\n firsty = ps[0].posy;\n }\n\n po[numplayersprites].ox = s->x;\n po[numplayersprites].oy = s->y;\n po[numplayersprites].oz = s->z;\n po[numplayersprites].oa = s->ang;\n po[numplayersprites].os = s->sectnum;\n\n numplayersprites++;\n if(j >= 0)\n {\n s->owner = i;\n s->shade = 0;\n s->xrepeat = 42;\n s->yrepeat = 36;\n s->cstat = 1+256;\n s->xoffset = 0;\n s->clipdist = 64;\n\n if( (g&MODE_EOL) !", "= MODE_EOL || ps[j].last_extra == 0)\n {\n ps[j].last_extra = max_player_health;\n s->extra = max_player_health;\n }\n else s->extra = ps[j].last_extra;\n\n s->yvel = j;\n\n if(s->pal == 0)\n {\n s->pal = ps[j].palookup = which_palookup;\n which_palookup++;\n if( which_palookup >= 17 ) which_palookup = 9;\n }\n else ps[j].palookup = s->pal;\n\n ps[j].i = i;\n ps[j].frag_ps = j;\n hittype[i].owner = i;\n\n hittype[i].bposx = ps[j].bobposx = ps[j].oposx = ps[j].posx = s->x;\n hittype[i].bposy = ps[j].bobposy = ps[j].oposy = ps[j].posy = s->y;\n hittype[i].bposz = ps[j].oposz = ps[j].posz = s->z;\n ps[j].oang = ps[j].ang = s->ang;\n\n updatesector(s->x,s->y,&ps[j].cursectnum);\n\n j = connectpoint2[j];\n\n }\n else deletesprite(i);\n i = nexti;\n }\n}\n\nvoid clearfrags(void)\n{\n short i;\n\n for(i = 0;i<MAXPLAYERS;i++)\n ps[i].frag = ps[i].fraggedself = 0;\n clearbufbyte(&frags[0][0],(MAXPLAYERS*MAXPLAYERS)<<1,0L);\n}\n\nvoid resettimevars(void)\n{\n vel = svel = angvel = horiz = 0;\n\n totalclock = 0L;\n cloudtotalclock = 0L;\n ototalclock = 0L;\n lockclock = 0L;\n ready2send = 1;\n}\n\n\nvoid genspriteremaps(void)\n{\n long j,fp;\n signed char look_pos;\n char *lookfn = \"lookup.dat\";\n char numl;\n\n fp = kopen4load(lookfn,0);\n if(fp !", "= -1)\n kread(fp,(char *)&numl,1);\n else\n gameexit(\"\\nERROR: File 'LOOKUP.DAT' not found.\");", "\n\n for(j=0;j < numl;j++)\n {\n kread(fp,(signed char *)&look_pos,1);\n kread(fp,tempbuf,256);\n makepalookup((long)look_pos,tempbuf,0,0,0,1);\n }\n\n kread(fp,&waterpal[0],768);\n kread(fp,&slimepal[0],768);\n kread(fp,&titlepal[0],768);\n kread(fp,&drealms[0],768);\n kread(fp,&endingpal[0],768);\n\n palette[765] = palette[766] = palette[767] = 0;\n slimepal[765] = slimepal[766] = slimepal[767] = 0;\n waterpal[765] = waterpal[766] = waterpal[767] = 0;\n\n kclose(fp);\n}\n\nvoid waitforeverybody()\n{\n long i;\n\n if (numplayers < 2) return;\n packbuf[0] = 250;\n for(i=connecthead;i>=0;i=connectpoint2[i])\n if (i !", "= myconnectindex)\n sendpacket(i,packbuf,1);\n\n playerreadyflag[myconnectindex]++;\n do\n {\n getpackets();\n for(i=connecthead;i>=0;i=connectpoint2[i])\n if (playerreadyflag[i] < playerreadyflag[myconnectindex]) break;\n } while (i >= 0);\n}\n\nvoid dofrontscreens(void)\n{\n long tincs,i,j;\n\n if(ud.recstat !", "= 2)\n {\n ps[myconnectindex].palette = palette;\n for(j=0;j<63;j+=7) palto(0,0,0,j);\n i = ud.screen_size;\n ud.screen_size = 0;\n vscrn();\n clearview(0L);\n\n rotatesprite(320<<15,200<<15,65536L,0,LOADSCREEN,0,0,2+8+64,0,0,xdim-1,ydim-1);\n\n if( boardfilename[0] !", "= 0 && ud.level_number == 7 && ud.volume_number == 0 )\n {\n menutext(160,90,0,0,\"ENTERING USER MAP\");\n gametextpal(160,90+10,boardfilename,14,2);\n }\n else\n {\n menutext(160,90,0,0,\"ENTERING\");\n menutext(160,90+16+8,0,0,level_names[(ud.volume_number*11) + ud.level_number]);\n }\n\n nextpage();\n\n for(j=63;j>0;j-=7) palto(0,0,0,j);\n\n KB_FlushKeyboardQueue();\n ud.screen_size = i;\n }\n else\n {\n clearview(0L);\n ps[myconnectindex].palette = palette;\n palto(0,0,0,0);\n rotatesprite(320<<15,200<<15,65536L,0,LOADSCREEN,0,0,2+8+64,0,0,xdim-1,ydim-1);\n menutext(160,105,0,0,\"LOADING...\");\n nextpage();\n }\n}\n\nvoid clearfifo(void)\n{\n syncvaltail = 0L;\n syncvaltottail = 0L;\n syncstat = 0;\n bufferjitter = 1;\n mymaxlag = otherminlag = 0;\n\n movefifoplc = movefifosendplc = fakemovefifoplc = 0;\n avgfvel = avgsvel = avgavel = avghorz = avgbits = 0;\n otherminlag = mymaxlag = 0;\n\n clearbufbyte(myminlag,MAXPLAYERS<<2,0L);\n clearbufbyte(&loc,sizeof(input),0L);\n clearbufbyte(&sync[0],sizeof(sync),0L);\n clearbufbyte(inputfifo,sizeof(input)*MOVEFIFOSIZ*MAXPLAYERS,0L);\n\n clearbuf(movefifoend,MAXPLAYERS,0L);\n clearbuf(syncvalhead,MAXPLAYERS,0L);\n clearbuf(myminlag,MAXPLAYERS,0L);\n\n// clearbufbyte(playerquitflag,MAXPLAYERS,0x01);\n}\n\nvoid resetmys(void)\n{\n myx = omyx = ps[myconnectindex].posx;\n myy = omyy = ps[myconnectindex].posy;\n myz = omyz = ps[myconnectindex].posz;\n myxvel = myyvel = myzvel = 0;\n myang = omyang = ps[myconnectindex].ang;\n myhoriz = omyhoriz = ps[myconnectindex].horiz;\n myhorizoff = omyhorizoff = ps[myconnectindex].horizoff;\n mycursectnum = ps[myconnectindex].cursectnum;\n myjumpingcounter = ps[myconnectindex].jumping_counter;\n myjumpingtoggle = ps[myconnectindex].jumping_toggle;\n myonground = ps[myconnectindex].on_ground;\n myhardlanding = ps[myconnectindex].hard_landing;\n myreturntocenter = ps[myconnectindex].return_to_center;\n}\n\nvoid enterlevel(char g)\n{\n short i,j;\n long l;\n char levname[256];\n\n if( (g&MODE_DEMO) !", "= MODE_DEMO ) ud.recstat = ud.m_recstat;\n ud.respawn_monsters = ud.m_respawn_monsters;\n ud.respawn_items = ud.m_respawn_items;\n ud.respawn_inventory = ud.m_respawn_inventory;\n ud.monsters_off = ud.m_monsters_off;\n ud.coop = ud.m_coop;\n ud.marker = ud.m_marker;\n ud.ffire = ud.m_ffire;\n\n if( (g&MODE_DEMO) == 0 && ud.recstat == 2)\n ud.recstat = 0;\n\n FX_StopAllSounds();\n clearsoundlocks();\n FX_SetReverb(0);\n\n i = ud.screen_size;\n ud.screen_size = 0;\n dofrontscreens();\n vscrn();\n ud.screen_size = i;\n\n#ifndef VOLUMEONE\n\n if( boardfilename[0] !", "= 0 && ud.m_level_number == 7 && ud.m_volume_number == 0 )\n {\n if ( loadboard( boardfilename,&ps[0].posx, &ps[0].posy, &ps[0].posz, &ps[0].ang,&ps[0].cursectnum ) == -1 )\n {\n sprintf(tempbuf,\"Map %s not found!\",boardfilename);\n gameexit(tempbuf);\n }\n }\n else if ( loadboard( level_file_names[ (ud.volume_number*11)+ud.level_number],&ps[0].posx, &ps[0].posy, &ps[0].posz, &ps[0].ang,&ps[0].cursectnum ) == -1)\n {\n sprintf(tempbuf,\"Map %s not found!\",level_file_names[(ud.volume_number*11)+ud.level_number]);\n gameexit(tempbuf);\n }\n\n#else\n\n l = strlen(level_file_names[ (ud.volume_number*11)+ud.level_number]);\n copybufbyte( level_file_names[ (ud.volume_number*11)+ud.level_number],&levname[0],l);\n levname[l] = 255;\n levname[l+1] = 0;\n\n if ( loadboard( levname,&ps[0].posx, &ps[0].posy, &ps[0].posz, &ps[0].ang,&ps[0].cursectnum ) == -1)\n {\n sprintf(tempbuf,\"Map %s not found!\",level_file_names[(ud.volume_number*11)+ud.level_number]);\n gameexit(tempbuf);\n }\n#endif\n\n clearbufbyte(gotpic,sizeof(gotpic),0L);\n\n prelevel(g);\n\n allignwarpelevators();\n resetpspritevars(g);\n\n cachedebug = 0;\n automapping = 0;\n\n if(ud.recstat !", "= 2) MUSIC_StopSong();\n\n cacheit();\n docacheit();\n\n if(ud.recstat !", "= 2)\n {\n music_select = (ud.volume_number*11) + ud.level_number;\n playmusic(&music_fn[0][music_select][0]);\n }\n\n if( (g&MODE_GAME) || (g&MODE_EOL) )\n ps[myconnectindex].gm = MODE_GAME;\n else if(g&MODE_RESTART)\n {\n if(ud.recstat == 2)\n ps[myconnectindex].gm = MODE_DEMO;\n else ps[myconnectindex].gm = MODE_GAME;\n }\n\n if( (ud.recstat == 1) && (g&MODE_RESTART) !", "= MODE_RESTART )\n opendemowrite();\n\n#ifdef VOLUMEONE\n if(ud.level_number == 0 && ud.recstat !", "= 2) FTA(40,&ps[myconnectindex]);\n#endif\n\n for(i=connecthead;i>=0;i=connectpoint2[i])\n switch(sector[sprite[ps[i].i].sectnum].floorpicnum)\n {\n case HURTRAIL:\n case FLOORSLIME:\n case FLOORPLASMA:\n resetweapons(i);\n resetinventory(i);\n ps[i].gotweapon[PISTOL_WEAPON] = 0;\n ps[i].ammo_amount[PISTOL_WEAPON] = 0;\n ps[i].curr_weapon = KNEE_WEAPON;\n ps[i].kickback_pic = 0;\n break;\n }\n\n //PREMAP.C - replace near the my's at the end of the file\n\n resetmys();\n\n ps[myconnectindex].palette = palette;\n palto(0,0,0,0);\n\n setpal(&ps[myconnectindex]);\n flushperms();\n\n everyothertime = 0;\n global_random = 0;\n\n ud.last_level = ud.level_number+1;\n\n clearfifo();\n\n for(i=numinterpolations-1;i>=0;i--) bakipos[i] = *curipos[i];\n\n restorepalette = 1;\n\n flushpackets();\n waitforeverybody();\n\n palto(0,0,0,0);\n vscrn();\n clearview(0L);\n drawbackground();\n\n clearbufbyte(playerquitflag,MAXPLAYERS,0x01010101);\n ps[myconnectindex].over_shoulder_on = 0;\n\n clearfrags();\n\n resettimevars(); // Here we go\n\n\n}\n\n/*\nDuke Nukem V\n\nLayout:\n\n Settings:\n Suburbs\n Duke inflitrating neighborhoods inf. ", "by aliens\n Death Valley:\n Sorta like a western. ", " Bull-skulls halb buried in the sand\n Military compound: Aliens take over nuke-missle silo, duke\n must destroy.", "\n Abondend Aircraft field\n Vegas:\n Blast anything bright! ", " Alien lights camoflauged.", "\n Alien Drug factory. ", "The Blue Liquid\n Mountainal Cave:\n Interior cave battles.", "\n Jungle:\n Trees, canopee, animals, a mysterious hole in the earth\n Penetencury:\n Good use of spotlights:\n Inventory:\n Wood,\n Metal,\n Torch,\n Rope,\n Plastique,\n Cloth,\n Wiring,\n Glue,\n Cigars,\n Food,\n Duck Tape,\n Nails,\n Piping,\n Petrol,\n Uranium,\n Gold,\n Prism,\n Power Cell,\n\n Hand spikes (Limited usage, they become dull)\n Oxygent (Oxygen mixed with stimulant)\n\n\n Player Skills:\n R-Left,R-Right,Foward,Back\n Strafe, Jump, Double Flip Jump for distance\n Help, Escape\n Fire/Use\n Use Menu\n\nProgramming:\n Images: Polys\n Actors:\n Multi-Object sections for change (head,arms,legs,torsoe,all change)\n Facial expressions. ", " Pal lookup per poly?", "\n\n struct imagetype\n {\n int *itable; // AngX,AngY,AngZ,Xoff,Yoff,Zoff;\n int *idata;\n struct imagetype *prev, *next;\n }\n\n*/\n" ]

{ "pile_set_name": "Github" }

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[ "Something which I have always found to be interesting about wrestlers is the longevity which their art can offer their mixed martial arts careers. ", "Of course there are plenty of examples of great wrestlers who have gone on to become train wrecks, such as Mark Kerr, but the more thought you give it the more you realize that it is the men who can control the action and avoid punishment with their clinches who are able to keep their health throughout lengthier careers than their peers.", "\n\nInspired by the retirement of the great Dan Severn at the ripe old age of 54, today I am going to be focusing (rather unusually) on wrestlers. ", "I should clarify that by \"wrestler\" I don't mean the Kevin Randleman breed of wrestler. ", "Randleman had a vertical jump which looked as if it could be used to escape the octagon, and could shoot a successful double from half way across PRIDE's thirty foot ring but he soon faded. ", "Instead I should substitute the word \"clinchers\" - for he who controls the clinch controls the match. ", "The man with the better clinching skills and technique in the clinch is able to avoid the punishment that strikes do to a fighter's body and brain.", "\n\nThis rule is universally true in combat sports from MMA to kickboxing to boxing. ", "Even in sports in which it is outlawed to clinch - such as karate and K-1 - it is a useful means to escape a bad situation and will likely be tolerated for a while before the referee does anything about it. ", "Take a quick gander at Floyd Mayweather's clinch heavy performance against Ricky Hatton. ", "Hatton made a career of taking punches to give punches because he hit harder than almost anyone in his division. ", "Every time Mayweather hit Hatton coming in, he was immediately chest to chest and the majority of the time had a good underhook and a good overhook - the classical tie up.", "\n\nFloyd Mayweather vs. Ricky Hatton (Highlights) (via Erraticsboxing)\n\nAs I said he who controls the clinch controls the fight and Mayweather uses it to pot shot his opponents before immediately getting chest to chest and preventing them from hitting him back. ", "Once the opponent is happy to clinch back, however, Mayweather begins his typical elbow tactics - cross facing his opponent and landing free shots as their hands are in a position of trying to clinch, rather than trying to box. ", "Notice how many times Floyd hits his opponents while he is doing more to wrestle them than boxing.", "\n\nThe first major exponent of Greco-Roman wrestling in MMA, Randy Couture remains one of the few fighters to make full use of opportunities to clinch and the damage which can be done to the opponent from there. ", "Bulldozing in worked on the casual martial artists who were still present in the earlier UFC events - but against Vitor Belfort, Couture showed that the clinch did not have to always be entered offensively - as is the method in wrestling - but could be entered in defence as it is in boxing but as a means to the wrestler's ends.", "\n\nHere Couture attacks with a back-handed jab which had little chance of landing or causing any damage - but he immediately moves away from the counter which Belfort swings at him.", "\n\nAs Belfort chases Couture back with strikes (as is his modus operandi) Couture stops retreating and smashes in chest to chest with Belfort, securing an underhook on Belfort's punching left hand.", "\n\nThe genius of Couture's approach was that when a fighter is on the defensive his elbows are tight to his body and there are few good handles to grab a hold of. ", "If an opponent holds this posture Couture can struggle to get a good clinch on them. ", "Against Brandon Vera, Couture kept taking a grip on Vera's neck and being shucked off as he tried to move Vera. ", "If the opponent's elbows can be lured away from his body, Couture can move in and grab at least one underhook with ease.", "\n\nHere Couture attempts to bait Belfort with a jab again but Belfort throws a left straight without committing and Couture leaves it at that.", "\n\nWhere a good boxer would slip and look to counter punch - Couture's counter is the clinch, where he can tire his opponents out, move them around and throw them. ", "Here's another example of Belfort attempting to punch Couture, then Randy leaning back and parrying before getting the underhooks on Belfort's swinging arms.", "\n\n\n\n\n\nCouture got hit a few times in his fights with Belfort because his method in the first was far from a science, more a strategy which he was developing. ", "Later in Couture's career, however, he began to jab into the clinch - but not in the way that most remember when they think of jabbing into the clinch. ", "In a recent piece I observed that UFC heavyweight champion, Cain Velasquez often jabs his way in with his head bolt upright and can be force fed straights while doing so - as Cheick Kongo managed to and as an exhausted and flustered Junior dos Santos attempted to do.", "\n\nCouture's method of moving to the clinch has evolved to be based around a dipping jab - taking his head offline to his right and inviting counter punches. ", "Notice that when Randy jabs he dips his head very low to his right side - ducking it down behind his left shoulder. ", "This sort of position invites counter punches but provides much more protection than his leaning back against Belfort did. ", "As soon as the opponent opens up with punches, Couture is chest to chest with them and they have given him an underhook.", "\n\nOne advantage of constantly using this dipping jab - other than avoiding most counter punches - is that Couture can invite counter punches with the opponents right hand which he can simply move inside of and underhook. ", "Against Gabriel Gonzaga, Couture began using his dipping jab early and hit Gonzaga hard while the latter was moving in.", "\n\nLater in the same fight, Gonzaga threw back a right hand in response to Couture's jab and Couture was able to underhook it and get a good punch off with his right hand. ", "Couture used the inside slip to get close enough to grab a hold of Tim Sylvia numerous times in their bout.", "\n\nA final advantage of Randy Couture's dipping jab over the traditional jab and run in to clinch method is that Couture dipped so low with his jab that it could be easily turned into a duck to get control of his opponent's hips (something which Floyd Mayweather even used against Hatton in the highlight mentioned earlier, and which Prince Naseem would do when he was in trouble). ", "After Gonzaga clipped Couture with a hard right hand, he chased Couture with another, Couture was able to slip to the side of and below the punch and get control of Gonzaga's hips - which he soon turned into the first takedown of the fight.", "\n\nRandy Couture was a man of many tricks in the octagon - but it is hard to find a Couture fight in his later career where he doesn't use this slower, dipping jab to draw attacks from his opponent or sneak his head in towards their chest or hips. ", "The techniques needed by a grappler to secure a good clinch needn't be many - Couture used his dipping jab and an occasional overhand, Fedor Emelianenko used his right hand lead almost exclusively - but anything beyond the traditional jab straight in method is a step forward in the evolution of wrestling in MMA, and a massive bolster to the success rate of wrestlers trying to obtain the clinch.", "\n\nEvery wrestler should be looking to simplify and improve his striking to work with his primary skill rather than in place of it when he can't get his techniques working. ", "After all, a many wrestlers never forget their old tricks once they get to the position where they can use them. ", "Here's to ageing gracefully in the brutal world of combat sports.", "\n\nLearn the techniques and strategies of effective striking in Jack Slack's BRAND NEW ebook: Elementary Striking.", "\n\n20 of the world's top strikers from boxing, kickboxing and MMA have their techniques dissected in Jack Slack's first ebook, Advanced Striking.", "\n\nJack can be found on Twitter, Facebook and at his blog; Fights Gone By.", "\n\n\n\n" ]

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[ "Q:\n\nPOST JSON Data from form without AJAX\n\nI am trying to POST data to a REST api without using AJAX. ", " I want to send the data in JSON format. ", " I have the following code but am stuck trying to figure out how to convert the input field and POST it to the server. ", " Here is my code attempt:\n<form id = \"myform\" method = \"post\">\nid: <input type = \"text\" id = \"user_id\" name = \"user_id\"> \ndata: <input type = \"text\" id = \"user_data\" name = \"user_data\"> \n<input type = \"button\" id = \"submit\" value = \"submit\" onClick='submitform()'>\n</form>\n\n<script language =\"javascript\" type = \"text/javascript\" >\nfunction submitform()\n{ \n var url = '/users/' + $('#user_id').val();\n $('#myform').attr('action', url);\n\n //\n // I think I can use JSON.stringify({\"userdata\":$('#user_data').val()}) \n // to get the data into JSON format but how do I post it using js? ", " \n //\n\n $(\"#myform\").submit();\n}\n\nA:\n\nYou can add a hidden input field with the json value, like this -\nfunction submitform() {\n var url = '/users/' + $('#user_id').val();\n $('#myform').attr('action', url);\n var data = JSON.stringify({\n \"userdata\": $('#user_data').val()\n })\n $('<input type=\"hidden\" name=\"json\"/>').val(data).appendTo('#myform');\n $(\"#myform\").submit();\n}\n\nYou can access your json using json parameter (name of hidden input)\n\n" ]

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[ "Q:\n\nCorrect way to use \"NOT IN\" Postgres\n\nI have two tables, People, and Vehicles. ", " Vehicles belongs to people. ", " Im trying to check if a person does not have a vehicle. ", " I was attempting to do this by joining People and Vehicles, and displaying the persons ID that is NOT IN Vehicles.person_id.", "\nThis is returning nothing, and has me wondering if there is something I did wrong, or if there is a more efficient way of doing this.", "\nQuery is below\nSelect People.id\nFrom People\nINNER JOIN Vehicles\non People.id=Vehicles.person_id\nwhere People.id NOT IN Vehicles.person_id;\n\nA:\n\nUse left join to figure out the persons with no vehicles\n Select distinct People.id \n From People \n LEFT JOIN Vehicles on People.id=Vehicles.person_id \n where Vehicles.person_id is NULL\n\n" ]

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[ "Q:\n\nCombine many notifications to one\n\nI'm re-creating my notification system to make it able to combine many notifications to one. ", "Instead of showing 9 rows that say \"Adam has answered your question\", it could say \"9 people have answered your question\".", "\nMy current notification table looks like this:\nnotification_id (PK) | recipient_id | sender_id | type | foreign_id | date | viewed\n\nRecipient_id is the one to get the notification. ", "Sender_id is the one it came from. ", "Type is what kind of notification, so I know what it should say, \"xx has answered\" or \"xx liked your post\". ", "Foreign_id could be the ID of a question.", "\nWhich would be the best way to create this \"combine\" system? ", "Should I update the notification when there is a new answer and say that another person has answered too, or should the query maybe count how many there is of the same foreign_id. ", "What do you think?", "\n\nA:\n\nTry this: \nSELECT foreign_id, SUM(IF (TYPE= 'like', 1, 0)) likecnt, \n SUM(IF(TYPE = 'answer', 1, 0)) anscnt \nFROM notification \nGROUP BY foreign_id\n\nEDIT: Check this\nSELECT n.foreign_id, n.sender_id\nFROM notifications n \nINNER JOIN (SELECT foreign_id, MAX(DATE) DATE \n FROM notifications \n GROUP BY foreign_id, user_id\n ) a ON n.foreign_id = a.foreign_id AND n.date = a.date\n\n" ]

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[ "2019 Mountain West Conference football season\n\nThe 2019 Mountain West Conference football season, part of that year's NCAA Division I FBS football season, is the 21st season of College Football for the Mountain West Conference (MW) since 2012. ", "12 teams have competed in the MW-football conference. ", "The season began on August 24, 2019 and will end on November 30. ", "The entire schedule was released on February 4, 2019.", "\n\nPrevious season\n\nIn 2018 Boise State won the Mountain Division with a 7–1 Conference record, while Fresno State won the West Division with a 7–1 Conference Record. ", "Fresno State beat Boise State 19–16 in Overtime in the 2018 Mountain West Conference Championship football game, held at Albertsons Stadium in Boise.", "\n\nPreseason\n\nMountain West Media\nThe Mountain West Media Days took place on July 23 and 24 at the Cosmopolitan.", "\n\nPreseason Polls\n\nFirst place votes in parenthesis\n\nPreseason All–Mountain West team\n\nPreseason Offensive Player of the Year:\nJordan Love, Jr., QB, Utah State\nPreseason Defensive Player of the Year:\nCurtis Weaver, Jr., DL, Boise State\nPreseason Special Teams Player of the Year:\nCooper Rothe, Sr., ", "PK, Wyoming\n\n(* – member of the 2018 All–Mountain West first team)\n\n(** – member of the 2018 All–Mountain West second team)\n\nRankings\n\nListed are the Mountain West teams who were ranked or received votes at some point during the season. ", "Colorado State, Nevada, New Mexico, San Jose State, and UNLV were never ranked nor received any votes.", "\n\nCoaches\n\nCoaching changes\nThere was only one coaching change in the Mountain West Conference for the 2019 season.", "\n\nGary Andersen, who previously coached at Utah State, returned to Logan to replace Matt Wells, who left to coach at Texas Tech.", "\n\nSchedule\nThe Regular season will begin on August 24 and will end on November 30.", "\n\nRegular season\n\nWeek One\n\nWeek Two\n\nWeek Three\n\nWeek Four\n\nWeek Five\n\nWeek Six\n\nWeek Seven\n\nWeek Eight\n\nWeek Nine\n\nWeek Ten\n\nWeek Eleven\n\nWeek Twelve\n\nWeek Thirteen\n\nWeek Fourteen\n\nChampionship Game\n\nWeek Fifteen (Mountain West Championship game)\n\nThe 2019 Mountain West Conference Championship Game was held on December 7 between the champions of the Mountain Division and the West Division.", "\n\nPostseason\n\nBowl games\n\n \n\nRankings are from CFP rankings. ", " All times Mountain Time Zone. ", " Mountain West teams shown in bold.", "\n\nSelection of teams\nBowl eligible: Air Force, Boise State, Hawaii, Nevada, San Diego State, Utah State, Wyoming\nBowl-ineligible: Colorado State, Fresno State, New Mexico, San Jose State, UNLV\n\nMountain West records vs Other Conferences\n2019–2020 records against non-conference foes:\n\nRegular Season\n\nPost Season\n\nMountain West vs Power Five matchups\n\nThis is a list of games the Mountain West has scheduled versus power conference teams (ACC, Big 10, Big 12, Pac-12, BYU/Notre Dame and SEC). ", "All rankings are from the current AP Poll at the time of the game.", "\n\nMountain West vs Group of Five matchups\nThe following games include Mountain West teams competing against teams from the American, C-USA, MAC or Sun Belt.", "\n\nMountain West vs FBS independents matchups\nThe following games include Mountain West teams competing against FBS Independents, which includes Army, Liberty, New Mexico State, or UMass.", "\n\nMountain West vs FCS matchups\n\nAwards and honors\n\nPlayer of the week honors\n\nMountain West Individual Awards\nThe following individuals received postseason honors as voted by the Mountain West Conference football coaches at the end of the season\n\nAll-conference teams\n\nAll Conference Honorable Mentions:\nAir Force: Milton Bugg III, Jr., DB; Jeremy Fejedelem, Sr., ", "DB; Jordan Jackson, Jr., DL; Timothy Jackson, So., ", "FB; Jake Koehnke, Sr., ", "PK; Zan Lewis, Sr., ", "DB; Geraud Sanders, Sr., ", "WR; Kadin Remsberg, Jr., RB\nBoise State: John Bates, Jr., TE; Sonatane Lui, Sr., ", "DL; Eric Quevedo, Sr., ", "OL; Khalil Shakir, So., ", "WR; Avery Williams, Jr., DB\nColorado State: Anthony Hawkins, Sr., ", "KR; Manny Jones, Jr., DL\nFresno State: Kevin Atkins, Jr., DL; Blake Cusick, Sr., ", "P; Juju Hughes, Sr., ", "DB\nHawaii: Cortez Davis, Jr., DB; Rojesterman Farris II, Sr., ", "DB; Solomon Matautia, Sr., ", "LB; Kaimana Padello, Sr., ", "DL; Gene Pryor, OL; Taaga Tuulima, Jr., OL; Blessman Ta'ala, So., ", "DL\nNevada: Daniel Brown, Sr., ", "DB; Romeo Doubs, WR, PR; Gabriel Sewell, Sr., ", "LB; Toa Taua, So., ", "RB\nNew Mexico: Ahmari Davis, Sr., ", "RB; Alex Hart, Sr., ", "LB; Teton Saltes, Jr., OL; Kyle Stapley, Jr., OL\nSan Diego State: Matt Araiza, Fr., ", "PK; William Dunkle, Fr., ", "OL; Darren Hall, So., ", "DB; Brandon Heicklen, Sr., ", "P; Dwayne Johnson Jr., Jr., DB\nSan Jose State: Bailey Gaither, Sr., ", "WR; Troy Kowalski, Sr., ", "OL; Matt Mercurio, Fr., ", "PK; Jack Snyder, Jr., OL\nUNLV: Julio Garcia, Jr., OL; Rayshad Jackson, Sr., ", "LB; Justin Polu, Sr., ", "OL; Javin White, Sr., ", "LB\nUtah State: Shaq Bond, Jr., DB; Gerold Bright, Sr., ", "RB; Jordan Love, Jr., QB; Caleb Repp, Sr., ", "TE; Christopher Unga, Sr., ", "DL\nWyoming: Tyler Hall, Sr., ", "DB; Logan Harris, Jr., OL; Cassh Maluia, Sr., ", "LB\n\nAll-Americans\n\nThe 2019 College Football All-America Teams are composed of the following College Football All-American first teams chosen by the following selector organizations: Associated Press (AP), Football Writers Association of America (FWAA), American Football Coaches Association (AFCA), Walter Camp Foundation (WCFF), The Sporting News (TSN), Sports Illustrated (SI), USA Today (USAT) ESPN, CBS Sports (CBS), FOX Sports (FOX) College Football News (CFN), Bleacher Report (BR), Scout.com, Phil Steele (PS), SB Nation (SB), Athlon Sports, Pro Football Focus (PFF) and Yahoo! ", "Sports (Yahoo!).", "\n\nCurrently, the NCAA compiles consensus all-America teams in the sports of Division I-FBS football and Division I men's basketball using a point system computed from All-America teams named by coaches associations or media sources. ", " The system consists of three points for a first-team honor, two points for second-team honor, and one point for third-team honor. ", " Honorable mention and fourth team or lower recognitions are not accorded any points. ", " Football consensus teams are compiled by position and the player accumulating the most points at each position is named first team consensus all-American. ", " Currently, the NCAA recognizes All-Americans selected by the AP, AFCA, FWAA, TSN, and the WCFF to determine Consensus and Unanimous All-Americans. ", "Any player named to the First Team by all five of the NCAA-recognized selectors is deemed a Unanimous All-American.", "\n\n*AFCA All-America Team (AFCA)\n*Walter Camp Football Foundation All-America Team (WCFF)\n*Associated Press All-America Team (AP)\n*The Sporting News All-America Team (TSN)\n*Football Writers Association of America All-America Team (FWAA)\n*Sports Illustrated All-America Team (SI)\n*Bleacher Report All-America Team (BR)\n*College Football News All-America Team (CFN)\n*ESPN All-America Team (ESPN)\n*CBS Sports All-America Team (CBS)\n*Athlon Sports All-America Team (Athlon)\n\nAll-Academic\n\nNational award winners\n2019 College Football Award Winners\n\nHome game attendance\n\nBold – Exceed capacity\n†Season High\n\nNotes\n\nReferences\n\nCategory:2019 Mountain West Conference football season" ]

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[ "On March 24th, Congress was set to take a vote on the repeal-and-replacement of the Affordable Care Act (ACA, also known as Obamacare) with the American Health Care Act (AHCA). ", "As the time for the vote approached, House Speaker Paul Ryan pulled the bill from the floor. ", "We applaud his decision to revisit this act and its design and encourage him and others in government to thoroughly vet their new design with experts and analyze the impact of their recommendations on all populations.", "\n\nPoliticians on both sides of the aisle and in many corners of government critiqued the replacement act as being unsound in design, lacking supporting data, and having a detrimental fiscal impact. ", "The removal of the act was in response to and on behalf of their constituents. ", "Constituents like you.", "\n\nThey heard you because you spoke up.", "\n\nIn one example, Illinois Representative Raja Krishnamoorthi reports that he received 1,959 calls from his district against AHCA, with only 30 in favor. ", "This happened all over the nation: voicemail inboxes were filled with messages and townhalls were crowded with American citizens dissatisfied with the bill and letting their representatives know.", "\n\nYour voice can affect policy. ", "Your actions can affect Washington. ", "You are an integral part of the democratic process.", "\n\nRecently, the White House proposed an immediate cut of $1.2 billion to the National Institutes of Health and cuts to many other science programs around the nation. ", "We covered why the initial proposed budget was so damaging to the protection of the American people; this funding cut only compounds those issues.", "\n\nThere’s much more you and we all can do. ", "Oppose the proposed 2017 and 2018 budget and their drastic cuts to important, research-based federal agencies. ", "Call your representatives and tell them what is worth protecting.", "\n\nWe’ve put together a short script to use in those calls:\n\nMy name is _____________, I am a constituent in ___________. ", "I oppose the proposed 2018 budget, and the proposed funding cuts to the NIH in 2017." ]

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[ "The present invention relates to a light-emitting device like a semiconductor laser device, and more particularly relates to a semiconductor light-emitting device for emitting radiation in the ultraviolet to blue regions. ", "The present invention also relates to a method for fabricating the semi-conductor light-emitting device and to an optical disk apparatus using the light-emitting device.", "\nIn recent years, semiconductor light-emitting devices that can emit radiation at short wavelengths ranging from the ultraviolet to blue regions, or semiconductor laser devices, in particular, have been researched and developed vigorously. ", "This is because such light-emitting devices are expected to further increase the recording density of optical disks or the resolution of laser printers and are applicable to optical measuring instruments, medical equipment, display devices, illuminators and so on.", "\nExamples of semiconductor materials that can emit radiation at such short wavelengths include Group III nitride semiconductors. ", "For instance, a semiconductor laser device with a multiple quantum well active layer, which is a stack of silicon (Si)-doped GaInN/GaInN layers, can oscillate continuously at a wavelength of about 401 nm and at room temperature and can operate for as long as about 3,000 hours under the conditions that the ambient temperature is 20° C. and the output power thereof is 2 mW. See Japanese Journal of Applied Physics, Vol. ", "36 (1997), pp. ", "1568-1571, for example.", "\nGroup III nitride semiconductor crystals are generally grown by a metalorganic vapor phase epitaxy (MOVPE) process. ", "For example, Japanese Laid-Open Publication No. ", "6-196757 discloses a method of growing a semiconductor layer of GaInN of excellent crystal quality on a semiconductor layer of GaN by using nitrogen as a carrier gas.", "\nThe known method of producing a Group III nitride semiconductor, however, is disadvantageous in that pits are created in the GaInN/GaN multiple quantum well structure thereof (to be an active layer) at as high a density as 108 to 109 cm−2 as described in Applied Physics Letters, Vol. ", "72 (1998), pp. ", "710-712, for example.", "\nThose pits adversely affect the operation characteristics of a light-emitting device, e.g., raises the threshold value, at which the laser device starts to oscillate, or lowers the reliability thereof. ", "This is because the existence of the pits not only decreases the luminous efficacy, but also causes localized levels by making the composition of In non-uniform, constitutes a source of diffusion of In being grown or results in scattering or absorption loss in an optical waveguide.", "\nTo obtain a Group III nitride semiconductor light-emitting device, or semiconductor laser device, in particular, with characteristics practically applicable to an optical disk apparatus, for example, the composition of In within the GaInN well layer thereof should be uniformized. ", "In addition, each multiple quantum well layer should be of uniform quality and be sufficiently planarized.", "\nMoreover, the structure of the device should be modified such that electrons, which are injected from an n-type conductive layer into the quantum well layer, can be injected into the active layer efficiently and uniformly without overflowing into a p-type conductive layer during the operation of the device." ]

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[ "I can’t even begin to explain how much fun I had in NYC last week for the final retreat with my amazing Inner Circle online business mastermind group that i’ve spent this past year with, led by the brilliant and most beautiful human being ever, James Wedmore! ", "I swear i’m not hiding in the pic […]" ]

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