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http://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/i/Indian_Standard_Time.htm	# Indian Standard Time Location of Mirzapur and the 82.5° E longitude that is used as the reference longitude for Indian Standard Time. Indian Standard Time (IST) is the time observed throughout India, with a time offset of UTC+5:30. India does not observe daylight saving time (DST) or other seasonal adjustments, although DST was used briefly during the Sino–Indian War of 1962, and the Indo–Pakistani Wars of 1965 and of 1971. In military and aviation time, IST is designated E* = Echo*. Indian Standard Time is calculated on the basis of 82.5 °E longitude which is just west of the town of Mirzapur, near Allahabad in the state of Uttar Pradesh. The longitude difference between Mirzapur and the United Kingdom's Royal Observatory at Greenwich translates to an exact time difference of 5 hours and 30 minutes. Local time is calculated from a clock tower at the Allahabad Observatory ( 25.15° N 82.5° E) though the official time-keeping devices are entrusted to the National Physical Laboratory, located in New Delhi. ## History One of the earliest descriptions of standard time in India appeared in the 4th century CE astronomical treatise Surya Siddhanta. Postulating a spherical earth, the book defined the prime meridian, or zero longitude, as passing through Avanti, the ancient name for the historic city of Ujjain ( 23°11′N 75°45′E), and Rohitaka, the ancient name for Rohtak ( 28°54′N 76°38′E), a city near the historic battle-field of Kurukshetra. Situated upon the line which passes through the haunt of the demons (equator and 76° E) and the mountain which is the seat of the gods (the North Pole), are Rohitaka and Avanti ... The sidereal day of ancient Indian astronomy began at sunrise at the prime meridian of Ujjain, and was divided into smaller time units in the following manner: Time that is measurable is that which is in common use, beginning with the prāṇa (or, the time span of one breath). The pala contains six prāṇas. The ghalikā is 60 palas, and the nakṣatra ahórātra, or sidereal day, contains 60 ghalikās. A nakṣatra māsa, or sidereal month, consists of 30 sidereal days. Taking a sidereal day to be 24 hours, it is easily computed that the smallest time unit, prāṇa, or one respiratory cycle, equals 4 seconds, a value consistent with the normal breathing frequency of 15 breaths/min used in modern medical research. The Surya Siddhanta also described a method of converting the local time of an observer to the standard time of Ujjain. However, despite these early advances, standard time was not widely used outside of astronomy. For most of India's history, ruling kingdoms kept their own local time, typically using the Hindu calendar in both lunar and solar units. For example, the Jantar Mantar observatory built by Maharaja Sawai Jai Singh in Jaipur in 1733 contains large sundials, up to 90 ft. high, which were used to accurately determine the local time. Astronomer John Goldingham is credited with the establishment of the current fractional time zone (UTC+5:30). In 1792, the British East India Company established the Madras Observatory in Chennai (then Madras), largely due to the efforts of the British sailor–astronomer Michael Topping. In 1802, John Goldingham, appointed as the first official astronomer of the Company in India, established the longitude of Madras ( 13°5′24″N, 80°18′30″E) as 5 hours and 30 minutes ahead of Greenwich Mean Time as the local standard time. This marked the first ever use of the current time zone, and departure from the earlier standard of the day beginning at sunrise; now it started at midnight. The clock in the observatory was attached to a gun that was fired at 8 p.m. daily to announce that "all was well" with IST. Time-keeping support for shipping activities in Bombay Harbour was provided by the Colaba Observatory in Bombay, which was established in 1826. Most of the towns in India retained their own local time until a few years after the introduction of the railways in the 1850s, when the need for a unified time–zone became apparent. As headquarters of the two largest Presidencies of British India, local time in Mumbai (then Bombay) and Kolkata (then Calcutta) assumed special importance, that was gradually adopted by the nearby provinces and princely states. In the 19th century, clocks at various locations were kept in synchronisation through telegraphic means – for example the railways synchronised their clocks thorough a time signal which was sent from the head office or the regional headquarters at a specified time every day. In 1884, the International Meridian Conference in Washington, D.C., set up uniform time zones across the world. It was decided that India would have two time zones, with Kolkata using the 90th east meridian and Bombay the 75th east meridian. Calcutta time was set at 5 hours, 30 minutes, and 21 seconds in advance of GMT, while Bombay time was 4 hours, 51 minutes ahead of GMT. By the late 1880s, many railway companies began to use the Madras time (known as " Railway time") as an intermediate time between the two zones. Another time zone, Port Blair mean time, was established at Port Blair, the capital of the Andaman and Nicobar Islands in the Bay of Bengal. The Port Blair mean time was set to 49 minutes and 51 seconds ahead of Madras time. British India did not officially adopt the standard time zones until 1905, when the meridian passing east of Allahabad at 82.5 degrees east longitude was picked as the central meridian for India, corresponding to a single time zone for the country. This came into force on January 1, 1906, and also applied to Sri Lanka (then Ceylon). However Calcutta time was officially maintained as a separate time zone until 1948. IST in relation with the bordering nations. In 1925, time synchronisation began to be relayed through omnibus telephone systems and control circuits to organisations that needed to know the precise time. This continued until the 1940s, when time signals began to be broadcast using the radio by the government. After independence in 1947, the Indian government established IST as the official time for the whole country, although Kolkata and Mumbai retained their own local time for a few more years. The Central observatory was moved from Chennai to a location near Mirzapur so that it would be as close to UTC +5:30 as possible. During the Sino–Indian War of 1962, and the Indo–Pakistani Wars of 1965 and 1971, daylight saving was briefly used to reduce civilian energy consumption. ## Problems A single, large time zone has been shown to cost more, and requires rescheduling of events to make them compatible with the rest of the zone or with the day's cycle. The country's east–west distance of more than 2,000 km (1,200 mi) covers over 28 degrees of longitude, resulting in the sun rising and setting almost two hours earlier on India's eastern border than in the Rann of Kutch in the far west. Inhabitants of the north–eastern states have long demanded a separate time zone to advance their clocks with the early sunrise and avoid the extra consumption of energy after daylight hours. In the late 1980s, a team of researchers proposed separating the country into two or three time zones to conserve energy. The binary system that they suggested involved a return to British–era time zones; the recommendations were not adopted. In 2001, the government established a four–member committee under the Department of Science and Technology to examine the need for multiple time zones and daylight saving. The findings of the committee, which were presented to Parliament in 2004 by the Minister for Science and Technology, Kapil Sibal, did not recommend changes to the unified system stating that "the prime meridian was chosen with reference to a central station, and that the expanse of the Indian State was not large." Though the government has consistently refused to split the country into multiple time zones, provisions in various Indian labour laws such as the Plantations Labour Act, 1951 do allow the Central and State governments to define and set the local time for a particular industrial area. ## Time signals In India, official time signals are generated by the Time and Frequency Standards Laboratory at the National Physical Laboratory in New Delhi, for both commercial and official use. The signals are based on atomic clocks and are synchronised with the worldwide system of clocks that support the Universal Coordinated Time. Features of the Time and Frequency Standards Laboratory include : • Four caesium and rubidium atomic clocks; • High frequency broadcast service operating at 10 MHz under call sign ATA to synchronise the user clock within a millisecond; • Indian National Satellite System satellite–based standard time and frequency broadcast service, which offers IST correct to ±10 microsecond and frequency calibration of up to $\pm 10^{-10}$; and • Time and frequency calibrations are made with the help of pico– and nanoseconds time interval frequency counters and phase recorders. To communicate the exact time to the people, the exact time is broadcast over the state–owned All India Radio and Doordarshan television network. Telephone companies have dedicated phone numbers connected to mirror timeservers that also relay the precise time. Another increasingly popular means of obtaining the time is through Global Positioning System (GPS) receivers.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27795907855033875, "perplexity": 3164.160479209807}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398474527.16/warc/CC-MAIN-20151124205434-00131-ip-10-71-132-137.ec2.internal.warc.gz"}
http://www.purplemath.com/learning/viewtopic.php?p=796	## solve inverse-cos(x) + inverse-cos(sq. rt. of 15x) = (pi/2) Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc. car.man Posts: 5 Joined: Tue Mar 24, 2009 6:18 pm ### solve inverse-cos(x) + inverse-cos(sq. rt. of 15x) = (pi/2) solve inverse-cos(x) + inverse-cos(square root of 15x) = (pi/2). No one was able to solve this problem. Please help! stapel_eliz Posts: 1738 Joined: Mon Dec 08, 2008 4:22 pm Contact: car.man wrote:solve inverse-cos(x) + inverse-cos(square root of 15x) = (pi/2). By definition of inverse cosine, the above means the following: . . . . .$\alpha\, +\, \beta\, =\, \frac{\pi}{2}$ ...for: . . . . .$\cos^{-1}(x)\, =\, \alpha\, \mbox{ and }\, \cos^{-1}(\sqrt{15}x)\, =\, \beta$ This is a sum of two angles, and involves cosines. What if we now take the cosine of both sides of the above? . . . . .$\cos(\alpha\, +\, \beta)\, =\, 0$ Using an angle-sum identity, we get: . . . . .$\cos(\alpha)\cos(\beta)\, -\, \sin(\alpha)\sin(\beta)\, =\, 0$ We can simplify the factors in the first product by using the definition of "inverse cosine": . . . . .$\cos\left(\cos^{-1}(x)\right)\cos\left(\cos^{-1}(\sqrt{15}x)\right)\, =\, (x)\left(\sqrt{15}x\right)\, =\, \sqrt{15}x^2$ To find the values of the sines, draw right triangles for each of the angles $\alpha$ and $\beta$. Using the Pythagorean Theorem, find the length of the "opposite" side for each triangle. Then read off the values of the sines. You should end up with an equation, after squaring, that looks like: . . . . .$15x^4\, =\, 1\, -\, 16x^2\, +\, 15x^4$ Then: . . . . .$0\, =\, 1\, -\, 16x^2\, =\, (1\, -\, 4x)(1\, +\, 4x)$ ...and so forth. car.man Posts: 5 Joined: Tue Mar 24, 2009 6:18 pm ### Re: solve inverse-cos(x) + inverse-cos(sq. rt. of 15x) = (pi/2) We got x=+-1/4 and x=+1/4 worked. Thank you!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 9, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9800466895103455, "perplexity": 3306.7729941721427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098072.11/warc/CC-MAIN-20150627031818-00191-ip-10-179-60-89.ec2.internal.warc.gz"}
http://publications.lib.chalmers.se/publication/198992-collective-symplectic-integrators	CPL - Chalmers Publication Library # Collective symplectic integrators Robert McLachlan ; Klas Modin (Institutionen för matematiska vetenskaper, matematik) ; Olivier Verdier Nonlinearity (0951-7715). Vol. 27 (2014), 6, p. 1525-1542. We construct symplectic integrators for Lie–Poisson systems. The integrators are standard symplectic (partitioned) Runge–Kutta methods. Their phase space is a symplectic vector space equipped with a Hamiltonian action with momentum map J whose range is the target Lie–Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by J. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on \$\mathfrak{so}(3)^*\$ . The method specializes in the case that a sufficiently large symmetry group acts on the fibres of J, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented. CPL Pubid: 198992 # Läs direkt! Länk till annan sajt (kan kräva inloggning) # Institutioner (Chalmers) Institutionen för matematiska vetenskaper, matematik (2005-2016)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9592896699905396, "perplexity": 5238.371200052775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718840.18/warc/CC-MAIN-20161020183838-00458-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-3-radian-measure-and-the-unit-circle-section-3-1-radian-measure-3-1-exercises-page-105/48	## Trigonometry (11th Edition) Clone Published by Pearson # Chapter 3 - Radian Measure and the Unit Circle - Section 3.1 Radian Measure - 3.1 Exercises - Page 105: 48 4.623 radian #### Work Step by Step $264.9\times \frac{\pi}{180} = 4.623$ radian After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8878000378608704, "perplexity": 2827.339715398191}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572215.27/warc/CC-MAIN-20220815235954-20220816025954-00724.warc.gz"}
http://mathoverflow.net/feeds/question/37021	Is Fourier analysis a special case of representation theory or an analogue? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:31:00Z http://mathoverflow.net/feeds/question/37021 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue Is Fourier analysis a special case of representation theory or an analogue? David Corwin 2010-08-29T03:47:43Z 2012-12-06T00:36:58Z <p>I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$."</p> <p>I've been introduced to the idea that Fourier analysis is related to representation theory. Specifically, when considering the representations of a finite abelian group $A$, these representations are all $1$-dimensional, hence correspond to characters $A \to \mathbb{R}/\mathbb{Z} \cong S^1 \subseteq \mathbb{C}$. On the other side, finite Fourier analysis is, in a simplistic sense, the study of characters of finite abelian groups. Classical Fourier analysis is, then, the study of continuous characters of locally compact abelian groups like $\mathbb{R}$ (classical Fourier transform) or $S^1$ (Fourier series). However, in the case of Fourier analysis, we have something beyond characters/representations: We have the Fourier series / transform. In the finite case, this is a sum which looks like $\frac{1}{n} \sum_{0 \le r < n} \omega^r \rho(r)$ for some character $\rho$, and in the infinite case, we have the standard Fourier series and integrals (or, more generally, the abstract Fourier transform). So it seems like there is something more you're studying in Fourier analysis, beyond the representation theory of abelian groups. To phrase this as a question (or two):</p> <p>(1) What is the general Fourier transform which applies to abelian and non-abelian groups?</p> <p>(2) What is the category of group representations we consider (and attempt to classify) in Fourier analysis? That is, it seems like Fourier analysis is more than just the special case of representation theory for abelian groups. It seems like Fourier analysis is trying to do more than classify the category of representations of a locally compact abelian group $G$ on vector spaces over some fixed field. Am I right? Or can everything we do in Fourier analysis (including the Fourier transform) be seen as one piece in the general goal of classifying representations?</p> <p>Let me illustrate this in another way. The basic result of Fourier series is that every function in $L^2(S^1)$ has a Fourier series, or in other words that $L^2$ decomposes as a (Hilbert space) direct sum of one dimensional subspaces corresponding to $e^{2 \pi i n x}$ for $n \in \mathbb{Z}$. If we encode this in a purely representation-theoretic fact, this says that $L^2(S^1)$ decomposes into a direct sum of the representations corresponding to the unitary characters of $S^1$ (which correspond to $\mathbb{Z}$). But this fact is not why Fourier analysis is interesting (at least in the sense of $L^2$-convergence; I'm not even worrying about pointwise convergence). Fourier analysis states furthermore an <em>explicit</em> formula for the function in $L^2$ giving this representation. Though I guess by knowing the character corresponding to the representation would tell you what the function is.</p> <p>So is Fourier analysis merely similar to representation theory, or is it none other than the abelian case of representation theory?</p> <p>(Aside: This leads into a more general question of mine about the use of representation theory as a generalization of modular forms. My question is the following: I understand that a classical Hecke eigenform (of some level $N$) can be viewed as an element of $L^2(GL_2(\mathbb{Q})\ GL_2(\mathbb{A}_{\mathbb{Q}})$ which corresponds to a subrepresentation. But what I don't get is why the representation tells you everything you would have wanted to know about the classical modular form. A representation is nothing more than a vector space with an action of a group! So how does this encode the information about the modular form?)</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/37031#37031 Answer by Yemon Choi for Is Fourier analysis a special case of representation theory or an analogue? Yemon Choi 2010-08-29T07:19:02Z 2010-08-30T00:02:55Z <p>Since David's asked or suggested that some remarks be written up as an answer, let me repeat what I said in the comments:</p> <ul> <li><p>in my opinion, the answer to the question in the title is "neither";</p></li> <li><p>my answer to Q1 would be "look up Fourier transform and noncommutative harmonic analysis" - if one is only interested in compact groups then there is a very satisfactory theory, outlined in Hewitt & Ross volume 2 for example;</p></li> <li><p>and my answer to Q2 would be "not applicable - the question is founded on a debatable premise".</p></li> </ul> <p>Actually, in the middle of the salvo that he's labelled as "Question 2", David asks</p> <blockquote> <p>That is, it seems like Fourier analysis is more than just the special case of representation theory for abelian groups. It seems like Fourier analysis is trying to do more than classify the category of representations of a locally compact abelian group $G$ on vector spaces over some fixed field. Am I right?</p> </blockquote> <p>and in my inexpert opinion, the answer is "yes". Why would it be 'just' a subtopic of the enterprise of classifying representations?</p> <hr> <p><b>Update:</b> to elaborate on my objections to the original questions, while not taking anything away from the informative comments and answers that other people have given: <em>there is more to Fourier analysis than constructing a Fourier transform between certain topological vector spaces and getting a Plancherel formula</em>. Hence being able to construct a generalization or analogue of the Fourier transform for nonabelian locally compact groups is not the be all and end all of the topic, unless the topic is "constructing a nonabelian Fourier transform". Looking in the literature on harmonic analysis, even in the abelian case, ought to bear this out.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/37071#37071 Answer by Theo Johnson-Freyd for Is Fourier analysis a special case of representation theory or an analogue? Theo Johnson-Freyd 2010-08-29T17:48:56Z 2010-08-30T03:08:03Z <p>This answer is essentially a comment, but slightly too long.</p> <p>As someone who is closer to representation theory than analysis, to me the word "Fourier analysis" means "Pontrjagin duality". You hint at this theorem in your question, but I will state (a version of) it for completeness:</p> <p><strong>Theorem:</strong> Let $G$ be a locally-compact topological abelian group. A <em>character</em> is a continuous homomorphism $G \to \mathbb S^1$ the circle. The set $G^\vee$ of characters is naturally an abelian group under $\otimes$, and has a canonical topological structure in which it is locally compact. The canonical pairing $G \times G^\vee \to \mathbb S^1$ induces a map $G \to (G^\vee)^\vee$, which is an isomorphism.</p> <p>A good reference is:</p> <ul> <li>André Joyal, Ross Street. An introduction to Tannaka duality and quantum groups. <em>Category Theory, Lecture Notes in Math</em>. 1991 vol. 1488 pp. 412–492.</li> </ul> <p>They describe the relationship between Pontrjagin duality and Fourier theory as a warm-up for various forms of Tannaka-Krein theory, which can be thought of as a "noncommutative" analogue of Pontrjagin duality.</p> <hr> <p>Incidentally, just as there are questions that Fourier analysts care about that aren't "just" representation theory of abelian groups, there are questions in the representation theory of abelian groups that aren't "just" Pontrjagin duality. For example, given a fixed vector space $V$, the collection of sets of $n$ commuting matrices $V \to V$ is naturally the same as the collection of representations of $\mathbb Z^n$ on $V$, i.e. $\operatorname{Hom}(\mathbb Z^n \to \operatorname{End}(V))$. Now, $\operatorname{GL}(V)$ acts on $\operatorname{End}(V)$ and hence on $\operatorname{Hom}(\mathbb Z^n \to \operatorname{End}(V))$ by conjugation. The corresponding moduli problem — find the moduli space of $n$ commuting matrices of fixed dimension — is hard, although I think it's solved.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/37078#37078 Answer by Dick Palais for Is Fourier analysis a special case of representation theory or an analogue? Dick Palais 2010-08-29T19:43:25Z 2010-08-29T20:06:26Z <p>First, I think it is better to restrict the term "Fourier Analysis" to refer to the process of expanding functions on a locally compact ABELIAN group $G$ as a "sum'' of the characters of the group. (I'll come back to that in a moment.) The generalization, when the group $G$ is not assumed to be abelian, should probably be better referred to as Harmonic Analysis". Regarding the latter, if the group $G$ is compact, then the Peter-Weyl Theorem gives an elegant and simple generalization to the theory of Fourier Series on the circle group---it shows how to write any $L^2$ function on $G$ as an series of (orthogonal) matrix elements of irreducible unitary representations of $G$. When $G$ is neither abelian nor compact, the theory becomes MUCH more complicated and sophisticated. BTW, note that when $G$ is abelian, then as you pointed out, the irreducble unitary representaions of $G$ are one-dimensional, so there is no difference between a matrix element and a character in this case and we are generalizing Fourier series on the circle group.</p> <p>OK, lets now restrict to the Fourier" case, where $G$ is locally compact and abelian. Note that an irreducible unitary character of $G$ is now just a group homomorphism of $G$ into the circle group $S= S^1$ (considered as the complex numbers of modulus one under multiplication). Since $G$ is abelian, the set $\hat G = Hom(G,S)$ is an abelian group, the character (or Pontrjagin dual) group of $G$, under pointwise multiplication. It is easy to see that $\hat G$ is locally compact (in the compact open topology) What Fourier analysis becomes in this case is a method for expressing an arbitrary element of $L^2(G)$ as an integral of the form $f(g) \sim \int \hat f(\chi)\chi(g) d\chi$, where $\hat f$, the Fourier transform of $f$ is defined dually by $\hat f(\chi) = \int f(g) \chi(g) dg$ (and the Haar measures on $G$ and $\hat G$ are suitably normalized). Note that if we take for $G$ the real line $R$ then this reduces to the classical Fourier transform. It is easy to show that the integral defining the Fourier transform $\hat f(\chi)$ is convergent when $f$ is in $L^1 \cap L^2$ and that then $\|\|\hat f\|\|_2 = \|\|f\|\|_2^2$, and since $L^1 \cap L^2$ is dense in $L^2$ it follows that the Fourier transform extends uniquely to a unitary map of $L^2(G)$ onto $L^2(\hat G)$.</p> <p>Now lets restrict further to the compact case, where characters, being continuous, are bounded and so integrable. As one can prove in a couple of lines (using the invariance of Haar measure), if $\chi$ is any character of $G$ then $\int \chi(g)\, dg = 0$ unless $\chi$ is the identity character in which case the integral is one (using normalized Haar measure on $G$). Since the complex conjugate of a character is its inverse in $\hat G$, it now follows trivially that the elements of $\hat G$ are orthonormal. In fact they form an orthonormal basis for $L^2(G)$, and the Fourier transform of the preceding paragraph becomes a formula for expanding any element of $L^2(G)$ as the sum of an infinite series in the characters of $G$, a direct generalization of the theory of Fourier series (the case when $G = S$).</p> <p>A good place to see all the details is Lynn Loomis' "Absract Harmonic Analysis". </p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/37098#37098 Answer by Kimball for Is Fourier analysis a special case of representation theory or an analogue? Kimball 2010-08-29T23:42:14Z 2010-08-29T23:42:14Z <p>As a complement to the other answers, the (Selberg or Arthur-Selberg) trace formula can be viewed as a generalization of Poisson summation. Harish-Chandra also generalized the Plancherel formula. Both of these can be carried out for connected reductive Lie groups and are important in representation theory. </p> <p>From this point of view, it is perhaps more evident what one should expect of a generalized Fourier transform, and its role is played by the Harish-Chandra/Selberg transform. For the simplest (but not really so simple) non-abelian case, see Iwaniec's "Spectral Theory of Automorphic Forms." </p> <p>As for what other groups one might be able to do this for, one can at least do the trace formula for finite groups (which can be viewed as a generalization of Frobenius reciprocity, cf. Arthur's "Trace formula and Hecke operators"---Arthur also has a Notices article which discusses the general Plancherel formula and Langlands' program), and perhaps it's not too hard to see what a generalization of the Fourier transform should be in nonabelian cases, but I'd need to think about it.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/37189#37189 Answer by Emerton for Is Fourier analysis a special case of representation theory or an analogue? Emerton 2010-08-30T19:40:03Z 2010-08-31T14:28:48Z <p>I would like to elaborate slightly on my comment. First of all, Fourier analysis has a very broad meaning. Fourier introduced it as a means to study the heat equation, and it certainly remains a major tool in the study of PDE. I'm not sure that people who use it in this way think of it in a particularly representation-theoretic manner.</p> <p>Also, when one thinks of the Fourier transform as interchanging position space and frequency space, or (as in quantum mechanics) position space and momentum space, I don't think that a representation theoretic view-point necessarily need play much of a role.</p> <p>So, when one thinks about Fourier analysis from the point of view of group representation theory, this is just one part of Fourier analysis, perhaps the most foundational part, and it is probably most important when one wants to understand how to extend the basic statements regarding Fourier transforms or Fourier series from functions on $\mathbb R$ or $S^1$ to functions on other (locally compact, say) groups.</p> <p>As I noted in my comment, the basic question is: how to decompose the regular representation of $G$ on the Hilbert space $L^2(G)$. When $G$ is locally compact abelian, this has a very satisfactory answer in terms of the Pontrjagin dual group $\widehat{G}$, as described in Dick Palais's answer: one has a Fourier transform relating $L^2(G)$ and $L^2(\widehat{G})$. A useful point to note is that $G$ is discrete/compact if and only if $\widehat{G}$ is compact/discrete. So $L^2(G)$ is always described as the Hilbert space direct integral of the characters of $G$ (which are the points of $\widehat{G}$) with respect to the Haar measure on $\widehat{G}$, but when $G$ is compact, so that $\widehat{G}$ is discrete, this just becomes a Hilbert space direct sum, which is more straightforward (thus the series of Fourier series are easier than the integrals of Fourier transforms).</p> <p>I will now elide Dick Palais's distinction between the Fourier case and the more general context of harmonic analysis, and move on to the non-abelian case. As Dick Palais also notes, when $G$ is compact, the Peter--Weyl theorem nicely generalizes the theory of Fourier series; one again describes $L^2(G)$ as a Hilbert space direct sum, not of characters, but of finite dimensional representations, each appearing with multiplicity equal to its degree (i.e. its dimension). Note that the set over which one sums now is still discrete, but is not a group. And there is less homogeneity in the description: different irreducibles have different dimensions, and so contribute in different amounts (i.e. with different multiplicities) to the direct sum.</p> <p>When G is locally compact but neither compact nor abelian, the theory becomes more complex. One would like to describe $L^2(G)$ as a Hilbert space direct integral of matrix coefficients of irreducible unitary representations, and for this, one has to find the correct measure (the so-called Plancherel measure) on the set $\widehat{G}$ of irreducible unitary representations. Since $\widehat{G}$ is now just a set, a priori there is no natural measure to choose (unlike in the abelian case, when $\widehat{G}$ is a locally compact group, and so has its Haar measure), and in general, as far as I understand, one doesn't have such a direct integral decomposition of $L^2(G)$ in a reasonable sense.</p> <p>But in certain situations (when $G$ is of "Type I") there is such a decomposition, for a uniquely determined measure, so-called Plancherel measure, on $\widehat{G}$. But this measure is not explicitly given. Basic examples of Type I locally compact groups are semi-simple real Lie groups, and also semi-simple $p$-adic Lie groups.</p> <p>The major part of Harish-Chandra's work was devoted to explicitly describing the Plancherel measure for semi-simple real Lie groups. The most difficult part of the question is the existence of atoms (i.e. point masses) for the measure; these are irreducible unitary representations of $G$ that embed as subrepresentations of $L^2(G)$, and are known as "discrete series" representations. Harish-Chandra's description of the discrete series for all semi-simple real Lie groups is one of the major triumphs of 20th century representation theory (indeed, 20th century mathematics!).</p> <p>For $p$-adic groups, Harish-Chandra reduced the problem to the determination of the discrete series, but the question of explicitly describing the discrete series in that case remains open.</p> <p>One important thing that Harish-Chandra proved was that not all points of $\widehat{G}$ (when $G$ is a real or $p$-adic semisimple Lie group) are in the support of Plancherel measure; only those which satisfy the technical condition of being "tempered". (So this is another difference from the abelian case, where Haar measure is supported uniformly over all of $\widehat{G}$.) Thus in explicitly describing Plancherel measure, and hence giving an explicit form of Fourier analysis for any real semi-simple Lie group, he <em>didn't</em> have to classify all unitary representations of $G$.</p> <p>Indeed, the classification of all such reps. (i.e. the explicit description of $\widehat{G}$) remains an open problem for real semi-simple Lie groups (and even more so for $p$-adic semi-simple Lie groups, where even the discrete series are not yet classified).</p> <p>This should give you some sense of the relationship between Fourier analysis in its representation-theoretic interpretation (i.e. the explicit description of $L^2(G)$ in terms of irreducibles) and the general classification of irreducible unitary representations of $G$. They are related questions, but are certainly not the same, and one can fully understand one without understanding the other.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/46041#46041 Answer by Marc Palm for Is Fourier analysis a special case of representation theory or an analogue? Marc Palm 2010-11-14T13:42:34Z 2010-11-14T14:55:23Z <p>Fourier Analysis on $\mathbb{R}$ has several similiar interpretations. The most important one is it realizes the Functional calculus for the rightregular representation.</p> <p>I can only be really sketchy here:</p> <p>We can see it as the realization of the functional calculus for the operator $D= - \mathrm{i} \frac{\partial}{\partial x}$. Observe that $\mathcal{F} D = M_x \mathcal{F}$. Here $M_x$ is multiplication by $x$, which is much easier to understand than taking derivative. That is the main reason, why the Fourier transform is so important in the theory of differential operators. A great generalization of this is the Gelfand transform, which identifies certain commutative Algebras with functions over topological spaces. In this theory, we identify the algebra $D$ given by normal closed operator with the continouos function on the spectrum of $D$.</p> <p>Analogues ideas in algebraic geometry have been introduced by Grothendieck, who associated to varieties over a commutative unital ring R also a spectrum. In the case of an algebraic group this spectrum can be seen as a certain the group ring.</p> <p>Since taking derivatives commutes with the right translations, which are exactly the right regular representation of $\mathbb{R}$. The Fourier analysis also realizes the functional calculus for this family of operators.</p> <p>The analysis of noncommutative groups is of course much more difficult since the right translation do not commute here, hence there is no functional calculus, since this is not available for non commutative algebras.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/56600#56600 Answer by saghar for Is Fourier analysis a special case of representation theory or an analogue? saghar 2011-02-25T06:02:56Z 2011-02-25T06:02:56Z <p>If you want to know what is the dual of a nonabelian locally compact group, you have to study about locally compact quantum groups. Then you can see that even we can define the fourier transform here as well.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/115355#115355 Answer by paul garrett for Is Fourier analysis a special case of representation theory or an analogue? paul garrett 2012-12-04T04:09:27Z 2012-12-04T04:09:27Z <p>In addition to the many other interesting and useful answers, and as evidence for the fruitfulness of the question (!), I do think there are a few other (maybe-interesting and maybe-useful) points to be made. </p> <p>First, to limit the scope, let's say we're talking about "Type I" groups, that is, groups which more-or-less have a reasonable/tractable representation theory, in the sense that "factor representations" are sums/integrals of irreducibles. This assumption deserves comment: as in one of the earlier good answers, there are many reasonable groups which do not fall into this class. (Dang...) The good news is that it is possible to understand this failing (e.g., see Alain Robert's wonderful LMS book), and that for many critical applications (in my own purview, to number-theoretic things) this is not an issue. Whew.</p> <p>It is likewise certainly true (as observed and documented in earlier answers) that the "full question" of determination of details about various Plancherel theorems (for reductive p-adic groups...) is still open... but, also, sufficiently-many examples are known that "we" feel some confidence in advancing in a certain way. It is important to note that many literal Plancherel formulas do not (as noted in other answers!) involve <em>all</em> (unitary) irreducibles, but only a nice (a.k.a. "tempered", in some contexts) subclass.</p> <p>A very educational case is the Gelfand-Naimark story from the late 1940's, addressing more-or-less reductive complex Lie groups, essentially proving that the decomposition of $L^2(G)$ needed only unitary principal series... [sic]</p> <p>At best, such an assertion is about $L^2$, not about pointwise convergence, etc. </p> <p>Harish-Chandra showed in the 1950s and '60s that things are (stunningly) more complicated for "real" Lie groups <em>not</em> obtained by the forgetful functor complex-to-real Lie group.</p> <p>Nevertheless, ... for applications to analytic number theory (!?), one would desire sharp estimates on convergence of spectral expansions of automorphic forms. Maass and Selberg initiated this study, but/and this line of thought has not-so-often interacted with the Schwartz-Grothendieck modern-analysis thinking... to all our loss.</p> <p>As David Farmer aptly quipped in Oklahoma at a lovely conference in Sept of this year, "convergence is tricky". </p> <p>... but/and in many interesting cases, the convergence of "an eigenfunction expansion" is exactly what a serious problem wants.</p> <p>(Some schools of thought study the individual "eigenfunctions", e.g., automorphic forms, ... but/and many applications require those pesky convergence considerations.)</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/115358#115358 Answer by Will Sawin for Is Fourier analysis a special case of representation theory or an analogue? Will Sawin 2012-12-04T05:28:35Z 2012-12-04T05:28:35Z <p>Here is a very quick, overly simplified answer:</p> <p>Fourier analysis = The representation theory of S^1 + Peter-Weyl for S^1.</p> <p>So it's not a special case of representation theory, but it is a special case of ( representation theory + Peter-Weyl ).</p> <p>The reason this is only a rough schematic is that there is Fourier analysis that isn't just about the $L^2$ spaces. But you can think of it as having an $L^2$, algebraic core, with all the other functions between Banach spaces with various properties "glued" to the core, meaning they determine each other.</p> http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue/115564#115564 Answer by Steve Huntsman for Is Fourier analysis a special case of representation theory or an analogue? Steve Huntsman 2012-12-06T00:36:58Z 2012-12-06T00:36:58Z <p>I just saw this and figured a more elementary and explicit answer discussing Fourier transforms on finite groups couldn't hurt, especially since it's a chance to use some of my notes.</p> <p>First, some scene-setting: harmonic analysis on a finite abelian group $G$ turns out to be a direct generalization of the theory of Fourier series. A function $f$ is decomposed according to</p> <p>$\hat f(\chi) := \lvert G \rvert^{-1/2} \sum_g f(g) \chi(g);$</p> <p>$f(g) = \lvert G \rvert^{-1/2} \sum_\chi \hat f(\chi) \chi(g^{-1}).$</p> <p>Here $\chi$ denotes a character. The factors of $\lvert G \rvert^{-1/2}$ are chosen for the sake of unitarity; the more general case of locally compact abelian $G$ is broadly similar.</p> <p>Nonabelian groups do not have enough unitary characters to enable a decomposition of the form above. Let $\rho: G \rightarrow GL(V)$ be a representation with dimension $d_\rho \equiv \dim V$; the corresponding character is $\chi_\rho(g) = \mbox{Tr} \rho(g)$.</p> <p>Two key identities express the orthogonality and completeness of representations, i.e.</p> <p><code>$\frac{d_\rho}{\lvert G \rvert} \sum_g \rho_{jk}(g^{-1}) \rho'_{\ell m}(g) = \delta_{\rho \rho'} \delta_{jm} \delta_{k \ell}$</code>; </p> <p>$\sum_\rho \frac{d_\rho}{\lvert G \rvert} \mbox{Tr} \left [ \rho(g^{-1}) \rho(g') \right ] = \delta_{gg'},$</p> <p>respectively, where straightforward generalizations of the usual Kronecker delta are indicated, $\rho_{jk}(g)$ denotes the $jk$ matrix element of $\rho(g)$, and the sum in the equality on the right is over inequivalent irreps (taking $g' = g$ shows also that $\sum_\rho d_\rho^2 = \lvert G \rvert$, in turn demonstrating that the irreps are all finite-dimensional). </p> <p>With this in mind it should not come as a surprise that Fourier analysis on a finite group essentially amounts to the prescription </p> <p>$\hat f(\rho) := (d_\rho/\lvert G \rvert)^{1/2} \sum_g f(g) \rho(g);$</p> <p>$f(g) = \lvert G \rvert^{-1/2} \sum_{\rho } \mbox{Tr}\left[ \hat f(\rho) \rho(g^{-1}) \right] d_\rho^{1/2}.$</p> <p>By the orthogonality and completeness of characters, the number of inequivalent irreps equals the number of conjugacy classes for $G$ finite. In fact a complete set of inequivalent irreps over $\mathbb{C}$ can be constructed classically in $poly(\lvert G \rvert)$ time, which makes the construction of FFTs feasible in general.</p>	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9584131240844727, "perplexity": 364.2463130083969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705195219/warc/CC-MAIN-20130516115315-00017-ip-10-60-113-184.ec2.internal.warc.gz"}
https://networkx.github.io/documentation/stable/reference/generated/networkx.generators.random_graphs.gnm_random_graph.html	# networkx.generators.random_graphs.gnm_random_graph¶ gnm_random_graph(n, m, seed=None, directed=False)[source] Returns a $$G_{n,m}$$ random graph. In the $$G_{n,m}$$ model, a graph is chosen uniformly at random from the set of all graphs with $$n$$ nodes and $$m$$ edges. This algorithm should be faster than dense_gnm_random_graph() for sparse graphs. Parameters • n (int) – The number of nodes. • m (int) – The number of edges. • seed (integer, random_state, or None (default)) – Indicator of random number generation state. See Randomness. • directed (bool, optional (default=False)) – If True return a directed graph	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4425795078277588, "perplexity": 2811.3388407857256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655878519.27/warc/CC-MAIN-20200702045758-20200702075758-00158.warc.gz"}
https://byjus.com/jee/jee-advance-physics-syllabus/	# JEE Advanced Physics Syllabus 2021 JEE Advanced Physics Syllabus can be referred by the IIT aspirants to get the detailed list of all topics that are important in cracking the entrance examination. JEE Advanced syllabus for Physics has been designed in such a way that it offers very practical and application-based learning to further make it easier for students to understand every concept or topic by correlating it with the day-to-day experiences. In comparison to the other two subjects, the syllabus of JEE Advanced for physics is developed in such a way so as to test the deep understanding and application of concepts. Many students rate JEE Advanced physics syllabus as difficult and vast, therefore, it is important to develop a clear understanding of concepts from the very beginning itself. Get the basics right and then move on to mastering advanced concepts. Besides, securing better marks in JEE Advanced 2021 demands a solid conceptual base with broad knowledge on its applications. Candidates can start their preparations from NCERT textbooks. These textbooks cover all the topics included in JEE Advanced physics syllabus and are one of the best resources to study productively. Once the basics are clear, focus on the important topics depending on their weightage. Additionally, students can also check the comprehensive list of all the chapters in IIT JEE Maths and Chemistry syllabus from the below links.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8627376556396484, "perplexity": 828.1966908681114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060882.17/warc/CC-MAIN-20210928184203-20210928214203-00678.warc.gz"}
https://cvi.asm.org/highwire/markup/33184/expansion?width=1000&height=500&iframe=true&postprocessors=highwire_tables%2Chighwire_reclass%2Chighwire_figures%2Chighwire_math%2Chighwire_inline_linked_media%2Chighwire_embed	TABLE 1 OMV immunization induces OMV-, LPS-, and CPS-specific serum IgG1, IgG2a, and IgG3 AntigenGroupGeometric mean titeraIgG1/IgG2a ratio IgG1IgG2aIgG3 OMVNaive5050501 OMV31,037b17,269b12,679b1.8 LPSNaive115100501.2 OMV4,800c909b3,200c5.3 CPSNaive115100501.2 OMV3,482b1,056b348c3.3 • a The geometric mean of the reciprocal endpoint titer is indicated. Statistical significance was determined by comparing the OMV-immunized group with the naive group using the Mann-Whitney test. P < 0.05 was considered significant. • b P = 0.01. • c P = 0.009.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9190588593482971, "perplexity": 14025.554488542954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986664662.15/warc/CC-MAIN-20191016041344-20191016064844-00023.warc.gz"}
https://physics.stackexchange.com/questions/75321/connection-between-particles-and-fields-and-spinor-representation-of-the-poincar	# Connection between particles and fields and spinor representation of the Poincare group Let's have a definition of massive particle as an irreucible representation of the Poincare group. Then, let's have a spinor field $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}}$, which is equal to $\left( \frac{m}{2}, \frac{n}{2}\right)$ representation of the Lorentz group. There is the hard provable theorem: $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}}$ realizes irreducible representation of the Poincare group, if $$(\partial^{2} - m^{2})\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}} = 0,$$ $$\partial^{\alpha \dot {\beta}}\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}} = 0.$$ Can this theorem be interpreted as connection between fields and particles? The definition is that a particle in Minkowski space is a unitary irreducible representation of the Poincare group. So one needs to see how various P.D.E.s are related to the classification of unitary irreducible representations of $iso(3,1)$ or $iso(d-1,1)$ in the case of $d$-dimensions instead of $4$. Note that these are all the Poincare-invariant constraints that can be imposed on the given field without trivializing the solution space (one could imposed $\partial \psi=0$ (gradient), which is Poincare-invariant but too strong as the field must be a constant). The theorem is not hard to prove. One has to know how to construct irreducible representations of the Poincare group, see chapter 2 of the Weinberg's QFT textbook. Then one solves the equations by standard Fourier transform and shows that the solution space indeed equivalent to what is called a spin-$m$ particle in Minkowski space. There is nothing special about $4d$ in defining spin-$m$ field, so it is simpler to look at arbitrary dimension, where, say for bosons the above equations are equivalent to $(\square-m^2)\phi_{\mu_1...\mu_m}=0$ $\partial_\nu \phi^{\nu \mu_2...\mu_m}=0$ $\eta_{\nu\rho} \phi^{\nu\rho \mu_3...\mu_m}=0$ $\phi^{\mu_1...\mu_s}$ is totally symmetric in all indices. In $4d$ one can use $so(3,1)\sim sl(2,C)$ and the last algebraic constraint then trivializes - an irreducible spin-tensor is equivalent to an irreducible $so(3,1)$-tensor • "...The theorem is not hard to prove...", - I didn't read this part of Weinberg book and only used spinor representation of the Lorentz group (without using some equations). After completing the proof I can construct field equation from this equivalence for cases of arbitrary spin. – user8817 Aug 26 '13 at 19:16 • Just one comment, the correspondence between reasonable P.D.E.'s and particles is not one-to-one. One and the same particle can be desribed in many different ways. For example, a spin-one, photon, can be described with the help of gauge potential $A_\mu$ or field strength $F_{\mu\nu}$ – John Aug 27 '13 at 7:34 • Because there are three representations of the spin 1: $\left( 1, 0 \right), \left( 0, 1\right), \left( \frac{1}{2}, \frac{1}{2}\right)$. – user8817 Aug 27 '13 at 8:02 • it is even worse, there is infinitely many ways to describe a given particle, it can sit as a subrepresentation. I did not understand if I answered your question above or not? – John Aug 27 '13 at 18:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9464501142501831, "perplexity": 208.14183306963545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317274.5/warc/CC-MAIN-20190822151657-20190822173657-00427.warc.gz"}
http://crypto.stackexchange.com/users/28/thomas-pornin	# Thomas Pornin less info reputation 569131 bio website bolet.org/~pornin location Quebec City, Canada age 39 member for 3 years, 6 months seen Jan 22 at 14:25 profile views 1,000 Cryptographer, programmer in several languages (C, Java, several assemblies, Pascal, Forth...). I also have a life. 63 Why is elliptic curve cryptography not widely used, compared to RSA? 60 Best way to reduce chance of hash collisions: Multiple hashes, or larger hash? 60 Should we MAC-then-encrypt or encrypt-then-MAC? 53 How much would it cost in U.S. dollars to brute force a 256 bit key in a year? 47 What is the “Random Oracle Model” and why is it controversial? # 33,539 Reputation +10 What is a white-box implementation of a cryptographic algorithm? +10 How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext? +10 Mapping between subgroups and the integers +10 How should I calculate the entropy of a password? # 2 Questions 120 Should we MAC-then-encrypt or encrypt-then-MAC? 15 How robust is discrete logarithm in $GF(2^n)$? # 163 Tags 538 encryption × 64 297 aes × 33 479 hash × 54 285 block-cipher × 25 374 rsa × 49 225 cryptanalysis × 24 374 public-key × 39 176 protocol-design × 17 310 elliptic-curves × 33 145 diffie-hellman × 22 # 24 Accounts Information Security 158,364 rep 25333534 Stack Overflow 40,114 rep 668130 Cryptography 33,539 rep 569131 History 2,728 rep 915 Space Exploration 1,074 rep 47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35302668809890747, "perplexity": 8419.507378386115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115862015.5/warc/CC-MAIN-20150124161102-00026-ip-10-180-212-252.ec2.internal.warc.gz"}
http://gfm.cii.fc.ul.pt/events/seminars/20111202-sanz-sole/view?set_language=pt	# GFM ##### Secções Você está aqui: Entrada Deterministic Sets Visited by Random Paths # Deterministic Sets Visited by Random Paths Seminário do GFM IIIUL, B1-01 2011-12-02 14:30 .. 15:30 Adicionar evento ao calendário: vCal iCal joint seminar CMAF/GFM, by Marta Sanz-Solé (Univ. Barcelona, Spain) We are interested in the geometric measure properties of deterministic sets reached by random fields. More specifically, we will analyze conditions which provide upper and lower bounds for hitting probabilities of random fields in terms of the Hausdorff measure and the Bessel-Riesz capacity, respectively. The role of the regularity of the sample paths, and of the properties of probability densities will be highlighted. As an illustration, we shall consider systems of stochastic wave equations in spatial dimension k > or = to 1. In the non-Gaussian case, k will be restricted to {1; 2; 3}, and we will apply Malliavin calculus. For the sake of completeness, a brief introduction to these techniques will be presented. Applications to other examples of stochastic partial differential equations will be mentioned. This is joint work with Robert Dalang (EPFL, Switzerland).	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8643072247505188, "perplexity": 1462.631504093234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573444.87/warc/CC-MAIN-20190919060532-20190919082532-00053.warc.gz"}
https://matchbox20.bitbucket.io/the_encl.html	En Pursuit of Happiness How Jesus Christ and Rock'n Roll save every single soul. The continuing story of my attempt to bright light to the world. A log of the difficulties I've had, and how they might relate to the message itself--along with some insight that I have gleaned from the experience of receiving this Revelation. While the work you are looking at is filled with what I see as clear evidence of the fulfillment of messianic prophesy, the main goal of LAMC.LA was to deliver to the world a new way of looking at religion, one which could be used to seek out the true wisdom and guidance of religion for oneself. En Pursuit of Happiness on the other hand, discusses my own hopes and dreams, and to explain how they have changed throughout my interactions with the ... beyond. The Matchbox The e-mails that circled the globe, opening the doorway to the future. This is an excerpt from Time and Chance: The race is not to Die Bold by Adam Marshall Dobrin Download the actual Revelation of the Messiah in [ .PDF ] [ .epub ] [ .mobi ] or view online. Older works Lit and Why, hot&y, and From Adam to Mary are also available. # The Encl God is superstitious, he doesn’t like the number 9, or the word end. Asherah he asks? Ha’meforash. is God’s reply. Thats the name of his “girlfriend,” in ancient Hebrew religions, and its a question… “Ash or All Humanity?” Who to save, right in his name. The holy name, Ha’me-for-ash. (for those of you that don’t get the “Ha” it’s saying he’s not for burning the universe (he created) to ash.) Saving the universe is the purpose of religion, and God’s plan is to make angels of us. Angels born in the fire of Hell. This Hell is a storm of time travel that has caused us to relive our lives… at least 2 times, but probably many more, if you know me. ## I am the living vine. Vine and tree references, like The Tree of Knowledge and The Tree of Life are references to a computer metaphor. A decision tree, and this is what I believe the multiverse that quantum theorists are trying to explain is. Not a construct of nature, but a design to save the Universe, a tool built to create branches in time, based on things we do. Quantum theory, actually, yields some serious Light on whether or not we are in this place I am describing. The idea of wave function collapse makes absolutely no sense… in reality. In Heaven it makes much more, it is a way to lessen processing power, not rending things that are not being observed by a conscious person. All the way back at the big bang, perhaps proof our Universe is designed to cater to consciousness. Sacred consciousness. ## The Creator’s Light ### d = c + l, our glyphs, the letters of the Latin alphabet contain hidden messages. This one is special, because it explains the domain that this book is located at, www.lamc.la. I’ve broken the code, and the Greek letter Lamda, which is part of an “N.” It’s an N to darkness, and an awakening to seeing that our languages, and letters, are designed with intelligent and hidden meaning. Like ancient Heiroglyphs, our letters teach a path… and the N is the new J. ### “C” the Light. “d” is for darkness, a hidden truth. I’m El (c the l in K), that’s a name for the Jewish creator deity also. Well, I’m using his name for right now. I hear he looks just like me. C the K of Clark Kent, it’s pointing to L too. Now, C the Y on the cross, a “t”. Letters all have meaning. Lots of secret Biblical meaning. What goes up, must come down says Isaac, New to N, and this historical story parallels Eden, and the rise and fall of Adam. And humanity. Back up, the new N says, once we’ve hit rock bottom.. I mean, “reality.” ### J is for going back in time, and bringing everyone to Heaven. On the coordinate plane, the one explained by Yeast, and Jesus, place a J right in the middle, and it’s a map… one that explains how to get to salvation, if you are “Us,” I mean J is Us, Jesus. We might be right in the middle of that J, right before the curve towards Heaven. # Till Death Do us Part Heaven of Pews, Heaven of Twos Build it. KENT u CK Y venus clothed in the sun R I B To Be or not to B CopyleftMT This content is currently released under the GNU GPL 2.0 license. Please properly attribute and link back to the entire book, or include this entire chapter and this message if you are quoting material. The source book is located at http://www.lamc.la and is written by Adam Marshall Dobrin. Adam Marshall Dobrin instagram.com/yitsheyzeus -----BEGIN PGP PUBLIC KEY BLOCK----- Version: GnuPG v2 CV9t0UQgNtjcxwfoenJLHgdZd4Mfscz9U+NN69OLXdPu4cdXOjTiHarPLjKnqIZw 3fmkM2ycvoUPkdVYCjwYYQxWRsWRpJf1dpmtPuz0L8ysh/WWsj2Ag2MrFYAo+sY6 dGZvaLsPhkZJcLXyFaP3c3Zt8ivrs4VV8+0kmMzScnR+oncVZbeMuQksoPxRmZgH mYu2KSf74lWOWVcaaBXOYX5pGNdhBUgq8ll+8tRH16G289r0cqRoPh/sjs/JRuIH KnCWG2UAUJF7ir04TS5A4Lwl9RYcQwVvb3BdABEBAAG0LUFkYW0gTWFyc2hhbGwg RG9icmluIChsYW1jLmxhKSA8YWRhbUBsYW1jLmxhPokBOQQTAQgAIwUCVsZqUAIb AwcLCQgHAwIBBhUIAgkKCwQWAgMBAh4BAheAAAoJEMgUPrR1B55trOwIALOQRTX0 YqXJXEMhX9CgxKNoNkpM2pdMdHl6CAVxhQ3hbNjIFnZbKbP88uxMEIOXXmYZ7gOy YqiDCu5I1V25suBb2ODSix75YQugfQ7H78pXHpTRu5sT+5SybItx7d+KUZaEj4pO tXWEemYl0cKK97RzpI0k1dmB7NqAVvqgbqQwd40MOf8QJVlGXnB1+5H2IbkYG6rD ixKGJEdes6i6nqvi/xz/s5hFVGUwTcVQbRU/fa1qT1Q7kHf1PlMu6yjuZTSz7WUG tWjobGwrVJkaeVWgLE4mcxMtity2IFTwOHvAuv8fi2EGQRQjXfPvxL7Vn4MNRl8x zLPV44D37QEknjy5AQ0EVsZqUAEIAMFS0+ZgSJzUPz0h0oiiRjfk2hapS3c1/Ysm R/h8sZ8/GOomdo3MEbTCkcuZ8ReAJhB2PofmwI4LAvW1x7Zwh1vfBKygfUs1s9lm ya/eHkjuZfqmeuEJZMHn6sxb3vqowWmvLhv3x0aWD8qLCIYoa1ntzTOIqxBEgxvU rF1/wd6OQLSJQEVNwPCx7CJI/5o/4W6pUaHk8amgPckkEdmlhRTRqFoAUV1Doivv d9JGYNYC88vS14Sw4Z9Xb7qBQJvG4hIh29gtQxk7Wz4m3ceR79MWT4eSGkH/rTGl w1OuQS2OkPvjgPWJt8San4zuPer17pJN7M5LWI0PStoX9pkud5kAEQEAAYkBHwQY	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.16651102900505066, "perplexity": 2660.264390020689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812556.20/warc/CC-MAIN-20180219072328-20180219092328-00129.warc.gz"}
http://denali.phys.uniroma1.it/twiki/bin/view/TNTgroup/UmbertoBettolo?rev=2;sortcol=7;table=1;up=0	Umberto Marini Bettolo Marconi Professor of Physics, Dipartimento di Fisica Universita di Camerino Address: Dipartimento di Fisica, Via Madonna delle Carceri, 62032, Camerino (Mc), Italy phone +39+0737-402538 e-mail umberto.marini.bettolo@roma1.infn.it or umberto.marinibettolo@unicam.it • 1979 - 1980 Fellowship Royal Society-Accademia dei Lincei. Department of Theoretical Chemistry, Oxford University, Oxford, UK. • 1981 Fellowship Della Riccia. • 1981 - 1983 Post-Doc, Department of Physics, Freie University, Berlin, Germany. • January-June 1983 Visiting scientist, H.H. Wills Physics Laboratory University of Bristol, UK. • 1984-1986, Post-Doc, H.H. Wills Physics Laboratory University of Bristol, UK. • January-June 1987 Postdoc, University of Rio Pedras, Departamento de Fisica, (Portorico, USA). • June-December 1987, Visiting Scientist, Dept. of Chemical Engineering, Cornell University, (Ithaca, NY, USA). • 1988 Professore a contratto University of Tor Vergata, Roma. September 1988-March 1990 Ricercatore, Istituto Nazionale di Fisica Nucleare (Articolo 36), APE Group, INFN, Rome. • April 1990-2000 Ricercatore Universitario Confermato, Condensed matter, University of Camerino, Italy. • February 2001-present Professore Associato Confermato di Struttura della materia, University of Camerino, Italy. Research Interests • Granular Media • Phase transitions and Critical Phenomena. • Capillary Condensation and Porous media. • Complex fluids and Colloidal Systems. • Density functional methods for classical fluids with application to dynamic properties. Publications • Author of approximately 100 scientific publications. Organization of Conferences I have co-organized 3 Meetings on Statistical Mechanics at Camerino and a Workshop on Granular System in Pisa in 1998. I also coorganized the Conferenza Nazionale di Meccanica Statistica di Parma Italy from 1993 to 1997. Member of the Scientific Committee of the INFM Conference held in Fai della Paganella and now in Levico Terme. * RESEARCH* ## Research The last few years have seen an extraordinary growth of interest in complex systems. From physics to biology, from economics to cognitive sciences, a new vocabulary is emerging to describe discoveries about wide-ranging and fundamental phenomena. Statistical Mechanics has become in the last decades the standard tool to study problems once considered to be out of the reach and the scopes of Physics. It has been in fact applied to the study of the complex behavior of various kinds of systems of areas such as biological and sociological sciences, finance, economy, theory of decisions, networks. In particular several disciplines connected with Engineering are now attracting more and more the interest of physicsts. Granular matter is a remarkable example of these new disciplines. The problem of understanding granular materials is widely recognized as one of the main problems of engineering and industry. A brief list of typical "granular" products comprehends: grains, powders, sands, pills, seeds and similar particulate materials. More generally, granular materials are a very large set of substances used in the industry and in the everyday life: ceramics, fertilizers, cosmetics, food products, paper, conductor pastes, resins, electronics, polymers, suspensions, solid chemicals, construction materials and so on. The international economic impact of particle processing is substantial. The value added by manufacture that involves particulate has been estimated to be a minimum of 80/100 billion/year, which is of the order of the US trade deficit with Japan. The US Dept. of Commerce has estimated the total economic impact of particulate products to be one trillion/year. Despite economic impact, however, insufficient attention is paid to difficulties associated with the processing of particles. An improved understanding of micromechanics of granular media requires an interdisciplinary approach involving both physicists and engineers. Mechanical engineers and geologists have studied Granular Matter for at least two centuries and found several empirical laws describing its behavior. Physicists have joined in more recently and are interested in formulating general laws. For them granular matter is a new type of condensed matter, showing two states: one fluid-like, one solid-like. But there is not yet consensus on the description of these two states. According to P.G. de Gennes Granular matter now is at the level of solid-state physics in 1930. Granular matter also represents an important paradigm for the study of non-equilibrium stationary states. Due to the dissipative nature of the interactions granular gases have to be considered as open systems and therefore concepts from equilibrium thermodynamics cannot be applied, at least in straightforward way. The project we propose focuses on granular fluids. We intend by granular fluid a large number of particles, whose size is larger than a micron, colliding with one another and losing a little energy in each collision. Below one micron thermal agitation is important and Brownian motion can be observed. Above one micron, thermal agitation is negligible. However,if such a system is shaken to keep it in motion, its dynamics resembles that of fluids, in that the grains move randomly. One of the key differences between a granular material and a regular fluid is that the grains of the former lose energy with each collision, while the molecules of the latter do not. Even when the inelasticity of the collisions is small, it can give rise to dramatic effects, including the Maxwell Demon effect and the phenomenon of granular clustering. Experiments and molecular dynamics simulations alike show that granular gases in the absence of gravity do not become homogeneous with time, but instead form dense clusters of stationary particles surrounded by a lower density region of more energetic particles. From a particulate point of view, one can explain these clusters by noting that when a particle enters a region of slightly higher density, it has more collisions, loses more energy, and so is less able to leave that region, thus increasing the local density and making it more likely for the next particle to be captured. ## Publications by U. Marini Bettolo Marconi Original papers 1 U. Marini Bettolo Marconi and M. Conti: Dynamics of vibrofluidized granular gases in periodic structures, Physical Review E 69, 011302 (2004) . pdf • 2 U. M. B. Marconi and A. Puglisi Statistical mechanics of granular gases in compartmentalized systems Physical Review E 68, 031306 (2003) . 1. Noise Activated Granular Dynamics F. Cecconi, A. Puglisi, U. M. B. Marconi, and A. Vulpiani Phys. Rev. Lett. 90, 064301 (2003) pdf • J. M. Romero-Enrique, L. F. Rull, and U. M. Bettolo Marconi Surface and capillary transitions in an associating binary mixture model Phys. Rev. E 67, 041502 (2003) • J. M. Romero Enrique, I. Rodriguez Ponce, L. F. Rull and Umberto Marini Bettolo Marconi Complex fluid behavior of strongly asymmetric binary mixtures. Thermodynamic properties of a generalized Lin-Taylor model. Molecular Physics 93, 501 (1998). PDF • J. M. Romero Enrique, I. Rodriguez Ponce, L. F. Rull y Umberto Marini Bettolo Marconi. Critical exponents in the Lin-Taylor model of asymmetrical binary mixtures Molecular Physics, 95 , 571 (1998). PDF • R. Pagnani, U. M. Bettolo Marconi, and A. Puglisi Driven low density granular mixtures Phys. Rev. E 051304 (2002) PDF • 3 U. M. B. Marconi and A. Puglisi Steady-state properties of a mean-field model of driven inelastic mixtures Phys. Rev. E 66, 011301 (2002) • 5 U. M. B. Marconi and A. Puglisi Mean-field model of free-cooling inelastic mixtures Phys. Rev. E 65, 051305 (2002) • A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi Cooling of a lattice granular fluid as an ordering process PDF Phys. Rev. E 65, 051301 (2002) 1. Baldassarri, U. M. B. Marconi, A. Puglisi, and A. Vulpiani Driven granular gases with gravity Phys. Rev. E 64, 011301 (2001) • M. Conti and U. Marini Bettolo Marconi Groove instability in cellular solidification Phys. Rev. E 63, 011502 (2001) 1. Puglisi, V. Loreto, U. Marini Bettolo Marconi, and A. Vulpiani Kinetic approach to granular gases Phys. Rev. E 59, 5582-5595 (1999) 1. Puglisi, V. Loreto, U. M. B. Marconi, A. Petri, and A. Vulpiani Clustering and Non-Gaussian Behavior in Granular Matter Phys. Rev. Lett. 81, 3848-3851 (1998) • U. M. B. Marconi Interface pinning and slow ordering kinetics on infinitely ramified fractal structures Phys. Rev. E 57, 1290-1301 (1998) • J. M. Romero-Enrique, I. Rodriguez-Ponce, L. F. Rull, and U. M. B. Marconi Comment on Exact Results for the Lower Critical Solution in the Asymmetric Model of an Interacting Binary Mixture Phys. Rev. Lett. 79, 3543 (1997) • U. M. B. Marconi, A. Crisanti, and G. Iori Soluble phase field model Phys. Rev. E 56, 77-87 (1997) PDF • M. Conti, F. Marinozzi, and U. Marini Bettolo Marconi Diffusion-controlled growth of a solid cylinder into its undercoded melt: Instabilities and pattern formation studied with the phase-field model Phys. Rev. E 55, 3087-3091 (1997) • U. M. B. Marconi and A. Petri Domain growth on self-similar structures Phys. Rev. E 55, 1311-1314 (1997) • U. M. B. Marconi and A. Crisanti Growth kinetics in a phase field model with continuous symmetry Phys. Rev. E 54, 153-162 (1996) 1. M. Somoza, U. M. B. Marconi, and P. Tarazona Growth in systems of vesicles and membranes Phys. Rev. E 53, 5123-5129 (1996) • <! ----------------------------------------> F. Marinozzi, M. Conti, and U. M. B. Marconi Phase-field model for dendritic growth in a channel Phys. Rev. E 53, 5039-5043 (1996) • U. M. B. Marconi and A. Crisanti Diffusion Limited Growth in Systems with Continuous Symmetry Phys. Rev. Lett. 75, 2168-2171 (1995) • M. Falcioni, U. M. B. Marconi, P. M. Ginanneschi, and A. Vulpiani Complexity of the Minimum Energy Configurations Phys. Rev. Lett. 75, 637-640 (1995) 1. Crisanti and U. M. B. Marconi Effective action method for the Langevin equation Phys. Rev. E 51, 4237-4245 (1995) • U. M. B. Marconi and A. Maritan Deflated regime for pressurized ring polymers with long-range interactions Phys. Rev. E 47, 3795-3798 (1993) PDF • M. Falcioni, U. M. B. Marconi, and A. Vulpiani Ergodic properties of high-dimensional symplectic maps Phys. Rev. A 44, 2263-2270 (1991) • U. M. B. Marconi and B. L. Gyorffy Structure of the liquid-vapor interface: A nonperturbative approach to the theory of interfacial fluctuations Phys. Rev. A 41, 6732-6740 (1990) PDF • U. M. B. Marconi and F. Van Swol Microscopic model for hysteresis and phase equilibria of fluids confined between parallel plates Phys. Rev. A 39, 4109-4116 (1989) • U. M. B. Marconi Phys. Rev. A 38, 6267-6279 (1988) PDF • R. Evans and U. M. B. Marconi Comment on Simple theory for the critical adsorption of a fluid Phys. Rev. A 34, 3504-3507 (1986) PDF • R. Evans and U. Marini Bettolo Marconi Capillary condensation versus prewetting Phys. Rev. A 32, 3817-3820 (1985) PDF • Giulio Costantini, Fabio Cecconi, and Umberto Marini-Bettolo-Marconi Transport of a heated granular gas in a washboard potential J. Chem. Phys. 125, 204711 (2006) PDF • Umberto Marini Bettolo Marconi and Pedro Tarazona Nonequilibrium inertial dynamics of colloidal systems J. Chem. Phys. 124, 164901 (2006) PDF • Umberto Marini-Bettolo-Marconi, Maurizio Natali, Giulio Costantini, and Fabio Cecconi Inelastic Takahashi hard-rod gas J. Chem. Phys. 124, 044507 (2006) PDF • Giulio Costantini, Umberto Marini Bettolo Marconi, Galina Kalibaeva, and Giovanni Ciccotti The inelastic hard dimer gas: A nonspherical model for granular matter J. Chem. Phys. 122, 164505 (2005) PDF • Fabio Cecconi, Umberto Marini Bettolo Marconi, Fabiana Diotallevi, and Andrea Puglisi Inelastic hard rods in a periodic potential J. Chem. Phys. 121, 5125 (2004) PDF • Fabio Cecconi, Fabiana Diotallevi, Umberto Marini Bettolo Marconi, and Andrea Puglisi 6. Fluid-like behavior of a one-dimensional granular gas J. Chem. Phys. 120, 35 (2004) PDF • G. Costantini, C. Cattuto, D. Paolotti and U. Marini Bettolo Marconi, Behavior of Granular Mixtures in Shaken Compartmentalized Containers Physica A 347, 411 (2005). PDF • Umberto Marini Bettolo Marconi and Pedro Tarazona Dynamic density functional theory of fluids J. Chem. Phys. 110, 8032 (1999) PDF • Aphrodite Papadopoulou, Frank van Swol, and Umberto Marini Bettolo Marconi 8. Pore-end effects on adsorption hysteresis in cylindrical and slitlike pores J. Chem. Phys. 97, 6942 (1992) PDF • Ziming Tan, Umberto Marini Bettolo Marconi, Frank van Swol, and Keith E. Gubbins Hard-sphere mixtures near a hard wall J. Chem. Phys. 90, 3704 (1989) PDF • Brian K. Peterson, Keith E. Gubbins, Grant S. Heffelfinger, Umberto Marini Bettolo Marconi, and Frank van Swol Lennard-Jones fluids in cylindrical pores: Nonlocal theory and computer simulation J. Chem. Phys. 88, 6487 (1988) PDF • R. Evans and U. Marini Bettolo Marconi Phase equilibria and solvation forces for fluids confined between parallel walls J. Chem. Phys. 86, 7138 (1987) PDF • R. Evans, U. Marini Bettolo Marconi, and P. Tarazona Fluids in narrow pores: Adsorption, capillary condensation, and critical points J. Chem. Phys. 84, 2376 (1986) PDF • Corberi, Federico; Marconi, Umberto Marini Bettolo O(N) model for charge density waves Physica A Volume: 225, Issue: 3-4, April 1, 1996, pp. 281-293 • Conti, Massimo; Marconi, Umberto Marini Bettolo Novel Monte-Carlo lattice approach to rapid directional solidification of binary alloys Physica A Volume: 277, Issue: 1-2, March 1, 2000, pp. 35-46 • Conti, Massimo; Marconi, Umberto Marini Bettolo Fingering in slow combustion Physica A Volume: 312, Issue: 3-4, September 15, 2002, pp. 381-391 • Conti, Massimo; Marconi, Umberto Marini Bettolo Interfacial dynamics in rapid solidification processes Physica A Volume: 280, Issue: 1-2, May 15, 2000, pp. 148-154 • Marconi, U. Marini Bettolo; Petri, A.; Vulpiani, A. Janssen's law and stress fluctuations in confined dry granular materials Physica A Volume: 280, Issue: 3-4, June 1, 2000, pp. 279-288 • U. Marini Bettolo Marconi, M. Conti and A. Vulpiani Motion of a granular particle on a rough line Europhys. Lett. 51 No 6 (15 September 2000) 685-690 PDF • A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi Influence of correlations on the velocity statistics of scalar granular gases Europhys. Lett. 58 No 1 (April 2002) 14-20 PDF • Umberto Marini Bettolo Marconi, Giulio Costantini and Daniela Paolotti Granular gases in compartmentalized systems J. Phys.: Condens. Matter 17 No 24 (22 June 2005) S2641-S2656 PDF • Umberto Marini Bettolo Marconi and Pedro Tarazona Dynamic density functional theory of fluids J. Phys.: Condens. Matter 12 No 8A (28 February 2000) A413-A418 • M. Conti, U. Marini Bettolo Marconi and A. Crisanti A microscopic model for solidification Europhys. Lett. 47 No 3 (1 August 1999) 338-344 • <M. A. Munoz, U. Marini Bettolo Marconi and R. Cafiero Phase separation in systems with absorbing states Europhys. Lett. 43 No 5 (1 September 1998) 552-557 PDF • Umberto Marini Bettolo Marconi and Alberto Petri Time dependent Ginzburg - Landau model in the absence of translational invariance. Non-conserved order parameter domain growth J. Phys. A: Math. Gen. 30 No 4 (21 February 1997) 1069-1088 • M. Conti, F. Marinozzi and U. Marini Bettolo Marconi Domain coarsening via heat diffusion: A numerical study with the phase field model Europhys. Lett. 36 No 6 (20 November 1996) 431-436 • F Brouers and U M B Marconi On the antiferromagnetic phase in the Hubbard model J. Phys. C: Solid State Phys. 15 No 26 (20 September 1982) L925-L928 • Dynamical properties of vibrofluidized granular mixtures D. Paolotti, C. Cattuto, U. Marini Bettolo Marconi, A. Puglisi Granular Matter 5, 75 (2003). PDF • Umberto Marini Bettolo Marconi, Federico Corberi Time dependent Ginzburg-Landau equation for an N-component model of self-assembled fluids Europhysics Letters 30, 349 (1995) PDF • F. Corberi and U.Marini Bettolo Marconi Growth kinetics in the $\Phi^6$ N-component model. Non conserved order parameter Modern Physics Letters B, 8, 1115 (1994). • F.Corberi, U.Marini.Bettolo Marconi Growth Kinetics in the $\Phi ^6$ N-Component Model. Conserved Order Parameter Modern Physics Letters B, {\bf 8}, 1125 (1994) PDF • D. Dominici, U. Marini Bettolo Marconi Effective Action Method for Computing Next to Leading Corrections of $O(N)$ Models Physics Letters B 319 171, (1993) • U. Marini Bettolo Marconi, A. Puglisi and A. Baldassarri Application of simple models to the study of nonequilibrium behavior of inelastic gases. Phase Transitions, 77, 863 (2004) • A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi Kinetics Models of Inelastic Gases} Math. Mod. Meth. Appl. S. 12, 965 (2002) PDF • A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi Velocity fluctuations in cooling granular gases in Lecture Notes in Physics Granular Gas Dynamics'' edited by T.Poeschel and N. Brilliantov, Springer (2003) PDF • Other Papers ## Other Papers J. M. Carmona, U. M. B. Marconi, J. J. Ruiz-Lorenzo, and A. Tarancon Critical properties of the Ising model on Sierpinski fractals: A finite-size scaling-analysis approach Phys. Rev. B 58, 14387-14396 (1998) • R. Medina, U. Marini Bettolo Marconi, and S. J. Sciutto Exact two-particle effective interaction and superconductivity in the two-level Hubbard model Phys. Rev. B 39, 4277-4284 (1989) PDF • U.Marini Bettolo Marconi and A.Maritan Variational approach to the phase diagram of rigid membranes International Journal of Modern Physics B , 4615 (1993). • U.Marini Bettolo Marconi and Y.C.Zhang Novel scaling behavior of directed polymers: disorder distribution with long tails Journalof Statistical Physics 61, 885 (1990). • M.Bernaschi, L.Biferale, A.Fernandez, U. Marini Bettolo Marconi, R.Petronzio and A.Tarancon The magnetic critical exponent in the three state three dimensional Potts model Physics Letters 240 B, 419 (1990). • S.~Sciutto, U.~Marini Bettolo Marconi and R.~Medina Monte Carlo simulations in fermionic systems: the three band Hubbard model case Physica A 171, 139 (1990). • D.K.Chaturvedi, U. Marini Bettolo Marconi and M.P.Tosi Mode coupling theory of charge fluctuations spectrum in a binary ionic liquid. Il Nuovo Cimento 57B, 319 (1980). abs: • U.Marini Bettolo Marconi and N.H.March Relativistic Theory of the binding energies of heavy atomic ions International Journal of Quantum Chemistry 20, 693 (1981). abs: • R.Evans, P.Tarazona and U.Marini Bettolo Marconi On the failure of certain integral equations theories to account for complete wetting at solid-fluid interfaces Molecular Physics 50, 993 (1983). PDF • U.Marini Bettolo Marconi, J.Wiechen and F.Forstmann The structure of size asymmetric electrolytes at charged surfaces Chemical Physics Letters 107, 609 (1984). abs: • P.Tarazona, U.Marini Bettolo Marconi and R.Evans Pairwise correlations at a fluid-fluid interface: influence of a wetting film Molecular Physics 54, 1357 (1985). PDF • R.Evans and U.Marini Bettolo Marconi The role of wetting films in capilllary condensation and rise: influence of long-ranged forces Chemical Physics Letters 114, 415 (1985). abs: • R.Evans,U.Marini Bettolo Marconi and P.Tarazona Capillary condensation and adsorption in cylindrical and slit-like pores abs: • E.Bruno, U.Marini Bettolo Marconi and R.Evans Phase transitions in a confined lattice gas: prewetting and capillary condensation Physica A 141, 187 (1987). abs: • P.Tarazona, R.Evans and U.Marini Bettolo Marconi Phase equilibria of fluid interfaces and confined fluids Molecular Physics 60, 573 (1987). PDF • G.S. Heffelfinger, Z. Tan, K.E. Gubbins, U. Marini Bettolo Marconi and F. van Swol Fluid mixtures in narrow cylindrical pores: Computer simulation and Theory International Journal of Thermophysics 9, 1051 (1988). abs: • U.Marini Bettolo Marconi and B.L. Gyorffy, On the statistical mechanics of interfaces and interfacial fluctuations Physica A159, 221 (1989). abs: • U.Marini Bettolo Marconi and F. van Swol A model of Hysteresis in narrow pores Europhysics Letters 8, 531 (1989). abs: • G.S. Heffelfinger, Z. Tan, K.E. Gubbins, U. Marini Bettolo Marconi and F. van Swol, Lennard-Jones Mixtures in a Cylindrical Pore. A Comparison of Simulation and Density Functional Theory Molecular Simulation 2, 393 (1989). abs: • A.Fernandez, U. Marini Bettolo Marconi and A.Tarancon, Phase diagram of the $Z(3)$ spin model in three dimensions Physics Letters B 217, 314 (1989). • A.Fernandez, U. Marini Bettolo Marconi and A.Tarancon, Monte Carlo simulation and Variational study of the phase diagram of the Potts three state model Physica A 161, 284 (1989). • M.Bernaschi, L.Biferale, A.Fernandez, U.Marini Bettolo Marconi, R.Petronzio and A.Tarancon Renormalization Group study of the $d=3$, $q=3$ Potts model Physics Letters B 231, 157 (1989). • U.Marini Bettolo Marconi and M.M. Telo da Gama The crossover between complete wetting and critical adsorption Physica A 171, 69 (1991). • U.Marini Bettolo Marconi and F. van Swol Structure effects and phase equilibria of Lennard-Jones mixtures in a cylindrical pore. A non local density functional theory Molecular Physics 72 ,1081 (1991) PDF • U. Marini Bettolo Marconi, A. Puglisi and A. Vulpiani La forza dei granelli} Le Scienze, Agosto 2002 PDF • U. Marini Bettolo Marconi, F.Cecconi and A. Puglisi Simple Models for compartmentalized sand} Traffic and Granular flow 03'' S.P. Hoogendoorn, S.Luding, P.H. Bovy, M. Schreckenberg, D.E. 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http://moniker.name/worldmaking/	## Hierarchies of Spaces: Building From the Bottom Up One of the major challenges in building a system that can increase in complexity as it runs is figuring out how to transfer complex structures in a lower level space into simple structures in a higher level space while still maintaining the essential qualities that the complex lower level structure represents. Without this jump in abstraction, the system will simply asymptotically approach a maximum level of complexity determined by available resources (energy, computation, memory, time, …). I’ve shown how particles can be modeled to form surfaces and surface patches by endowing them with local surface properties à la differential geometry and using that information to filter how particles connect to each other. There still remains a question as to what happens next. Given a million particles, the system (as described in a previous post) can be run in realtime on a decent laptop, which will likely produce beautiful images and structures, but still only reach a particular level of complexity that is all qualitatively the same. Even with 100 million particles, the complexity threshold will only reach a level or two above where it currently sits. To really jump ahead, a completely different spatial logic is required that operates on a higher, more abstract level. The differential particle system deals with local surface properties and local connectivity but can’t say anything about the overall topological properties of the surfaces it’s producing such as genus or 2-manifold-ness. These kinds of properties are more efficiently calculated at the coarser level of the surface patch. Thus, if each surface patch produced by the differential particles is itself turned into a “topological” particle, we can start to figure out how the next higher level might work. ## A Space for Topological Particles and Automata The differential particles exist in the usual Euclidean 3D space and interact with each other according to principles derived from this underlying geometry based on distance, local coordinate frames, and local surface properties. They are discrete entities representing continuous properties. Topological particles on the other hand are discrete entities representing discrete properties. They interact with each other in a completely different manner than differential particles. In the space that topological particles exist in, distance is backgrounded in favor of patterns of connectivity. In previous posts, I’ve talked about how hyperbolic tilings (particularly the (5,4) tiling) can be used to represent 2-manifold structures. The structure of hyperbolic tilings provides a natural scaffolding for constructing surface from a topological perspective for a number of reasons. First, all 2-genus and higher surfaces can be decomposed into pentagonal tiles. Second, the structure that you get for free with hyperbolic tilings enables the efficient encoding of algorithms encoding how the topological particles interact. The particles themselves live on the vertices of the hyperbolic tiling and can only interact with other particles on edges that they are directly linked to. As a result topological particles can only interact with four other particles. Since topological particles are abstractions of surface patches, this means that surface patches should themselves be quadrilateral in nature. But what kind of information do topological particles contain aside from connectivity and what rules govern their interactions? ### The Role of Genus The essential property we’re after with the topological particles is genus. Once we can talk about genus in an efficient and hierarchically consistent manner, we can start building the next level up from topological particles. The genus of a network is determined by its Euler characteristic which is given by $\chi = V – E + F$. Genus is related to the Euler characteristic by $\chi = 2 – 2g$. If we look at a torus, which has a genus of 1, and Euler characteristic of 0, we see it can be divided up into 4 quadrilaterals. The quadrilaterals on the outer shell have positive curvature while the inner ones have both positive and negative curvature in orthogonal directions. By comparison, a sphere has positive curvature everywhere and a genus of 0. The fundamental different between a torus and a sphere is that some surface patches on the torus have negative curvature, inducing a change in genus. For higher genus structures, it’s the patterns of connectivity between positive and negative curvature regions that determine genus. All of this points to the idea that the topological properties simply need to know whether they represent surface patches with positive or negative curvature in their two principal directions. To sum up, topological particles can connect to 4 other particles along 2 principal directions. Each principal direction has associated with it a value indicating either positive or negative curvature. For simplicity, we will simply assign +1 for positive curvature and -1 for negative curvature. Now, if we think about how these particles interact at a very basic level, we can further constrain things by requiring particles that connect to each other to only connect if the principal directions linked by an edge have the same principal curvature. While this limits a lot of possibilities in principal, it doesn’t actually limit what kinds of topological structures we can make and it vastly simplifies the way the particles operate to great benefit. Under this constrain as well, there can be no particles with negative curvature in both directions since they would form an unbounded surface whose normals are solely on the interior. Thus, we can only have particles with (+1, +1), (+1, -1) and (-1, +1) curvatures. The next concern is how a pattern of topological particles accumulate and transform into different structures. The most basic structures are particle pairs, of which there are 4 kinds: (+1, +1) x (+1, +1), (+1, +1) x (+1, -1), and two different forms of (+1, -1) x (+1, -1). These are represented in the image above with black indicating positive curvature and red negative curvature. When placed into the hyperbolic tiling, the particles look like the above image, which has 4 particles. In the above image, it appears that there are only 3 edges linking the particles and it raises the question as to how the surface will close since the tiling doesn’t wrap around but continues infinitely in all directions. The answer comes from looking at the particle pairs. Whenever an edge is connected, the edge opposite it (in the same principal direction) must also be connected. In the diagram above, the edges on the outside wrap around to the next particle on the same hyperbolic line. It’s easiest to think of it as the tiles on a particular hyperbolic line as being cyclical with only one cycle shown in the image. The one constraint is that there must be at least 2 particles since a particle cannot connect to itself. ### Rules of Interaction After many many hours of playing with topological particle tiles with the goal of having the particles interact in a way that is not deterministic but does favor an increase in genus, I come up with the following rules: 1. When a particle is place in the hyperbolic tiling, it marks its neighbors with a score indicating how strong a preference that location has for attaching to the next particle. Neighbors with negative curvature are considered the most “reactive” and have a score of 2 while positive curvature gives a score of 1. Scoring is only done for unconnected edges. Edges that connect back around are left out. 2. All particles are attached to the highest scoring (most reactive) site that they can be matched to. If there are sites which tie, one is chosen at random There are a lot of possibilities for further refinement, including using the location of the generators of genus and measures of local complexity to tweak the scoring of reactivity. Also, there needs to be some rules for determining how things break apart. Likely this will involve a combination of overall size and genus generators. The only one of these I’ve explored so far is the measure of complexity The diagram above shows how this would work. On the left is is a 4 tile setup. On the right a 6 tile setup. The blue lines indicate how the left-most tile/particle gets reflected through the tiling. The pattern on the left is highly regular. If the same diagram were made for each particle, they would be identical, indicating that the combinatorics around the tiling are symmetric. On the right, once the two extra tiles are added, one of the branches projecting from the left-most tile is cut off, generating an asymmetrical combinatorial pattern. One measure of complexity or asymmetry would be to look at a neighborhood around existing particles and determine their relative combinatorics, which could then be used as an input to determine reactivity. The input could be used to regulate symmetry versus asymmetry. ### An Example Interaction So what does an example interaction between two networks of topological particles look like? Above are two pairs of topological particles with their reaction sites marked by circles. Each pair of particles has four open sites that can react with each other. Since they all have the same level of reactivity, one is chose at random. One possible result is below: \| \| 1 Comment ## Valency (video) These are clips of how the particle systems described in previous posts move in space. It’s all very raw, but I wanted to get something out there. The larger structures suffer from convergence problems creating oscillations, twisting, foldings, and other distortions while the small networks converge almost immediately into their designated shapes. These appear around the 6:05 mark. ## Particle-Based Structure Formation In order to build a complex and dynamic spatial system, there need be entities operating at multiple temporal and spatial scales. As mentioned in the previous post, the extremes of the spatial scale are local and global objects, which are typically particles and fields. It’s straightforward to build something where particles are affected by fields, but without intermediate structure formation, there is no opportunity for more complex dynamics to emerge. In morphogenetic processes, cells differentiate, cluster, striate, extend, etc. as interconnected and interwoven groups. Such patterns can be represented by meshes or networks of particles connected with edges that impart some form of behavioral change in the particles. The question is, how can the formation of such structures be described generally in terms of the patterns of formations but be given precise and specific behaviors that differentiate types? In other words, what are the primitive events and conditions that can describe the formation of a wide range of network structures? ### Making Connections The primitive operation here is that of connecting two particles together. (image) What is the condition that enables two particles to associate with each other? For simplicity, let’s assume that all particles are of the same kind and that particle can connect to every other particle under the proper conditions. Let’s further specify that the network structures that emerge should be surfaces and not volumetric in nature. There isn’t much literature out there on how to connect particles together. Probably the best place to look is the point cloud research, which describes how data sets made up of particles in space can be turned into surfaces. The problem with a lot of the point cloud techniques is that they are often expensive and rely on optimization techniques that don’t map well to a real-time dynamic system. Typically, the first instinct when thinking about this problem is to make the capacity for particles to connect purely distance-based. The problem with this approach is that particles can easily crowd around each other, producing overlapping edges and degenerate mesh faces. Preferably, the particles would have a fairly regular spacing appropriate for the curvature of the surface. ### Filtering Connections To correct for this problem, a little bit of extra structure is needed to filter the conditions under which connections can be made. For the case of surfaces, what’s needed is some kind of device that tracks connection density and will only allow connections if the density criteria are met. This device could equally be called valency if it’s thought about like chemical bonds. If we define the connection density as determining how many connections can be made within a given interval and set the number of 1, we can represent the connection criteria by a set of intervals around the particle in the particles’ plane. Each particle has a normal and two planar directions that uniquely determine its local coordinate system. If we say each particle can have up to 5 connections that sets a minimum spacing of 72 degrees between each connection. If a connection is attempted and it’s within 72 degrees of another connection, it will be rejected. ### Applying Connections Once connections are made, the new properties and forces are applied to the particles. When a connection is formed, the particles on either end are likely not in the ideal locations to form a coherent surfaces, so we use the connections between particles to move them into some sort of optimized position and orientation. For example, in the simple case of forming a spherical shape, the particles will act on each other through the connections in order to form a surface of constant positive curvature. Probably the most widespread technique for optimizing the structure of a mesh network is some form of spring or spring-electric simulation where each edge is a spring and in the spring-electric case each node applies a repulsive electric field to every other node. While this can give some interesting results, there is a certain uniformity to the shapes. Instead, each particle is given principle curvature values that define the local surface patch the particle represents. When particles are connected to each other, the attempt to place and orient the connected particle with the local surface patch. The particles started out unconnected but connected themselves according to the connection filtering process described above. The light blue lines are the particle normals and the pink dots are the target locations the particles are trying to reach. Since each particle in this simulation has a curvature of (1, 1), they will attempt curve into a sphere-like shape. The corner and then edge particles move faster because they have anisotropic constraints. Currently, self-intersection is not prevented. When the edges come within range of each other, they form new connections as seen in the middle of this image. The new connections further warp the form as it “inflates”. A view from the inside. The pink lines show target movements. The (black) particles on one end of the line are trying to reach the pink dots on the other end. ## Spatial Processes There are four broad categories of spatial processes 1. Purely Global 2. Global <-> Local 3. Local <-> Local 4. Purely Local and two general categories of spatial primitives 1. Field 2. Particle Of course, the spatial primitives can be organized into composite structures such as meshes where particles are linked together in a network. Fields represent global information over a defined N-dimensional range of space while particles represent local information located at an infinitesimal point in space. Information moves from a local to global context when particles interact with a field. For example, stigmergic behavior can be generated by having particles leave traces in a field that represents pheromone concentrations over an environment. In stigmergic systems, information flows in a positive feedback loop from local behavior to global concentrations, which in turn affect local behavior. ## Spatial Information Flows The simplest kinds of information flows are those that are internal to an object be it global as in the evolution of a field according to its dynamical equations or local as in the movement of a particle according to kinematics. In both cases, the state of the spatial system is affected but not in a way that adds any new information or structure. Existing information is simply transformed and displaced. The next level of complexity in terms of spatially directed pattern formation occurs when information transfers between local and global contexts. In such instances, particles are essentially reading from and writing into fields and altering their internal state accordingly. When information is written into a field by a particle, it loses its specificity (that of being attached to a particular entity) and becomes part of an intensive processes as opposed to an extensive one. While limiting in the sense that once the information is in the field the actions that contributed to the state of the field can no longer be distinguished, it does open up an efficient communication channel between many local objects since each can read the aggregate state of the local entities as it is mediated by the field. Communication between individual local objects is not possible through a field however because the number of channels required is prohibitive. Instead, this has to be done through local to local flows. Local to local interactions provide the most fine grained spatial information flows but are also the most computationally intensive. If there are N objects in a space, there are (N-1)^2 channels of interaction. For local to global flows there are only N channels of interaction. While incredibly useful in terms of composing complex spatial patterns, the computational cost of local to local interactions can make them impractical. The classic example of local to local interactions is the N-body problem, which occurs in gravitational and electric field simulations among others. Both of these problems deal with particles generating fields that affect every other particle. To simplify the computational complexity of the problem, the local fields of each particle are accumulated into a hierarchy of fields that can then be precisely applied to each particle such that the total runtime is N log(N). This technique is called the Fast Multipole Method. While having every particle able to communicate efficiently with every other particle might be desirable in the ideal case, practically speaking it’s often good enough to have particles communicate with each other within a particular range or with the nearest N particles. Such a system can be implemented by binning particles according to their location and then operating over the binned particles to realize local to local communication channels. The fastest algorithm I’ve found for spatial binning and neighborhood calculation is Query Sphere Indexing. ### Spatial Hashing Query sphere indexing is a form of spatial whereby a spatial location is given an integer value that can be used as a key to look up the position quickly from a table. In the query sphere technique, the space is divided into a finite number of discrete cells and each cell is given an integer hash value based on its coordinate. Any particle falling within the cell is binned to it and associated with the cell’s integer hash value. The speed of the query sphere technique comes from precomputing as much information as possible about neighborhoods. When the data structure is created, a table of relative coordinates in terms of hash indices is calculated for all the possible distances and ordered according to nearest to furthest offset. For example, if you want to know how many objects are within radius 1 of a point in space, the offset lookup table will provide a list of cell offsets that can be added to the query location efficiently and all of the cells within radius 1 retrieved for processing. The calculations in the Query Sphere paper are incredibly efficient since they’re all operating on integers using addition and bit manipulation. In a version of the algorithm I’ve implemented for example, I can hash 302,500 (550×550) points at 33fps and query thousands of neighborhoods for nearest neighbors. As a result, local to local communication is incredibly efficient and used judiciously can add a lot more detail and complex pattern formation to the evolution of a spatial system. For instance, once local to local communication becomes efficient, it becomes to describe the formation of aggregate structures such as meshes in a straightforward manner, which can in turn introduce new behaviors and trajectories into the system. The cubes in the image above are the bins holding particles. The pink dots are query locations with lines connecting the nearest 16 neighbors within a particular radius. Posted in Software Engineering, Space and Geometry \| \| 1 Comment ## Transversion and the Torus As a first test of the efficacy of the new Transversion Automaton design, I set it up to carve out a simple torus. Since a torus only has two loops whose product generates the surface, it’s easy to model simply by mapping the two principle directions of the Transversion Automaton to the loops. The Transversion Automaton consists of a central unit sphere (in grey) whose center is a point on the surface being generated and two surface coordinate (UV-coordinate) generating lines. At each step, the automaton inverts the lines through the central unit sphere to generate the UV osculating circles. These circles represent the instantaneous curvature at the surface point in the principle directions. Next, the automaton is diffused through space along both principle circles and the locations of the generating lines is adjusted to properly account for the curvature values at the next location. Above are frame capturing the automaton’s motion around one of the cycles of the torus. The total movement through the nine frames is half a rotation (180 degrees) where the curvature of the direction given by the red line changes from 1 (spherical) on the outside to -1 (hyperbolic) on the inside. Notice that in frame 5, the circle generated by inverting the red line into the central unit sphere is actually a line, or a circle of infinite radius. This is because at the top of the torus, the lateral curvature is zero or Euclidean, which makes sense since for the curvature to change form 1 to -1 it has to pass through 0. These images show different points of view on the set of circles generated in one cycle of the torus. Notice that there are two lines indicating the two locations of Euclidean curvature. These images were generated simply by accumulating all of the circles and drawing them together. Overall, this iteration of the Transversion Automaton is much simpler to control and move through space. The logic of its structure is now super clear. One of the aspects I appreciate most is how its internal structure dictates its own transformation. In other words, it’s a self-modifying spatial automaton. All it does is read off a couple of curvature values and it’s able to move itself and adjust its relative weights accordingly. The next step is to figure out how to get the Transversion Automaton to operate in a panelizing mode where it traces out surface patches instead of entire cycles. This is the only way it will be able to generate more complex surfaces. ## Hyperbolic Tilings Again I’m revisiting a lot of the techniques I’ve implemented for working with hyperbolic geometry as I’m rebuilding the Transversion Automaton in order to make the simpler and clearer. The most basic structure that underlies the topology of the surfaces generated by the Transversion Automaton is the hyperbolic tiling. I recently found a much simpler derivation for the fundamental polygon that seeds the tiling and implemented it. In a regular hyperbolic tiling, each tile is exactly the same with the tile located at the center of the Poincaré disc called the fundamental polygon. By virtue of the expansive nature of hyperbolic space, there are an infinite number of regular tilings of the hyperbolic plane since the polygons can simply be resized until the rate at which they fill space matches the rate at which the hyperbolic plane expands. Thus, the key value to calculate when generating regular hyperbolic tilings is the radius of the fundamental polygon. The value that determines how big the radius is is the angle of parallelism of the edges of the fundamental polygon. The angle of parallelism essentially says how close to Euclidean space a particular tiling is. The closer to 90 degrees the angle of parallelism, the closer to Euclidean it is. For example, in the image above, the fundamental polygon on the far left is for the (3, 7) tiling. Notice how the radius of the pink circle is large and the arc that’s visible approaches a straight line. The angle that the line tangent to vertex D makes with the horizontal axis is the angle of parallelism. As D moves toward the top of the Poincaré disc circle, this angle approaches 90 degrees. If we changed the tiling to (3, 6), we would in fact have a 90 degree angle of parallelism and a Euclidean tiling. As the fundamental polygon increases in sides and number of polygons meeting at a vertex, the angle of parallelism decreases toward 0 as both values approach infinity. The diagrams above show how the angle of parallelism varies with the number if polygons incident to a vertex (left) and the number of sides (right). Calculating the location of the fundamental polygon’s vertices is equivalent to calculating the location of the circles that the polygon’s arcs lie on. Since they’re all equidistant from the origin, we just need to figure out how far away the circle is. To do this, we use right triangles AEF and ABF along with the fact that angles $\angle{}BAF = \frac{\pi}{p}$ and $\angle{}ABF = \frac{\pi}{q}$ are known to derive the fact that: 1. $sin(\angle{}EBF) = \frac{\overline{EF}}{\overline{BF}}$ and $sin(\angle{}BAF) = \frac{\overline{EB}}{\overline{AF}}$ 2. $1+\overline{BF}^2 = \overline{AF}^2$ 3. $sin(\angle{}EBF) = \frac{\pi}{2} – \frac{\pi}{q}$ 4. $sin(\angle{}BAF) = \frac{\pi}{p}$ 5. $\sqrt{\overline{AF}^2-1} \dot{} sin(\frac{\pi}{2} – \frac{\pi}{q}) = \overline{AF} \dot{} sin(\frac{\pi}{2})$ 6. $\overline{AF} = \sqrt{\frac{sin^2(\frac{\pi}{2}- \frac{\pi}{q})}{sin^2(\frac{\pi}{2}- \frac{\pi}{q}) – sin^2(\frac{\pi}{p})}}$ ## Thoughts on the Original Design The motivation for constructing a transversion-based spatial automaton was to construct a surface in 3D Euclidean space from as little local geometric information as possible. It was the automaton’s role to expand the information into a fully specified surface of arbitrary complexity. Essentially, this is a problem of differential geometry with a little bit of analysis (particularly complex harmonic functions) thrown in. The original design was essentially a straight translation into (Geometric Algebra) GA elements of the standard approaches differential geometry uses to analyze local surface properties and particularly curvature. The idea was to have two spheres, each representing one of the two principal surface coordinates and to relate the size of the spheres to instantaneous curvature. By inverting a differential surface element into the two spheres, the Transversion Machine incrementally constructed patches of the surface. While the design worked, it was unwieldy and suffered from some interpolation issues that lead to closed paths not actually closing like they should. The primary issue was figuring out how to update the automaton on each step. Essentially, this involved rotating and translating the entire machine to be centered at the new surface point in order to continue extended the surface while simultaneously warping with a transversion operation the two spheres such that their radii corresponded to the new local curvature values. In the end, there were just too many loosely connected transformations to keep in sync, so the mechanism felt cobbled together and inelegant. Furthermore, it wasn’t at all clear how the input curvature values parameterizing the Transversion Machine mapped to its internal transformations. ## Hyperbolic Inspiration In the back of my mind, I kept thinking that there had to be another way to organize things such that it just worked, every element was tightly integrated, and the input parameters mapped in a straightforward manner to the controls of the machine. Fortunately, inspiration struck while I was working on understanding hyperbolic translations. Hyperbolic translations are just like Euclidean translations in that they move along a line. The difference is that lines in hyperbolic space are actually Euclidean circles when the Poincaré model of hyperbolic geometry is used. The one exception is for lines that pass through the center of the unit disc. These hyperbolic lines are circles of infinite radius and thus also Euclidean lines. I then wondered how this fact could be used to redesign the Transversion Machine since it also needs to be able to smoothly move between rounds such as circles and spheres to flats such as lines and planes. This kind of behavior allows it to smoothly model changes in curvature from positive to flat to negative curvature. The only question is how to make such a mapping. ## Test 1: Real and Imaginary Spheres One of the unique features of GA is that many objects come in both real and imaginary forms. For instance, spheres are called real if they have a positive radius and imaginary if they have a negative radius. While it may seem strange for a sphere to have a negative radius, it has profound consequences for the interpretation of the geometry of a particular arrangement of geometry objects. For instance, when two spheres intersect, they generate a real circle. If they don’t intersect, they generate an imaginary circle. Importantly, the magnitude of the imaginary circle indicates how far apart the two spheres are. Since the curvature of a surface varies from positive through zero to negative values, I wondered if some way of transforming spheres between real and imaginary states could be used to implement the Transversion Machine. As a first test, I placed a line tangent to both a real and imaginary sphere colocated in space and with radii of the same magnitude but of opposite sign. The two spheres are given by $S_{real} = o – \frac{\infty}{2}$ and $S_{imaginary} = o + \frac{\infty}{2}$. On the left is a single instance of inverting a line by a sphere. The pink circle is generated by the real sphere while the blue is generated by the imaginary sphere. The inversion is described by $Circle = SLS^{-1}$. What’s curious is how the imaginary and real spheres result in exactly the same shape but on opposite sides of the center of the inversion sphere. When I saw this image, my first thought was that I could map the inversion sphere’s radius to surface curvature with the convenient mapping of imaginary spheres describing negative curvature (hyperbolic patches). The only question then is how to smoothly vary the spheres. First, however, I did the easy thing and varied the position of the line. The image on the right shows what happens when the line is varied from the center of the inversion sphere to the right. When the line is in the center, the inversion is an identity operation mapping the line to itself. As it moves to the right, the resulting circles vary in radius, smoothly shrinking to a point if the line was infinitely far away. Notice that the is exclusively to the right hand side of the inversion sphere’s center. If it was on the left, the real and imaginary circles would swap sides too. This is the answer to the question as to how to vary the size of the curvature spheres. We can simply move a plane (line) in space to generate the appropriate sized sphere. What’s really nice about this setup is it’s basically a spatial slider for controlling curvature that generates exactly the kind of geometry we need and takes precisely the kind of input we have, a scalar curvature value. We can map this curvature value to displacement along an axis running through the center of the inversion sphere. As a further bonus, the current point on the surface that the Transversion Machine is operating from is located at the center of the inversion sphere. In the end, we don’t actually need the imaginary sphere, but it’s nice to know how it operates with respect to spherical inversion for future problems. ## Test 2: Circular Rigid Body Motion The second problem in the original design was interpolating the motion of the Transversion Machine according to how the surface bends at each point. The transformation describing the motion is a rigid body motion through space that rotates the machine around a circle. The transformation is given by $V = exp(\infty\rfloor{}\frac{L}{2})$. The image above show the action of the circular rigid body motion. The diagrams are slightly skewed to show the axis of the circle since it comes out of the page. The axis is a dual line, which is a bivector inducing the motions in the images. Of course, the motion needed by the Transversion Machine is not arbitrary. It needs to be scaled such that it moves the Transversion Machine around the circle in fixed number of steps. To do this, we have to scale the circle such that its norm is equal to the distance travelled by each step around the circle. The following algorithm generates a motion around the circle in N steps from a line $L$ and an inversion sphere $S$: • Generate the circle that will move the Transversion Machine: $C = SLS^{-1}$. • Normalize the circler: $C_n = \frac{C}{\sqrt{CC}}$. • Calculate the versor representing the transformation and scale by the length of the circular arc: $V = exp(-\infty\rfloor{}\frac{C_n}{2}\frac{2\pi{}R}{N})$. • Transform the Transversion Machine inversion sphere and coordinate axis line: $S = VSV^{-1}, L = VSV^{-1}$. ## Hyperbolic Translations One of the crucial issues in the construction of the implementation of the transversion automaton is how it interpolate its geometric relationships (transversions and rigid body motions) through space to generate a surface in 3D. The input data is pentagonal tiling of the surface unfolded into a (5, 4) hyperbolic tiling with the state of the transversion automaton defined at the vertices of the tiling. Interpolation the state of the automaton across the tiling is subject to the constraint that any given arc through the tiling represents a complete cycle around some part of the surface and must therefore be a harmonic function of the transversion state. Ideally, the interpolation scheme chosen will evenly sample the space over which the interpolation is performed in order describe the most surface for the least number of interpolation points. In the case of the transversion automaton, the interpolation space is hyperbolic. As a result, our familiar Euclidean intuition about what constitutes evenly spaced samples doesn’t apply. The most straightforward way to interpolate hyperbolically is to work with a mapping of hyperbolic space into Euclidean space and develop the geometry of interpolation through that mapping. In this case, the mapping is the Poincaré disc model of the hyperbolic plane. ## The Poincaré Disc in Geometric Algebra In the conformal model of Geometric Algebra (GA), the usual Euclidean axes are augmented with two additional axes representing the point at infinity and the point at the origin. In other words, the point at infinity is explicitly represented in the model of Euclidean space. As a result, new symmetries appear enable exotic transformations that smoothly interpolate between circles and lines or spheres and planes since lines are essentially circles of infinite radius and planes are spheres of infinite radius. Furthermore, translations and rotations collapse into a single representation since translations are really just rotations about the infinite point. In GA, the particular representation of infinity is fungible and depending on which representation is chosen, a particular geometry will arise. For Euclidean geometry, infinity is taken as a point. For spherical and hyperbolic geometry, infinity is typically taken as the unit sphere where the only difference is that spherical geometry is given by the imaginary dual unit sphere and hyperbolic geometry by the real dual unit sphere. Imaginary spheres have a negative radius while real spheres have a positive radius. In the Poincaré disc model of hyperbolic space, infinity is also represented by a unit sphere in 3D and a 2D sphere (aka a circle) in 2D space. The beauty of all this is that all of our previously acquired knowledge about working with spatial transformations in the conformal model of Euclidean space still apply to hyperbolic space. All that needs to be done is to swap the usual $\infty$ with $o – \frac{\infty}{2}$. For example, given two points $p$ and $q$, the line through them is given by $L = p\wedge{}q\wedge{}\infty$. The hyperbolic line through them is given by $L = p\wedge{}q\wedge{}(o – \frac{\infty}{2})$. Of course, the hyperbolic line is not a line in the Euclidean sense but instead a circular arc. The reason for this difference is that hyperbolic space expands as you move away from the center, so the mapping to Euclidean space squishes it toward the unit circle, bending the line into a circle. ## Translations Translations in hyperbolic space also occur along circular arcs and from a Euclidean point of view move quickly through the center of the disc and slowly toward the perimeter. In Euclidean space, translations along lines are described by $exp(-\infty{}\rfloor{}\frac{L}{2})$ with the equivalent hyperbolic expression being $exp(-e\rfloor{}\frac{L}{2})$ where $e = o – \frac{\infty}{2}$. The real trick with hyperbolic translations is how to get them into a form that is useful for interpolation purposes. Essentially, we want to calculate a hyperbolic translation that will move between two points in a fixed number of steps. The form of the translation given above won’t do this because the line is not normalized and thus scales the distance covered by translation by a factor. To calculate a hyperbolic translation that moves between two points in N steps, the term $-e\rfloor{}L$ has to be normalized and then scaled by half the hyperbolic distance between the points. Hyperbolic distance can be calculated in GA by first normalizing the given points in terms of hyperbolic space. The condition for normalization is $-e\dot{}p = -1$. The distance between two points is then $d(p, q) = acosh(1 – \frac{-p}{e\dot{}p}\dot{}\frac{-q}{e\dot{}q})$. The algorithm for computing the interpolating translation is: 1. Construct a translation bivector between $p$ and $q$ as $T = e\rfloor{}(p\wedge{}q\wedge{}e)$. 2. Normalize it $T_n = \frac{T}{\sqrt{TT}}$. 3. Calculate the distance between $p$ and $q$ using $d(p, q) = acosh(1 – \frac{-p}{e\dot{}p}\dot{}\frac{-q}{e\dot{}q})$. 4. Scale the normalized bivector and exponentiate it $V = exp(T_n\frac{d}{2N})$ where N is the number of steps to take between the points. V is then the versor used to translate $p$ into $q$ over N steps. In the image above, the edges of the pentagon are drawn by interpolating between the vertices 20 steps. The pink points are midpoints while the blue divide the edges into quarters. The pink and blue points were calculated with a hyperbolic lerp constructed from the hyperbolic translation algorithm. ## Background The Meet operation in Geometric Algebra (GA) calculates the intersection of two subspaces. In set theoretical terms, the intersection is given by the relationship $M = A\cap{}B$ where M is the intersection (or Meet). Frequently in GA papers and books, you’ll find the elegant looking equation $M = A\cap{}B = (B^{}\wedge{}A^{})^{}$ where $A^{}$ indicates the dual of A. Unfortunately, calculating Meet is not so simple. Related to the intersection of two subspaces, we can define a second quantity that is the difference between their union and intersection, which is called the symmetric difference or delta product. Using this product, the intersection can be rewritten as $A\cap{}B = \frac{A + B – A\Delta{}B}{2}$. In Geometric Algebra, the delta product of blades is defined as the highest grade part of the geometric product of A and B $A\cap{}B = \langle{AB}\rangle{}_{max}$. The grade operation in the delta product stems from the way the geometric product automatically eliminates dependent factors. Furthermore, the grade of the Meet can equally be formulated in terms of the delta product as $grade(A\cap{}B) = \frac{grade(A) + grade(B) – grade(A\Delta{}B)}{2}$. ## The Algorithm ### Approach Through the delta product, we now have a proper way of expressing how the two subspaces A and B relate to each other and can formulate an algorithm to compute it. In the diagrams above, notice that the delta product includes both A and B but not their intersection while the dual of the delta product includes both the surrounding space as well as the intersection. We can take advantage of this fact by computing the meet by projecting the dual of the delta product into the subspace A since whatever projects into A from the dual delta product is the intersection because the intersection is by definition a subspace of A. Since the grade of the meet is not necessarily equal to the grade of $(A\cap{}B)^{}$, we must first factor it into an orthogonal basis before projecting it into A. Factoring a blade essentially involves probing the blade with a candidate basis vector and iteratively taking the basis vectors out of the blade until only one is left. In order to start the process, a good candidate basis vector must first be selected. Since the specific factorization doesn’t matter, we can simply select the element of the blade with the largest magnitude and go from there. Since we are dealing with a blade in this case, we can simply project the vectors of the element chosen into the blade and iteratively remove those projections. The algorithm looks like this given a blade B of grade k: 1. Calculate the norm of B $s = \Vert{}B\Vert{}$ 2. Find the basis element E in B with the largest coordinate 3. Determine the vectors $e_{i}$ of E 4. Normalize B $B_{c} = B/s$ 5. For each $e_{i}$ except for 1 • Project $e_{i}$ onto $B_{c}$ : $f_{i} = (e_{i}\rfloor{}B_{c})B_{c}^{-1}$ • Normalize $f_{i}$ • Remove $f_{i}$ from $B_{c}$ : $B_{c} \leftarrow{} f_{i}^{-1}\rfloor{}B_{c}$ 6. The last $f_{i}$ factor is set to $B_{c}$ : $f_{k} = B_{c}$ and normalized. ### The Basic Meet Algorithm The meet can be taken between two blades A and B of different grades and here we’ll assume blade A is of equal or greater grade than blade B. Given blades A and B: 1. Compute the dual delta product $S = (A\Delta{}B)^{}$ 2. Apply the factoring algorithm to S to get the factors $s_{i}$ 3. Compute the grade of the meet $grade(A\cap{}B) = \frac{grade(A) + grade(B) – grade(A\Delta{}B)}{2}$ 4. Set the meet result M to 1: $M = 1$ 5. For each $s_{i}$ • Project $s_{i}$ into A: $p_{i} = (s_{i}\rfloor{}A)\rfloor{}A^{-1}$ • If the projection $p_{i}$ is not zero, accumulate it into M: $M \leftarrow{} M\wedge{}p_{i}$ • If $grade(M)$ is equal to that computed in step 3, break the loop ### Circle Intersection A simple example of the meet operation in use is the case of circle intersections. If we define two circles. If we take two circles C1 and C2 at (1, 0, 0) and (-1, 0, 0) respectively, each with a radius of 2, the meet operation will return a point pair. In other words, the meet gives back a single object describing the two intersection points of the circle. • $C1 = oe_1e_2 – oe_2\infty + 1.5e_1e_2\infty$ • $C2 = oe_1e_2 + oe2\infty + 1.5e_1e_2\infty$ • $S = C1\Delta{}C2 = -2oe_1 – 3e_1\infty$ • $S^* = -2oe_2e_3 – 3e_2e_3\infty$ • $s_1 = e_2, s_2 = e_3, s_3 = – \frac{2}{\sqrt{12}}o – \frac{3}{\sqrt{12}}\infty$ • $A\cap{}B = \frac{2}{\sqrt{12}}oe_2 – \frac{3}{\sqrt{12}}e_2\infty$ The result of the meet as calculated above is a point pair. The points can be extracted by a simple equation that extracts the point pair’s axis and then pushes out a distance along this axis from their center point. The equation is given by: $P = p_{-}\wedge{}p_{+}$ where P is the point pair and $p_{\pm} = \frac{P\mp\sqrt{P^2}}{-\infty\rfloor{}P}$ are the points. For the case above, the points are located at $p_{\pm} = o \mp \sqrt{3}e_2 + 1.5\infty$. see Geometric Algebra for Computer Science by Dorst et al, p. 536 \| Tagged , , \| 2 Comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8148736357688904, "perplexity": 679.6620539264201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510270399.7/warc/CC-MAIN-20140728011750-00327-ip-10-146-231-18.ec2.internal.warc.gz"}
https://jeetblogs.org/post/filesystem-management-with-the-full-power-of-vim/	# Filesystem Management with the Full Power of Vim Just discovered vidir, a way to manipulate filenames inside your favorite text editor (better known as Vim). Previously, I would use complex and cumbersome BASH constructs using “for;do;done”, “sed”, “awk” etc., coupled with the operation itself: $for f in .txt; do mv$f $(echo$f \| sed -e 's/foo$\d\+$_$.$\.txt/bar_\1_blah_\2.txt/'); done Which almost always involved a “pre-flight” dummy run to make sure the reg-ex’s were correct: $for f in .txt; do echo mv$f $(echo$f \| sed -e 's/foo$\d\+$_$.$\.txt/bar_\1_blah_\2.txt/'); done This approach works well enough, but apart from being a little cumbersome (especially if there is any fiddling to get the reg-ex’s right), is a little error-prone. Now, the whole directory gets loaded up into the editor’s buffer (the “editor” being whatever ‘\$EDITOR‘ is set to), for you to edit as you wish. When done, saving and exiting results in all the changes being applied to the directory. This allows you to bring the full power of regular expressions, line- and column-based selections etc., to bulk renaming and other operations. Best of all, you get to see the final set of proposed changes in your text editor before committing to the actual operation. All of this can be rephrased as: ‘[fantastic!]’ Share	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49434173107147217, "perplexity": 3173.6818640954166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573284.48/warc/CC-MAIN-20190918110932-20190918132932-00131.warc.gz"}
http://mathhelpforum.com/calculus/104653-trig-integration-secant-tangent-problem-print.html	# Trig Integration -- secant and tangent problem • Sep 27th 2009, 02:26 PM bgonzal8 Trig Integration -- secant and tangent problem Indefinite integral of (sec(x))^1/2(tan^3(x)).... I know that tan^3(x) becomes tan^2(x) and tanx.... so it will be the Indefinite integral of (sec(x))^-1/2 (tan^2(x)) secx tanx dx ... I dont understand why my book says that there will be a secx in there, why isn't it just (sec(x))^1/2 (tan^2(x)) tanx dx? • Sep 27th 2009, 03:12 PM bgonzal8 Quote: Originally Posted by bgonzal8 Indefinite integral of (sec(x))^1/2(tan^3(x)).... I know that tan^3(x) becomes tan^2(x) and tanx.... so it will be the Indefinite integral of (sec(x))^-1/2 (tan^2(x)) secx tanx dx ... I dont understand why my book says that there will be a secx in there, why isn't it just (sec(x))^1/2 (tan^2(x)) tanx dx? Nevermind I figured it out... sec-1/2 (x) +sec x = -1/2 +1 = 1/2 .. sec^1/2 (x) is the original factor... • Sep 27th 2009, 03:26 PM DeMath Quote: Originally Posted by bgonzal8 Indefinite integral of (sec(x))^1/2(tan^3(x)).... I know that tan^3(x) becomes tan^2(x) and tanx.... so it will be the Indefinite integral of (sec(x))^-1/2 (tan^2(x)) secx tanx dx ... I dont understand why my book says that there will be a secx in there, why isn't it just (sec(x))^1/2 (tan^2(x)) tanx dx? Hint: $\int {{{\sec }^{1/2}}x\,{{\tan }^3}x\,dx} = \int {\frac{{{{\tan }^3}x}}{{{{\cos }^{1/2}}x}}\,dx} = \int {\frac{{{{\sin }^3}x}} {{{{\cos }^{1/2}}x\,{{\cos }^3}x}}\,dx} = \int {\frac{{{{\sin }^3}x}} {{{{\cos }^{7/2}}x}}\,dx} =$ $= - \int {\frac{{\sin x\left( {{{\cos }^2}x - 1} \right)}} {{{{\cos }^{7/2}}x}}\,dx} = \left\{ \begin{gathered} \cos x = u, \hfill \\ - \sin x\,dx = du \hfill \\ \end{gathered} \right\} = \int {\frac{{{u^2} - 1}} {{{u^{7/2}}}}\,du} = \ldots$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.985262930393219, "perplexity": 5764.262115406919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542668.98/warc/CC-MAIN-20161202170902-00013-ip-10-31-129-80.ec2.internal.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=1975_AHSME_Problems&oldid=64579	# 1975 AHSME Problems ## Problem 1 The value of ${\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}$ (Error compiling LaTeX. ! Missing } inserted.) is ## Problem 2 For which real values of m are the simultaneous equations y = mx + 3 \\ \[ y = (2m - 1)x + 4 (Error compiling LaTeX. ! LaTeX Error: \begin{equation*} on input line 20 ended by \end{document}.) satisfied by at least one pair of real numbers ? ## Problem 3 Which of the following inequalities are satisfied for all real numbers which satisfy the conditions , and ? ## Problem 4 If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? ## Problem 5 The polynomial is expanded in decreasing powers of . The second and third terms have equal values when evaluated at and , where and are positive numbers whose sum is one. What is the value of ? ## Problem 6 The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is ## Problem 7 For which non-zero real numbers is a positive integers? ## Problem 8 If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true? I. All shirts in this store are at non-sale prices. II. There is some shirt in this store not on sale. III. No shirt in this store is on sale. IV. Not all shirts in this store are on sale. ## Problem 9 Let and be arithmetic progressions such that , and . Find the sum of the first hundred terms of the progression ## Problem 10 The sum of the digits in base ten of , where is a positive integer, is ## Problem 11 Let be an interior point of circle other than the center of . Form all chords of which pass through , and determine their midpoints. The locus of these midpoints is ## Problem 12 If , and , which of the following conclusions is correct? The equation has ## Problem 14 If the is when the is and the and is , what is the when the is , the and is and the is is two ( and are variables taking positive values)? ## Problem 15 In the sequence of numbers each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is ## Problem 16 If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is , then the sum of the first two terms of the series is ## Problem 17 A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of x working days, the man took the bus to work in the morning times, came home by bus in the afternoon times, and commuted by train (either morning or afternoon) times. Find . ## Problem 18 A positive integer with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that is an integer is ## Problem 19 Which positive numbers satisfy the equation ? ## Problem 20 In the adjoining figure is such that and . If is the midpoint of and , what is the length of ? ## Problem 21 Suppose is defined for all real numbers ; for all , and for all and . Which of the following statements is true? ## Problem 22 If and are primes and has distinct positive integral roots, then which of the following statements are true? ## Problem 23 In the adjoining figure and are adjacent sides of square ; is the midpoint of ; is the midpoint of ; and and intersect at . The ratio of the area of to the area of is ## Problem 24 In , and , where . The circle with center and radius intersects at and intersects , extended if necessary, at and at ( may coincide with ). Then ## Problem 25 A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player? ## Problem 26 In acute the bisector of meets side at . The circle with center and radius intersects side at ; and the circle with center and radius intersects side at . Then it is always true that ## Problem 27 If and are distinct roots of , then equals ## Problem 28 In shown in the adjoining figure, is the midpoint of side and . Points and are taken on and , respectively, and lines and intersect at . If then equals ## Problem 29 What is the smallest integer larger than ? ## Problem 30 Let . Then equals	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8712334632873535, "perplexity": 702.4232568950654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141188800.15/warc/CC-MAIN-20201126142720-20201126172720-00718.warc.gz"}
https://answers.opencv.org/question/186712/find-an-image-by-nearest-color/	# Find an image by nearest color? edit Hi OpenCV’ers! I’ve got a project in mind and need some expert input. I have a collection of about 1100 abstract images (desktop wallpaper tiles) that I want to be searchable by predominant color. I’ve already found Python code to help me use the OpenCV libraries to do k-means clustering and extract the top 3 (for example) colors in each image. From that, I can get the three colors as RGB values and the percentage of each. I plan to stuff all that data into a database for searching. Ideally, I want the program user to select a color (using a standard color-picker dialog box), and then query the database for the top X images with the “closest matching” predominant color. The problem is that I don’t know how to store and search this data effectively. That’s where I need your help. The math involved here is miles over my head. From doing some reading (Wikipedia), I have a hunch that storing and searching HSV values makes more sense than RGB values. I can’t fully explain why, except that it seems easier to find “more or less blue-ish” using HSV. Is anyone willing to help me untangle this? I thought this project would be straightforward, until I started digging into image analysis. This led me to k-means clustering, which led me here. edit retag close merge delete Using kmeans you get a features for each images. If you keep N colors you will get a feature 4N (3 for rgb +1 for percentage). You can use this post to build data space and try to get nearest image ( 2018-03-14 11:05:06 -0500 )edit Sort by » oldest newest most voted How to store these values? Just store the raw HSV values of each colour. This would make it easier especially on finding the closest match. In general, you have a distance minimisation problem. When the user selects their colour from the dialog box, you want to search through your database looking for a colour whose Euclidean distance is the smallest. If we assume that a database does not exist, then your function would look something like this, import sys all_colours_in_hsv = [colour1, colour2, colour3, colour4, colour5] def closest_colour(selected_colour): # set the distance to be a reallly big number # initialise closest_colour to empty shortest_distance, closest_colour = sys.max(), None # iterate through all the colours # for each colour in the list, find the Euclidean distance to the one selected by the user for colour in all_colours_in_hsv: # since your colours are in 3D space, perform the calculation in each respective space current_distance = sqrt(pow(colour.H - selected_colour.H, 2) + pow(colour.S - selected_colour.S, 2) + pow(colour.V - selected_colour.V, 2)) # unless you truly care about the exact length, then you don't need to perform the sqrt() operation. # it is a rather expensive one so you can just do this instead # current_distance = pow(colour.H - selected_colour.H, 2) + pow(colour.S - selected_colour.S, 2) + pow(colour.V - selected_colour.V, 2) # update the distance along with the corresponding colour if current_distance < shortest_distance: shortest_distance = current_distance closest_colour = colour return shortest_distance, closest_colour Since you have a database and have not told us what kind it is (MySQL, MS. Access, NoSQL etc), then it is hard to tell you exactly what you need to do but the general query would be something like Select top 3 from table colours order by Euclidean Distance calculation more Thank you. It seems you understand what I'm trying to do. My database will be SQLite, so I can easily pack the database in with the application and not need to worry about a client/server setup. I plan to pre-populate the tables with the color values, running k-means on each image and storing the results in columns like "color1_H", "color1_S", and "color1_V". Repeat for colors 2 and 3. I also plan to store the percentage of each color so I have another option for ranking the results. I'm going to use Python for the k-means crunching and loading the database. The end application will be C++, so hopefully the distance calculation(s) will be fast. I've got a follow-up question, but I'm running out of characters. Please see the next comment! Thanks! ( 2018-03-14 11:49:38 -0500 )edit Can I also use the distance calculation you described to determine the level of "contrast" between the top 3 colors in the image? By "contrast" I mean how far apart the top 3 are on the spectrum. If the image is all shades of blue, I presume your algorithm would produce what amounts to a "low difference" between the three colors. But if the image is blue, yellow, and red, the distances would be much larger. Am I understanding this correctly? That way, I could pick images that are close to the chosen color AND contain low-contrast companion colors. Or the chosen color and high-contrast companion colors. If you're curious to see what I'm talking about, or if it makes the situation clearer, here's my plan: http://www.bundito.com/index.php/2018... Thanks! ( 2018-03-14 11:57:12 -0500 )edit @bundito Your website try to extract fingerprint : it is not very fair "It seems you understand what I'm trying to do." May be you should try to understand what is flann ( 2018-03-14 12:26:01 -0500 )edit A couple things. 1: The method suggested by @LBerger does the exact thing I mentioned above (I didn't realize his comment until I posted this).It actually is better because it is optimized for image operations hence faster. All that I did is give you an introductory review of your problem and how to solve it on a small scale. flann not only implements this, but also gives you an array of other optimised operations. There are many ways to tackle distance minimisation. 2: I am still not 100% sure regarding the contrast measure because there are different types of contrasts. My advise, get the Euclidean stuff implemented then run experiments to see what conclusions you can draw about their contrasts as well. ( 2018-03-14 12:58:57 -0500 )edit If no conclusion can be drawn, then refer to the different contrast calculations and see which one best suites your needs. If you are relatively new to all this, then I'd recommend you first implement my suggested method and run it on a couple images before even connecting it to your DB. Once you fully understand and can interpret the results, switch to flann for scalability. ( 2018-03-14 13:03:56 -0500 )edit 2 @LBerger: I wasn't aware of the device fingerprinting going on. It must be part of the generic WordPress install. I'll find it and kill it. I don't like it any more than you do. Sorry about that, but thanks for pointing it out. @LBerger & @eshirima: I'll read up on flann and try to make sense of it. I appreciate the help. I know what I'm trying to accomplish, but I never got enough mathematics education to understand these kinds of calculations. But as I'm wiring in a new blog post, this is an open-source project, and open-source projects almost always have helpful community members. It's true once more. I really appreciate it. ( 2018-03-14 13:13:40 -0500 )edit I think you should real also hash module in opencv_contrib. If you want to use it in python you will need need to build opencv and opencv_contrib ( there is no official package for opencv_contrib) ( 2018-03-14 13:55:48 -0500 )edit Hello again @LBerger and @eshirima ! I successfully ran all my images through a K-Means implementation from the sklearn library. I was also able to successfully search for the nearest color using the formula @eshirima outlined above. It worked great and ran quickly. Thank you both for helping me understand how that works. But I've got a problem I don't fully understand. My extracted colors are "muddier" and darker than what an image-processing website generates. Theirs are more in line with what I want, and what I'd expect. Rather than cluttering up this helpful answer, I posted my dilemma on Stack Overflow. Can someone take a look and take a stab at an answer? https://stackoverflow.com/q/49336210/... Thanks. Very much appreciated. ( 2018-03-17 12:43:31 -0500 )edit Official site GitHub Wiki Documentation	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19672071933746338, "perplexity": 1275.1394011011039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527000.10/warc/CC-MAIN-20190721123414-20190721145414-00451.warc.gz"}
http://www.gnu.org/software/auctex/manual/reftex/reftex_62.html	[ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] ## 8.7 Viewing Cross-References User Option: reftex-view-crossref-extra Macros which can be used for the display of cross references. This is used when reftex-view-crossref' is called with point in an argument of a macro. Note that crossref viewing for citations, references (both ways) and index entries is hard-coded. This variable is only to configure additional structures for which crossreference viewing can be useful. Each entry has the structure (macro-re search-re highlight). macro-re is matched against the macro. search-re is the regexp used to search for cross references. ‘%s’ in this regexp is replaced with the macro argument at point. highlight is an integer indicating which subgroup of the match should be highlighted. User Option: reftex-auto-view-crossref Non-nil means, initially turn automatic viewing of crossref info on. Automatic viewing of crossref info normally uses the echo area. Whenever point is idle for more than reftex-idle-time seconds on the argument of a \ref or \cite macro, and no other message is being displayed, the echo area will display information about that cross reference. You can also set the variable to the symbol window. In this case a small temporary window is used for the display. This feature can be turned on and off from the menu (Ref->Options). User Option: reftex-idle-time Time (secs) Emacs has to be idle before automatic crossref display or toc recentering is done. User Option: reftex-cite-view-format Citation format used to display citation info in the message area. See the variable reftex-cite-format for possible percent escapes. User Option: reftex-revisit-to-echo Non-nil means, automatic citation display will revisit files if necessary. When nil, citation display in echo area will only be active for cached echo strings (see reftex-cache-cite-echo), or for BibTeX database files which are already visited by a live associated buffers. User Option: reftex-cache-cite-echo Non-nil means, the information displayed in the echo area for cite macros (see variable reftex-auto-view-crossref`) is cached and saved along with the parsing information. The cache survives document scans. In order to clear it, use M-x reftex-reset-mode. [ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ] This document was generated by Ralf Angeli on August, 9 2009 using texi2html 1.78.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.59770667552948, "perplexity": 7683.3596751965915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400380464.40/warc/CC-MAIN-20141119123300-00110-ip-10-235-23-156.ec2.internal.warc.gz"}
http://scholarpedia.org/article/Talk:Hopfield_network	Notice: Undefined offset: 248 in /var/www/scholarpedia.org/mediawiki/includes/parser/Parser.php on line 5961 Talk:Hopfield network - Scholarpedia # Talk:Hopfield network This is an excellent summary, written with the author's usual succinctness. I have just one minor point to raise: For large networks, it makes a big difference whether the barriers confining a system to the neighborhood of a particular fixed point grow with the size of the network. For example, for the case with all T_jk = T_0 > 0, there are just two fixed points (all up and all down) and the barriers grow proportional to the size of the system. On the other hand, if the couplings have the opposite sign, there can be many fixed points, with small barriers between them. With finite noise, this difference is important: We are generally interested in large networks which are stabilized by their size (this is implicit in the notion of "collective computation"). This point seems worth mentioning, if space allows. This is a very pleasantly written, succinct review. Just two comments: 1) Would it be possible to include more precise citations in the text as to which article in the reference list discusses which issue (binary vs. continuous variables, optimization etc.)? 2) Also, one might consider including some reference to the related area of automata networks (e.g. Fogelman-Robert-Tchuente, "Automata Networks in Computer Science", Princeton UP 1987, or Goles-Martinez, "Neural and automata networks", Kluwer 1990). There very similar discrete-time dynamical models have been studied, starting from different considerations but concluding with quite analogous techniques and results. (Only with respect to analysis of the dynamics, of course, not for associative memory or optimization.)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8305839896202087, "perplexity": 1113.913278524914}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304883.8/warc/CC-MAIN-20220129092458-20220129122458-00622.warc.gz"}
https://tex.stackexchange.com/questions/127539/labelling-commands-for-quick-use	# Labelling commands for quick use I have a certain command in LaTeX that I have used a lot in my current work: {\bf y} # this creates a bold letter y I want to create a shortcut e.g. \by that will automatically create a bold letter y. I have similar expressions I would like to simplify as my equations are getting pretty long. Any advice would be appreciated. • Use a good editor with good shortcuts if you want to speed up your editing, don't write unreadble code. New LaTeX macros are useful to keep consistency throughout your documents, not to avoid typing. Aug 9, 2013 at 14:36 • Please throw away your LaTeX documentation: \bf has been deprecated for two decades now. Aug 9, 2013 at 20:55 This is done with \newcommand, i.e.: \newcommand{\by}{\textbf{y}} There is more functionality in it than just substituting, i.e., you can have arguments there. Please find more info and some examples here. • Cheers, that's great! – Jonny Phelps Aug 9, 2013 at 14:14 • Problem I'm having is that when I write $\by x$ I get both the y and the x as bold. Any way to resolve this and only have the y bold? – Jonny Phelps Aug 9, 2013 at 14:28 • Just put it in brackets, as edited. \bf is actually deprecated, so use \textbf instead Aug 9, 2013 at 14:31 • If the command is to be used in math mode, perhaps \mathbf{y} is preferable. Aug 9, 2013 at 21:39 \newcommand is the command you look for (see sashkello answer) There are some things you should look for: \documentclass{scrartcl} \newcommand{\by}{\textbf{y}} \begin{document} Now we write a \by and a x. \end{document} The y is direct connected to the and. You may add a space in \newcommand to add a space before the and: \documentclass{scrartcl} \newcommand{\by}{\textbf{y} } %<- space before the closing } \begin{document} Now we write a \by and a x. Now we write a \by. The next sentence. \end{document} The next problem: There is an additional space when you start a new sentence. For this problem you may use \xspace: \documentclass{scrartcl} \usepackage{xspace} \newcommand{\by}{\textbf{y}\xspace} \begin{document} Now we write a \by and a x. Now we write a \by. The next sentence. \end{document} The result:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7442063689231873, "perplexity": 2236.2925111678524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104683020.92/warc/CC-MAIN-20220707002618-20220707032618-00097.warc.gz"}
http://mathhelpforum.com/differential-geometry/193504-convergence-r.html	# Math Help - convergence in R 1. ## convergence in R Let $(x^{(n)})^\infty_{n=1}= ((x_k^{(n)})_{k=1}^{\infty})_{n=1}^{\infty}$be a sequence of elements of $l_1$. Prove that if $(x^{(n)})_{n=1}^{\infty}$ converges in $l_1$ to $x=(x_k)_{k=1}^{\infty}$, then for every $K \in N$, the sequence $(x_k^{(n)})_{n=1}^{\infty}$ converges in $R$ to $x_k$. Show by example that the converse is not true. 2. ## Re: convergence in R $\|x_k^{(n)}-x_k\|\leq \|x^{(n)}-x\|$ and taking $x^{(k)}=e^{(k)}$ where $e_n^{(k)}=\begin{cases}1&\mbox{ if }n=k\\0&\mbox{ otherwise}\end{cases}$ we get a counter-example for the converse.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9962117075920105, "perplexity": 130.39357928317995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928864.73/warc/CC-MAIN-20150521113208-00155-ip-10-180-206-219.ec2.internal.warc.gz"}
http://scitation.aip.org/content/aip/journal/jap/99/11/10.1063/1.2199753	• journal/journal.article • aip/jap • /content/aip/journal/jap/99/11/10.1063/1.2199753 • jap.aip.org 1887 No data available. No metrics data to plot. The attempt to plot a graph for these metrics has failed. Flat panel displays for ubiquitous product applications and related impurity doping technologies USD 10.1063/1.2199753 View Affiliations Hide Affiliations Affiliations: 1 Sumitomo Eaton Nova Corporation, SBS Tower 9F, 10-1 Yoga 4 Chome, Setagaya-ku, Tokyo, 158-0097 Japan a) Electronic mail: suzuki_tsh@senova.co.jp J. Appl. Phys. 99, 111101 (2006) /content/aip/journal/jap/99/11/10.1063/1.2199753 http://aip.metastore.ingenta.com/content/aip/journal/jap/99/11/10.1063/1.2199753 ## Figures FIG. 1. Process scheme of two shot-SLS technology (see Ref. 55). FIG. 2. The cross section of the excimer laser annealed poly-Si film (see Ref. 25). FIG. 3. Interface state density vs energy in the silicon band gap (see Ref. 76). FIG. 4. Summary of (a). and (b). reported on TEOS oxide and silane based oxide. [a: S. Higashi et al., b: T. Strutz et al., c: M. C. Lee et al., d: N. Stydlo et al., e: Y. Nishi et al., f: F. Farmakis et al., g: T. Strutz et al., and h: T. Strutz et al. (see Refs. 25 and 72–78)]. FIG. 5. FE-SEM image (plane view and birds view). Plane view was observed after Secco etching. Improvement of gate leakage current (see Ref. 43). FIG. 6. Current density vs electric field characteristics of the gate oxide for the conventional TEOS oxide and proposed stack oxide poly-Si TFTs (see Ref. 80). FIG. 7. Typical implantation conditions for LTPS TFT. FIG. 8. Dependence of field-effect mobility on the number of grain boundaries crossing the current direction in active channel (see Ref. 24). FIG. 9. (a) As the grain size increases the trap density at grain boundaries in the channel decreases. (b) Number of grain boundaries in a channel deviates as the channel length approaches the grain size (see Ref. 98). FIG. 10. Comparison of the calculated average carrier concentration vs doping concentration with the room-temperature Hall-measurement data. The solid line is the theoretical curve (see Ref. 48). FIG. 11. Schematic diagram of an ICP ion source (see Ref. 118). FIG. 12. Schematic diagram of a mass analyzed ion implanter (see Ref. 120). FIG. 13. Compensation deflector array (see Ref. 121). FIG. 14. Cross-sectional TEM image of a laser-activated TFT showing amorphous region at source/drain junction (see Ref. 134). FIG. 15. Example of pixel circuit to compensate dispersion of TFT characteristics (Ref. 135). FIG. 16. Field-effect mobility values observed from TFTs fabricated on imprint-grown single-grain films and SPC poly-Si films. The mobility values observed for MOSFET on (111) and (100) Si single-crystal surface are also indicated. The metal-imprint-grown single-grain films have a (111) crystal surface (see Ref. 144). FIG. 17. (a) Schematic viewgraph of the -Czochralski (grain-filter) process. (b) Slant SEM image after irradiation of excimer laser (see Ref. 146). FIG. 18. EBSD orientation mapping of a grid of grains by the -Czochralski process overlapping the SEM image. The curved lines indicate boundaries 3 (red), 9 (yellow) CSL boundaries, and random boundaries (white) (see Ref. 147). ## Tables Table I. System integration trend of LTPS TFT LCD (see Ref. 29). Table II. Typical top gate LTPS-TFT process. Table III. Improvement of gate oxide leak current by smoothing the poly-Si surface. , , (see Ref. 43). /content/aip/journal/jap/99/11/10.1063/1.2199753 2006-06-01 2014-04-17 Article content/aip/journal/jap Journal 5 3 ### Most cited this month More Less This is a required field	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9106096625328064, "perplexity": 25344.211501396556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00596-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.enotes.com/homework-help/find-values-h-that-makes-following-system-454757	# Find the values of h that makes the following system inconsistent: x + 2y + hz = hx + 3y + 4z = h2x + 4y + 5z = 5 Tibor Pejić \| Certified Educator calendarEducator since 2012 starTop subjects are Math, Science, and History There are number of different ways to solve this: using Cramer's ruleRouché–Capelli theorem, some forms of matrix decomposition... But probably the best way for you would be to simply thy to solve the system and see what you get. `x+2y+hz=h` `x+3y+4z=h` `2x+4y+5z=5` Now 1) subtract first row from the second row and 2) subtract first row multiplied by 2 from the third row `x+2y+hz=h` `0+y+(4-h)z=0` `0+0+(5-2h)z=5-2h` Now, only way we would have some problem with solving this system of equations is if `5-2h=0=>h=5/2` (then the last row would be only zeros). If we put that value of `h` back into our system we would get infinitely many solutions, to be exact solutions would be defined with `y=15/2-3x` `z=-5+2x,` `x in RR` In all the other cases the system has unique solution. And since a system of linear equations is inconsistent when it has no solution there exist no real number h for which this system is inconsistent. check Approved by eNotes Editorial	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8561403155326843, "perplexity": 529.474211502035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400191780.21/warc/CC-MAIN-20200919110805-20200919140805-00288.warc.gz"}
http://www.sciencemadness.org/talk/viewthread.php?tid=22145#pid265507	Sciencemadness Discussion Board » Fundamentals » Chemistry in General » Extremely strong bases for enolization Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues Author: Subject: Extremely strong bases for enolization Doc B Hazard to Others Posts: 107 Registered: 5-4-2012 Member Is Offline Mood: No Mood Extremely strong bases for enolization Is there any suggestions for a convenient extremely strong base, obtainable for the hobbiest, that is able to tautomerise acetone to the enol? http://en.wikipedia.org/wiki/Keto-enol_tautomerization sargent1015 National Hazard Posts: 315 Registered: 30-4-2012 Location: WI Member Is Offline Mood: Relaxed Is NaOH not an option? The Home Chemist Book web page and PDF. Help if you want to make Home Chemist history! http://www.bromicacid.com/bookprogress.htm UnintentionalChaos International Hazard Posts: 1454 Registered: 9-12-2006 Location: Mars Member Is Offline Mood: Nucleophilic Quote: Originally posted by sargent1015 Is NaOH not an option? Not if you want quantitative conversion. Probably sodium amide is a solid choice. Department of Redundancy Department - Now with paperwork! 'In organic synthesis, we call decomposition products "crap", however this is not a IUPAC approved nomenclature.' -Nicodem Doc B Hazard to Others Posts: 107 Registered: 5-4-2012 Member Is Offline Mood: No Mood SGT: no NaOH's conjugation pKa is not high enough. UC: the sodium aide is defiantly an option, however it's acquisition &/or synthesis (al la Vogel) is currenty precluding it. kristofvagyok International Hazard Posts: 659 Registered: 6-4-2012 Location: Europe Member Is Offline Mood: No Mood Sodium-isopropoxide or sodium-tertbutoxide will do it. NaOH is a weak base fot this. I have a blog where I post my pictures from my work: http://labphoto.tumblr.com/ -Pictures from chemistry, check it out(: "You can’t become a chemist and expect to live forever." DJF90 International Hazard Posts: 2266 Registered: 15-12-2007 Location: At the bench Member Is Offline Mood: No Mood Sodium hydride is a good choice, as is sodamide. What do you want quantitative enolate formation for? What reaction are you doing? Have you tried using an enolate equivalent, e.g. enamine? Hydroxide is actually a very strong base its just the fact most reactions are performed using it in aqueous solution. In methanol solution, I've used KOH to deprotonate an acetylene (pKa of phenylacetylene is ~28) prior to reaction of the anion. Phase transfer catalysis in say toluene has also been used to deprotonate very non-acidic protons, those that would typically be taken using sodamide or LDA or KHMDS. I think the problem with this application is the fact that quantitative conversion needs to occur rapidly, to avoid self condensation. I don't know how such systems stand up on this basis. AndersHoveland Hazard to Other Members, due to repeated speculation and posting of untested highly dangerous procedures! Posts: 1986 Registered: 2-3-2011 Member Is Offline Mood: No Mood Quote: Originally posted by UnintentionalChaos Probably sodium amide is a solid choice. I have a feeling that sodium amide would cause a different reaction, particularly in the absence of any other reactant. "The cleavage of ketones by sodium amide was discovered in 1906 by Semmler" see the Haller-Bauer reaction R-CO-R' --> R-CO-NH2 and H-R' not sure if this only applies to non-enolisable ketones [Edited on 6-11-2012 by AndersHoveland] I'm not saying let's go kill all the stupid people...I'm just saying lets remove all the warning labels and let the problem sort itself out. zed International Hazard Posts: 2247 Registered: 6-9-2008 Location: Great State of Jefferson, City of Portland Member Is Offline Mood: Semi-repentant Sith Lord Ummm. Enolize Acetone....Might be difficult to do by ordinary methods. A relatively useful method for forming the enol of Acetone, is reacting acetone with Ketene. The resulting product is Isopropenyl Acetate. The Acetate Ester of Acetone Enol. The product is commercially available at less than $100.00 U.S./Liter. Ketene itself, isn't too hard to generate, but its use is fairly dangerous. Given the chance, it will rip though your lungs like a raging Roto-Rooter. Doc B Hazard to Others Posts: 107 Registered: 5-4-2012 Member Is Offline Mood: No Mood The MeOH sounds the most convenient. Ketene is another good option and easy to obtain for the hobbyist. However being that my eyes still feel like sand paper from my last experiment synthesising formic acid and alkyl alcohol, I'm not sure I'm an enthusiastic enough hobbyist to risk a good waft of ketene at the moment. The enol is theorised to be used in a halo condensation. Most of the super bases mentioned will work but the acquisition is out of my range at present. I'll give the MeOH a go first and report back. Cheers DJF90 International Hazard Posts: 2266 Registered: 15-12-2007 Location: At the bench Member Is Offline Mood: No Mood A halo condensation? You mean halogenation? Depending on your substrate, you may suffer from regioselectivity issues. If the molecule is symmetrical (e.g. acetone) then this is of little concern. Monohalides are typically formed under acidic conditions, if that helps. Doc B Hazard to Others Posts: 107 Registered: 5-4-2012 Member Is Offline Mood: No Mood Quote: Originally posted by DJF90 A halo condensation? You mean halogenation? Depending on your substrate, you may suffer from regioselectivity issues. If the molecule is symmetrical (e.g. acetone) then this is of little concern. Monohalides are typically formed under acidic conditions, if that helps. The enol displaces the halogen, I'm not exactly sure of the reactions name. I guess it's an Adol condensation? SM2 National Hazard Posts: 359 Registered: 8-5-2012 Location: the Irish Springs Member Is Offline Mood: Affect Quote: Originally posted by zed Ummm. Enolize Acetone....Might be difficult to do by ordinary methods. A relatively useful method for forming the enol of Acetone, is reacting acetone with Ketene. The resulting product is Isopropenyl Acetate. The Acetate Ester of Acetone Enol. The product is commercially available at less than$100.00 U.S./Liter. Ketene itself, isn't too hard to generate, but its use is fairly dangerous. Given the chance, it will rip though your lungs like a raging Roto-Rooter. if you filled a flower jar w/ acetone, and had a nice coated Pt catylist floating and fixed above, could you just heat the catylist, and then, it would be exothermic, with the evaporation sendingg a stream of acetone over the glowing Pt? ScienceSquirrel International Hazard Posts: 1863 Registered: 18-6-2008 Location: Brittany Member Is Offline Mood: Dogs are pets but cats are little furry humans with four feet and self determination! Quote: Originally posted by Fennel Ass Ih Tone Quote: Originally posted by zed Ummm. Enolize Acetone....Might be difficult to do by ordinary methods. A relatively useful method for forming the enol of Acetone, is reacting acetone with Ketene. The resulting product is Isopropenyl Acetate. The Acetate Ester of Acetone Enol. The product is commercially available at less than \$100.00 U.S./Liter. Ketene itself, isn't too hard to generate, but its use is fairly dangerous. Given the chance, it will rip though your lungs like a raging Roto-Rooter. if you filled a flower jar w/ acetone, and had a nice coated Pt catylist floating and fixed above, could you just heat the catylist, and then, it would be exothermic, with the evaporation sendingg a stream of acetone over the glowing Pt? That sounds like an excellent way to get a good solvent fire going. Acetone is quite volatile and flammable. This is how you should go about it. http://www.orgsyn.org/orgsyn/orgsyn/prepContent.asp?prep=cv1... Salmo Harmless Posts: 42 Registered: 20-9-2012 Member Is Offline Mood: No Mood I think you need a really strong base to move the equilibrium between the two tautomers, LDA maybe? i think sodium amide would be a nucleophile other than a base. sodium/potassium tertbutoxide would work too, something that is a strong base but too big to be a good nucleophile too.. DJF90 International Hazard Posts: 2266 Registered: 15-12-2007 Location: At the bench Member Is Offline Mood: No Mood Its alpha-halogenation of a ketone/aldehyde. Like I said, if you want mono-halogenation, a good method is to use e.g. bromine in acetic acid. Under basic conditions, halogenation tends to be exhaustive e.g. methyl ketones undergo the haloform reaction. Complete formation of the enolate and reaction with an electrophilic halogen source (e.g. N-Halosuccinimide) will also work. Again, I'll mention the regioselectivity issue you may or may not have; you'll get more help if you can be more transparent regards your substrate. chemrox International Hazard Posts: 2942 Registered: 18-1-2007 Location: UTM Member Is Offline Mood: psychedelic Na shot in xylene - it's cheap and convenient "Ignorance is the Mother of Devotion." — Robert Burton. DJF90 International Hazard Posts: 2266 Registered: 15-12-2007 Location: At the bench Member Is Offline Mood: No Mood You sure you wouldn't get competing pinacol coupling, Chemrox? TheChemINC Hazard to Self Posts: 61 Registered: 22-2-2012 Member Is Offline Mood: fuming what about something like sodium methoxide? Sciencemadness Discussion Board » Fundamentals » Chemistry in General » Extremely strong bases for enolization Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32119691371917725, "perplexity": 23012.178123641537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301670.75/warc/CC-MAIN-20220120005715-20220120035715-00214.warc.gz"}
https://search.datacite.org/works/10.17605/osf.io/8zupj	### Experiment 3b Matthias Raemaekers, Ian Hussey & Jan De Houwer A replication of Experiment 3, to which two changes were made: (1) participants only completed one phase, simply to shorten the experiment and reduce costs, and (2) the instructions for the one of the blocks were slightly altered to investigate whether this might have an effect on the results. This data repository is not currently reporting usage information. For information on how your repository can submit usage information, please see our documentation.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.849947988986969, "perplexity": 2449.8933919186056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710933.89/warc/CC-MAIN-20221203143925-20221203173925-00154.warc.gz"}
https://math-faq.com/chapter-14/section-14-1/	# The Substitution Method In Chapter 13, we reversed the derivative process for basic functions like power and exponential functions. By taking the antiderivative of a power function, we were able to find the original function we had taken the derivative of. In this question, we continue to find antiderivatives of function. For example, suppose we want to take the derivative Since the express is a composition of two functions, we must use the chain rule to take this derivative. Start by identifying the inside and outside functions of the composition, The derivatives of these functions are Using the chain rule, This derivative can also be written as the antiderivative, If we have already carried out the derivative, writing out the antiderivative is just the reverse process. However, it is rarely the case that we have the derivative available to help us evaluate the antiderivative. For integrands involving compositions, Substitution Method may help us to find the antiderivative.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9435266256332397, "perplexity": 226.93234328324905}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178359082.48/warc/CC-MAIN-20210227174711-20210227204711-00303.warc.gz"}