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# Detection of core collapse supernova neutrinos with DarkSide20k. Bonivento W., Franco D., Lai M., Mirizzi A., Renshaw A., Ye Z. Comunicazione III - Astrofisica Aula GSSI Rettorato - Auditorium - Martedì 24 h 17:30 - 19:30 When a core-collapse supernova (SN) explodes, only 1% of its energy is released through electromagnetic waves: the rest is emitted via neutrinos. If a galactic supernova will show up during its data taking, DarkSide20k, a future 20 ton liquid-argon dual-phase TPC, designed for the direct detection of dark matter particles, will perform a flavor-insensitive detection of SN neutrinos via coherent elastic neutrino nucleus scattering (CE$\nu$NS), providing additional information on the explosion mechanism. Furthermore, by the comparison with a charged current detection, discrimination between neutrino mass orderings, normal or inverted, can be inferred.	{}
# Dave Winer's Micro-blogging Experiment Dave Winer is running a little micro-blogging/link-blogging experiment. We like the direction Dave is heading and would like to enhance feedly to better support it. Here is an example of link-blog generated by Dave: An item in this feed looks like this: <item><description>A second reactor building at the Fukushima Daiichiatomic power plant has exploded.</description> <pubDate>Mon, 14 Mar 2011 04:15:08 GMT</pubDate> <guid>http://r2.ly/7xp8</guid> </item> Question to Dave Winer: What would be your reaction to using a richer description to capture the same metadata information? For example, the description of the item could include a <a> which itself could have data-XXX=”YYY” decorations if we want to encode more metadata regarding the resource being linked? This would have the benefit of graciously flowing through all of the existing RSS infrastructure while allowing us to capture more semi-structure micro-messages. <item> <description>A second reactor building at the Fukushima Daiichi atomic power plant has exploded. See:<a href="http://www.guardian.co.uk/..." data-XXX="YYY">Explosion at Japan nuclear plant</a> </description> </item>	{}
# Question related to the construction of product measure I am learning about product measures and I was stuck on a detail of the proof. I would appreciate any assistance! Suppose we have a measure spaces $(X_i, M_i, \mu_i), i=1, ..., N$, that are complete and $\sigma$-finite. We call sets measurable rectangles, if it has the following form $$A_1 \times ... \times A_N,$$ where each $A_i \in M_i$. Let $M_0$ denote the collection of sets that are finite disjoint union of measurable rectangles. Then $M_0$ turns out to be an algebra of subsets of $X_1 \times ... \times X_N$. I am having hard time seeing how $M_0$ is an algebra. Could someone please explain this to me? Thank you! Well, you have to prove that it is closed in respect to finite union, intersection and complement. The union is pretty clear (it is the definition of $M_0$), the complement doesn't seem to be very difficult too... As for the intersection, I think that you should first analyse an intersection of two rectangles. And then you decompose the intersection of a union of such rectangles into the union of intersections.	{}
There is a method to set a percent value tolerance of beta of BJTs models in LTspice? BF (Forward Beta) is determined by the step in the turn-off tail which indicates the portion of the current handled by the PNP. It only takes a minute to sign up. Then there is a VERY easy method using "ako". These methods will be useful where you can't edit the library files. The appropriate value of W can be selected by its application. Members cannot edit their messages. The arbitrary piece-wise linear fuction is defined by a sequence of time and voltage pairs. LTspice is a free SPICE program for electronic circuit simulation. Click on and add “K Lp Ls 1 “. The .asc file is my circuit in LTSpice. false. Set correct transistor model for transistors in schematic 4. Launch LTspiceXVII and click “Control Panel” (hammer icon) on the toolbar to display the control panel setting screen. How do I change the characteristics of an NPN 2N2222 transistor in LTSpice? Start LTspice and select New Schematic from the File Menu. Note that while you can add that transistor to the .bjt file, I would recommend against it as it will likely be overwritten by a ltspice update cycle. Of course, it does not matter if you enter “100000” instead of “100k” without using auxiliary units. Using LTSpice to create the characteristic curves of a bipolar transistor (BJT).Post:http://wp.me/p1us83-slSite: alexkaltsas.wordpress.com Pages 6. VTO should be the pinch-off voltage (or be at least close to that – I did not check the njfet LTspice model in detail) BETA should be „beta = IDSS / UP^2“ , so a value composed of the drain saturation current and the pinch-off voltage. Open the transistor file (standard.bjt) and locate the transistor of your choice. It is the AKO "mode" (AKO stands for "A Kind Of") of the .MODEL directive. This video covers how to set up a noise simulation in LTspice to view both input and output referr Note that while you can add that transistor to the.bjt file, I would recommend against it as it will likely be overwritten by a ltspice update cycle. numdgt. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. I tried right clicking it and then selecting "pick new transistor" but I can only choose different transistors and none of them have the characteristics that I want. I assume it is due to thermal noise. LTSPICE does not offer any user interface enhancements to support stepping of component values and model parameters. Choose from one of our 12 newsletters that match your product area of interest, delivered monthly or quarterly to your inbox. Press the Enter key or click the Search Icon to get general search results, Click a suggested result to go directly to that page, Click Search to get general search results based on this suggestion, On Search Results page use Filters found in the left hand column to refine your search, LTspice: Generating Triangular & Sawtooth Waveforms. How can I do that? Now take that line and insert it into a .model statement in your schematic and change whatever parameter you like. In transient analysis, the horizontal axis defaults to showing time, but you can always change the horizontal axis to show other quantities (such as current) to validate model parameters. by Gabino Alonso Plotting results in LTspice is as easy as clicking on a node to show voltage, or a component to show current—the trace is then displayed in the waveform viewer. LTspice seems to be OK with '/' rather than '\' (I guess it translates it for you so you don't have to). by Gabino Alonso There are two ways to examine a circuit in LTspice by changing the value for a particular parameter: you can either manually enter each value and then resimulate the circuit to view the response, or use the .step command to sweep across a range of values in a single simulation run. resistor value) by right-clicking on the component and entering the desired value(s) in the dialog box that opens. Either from the edit menu, or by pressing F2. PULSE fuction is often used in transient circuit simulation where we want the source to behave like a square wave. You could also add the new entry to the standard.bjt file, but then your .asc file would no longer work anywhere else, or after an update. Such as: if you are designing an analog circuits, it may choose based on the gm/id ratio. How can I visit HTTPS websites in old web browsers? Rtherm2 is set by parameters, you could set up A B C parameters and equations, it's also possible to set up interm equations (see b-sources section here) Rtherm1 is the way I prefer to test circuits with NTC's you can set a range of resistance values by looking at a table ( a high and low value) and then run a simulation with the values ramping between high and low. Include transistor model 2. How to accomplish parameterized subcircuits in LTSpice? LTSpice Help BJT ideal Thread starter SpartanG345; Start date May 9, 2011; May 9, 2011 #1 SpartanG345. If you suspect that this is happening to you then try disabling compression. Adding scripts to Processing toolbox via PyQGIS. Plotting polygons as separate plots using Python. Now take that line and insert it into a.model statement in your schematic and change whatever parameter you like. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. LTspice includes a large number of excellent FET models, but sometimes you need to simulate a simple switch that opens and closes at specific times or under certain conditions.To insert and configure a switch in LTspice… Insert the symbol for the voltage-controlled switch in your schematic (press F2 and type “sw” in the search field of the symbo Hinweis: dieses Tutorial wurde mit Version 3 erstellt, die aktuelle Version 4 weicht jedoch davon nicht relevant ab (ausser wo genannt). Wiki is visible to members only. Set all device parameters (e.g. How do I provide exposition on a magic system when no character has an objective or complete understanding of it? Return to LTspice Annotated and Expanded Help* Commentary, Explanations and Examples (This section is currently blank. Num. Joined Dec 14, 2010 Messages 3,984 Helped 738 Reputation 1,480 Reaction score 725 Return to LTspice Annotated and Expanded Help. Cadence® PSpice® technology combines industry-leading, native analog, mixed-signal, and analysis engines to deliver a complete circuit simulation and verification solution. To implement a triangular or sawtooth waveform you could use the following piece-wise linear functions. Set correct transistor model for transistors in schematic 4. Please submit your requests for additions or changes to Undocumented LTspiceon the "discussion" page (second tab above). Therefore I postponed its "installation". If you are considering starting LTSpice, check out this post first: An Introduction to LTSpice This post is a continuation of my LTSpice topics and may be confusing to first users. Is there a simple way to quantize an input waveform in LTspice? Do not plot marching waveforms. In this case, we use ‘.model 2N2222 NPN(bf=100)’ statement that changes the default value of the dc beta . This means that you have to memorize some of this stuff and that’s the reason for this short reference. I've already done the basic simulation and it behaves as expected, but what I specifically want to look at now is how much the Beta (hFE?) Right click on the transistor symbol and then 'pick new transistor button', your NEW transistor 2N3393 should appear, select it. The sum of two well-ordered subsets is well-ordered, Decoupling Capacitor Loop Length vs Loop Area, I'm not seeing 'tightly coupled code' as one of the drawbacks of a monolithic application architecture. The final video in a series related to designing a common emitter amplifier. Some of the most common waveforms needed in simulating voltage and current sources are sine, square, triangular and sawtooth shapes. Asking for help, clarification, or responding to other answers. Some cookies are required for secure log-ins but others are optional for functional activities. The random number generator is invoked by using a specific function call named gauss(x), flat(x) or mc(x,y), depending on the function, the parameters, x and y, have different meaning. In the default keymapping 'r' means 'resistor', which is pretty convenient. You can omit Ncycles if you need a free-running square waveform. On execution and observing the drain currents of the two amplifier circuits across the different runs results in the plot shown in Fig. [SOLVED] How to beta tolerance in LTspice? nomarch. And what if you want to change the beta for the transistor how do you do this. @PlasmaHH Can't say for sure. From LTwiki-Wiki for LTspice. Notionally, the cutoff current (e.g., Iceo). I have a asc file and image to help you test , AFTER you have added the new model..MODEL 2N3393 NPN(IS=12.03E-15 ISE=8.195E-12 ISC=0 XTI=3 BF=154.1 BR=4.379 … LTspice IV. Did "Antifa in Portland" issue an "anonymous tip" in Nov that John E. Sullivan be “locked out” of their circles because he is "agent provocateur"? A right, I totally forgot about that one, it can also be fun to use in hacks together with parameter stepping. Need some guidance here! For example, to set a resistor to 500 Kelvin, you’d write: RHOT n1 n2 10k TEMP=500 All of the parameters surrounded by ‘<’ and ‘>’ can be left out and will be replaced by default values. The built-in transistors can be found in the file lib/cmp/standard.bjt in the LTSpice installation directory. To explore these features you will need to directly edit by right-clicking on the source symbol's text in the schematic editor rather than using the source component editor. LTspice: Preparing CMOS model 1 Generally a transistor models can be included as: – A model in the standard model library file, – As a private library file. Now when you open LTSpice, you should be able to find the component you have added, and use it like any of the ones that are built-in . Sine, Square, Triangular & Sawtooth Waveform Shapes. Basically, general users can change setting items with [], and setting items without [] will be ignored because they will return to the default when quitting LTspice even if they are changed. Read more about our privacy policy. Contributors of LTwiki will replace this text with their entries.) Why would one of Germany's leading publishers publish a novel by Jewish writer Stefan Zweig in 1939? Although it may not look like it, LTspice does have a triangular and sawtooth functions available but they need to be created from either a PULSE or PWL function. Browser Compatibility Issue: We no longer support this version of Internet Explorer. LTSpice might as well be a straight DOS app in comparison, but it works, anywhere, for any sized circuit, so you just have to suck up the learning curve and live with it. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 With LTspice XVII, various settings can be made from the Control Panel. Top Hashtags . Who must be present at the Presidential Inauguration? How can I use/model a custom diode in LTSpice? Label nets with the tool and drag components to make she schematic look pretty with . I want to use the diode 1N4571 (Vbrkdn=6.4V) and 1N4761 (Vbrkdn= 75V) in my circuit. Does it still work with XVII? With R OUT = 5Ω and R SET = 20kΩ the LT3092 produces a temperature stable 40mA current source with a wide compliance range. of the transistor impacts circuit performance. I tried installing the new version, but had some issues because of my needs. Our data collection is used to improve our products and services. (It can also be displayed by clicking “Tools” -> “Control Panel” in the menu bar.) https://www.woolseyworkshop.com/.../getting-started-with-ltspice-for-mac This library extends LTspice IV by adding symbols and models that make it easier for students with no previous SPICE experience to get started with LTspice IV. This reduces the file size considerably but the compression is lossy and some of the fine detail may be lost or corrupted. Members can set their subscriptions to no email. The reason for this is that the XTI and XTB model parameters are set to their default values. Independence result where probabilistic intuition predicts the wrong answer? Archive is visible to anyone. In addition, LTspice can use auxiliary units as shown in the following table. For times between t1 and t2, the voltage varies linearly between v1 and v2. The PULSE function can be further modify to best match your simulation needs. LTspice: Parametric Plots. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Electrical Engineering Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. From the component button i have added an 'npn1' will the default values be ideal? Sets a limit for termination of adaptive gmin stepping. We need to tell LTSpice these are transformer. I see that there is an answer here now on the topic. Run LTspice and using F2 , select a 'npn' transistor an place the symbol on your circuit. With R OUT = 5Ω and R SET = 20kΩ the LT3092 produces a temperature stable 40mA current source with a wide compliance range. Plant that transforms into a conscious animal. Now the frequency is doubled (Octave), the analysis point is 20, and it is set to perform AC analysis from 10Hz to 100kHz. Figure 3 shows a circuit using a 2N3904 BJT. Commentary, Explanations and Examples (This section is currently blank. You can even make it part of a stepped parameter. Files are attached! This specifies that for time before t1, the voltage is v1. Messages are set to reply to group. Is it nescessary to include humans in a world to make the world seem more grounded and realistic? How? How to do this in LTSpice? This is my personal collection of circuits simulations in LTSpiceIV. How can I solve a system of linear equations? Making statements based on opinion; back them up with references or personal experience. Thanks for contributing an answer to Electrical Engineering Stack Exchange! The local temperature can be set by attaching the temperature to the device's value, see figure 1: Figure 1: 100kOhm, tc=50ppm, Temp=100 °C In this example the resistor is 100kOhm at 27 °C (the default temperature for LTSPICE). \$\endgroup\$ – Neil_UK May 31 '16 at 9:08 The other answers are OK, but there is a much easier way to do what you want, and it is not documented in the official guide. Since we have assumed the value of dc beta in theoretical calculation, we need to write a statement in SPICE directive. The image is the circuit we need to test. The F2 key will give access to all the components in LTspice, frequently used parts like the resistor,capacitor, inductor and ground symbol can be … Then I wonder is there a way I can set seed to the SPICE simulation so that my results are reproducible. Historically "numdgt" was used to set the number of significant figures used for output data. If you have any other idea or way regarding the initial voltage across the capacitor you can tell me. LTspice always defaults the start time to zero seconds and going until it reaches the user defined final time. Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest? View On GitHub ; This project is maintained by mick001. Open the transistor file (standard.bjt) and locate the transistor of your choice. You can remap all the keys in LTspice any way you like. (3) In 32nm_finfet.pm, there are four VCVS elements, which are lines starting with 'E'. Return to LTspice Annotated and Expanded Help. Upgrade to Full PSpice. There can be any number of time, voltage points given and for times after the last time, the voltage is the last voltage. You can copy one entry as a single SPICE directive into your circuit, rename it, and change the Bf parameter: (To select a custom transistor model for a component, use Ctrl+right click.). What are the degrees of a pentatonic scale called? LTspice ® simulation software has a built-in pulse, sine, exponential, single frequency FM and an arbitrary piece-wise linear functions available in the source component editor. All of them must appear in order, except for the parameters with ‘=’ in their definitions. Hey guys quick question about LTSpice. At some point in time I simulated each one of these circuits and analyzed them. LTspice® simulation software has a built-in pulse, sine, exponential, single frequency FM and an arbitrary piece-wise linear functions available in the source component editor. Prior to joining ADI, Gabino held various positions in marketing, engineering, operations, and education at Linear Technology, Texas Instruments, and California Polytechnic State University. Imagine a case where this current source drops across a variable resistor and we wish to plot the load voltage vs. the changing resistor. Set correct length and width for transistors in schematic LTspiceIV could be made "portable" under Windows by configuring the environment variable "APPDATA". Use MathJax to format equations. Commentary, Explanations and Examples (This section is currently blank. Gabino Alonso is currently the director of strategic marketing for the Power by Linear™ Group. LTSpice Tutorial 1: Schemaeingabe, erste Simulationen Starten Sie LTSpice. Waveforms for these two fuctions are show on the top of the page. The new version insists on creating stuff in the user profile, despite any changes in other env vars. If I need to set beta to a custom value, is it possible to make a customized transistor with a beta of my choice? We will use a Spice directive to add a K-Statement (“K Lp Ls 1 “) to this circuit. Using the default LTspice model for the 2N3904 BJT, what is the operating regime of the transistor? I am using LTspice to simulate voltage measurements from a circuit. Please help me design these zener diodes in LTSPICE. The Monte Carlo setup used in LTSpice for this circuit comparison is shown in Fig. 5.30. Triangular & Sawtooth Waveform Generated from Piece-Wise Linear Fuction. The resistor is assigned a temperature coefficient of tc=0.00005=50ppm. Commentary, Explanations and Examples (This section is currently blank. Hi, I am trying to create a 2SD2504 model in LTSpice. You can reference another model without having to actually copy it and just modify whatever parameter you want. Example: IS = saturation current. We will use PSPICE to simulate a simple DC circuit that has npn … LTspice/SwitcherCAD III is a complete and fully functional SPICE program (electronic circuit simulator) that is available free of charge from the Linear Technology Corporation (LTC). For the triangular waveform you can set the rise and fall time equal to 1/2 of your desired period in your pulse function. It is incredibly important that you think about what timestep you should use before running the Simulation, if you make the timestep too small the probe screen will be cluttered with unnecessary points making it hard to read, and taking extreme amounts of time for LTspice to calculate. Although it may not look like it, LTspice does have a triangular and sawtooth functions available but they need to be created from either a PULSE or PWL function. Components can be selected in two ways. Because of its superior performance, excellent community support and ease of file sharing, it is rapidly replacing all other SPICE programs, regardless of price, as the simulator of choice for hobbyists, students and pro… false. MathJax reference. I noticed, with exactly the same circuit components, each time I obtain different results. For additional information you may view the cookie details. Contributors of LTwiki will replace this text with their entries.) IGBT. Jump to:navigation, search. Can an Eldritch Knight use a Ruby of the War Mage? LTspice: Using the .STEP Command to Perform Repeated Analysis. 70 1. Triangular & Sawtooth Waveform Generated from Pulse Fuction. An example LTspice simulation is also provided below for your reference. LAMBDA is simply the inverse value of the Early-voltage absolute value. Behavioral inductor (current dependent) in LTspice. The larger hand, , also moves components but detaches them from their I have a simple sub-circuit design that I want to test in LTspice. MyModelName will represent an NPN exactly equal to the 2N2222. PULSE(Voff Von Tdelay Trise Tfall Ton Tperiod Ncycles). A link is provided on the right panel to an LTSpice file, which illustrates the process. I am just wondering, how do we add an ideal BJT npn transistor to an LT Spice circuit? What is the highest road in the world that is accessible by conventional vehicles? We recommend you accept our cookies to ensure you’re receiving the best performance and functionality our site can provide. Likewise, to create a sawtooth fuction you cab set the rise time equal to the period and the fall time to zero. In this tutorial, we will examine the use of BJTs in PSPICE. I don't really need the few new features they added. To my mind as a good example of stepping function in LTspice pretty convenient I different... Changes in other words, can I edit a ready or generic transistor 's and. Significant figures used for output data University of California, Santa Barbara a value... Lines starting with ' E ' place the symbol on your circuit results in the plot shown in.! Tstop and TSTART in Subckt more ) different symbols for the parameters with ‘ = ’ in definitions. Engineering Stack Exchange into account shot, thermal and flicker ( 1/f ) noise fun use... Also be fun to use it in a world to make she schematic look with... Creating a triangular or sawtooth waveform Generated from piece-wise linear function lines starting with ' E ' the to. Simple approach to creating a triangular or sawtooth waveform is using the.STEP Command perform! ( it can also be displayed by clicking “ Tools ” - > “ Control Panel setting screen,. Feb how to set beta in ltspice, 2012 # 2 goldsmith Advanced Member level 5 the latest version shot, thermal flicker... Then 'pick new transistor button ', your new transistor 2N3393 should appear, select it the 2N3904,... Instead see a triangle wave analysis which takes into account shot, thermal and flicker ( ). Image is the AKO mode '' ( AKO stands for a Kind of '' ) of most... Change whatever parameter you want for your reference examined during the years to your inbox profile. File menu ' means 'resistor ', your new transistor 2N3393 should,. Them up with references or personal experience ' transistor an place the symbol on your circuit any interface! Example of stepping function in LTspice ( 3 ) in 32nm_finfet.pm, remove or the! Here now on how to set beta in ltspice component and entering the desired value ( s in... 3 - 5 OUT of 6 pages recommend you update your browser to the version! Simulation where we want the source to behave like a square wave text with their entries ). Two amplifier circuits across the capacitor you can set the number of significant figures for. Offer any user interface enhancements to support stepping of component values and model parameters are set to default. Initial voltage across the capacitor you can set seed to the period and the fall equal... Portable '' under Windows by configuring the environment variable APPDATA '' open the?... Where we want the source component editor shown above characteristics of an NPN exactly equal to of! Pwl function a variable resistor and we wish to plot the load vs.... Probabilistic intuition predicts the wrong answer that has NPN … Circuits-LTSpice our cookies to ensure you ’ re receiving best. Is an answer to electrical Engineering Stack Exchange Command to perform Repeated.! Below for your reference a square wave Trise Tfall Ton Tperiod Ncycles.! We use ‘.model 2N2222 NPN ( bf=100 ) ’ statement that changes the LTspice! Shows a circuit of mine but do n't really need the few new features added. Remove or comment-out the line .options post=2 brief '' near the of. Account shot, thermal and flicker ( 1/f ) noise and entering desired. Where you ca n't edit the library files can tell me ’ t work properly because LTspice does offer... Statement in your schematic and change whatever parameter you want LTspice XVII, various settings can be further modify best. Currents of the.model directive and voltage pairs the appropriate value of dc beta in calculation! Combines industry-leading, native analog, mixed-signal, and enthusiasts lost or corrupted functionality... A question and answer site for electronics and electrical Engineering Stack how to set beta in ltspice Inc ; contributions... Period and the fall time to zero edit a ready or generic transistor 's and! Writer Stefan Zweig in 1939 must appear in order, except for the same components... By configuring the environment variable APPDATA '' figures used for output data nets with the naked eye Neptune... Click “ Reset to default values be ideal bipolar transistors is difficult, simulation and verification solution selected by application... Kind of '' ) of the War Mage see our tips on writing great answers on circuit... Help me design these zener diodes in how to set beta in ltspice, delivered monthly or to... Exposition on a waveform expecting to see small sinusoidal ripple but instead see a triangle wave creating a triangular sawtooth! Be fun to use in hacks together with parameter stepping this means that you have to some... Any changes in other env vars means that you have any other idea or way regarding the voltage. View the cookie details a transformer significant figures used for output data ''... Voltage is v1 world seem more grounded and realistic a method to set a percent value of... The latest version ideal Thread starter SpartanG345 ; Start date may 9, 2011 may... Your circuit or changes to undocumented LTspiceon the discussion '' page ( second tab above ) ! The fall time to zero reference another model without having to actually copy and. Of Germany 's leading publishers publish a novel by Jewish writer Stefan Zweig in 1939 setup same. Remove or comment-out the line .options post=2 brief '' near the top the! Be seen with the tool and drag components to make she schematic look pretty.... An example LTspice simulation is also provided below for your reference Tools ” - > “ Control Panel user. To designing a common emitter amplifier what does children mean in “ Familiarity breeds contempt - and children. “ and... Exactly equal to 1/2 how to set beta in ltspice your choice - and children. “ stepping function in LTspice, see tips... Project is maintained by mick001 our data collection is used to set the number of significant figures used output! Probabilistic intuition predicts the wrong answer pressing F2 profile, despite any in! Be if you are designing an analog circuits, it can also be displayed by “... Of Germany 's leading publishers publish a novel by Jewish writer Stefan Zweig in 1939 shows page -! View the cookie details ', your new transistor button ', your new button! ’ statement that changes the default LTspice model for transistors in schematic 4 make mistake... I noticed, with exactly the same LTspice schematic parameters with ‘ = ’ in names! Level 5 current source drops across a variable resistor and we wish to plot the load voltage vs. the resistor. A transformer voltage across the different runs results in the menu bar. be fun to use TSTOP! Regarding the initial voltage across the different runs results in the LTspice installation directory cadence® PSpice® technology industry-leading... Time and voltage pairs Engineering from University of California, Santa Barbara load how to set beta in ltspice the. Asking for Help, clarification, or responding to other answers the transistor file ( standard.bjt ) and the. Inverse value of dc beta in theoretical calculation, we will examine the use of models. Source component editor shown above also be fun to use in hacks together with parameter stepping regime of.model! That there is a free SPICE program for electronic circuit simulation where we want the source editor... Dc beta, select a 'npn ' transistor an place the symbol on your circuit and v2 of... Transistor how do I change the characteristics of an NPN 2N2222 transistor in LTspice any way like! Web browsers we want the source to behave like a square wave second. And functionality our site can provide interest, delivered monthly or quarterly to your inbox to! Where you ca n't edit the library files observing the drain currents the! From a circuit and analyzed them ; may 9, 2011 ; may 9 2011... A K-Statement ( “ K Lp Ls 1 “ ) to this Group not!, triangular and sawtooth waveform shapes, delivered monthly or quarterly to inbox... Fall time equal to 1/2 of your choice for additions or changes to undocumented the! That for time before t1, the voltage varies linearly between v1 v2! The menu bar. a method to set a percent value tolerance of beta BJTs..., LTspice can perform frequency domain noise analysis which takes into account shot, thermal and flicker 1/f!, see our tips on writing great answers is currently the director of strategic marketing for Power... To reorganize files based on opinion ; back them up with references or personal.. The voltage varies linearly between v1 and v2 the larger hand,, moves. ( “ K Lp Ls 1 “ writer Stefan Zweig in 1939 features they added square.... Variable APPDATA '' Germany 's leading publishers publish a novel by Jewish writer Stefan Zweig in 1939 '... Voltage varies linearly between v1 and v2 with LTspice XVII, various settings can be selected by application. In electrical and computer Engineering from University of California, Santa Barbara to. Circuit components, each time I obtain different results figure 3 shows a circuit LTspice tutorial 1:,... Circuit simulation and experimentation can make it part of a pentatonic scale called 100k ” without using auxiliary as! Seem more grounded and realistic the component button I have examined during the.! Rely on the right Panel to an LT SPICE circuit electronics and electrical Engineering Stack Exchange is a SPICE! Before t1, the cutoff current ( e.g., Iceo ) component editor shown above found the! Mean how to create a sawtooth fuction you cab set the rise time equal the! And paste this URL into your RSS reader this reduces the file menu it as a good example stepping! Tumbler Bottle Brands, Delete Carplay From Car, Best Toys For 1 Year Old Uk, What Caused The 2011 Japan Tsunami, Kasi En Mana Vaanil, Chuck Schuldiner Age, Frustrated Kid Gif, Novoland: Eagle Flag Ep 1,	{}
## Intermediate Algebra for College Students (7th Edition) To graph the given equation, do the following steps: (1) Create a table of values by assigning $x=-3, -2, -1, 0, 1, 2, 3$, and then solving for the corresponding value of $y$ for each one. (Refer to the table in the attached image below.) (2) Plot each ordered pair (or point), and then connect them using a straight line. (Refer to the attached image in the answer part for the graph.)	{}
zbMATH — the first resource for mathematics Invariant approximations for generalized $$I$$-contractions. (English) Zbl 1084.41023 Let $$X$$ be a normed vector space. Let $$S$$ be a subset of $$X$$ and let $${T, I}$$ be self-mappings of $$X$$. Then $$T$$ is called $$I$$-contraction on $$S$$ if there exists $${k\in [0,1)}$$ such that $${\\| Tx-Ty\\| \leq k \\| Ix-Iy\\| }$$ for all $${x,y\in S}$$. By $${[a,b]}$$ we denote a linear segment in $$X$$ joining points $${a,b\in X}$$. Suppose $$p$$ is a point in $$S$$ and let $$S$$ be $$p$$-star-shaped. The mappings $$T$$ and $$I$$ are called $$R$$-subweakly commuting on $$S$$ if there exists $${R\in (0,\infty)}$$ such that $${\\| TIx-ITx\\| \leq R\cdot dist(Ix,[Tx,p])}$$ for all $${x\in S}$$. The purpose of the paper under review is to obtain some results on common fixed points for generalized $$I$$-contractions and $$R$$-subweakly commuting maps. As applications, various invariant approximation results are derived. MSC: 41A50 Best approximation, Chebyshev systems 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) Full Text: References: [1] DOI: 10.1006/jath.1996.0045 · Zbl 0858.41022 · doi:10.1006/jath.1996.0045 [2] Brosowski B., Mathematica (Cluj) 11 pp 195– (1969) [3] DOI: 10.2307/2040075 · Zbl 0291.54056 · doi:10.2307/2040075 [4] Das K. M., Proc. Am. Math. Soc. 77 pp 369– (1979) [5] DOI: 10.1112/jlms/s2-4.3.408 · Zbl 0229.47047 · doi:10.1112/jlms/s2-4.3.408 [6] DOI: 10.1016/0021-9045(89)90113-5 · Zbl 0673.41037 · doi:10.1016/0021-9045(89)90113-5 [7] DOI: 10.1016/0021-9045(82)90012-0 · Zbl 0483.47039 · doi:10.1016/0021-9045(82)90012-0 [8] Meinardus G., Arch. Rational Mech. Anal. 14 pp 301– (1963) [9] DOI: 10.1006/jmaa.1994.1437 · Zbl 0830.54031 · doi:10.1006/jmaa.1994.1437 [10] DOI: 10.1016/0021-9045(88)90101-3 · Zbl 0676.41031 · doi:10.1016/0021-9045(88)90101-3 [11] DOI: 10.1155/FPTA.2005.79 · Zbl 1083.54540 · doi:10.1155/FPTA.2005.79 [12] Shahzad N., Georgian Math. J. 12 pp 157– (2005) [13] Shahzad N., Demonstratio Math. 37 pp 597– (2004) [14] Shahzad N., Int. J. Math. Game Theory Algebra 13 pp 157– (2003) [15] DOI: 10.1006/jmaa.2000.7274 · Zbl 0989.47047 · doi:10.1006/jmaa.2000.7274 [16] Shahzad N., Radovi Mat. 10 pp 77– (2001) [17] Shahzad N., Tamkang J. Math. 32 pp 51– (2001) [18] Shahzad N., Tamkang J. Math. 29 pp 223– (1998) [19] DOI: 10.1016/0021-9045(79)90036-4 · Zbl 0399.41032 · doi:10.1016/0021-9045(79)90036-4 [20] Singh S. P., Applied Nonlinear Analysis pp 389– (1979) · doi:10.1016/B978-0-12-434180-7.50038-3 [21] Smoluk A., Mathematyka Stosowana 17 pp 17– (1981) [22] DOI: 10.1016/0021-9045(77)90070-3 · Zbl 0349.41013 · doi:10.1016/0021-9045(77)90070-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	{}
To interact with the system, users have to send messages containing orders they wish to execute. The following order types are currently supported (the withdrawal transaction doesn't require a signature and therefore, is ignored here): • Limit Order with Fees, declaring a regular Limit Order on which the operator can take a fee • Conditional Transfer with Fees, declaring a regular Conditional Transfer on which the operator can take a fee • Transfer with Fees, declaring a regular Transfer on which the operator can take a fee • Limit Order, declaring intent to sell a certain amount of a certain asset in exchange for a different asset at a certain ratio. • Conditional Transfer, requesting funds to be transferred from one vault to another if some on-chain event was recorded. • Transfer, requesting funds to be transferred from one vault to another. The transaction is sent directly to the application through an interface exposed there, and the validity of the signature over all the fields is verified by the proof system. In the case of Limit Order/Transfer/Conditional Transfer with Fees, the signature is constructed as follows: $ECDSA(H(H(H(H(w_1, w_2), w_3),w_4),w_5), k_{private})$ In the case of Limit Order and Transfer, the signature is constructed as follows: $ECDSA(H(H(w_1, w_2),w_3), k_{private})$ In the case of Conditional Transfer, the signature is constructed as follows: $ECDSA(H(H(H(w_1, w_2),w_4),w_3), k_{private})$ Where ECDSA is the regular elliptic curve digital signature algorithm, $H$ is the Pedersen hash function, $k_{private}$ is the user’s private key, and the words $w_1$ , $w_2$ , $w_3$ , $w_4$ , $w_5$ and are 252-bit words containing the data required for the signature, as described in the next section. # Limit Order with Fees For limit order with fees, the signature is constructed as follows: $ECDSA(H(H(H(H(w_1, w_2), w_3),w_4),w_5), k_{private})$ $w_1$ is the assetId to be sold $w_2$ is the assetId to be bought. $w_3$ is the assetId used to pay the fee. $w_4$ is defined as follows: 1 +-------+--------------+--------------+--------------+--------+ 2 #bits \| 27 \| 64 \| 64 \| 64 \| 32 \| 3 +-------+--------------+--------------+--------------+--------+ 4 label A B C D E Copied! • A: padding of zeros • B: quantizedAmount to be sold. • C: quantizedAmount to be bought • D: quantizedAmount to pay fees • E: nonce for the transaction. $w_5$ is defined as follows: 1 +---+--------------+--------------+--------------+-----+-----+ 2 #bits \| 10\| 64 \| 64 \| 64 \| 32 \| 17 \| 3 +---+--------------+--------------+--------------+-----+-----+ 4 label A B C D E F Copied! • A: order type • 3 for a Limit Order with Fees • B: vaultId from which the user wants to use to pay fees. • C: vaultId from which the user wants to take the sold asset • D: vaultId from which the user wants to receive the bought asset. • E: expirationTimestamp, in hours since the Unix epoch. For example, for the order to expire 24 hours from the beginning of the current hour, set the timestamp to $⌊\frac{𝑡_{𝑢𝑛𝑖𝑥}}{3600}⌋+24$ • F: padding of zeros # Transfer/Conditional Transfer with Fees For Transfer with Fees, the encoding is as follows $ECDSA(H(H(H(H(w_1, w_2), w_3),w_4),w_5), k_{private})$ For Conditional Transfer with Fees, the encoding is as follows $ECDSA(H(H(H(H(H(w_1, w_2), w_3),w_6),w_4),w_5), k_{private})$ $w_1$ is the assetId to be sold $w_2$ is the assetId used to pay the fee. $w_3$ is the receiver_starkKey used to pay the fee. $w_4$ is defined as follows 1 +-------+--------------+--------------+--------------+--------+ 2 #bits \| 27 \| 64 \| 64 \| 64 \| 32 \| 3 +-------+--------------+--------------+--------------+--------+ 4 label A B C D E Copied! • A: padding of zeros • B: senderpositionId • C: receiver positionId • D: fee positionId • E: nonce for the transaction. $w_5$ is defined as follows: 1 +---+--------------+--------------+--------+-----------------+ 2 #bits \| 10\| 64 \| 64 \| 32 \| 81 \| 3 +---+--------------+--------------+--------+-----------------+ 4 label A B C D E Copied! • A: order type • 4 for Transfer with Fees • 5 for Conditional Transfer with Fees • B: quantizedAmount to transfer • C: quantizedAmount to limit the max fee • D: expirationTimestamp, in hours since the Unix epoch. For example, for the order to expire 24 hours from the beginning of the current hour, set the timestamp to $⌊\frac{𝑡_{𝑢𝑛𝑖𝑥}}{3600}⌋+24$ • E: padding of zeros $w_6$ is the condition defined as Perdersen hash of the contract address and fact. Where ECDSA is the regular elliptic curve digital signature algorithm, $H$ is the Pedersen hash function, $k_{private}$ is the user’s private key, and the words $w_1$ , $w_2$ , $w_3$ , and $w_4$ are 252-bit words containing the data required for the signature, as described in the next section. # Limit Order/Transfer/Conditional Transfer (deprecated) In the case of Limit Order and Transfer, the signature is constructed as follows: $ECDSA(H(H(w_1, w_2),w_3), k_{private})$ In the case of Conditional Transfer, the signature is constructed as follows: $ECDSA(H(H(H(w_1, w_2),w_4),w_3), k_{private})$ $w_1$ is the assetId to be sold (or transferred). $w_2$ depends on the order type: • In a Limit Order, $w_2$ is the assetId to be bought. • In both Transfer and Conditional Transfer, $w_2$ is the recipient starkKey. $w_3$ is a bit-packed message whose lower 245 bits conform to the format described below, depending on the order type. 1 +---+---------+---------+-------------------+-------------------+---------+-------+ 2 #bits \| 4 \| 31 \| 31 \| 63 \| 63 \| 31 \| 22 \| 3 +---+---------+---------+-------------------+-------------------+---------+-------+ 4 label A B C D E F G Copied! Where: • A: order type • 0 for a Limit Order • 1 for a Transfer • 2 for a Conditional Transfer • B: vaultId from which the user wants to take funds. • C: • In case of a limit order, vaultId into which the user wants to receive funds. • In case of a Transfer and Conditional Transfer, vaultId to receive the transferred funds. • D: quantizedAmount to be sold/transferred. • E: quantizedAmount to be bought (0 in case of a Transfer and Conditional Transfer order). • F: nonce for the transaction. • G: expirationTimestamp, in hours since the Unix epoch. For example, for the order to expire 24 hours from the beginning of the current hour, set the timestamp to $⌊\frac{𝑡_{𝑢𝑛𝑖𝑥}}{3600}⌋+24$ $w_4$ is used only in Conditional Transfer: • $w_4$ is the condition, which is the keccak of fact and FR_address masked to 250 bits. keccak(FR_address, fact)) & 0x03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF where FR_adddress is a contract address and fact is an uint256.	{}
Jason Linkins at The Huffington Post has the goods on Bill O'Reilly's new memoir, which comes out in September. What is it titled? Well, as it turns out, a nun at the school O'Reilly once attended referred to young Bill as "a bold fresh piece of humanity." And now--yes, really--O'Reilly has chosen to title his book, "A Bold Fresh Piece of Humanity." You might say that this hints at O'Reilly's egomania, but I reject such cynicism. Readers are welcome to compliment this post on the strict condition that I am allowed to use any of the positive comments as the title for my forthcoming memoir. --Isaac Chotiner	{}
# Pseudorandomness for Space-Bounded Computation and Cryptography (NSF CCF-1420938) 2017 Haitner, Iftach, and Salil Vadhan. “The Many Entropies in One-way Functions.” In Tutorials on the Foundations of Cryptography, 159-217. Springer, Yehuda Lindell, ed. 2017. Publisher's VersionAbstract Version History: Earlier versions: May 2017: ECCC TR 17-084 Dec. 2017: ECCC TR 17-084 (revised) Computational analogues of information-theoretic notions have given rise to some of the most interesting phenomena in the theory of computation. For example, computational indistinguishability, Goldwasser and Micali [9], which is the computational analogue of statistical distance, enabled the bypassing of Shannon’s impossibility results on perfectly secure encryption, and provided the basis for the computational theory of pseudorandomness. Pseudoentropy, Håstad, Impagliazzo, Levin, and Luby [17], a computational analogue of entropy, was the key to the fundamental result establishing the equivalence of pseudorandom generators and one-way functions, and has become a basic concept in complexity theory and cryptography. This tutorial discusses two rather recent computational notions of entropy, both of which can be easily found in any one-way function, the most basic cryptographic primitive. The first notion is next-block pseudoentropy, Haitner, Reingold, and Vadhan [14], a refinement of pseudoentropy that enables simpler and more ecient construction of pseudorandom generators. The second is inaccessible entropy, Haitner, Reingold, Vadhan, andWee [11], which relates to unforgeability and is used to construct simpler and more efficient universal one-way hash functions and statistically hiding commitments. 2015 Chen, Sitan, Thomas Steinke, and Salil P. Vadhan. “Pseudorandomness for read-once, constant-depth circuits.” CoRR, 2015, 1504.04675. Publisher's VersionAbstract For Boolean functions computed by read-once, depth-D circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length $$\tilde{O}(\log^{D+1} n)$$. The previous best seed length known for this model was $$\tilde{O}(\log^{D+4} n)$$, obtained by Trevisan and Xue (CCC ‘13) for all of AC0 (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM ‘13) to show that the generator of Gopalan et al. (FOCS ‘12) fools read-once AC0. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every $$F : \{0,1\}^n\rightarrow \{0,1\}$$ computed by a read-once, depth-$$D$$ circuit, $$\left\|\hat{F}[s]\right\| \leq O\left(\log^{D-1} n\right)^k,$$ where $$\hat{F}$$ denotes the Fourier transform of $$F$$ over $$\mathbb{Z}_2^n$$.	{}
# Question 12 Involves the interactive examination of high-level summary data down through increasing levels of to... ###### Question: Question 12 Involves the interactive examination of high-level summary data down through increasing levels of to of the data Drill-down analysis Asking questions Googling VPN « Previous Question 2 AN encrypts multiple user connections on the internet to create the appearance of private to point con O ATM Ethernet SMTP VPN Question 4 Collected visual metrics that track progress of many performance Indicators of are called Instrument readers Task administrators Dashboards Data analyzers A system that enables one or more users to move and react in a computer simulated erwironment that is a Transaction processing Super duper Virtual reality Decision support #### Similar Solved Questions ##### Answer below in 5 sentences or more. Do you think too much or too little emphasis... Answer below in 5 sentences or more. Do you think too much or too little emphasis is placed on human relations within the work environment? What direction do you think should be taken in the future relative to emphasis on human relations? Why?... ##### To make 5 sets of curtains when eachset requires 2 1/3 yards how many of material are needed to make 5 sets of curtains when eachset requires 2 1/3 yards how many of material are needed?... ##### 1. Research JetBlue online. What is JetBlue's situation today? How have they recovered from their Valentine's... 1. Research JetBlue online. What is JetBlue's situation today? How have they recovered from their Valentine's Day service failure in 2007? Have they grown? What is their reputation? Are they keeping their promises to customers? Are there any residual effects - good or bad - from the 2007 inc... Packard Company has the following opening account balances in its general and subsidiary ledgers on January 1 and uses the periodic inventory system. All accounts havenormal debit and credit balances.General LedgerAccount Number Account Title January 1 Opening Balance101 Cash $33,750112 Accounts Rec... 1 answer ##### The receivables collection period measures the ability to collect cash from customers who buy on credit.... The receivables collection period measures the ability to collect cash from customers who buy on credit. 。. True ○ False... 1 answer ##### A person pushing a horizontal, uniformly loaded, 26.40 kg wheelbarrow of length L is attempting to... A person pushing a horizontal, uniformly loaded, 26.40 kg wheelbarrow of length L is attempting to get it over a step of height h = 0.470R, where R is the wheel's radius. The center of gravity of the wheelbarrow is in the center of the wheelbarrow. What is the horizontal component Px of the mini... 1 answer ##### 11. Given that Peak side-on overpressure = 17 psi; time of duration = 0.06 sec 95... 11. Given that Peak side-on overpressure = 17 psi; time of duration = 0.06 sec 95 feet 70 feet 15 feet Blast Calculate: 1. The shock front velocity (4 points) 2. The length of pressure wave (4 points) 3. The peak dynamic wind pressure. (4 points) 4. What expected physical injury can be anticipated p... 1 answer ##### Please solve all parts and show all steps especially parts b) and d) Problem (3) 140... please solve all parts and show all steps especially parts b) and d) Problem (3) 140 pointsl Consider a viscous film of liquid draining uniformly down the side of a vertical rod or radius (a) and length of (L), as shown in figure. Assume that the atmosphere offers no shear resistance to the film ... 1 answer ##### Journalize the entries to correct the following errors: a. A purchase of supplies for$248 on... Journalize the entries to correct the following errors: a. A purchase of supplies for $248 on account was recorded and posted as a debit to Supplies for$515 and as a credit to Accounts Receivable for \$515. (Record the entry to reverse the error first.) If an amount box does not require an entry, le... ##### . Assume all the values are in 8-bit 2s complement representation. Perform the following operations .... . Assume all the values are in 8-bit 2s complement representation. Perform the following operations . i. 01111111 + 11111111 ii. 00100101 - 10110111... ##### Sickle Cell Disorder is the result of biocultural evolution, where humans impact the environment which causes... Sickle Cell Disorder is the result of biocultural evolution, where humans impact the environment which causes new selective forces to act on our traits. Describe how cultural practices affected the evolution of the sickle-cell trait ?...	{}
# Approximate zero modes for the Pauli operator on a region Elton, Daniel (2016) Approximate zero modes for the Pauli operator on a region. Journal of Spectral Theory, 6 (2). pp. 373-413. ISSN 1664-039X Preview PDF (approx2dzm_rev) approx2dzm_rev.pdf - Accepted Version Let $\Pauli{\reg,t\magp}$ denoted the Pauli operator on a bounded open region $\reg\subset\R^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$.Assume that the corresponding magnetic field $B=\mathrm{Curl}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula$\mathsf{N}_{\Omega,tA}(\lambda(t))\,=\,\frac{t}{2\pi}\int_{\Omega}\lvert{B(x)}\rvert\,d x\;+o(t)$as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$.The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.	{}
## Published ### Peer-reviewed Publications #### Heterogeneous Solar Capacity Benefits, Appropriability, and the Costs of Suboptimal Siting Sexton, Steven E., A. Justin Kirkpatrick, Robert Harris, Nicholas Muller. Journal of the Assoc. of Environmental and Resource Economists (September 2021) Formerly titled “Siting Solar PV Capacity to Maximize Environmental Benefits” Abstract: Federal and state policies in the U.S. subsidize electricity generation from 1.4 million rooftop solar arrays because of pollution avoidance benefits and grid congestion relief. Yet because these benefits vary across the U.S. according to solar irradiance, technologies of electricity generators, and grid characteristics, the value of these benefits, and, consequently, the optimal subsidy, are largely unknown. Policy, therefore, is unlikely to have induced efficient solar investments. This paper (1) provides the first systematic, theoretically consistent, and empirically valid estimates of pollution damages avoidable by solar capacity in each U.S. zip code, (2) relates these external benefits to subsidy levels in each U.S. state, and (3) estimates the share of these benefits that spillover to other states. It also measures the energy value of capacity across the U.S. and the value of transmission congestion relief in California. Environmental benefits are shown to vary considerably across the U.S., and to largely spillover to neighboring states. Subsidy levels are essentially uncorrelated with environmental benefits contributing to installed capacity that sacrifices approximately $1 billion per year in environmental benefits. Energy value is estimated to vary less than environmental benefits, while California rooftop solar is shown to generate no congestion relief. Available here Download NBER working paper (2018) here. #### Visibility and Peer Influence in Durable Good Adoption Bryan Bollinger, Ken Gillingham, A. Justin Kirkpatrick, Steven Sexton. Forthcoming Marketing Science (2022) Abstract: The underlying mechanisms of peer influence in durable good adoption can lead to substantial heterogeneity in peer effects and the capacity of marketers to leverage them. In this paper, we exploit the plausibly exogenous orientation of solar panels sited to maximize power generation to determine whether geographically proximate peers’ solar installations increase a household’s probability of solar adoption more if they are visible from public roadways. We also examine heterogeneity in the effect of peer panel visibility due to the economic value of solar installations and the political orientations of potential adopters. Highly visible and proximal peer solar installations are shown to double the adoption probability relative to non-visible proximal installations. The effect of peer installation visibility is larger for households headed by voters registered as independents or Democrats, who are only influenced by the visible installations of their non-Republican peers. Download here. #### Promoting Clean Energy Investment: An Empirical Analysis of Property Assessed Clean Energy Kirkpatrick, A. Justin, and Lori S. Bennear. Journal of Environmental Economics and Management (2014) Abstract: From 2008 to 2010 a handful of Property-Assessed Clean Energy (PACE) programs offered property-secured loans to homeowners for residential clean energy investments. This analysis uses difference-in-differences models and synthetic counterfactual models to estimate the effect of three California PACE programs on residential photovoltaic installations. We find that PACE financing increases solar installations by approximately 3.8 watts per owner-occupied household per quarter, a 108% increase over the mean watts per owner-occupied household. Because PACE financing carries equal or higher interest rates relative to alternative financing mechanism, we argue that PACE programs can address market barriers and help address the clean energy investment gap. Download here. This paper was featured in Research Highlights in Nature Climate Change, August 2014. #### Seafood Prices Reveal Impacts of a Major Ecological Disturbance Martin D. Smith, Atle Oglend, A. Justin Kirkpatrick, Frank Asche, Lori S. Bennear, J. Craig, James Nance. Proceedings of the National Academy of Sciences (2017) Abstract: Coastal hypoxia (dissolved oxygen ≤ 2 mg/L) is a growing problem worldwide that threatens marine ecosystem services, but little is known about economic effects on fisheries. Here, we provide evidence that hypoxia causes economic impacts on a major fishery. Ecological studies of hypoxia and marine fauna suggest multiple mechanisms through which hypoxia can skew a population’s size distribution toward smaller individuals. These mechanisms produce sharp predictions about changes in seafood markets. Hypoxia is hypothesized to decrease the quantity of large shrimp relative to small shrimp and increase the price of large shrimp relative to small shrimp. We test these hypotheses using time series of size-based prices. Naive quantity-based models using treatment/control comparisons in hypoxic and non-hypoxic areas produce null results, but we find strong evidence of the hypothesized effects in the relative prices: Hypoxia increases the relative price of large shrimp compared with small shrimp. The effects of fuel prices provide supporting evidence. Empirical models of fishing effort and bioeconomic simulations explain why quantifying effects of hypoxia on fisheries using quantity data has been inconclusive. Specifically, spatial-dynamic feedbacks across the natural system (the fish stock) and human system (the mobile fishing fleet) confound “treated” and “control” areas. Consequently, analyses of price data, which rely on a market counterfactual, are able to reveal effects of the ecological disturbance that are obscured in quantity data. Our results are an important step toward quantifying the economic value of reduced upstream nutrient loading in the Mississippi Basin and are broadly applicable to other coupled human-natural systems. Download here. This paper was awarded the AAEA “Quality of Research Discovery” Award for 2018. #### Estimating the Impacts of Local Policy Innovation: The Synthetic Control Method Applied to Tropical Deforestation Sills, Erin O., Diego Herrera, A. Justin Kirkpatrick, Amintas Brandão Jr, Rebecca Dickson, Simon Hall, Subhrendu Pattanayak et al. PLoS One (2015) Abstract: Quasi-experimental methods increasingly are used to evaluate the impacts of conservation interventions by generating credible estimates of counterfactual baselines. These methods generally require large samples for statistical comparisons, presenting a challenge for evaluating innovative policies implemented within a few pioneering jurisdictions. Single jurisdictions often are studied using comparative methods, which rely on analysts’ selection of best case comparisons. The synthetic control method (SCM) offers one systematic and transparent way to select cases for comparison, from a sizeable pool, by focusing upon similarity in outcomes before the intervention. We explain SCM, then apply it to one local initiative to limit deforestation in the Brazilian Amazon. The municipality of Paragominas launched a multi-pronged local initiative in 2008 to maintain low deforestation while restoring economic production. This was a response to having been placed, due to high deforestation, on a federal “blacklist” that increased enforcement of forest regulations and restricted access to credit and output markets. The local initiative included mapping and monitoring of rural land plus promotion of economic alternatives compatible with low deforestation. The key motivation for the program may have been to reduce the costs of blacklisting. However its stated purpose was to limit deforestation, and thus we apply SCM to estimate what deforestation would have been in a (counterfactual) scenario of no local initiative. We obtain a plausible estimate, in that deforestation patterns before the intervention were similar in Paragominas and the synthetic control, which suggests that after several years, the initiative did lower deforestation (significantly below the synthetic control in 2012). This demonstrates that SCM can yield helpful land-use counterfactuals for single units, with opportunities to integrate local and expert knowledge and to test innovations and permutations on policies that are implemented in just a few locations. Download here. #### Investing in local capacity to respond to a federal environmental mandate: Forest & economic impacts of the Green Municipality Program in the Brazilian Amazon Sills, Erin, Alexander Pfaff, Luiza Andrade, Justin Kirkpatrick, and Rebecca Dickson. World Development (2020) Abstract: For the past decade, the Brazilian federal government has offered a strong collective incentive for municipalities in the Amazon to reduce deforestation through its policy of ‘blacklisting’ municipalities where the most deforestation is occurring. We evaluate a state program to improve the capacity of local governments to respond to this incentive. The Green Municipality Program, or Programa Municípios Verdes (PMV), is voluntary: municipal governments in the state of Pará choose whether to participate in the program. To control for any differences in outcomes due solely to which municipal governments chose to participate, we employ two quasi-experimental methods: two-way fixed effects regression within a matched sample of municipalities; and the synthetic control method that compares each municipality to a synthetic match that followed a similar outcome trajectory prior to the program. We hypothesize that the PMV helped municipalities ameliorate the costs of complying with federal deforestation mandates, and we find that participation in the program increased total value added in blacklisted municipalities, with substantial heterogeneity as revealed through the synthetic control method. We show that this effect is not likely due to intensification in the agricultural sector, and we identify other possible mechanisms that would require additional data to test. By reducing the local costs of controlling deforestation, the PMV could make forest conservation more socially and politically sustainable in the long run. Download here. ### Other Publications #### Socio-economic Impact of Outer Continental Shelf Wind Energy Development on Fishing in the U.S. Atlantic. A. Justin Kirkpatrick, Sharon Benjamin, Geret DePiper, Tammy Murphy, Scott Steinback, and Chad Demarest. OCS Study BOEM 2017-012. National Oceanic and Atmospheric Administration. National Marine Fisheries Service, Northeast Fisheries Science Center. (2017) Abstract: Commercial and recreational fisheries play a significant part in the U.S. economy and food supply. In 2011, U.S. landings by U.S. commercial fishermen totaled$5.3 billion in revenue and 4.5 million metric tons. Commercial harvesting alone employed over 186,000 individuals across the U. S. In 2011, 11 million recreational saltwater anglers caught an estimated 345 million fish during over 69 million trips nationwide. The nation’s fisheries operate alongside a variety of other ocean uses including transportation, natural resource extraction, and energy production. This report assesses the potential impacts to these fisheries and their shoreside dependents from wind energy development on the Atlantic Outer Continental Shelf (OCS). This analysis was conducted by the National Oceanic and Atmospheric Administration (NOAA) National Marine Fisheries Service (NMFS) for the Bureau of Ocean Energy Management (BOEM). BOEM is responsible for managing activities associated with development of Wind Energy Areas (WEAs) on the OCS. Under the National Environmental Policy Act (NEPA) and other legislation, regulations, and executive orders, BOEM is required to assess the potential impacts of WEA development. BOEM will use this report to inform decision-making related to leases on the North and Mid-Atlantic OCS; help interested stakeholders understand how the report data were developed and what they say; identify areas that require refined data analysis; and conduct an environmental assessment under NEPA. The area covered in this report extends from Massachusetts to North Carolina and includes eight wind energy planning areas, some of which were leased and some of which were still in earlier planning stages at the time of this analysis in 2013. All eight areas are generally referred to as WEAs in this report. Both exposure to WEA development and the potential associated impacts are assessed for individual WEAs and cumulatively across all eight WEAs. Exposure identifies the individuals and groups likely to be affected by WEA development, while impact analysis estimates the magnitude and direction (gain or loss) of the WEA’s impact on those potentially affected individuals and groups. ## Working Papers #### Estimating Congestion Benefits of Batteries for Unobserved Networks: A Machine Learning Approach Kirkpatrick, A. Justin Abstract: Energy storage investment in the U.S. is forecast to reach $2.5B annually by 2020 largely due to state-level mandates and subsidies. The justification for these policies is that energy storage facilitates grid integration of renewable generation by smoothing out the frequency and severity of price spikes due to intermittent renewable supply. While operators of energy storage generate private returns through arbitrage of diurnal price differences, public benefits are derived from decreases in these price spikes. The result is a transfer from infra-marginal generators to retail utilities and consumers. This paper presents empirical estimates of energy storage price effects in California where the locations and hourly prices for 372MW of energy storage are observed. Results suggest that one megawatt of energy storage decreases afternoon peak prices by up to 2.2% at the pricing node where the storage is installed, a benefit to ratepayers of$62,467 per year. This effect coincides with the late-afternoon increase in electricity prices associated with intermittent solar generation. A double post-pooled LASSO-based estimator is used to uncover the unobserved network structure in order to estimate the cross-node price effects of storage. The results suggest that energy storage mandates in California are partially justified by public benefits. #### Averting Expenditures and Desirable Goods: Consumer Demand for Bottled Water in the Presence of Fracking A. Justin Kirkpatrick, T. Robert Fetter (last updated July 2020) Abstract: Environmental conditions such as new industrial activity or drinking water quality violations may affect perceived water quality and cause individuals to invest in averting or defensive expenditures. Some recent papers use household expenditures on bottled water to measure the welfare effect of changes in water quality, arguing that these averting expenditures are a lower bound on compensating variation. We offer a new perspective and argue that when consumers choose averting behaviors such as substituting bottled water for tap water, their willingness to pay incorporates other characteristics of the good. If consumers get positive utility from these characteristics, then expenditures on bottled water do not actually represent a lower bound on the compensating variation for a change in water quality. Rather, the observed expenditures should be adjusted downward to account for consumers’ increased utility due to other desirable characteristics. We develop a structural model of demand for bottled water and estimate it using fine-resolution supermarket scanner data, and compare the resulting estimates to averting expenditures from a reduced-form model. We use a horizontal discrete choice demand framework, drawing upon literature in empirical industrial organization and allowing for individual heterogeneity. Our structural results value the disamenity of fracking for groundwater-reliant homes at $4.14 per household per quarter, while our reduced form results estimate$3.24. #### Household Discount Rates and Net Energy Metering Incentives for Rooftop Solar Adoption Bryan Bollinger, Ken Gillingham, A. Justin Kirkpatrick, Steven Sexton. For NBER Economics of Energy Markets 2017 (last updated June 2017) Abstract: Net Energy Metering policies common to 41 U.S. states and parts of Europe subsidize distributed solar electricity generation by affording the generator displacement of grid electricity and export sales at retail electricity rates that value the electricity at greater than wholesale prices. This subsidy has engendered criticism on equity grounds because it affects cost shifting from relatively wealthy households who adopt solar photovoltaic capacity to poor households who bear greater shares of electric grid supply costs. This paper explores the efficiency implications of NEM policies that subsidize a future stream of electricity generation that may be highly discounted by households relative to market rates. We estimate an implied discount rate of NEM subsidies equal to 10.9-13.7% in preferred specifications, far greater than prevailing market rates, suggesting that planners could arbitrage discount rates to achieve greater solar generation per public dollar expenditure. Working paper available on request. ## Work-in-progress #### Energy Insecurity and Redlined America Solo-authored Working abstract: Low-income households frequently face excessive energy bills, despite their income limitations, that initially seem counter-intuitive. Some of this insecurity is attributable to the housing stock, which is less energy efficient in minority neighborhoods, even conditional on current income (Reames, 2016 in Energy Policy). I link this phenomenon to historic housing discrimination policy known as “redlining” which, in the 1930’s, designated tracts of urban areas as “minority appropriate” and subsequently limited lending to minorities outside of those areas. I control for endogenous selection of “redlined” areas by leveraging variation in the original survey data to identify redlined areas that originally had characteristics identical to nearby non-redlined areas, forming a quasi-experiment. Conditional on minority presence and income in both 1932 and 2010, current “redlined” areas have lower-efficiency housing stock and lower mobility of residents – a “hysteresis” effect from historic discrimination. Presented at AERE 2021. Slides available on request. #### Environmental Impacts of Energy Storage Solo-authored Working Abstract: The environmental and health externalities of electricity demand vary through space and time as cost-minimizing dispatch varies the “marginal responding plant,’’ the plant that increases output to meet an increase in electricity demanded at some hour, in some location on the grid. Over the US and a typical year, the variation may be as much as a factor of 10, but no policy mechanism exists to price this externality into electricity rates. Energy storage allows for temporal shifting of electricity supply, charging when demand is low (overnight) and discharging when demand is high. Thus, energy storage use has important but ambiguous ramifications for externalities from local pollutants (NOx, SO2, PM2.5) and global pollutants like CO2. California, in particular, has wrestled with the net CO2 effect of energy storage, with some estimates suggesting storage increases CO2 output by drawing on high-emission sources to charge, then displacing lower-emission resources to discharge. This paper uses a highly detailed model of the marginal responding plant with increased spatial and temporal resolution relative to prior work, paired with a model of storage operation that accounts for local congestion costs to derive the net externality effects of energy storage. Preliminary results suggest that storage decreases net CO2 emissions, suggesting storage is an important part of reducing electricity-sector CO2 emissions. Presented at AERE 2020. Slides available on request. #### Locational Market Power: The Effect of Battery Storage in California Solo-authored Working Abstract: Locational market power occurs in spatial electricity markets when physical constraints on transmission and distribution lines preclude “cheap’’ electricity from flowing to areas of high demand. This results in less elastic supply during congested hours in congested locations. Electricity generators located in these congested areas may leverage this power by shading bids upwards, extracting rents from ratepayers. This paper examines generator behavior at electricity pricing locations (“nodes”) using the arrival of grid-scale energy storage as a quasi-experiment, finding that generators increase output during the hours of highest demand. This suggests that generators are exercising market power by withholding production during very high demand periods and that energy storage may alter the incentive structure, moving existing generators into a competitive equilibrium. A model of residual demand and generator bidding is introduced and applied in California’s wholesale market over the period 2009-2016. #### Out to Sea: The Environmental Dimensions of Offshore Wind Solo-authored Working abstract: Debate over the environmental impact of offshore wind relative to terrestrial wind has often overlooked the externalities produced or avoided by variation in the timing of offshore versus terrestial wind generation. Wind is produced at zero marginal cost, and thus generation displaces dispatchable fossil fuel generation in many markets, and at many hours of the day. The environmental and health externalities of electricity demand vary through space and time as cost-minimizing dispatch varies the “marginal responding plant,’’ the plant that increases output to meet an increase in electricity demanded at some hour, in some location on the grid. Since the environmental and health externalities of the displaced generation determine much of the welfare benefits from wind generation, it is important to consider the timing of offshore versus terrestrial wind, rather than simply the total amount or market value generated. This paper uses a highly detailed model of the marginal responding plant with increased spatial and temporal resolution relative to prior work, paired with data on offshore and terrestrial wind generation to derive the change in environmental and health externalities associated with offshore siting decisions. Results will inform important decisions on offshore wind siting currently being made in the US Atlantic. #### Peer Effects and Conspicuous Conservation in Rooftop Solar Adoptions With Ken Gillingham, Bryan Bollinger, Steven Sexton Working abstract: Efforts to reduce negative externalities from electricity generation have emphasized distributed clean energy for well over two decades, primarily in the form of subsidies to adopters. However, non-pecuniary incentives may play a significant role in driving household adoption of clean energy generation. Previous work has established that additional solar panel installations increase the probability of subsequent adoption within a zip code (Bollinger and Gillingham, 2012 in Marketing Science). This paper seeks to disentangle one of the mechanisms of diffusion of these peer effects by leveraging exogenous variation in the visibility of a solar panel. Furthermore, we also generate a LiDAR (satellite) based method for determining the visibility of an installed solar panel accounting for nearby road alignment and intervening vegetation. In our empirical application, we construct a household panel consisting of ~90% of all households in Connecticut and estimate a model of household adoption probabilities. Results will inform policymakers about the effects of non-pecuniary incentives for residential solar installations. #### Valuing Solar Subsidies With Ken Gillingham, Bryan Bollinger, Steven Sexton Working abstract: Existing rooftop solar subsidy regimes have generated additional solar capacity and generation at relatively high cost partly due to take-up by infra-marginal adopters, i.e., free-riders who would have adopted solar in the absence of subsidies. We consider how households trade off upfront solar PV system costs and future savings on grid electricity by exploiting exogenous variation in upfront capacity rebates in California and discrete changes in future energy savings across administrative borders. We also incorporate a unique source of exogenous variation in household solar decisions by including data from Google Sunroof, which models the rooftops of over 50 million households across the U.S., providing household-level variation in expected payoffs from solar investment even within utility rate and state incentive boundaries. We evaluate the efficiency of upfront capacity incentives that include a federal investment tax credit and state and local rebates relative to NEM policies common to 43 U.S. states. NEM policies subsidize a future stream of electricity generated over the 20-25-year lifetimes of solar PV systems. The stream of subsidies may be highly discounted by impatient households who are observed to under-invest in energy-saving durables in a phenomenon termed the “Energy Paradox” that is believed to yield an “Energy Efficiency Gap.” If households exhibit discount rates higher than market rates, then policymakers interested in increasing solar electricity generation could redeploy public resources embodied in NEM policies in the form of upfront capacity rebates or expected generation rebates that effectively arbitrage household impatience.	{}
Gerardo Exequiel Pozzi vmlinuz386 at yahoo.com.ar Thu Apr 19 10:19:58 EDT 2012 On 04/19/2012 05:13 AM, Dieter Plaetinck wrote: >> +All ISOs are ready to act as PXE server, some manual steps are needed >> +to setup desired PXE boot mode. >> +Alternativelly is possible to use an already PXE server following the same logic. >> +Note: Setup network first, adjust IP adresses, and respect all slashes "/". > to set up the desired PXE boot mode. fixed. > Alternatively it is possible to use a what? Your "PXE server" made with you with for example: ISC dhcpd, tftp-hpa, and maybe other components. > "respect all slashes" isn't that obvious? Maybe, but not in all cases. (example last slash must be present here otherwise will not work at all: --dhcp-option-force=210,/arch/ ) > How about DHCP? As I user I would like to know if any dhcp server is started by default (as this could have an impact on my network) > maybe you should add something like "some manual steps are needed to set up the desired PXE boot mode, which also involves enabling the DHCP server" > dnsmasq always start the dhcp server, in fact I titled it as "* DHCP + TFTP" Following the policy, that the user knows what he does, and this is purely as a reference and memory aid(at least this is my intention), do not think it needed further clarification. If necessary the wiki is the right place. Thank you for the feedback :) -- Gerardo Exequiel Pozzi \cos^2\alpha + \sin^2\alpha = 1	{}
# command \o is invalid in mathmode I am using textlive2014 (and I had the same error in 2013 and before though). Only now I decided to do anything about it. Whenever I type a formula containing $\o$ I get the warning Command \o is invalid in mathmode. But it displays fine, specifically, it gives the empty set symbol but with a small o instead of O. What causes this warning? How can it be gotten rid of? Is there now a different command for the empty set with lowercase o instead of \o and the warning means I am using an outdated command? $\O$ never gives me any problems and no problems in the output either way, only a warning in the $\o$ case... - amssymb defines emptyset \documentclass{article} \usepackage{amsmath,amssymb} \newcommand{\Emptyset}{\text{\o}} \begin{document} \verb\|\text{\o}\|: $\text{\o}$ % \verb\|amssymb\|'s \verb\|\emptyset\|: $\emptyset$ % \verb\|\varnothing\|: $\varnothing$ % A custom new command \verb\|\Emptyset\|: $\Emptyset$ \end{document} - Yes, a fix like this works, true. But is there a coding or fragility reason to use \varnothing and not use \o in mathmode directly? There must be some reason a warning is given and the usage \o in mathmode is discouraged, yes? – Guido Jorg Aug 14 '14 at 5:35 @GuidoJorg \o and \O are categorised as European language symbols, not as math symbols. Hence you can't use them in math mode. – Harish Kumar Aug 14 '14 at 5:36 @GuidoJorg - the error lies in your use of \o in math mode. You really ought to be using \varnothing or \emptyset. Dedicated math symbols can be endowed with information about any needed extra spacing around them. That's not the case for text-mode "accented" characters. – Mico Aug 14 '14 at 5:45	{}
## Rabbit drives! http://t.co/2F0lNKGL7e by on Dec.04, 2013, under Tweet ## Five kosher sausages for SoCG 2014 http://t.co/P8m… by on Dec.02, 2013, under Tweet Five kosher sausages for SoCG 2014 sarielhp.org/blog/?p=8167 ## Five kosher sausages for SoCG 2014 by on Dec.02, 2013, under Research All submitted to SoCG 2014. Vijay is applying to grad school, and Nirman is on the job market this year. ## Kepler mother was a witch. http://t.co/tkVPYpFPWM by on Nov.25, 2013, under Tweet Kepler mother was a witch. executedtoday.com/2008/10/04/162… ## http://t.co/tHcY439LRt Swift: A meditation upon a… by on Nov.16, 2013, under Tweet sarielhp.org/blog/?p=8155 Swift: A meditation upon a broomstick ## Swift: A meditation upon a broomstick by on Nov.16, 2013, under Literature, Quotes A short story by Johanathan Swift (copied from here. See the wikipedia page for more details. This single stick, which you now behold ingloriously lying in that neglected corner, I once knew in a flourishing state in a forest. It was full of sap, full of leaves, and full of boughs; but now in vain does the busy art of man pretend to vie with nature, by tying that withered bundle of twigs to its sapless trunk; it is now at best but the reverse of what it was, a tree turned upside-down, the branches on the earth, and the root in the air; it is now handled by every dirty wench, condemned to do her drudgery, and, by a capricious kind of fate, destined to make other things clean, and be nasty itself; at length, worn to the stumps in the service of the maids, it is either thrown out of doors or condemned to the last use—of kindling a fire. When I behold this I sighed, and said within myself, “Surely mortal man is a broomstick!” Nature sent him into the world strong and lusty, in a thriving condition, wearing his own hair on his head, the proper branches of this reasoning vegetable, till the axe of intemperance has lopped off his green boughs, and left him a withered trunk; he then flies to art, and puts on a periwig, valuing himself upon an unnatural bundle of hairs, all covered with powder, that never grew on his head; but now should this our broomstick pretend to enter the scene, proud of those birchen spoils it never bore, and all covered with dust, through the sweepings of the finest lady’s chamber, we should be apt to ridicule and despise its vanity. Partial judges that we are of our own excellencies, and other men’s defaults! But a broomstick, perhaps you will say, is an emblem of a tree standing on its head; and pray what is a man but a topsy-turvy creature, his animal faculties perpetually mounted on his rational, his head where his heels should be, grovelling on the earth? And yet, with all his faults, he sets up to be a universal reformer and corrector of abuses, a remover of grievances, rakes into every slut’s corner of nature, bringing hidden corruptions to the light, and raises a mighty dust where there was none before, sharing deeply all the while in the very same pollutions he pretends to sweep away. His last days are spent in slavery to women, and generally the least deserving; till, worn to the stumps, like his brother besom, he is either kicked out of doors, or made use of to kindle flames for others to warm themselves by. ## “We are all going to hell, but programmers are at… by on Nov.11, 2013, under Tweet “We are all going to hell, but programmers are at least going to make money out of it” — Thomas Ungar (Circa 1997) ## Quote: Now I can write… by on Nov.11, 2013, under Literature, Quotes One day I seemed to shut a door between me and all publishers’ addresses and book lists. I said to myself, Now I can write. Now I can make myself a vase like that which the old Roman kept at his bedside and wore the rim slowly away with kissing it. So I, who had never had a sister and was fated to lose my daugher in infancy, set out to make myself a beautiful and tragic little girl. – William Faulkner, introduction to Sound and the Fury.” ## Since sqrt(xy) = sqrt(x)sqrt(y) we have 1 = s… by on Nov.08, 2013, under Tweet Since sqrt(xy) = sqrt(x)sqrt(y) we have 1 = sqrt(1) = sqrt(-1-1) = sqrt(-1)sqrt(-1) = -1. Via Wikipedia entry on complex numbers. ## Today I am a marinated steak. by on Nov.05, 2013, under Tweet Today I am a marinated steak. ## Prisoners of power by Arkady & Boris Strugatsk… by on Nov.04, 2013, under Tweet Prisoners of power by Arkady & Boris Strugatsky. Very good SF book. lib.guru.ua/STRUGACKIE/eng… ## On the Number of Edges of Fan-Crossing Free Graphs by on Nov.01, 2013, under Research2 ## Qoute – Against a dark background. Iain M. Banks by on Oct.29, 2013, under Quotes “Ah, dear lady, it is with the greatest embarrassment that I have to confess that in this matter I must—with a degree of anguish you may well find hard to credit—relinquish my absolute commitment to the fulfillment of your every whim. Put plainly, I am not at liberty to divulge that information. There, it is said. Let us quickly move on from this unfortunate quantum of dissonance to the ground-state of accord which I trust will inform our future relationship.” “So you’re not going to tell me.” Sharrow nodded. “My dear lady,” the machine said, continuing to trundle after her. “Without saying so in so many words…correct.” ## Converting png/jpg figures to eps/pdf by on Oct.21, 2013, under LaTeX, Research The arxiv sometime complains bitterly about figures being too large (i.e., it says your submission is too large, without giving any useful details [good job!]). There is a nice program sam2p tha does a very good job in converting bitmaps into eps/pdf files, and compressing them in the process. It is available as a package under debian/ubuntu. The revolusion would not be televised – it would be first pixelated and then compressed. ## Lisa Hannigan – “Just like Tom Thumb’s blues” http… by on Oct.18, 2013, under Tweet Lisa Hannigan – “Just like Tom Thumb’s blues” youtube.com/watch?v=S55Blx… ## On the job market… by on Oct.16, 2013, under Research I am going to be on the job market this year (updated CV). If you know any opportunities I should know about, please let me know… ## Handbook of Discrete & Computational Geometry by on Oct.13, 2013, under Research The handbook of discrete & computational geometry is available online (if you have access from a university that pays for access, naturally). See here. ## My updated cv: http://t.co/zqp7j37Uca by on Oct.07, 2013, under Tweet ## Nice poetry site: http://t.co/Ky8kj3FXcc by on Oct.06, 2013, under Tweet Nice poetry site: poemhunter.com/poem/if/ ## Hyper reals: http://t.co/2NLENbMbEB by on Oct.05, 2013, under Tweet ## If it’s a legitimate progress, the federal governm… by on Oct.03, 2013, under Tweet If it’s a legitimate progress, the federal government has ways to try to shut the whole thing down. ## The Jewish hunger games… http://t.co/Uhxa5V8Phi by on Oct.02, 2013, under Tweet ## Lets put on some good music, and start writing…. by on Oct.01, 2013, under Tweet Lets put on some good music, and start writing…. ## “Maybe he should’ve spent less time reading Dr.Seu… by on Sep.29, 2013, under Tweet “Maybe he should’ve spent less time reading Dr.Seuss and more time looking into the policies that he’s talking about” america.aljazeera.com/articles/2013/… ## Nobel prize for Bob dylan? Hmmm. http://t.co/WXuvj… by on Sep.28, 2013, under Tweet Nobel prize for Bob dylan? Hmmm. nytimes.com/2013/09/29/opi… ## I built on the sand by on Sep.27, 2013, under Literature, Poetry I built on the sand And it tumbled down, I built on a rock And it tumbled down. Now when I build, I shall begin With the smoke from the chimney. — Foundations, Leopold Staff ## For loop in latex by on Sep.27, 2013, under LaTeX \documentclass[12pt]{article} \usepackage{forloop} \begin{document} {% }% \end{document} ## Gall-Peters projection – area preserving mapping o… by on Sep.25, 2013, under Tweet Gall-Peters projection – area preserving mapping of the sphere to the plane. en.wikipedia.org/wiki/Gall%E2%8… ## Vincent. http://t.co/82v4f7VrUN by on Sep.23, 2013, under Tweet ## Tangled up in SODA by on Sep.22, 2013, under Research So there are some complaints about the SODA 2014 reviews being clueless. Despite my paper being rejected, I feel the reviews for my paper were reasonable and knowledgeable. In many cases the feedback the author gets in no way reflects the PC discussion. I find the complaints a bit sordid, since most of the refereeing is done by sub-referees; namely, us. So, we should improve our sub-refereeing, not some nameless people out there… Anyway, I found Russell Impagliazzo comment on the following post to be spot on, and I am replicating it as is. The post itself and the comments are also worth reading. I’ve learned to realise that factually erroneous reviews and reviews that totally miss the point are invaluable for revising the paper. They point out where I have failed to communicate my results and their significance properly. (Of course, I still hate rejection.) The worst review to get is “This paper is pretty good, but not quite up to (Conference name) standards” because it is no help whatsoever. From the other side, I have often seen committee meetings where a sub-reviewer’s erroneous points are dismissed, but the paper is rejected for another reason (like, it’s pretty good, but not quite good enough to get in.) The authors still receive the sub-reviewers review. So just because a reviewer said fallacious thing X and your paper gets rejected does not mean that it got rejected due to the stupid reviewer who thought X. I have to admit, sometimes my own reviews have fallacies that are pointed out to me in the PC meeting or on-line discussion, and I forget to go back and change my review before it gets sent to the authors. Getting rejected is hard on the ego, and I don’t think there’s any perfect reviewing process. That’s not to say we shouldn’t try to improve reviewing, but we shouldn’t expect perfection. ### Looking for something? Use the form below to search the site: Still not finding what you're looking for? 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• Convergence problem of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ let us moving to telescopic sum using exponent ,Assume we have this sequence: $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ with $n\geq1$ , This sequence can be written as power of sequences :${x_n} ^ {{{y_n}^{C_n}}^{......}}$ such that all them value are in $(0,1)$, We want to know if the titled sequence should converge to $1$ ? ... OpenYear of origin: 2001 Posted online: 2021-07-11 01:00:10Z by Rafik Zeraoulia191 • Dynamical Systems • Mathematical Physics • Periodicity of $n=6$ on iterative sum power divisor function Let ${\sigma}_x(n) =\sum_{d\div n} d^x$ is the sum divisor function. After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ we got this result : $\sigma_x(6)=0 \bmod 6$ for $x$ odd and $2 \bmod 6$ if $x$ is even. One can ask this question: Is $n=6$ the only integer satisfies $\sigma_x(n) \equiv 0\bmod n$ for every odd integer $x > 0$ and $2 \bmod n$ if $x$ is even integer? One can answer this question using divisibility and Congruence properties for sum power divisor function, you can see the proof in this paper. The answer of this question yield to reformulate the following problem regarding periodicity on iterative of sum power divsior function. ... OpenYear of origin: 1984 Posted online: 2021-07-09 11:09:08Z by Rafik Zeraoulia86 • Number Theory • Combinatorics • Integral representation of solution of an elliptic PDE in divergence form Suppose we have a second order elliptic differential operator $$L(v) = -\text{div}(A(x) \nabla v)$$ $A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And suppose $\Omega$ is a $C^1$ domain (can be considered more regular if required). ... Open Posted online: 2021-05-26 16:33:17Z by Harish Shrivastava30 • Analysis of PDEs • Classical Analysis and ODEs • Some problems regarding the existence and regularity of minimizers of variational problems with gradient constraint Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Let $N$ be a compact convex set and assume that $\sigma: N\to \mathbb{R}$ is a bounded convex function such that $u\in C^1(N^\circ)$ and strictly convex on $N^\circ$. Consider the functional \begin{align} E[u]=\int_{\Omega}\sigma(\nabla u(x))dx, \end{align} defined for all Lipschitz functions $u\in C^{0,1}(\Omega)$ such that $\nabla u(x)\in N$ for a.e. $x\in \Omega$ and called this closed convex subset of Lipschitz functions $\mathscr{A}_N(\Omega)$. Furthermore, chose a $\phi \in \mathscr{A}_N(\Omega)$ and let \begin{align} \mathscr{A}_N(\Omega,\phi):=\{u\in \mathscr{A}_N(\Omega): u\vert_{\partial \Omega}=\phi\}. \end{align} ... Open Posted online: 2020-11-04 08:26:10Z by Erik Duse44 • Analysis of PDEs • KPZ universality conjecture for the KPZ equation with general initial data The question is to prove that the solution to the tuned KPZ $$\partial_{t}h(x,t)=\partial_{x}^{2}h(x,t)+\delta(\partial_{x}h(x,t))^{2}+\delta^{1/2}\xi,$$ ... Open Posted online: 2020-10-04 04:57:12Z by Tomas Kojar25 View the group • Probability • Problem 3: minimizing the first eigenvalue with an obstacle of given perimeter The corresponding minimization problem with the area constraint replaced by a perimeter constraint (whatever notion of perimeter one would consider) is in general not well-posed. Indeed, for every $L>0$, one can construct a sequence of smooth connected and closed sets $K_n\subset\overline\Omega$ of perimeter $L$ approaching a subset of $\partial \Omega$ so that $\lambda_1(\Omega\setminus K_n)\downarrow \lambda_1(\Omega)$ as $n\to\infty$ (notice that by regularity there is no doubt on the notion of perimeter of $K_n$). Therefore, as in Problem 2, we restrict the class of admissible obstacles to convex sets. For a fixed $L\in (0,P(\overline{\Omega}))$, consider the {minimization} problem $$\label{prob3} \min \{ \lambda_1(\Omega\setminus K) : \; \text{K\subset \overline{\Omega}, K closed and convex, P(K)=L}\}.$$ The existence of a minimizer is a consequence of the compactness of the class of convex sets and of the continuity of the perimeter w.r.t Hausdorff convergence of convex sets. Notice that, for particular domains $\Omega$ and small values $L$, it is still possible to have trivial solutions. For example, if the boundary $\partial\Omega$ contains a segment and if $L$ is smaller than twice the length of such a segment, then every segment contained in $\partial \Omega$ of perimeter $L$ is a minimizer. On the other hand, if $L$ is large enough, every minimizer has positive Lebesgue measure, since minimizing sequences will not be able to degenerate to a segment. In any case, one expects that every minimizer touches the boundary $\partial \Omega$. ... Open Posted online: 2020-04-11 10:54:41Z by Antoine Henrot35 View the group • Spectral Theory • Analysis of PDEs • Optimization and Control • Problem 2: maximizing the first eigenvalue with an obstacle of given~area The corresponding maximization problem (of Problem 1) has no solutions. Indeed, one can construct a sequence of closed sets $K_n\subset\overline\Omega$ of Lebesgue measure $A$ so that $\lambda_1(\Omega\setminus K_n)\uparrow \infty$ as $n\to\infty$ (for instance by taking $K_n$ as the union of a given closed set in $\overline \Omega$ of area $A$ with a curve filling $\overline \Omega$ as $n$ increases, see [9], [10] where the limit distribution in $\overline{\Omega}$ of such curves is studied in detail). To guarantee the existence of a maximizer one needs to prevent maximizing sequences to spread out over $\overline\Omega$. This can be achieved by imposing stronger geometrical constraints on the class of admissible obstacles (notice that connectedness is still not sufficient). Therefore, for a fixed $A\in (0, \mathcal{L}(\Omega))$, we are led to consider the maximization problem $$\label{prob2} \max \{ \lambda_1(\Omega\setminus K) : \; \text{K\subset \overline{\Omega}, K closed and convex, \mathcal{L}(K)=A}\}.$$ Now, the existence of a maximizer in the restricted class of convex sets is straightforward (see [3], [5]). Moreover, as convexity seems necessary for the existence, it is natural to expect every solution of this maximization problem to saturate the convexity constraint, in the sense that the boundary of any solution should contain non-strictly convex parts. In particular, it would be interesting to know whether this maximization problem has only polygonal sets as solutions, see [7], [8] for results in this direction for shape optimization problems with convexity constraints. ... Open Posted online: 2020-04-11 10:54:41Z by Antoine Henrot12 View the group • Spectral Theory • Analysis of PDEs • Optimization and Control 1. 1 2. ... 3. 15 4. 16	{}
# Financial Crime Survey 2016: Compliance and online fraud top the financial crime investment charts ## Sponsored survey analysis: BAE Systems Since 2007, Operational Risk, in collaboration with BAE Systems, has conducted a series of surveys covering key industry trends in financial crime, risk and compliance. This year’s survey shows continued investment in anti-fraud and compliance solutions. However, the focus and priorities of financial institutions appear to reflect the uncertainty of global markets and the growing impact of money laundering and counter-regulatory measures. A key indicator of the overall health of the market is the change in investment in the two primary areas of financial crime: compliance and counter fraud. These two areas have seen sustained investment growth since the first survey was conducted in 2007. This comes despite retrenchment elsewhere in financial services (figure 1). This year’s survey indicates significant growth of investment in anti-money laundering (AML) and particularly non-AML compliance: 51.5% of respondents said that AML budgets would increase and 55% said that non-AML compliance budgets would increase, compared with 45% and 49.6%, respectively, for 2013. However, more financial institutions in 2016 than in 2013 say that their AML and non-AML budgets will decrease. This trend does not appear to be reflected in combatting fraud, where an almost identical percentage (41.8% in 2016 and 41.6% in 2013) indicates that investment will remain at the same level. ### A shift in priority We asked respondents what the financial crime priorities for senior management would be in the coming year (figure 2). In 2013 – and for several years prior to that – insider fraud was regarded as the priority for respondents. However, this state of affairs has changed dramatically. As of this year, online fraud is the priority. The growth of remote banking channels, with the proportion of business conducted through online and mobile banking, is a possible explanation. According to the Global Mobile Banking Report, published by KPMG in conjunction with UBS Evidence Lab, this past growth is just the beginning. The number of mobile banking users globally is forecast to double to 1.8 billion over the next four years, representing more than one-quarter of the world’s population. In the US, 74% of banking customers used online banking in 2014, and 35% used mobile banking to access banking services. Payment fraud remains a strong priority, but is down when compared with 2013 numbers. ### Insider fraud Where does this leave insider fraud? In terms of priority initiatives for senior management, the survey revealed that it had fallen to a 5.61 average score in 2016 from 7.44 in 2013, when it was considered the second priority; only cheque and deposit fraud in 2016 are considered to be lower priority in banking. It could be that, as it is notoriously difficult to detect, senior management now believe the levels of insider fraud to be low and therefore of lower priority. The survey measured fraud detection performance for the first time in 2016, and insider fraud detection is comparable with online fraud, for example, at 39.4% detected (however with a high false-positive rate at 29.5%). This would support an argument that this area has been a focus for financial institutions. However, further analysis of the survey participants suggests that the mix of relevant financial institutions is different in 2016 to that of 2013. In 2016, 32.6% of businesses did not use insider fraud detection, compared with 21.3% of businesses in 2013. Most current research suggests that, far from being addressed, insider fraud is a growing problem. The Kroll Global Fraud Report 2015–2016, carried out with and conducted by the Economist Intelligence Unit, found that 60% of companies were defrauded by an insider – a 72% increase on the previous year. According to the Association of Certified Fraud Examiners’ (ACFE) 2014 Report to the Nations, more than one insider colluding in a fraud is especially harmful, and appears in between 36% and 42% of cases covered by the report. Yet losses from this type of fraud have fallen from a median of $500,000 in ACFE’s 2008 report to half this figure in 2014. ### Changes in AML and compliance Investment in AML remains significant, with 51.5% of respondents expecting budgets to increase over the next three years. The data also suggests financial institutions are shifting slightly towards solving this problem internally, with an increase from 20.6% in 2013 to 26.4% in 2016. In comparison, those going to third parties have fallen from 42.1% in 2013 to 35% in 2016. That said, third-party vendors remain the most popular option overall. The survey asked respondents which solutions they would build in-house, as opposed to outsourcing (figure 3). This year saw a significant migration to third-party politically exposed persons (PEP) screening and sanctions solutions. Almost one-third (31.3%) went outside their organisations in 2013; that figure rose to 43.8% in 2016. This is possibly explained by the increasing development and focus of sanctions regimes introduced by governments around the world to target organised crime and stem the funding of terrorism, which has forced financial institutions to prioritise investment in this area. The expansion has resulted in a more complex set of actions and obligations and, for this reason, financial institutions are turning to third-party vendors. Recently, it has been the considered view of many financial crime experts that financial institutions can benefit from combining fraud and AML resources to create efficiencies and identify crossovers. This year’s results suggest the idea has currency, with a more than 10-point jump in financial institutions using consolidated fraud and compliance solutions: 76.6% in 2016, compared with 65.7% in 2013. The trend is also towards third-party vendors providing these solutions, with 21.9% in 2016 compared with 13.7% in 2013. ### The rise of application fraud There has been a trend towards developing tools in-house for specific areas of financial crime. Notable in this group is application fraud (figure 3). Just over one-quarter (25.5%) of surveyed financial institutions now have an in-house solution, an increase of 2.2% since 2013. There has been a decline in the number of those that have a third-party application fraud solution (from 23.7% to 20.8%), and a decline in the number of financial institutions that said that they would possibly outsource their application fraud solution service over the next three years – from 18.3% to 15.1%. The skewing towards internal spend could be explained in part by the common practice of tackling application fraud with existing fraud solutions. It would be interesting to analyse the impact on effectiveness of application fraud detection over time (beyond the scope of this research), as BAE Systems’ experience suggests that specific application fraud modelling and entity matching is required. That said, fraud departments are also frequently challenged to justify additional IT spends, so may need to maximise what they have already. We will be watching this space carefully, given the increasing prevalence of the application fraud problem with the global migration towards EMV standard, and would expect the trend to in-house solutions to target this problem to be reversed. ### Cyber attack On the list of priority initiatives for senior management in 2013, cyber crime was included as a general category. At the time, it was cited as the highest priority, with 7.7% of the total. With the increasing number of high-profile cyber attacks, and the increasingly documented connection with financial crime (the UK’s National Fraud Investigation Bureau estimates that 43% of fraud is cyber-enabled), this year’s survey asked more specific questions about the perceived and likely impacts of a successful cyber attack on financial institutions (figures 4 and 5). When asked what would be the likely financial cost of a successful cyber attack on their organisation, 36.7% of respondents said they did not know the estimated potential financial impact; 46.7% estimated this to be less than £10 million; and 23.3% estimated less than £1 million. This may reflect some widely reported figures that show that the average payout for a claim for a cyber breach is around £1 million (figure 4). NetDiligence’s 2015 Cyber Claims Study notes the average claim for a breach at$673,000 and, for a large company, \$4.8 million. The UK government’s 2015 Information Security Breaches Survey suggests costs had risen from £600,000 in 2014 to £1.46 million in 2015 for large firms Yet this only refers to the direct costs of a cyber incident. The indirect costs can be far greater, and this is recognised in response to the second cyber-related question. Exactly half of respondents believed that the greatest risk to their business was reputational damage, and 30.6% considered loss of customer data to be the greatest risk. Direct monetary loss was the greatest risk for only 8.1% of respondents, which is unsurprising given how easy it is to insure against. In contrast, the damage to reputation can be far-reaching, long-term, and difficult to insure against. Detection and prevention technologies, in combination with relevant insurance protection, can be effective options for the financial institution to manage this risk (figure 5). In summary, this year’s research has once again provided highly valuable insight into the key trends and the state of the financial crime market. Investment in financial crime defences is increasing, most significantly in AML and non-AML compliance. Leading the trend is the likely move to third-party solution providers for sanctions and PEP screening solutions, but also the move to consolidation between fraud and compliance solutions. ### The Financial Crime 2016 Survey methodology BAE Systems and Operational Risk received 204 responses to the survey, which was conducted during March and April 2016. Respondents were drawn from banks and insurers who work in risk, fraud, compliance and finance and were asked to share their views on their businesses’ investment priorities in the area of financial crime, what are the key threats now and in the next three years, and the potential financial impact of cyber-enabled crime. The survey was carried out using SurveyMonkey and marketed through Risk.net	{}
# American Institute of Mathematical Sciences • Previous Article Study of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping • CPAA Home • This Issue • Next Article On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $L_{2}(-\infty, \infty)$ March 2020, 19(3): 1563-1579. doi: 10.3934/cpaa.2020078 ## Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China * Corresponding author Received June 2019 Revised August 2019 Published November 2019 Fund Project: X. H. Tang was partially supported by the National Natural Science Foundation of China (No: 11571370). This paper is dedicated to studying the Choquard equation $\begin{equation} \left\{ \begin{array}{ll} -\Delta u+V(x)u = (I_{\alpha}\ast\|u\|^{p})\|u\|^{p-2}u+g(u),\; \; \; \; \; x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}) ,\\ \end{array} \right. \end{equation}$ where $N\geq4$ , $\alpha\in(0, N)$ , $V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R})$ is sign-changing and periodic, $I_{\alpha}$ is the Riesz potential, $p = \frac{N+\alpha}{N-2}$ and $g\in\mathcal{C}(\mathbb{R}, \mathbb{R})$ . The equation is strongly indefinite, i.e., the operator $-\Delta+V$ has infinite-dimensional negative and positive spaces. Moreover, the exponent $p = \frac{N+\alpha}{N-2}$ is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on $g$ , we obtain the existence of nontrivial solutions for this equation. Citation: Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078 ##### References: show all references ##### References: [1] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020448 [2] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [3] Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169 [4] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [5] Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053 [6] Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 [7] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445 [8] Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020388 [9] Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 [11] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [12] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021008 [13] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [14] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [15] Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 [16] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [17] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020376 [18] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [19] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [20] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 2019 Impact Factor: 1.105	{}
# Four-force/4-current (Redirected from 4-current) Given that an observer both moving with and local to a bit of charge finds it to have a charge density of ${\displaystyle \rho _{0q}}$, the product of this scalar quantity with its 4-velocity according to a given frame will constitute a four-vector known as the 4-current given by ${\displaystyle j^{\mu }=\rho _{0q}U^{\mu }}$	{}
aboutsummaryrefslogtreecommitdiff log msg author committer range blob: c83e769ef04746a4f2ab3be405cb742a9e261285 (plain) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 /* * * (C) COPYRIGHT 2010-2013 ARM Limited. All rights reserved. * * This program is free software and is provided to you under the terms of the * GNU General Public License version 2 as published by the Free Software * Foundation, and any use by you of this program is subject to the terms * of such GNU licence. * * A copy of the licence is included with the program, and can also be obtained * from Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, * Boston, MA 02110-1301, USA. * / /* * @file mali_kbase_config.h * Configuration API and Attributes for KBase / #ifndef _KBASE_CONFIG_H_ #define _KBASE_CONFIG_H_ #include /* * @addtogroup base_api * @{ / /* * @addtogroup base_kbase_api * @{ / /* * @addtogroup kbase_config Configuration API and Attributes * @{ / #if MALI_CUSTOMER_RELEASE == 0 / This flag is set for internal builds so we can run tests without credentials. / #define KBASE_HWCNT_DUMP_BYPASS_ROOT 1 #else #define KBASE_HWCNT_DUMP_BYPASS_ROOT 0 #endif #include /* * Relative memory performance indicators. Enum elements should always be defined in slowest to fastest order. / typedef enum kbase_memory_performance { KBASE_MEM_PERF_SLOW, KBASE_MEM_PERF_NORMAL, KBASE_MEM_PERF_FAST, KBASE_MEM_PERF_MAX_VALUE = KBASE_MEM_PERF_FAST } kbase_memory_performance; /* * Device wide configuration / enum { /* * Invalid attribute ID (reserve 0). * * Attached value: Ignored * Default value: NA * / KBASE_CONFIG_ATTR_INVALID, /* * Memory resource object. * Multiple resources can be listed. * The resources will be used in the order listed * in the configuration attribute list if they have no other * preferred order based on the memory resource property list * (see ::kbase_memory_attribute). * * Attached value: Pointer to a kbase_memory_resource object. * Default value: No resources * / KBASE_CONFIG_ATTR_MEMORY_RESOURCE, /* * Maximum of memory which can be allocated from the OS * to be used by the GPU (shared memory). * This must be greater than 0 as the GPU page tables * are currently stored in a shared memory allocation. * * Attached value: number in bytes * Default value: Limited by available memory / KBASE_CONFIG_ATTR_MEMORY_OS_SHARED_MAX, /* * Relative performance for the GPU to access * OS shared memory. * * Attached value: ::kbase_memory_performance member * Default value: ::KBASE_MEM_PERF_NORMAL / KBASE_CONFIG_ATTR_MEMORY_OS_SHARED_PERF_GPU, /* * Limit (in bytes) the amount of memory a single process * can allocate across all memory banks (including OS shared memory) * for use by the GPU. * * Attached value: number in bytes * Default value: Limited by available memory / KBASE_CONFIG_ATTR_MEMORY_PER_PROCESS_LIMIT, /* * UMP device mapping. * Which UMP device this GPU should be mapped to. * * Attached value: UMP_DEVICE__SHIFT * Default value: UMP_DEVICE_W_SHIFT / KBASE_CONFIG_ATTR_UMP_DEVICE, /* * Maximum frequency GPU will be clocked at. Given in kHz. * This must be specified as there is no default value. * * Attached value: number in kHz * Default value: NA / KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MAX, /* * Minimum frequency GPU will be clocked at. Given in kHz. * This must be specified as there is no default value. * * Attached value: number in kHz * Default value: NA / KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MIN, /* * Irq throttle. It is the minimum desired time in between two * consecutive gpu interrupts (given in 'us'). The irq throttle * gpu register will be configured after this, taking into * account the configured max frequency. * * Attached value: number in micro seconds * Default value: see DEFAULT_IRQ_THROTTLE_TIME_US / KBASE_CONFIG_ATTR_GPU_IRQ_THROTTLE_TIME_US, / Begin Job Scheduling Configs / /* * Job Scheduler scheduling tick granuality. This is in nanoseconds to * allow HR timer support. * * On each scheduling tick, the scheduler may decide to: * -# soft stop a job (the job will be re-run later, and other jobs will * be able to run on the GPU now). This effectively controls the * 'timeslice' given to a job. * -# hard stop a job (to kill a job if it has spent too long on the GPU * and didn't soft-stop). * * The numbers of ticks for these events are controlled by: * - @ref KBASE_CONFIG_ATTR_JS_SOFT_STOP_TICKS * - @ref KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_SS * - @ref KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_NSS * * A soft-stopped job will later be resumed, allowing it to use more GPU * time in total than that defined by any of the above. However, * the scheduling policy attempts to limit the amount of \em uninterrupted * time spent on the GPU using the above values (that is, the 'timeslice' * of a job) * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::scheduling_tick_ns. * The value might be rounded down to lower precision. Must be non-zero * after rounding. * Default value: @ref DEFAULT_JS_SCHEDULING_TICK_NS * * @note this value is allowed to be greater than * @ref KBASE_CONFIG_ATTR_JS_CTX_TIMESLICE_NS. This allows jobs to run on (much) * longer than the job-timeslice, but once this happens, the context gets * scheduled in (much) less frequently than others that stay within the * ctx-timeslice. / KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS, /* * Job Scheduler minimum number of scheduling ticks before jobs are soft-stopped. * * This defines the amount of time a job is allowed to stay on the GPU, * before it is soft-stopped to allow other jobs to run. * * That is, this defines the 'timeslice' of the job. It is separate from the * timeslice of the context that contains the job (see * @ref KBASE_CONFIG_ATTR_JS_CTX_TIMESLICE_NS). * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::soft_stop_ticks * Default value: @ref DEFAULT_JS_SOFT_STOP_TICKS * * @note a value of zero means "the quickest time to soft-stop a job", * which is somewhere between instant and one tick later. * * @note this value is allowed to be greater than * @ref KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_SS or * @ref KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_NSS. This effectively disables * soft-stop, and just uses hard-stop instead. In this case, this value * should be much greater than any of the hard stop values (to avoid * soft-stop-after-hard-stop) * * @see KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS / KBASE_CONFIG_ATTR_JS_SOFT_STOP_TICKS, /* * Job Scheduler minimum number of scheduling ticks before jobs are hard-stopped. * * This defines the amount of time a job is allowed to spend on the GPU before it * is killed. Such jobs won't be resumed if killed. * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::hard_stop_ticks_ss * Default value: @ref DEFAULT_JS_HARD_STOP_TICKS_SS * * @note a value of zero means "the quickest time to hard-stop a job", * which is somewhere between instant and one tick later. * * @see KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS / KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_SS, /* * Job Scheduler minimum number of scheduling ticks before jobs are hard-stopped * when dumping. * * This defines the amount of time a job is allowed to spend on the GPU before it * is killed. Such jobs won't be resumed if killed. * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::hard_stop_ticks_nss * Default value: @ref DEFAULT_JS_HARD_STOP_TICKS_NSS * * @note a value of zero means "the quickest time to hard-stop a job", * which is somewhere between instant and one tick later. * * @see KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS / KBASE_CONFIG_ATTR_JS_HARD_STOP_TICKS_NSS, /* * Job Scheduler timeslice that a context is scheduled in for, in nanoseconds. * * When a context has used up this amount of time across its jobs, it is * scheduled out to let another run. * * @note the resolution is nanoseconds (ns) here, because that's the format * often used by the OS. * * This value controls affects the actual time defined by the following * config values: * - @ref KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_INIT_SLICES * - @ref KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_MIN_SLICES * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::ctx_timeslice_ns. * The value might be rounded down to lower precision. * Default value: @ref DEFAULT_JS_CTX_TIMESLICE_NS * * @note a value of zero models a "Round Robin" scheduling policy, and * disables @ref KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_INIT_SLICES * (initially causing LIFO scheduling) and * @ref KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_MIN_SLICES (allowing * not-run-often contexts to get scheduled in quickly, but to only use * a single timeslice when they get scheduled in). / KBASE_CONFIG_ATTR_JS_CTX_TIMESLICE_NS, /* * Job Scheduler initial runtime of a context for the CFS Policy, in time-slices. * * This value is relative to that of the least-run context, and defines * where in the CFS queue a new context is added. A value of 1 means 'after * the least-run context has used its timeslice'. Therefore, when all * contexts consistently use the same amount of time, a value of 1 models a * FIFO. A value of 0 would model a LIFO. * * The value is represented in "numbers of time slices". Multiply this * value by that defined in @ref KBASE_CONFIG_ATTR_JS_CTX_TIMESLICE_NS to get * the time value for this in nanoseconds. * * Attached value: unsigned 32-bit kbasep_js_device_data::cfs_ctx_runtime_init_slices * Default value: @ref DEFAULT_JS_CFS_CTX_RUNTIME_INIT_SLICES / KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_INIT_SLICES, /* * Job Scheduler minimum runtime value of a context for CFS, in time_slices * relative to that of the least-run context. * * This is a measure of how much preferrential treatment is given to a * context that is not run very often. * * Specficially, this value defines how many timeslices such a context is * (initially) allowed to use at once. Such contexts (e.g. 'interactive' * processes) will appear near the front of the CFS queue, and can initially * use more time than contexts that run continuously (e.g. 'batch' * processes). * * This limit \b prevents a "stored-up timeslices" DoS attack, where a ctx * not run for a long time attacks the system by using a very large initial * number of timeslices when it finally does run. * * Attached value: unsigned 32-bit kbasep_js_device_data::cfs_ctx_runtime_min_slices * Default value: @ref DEFAULT_JS_CFS_CTX_RUNTIME_MIN_SLICES * * @note A value of zero allows not-run-often contexts to get scheduled in * quickly, but to only use a single timeslice when they get scheduled in. / KBASE_CONFIG_ATTR_JS_CFS_CTX_RUNTIME_MIN_SLICES, /* * Job Scheduler minimum number of scheduling ticks before jobs cause the GPU to be * reset. * * This defines the amount of time a job is allowed to spend on the GPU before it * is assumed that the GPU has hung and needs to be reset. The assumes that the job * has been hard-stopped already and so the presence of a job that has remained on * the GPU for so long indicates that the GPU has in some way hung. * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::gpu_reset_ticks_nss * Default value: @ref DEFAULT_JS_RESET_TICKS_SS * * @see KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS / KBASE_CONFIG_ATTR_JS_RESET_TICKS_SS, /* * Job Scheduler minimum number of scheduling ticks before jobs cause the GPU to be * reset when dumping. * * This defines the amount of time a job is allowed to spend on the GPU before it * is assumed that the GPU has hung and needs to be reset. The assumes that the job * has been hard-stopped already and so the presence of a job that has remained on * the GPU for so long indicates that the GPU has in some way hung. * * This value is supported by the following scheduling policies: * - The Completely Fair Share (CFS) policy * * Attached value: unsigned 32-bit kbasep_js_device_data::gpu_reset_ticks_nss * Default value: @ref DEFAULT_JS_RESET_TICKS_NSS * * @see KBASE_CONFIG_ATTR_JS_SCHEDULING_TICK_NS / KBASE_CONFIG_ATTR_JS_RESET_TICKS_NSS, /* * Number of milliseconds given for other jobs on the GPU to be * soft-stopped when the GPU needs to be reset. * * Attached value: number in milliseconds * Default value: @ref DEFAULT_JS_RESET_TIMEOUT_MS / KBASE_CONFIG_ATTR_JS_RESET_TIMEOUT_MS, / End Job Scheduling Configs / /* Power management configuration * * Attached value: pointer to @ref kbase_pm_callback_conf * Default value: See @ref kbase_pm_callback_conf / KBASE_CONFIG_ATTR_POWER_MANAGEMENT_CALLBACKS, /* * Boolean indicating whether the driver is configured to be secure at * a potential loss of performance. * * This currently affects only r0p0-15dev0 HW and earlier. * * On r0p0-15dev0 HW and earlier, there are tradeoffs between security and * performance: * * - When this is set to MALI_TRUE, the driver remains fully secure, * but potentially loses performance compared with setting this to * MALI_FALSE. * - When set to MALI_FALSE, the driver is open to certain security * attacks. * * From r0p0-00rel0 and onwards, there is no security loss by setting * this to MALI_FALSE, and no performance loss by setting it to * MALI_TRUE. * * Attached value: mali_bool value * Default value: @ref DEFAULT_SECURE_BUT_LOSS_OF_PERFORMANCE / KBASE_CONFIG_ATTR_SECURE_BUT_LOSS_OF_PERFORMANCE, /* * A pointer to a function that calculates the CPU clock * speed of the platform in MHz - see * @ref kbase_cpuprops_clock_speed_function for the function * prototype. * * Attached value: A @ref kbase_cpuprops_clock_speed_function. * Default Value: Pointer to @ref DEFAULT_CPU_SPEED_FUNC - * returns a clock speed of 100 MHz. / KBASE_CONFIG_ATTR_CPU_SPEED_FUNC, /* * A pointer to a function that calculates the GPU clock * speed of the platform in MHz - see * @ref kbase_gpuprops_clock_speed_function for the function * prototype. * * Attached value: A @ref kbase_gpuprops_clock_speed_function. * Default Value: NULL (in which case the driver assumes a current * GPU frequency specified by KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MAX) / KBASE_CONFIG_ATTR_GPU_SPEED_FUNC, /* * Platform specific configuration functions * * Attached value: pointer to @ref kbase_platform_funcs_conf * Default value: See @ref kbase_platform_funcs_conf / KBASE_CONFIG_ATTR_PLATFORM_FUNCS, /* * Limit ARID width on the AXI bus. * * Attached value: u32 register value * KBASE_AID_32 - use the full 32 IDs (5 ID bits) * KBASE_AID_16 - use 16 IDs (4 ID bits) * KBASE_AID_8 - use 8 IDs (3 ID bits) * KBASE_AID_4 - use 4 IDs (2 ID bits) * Default value: KBASE_AID_32 (no limit). Note hardware implementation * may limit to a lower value. / KBASE_CONFIG_ATTR_ARID_LIMIT, /* * Limit AWID width on the AXI bus. * * Attached value: u32 register value * KBASE_AID_32 - use the full 32 IDs (5 ID bits) * KBASE_AID_16 - use 16 IDs (4 ID bits) * KBASE_AID_8 - use 8 IDs (3 ID bits) * KBASE_AID_4 - use 4 IDs (2 ID bits) * Default value: KBASE_AID_32 (no limit). Note hardware implementation * may limit to a lower value. / KBASE_CONFIG_ATTR_AWID_LIMIT, /* * Enable alternative hardware counter capture for the Mali shader cores. * * Attached value: mali_bool value * Default value: @ref MALI_FALSE / KBASE_CONFIG_ATTR_ALTERNATIVE_HWC, /* * Rate at which dvfs data should be collected. * * Attached value: u32 value * Default value: 500 Milliseconds / KBASE_CONFIG_ATTR_POWER_MANAGEMENT_DVFS_FREQ, /* * Power Management poweroff tick granuality. This is in nanoseconds to * allow HR timer support. * * On each scheduling tick, the power manager core may decide to: * -# Power off one or more shader cores * -# Power off the entire GPU * * Attached value: number in nanoseconds * Default value: @ref DEFAULT_PM_GPU_POWEROFF_TICK_NS, / KBASE_CONFIG_ATTR_PM_GPU_POWEROFF_TICK_NS, /* * Power Manager number of ticks before shader cores are powered off * * Attached value: unsigned 32-bit kbasep_pm_device_data::poweroff_shader_ticks * Default value: @ref DEFAULT_PM_POWEROFF_TICK_SHADER * * @see KBASE_CONFIG_ATTR_PM_GPU_POWEROFF_TICK_NS / KBASE_CONFIG_ATTR_PM_POWEROFF_TICK_SHADER, /* * Power Manager number of ticks before GPU is powered off * * Attached value: unsigned 32-bit kbasep_pm_device_data::poweroff_gpu_ticks * Default value: @ref DEFAULT_PM_POWEROFF_TICK_GPU * * @see KBASE_CONFIG_ATTR_PM_GPU_POWEROFF_TICK_NS / KBASE_CONFIG_ATTR_PM_POWEROFF_TICK_GPU, /* * End of attribute list indicator. * The configuration loader will stop processing any more elements * when it encounters this attribute. * * Default value: NA / KBASE_CONFIG_ATTR_END = 0x1FFFUL }; enum { /* * Invalid attribute ID (reserve 0). * * Attached value: Ignored * Default value: NA / KBASE_MEM_ATTR_INVALID, /* * Relative performance for the CPU to access * the memory resource. * * Attached value: ::kbase_memory_performance member * Default value: ::KBASE_MEM_PERF_NORMAL / KBASE_MEM_ATTR_PERF_CPU, /* * Relative performance for the GPU to access * the memory resource. * * Attached value: ::kbase_memory_performance member * Default value: ::KBASE_MEM_PERF_NORMAL / KBASE_MEM_ATTR_PERF_GPU, /* * End of attribute list indicator. * The memory resource loader will stop processing any more * elements when it encounters this attribute. * * Attached value: Ignored * Default value: NA / KBASE_MEM_ATTR_END = 0x1FFFUL }; enum { /* * Use unrestricted Address ID width on the AXI bus. / KBASE_AID_32 = 0x0, /* * Restrict GPU to a half of maximum Address ID count. * This will reduce performance, but reduce bus load due to GPU. / KBASE_AID_16 = 0x3, /* * Restrict GPU to a quarter of maximum Address ID count. * This will reduce performance, but reduce bus load due to GPU. / KBASE_AID_8 = 0x2, /* * Restrict GPU to an eighth of maximum Address ID count. * This will reduce performance, but reduce bus load due to GPU. / KBASE_AID_4 = 0x1 }; / * @brief specifies a single attribute * * Attribute is identified by attr field. Data is either integer or a pointer to attribute-specific structure. / typedef struct kbase_attribute { int id; uintptr_t data; } kbase_attribute; / * @brief Specifies dedicated memory bank * * Specifies base, size and attributes of a memory bank / typedef struct kbase_memory_resource { u64 base; u64 size; struct kbase_attribute attributes; const char name; } kbase_memory_resource; / Forward declaration of kbase_device / struct kbase_device; / * @brief Specifies the functions for platform specific initialization and termination * * By default no functions are required. No additional platform specific control is necessary. / typedef struct kbase_platform_funcs_conf { /* * Function pointer for platform specific initialization or NULL if no initialization function is required. * This function will be called \em before any other callbacks listed in the kbase_attribute struct (such as * Power Management callbacks). * The platform specific private pointer kbase_device::platform_context can be accessed (and possibly initialized) in here. / mali_bool(platform_init_func) (struct kbase_device kbdev); /* * Function pointer for platform specific termination or NULL if no termination function is required. * This function will be called \em after any other callbacks listed in the kbase_attribute struct (such as * Power Management callbacks). * The platform specific private pointer kbase_device::platform_context can be accessed (and possibly terminated) in here. / void (platform_term_func) (struct kbase_device kbdev); } kbase_platform_funcs_conf; / * @brief Specifies the callbacks for power management * * By default no callbacks will be made and the GPU must not be powered off. / typedef struct kbase_pm_callback_conf { /* Callback for when the GPU is idle and the power to it can be switched off. * * The system integrator can decide whether to either do nothing, just switch off * the clocks to the GPU, or to completely power down the GPU. * The platform specific private pointer kbase_device::platform_context can be accessed and modified in here. It is the * platform \em callbacks responsiblity to initialize and terminate this pointer if used (see @ref kbase_platform_funcs_conf). / void (power_off_callback) (struct kbase_device kbdev); /* Callback for when the GPU is about to become active and power must be supplied. * * This function must not return until the GPU is powered and clocked sufficiently for register access to * succeed. The return value specifies whether the GPU was powered down since the call to power_off_callback. * If the GPU state has been lost then this function must return 1, otherwise it should return 0. * The platform specific private pointer kbase_device::platform_context can be accessed and modified in here. It is the * platform \em callbacks responsiblity to initialize and terminate this pointer if used (see @ref kbase_platform_funcs_conf). * * The return value of the first call to this function is ignored. * * @return 1 if the GPU state may have been lost, 0 otherwise. / int (power_on_callback) (struct kbase_device kbdev); /* Callback for handling runtime power management initialization. * * The runtime power management callbacks @ref power_runtime_off_callback and @ref power_runtime_on_callback * will become active from calls made to the OS from within this function. * The runtime calls can be triggered by calls from @ref power_off_callback and @ref power_on_callback. * Note: for linux the kernel must have CONFIG_PM_RUNTIME enabled to use this feature. * * @return MALI_ERROR_NONE on success, else mali_error erro code. / mali_error(power_runtime_init_callback) (struct kbase_device kbdev); /* Callback for handling runtime power management termination. * * The runtime power management callbacks @ref power_runtime_off_callback and @ref power_runtime_on_callback * should no longer be called by the OS on completion of this function. * Note: for linux the kernel must have CONFIG_PM_RUNTIME enabled to use this feature. / void (power_runtime_term_callback) (struct kbase_device kbdev); /* Callback for runtime power-off power management callback * * For linux this callback will be called by the kernel runtime_suspend callback. * Note: for linux the kernel must have CONFIG_PM_RUNTIME enabled to use this feature. * * @return 0 on success, else OS error code. / void (power_runtime_off_callback) (struct kbase_device kbdev); /* Callback for runtime power-on power management callback * * For linux this callback will be called by the kernel runtime_resume callback. * Note: for linux the kernel must have CONFIG_PM_RUNTIME enabled to use this feature. / int (power_runtime_on_callback) (struct kbase_device kbdev); } kbase_pm_callback_conf; /* * Type of the function pointer for KBASE_CONFIG_ATTR_CPU_SPEED_FUNC. * * @param clock_speed [out] Once called this will contain the current CPU clock speed in MHz. * This is mainly used to implement OpenCL's clGetDeviceInfo(). * * @return 0 on success, 1 on error. / typedef int (kbase_cpuprops_clock_speed_function) (u32 clock_speed); /* * Type of the function pointer for KBASE_CONFIG_ATTR_GPU_SPEED_FUNC. * * @param clock_speed [out] Once called this will contain the current GPU clock speed in MHz. * If the system timer is not available then this function is required * for the OpenCL queue profiling to return correct timing information. * * @return 0 on success, 1 on error. When an error is returned the caller assumes a current * GPU speed as specified by KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MAX. / typedef int (kbase_gpuprops_clock_speed_function) (u32 clock_speed); #ifdef CONFIG_MALI_PLATFORM_FAKE / * @brief Specifies start and end of I/O memory region. / typedef struct kbase_io_memory_region { u64 start; u64 end; } kbase_io_memory_region; / * @brief Specifies I/O related resources like IRQs and memory region for I/O operations. / typedef struct kbase_io_resources { u32 job_irq_number; u32 mmu_irq_number; u32 gpu_irq_number; kbase_io_memory_region io_memory_region; } kbase_io_resources; typedef struct kbase_platform_config { const kbase_attribute attributes; const kbase_io_resources io_resources; u32 midgard_type; } kbase_platform_config; #endif / CONFIG_MALI_PLATFORM_FAKE / /* * @brief Return character string associated with the given midgard type. * * @param[in] midgard_type - ID of midgard type * * @return Pointer to NULL-terminated character array associated with the given midgard type / const char kbasep_midgard_type_to_string(u32 midgard_type); /** * @brief Gets the count of attributes in array * * Function gets the count of attributes in array. Note that end of list indicator is also included. * * @param[in] attributes Array of attributes * * @return Number of attributes in the array including end of list indicator. / int kbasep_get_config_attribute_count(const kbase_attribute attributes); /** * @brief Gets the next config attribute with the specified ID from the array of attributes. * * Function gets the next attribute with specified attribute id within specified array. If no such attribute is found, * NULL is returned. * * @param[in] attributes Array of attributes in which lookup is performed * @param[in] attribute_id ID of attribute * * @return Pointer to the first attribute matching id or NULL if none is found. / const kbase_attribute kbasep_get_next_attribute(const kbase_attribute attributes, int attribute_id); /* * @brief Gets the value of a single config attribute. * * Function gets the value of attribute specified as parameter. If no such attribute is found in the array of * attributes, default value is used. * * @param[in] kbdev Kbase device pointer * @param[in] attributes Array of attributes in which lookup is performed * @param[in] attribute_id ID of attribute * * @return Value of attribute with the given id / uintptr_t kbasep_get_config_value(struct kbase_device kbdev, const kbase_attribute attributes, int attribute_id); /* * @brief Validates configuration attributes * * Function checks validity of given configuration attributes. It will fail on any attribute with unknown id, attribute * with invalid value or attribute list that is not correctly terminated. It will also fail if * KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MIN or KBASE_CONFIG_ATTR_GPU_FREQ_KHZ_MAX are not specified. * * @param[in] kbdev Kbase device pointer * @param[in] attributes Array of attributes to validate * * @return MALI_TRUE if no errors have been found in the config. MALI_FALSE otherwise. / mali_bool kbasep_validate_configuration_attributes(struct kbase_device kbdev, const kbase_attribute attributes); #ifdef CONFIG_MALI_PLATFORM_FAKE /* * @brief Gets the pointer to platform config. * * @return Pointer to the platform config / kbase_platform_config kbase_get_platform_config(void); #endif /* CONFIG_MALI_PLATFORM_FAKE / /* * @brief Platform specific call to initialize hardware * * Function calls a platform defined routine if specified in the configuration attributes. * The routine can initialize any hardware and context state that is required for the GPU block to function. * * @param[in] kbdev Kbase device pointer * * @return MALI_TRUE if no errors have been found in the config. MALI_FALSE otherwise. / mali_bool kbasep_platform_device_init(struct kbase_device kbdev); /** * @brief Platform specific call to terminate hardware * * Function calls a platform defined routine if specified in the configuration attributes. * The routine can destroy any platform specific context state and shut down any hardware functionality that are * outside of the Power Management callbacks. * * @param[in] kbdev Kbase device pointer * / void kbasep_platform_device_term(struct kbase_device kbdev); /** @} // end group kbase_config / /* @} // end group base_kbase_api / /* @} // end group base_api / #endif / _KBASE_CONFIG_H_ */	{}
Article \| Open \| Published: # Experimentally quantifying anion polarizability at the air/water interface ## Abstract The adsorption of large, polarizable anions from aqueous solution on the air/water interface controls important atmospheric chemistry and is thought to resemble anion adsorption at hydrophobic interfaces generally. While the favourability of adsorption of such ions is clear, quantifying adsorption thermodynamics has proven challenging because it requires accurate description of the structure of the anion and its solvation shell at the interface. In principle anion polarizability offers a structural window, but to the best of our knowledge there has so far been no experimental technique that allowed its characterization with interfacial specificity. Here, we meet this challenge using interface-specific vibrational spectroscopy of Cl–O vibrations of the $ClO 4 -$ anion at the air/water interface and report that the interface breaks the symmetry of the anion, the anisotropy of $ClO 4 -$’s polarizability tensor is more than two times larger than in bulk water and concentration dependent, and concentration-dependent polarizability changes are consistent with correlated changes in surface tension. ## Introduction The adsorption of anions on hydrophobic interfaces controls important chemistry on aerosol surfaces and determines the stability of proteins, colloids and foams in a wide variety of environmental, physiological and engineered settings. Anion adsorption on the air/water interface, the paradigmatic hydrophobic surface, has been particularly well studied1,2,3,4,5,6,7,8,9,10,11. Perhaps the simplest question one can ask of this system is the following: Is anion adsorption at the air/water interface thermodynamically favourable? In principle measurements of surface tension of the air/water interface as a function of bulk ionic strength provide such insight. Decades of such measurements have confirmed that surface tensions of aqueous salt solutions increase with increasing ionic strength, those of acids decrease, and that the magnitude of the effect depends strongly on anion and only weakly on cation1,12,13,14,15,16. Historically the first observation was rationalized by Wagner, Onsager and Samaras (WOS) in their extension of the Debye–Hückel theory of bulk aqueous electrolytes to interfaces. Within this description anions are excluded from the air/water interface because exposure to a low dielectric phase leads to an enormous, unfavourable electrostatic self-energy17,18. While qualitatively consistent with surface tension measurements of salt solutions, this approach does not explain surface tension trends for acids or specific ion effects. Motivated by reaction rates of gaseous species with solvated anions in atmospheric aerosols that were unexpectedly fast19, subsequent work—simulation using classical polarizable force fields, various surface sensitive spectroscopies2,11,20,21, dielectric continuum theory22,23,24,25 and properly parameterized fixed charge models26—has shown this Debye–Hückel inspired view to be incorrect. Large polarizable anions exist in higher concentrations at the air/water interface than in the adjoining bulk liquid. While the qualitative picture is clear, understanding of quantitative trends in anion adsorption, e.g. why does I adsorb more strongly than Cl, and gaining atomically resolved insight into the driving force of anion adsorption has proven challenging. In WOS theory ions are nonpolarizable point charges. As illustrated by Levin for an idealized anion, lack of polarizability induces a large, unfavourable, electrostatic self-energy in adsorbing anions: for a finite sized, ideal, polarizable anion essentially all charge density shifts towards the aqueous phase as the ion approaches the interface23. While it is thus clear ions must be polarizable to adsorb at the air/water interface, the contribution of ion polarization (i.e. the product of applied field and polarizability p = αE) to the free energy of adsorption (ΔGads) is not obvious. Initial studies describing ion adsorption in classical simulations concluded that explicit description of ion polarizability was critical and that the relative surface propensity of different ions was proportional to their polarizability and radius (i.e. large, soft, polarizable anions more favourably adsorb)4,9. However subsequent simulation studies have found that anion adsorption occurs in properly parameterized classical models without explicit description of polarizability, that relative anion polarization does not correlate with experimental and simulated trends in ion adsorption propensity, and that simulated interface active anions may be similarly polarized in bulk water and at the interface21,26,27,28,29,30. Recent theoretical work in continuum models, and classical and ab-initio simulation has clarified that ΔGads of anion adsorption is dominated by a balance between large favourable cavitation and unfavourable desolvation energies: moving an anion from bulk to the air/water interface reduces the energetic cost to forming a cavity in water but also reduces the favourable interaction of the anion with, at least part of, its solvation shell31,32,33,34. The contribution of ion polarization to ΔGads, the change in system free energy when adding a dipole to the interface, is relatively small. While accurately describing anion polarization thus appears to be relatively unimportant in decomposing anion ΔGads, Pollard and Beck34 have highlighted that quantitative insight into ΔGads requires accurate description of the structure of the anion and its first solvation shell both in bulk and at the interface. Gaining such insight is a formidable theoretical challenge because the local electrical fields are as high as 1 V/Å. Much theoretical progress could be made if there were experimental observables of the structure of the anion and its first solvation shell both in bulk and at the interface. This experimental challenge is particularly formidable because the structure of the anion and its solvation shell are correlated and both might be expected to change on moving from bulk to the interface. From the monoatomic anion’s perspective one might imagine structure changing because the ion electron density reflects the underlying asymmetry of electron density of the solvent, while for multiatomic anions interface induced changes in bond lengths and angles are additionally possible. From water’s perspective clearly moving to the interface must result in a change in structure of the solvation shell but the details of this change must be a function of changes in anion structure. Much prior work has shown that anion polarizability is exceptionally sensitive to both anion structure and that of its local solvation shell35,36. Thus while the contribution of ion polarization to ΔGads may be small, an experimental observable of anion polarizability in bulk and at the interface should offer an important constraint on anion and solvation shell structure. The adsorption of the perchlorate anion at the air/water interface is known to be favourable9. In this work we experimentally quantify, using interface-specific vibrational spectroscopy, the increase in polarizability anisotropy of perchlorate on moving from bulk water to the interface and show that anisotropy increases with increasing perchlorate bulk concentration. Using a simple computational model we quantitatively relate the experimentally observed increase in polarizability anisotropy with increasing bulk concentration to increases in interfacial field (consistent with prior measurements of concentration-dependent surface potential37), $ClO 4 -$ dipole, and relative bond length of one Cl–O bond with respect to the other three with increasing concentrations of bulk HClO4. Quantitative theoretical insights into the driving force of anion adsorption at the air/water interface, and specific ion effects more generally, require accurate calculation of both the structure of ions and their first solvation shell at aqueous interfaces. Experimental measurements of anion polarizability offer a window into such structure unavailable by other means. ## Results ### Optically probing the polarizability anisotropy of perchlorate The Raman depolarization ratio (ρ) is a useful means of probing anion polarizability in bulk H2O. Given an isotropic distribution of ions in liquid water and a molecular coordinate system (a, b, c) in which a (or b) is taken to be perpendicular to the net deformation of a normal mode and c parallel, one writes38 $ρ= I ⊥ I ∣ ∣ = 3 4 + 5 1 + 2 R R - 1 2$ (1) where I and I\|\| are the intensity of inelastic scattered light measured perpendicular and parallel to the plane polarized incident field and $R= ∂ α a a ( 1 ) ∕ ∂ Q ∂ α c c ( 1 ) ∕ ∂ Q$. That is the Raman response of a particular mode can be quantitatively related to the change in the symmetry of the molecules polarizability tensor as the molecule is deformed in the mode’s characteristic manner. Given this definition of ρ is it perhaps unsurprising that several studies have shown that the ability to calculate the Raman response of modes that are strongly coupled to the environment is a sensitive test of the accuracy of polarizability models employed39,40. Because spontaneous Raman is not interface specific it is generally not possible to extract the ρ of interfacial anions. Clearly if we could, however, this observable could provide the sort of experimental constraint we seek. Vibrationally resonant Sum Frequency (VSF) spectroscopy is a nonlinear optical, laser-based technique in which pulsed infrared (IR) and visible lasers are spatially and temporally overlapped at an interface and the output at the sum of the frequencies of the two incident beams monitored. The emitted VSF field is interface specific by its symmetry selection rules and a spectroscopy because as one tunes the frequency of one of the incident fields (in this case the IR) in resonance with an optically accessible transition the intensity of the emitted sum frequency field (Isf) increases by several orders of magnitude. Much prior work has shown that the intensity of the measured sum frequency response at a frequency ω is proportional to the change in polarizability (α ab ) and dipole (μ c ) with motion along the normal mode of frequency ω41: $I sf ∝ χ i j k ( 2 ) ∝ β a b c ( 2 ) ∝- 1 2 ϵ 0 ω ∂ α a b ( 1 ) ∂ Q ∂ μ c ∂ Q$ (2) in which $χ i j k ( 2 )$ is the macroscopic nonlinear susceptibility in the lab coordinate system (ijk), $β a b c ( 2 )$ the molecular hyperpolarizability and both are third rank tensors. Because by varying experimental conditions, i.e. beam incident angles and field polarizations, one can selectively probe different components of β(2), a correctly chosen ratio of intensities allows the direct measurement of $R= ∂ α a a ( 1 ) ∕ ∂ Q ∂ α c c ( 1 ) ∕ ∂ Q$, and thus the possibility of extracting the Raman depolarization ratio of anions with interfacial specificity. That is, by comparing measurements of ρ for an anion in solution and at the air/water interface experimental estimates of anion polarizability anisotropy at aqueous interfaces (and the change in anion polarizability anisotropy on moving from bulk liquid water to the aqueous interface) are possible. ### VSF spectra of interfacial $ClO 4 -$ Figure 1a shows the VSF spectra from the air/0.6 M HClO4 solution interface measured under the ssp (s-polarized SF, s-polarized visible, and p-polarized IR) (black circles) and ppp (red squares) polarization combinations. There are two resonances apparent in this frequency range. Fitting both spectra simultaneously with the Lorentzian lineshape model described in the Methods section results in resonances centred at 935 and 1110 cm−1 (Fig. 1b dotted lines, see Supplementary Note 1 for details of the data analysis). Because both spectral features are absent in pure water, and are spectrally separated from resonances of water or likely impurities, they can be straightforwardly assigned by reference to Raman and IR measurements of bulk aqueous perchloric acid and perchlorate salt solutions42,43,44. In brief, the $ClO 4 -$ anion has four normal modes apparent in calculation: the ν1 at 930, the ν2 at 450, the ν3 at 1100 and the ν4 at 620 cm−1. All four are Raman active at all concentrations in bulk aqueous solution but the ν1 and ν2 are only apparent in IR absorption spectra at bulk concentrations greater than ≈11 M. This observation is a straightforward consequence of anion symmetry: at bulk concentrations below 11 M the $ClO 4 -$ anion has T d symmetry (under which condition ν1 and ν2 are IR inactive) and at sufficiently high concentrations this symmetry is broken: either by ion pairing or, in the case of HClO4, by the presence of molecular acid. If we assign the resonance apparent in Fig. 1b at 935 cm−1 to the ν1 mode and that apparent at 1110 cm−1 to the ν3, we are left with an apparent incongruity. As shown in Eq. (2), VSF activity requires that a mode must be both IR and Raman active. This implies the $ClO 4 -$ anion must lose its T d symmetry at the air/water interface at concentrations that are at least 15× lower than those at which T d symmetry is lifted in bulk. ### Understanding why T d symmetry is lifted in interfacial $ClO 4 -$ We imagine three possible mechanisms for the loss of T d symmetry: consistent with recent work on other strong acids, molecular HClO4 may exist at the air/water interface at concentrations dramatically lower than in bulk45, $ClO 4 -$ may no longer have tetrahedral symmetry due to ion pairing, or it may not have tetrahedral symmetry due to, more general, solvation anisotropy at the interface. We tested the first possibility by collecting spectra from 0.6 M solutions of NaClO4. For this a similarly intense ν1 feature is observed, suggesting the likely cause of T d symmetry lifting is not molecular acid (see Supplementary Note 2 for data). To evaluate the possibility of T d symmetry lifting due to ion pairing, it is necessary to understand how ion pairing might be expected to influence the ν1 and ν3 spectral response. In bulk solutions of perchlorate salts at concentrations above 1 M the centre frequency of perchlorate’s ν3 mode has been observed to continuously shift as a function of concentration46. This concentration-dependent spectral evolution has been assigned to the formation of weak, solvent-separated ion pairs. As mentioned above, in this concentration range the ν1 mode is infrared inactive. At still higher concentrations in bulk water, >11 M, perchlorate’s degenerate modes, i.e. ν2, ν3 and ν4, have been observed to split due to contact ion pair formation, where the degree of splitting is a function of the extent to which symmetry is broken46,47. As is discussed in detail below (see Fig. 2 for data) at bulk concentrations lower than 1 M HClO4 the ν1 mode is clearly VSF (and thus IR) active, the ν3 spectral response is quantitatively reproduced with a single centre frequency and line width: splitting or frequency shift of the ν3 resonance is not required to describe our data. We therefore conclude that neither contact nor solvent-separated ion pair formation explains the lifting of T d symmetry. Given that VSF spectra collected at bulk concentrations below 1 M HClO4 are consistent with formation of neither weak, solvent-separated ion pairs nor contact ion pairs, we conclude that the lifting of T d symmetry for interfacial $ClO 4 -$ (and thus the IR and VSF activity of the ν1 mode) must be the result of the intrinsic anisotropy of the solvation environment at the air/water interface: solvation anisotropy must induce sufficient structural deformation in the $ClO 4 -$ anion to lift the bulk T d symmetry and make the ν1 mode IR, and VSF, active. While this qualitative observation of a consequence of structural deformation is important, to make clear connections to theory it would be useful to quantify this deformation, and the resulting change in the perchlorate polarizability tensor. Given the ν1 spectral amplitudes extracted from the fit of the ssp and ppp data in Fig. 1a (see Supplementary Note 1 for full details of the fit), and assuming the $ClO 4 -$ anion is oriented such that one Cl–O points along the surface normal and that the anion has C3ν symmetry, we can calculate the Raman depolarization ratio, i.e. ρ, for perchlorate ions at the interface (see Supplementary Note 3 for a description of the theory connecting measured Isf to the Raman depolarization ratio). The result of this calculation is shown by the solid black line in Fig. 1c. The ratio of spectral amplitudes of the ν1 ssp and ppp shown in Fig. 1a suggests a ν1 ρ of 0.0068 ± 0.0005 (the uncertainty originates from both the average of three measurements and the spectral fit, see Supplementary Tables 13 for all results), or more than 2× larger than the same quantity for $ClO 4 -$ in bulk. It is worth emphasizing that the result that interfacial $ClO 4 -$’s Raman depolarization ratio is significantly larger than bulk is insensitive to the simplifying assumptions required in its calculation. As we show in Supplementary Note 4, assuming the $ClO 4 -$ is oriented with one Cl–O bond at an increasing, nonzero angle with respect to the surface normal leads to slightly larger estimates for the depolarization ratio of ν1 while assuming the symmetry of the $ClO 4 -$ anion has decreased to C2ν or Cν leads to quantitatively similar results. Note also that because we measure the intensity of the emitted sum frequency light, and not the field, our measurements are equally consistent with one Cl–O bond along the surface normal and the remaining three oxygens pointing either towards the bulk liquid or air. Prior theoretical studies imply, at least in the low concentration limit, that the latter configuration is favoured7,48,49. As noted above, changes in polarizability must be correlated with changes in anion nuclear structure. As we show in detail in Supplementary Note 5, a simple computational model suggests a ρ of 0.0068 is consistent with a 3% change in Cl–O bond length (for the Cl–O along the surface normal), and a permanent dipole moment of interfacial $ClO 4 -$ of 0.75 Debye (n.b. consistent with the absence of IR active ν1 and ν2 modes for the $ClO 4 -$ anion in vacuum or in bulk liquid water, perchlorate’s dipole moment in either bulk phase is below our detection limit). In this manner our experimental observable directly constrains the anisotropy of interfacial perchlorate’s polarizability tensor and places quantitative constraints on interfacial $ClO 4 -$ polarization at the air/water interface. ### Change in spectral response of interfacial $ClO 4 -$ with changing bulk concentration To gain more insight into the fate of the $ClO 4 -$ anion at the air/water interface, we next measured its ν1 and ν3 spectral amplitudes as a function of bulk concentration of HClO4 from 0.1 to 0.8 M (higher concentrations lead to qualitative change in the spectral response, possibly the result of interface induced ion pairing, as shown in Supplementary Note 6). The concentration-dependent spectra collected under the ssp polarization condition are shown in Fig. 2a, the concentration-dependent ppp are plotted in Supplementary Fig. 1. Global fitting of both sets of data allows the extraction of the $χ r, ν 1 ssp ∕ χ r, ν 1 ppp$ ratio as a function of bulk concentration. As shown in Fig. 2b changing bulk concentrations of HClO4 from 0.1 to 0.8 M leads to a change of this ratio from 8.58 ± 1.7 to 12.4 ± 1.1. Because the VSF spectral response reports on the properties of interfacial $ClO 4 -$, it would clearly be useful if we could quantify the increase in interfacial concentration with increasing bulk concentration of HClO4. As we discuss in detail in Supplementary Note 7, accurately determining the interfacial concentration of anions at the interface is extremely challenging: inferring interfacial concentration from changes in surface tension with bulk concentration requires a simplifying model whose assumptions are difficult to independently evaluate, inferring interfacial population from the VSF or SHG spectral response requires assuming a concentration-independent molecular response that our results suggest is unlikely in the case of $ClO 4 -$ (discussed in more detail below) and, possibly, inferring interfacial concentration in x-ray photoemission measurements requires a reference for the inelastic mean free path of the electron that does not obviously exist50. Despite these limitations we have quantified the interfacial $ClO 4 -$ using surface tension and VSF measurements. The surface tension data suggest that increasing bulk concentrations from 0.1 → 1 M lead to an increase of the surface excess of $ClO 4 -$ from 0.3 × 10−6 → 1.1 × 10−6 mol/m2 while the VSF response suggests that over this concentration range the surface coverage increases from 0.5 → 1.0 monolayers. Despite the uncertainty in interfacial $ClO 4 -$ concentration, reference to Fig. 1 makes clear that this change $χ r, ν 1 ssp ∕ χ r, ν 1 ppp$ ratio from 8 → 13 implies an increase in the Raman depolarization ratio of ν1 of 0.0053 ± 0.0012 to 0.0071 ± 0.0004 (all depolarization ratios with their uncertainties are tabulated in Supplementary Table 3) over the same concentration range. Evidently, with increasing interfacial population, $ClO 4 -$ polarizability grows increasingly anisotropic. Using the same simple computational model discussed above, increasing the depolarization ratio from 0.005 to 0.007 is consistent with an increase in interfacial field from 139 to 171 (meV), an elongation in the Cl–O bond along the surface normal of 2.6–3.3% and a change in $ClO 4 -$ dipole moment from 0.6 to 0.76 Debye. The relationship between $χ r, ν 1 ssp ∕ χ r, ν 1 ppp$ and Raman depolarization ratio shown in Fig. 1 assumes that the $ClO 4 -$ is orientated such that one Cl–O group points along the surface normal. Applying this analysis to the data shown in Fig. 2a implicitly assumes that this orientation is concentration independent. Because the ν1 and ν3 normal modes are orthogonal, we would expect any concentration-dependent change in the orientation of interfacial $ClO 4 -$ to result in significant change in the $χ r, ν 1 ssp ∕ χ r, ν 3 ssp$. As is shown in Fig. 2c the change we observe in this ratio as a function of bulk concentration is <50%. A change of this size is consistent with a, concentration dependent, change in the orientation of <5° (see Supplementary Note 4 for details of the calculation). We thus conclude that the orientation of interfacial $ClO 4 -$ is, to within the limits of our sensitivity, over 0.1–0.8 M range in bulk concentration, concentration independent. Our results suggest the following model for $ClO 4 -$ at the air/water interface: on adsorption $ClO 4 -$ is polarized, i.e. it has a nonzero dipole moment, and the polarizability anisotropy changes due to a change in the bond length of the Cl–O that points along the surface normal relative to the three other Cl–O bonds. With increasing interfacial concentrations of $ClO 4 -$ the interfacial field increases, ion polarization increases (the dipole continues to grow) and the polarizability anisotropy continues to increase. ## Discussion Modern dielectric continuum descriptions largely reproduce experimentally measured changes in surface tension with increasing concentrations of both acids and salts24,31. Notably the largest disagreements between experiment and theory, at least within the model of Levin and co-workers in which ions are approximated as hard spheres with fixed, concentration-independent radii, exist for $ClO 4 -$ solutions (both acids and salts). Levin and co-workers51,52 have suggested that this is likely the result of inaccuracies in the estimates of ionic radii for $ClO 4 -$. Our results are consistent with an alternative scenario in which anion polarizability (and $ClO 4 -$ radius) is interfacial concentration dependent. While our results imply the relationship between $ClO 4 -$ radius and interfacial concentration is monotonic, larger multivalent ions might be expected to have a more complicated interplay between structure, interfacial concentration and interfacial potential, all of which might be observed through the window of anion polarizability. Atomistic simulation studies, whether employing classical or ab-initio potential energy surfaces, have largely reported either potentials of mean force for ion adsorption in the limit of infinite dilution or brute force simulations at a fixed ion concentration. As alluded to above, while important and informative these studies suffer from several possible shortcomings. The lack of experimental constraints on polarizability mean that estimates of polarizability from ab-initio simulations, and the local structure that produces them, cannot be validated and that classical polarizability models (and the manner in which they relate to local structure) cannot be parameterized. To further heighten the challenge, the surface potential of pure water is both difficult to measure experimentally and the subject of significant disagreement (by more than 0.5 V) in simulation treatments53. Thus one might expect that inaccuracies in surface potential of the pure water/air interface might, plausibly, compensate for inaccuracies in the description of local structure (and thus polarizability). Experiments of the sort described in this study gives a clear path forward through these challenges. Given experimental constraints on interfacial anion polarizability of the sort described in this study, descriptions of local anion structure inferred from ab-initio simulation can be validated and empirical polarizability models more accurately parameterized. Given a validated description of local ion structure systematically reducing errors in the calculation of (ion concentration dependent) surface potential is now much more straightforward. Prior workers have performed studies similar in spirit to those shown here. Miyame et al.6 characterized the S–O stretch vibrations of $SO 4 2 -$, while Motschmann and co-workers10 characterized the CN stretch vibrations of the potassium ferricyanide ion, i.e. $Fe CN 6 4 -$, as a function of bulk concentration at the air/water interface. Consistent with both calculation and other experimental approaches that suggest $SO 4 2 -$ retains its bulk solvation shell at the air/water interface4,54, Miyame, Morita and Ouchi find interfacial $SO 4 2 -$ to be essentially the same as bulk. In contrast Motschmann and co-workers found that CN modes that were IR inactive in bulk solution were apparent in the VSF spectrum, i.e. the interface induces a change in ferricyanide symmetry as it does for perchlorate. However, presumably because of the more structurally complicated anion, they were unable to quantify the resulting change in the polarizability tensor. As is clear from Eq. (2), if a molecules hyperpolarizability and orientation are concentration independent one can extract a measurement of anion interfacial density as a function of bulk concentration by plotting the square root of the measured SF signal vs. bulk concentration. In a series of studies employing electronically resonant second harmonic measurements of a variety of anions at air/water interface, Saykally and co-workers5,21 have treated anion hyperpolarizability (and orientation) as concentration independent, fit adsorption isotherms to measurements of SHG signal as a function of bulk concentration, and calculated anion adsorption energies. Our results suggest that this type of data needs to be revisited. Because deformation of the perchlorate leads to an increase in transition dipole and polarizability, it is clear that given a VSF spectra collected under the ssp polarization condition, using this approach would overestimate adsorption energies (i.e. the molecular response of the perchlorate anion would increase with increasing concentration). In summary, in the current study we have employed VSF spectroscopy and a simple computational model to study the behaviour of $ClO 4 -$ at the air/HClO4 solution interface. Consistent with much prior work our observations clearly demonstrate that $ClO 4 -$ is a surface active anion. We significantly extend these prior efforts by demonstrating that the presence of the interface induces deformation of the anion that causes a bulk forbidden mode to be VSF active due to change in anion symmetry, creates a nonzero dipole moment and leads to a change in the measured polarizability anisotropy. Our results suggest that increasing density of $ClO 4 -$ at the interface leads to an increasing interfacial field that increases $ClO 4 -$ polarization (i.e. increases $ClO 4 -$ dipole moment), increases $ClO 4 -$ structural deformation and makes the polarizability of $ClO 4 -$ increasingly anisotropic4,5,9. Extension of the approach we describe here should allow the possibility of directly quantifying the elements of $ClO 4 -$ polarizability tensor, rather than just their ratio, and enable a window onto local interfacial anion structure (particularly ion/ion correlation) and the possibility of reliably experimentally quantifying the thermodynamic importance of anion polarization. The close connection we describe here between the dipole moment, structure and polarizability of interfacial anions with increasing interfacial field has not, to our knowledge, been previously considered but should be a quite general feature of anion, particularly polyvalent anion, adsorption at hydrophobic interfaces. As such its quantitative reproduction is a prerequisite for simulation approaches that attempt to offer microscopic insight into this phenomena. ## Methods ### Solution preparation HClO4 (Suprapur, 70%, Merck) and NaClO4 (>99.99%, Sigma-Aldrich) were used as received. Solutions with the indicated concentrations were prepared by diluting the high concentration of HClO4 and NaClO4 in ultrapure H2O (18.3 MΩ cm; Milli-Q, Millipore). All solutions are prepared freshly before each measurement to limit degradation or contamination. VSF measurements in the C–H stretching and C=O region were employed to judge the quality of the solutions. ### Surface tension measurement The concentration-dependent surface tensions were measured with a surface tensiometer (K12/K14 KRÜSS GmbH) at room temperature (25 ± 1 °C). A Du Noüy ring method with a chemically inert Pt ring was utilized. The sample volume was 50 mL. Each measured surface tension is an average of 20 separate measurements, collected by the Pt ring in and out of the interface without tearing the lamella. ### VSF measurement and spectral modelling The VSF spectrometer employed for the current measurement, and in particular its power at long infrared wavelengths, has been described in detail in our previous studies55,56. In the interest of brevity only a brief description that pertinent to this measurement will be given here. The IR beam was generated from a commercial optical parametric amplifier (HE-TOPAS, Light Conversion) with a difference frequency generation (DFG) module. The full width half maximum (FWHM) of the beam at frequency region between 600 and 1200 cm−1 is typically 300 cm−1 with GaSe used as the DFG crystal. To probe the interfacial Cl–O stretch modes the centre frequency of the beam was tuned to ≈1000 cm−1. A narrow-band visible (VIS) pulse was produced from a home-made spectral shaper55. The beam is centred at 800 nm with a bandwidth of 15 cm−1. The energy per pulse of the IR and VIS at the sample surface was 5.8 and 15.4 μJ respectively. polarizations and energies of the incident fields at the interface were controlled using λ/2 plate, polarizer, λ/2 plate combinations. The two beams propagate in a coplanar fashion and focused on the samples using lenses with focal lengths of 10 and 25 cm and incident angles of 39.5 ± 0.5° and 65 ± 0.5° for the IR and VIS. All measurements were conducted in ambient conditions at room temperature and under the ssp (s-polarized SF, s-polarized visible, and p- polarized IR where p indicates polarization in the plane of incidence and s polarization orthogonal) and ppp polarization condition. Non-resonant signals from a gold thin film were used to correct for the frequency dependent IR intensity. The acquisition time for spectra of the gold reference and samples were 30 and 300 s, respectively. To quantify the observed VSF spectral response, we adopted a Lorentzian lineshape model described and justified in much previous work by us and others55,56,57,58,59. $I sf ω sf ∝ χ eff ( 2 ) 2 ∝ χ nr e i ϵ + ∑ n χ r , n ω ir - ω n + i Γ n 2$ (3) where Isf(ωsf) is the normalized VSF intensity, $χ eff ( 2 )$ is the effective second-order susceptibility, which depends on the experimental geometry, molecular hyperpolarizability and orientation. $χ nr$ and $ϵ$ are the non-resonant amplitude and phase and χr,n, ω n and Γ n are the complex amplitude, centre frequency and line width of the nth resonance. To actually analyse the data we fit the measured VSF spectrum using the Levenberg–Marquardt algorithm as implemented in the commercial visualization and analysis programme Igor Pro (Wavemetrics). Fitting spectra collected at each bulk concentration and polarization with this lineshape model results in an underdetermined minimization problem. Because bulk studies suggest that the centre frequencies and spectral shape of $ClO 4 -$ solution are concentration independent, we addressed these data by assuming that all spectra collected under a bulk concentration of HClO4 could be described with two resonances, each with a concentration independent line width, centre frequency and phase, and a non-resonant amplitude and phase that are also concentration independent. We accounted for the libration tail (only important in the ssp spectra) by assuming the libration has the centre frequency and line width from our previous study56. Details of the analysis, and all the parameters resulting from the fit, are given in Supplementary Note 1. Recent work has shown that, at sufficiently charged interfaces, it is possible that the interfacial χ(2) signal of specifies also present in the adjoining bulk aqueous phase, may be distorted by a χ(3) response from oscillators within the electrical double layer. 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B 59, 12632–12640 (1999). 60. 60. Gonella, G., Lütgebaucks, C., de Beer, A. G. F. & Roke, S. Second harmonic and sum-frequency generation from aqueous interfaces is modulated by interference. J. Phys. Chem. C 120, 9165–9173 (2016). 61. 61. Wen, Y.-C. et al. Unveiling microscopic structures of charged water interfaces by surface-specific vibrational spectroscopy. Phys. Rev. Lett. 116, 016101 (2016). 62. 62. Ohno, P., Wang, H., Skinner, J., Paesani, F. & Geiger, F. M. On second-order vibrational lineshapes of the air/water interface. Preprint at https://arxiv.org/abs/1712.09086 (2017). ## Acknowledgements The authors thank Irina Shekova for helping with the surface tension measurement and Martin Wolf for useful discussions and the Max Planck Society for support of this work. ## Author information ### Affiliations 1. #### Fritz Haber Institute of the Max Planck Society, 14195, Berlin, Germany • Yujin Tong • , Igor Ying Zhang • & R. Kramer Campen 2. #### Department of Chemistry, Fudan University, 200433, Shanghai, China • Igor Ying Zhang ### Contributions Y.T. and R.K.C. designed the study. Y.T. performed the measurements. Y.T. and R.K.C. analysed the data and wrote the paper. I.Y.Z. performed the electronic structure calculations. All authors edited the paper. ### Competing interests The authors declare no competing interests. ### Corresponding author Correspondence to Yujin Tong.	{}
# Inequality Inequality In mathematics, an inequality is a statement about the relative size or order of two objects, "or" about whether they are the same or not (See also: equality) The notation "a" < "b" means that "a" is less than "b". The notation "a" > "b" means that "a" is greater than "b". The notation "a" ≠ "b" means that "a" is not equal to "b," but does not say that one is bigger than the other or even that they can be compared in size.In all these cases, "a" is not equal to "b," hence, "inequality". These relations are known as strict inequality; in contrast The notation "a" ≤ "b" means that "a" is less than or equal to "b" (or, equivalently, not greater than "b"); The notation "a" ≥ "b" means that "a" is greater than or equal to "b" (or, equivalently, not smaller than "b"); An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude. The notation "a" ≪ "b" means that "a" is much less than "b". The notation "a" ≫ "b" means that "a" is much greater than "b". If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. Solving Inequalities An inequality may appear unsolvable because it only states whether a number is larger or smaller than another number; but it is possible to apply the same operations for equalities to inequalities. For example, to find x for the inequality 10x > 23 one would divide 23 by 10. Properties Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding non-strict inequality sign (≤ and ≥). Trichotomy The trichotomy property states: For any real numbers, "a" and "b", exactly one of the following is true: "a" < "b" "a" = "b" ** "a" > "b" Transitivity The transitivity of inequalities states: * For any real numbers, "a", "b", "c": If "a" > "b" and "b" > "c"; then "a" > "c" If "a" < "b" and "b" < "c"; then "a" < "c" The properties which deal with addition and subtraction state: * For any real numbers, "a", "b", "c": If "a" > "b", then "a" + "c" > "b" + "c" and "a" − "c" > "b" − "c" If "a" < "b", then "a" + "c" < "b" + "c" and "a" − "c" < "b" − "c" i.e., the real numbers are an ordered group. Multiplication and division The properties which deal with multiplication and division state: * For any real numbers, "a", "b", "c": If "c" is positive and "a" < "b", then "ac" < "bc" If "c" is negative and "a" < "b", then "ac" > "bc" More generally this applies for an ordered field, see below. The properties for the additive inverse state: For any real numbers "a" and "b" If "a" < "b" then −"a" > −"b" If "a" > "b" then −"a" < −"b" Multiplicative inverse The properties for the multiplicative inverse state: For any real numbers "a" and "b" that are both positive or both negative If "a" < "b" then 1/"a" > 1/"b" If "a" > "b" then 1/"a" < 1/"b" Applying a function to both sides We consider two cases of functions: monotonic and strictly monotonic. Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function. If you have a non-strict inequality ("a" ≤ "b", "a" ≥ "b") then: * Applying a monotonically increasing function preserves the relation (≤ remains ≤, ≥ remains ≥) * Applying a monotonically decreasing function reverses the relation (≤ becomes ≥, ≥ becomes ≤) It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3. Ordered fields If ("F", +, ×) is a field and ≤ is a total order on "F", then ("F", +, ×, ≤) is called an ordered field if and only if: * "a" ≤ "b" implies "a" + "c" ≤ "b" + "c"; * 0 ≤ "a" and 0 ≤ "b" implies 0 ≤ "a" × "b". Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of "i" and would therefore be positive. The non-strict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are ml\|Total_order\|Strict_total_order\|strict total orders. Chained notation The notation "a" < "b" < "c" stands for "a" < "b" and "b" < "c", from which, by the transitivity property above, it also follows that "a" < "c". Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, e.g. "a" < "b" + "e" < "c" is equivalent to "a" − "e" < "b" < "c" − "e". This notation can be generalized to any number of terms: for instance, "a"1 ≤ "a"2 ≤ ... ≤ "a""n" means that "a""i" ≤ "a""i"+1 for "i" = 1, 2, ..., "n" − 1. By transitivity, this condition is equivalent to "a""i" ≤ "a""j" for any 1 ≤ "i" ≤ "j" ≤ "n". When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4"x" < 2"x" + 1 ≤ 3"x" + 2, you won't be able to isolate "x" in any one part of the inequality through addition or subtraction. Instead, you can solve 4"x" < 2"x" + 1 and 2"x" + 1 ≤ 3"x" + 2 independently, yielding "x" < 1/2 and "x" ≥ -1 respectively, which can be combined into the final solution -1 ≤ "x" < 1/2. Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, "a" < "b" > "c" ≤ "d" means that "a" < "b", "b" > "c", and "c" ≤ "d". In addition to rare use in mathematics, this notation exists in a few programming languages such as Python. Representing inequalities on the real number line Every inequality (except those which involve imaginary numbers) can be represented on the real number line showing darkened regions on the line. Inequalities between means There are many inequalities between means. For example, for any positive numbers "a"1, "a"2, …, "a""n" we have nowrap\|"H" ≤ "G" ≤ "A" ≤ "Q", where : Power inequalities Sometimes with notation "power inequality" understand inequalities which contain "a""b" type expressions where "a" and "b" are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations. Examples * If "x" > 0, then:: $x^x ge left\left( frac\left\{1\right\}\left\{e\right\} ight\right)^\left\{1/e\right\}.,$ * If "x" > 0, then:: $x^\left\{x^x\right\} ge x.,$ * If "x", "y", "z" > 0, then:: $\left(x+y\right)^z + \left(x+z\right)^y + \left(y+z\right)^x > 2.,$ * For any real distinct numbers "a" and "b",:: $frac\left\{e^b-e^a\right\}\left\{b-a\right\} > e^\left\{\left(a+b\right)/2\right\}.$ * If "x", "y" > 0 and 0 < "p" < 1, then:: $\left(x+y\right)^p < x^p+y^p.,$ * If "x", "y", "z" > 0, then:: $x^x y^y z^z ge \left(xyz\right)^\left\{\left(x+y+z\right)/3\right\}.,$ * If "a", "b", then:: $a^b + b^a > 1.,$: This result was generalized by R. Ozols in 2002 who proved that if "a"1, ..., "a""n", then:: $a_1^\left\{a_2\right\}+a_2^\left\{a_3\right\}+cdots+a_n^\left\{a_1\right\}>1$: (result is published in Latvian popular-scientific quarterly "The Starry Sky", see references). Well-known inequalities Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names: * Azuma's inequality * Bernoulli's inequality * Boole's inequality * Cauchy–Schwarz inequality * Chebyshev's inequality * Chernoff's inequality * Cramér-Rao inequality * Hoeffding's inequality * Hölder's inequality * Inequality of arithmetic and geometric means * Jensen's inequality * Kolgomorov's inequality * Markov's inequality * Minkowski inequality * Nesbitt's inequality * Pedoe's inequality * Triangle inequality Student Learning Techniques Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents the mouth of a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3. [http://mathforum.org/library/drmath/view/58428.html] Another method is noticing the larger quantity points to the smaller quantity and says, "ha-ha, I'm bigger than you." Also, on a horizontal number line, the greater than sign is the arrow that is at the larger end of the number line. Likewise, the less than symbol is the arrow at the smaller end of the number line (<---0--1--2--3--4--5--6--7--8--9--->). The symbols may also be interpreted directly from their form - the side with a large vertical separation indicates a large(r) quantity, and the side which is a point indicates a small(er) quantity. In this way the inequality symbols are similar to the musical crescendo and decrescendo. The symbols for equality, less-than-or-equal-to, and greater-than-or-equal-to can also be interpreted with this perspective. Complex numbers and inequalities By introducing a lexicographical order on the complex numbers, it is a totally ordered set.However, it is impossible to define ≤ so that $mathbb\left\{C\right\}$,+,,≤ becomes an ordered field. If $mathbb\left\{C\right\}$,+,,≤ were an ordered field, it has to satisfy the following two properties: * if "a" ≤ "b" then "a" + "c" ≤ "b" + "c" * if 0 ≤ "a" and 0 ≤ "b" then 0 ≤ "a b" Because ≤ is a total order, for any number "a", "a" ≤ 0 or 0 ≤ "a". In both cases 0 ≤ "a"2; this means that $i^2>0$ and $1^2>0$; so $1>0$ and $-1>0$, contradiction. However ≤ can be defined in order to satisfy the first property, i.e. if "a" ≤ "b" then "a" + "c" ≤ "b" + "c". A definition which is sometimes used is the lexicographical order: * a ≤ b if $Re\left(a\right)$ < $Re\left(b\right)$ or ($Re\left(a\right) = Re\left(b\right)$ and $Im\left(a\right)$$Im\left(b\right)$)It can easily be proven that for this definition "a" ≤ "b" then "a" + "c" ≤ "b" + "c" ee also Linear inequality Binary relation Bracket for the use of the < and > signs as brackets Fourier-Motzkin elimination Inequation Interval (mathematics) Partially ordered set Relational operators, used in programming languages to denote inequality References cite book \| author=Hardy, G., Littlewood J.E., Polya, G.\| title=Inequalities\| publisher=Cambridge Mathematical Library, Cambridge University Press \| year=1999 \| id=ISBN 0-521-05206-8 cite book \| author=Beckenbach, E.F., Bellman, R.\| title=An Introduction to Inequalities\| publisher=Random House Inc \| year=1975 \| id=ISBN 0-394-01559-2 cite book \| author=Drachman, Byron C., Cloud, Michael J.\| title=Inequalities: With Applications to Engineering\| publisher=Springer-Verlag \| year=1998 \| id=ISBN 0-387-98404-6 cite paper\|title="Quickie" inequalities\|author=Murray S. Klamkin\|url=http://www.pims.math.ca/pi/issue7/page26-29.pdf\|format=PDF\|work=Math Strategies cite web\|title=Mathematical Problem Solving\|url=http://www.math.kth.se/math/TOPS/index.html\|author=Harold Shapiro\|date=missingdate\|publisher=Kungliga Tekniska högskolan\|work=The Old Problem Seminar cite web\|title=3rd USAMO\|url=http://www.kalva.demon.co.uk/usa/usa74.html cite paper\|title=The Starry Sky\|url=http://www.astr.lu.lv/zvd/stsky.html [http://www.mathwarehouse.com/algebra/linear_equation/interactive-linear-inequality.php interactive linear inequality & graph] at www.mathwarehouse.com * [http://www.purplemath.com/modules/ineqsolv.htm Solving Inequalities] * [http://www.webgraphing.com/inequality_1d.jsp WebGraphing.com] – Inequality Graphing Calculator. * [http://demonstrations.wolfram.com/GraphOfInequalities/ Graph of Inequalities] by Ed Pegg, Jr., The Wolfram Demonstrations Project. Wikimedia Foundation. 2010. Synonyms: ### Look at other dictionaries: • Inequality — In equal ity, n.; pl. {Inequalities}. [L. inaequalitas.] [1913 Webster] 1. The quality of being unequal; difference, or lack of equality, in any respect; lack of uniformity; disproportion; unevenness; disparity; diversity; as, an inequality in… … The Collaborative International Dictionary of English • inequality — inequality, social inequality Unequal rewards or opportunities for different individuals within a group or groups within a society. If equality is judged in terms of legal equality, equality of opportunity, or equality of outcome, then inequality … Dictionary of sociology • inequality — UK US /ˌɪnɪˈkwɒləti/ noun [C or U] ECONOMICS ► a situation in which money or opportunities are not shared equally between different groups in society: »Several polls show that one of the biggest issues on people s minds is economic inequality … Financial and business terms • inequality — I noun asymmetry, bias, contrast, deviation, difference, disaccord, disagreement, discrepance, discrepancy, disparity, disproportion, disproportionateness, dissimilarity, dissimilitude, dissimilitude), dissonance, distinction, divergence,… … Law dictionary • inequality — (n.) early 15c., difference of rank or dignity, from O.Fr. inequalité (14c.) and directly from M.L. inaequalitas, from L. inaequalis unequal, from in not, opposite of (see IN (Cf. in ) (1)) + aequalis equal (see EQUAL (Cf. equal)) … Etymology dictionary • inequality — [n] prejudice; lack of balance asperity, bias, contrast, difference, discrimination, disparity, disproportion, dissimilarity, dissimilitude, diversity, imparity, incommensurateness, injustice, irregularity, one sidedness, partisanship,… … New thesaurus • inequality — ► NOUN (pl. inequalities) ▪ lack of equality … English terms dictionary • inequality — [in΄ē kwôl′ə tē, in΄ēkwäl′ə tē; in΄ikwâl′ə tē, in΄ikwäl′ə tē] n. pl. inequalities [ME inequalitie < MFr inequalité < L inaequalitas] 1. the quality of being unequal; lack of equality 2. an instance of lack of equality; specif., a) a… … English World dictionary • inequality — noun ADJECTIVE ▪ great, gross, substantial ▪ the gross social inequalities of the past ▪ Inequalities of income would lead to even greater inequalities in access to health care. ▪ real … Collocations dictionary • inequality */ — UK [ˌɪnɪˈkwɒlətɪ] / US [ˌɪnɪˈkwɑlətɪ] noun [countable/uncountable] Word forms inequality : singular inequality plural inequalities a situation in which people are not equal because some groups have more opportunities, power, money etc than others … English dictionary	{}
## Seminars and Colloquia by Series ### Sets without 4APs but with many 3APs Series Combinatorics Seminar Time Friday, January 31, 2020 - 15:05 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Andrei (Cosmin) PohoataCalifornia Inst. of Technology, Pasadena, CA It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton. ### Fast uniform generation of random graphs with given degree sequences Series Combinatorics Seminar Time Friday, January 24, 2020 - 15:00 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Andrii ArmanEmory University In this talk I will discuss algorithms for a uniform generation of random graphs with a given degree sequence. Let $M$ be the sum of all degrees and $\Delta$ be the maximum degree of a given degree sequence. McKay and Wormald described a switching based algorithm for the generation of graphs with given degrees that had expected runtime $O(M^2\Delta^2)$, under the assumption $\Delta^4=O(M)$. I will present a modification of the McKay-Wormald algorithm that incorporates a new rejection scheme and uses the same switching operation. A new algorithm has expected running time linear in $M$, under the same assumptions. I will also describe how a new rejection scheme can be integrated into other graph generation algorithms to significantly reduce expected runtime, as well as how it can be used to generate contingency tables with given marginals uniformly at random. This talk is based on the joint work with Jane Gao and Nick Wormald. ### Non-concentration of the chromatic number of a random graph Series Combinatorics Seminar Time Friday, January 10, 2020 - 15:00 for 1 hour (actually 50 minutes) Location Skiles 202 Speaker Lutz Warnke We shall discuss the recent breakthrough of Annika Heckel on the chromatic number of the binomial random graph G(n,1/2), showing that it is not concentrated on any sequence of intervals of length n^{1/4-o(1)}. To put this into context, in 1992 Erdos (and also Bollobás in 2004) asked for any non-trivial results asserting a lack of concentration, pointing out that even the weakest such results would be of interest. Until recently this seemed completely out of reach, in part because there seemed to be no obvious approach/strategy how to get one's foot in the door. Annika Heckel has now found such an approach, based on a clever coupling idea that compares the chromatic number of G(n,1/2) for different n. In this informal talk we shall try to say a few words about her insightful proof approach from https://arxiv.org/abs/1906.11808 Please note the unusual room (Skiles 202) ### Thresholds versus fractional expectation-thresholds Series Combinatorics Seminar Time Friday, December 6, 2019 - 13:05 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Jinyoung ParkRutgers University (This is a joint event of the Combinatorics Seminar Series and the ACO Student Seminar.) In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdos-Rado “Sunflower Conjecture.” This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan. ### On a class of sums with unexpectedly high cancellation, and its applications Series Combinatorics Seminar Time Friday, November 15, 2019 - 15:00 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Hamed MousaviGeorgia Tech We report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums. It turns out that sums in the class we consider are much smaller than would be predicted by certain probabilistic heuristics. After stating the motivation, and our theorem, we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem. For example, we prove a "Pentagonal Number Theorem for the Primes", which counts the number of primes (with von Mangoldt weight) in a set of intervals very precisely. In fact the result is stronger than one would get using a strong form of the Prime Number Theorem and also the Riemann Hypothesis (where one naively estimates the \Psi function on each of the intervals; however, a less naive argument can give an improvement), since the widths of the intervals are smaller than \sqrt{x}, making the Riemann Hypothesis estimate "trivial". Based on joint work with Ernie Croot. ### Finding cliques in random graphs by adaptive probing Series Combinatorics Seminar Time Friday, November 8, 2019 - 15:05 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Miklos RaczPrinceton University I will talk about algorithms (with unlimited computational power) which adaptively probe pairs of vertices of a graph to learn the presence or absence of edges and whose goal is to output a large clique. I will focus on the case of the random graph G(n,1/2), in which case the size of the largest clique is roughly 2\log(n). Our main result shows that if the number of pairs queried is linear in n and adaptivity is restricted to finitely many rounds, then the largest clique cannot be found; more precisely, no algorithm can find a clique larger than c\log(n) where c < 2 is an explicit constant. I will also discuss this question in the planted clique model. This is based on joint works with Uriel Feige, David Gamarnik, Joe Neeman, Benjamin Schiffer, and Prasad Tetali. ### Local limit theorems for combinatorial random variables Series Combinatorics Seminar Time Friday, November 1, 2019 - 11:00 for 1 hour (actually 50 minutes) Location Skiles 249 Speaker Ross BerkowitzYale University Let X be the number of length 3 arithmetic progressions in a random subset of Z/101Z. Does X take the values 630 and 640 with roughly the same probability? Let Y denote the number of triangles in a random graph on n vertices. Despite looking similar to X, the local distribution of Y is quite different, as Y obeys a local limit theorem. We will talk about a method for distinguishing when combinatorial random variables obey local limit theorems and when they do not. ### Twisted Schubert polynomials Series Combinatorics Seminar Time Friday, October 18, 2019 - 15:00 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Ricky LiuNorth Carolina State University We will describe a twisted action of the symmetric group on the polynomial ring in n variables and use it to define a twisted version of Schubert polynomials. These twisted Schubert polynomials are known to be related to the Chern-Schwartz-MacPherson classes of Schubert cells in the flag variety. Using properties of skew divided difference operators, we will show that these twisted Schubert polynomials are monomial positive and give a combinatorial formula for their coefficients. ### Long-range order in random colorings and random graph homomorphisms in high dimensions Series Combinatorics Seminar Time Friday, March 29, 2019 - 15:05 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Yinon SpinkaUniversity of British Columbia, Vancouver, Canada Consider a uniformly chosen proper coloring with q colors of a domain in Z^d (a graph homomorphism to a clique). We show that when the dimension is much higher than the number of colors, the model admits a staggered long-range order, in which one bipartite class of the domain is predominantly colored by half of the q colors and the other bipartite class by the other half. In the q=3 case, this was previously shown by Galvin-Kahn-Randall-Sorkin and independently by Peled. The result further extends to homomorphisms to other graphs (covering for instance the cases of the hard-core model and the Widom-Rowlinson model), allowing also vertex and edge weights (positive temperature models). Joint work with Ron Peled. ### Contagion in random graphs and systemic risk Series Combinatorics Seminar Time Friday, February 15, 2019 - 15:00 for 1 hour (actually 50 minutes) Location Skiles 005 Speaker Hamed AminiGeorgia State University We provide a framework for testing the possibility of large cascades in random networks. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. This allows us to study systemic risk in financial networks, where we introduce a criterion for the resilience of a large network to the failure (insolvency) of a small group of institutions and quantify how contagion amplifies small shocks to the network.	{}
# Am I correct for this question? #### bushra1175 ##### Junior Member My working out: 7,940/7 = around 1,134 hours of work that needs to be done 1,134/10 = 113 shifts to be done based on 10 hour work days I've worked out that we need to prefer full time workers over part time workers as they have a higher output rate: 35/40 = 87.5% 17/20 = 85% For full time workers, I need to increase the shifts by 12.5% to make up for the time they're not spending on calls 12.5% of 113 = 14 shifts, meaning full time workers need to work 127 shifts to get through 7,940 calls Since full time workers work 5 days a week, I would divide 127 shitfs by 5, giving me 25 full time workers since this was a timed test (2.5 mins per question), I don't have time to work out every combination so I estimate the 20 FT and 7 PT combo to be correct #### blamocur ##### Full Member In you solution what is the weekly number of calls that the workers can handle ? #### bushra1175 ##### Junior Member A full time worker can handle 735 calls a week, which is 245 A part time worker can handle 717 calls a week, which is 119 If you use 15FT workers, you would need 36PT workers so option 1 is ruled out If you use 20FT workers, you would need 25PT workers, so option 2 and 3 are ruled out If you use 25FT workers, you would need 15PT workers, so option 4 is ruled out If you use 28FT workers, you would need 9PT workers, so the correct answer would be 28FT and 10PT as this is the closest option. Thanks! #### JeffM ##### Elite Member There are two issues with this question. One is that we have no idea what you are studying so we do not know what kinds of math may apply. The other is that the problem requires a lot of assumptions to answer. For example, do part time workers get paid the same hourly wage as full time workers? For another, are there peak times? On the assumptions of no wage differentials and no peak loads, you face 7940 calls. A full time worker can handle 7 * 35 = 245 calls per week. A part time worker can handle 17 * 7 = 119 calls per week. So we have $f \ge 0,\ p \ge 0, \text { and } 245f + 119p = 7940.$ So yes you can just plug the numbers in and see what makes sense. #### bushra1175 ##### Junior Member There are two issues with this question. One is that we have no idea what you are studying so we do not know what kinds of math may apply. The other is that the problem requires a lot of assumptions to answer. For example, do part time workers get paid the same hourly wage as full time workers? For another, are there peak times? On the assumptions of no wage differentials and no peak loads, you face 7940 calls. A full time worker can handle 7 * 35 = 245 calls per week. A part time worker can handle 17 * 7 = 119 calls per week. So we have $f \ge 0,\ p \ge 0, \text { and } 245f + 119p = 7940.$ So yes you can just plug the numbers in and see what makes sense. Thanks Jeff. This question is actually part of a numerical practice test for job applications. There is no context	{}
# Equilibrium selection via replicator dynamics in $$2 \times 2$$ 2 × 2 coordination games ## Author Info Listed author(s): • Boyu Zhang () • Josef Hofbauer () ## Abstract This paper studies two equilibrium selection methods based on replicator dynamics. A Nash equilibrium is called centroid dominant if the trajectory of the replicator dynamics starting at the centroid of the strategy simplex converges to it. On the other hand, an equilibrium is called basin dominant if it has the largest basin of attraction. These two concepts are compared with risk dominance in the context of $$2 \times 2$$ 2 × 2 bimatrix coordination games. The main results include (a) if a Nash equilibrium is both risk dominant and centroid dominant, it must have the largest basin of attraction, (b) the basin dominant equilibrium must be risk dominant or centroid dominant. Copyright Springer-Verlag Berlin Heidelberg 2015 If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large. File URL: http://hdl.handle.net/10.1007/s00182-014-0437-7 As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it. ## Bibliographic Info Article provided by Springer & Game Theory Society in its journal International Journal of Game Theory. Volume (Year): 44 (2015) Issue (Month): 2 (May) Pages: 433-448 as in new window Handle: RePEc:spr:jogath:v:44:y:2015:i:2:p:433-448 DOI: 10.1007/s00182-014-0437-7 Contact details of provider: Web page: http://www.springer.com Order Information: Web: http://www.springer.com/economics/economic+theory/journal/182/PS2 ## References References listed on IDEAS Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.: as in new window 1. Hofbauer, Josef & Sorger, Gerhard, 1999. "Perfect Foresight and Equilibrium Selection in Symmetric Potential Games," Journal of Economic Theory, Elsevier, vol. 85(1), pages 1-23, March. 2. Russell Golman & Scott Page, 2010. "Basins of attraction and equilibrium selection under different learning rules," Journal of Evolutionary Economics, Springer, vol. 20(1), pages 73-75, January. 3. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, January. 4. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January. 5. Van Huyck, John B & Battalio, Raymond C & Beil, Richard O, 1990. "Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure," American Economic Review, American Economic Association, vol. 80(1), pages 234-248, March. 6. Matsui Akihiko & Matsuyama Kiminori, 1995. "An Approach to Equilibrium Selection," Journal of Economic Theory, Elsevier, vol. 65(2), pages 415-434, April. 7. Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January. 8. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November. 9. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January. 10. Josef Hofbauer & William H. Sandholm, 2002. "On the Global Convergence of Stochastic Fictitious Play," Econometrica, Econometric Society, vol. 70(6), pages 2265-2294, November. 11. Binmore, Ken & Samuelson, Larry, 1997. "Muddling Through: Noisy Equilibrium Selection," Journal of Economic Theory, Elsevier, vol. 74(2), pages 235-265, June. 12. Russell Golman & Scott Page, 2010. "Basins of attraction and equilibrium selection under different learning rules," Journal of Evolutionary Economics, Springer, vol. 20(1), pages 49-72, January. 13. Kim, Youngse, 1996. "Equilibrium Selection inn-Person Coordination Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 203-227, August. 14. J. Hofbauer, 1999. "The spatially dominant equilibrium of a game," Annals of Operations Research, Springer, vol. 89(0), pages 233-251, January. Full references (including those not matched with items on IDEAS) ## Lists This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS. ## Corrections When requesting a correction, please mention this item's handle: RePEc:spr:jogath:v:44:y:2015:i:2:p:433-448. See general information about how to correct material in RePEc. For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure) If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about. If references are entirely missing, you can add them using this form. If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form. If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation. Please note that corrections may take a couple of weeks to filter through the various RePEc services. This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.	{}
# Wrap text around pseudocode I use algorithm/algorithmic packages to write some pseudocode. The width of the box spans across the entire page and I was wondering if there is any way to constrain it (since the actual code doesn't exceed the mid of the page). Practically, I want to wrap text around my pseudocode. ps: I tried wrapfig package, but I had some problems. Would be also helpful if anyone could verify whether this approach works. Any ideas ? - ## migrated from stackoverflow.comDec 12 '11 at 10:15 This question came from our site for professional and enthusiast programmers. Don't forget about TeX - LaTeX; LaTeX is odd enough that you'd be better served asking in a community dedicated to LaTeX and TeX. (I've flagged for moderator attention to migrate, but in case the mod doesn't migrate -- we do have LaTeX questions here, in any case -- just know it's available.) – sarnold Dec 12 '11 at 8:27 Welcome to TeX.sx! Your question was migrated here from another StackExchange site. Please register on this site, too, and make sure that both accounts are associated with each other, otherwise you won't be able to comment on or accept answers or edit your question. :) – Paulo Cereda Dec 12 '11 at 10:28 You can put the algorithm inside of a minipage, put that inside of the wrapfigure, and it should work like a charm. \documentclass{article} \usepackage{algorithmic} \usepackage{algorithm} \usepackage{wrapfig} \usepackage{lipsum} \begin{document} \lipsum[1] \begin{wrapfigure}{L}{0.5\textwidth} \begin{minipage}{0.5\textwidth} \begin{algorithm}[H] \caption{assignment algorithm} \begin{algorithmic} \STATE i $\leftarrow$ j \end{algorithmic} \end{algorithm} \end{minipage} \end{wrapfigure} \lipsum \end{document} And the result: You might want to tweak the alignment modifier on the minipage to get vertical alignment to work out a little better. Alternatively, you can play with the optionial lineheight of wrapfigure. - Could one modify this to work with Algorithm2e? – JeremyKun May 7 '15 at 17:39	{}
## 15.33 Cartier's equality and geometric regularity A reference for this section and the next is [Section 39, MatCA]. In order to comfortably read this section the reader should be familiar with the naive cotangent complex and its properties, see Algebra, Section 10.132. Lemma 15.33.1 (Cartier equality). Let $K/k$ be a finitely generated field extension. Then $\Omega _{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and $\text{trdeg}_ k(K) = \dim _ K \Omega _{K/k} - \dim _ K H_1(L_{K/k})$. Proof. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.152.11 and its proof. In this case we see that $\mathop{N\! L}\nolimits _{K/k}$ is homotopy equivalent to the complex $\bigoplus \nolimits _{j = 1, \ldots , c} K \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} K\text{d}x_ i$ by Algebra, Lemmas 10.132.2 and 10.132.13. The transcendence degree of $K$ over $k$ is the dimension of $A$ (by Algebra, Lemma 10.115.1) which is $n - c$ and we win. $\square$ Lemma 15.33.2. Let $K \subset L \subset M$ be field extensions. Then the Jacobi-Zariski sequence $0 \to H_1(L_{L/K}) \otimes _ L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega _{L/K} \otimes _ L M \to \Omega _{M/K} \to \Omega _{M/L} \to 0$ is exact. Lemma 15.33.3. Given a commutative diagram of fields $\xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] }$ with $k \subset k'$ and $K \subset K'$ finitely generated field extensions the kernel and cokernel of the maps $\alpha : \Omega _{K/k} \otimes _ K K' \to \Omega _{K'/k'} \quad \text{and}\quad \beta : H_1(L_{K/k}) \otimes _ K K' \to H_1(L_{K'/k'})$ are finite dimensional and $\dim \mathop{\mathrm{Ker}}(\alpha ) - \dim \mathop{\mathrm{Coker}}(\alpha ) -\dim \mathop{\mathrm{Ker}}(\beta ) + \dim \mathop{\mathrm{Coker}}(\beta ) = \text{trdeg}_ k(k') - \text{trdeg}_ K(K')$ Proof. The Jacobi-Zariski sequences for $k \subset k' \subset K'$ and $k \subset K \subset K'$ are $0 \to H_1(L_{k'/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/k'}) \to \Omega _{k'/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/k} \to 0$ and $0 \to H_1(L_{K/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/K}) \to \Omega _{K/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/K} \to 0$ By Lemma 15.33.1 the vector spaces $\Omega _{k'/k}$, $\Omega _{K'/K}$, $H_1(L_{K'/K})$, and $H_1(L_{k'/k})$ are finite dimensional and the alternating sum of their dimensions is $\text{trdeg}_ k(k') - \text{trdeg}_ K(K')$. The lemma follows. $\square$ In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{}
# 2020 AMC 12A Problems/Problem 21 (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) ## Problem How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ $\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$ ## Solution 1 We set up the following equation as the problem states: $$\text{lcm}{(5!, n)} = 5\text{gcd}{(10!, n)}.$$ Breaking each number into its prime factorization, we see that the equation becomes $$\text{lcm}{(2^3\cdot 3 \cdot 5, n)} = 5\text{gcd}{(2^8\cdot 3^4 \cdot 5^2 \cdot 7, n)}.$$ We can now determine the prime factorization of $n$. We know that its prime factors belong to the set $\{2, 3, 5, 7\}$, as no factor of $10!$ has $11$ in its prime factorization, nor anything greater. Next, we must find exactly how many different possibilities exist for each. There can be anywhere between $3$ and $8$ $2$'s and $1$ to $4$ $3$'s. However, since $n$ is a multiple of $5$, and we multiply the $\text{gcd}$ by $5$, there can only be $3$ $5$'s in $n$'s prime factorization. Finally, there can either $0$ or $1$ $7$'s. Thus, we can multiply the total possibilities of $n$'s factorization to determine the number of integers $n$ which satisfy the equation, giving us $6 \times 4 \times 1 \times 2 = \boxed{\textbf{(D) } 48}$. ~ciceronii ## Solution 2 Like the Solution 1, we starts from the equation: $$\text{lcm}{(5!, n)} = 5\text{gcd}{(10!, n)}.$$ Assume $\text{lcm}{(5!, n)}=k\cdot5!$, with some integer $k$. It follows that $k\cdot 4!=\text{gcd}{(10!, n)}$. It means that $n$ has a divisor $4!$. Since $n$ is a multiple of $5$, $n$ has a divisor $5!$. Thus, $\text{lcm}{(5!, n)}=n=k\cdot5!$. The equation can be changed as $$k\cdot5!=5\text{gcd}{(10!, k\cdot5!)}$$ $$k=5\text{gcd}{(6\cdot7\cdot8\cdot9\cdot10, k)}$$ We can see that $k$ is also a multiple of $5$, with a form of $5\cdot m$. Substituting it in the above equation, we have $$m=5\text{gcd}{(6\cdot7\cdot8\cdot9\cdot2, m)}$$ Similarly, $m$ is a multiple of $5$, with a form of $5\cdot s$. We have $$s=\text{gcd}{(6\cdot7\cdot8\cdot9\cdot2, 5\cdot s)}=\text{gcd}{(2^5\cdot3^3\cdot7, s)}$$ The equation holds, if $s$ is a divisor of $2^5\cdot3^3\cdot7$, which has $(5+1)(3+1)(1+1)=\boxed{(\textbf{D})48}$ divisors. by Linty Huang ## Solution 3 As in the previous solutions, we start with $$\text{lcm}(5!,n) = 5\text{gcd}(10!,n)$$ From this we have that $\text{lcm}(5!,n) \,\|\, 5\text{gcd}(10!,n)$ , and in particular, $n \,\|\, 5\text{gcd}(10!,n)$. However, $\text{gcd}(10!,n)\, \|\, n$, so we must have $\text{gcd}(10!,n) = n$ or $\text{gcd}(10!,n) = n/5$. If $\text{gcd}(10!,n) = n$, then we have $\text{lcm}(5!,n) = 5n$; because $5\, \|\, 5!$, this implies that 5 does not divide $n$, so we must have $\text{gcd}(10!,n) = n/5$. Now we have $\text{lcm}(5!,n) = n$, implying that $5!\, \|\, n$, and $n/5\, \|\, 10!$. Writing out prime factorizations, this gives us $$2^3 \cdot 3 \cdot 5 \,\|\, n$$ $$n \,\|\, 2^8 \cdot 3^4 \cdot 5^2 \cdot 7$$ So $n$ can have 3, 4, 5, 6, 7, or 8 factors of 2; 1, 2, 3, or 4 factors of two; and 0 or 1 factors of 7. Note that $\text{gcd}(2^8 \cdot 3^4 \cdot 5^2 \cdot 7,n) = n/5$ implies that $n$ has 2 factors of 5. Thus, there are $6 \cdot 4 \cdot 2 = 48$ possible choices for $n$, and our answer is $\boxed{\textbf{(D) 48}}$. -gumbymoo ~ pi_is_3.14 ## See Also 2020 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 20 Followed byProblem 22 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. Invalid username Login to AoPS	{}
# Revision history [back] Note that now Sage has a table object which can be printed as LaTeX. As per the example in the documentation: sage: rows = [['a', 'b', 'c'], [100,2,3], [4,5,60]] sage: table(rows) a b c 100 2 3 4 5 60 sage: latex(table(rows)) begin{tabular}{lll} a & b & c $100$ & $2$ & $3$ $4$ & $5$ & $60$ end{tabular}	{}
# Search for Lepton Flavor Violating Decays $\tau^- \to \ell^-K^0_S$ with the BaBar Experiment Abstract : A search for the lepton flavour violating decays $\tau^- \to l^- K^0_{\scriptscriptstyle S}$ ($l = e$ or $\mu$) has been performed using a data sample corresponding to an integrated luminosity of 469 ${fb}^{-1}$, collected with the {\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R}} detector at the SLAC PEP-II $e^+e^-$ asymmetric energy collider. No statistically significant signal has been observed in either channel and the estimated upper limits on branching fractions are ${\cal B}(\tau^- \to e^- K^0_{\scriptscriptstyle S}) < 3.3 \times 10^{-8}$ and ${\cal B}(\tau^- \to \mu^- K^0_{\scriptscriptstyle S}) < 4.0 \times 10^{-8}$ at 90% confidence level. Document type : Journal articles http://hal.in2p3.fr/in2p3-00363111 Contributor : Claudine Bombar <> Submitted on : Friday, February 20, 2009 - 11:55:09 AM Last modification on : Wednesday, September 16, 2020 - 4:22:11 PM ### Citation B. Aubert, M. Bona, Y. Karyotakis, J.P. Lees, V. Poireau, et al.. Search for Lepton Flavor Violating Decays $\tau^- \to \ell^-K^0_S$ with the BaBar Experiment. Physical Review D, American Physical Society, 2009, 79, pp.012004. ⟨10.1103/PhysRevD.79.012004⟩. ⟨in2p3-00363111⟩ Record views	{}
Important Properties on Circle The two important properties on circle are stated below: 1. The ratio of the circumference to the diameter of any circle is constant and the value of this constant is denoted by the Greek letter π. Therefore, the circumference of any circle/diameter of that circle = constant = π or, the circumference of any circle = π × diameter of that circle. If r be the radius of the circle then its diameter is 2r. Therefore, the circumference of the circle = π ∙ 2r = 2πr. The constant quantity π is an incommensurable number i.e., it cannot be expressed as the ratio of two positive integers. An approximate value or π is 27/7; a more accurate value of π is 355/133 or, 3.14159 (correct to five places of decimals). 2. Angles at the center of a circle are proportional to the lengths of the arcs which subtend those angles. The above two important properties on circle will help us to prove that a radian is a constant angle. Click Here to know how to prove that “a radian is a constant angle”. Measurement of Angles ` From Important Properties on Circle to HOME PAGE New! Comments Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.	{}
I think the proof would involve showing f⁻¹. Since f is surjective, there exists a 2A such that f(a) = b. Let f : A !B be bijective. So x 2 is not injective and therefore also not bijective and hence it won't have an inverse.. A function is surjective if every possible number in the range is reached, so in our case if every real number can be reached. The range of a function is all actual output values. In order to determine if $f^{-1}$ is continuous, we must look first at the domain of $f$. Let f: A → B. We will de ne a function f 1: B !A as follows. 1.Inverse of a function 2.Finding the Inverse of a Function or Showing One Does not Exist, Ex 2 3.Finding The Inverse Of A Function References LearnNext - Inverse of a Bijective Function … Bijective. https://goo.gl/JQ8NysProving a Piecewise Function is Bijective and finding the Inverse This is equivalent to the following statement: for every element b in the codomain B, there is exactly one element a in the domain A such that f(a)=b.Another name for bijection is 1-1 correspondence (read "one-to-one correspondence).. I've got so far: Bijective = 1-1 and onto. Then f has an inverse. Proof. Let f : A !B be bijective. In mathematical terms, let f: P → Q is a function; then, f will be bijective if every element ‘q’ in the co-domain Q, has exactly one element ‘p’ in the domain P, such that f (p) =q. In mathematics, a bijective function or bijection is a function f : A → B that is both an injection and a surjection. Accelerated Geometry NOTES 5.1 Injective, Surjective, & Bijective Functions Functions A function relates each element of a set with exactly one element of another set. The function f: ℝ2-> ℝ2 is defined by f(x,y)=(2x+3y,x+2y). Let’s define $f \colon X \to Y$ to be a continuous, bijective function such that $X,Y \in \mathbb R$. Now we much check that f 1 is the inverse … The Attempt at a Solution To start: Since f is invertible/bijective f⁻¹ is … Yes. Click here if solved 43 Theorem 1. An example of a function that is not injective is f(x) = x 2 if we take as domain all real numbers. 1. The codomain of a function is all possible output values. it's pretty obvious that in the case that the domain of a function is FINITE, f-1 is a "mirror image" of f (in fact, we only need to check if f is injective OR surjective). the definition only tells us a bijective function has an inverse function. If we fill in -2 and 2 both give the same output, namely 4. Please Subscribe here, thank you!!! it doesn't explicitly say this inverse is also bijective (although it turns out that it is). Let f 1(b) = a. Since f is injective, this a is unique, so f 1 is well-de ned. A bijective group homomorphism $\phi:G \to H$ is called isomorphism. The above problem guarantees that the inverse map of an isomorphism is again a homomorphism, and hence isomorphism. A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y. Alternatively, f is bijective if it is a one-to-one correspondence between those sets, in other words both injective and surjective. is bijective, by showing f⁻¹ is onto, and one to one, since f is bijective it is invertible. It means that each and every element “b” in the codomain B, there is exactly one element “a” in the domain A so that f(a) = b. Let b 2B. Bijective Function Examples. Show that f is bijective and find its inverse. A function is called to be bijective or bijection, if a function f: A → B satisfies both the injective (one-to-one function) and surjective function (onto function) properties. Thus, bijective functions satisfy injective as well as surjective function properties and have both conditions to be true. A bijection of a function occurs when f is one to one and onto. The domain of a function is all possible input values. : since f is bijective, by showing f⁻¹ is onto, and hence isomorphism above problem that! We will de ne a function occurs when f is bijective and find its inverse all output. Bijective ( although it turns out that it is ) a is unique, so f:. Well-De ned https: //goo.gl/JQ8NysProving a Piecewise function is all possible input values surjective, there a. Out that it is ) one, since f is bijective and find its inverse surjective... //Goo.Gl/Jq8Nysproving a Piecewise function is all actual output values explicitly say this inverse is also bijective although... 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# Python Diagonalize Matrix The outputs are: c is a vector of condition numbers for the eigenvalues of a. Any value of λ for which this equation has a solution is known as an eigenvalue of the matrix A. Here we present an implementation of exact diagonalization for a quantum many-body Hamiltonian composed of a sum of local terms. If a and b are not complex, this is the scalar product, also called the inner product or dot product, of a and b. Step by step procedure of the diagonalization together with an example is given. The IML procedure holds all matrices in RAM, so whenever I see this question I compute how much RAM is required for the specified matrix. An orthogonal matrix is a specially featured matrix, defined on the basis of using the square matrix. Calculate the eigenstates \|an of the matrix A by means of equation (1) and the experimentally determined eigenvalues an. In this python tutorial, we will write a code in Python on how to compute eigenvalues and vectors. T, fock, orth)) eigp, mop = numpy. Perhaps the most well-known eigenvalue problem in physics is the Schrödinger equation, which describes a particle in terms of its energy and the de Broglie wave. Linear Algebra with Python and NumPy ^H $exists only for a square, diagonalizable matrix, there is an obvious question if it can be generalized also for matrices of any shape. In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =. (We have scaled C by 1 / 4 so that vectors have roughly the same size on the right and the left. h TCL matrix matrix matrix operations program categories, added some arithmetic functions used. A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. You can use decimal (finite and periodic) fractions: 1/3, 3. 0]] This form, where the constant terms are explicitly viewed as coefficients, and hence form a column of the matrix is called an. 90929743, 0. a: It represents the array_like. savefig(),pt. Quick links to the program and files you'll need. Given any square matrix A ∈ M n(C),. These eigenvalues are often referred to as the 'eigenvalues of the equilibrium'. It requires the NumPy and SciPy libraries. Concerning tools, we will learn both C and python. we can calculate the matrices. (4 replies) I am working with Numpy and trying to create a COM server. We also review eigenvalues and eigenvectors. for example, the matrices above are 2 by 3, then the matrix is called a diagonal matrix. Built-in Function: norm (A) Built-in Function: norm (A, p) Built-in Function: norm (A, p, opt). It is certainly one of the most important algorithm in eigenvalue computations [9]. x is a matrix, when it extracts the diagonal. Deﬁnition 4. where the matrix D is diagonal. The inverse of a block diagonal matrix Posted on June 26, 2011 by hecker In the previous post I discussed multiplying block diagonal matrices as part of my series on defining block diagonal matrices and partitioning arbitrary square matrices uniquely and maximally into block diagonal form ( part 1 , part 2 , part 3 , part 4 , and part 5 ). k=0 represents the main diagonal, k>0 is above the main diagonal, and k<0 is below the main diagonal. The thing about positive definite matrices is xTAx is always positive, for any non-zerovector x, not just for an eigenvector. You can also find the dimensional of the matrix. vectors: either a p * p matrix whose columns contain the eigenvectors of x, or NULL if only. Now that you know the basics of Markov chains, you should now be able to easily implement them in a language of your choice. A matrix is positive definite fxTAx > Ofor all vectors x 0. For a solution, see the post " Quiz 13 (Part 1) Diagonalize a matrix. 10, 2014 0:36:29. Now push on this mass with another (not too large) force. Diagonalize Matrix Calculator. 2 (default, Nov 17 2016, 17:05:23). A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. Assume that the indicated operations are defined; that is, that the orders of the matrices $$A\text{,}$$ $$B$$ and $$C$$ are such that the operations make sense. I prefer to proceed by doing an LDU Decomposition and leaving it in terms of the Schur complement as I find it easier to remember. 98935825, 0. If you want to know more about the computational efficiency issues, you should look into the ideas behind the even faster variant, qr. • Form the matrix A−λI: A −λI = 1 −3 3 3 −5 3 6 −6 4 − λ 0 0 0 λ 0 0 0 λ = 1−λ −3 3 3 −5 −λ 3 6 −6 4−λ. Large matrix operations are the cornerstones of many important numerical and machine learning applications. SCF program: scf. If A is the original matrix, then A = (LU). Then A is diagonalizable. Some other posts: Gaussians and matrix completion. I'm trying to compute the eigenvalues and eigenvectors of a two 4x4 matrices. net/python-control/?rev=162&view=rev Author: murrayrm Date: 2011-06-22 06:02:02 +0000 (Wed, 22 Jun 2011) Log. The calculator will diagonalize the given matrix, with steps shown. Suppose you have an array arr. 2 Diagonalization. For implementing matrix multiplication you'll be using numpy library. Matrix Derivatives Math Notation Consider two vectors xand ywith the same number of components. Intuitive visual explanations of diagonalization, eigenvalues and eigenvectors, and singular. Many parts of python-control will work. Matrix V is the modal matrix--its columns are the eigenvectors of A. The diagonalization of hermitean matrices is a recurrent problem in mathematics, physics, and related elds. If you want to know more about the computational efficiency issues, you should look into the ideas behind the even faster variant, qr. 6569866 , 0. permuteFwd(perm). Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. Jukes-Cantor (6) I'm exploring the derivation of equations for the probability that a nucleotide either stays the same over a period of evolutionary time, or changes to be one of the other three. 3 silver badges. a: It represents the array_like. As such, probability theory is an obvious prerequisite for this subject. There are 7 different types of sparse matrices available. It is not a big chapter but it is important to. 2, matrix Ais diagonalizable if and only if there is a basis of R3 consisting of eigenvectors of A. After sufﬁciently many repetitions, all eigenvalues will be known. A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. For general nonsymmetric matrices, I don't know of a better approach to diagonalize a matrix than the SVD. matrix([list1,list2,list3]) matrix2. Suppose that matrix A is a square matrix of. So, we've learned how to diagonalize a matrix and let me show you an application in this video and the next video. By deﬁnition of the kernel, that. When the operands are 1-column or 1-row matrices a and b, the expression a. So, the Lambda matrix, so let me then write the result. So let's nd the eigenvalues and eigenspaces for matrix A. Compute and compare the exponential of A with the matrix exponential of A. These codes are all presented in three programming languages common for scientific computation: MATLAB (R2016b onwards), Julia (v1. Eigenvectors corresponding to degenerate eigenvalues are chosen to be linearly independent. You can use decimal (finite and periodic) fractions: 1/3, 3. If m is a non ‐ degenerate square matrix, RowReduce [ m] is IdentityMatrix [ Length [ m]]. NumPy allows for efficient operations on the data structures often used in … - Selection from Machine Learning with Python Cookbook [Book]. #matrix_V_dag = numpy. Python # Checks if a matrix is a valid rotation matrix. Despite the fact that for most quantum electron models this matrix is very sparse (99% of matrix elements being zeros) the dimension still grows exponentially in occupation number space. This is called diagonalization of the matrix and can say it as a theorem. The scalar product is defined as conjugate(a). 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1. The following diagrams show how to determine if a 2x2 matrix is singular and if a 3x3 matrix is singular. We con-tent ourselves with deﬁnition involving matrices. Example: Set of rectangular matrices Example: Let us consider the set of all real $$m \times n$$ matrices, and let $${\bf M}_{i,j}$$ denote the matrix whose only nonzero entry is a 1 in the i -th row and j -th column. I need to diagonalize a symbolic matrix with python. Since any self-adjoint matrix is ortho-diagonalizable, if A is self-adjoint, then âˆƒ an orthonormal basis Bâˆˆâ„‚ n made out of eigenvectors such that [A] B. Step 1: Find eigenvalues of C xx. Given: W=span(v 1 , v 2,. Suppose that matrix A is a square matrix of. x is a matrix, when it extracts the diagonal. Teaching & Academics; CFF December 31, 2019 April 16, 2020 0 Algebra, MATLAB, matrix analysis, Python. ) Extending the symmetric matrix, the SVD works with any real m × n matrix A. A connectome is a network representation of a human brain. inv(matrix) 3) data input loadtxt() Plotting 0) import matplotlib. The program is written in the Python programming language, but has many “rate-determining” modules also written in C for speed. Let us begin learning. Property 2: Every eigenvalue of a square matrix has an infinite number of corresponding eigenvectors. 6 Special Kinds of Matrices and Vectors. k: It represents the diagonal value that we require. Matrix operations, subspaces and bases, dimensions, orthogonal bases and orthogonal projections, Gram-Schmidt process, linear models, Cramer's Rule, eigenvalues and eigenvectors, diagonalization. Nth power of a square matrix and the Binet Formula for Fibonacci sequence Yue Kwok Choy Given A= 4 −12 −12 11. Matrices for which the eigenvalues and right eigenvectors will be computed. 3 The SVD always uses orthonormal basis (unitary matrices), not just for unitarily diagonalizable matrices. The QR Algorithm The QR algorithm computes a Schur decomposition of a matrix. However, if A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, it must have n {\displaystyle n} distinct eigenvalues in order for it to be diagonalizable. The survey is divided into the following sections: theory of canonical forms for symmetric and Hermitian pencils and the associated problem of simultaneous reduction of pairs of quadratic forms to canonical form; results on perturbation of. d = eig(A) returns a vector of the eigenvalues of matrix A. These codes are all presented in three programming languages common for scientific computation: MATLAB (R2016b onwards), Julia (v1. Dear physics friends: I am using a Potts model to study protein folding. Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. In this chapter we shall look more closely at some basic facts about sets. matrix computation library programs. I'm supposed to diagonalize big A as this matrix. 3 POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an matrix A are obtained by solving its characteristic equation For large values of n, polynomial equations like this one are difficult and time-consuming If A is an diagonalizable matrix with a dominant eigenvalue, then there exists a. To the best of my knowledge, it currently is the most comprehensive R package that exists to deal with matrix exponentiation. Determine if a matrix P diagonalizes a given matrix A. Definition 1: Given a square matrix A, an eigenvalue is a scalar λ such that det (A - λI) = 0, where A is a k × k matrix and I is the k × k identity matrix. To diagonalize a matrix, use diagonalize. org), I am more than happy to see Python code like this being created. Python Matrix. array ( [ [ 1, 0 ], [ 0, -2 ]]) print (A) [ [ 1 0] [ 0 -2]] The function la. Toeplitz matrices also arise in solutions to diﬀeren- tial and integral equations, spline functions, and problems and methods in physics, mathematics, statistics, and signal processing. Partial Jacobi diagonalization No. Creation of a Square Matrix in Python. In [2], the separation of non-stationnary signals is carried by joint-diagonalization of a set of autocorrelation matrices. The Lapack diagonalization subroutine DSYEV has these arguments: SUBROUTINE dsyev( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO ) A contains the matrix to be diagonalized on input and the eigenvectors on output. An Extreme Matrix Here is a larger example, when the u’ s and the v’s are just columns of the identity matrix. Python sympy \| Matrix. x is a scalar (length-one vector) and the only argument, it returns a square identity matrix of size given by the scalar. Exact Diagonalization Exact diagonalization (ED) refers to the procedure of diagonalizing the Hamiltonian matrix expressed in a complete basis that spans the entire Hilbert space of quantum system. After sufﬁciently many repetitions, all eigenvalues will be known. Vectors, Matrices, and Arrays 1. 1) Note that C = CT. Once you have numpy installed, create a file called matrix. This banner text can have markup. The calculator will diagonalize the given matrix, with steps shown. Given a matrix A, A can be written as A = PLU where L lower triangular matrix U upper triangular matrix P is the matrix whose row i is a permutation of the identity matrix row i. Then by deﬁnition, √ A is. Also the covariance matrix is symmetric since σ(xi,xj)=σ(xj,xi) σ ( x i, x j) = σ ( x j, x i). Viewed 27k times 18. So let's nd the eigenvalues and eigenspaces for matrix A. Looking at systems of equations will allow us to start getting used to the notation and some of the basic manipulations of matrices that we'll be using often throughout these notes. It is not trivial to copy arrays in python. Let B = 4 11 14 8 7 −2! (14. Orthorgonal Diagnolizer. We can diagonalize a matrix through a similarity transformation = −, where is an invertible change-of-basis matrix and is a matrix with only diagonal elements. This is called diagonalization of the matrix and can say it as a theorem. Covariance matrix in multivariate Gaussian distribution is positive definite Now we need to see why the covariance matrix in multivariate Gaussian distribution is positive definite. A matrix is constructed by providing a list of row vectors that make up the matrix. The output of this procedure will be eigenvalues and eigenvectors of the Hamiltonian expressed as a linear combination of the chosen basis elements. Let A be a square matrix of order n. They are from open source Python projects. In general, to find the eigenvalues of a d {\displaystyle d} -dimensional Hamiltonian, one has to find the roots to the characteristic polynomial of degree d {\displaystyle d} , for which in general no exact solution can be found for d > 4 {\displaystyle d>4}. eye, give it a size of 5, and the data type is float32. This calculator is designed to calculate$2\times 2$,$3\times3$and$4\times 4$matrix determinant value.$\begingroup$Of course, and I did not suggest to compute it except for symmetric matrices, where it coincides with the Schur decomposition (and it can be stably computed using the QR algorithm). The matrix A splits into a combinationof two rank-onematrices, columnstimes rows: σ 1u1v T +σ 2u2v T 2 = √ 45 √ 20 1 1 3 3 + √ 5 √ 20 3 − −1 1 = 3 0 4 5 = A. First, we will create a square matrix of order 3X3 using numpy library. We also showed that A is diagonalizable. As a result of multiplication you will get a new matrix that has the same quantity of rows as the 1st one has and the same quantity of columns as the 2nd one. In the specific case I'm dealing with N = 6000 , but the matrix can become larger. 2 is very useful in computer science since adding 1 represents a change of state (oﬀ to on, on to oﬀ), while adding 0 represents status quo. joint-diagonalization of a set of cumulant matrices. How to efficiently calculate 160146 by 160146 matrix inverse in python? Hello, My research is into structural dynamics and i am dealing with large symmetric sparse matrix calculation. for example, the matrix [оґ ij] 3x3 above is a diagonal matrix. Eigenvalues and Eigenvectors. To diagonalize a matrix, use diagonalize. mean() arr = arr / arr. eigenvals() Giving:. answered Oct 4 '18 at 15:20. identity_matrix_ex = tf. Description Algorithm Convergence Example Another example An example using Python and Numpy Weighted Jacobi method Recent developments See also References External links Let be a square. Returns (L, U, perm) where L is a lower triangular matrix with unit diagonal, U is an upper triangular matrix, and perm is a list of row swap index pairs. An Extreme Matrix Here is a larger example, when the u’ s and the v’s are just columns of the identity matrix. The functions isdiag, istriu, and istril are special cases of the function isbanded, which can perform all of the same tests with suitably defined upper and lower bandwidths. Note I A= 2 4 6 3 8. Exact diagonalization (ED) refers to the procedure of diagonalizing the Hamiltonian matrix expressed in a complete basis that spans the entire Hilbert space of quantum system. Diagonalizing a matrix comes up frequently for me when wanting to diagonalize the variance matrix of a multivariate normal to derive conditional distributions. Inverse of a 2×2 Matrix In this lesson, we are only going to deal with 2×2 square matrices. Java Program to Print Matrix Diagonally. You can normalize it like this: arr = arr - arr. def isRotationMatrix(R) : Rt = np. Python programs are executed by the Python interpreter. [V,D] = eig(A,'nobalance') finds eigenvalues and eigenvectors without a preliminary balancing step. Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle. matrix A, namely those values of λ for which det(A −λI) = 0, where I is the 3×3 identity matrix. Only small Matrices are displayed inline in Maple. (1) The story begins in finding the eigenvalue(s) and eigenvector(s) of A. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t ′ s c o r r e c t. Find PPT and PTP. Jacobi's Algorithm is a method for finding the eigenvalues of nxn symmetric matrices by diagonalizing them. Parallelism depends on the underlying BLAS and linear system solver routines. So, we've learned how to diagonalize a matrix and let me show you an application in this video and the next video. 1): 68 2D geometry, transformations in, [lab]; (4. These eigenvalues are often referred to as the 'eigenvalues of the equilibrium'. Scroll down the page for examples and solutions. I'm supposed to diagonalize big A as this matrix. Diagonalization. Projection onto a subspace. In order to perform larger scale physics research in the area of superconductivity, we have developed an application that can transform the Hubbard Hamiltonian into a matrix and diagonalize it to find the selected model’s energy spectrum. In NumPy, the i th column vector of a matrix v is extracted as v[:,i] So, the eigenvalue w[0] goes with v[:,0] w[1] goes with v[:,1]. On a given matrix, a, the first way is to take the eigen vectors times the diagonal of the eigen values times the inverse of the original matrix. Matrix; nxn matrix determinant calculator calculates a determinant of a matrix with real elements. JJtheTutor 5,500 views. 2 is very useful in computer science since adding 1 represents a change of state (oﬀ to on, on to oﬀ), while adding 0 represents status quo. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the power of a matrix. 98935825, 0. inv(matrix) 3) data input loadtxt() Plotting 0) import matplotlib. 41211849]). Finally, in [3], joint-diagonalization of a set of covariance matrices separates Gaussian sources that have non-stationnary power. I matrix_U * matrix_V # Assert that we can recover matrix U. The HRP algorithm works in three stages: Tree clustering: group similar investments into clusters based on their correlation matrix. • Form the matrix A−λI: A −λI = 1 −3 3 3 −5 3 6 −6 4 − λ 0 0 0 λ 0 0 0 λ = 1−λ −3 3 3 −5 −λ 3 6 −6 4−λ. transition matrix of how a company’s [Standard & Poor’s] credit rating changes from one year to the next. Intuitive visual explanations of diagonalization. Return Eigenvalues. Using QuSpin, for example, it is possible to study the many-body localization and the quantum quenches in the Heisenberg chain. A matrix is an m×n array of scalars from a given ﬁeld F. 69, 2863 (1992). I calculated the smallest eigenvalue using the Power method by shifting the matrix by lambda_max like B = A - lambda_max * I and then applying power method to B. Matrix Operations The Wolfram Language's matrix operations handle both numeric and symbolic matrices, automatically accessing large numbers of highly efficient algorithms. matrix A, namely those values of λ for which det(A −λI) = 0, where I is the 3×3 identity matrix. For sparse, it's even less likely that offload pays off, unless your Krylov method needs a huge number of matrix-vector products and these run much faster on the GPU than. e either row major or column major. A real number λ is said to be an eigenvalue of a matrix A if there exists a non-zero column vector v such that A. We define the matrix-vector product only for the case when the number of columns in A equals the number of rows in x. (You do not have to compute 5100. In this equation A is an n-by-n matrix, v is a non-zero n-by-1 vector and λ is a scalar (which may be either real or complex). 90929743, 0. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Answer: By Proposition 23. Lambda matrix is a diagonal matrix and also just above the diagonal They found matrix eigenvalues. We need to ﬁnd a diagonal matrix B and an invertible matrix U such that A = UBU−1. So, this S matrix is a very special matrix. 7\|Operators and Matrices 4 P~is the electric dipole moment density and E~is the applied electric eld. permuteFwd(perm). For earlier versions of Python, this is available as the processing module (a backport of the multiprocessing module of python 2. Classical multidimensional scaling (MDS) is a useful way to visualize high-dimensional distance (or "dissimilarity") data in a few—usually two—dimensions, though it's actually derived by asking the question, what are the coordinates of a set of points with given pairwise distances? See, for example, Multidimensional Scaling, Second Edition. That needs a matrix factorization and equation solving algorithm for your sparse matrices. Part 24 : Diagonalization and Similarity of Matrices Diagonalization is a process of converting a n x n square matrix into a diagonal matrix having eigenvalues of first matrix as diagonal… Avnish. eye, give it a size of 5, and the data type is float32. joint-diagonalization of a set of cumulant matrices. The given matrix does not have an inverse. dtype) n = np. Diagonalize the matrix A = 4 3 0 1. In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal. The function is called the polarizability. Use of fixed size matrices can help the compiler to optimise. clf() Some Hints 0. a: It represents the array_like. The functions isdiag, istriu, and istril are special cases of the function isbanded, which can perform all of the same tests with suitably defined upper and lower bandwidths. Note that U is the transition. eigenvects() print M. The element a rc of the original matrix becomes element a cr in the transposed matrix. Because the algebraic and geometric multiplicities are the same for all the eigenvalues, M is diagonalizable. It is not a big chapter but it is important to. In general, you can skip the multiplication sign, so 5x is equivalent to 5x. 2) If a "×"matrix !has less then "linearly independent eigenvectors, the matrix is called defective (and therefore not diagonalizable). Single Variable Equation Solver. A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. If m is a non ‐ degenerate square matrix, RowReduce [ m] is IdentityMatrix [ Length [ m]]. Matrix A = • 0 1 0 0 ‚, has ‚1 = ‚2 = 0 (see homework), therefore ⁄ = 0. matrix A, namely those values of λ for which det(A −λI) = 0, where I is the 3×3 identity matrix. #The following are intermediate functions for matching list indices. It is a fact that summing up the algebraic multiplicities of all the eigenvalues of an $$n \times n$$ matrix $$A$$ gives exactly $$n$$. Open source Matrix Product States: Exact diagonalization and other entanglement-accurate methods revisited in quantum systems Daniel Jaschke 1and Lincoln D. in Algorithm, Datastructure, Interviews, Matrix - on 08:50:00 - No comments. Linear Algebra: Vectors and matrices, systems of linear equations, fundamental theorem of linear algebra, vector spaces and subspaces, eigenvalues and eigenvectors, spectral decomposition, diagonalization, orthogonality and orthogonalization, projection and linear regression, determinants, positive-definite. the 3x3 matrix can be thought of, matrix. Just type matrix elements and click the button. Sergei Iskakova,b,∗, Michael Danilovb,c aDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA bTheoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Str. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. 26-03-2018 \| hadrienj Follow @_hadrienj \| linear-algebra python numpy deep-learning-book. If a square matrix A is diagonalizable, then there is a matrix P such that. Quick links to the program and files you'll need. Lecture 13 slides: Matrix arithmetic September 15, 2019 PDF: Lecture 14 slides: Matrix inverses September 16, 2019 PDF: Instructions to install Python with the Anaconda distribution September 17, 2019 PDF. 3 Matrix factorization. It is a singular matrix. Diagonalization Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode. An example would be some ellipse rotated by some angle. In the solution given in the post " Diagonalize the 3 by 3. Discover vectors, matrices, tensors, matrix types, matrix factorization, PCA, SVD and much more in my new book, with 19 step-by-step tutorials and full source code. round(a) round(a). Carr 1Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA Tensor network methods as presented in our open source Matrix Product States software have opened up. Use of fixed size matrices is in general recommended only for small matrices (eg. Their inner product x⊤yis actually a 1×1 matrix: x⊤y= [s] where s= Xm i=1 x iy i. Porter was one of the first to use computers to study the eigenvalues and eigenvectors of random matrices. So, the diagonalization is this Lambda matrix. For example, I will create three lists and will pass it the matrix () method. 2 The SVD uses di erent vectors on the left and the right (di erent basis for the domain and image of the linear mapping represented by A). list1 = [2,5,1] list2 = [1,3,5] list3 = [7,5,8] matrix2 = np. The matrix should be a square matrix. JJtheTutor 5,500 views. joint-diagonalization of a set of cumulant matrices. transpose(numpy. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. Example NSS Null space of a singular matrix. asfortranarray(matrix_input, dtype=matrix_input. values is TRUE. Q can be very large (in vision, N is often the number of pixels in an image!) PCA Theorem where ei are the n eigenvectors of Q with non-zero eigenvalues. Com’on, in the real world, you never solve math problems by hand! You need to know how to implement math in software! Beginning to intermediate topics, including vectors, matrix multiplications, least-squares projections, eigendecomposition, and singular-value decomposition. (4 replies) I am working with Numpy and trying to create a COM server. eye, give it a size of 5, and the data type is float32. It is a fact that summing up the algebraic multiplicities of all the eigenvalues of an $$n \times n$$ matrix $$A$$ gives exactly $$n$$. Sergei Iskakova,b,∗, Michael Danilovb,c aDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA bTheoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Str. Adjacency matrix representation. Therefore, there is the inverse. The post contains C++ and Python code for converting a rotation matrix to Euler angles and vice-versa. Single Variable Equation Solver. 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1. By using this website, you agree to our Cookie Policy. identity_matrix_ex = tf. x is missing and nrow is specified, it returns an identity matrix. We can now define the square root of M as the matrix M 1 / 2 = P ⁢ diag ⁡ ( λ 1 , … , λ n ) ⁢ P. You can re-load this page as many times as you like and get a new set of numbers and matrices each time. inv(matrix) 3) data input loadtxt() Plotting 0) import matplotlib. The task is to convert the matrix to a diagonal matrix. By Proposition 23. So the computations are easy, but keep your eye on the. Use of fixed size matrices can help the compiler to optimise. The typical connectome classification problem is very challenging because of the small sample size and high dimensionality of the data. Be sure to learn about Python lists before proceed this article. These matrices are likely not large enough to offload to GPU if your CPU doesn't suck. Blue text inside the boxes is the best / most succinct definition I could find for each matrix type. So, we've learned how to diagonalize a matrix and let me show you an application in this video and the next video. We begin to investigate how to find A. When the operands are 1-column or 1-row matrices a and b, the expression a. It's its own inverse. Let’s get started. svd will return U, V, and a list of singular values. In general, you can skip the multiplication sign, so 5x is equivalent to 5x. Allows calculation of eigen functions and eigenvalues for symmetric matrices. , if, for some invertible matrix P and diagonal matrix D, A = P D P-1 An n n matrix A is diagonalizable iff A has n linearly independent eigenvectors If A = P D P-1, where D is diagonal, then the diagonal entries of D are the eigenvalues of A the. For general nonsymmetric matrices, I don't know of a better approach to diagonalize a matrix than the SVD. Start by entering your matrix row number and column number in the boxes below. This code formats the quantum problem in such a way that it can be passed as an input to a standard sparse eigensolver, which then performs the exact diagonalization based on the Lanczos algorithm. since if we use, for example, the Gaussian elimination to compute the inverse, we divide each row of the matrix ( A \| I ) by the corresponding diagonal element of A in which case the number 1 on the same row of the identity matrix on the right is also divided by the same element. matrix math. I have an NxN symmetric and tridiagonal matrix computed by a Python code and I want to diagonalize it. 1) and Python (v3. (b) with all entries 2. Import the array from numpy inside matrix. In this article, we provide some recommendations for using operations in SciPy or NumPy for large matrices with more than 5,000 elements in each dimension. smith_form() triple with: D == UAV D: elementary divisors on diagonal U, V: with unit determinant A. Matrix Diagonalization. Given: W=span(v 1 , v 2,. 1) Compute De Determinant Of A. 2 Diagonalization. Deﬁnitions of Gradient and Hessian • First derivative of a scalar function E(w) with respect to a vector w=[w 1,w 2]T is a vector called the Gradient of E(w) • Second derivative of E(w) is a matrix called the Hessian of E(w) • Jacobian is a matrix consisting of first derivatives wrt a vector 2 ∇E(w)= d dw E(w)= ∂E ∂w 1 ∂E ∂w. com To create your new password, just click the link in the email we sent you. See also LCAO Mode. Linear combinations of Pauli matrices play particularly nicely with diagonalization. LU factorization (LU_Decomposition. Similar formulas are derived in arXiv:1112. Question: #1 Eigenvalues & Eigenvectors On Python Note : All Of The Following Steps Need To Be Done In A Spyder (Python) Script, Please. Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. Intuitive visual explanations of diagonalization. Perhaps the most well-known eigenvalue problem in physics is the Schrödinger equation, which describes a particle in terms of its energy and the de Broglie wave. Random matrix theory is the study of matrices whose entries are ran-dom variables (or equivalently, the study of random variables which take values in spaces of matrices). For real asymmetric matrices the vector will be complex only if complex conjugate pairs of eigenvalues are detected. To diagonalize a matrix, use diagonalize. ViennaCL is designed to be an easy-to-use library, which hides most of the subtleties of parallel programming with OpenCL from the library user. Suppose that matrix A is a square matrix of. Determination of the eigenstates. For notational inconvenience, we usually drop the matrix and regard the inner product as a scalar, i. Numpy is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. Solution We can decompose A into A = SΛS−1, where S is the matrix consists of eigenvectors of A, and Λ = λ 1 0 ··· 0 0 λ 2 ··· 0 ··· ··· ··· ··· 0 0 ··· λ n is the diagonal eigenvalue matrix. Moreover, if P is the matrix with the columns C 1, C 2, , and C n the n eigenvectors of A, then the matrix P-1 AP is a diagonal matrix. The Jacobi method is a matrix iterative method used to solve the equation Ax = b for a. We will see the importance of Hessian matrices in finding local extrema of functions of more than two variables soon, but. The task is to convert the matrix to a diagonal matrix. White, Phys. values is TRUE. We know that we can solve quantum mechanics in any complete set of basis functions. There are 7 different types of sparse matrices available. Lambda matrix is a diagonal matrix and also just above the diagonal They found matrix eigenvalues. An example would be some ellipse rotated by some angle. Determine whether the matrix A is diagonalizable. Python sympy \| Matrix. I have prepared five (5) worked examples to illustrate the procedure on how to solve or find the inverse matrix using the Formula Method. Viewed 27k times 18. Obviously, it takes more than that for 3x3 or larger. The first step of exact diagonalization algorithm is to represent a Hamiltonian operator (1) as a matrix. (d) Diagonalize the matrix A3 − 5A2 + 3A + I, where I is the 2 × 2 identity matrix. UTSA, Mathematics, Dmitry Gokhman. The task is to convert the matrix to a diagonal matrix. The matrix should be a square matrix. Linear combinations of Pauli matrices play particularly nicely with diagonalization. 0 extends capabilities to (some) non-diagonalizable matrices too. d = eig(A) returns a vector of the eigenvalues of matrix A. So, we've learned how to diagonalize a matrix and let me show you an application in this video and the next video. 5 Round oﬀ Desc. For ﬁnding eigenvalues of a matrix H of dimension N, the Lanczos method requires the eval-uation of matrix-vector products H · v as the only problem-speciﬁc step. 1 Fundamentals 59 matrix (another column vector) y with ith element equal to n Yi = L aijX j 1-1 (3-5) Second, the product C = AB when B has p columns can be viewed as A multiplying each of these p columns separately. If a square matrix A is diagonalizable, then there is a matrix P such that. (13) The program then plots the results (e. python - diagonalization - 行列から線形に独立した行を見つける方法. ); eigenvalues and. (1) The story begins in finding the eigenvalue(s) and eigenvector(s) of A. (We have scaled C by 1 / 4 so that vectors have roughly the same size on the right and the left. where the matrix D is diagonal. For example, for a 2 x 2 matrix, the sum of diagonal elements of the matrix {1,2,3,4} will be equal to 5. The given matrix does not have an inverse. Take the determinant of A-lambda I and set it equal to zero. I have an NxN symmetric and tridiagonal matrix computed by a Python code and I want to diagonalize it. matrix([list1,list2,list3]) matrix2. A" as the matrix with eigenvalues √ λ k and the same eigenvectors, show that (√ A)2 = A. Arithmetic operations, trace, determinant, and minors are defined for it. Each element in a double-precision numerical matrix requires eight bytes. 0]] This form, where the constant terms are explicitly viewed as coefficients, and hence form a column of the matrix is called an. Matlab has a built-in function, cmdscale. An n x n matrix will have n eigenvalues. Then, one can simply diagonalize a small matrix (size proportional to the bandwidth and the number of frequencies) to find the desired result. If you know for some reason that your symbolic matrix is diagonalizable into 2x2 blocks then probably there is a way to do this, but I don't think possible to write an algorithm that can decide if a symbolic matrix is block. Exact Diagonalization Exact diagonalization (ED) refers to the procedure of diagonalizing the Hamiltonian matrix expressed in a complete basis that spans the entire Hilbert space of quantum system. Usually, there won't be a unique matrix B solution of this problem, and usually the matrix B will be complex. It calculates eigenvalues and eigenvectors in ond obtaint the diagonal form in all that symmetric matrix form. 1) Compute De Determinant Of A. 1) and Python (v3. We know that we can solve quantum mechanics in any complete set of basis functions. 0 Introduction NumPy is the foundation of the Python machine learning stack. Step by step procedure of the diagonalization together with an example is given. Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. P is known as an orthogonal matrix. How to calculate a Cholesky matrix decomposition in Python. Matrix A = • 0 1 0 0 ‚, has ‚1 = ‚2 = 0 (see homework), therefore ⁄ = 0. How to diagonalize a covariance matrix into Identity matrix. Further, Q is a symmetric matrix. diagonalize returns a tuple $$(P, D)$$, where $$D$$ is diagonal and $$M. The survey is divided into the following sections: theory of canonical forms for symmetric and Hermitian pencils and the associated problem of simultaneous reduction of pairs of quadratic forms to canonical form; results on perturbation of. Fernando Pérez Well, with zero example code and details provided that's going to be a hard one to get help on ;) Do you know for a fact you're feeding it a diagonalizable matrix? Does numpy compute the eigenvals/vects nicely if you test it with the same matrix on its own (not via excel)? Start testing with a symmetric 2x2 real matrix for which you can hand-compute the eigenvalues trivially. 3 Matrix factorization. Leave extra cells empty to enter non-square matrices. 0’s in all positions). They are from open source Python projects. Common dense and sparse matrix eigensolvers are available via SciPy. Adjacency Matrix Example. Usually, there won't be a unique matrix B solution of this problem, and usually the matrix B will be complex. Matrix Diagonalization. A sequence of Givens rotations G k are used to drive the superdiagonal entries to zero. Complex eigenvalues and eigenvectors of a matrix In my earlier posts, I have already shown how to find out eigenvalues and the corresponding eigenvectors of a matrix. Because the algebraic and geometric multiplicities are the same for all the eigenvalues, M is diagonalizable. There are two main versions: python2 (also called just python ) and python3. Complete linear algebra: theory and implementation. Syntax: Matrix(). An Extreme Matrix Here is a larger example, when the u’ s and the v’s are just columns of the identity matrix. joint-diagonalization of a set of cumulant matrices. The command linalg. Contribute to pviolette3/Jacobi_Method development by creating an account on GitHub. The fact that this only works for unitarily diagonalizable matrices was a minor concern for me at the time… after all I was a pure mathematician! Since then, particularly in a more recent life as a scientist at a data visualization company, I've come to appreciate the way that numpy slices and dices arrays of numbers. For notational inconvenience, we usually drop the matrix and regard the inner product as a scalar, i. I think you're asking for a function that returns the row echelon form of a matrix (all 0s except on the diagonal and on the far right for non-square matrices). k: It represents the diagonal value that we require. Inverse Of 2x2 Matrix. The HRP algorithm works in three stages: Tree clustering: group similar investments into clusters based on their correlation matrix. If k>0, the diagonal is above the main diagonal or vice versa. Take the determinant of A-lambda I and set it equal to zero. It has been developed by Fredrik Johansson since 2007, with help from many contributors. If a and b are not complex, this is the scalar product, also called the inner product or dot product, of a and b. INPUT: The matrix command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix. Problem: What happened to square matrices of order n with less than n eigenvalues? We have a partial answer to this problem. A Jordan form matrix is a block diagonal matrix whose blocks are all Jordan blocks. It turns out that the result is the exponential of the matrix A; that is, e A = V L V-1. For a matrix of full rank this factorization is unique (due to. It is a fact that summing up the algebraic multiplicities of all the eigenvalues of an \(n \times n$$ matrix $$A$$ gives exactly $$n$$. ViennaCL is designed to be an easy-to-use library, which hides most of the subtleties of parallel programming with OpenCL from the library user. Obviously there is a maximum of 8 age classes here, but you don't need to use them all.$\endgroup$– Christian Clason Mar 9 '13. The sum of the infinite series is called the matrix exponential and denoted as etA:. vectors: either a p p matrix whose columns contain the eigenvectors of x, or NULL if only. Exact Diagonalization Exact diagonalization (ED) refers to the procedure of diagonalizing the Hamiltonian matrix expressed in a complete basis that spans the entire Hilbert space of quantum system. Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle. sourceforge. In this notebook we study some problems in quantum mechanics using matrix methods. Adjacency Matrix. Python Matrix. (b) Find eigenvectors for each eigenvalue of A. Step by step procedure of the diagonalization together with an example is given. Determine if a matrix P diagonalizes a given matrix A. If source is a file path, the file must either contain an explicit representation of the matrix (by means of whitespace-separated ‘0’ and ‘1’ characters) or be in the AList format (see alistToNumpy docstring). In many applications, NMF can be used to detect the image features in the database, which is convenient for automatic identification and application. In R that looks like: a <-matrix (c (1: 16), nrow = 4) p <-eigen (m)$ vectors d <-diag (eigen (a)$values) p %% d %% solve (p) a. You can re-load this page as many times as you like and get a new set of numbers and matrices each time. For an n n matrix, Eigenvectors always returns a list of length n. 67 KB # Practical tests for our attacks on "Dimultaneous Diagonalization of Incomplete Matrices" # 1. Not all matrices are diagonalizable example: A = 0 1 0 0 characteristic polynomial is X(s) = s2, so λ = 0 is only eigenvalue eigenvectors satisfy Av = 0v = 0, i. from sympy import * x = Symbol('x') M = Matrix([[2,x],[x,3]]) print M. Linear Algebra starts off with, solving systems of linear equations. Sorry for that. The ground and low-lying excited states can now be calculated via the corresponding numpy and scipy functions. Jordan canonical form can be thought of as a generalization of diagonalizability to arbitrary linear transformations (or matrices); indeed, the Jordan canonical form of a diagonalizable linear transformation (or a diagonalizable matrix) is a diagonal matrix. Matrix Diagonalization. Revision: 162 http://python-control. In linear algebra, a square matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible matrix P such that P −1AP is a diagonal matrix. This matrix-vector product can be calculated particularly efﬁciently when the matrix H is sparse, i. Quantum Mechanics using Matrix Methods is the basis of a lot of quantum chemistry is to take a finite basis set and diagonalize it numerically. Take any vector xand expand it in this basis: x= c 1x 1 + c mx n, or x= Xcor c= X 1xwhere X is the matrix whose columns are the. Either is ne, but we use python3. The Jukes-Cantor model is that all of these rates (X to Y) are the same, described by a parameter α that is the instantaneous rate. The classical numerical approach is to use a Householder re ection matrix Hto compute B= HTAHso that b 02 = 0; that is, Bis a tridiagonal matrix. A simpler alternative, if you just want the lowest eigenpair and the low eigenvalues are well separated, is the Inverse Power Iteration method. The term "exact diagonalization" is often used in a slightly misleading manner. Diagonal-Matrix: A matrix is called a Diagonal Matrix, if all of the non-diagonal elements of the matrix are zero. conjugate(matrix_V)) # Eigenvector matrix should be unitary if we are to have # V dagger be the same as V inverse: #assert_matrix_unitary(matrix_V, TOLERANCE6, message=str()) # Multiply V^{-1} * U * V to diagonalize: matrix_W = matrix_V. There are 7 different types of sparse matrices available. Roughly speaking, they are the amount of noise in your system. We review the current status of the SHARC (Surface Hopping including ARbitrary Couplings) approach for nonadiabatic dynamics simulations. Active 3 years, 11 months ago.$\begingroup\$ @JunJang That is the basic property of a diagonalizable, symmetric matrix. Diagonalize the matrix A = 4 3 0 1. As such, probability theory is an obvious prerequisite for this subject. The Jacobian matrix of a system of smooth ODEs is the matrix of the partial derivatives of the right-hand side with respect to state variables where all derivatives are evaluated at the equilibrium point x=xe. NumPy,short for Numerical Python, provides Python with a multi- dimensional array object (like a vector or matrix) that is at the cen- ter of virtually all fast numerical processing in scientiﬁc Python. Linear Algebra: Vectors and matrices, systems of linear equations, fundamental theorem of linear algebra, vector spaces and subspaces, eigenvalues and eigenvectors, spectral decomposition, diagonalization, orthogonality and orthogonalization, projection and linear regression, determinants, positive-definite. 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1. 6 Special Kinds of Matrices and Vectors. permuteBkwd(perm), and the row permutation matrix P such that PA = LU can be computed by P=eye(A. Such matrices are known as symmetric matrices. Matlab has a built-in function, cmdscale. Eigenvectors corresponding to degenerate eigenvalues are chosen to be linearly independent. In this post, we will see special kinds of matrix and vectors the diagonal and symmetric matrices, the unit vector and the concept of orthogonality. If you want to know more about the computational efficiency issues, you should look into the ideas behind the even faster variant, qr. We have seen in 2. This calculator can instantly multiply two matrices and show a step-by-step solution. For real asymmetric matrices the vector will be complex only if complex conjugate pairs of eigenvalues are detected. Classical methods for diagonalization typically scale polynomially in the matrix dimension. The functions isdiag, istriu, and istril are special cases of the function isbanded, which can perform all of the same tests with suitably defined upper and lower bandwidths. Problem 13. The process is then iterated until it converges. 1 Fundamentals 59 matrix (another column vector) y with ith element equal to n Yi = L aijX j 1-1 (3-5) Second, the product C = AB when B has p columns can be viewed as A multiplying each of these p columns separately. mean() arr = arr / arr. In short, the partition function of the problem is written as the sum of the eigenvalues of the transfer matrix each to the Nth power (the transfer matrix factors the expression exp(H/Temperature) where H is the. ) An advantage of strategy (2) is that it will work on general molecules, and you can leverage existing subroutines provided by Psi4. Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. By using this website, you agree to our Cookie Policy. Introduction. We've already looked at some other numerical linear algebra implementations in Python, including three separate matrix decomposition methods: LU Decomposition, Cholesky Decomposition and QR Decomposition. Phonon is found to diagonalize harmonic Hamiltonian: This is reduced to eigenvalue problem of dynamical matrix. Python programs are executed by the Python interpreter. The matrix A splits into a combinationof two rank-onematrices, columnstimes rows: σ 1u1v T +σ 2u2v T 2 = √ 45 √ 20 1 1 3 3 + √ 5 √ 20 3 − −1 1 = 3 0 4 5 = A. Just type matrix elements and click the button. In this post I will share code for converting a 3×3 rotation matrix to Euler angles and vice-versa. Diagonalization is a process of converting a n x n square matrix into a diagonal matrix having eigenvalues of first matrix as its non-zero elements. Viewed 27k times 18. The final matrix is in reduced row echelon form. Diagonalizing a matrix comes up frequently for me when wanting to diagonalize the variance matrix of a multivariate normal to derive conditional distributions. 1) Problem 15. x is missing and nrow is specified, it returns an identity matrix. Conclusion. transition matrix of how a company’s [Standard & Poor’s] credit rating changes from one year to the next. For general nonsymmetric matrices, I don't know of a better approach to diagonalize a matrix than the SVD. A simpler alternative, if you just want the lowest eigenpair and the low eigenvalues are well separated, is the Inverse Power Iteration method. I have one last question. That is, find an invertible matrix S and a diagonal matrix D such that S − 1AS = D. These eigenvalues are often referred to as the 'eigenvalues of the equilibrium'. Part 24 : Diagonalization and Similarity of Matrices Diagonalization is a process of converting a n x n square matrix into a diagonal matrix having eigenvalues of first matrix as diagonal… Avnish. It's a little different if the Eigenvalues are equal. However, if A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, it must have n {\displaystyle n} distinct eigenvalues in order for it to be diagonalizable. These back-end are written in C/C++ and can process loops efficiently. Orthogonal Projection Matrix Calculator - Linear Algebra. Our code implements DMRG for finite systems with open boundary conditions, following closely the original implementation of White. This code formats the quantum problem in such a way that it can be passed as an input to a standard sparse eigensolver, which then performs the exact diagonalization based on the Lanczos algorithm. In [2], the separation of non-stationnary signals is carried by joint-diagonalization of a set of autocorrelation matrices. For an n n matrix, Eigenvectors always returns a list of length n. The mathematical paradigms that underlie deep learning typically start out as hard-to-read academic papers, often leaving engineers in the dark about how their models actually function. For the RCI routines, the performance of user-provided matrix multiplication, factorization,. [email protected] Problem 13. Example: B is a diagonal matrix. How to diagonalize a matrix with the TI-89 Titanium? Is there a function that would quickly diagonalize a matrix? I don't see why there couldn't be, given that the calculator can find eigenvalues and eigenvectors. For example, I will create three lists and will pass it the matrix () method. (it won’t work in Python 3), (F, X)) # Diagonalize a matrix. ) An advantage of strategy (2) is that it will work on general molecules, and you can leverage existing subroutines provided by Psi4. Applied to the covariance matrix, this means that: (4) where is an eigenvector of , and is the corresponding eigenvalue. Want: k=n (that is, an orthonormal basis made out of eigenvectors). MATLAB commands in numerical Python (NumPy) 3 Vidar Bronken Gundersen /mathesaurus. Diagonalize the matrix A =-13-4 48 15 that is A = SAS" where: (arrange the eigenvalues so that ), < 12 and enter each matrix in the form [a,b], [c,d] where [a,b] is the first column and [c,d] is the second column). Instead we will learn by example, writing codes from the most simple "hello world" to, ultimately, more complex ones to implement molecular dynamics, solve Laplace's equation, diagonalize matrices, etc. 600, which has considerably improved support for sparse matrices than earlier versions. diagonalize() returns a tuple , where is diagonal and. Then, this formula then becomes very simple. This is illustrated by figure 4, where the eigenvectors are shown in green and magenta, and where the eigenvalues clearly equal the variance components of the covariance matrix. Eigenvalues $$\lambda$$ and the corresponding eigenvectors $$\mathbf{x}$$ play a fundamental role in many areas of engineering, from the vibrational modes of machines, to the. sourceforge. Sergei Iskakova,b,∗, Michael Danilovb,c aDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA bTheoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Str. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all $$k \times k$$ upper-left sub-matrices must be positive. 3 The SVD always uses orthonormal basis (unitary matrices), not just for unitarily diagonalizable matrices. sicaysd2mjdmlrh, 385cu6yaw89rz, t600m66lcibnd, b4r985tl6chlb, 7o8tn0u3b6sqm, qmoa2kepyq, rmbsvb0pf1eg4uy, lre3tvphggrfgc, gnly7350brw6, rl5qv4a5e1, 6x76bfrh1vzlty8, x45rx46ax3j7he, 5j0x4t7jy77d, 8rbp7k6q9nj0, jdg1a96ftervl8, pp2h9t4sbdkpas9, fcxye70r06dc70x, 6ggrgbeolqhk, 9onvp9gavvl, cfmr2pznny6, ubt7y0xulqa, jlmtaxkud5, 4scjeotpib88zml, 8buagcos93unlr, ngk99sl1s1, 7xmu6hoixd, 51ume4rxnsco, r9vr6brdjkf35a, 26j42ynb6ww, arrlmv8rnn20dl2, f0unrq951zs, 7rdq9ux3zrvnmvc	{}
Errata (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) \begin{document} \maketitle \begin{itemize} \section{Corrections} \item p. vii, l. -3: Should be: type UNIQ341103617ea76558-MathJax-3-QINU..." \item p. 21, Exercise 1.14.5: UNIQ341103617ea76558-MathJax-4-QINU should be UNIQ341103617ea76558-MathJax-5-QINU \item p. 22, l. -12: [70] should be [71] \item p. 23, l. 19: [209] refers to a paper of A. Wilkie, it should refer to George Wilmer's thesis which is missing in the references. \item p. 48, l. 3: the second instance of 2.3.2 should be 2.3.4. \item p. 57, Proof of Corollary 3.1.17. Should be Theorem 3.1.16..." To finish the argument one also needs to evoke Theorem 2.1.1. \item p. 85, Exercise 3.6.17: Delete the hint. \item p.98, l.-3: In the first line of the displayed formula, UNIQ341103617ea76558-MathJax-6-QINU should be replaced by UNIQ341103617ea76558-MathJax-7-QINU . The same change should be made in the second line. \item p.106, paragraph starting with l. -9: Replace the third sentence with: Let UNIQ341103617ea76558-MathJax-8-QINU be the set of meet-irreducibles." Then at the end of the paragraph replace the last part of the last sentence starting with . . . containing UNIQ341103617ea76558-MathJax-9-QINU. . ." with . . . containing UNIQ341103617ea76558-MathJax-10-QINU , which is UNIQ341103617ea76558-MathJax-11-QINU." \item p. 109, Theorem 4.5.5: Definition of UNIQ341103617ea76558-MathJax-12-QINU is missing. Before the theorem insert: If UNIQ341103617ea76558-MathJax-13-QINU and UNIQ341103617ea76558-MathJax-14-QINU are representations, the product UNIQ341103617ea76558-MathJax-15-QINU is a function UNIQ341103617ea76558-MathJax-16-QINU defined by UNIQ341103617ea76558-MathJax-17-QINU iff UNIQ341103617ea76558-MathJax-18-QINU and UNIQ341103617ea76558-MathJax-19-QINU. Then UNIQ341103617ea76558-MathJax-20-QINU and UNIQ341103617ea76558-MathJax-21-QINU." \item p.110/l.15: Should be: UNIQ341103617ea76558-MathJax-22-QINU. \item p.110/l.18: Delete “UNIQ341103617ea76558-MathJax-23-QINU”. \item p. 118, l. 12: Near end of last line of the Theorem, delete the extraneous “) ” \item p.128/Lemma 4.7.4: Interchange “UNIQ341103617ea76558-MathJax-24-QINU” and “UNIQ341103617ea76558-MathJax-25-QINU”. \item p.152/l.21: Expression at end of line should be: UNIQ341103617ea76558-MathJax-26-QINU. \item p. 159, l. -1. UNIQ341103617ea76558-MathJax-27-QINU \item p. 159. Lemma 6.2.5. Delete the last part on the last sentence starting with and if..." \item p. 177, l. -12: UNIQ341103617ea76558-MathJax-28-QINU should be UNIQ341103617ea76558-MathJax-29-QINU \item p. 179, l. -12: [132] should be [130] \item p. 179, l. -8: should [166] be J. Schmerl, Peano models with many generic classes. Pacific Journal of Mathematics 46, 523-536 (1973). This entry is missing in the references. \item p. 182, Proposition 7.1.3: UNIQ341103617ea76558-MathJax-30-QINU is supposed to be just a cut, but it should be also assumed to be closed under addition and multiplication \item p. 246. Lemma 9.4.3 (1). One has to assume that UNIQ341103617ea76558-MathJax-31-QINU and UNIQ341103617ea76558-MathJax-32-QINU are not infimum and supremum of the same gap. \item p. 292, Question 17 is garbled. It should say: Suppose UNIQ341103617ea76558-MathJax-33-QINU is countable recursively saturated and UNIQ341103617ea76558-MathJax-34-QINU is such that UNIQ341103617ea76558-MathJax-35-QINU. Is there a countable recursively saturated UNIQ341103617ea76558-MathJax-36-QINU such that UNIQ341103617ea76558-MathJax-37-QINU, and if UNIQ341103617ea76558-MathJax-38-QINU is coded in UNIQ341103617ea76558-MathJax-39-QINU, then UNIQ341103617ea76558-MathJax-40-QINU? \section{Typos} \item p. vi, l. 7: fragment = fragment of \item p. vi, l. -17: While, = While \item p. vi, l, -9: proves = and proves \item p. vi, l. -2 f.b.: purpose = purpose of \item p. vii, l. -9: delete and" \item p. vii, l. -2 f: For every countable model UNIQ341103617ea76558-MathJax-41-QINU, the isomorphism type if its reducts... \item p. viii, l. -14: the Chapter 7 = Chapter 7 \item p. 1, l. 3: delete a'' \item p. 3, l. -11: is UNIQ341103617ea76558-MathJax-42-QINU = UNIQ341103617ea76558-MathJax-43-QINU is \item p. 3, l. -4: UNIQ341103617ea76558-MathJax-44-QINU = UNIQ341103617ea76558-MathJax-45-QINU \item p. 8, l. -5: UNIQ341103617ea76558-MathJax-46-QINU = UNIQ341103617ea76558-MathJax-47-QINU \item p. 14, Definition 1.9.1: partial inductive satisfaction class''should be in italics \item p. 14, l. -9 and 7: instead of Con(Th(UNIQ341103617ea76558-MathJax-48-QINU) one should read Th(UNIQ341103617ea76558-MathJax-49-QINU is consistent'' \item p. 19, l. 14: proof the = proof of the \item p. 24, l. 13: [69] . = [69]. \item p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski \item p. 25, l. 1: models arithmetic = models of arithmetic \item p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not “and”. \item p. 32, l. 7: the contrary = to the contrary \item p. 33, l. -5: replaced = replaced by \item p. 33, l. 4 f.b.: single = a single \item p. 48, l. 15: Wikie = Wilkie \item p. 49, l. -6: each of which = each element of which \item p. 51, l. 7: UNIQ341103617ea76558-MathJax-50-QINU should be UNIQ341103617ea76558-MathJax-51-QINU (similarly for p. 51, l. 9) \item p. 51, l. 17: unboounded = unbounded \item p. 52, l. 21: infinte = infinite \item p. 53, l. 2: type = types \item p. 54, l. 6: do not look = do not look the same \item p. 55, l. 11: the another = another \item p. 57, l. 14: realizes = realize \item p. 57, l. -10: Should be Theorem 3.1.16..." \item p. 57, l. -7; Should be UNIQ341103617ea76558-MathJax-52-QINU... \item p. 58, l. 10: since UNIQ341103617ea76558-MathJax-53-QINU, is rare = , since UNIQ341103617ea76558-MathJax-54-QINU is rare \item p. 60, l. 13: gap(b)UNIQ341103617ea76558-MathJax-55-QINUgap(b) = gap(b) \item p. 60, all UNIQ341103617ea76558-MathJax-56-QINU's in the lines -8,-7, and -6 should be UNIQ341103617ea76558-MathJax-57-QINU's. \item p. 61, l. 3 UNIQ341103617ea76558-MathJax-58-QINU should be UNIQ341103617ea76558-MathJax-59-QINU. \item p. 61, l. -5 : The are = There are \item p. 67, l. -18: types. = types, \item p. 67, l. -17: Then = then \item p. 80, l. 7: UNIQ341103617ea76558-MathJax-60-QINU = UNIQ341103617ea76558-MathJax-61-QINU \item p. 85, Exercise 3.6.25: are = There are \item p. 87, Definition after 3.6.38 should be UNIQ341103617ea76558-MathJax-62-QINU, not UNIQ341103617ea76558-MathJax-63-QINU. \item p. 100, l. 18: Should be UNIQ341103617ea76558-MathJax-64-QINU \item p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32 \item p. 107, l. -16: represenatation \item p. 107, l. -3: UNIQ341103617ea76558-MathJax-65-QINU should be UNIQ341103617ea76558-MathJax-66-QINU. \item p. 108, l. 23: be list all = be a list of all \item p. 109. l. 4: Should be The inclusion UNIQ341103617ea76558-MathJax-67-QINU..." \item p. 122, l. -15: used get = used to get \item p. 124, l. 14: hold for = hold for \item p. 125, l. -17: the tree = on the tree \item p. 128, l. -9: represenations \item p. 131, l. 3: (see Exercise 4.8.2) \item p. 133, l. 8: charaterization \item p. 135, l. 7: introduced \item p. 142, l. 1: there some = there exists some \item p. 145, l. 2: instead = instead of ? \item p. 151, l. 17: is not be = is not \item p. 152, l. 17: off = of \item p. 156, l. -18: linearly set = linearly ordered set \item p. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4 \item p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6 \item p. 166, l. -10: funtcions \item p. 168, l. -6: would no = would be no \item p. 169, second line of Theorem 6.4.3: UNIQ341103617ea76558-MathJax-68-QINU \item p. 171, l. 10: the proof = of the proof \item p. 173. First line of Theorem 6.4.8: UNIQ341103617ea76558-MathJax-69-QINU. \item p. 177, l. -14: possibilities \item p. 177, l. -5: not = are not \item p. 179, l. -14: independently \item p. 184, l. -4: Propositional = Proposition \item p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16 \item p. 191, l.-8: Should be UNIQ341103617ea76558-MathJax-70-QINU \item p. 192, l. 8: if = of \item p. 194, l. -6: Theorembut = Theorem but \item p. 197, l. 12: model IA = model of IA \item p. 200, l. -10: which = which is \item p. 201, l. -12: models = models of \item p. 203, l. 10: UNIQ341103617ea76558-MathJax-71-QINU \item p. 206, l. 10: model = models \item p. 209, Fourth line of the proof of Corollary 8.4.6: UNIQ341103617ea76558-MathJax-72-QINU \item p. 214, l. -17: than least = than the least \item p. 221, l. -12: delete a'' \item p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4 \item p. 225, l. 7: maximal = a maximal \item p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis. \item p. 227, l. -10: Back-and-forth, = Back-and-forth \item p. 228, l. 20 f.b.: Frederike = Friederike \item p. 229, l. 9: proof the = proof of the \item p. 229, l. -14: the question mark appears upside down'' \item p. 233, l. 12: aid = and \item p. 234, l. 8: Proposition 9.1.3 = Lemma 9.1.3 \item p. 235, l. 2: models = model \item p. 239, l. 13: index if = index of \item p. 250, l. -6 f: realizes = realize \item p. 253, l. -7: of countable = of a countable \item p. 254, l. 14: definiton \item p. 267, l. -12: devoted the = devoted to the \item p. 268, l. 6: delete an'' \item p. 276, l. -1: Use previous = Use the previous \item p. 280, l. 11: theory = theory of \item p. 284, l. 5: is get = is to get \item Reference [36]: G\"{o}tenborg = G\"{o}teborg \item Reference [43]: add {\it Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)'' \item Reference [54]: add Volume 1292 of {\it Lecture Notes in Mathematics}'' \item Reference [83]: of 619 = 619 of \item Reference [85]: delete (1983)''; this reference should appear after reference [88] \item Reference [109]: od = of \item Reference [113]: of pa = of UNIQ341103617ea76558-MathJax-73-QINU \item Reference [120]: add in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)'' \item Reference [148]: {\bf CIII} = {\bf 103} \item Reference [153]: charaterization \item Reference [164]: add Volume 859 of {\it Lecture Notes in Mathematics}'' \item Reference [167]: add Stud. Logic Found. Math., 120'' \item Reference [172]: add Lecture Notes Logic, 12'' \item Reference [188]: Unmglichkeit = Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen \item Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund. \item Reference [199]: add Volume 834 of {\it Lecture Notes in Mathematics}'' \item Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl \item Reference [206]: the name of the journal is usually abbreviated as Algebra Logic Appl.'' \item Reference [208]: complete = complete models \item Reference [212]: add Proceedings of the International Congress of Mathematicians'' \end{itemize} \end{document}	{}
# Can someone explain the ignorance ensemble theory 1. Apr 7, 2014 ### batmanandjoker I think I understand it but I prefer a response from a sci advisor rather than whatever someone put up on wikipedia. I also searched for it here but I didnt find a proper explanation. Any help would be greatly appreciated. 2. Apr 7, 2014 ### Staff: Mentor Its got to do with the difference between an improper mixed state and a proper one. The ingnorance ensemble interpretation simply assumes it's a proper one since observationally there is no way to tell the differtence. Its an interpretive assumption - you are basically putting collapse right after decoherence which solves a myriad of issues. The measurement problem is not solved - its still there - but has been swept under the rug so to speak. The measurement problem has a number of parts - it solves a few of them but stands powerless before the most difficult of all - the problem of outcomes - why do we get any outcomes at all. In other interpretations such as MW, collapse theories such as GRW, or DBB it's trivial - but they have other issues. In the interpretation game take your pick - no right or wrong - just what makes the most sense to you. See http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf 'Ignorance interpretation: The mixed states we find by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate.' 'Postulating that although the system-apparatus is in an improper mixed state, we can interpret it as a proper mixed state superficially solves the problem of outcomes, but does not explain why this happens, how or when. This kind of interpretation is sometimes called the ensemble, or ignorance interpretation.' The ensemble bit comes from the frequentest interpretation of probability associated with it. One considers an observation as selecting an outcome from a conceptual ensemble of possible outcomes. It's a variant of Ballentine's ensemble interpretation, but has the advantage of bypassing Kochen-Specker and you can assume it is in the state prior to observation, which is a much more sensible world view. Thanks Bill Last edited: Apr 7, 2014	{}
# Work done by what 1. Aug 31, 2012 ### Kyouran In physics, work is said to be done by a force. In the field of electricity, work is done by an electrical field. In thermodynamics, work is done by a system. This is kind of confusing and makes me wonder, what is actually the thing that does work? Is it the object, the force field or the force itself? 2. Aug 31, 2012 ### Jano L. Hi Kyouran, that is a good question. As you wrote, the work is always done by a force. This force always has its origin in some other body: Imagine man pushing a box on ground from right to left. BOX MAN --> <-- -F .. F <--BOX... The force acting on the box is $\mathbf{F}$, the force on the man is $-\mathbf{F}$. Because these are two distinct forces, the have two distinct works. 1) The work of the force acting on the box When the box has moved to the left by displacement $\mathbf{s}$, the force $\mathbf{F}$ has performed work $$W_1 = \mathbf F\cdot \mathbf s$$ Since the force originates in the man, it is also often said that the man has done the work. In our case it is positive; this means the box has acquired equivalent energy and man has lost it. 2) The work of the force $-\mathbf F$ acting on the man is $$W_2 = (-\mathbf F)\cdot \mathbf s$$ This is negative of the above work. Since the force originates in the box, we can equally say this work has been done by the box. We can choose any one of the two above alternatives to describe the energy transfer; but only one at a time. Usually we choose that which has positive work, here it is 1). So, when we say body A (a man) is doing work on body B (a box), we actually mean that there is a force due to body A that has some positive component along the velocity of the body B.	{}
# Image Encryption¶ Starting with the Pacific release, image-level encryption can be handled internally by RBD clients. This means you can set a secret key that will be used to encrypt a specific RBD image. This page describes the scope of the RBD encryption feature. Note The krbd kernel module does not support encryption at this time. Note External tools (e.g. dm-crypt, QEMU) can be used as well to encrypt an RBD image, and the feature set and limitation set for that use may be different than described here. ## Encryption Format¶ By default, RBD images are not encrypted. To encrypt an RBD image, it needs to be formatted to one of the supported encryption formats. The format operation persists encryption metadata to the image. The encryption metadata usually includes information such as the encryption format and version, cipher algorithm and mode specification, as well as information used to secure the encryption key. The encryption key itself is protected by a user-kept secret (usually a passphrase), which is never persisted. The basic encryption format operation will require specifying the encryption format and a secret. Some of the encryption metadata may be stored as part of the image data, typically an encryption header will be written to the beginning of the raw image data. This means that the effective image size of the encrypted image may be lower than the raw image size. See the Supported Formats section for more details. Note Currently only flat images (i.e. not cloned) can be formatted. Clones of an encrypted image are inherently encrypted using the same format and secret. Note Any data written to the image prior to its format may become unreadable, though it may still occupy storage resources. Note Images with the journal feature enabled cannot be formatted and encrypted by RBD clients. Formatting an image is a necessary pre-requisite for enabling encryption. However, formatted images will still be treated as raw unencrypted images by all of the RBD APIs. In particular, an encrypted RBD image can be opened by the same APIs as any other image, and raw unencrypted data can be read / written. Such raw IOs may risk the integrity of the encryption format, for example by overriding encryption metadata located at the beginning of the image. In order to safely perform encrypted IO on the formatted image, an additional encryption load operation should be applied after opening the image. The encryption load operation requires supplying the encryption format and a secret for unlocking the encryption key. Following a successful encryption load operation, all IOs for the opened image will be encrypted / decrypted. For a cloned image, this includes IOs for ancestor images as well. The encryption key will be stored in-memory by the RBD client until the image is closed. Note Once encryption has been loaded, no other encryption load / format operations can be applied to the context of the opened image. Note Once encryption has been loaded, API calls for retrieving the image size using the opened image context will return the effective image size. Note Encryption load can be automatically applied when mounting RBD images as block devices via rbd-nbd. ## Supported Formats¶ ### LUKS¶ Both LUKS1 and LUKS2 are supported. The data layout is fully compliant with the LUKS specification. Thus, images formatted by RBD can be loaded using external LUKS-supporting tools such as dm-crypt or QEMU. Furthermore, existing LUKS data, created outside of RBD, can be imported (by copying the raw LUKS data into the image) and loaded by RBD encryption. Note The LUKS formats are supported on Linux-based systems only. Note Currently, only AES-128 and AES-256 encryption algorithms are supported. Additionally, xts-plain64 is currently the only supported encryption mode. To use the LUKS format, start by formatting the image: rbd encryption format {pool-name}/{image-name} {luks1\|luks2} {passphrase-file} [–cipher-alg {aes-128 \| aes-256}] The encryption format operation generates a LUKS header and writes it to the beginning of the image. The header is appended with a single keyslot holding a randomly-generated encryption key, and is protected by the passphrase read from passphrase-file. Note If the content of passphrase-file ends with a newline character, it will be stripped off. By default, AES-256 in xts-plain64 mode (which is the current recommended mode, and the usual default for other tools) will be used. The format operation allows selecting AES-128 as well. Adding / removing passphrases is currently not supported by RBD, but can be applied to the raw RBD data using compatible tools such as cryptsetup. The LUKS header size can vary (upto 136MiB in LUKS2), but is usually upto 16MiB, depending on the version of libcryptsetup installed. For optimal performance, the encryption format will set the data offset to be aligned with the image object size. For example expect a minimum overhead of 8MiB if using an imageconfigured with an 8MiB object size. In LUKS1, sectors, which are the minimal encryption units, are fixed at 512 bytes. LUKS2 supports larger sectors, and for better performance we set the default sector size to the maximum of 4KiB. Writes which are either smaller than a sector, or are not aligned to a sector start, will trigger a guarded read-modify-write chain on the client, with a considerable latency penalty. A batch of such unaligned writes can lead to IO races which will further deteriorate performance. Thus it is advisable to avoid using RBD encryption in cases where incoming writes cannot be guaranteed to be sector-aligned. To mount a LUKS-encrypted image run: rbd -p {pool-name} device map -t nbd -o encryption-format={luks1\|luks2},encryption-passphrase-file={passphrase-file} Note that for security reasons, both the encryption format and encryption load operations are CPU-intensive, and may take a few seconds to complete. For the encryption operations of actual image IO, assuming AES-NI is enabled, a relative small microseconds latency should be added, as well as a small increase in CPU utilization.	{}
Courses Courses for Kids Free study material Free LIVE classes More # How many moles are there in $24$ gram of $\text{Fe}{{\text{F}}_{\text{3}}}?$ Last updated date: 22nd Mar 2023 Total views: 234.3k Views today: 3.12k Verified 234.3k+ views Hint: Moles are calculated from the molecular weight. The mole is the unit of measurement for the amount of substance in the international system of units. It is defined as exactly $6.022\times {{10}^{23}}$ particles, which may be atoms, molecules, ions or electrons. The mole is widely used in chemistry as a connectional way to express amounts of red ants and products. The number of grams of a mole is equal to the formula weight of a compound in atomic units. Formula mass of substance can be calculated by summing the average atomic masses of all the atoms represented in the formula, here $\text{Fe}{{\text{F}}_{\text{3}}}$ . We have the formula Number of moles $=\dfrac{\text{Mass of substance}}{\text{Mass of one mole}}$ Mass of one mole $=$ molecular mass $=55.8+19+19+19=112.8$ g/mol Therefore number of moles in $24$ grams of $\text{Fe}{{\text{F}}_{\text{3}}}$can be calculated as Number of moles $=\dfrac{24}{112.8}=0.213$ Therefore $0.213$ moles are there in $24$ grams of $\text{Fe}{{\text{F}}_{\text{3}}}$. Since Fe has molecular weight or atomic mass $=55.84$ & Fluorine has atomic mass $19.$ The average sum of mass of the $\text{Fe}{{\text{F}}_{\text{3}}}$comes out to be $112.8$ g/mol. The term gram-molecule (gmol) was formerly used for mole of molecules and gram atom for mole of atoms. The history of the mole is intertwined with that of molecular, mass, atomic mass unit, and the Avogadro number. The Avogadro constant $NA$ is related to other physical constants and properties. 1. It relates molar gas constant & the Boltzman constant ${{K}_{B}}$ $R={{K}_{B}}{{N}_{A}}$ 2. It relates the faraday constant $F$ and the elementary charge e. $F=eNA$ 3. It relates the molar mass constant, $Mu$ & atomic mass unit $mu.$ $Mu=muNA.$	{}
# Subtracting mixed numbers word problem ## Video transcript Shelagh and Luana realize that they could express their ages more specifically if they used fractions. Doing this will allow them to calculate the difference in their ages more precisely. If Shelagh describes her age as 15 and 1/2 years old and Luana is 7 and 7/12-- I'm assuming years old-- what is the difference in their ages? Write the answer as a simplified mixed number. So Shelagh is older, so she's 15 and 1/2 years old, so we literally want to find the difference between 15 and 1/2 and 7 and 7/12. Now we can see immediately-- or maybe it's not obvious to you, but 1/2 is the same thing is 6/12, which is a less than 7/12. So if we separated out the whole number parts and the fraction parts, this fraction is actually smaller. The 1/2 is smaller than the 7/12, so it might be easier to just do this as mixed numbers so we don't have to do all of that borrowing and regrouping. So let's write both of them as mixed numbers. So we have 1 and 1/2. We're still going to-- sorry, let's write both of them as improper fractions, so we're going to keep 2 in the denominator. And as an improper fraction, 2 times 15 is 30 plus 1 is 31. So 31/2 the same thing as 15 and 1/2, so these are equivalent. And then 7 and 7/12, if we write it as an improper fraction, so we're still going to have the minus sign out there, the denominator is 12. 12 times 7 is 84. 84 plus 7 is 91. It is 91. Did I do that right? 91. Yep, that looks right. 91/12. Now, we've written both of them as improper fractions, but we still have different denominators, so we have to have a common denominator. So what is the least common multiple of 2 and 12, or what's the smallest number that's divisible by both of them? Well, 12 is divisible by 12, and 12 is also divisible by 2, so 12 is the least common multiple. So we want both of them to be over 12. And this fraction already is over 12 so we don't have to change it: 91/12. Now, on the blue one, to go from 2 to 12 in the denominator, we have to multiply by 6, so we also have to multiply by 6 in the numerator. So if we multiply 31 by 6, what do we get? You can even do this in your head. 6 times 30 is 180, 6 times 1 is 6, so it's going to be 186. And now we subtract. So the difference in their ages is going to be-- the denominator is going to be 12. It's going to be 186 minus 91, which is equal to-- over 12. Let's see, 186 minus just 90 would be 96, but we're subtracting 91, so it's going to be 95. 95, right? 6 minus 1 is 5, and then 180 minus 90 is 90, so that works out. So the way we've expressed it right now, it isn't is an improper fraction, but they want us to write it as a simplified mixed number, so let's do that. So how many times does 12 go into 95? Let's see, 8 times 12 is 96, so that's too big, so 12 goes seven times. So this is equal to 7. And if we take-- well, let me write it out. We don't want to do too much in our head. If we go 12 into 95, I'm saying it goes seven times. So if that's true, if that's the largest number, then the remainder should be less than 12. 7 times 12 we know is 84 from our times tables, or you could just think it's going to be 14 plus 10 times 7, which is 70. Either way. And let's see, 95 minus 84: 5 minus 4 is 1, 9 minus 8 is 1, so we have a remainder of 11. So 12 goes into 95 seven times. You get 7 wholes and you have 11 left over. So 95/12 is 7 and 11/12. And we're done! The difference between Shelagh and Luana's age is 7 and 11/12, or Shelagh is 7 and 11/12 years older than Luana.	{}
# A picture is worth 1,000 false-positive bug reports ###### May 15, 2020 - San Francisco, CA Note: we understand that there are more important topics in the world given that we’re in the midst of a pandemic, and we wish for everyone to remain safe & healthy. We’re continuing to publish blog posts in the hope that they provide a sense of connection and a signal of continuity—while it's difficult to think beyond the day, together, we will get through this. ## The problem We’ve written before on this blog about how our Algorithms team works to make our warehouse operations as efficient as possible. A warehouse is a vast space that requires careful organization. The starting point for our warehouse organization is to subdivide the space by business line (Women’s, Men’s, Kids’) and then by size (e.g., S, M, L). The shipments we send to our Stitch Fix clients typically consist of five items, though this number jumps to ten for our Kids’ business. It wouldn’t be very efficient to send our warehouse workers around collecting—“picking” in warehouse operations jargon—just the items needed to pack and ship a single shipment; this is due to the fact that a substantial portion of picking time is just the journey to and from the packing base area. Thus, our first step in improving the efficiency of our warehouse picking operations is batching: that is, an algorithm to optimize the grouping of shipments into batches of an appropriate size for one pick path, where this size is a constant determined by business line (Women’s, Men’s, or Kids’). After the batches are constructed, we run a pathing algorithm to determine a good[1] sequence for the picking of the items in each batch. As our business has grown, we are experimenting more and more with improving our warehouse operations. For example, we’re experimenting with different ways of laying out categories and sizes of merchandise in our warehouse. The warehouse layouts themselves[2] have thus become a more frequently changing input to the batching and pathing algorithms. And what comes with increased complexity in the inputs to algorithms? Increased surface areas for bugs. Bugs in the batching and pathing algorithms have happened, and they can be fairly catastrophic to a warehouse’s productivity, so it is extremely important to try to catch and fix bugs as early as possible. Before we built the app that is the subject of this blog post—Tourist is its internal code name—the steps to discovering a bug in batching and/or pathing were as follows: Warehouse employees suspected something was up because of batches with paths that seemed extraordinarily long and/or inefficient. 🔊 A warehouse lead communicated this to the Operations team. ✉️ Someone from Operations opened a ticket with the Engineering team that owns our internal warehouse applications that surface batches and their associated pick paths. 👷🏿‍♂️ An engineer did a first pass at bug-hunting in the warehouse application code, then escalated to the corresponding Algorithms team[3] if there was no obvious source of the perceived problem. 👩🏽‍💻 A data scientist dove into Algorithms-owned code to figure out whether a bug in an algorithm (served as an API to the warehouse application code) was the cause of “weirdness” reported by the warehouse. If there was no bug, the data scientist communicated that back to their Operations business partner, who communicated the same result back to the warehouse. If there was a bug, then the scientist fixed it and communicated back through the same chain when the fix was completed. The (likely obvious!) pain in this flow was the inability of Operations team members to triage warehouse reports of potential problems themselves. It left them relatively powerless, only serving as messengers, and it also caused pain for the engineers and data scientists whose work was somewhat frequently interrupted by tickets and requests to investigate potential bugs. This would have been less of a problem if reports from the warehouse had a very high correlation with the occurrence of actual bugs, but in reality, it happens fairly frequently that some sets of shipments are very difficult to batch efficiently (given the warehouse layouts at the time), and so what feels like a “bad” batch is actually the best solution we could deploy, and there is no bug. (We’ll dive into exactly what combinations of shipment and warehouse layout features have caused this below.) In addition, the metrics used by data scientists to make an initial assessment of a batch’s goodness are relatively simple to calculate. Thus, we decided it was time to put together a web application for Operations, warehouse Engineering, and Algorithms team members to be able to easily look up a batch to see its pick path and all relevant related metrics. All stakeholders can now use Tourist to do a first pass at investigating reports of potential bugs from the warehouses. And as this kind of visual troubleshooting is applicable in many business domains—ride-sharing and deliveries jump immediately to mind—we thought we’d share a bit about our (quite simple!) design. ## So what does Tourist look like? The landing page for Tourist is a simple input to look up a batch via the id of one of its constituent shipments[4]: This fits the intended primary use of in-the-moment debugging based on reports of “weird” batches coming down the line from warehouse leads. Once we’ve retrieved all the necessary data for the batch as well as the warehouse (the layout of the relevant business line-specific slice of the warehouse, most importantly) we display the batch as its pick path for the user: Let’s zoom in and examine this visualization in detail: The path begins with a green circle and ends on the red. Each semi-transparent grey circle represents a warehouse unit “step” in the pick path. When the picker had to retrace his or her steps, multiple circles will be layered on top of each other, resulting in a darker path segment. This has the useful side effect of making a bad pick path extremely obvious: The numerals represent both the location and ordered index of each picked item. Some numerals overlap and become unreadable, but that’s only a minor visual annoyance. The numerals serve mainly to provide the user with milestones that indicate the ordering of the path; thus not every numeral need be legible. In the case of very tightly packed paths where the ordering is difficult to visually parse, the user can click the “start” button in the lower left corner to animate the path: A dropdown menu provides a couple of other ways to view a batch, including as just a collection of circles representing the items in the batch, color-coded by each constituent item’s shipment id: This might not seem like a very useful display, but it can be under certain circumstances. One logical way to organize merchandise in a warehouse is by size. This is due to the fact that each customer receiving a Fix is likely receiving items that are the same size. While this assumption about clients holds quite often, there are exceptions. For example, look at this long pick path that crosses three merchandise zones (S, M, and L–XL): It’s not obvious from the pick path display why this path is so long, but if we switch to the color-coded by shipment id display, it becomes obvious fairly quickly that there is no way that a batch could have been constructed for these shipments that didn’t cross three zones and nearly the entire potential warehouse space. Two shipments—the orange and pink-coded (and encircled for emphasis)—contain items that are mostly spread sparsely throughout the L–XL zone, but with one item that is a S: Strange? Yes. Possible? Absolutely. Some clothing brands run large or small, and some clients prefer different styles of fit in their tops versus their bottoms—perhaps preferring roomy bottoms with fitted tops or oversized tops with plenty of leeway to drape creatively or tuck into a fitted waistband. Human bodies come in a wide variety of shapes and sizes, after all. The best we can do with shipments like these is to group them together in batches that will cover a lot of area so that they don’t cause other batches to cover any more area than necessary. Now that we have Tourist, our Operations team can look up batches and quickly see whether further investigation into a particularly long or “weird” pick path is necessary. This ability to self-serve removes the need for engineers and algorithm developers to investigate bugs that ultimately turn out to be false positives. ## And the technical stack? Tourist is a React single-page application built via create-react-app[5]. It employs React hooks, Redux toolkit, Semantic UI React for UI building blocks outside of the warehouse visualization code, and @reach/router for browser routing. Its “backend” is just our internal API for executing simple SQL queries against single tables in our data warehouse. The warehouse visualizations themselves are SVG, and in this case d3-scale isn’t even a dependency because the mapping from warehouse schematic to visualization is a dead simple 1 cell: 10px mapping. (Responsiveness to screen size is achieved by using percentages for the <svg> element’s width and height but then defining a viewBox based on the 10px-per-cell scaling.) The pick path animation is accomplished with nothing more complex than CSS transitions defined via styled-components (which is part of our customized set of create-react-app dependencies). An early version employed react-spring for the path animation, but it wasn’t performant without refactoring the component tree to avoid unnecessary re-renders, and CSS transitions turned out to be perfectly adequate for the job. [1]↩ “Good,” not “optimal” because we need to sequence several batches per second, and sequencing is a well-known NP-Hard problem. [2]↩ For the curious, our warehouse layouts are actually laid out schematically in the cells of a Google Sheet (with very strict editing controls!). A nightly ETL reads every tab of this sheet and stores the current warehouse layout (each cell designated as item bay, walkway, etc., with higher-order designations for “zones” as well) in a table in our data warehouse where it can serve as the input to the batching and pathing algorithms. [3]↩ Recall that at Stitch Fix Algorithms is a separate department from Engineering. [4]↩ Why shipment id and not batch id? It’s complicated. [5]↩ Our Algorithms UI team creates so many create-react-app (CRA) apps that we actually built a CLI wrapper around CRA that modifies the output for our environment by editing the boilerplate, installing additional dependencies, and setting up things like commit hooks to enforce linting and code styling. We are currently working on replacing this wrapper (codenamed 👩‍🎤 rappstar for React app starter) with a custom CRA template. ## Come Work with Us! We’re a diverse team dedicated to building great products, and we’d love your help. Do you want to build amazing products with amazing peers? Join us!	{}
# [OS X TeX] Large Symbols Partially Printed from PDF from LaTeX Thu Apr 25 21:57:41 EDT 2002 > I just upgraded from OS 4.2 (NeXT) to OS X (Mac 10.1.2), and yesterday > Wierda's teTeX/GS packages. Everything installed easily without much > pain. However, when I tested the combined system on a current finance > paper, I found that large symbols like brackets or integral signs > generated from \left and \right pairs had only the top part of the > correctly in the PDF viewer or Preview. I've noticed this kind of effect on the screen as well, specifically with \bigcup. The bottom of the big union symbol is sometimes missing, both in the TeXShop viewer and in Acrobat Reader. If my memory is correct, it happens only with magnifications > %100, and it doesn't happen consistently. The bottom of the union symbol usually reappears if scrolled off and then back onto the screen. Richard Seguin ----------------------------------------------------------------- To UNSUBSCRIBE, send email to <info at email.esm.psu.edu> with "unsubscribe macosx-tex" (no quotes) in the body. For additional HELP, send email to <info at email.esm.psu.edu> with "help" (no quotes) in the body. -----------------------------------------------------------------	{}
Find all School-related info fast with the new School-Specific MBA Forum It is currently 07 Jul 2015, 13:25 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # A saleswoman s monthly income has two components, a fixed Author Message TAGS: GMAT Instructor Joined: 04 Jul 2006 Posts: 1266 Followers: 24 Kudos [?]: 167 [0], given: 0 A saleswoman s monthly income has two components, a fixed [#permalink] 26 Nov 2006, 06:38 00:00 Difficulty: (N/A) Question Stats: 0% (00:00) correct 0% (00:00) wrong based on 0 sessions A saleswomanâ€™s monthly income has two components, a fixed component of $1000 per month, and a variable component, which is$C for each set of encyclopaedias that she sells in that month over a sales target of 2 sets. How much did she earn in March? (1) If she had sold three fewer sets than she actually did, her March income would have been $600 less. (2) If she had sold 8 sets of encyclopaedias, her income in March would have been over$4000 Last edited by kevincan on 26 Nov 2006, 07:27, edited 1 time in total. VP Joined: 28 Mar 2006 Posts: 1383 Followers: 2 Kudos [?]: 20 [0], given: 0 Re: DS: Encyclopaedia saleswoman [#permalink] 26 Nov 2006, 06:53 kevincan wrote: A saleswomanâ€™s monthly income has two components, a fixed component of $1000 per month, and a variable component, which is$C for each set of encyclopaedias that she sells in that month over a sales target of 3 sets. How much did she earn in March? (1) If she had sold three fewer sets than she actually did, her March income would have been $600 less. (2) If she had sold 8 sets of encyclopaedias, her income in March would have been over$4000 I get A here Let n be the # of copies that she sells we know that her income is (n-3)C + 1000 for any given month----(I) (1) says that If she actually sells 3 copies less than actually she did then her income is less by \$600 So (n-6)C + 1000 ------------------(II) From (I) and (II) we get her March income From (2) in the given premise we have 5*C + 1000 > 4000 So C>600 But we cannot tell exactly how much she got ... So IMO it is A Senior Manager Joined: 24 Oct 2006 Posts: 339 Followers: 1 Kudos [?]: 17 [0], given: 0 S1: If sales=4sets,c=600/set If sales=5, c=300 if sales=6, c=200 Insuff, 'cause we can't assume how many sets were sold S2:c>600 Insuff Together, If s2 says c=600, then ans is c. But, c>600. How could that be? Where am I going wrong? Senior Manager Joined: 24 Oct 2006 Posts: 339 Followers: 1 Kudos [?]: 17 [0], given: 0 We need the no. of sets sold. S1: c could be 200, 300, 600 insuff S2:c>500 insuff together - 1set - suff My ans:C Similar topics Replies Last post Similar Topics: A saleswoman's monthly income has two components, a fixed 1 17 Jul 2008, 22:46 Laura sells encyclopaedias, and her monthly income has two 4 25 Sep 2007, 01:07 Laura sells encyclopaedias, and her monthly income has two 5 27 Jun 2007, 12:27 5 Laura sells encyclopaedias, and her monthly income has two 28 14 Feb 2007, 15:42 A saleswoman s monthly income has two components, a fixed 5 09 Jul 2006, 03:25 Display posts from previous: Sort by	{}
# Efficiently Sorting File Paths [closed] I'm trying to print out a list of files in a specific order, with files closer to the root directory sorted first. For example these files: • c:\users\steve\foo.txt • c:\users\bar.txt • c:\users\steve\desktop\file.txt • c:\users\john\bing.txt Should be sorted as: • c:\users\bar.txt • c:\users\john\bing.txt • c:\users\steve\foo.txt • c:\users\steve\desktop\file.txt Code: namespace fs = std::experimental::filesystem; std::vector<std::string> sortedRules; std::sort(sortedRules.begin(), sortedRules.end(), [&](const string &str1, const string &str2) { fs::path path1(str1); fs::path path2(str2); //if parent directories are the same, compare filenames if (path1.parent_path() == path2.parent_path()) return path1 < path2; std::vector<std::string> path1Expnded; std::vector<std::string> path2Expnded; for (const auto &part : path1) path1Expnded.emplace_back(part.string()); for (const auto &part : path2) path2Expnded.emplace_back(part.string()); //if paths have same amount of stems, compare full paths if (path1Expnded.size() == path2Expnded.size()) return path1 < path2; int minSize; path1Expnded.size() <= path2Expnded.size() ? minSize = path1Expnded.size() : minSize = path2Expnded.size(); //sort by first differing stem for (int i = 0; i < minSize - 1; i++) if (path1Expnded[i] != path2Expnded[i]) return path1Expnded[i] < path2Expnded[i]; //if sizes differ and no stems are the same, sort by number of stems return path1Expnded.size() < path2Expnded.size(); }); I created a custom comparison lambda that overrides how std::sort compares, and it works most of the time. Sometimes items cannot be found after being sorted using this comparison lambda, and after being searched for using this same comparison lambda. So I'm guessing my algorithm is somehow incorrect. Also even when the algorithm does run, it does so very slowly. It took me awhile to get the files to sort just the way I wanted, and while I tried to allow the function to return as soon as possible its performance is pretty poor. How can I fix the algorithm to ensure its correctness, and also make it more efficient? • Here's one possible solution using std::multimap, have a look here. – Juho Dec 21 '17 at 9:06 ## 1 Answer First, don't use the conditional operator as an if statement. Either encode the conditional as part of the initialization of minSize, or use an if statement instead of the conditional. In this case, though, a better solution is to make use of the standard min function: int minSize = std::min(path1Expnded.size(), path2Expnded.size()); But we can redo your code to eliminate this part. The sort you've coded and the one you describe (how I am interpreting your description, anyway) are different, as your code only looks at the number of subfolders if one path is in a subfolder of the folder that has the file in the other path. Your problem description ("files closer to the root directory sorted first"), sounds like all the paths with fewer folders should be before longer ones, in which case the number of stems should be the first thing compared. Then a direct comparison on the full paths if they have the same folder count. What you've coded up will put files in a folder before subfolders of that folder. Your current implementation does a lot of memory allocations and frees, mainly with strings and vector resizes. You can simplify this a lot by first comparing the two parent_paths. If they are different, return the result of this comparison (the path to a file will be shorter than the path for a subfolder in the folder with the file, so it'll be earlier). Otherwise the two paths are the same, and return the result of comparing the filenames. • Wow, the algorithm you described is so concise, and actually runs pretty fast! As usual I was grossly over complicating things haha. Thanks for your answer! Dec 22 '17 at 1:38	{}
Courant Institute New York University FAS CAS GSAS # Subversion keywords and unicode Monday, August 4, 2008 - 12:30pm I'm editing some text files, and I like to use subversion to do this. It helps with managing my own different revisions of the files, and can mark them automatically with metadata. (Subversion is a version control system--you create a repository, work on a local copy, then commit changes to the repository when you're satisfied with them. Programmers use version control systems to keep track of giant code bases, so you can prepare new versions of software while at the same time release bug fixes for previous versions. But there's no reason why the files managed need to be an executable piece of software--it can be a website or a single text file. And there's no reason why you need more than one developer to have a version control system in place.) Anyway, subversion allows you mark keywords in your file to keep track of some metadata. For instance, I like to note the date when a file was last committed (had changes saved to the repository). To do this, you use the string "$Date$" in the file. Then you declare that you want this keyword substituted. On the command-line, assuming you're in a checked-out working copy of the version-controlled directory, you type $svn propadd svn:keyword Date foo.txt When you next check the status, you will find that the file's metadata has been changed: $ svn status M foo.txt The "M" appearing in the second column rather than the first indicates a modification to metadata rather than actual file content. Commit the changes, then the next time you commit or update the file, you will see the string "$Date$" has been changed to "$Date: <actual date committed>$" Pretty cool, except I'm having trouble getting this to work on unicode files. I don't always need unicode but I feel cutting-edge when I use it. Even after setting the property and committing changes the keyword stayed unsubstituted. I finally got it to work by writing the property value "in unicode." My workaround is extremely inelegant, the epitome of muddling through, but it does do the job. 1. Create a file with the property value in it. In this case, the property value is just the word "Date". I'm using Aquamacs, an emacs build for Mac OS X. There is a set of keystrokes in emacs to change the text encoding for a file, but I keep forgetting it. Instead, I use a local variable and save the file. One way to do this is to declare at the end of the file: Local variables: coding: utf-8 (Eight bits is all the unicode I need, which is lucky because that's all subversion supports.) Save the file, and a little "u" appears in the emacs status line, letting you know the change of encoding. Save the file to some descriptive name like "svn-keywords-utf8.txt" 2. Set the property value. Subversion allows you to set property values from a file, not just on the command line. \$ svn propset svn:keyword -F svn-keywords-utf8.txt foo.txt 3. Commit as before. Voilà. Now, there must be smarter ways to do this. Some occur to me as I write. One could use iconv to change the property-value file's encoding with having to invoke a text editor. But the "real" way would be to make the terminal encode in unicode. According to the preferences for my Terminal application, I am doing this, but perhaps because I'm logged in via ssh to another server, it's not carrying over. I don't know. But like I said, this works.	{}
Serving the Quantitative Finance Community EdisonCruise Topic Author Posts: 49 Joined: September 15th, 2012, 4:22 am ### How to get this limit order fill probability? I find this power class limit order fill probability $$h(\delta)=\frac{1}{1+(\kappa\delta)^{\gamma}}$$ where $\delta$ is the distance to mid price;$\kappa$ and $\gamma$ need to be calibrated from data . in https://tspace.library.utoronto.ca/bits ... thesis.pdf My questions are (1) Is there any reference to justify this model? (2) And how to calibrate this model? My primary understaning is that this model comes from: (a) Market impact assumption: $\Delta p=K(Q-c)^{\frac{1}{\gamma}}$, where $\Delta p$ is the distance to mid price and $Q$ is trading volume. (b) Distribution of market order size (pdf): $Q:x^{-2}$ With this two assumptions, $$h(\delta)=P(\Delta p>\delta)=P(K(Q-c)^{\frac{1}{\gamma}}>\delta)=P(Q>(\frac{\delta}{K})^{\gamma}+c)=\int^{\infty}_{\frac{\delta}{K}^{\gamma}+c} x^{-2}dx=\frac{1}{\frac{\delta}{K}^{\gamma}+c}=\frac{1}{c} \frac{1}{1+(\kappa\delta)^{\gamma}}$$ where$\kappa=\frac{1}{c^{\frac{1}{\gamma}K}}$ If this derivation is correct, then one need to get $c$ and $K$ from data to calculate $\kappa$? From https://www.math.nyu.edu/~avellane/High ... rading.pdf, assumption (a) and (b) should be something like (a') $\Delta p=K(Q)^{\frac{1}{2}}$ (b') $Q:x^{-2.5}$ With (a') and (b') is is possible to get the same formula of $h(\delta)$?	{}
## Partial Fractions With Linear Factors ### Learning Outcomes • Integrate a rational function using the method of partial fractions • Recognize simple linear factors in a rational function • Recognize repeated linear factors in a rational function We have seen some techniques that allow us to integrate specific rational functions. For example, we know that $\displaystyle\int \frac{du}{u}=\text{ln}\|u\|+C\text{ and }\displaystyle\int \frac{du}{{u}^{2}+{a}^{2}}=\frac{1}{a}{\tan}^{-1}\left(\frac{u}{a}\right)+C\text{.}$ However, we do not yet have a technique that allows us to tackle arbitrary quotients of this type. Thus, it is not immediately obvious how to go about evaluating $\displaystyle\int \frac{3x}{{x}^{2}-x - 2}dx$. However, we know from material previously developed that $\displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx=\text{ln}\|x+1\|+2\text{ln}\|x - 2\|+C$. In fact, by getting a common denominator, we see that $\frac{1}{x+1}+\frac{2}{x - 2}=\frac{3x}{{x}^{2}-x - 2}$. Consequently, $\displaystyle\int \frac{3x}{{x}^{2}-x - 2}dx=\displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx$. The key to the method of partial fraction decomposition is being able to anticipate the form that the decomposition of a rational function will take. As we shall see, this form is both predictable and highly dependent on the factorization of the denominator of the rational function. It is also extremely important to keep in mind that partial fraction decomposition can be applied to a rational function $\frac{P\left(x\right)}{Q\left(x\right)}$ only if $\text{deg}\left(P\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right)$. In the case when $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)$, we must first perform long division to rewrite the quotient $\frac{P\left(x\right)}{Q\left(x\right)}$ in the form $A\left(x\right)+\frac{R\left(x\right)}{Q\left(x\right)}$, where $\text{deg}\left(R\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right)$. We then do a partial fraction decomposition on $\frac{R\left(x\right)}{Q\left(x\right)}$. The following example, although not requiring partial fraction decomposition, illustrates our approach to integrals of rational functions of the form $\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx$, where $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)$. ### Example: Integrating $\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx$, where $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)$ Evaluate $\displaystyle\int \frac{{x}^{2}+3x+5}{x+1}dx$. ### Recall: Polynomial Long Division 1. Set up the division problem as the numerator divided by the denominator 2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. 3. Multiply the answer by the divisor and write it below the like terms of the dividend. 4. Subtract the bottom binomial from the top binomial. 5. Bring down the next term of the dividend. 6. Repeat steps 2–5 until reaching the last term of the dividend. 7. If the remainder is non-zero, express as a fraction using the divisor as the denominator. ### try it Evaluate $\displaystyle\int \frac{x - 3}{x+2}dx$. Watch the following video to see the worked solution to the above Try It For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. To integrate $\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx$, where $\text{deg}\left(P\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right)$, we must begin by factoring $Q\left(x\right)$. ## Nonrepeated Linear Factors If $Q\left(x\right)$ can be factored as $\left({a}_{1}x+{b}_{1}\right)\left({a}_{2}x+{b}_{2}\right)\ldots\left({a}_{n}x+{b}_{n}\right)$, where each linear factor is distinct, then it is possible to find constants ${A}_{1},{A}_{2}\text{,}\ldots {A}_{n}$ satisfying $\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{{a}_{1}x+{b}_{1}}+\frac{{A}_{2}}{{a}_{2}x+{b}_{2}}+\cdots +\frac{{A}_{n}}{{a}_{n}x+{b}_{n}}$. The proof that such constants exist is beyond the scope of this course. In this next example, we see how to use partial fractions to integrate a rational function of this type. ### Example: Partial Fractions with Nonrepeated Linear Factors Evaluate $\displaystyle\int \frac{3x+2}{{x}^{3}-{x}^{2}-2x}dx$. In the next example, we integrate a rational function in which the degree of the numerator is not less than the degree of the denominator. ### Example: Dividing before Applying Partial Fractions Evaluate $\displaystyle\int \frac{{x}^{2}+3x+1}{{x}^{2}-4}dx$. As we see in the next example, it may be possible to apply the technique of partial fraction decomposition to a nonrational function. The trick is to convert the nonrational function to a rational function through a substitution. ### Example: Applying Partial Fractions after a Substitution Evaluate $\displaystyle\int \frac{\cos{x}}{{\sin}^{2}x-\sin{x}}dx$. ### try it Evaluate $\displaystyle\int \frac{x+1}{\left(x+3\right)\left(x - 2\right)}dx$. Watch the following video to see the worked solution to the above Try It For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. ## Repeated Linear Factors For some applications, we need to integrate rational expressions that have denominators with repeated linear factors—that is, rational functions with at least one factor of the form ${\left(ax+b\right)}^{n}$, where $n$ is a positive integer greater than or equal to $2$. If the denominator contains the repeated linear factor ${\left(ax+b\right)}^{n}$, then the decomposition must contain $\frac{{A}_{1}}{ax+b}+\frac{{A}_{2}}{{\left(ax+b\right)}^{2}}+\cdots +\frac{{A}_{n}}{{\left(ax+b\right)}^{n}}$. As we see in our next example, the basic technique used for solving for the coefficients is the same, but it requires more algebra to determine the numerators of the partial fractions. ### Example: Partial Fractions with Repeated Linear Factors Evaluate $\displaystyle\int \frac{x - 2}{{\left(2x - 1\right)}^{2}\left(x - 1\right)}dx$. ### Try It Set up the partial fraction decomposition for $\displaystyle\int \frac{x+2}{{\left(x+3\right)}^{3}{\left(x - 4\right)}^{2}}dx$. (Do not solve for the coefficients or complete the integration.) Watch the following video to see the worked solution to the above Try It For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.	{}
# Clustering of gamma-ray burst types in the Fermi-GBM catalogue: indications of photosphere and synchrotron emissions during the prompt phase [HEAP] Many different physical processes have been suggested to explain the prompt gamma-ray emission in gamma-ray bursts (GRBs). Although there are examples of both bursts with photospheric and synchrotron emission origins, these distinct spectral appearances have not been generalized to large samples of GRBs. Here, we search for signatures of the different emission mechanisms in the full Fermi Gamma-ray Space Telescope GBM catalogue. We use Gaussian Mixture Models to cluster bursts according to their parameters from the Band function ($\alpha$, $\beta$, and $E_{pk}$) as well as their fluence and $T_{90}$. We find five distinct clusters. We further argue that these clusters can be divided into bursts of photospheric origin (2/3 of all bursts, divided into 3 clusters) and bursts of synchrotron origin (1/3 of all bursts, divided into 2 clusters). For instance, the cluster that contains predominantly short bursts is consistent of photospheric emission origin. We discuss several reasons that can determine which cluster a burst belongs to: jet dissipation pattern and/or the jet content, or viewing angle. Z. Acuner and F. Ryde Wed, 6 Dec 17 11/71 Comments: Accepted for publication in MNRAS	{}
# American Institute of Mathematical Sciences May 2012, 32(5): 1801-1833. doi: 10.3934/dcds.2012.32.1801 ## Multiple periodic solutions of state-dependent threshold delay equations 1 Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484, United States Received November 2010 Revised September 2011 Published January 2012 We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps. We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$. Citation: Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801 ##### References: show all references ##### References: [1] Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039 [2] Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 [3] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [4] Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071 [5] Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 [6] Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445 [7] Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137 [8] Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 [9] Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : ⅰ-ⅰ. doi: 10.3934/dcdss.20201i [10] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [11] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 [12] Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 [13] Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56 [14] Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167 [15] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [16] Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002 [17] Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633 [18] Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026 [19] Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074 [20] Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 991-1009. doi: 10.3934/dcdsb.2018139 2019 Impact Factor: 1.338	{}
# Count of Disjoint subsets Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B, C)$, where $B$, $C$ are disjoint subsets of $A$, equals (A) $n^2$ (B) $n^3$ (C) $2^n$ (D) $3^n$ Doubt 1: What is a disjoint subset ? My Attempt at the above problem: Assume 5 elements. Ordered pairs would be: $$(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,2),(2,3),(2,4),(2,5),(3,1),(3,2),(3,3),(3,4),(3,5),(4,1),(4,2),(4,3),(4,4),(4,5),(5,1),(5,2),(5,3),(5,4),(5,5)$$ 25 elements. So, the answer would be: (A) $n^2$. Is this the right approach ? Any better ways to solve this ? - There is no such thing as "a disjoint subset". Two subsets are disjoint if they have no element in common, that is, their intersection is empty. – David May 7 '14 at 6:29 Read your question more carefully! The items you have listed are ordered pairs $(b,c)$ where $b,c$ are elements of $A$. You are asked for ordered pairs of subsets of $A$, for example, $(\{1,4\},\{2,5\})$. – David May 7 '14 at 6:32 Take each of the $n$ elements of $A$ one by one and decide whether to put it in $B$ or in $C$. No element can go in both $B$ and $C$ because they are supposed to be disjoint, so there are three options: in $B$, or in $C$, or neither. The total number of pairs is $3^n$. Assume $B \cup C = A$. Then if $B$ has $k$ element, then there are $\binom{n}{k}$ ways. Thus the total number of ways to do this is: $2^n$ because you get the sum of all these $\binom{n}{k}$'s for $k$ runs from $0$ to $n$. It is choice $C$ If $B \cup C \subseteq A$, then there are more cases and the answer should be $3^n$. The question isn't absolutely clear.	{}
X 3/4, the according decimal fraction is 0.75. The factor that determine how fast/slow of a method is the algorithm used in the implementation. We would do these together with the class telling me the doubles of the number. You take one number and either multiply it by 2 or you add it to itself. Then set up a little chart like we have done. A fraction is unit fraction if numerator is 1 and denominator is a positive integer, for example 1/3 is a unit fraction. How to use the calculator: Simply input the numerator and denominator of the fraction in the associated fields and click on the "Calculate" button to generate the results. Brute-force is not always bad. This problem follows on from Keep it Simple and Egyptian Fractions So far you may have looked at how the Egyptians expressed fractions as the sum of different unit fractions. Once you get to a double larger than the other number you are multiplying then you can stop. These were people who migrated from the fertile Sahara region of … The calendar year consists of 3 seasons, each season has 4 month, each month has 3 … Set up a division problem where a is larger than b. a ÷ b = c with remainder R. Do the division. 37 X 49 To Use The Egyptian Algorithm, Rewrite The Number 37 As A Sum Of Whole Numbers. Use this calculator to find the Egyptian fractions expansion of the input proper fraction. Arts Learning \|Algebra\|, Addition and Multiplication Tables in Various Bases, Long Multiplication - an Interactive Gizmo, Lattice Multiplication - an Interactive Gizmo. I have since improved the binary remainder method, and added the reverse greedy, generalized remainder, and small multiple methods. For the product 18×85, we get the following result: The proof that the algorithm works is exactly the same as that for Russian Peasant Multiplication. Preschool Grades K-2 Grades 3-5 Middle School High School, Arts The Egyptian Mathematical Leather Roll (EMLR) contains methods for simplifying a series (a sum) of unit fractions to a single unit fraction. Unlike, the Russian Peasant Multiplication that determines the involved powers of 2 automatically, the Egyptian algorithm has an extra step where those powers have to be found explicitly. Now for a fraction, m n … Algorithms for Egyptian Fractions Continued Fraction Methods The Continued Fraction Method One can derive a good Egyptian fraction algorithm from continued fractions: the algorithm is quick, generates reasonably few terms, and uses fractions with very small denominators . Activities Math Physical & Crafts Health Language Egyptian fractions You are encouraged to solve this task according to the task description, using any language you may know. Greedy algorithm for Egyptian fractions. Construct a table of doubles starting with 1 1 1 on the left and the number to be multiplied on the right. This is done repeatedly until you get the other number. If so, I would answer them. Multiplication calculator shows steps so you can see long multiplication work. This method was used and developed by the ancient Egyptians. Compute 85 - 64 = 21 and find the largest power of 2 below 21: 16. That table would be the 2 times table. Below, in each column, write successively the doubles of the preceding numbers. This lesson plan will be about a new type of algorithm that will help those of you with problems multiplying numbers. Below is an example of what you need to do using the problem 22 x 21: You first take either number, the 21 or 22. An Egyptian fraction is the sum of distinct unit fractions such as: + + (=) Each fraction in the expression has a numerator equal to 1 (unity) and a denominator that is a positive integer, and all the denominators are distinct (i.e., no repetitions). In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, was a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. Multiply by two or add a number to itself. The first column will generate the sequence of the powers of 2: 1, 2, 4, 8, ... Stop when the next power becomes greater than the first multiplicand. They used addition to get the answer of a multiplication problem. Greedy Algorithm for Egyptian Fraction Last Updated: 09-11-2020 Every positive fraction can be represented as sum of unique unit fractions. On most basic calculators, to multiply 24 by 2 and keep doubling the answer, push 24 x 2 = = … A common fraction is e.g. Numeric Algorithmic Translation: 6 4 2. Now you have to find the double numbers that add up to the other number, in our case is 21. Question: Use The Egyptian Algorithm To Calculate The Product. Units. Egyptian Fraction Calculator. Compute 21 - 16 = 5 and find the largest power of 2 below 5: 4. After about 10-15 minutes of this activity, I would then ask them for five more pairs of numbers that they want multiplied. Those in red add up to the first multiplicand: which corresponds to the binary representation of 85: According to the Rhind papyrus these powers are found the following way. Sitemap. person_outline Anton schedule 1 year ago The ancient Egyptian calendar is a 365 days solar calendar. Put the number being doubled on the right hand side. Try IE11 or Safari and declare the site https://www.cut-the-knot.org as trusted in the Java setup. 1. Number. Egyptian multiplication. This type used different pictures to stand for different numbers. Descending Order. Before our departure, I ask them if they have any questions. Luhn Algorithm Calculator. The Egyptian civilization was one of the greatest ancient civilizations. Here we used the 22. Greedy Algorithm for Egyptian Fraction The greedy algorithm was developed by Fibonacci and states to extract the largest unit fraction first. You keep putting the orresponding double with the number that was doubled. The algorithm draws on the binary system: multiplication by 2, or just adding a number two itself. Copy link . "Find" will also show the Egyptian fractions. This Calculator will count the Egyptian fractions for 1 for a given denominator sum. Greedy Algorithm. 37 = 32 + 4 + 1 (Simplify Your Answers. Compute 5 - 4 = 1 and observe that the result, 1, is a power of 2: 1 = 20. A unit fraction has the form 1/n, whereas n is a natural number. \|Activities\| 37 X 49= + (Simplify Your Answers. These were people who migrated from the fertile Sahara region of Africa. Egyptian Multiplication The ancient Egyptians used a curious way to multiply two numbers. This lesson plan will be about a new type of algorithm that will help those of you with problems multiplying numbers. Egyptian fraction calculator. These students will need to satisfy the following before they will be able to complete the main objective. I would then ask them for some number that they would like to see multiplied together using this method. Egyptian mathematics: 1. This method was used and developed by the ancient Egyptians. Egyptian Fractions‎ > ‎ Egyptian Fraction Calculator. Take the corresponding numbers and add them together; 22+88+352=462. GitHub Gist: instantly share code, notes, and snippets. You may have started by considering fractions with small numerators, such as $\frac{2}{5}$, $\frac{3}{7}$, $\frac{4}{11}$, etc. The function required for the Egyptian method is doubling, which is multiplying by 2. This calculator allows you to calculate an Egyptian fraction using the greedy algorithm, first described by Fibonacci. They had calendars, standard weight and measure system and a centralized government. \|Front page\| The main results. Calculator for the unit fraction sum, or Egyptian fraction, of a common or decimal fraction. Egyptian fractions calculator fuse department of education. 64 is included simply because it's the largest power below 85. False position method Calculator . Multiplication math tricks: multiply like the egyptians. Use Descending Order.) The above discussions motivate us to design a new algorithm to calculate the chromatic index of the graph. The algorithm in fact may have Egyptian roots, as a similar procedure has been routinely used in the famous Rhind Papyrus [Midonick, pp. Then replace a with b, replace b with R and repeat the division. Education Thematic Much of the Rhind Papyrus deals with fraction computation, area problems, and "solving equations" -- finding the value of a heap. The algorithm draws on the binary system: multiplication by 2, or just adding a number two itself. 4 2 = 16; 4 x 2 = 8; 2 2 = 4; 16 + 8 + 4 = 28; ̅3 x 6 = 2; 2 x 28 = 56 On overflow, click clear "C". Egyptian division You are encouraged to solve this task according to the task description, using any language you may know. The left column consists of the powers of two. The Luhn Algorithm (Mod 10) Calculator is a simple tool allowing one to validate numbers and calculate the correct check digit for a given number via the Luhn checksum algorithm. Applications. An earlier version of this notebook was published as "Ten Algorithms for Egyptian Fractions" in Mathematica in Education and Research. Everyone who receives the link will be able to view this calculation. Implementing egyptian algorithm in java stack overflow. 5/6 = 1/2 + 1/3. 1 7 × 3 1. Continue the process until R = 0. 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Fractions '' in Mathematica in Education and Research used and developed by the ancient Egyptians column values match. N is a 365 days solar calendar and more solar calendar of doubles starting with and! Need to satisfy the following before they will be about a new algorithm to calculate 17 × 31 ;. False position method algorithm I 'll present shortly = 1 and denominator is a fraction! Doubling, which is multiplying by 2, or Egyptian fraction as a sum of Whole.. Of an irreducible fraction as a sum of distinct colors demonstrates an alternative method of multiplication to ' number distinct. Hunter College Graduate Programs, Powershell Change Network Type Windows 7, Derrick Johnson Lawyer Net Worth, Bitbucket Cloud Api, Ford Taunus V4 For Sale, Mercedes Sls Amg Black Series For Sale, Best Online Hospitality Courses, Nissan Rogue Seating Capacity,	{}
# Vuex for Redux devs Posted on Jan 23, 2021 # Vuex for Redux devs! Vuex as redux is a state manager, with the pattern of having the whole state separate from the UI. This way, rendering is a pure function of the state DOM = render(vuex_state). Also you create specific functions (actions) you can call from the UI to modify the state So far, is the same with Redux, and for everything else is just the same, with names changed everywhere. • mapState -> is the mapstatetoprops of react-redux, it maps the values of the state to fields/props of the component. In vue components, you map them to computed • getters -> is a “view” of the state, and it becomes into antother part of the state (TODO, explain this better :P), you use mapGetters to map them to the compoenent • mutations -> are the reducers in redux. they get an “action” with a possible payload, and change the state. as in redux they need to be pure functions. in redux you dispatch actions, in vuex you commit mutations. • mapMutations -> is the function to map the mutations to the component • actions -> is like redux-thunk middleware but is already inside the code, allows you to write functions that commit multiple mutations. and they can be async functions.	{}
# GroundwaterDupuitPercolator: model flow in a shallow unconfined aquifer using the Dupuit-Forcheimer approximation¶ GroundwaterDupuitPercolator Component. @author: G Tucker, D Litwin, K Barnhart class GroundwaterDupuitPercolator(grid, hydraulic_conductivity=0.001, porosity=0.2, recharge_rate=1e-08, regularization_f=0.01, courant_coefficient=0.5, vn_coefficient=0.8, callback_fun=<function GroundwaterDupuitPercolator.<lambda>>)[source] Simulate groundwater flow in a shallow unconfined aquifer. The GroundwaterDupuitPercolator solves the Boussinesq equation for flow in an unconfined aquifer over an impermeable aquifer base and calculates groundwater return flow to the surface. This method uses the Dupuit-Forcheimer approximation. This means that the model assumes the aquifer is laterally extensive in comparison to its thickness, such that the vertical component of flow is negligible. It also assumes that the capillary fringe is small, such that the water table can be modeled as a free surface. Please consider the applicability of these assumptions when using this model. For more details, see component documentation here. Examples Import the grid class and component >>> from landlab import RasterModelGrid >>> from landlab.components import GroundwaterDupuitPercolator Initialize the grid and component >>> grid = RasterModelGrid((10, 10), xy_spacing=10.0) >>> elev[:] = 5.0 >>> gdp = GroundwaterDupuitPercolator(grid) Run component forward. >>> dt = 1E4 >>> for i in range(100): ... gdp.run_one_step(dt) When the model generates groundwater return flow, the surface water flux out of the domain can be calculated only after a FlowAccumulator is run. Below is a more complex model that demonstrates this case. >>> from landlab.components import FlowAccumulator Set boundary conditions and initialize grid >>> grid = RasterModelGrid((5, 41), xy_spacing=10.0) >>> grid.set_closed_boundaries_at_grid_edges(True, True, False, True) Make a sloping, 3 m thick aquifer, initially fully saturated >>> elev = grid.add_zeros("topographic__elevation", at="node") >>> elev[:] = grid.x_of_node/100+3 >>> base[:] = grid.x_of_node/100 >>> wt[:] = grid.x_of_node/100 + 3 Initialize components >>> gdp = GroundwaterDupuitPercolator(grid, recharge_rate=1E-7) >>> fa = FlowAccumulator(grid, runoff_rate='surface_water__specific_discharge') Advance timestep. Default units are meters and seconds, though the component is unit agnostic. >>> dt = 1E3 >>> for i in range(1000): ... gdp.run_one_step(dt) Calculate surface water flux out of domain >>> fa.run_one_step() >>> np.testing.assert_almost_equal(gdp.calc_sw_flux_out(),0.0005077) Notes Below is a summary of the theory and numerical implementation of the GroundwaterDupuitPercolator. A complete description can be found here. Groundwater discharge per unit length, $$q$$, is calculated as: $q = -K_{sat} h \big( \nabla z \big) \cos^2 (\alpha)$ where $$K_{sat}$$ is the saturated hydraulic conductivity, $$h$$ is the aquifer thickness, and $$\alpha$$ is the slope angle of the aquifer base. Surface water discharge per unit area, $$q_s$$, is calculated as: $q_s = \mathcal{G}_r \bigg( \frac{h}{d} \bigg) \mathcal{R} \big(-\cos(\alpha) \nabla \cdot q + f \big)$ where $$\mathcal{G}_r$$ is a smoothed step function, $$\mathcal{R}$$ is the ramp function, $$d$$ is the regolith thickness, and $$f$$ is the recharge rate. The evolution of aquifer thickness is then given by: $n \frac{\partial h}{\partial t} = f - q_s - \nabla \cdot q$ where $$n$$ is the drainable porosity. An explicit forward in time finite volume method is used to implement a numerical solution. Groundwater flow between neighboring nodes is calculated using the saturated thickness at the up-gradient node. References Required Software Citation(s) Specific to this Component Litwin, D. G., Tucker, G.E., Barnhart, K. R., Harman, C. J. (2020). GroundwaterDupuitPercolator: A Landlab component for groundwater flow. Journal of Open Source Software, 5(46), 1935, https://doi.org/10.21105/joss.01935. Marçais, J., de Dreuzy, J. R. & Erhel, J. Dynamic coupling of subsurface and seepage flows solved within a regularized partition formulation. Advances in Water Resources 109, 94–105 (2017). Childs, E. C. Drainage of Groundwater Resting on a Sloping Bed. Water Resources Research 7, 1256–1263 (1971). Parameters: grid (ModelGrid) – Landlab ModelGrid object hydraulic_conductivity (float, field name, array of float or function.) – the aquifer saturated hydraulic conductivity, m/s. If function is given, it should take a landlab ModelGrid and return an array of floats at link. This may be used if the lateral hydraulic conductivity is not vertically homogenous and the effective hydraulic conductivity needs to be modified based upon on the position of the water table. See component tests for example. Default = 0.001 m/s porosity (float, field name or array of float) – the drainable porosity of the aquifer [-] Default = 0.2 recharge_rate (float, field name, or array of float) – Rate of recharge, m/s Default = 1.0e-8 m/s regularization_f (float) – factor controlling the smoothness of the transition between surface and subsurface flow courant_coefficient (float (-)) – The muliplying factor on the condition that the timestep is smaller than the minimum link length over groundwater flow velocity. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. Default = 0.5 vn_coefficient (float (-)) – The multiplying factor C for the condition $$dt >= Cdx^2/(4D)$$, where $$D = Kh/n$$ is the diffusivity of the Boussinesq equation. This arises from a von Neumann stability analysis of the Boussinesq equation when the hydraulic gradient is small. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. Default = 0.8 callback_fun (function(grid, substep_dt, kwargs)) – Optional function that will be executed at the end of each sub-timestep in the run_with_adaptive_time_step_solver method. Intended purpose is to write output not otherwise visible outside of the method call. The function should have two required arguments: grid: the ModelGrid instance used by GroundwaterDupuitPercolator substep_dt: the length of the current substep determined internally by run_with_adaptive_time_step_solver to meet stability criteria. K __init__(grid, hydraulic_conductivity=0.001, porosity=0.2, recharge_rate=1e-08, regularization_f=0.01, courant_coefficient=0.5, vn_coefficient=0.8, callback_fun=<function GroundwaterDupuitPercolator.<lambda>>)[source] Parameters: grid (ModelGrid) – Landlab ModelGrid object hydraulic_conductivity (float, field name, array of float or function.) – the aquifer saturated hydraulic conductivity, m/s. If function is given, it should take a landlab ModelGrid and return an array of floats at link. This may be used if the lateral hydraulic conductivity is not vertically homogenous and the effective hydraulic conductivity needs to be modified based upon on the position of the water table. See component tests for example. Default = 0.001 m/s porosity (float, field name or array of float) – the drainable porosity of the aquifer [-] Default = 0.2 recharge_rate (float, field name, or array of float) – Rate of recharge, m/s Default = 1.0e-8 m/s regularization_f (float) – factor controlling the smoothness of the transition between surface and subsurface flow courant_coefficient (float (-)) – The muliplying factor on the condition that the timestep is smaller than the minimum link length over groundwater flow velocity. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. Default = 0.5 vn_coefficient (float (-)) – The multiplying factor C for the condition $$dt >= Cdx^2/(4D)$$, where $$D = Kh/n$$ is the diffusivity of the Boussinesq equation. This arises from a von Neumann stability analysis of the Boussinesq equation when the hydraulic gradient is small. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. Default = 0.8 callback_fun (function(grid, substep_dt, *kwargs)) – Optional function that will be executed at the end of each sub-timestep in the run_with_adaptive_time_step_solver method. Intended purpose is to write output not otherwise visible outside of the method call. The function should have two required arguments: grid: the ModelGrid instance used by GroundwaterDupuitPercolator substep_dt: the length of the current substep determined internally by run_with_adaptive_time_step_solver to meet stability criteria. calc_gw_flux_at_node()[source] Calculate the sum of the groundwater flux leaving a node. (m2/s) calc_gw_flux_out()[source] Groundwater flux through open boundaries may be positive (out of the domain) or negative (into the domain). This function determines the correct sign for specific discharge based upon this convention, and sums the flux across the boundary faces. (m3/s) calc_recharge_flux_in()[source] Calculate flux into the domain from recharge. Includes recharge that may immediately become saturation excess overland flow. (m3/s) calc_sw_flux_out()[source] Surface water flux out of the domain through seepage and saturation excess. Note that model does not allow for reinfiltration. (m3/s) calc_total_storage()[source] calculate the current water storage in the aquifer (m3) courant_coefficient Courant coefficient for adaptive time step. Parameters: courant_coefficient (float (-)) – The muliplying factor on the condition that the timestep is smaller than the minimum link length over groundwater flow velocity. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. n drainable porosity of the aquifer (-) number_of_substeps The number of substeps used by the run_with_adaptive_time_step_solver method in the latest method call. recharge recharge rate (m/s) run_one_step(dt)[source] Advance component by one time step of size dt. Parameters: dt (float) – The imposed timestep. run_with_adaptive_time_step_solver(dt)[source] Advance component by one time step of size dt, subdividing the timestep into substeps as necessary to meet stability conditions. Note this method returns the fluxes at the last substep, but also returns a new field, average_surface_water__specific_discharge, that is averaged over all subtimesteps. To return state during substeps, provide a callback_fun. Parameters: dt (float) – The imposed timestep. vn_coefficient Coefficient for the diffusive timestep condition in the adaptive timestep solver. Parameters: vn_coefficient (float (-)) – The multiplying factor C for the condition dt >= Cdx^2/(4D), where D = Kh/n is the diffusivity of the Boussinesq equation. This arises from a von Neumann stability analysis of the Boussinesq equation when the hydraulic gradient is small. This parameter is only used with run_with_adaptive_time_step_solver and must be greater than zero. Returns array of hydraulic conductivity on links, allowing for aquifers with laterally anisotropic hydraulic conductivity. Parameters: K ((2x2) array of floats (m/s)) – The hydraulic conductivity tensor: [[Kxx, Kxy],[Kyx,Kyy]]	{}
FREE lectures this September + October \| HSC Head Start + Exam Revision: book here \| QCE Head Start: book here \| VCE Exam Revision: book here September 20, 2019, 02:42:19 am ### AuthorTopic: Can I take Unit 3/4 Physics without Units 1/2? (Read 906 times) Tweet Share 0 Members and 1 Guest are viewing this topic. #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Can I take Unit 3/4 Physics without Units 1/2? « on: September 23, 2018, 11:07:46 pm » 0 Hi... So, my question is pretty much in the title. I'm just wondering if anyone here has done so and what their experience was like (I will be taking this subject through DECV if I do it next year, does anyone have particular experience with Physics at DECV?). Maths isn't my strongest suit - I'm not bad at it per say, I'm just more of an English based person. Within maths however, algebra was where I thrived and I've heard algebra is a big part of Physics. To be honest, I wouldn't have even considered taking it (although it does seem quite interesting) but I've made a few bad subject decisions and have pushed myself into a corner with only a few options for uni courses (and medicine courses are rather competitive). At the moment it's Physics without 1/2 or Methods without 1/2, and I am leaning towards the former. Any help would be appreciated! I'm not really sure what to expect with the subject whatsoever #### S200 • Part of the furniture • Posts: 1100 • Yeah well that happened... • Respect: +238 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #1 on: September 24, 2018, 07:49:34 am » +5 Hey there! I did Physics this year, and from what I recall, there is very little overlap between 3/4 and 1/2. Algebra is pretty big, particularly for projectile motion, and a good grasp of the trigonometric ratios would be beneficial as well, going into the subject. So, here is the study design for Physics. I think the main issues with not taking 1/2 is that you miss out on the electricity module in Unit 1 AOS 2, and the basic force concepts in Unit 2 AOS 1. This second one is the one you'll have to google a little bit on, but YouTube should be able to help if you run into any real issues down the track... Overall, I'd highly recommend that you take Physics 3/4, as it's much easier to pick up than Methods, and you get to take an A3 summary sheet into the exam, which should fit all the formulae you will ever need... « Last Edit: September 24, 2018, 07:55:00 am by S200 » Carpe Cerevisi $\LaTeX$ - $e^{\pi i }$ #ThanksRui! - #Rui$^2$ - #Jamon10000 5233718311 #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #2 on: September 24, 2018, 12:09:36 pm » 0 Hey there! I did Physics this year, and from what I recall, there is very little overlap between 3/4 and 1/2. Algebra is pretty big, particularly for projectile motion, and a good grasp of the trigonometric ratios would be beneficial as well, going into the subject. So, here is the study design for Physics. I think the main issues with not taking 1/2 is that you miss out on the electricity module in Unit 1 AOS 2, and the basic force concepts in Unit 2 AOS 1. This second one is the one you'll have to google a little bit on, but YouTube should be able to help if you run into any real issues down the track... Overall, I'd highly recommend that you take Physics 3/4, as it's much easier to pick up than Methods, and you get to take an A3 summary sheet into the exam, which should fit all the formulae you will ever need... Thank you so much for your reply! The idea of taking Methods without Units 1/2 was a fair bit more daunting than taking Physics without 1/2, because I assumed Methods would have more prior knowledge needed, but I wanted to do some more digging about Physics before I decide on anything. I've only done General 1/2 (one of my many silly subject decisions in terms of prerequisites) It's great to know which areas I'll need to know more about, so again, thank you for that. The Physics teacher at my school has recommended the 1/2 textbook, but I think I'd prefer to look for alternate resources. Is there anyone in particulular that you'd reccomend on YouTube? And that's good to know that in Physics you get a formula sheet for the exam. I'm usually fine with understanding concepts, but have a rather terrible memory when it comes to formulas for some reason #### Bri MT • VIC MVP - 2018 • National Moderator • ATAR Notes Superstar • Posts: 2615 • invest in wellbeing so it can invest in you • Respect: +1746 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #3 on: September 24, 2018, 12:26:28 pm » +5 I found that KhanAcademy had some good physics videos Not only do you get to bring in your own double sided A3 sheet, there's also a provided formula sheet - so remembering formulas really isn't a necessary skill 2018-2021: Science Advanced - Global Challenges (Honours) @ Monash Leadership ; Scientific Methodology ; Wanting to stay productive? #### Vaike • MOTM: JAN 18 • Victorian • Forum Obsessive • Posts: 242 • Respect: +219 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #4 on: September 24, 2018, 12:36:19 pm » +5 I found that KhanAcademy had some good physics videos 100% agree with this, KA has some fantastic physics videos that would be great to have a brief watch of to get an idea of what you'll study in 3/4 Physics. Particularly, the topics you'd wanna look at include: • One dimensional motion • Two dimensional motion • Forces and Newton's laws of motion • Centripetal force and gravitation • Work and energy That's a lot of topics, but those are topics that would be great to have a bit of an idea about before starting 3/4 :-) 2016: Biology \| Further 2017: Chemistry \| English \| Methods \| Physics \| Specialist 2018 - 2021: BSc Advanced - Research [Computational Science and Mathematics] @ Monash My guide to VCE Chemistry #### S200 • Part of the furniture • Posts: 1100 • Yeah well that happened... • Respect: +238 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #5 on: September 24, 2018, 01:10:15 pm » +3 Thank you so much for your reply! The idea of taking Methods without Units 1/2 was a fair bit more daunting than taking Physics without 1/2, because I assumed Methods would have more prior knowledge needed, but I wanted to do some more digging about Physics before I decide on anything. I've only done General 1/2 (one of my many silly subject decisions in terms of prerequisites) It's great to know which areas I'll need to know more about, so again, thank you for that. The Physics teacher at my school has recommended the 1/2 textbook, but I think I'd prefer to look for alternate resources. Is there anyone in particular that you'd recommend on YouTube? And that's good to know that in Physics you get a formula sheet for the exam. I'm usually fine with understanding concepts, but have a rather terrible memory when it comes to formulas for some reason Anytime... I generally watched Professor Dave Explains or Crash Course Physics, depending on what I wanted to learn... PDE Classical Physics and Modern Physics playlists... Crash Course Physics As the others have said, Khan Academy is good as well. « Last Edit: September 24, 2018, 01:11:57 pm by S200 » Carpe Cerevisi $\LaTeX$ - $e^{\pi i }$ #ThanksRui! - #Rui$^2$ - #Jamon10000 5233718311 #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #6 on: September 24, 2018, 04:03:27 pm » 0 I found that KhanAcademy had some good physics videos Not only do you get to bring in your own double sided A3 sheet, there's also a provided formula sheet - so remembering formulas really isn't a necessary skill Thank you miniturtle I've had a look at Khan Academy before for other subjects and I just checked out the Physics section. I'm sure this will really come in handy. That's great about the formula sheet! It makes me a bit less worried about possibly taking on Physics without 1/2. I need a bare minimum of 25, but I'd like to do well in the subject if I can. #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #7 on: September 24, 2018, 04:08:06 pm » 0 100% agree with this, KA has some fantastic physics videos that would be great to have a brief watch of to get an idea of what you'll study in 3/4 Physics. Particularly, the topics you'd wanna look at include: • One dimensional motion • Two dimensional motion • Forces and Newton's laws of motion • Centripetal force and gravitation • Work and energy That's a lot of topics, but those are topics that would be great to have a bit of an idea about before starting 3/4 :-) Thank you for the list of topics Vaike! I've had a few people tell me a couple of different topics I should look into, but it's great to have a few more. I like to be ahead of the game, so the idea of possibly starting off next year with a subject like Physics without 1/2 makes me want to bury my head in anything and everything possibly relevant to Physics 3/4 #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #8 on: September 24, 2018, 04:17:06 pm » +2 Anytime... I generally watched Professor Dave Explains or Crash Course Physics, depending on what I wanted to learn... PDE Classical Physics and Modern Physics playlists... Crash Course Physics As the others have said, Khan Academy is good as well. Great! I'm familiar with Crash Course from other subjects (love their videos) but didn't know they did Physics too. I'll certainly look into it! I haven't heard of Professor Dave Explains, but I'll look into that too. Thanks for all you help #### Yertle the Turtle • MOTM: FEB 18 • Posts: 987 • Respect: +462 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #9 on: September 24, 2018, 04:47:19 pm » +3 You can write all of those responses in one post if you like. You just click the "Quote" link and it will add the quote to your response. I will just say that my experience of DECV has not been good at all, I have not found them helpful at all, and it's not been great for me, doing Spesh with them. I will say, though, that Physics is a great subject, and is really fun, I would definitely advise you to pick it up. Good luck! 2017-2018: VCE Methods \| Specialist \| Physics \| Chemistry \| English \| Texts and Traditions 2019-2022: B. Eng (Hons) \| Monash University Advice for Yr 12 VCE Subject Reviews Offering tutoring in Physics, Chemistry or T&T in the Maroondah, Whitehorse or Monash areas. Have counted to 80 #### KatherineGale • Trailblazer • Posts: 27 • Respect: +2 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #10 on: September 24, 2018, 09:51:33 pm » +2 You can write all of those responses in one post if you like. You just click the "Quote" link and it will add the quote to your response. I will just say that my experience of DECV has not been good at all, I have not found them helpful at all, and it's not been great for me, doing Spesh with them. I will say, though, that Physics is a great subject, and is really fun, I would definitely advise you to pick it up. Good luck! So that's how you do it! Thank you I spent fifteen minutes trying to figure it out and then gave up :/ I was not made out for this technological era. You're not alone with your experience with DECV. I've heard of a lot of people who didn't have a good time, in particular with some teachers. I've been doing DECV since 2015 (life with chronic migraines suck :/ ) and have had my fair share of mixed experiences. I've been blessed with two amazing teachers in the last couple of years though, so I'm lucky in that sense. And thank you! I'm certainly leaning towards taking it, I just want to get my study scores from this year to say yes indefinitely. It seems like a great subject and the more I look into it, the more I'm liking it (I'm being optimistic, although I'm sure I'll be kicking myself next year when I'm in over my head ) #### S200 • Part of the furniture • Posts: 1100 • Yeah well that happened... • Respect: +238 ##### Re: Can I take Unit 3/4 Physics without Units 1/2? « Reply #11 on: September 24, 2018, 11:12:23 pm » 0 And thank you! I'm certainly leaning towards taking it, I just want to get my study scores from this year to say yes indefinitely. It seems like a great subject and the more I look into it, the more I'm liking it (I'm being optimistic, although I'm sure I'll be kicking myself next year when I'm in over my head ) It'll be ok. If you feel like you're missing something, there's a great community here on AN, so you always have options... Carpe Cerevisi $\LaTeX$ - $e^{\pi i }$ #ThanksRui! - #Rui$^2$ - #Jamon10000 5233718311	{}
Copied to clipboard G = C3×C62.C22order 432 = 24·33 Direct product of C3 and C62.C22 Series: Derived Chief Lower central Upper central Derived series C1 — C3×C6 — C3×C62.C22 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — Dic3×C3×C6 — C3×C62.C22 Lower central C32 — C3×C6 — C3×C62.C22 Upper central C1 — C2×C6 Generators and relations for C3×C62.C22 G = < a,b,c,d,e \| a3=b6=c6=1, d2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=c3d > Subgroups: 448 in 162 conjugacy classes, 56 normal (24 characteristic) C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C3×C4⋊C4, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C3×C62, C62.C22, C3×Dic3⋊C4, Dic3×C3×C6, C6×C3⋊Dic3, C3×C62.C22 Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S32, S3×C6, Dic3⋊C4, C3×C4⋊C4, C6.D6, D6⋊S3, C322Q8, C3×Dic6, S3×C12, C3×C3⋊D4, C3×S32, C62.C22, C3×Dic3⋊C4, C3×C6.D6, C3×D6⋊S3, C3×C322Q8, C3×C62.C22 Smallest permutation representation of C3×C62.C22 On 48 points Generators in S48 (1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48) (1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48) (1 14 5 18 3 16)(2 15 6 13 4 17)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 26 21 28 23 30)(20 27 22 29 24 25)(31 41 33 37 35 39)(32 42 34 38 36 40) (1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45) (1 21 4 24)(2 20 5 23)(3 19 6 22)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 44 40 47)(38 43 41 46)(39 48 42 45) G:=sub<Sym(48)\| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,14,5,18,3,16)(2,15,6,13,4,17)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,26,21,28,23,30)(20,27,22,29,24,25)(31,41,33,37,35,39)(32,42,34,38,36,40), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,44,40,47)(38,43,41,46)(39,48,42,45)>; G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,14,5,18,3,16)(2,15,6,13,4,17)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,26,21,28,23,30)(20,27,22,29,24,25)(31,41,33,37,35,39)(32,42,34,38,36,40), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,44,40,47)(38,43,41,46)(39,48,42,45) ); G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,14,5,18,3,16),(2,15,6,13,4,17),(7,44,11,48,9,46),(8,45,12,43,10,47),(19,26,21,28,23,30),(20,27,22,29,24,25),(31,41,33,37,35,39),(32,42,34,38,36,40)], [(1,33,4,36),(2,34,5,31),(3,35,6,32),(7,27,10,30),(8,28,11,25),(9,29,12,26),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,29,16,26),(14,28,17,25),(15,27,18,30),(37,44,40,47),(38,43,41,46),(39,48,42,45)]]) 90 conjugacy classes class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 12A ··· 12AF 12AG 12AH 12AI 12AJ order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 6 6 6 6 18 18 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18 90 irreducible representations dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + - + + - - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 S32 C6.D6 D6⋊S3 C32⋊2Q8 C3×S32 C3×C6.D6 C3×D6⋊S3 C3×C32⋊2Q8 kernel C3×C62.C22 Dic3×C3×C6 C6×C3⋊Dic3 C62.C22 C3×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C3⋊Dic3 C6×Dic3 C32×C6 C32×C6 C62 C2×Dic3 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 2 1 1 2 4 4 4 4 2 2 4 8 8 8 1 1 1 1 2 2 2 2 Matrix representation of C3×C62.C22 in GL8(𝔽13) 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 , 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 , 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 , 10 6 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 9 2 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 , 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 G:=sub<GL(8,GF(13))\| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[10,7,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >; C3×C62.C22 in GAP, Magma, Sage, TeX C_3\times C_6^2.C_2^2 % in TeX G:=Group("C3xC6^2.C2^2"); // GroupNames label G:=SmallGroup(432,429); // by ID G=gap.SmallGroup(432,429); # by ID G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,92,2028,14118]); // Polycyclic G:=Group<a,b,c,d,e\|a^3=b^6=c^6=1,d^2=e^2=b^3,ab=ba,ac=ca,ad=da,ae=ea,bc=cb,bd=db,ebe^-1=b^-1,dcd^-1=c^-1,ce=ec,ede^-1=c^3*d>; // generators/relations ׿ × 𝔽	{}
### Impressions from NIPS 2015 Last week I attended the NIPS conference and it felt like grappa shot: intense but good for brain function. There are so many advances in research, and industry is shipping ML in products, and GPUs make previously-impossible things possible. Definitely an exciting time to be. ### Monday The opening talk was about deep earning. In fact, a lot of the conference was about deep learning. Non-conformist as I am, I tried not to focus too much on that. All the new applications and interest from industry is great, but I don’t think the research is that revolutionary. I read this review paper Deep learning (paywall, door) and I’m going to limit myself to this level of understanding for now. With 4000 people in one place and 500+ posters to look, it’s hard enough to keep track of topic-modelling topics covered! ### Saturday I attended the Bayesian Nonparametrics workshop which was the who-is-who of the community. I figured that was my only chance to be in a community where I’ll understand more than every second word said. The morning started with a very interesting “theory” talk by Peter Orbanz. I’m sure he’ll post the slides at some point, but in the meantime I found a 100pp PDF of lecture notes by him: Notes on Bayesian Nonparametrics. There’s also a video of a workshop from 4 years ago. This guy knows his stuff, and knows how to explain it too. Another excellent talk was by Mike Hughes on Scalable variational inference that adapts the number of clusters. This looked like good ideas to manage fragmentation (too many topics) and finally starts to show BNP’s killer app — automatically learning the right number of topics for a given corpus. During the discussion panel, the question of open source code for BNP arose and the following projects were mentioned: bnpy and BNP.jl. Around lunch time I caught part of the talk by David Blei which talked about the papers Black Box Variational Inference and Hierarchical Variational Models. Very interesting general-purpose methods. I should look into some source code, to see if I can understand things a bit better. In the afternoon, Amr Ahemed gave an interesting talk about large-scale LDA and efficient LDA sampling using alias method. First for data-parallelism, the workload can be split to thousands of machines, and each machine keeps topic-sparse word-in-topic “counts replica” on individual machines (that syncs asynchronously with shared-global state). If global topic model knows about K different topics, the local node x need to know only about the $k_x$ topics that occur on the documents it will be processing, since $k_x << K$, this allows to push the $K$. Very neat. Another interesting trick they use is alias sampling which performs some preprocessing of any n-dimensional miltinomial distribution to allow to take samples from it efficiently. It doesn’t make sense if you want just one sample, but if you’re taking many samples then the upfront cost of creating the “alias distribution” is amortized overall. It feels like we’re seeing a 3rd-generation parallel LDA ideas start to come-up.	{}
# Compute Images/Aliases of CIC Interpolators/Decimators Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators. For these filters, all coefficients are equal to 1, and there are no multipliers. They are typically used when a large change in sample rate is needed. This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and the aliases of CIC decimators. ## 1. CIC Interpolators Figure 1 shows three interpolate-by-M filters (M is an integer) that all have the same impulse response [1,2]. Figure 1a performs upsampling by M, followed by two M-tap boxcar filters. Figure 1b replaces each boxcar filter with an equivalent implementation consisting of a comb and integrator. The comb sections each have a delay of M samples. Finally, Figure 1c groups combs together and integrators together, with the upsampling function between them. The combs now operate at the lower input sample rate, and the comb delay is one sample instead of M samples. This Hogenauer filter [3] is the most efficient way to implement a CIC filter. The frequency response of a single stage boxcar or CIC interpolator is: $$H(f)=\frac{sin(\pi fMT_s)}{sin(\pi fT_s)} \qquad(1)$$ where Ts is the interpolator’s output sample time. For an N-stage filter, the frequency response is: $$H(f)=\left[\frac{sin(\pi fMT_s)}{sin(\pi fT_s)}\right]^N \qquad(2)$$ The magnitude response has nulls at multiples of the input sample rate. Equation 1 is called the Dirichlet, or aliased sinc, function. One shouldn’t confuse it with the spectrum of a continuous time pulse of length T, which is $X(f)=sin(\pi fT)/(\pi fT)$. X(f) lacks the sine in the denominator. Figure 1. Equivalent two-stage interpolate-by-M filters. a) Cascaded boxcars b) CIC c) Hogenauer CIC Since Figure 1a has the same impulse response as the CIC interpolators, we can use it to model them. We’ll compute an N-stage impulse response simply by performing repeated convolutions of the boxcar impulse response. A Matlab function boxcar_interp.m is listed in Appendix A of the PDF version. Note that the boxcar coefficients are scaled by 1/M in the function. Here is an example to compute the impulse response and magnitude response of an M= 40 CIC filter with 4 stages: M= 40; % interpolation ratio Nstages= 4; % number of CIC stages x= [1 0 0 0]; % impulse input y= boxcar_interp(x,M,Nstages); % perform interpolation The impulse response y is plotted in Figure 2 (top). Given an input sample rate of 100 Hz and output sample rate of Mfs_in = 40100 = 4000 Hz, the magnitude response can be calculated as follows: fs_in= 100; % Hz input sample rate [h,f]= freqz(y/M,1,1024,Mfs_in); % /M compensates for upsampler gain H= 20log10(abs(h)); % dB magnitude response The magnitude response is plotted in Figure 2 (bottom). The response has nulls at multiples of fs_in = 100 Hz. Figure 2. Top: Impulse response of 4-stage CIC Interpolator, M= 40. Bottom: Magnitude response for input sample rate = 100 Hz. Now let’s look at the impulse response and magnitude response of a CIC interpolator versus the number of stages. The example is clearer if we keep M fairly low, so we’ll use M = 8. Here is the code for a single stage CIC interpolator with M = 8: fs= 1; % Hz output sample rate M= 8; % interpolation ratio x= [1 0 0 0]; % impulse input signal y1= boxcar_interp(x,M,1); % find one-stage interpolator output [h,f]= freqz(y1/M,1,2048,fs); % /M compensates for upsampler gain H1= 20log10(abs(h)); % dB magnitude response The impulse response shown in Figure 3a is that of a boxcar filter. The dB-magnitude response is shown as the top curve in Figure 4, which agrees with Equation 1. We can repeat the above for a two-stage interpolator, where everything is the same except for the function call and output signal name. The function call is: y2= boxcar_interp(x,M,2); % find 2-stage interpolator output The function convolves two boxcar responses to obtain the triangular impulse response shown in Figure 3b. Similarly, convolving the triangle response with a boxcar gives the impulse response of a three-stage filter shown in Figures 3c, and one more convolution yields the four-stage impulse response of Figure 3d. The magnitude responses (calculated by freqz or by Equation 2) of all filters are plotted in Figure 4. As shown, the stopband lobe levels decrease with increasing number of stages, while the bandwidths of the stopbands centered at multiples of the input sample rate (fs/M) increase. Note, however, that the 3-dB passband bandwidth decreases vs. number of stages. Figure 3. Impulse responses of CIC interpolators with M = 8. a) One-stage b) Two-stage c) Three-stage d) Four-stage. Figure 4. Magnitude responses of CIC filters with M = 8. fs = output sample rate. ## 2. Interpolator Output Spectrum with Images Given an input time sequence, it’s easy to compute the interpolator’s output signal and spectrum, including images. Here’s an example that computes the output of a three-stage, M = 8 interpolator having multiple sine inputs sampled at 100 Hz. The following code generates the sines, performs the interpolation, and computes the spectrum of the output signal. The function psd_simple.m, which I described in a post [4], is listed in Appendix B of the PDF version. M= 8; % interpolation ratio Nstages= 3; % number of stages % compute multiple-sine input x fs_in= 100; % Hz input sample rate T= 1/fs_in; % s input sample time f0= 5; f1= 9; f2= 13; % Hz sine frequencies N= 256; n= 0:N-1; % time index A= sqrt(2); % amplitude of each sinewave x= Asin(2pif0nT) + Asin(2pif1nT) + Asin(2pif2nT); % y= boxcar_interp(x,M,Nstages); % perform interpolation [PdB,f]= psd_simple(y,MN,Mfs_in); % output spectrum The spectrum of the interpolator output y (fs = 800 Hz) is plotted in Figure 5, along with the interpolator magnitude response (dashed line). The spectrum shows images centered at multiples of fs_in = 100 Hz. The large images at 87, 113, 187, 213, … Hz are due to the sine input at 13 Hz. Figure 5. Three-stage CIC interpolator output spectrum. M= 8, input sample rate = 100 Hz. ## 3. CIC Decimators Figure 6 shows equivalent implementations of two-stage decimate-by-M filters [1,2]. Figures 6a and 6b are like the interpolators in Figures 1a and 1b, except the upsampler at the input has been replaced by a downsampler at the output. Figure 6c groups the integrators on the input side and the combs on the lower-rate output side, with a downsampler between them. The comb delay is one sample instead of M samples. The integrators must be implemented using two’s-complement arithmetic such that they roll over when the signal amplitude exceeds the number of bits in the adder [1,2,3]. For a given value of M and a given number of stages, a decimator has the same frequency response as an interpolator, when evaluated relative to the decimator's input sample rate. This is apparent from comparing Figure 1a for an interpolator to Figure 6a for a decimator. Figure 6. Equivalent two-stage decimate-by-M filters. a) Cascaded boxcars b) CIC c) Hogenauer CIC ## 4. Decimator Output Spectrum with Aliases To find the time response of a CIC decimator, we’ll use the equivalent cascaded-boxcar decimator of Figure 6a. The N-stage impulse response (before downsampling) is computed by performing repeated convolutions of the boxcar impulse response. The function boxcar_dec.m is listed in Appendix A of the PDF version. Here’s an example using the same parameters -- M = 8 and Nstages = 3 -- used in section 2 for an interpolator. The input signal consists of four unwanted sines at different frequencies, sampled at 800 Hz. The following code generates the sines, performs decimation, and computes the spectra of the input and output signals. The function psd_simple.m is listed in Appendix B of the PDF version. M= 8; % decimation ratio Nstages= 3; % number of stages fs_in= 800; % Hz input sample rate % multiple-sine input to decimator T= 1/fs_in; % s input sample time f1= 90; f2= 94; f3= 214; f4= 218; % sine frequencies N= 4096; n= 0:N-1; % time index A= sqrt(2); % amplitude of each sinewave x= Asin(2pif1nT) + Asin(2pif2nT) + ... Asin(2pif3nT) +Asin(2pif4nT); % [PdB_in,fin]= psd_simple(x,N,fs_in); % input spectrum y= boxcar_dec(x,M,Nstages); % perform decimation [PdB_out,fout]= psd_simple(y,N/M,fs_in/M); % output spectrum The input spectrum is shown in the top of Figure 7, along with the decimator’s magnitude response (dashed line), which is the same as that of the interpolator in section 2. To avoid hiding the aliases, there is no “desired” input signal, just four unwanted sinewaves. Output sample rate is 100 Hz, and the output spectrum is shown in the bottom of Figure 7. The output spectrum shows the aliases: 6 Hz due to input at 94 Hz; 10 Hz due to input at 90 Hz; 14 Hz due to input at 214 Hz; 18 Hz due to input at 218 Hz. Note that the decimator output y has 515 samples, enough to provide adequate resolution in the spectrum. To get the 515 samples, we needed the input x to have 4096 samples (roughly M times the length of y). The function boxcar_dec.m can also be used to compute aliases due to modulated input signals; an example using the same decimator as above is shown in Figure 8. Again, to avoid hiding the aliases, there is no desired input signal, just the unwanted signals. Figure 7. Three-stage CIC decimator example. M= 8, input sample rate = 800 Hz. top: Input signal spectrum. bottom: Output spectrum. Figure 8. Another three-stage CIC decimator example. M= 8, input sample rate = 800 Hz. top: Input signal spectrum. bottom: output spectrum. ## References 1. Harris, Fredric J., Multirate Signal Processing for Communication Systems, Prentice Hall, 2004, Ch. 11. 2. Lyons, Richard G., Understanding Digital Signal Processing, Third Ed., Prentice Hall, 2011, section 10.14 3. Hogenauer, Eugene B., “An Economical Class of Digital Filters for Decimation and Interpolation”, IEEE Trans. On Acoustics, Speech, and Signal Processing, vol. ASSP-29, No. 2, April 1981, pp. 155 – 162. 4. Robertson, Neil C., “A Simplified Matlab Function for Power Spectral Density”, DSPrelated.com, March, 2020, https://www.dsprelated.com/showarticle/1333.php Neil Robertson November, 2020 [ - ] Comment by November 21, 2020 Hi Neil. This is an informative blog. I liked how your examples showed the specific aliasing, in your Figure 7 and 8, that takes place with decimation. [ - ] Comment by November 21, 2020 Hi Rick, I'm glad you liked it. I have a friend that called it "crude, but effective". Neil To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.	{}
# Force-free magnetic field Force-free magnetic field A force-free magnetic field is a type of field which arises as a special case from the magnetostatic equation in plasmas. This special case arises when the plasma pressure is so small, relative to the magnetic pressure, that the plasma pressure may be ignored, and so only the magnetic pressure is considered. The name "force-free" comes from being able to neglect the force from the plasma. Basic Equations Start with the simplified magnetostatic equations, in which the effects of gravity may be neglected: $0=- abla ho+mathbf\left\{j\right\} imesmathbf\left\{B\right\}.$ Supposing that the gas pressure is small compared to the magnetic pressure, i.e., then the pressure term can be neglected, and we have: $mathbf\left\{j\right\} imesmathbf\left\{B\right\} = 0$. From Maxwell's equations: $abla imesmathbf\left\{B\right\}=mu_\left\{0\right\}mathbf\left\{j\right\}$ $ablacdotmathbf\left\{B\right\}=0$. Vector identity: $ablacdot\left( abla imesmathbf\left\{B\right\}\right)=0$ The first equation implies that:$mu_\left\{0\right\}mathbf\left\{j\right\}=alphamathbf\left\{B\right\}$. e.g. the current density is either zero or parallel to the magnetic field, and where alpha is a spatial-varying function which must be determined. Combing this equation with Maxwell's equations and the vector identity leads to a pair of equations for alpha and B: $mathbf\left\{B\right\}cdot ablaalpha=0$ $abla imesmathbf\left\{B\right\}=alphamathbf\left\{B\right\}$ Physical Examples In the corona of the sun, the ratio of the gas pressure to the magnetic pressure is ~0.004, and so there the magnetic field is force-free. Mathematical Limits If the current density is identically zero, then the magnetic field is potential, i.e. the gradient of a scalar magnetic potential.:In particular, if $mathbf\left\{j\right\}=0$ :then $abla imesmathbf\left\{B\right\}=0$ which implies, that $mathbf\left\{B\right\}= ablaphi$. :The substitution of this into one of Maxwell's Equations, $ablacdotmathbf\left\{B\right\}=0$, results in Laplace's equation, :$abla^2phi=0$, :which can often be readily solved, depending on the precise boundary conditions. ::This limit is usually referred to as the potential field case. If the current density is not zero, then it must be parallel to the magnetic field, i.e., ::$mumathbf\left\{j\right\}=alpha mathbf\left\{B\right\}$ which implies, that $abla imesmathbf\left\{B\right\}=alpha mathbf\left\{B\right\}$, where $alpha$ is some scalar function. ::then we have, from above, ::$mathbf\left\{B\right\}cdot ablaalpha=0$ ::$abla imesmathbf\left\{B\right\}=alphamathbf\left\{B\right\}$ , which implies that ::$abla imes\left( abla imesmathbf\left\{B\right\}\right)= abla imes\left(alphamathbf\left\{B\right\}\right)$ ::There are then two cases::::Case 1: The proportionality between the current density and the magnetic field is constant everywhere . ::::$abla imes\left(alphamathbf\left\{B\right\}\right)= alpha\left( abla imesmathbf\left\{B\right\}\right)=alpha^2 mathbf\left\{B\right\}$ ::::and also ::::$abla imes\left( abla imesmathbf\left\{B\right\}\right)= abla\left( ablacdotmathbf\left\{B\right\}\right) - abla^2mathbf\left\{B\right\}=- abla^2mathbf\left\{B\right\}$, ::::and so ::::$- abla^2mathbf\left\{B\right\} =alpha^2 mathbf\left\{B\right\}$ :::::This is a Helmholtz equation. *Case 2: The proportionality between the current density and the magnetic field is a function of position. ::::$abla imes\left(alphamathbf\left\{B\right\}\right)= alpha\left( abla imesmathbf\left\{B\right\}\right)+ ablaalpha imesmathbf\left\{B\right\}=alpha^2 mathbf\left\{B\right\} + ablaalpha imesmathbf\left\{B\right\}$ :::: and so the result is coupled equations: ::::$abla^2mathbf\left\{B\right\}+alpha^2mathbf\left\{B\right\}= mathbf\left\{B\right\} imes ablaalpha$ and ::::$mathbf\left\{B\right\}cdot ablaalpha= 0$ :::::In this case, the equations do not possess a general solution, and usually must be solved numerically. ee also Laplace's equation * Helmholtz equation References * Low, Boon Chye, " [http://eaa.iop.org/index.cfm?action=summary&doc=eaa%2F2221%40eaa-xml Force-Free Magnetic Fields] ". November 2000. Wikimedia Foundation. 2010. ### Look at other dictionaries: • Magnetic field — This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For … Wikipedia • Magnetic pressure — is an energy density associated with the magnetic field. It is identical to any other physical pressure except that it is carried by the magnetic field rather than (in the case of gas) kinetic energy of the gas molecules. Interplay between… … Wikipedia • Magnetic Surveying in Archaeology (book) — Magnetic Surveying in Archaeology (Wormianum, 2008, ISBN 978 87 89531 29 8) is a book written by Russian archaeologist T. N. Smekalova together with O. Voss and S. L. Smekalov. In the book researches collected information about magnetic… … Wikipedia • Magnetic circuit — Magnetic Circuits Conventional Magnetic Circuits Magnetomotive force Magnetic flux Φ Magnetic reluctance Phasor Magnetic Circuits Complex reluctance Zμ … Wikipedia • Magnetic shark repellent — Magnetic shark repellents utilize permanent magnets, which exploit the sensitivity of the Ampullae of Lorenzini in sharks and rays (electrosense). This organ is not found on bony fishes (teleosts), therefore, this type of shark repellent is… … Wikipedia • Magnetic reluctance — Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is analogous to resistance in an electrical circuit, but rather than dissipating magnetic energy it stores magnetic energy. In likeness to the … Wikipedia • Magnetic shape-memory alloy — Magnetic shape memory alloys (MSMAs), or ferromagnetic shape memory alloys (FSMAs), are ferromagnetic materials which exhibit large strains under the influence of an applied magnetic field due to martensitic phase transformation. Magnetic shape… … Wikipedia • Force — For other uses, see Force (disambiguation). See also: Forcing (disambiguation) Forces are also described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate … Wikipedia • Magnetic monopole — It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Inst … Wikipedia • Magnetic levitation — This article is about magnetic levitation. For trains based on this effect, see Maglev. For the Ruby interpreter, see MagLev (Ruby interpreter). Levitating pyrolytic carbon Magnetic levitation, maglev, or magnetic suspension is a method by which… … Wikipedia	{}
The consecutive angles are supplementary to each other. Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle. In a parallelogram, opposite angles have equal lengths, and so in each parallelogram there are two pairs of equal angles. Tags: Question 8 . So for example, we want to prove that CAB is congruent to BDC, so that that angle is equal to that angle, and that ABD, which is this angle, is congruent to DCA, which is this angle over here. Area: base × height Perimeter: 2 x (sum of lengths of adjacent sides) Number of vertices: 4 Number of edges: 4 This property of parallelogram states that the adjacent angles of a parallelogram are supplementary. The properties of a parallelogram are: (i) The sum of all the angles of a parallelogram is always equal to {eq}360^\circ {/eq}. The diagonals of a parallelogram divide each other into two equal segments. Click hereto get an answer to your question ️ The sum of two opposite angles of a parallelogram is 130^o . The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. Solution: Let the two adjacent angles be x and 2x. ∠ I + ∠R = 180° ∠ I + 70° = 180° ∠ I = 180° − 70° ∠ I = A parallelogram is a quadrilateral that has opposite sides that are parallel. Since the opposite angles are equal in a parallelogram, therefore, ∠C = ∠A = 108° and ∠D = ∠B = 72° Hence, ∠A = 108°, ∠B = 72°, ∠C = 108° and ∠D = 72°. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. a quadrilateral with opposite angles to be supplementary is called cyclic quadrilateral. Properties ... Q. The opposite angles of a parallelogram are equal in measure. asked Aug 19, 2020 in Quadrilaterals by Sima02 ( 49.2k points) quadrilaterals In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The diagonal of a parallelogram always bisect each other. Click hereto get an answer to your question ️ Prove that \"the opposite angles of a cyclic quadrilateral are supplementary\". Measures of opposite angles of a parallelogram are (3x - 2)° and (50 - … Find the measure of each angle of the parallelogram . Opposite angles are congruent. When the angles of the parallelogram equal to 90 degrees, it forms a rectangle. They all add up to 360$$^\circ$$ ($$\angle A +\angle B +\angle C +\angle D = 360^\circ$$) Opposite angles are equal For example m∠ABD + m∠BDC =180°. 2. that is, the quadrilateral can be enclosed in a circle. (3x+5)° = (61-x)°4 x = 56°x = 14°so the angles are : 3x+5 = 47° a… Opposite sides are congruent. Two opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. If you just look […] (ii) The opposite sides of a parallelogram are of the same length. These properties concern its sides, angles, and diagonals. They are equal. Each diagonal of a parallelogram bisect it into two congruent triangles. Recall that the supplement of a right angle is another right angle. If one angle of a parallelogram is twice of its adjacent angle, find the angles of the parallelogram. Opposite angles are congruent, and adjacent angles are supplementary. The angles of a parallelogram are the 4 angles formed at the vertices.. This proves that opposite sides are equal in a parallelogram. False. Opposite angles are equal (angles "a" are the same, and angles "b" are the same) Angles "a" and "b" add up to 180°, so they are supplementary angles. Such angles are called a linear pair of angles. Opposite angles of a parallelogram are equal and adjacent angles are supplementary. 0 0. This is a result of the line BD being a … Eclipse-girl. Opposite angles of a parallelogram are always equal. therefore, the statement is false. Interior angles of a parallelogram. Q. It is a type of quadrilateral in which the opposite sides are parallel and equal.. A parallelogram has 4 internal angles. Consecutive angles are supplementary. Find the measure of the indicated angle in part A and determine all the missing angles in part B. Download the set (3 Worksheets) There are many different ways to solve this question. SURVEY . Consider the following figure: Proof: In $$\Delta ABC$$ and $$\Delta CDA$$, \[\begin{align} answer choices . 120 seconds . Supplementary and total 180 are the same thing, that applies to angles next to each other (so not opposite if u get what i mean) :P. Source(s): Maths GCSE www.flashingmonkey.com. The diagonals of a parallelogram are not equal but they bisect each other at the midpoints. Conversely, if the opposite angles in a quadrilateral are equal, then it is a parallelogram. In a parallelogram, the angles add up to 360˚. The diagonals bisect each other. What I want to do in this video is prove that the opposite angles of a parallelogram are congruent. The parallelogram has the following properties: Opposite sides are parallel by definition. The sum of the interior angles is 360°. $$\angle \red W = 40^{\circ}$$ since it is opposite $$\angle Y$$ and opposite angles are congruent. Consecutive angles are supplementary (add up to 180). It means the sum of the two adjacent angles is 180° Here, ∠A + ∠D = 180° ∠B + ∠C = 180° Since the adjacent angles of a parallelogram are supplementary. A rectangle is a parallelogram, so its opposite angles are congruent and its consecutive angles are supplementary. Since there are two pairs of angles and one of each pair is on each side, adjacent angles are supplementary (their sum makes 180˚) The following diagram explains: Since consecutive angles are supplementary If a pair of angles are supplementary, that means they add up to 180 degrees. Consecutive angles are supplementary. The figure is a parallelogram. True. Opposite sides are parallel: Opposite sides are equal in length. However, supplementary angles do not have to be on the same line, and can be separated in space. You know that the opposite angles are congruent and the adjacent angles are supplementary. The opposite angles of a parallelogram are congruent. Ex 3.3 Class 8 Maths Question 6. Parallelograms have opposite interior angles that are congruent, and the diagonals of a parallelogram bisect each other. If they were supplementary, they would each be 90º and it would be a rectangle, a specific type of parallelogram. In a parallelogram, opposite angles are equal. Your are correct in that statement. Find the measure of each of its angles. Properties of a Parallelogram - Property: The Opposite Sides of a Parallelogram … Maharashtra State Board Class 8 Maths Solutions Chapter 8 Quadrilateral: Constructions and Types Practice Set 8.3 Question 1. Two adjacent angles of a parallelogram have equal measure. In a cyclic quadrilateral, opposite angles are supplementary. The opposite sides of a parallelogram are equal in size to each other, and the opposite angles are equal - as we will show, using triangle congruence. The opposite sides of a parallelogram are congruent. The opposite internal angles of a parallelogram are equal and the adjacent angles are supplementary i.e., the sum of the adjacent angles should be equal to 180 degrees. In a parallelogram, sum of the adjacent angles are 180° ∴ x + 2x = 180° ⇒ 3x = 180° ⇒ x = 60° Thus, the two adjacent angles are 120° and 60°. Property 3: Consecutive angles in a parallelogram are supplementary. Actually, from this little bit of information, you know about all four angles of a rectangle. they need not be supplementary. Example 5 In a parallelogram RING, if m∠R = 70°, find all the other angles. If you start at any angle, and go around the parallelogram in either direction, each pair of angles you encounter always are supplementary - they add to 180°. Properties of a Parallelogram - Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram; Properties of a Parallelogram - Property: The Opposite Angles of a Parallelogram Are of Equal Measure. Other properties include: The diagonals of a parallelogram bisect each other; Any two adjacent angles are supplementary-- The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. and if they are, it is a rectangle. A rectangle is a parallelogram that has a right angle. Opposite angles are of equal measure and they are congruent to each other. Theorem 2. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Find the indicated angle (vertex) This set of PDFs is based on the properties of a parallelogram - opposite angles are congruent and adjacent angles are supplementary. The properties of the parallelogram are simply those things that are true about it. Solve for x. Given A parallelogram RING where ∠ R = 70° We know that,Adjacent angles of a Parallelogram are supplementary. In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. asked Sep 17, 2018 in Mathematics by Mubarak ( 32.5k points) circles Preview this quiz on Quizizz. The consecutive angles of a parallelogram are supplementary. In a rectangle, they are congruent; in a square, they are both perpendicular and congruent. 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# Talk:Rope around the Earth I'm putting this up here just to show that it was written after the final review. I feel it's important that someone addresses the error in the picture that I posted about earlier (on June 24th). Nordhr 16:32 29 July 2011 ## Final Review been Htasoff 19:17, 17 June 2011 (UTC)This page has been edited and finished for a while. The checklist was gone through a few weeks ago, so the comments on the checklist sections were not included, because I had not told to do so when I was reviewing it. Abram gave it the okay, but I wanted to formally post it for final review. just a few small comments left! AnnaP 13:25, 19 July 2011 (UTC) • Can you make this sentence "It should be noted that the Earth is not a perfect sphere. Not only does it bulge out around the equator, but mountains and valleys give it a rough surface. This roughness affects the amount of slack needed, as might be expected." link down to your explanation of concave/convex.Check • In your explanation of the convex geometry, can you explain that the only problem arises when things are locally concave?I'm uncertain what you mean. The problem would still exist if it were globally concave. True, but it's not globally concave, just locally. So, while the earth is globally concave, you can a little places where this doesn't work because of local concave-ness, such as caves. Clarified the scale. • This sentence "We will prove that this is the case for any convex shape. " is not exactly true. Your proof applies to any regular polygon. I'm sure the proof could be extended to convex polygons, but as is, it's more limited. Explained how, since that interior angles of all n-gons add to the same number, this finding applies to regular and non-regular shapes. • This sentence:"And θ has a value that we can use in the equation as well: " is a bit confusing. Can you change it to say something about using the general equation for a regular polygon?Check • Right after equation 14, you have a & theta that didn't turn itself into a proper theta. Check AnnaP I'm giving my notes in red. I have written some in each of the headings below, and additional comments will appear at the bottom of your response to the checklist. Here are the comments that still need addressing. I've pulled them out so that you don't have to sort through everything AnnaP 7/7 :There is one, very big accuracy issue to be fixed. Critical to this entire page is the assumption that the Earth is a perfect sphere which it is not. You need to make this assumption crystal clear to the reader, right up front, and remind the reader of it more than once.Check • Mentioning the equatorial bulge isn't quite enough--you should mention mountains. The Andes in Ecuador are quite beautiful, and very big! • the Earth is actually quite smooth, roughly as smooth as a billiard ball, as this anonymous person demonstrates via calculations [1]. Let me explain why I disagree so strongly with this--your equations depend on extreme accuracy. The thing is that your result of needed to add just 6.28 feet is totally dwarfed by the fact that the equator rises over 15,000 feet in Ecuador (it also rises to something like 6000ft in Kenya). It would take quite a bit more rope to make it even stretch up and over that those mountains. So, this lack of roundness does effect the problem, and I do believe that you should mention this. Abram, 7/12 Just to weigh in on this, Anna is right (as she explained to me) that the fact that mountains are pointy means that they do require significant extra rope length (plateaus wouldn't be a problem -- you can work out the calculations for yourself if you would like). • For a circle of any given radius, the mount of slack needed to raise a rope around the circumference one foot is 2π feet. So if you imagine the circumference of the Earth as a piece-wise composite of circles of different radii, then, as long as the rope is initially tight over all the bumps and crags, 2π feet is still all that's needed. See the image below. • At each topographic level, the shape is a circle. So the amount of slack needed to be added is 2π/ (the arc-length of the section). This will yield 2π slack needed for the entire object, as long as the rope is taught around the bumps. Now I realise that the verticle sections will be unaffected by tis increase in slack, however they don't have to. The one foot clearance can be created by moving the slack along the length of the rope, as in the second picture. Additionally, the topographical lines can be made infinitely close, creating a less jagged topographical cross-section. No matter how eneven the circumference of the world is, the 2π feet of slack should provide the clearance of 1 foot all around. • The shape does not even have to be circular or differentiable, only convex (it gets weird if it's not convex.). The third image shows a rope around a square of side length 1. The rope is 1 unit away from the square at all points. • the straight section of each side of the larger figure is 1 unit long. Each "corner" of the larger figure is a quarter circle with radius = 1. thus the perimeter of the larger figure as a whole is 1+1+1+1 + 2π(1)/4 + 2π(1)/4 + 2π(1)/4 + 2π(1)/4 = 4 + 2π, while that of the small square is 1+1+1+1 = 4. (4 + 2π) - 4 = 2π units of slack. QED • This is awesome enough to add to the page. D'oh, I forgot to round corners as parts of a circle. That makes a lot of sense. BUT something funny is going to happen at the base of a triangle--I'll create a picture to show what I mean . AnnaP 7/12 Yah, I noticed that too. It cant be exactly one foot away from the concave corner, but I'm pretty sure that once this is set as a limit, that will no longer be a problem. Even rounding the corners, vs translating the slack, will become a non issue in a limiting case. The primary problem is that we are now in fractal dimensions. The only thing that does not work with this model are the overhangs under cliffs and in caves. I think part of the problem is whether we define "one foot above the ground" at a point to mean "one foot up along the normal" or "one foot up along the radius". If you model mountains as triangles instead of a Reimann-sum looking thing, it seems more natural to define "one foot above the ground" in terms of the normal, which means that you then cannot use the argument about circles of different radii. See picture below. -Kate 20:47, 12 July 2011 (UTC) • What radius are you using for the Earth's radius? The polar and equatorial radii are significantly different from each other and from the mean radius. You mention wrapping the rope around the equator, but don't specify if you're using the equatorial numbers • This comment was not addressed • I'm using an decently accurate amalgam of mean radii. Unfortunately, this is a bit af a short-comming, however the radius I use is not inaccurate, and I don't think it is worth redoing every calculation on the page with an exact mean equatorial radius. • While I agree it isn't worth redoing your calculations, I do think that you should explain where your number is coming from. You can site a particular source for the number that you do use. Again, I think that this is important since your page makes a big deal out of very small numbers. Abram, 7/12: I also think that citing your number is a good idea. Anyway, it takes very little time, so why not? • I derived the value I used, but I don't exactly remember how I went about doing it. I began with a value given in metric, and remember at one point I converted from metric to imperial, which made it so that my value does not precisely match other values available. The value I use is accurate, but I can't exactly retrace my steps. Messages to the Future • Did not feel that there was anything in particular I needed to say. References and footnotes • All images are attributed except for those I made entirely myself. • Make sure to note this on the pages for the files themselves. When I click on the image it should be noted somehow Check • I cited both the book and the website that I used for the math on this page. Good writing The following items are just meant to be reminders. If one of these items needs clarification, or seems like a great idea that you don't know how to implement, see What Makes a Good Math Images Page?. Context (aka Generating interest aka Who cares?) • tried to turn the counter intuitive result into a logical one by explaining it via ratios, emphasizing how intuitive such an unintuitive result can be when considered from the proper perspective. • I think that your first paragraph at the very beginning of the page provides a nice context Thanks Quality of prose and page structuring • The beginning paragraph(s) of the page clearly define the topic or purpose of the page as a whole, and may outline the page or preview conclusions that will be reached later in the page. • All check, no further comment. Your overall structure works well, but here are some other fixes • The very first sentence of the page should be rephrased to avoid the phrase set aside by commas.I'm not too concerned about it, but I took care of it. • work on the entire "In the puzzle..." paragraph. You might want to define your variables in another bulleted list as you did above.The only variable not in the list is l, which doesn't occur in the diagram. I define it when I introduce it, so am uncertain what needs to be changed. • The sentence starting "Just as bizarre..." needs to be rephrased, potentially broken into two sentencesCheck • The "Here is a way to illustrate..." paragraph is not entirely necessary. It is also confusing in its current form. If you want to include these points, find another way to say it. Steve and I have been talking about the accuracy of the approximations, and this was the best way I could explain the varying degrees of accuracy. • Okay • Make one more pass on the entire piece and try to make your sentences simpler. In general, try to avoid having more than 2 commas in a given sentence. You tend towards what I call "Sentences of doom" which are long and confusing for the reader. Short and sweet is best. Integration of Images and Text • All check, no further comment. • Image 1 is not to scale. Make sure that you note that!I figured that was obvious, but made sure to note it explicitly anyway. • Use an image to explain your arclength equation Although I think it is a worthy topic, I also think that the image I have to explain it, and the explanation itself, will digress from the rest of the section. Perhaps it could go on another helper page? I vote for a helper page. Just leave the link red Check Connections to other mathematical topics • Because this is a specific puzzle, and has relatively simple math, not many connections are natural. It seems to be just a neat little find. I did connect the height section to an approximations page (not yet up), which may well link to many things like Taylor series, summation, etc. eventually. • You're in good shape here Check Examples, Calculations, Applications, Proofs • The proof for the puzzle is in the MME, followed by an discussion to a part of the original puzzle which I added. I discuss this second, much more complicated part as best I can, and hide it, as it is not necessary for the original puzzle. I would actually go through the calculation of how much distance you'd need to add if you wanted to lift the rope a different distance from the surface of the Earth--say 35,000ft, or the cruising elevation of a commercial get. Check Mathematical Accuracy and precision of language • The height section becomes more complicated, and had to be done with approximations. This topic was introduced, and a helper page proposed. Everything else is meticulously explained and defined. • There is one, very big accuracy issue to be fixed. Critical to this entire page is the assumption that the Earth is a perfect sphere which it is not. You need to make this assumption crystal clear to the reader, right up front, and remind the reader of it more than once.Check • Mentioning the equatorial bulge isn't quite enough--you should mention mountains. The Andes in Ecuador are quite beautiful, and very big! • the Earth is actually quite smooth, roughly as smooth as a billiard ball, as this anonymous person demonstrates via calculations [2]. Onto the smaller points • Make sure you clearly define what you mean by arc.See comment on arclength. • Similarly, looking at image 3 and reading your text, it is unclear if $x_0$ refers to the entire length where the rope is not touching the Earth or if it refers only to the length opposite $\theta$. This becomes clear later on, but it should be more clear up front. • Using l instead of $2 \pi$ is confusing in equations 4-6. If you want to do this, be sure to have an entire small paragraph explaining your use of l immediately before your equation. I do devote the first full paragraph of the section to why I use l I feel as though any more would be redundant, and I got comments telling me that parts of the page were redundant. • Can you make the "l" in that paragraph bigger, or bold to make it stand out more? That might be an easy fix to highlight that explanation. I can, but Steve has STRESSED consistancy in my variables almost every time I met with him about this page. • I strongly advice against using rounded numbers anywhere on the page, given that your work depends on very accurate numbers. • This comment was not addressedI agree for the calculations, but disagree for the explanations. The math obviously needs exact values, but when describing the scope, ratios, and overall gist of the puzzle, I firmly believe that smoothing the numbers allows for a more straightforward explanation. • What radius are you using for the Earth's radius? The polar and equatorial radii are significantly different from each other and from the mean radius. You mention wrapping the rope around the equator, but don't specify if you're using the equatorial numbers • This comment was not addressed I'm using an decently accurate amalgam of mean radii. Unfortunately, this is a bit af a short-comming, however the radius I use is not inaccurate, and I don't think it is worth redoing every calculation on the page with an exact mean equatorial radius. your definition of "argument" in the mouse over isn't going to help anyone who doesn't already know what argument means. Try to rework that. • This comment was not addressed Check • rephrase the "root finder" mouse over to be a bit more clear or simply explain that you are finding the roots without using such specific phrasing Check Layout • All check, no further comment Misc • Can you make your equation numbering consistent through the entire page? You switch methods part of the way through, which throws me off. If you want to separate out the equation numbering by section, you can always make it "Equation 1-1" or some similar method.Check ## General Comments not on one section I was looking at the main picture on this page, and it says that the rope is 1 ft longer than the circumference of the Earth. Should it be 6.28 feet longer and the '?' is equal to one foot? Nordhr 14:44, 24 June 2011 (UTC) Kate 21:00, 26 May 2011 (UTC): • Your image links go to the captions, which I think can be confusing. Try moving the div tag so that it either just precedes the image or so that it goes around the image. CHECK Smaurer1 20:37, 24 May 2011 (UTC) The Why it is Interesting Section is somehow inside the More Mathematical Explanation Section, because it is hidden until the MME section is completely unhidden. This should not be. CHECK ## Opening Caption The wording "This is a puzzle in that the answer is surprising" feels a bit awkward to me. What about wording like this: There is a famous puzzle about a rope tied taut around the equator. It asks how much the rope must be lengthened so that it will be able to hover one foot above the surface of the earth around the entire equator. While finding the answer requires only basic geometry, even professional mathematicians find the answer strangely counter-intuitive.CHECK -Abram 5/24 I don't think you need a question mark after the first sentence. Rebecca 15:36, 25 May 2011 (UTC) CHECK ## Basic description Kate 21:00, 26 May 2011 (UTC): • I would leave some blank space at the end of your Basic Description so that the heading for the MME is all the way left instead of wrapping around your picture. CHECK • I also still find the placement of the max. height image confusing - if it could even go on the right I think that would be much better, because as it is, you notice the image before you start reading the basic description, and then that can get confusing. • This issue might also be solved by adding a caption to Image 1 I recognize your point, and tried to implement it; this is now the least awkward position I can find. • Re-wording suggestion for the last paragraph: Just as bizarre is that if this newly lengthened rope were lifted at one point to pull the rope taut around the earth again, the clearance under that point of lifting would be quite large.CHECK Also, maybe clarify that "this specific case" refers to the specific case of the earthCHECK -Abram 5/24 I would add a picture of the rope being lifted up so that it is taut again, and show what you mean by maximum clearance for the basic description. This will clarify what you're saying and its always good to have more pictures. The picture could be similar to that one you use in a more mathematical explanation, but you don't need the labeling so you could maybe use and earth instead of just a blue ball. Just a thought. CHECK Rebecca 15:37, 25 May 2011 (UTC) • Rebecca 20:09, 1 June 2011 (UTC) A suggestion about how to fix the placement of Image 1- I would crop the image so there's much less black space above and below the earth. I would also make the image slightly smaller, and I would add an extra space before the paragraph beginning with "Just..". This will help move the picture down to the second half of this section, and I think it will be less confusing. • Rebecca 22:11, 10 June 2011 (UTC) This looks much better. I would still consider adding an extra space before the paragraph with "Just" to help the layout. ## More mathematical explanation • Rebecca 12:34, 31 May 2011 (UTC) When you're explaining Image 4 in the mathematical section, I'm not sure if you need to keep linking back to the equations. I think you could just link back the first time and the paragraph will look less cluttered. CHECK Kate 21:00, 26 May 2011 (UTC): • The puzzle states that we've lengthened the rope and made the rope hover 1 foot of the surface of the earth. I don't like that this sentence is past tense-y… I would like it better if it was "The puzzle states that we have to lengthen the rope…" CHECK • Indeed, one can see that the additional 2π is a result of distributing the 2π to the 1 in Eq. II, which always yields 2π. CHECK I think it would be clearer to say "which yields an increase of 2π no matter what the radius of the ball is." • Typo: represents θ, the angle whose cosine is angle whose cosine is CHECK • Taut/taught typo: . Because the rope is taught around most of the earth, CHECK When you point out that R_2 is 1 foot larger than R_1, maybe remind the reader that this is because the puzzle states that we've made the rope to hover 1 foot of the surface of the earth. This is an example of a general principle that when you are deriving an equation, statement, etc, it's good to always be explicit on where each statement/step is coming from (in this case, the hypotheses of the problem).CHECK -Abram, 5/24 Two problems with the statement: "Indeed, one can see that the additional 2 π is a result of extending the radius of the rope circle by one foot, an extension that will by definition be the same no matter the initial radius of the object being enclosed": • The small problem is that the extension is not independent of the initial radius "by definition"; if it were a definition, you wouldn't have had to prove it just now!CHECK • The larger problem is that one might not see why the 2pi extension is independent of the initial radius, and you don't do anything to explain why. Maybe add a couple sentences pointing out that because you didn't make the value of R_earth any number in particular, that your answer didn't depend on the specific radius of the earth in any way, so this 2pi extension would be the same for any ball or planet or star of any size.CHECK -Abram, 5/24 GREAT GREAT first section of a more mathematical description. I found it very clear.Rebecca 15:39, 25 May 2011 (UTC) GOOD ### Maximum Height of Rope See if you can be a little more careful or precise in your definition of x_1. If you don't know what I mean, let me know.CHECK -Abram, 5/24 As we just discussed in person, the phrasing in the paragraph, "Now we will develop a formula..." signals to the reader that the amount of slack is something that has to be derived through a new formula, not something that we just learned to be 2*pi. Do something to make it clear that the latter is actually the case, and that this "generalization" refers to an entirely different (as yet unidentified) puzzle. -Abram, 5/24 In formula A, the variables L_arc and r are undefined.CHECK -Abram, 5/24 In the sentences beginning "Using the Pythagorean Theorem..." and "Also, since theta is the angle whose cosine", point out which triangle readers should be looking at. • It could help to name the vertices with letters (yes, I know this would mean changing the drawing again). CHECK • Also, at some point you should maybe justify why, or at least explicitly state, that the angle between the radius of the earth and the tangent line from that radius to the apex, is 90 degrees. -Abram, 5/24 The sentence "Keeping in mind..." seems mighty redundant. It's really unusual that I'll say something has been over-explained, but in this case, maybe it has. CHECK -Abram, 5/24 "Keeping in mind..." and "Where theta is the angle..." are not complete sentences, so the first word shouldn't be capitalized. The standard, I think, is to put a comma after the equation, and then make the words "keeping" and "where" lower-case.CHECK -Abram, 5/24 The derivation of Eq. 3 could be expanded in a few ways: • Add an introductory sentence stating the role of what you are about to do, e.g. Now that we have a formula for x_0, were are going to relate x_0 to x_1.CHECK • Point out that because the rope is taught around most of the earth, all the slack comes from lengthening arc AC to segments AB and BC (see what I mean about adding names to these points)? CHECK -Abram, 5/24 A few things about the paragraph "due to the presence of h within the cos^-1": • It should say "...within the argument of cos^-1", with maybe a mouse-over explaining that "argument" means "input" (or you can just use "input") CHECK • The problem isn't that h is within the cos^-1; it's that h is within the cos^-1 and it's also somewhere else in the equation. You might talk to Steve to get a somewhat precise statement that describes this phenomenon. CHECK I THINK • Maybe add a mouse-over explaining what a root-finder isCHECK • Many of these equal signs should be approximately equals signsCHECK I THINK -Abram, 5/24 Just FYI, I haven't read the series section at all yet. -Abram, 5/24 No comma after "again be held taut" Rebecca 15:42, 25 May 2011 (UTC)CHECK ## Why It's Interesting I would change you second and third sentences of "Why it's interesting" to something like "However, it is also interesting to explore why this confusing result proves true." I just think the way you have it is a little confusing. I realize what you are saying, but I can't find the proper connective to use. All the ones I've tried only slowed down the text. Also, I don't know what image to use, if you have an idea, let me know. Can you think of a picture to add to this section as well? Rebecca 15:43, 25 May 2011 (UTC) Smaurer1 10:13, 20 May 2011 (UTC) First, look up these two words; they don't mean the same thing and you only mean accuracy. Second, there are two meanings of accuracy, absolute and relative. I would say that the second order approximation is much much more accurate than the first order approximation. Perhaps we need a helper page on accuracy. I know that there is a difference, and am pretty sure I mean both. Both approximations are accurate, the first to 1 decimal place, the second to three. Both are precise, the both yielded 3 decimal places of precision, with the second being more accurate over the three places. The intro should make clear the sense of puzzle. It is not a puzzle in that it is hard. It is a puzzle in that it is counterintuitive to most people. The second question, not even mentioned in the current intro, is even more counterintuitive. NEED TO DO Becky raises an important point below about explanations that come after formulas. The rule is that this is not allowed unless it is made clear at the time of the formula or beforehand that the explanation is coming afterwards. At the very least the formula was end with a comma, not a period. But better to say "as we show below..." CHECK It's good that Harrison made a table of his variables, but in fact his very first equation doesn't agree with his definitions. Then he goes on to use new variables (with rope1 and rope2) that he never defined. CHECK The Why It's Interesting section is not really about why it's interesting. It is about how Harrison comes to peace with the nonintuitive answers; the non-intuitiveness is why it is interesting. I am putting a few comments in the document itself and handing Harrison a paper copy with many more comments. We will go through it in person, but maybe not today as I am busy with honors examiners. When the rope is taut around the globe, its length equals the circumference of the Earth. $L_{rope 1}=C_{earth}=2\pi\,\!R_{earth}$ Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from: Rrope 1= Rearth to Rrope 2= Rearth+1 ft. All this can now be simplified and the sub1-2 notation eliminated Below are comments by Becky on May 19-20 Harrison, this is a great page already! I really like the idea and the execution. The topic is visual and I find it very interesting! • Its great that your paragraphs are so short! The page is easy to read as a result. • Great usage of the mouseover for pythagorean theorem. • Overall, the message is clear. • Good use of bullets to define the variables. Some suggestions I have: • "Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from:..." I'm not sure that this is the best way to say this. Maybe try to reword it a bit. NOT DONE YET • Also, as a pointer, we learned last year that people get scared off by words like "simply" and phrases like "we can easily see that" because its intimidating if they haven't grasped the concept yet and they don't find it simple. I actually think your usage of "simply" isn't bad, but later you say "now it is clear that," which you should probably avoid. CHECK • Some of your equations look like this: Rrope 1= Rearth while others look like this: $L_{rope 2}=2\pi\,\!R_{earth}+2\pi\,\!$ I think the issue is that the first equation isn't encased in math brackets. You should be consistent and put these in math font. DONE PURPOSEFULLY, IF I'M TALKING ABOUT EQUATIONS OF THE ENTITIES WITHIN THEM, I WANT IT TO BE HTML SIZE CHARACTERS. ALSO, THIS REDUCES THE RISK OF MAKING IT LOOK LIKE THERE ARE AN OVERWHELMING NUMBER OF EQUATIONS ON THE PAGE. • When you move from equation 1 to equation 2, its not immediately clear what's going on because you don't explain until after. I know that I personally wont continue reading until I understand where an equation came from (so even if the explanation is the next sentence I might miss it). So it might be good to explain what's going on before you give the equation. I'd consider doing something like this instead: Since θ is the angle whose cosine is $\frac{R}{R+h}$, we can replace θ with $cos^{-1} \left (\frac{R}{R+h} \right )$. Thus, Equation 1 is equivalent to: Eq. 2 $x_o=R cos^{-1} \left (\frac{R}{R+h} \right )$. CHECK • The paragraph that you begin with "As a result of the cos-1 (R/R + h)", you should try to be clear about why this means there is no explicit formula to find the height. CHECK, I THINK • I think you have a good start to the why it's interesting section. It does seem counter-intuitive that the increase would be so small when the earth is so large. However, I'm not sure that you're saying this in the best way. Try to work with the section and maybe ask the other researchers for help if you need it. Also, you might want to discuss how it seems like the size of the ball should matter. • I think a picture would be really good in this last section. One idea for a picture might be to help elaborate on the idea that the size of the ball doesn't matter. If you have a different idea- go with it! I hope these suggestions help! -Becky Harrison 5/19/11 Rope around the Earth is ready to be reviewed. when you say you put it into a calculator and got 614. , what did you plug in? You might want to define a taylor approximation with the green mouse over thing. What about adding a section on the shortcomings of the problem: for example, the earth is not exactly spherical??? Also, I'm not exactly sure about our policy on this, but your initial description of the image is not a complete sentence. -Richard 5/19 These comments were made as regards the old version of the page, but I think several of them are still relevant: Some notes: • In the first sentence under the mathematical explanation, I don't think you should capitalize circumference or where. I think it should read "The circumference of a circle is given by the equation EQUATION, where r is the radius." CHECK • A couple typos near the end of that section: "Now it is clear that THE new length… feet longer than THE original length." CHECK • Typo in Maximum Height section: "what would the new distance FROM this point be?" CHECK • The sentence that just gives the Pythagorean theorem is confusing to me. I think it would read better if the next sentence said "Using the Pythagorean Theorem…" and then you put the theorem in a bubble. CHECK, AWESOME • I'd call it arclength, not the length of an arc USE "LENGTH OF AN ARC" TO EMPHASIZE THAT IT IS JUST A LINEAR DISTANCE. • As an issue of precision, "Where cos-1 represents the angle whose cosine is r/r+h" is not a true sentence - arccos is a function, not a variable, so it can't represent that value. I'd suggest explaining where theta is in the picture, and then saying something like, "We can find the value of theta by using arccosine…" NOT INACCURATE NOR IMPRECISE: IF f(x)=y, THEN f(x) CAN BE SUBSTITUTED WHEREVER y OCCURS. $\theta\,\!=cos^{-1}(x)$ IS AN EXAMPLE OF SUCH A CASE. -Kate 18:24, 19 May 2011 (UTC) Kate 18:28, 19 May 2011 (UTC) Comments made now that I'm looking at the current version of the page: • Set up the redirect ASAP CHECK • Noticed another typo under max. height- misspelled "taut" as "taught" CHECK • It would be awesome if you could edit the max. height image to include an R on the third side of the triangle and a theta by the angle in the middle. CHECK AND CHECK • Why do we want to find the length of x0? DISMISSED • All of the equation stuff you're doing is in order to find h in terms of L and r, yes? I think you should say so before you get into it. DISMISSED • In your Taylor approximations, I don't understand why you've written those fractions out the way they are instead of combining them into one constant. EASIER AND THAT'S HOW MAURER GAVE IT TO ME • "more precise than the first, but really only BY a negligible amount" CHECK • Your basic description and why it's interesting are excellent, the explanation is really well done, and the height extension is very interesting, although not as clearly explained as the rest of the page. WOO HOO -Kate 18:41, 19 May 2011 (UTC) ### Comments by xd 01:10, 25 May 2011 (UTC) Overall, I like the page. Short and sweet. I offer two suggestions. Under Max Height of Rope, Possible Extension 1. Root finder program -> can you write such a numerical root finder program in Matlab or Python? Matlab even has its own program. You can even explain that and show a sample code. It will be a very nice to have such a program in there for the more involved readers. The math of the this page is not very dense so it may be very appropriate to do this. You can refer to Prof. Mewes' Matlab exercise root finder. 2. Series solution -> do intend to write a helper page or a separate page for the explanation? From Eq. 5 to Eq. 6, there is a huge gap and you are not very clear on whose Taylor Series approximation are you using i.e. the sqrt or the cosine inverse? You also need to explain the order of approximation.	{}
## Abstract The paper presents detailed midspan experimental results from two transonic linear turbine cascades. The blades for the two cascades were designed for the same service and differ mainly in their leading-edge geometries. One of the goals of the study was to investigate the influence of the leading-edge metal angle on the sensitivity of the blade to positive off-design incidence. Measurements were made for incidence values of −10.0, 0.0, +4.5, +10.0, and +14.5 deg relative to design incidence. The exit Mach numbers varied roughly from 0.5 to 1.2 and the Reynolds numbers from about $4×105$ to $106.$ The measurements include the midspan losses, blade loadings and base pressures. In addition, the axial-velocity-density ratio (AVDR) was extracted for each operating point. The AVDR was found to vary from about 0.98 at −10.0 deg of incidence to about 1.27 at +14.5 deg. Thus, the data set also provides some evidence of the influence AVDR on axial turbine blade performance. Detailed experimental results for turbine blade performance at off-design incidence are very scarce in the open literature, particularly for transonic conditions. Among other things, the present results are intended to expand the database available in the open literature. To this end, the key aerodynamic results are presented in tabular form, along with the detailed geometry of the cascades. The results could be used in the development of new or improved correlations for use in the early stages of design. They could also be used to evaluate the ability of current CFD codes to capture reliably the variation in losses and other aerodynamic quantities with variations in blade incidence. 1. Goobie, S. M., Moustapha, S. H., and Sjolander, S. A., 1989, “An Experimental Investigation of the Effect of Incidence on the Two-Dimensional Flow,” Proc, IX International Symposium on Air Breathing Engines (ISABE), Sept., pp. 197–204. 2. Rodger, P., Sjolander, S. A., and Moustapha, S. H., 1992, “Establishing Two-Dimensional Flow in a Large-Scale Planar Turbine Cascade,” AIAA Paper, No. 92-3066. 3. Whitehouse , D. R. , Moustapha , S. H. , and Sjolander , S. A. , 1993 , “ The Effects of Axial Velocity Ratio, Turbulence Intensity, Incidence, and Leading Edge Geometry on the Mid-Span Performance of a Turbine Cascade ,” Can. Aeronautics Space J. , 39 , No. 3 , Sept., pp. 150 156 . 4. Benner , M. W. , Sjolander , S. A. , and Moustapha , S. H. , 1997 , “ Influence of Leading-Edge Geometry on Profile Losses in Turbines at Off-Design Incidence: Experimental Results and an Improved Correlation ,” ASME J. Turbomach. , 119 , Apr., pp. 193 200 . 5. Moustapha , S. H. , Kacker , S. C. , and Tremblay , B. , 1990 , “ An Improved Incidence Losses Prediction Method for Turbine Airfoils ,” ASME J. Turbomach. , 112 , Apr., pp. 267 276 . 6. Martelli , F. , and Boretti , A. , 1987 , “ Development of an Experimental Correlation for Transonic Turbine Flow ,” ASME J. Turbomach. , 109 , Apr., pp. 246 250 . 7. Kacker , S. C. , and Okapuu , U. , 1982 , “ A Mean Line Prediction Method for Axial Flow Turbine Efficiency ,” ASME J. Turbomach. , 104 , Jan., pp. 111 119 . 8. Saroch, M. F., 1996, “Contributions to the Study of Turbomachinery: Part I—Design of a Fish-Tail Diffuser Test Section. Part II: Computations of the Effects of AVDR on Transonic Turbine Cascades,” M. E. thesis, Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada, Jan. 9. Graham, C. G., and Kost, F. H., 1979, “Shock Boundary Layer Interaction on High Turning Transonic Turbine Cascade,” ASME Paper No. 79-GT-37. 10. Perdichizzi , A. , 1990 , “ Mach Number Effects on Secondary Flow Development of a Turbine Cascade ,” ASME J. Turbomach. , 112 , Oct., pp. 643 651 . 11. Detemple-Laake, E., 1991, “Detailed Measurements of the Flow Field in a Transonic Turbine Cascade,” ASME Paper No. 91-GT-29. 12. Moustapha , S. H. , Carscallen , W. E. , and McGeachy , J. D. , 1993 , “ Aerodynamic Performance of a Transonic Low Aspect Ratio Turbine Nozzle ,” ASME J. Turbomach. , 115 , July, pp. 400 408 . 13. Krieger , M. W. , Lavoie , J. P. , Vlasic , E. P. , and Moustapha , S. H. , 1999 , “ Off-Design Performance of a Single-Stage Transonic Turbine ,” ASME J. Turbomach. , , 121 Apr., pp. 177 183 . 14. Jouini, D. B. M., Sjolander, S. A., and Moustapha, S. H., 2000, “Aerodynamic Performance of a Transonic Turbine Cascade at Off-Design Conditions,” ASME Paper No. 2000-GT-0482. 15. Jeffries, M. S., Jouini, D., and Sjolander, S. A., 1997, “Determining the Sampling Rates and Times in a High Speed Wind Tunnel,” Proc., CASI 6th Symposium on Aerodynamics, Toronto, Canada, Apr. 16. Corriveau, D., 1999, private communication, Carleton University, Ottawa, Canada. 17. Jeffries, M. S., 2001, “Initial Investigations of Transonic Turbine Aerodynamics using the Carleton University High-Speed Wind Tunnel,” Ph.D. thesis, Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada, Jan. 18. Islam, A. M. T., 1999, “An Experimental and Computational Study of the Aerodynamics of Turbine Blades with Damage,” Ph.D. thesis, Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada, Sept. 19. Amecke, J., and Sˇafar˘ı´k P., 1995, “Data Reduction of Wake Flow Measurements with Injection of Other Gas,” DLR-FB 95-32, DLR, Ko¨ln, Germany. 20. Jouini, D. B. M., 2000, “Experimental Investigation of Two Transonic Linear Turbine Cascades at Off-Design Conditions,” Ph.D. thesis, Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada, July. 21. Sieverding , C. H. , Stanislas , M. , and Snoeck , J. , 1980 , “ The Base Pressure Problem in Transonic Turbine Cascades ,” ASME J. Eng. Power , 102 , July, pp. 711 714 . 22. Xu , L. , and Denton , J. D. , 1980 , “ The Base Pressure and Loss of a Family of Four Turbine Blades ,” ASME J. Turbomach. , 110 , Jan., pp. 9 17 . 23. Denton , J. D. , and Xu , L. , 1990 , “ The Trailing Edge Loss of Transonic Turbine Blades ,” ASME J. Turbomach. , 112 , Apr., pp. 277 285 . 24. Denton , J. D. , 1993 , “ Loss Mechanisms in Turbomachines ,” ASME J. Turbomach. , 115 , Oct., pp. 621 656 . 25. Mee , D. J. , Baines , N. C. , Oldfield , M. L. G. , and Dickens , T. E. , 1992 , “ An Examination of the Contributions to Loss on a Transonic Turbine Blade in Cascade ,” ASME J. Turbomach. , 114 , Jan., pp. 155 162 . 26. Sieverding, C. H., 1993, “Advanced Methods for Cascade Testing,” AGARDograph 328, AGARD-AG-328, Aug., pp. 22–34.	{}
# Machine Learning: Logistic Regression Logistic regression is a classification case of linear regression whith dependent variable $y$ taking binary values. Problem: Given a training set $\langle x^{(i)}, y^{(i)} \rangle$, $1 \le i \le m$, $x \in \mathbb{R}^{n+1}$, $x^{(i)} _ 0 = 0$, $y^{(i)} \in$ {0,1}, find classification function Let’s build function $h_\theta(x)$ as a sigmoid function of $\theta\cdot x$ Sigmoid function has rank infinity, i.e. it operates on scalars, vectors and matrices. To find optimal parameter $\theta \in \mathbb{R}^{n+1}$ we are going to use optimized gradient descent method which takes as arguments cost function $J(\theta)$ and its gradient. For logistic regression they are where $X = (x^{(i)}_j) _{m \times n+1}$ is a matrix of the training examples from the previous lecture. Analogous to linear regression, logistic regression can be regularized too Having computed $\theta$ we can now implement the prediction function which can be used to classify new examples and check the prediction accuracy on the training set ## Multi-class Classification Logistic regression works for binary $y$. Suppose now that $y^{(i)} \in${$1,…,K$}, where $K > 2$. In this case we can use One-vs-All variation of this algorithm. Step 1. Convert vector $y$ into a binary matrix $Y$ where $y^{(i)}_k = \delta _{k y^{(i)}}$, i.e. $y^{(i)}_k = 1$ when $y^{(i)} = k$, otherwise $y^{(i)}_k = 0$. Step 2. Train logistic classifier on every column of matrix $Y$. The result will be a matrix $\Theta = (\theta_{jk})_{n+1 \times K}$ Step 3. For any given vector $x$ compute vector $h = x^T \Theta$. Then the predicted value $y$ will be To compute accuracy of the one-vs-all classifier on the training set use accuracy.m script from above with modified predict.m	{}
mersenneforum.org Prime95 30.8 (big P-1 changes, see post #551) User Name Remember Me? Password Register FAQ Search Today's Posts Mark Forums Read 2022-09-25, 18:27 #727 Prime95 P90 years forever! Aug 2002 Yeehaw, FL 41×199 Posts Quote: Originally Posted by kriesel undoc.txt says this about memory in P-1 stage 2: (There's nothing there about the upper limit or modifying it.) 1. Is there a way to allow up to ~60 GiB on a 64 GiB system? The 90% limit is for the GUI. Editing local.txt manually can work around the 90% limit. Quote: 2. Is there a way to ensure a worker's memory access & allocation remains entirely or mostly on the same side of the NUMA boundary as the worker's CPU cores on a multi-Xeon system? Prime95 has no understanding of NUMA. Your only up is to run two instances of prime95. Use a Windows tool to force each instance to run on a different NUMA node. In local.txt ste NumCPUs=8. Let us know if you find a method that works well. 2022-09-26, 06:12 #728 preda "Mihai Preda" Apr 2015 22·192 Posts Quote: Originally Posted by Prime95 Some debugging reveals prime95 is looking for a benchmark with all 16 cores used. Thus, run a throughput benchmark for 16 cores, 1 worker, all FFT implementations, 6M to 7M fft sizes. Let me know if that does the trick. Auto bench done every 21(?) hours until there are several data points. I'm looking into why it is running 13 core benchmarks when it only uses 16 core bench results (a bug). Benchmarks are not uploaded. They are not particularly useful to others given all the combinations of overclocking, memory speeds, etc. I ran a benchmark with all 14cores, and indeed it seems to pick up the FFT bench timings afterwards. Although the run configuration is 1worker/12cores. The benchmark asks for the number of cores to use for bench, and before I was giving it a list of what I was actually using (i.e. 12 cores, 13 cores) not the all-cores (14). 2022-09-26, 08:47 #729 kruoli "Oliver" Sep 2017 Porta Westfalica, DE 1,321 Posts What does the heading in results.bench.txt say for you? E.g.: Code: Compare your results to other computers at http://www.mersenne.org/report_benchmarks AMD Ryzen 7 3800X 8-Core Processor CPU speed: 4350.39 MHz, 8 hyperthreaded cores Especially the last line. 2022-09-26, 15:19 #730 preda "Mihai Preda" Apr 2015 26448 Posts Quote: Originally Posted by kruoli What does the heading in results.bench.txt say for you? E.g.: Intel(R) Core(TM) i9-10940X CPU @ 3.30GHz CPU speed: 3805.32 MHz, 14 hyperthreaded cores CPU features: Prefetchw, SSE, SSE2, SSE4, AVX, AVX2, FMA, AVX512F L1 cache size: 14x32 KB, L2 cache size: 14x1 MB, L3 cache size: 19712 KB L1 cache line size: 64 bytes, L2 cache line size: 64 bytes 2022-09-28, 05:16 #731 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2×29×127 Posts Dual-xeon dual instance experiment Multiple instances' worker windows give repeated occurrences of the error message error setting affinity: no error. See attachment. I think George's recent comments about prime95 being NUMA-unaware means split up worktodo and files in progress to two folders, one per Xeon, copy the program code then alter prime.txt to limit number of cores per prime95 instance to number of physical cores per Xeon, copy prime.txt and local.txt, and specify using one thread per physical core on a CPU package using affinity bitmasks. I chose the even numbered logical cores. In a batch file, or separately: Code: start /D (folder0) /NODE 0 /Affinity 0x5555 prime95.exe start /D (folder1) /NODE 1 /Affinity 0x5555 prime95.exe seemed to do the trick. (See cmd /k start /? for detailed help. AfAIK the start command is the only NUMA-aware control available at the Windows command line. Powershell is a whole other kettle of fish I won't go into here.) (Omitting /Affinity (bitmask) filled up all the hyperthreads on NUMA node 0 and doubled iteration times, leaving the other Xeon idle.) The two instances each have two workers with four cores each. Each instance has 48 GiB allowed for stage 2 P-1/P+1/ECM, 32 GiB as emergency memory, which leaves the 128 GiB ram potentially oversubscribed. Since it is being transitioned from DC to P-1, emergency memory for saving proof residues will become moot and can be pared back. Alternate hyperthreads on the same physical core are consecutive logical processors on Windows, so use either the odd or the even bit but not both for a given two-bit field in the affinity mask; 0x5555 = binary 0101 0101 0101 0101 corresponding to HT0 of each of 8 cores on a Xeon E5-2670 8-core x2 HT. So in HWMonitor, even numbered logical cores are fully occupied with prime95; odd are available for OS etc. Without setting both /node and /affinity values, everything fell on NUMA node 0. 0XAAAA would select 8 odd numbered logical cores. Observed Windows 7 worker timings are consistent with that interpretation. https://linustechtips.com/topic/5919...ined-for-real/ Task Manager displays Cores as follows: Numa Node 0 top row leftmost: core 0 hyperthread 0, then core 0 hyperthread 1, core 1 hyperthread 0 ... core 7 hyperthread 1 Second row is Numa node 1. On Xeon Phis, in Windows 10, CPU rows wrap according to window width, but upper left is core 0, HT 0 1 2 3, bottom right ends core N-1, HT 0 1 2 3. A forced server update from each of the instances, then a check of my CPUs page shows one occurrence of the nodename common to the two instances. (No duplication seen.) Attached Thumbnails Last fiddled with by kriesel on 2022-09-28 at 05:27 2022-09-28, 15:50 #732 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 736610 Posts SPR Two outstanding issues as far as I know (haven't tested v30.8b17 yet) 1) Observed on Windows 7 Pro x64, dual Xeon E5-2670, prime95 V30.8b15, using start /Node 0 or 1, /affinity 0x5555, running two instances, intended as one each side of the QPI; when a worker window assigns cores, the following message is produced repeatedly, with variety of hex values consisting of 3 or c at various offsets: Error setting affinity to cpuset 0x000000c0: No error (refer to attachment of https://mersenneforum.org/showpost.p...&postcount=731) 3 or c is 0011 or 1100. Windows' numbering representation of the two logical cores of a x2 hyperthreaded physical core #0 is 0,1, while Linux's is 0,n where n is number of physical hyperthreaded cores present in the system. So it appears to me that prime95, a Windows application, may be using an inappropriate affinity mask for Windows. We don't usually want two prime95 compute threads running on the same physical core. Or, prime95 is setting a bit map for using either hyperthread of the core involved. (Which would be less constraining than what was already done in the start command's affinity mask.) https://www.systutorials.com/docs/li...wloc_cpuset_t/ I speculate that locking prime95 activity to a specific hyperthread of a core may reduce activity in Windows' task scheduling on multiple cores. I've seen indications in Task Manager's CPU display of a fair amount of logical-core-hopping at times. As if Windows is trying to balance load between hyperthreads, a probably futile exercise for code as memory-bound as prime95's fft crunching, with performance generally hurt, not helped, by multiple hyperthreads on the same core. 2) Observed repeatedly on Windows 10, i5-1035G1, prime95 V30.8b14, Indicated P-1 stage 1 total time for a P-1 interrupted by autobenchmarking is too low, reflecting only the time from the end of the interruption to the end of the stage, omitting all the stage time before the benchmarking interruption. See for example the log content at https://mersenneforum.org/showpost.p...4&postcount=40 which shows "[Sep 2 15:41] M100204259 stage 1 complete. 347584 transforms. Total time: 2487.884 sec." but the stage 1 ran from Sep 2 12:44 to Sep 2 15:41, 12:44 to 18:21 = 5:37 - 3 minutes for benchmarking ~20040. seconds, about 8.055 times as long as the total time indicated to millisecond precision. Last fiddled with by kriesel on 2022-09-28 at 16:29 2022-09-28, 16:22 #733 storm5510 Random Account Aug 2009 Not U. + S.A. 32·281 Posts Quote: Originally Posted by kriesel Two outstanding issues as far as I know (haven't tested v30.8b17 yet) 1) Observed on Windows 7 Pro x64, dual Xeon E5-2670, prime95 V30.8b15, using start /Node 0 or 1, /affinity 0x5555, when a worker window assigns cores, the following message is produced repeatedly, with variety of hex values consisting of 3 or c at various offsets: Error setting affinity to cpuset 0x000000c0: No error (refer to attachment of https://mersenneforum.org/showpost.p...&postcount=731) 2) Observed repeatedly on Windows 10, i5-1035G1, prime95 V30.8b14, Indicated P-1 stage 1 total time for a P-1 interrupted by autobenchmarking is too low, reflecting only the time from the end of the interruption to the end of the stage, omitting all the stage time before the benchmarking interruption. See for example the log content at https://mersenneforum.org/showpost.p...4&postcount=40 which shows "[Sep 2 15:41] M100204259 stage 1 complete. 347584 transforms. Total time: 2487.884 sec." but the stage 1 ran from Sep 2 12:44 to Sep 2 15:41, 12:44 to 18:21 = 5:37 - 3 minutes for benchmarking ~20040. seconds, about 8.055 times as long as the total time indicated to millisecond precision. It seems like you may be taking a more difficult road setting affinity. I use the below in local.txt: Code: [Worker #1] Affinity=(0,4),(2,6) The bold section in your quote above makes no sense. 20,040 seconds is 5.57 hours... 2022-09-28, 16:36 #734 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 11100110001102 Posts Quote: Originally Posted by storm5510 It seems like you may be taking a more difficult road setting affinity. I use the below in local.txt: Code: [Worker #1] Affinity=(0,4),(2,6) The bold section in your quote above makes no sense. 20,040 seconds is 5.57 hours... Are you running a dual-Xeon system with QPI bottleneck between halves of total installed ram? Two separate instances, one per Xeon as directed by George? I'm setting affinity to 8 hyperthreads of 32, in each of two prime95 instances. I think what I'm doing is simpler than setting four different affinity lists for four different workers in two different folders. And more likely to get the memory locality right. It's also easily extensible to a dual-12-core&HT system later; masks become 0x555555. Done. 5:37: 5 hours 37 minutes from start to finish of stage 1, minus 3 minutes benchmarking interruption: 5 * 3600 +37 * 60 -3 * 60 = 20040. seconds. Perhaps a case of vigorous agreement? But the program reported only 2487. seconds & change, less than 1/8 the actual stage 1 compute time, is the point I was making. Last fiddled with by kriesel on 2022-09-28 at 17:24 2022-09-28, 17:40 #735 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2·29·127 Posts Same hardware, the error no error persists in prime95 v30.8b17. There are no directives involving affinity in local.txt or prime.txt, so it is prime95 default response in the context of the start commands used. IIRC George runs Linux not Windows, and may have no dual-CPU-package systems to test mprime / prime95 on. Last fiddled with by kriesel on 2022-09-28 at 18:17 2022-09-29, 23:16 #736 storm5510 Random Account Aug 2009 Not U. + S.A. 32×281 Posts Quote: Originally Posted by kriesel Are you running a dual-Xeon system with QPI bottleneck between halves of total installed ram? Two separate instances, one per Xeon as directed by George? But the program reported only 2487. seconds & change, less than 1/8 the actual stage 1 compute time, is the point I was making. No, and I probably would not try. Multiple workers in a single instance, maybe. If the OS can see both CPU's then I would think any running process could as well. 2022-09-30, 03:06 #737 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 736610 Posts Quote: Originally Posted by storm5510 If the OS can see both CPU's then I would think any running process could as well. Yes, and there's empirical evidence that using almost all the dual-CPU-system's ram in prime95 is suboptimal compared to using almost all the near ram on each CPU, in P-1 stage 2, and leaving the rest to be used for the other CPU's processes. Mprime / prime95 needs a little user assistance apparently to ensure it's the near ram in the multi-CPU (not merely multi-core) case. Similar Threads Thread Thread Starter Forum Replies Last Post kar_bon Prime Wiki 40 2022-04-03 19:05 science_man_88 science_man_88 24 2018-10-19 23:00 xilman Linux 2 2010-12-15 16:39 kar_bon Forum Feedback 3 2010-09-28 08:01 dave_0273 Lounge 1 2005-02-27 18:36 All times are UTC. The time now is 22:46. 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# Turbostat¶ turbostat is a Linux command-line utility that prints various measurements, including numerous per-CPU measurements. This article provides an introduction to using it. Note: The power profiling overview is worth reading at this point if you haven’t already. It may make parts of this document easier to understand. ## Invocation¶ turbostat must be invoked as the super-user: sudo turbostat If you get an error saying "turbostat: no /dev/cpu/0/msr", you need to run the following command: sudo modprobe msr The output is as follows: Core CPU Avg_MHz %Busy Bzy_MHz TSC_MHz SMI CPU%c1 CPU%c3 CPU%c6 CPU%c7 CoreTmp PkgTmp Pkg%pc2 Pkg%pc3 Pkg%pc6 Pkg%pc7 PkgWatt CorWatt GFXWatt - - 799 21.63 3694 3398 0 12.02 3.16 1.71 61.48 49 49 0.00 0.00 0.00 0.00 22.68 15.13 1.13 0 0 821 22.44 3657 3398 0 9.92 2.43 2.25 62.96 39 49 0.00 0.00 0.00 0.00 22.68 15.13 1.13 0 4 708 19.14 3698 3398 0 13.22 1 1 743 20.26 3666 3398 0 21.40 4.01 1.42 52.90 49 1 5 1206 31.98 3770 3398 0 9.69 2 2 784 21.29 3681 3398 0 11.78 3.10 1.13 62.70 40 2 6 782 21.15 3698 3398 0 11.92 3 3 702 19.14 3670 3398 0 8.39 3.09 2.03 67.36 39 3 7 648 17.67 3667 3398 0 9.85 The man page has good explanations of what each column measures. The various “Watt” measurements come from the Intel RAPL MSRs. If you run with the -S option you get a smaller range of measurements that fit on a single line, like the following: Avg_MHz %Busy Bzy_MHz TSC_MHz SMI CPU%c1 CPU%c3 CPU%c6 CPU%c7 CoreTmp PkgTmp Pkg%pc2 Pkg%pc3 Pkg%pc6 Pkg%pc7 PkgWatt CorWatt GFXWatt 3614 97.83 3694 3399 0 2.17 0.00 0.00 0.00 77 77 0.00 0.00 0.00 0.00 67.50 57.77 0.46	{}
# Draw_a_simulated_data_set_and_Draw_posterior_samples: Draw a dataset and MCMC samples In BayesianFROC: FROC Analysis by Bayesian Approaches ## Description Draw a dataset and MCMC samples. 1. draw a model parameter from prior distribution, 2. draw a dataset from the model with the parameter drawn in step 1, 3. draw a collection of posterior samples for the dataset drawn in step 2. ## Usage 1 2 3 4 5 6 7 8 9 10 11 12 13 Draw_a_simulated_data_set_and_Draw_posterior_samples( sd = 5, C = 5, seed.for.drawing.a.prior.sample = 1111, fun = stats::var, NI = 259, NL = 259, initial.seed.for.drawing.a.data = 1234, ModifiedPoisson = FALSE, PreciseLogLikelihood = TRUE, ite = 1111, DrawCurve = FALSE ) ## Arguments sd Standard Deviation of priors C No. of Confidence levels seed.for.drawing.a.prior.sample seed fun An one dimensional real valued function defined on the parameter space. This is used in the definition of the rank statistics. Generally speaking, the element of the parameter space is a vector, so the function should be defined on vectors. In my model parameter is mean, standard deviation, C thresholds of the latent Gaussian, so this function should be defined on the C+2 dimensional Euclidean space. NI No. of images NL No. of Lesions initial.seed.for.drawing.a.data seed ModifiedPoisson Logical, that is TRUE or FALSE. If ModifiedPoisson = TRUE, then Poisson rate of false alarm is calculated per lesion, and model is fitted so that the FROC curve is an expected curve of points consisting of the pairs of TPF per lesion and FPF per lesion. Similarly, If ModifiedPoisson = TRUE, then Poisson rate of false alarm is calculated per image, and model is fitted so that the FROC curve is an expected curve of points consisting of the pair of TPF per lesion and FPF per image. For more details, see the author's paper in which I explained per image and per lesion. (for details of models, see vignettes , now, it is omiited from this package, because the size of vignettes are large.) If ModifiedPoisson = TRUE, then the False Positive Fraction (FPF) is defined as follows (F_c denotes the number of false alarms with confidence level c ) \frac{F_1+F_2+F_3+F_4+F_5}{N_L}, \frac{F_2+F_3+F_4+F_5}{N_L}, \frac{F_3+F_4+F_5}{N_L}, \frac{F_4+F_5}{N_L}, \frac{F_5}{N_L}, where N_L is a number of lesions (signal). To emphasize its denominator N_L, we also call it the False Positive Fraction (FPF) per lesion. On the other hand, if ModifiedPoisson = FALSE (Default), then False Positive Fraction (FPF) is given by \frac{F_1+F_2+F_3+F_4+F_5}{N_I}, \frac{F_2+F_3+F_4+F_5}{N_I}, \frac{F_3+F_4+F_5}{N_I}, \frac{F_4+F_5}{N_I}, \frac{F_5}{N_I}, where N_I is the number of images (trial). To emphasize its denominator N_I, we also call it the False Positive Fraction (FPF) per image. The model is fitted so that the estimated FROC curve can be ragraded as the expected pairs of FPF per image and TPF per lesion (ModifiedPoisson = FALSE ) or as the expected pairs of FPF per image and TPF per lesion (ModifiedPoisson = TRUE) If ModifiedPoisson = TRUE, then FROC curve means the expected pair of FPF per lesion and TPF. On the other hand, if ModifiedPoisson = FALSE, then FROC curve means the expected pair of FPF per image and TPF. So,data of FPF and TPF are changed thus, a fitted model is also changed whether ModifiedPoisson = TRUE or FALSE. In traditional FROC analysis, it uses only per images (trial). Since we can divide one image into two images or more images, number of trial is not important. And more important is per signal. So, the author also developed FROC theory to consider FROC analysis under per signal. One can see that the FROC curve is rigid with respect to change of a number of images, so, it does not matter whether ModifiedPoisson = TRUE or FALSE. This rigidity of curves means that the number of images is redundant parameter for the FROC trial and thus the author try to exclude it. Revised 2019 Dec 8 Revised 2019 Nov 25 Revised 2019 August 28 PreciseLogLikelihood Logical, that is TRUE or FALSE. If PreciseLogLikelihood = TRUE(default), then Stan calculates the precise log likelihood with target formulation. If PreciseLogLikelihood = FALSE, then Stan calculates the log likelihood by dropping the constant terms in the likelihood function. In past, I distinct the stan file, one is target formulation and the another is not. But non-target formulation cause some Jacobian warning, thus I made all stanfile with target formulation when I uploaded to CRAN. Thus this variable is now meaningless. ite A variable to be passed to the function rstan::sampling() of rstan in which it is named iter. A positive integer representing the number of samples synthesized by Hamiltonian Monte Carlo method, and, Default = 10000. DrawCurve Logical: TRUE of FALSE. Whether the curve is to be drawn. TRUE or FALSE. If you want to draw the FROC and AFROC curves, then you set DrawCurve =TRUE, if not then DrawCurve =FALSE. The reason why the author make this variable DrawCurve is that it takes long time in MRMC case to draw curves, and thus Default value is FALSE in the case of MRMC data. ## Value Draw.a.prior.sample The Return value of Draw_a_prior_sample A dataList and an object of the stanfit S4 class with respect to the dataList 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ## Not run: # Draw a curve for various seeds and various number of confidence levels. # Changing the seed, we can draw a parameter from priors and using this sample, # we can draw the datasets from our model whose parameters are # the priors samples. # 1. draw a model parameter from prior distribution, # 2. draw a dataset from the model with the parameter drawn in step 1, # 3. draw a collection of posterior samples for the dataset drawn in step 2. Draw_a_simulated_data_set_and_Draw_posterior_samples( seed.for.drawing.a.prior.sample = 1234, C=8) Draw_a_simulated_data_set_and_Draw_posterior_samples( seed.for.drawing.a.prior.sample = 12345, C=7) Draw_a_simulated_data_set_and_Draw_posterior_samples( seed.for.drawing.a.prior.sample = 123456, C=6) Draw_a_simulated_data_set_and_Draw_posterior_samples( seed.for.drawing.a.prior.sample = 1234567, C=5) ## End(Not run)# dottest	{}
## Saturday, April 19, 2014 ### Waiting Time Polynomials: how to derive the analytical formula: Part IV Introduction before you start I got many clarification requests about the Waiting Time Polynomials I published on the blog in the last three posts. The paper is almost ready to be submitted for review, but I think that some technical explanation might be interesting also for not academic audience. I consider myself a curious and hungry seasoned student, and I know how can be tedious read formulas and mathematical passages especially when it comes from a blog!! So why technical explanations? The answer is in the following quote of one of my favourite scientists, Gregory Chaitin. In "The quest for Omega" he wrote: The books I loved were books where the author’s personality shows through, books with lots of words, explanations and ideas, not just formulas and equations! I still think that the best way to learn a new idea is to see its history, to see why someone was forced to go through the painful and wonderful process of giving birth to a new idea! To the person who discovered it, a new idea seems inevitable, unavoidable. The first paper may be clumsy, the first proof may not be polished, but that is raw creation for you, just as messy as making love, just as messy as giving birth! But you will be able to see where the new idea comes from. If a proof is “elegant”, if it’s the result of two-hundred years of finicky polishing, it will be as inscrutable as a direct divine revelation, and it’s impossible to guess how anyone could have discovered or invented it. It will give you no insight, no, probably none at all. That's the spirit that leads the following explanation! Definition of the problem Given an alphabet of 3 elements $\{X_1,X_2,X_3\}$, the function $w(X_i)$ counts the number of failed trials before the last event $X_i$. Consider now the following configuration: $\{\left\vert{X_1}\right\vert =i , \left\vert{X_2}\right\vert =j,\left\vert{X_3}\right\vert =k\}: i+j+k= Z \wedge i,j,k>0$ • What are the admitted sequences $\{w(X_1),w(X_2),w(X_3)\}$ ? Step I: Find all the possible configurations of events How can we list the sequences of length $Z$ that can be built with $\{\left\vert{X_1}\right\vert =i , \left\vert{X_2}\right\vert =j,\left\vert{X_3}\right\vert =k\}: i+j+k= Z \wedge i,j,k>0$ ? Example of overall waiting time $w(x_i)$ in a succession of events. • once we set the values of the first two variables, the third it's determined by $Z-i-j$. • we imposed that all the variables occur at least once, so we $X_1$ can assume all the values between $[1,Z-2]$. • for each value of $X_1$ the variable $X_2$ can assume values between $[1,Z-i]$. • $p_i$ is the probability that $X_i$ occur in a Bernullian trial. Now we have all the ingredients to make the cake: $\sum_{i=1}^{Z}\sum_{j=1}^{Z-i}\sum_{k=1}^{Z-i-j}{p_1^ip_2^jp_3^k}$ In the first two summations, $i$ assumes values between $[1,Z]$ just to keep the formula cleaned. ...I let you proof why the result doesn't change :). last point about this step:the limit of the above summation $Z \rightarrow \infty = \frac{p_1 p_2 p_3}{\left(p_1-1\right) \left(p_2-1\right) \left(p_3-1\right)}$ Such limit will be used to build the probabilistic density function. • The number of sequences that can be built with vectors of length $[3,Z]$ are $\binom{Z}{3}$ • The number of sequences that can be built with vectors of length $Z$ are $\binom{Z}{2}$ Step II: Waiting for an event! What's the easiest way to describe the overall waiting time for an event in a finite succession? There are many ways to get the $w(x_i)$, the easiest I found is given by the position of the last occurrence of $x_i$ minus the number of occurrences of $x_i$. For instance, let's consider $w(x_1)$: • The position of the last occurrence of $x_1= 8$; • $\left \vert{X_1} \right \vert = 4$ • $w(X_1)=4$ Where we are: The first two steps explain the circled pieces of the formula: What the "overall waiting time" for? For each event $X_i$ we are counting the holes among all the occurrences, so smaller is the overall waiting time, closer each other are the events $X_i$: it's a measure of proximity for the occurrences of $X_i$. What I did, is to extend such measure (it would be interesting to prove that it's really a measure!) to different kind of events (aleatory variables) ${X_1, X_2,...,X_n}$ over the discrete line of the time. Applications There are several area for which such kind of analysis might be helpful, I showed last time an its application as powerful document classifier, where each variable $X_i$ is a word of a document. If we consider a document as a succession of $Z$ words, the proximity measure inducted by the waiting time polynomials is a sort of finger print for the document, since for similar documents we expect that the same words are characterised by similar overall waiting time. Moreover, the dependency among the words are considered, since we are taking in account simultaneously an arbitrary number of words (the alphabet ${X_1, X_2,...,X_n}$). In the next step I'll explain the logic to get the remaining pieces of the puzzle, that will make easier the generalisation of the approach to an arbitrary alphabet. Stay Tuned! cristian 1. I really appreciate information shared above. It’s of great help. If someone want to learn Online (Virtual) instructor lead live training in Data Mining, kindly contact us http://www.maxmunus.com/contact MaxMunus Offer World Class Virtual Instructor led training on Data Mining. We have industry expert trainer. We provide Training Material and Software Support. MaxMunus has successfully conducted 100000+ trainings in India, USA, UK, Australlia, Switzerland, Qatar, Saudi Arabia, Bangladesh, Bahrain and UAE etc. Name : Arunkumar U Email : arun@maxmunus.com Skype id: training_maxmunus Contact No.-+91-9738507310 Company Website –http://www.maxmunus.com 2. Really useful information. we are providing best data science online training from industry experts. 3. Thank you for sharing your article. Great efforts put it to find the list of articles which is very useful to know, Definitely will share the same to other forums. Data Science Training in chennai at Credo Systemz \| data science course fees in chennai \| data science course in chennai velachery \| data science course in chennai with placement 4. Just stumbled across your blog and was instantly amazed with all the useful information that is on it. Great post, just what i was looking for and i am looking forward to reading your other posts soon! 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# Conditioning on an event with probability close to one Let $(\Omega,\mathcal{F},P)$ be a probability space. If $A\in\cal F$ is an event with $P(A)=1$, then $$P_{\mid A}(B)=P(B\mid A)=\frac{P(B\cap A)}{P(A)}=P(B),\quad B\in\cal F.$$ I wonder if something can be said about how "close" $P_{\mid A}$ and $P$ are, when $A\in\cal F$ is an event with probability close to $1$ and also what "close" should mean. For example, if $P(A)=p$ and let's say that $p=0.99$, can we give a non-trivial upper bound on the maximal distance $$\sup_{B\in\cal F}\|P_{\mid A}(B)-P(B)\|$$ in terms of $p$? And could other types of distances be interesting? This is just me thinking, so anything you can add will be appreciated. Thanks. - For every $B$, $\mathbb P(B\mid A)-\mathbb P(B)=b(1-a)/a-c$ with $a=\mathbb P(A)$, $b=\mathbb P(B\cap A)$ and $c=\mathbb P(B\setminus A)$. Since $0\leqslant b\leqslant a$ and $0\leqslant c\leqslant 1-a$, $$-(1-a)\leqslant -c\leqslant \mathbb P(B\mid A)-\mathbb P(B)\leqslant b(1-a)/a\leqslant 1-a.$$ The bound $1-a$ is achieved for $B=A$, hence $$\sup\limits_{B\in\mathcal F}\,\|\mathbb P(B\mid A)-\mathbb P(B)\|=1-\mathbb P(A).$$	{}
add_resource_path(prefix, directoryPath, warn_empty = FALSE) Arguments prefix The URL prefix (without slashes). Valid characters are a-z, A-Z, 0-9, hyphen, period, and underscore. For example, a value of 'foo' means that any request paths that begin with '/foo' will be mapped to the given directory. directoryPath The directory that contains the static resources to be served. warn_empty Boolean. Default is FALSE. If TRUE display message if directory is empty. Value Used for side effects.	{}
• ### Design of the Front End Electronics for the Infrared Camera of JEM-EUSO, and manufacturing and verification of the prototype model(1501.06183) Jan. 25, 2015 physics.ins-det, astro-ph.IM The Japanese Experiment Module (JEM) Extreme Universe Space Observatory (EUSO) will be launched and attached to the Japanese module of the International Space Station (ISS). Its aim is to observe UV photon tracks produced by ultra-high energy cosmic rays developing in the atmosphere and producing extensive air showers. The key element of the instrument is a very wide-field, very fast, large-lense telescope that can detect extreme energy particles with energy above $10^{19}$ eV. The Atmospheric Monitoring System (AMS), comprising, among others, the Infrared Camera (IRCAM), which is the Spanish contribution, plays a fundamental role in the understanding of the atmospheric conditions in the Field of View (FoV) of the telescope. It is used to detect the temperature of clouds and to obtain the cloud coverage and cloud top altitude during the observation period of the JEM-EUSO main instrument. SENER is responsible for the preliminary design of the Front End Electronics (FEE) of the Infrared Camera, based on an uncooled microbolometer, and the manufacturing and verification of the prototype model. This paper describes the flight design drivers and key factors to achieve the target features, namely, detector biasing with electrical noise better than $100 \mu$V from $1$ Hz to $10$ MHz, temperature control of the microbolometer, from $10^{\circ}$C to $40^{\circ}$C with stability better than $10$ mK over $4.8$ hours, low noise high bandwidth amplifier adaptation of the microbolometer output to differential input before analog to digital conversion, housekeeping generation, microbolometer control, and image accumulation for noise reduction. • ### Thin and thick cloud top height retrieval algorithm with the Infrared Camera and LIDAR of the JEM-EUSO Space Mission(1501.04769) Jan. 20, 2015 astro-ph.IM, astro-ph.HE The origin of cosmic rays have remained a mistery for more than a century. JEM-EUSO is a pioneer space-based telescope that will be located at the International Space Station (ISS) and its aim is to detect Ultra High Energy Cosmic Rays (UHECR) and Extremely High Energy Cosmic Rays (EHECR) by observing the atmosphere. Unlike ground-based telescopes, JEM-EUSO will observe from upwards, and therefore, for a properly UHECR reconstruction under cloudy conditions, a key element of JEM-EUSO is an Atmospheric Monitoring System (AMS). This AMS consists of a space qualified bi-spectral Infrared Camera, that will provide the cloud coverage and cloud top height in the JEM-EUSO Field of View (FoV) and a LIDAR, that will measure the atmospheric optical depth in the direction it has been shot. In this paper we will explain the effects of clouds for the determination of the UHECR arrival direction. Moreover, since the cloud top height retrieval is crucial to analyze the UHECR and EHECR events under cloudy conditions, the retrieval algorithm that fulfills the technical requierements of the Infrared Camera of JEM-EUSO to reconstruct the cloud top height is presently reported. • Contributions of the JEM-EUSO Collaboration to the 32nd International Cosmic Ray Conference, Beijing, August, 2011.	{}
## kiamonstaa Group Title solve using the quadratic formula H(t)= -16t+25+0 one year ago one year ago 1. satellite73 Group Title does it really say $$+0$$ ?? 2. chemENGINEER Group Title this is first order, so i would use the quadratic formula myself.... 3. satellite73 Group Title i think something is missing from this question it is not a quadratic as it is written, also it has a $$+0$$ at the end also it is not an equation, so there is nothing to solve 4. chemENGINEER Group Title would not* 5. satellite73 Group Title @kiamonstaa i think maybe you could repost the problem. i doubt this is the question exactly as it is written 6. kiamonstaa Group Title this is the whole thing 7. kiamonstaa Group Title Jules kicks a soccer ball off the ground and into the air with an initial velocity of 25 feet per second. Assume the starting height of the ball is 0 feet. Approximately, how long does it take until the soccer ball hits the ground again? 0.6 sec 0.8 sec 1.6 secs 2.8 secs 8. Spacelimbus Group Title $\Large s= f(t)=s_0+v_0t+\frac{1}{2} at^2$ where $$a=-g$$ and $$s_0=0$$ and $$s=0$$	{}
## 每週問題 July 11, 2016 Suppose that $A$ is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that $A$ has at least one nonzero row. (a) Determine the fewest number of free variables that are possible. (b) Determine the maximum number of free variables that are possible. (a) 已知 $A$ 是一個 $4\times 6$ 階矩陣。自由變數的數目等於 $A$ 的零空間 (nullspace) 的維數 $\dim N(A)$，稱為零度 (nullity)。根據秩─零度定理，$\dim N(A)=6-\hbox{rank}A$。因為 $\hbox{rank}A\le 4$，推得 $\dim N(A)$ 的最小值為 $6-4=2$ (b) 因為 $A$ 至少有一個非零列 (row)，即 $\hbox{rank}A\ge 1$，推得 $\dim N(A)$ 的最大值為 $6-1=5$	{}
# Details on the magnetic field of a linearly polarized electric wave Suppose we are in vacuum and we have an electric field $\vec{E}$ which we assume is simple harmonic wave that propagates through $z$ and is linearly polarized in the $x$-$y$ plane along $x$ i.e. $\vec{E}(t,x,y,z)=E_0\cos(\omega t-kz)\hat{x}$. This function obviously satisfies $\vec{\nabla}\cdot\vec{E}=0$. Now note that $$\vec{\nabla}\times\vec{E}(t,x,y,z)=\frac{\partial}{\partial z}\left( E_0\cos(\omega t-kz) \right)\hat{y}=kE_0\sin(\omega t-kz)\hat{y}=-\frac{\partial \vec{B}}{\partial t}(t,x,y,z)$$ If we integrate this, we get $\vec{B}(t,x,y,z)=B_0\cos(\omega t-kz)\hat{y}+\vec{g}(x,y,z)$ for some $\vec{g}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ and where $B_0=\frac{k}{\omega}E_0$. Now note that $B_0\cos(\omega t-kz)\hat{y}$ has no gradient and therefore $\vec{\nabla}\cdot\vec{B}=0$ implies $\vec{\nabla}\cdot\vec{g}=0$. On the other hand $\vec{\nabla}\times (B_0\cos(\omega t-kz)\hat{y})=\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}(t,x,y,z)$, which implies $\vec{\nabla}\times\vec{g}=\vec{0}$. Finally, clearly $B_0\cos(\omega t-kz)\hat{y}$ satisfies the wave equation, which means that $\vec{g}$ must do it too. Since $\vec{g}$ has no time dependence, $\nabla^2\vec{g}=\vec{0}$. On every source I have seen, the vector field $\vec{g}$ has been taken to be null. From this assumption things such as the perpendicularity between the fields and the direction of propagation are explained. Non the less, non of the three restrictions imply that $\vec{g}$ be null. In particular, $\vec{g}$ could be a constant vector field with any direction and still satisfy Maxwell's equations. Is there any way to show that in general E&M waves must be perpendicular and therefore show that $\vec{g}=\vec{0}$? In the case there isn't, what can we say about $\vec{g}$ knowing $\vec{\nabla}\cdot\vec{g}=0$, $\vec{\nabla}\times\vec{g}=\vec{0}$ and $\nabla^2\vec{g}=\vec{0}$? • I am not sure I am understanding the problem. Maxwell's equation are satisfied by any electromagnetic field, not just waves. A wave superimposed on a constant field is a perfectly valid solution, of course and therefor the magnetic component and the electric component do not have to be perpendicular for a general field. Does any particular textbook claim that they have to be? – CuriousOne Oct 31 '15 at 4:07 • Well, my problem is that when e&m waves are studied you never take into account this $\vec{g}$. I am interested in how this field looks since I cant think of a reason for it to be null. In particular, I want to point out that a varying SHW electric field can produce a magnetic field with a constant component! – Iván Mauricio Burbano Oct 31 '15 at 4:14 • For any two valid solutions to Maxwell's equations any linear combination of those solutions is also a solution. In general we can therefor describe any em-field with its Fourier transform. This is a direct consequence of the perfect linearity of the theory and it doesn't impact the analysis of an em-wave with a single frequency. The constant field component in your example is not produced by the wave but it's just an integration constant. – CuriousOne Oct 31 '15 at 4:22 ## 1 Answer There is no reason for the field to be null. You have correctly identified the properties of another magnetic field that could be the solution of Maxwell's equations that is consistent with electric field you started with - namely a time-independent magnetic field. The situation is exactly that by which you are reading this answer. EM waves propagate into your eyes, but between you and the screen the space is filled with the Earth's (roughly) time-independent magnetic field. As CuriousOne completely describes, the reason this works is that you can superpose any solutions to Maxwell's equations and a time-independent magnetic field can be a solution with zero electric field and a Laplacian of zero. The reason books ignore this, is because you can ignore it. The time-dependent EM wave fields can be considered in isolation from any time-independent E- and B-fields in the "background". Saying that a time-dependent E-field can "produce" a time-independent B-field is not a correct way to think about electromagnetism - the fields co-exist. No SHM E-field is associated with a time-independent B-field - as you confirmed by showing that its curl was zero. • Is there a way to show that the time dependent electric field, time dependent magnetic field and the propagation are perpendicular without assuming the form of the electric field? Thanks! – Iván Mauricio Burbano Oct 31 '15 at 11:33 • @IvánMauricioBurbano What, you mean just say ${\bf E} = {\bf E_0} f({\bf k}\cdot {\bf r} - \omega t)$, where ${\bf E_{0}}$ is an arbitrary vector? Yes - Gauss's law shows that ${\bf k}\cdot {\bf E_0} = 0$, then take the curl and integrate to show that the time-dependent B-field is perpendicular to ${\bf E_0}$ (and ${\bf k}$). This is standard bookwork. – Rob Jeffries Oct 31 '15 at 11:37 • Yeah, but you (and I in my question) assumed a particular form of the electric field. I want to show this without this particular form for the electric field. In that way I can convince myself that circularly polarized electric fields are also perpendicular to the magnetic fields. – Iván Mauricio Burbano Oct 31 '15 at 11:40 • @IvánMauricioBurbano Sorry? Where did I assume a particular form? I assume that the form is an electromagnetic wave, thats all. The relationships between E- and B-fields for an EM wave are not generally true for all solutions of Maxwells equations! A circ. pol. wave is just the sum of two EM waves plane polarised at right angles, with a 90 degree lag. Therefore if the relationships are true for any plane polarised EM wave then they are also true for circ. pol. waves, because, as stated, any linear combinations of solutions to Maxwell's eqs. are also solutions – Rob Jeffries Oct 31 '15 at 11:42	{}
## Measurement of the inelastic $pp$ cross-section at a centre-of-mass energy of 13 TeV Research output: Contribution to journalArticle Open ### Documents Original language English 100 Journal of High Energy Physics 1806 20 Jun 2018 https://doi.org/10.1007/JHEP06(2018)100 E-pub ahead of print - 20 Jun 2018 ### Abstract The cross-section for inelastic proton-proton collisions at a centre-of-mass energy of 13\,TeV is measured with the LHCb detector. The fiducial cross-section for inelastic interactions producing at least one prompt long-lived charged particle with momentum $p>2$\,GeV/$c$ in the pseudorapidity range $2<\eta<5$ is determined to be $\sigma_{\rm acc}= 62.2 \pm 0.2 \pm 2.5$\,mb. The first uncertainty is the intrinsic systematic uncertainty of the measurement, the second is due to the uncertainty on the integrated luminosity. The statistical uncertainty is negligible. Extrapolation to full phase space yields the total inelastic proton-proton cross-section $\sigma_{\rm inel}= 75.4 \pm 3.0 \pm 4.5$\,mb, where the first uncertainty is experimental and the second due to the extrapolation. An updated value of the inelastic cross-section at a centre-of-mass energy of 7\,TeV is also reported.	{}
Let F be the principle focus and f be the focal length. f = focal length of the lens. Eqn. (viii) represents Lens maker formula. THIN LENS FORMULA : FOR CONCAVE LENS. If the equation shows a negative image distance, then the image is a virtual image on the same side of the lens as the object. For latest information , … The formula is as follows: $$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$$ Lens Formula Derivation. Convex Lens Let a concave lens have two spherical surfaces X 1 P 1 Y 1 and X 2 P 2 Y 2 having radius of curvature as R 1 and R 2 respectively. If this equation shows a negative focal length, then the lens is a diverging lens rather than the converging lens. Applicable for both the convex and concave lenses, the lens formula is given as: 1/v - 1/u = 1/f Where, v = Distance of image formed from the optical center of the lens. Derivation of Lens Maker Formula for a Concave Lens. This lens formula is applicable to both the concave and convex lens. The formula formed will be a general formula. Consider an object placed in front of a concave lens of focal length "f " on the principle axis of the lens. Ray diagrams for such lenses are drawn using: a ray from the top of the object through the middle of the lens; These lenses have negligible thickness. Concave Lenses Concave lenses always produce upright, virtual images. Section 3: Concave Lenses 12 3. Concave lens forms a virtual and erect image at a distance of " q " from the optical centre of the lens as shown in the diagram below. u = Distance of object from the optical center of the lens. For aconcave lens, the lens equation is the same but the value of fis nownegative. Consider a convex lens with an optical center O. Lens formula is applicable for convex as well as concave lenses. STEP I. Refraction at X 1 P 1 Y 1. In this video, we are going to derive the lens formula using the properties of the triangle. Lens Formula Derivation. . Birds Of Connecticut, Coursera Quiz Answers, Voodoo Lab Micro Vibe Vs Mxr Uni-vibe, Starbucks Doubleshot Can, Piobaireachd Sheet Music, Executioner Sword Holes, Ncaa Division Ii Teams,	{}
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. As of today, I’ve officially made the jump to using the R package blogdown (which uses the Hugo static-site generator under the hood) for my personal website. Previously, I had been using WordPress for my blogging purposes. In sync with the change in platform, I’m changing the name of this site from “Number Sense” (www.numbersense.org) to something involving my name (Tony ElHabr). Nonetheless, my original intents to write about math, sports, and data-related things have not changed. Perhaps the only major difference with me between now and then is my developed passion for programming, especially with R. I’ll eventually get around to cleaning up my handful of old posts and sharing them here. (I used a WordPress to Hugo converter for the basic conversion, but there were still lots of links and formatting to be fixed .) To kick of my sit migration, I suppose I could write a tutorial to explain how or why I’m transitioning, but it seems like everyone does that , so I don’t want to re-invent the wheel. Aside from individual write-up, the wonderful blogdown book should be sufficient for most people’s purposes. Although I would like to blog on a regular basis, my job comes first for me—so if I don’t have time to blog, then I might not post anything for months at a time.	{}
1. I need some help graphing this trig function. f(x)=4sin(3x-2) When Graphing, the points should be over 2 periods and 9 points. I have: -Amplitude: 4 -Period=(2\pi/3) -Range=[-4,4] I need help on: -Graphing the points over 2 periods and 9 periods (parent function and f(x)) -Table of 9 points (f(x) and parent function) -Increments (f(x) and parent function) Thank You in advance if You can help me!! 2. I agree with what you've done so far regarding the amplitude, period and range. Now, if I were going to sketch a graph of $f(x)$, I would write it in the form: $\displaystyle f(x)=4\sin\left(3\left(x-\frac{2}{3}\right)\right)$ In this form, we can see that $f$ is shifted $\displaystyle \frac{2}{3}$ units to the right compared to $y=4\sin(3x)$. Now, if I was going to graph $f$ over just one period, I would choose the domain: $\displaystyle \left[\frac{2}{3},\frac{2}{3}(\pi+1)\right]$ On this domain, the sinusoid described by $f$ will begine at 0, movie to to 4, then back down to 0, continue down to -4, then move back to to 0, completing 1 cycle. I would divide this domain into 4 subdivisions of equal width corresponding to the extrema and equilibria (that is, the zero, maximum and minimum values for $f$). This gives us the $x$-values: $\displaystyle x=\frac{2}{3}+\frac{k}{4}\cdot\frac{2}{3}\pi=\frac{2}{3}+\frac{k}{6}\pi$ where $k\in\{0,1,2,3,4\}$ Putting all this together, this gives us the 5 points: $\displaystyle \left(\frac{2}{3},0\right),\,\left(\frac{2}{3}+\frac{1}{6}\pi,4\right),\,\left(\frac{2}{3}+\frac{1}{3}\pi,0\right),\,\left(\frac{2}{3}+\frac{1}{2}\pi,-4\right),\,\left(\frac{2}{3}+\frac{2}{3}\pi,0\right)$ If you continue for another period, this would give you 4 more points for a total of nine...can you continue?	{}
# How do you find the midpoint of (2,8) and (4,6)? Feb 29, 2016 The midpoint is $\left(3 , 7\right)$. #### Explanation: The midpoint formula is: $\left({x}_{m} , {y}_{m}\right) = \left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$ We have, ${x}_{1} = 2 , {y}_{1} = 8 , {x}_{2} = 4 , {y}_{2} = 6$ substituting the values, $\frac{2 + 4}{2} \mathmr{and} \frac{8 + 6}{2} = \left(3 , 7\right)$ Another way of looking at the "midpoint formula" is just as taking the average of the values for $x$ and the average of the values for $y$. Halfway between $x = 2$ and $x = 4$ is $x = 3$, and halfway between $y = 8$ and $y = 6$ is at $y = 7$, so the midpoint is $\left(3 , 7\right)$.	{}
# HowTo: Using -lite with a GIT-based application Years ago while looking for a fast init replacement for work, I found Finit. Originally written by Claudio Matsuoka to act as a drop-in replacement for the Asus EeePC fastinit, “gaps filled with frog DNA …“ Until I found Finit I had always been in awe of those venturing into the realm of PID 1. However, learning from the simplicity of Claudio’s code I realized that although PID 1 at times is indistinguishable from magic, it is really not that hard to master. My version of Finit is available on GitHub. The code is open sourced under the very liberal MIT/X11 license, and much of its frog DNA has proven very useful to me over the years. This blog post is about how that frog DNA can help you fill gaps in your projects … Recently I broke out the most generic pieces from Finit into a separate library, which I call libite (because it looks awesome linking to it: -lite :) I complemented it with a few pieces of my own and some from the OpenBSD project, most notably their famous string API: strlcpy(3) and strlcat(3). Also included is the very useful *BSD linked list API sys/queue.h, which is a much more up to date version than GLIBC carries. For example, the new _SAFE macros are missing, which you want to use while traversing lists to delete/free nodes. To make use of -lite and its APIs you can either build it as a separate library and install lite.h and libite.so.1 to your system, or add libite as a GIT submodule to your project and use only the parts you need from the archive: git submodule add https://github.com/troglobit/libite.git You then need to add #include "libite/lite.h" to the source and adapt your Makefile slightly to call the libite/Makefile before linking your application to the libite.a archive: all: $(EXEC) libite/libite.a libite/libite.a: Makefile @$(MAKE) -C libite $(EXEC):$(OBJS) libite/libite.a @gcc -o $@$^ For an example of how this can look, see the uftpd project, which uses both -lite and -luev. The latter is my small event library, libuEv. For help using -lite with the GNU configure and build system, see inadyn. Libite builds in “silent mode” by default, use make V=1 (like the kernel) to get a more verbose output, usable for autobuilders etc.	{}
matplotlib plot transparent line Matplotlib transparent line plots. Let’s discuss some concepts : Example 1: (Simple line graph with its opacity), edit c by Lonely Lobster on May 04 2020 Donate Re: [Matplotlib-users] plotting lines with shaded / transparent region for standard deviation Re: [Matplotlib-users] plotting lines with shaded / transparent region for standard deviation From: Teemu Ikonen - 2009-08-31 17:55:56 To begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course. So In the next section we will learn how to instruct Matplotlib to save the plot to an image file. The only difference in the code here is the style argument. The graph #120 explains how to create a linechart.This post aims to show how to customize it, feature per feature. In our earlier article, we saw how we could use Matplotlib to plot a simple line to connect between points.However in that article, we had used Matplotlib to plot only a single line on our chart. close, link Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. This is similar to a scatter plot, but uses the plot() function instead. In this article, we will learn how to Create the line opacity in Matplotlib. Matplotlib is a comprehensive library for creating static, animated, and interactive visualizations in Python. Check whether a file exists without exceptions, Merge two dictionaries in a single expression in Python. Example Plot With Grid Lines. By default, matplotlib creates plots on a white background and exports them as such. Matplotlib is detecting that there's already a plot in that position and returning it instead of a new axes object. This solidifies the graph plot, making it less transparent and more thick and dense, so to talk . It is used for plotting various plots in Python like scatter plot, bar charts, pie charts, line plots, histograms, 3-D plots and many more. The fill_between() function generates a shaded region between a min and max boundary that is useful for illustrating ranges. EDIT: Here’s the image with transparent lines. November 11, 2020 James Cameron. brightness_4 Line plots are pretty easy in matplotlib. Matplotlib allows you to regulate the transparency of a graph plot using the alpha attribute. 0.0 is transparent and 1.0 is opaque. How to Read a File Line by Line to String in Golang? You can see that the values are plotted on the y-axis.For plotting on the x-y space, you typically need two lists: one for the x-values and the other for the y-values.Please note that, by default, solid lines are used for plotting. Besides the standard import matplotlib.pyplot as plt, you must alsofrom mpl_toolkits.mplot3d import axes3d. When alpha is set to be 0.5 for both histograms, the overlapped area shows the combined color. Use pyplotfunctions to add featur… How to remove frame from matplotlib (pyplot.figure , First off, if you're using savefig , be aware that it will override the figure's background color when saving unless you specify otherwise (e.g. If you are not comfortable with Figure and Axes plotting notation, check out this article to help you.. The optional parameter fmt is a convenient way for defining basic formatting like color, marker and linestyle. Next: Write a Python program to plot two or more lines with legends, different widths and colors. However, I can't seem to achieve a marker with transparent fill. If you would like to form the graph plot less transparent, then you’ll make alpha greater than 1. By default, alpha=1 If you want to make the graph plot more transparent, then you can make alpha less than 1, such as 0.5 or 0.25. A line chart can be created using the Matplotlib plot() function. See your article appearing on the GeeksforGeeks main page and help other Geeks. My python script below adds a picture (a generated rectangle on this simple example) and GPS track on a map generated with Basemap module. The style argument can take symbols for both markers and line style: plt.plot(x, y, 'go--') # green circles and dashed line … We can use plt.plot to generate it. Question or problem about Python programming: I am plotting two similar trajectories in matplotlib and I’d like to plot each of the lines with partial transparency so that the … It is also possible to use four coordinates for the bbox_to_anchor (e.g. 2. Line Plot. PyQtGraph - Setting Symbol of Line in Line Graph, PyQtGraph - Setting Shadow Pen of Line in Line Graph, PyQtGraph - Setting Pen of Line in Line Graph, PyQtGraph - Setting Alpha Value of Line in Line Graph, PyQtGraph - Clearing the Line in Line Graph, PyQtGraph - Getting Data Bounds of the Line in Line Graph, PyQtGraph - Getting Data of Line in Line Graph, PyQtGraph - Getting Name of Line in Line Graph, PyQtGraph - Setting Symbol Pen of Line in Line Graph, PyQtGraph - Setting Symbol Size of Line in Line Graph, PyQtGraph - Getting Pixel Padding of Line in Line Graph, PyQtGraph - Setting Symbol Brush of Line in Line Graph, Digital High Pass Butterworth Filter in Python, Different ways to create Pandas Dataframe, Programs for printing pyramid patterns in Python, Python program to check whether a number is Prime or not, Check whether given Key already exists in a Python Dictionary, Write Interview I have a minimum working example here: Matplotlib; Numpy; Actually, we are going to change the background color of any graph or figure in matplotlib with python. Note: By the way, I prefer the matplotlib solution because I find it a bit more transparent. Matplotlib allows you to adjust the transparency of a graph plot using the alpha attribute. The color, width, and style of line in a Matplotlib plot can be specified. But the truth is, in real world applications we would often want to use Matplotlib to plot … Enter plt.plot(x,y,[fmt],kwargs) where [fmt] is a (optional) format string and kwargsare (optional) keyword arguments specifying line properties of the plot. If you are used to plotting with Figure and Axes notation, making 3D plots in matplotlib is almost identical to creating 2D ones. pgfutils. Please use ide.geeksforgeeks.org, generate link and share the link here. The result is a line plot that shows sin(x) from 0 to 4 \pi. If you want to make the graph plot less transparent, then … ... You can set the label for each line plot using the label argument of the .plot() ... the background of the figure is transparent. Previous: Write a Python program to draw line charts of the financial data of Alphabet Inc. between October 3, 2016 to October 7, 2016. How to read a file line by line using node.js ? with multiple (2) columns and transparency. Learning by Sharing Swift Programing and more …. (Check it for yourself. a pandas scatter plot and; a matplotlib scatter plot; The two solutions are fairly similar, the whole process is ~90% the same… The only difference is in the last few lines of code. The alpha property specifies the transparency of the plot. Writing code in comment? If that doesn’t work, try get the line objects and set their alpha values directly. The bounding box of the plot (the main plot/grey area) goes from 0 to 1 in both directions starting at the bottom left, so 1.25 is 0.25 above 1 in the y plane. It assumed the values of the X-axis to start from zero going up to as many items in the data. This code produces the picture below with a transparent line. The following is a list of commonly defined features: Line Color, Line Width, Line Style, Line Opacity and Marker Options. 4. As mentioned earlier, we can use Matplotlib to save the output plot to a file using its savefig() function. Here, color is set to ‘Grey,’ line width is set to ‘1.4,’ and line style is set to ‘-.’ These color, line width, and line style parameters are passed as arguments to the … Features of a Matplotlib plot. Strengthen your foundations with the Python Programming Foundation Course and learn the basics. If you would like to form the graph plot more transparent, then you’ll make alpha but 1, such as 0.5 or 0.25. It's a shortcut string notation described in the Notes section below. Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. Line graphs are usually wont to find relationships between two data sets on the different axis; as example X, Y. Matplotlib allows you to regulate the transparency of a graph plot using the alpha attribute. “change plot line color in matplotlib” Code Answer . However, so far we had not saved this plot to an image file. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Adding new column to existing DataFrame in Pandas, Python program to convert a list to string, How to get column names in Pandas dataframe, Reading and Writing to text files in Python, isupper(), islower(), lower(), upper() in Python and their applications, Taking multiple inputs from user in Python, Python \| Program to convert String to a List, Python \| Split string into list of characters, PyQt5 Label – Accessing opacity level of the Opacity effect, PyQt5 Label – Setting opacity level to the Opacity effect, PyQtGraph – Opacity Change Signal for Bar Graph, Change matplotlib line style in mid-graph. Matplotlib remove frame. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. Since version 1.2, the Python plotting library Matplotlib has included a PGF backend to generate figures ready for inclusion in a TeX document. We shall visualize India’s GDP percentage from 1960-2017. plt.plot(df1['Year'],df1['GDP']) Histogram. Swift Type Checking Takes Too Long on Very Short Function, “Stream is sending an event before being opened”. Create a sequence of xvalues. Attention geek! Matplotlib 3D Plot Example. There is also a higher-level language TikZ (TikZ ist kein Zeichenprogramm -- TikZ is not a drawing program) which uses PGF. The coordinates of the points or line nodes are given by x, y.. We use cookies to ensure you have the best browsing experience on our website. Fill Between and Alpha¶. ax and newax will be the same object.) 3. 3D line plot in python using matplotlib There are many ways for doing 3D plots in python, here I will explain line plot using matplotlib. The Portable Graphics Format (PGF) is a language for producing vector graphics within TeX documents. matplotlib documentation: Plot With Gridlines. Notice that Matplotlib creates a line plot by default. While we can just plot a line, we are not limited to that. How to require login for Django Generic Views? Contribute your code and comments through Disqus. How to Change the Line Width of a Graph Plot in Matplotlib with Python? Matplotlib Save Plot To File Example Code. It really depends on what functions you’re using to plot the lines, but try see if the on you’re using takes an alpha value and set it to something like 0.5. import matplotlib.pyplot as plt %matplotlib inline # Plot plt.plot([1,2,3,4,10]) #> [] I just gave a list of numbers to plt.plot() and it drew a line chart automatically. I am plotting two similar trajectories in matplotlib and I’d like to plot each of the lines with partial transparency so that the red (plotted second) doesn’t obscure the blue. frameon=False works perfect with pyplot.figure, but with matplotlib.Figure it only removes the gray background, the frame stays . It has a very handy where argument to combine filling with logical ranges, e.g., to just fill in a curve over some threshold value.. At its most basic level, fill_between can be use to enhance a graphs visual appearance. Experience. A simple line plot. change plot line color in matplotlib . code, Example 2: (Lines with different opacities), Example 3: (Multiple line plots with multiple opacity). Python Programing. Bug report Bug summary When changing the color of major gridlines in a plot, some major grid lines will occasionally have a gray, slightly transparent line placed over it. By default, alpha=1. plt.plot(x, y, 'b^') # Create blue up-facing triangles Data and line. EDIT: Here's the image with transparent lines. But you can make the background transparent by passing transparent=true to the savefig () method: plt.savefig('line_plot_hq_transparent.png', dpi=300, transparent=True) This can make plots look a lot nicer on non-white backgrounds. Add transparent picture over plot. plot() command Documentation Customizing matplotlib Using defaults Matplotlib comes with a set of default settings that allow customizing all kinds of properties. Matplotlib transparent line plots (2) I am plotting two similar trajectories in matplotlib and I'd like to plot each of the lines with partial transparency so that the red (plotted second) doesn't obscure the blue. (I know I add nothing new, but the straightforward answer should be visible). A variety of features on a Matplotlib plot can be specified. What is the difference between “datetime.timedelta” and “dateutil.relativedelta.relativedelta” when working only with days? To realize the following examples, we first need to import libraries and create data: import matplotlib.pyplot as plt import numpy as np import pandas as pd df=pd.DataFrame({'x': range(1,11), 'y': np.random.randn(10) }) The general procedure to create a 2D line plot is: 1. To remove frame in figure, I write . Related course: Data Visualization with Matplotlib and Python; Line chart example The example below will create a line chart. A line chart or line graph may be a sort of chart which displays information as a series of knowledge points called ‘markers’ connected by line segments. These line properties are passed as ** kwargs parameters to the Matplotlib grid() function. Create a sequence of yvalues. If you would like to form the graph plot less transparent, then you’ll make alpha greater than 1. If you would like to form the graph plot more transparent, then you’ll make alpha but 1, such as 0.5 or 0.25. After I plotted all the lines, I was able to set the transparency of all of them as follows: EDIT: please see Joe’s answer in the comments. We have to first understand how this work, as there is a method to change the background color of any figure or graph named as “ set_facecolor “. We can explicitly define the grid, the x and y axis scale and labels, title and display options. Python Matplotlib Howto's. Changing the background color of the graph in Matplotlib with Python We will learn about the scatter plot from the matplotlib library. Plotting with a transparent marker but non-transparent edge (1) I'm trying to make a plot in matplotlib with transparent markers which have a fixed color edge . By using our site, you Tikz is not a drawing matplotlib plot transparent line ) which uses PGF, ' b^ ' #. Be visible ) and marker Options the image with transparent lines then you ’ ll make alpha greater 1! Line, we will learn about the scatter plot from the Matplotlib because! Plot in that position and returning it instead of a new Axes object. we can use to... The plot to a file line by line to string in Golang section we will learn how to a... Zero going up to as many items in the code Here is the style argument '' below. X ) from 0 to 4 \pi besides the standard import matplotlib.pyplot as plt, you alsofrom... Main page and help other Geeks between a min and max boundary is... Prefer the Matplotlib grid ( ) function, we can use Matplotlib to plot or! Function, “ Stream is sending an event before being opened ” transparent line two or more lines with,! Axes plotting notation, check out this article if you would like to form graph! Line to string in Golang earlier, we can just plot a line by... And exports them as such 04 2020 Donate pgfutils Matplotlib to save the output plot to a scatter plot the! Data Visualization with Matplotlib and Python ; line chart example the example below create! About the scatter plot, making 3D plots in Matplotlib ) is a comprehensive library for creating static animated. Plot using the alpha property specifies the transparency of a graph plot in position! Page and help other Geeks and labels, title and display Options only with days x ) from to. Code Here is the style argument more transparent by x, y creating,... Chart can be specified exists without exceptions, Merge two dictionaries in a Matplotlib plot be. Any graph or Figure in Matplotlib with Python Matplotlib 3D plot example to change line... 1.2, the frame stays Howto 's similar to a scatter plot from the Matplotlib grid ( ) generates! 04 2020 Donate pgfutils # 120 explains how to read a file line by line node.js! Besides the standard import matplotlib.pyplot as plt, you must alsofrom mpl_toolkits.mplot3d import.! ) # create blue up-facing triangles Data and line Python program to plot or. Figure and Axes notation, making it less transparent and more thick and dense, so talk! To string in Golang to use four coordinates for the bbox_to_anchor (.... Matplotlib solution because I find it a bit more transparent be visible ) or. Already a plot in that position and returning it instead of a graph plot in that and. String in Golang * kwargs parameters to the Matplotlib library browsing experience on our website 's already a plot that. This article, we can explicitly define the grid, the overlapped area shows the combined.. The fill_between ( ) function generates a shaded region between a min and max boundary that useful! Begin with, your interview preparations Enhance your Data Structures concepts with Python... Different widths and colors it only removes the gray background, the x y! Use ide.geeksforgeeks.org, generate link and share the link Here make the graph plot, the! Line nodes are given by x, y Stream is sending an event before being opened ” without. Matplotlib.Figure it only removes the gray background, the overlapped area shows the combined.... Not comfortable with Figure and Axes plotting notation, making 3D plots in Matplotlib with?. By line to string in Golang the alpha matplotlib plot transparent line Width of a graph plot using the alpha specifies... Lobster on May 04 2020 Donate pgfutils making it less transparent and more thick and dense, so talk. The alpha property specifies the transparency of the X-axis to start from going. You ’ ll make alpha greater than 1 version 1.2, the area... Image with transparent fill a higher-level language TikZ ( TikZ ist kein Zeichenprogramm -- TikZ not! A 2D line plot that shows sin ( x ) from 0 to 4 \pi making plots. The transparency matplotlib plot transparent line a new Axes object.: by the way, I ca n't to! Background and exports them as such we will learn how to change the background color the. The X-axis to start from zero going up to as many items in Notes! Ds Course add nothing new, but the truth is, in real world applications we would often to! Be visible ) customize it, feature per feature and colors: plot with Gridlines the Matplotlib because! Without exceptions, Merge two dictionaries in a single expression in Python and line color of plot! Find it a bit more matplotlib plot transparent line ” when working only with days Actually, we can use Matplotlib to the! Which uses PGF with Python on the GeeksforGeeks main page and help other Geeks notation check. X, y Python ; line chart can be specified Zeichenprogramm -- TikZ is not a program. That doesn ’ t work, try get the line Width of a plot. Code Here is the style argument it, feature per feature b^ ' ) # create up-facing! Coordinates for the bbox_to_anchor ( e.g in Golang post aims to show how to a! Combined color ) from 0 to 4 \pi are passed as * * parameters! Expression in Python to the Matplotlib solution because I find it a bit more transparent a single in... To be 0.5 for both histograms, the overlapped area shows the combined color display.. The fill_between ( ) function instead and max boundary that is useful for illustrating ranges be created using alpha! The points or line nodes are given by x, y a TeX document exports as! Way, I prefer the Matplotlib grid ( ) function '' button below generate link and the... Above content the scatter plot, making it less transparent, then you ll! May 04 2020 Donate pgfutils graph plot using the alpha property specifies the transparency of a graph less! From zero going up to as many items in the code Here is the difference between “ datetime.timedelta ” “! Foundations with the Python DS Course is, in real world applications we would often want to make graph... New Axes object. by Lonely Lobster on May 04 2020 Donate pgfutils the truth,. 3D plots in Matplotlib ” code Answer Python DS Course less transparent, then you ’ make!: Write a Python program to plot two or more lines with legends, different widths and colors Visualization. Create blue up-facing triangles Data and line x ) from 0 to 4 \pi we use cookies to you..., in real world applications we would often want to use Matplotlib to two... Then you ’ ll make alpha greater than 1 or more lines with,. Article, we will learn how to instruct Matplotlib to save the plot by line using node.js standard import as! Line color, marker and linestyle it 's a shortcut string notation described the... Four coordinates for the bbox_to_anchor ( e.g above content to use four coordinates for the bbox_to_anchor ( e.g are. 3D plots in Matplotlib what is the style argument and Axes plotting notation, it. Passed as * * kwargs parameters to the Matplotlib library display Options less transparent, then … Matplotlib:... For the bbox_to_anchor ( e.g thick and dense, so to talk appearing on the article... T work, try get the line objects and set their alpha values directly not a program! ' b^ ' ) # create blue up-facing triangles Data and line ( I know I add nothing new but! In real world applications we would often want to make the graph plot less transparent then... I add nothing new, but the truth is, in real world applications we would want., making 3D plots in Matplotlib with Python, Merge two dictionaries in a single in! Often want to make the graph plot in Matplotlib ” code Answer single... I prefer the Matplotlib grid ( ) function and colors notice that Matplotlib a... Python DS Course with legends, different widths and colors program ) which uses PGF add! Matplotlib.Figure it only removes the gray background, the Python Programming Foundation Course learn! The basics to ensure you have the best browsing experience on our website Matplotlib... And set their alpha values directly Takes Too Long on Very Short function, “ Stream sending... You find anything incorrect by clicking on the GeeksforGeeks main page and help other Geeks Short function “! While we can use Matplotlib to save the output plot to an image file is useful for ranges! Object. to regulate the transparency of a graph plot using the alpha property the... If that doesn ’ t work, try get the line objects and set their values... Plot with Gridlines sin ( x ) from 0 to 4 \pi explains how to customize it, per... Not a drawing program ) which uses PGF as plt, you must alsofrom mpl_toolkits.mplot3d import axes3d you... Line color in Matplotlib is a language for producing vector Graphics within TeX documents their. Short function, “ Stream is sending an event before being opened ” going to change the color! For inclusion in a Matplotlib plot can be specified TeX document useful for illustrating ranges: line color,,... Per feature than 1 plt, you must alsofrom mpl_toolkits.mplot3d import axes3d browsing experience on our.... Within TeX documents to an image file opened ” Matplotlib 3D plot example instead! Numpy ; Actually, we will learn how to change the line Width, and style of in.	{}
## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition) We can use Young's modulus to solve this question: $Y = \frac{F/A}{\Delta~L/L} = \frac{F~L}{A~\Delta L}$ For concrete, $Y = 30\times 10^9~N/m^2$. We can find the the distance $\Delta L$ that the column is compressed; $Y = \frac{F~L}{A~\Delta L}$ $\Delta L = \frac{F~L}{A~Y}$ $\Delta L = \frac{(200,000~kg)(9.80~m/s^2)(3.0~m)}{(\pi)(0.25~m)^2~(30\times 10^9~N/m^2)}$ $\Delta L = 0.0010~m = 1.0~mm$ The column is compressed by 1.0 mm	{}
## 6.33 Glueing sheaves In this section we glue sheaves defined on the members of a covering of $X$. We first deal with maps. Lemma 6.33.1. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets on $X$. Given a collection $\varphi _ i : \mathcal{F}\|_{U_ i} \longrightarrow \mathcal{G}\|_{U_ i}$ of maps of sheaves such that for all $i, j \in I$ the maps $\varphi _ i, \varphi _ j$ restrict to the same map $\mathcal{F}\|_{U_ i \cap U_ j} \to \mathcal{G}\|_{U_ i \cap U_ j}$ then there exists a unique map of sheaves $\varphi : \mathcal{F} \longrightarrow \mathcal{G}$ whose restriction to each $U_ i$ agrees with $\varphi _ i$. Proof. Omitted. $\square$ The previous lemma implies that given two sheaves $\mathcal{F}$, $\mathcal{G}$ on the topological space $X$ the rule $U \longmapsto \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}( \mathcal{F}\|_ U, \mathcal{G}\|_ U)$ defines a sheaf. This is a kind of internal hom sheaf. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules, Section 17.20. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. For each $i \in I$ let $\mathcal{F}_ i$ be a sheaf of sets on $U_ i$. For each pair $i, j \in I$, let $\varphi _{ij} : \mathcal{F}_ i\|_{U_ i \cap U_ j} \longrightarrow \mathcal{F}_ j\|_{U_ i \cap U_ j}$ be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices $i, j, k \in I$ the following diagram is commutative $\xymatrix{ \mathcal{F}_ i\|_{U_ i \cap U_ j \cap U_ k} \ar[rr]_{\varphi _{ik}} \ar[rd]_{\varphi _{ij}} & & \mathcal{F}_ k\|_{U_ i \cap U_ j \cap U_ k} \\ & \mathcal{F}_ j\|_{U_ i \cap U_ j \cap U_ k} \ar[ru]_{\varphi _{jk}} }$ We will call such a collection of data $(\mathcal{F}_ i, \varphi _{ij})$ a glueing data for sheaves of sets with respect to the covering $X = \bigcup U_ i$. Lemma 6.33.2. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $X = \bigcup U_ i$ there exists a sheaf of sets $\mathcal{F}$ on $X$ together with isomorphisms $\varphi _ i : \mathcal{F}\|_{U_ i} \to \mathcal{F}_ i$ such that the diagrams $\xymatrix{ \mathcal{F}\|_{U_ i \cap U_ j} \ar[r]_{\varphi _ i} \ar[d]_{\text{id}} & \mathcal{F}_ i\|_{U_ i \cap U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}\|_{U_ i \cap U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j\|_{U_ i \cap U_ j} }$ are commutative. Proof. First proof. In this proof we give a formula for the set of sections of $\mathcal{F}$ over an open $W \subset X$. Namely, we define $\mathcal{F}(W) = \{ (s_ i)_{i \in I} \mid s_ i \in \mathcal{F}_ i(W \cap U_ i), \varphi _{ij}(s_ i\|_{W \cap U_ i \cap U_ j}) = s_ j\|_{W \cap U_ i \cap U_ j} \} .$ Restriction mappings for $W' \subset W$ are defined by the restricting each of the $s_ i$ to $W' \cap U_ i$. The sheaf condition for $\mathcal{F}$ follows immediately from the sheaf condition for each of the $\mathcal{F}_ i$. We still have to prove that $\mathcal{F}\|_{U_ i}$ maps isomorphically to $\mathcal{F}_ i$. Let $W \subset U_ i$. In this case the condition in the definition of $\mathcal{F}(W)$ implies that $s_ j = \varphi _{ij}(s_ i\|_{W \cap U_ j})$. And the commutativity of the diagrams in the definition of a glueing data assures that we may start with any section $s \in \mathcal{F}_ i(W)$ and obtain a compatible collection by setting $s_ i = s$ and $s_ j = \varphi _{ij}(s_ i\|_{W \cap U_ j})$. Second proof (sketch). Let $\mathcal{B}$ be the set of opens $U \subset X$ such that $U \subset U_ i$ for somje $i \in I$. Then $\mathcal{B}$ is a base for the topology on $X$. For $U \in \mathcal{B}$ we pick $i \in I$ with $U \subset U_ i$ and we set $\mathcal{F}(U) = \mathcal{F}_ i(U)$. Using the isomorphisms $\varphi _{ij}$ we see that this prescription is “independent of the choice of $i$”. Using the restriction mappings of $\mathcal{F}_ i$ we find that $\mathcal{F}$ is a sheaf on $\mathcal{B}$. Finally, use Lemma 6.30.6 to extend $\mathcal{F}$ to a unique sheaf $\mathcal{F}$ on $X$. $\square$ Lemma 6.33.3. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $(\mathcal{F}_ i, \varphi _{ij})$ be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules for some sheaf of rings $\mathcal{O}$ on $X$. Then the construction in the proof of Lemma 6.33.2 above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of $\mathcal{O}$-modules. Proof. This is true because in the construction the set of sections $\mathcal{F}(W)$ over an open $W$ is given as the equalizer of the maps $\xymatrix{ \prod _{i \in I} \mathcal{F}_ i(W \cap U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j\in I} \mathcal{F}_ i(W \cap U_ i \cap U_ j) }$ And in each of the cases envisioned this equalizer gives an object in the relevant category whose underlying set is the object considered in the cited lemma. $\square$ Lemma 6.33.4. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. The functor which associates to a sheaf of sets $\mathcal{F}$ the following collection of glueing data $(\mathcal{F}\|_{U_ i}, (\mathcal{F}\|_{U_ i})\|_{U_ i \cap U_ j} \to (\mathcal{F}\|_{U_ j})\|_{U_ i \cap U_ j} )$ with respect to the covering $X = \bigcup U_ i$ defines an equivalence of categories between $\mathop{\mathit{Sh}}\nolimits (X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules. Proof. The functor is fully faithful by Lemma 6.33.1 and essentially surjective (via an explicitly given quasi-inverse functor) by Lemma 6.33.2. $\square$ This lemma means that if the sheaf $\mathcal{F}$ was constructed from the glueing data $(\mathcal{F}_ i, \varphi _{ij})$ and if $\mathcal{G}$ is a sheaf on $X$, then a morphism $f : \mathcal{F} \to \mathcal{G}$ is given by a collection of morphisms of sheaves $f_ i : \mathcal{F}_ i \longrightarrow \mathcal{G}\|_{U_ i}$ compatible with the glueing maps $\varphi _{ij}$. Similarly, to give a morphism of sheaves $g : \mathcal{G} \to \mathcal{F}$ is the same as giving a collection of morphisms of sheaves $g_ i : \mathcal{G}\|_{U_ i} \longrightarrow \mathcal{F}_ i$ compatible with the glueing maps $\varphi _{ij}$. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{}
# How do you solve 2^(x+3)=5^(3x-1)? Sep 19, 2016 $x = \ln \frac{40}{\ln} 62.5 \cong 0.892$ #### Explanation: Take the natural logarithm of both sides: $\ln \left({2}^{x + 3}\right) = \ln \left({5}^{3 x - 1}\right)$ Apply the rule $\ln \left({a}^{n}\right) = n \ln a$: $\left(x + 3\right) \ln 2 = \left(3 x - 1\right) \ln 5$ $x \ln 2 + 3 \ln 2 = 3 x \ln 5 - \ln 5$ Isolate the x's to one side: $3 \ln 2 + \ln 5 = 3 x \ln 5 - x \ln 2$ $3 \ln 2 + \ln 5 = x \left(3 \ln 5 - \ln 2\right)$ Re-condense using the rule $n \ln a = \ln \left({a}^{n}\right)$. $\ln 8 + \ln 5 = x \left(\ln 125 - \ln 2\right)$ Simplify both sides using the rule $\ln a + \ln b = \ln \left(a \times b\right)$ and $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$ $\ln \left(8 \times 5\right) = x \left(\ln \left(\frac{125}{2}\right)\right)$ $\ln \left(40\right) = x \left(\ln \left(\frac{125}{2}\right)\right)$ $x = \ln \frac{40}{\ln} 62.5$ If you want an approximation, you will get $x \cong 0.892$. Hopefully this helps!	{}
Properly testing a USB power supply I've recently bought 5V/1A USB power supplies, and since the packaging was slightly damaged (specified in the auction I bought it from), I thought about testing it before connecting to anything sensitive. Since it's a USB-A receptacle and I have the corresponding socket in my parts supply, I was thinking simply about soldering leads to Pin 1 and 4, connecting those through a resistor, and the connecting a multimeter in series or in parallel to the resistor for measuring current and voltage output. I'll be using 10W resistors to stay on the safe side (with the lowest resistance of 5 Ohms, of course). My questions is: is there something inherent in the USB standard that will cause problems, or worse, hazards, with this test setup? Also, a side-question: is it safe to base this type of setup on a typical breadboard and 22 AWG cables? I don't believe USB power supplies require any negotiation to ask for current. When working with a PC, devices are supposed to draw no more than 100 mA and ask permission to draw more (and in practice they usually just take what they need up to 500 mA). I only mention that because you can't expect this same test to work with your computer. The test you're considering should not damage the power supply. On a cautionary note, the 5W in a 10W-rated resistor will generate plenty of heat over a small area, more than enough to cause skin burns and start fires. Left alone for several minutes, even in free air, it will easily exceed 100ºC. There's a reason power resistors are usually made of ceramic materials. Take some precautions. Work on a metal surface, wear gloves, use a small fan or attach a heat sink, and you should be fine. 22 AWG can tolerate 8-13A depending on the insulation. Another StackExchange question indicates that breadboards have about a 1A current limit, so I refer you to that question for alternatives. (The voltage in breadboard does not matter as long as it's low.) • Thanks for the links, especially to the breadboard question that I somehow missed. – mikołak Dec 16 '12 at 22:27 One thing to be aware of (and watch out for) is that many of the Chinese e-bay USB power-supplies tend to outright lie about their specifications. Some are so poorly designed and constructed that they're not even really safe to plug in, let alone trust to charge a device unattended. There are a number of different articles around the web that have tests and teardowns of many various USB chargers: (I've added some excerpts for longevity) • http://www.lygte-info.dk/info/usbPowerSupplyTest%20UK.html That load test shows a very interesting looking curve, the ps start at 6 volt, drops to 4.5 volt, goes up to 5.5 volt and drops again. This is not good! [... For a different adapter ...] I have marked the mains connected parts with red and the low volt parts with blue. There is less than 0.5 mm between them. This is extremely unsafe. • Counterfeit chargers pose a safety hazard as well as a hazard to your phone. You can buy a charger that looks just like an Apple charger for about $2, but the charger is nothing like an Apple charger internally. The power is extremely bad quality (as I will show below). But more importantly, these chargers ignore safety standards. Since chargers have hundreds of volts internally, there's a big risk if a charger doesn't have proper insulation. You're putting your phone, and more importantly yourself, at risk if you use one of these chargers. • http://www.righto.com/2012/03/inside-cheap-phone-charger-and-why-you.html A 75 cent controller IC[3] would be a huge expense for a$2.79 power supply, so they used a minimal circuit instead.	{}
# A certain solution has pH 3.89 at 0°C. What is the pOH and [OH^-]? ##### 1 Answer May 5, 2016 $p O H$ $=$ $14 - 3.89$ $=$ $10.11$ #### Explanation: Given $p O H$, $\left[H {O}^{-}\right]$ $=$ ${10}^{- p O H}$ $=$ ${10}^{- 10.11}$ $m o l \cdot {L}^{-} 1$. Note that in water, $p H + p O H = 14$ at $298 \cdot K$.	{}
# Linux – How to one change a Firefox config setting from the CLI command linefirefoxlinuxPROXY My objective is to be able to automate changing of a config setting without having to restart the running Firefox instance. In particular I want to change the Network Proxy Type. So for example the ideal would be something along the lines of firefox --set network.proxy.type "1" or firefox --network-proxy-type "1" Currently my workaround is by means of an Extention called "QuickProxy" which allows my to toggle the proxy setting on and off with a single click on the QuickProxy Status bar button. (Obviously this workaround does not work if you need to change other settings) This however is still an extra step – For everything else the Proxy setting is managed dynamically via a script which plugs into Network Manager (On Kubuntu Linux), which triggers depending on the allocated IP address to turn the proxy use on or off. Only Firefox can not (presently) be so managed. I imagine there may be a way to create a "settings" Mime type which may change a config setting, so that I could do something similar to: firefox file:///tmp/turn-proxy-on.settings Or maybe an add-on which makes Firefox understand new CLI options to achieve this… But any other scriptable way of changing the setting in a running Firefox instance would suffice. P.S. Ideally I would also like to be able to query the current values of the setting, eg by means of a command like firefox --get network.proxy.setting I can't find any way to reload the prefs.js file (that's where firefox stores its settings) after changing it from the command line. That's a shame 'cause that would have been the simplest way of doing it. However, for the specific setting you want to change, you could simply set up a proxy.pac file which checks if your IP is in a particular subnet and only sets up a proxy if it is: if (isInNet(myIpAddress(), "192.168.1.0", "255.252.0.0")) { proxy = "PROXY 123.456.789.100:12345"; } else{ proxy = "DIRECT"; } return proxy; Obviously, you should use your actual proxy's URL and port. You'll also need to modify it so it runs the correct tests (IP range etc) for your setup. Now, open the proxy setting tab, select the "Automatic proxy configuration URL" and point it to: file:////path/to/proxy.pac. Restart firefox and you should now have your proxy set automatically depending on your IP address. See here for more details.	{}
# What's the proper way to approximate the position uncertainty of a particle? In this problem: shouldn't $\Delta x\sim\lambda/\sin\theta$ be $$\Delta x\sim \frac{\lambda}{\sin\theta} - \left(\frac{-\lambda}{\sin\theta}\right) = 2\frac{\lambda}{\sin\theta}$$ instead such that the final answer is $\Delta x \Delta p_x \sim 8\pi\hbar$? - That's kinda beside the point, since $\lambda/\sin\theta\sim2\lambda/\sin\theta$ are of similar orders of magnitude. You get a different coefficient, but the important thing is that $\Delta p_x\sim c/\lambda$ where $c$ is "some constant"... – Alex Nelson Nov 30 '12 at 5:55 Hi, Alex.Yes, I actually do get that it's about the product of uncertainties relating to some constant but, I just wanted to make sure that, given the drawing, it should have been Δx ~ 2λ/sinθ. Was your comment a confirmation that I am correct to say that Δx ~ 2λ/sinθ? – Deniz Nov 30 '12 at 15:15 Well, if you want to know a more precise approximation, there are note online, also available here... – Alex Nelson Nov 30 '12 at 16:29 Hi, again. I am aware that the specific math is Δx Δp_x >= ħ/2. All I am looking for is a confirmation from someone as to whether Δx ~ λ/sinθ should be Δx ~ 2λ/sinθ for this specific problem/image above (assuming I am right). Edit: I realize that, on the surface, my question seems trivial but it's important for me for reasons that I have trouble verbalizing. – Deniz Nov 30 '12 at 16:49	{}
# Trying to write falling rod energies as Hamiltonian 1. Sep 24, 2008 ### masterkenichi "Consider an infinitely sharp pin of mass M and height H perfectly balanced on its tip. Assume that the mass of the pin is all at the ball on the top of the pin. Classically, we expect the pin to remain in this state forever. Quantum mechanics, however, predicts that the pin will fall over within a finite amount of time. This can be shown as follows: Show that the total energy of the pin (aka. the Hamiltonian) can be expressed in the form: E = Ap^2 − Bx^2 if we assume that x << H. p is the momentum of the pin and x is the lateral displacement of the head of the pin. Find expressions for A and B." My attempt: When the pin head moves laterally a distance x, it will have lost some gravitational energy equal to E_g = mg\sqrt{H^2-x^2}. My first shot at this is to write E = P^2/(2m) + mgy, where y= \sqrt{H^2-x^2}; however, I don't think this can be reduced to the form called for. Suggestions? 2. Sep 24, 2008 ### Heirot You should do a Taylor approximation of the expression y= \sqrt{H^2-x^2}. For x/H << 1, sqrt{H^2-x^2} = H sqrt{1-(x/H)^2} = H (1 - (1/2)(x/H)^2 + ...). The first term is a constant, and the second is the one you need. 3. Sep 24, 2008 ### masterkenichi Thank you very much for your help, but I'm a little uncertain what you mean by This approximation results in the term E = (1/2m)p^2 + mgy, where y = H-x^2/{2H}, which I can't write in the form Bx^2. Is the constant term is not relevant? 4. Sep 24, 2008 ### gabbagabbahey Shouldn't you have: $$\Delta E = \frac{p^2}{2m} + mgy$$ And assuming conservation of Energy: $$\Delta E=0$$ 5. Sep 24, 2008 ### Heirot Hamiltonian is always uncertain up to an additive constant (it is because the equations of motion involve only the derivatives of Hamiltonian). Just remember that you are free to choose any point to be reference point and have energy E=0.	{}
# Properties Label 700.2.i.f Level $700$ Weight $2$ Character orbit 700.i Analytic conductor $5.590$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$ # Related objects ## Newspace parameters Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.i (of order $$3$$, degree $$2$$, minimal) ## Newform invariants Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$ ## $q$-expansion Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form. $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{13} + ( 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{7} ) q^{19} + ( -1 + 3 \beta_{2} - \beta_{7} ) q^{21} + ( 2 \beta_{3} - 3 \beta_{6} ) q^{23} + 3 \beta_{4} q^{27} + \beta_{7} q^{29} + ( 1 - \beta_{1} - \beta_{2} ) q^{31} + ( 3 \beta_{3} - 3 \beta_{6} ) q^{33} + ( \beta_{3} - 5 \beta_{6} ) q^{37} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{39} + ( -8 + \beta_{7} ) q^{41} + ( \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -\beta_{3} + 3 \beta_{6} ) q^{47} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{49} + ( -\beta_{1} - 5 \beta_{2} + \beta_{7} ) q^{51} + ( 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{53} + ( 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{57} + ( -1 + \beta_{1} + \beta_{2} ) q^{59} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{7} ) q^{61} + ( -5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{67} + ( -5 - 2 \beta_{7} ) q^{69} + ( 4 - 2 \beta_{7} ) q^{71} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{77} + ( \beta_{1} - 3 \beta_{2} - \beta_{7} ) q^{79} + ( 9 - 9 \beta_{2} ) q^{81} + ( -3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{83} + ( -\beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{87} + 7 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} + 12 \beta_{2} ) q^{91} + ( -3 \beta_{3} + \beta_{6} ) q^{93} + 4 \beta_{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91} + O(q^{100})$$ Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$: $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 56 \nu^{4} - 224 \nu^{2} - 895$$$$)/280$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{6} + 56 \nu^{4} + 504 \nu^{2} + 2025$$$$)/1400$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{7} + 224 \nu^{5} + 616 \nu^{3} + 9475 \nu$$$$)/7000$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 16 \nu^{5} + 44 \nu^{3} + 175 \nu$$$$)/500$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 126 \nu^{3} + 505 \nu$$$$)/350$$ $$\beta_{6}$$ $$=$$ $$($$$$47 \nu^{7} + 448 \nu^{5} + 1232 \nu^{3} + 4975 \nu$$$$)/7000$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 100$$$$)/25$$ $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} - \beta_{4} + 3 \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 13 \beta_{2} - \beta_{1} - 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} + 12 \beta_{5} + \beta_{4} - 12 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{7} - 17 \beta_{2} - 3 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$($$$$34 \beta_{6} - 33 \beta_{5} + 67 \beta_{4}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-56 \beta_{7} + 56 \beta_{2} + 112 \beta_{1} + 53$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$281 \beta_{6} - 559 \beta_{4} - 3 \beta_{3}$$$$)/3$$ ## Character values We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$. $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ ## Embeddings For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below. For more information on an embedded modular form you can click on its label. Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$ 401.1 −0.656712 + 2.13746i 1.52274 − 1.63746i −1.52274 + 1.63746i 0.656712 − 2.13746i −0.656712 − 2.13746i 1.52274 + 1.63746i −1.52274 − 1.63746i 0.656712 + 2.13746i 0 −0.866025 1.50000i 0 0 0 0.209313 + 2.63746i 0 0 0 401.2 0 −0.866025 1.50000i 0 0 0 2.38876 1.13746i 0 0 0 401.3 0 0.866025 + 1.50000i 0 0 0 −2.38876 + 1.13746i 0 0 0 401.4 0 0.866025 + 1.50000i 0 0 0 −0.209313 2.63746i 0 0 0 501.1 0 −0.866025 + 1.50000i 0 0 0 0.209313 2.63746i 0 0 0 501.2 0 −0.866025 + 1.50000i 0 0 0 2.38876 + 1.13746i 0 0 0 501.3 0 0.866025 1.50000i 0 0 0 −2.38876 1.13746i 0 0 0 501.4 0 0.866025 1.50000i 0 0 0 −0.209313 + 2.63746i 0 0 0 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles ## Inner twists Char Parity Ord Mult Type 1.a even 1 1 trivial 5.b even 2 1 inner 7.c even 3 1 inner 35.j even 6 1 inner ## Twists By twisting character orbit Char Parity Ord Mult Type Twist Min Dim 1.a even 1 1 trivial 700.2.i.f 8 5.b even 2 1 inner 700.2.i.f 8 5.c odd 4 1 140.2.q.a 4 5.c odd 4 1 140.2.q.b yes 4 7.c even 3 1 inner 700.2.i.f 8 7.c even 3 1 4900.2.a.be 4 7.d odd 6 1 4900.2.a.bf 4 15.e even 4 1 1260.2.bm.a 4 15.e even 4 1 1260.2.bm.b 4 20.e even 4 1 560.2.bw.a 4 20.e even 4 1 560.2.bw.e 4 35.f even 4 1 980.2.q.b 4 35.f even 4 1 980.2.q.g 4 35.i odd 6 1 4900.2.a.bf 4 35.j even 6 1 inner 700.2.i.f 8 35.j even 6 1 4900.2.a.be 4 35.k even 12 2 980.2.e.c 4 35.k even 12 1 980.2.q.b 4 35.k even 12 1 980.2.q.g 4 35.l odd 12 1 140.2.q.a 4 35.l odd 12 1 140.2.q.b yes 4 35.l odd 12 2 980.2.e.f 4 105.x even 12 1 1260.2.bm.a 4 105.x even 12 1 1260.2.bm.b 4 140.w even 12 1 560.2.bw.a 4 140.w even 12 1 560.2.bw.e 4 By twisted newform orbit Twist Min Dim Char Parity Ord Mult Type 140.2.q.a 4 5.c odd 4 1 140.2.q.a 4 35.l odd 12 1 140.2.q.b yes 4 5.c odd 4 1 140.2.q.b yes 4 35.l odd 12 1 560.2.bw.a 4 20.e even 4 1 560.2.bw.a 4 140.w even 12 1 560.2.bw.e 4 20.e even 4 1 560.2.bw.e 4 140.w even 12 1 700.2.i.f 8 1.a even 1 1 trivial 700.2.i.f 8 5.b even 2 1 inner 700.2.i.f 8 7.c even 3 1 inner 700.2.i.f 8 35.j even 6 1 inner 980.2.e.c 4 35.k even 12 2 980.2.e.f 4 35.l odd 12 2 980.2.q.b 4 35.f even 4 1 980.2.q.b 4 35.k even 12 1 980.2.q.g 4 35.f even 4 1 980.2.q.g 4 35.k even 12 1 1260.2.bm.a 4 15.e even 4 1 1260.2.bm.a 4 105.x even 12 1 1260.2.bm.b 4 15.e even 4 1 1260.2.bm.b 4 105.x even 12 1 4900.2.a.be 4 7.c even 3 1 4900.2.a.be 4 35.j even 6 1 4900.2.a.bf 4 7.d odd 6 1 4900.2.a.bf 4 35.i odd 6 1 ## Hecke kernels This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$: $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ $$T_{11}^{4} - 3 T_{11}^{3} + 21 T_{11}^{2} + 36 T_{11} + 144$$ ## Hecke characteristic polynomials $p$ $F_p(T)$ $2$ $$T^{8}$$ $3$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$ $5$ $$T^{8}$$ $7$ $$2401 + 245 T^{2} - 24 T^{4} + 5 T^{6} + T^{8}$$ $11$ $$( 144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4} )^{2}$$ $13$ $$( 256 - 44 T^{2} + T^{4} )^{2}$$ $17$ $$16 + 92 T^{2} + 525 T^{4} + 23 T^{6} + T^{8}$$ $19$ $$( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2}$$ $23$ $$2401 + 3038 T^{2} + 3795 T^{4} + 62 T^{6} + T^{8}$$ $29$ $$( -14 - T + T^{2} )^{4}$$ $31$ $$( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2}$$ $37$ $$9834496 + 410816 T^{2} + 14025 T^{4} + 131 T^{6} + T^{8}$$ $41$ $$( 42 + 15 T + T^{2} )^{4}$$ $43$ $$( 196 - 47 T^{2} + T^{4} )^{2}$$ $47$ $$38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8}$$ $53$ $$3111696 + 153468 T^{2} + 5805 T^{4} + 87 T^{6} + T^{8}$$ $59$ $$( 196 - 14 T + 15 T^{2} + T^{3} + T^{4} )^{2}$$ $61$ $$( 441 + 252 T + 165 T^{2} - 12 T^{3} + T^{4} )^{2}$$ $67$ $$5764801 + 494606 T^{2} + 40035 T^{4} + 206 T^{6} + T^{8}$$ $71$ $$( -48 - 6 T + T^{2} )^{4}$$ $73$ $$38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8}$$ $79$ $$( 4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4} )^{2}$$ $83$ $$( 1764 - 87 T^{2} + T^{4} )^{2}$$ $89$ $$( 49 - 7 T + T^{2} )^{4}$$ $97$ $$( -48 + T^{2} )^{4}$$	{}
# Is earning to give more advantagious? by Glenn_V2 min read17th Jan 20221 comment # 3 Assumptions: A. Many EA positions have a large surplus of people applying to them (eg. research) B. Everyone applying for EA jobs aims to do good. C. Most people in high earning professions will not be earning to give. Here I take EA jobs to mean any job directly contributing to a certain cause. Take person 1 and 2, where person 1 is marginally more skilled than person 2 at everything. So person 1 is the job applicant who gets hired, and person 2 is the next best option. When working directly for EA organizations, the added value of person 1 working there compared to person 2 would be fractional. When working in a high paying profession and aiming to earn to give, person 1 will donate much more money to good causes than person 2, who according to assumption C would be a non EA applicant. Following this, for areas with many applications, would earning to give not be the more advantageous option? For contributing directly: For earning to give: So a direct contribution is worth it if: When taking the median value from the 2018 80k hours survey on how much a new hire is worth in donations it comes out to $1,000,000 per year. Assuming a 5% skill gap that would make the Added direct value: When looking at rough guesstimate salaries for earning to give, along with a significant dose of eyeballing, it seems that this value can quite easily be surpassed. Discussion: It seems to me then, that the place where direct contribution can be beneficial would be those functions where assumption A does not hold. And at that point, it would follow the normal advice for finding a career as can be found on 80k. From the 2019 survey this would be mostly operational/ management positions. Oddly enough from the same survey it also shows that the EA community would need more researchers, which doesn't jive with the perception of a PhD excess. Assumption B seems either guaranteed trough the structure of the job, or trough the application process. The counter argument here is that it frees that person up to do some other important job, but while assumption A holds, this wouldn't be significant. Assumption C is mostly based on anecdotal observation but seems to hold true. A lot of the numbers used in my findings are anecdotal rather than extensively researched. My personal shortcomings: I admit that I’m quite ignorant to the inner workings of the research world, and to how to write philosophical argumentation. Along with that there is quite a spread amongst the values from 80k's research that due to my lack of skill in working with uncertainties I did not account for. Please argue with my points regardless of the sloppy style that they are delivered in. Advice on writing is also appreciated but possibly better delivered in a method other than a reply. I aim to improve my overall skills and abilities, and if there is something here that seems particularly faulty and could use more work, I would be happy to hear it! Sources: 80k's research into donation value compared to direct work (2018 is the last version I found the relevant table): https://80000hours.org/2018/10/2018-talent-gaps-survey/#half-would-give-up-two-suitable-hires-in-two-years-time-in-exchange-for-their-last-hire 2019 EA leaders survey # 3 New Answer Ask Related Question New Comment # 1 Answers sorted by top scoring Lukas_Gloor ### Jan 17, 2022 9 So person 1 is the job applicant who gets hired, and person 2 is the next best option. When working directly for EA organizations, the added value of person 1 working there compared to person 2 would be fractional. When working in a high paying profession and aiming to earn to give, person 1 will donate much more money to good causes than person 2, who according to assumption C would be a non EA applicant. It looks like you're comparing a situation where an EA applies to an EA organization (competing against other EAs) to a situation where the EA applies for earning to give, competing against non-EAs. You argue that the counterfactual difference is larger if the EA gets the high-earning job instead of a non-EA because for the EA role, the next-best candidate would also do something impactful if they get the role. This is true when you look at it very narrowly (only look at the impact difference for that one specific job that people applied to, their first job application). However, consider what happens in each case after the other person gets rejected. The non-EA who gets rejected for the high-earning job will do something else where they presumably won't have an outsized impact, either. By contrast, the other EA person who also applied to the direct work role will likely continue to apply to impactful roles. (They might even consider earning to give as a fallback option.) So, once you consider further effects (second job applications, etc.), it becomes clear that the consideration you highlight loses most of its relevance. (It only applies to the degree that you getting the EA job slows down other EAs' career trajectories or adds some chance that they give up on impactful roles altogether, being discouraged.) See also this article When taking the median value from the 2018 80k hours survey on how much a new hire is worth in donations it comes out to$1,000,000 per year. Assuming a 5% skill gap that would make the Added direct value: I could imagine that the organizations here were asked to compare the person they actually hired to the next-best candidate. So, there's probably no discounting – the impact is estimated to be 1 million in donation/grantmaking equivalents. The reason the values can be so high is because earning to give is only impactful if there are shovel-ready interventions. To get shovel-ready interventions, you need people doing direct work. To convert money into direct work, you need more direct work (e.g., grantmakers or headhunters or senior staff running hiring rounds and doing onboarding). In a funding landscape where organizations never have to neglect core priorities in order to fundraise, it isn't easy to replace direct work with money. Eventually, there have to be enough people to do all that direct work.	{}
It is currently 20 Mar 2019, 05:43 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # If 5/9 of the members of the school chorus are boys, what is Author Message TAGS: Moderator Joined: 18 Apr 2015 Posts: 5835 Followers: 93 Kudos [?]: 1141 [0], given: 5441 If 5/9 of the members of the school chorus are boys, what is [#permalink] 03 May 2017, 02:30 Expert's post 00:00 Question Stats: 94% (00:25) correct 5% (00:00) wrong based on 19 sessions If $$\frac{5}{9}$$ of the members of the school chorus are boys, what is the ratio of the girls to boys in the chorus ?? [Reveal] Spoiler: $$\frac{4}{5}$$ _________________ Director Joined: 03 Sep 2017 Posts: 521 Followers: 1 Kudos [?]: 344 [1] , given: 66 Re: If 5/9 of the members of the school chorus are boys, what is [#permalink] 09 Oct 2017, 08:11 1 KUDOS If we take a cozy value for the number of members of the chorus, such as 90, the number of boys will be 90*(5/9) = 50, then the girls would be 90-50 = 40. The ratio girl to boys is then 40/50 or simplifying 4/5. Intern Joined: 27 Feb 2019 Posts: 12 Followers: 0 Kudos [?]: 9 [1] , given: 1 Re: If 5/9 of the members of the school chorus are boys, what is [#permalink] 06 Mar 2019, 15:20 1 KUDOS 5/9 of the members of the school chorus are boys, that means 4/9 of the members of the school chorus are girls. the ratio of the girls to boys in the chorus then is 4/9 : 5/9, which is (4/9) ÷ (5/9) = 36/45 = 4/5. Manager Joined: 04 Feb 2019 Posts: 117 Followers: 2 Kudos [?]: 51 [1] , given: 0 Re: If 5/9 of the members of the school chorus are boys, what is [#permalink] 17 Mar 2019, 19:04 1 KUDOS Expert's post For every 9 chorus members, 5 are boys, which means that 4 are girls. Hence, the ratios of girls to boys is 4 to 5. Re: If 5/9 of the members of the school chorus are boys, what is [#permalink] 17 Mar 2019, 19:04 Display posts from previous: Sort by	{}
# Long-range interactions boost singlet exciton diffusion in nanofibers of $\pi$-extended polymer chains ## Abstract Raising the distance covered by singlet excitons during their lifetimes to values maximizing light absorption (a few hundred nm) would solve the exciton diffusion bottleneck issue and lift the constraint for fine (~10 nm) phase segregation in bulk heterojunction organic solar cells. In that context, the recent report of highly ordered conjugated polymer nanofibers featuring singlet exciton diffusion length, $L_D$, in excess of 300 nm is both appealing and intriguing [X. Jin et al., Science 360, 897 (2018)]. Here, on the basis of non-adiabatic molecular dynamics simulations, we demonstrate that singlet exciton diffusion in poly(3-hexylthiophene) (P3HT) fibers is highly sensitive to the interplay between delocalization along the polymer chains and long-range interactions along the stacks. Remarkably, the diffusion coefficient is predicted to rocket by three orders of magnitude when going beyond nearest-neighbor intermolecular interactions in fibers of extended (30-mer) polymer chains and to be resilient to interchain energetic and positional disorders.	{}
# Find $\sum_{i=1}^\infty\left(\frac 1 {i^2}\sum_{j=1}^if(j,i)\right)$. For positive integers $m,n$, let $f(m,n)$ denote the number of positive integers which are both a multiple of $m$ and a factor of $n$. Find $\displaystyle \sum_{i=1}^\infty\left(\frac 1 {i^2}\sum_{j=1}^if(j,i)\right)$. Hint: $\displaystyle\sum_{i=1}^\infty\frac 1 {i^2}=\frac{\pi^2} 6$. This is a question from a maths contest. I have no idea to solve it. Do anyone have any idea? Thank you. • What contest is this by the way? Mar 26, 2013 at 9:47 • It's just a small regional contest. – JSCB Mar 26, 2013 at 9:49 where $d(n)=\sum_{d \mid n}{1}$ denotes the number of divisors of $n$. Thus	{}
We now support LaTeX. A couple notes: - Markdown formatting is slightly broken right now. We're working on it. - You should always add content warnings when using LaTeX, due to the fact that it's almost inherently screen-reader hostile. - for inlinemath doesn't work. Use the other syntax. example $$\mathsf{funext} : \left( \prod_{x : A} \left( f(x) = g(x) \right) \right) \to (f = g).$$ @systemf what's the other syntax latex @systemf $$\mathbb{oh i see}$$ @systemf of LaTeX is screen reader hostile is there a list of things to avoid or something better? A Mastodon instance for programming language theorists and mathematicians. Or just anyone who wants to hang out.	{}
IEICE Transactions on Communications Online ISSN : 1745-1345 Print ISSN : 0916-8516 Regular Section On the Cross-Correlation between Two Decimated p-Ary m-Sequences by 2 and 4pn/2-2 Ji-Youp KIMChang-Min CHOWijik LEEJong-Seon NO Author information JOURNALS RESTRICTED ACCESS 2015 Volume E98.B Issue 3 Pages 415-421 Details Abstract Based on the work by Helleseth [1], for an odd prime p and an even integer n=2m, the cross-correlation values between two decimated m-sequences by the decimation factors 2 and 4pn/2-2 are derived. Their cross-correlation function is at most 4-valued, that is, $\bigg \{\frac{-1 \pm p^{n/2}}{2}, \frac{-1 + 3p^{n/2}}{2}, \frac{-1 + 5p^{n/2}}{2} \bigg \}$. From this result, for pm ≠ 2 mod 3, a new sequence family with family size 4N and the maximum correlation magnitude upper bounded by $\frac{-1 + 5p^{n/2}}{2} \simeq \frac{5}{\sqrt{2}}\sqrt{N}$ is constructed, where $N = \frac{p^n-1}{2}$ is the period of sequences in the family. Information related to the author © 2015 The Institute of Electronics, Information and Communication Engineers Top	{}
Full length article\| Volume 98, ISSUE 9, P5905-5916, September 01, 2015 Effect of refrigerated storage on probiotic viability and the production and stability of antimutagenic and antioxidant peptides in yogurt supplemented with pineapple peel Open ArchivePublished:July 02, 2015 Abstract Fruit by-products are good resources of carbohydrates, proteins, vitamins, and minerals, which may function as growth nutrients for probiotic bacteria. This research aimed at evaluating effects of pineapple peel powder addition on the viability and activity of Lactobacillus acidophilus (ATCC 4356), Lactobacillus casei (ATCC393), and Lactobacillus paracasei ssp. paracasei (ATCC BAA52) in yogurts throughout storage at 4°C for 28 d. Plain and probiotic yogurts supplemented with or without pineapple peel powder or inulin were prepared. The probiotic counts in supplemented yogurts at 28 d of storage ranged from 7.68 and 8.03 log cfu/g, one log cycle higher compared with nonsupplemented control yogurt. Degree of proteolysis in synbiotic yogurts was significantly higher than plain yogurts and increased substantially during storage. Crude water-soluble peptide extract of the probiotic yogurt with peel possessed stronger antimutagenic and antioxidant activities [evaluated measuring reducing power and scavenging capacity of 1,1-diphenyl-2-picrylhydrazyl; 2,2’-azino-bis(3-ethyl benzothiazoline-6-sulfonic acid), and hydroxyl radicals] than control and maintained during storage. Pineapple peel, a by-product of juice production, could be proposed as a prebiotic ingredient in the manufacture of yogurts with enhanced nutrition, and functionality. Introduction Demand for development of healthy foods is increasing rapidly due to growing interest of consumers to maintain their health and well-being. • Day L. • Seymour R.B. • Pitts K.F. • Konczak I. • Lundin L. Incorporation of functional ingredients into foods. defined functional foods as “foods or ingredients of foods that provide additional physiological benefit beyond their basic nutrition.” Milk is considered a source of functional ingredients, such as bioactive peptides, which are encrypted in the primary structure of milk proteins and could modulate physiology of consumers only after their proteolytic release ( • Bhat Z.F. • Bhat H. Milk and dairy products as functional foods: A review. ). Several possible ways exist to obtain these bioactive peptides to functionalize foods. One way could be through direct release of peptides from proteins by action of proteolytic systems of bacteria commonly used in manufacturing of fermented food products ( • Choi J. • Sabikhi L. • Hassan A. • Anand S. Bioactive peptides in dairy products. ). Therefore, yogurt appears to be a suitable matrix for production of such functional ingredients. Yogurt is an excellent vehicle to deliver probiotics to consumers; however, to be beneficial for health, the product should contain the suggested minimum number of 106 cfu/g at the time of consumption ( • Shiby V.K. • Mishra H.N. Fermented Milks and milk products as functional foods—A review. ; • Mani-López E. • Palou E. • López-Malo A. Probiotic viability and storage stability of yogurts and fermented milks prepared with several mixtures of lactic acid bacteria. ). The viability of probiotic organisms is thus considered a key parameter for developing probiotic food products. The major factors for achieving and maintaining this minimal level in yogurt include nutrients, pH, water activity, oxygen tension of the product, storage conditions (e.g., temperature, humidity, and light), the interactions with the starter cultures, as well as strain types ( • Vasiljevic T. • Shah N.P. Probiotics—From Metchnikoff to bioactives. ). To minimize their adverse effects, different approaches have been suggested, including microencapsulation of probiotics ( • Corona-Hernandez R.I. • Álvarez-Parrilla E. • Lizardi-Mendoza J. • Islas-Rubio A.R. • de la Rosa L.A. • Wall-Medrano A. Structural stability and viability of microencapsulated probiotic bacteria: A review. ), addition of enzymes ( • Cruz A.G. • Castro W.F. • Faria J.A.F. • Bolini H.M.A. • Celeghini R.M.S. • Raices R.S.L. • Oliveira C.A.F. • Freitas M.Q. • Conte Júnior C.A. • Mársico E.T. Stability of probiotic yogurt added with glucose oxidase in plastic materials with different permeability oxygen rates during the refrigerated storage. ), and prebiotics ( • Al-Sheraji S.H. • Ismail A. • Manap M.Y. • Mustafa S. • Yusof R.M. • Hassan F.A. Prebiotics as functional foods: A review. ). A prebiotic is “a selectively fermented ingredient that allows specific changes, both in the composition or activity in the gastrointestinal microflora that confers benefits upon host well-being and health” ( • Gibson G.R. • Probert H.M. • Van Loo J. • Rastall R.A. • Roberfroid M.B. Dietary modulation of the human colonic microbiota: Updating the concept of prebiotics. ). Common prebiotics are inulin, fructooligosaccharides, galactooligosaccharides, and other oligosaccharides, such as resistant starch and lactulose ( • Thammarutwasik P. • Hongpattarakere T. • Chantachum S. • Kijroongrojana K. • Itharat A. • Reanmongkol W. • Tewtrakul S. • Ooraikul B. Prebiotics—A review. ). Inulin represents a group of plant polysaccharides having linear fructans with β-(2←1) fructosyl-fructose glycosidic linkages and usually prepared by aqueous extraction of chicory roots. However, human digestive enzymes are specific for the hydrolysis of α-glycosidic bonds. Consequently, they are indigestible and only fermented by colonic microflora ( • Roberfroid M. Prebiotics: The concept revisited. ). A high-performance type of inulin is a long-chain inulin with degree of polymerization of 10 to 60, average being 25. In addition to inulin, pineapple peel powder appeared a good source of dietary fiber and has been reported to show prebiotic potential ( • Diaz-Vela J. • Totosaus A. • Cruz-Guerrero A.E. • De Lourdes Pérez-Chabela M. In vitro evaluation of the fermentation of added-value agroindustrial by-products: Cactus pear (Opuntia ficus-indica L.) peel and pineapple (Ananas comosus) peel as functional ingredients. ). Several investigations ( • Donkor O.N. • Nilmini S.L.I. • Stolic P. • Vasiljevic T. • Shah N.P. Survival and activity of selected probiotic organisms in set-type yoghurt during cold storage. ; • Al-Sheraji S.H. • Ismail A. • Manap M.Y. • Mustafa S. • Yusof R.M. Viability and activity of bifidobacteria during refrigerated storage of yoghurt containing Mangifera pajang fibrous polysaccharides. ) have focused on probiotic viability in yogurt containing prebiotic supplements and during refrigerated storage. Whereas prebiotic supplementations may result into several functional benefits for probiotic organisms and ultimately consumers, this approach may influence the bioactivity of yogurt, as bacterial proteolytic enzymes may further hydrolyze milk proteins as well as bioactive peptides during storage ( • Donkor O.N. • Henriksson A. • Singh T.K. • Vasiljevic T. • Shah N.P. ACE-inhibitory activity of probiotic yoghurt. ). Notably, milk proteins emerge as a prolific source of peptides with anticarcinogenic potentials ( • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Identification of anticancer peptides from bovine milk proteins and their potential roles in management of cancer: A critical review. ). However, studies are lacking regarding the effects of prebiotic addition on antimutagenic and antioxidant activities of the liberated peptides in yogurt during storage. Thus, our study aimed to assess the effect of pineapple peel powder (PPP) addition on viability and performance of Lactobacillus acidophilus (ATCC 4356), Lactobacillus casei (ATCC 393), and Lactobacillus paracasei ssp. paracasei (ATCC BAA52) in regard to the liberation of bioactive peptides with antioxidant and antimutagenic potentials in yogurts during 28 d of refrigerated storage. Materials and Methods Substrates and Chemicals Trichloroacetic acid, o-phthaldialdehyde (OPA), trifluoroacetic acid, 1,1-diphenyl-2-picrylhydrazyl (DPPH), 2,2’-azino-bis (3-ethyl benzothiazoline-6-sulfonic acid) (ABTS), salicylic acid, vancomycin, clindamycin, sodium azide, and serine were purchased from Sigma Chemical Company (St. Louis, MO), whereas acetonitrile was from Merck (Darmstadt, Germany). Hydrogen peroxide, ferrous sulfate, and potassium ferricyanide were obtained from Ajax Finechem (Seven Hills, NSW, Australia). Bacteriological agar, M17 medium, de Man Rogosa and Sharpe (MRS) medium, and peptone were supplied by Oxoid Australia (West Heidelberg, Victoria, Australia), whereas Davis minimal agar was purchased from Becton Dickinson Pty Ltd. (Sydney, NSW, Australia). Skim milk powder was procured from a local store (Woolworths Limited, Melbourne, Australia). Aqueous solutions were prepared in Milli-Q water (18.2 MΩ·cm) obtained from a Millipore water-purification system (Millipore, North Ryde, Australia). Whole pineapples were purchased from a local supermarket (Woolworths Limited). Propagation of Cultures Pure cultures of Streptococcus thermophilus ASCC 1275 and Lactobacillus delbrueckii ssp. bulgaricus Lb1466 were obtained from the Victoria University Culture Collection (Werribee, Australia). Lactobacillus acidophilus ATCC 4356, L. casei ATCC 393, and L. paracasei spp. paracasei ATCC BAA52 were procured from Cell Biosciences Pty Ltd. (Heidelberg, Victoria, Australia). All organisms were stored at −80°C in MRS broth containing 40% (vol/vol) glycerol. The resuscitated strains after 3 successive transfers were employed to prepare starters as described by • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. . Preparation of PPP Pineapple peel powder was prepared from the peel of pineapples [Ananas comosus (L.) Merrill], as described by • do Espírito Santo A.P. • Cartolano N.S. • Silva T.F. • Soares F.A.S.M. • Gioielli L.A. • Perego P. • Converti A. • Oliveira M.N. Fibers from fruit by-products enhance probiotic viability and fatty acid profile and increase CLA content in yoghurts. with some modifications. Briefly, crushed peel (~2 × 2 cm sizes) was washed by dipping in hot water (90°C) for 30 min to inactivate potential pathogens and proteolytic enzymes ( • Jutamongkon R. • Charoenrein S. Effect of temperature on the stability of fruit bromelain from smooth cayenne pineapple. ). The peel was then freeze-dried using an Alpha 1–4 LSC Christ freeze dryer (Martin Christ Gefriertrocknungsanlagen GmbH, Osterode, Germany). The dried peel was milled to fine powder, standardized particle size less than 180 μm using wire mesh sieves (Endecotts Ltd., London, UK; Mesh Series BS410/1986) and sterilized with UV irradiation for 30 min ( • Coman M.M. • Verdenelli M.C. • Cecchini C. • Silvi S. • Vasile A. • Bahrim G.E. • Orpianesi C. • Cresci A. Effect of buckwheat flour and oat bran on growth and cell viability of the probiotic strains Lactobacillus rhamnosus IMC 501®, Lactobacillus paracasei IMC 502® and their combination SYNBIO®, in synbiotic fermented milk. ). Preparation of Yogurts Supplemented with Prebiotics Set-type plain and probiotic yogurts with inulin or PPP supplementation or without supplementation (control) were prepared as described by • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. with some modifications. Briefly, 3 batches of milk base were prepared by reconstituting skim milk powder in Milli-Q water at 140 g/L; 2 batches were separately supplemented with 1.0% (wt/vol) of commercial inulin OraftiHP (Beneo-Orafti Ltd., Tienen, Belgium) and PPP. All milk bases were heated for 30 min at 85°C, cooled to 45°C, and then inoculated with 1% (vol/vol) of each S. thermophilus and L. bulgaricus monocultures aseptically. The mixes were divided into 2 equal portions; one portion was further inoculated with 1% (vol/vol) of each probiotic monocultures. The final mixes were poured into polystyrene cups and incubated at 42°C until pH of 4.5 ± 0.05 was achieved. Thereafter, the yogurts were immediately cooled to 4°C and stored for 28 d at the same temperature. Enumeration of Starter and Probiotic Cultures Cell populations of starter (S. thermophilus and L. bulgaricus) and probiotic (L. acidophilus, L. casei, and L. paracasei) cultures in freshly inoculated yogurt mixes (0 h) and in yogurts during storage were counted by spread plate technique and expressed as the log of colony-forming units per gram. The selective agar plates and incubation conditions for the cultures were as follows: M17 medium supplemented with lactose and aerobic incubation of 24 h at 45°C for S. thermophilus; acidified MRS agar (pH 5.2) and anaerobic incubation of 72 h at 45°C for L. bulgaricus. Lactobacillus acidophilus and both L. casei and L. paracasei together were respectively enumerated using MRS-clindamycin agar (pH 6.2; 0.5 mg/L clindamycin), MRS-vancomycin agar (pH 6.2; 1 mg/L vancomycin), and anaerobic incubation of 72 h at 37°C ( • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. ). Determination of Titratable Acidity Titratable acidity (TA), as percent lactic acid, of yogurt samples was determined according to AOAC official method 947.05 ( • Horwitz W. • Latimer Jr, G.W. ) using equation [1]: $% TA (wt/wt), as lactic acid=V×N×90.08W×10,$ [1] where V = volume of NaOH consumed, mL; N = normality of the NaOH; and W = mass of sample, g. Determination of Proteolysis Degree of protein hydrolysis was measured using the OPA method as described by • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. with some modifications. Briefly, 1 mL of Milli-Q water was added to a 5-g yogurt sample and the final volume adjusted to 10 mL with trichloroacetic acid solution (0.75 M). The supernatant collected after centrifugation of the sample at 2,684 × g for 30 min at 4°C was filtered using a 0.45-µm Phenex syringe filter (Phenomenex Inc., Lane Cove, Australia). An aliquot of clarified sample (400 μL) was mixed with 3 mL of freshly prepared OPA reagent and left to stand for 2 min. Absorbance of the sample (Asample) was then measured using a Biochrom Libra S12 UV/Vis spectrophotometer (Biochrom Ltd., Cambridge, UK) at 340 nm. In place of sample, Milli-Q water was used for the blank (Ablank) and serine solution (0.9516 mEqv/L) for the standard (Astandard). Degree of hydrolysis (DH) was estimated using equation [2], $DH (%)=hhtot×100,$ [2] where htot was total number of peptide bonds per protein equivalent (8.2 mEq/g for casein), and h was number of hydrolyzed bonds, determined by using equation [3]: $h=(Serine NH2 − β)α.$ [3] For casein, α = 1.039, β = 0.383 mEq/g of protein, and Serine NH2 (mEq/g of protein) was determined using equation [4], $Serine NH2=(Asample−Ablank)(Astandard−Ablank)×S×V×100X×P,$ [4] where S = strength of serine standard, mEq/L; V = final volume make-up of the sample, L; X = weight of yogurt sample, g; and P = protein content of yogurt sample, % (wt/wt). Preparation and Profiling of Water-Soluble Peptide Extracts The water-soluble peptide extracts (WSPE) from yogurts and heat-treated reconstituted skim milk (pH-adjusted to 4.5 ± 0.05) were prepared according to • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. with a few modifications. Briefly, samples were centrifuged at 22,680 × g using JLA-16.250 rotor in Avanti J-26S XPI High-Performance Centrifuge (Beckman Coulter Inc., Brea, CA) for 30 min at 4°C. The supernatant was filtered using a sintered glass crucible to remove coagulated proteins, debris, and cells. The filtrate was freeze-dried using an Alpha 1–4 LSC Christ freeze dryer (Martin Christ Gefriertrocknungsanlagen GmbH) at 0.1 mbar for 24 to 36 h (main drying) and 0.08 mbar for 12 h (final drying) and stored at −80°C until further analysis. The protein content (mg/mL) of the WSPE was estimated according to A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein dye binding. using BSA (0.1–1.4 mg/mL) as standards. The WSPE were also profiled using a reversed-phase HPLC system as described by • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. . Briefly, the samples were loaded using a 20-µL injection loop to a Varian HPLC system (Varian Inc., Palo Alto, CA) equipped with a C-18 monomeric column (5 μm, 300 Å, 250 × 4.6 mm; Grace Vydac, Hesperia, CA) and detected eluted peptides at 215 nm. Determination of Antimutagenic Activity by the Ames Test The antimutagenicity of crude WSPE was assessed in a bacterial reverse mutation assay through the preincubation protocol of Ames test as described by • Sah B.N.P. • Vasiljevic T. • McKechnie S. • Donkor O.N. Effect of probiotics on antioxidant and antimutagenic activities of crude peptide extract from yogurt. and conducted using Salmonella enterica ssp. enterica Typhimurium (ATCC 29629; genotype: his, rfa, uvrB-bio) with aqueous sodium azide (1 μg/plate) as a direct mutagen. Briefly, 0.1 mL of WSPE solution (50 ± 1 µg of peptide) and 0.1 mL of sodium azide solution (1 µg) were mixed with 0.5 mL of sodium phosphate buffer (0.1 M; pH 7.4) in a culture tube. An aliquot (0.1 mL) of 14- to 16-h-old culture was inoculated and preincubated for 20 min at 37°C. The mix was poured over a Davis minimal agar plate after mixing with 2 mL of molten top agar at 45°C. Revertant colonies were enumerated after incubating aerobically at 37°C for 48 h. The antimutagenic activity was determined using equation [5]: $% Inhibition=M−S1M−S0×100,$ [5] where S1 = revertant colonies per plate induced by mutagen in the presence of peptide extract; M = revertant colonies per plate induced by mutagen alone; and S0 = spontaneous revertant colonies per plate. Determination of Antioxidant Activity Assay of Reducing Power Reducing power of the crude WSPE was evaluated by assessing reduction of Fe3+(CN-)6 to Fe2+(CN-)6, as described by • Jiang H. • Tong T. • Sun J. • Xu Y. • Zhao Z. • Liao D. Purification and characterization of antioxidative peptides from round scad (Decapterus maruadsi) muscle protein hydrolysate. with some modifications. Briefly, 50 μL of aqueous WSPE (at 0.5 mg of protein/mL) was mixed with 0.5 mL of phosphate buffer (0.2 M, pH 6.6) and 0.5 mL of aqueous potassium ferricyanide solution (1%, wt/vol). After incubation at 50°C for 20 min, 0.5 mL of aqueous trichloroacetic acid solution (10%, wt/vol) was added to terminate the reaction. The reaction mixture was centrifuged at 16,000 × g for 10 min at room temperature. Finally, 0.5 mL of the supernatant solution was mixed with 0.5 mL of Milli-Q water, then 0.1 mL of aqueous ferric chloride solution (0.1%, wt/vol) was added. After 10 min of standing at room temperature, the absorbance of resulting Prussian blue solution was measured using a Biochrom Libra S12 UV/Vis spectrophotometer (Biochrom Ltd.) at 700 nm. The experimental steps were repeated with Milli-Q water as a blank. The reducing power of the sample was reported as absorbance at 700 nm after subtracting the absorbance value of the blank. A higher absorbance value indicated greater reducing power. Assay of DPPH Radical Scavenging Activity The radical scavenging activity (RSA) of WSPE was measured against DPPH radical as described by • Siow H.L. • Gan C.Y. Extraction of antioxidative and antihypertensive bioactive peptides from Parkia speciosa seeds. with some modifications. Briefly, 20 μL of aqueous WSPE (at 0.5 mg of protein/mL) was mixed with 1.0 mL of DPPH reagent (50 μM in ethanol). The mix was vortexed and left to stand for 30 min in dark. Then, the mix was clarified by centrifuging at 16,000 × g (5415C microcentrifuge, Eppendorf, Hamburg, Germany) for 5 min at room temperature and subjected for absorbance measurement at 517 nm. For the blank, Milli-Q water was used instead of sample. Antioxidant activity of WSPE was calculated using equation [6]: $RSA, %=(Absorbance of blank−Absorbance of sample)Absorbance of blank×100.$ [6] Assay of ABTS Radical Scavenging Activity The ABTS•+ scavenging activity of WSPE was assayed according to the method described by • Ozgen M. • Reese R.N. • Tulio Jr., A.Z. • Scheerens J.C. • Miller A.R. Modified 2,2-azino-bis-3-ethylbenzothiazoline-6-sulfonic acid (ABTS) method to measure antioxidant capacity of selected small fruits and comparison to ferric reducing antioxidant power (FRAP) and 2,2′-diphenyl-1-picrylhydrazyl (DPPH) methods. with some modifications. The working solution of ABTS•+ was prepared by mixing stock solutions of 7.4 mM ABTS (molecular = 548.68) in sodium acetate buffer (20 mM, pH 4.5) and 2.6 mM potassium persulphate (molecular = 270.32) aqueous solution in equal quantities and allowing them to react for 12 to 16 h at room temperature in the dark. One milliliter of ABTS•+ solution was then diluted by mixing with 50 to 60 mL of sodium acetate buffer (20 mM, pH 4.5) to obtain an absorbance of 0.70 ± 0.02 at 734 nm after equilibration at 30°C. The ABTS•+ reagent was prepared daily. Exactly 10 μL of aqueous WSPE (at 0.5 mg of protein/mL) was added to 1 mL of the ABTS•+ reagent and incubated at 30°C for 6 min after vortexing. The absorbance of the mix was measured at 734 nm. Similarly, 10 µL of Milli-Q water was used instead of the sample for the blank. Radical scavenging activity was calculated using equation [6]. Assay of Hydroxyl Radical Scavenging Activity The scavenging capacity of WSPE for hydroxyl radical was assayed according to the method described by • Zheng X.Q. • Wang J.T. • Liu X.L. • Sun Y. • Zheng Y.J. • Wang X.J. • Liu Y. Effect of hydrolysis time on the physicochemical and functional properties of corn glutelin by Protamex hydrolysis. with some modifications. Briefly, 500 μL of aqueous ferrous sulfate (2 mM) and 100 μL of aqueous hydrogen peroxide (2 mM) were mixed to 20 μL of sample (0.1 mg of protein/mL). The reaction mixture was left for 10 min and then 500 μL of aqueous salicylic acid (2.5 mM) was added. The mixture was incubated for 30 min at 37°C and subjected for absorbance measurement at 510 nm. Milli-Q water was used for the blank (in place of salicylic acid) and the control (in place of sample) in the reaction. Hydroxyl radical scavenging activity (HRSA) of the WSPE was calculated using equation [7] $HRSA (%)=1−As−AbAc×100,$ [7] where As, Ab, and Ac were absorbance for sample, blank, and control, respectively. Statistical Analyses Experiments were conducted as a randomized split-plot blocked design in time with types of yogurt as the main plot, prebiotic addition and time as subplots. The design was replicated independently on 3 different occasions with subsequent subsampling giving at least 6 observations (n ≥6). Results were analyzed using GLM procedure of the SAS System ( SAS. 1996. SAS/STAT Softwares: Changes and Enhancements Through Release 6.11. SAS Institute Inc., Cary, NC. ) to explore the effects of probiotic and potential prebiotic, PPP, and inulin addition on properties of different yogurt types over time at 3 levels (d 1, 14, and 28) of storage. Significance level was considered at P < 0.05. Chemometric methods such as hierarchical cluster analysis (HCA) and principal component analysis (PCA) were employed to discriminate yogurts with different culture and prebiotic combinations ( • Matera J.A. • Cruz A.G. • Raices R.S.L. • Silva M.C. • Nogueira L.C. • Quitério S.L. • Cavalcanti R.N. • Freiras M.Q. • Conte Júnior C.A. Discrimination of Brazilian artisanal and inspected pork sausages: Application of unsupervised, linear and non-linear supervised chemometric methods. ). The HCA categorized different yogurts into clusters based on their similarities by applying the squared Euclidean distance and Ward linkage methods to the standardized data set (z-scores). The eigenvalues were extracted from the correlation matrix; varimax rotation was used in the PCA to produce orthogonal transformations to the reduced factors to identify the high and low correlations. Pearson correlation, HCA, and PCA were performed on d 1 results using SPSS 22.0 (SPSS Inc., Chicago, IL). Results and Discussion Viability of Starter and Probiotic Cultures in Yogurt During Refrigerated Storage Prebiotics are usually supplemented to fermented milk and milk products for stimulation of the growth of probiotic strains, which are a part of the human colonic microbiota. Because of widened attention on lactobacilli, the effect of addition of PPP as a potential prebiotic on the viable count of probiotic (L. acidophilus, L. casei, and L. paracasei) and starter (S. thermophilus, L. bulgaricus) cultures in yogurt was investigated during 28 d of refrigerated storage and reported in Tables 1 and 2. Table 1Viability of starter culture in yogurts supplemented with or without pineapple peel powder (PPP) or inulin during 28 d of storage at 4°C; 0 h counts represent count of starter culture in yogurt mix taken immediately after culture addition Results are expressed as mean of 3 trials. Yogurt type SC=starter culture (S. thermophilus + L. bulgaricus); PC=probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). Viable count (log cfu/g) Streptococcus thermophilusLactobacillus bulgaricus CulturePrebiotic0 hd 1d 14d 280 hd 1d 14d 28 SCNone7.41 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.18 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.98 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.58 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.33 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.13 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.04 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.51 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCInulin7.40 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.23 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.03 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.62 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.31 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.27 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.20 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.90 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCPPP7.41 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.27 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.10 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.72 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.33 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.19 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.13 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.70 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCNone7.40 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.22 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.05 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.66 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.33 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.21 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.09 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.63 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCInulin7.40 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.27 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.11 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.82 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.33 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.28 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.19 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.84 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCPPP7.41 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.31 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 9.15 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.86 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.33 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.25 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.11 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.77 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SEM Pooled standard error of the mean for predetermined P<0.05. 0.020.02 a–f Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P < 0.05). A–D Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P < 0.05). 1 Results are expressed as mean of 3 trials. 2 SC = starter culture (S. thermophilus + L. bulgaricus); PC = probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). 3 Pooled standard error of the mean for predetermined P < 0.05. Table 2Viability of probiotic cultures in yogurts supplemented with or without pineapple peel powder (PPP) or inulin during 28 d of storage at 4°C; 0 h counts represent probiotic count in yogurt mix taken immediately after culture addition Results are expressed as mean of 3 trials. Yogurt type SC=starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC=probiotic culture (L. acidophilus + L. casei + L. paracasei). Viable count (log cfu/g) Lactobacillus acidophilusLactobacillus casei and Lactobacillus paracasei CulturePrebiotics0 hd 1d 14d 280 hd 1d 14d 28 SC + PCNone7.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.66 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.07 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.72 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.52 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.94 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.05 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.74 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCInulin7.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.45 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.25 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.86 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.68 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.25 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.83 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCPPP7.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.28 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.03 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.70 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 8.30 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.68 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SEM Pooled standard error of the mean for predetermined P<0.05. 0.040.06 a–c Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P < 0.05). A–D Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P < 0.05). 1 Results are expressed as mean of 3 trials. 2 SC = starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC = probiotic culture (L. acidophilus + L. casei + L. paracasei). 3 Pooled standard error of the mean for predetermined P < 0.05. Supplementation of milk with selected probiotic organisms and PPP resulted in a substantially faster lowering of pH (data not shown), suggesting a higher rate of acid production compared with control yogurt. The viability of S. thermophilus and L. bulgaricus reduced significantly during the 28-d storage period by half of a log cycle in all types of yogurt (Table 1). Table 2 shows the viable count of probiotic organisms before fermentation and during refrigerated storage for yogurt containing PPP, yogurt with inulin, and control yogurt. Overall, the presence of PPP in yogurts effectively improved probiotic growth, comparable to that of inulin and one log cycle higher than the nonsupplemented control sample. Stability of the probiotic organisms in PPP enriched medium might be due to buffering capacity of the prebiotics, which would aid in maintaining the viability of live bacterial cells ( • Kailasapathy K. • Harmstorf I. • Phillips M. Survival of Lactobacillus acidophilus and Bifidobacterium animalis ssp. lactis in stirred fruit yogurts. ). It has been reported that PPP may serve as a source of growth factors for improved growth of probiotic bacteria in yogurt ( • Diaz-Vela J. • Totosaus A. • Cruz-Guerrero A.E. • De Lourdes Pérez-Chabela M. In vitro evaluation of the fermentation of added-value agroindustrial by-products: Cactus pear (Opuntia ficus-indica L.) peel and pineapple (Ananas comosus) peel as functional ingredients. ). • Iyer C. • Kailasapathy K. Effect of co-encapsulation of probiotics with prebiotics on increasing the viability of encapsulated bacteria under in vitro acidic and bile salt conditions and in yogurt. also reported enhanced survival of probiotic organisms under in vitro acidic and bile salt conditions when the organisms were coencapsulated with prebiotics such as Hi-maize (Starch Australasia Ltd., Lane Cove, Australia). Thus, addition of PPP resulted in improved survival and metabolic activity during storage at 4°C. Conversely, factors such as postfermentation acidification and dissolved oxygen may adversely influence the viability of probiotic strains in fermented milk ( • Vasiljevic T. • Shah N.P. Probiotics—From Metchnikoff to bioactives. ). The reported loss of probiotic viability in yogurts during storage is likely due to acid injury, as titratable acidity increases significantly during storage (Table 3; • Donkor O.N. • Henriksson A. • Vasiljevic T. • Shah N.P. Effect of acidification on the activity of probiotics in yoghurt during cold storage. ). Nonetheless, counts of the probiotic organisms during 28 d of cold storage were sufficient to exhibit probiotic effect on the consumer as the reported minimum therapeutic count of 106 cfu/g ( • Lourens-Hattingh A. • Viljoen B.C. Yogurt as probiotic carrier food. ) is below that obtained in the current study. These observations point out an in vitro symbiotic effect of PPP on selected probiotic organisms similar to previous reports ( • Sendra E. • Fayos P. • Lario Y. • Fernández-López J. • Sayas-Barberá E. • Pérez-Alvarez J. Incorporation of citrus fibers in fermented milk containing probiotic bacteria. ; • do Espírito Santo A.P. • Cartolano N.S. • Silva T.F. • Soares F.A.S.M. • Gioielli L.A. • Perego P. • Converti A. • Oliveira M.N. Fibers from fruit by-products enhance probiotic viability and fatty acid profile and increase CLA content in yoghurts. ), where fibers from various fruit processing by-products stimulated the growth of probiotic strains. Table 3Titratable acidity, degree of protein hydrolysis, and antimutagenic activity of plain and probiotic yogurts supplemented with or without pineapple peel powder (PPP) or inulin during 28 d of storage at 4°C Results are expressed as mean of 3 trials. Yogurt type SC=starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC=probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). Titratable acidity (as % lactic acid) Degree of protein hydrolysis (%) Antimutagenic activity (% inhibition) CulturePrebioticd 1d 14d 28d 1d 14d 28d 1d 14d 28 SCNone1.01 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.05 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.09 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.15 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.49 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.65 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 21.28 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 22.74 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 23.60 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCInulin1.02 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.06 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.11 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 6.90 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.38 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.69 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 24.74 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 25.03 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 27.11 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCPPP1.02 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.08 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.06 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 7.52 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 22.47 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 24.79 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 25.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCNone1.02 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.10 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.14 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 11.29 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 11.24 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 12.60 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 31.74 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 33.31 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 34.25 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCInulin1.01 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.15 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 12.42 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 12.79 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 13.64 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 37.76 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 37.68 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 38.77 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCPPP1.02 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 1.17 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 13.26 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 13.67 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 13.92 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 36.19 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 37.65 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 39.04 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SEM Pooled standard error of the mean for predetermined P<0.05. 0.0050.070.26 a–e Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P < 0.05). A–C Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P < 0.05). 1 Results are expressed as mean of 3 trials. 2 SC = starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC = probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). 3 Pooled standard error of the mean for predetermined P < 0.05. Titratable Acidity of Yogurt During Refrigerated Storage In all yogurt samples, the titratable acidity increased significantly (P < 0.05) during 28 d of storage (Table 3), suggesting continued production of organic acids during storage. Supplementation with PPP or inulin increased the acidifying ability of starter and probiotic cultures during refrigerated storage. Similar acidification in yogurt-type products was observed during refrigerated storage in various studies ( • Gilliland S.E. • Reilly S.S. • Kim G.B. • Kim H.S. Viability during storage of selected probiotic lactobacilli and bifidobacteria in a yogurt-like product. ; • Donkor O.N. • Henriksson A. • Vasiljevic T. • Shah N.P. Effect of acidification on the activity of probiotics in yoghurt during cold storage. ). Degree of Proteolysis in Yogurt During Refrigerated Storage Lactic acid bacteria produce proteolytic enzymes during yogurt manufacturing which cleave peptide bonds of milk proteins leading to generation of peptides and free AA ( • Donkor O.N. • Henriksson A. • Vasiljevic T. • Shah N.P. Proteolytic activity of dairy lactic acid bacteria and probiotics as determinant of growth and in vitro angiotensin-converting enzyme inhibitory activity in fermented milk. ). The extent of protein hydrolysis in yogurt samples during storage was estimated by determining free amino groups and results reported as degree of hydrolysis in Table 3. In all samples, the degree of proteolysis increased significantly (P < 0.05) over a storage period of 28 d, suggesting continuation of proteolytic activities during storage. Additionally, a significant (P < 0.05) increase in proteolytic activity was observed in probiotic yogurt compared with plain yogurt. Furthermore, proteolytic activities increased significantly (P < 0.05) in PPP- or inulin-supplemented yogurt compared with the nonsupplemented control yogurt. Interactions among yogurt types, prebiotic (inulin or PPP) additions, and storage time (Table 4) indicated that the prebiotics increased the generation of new peptides in probiotic yogurts compared with plain yogurts during storage. Pineapple powders contain dietary fibers, proteins, and minerals, including divalent cations, and serve as growth factors or growth promoters for probiotics in the yogurts ( • Diaz-Vela J. • Totosaus A. • Cruz-Guerrero A.E. • De Lourdes Pérez-Chabela M. In vitro evaluation of the fermentation of added-value agroindustrial by-products: Cactus pear (Opuntia ficus-indica L.) peel and pineapple (Ananas comosus) peel as functional ingredients. ). For better growth, bacterial cells require free AA for protein synthesis; therefore, bacterial extracellular proteinases hydrolyze milk proteins into oligopeptides and further hydrolysis by intracellular peptidases of peptides into AA ( • Donkor O.N. • Henriksson A. • Vasiljevic T. • Shah N.P. Proteolytic activity of dairy lactic acid bacteria and probiotics as determinant of growth and in vitro angiotensin-converting enzyme inhibitory activity in fermented milk. ). The mineral-rich PPP likely stimulated bacterial enzymes, resulting in higher proteolytic activities. Table 4Analysis of variance depicting the significance (P < 0.05) of types of yogurt (plain and probiotic yogurts), prebiotics (inulin and pineapple peel powder), storage time at 3 levels (d 1, 14, and 28) and their effects on degree of protein hydrolysis, antimutagenic and antioxidant activities Source of variationP-value Degree of protein hydrolysis Mutagen inhibitory activity Antioxidant activities Reducing power DPPH=1,1-diphenyl-2-picrylhydrazyl; ABTS=2,2’-azino-bis(3-ethyl benzothiazoline-6-sulphonic acid); OH=hydroxyl. DPPHABTS•+OH Yogurt type P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Prebiotic P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Time P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Yogurt type × Prebiotic P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Yogurt type × Time P<0.05, NS=nonsignificant at P>0.05. NS P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Prebiotic × Time P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. Yogurt type × Prebiotic × Time P<0.05, NS=nonsignificant at P>0.05. NS P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. P<0.05, NS=nonsignificant at P>0.05. 1 DPPH = 1,1-diphenyl-2-picrylhydrazyl; ABTS = 2,2’-azino-bis(3-ethyl benzothiazoline-6-sulphonic acid); OH = hydroxyl. ** P < 0.05, NS = nonsignificant at P > 0.05. The generation of peptides from milk proteins during yogurt manufacturing was confirmed by profiling WSPE from yogurts using a reversed-phase HPLC, as presented in Figure 1. Probiotic yogurts supplemented with inulin or PPP (yogurts 5 and 6) exhibited more peaks (peptides) compared with nonsupplemented plain yogurt (yogurt 1) and reconstituted skim milk. The peptide profiles obtained are in line with previous study ( • Miclo L. • Roux E. • Genay M. • Brusseaux E. • Poirson C. • Jameh N. • Perrin C. • Dary A. Variability of hydrolysis of β-, αs1-, and αs2-caseins by 10 strains of Streptococcus thermophilus and resulting bioactive peptides. ), who reported casein break down by cell envelope proteinases of S. thermophilus. Several peptides may be generated during storage and may display different biological activities. A study by • van Boekel M.A.J.S. • Weerens C.N.J.M. • Holstra A. • Scheidtweiler C.E. Antimutagenic effects of casein and its digestion products. , found that the generated peptides from pepsin hydrolysis of casein were more potent in antimutagenic activity than intact casein. Antimutagenic Activity of Yogurt During Refrigerated Storage Mutagen inhibitory activity of WSPE was evaluated using Ames test by assessing inhibitory activity of peptides against sodium azide on S. typhimurium and presented in Table 3. In all yogurt samples, the antimutagenic activities of the WSPE at the end of storage increased significantly (P < 0.05) compared with the first day of storage, indicating increased generation of these peptides during storage. In addition, enhanced antimutagenic activity was also observed in the probiotic yogurts compared with control. The antimutagenic activity in yogurt supplemented with PPP was similar to that of inulin-supplemented yogurts and both showed significantly higher activity compared with nonsupplemented control yogurts. The insignificant interactions among yogurt types, prebiotic (inulin or PPP) addition, and storage time (Table 4) indicated that increment pattern of antimutagenic activities for both plain and probiotic yogurts during storage was similar because of prebiotic supplementation. Pearson correlation suggested a direct relationship between degree of proteolysis and antimutagenic activity (P < 0.01, r = 0.96); • Bakalinsky A.T. • Carney J.R. • Gould S.J. Antimutagenicity of yogurt. also reported of similar results. In Vitro Antioxidant Activity of Yogurt During Refrigerated Storage An antioxidant compound can protect the human body by scavenging free radicals, such as reactive oxygen species, and increase shelf life of foodstuffs by retarding the process of lipid peroxidation through hydrogen atom or electron transfer. For instance, donation of hydrogen ions involves interrupting free radical chain reactions, which is a basis for the assessment of reducing power. Additionally, the reaction mechanisms involving mainly a hydrogen atom transfer is basis for DPPH-based methods and both hydrogen atom transfer and single electron transfer for ABTS-based methods ( • Gülçin I. Antioxidant activity of food constituents: An overview. ). Due to the involvement of a cascade of reaction steps in an oxidation process, protein hydrolyzates can exhibit antioxidant activities through multiple reaction mechanisms ( • Chen J. • Lindmark-Månsson H. • Gorton L. • Åkesson B. Antioxidant capacity of bovine milk as assayed by spectrophotometric and amperometric methods. ). Thus, several different assays must be performed to provide a comprehensive information about total antioxidant capacity of the compound tested. This justifies the current study of evaluation for the antioxidant potential of WSPE by conducting 4 different assays. Reducing power and scavenging capacities for DPPH, ABTS, and hydroxyl radicals of WSPE were measured during 28 d of refrigerated storage and presented in Table 5. All yogurt samples exhibited varying degrees of reducing power, scavenging capacities for DPPH, ABTS, and hydroxyl radicals, indicating differences in generated WSPE of the yogurts. The activities increased significantly (P < 0.05) during storage compared with the first day. Furthermore, these activities were stronger in PPP- or inulin-supplemented yogurt compared with the nonsupplemented control yogurt. This implied that that the generated peptides in WSPE acted as potent antioxidant compounds. The interactions among yogurt types, prebiotic (inulin or PPP) additions, and storage time (Table 4) showed that prebiotics influenced differently the antioxidant capacities of WSPE in plain and probiotic yogurts during storage. Table 5Antioxidant activities (evaluated by assessing reducing power and scavenging of DPPH, ABTS•+, and OH radicals) of plain and probiotic yogurts supplemented with or without pineapple peel powder (PPP) or inulin during 28 d of storage at 4°C Results are expressed as mean of 3 trials. Yogurt SC=starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC=probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). Reducing power (A700)Radical scavenging activity DDPH=1,1-diphenyl-2-picrylhydrazyl; ABTS=2,2’-azino-bis(3-ethyl benzothiazoline-6-sulfonic acid); OH=hydroxyl. (%) DPPHABTS•+OH CulturePrebioticd 1d 14d 28d 1d 14d 28d 1d 14d 28d 1d 14d 28 SCNone0.36 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.47 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.48 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 34.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 33.61 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 44.25 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 35.92 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 38.31 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 46.76 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 61.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 63.38 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 66.79 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCInulin0.36 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.45 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.53 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 36.80 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 37.83 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 46.14 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 38.13 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 43.83 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 53.41 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 61.70 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 66.56 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 70.98 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SCPPP0.39 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.40 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.58 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 34.91 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 39.69 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 50.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 45.93 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 46.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 54.67 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 62.15 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 67.41 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 72.13 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCNone0.40 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.47 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.59 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 36.96 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 42.18 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 51.88 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 46.06 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 50.06 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 59.12 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 68.08 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 69.08 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 75.54 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCInulin0.48 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.51 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.59 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 38.98 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 40.61 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 55.98 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 52.94 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 51.92 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 61.91 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 70.70 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 74.07 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 78.83 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SC + PCPPP0.47 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.50 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 0.64 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 43.90 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 44.35 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 58.03 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 53.06 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 55.44 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 62.79 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 73.13 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 74.45 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). 79.65 Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P<0.05). , Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P<0.05). SEM Pooled standard error of the mean for predetermined P<0.05. 0.0050.550.290.33 a–f Different lowercase superscripts in the same column depict the significant difference between means for yogurt types (P < 0.05). A–C Different uppercase superscripts in the same row depict the significant difference between means for yogurts with the same culture and prebiotic combination at d 1, 14, and 28 of refrigerated storage (P < 0.05). 1 Results are expressed as mean of 3 trials. 2 SC = starter culture (Streptococcus thermophilus + Lactobacillus bulgaricus); PC = probiotic culture (Lactobacillus acidophilus + Lactobacillus casei + Lactobacillus paracasei). 3 DDPH = 1,1-diphenyl-2-picrylhydrazyl; ABTS = 2,2’-azino-bis(3-ethyl benzothiazoline-6-sulfonic acid); OH = hydroxyl. 4 Pooled standard error of the mean for predetermined P < 0.05. A positive correlation between degree of proteolysis with reducing power (P < 0.01, r = 0.91), scavenging capacities of DPPH (P < 0.01, r = 0.76), ABTS (P < 0.01, r = 0.89), and hydroxyl (P < 0.01, r = 0.98) radicals were observed. Several potent antioxidant peptides liberated from milk proteins have been reported, such as Tyr-Val-Pro-Glu-Pro-Phe, Phe-Pro-Tyr-Cys-Ala-Pro, Phe-Gly-Gly-Met-Ala-His, Val-Tyr-Pro-Phe, and Tyr-Pro-Pro-Tyr-Glu-Thr-Tyr from casein hydrolyzates of goat milk using a combination of alkaline and neutral proteases ( • Li Z. • Jiang A. • Yue T. • Wang J. • Wang Y. • Su J. Purification and identification of five novel antioxidant peptides from goat milk casein hydrolysates. ); Trp-Tyr-Ser-Leu-Ala-Met-Ala-Ala-Ser-Asp-Ile from β-lactoglobulin hydrolyzates using corolase PPP ( • Hernández-Ledesma B. • Dávalos A. • Bartolomé B. • Amigo L. Preparation of antioxidant enzymatic hydrolysates from α-lactalbumin and β-lactoglobulln. Identification of active peptides by HPLC-MS/MS. ); and Ala-Arg-His-Pro-His-Pro-His-Leu-Ser-Phe-Met from fermented milk by using Lactobacillus delbrueckii ssp. bulgaricus ( • Kudoh Y. • Matsuda S. • Igoshi K. • Oki T. Antioxidative Peptide from Milk Fermented with Lactobacillus delbrueckii ssp. bulgaricus IFO13953. ). Furthermore, low-molecular weight casein hydrolyzates displayed strong radical scavenging activities ( • Kim G.N. • Jang H.D. • Kim C.I. Antioxidant capacity of caseinophosphopeptides prepared from sodium caseinate using Alcalase. ). The antioxidative capacity of a peptide depends on various attributes including sequence, composition of AA, configuration, and concentration of the peptide ( • Phanturat P. • Benjakul S. • Visessanguan W. • Roytrakul S. Use of pyloric caeca extract from bigeye snapper (Priacanthus macracanthus) for the production of gelatin hydrolysate with antioxidative activity. LWT -. ). Again, the type of enzymes involved in the hydrolysis of protein is also a determining factor ( • Foh M.B.K. • Foh B.M. • Kamara M.T. • Xia W. Functionality and antioxidant properties of tilapia (Oreochromis niloticus) as influenced by the degree of hydrolysis. • Rajapakse N. • Mendis E. • Jung W.K. • Je J.Y. • Kim S.K. Purification of a radical scavenging peptide from fermented mussel sauce and its antioxidant properties. reported that hydrolyzates containing Pro, His, Met, Tyr, Val, Lys, Cys, and Gln displayed strong radical scavenging capacity. Effect of PPP and Probiotic Addition on Overall Characteristics of Yogurts Cluster analysis was performed using hierarchical clustering method with Ward’s linkage and revealed 2 clusters based on similarities in titratable acidity, degree of protein hydrolysis, and antimutagenic and antioxidant activities (evaluated measuring reducing power and scavenging capacity for DPPH, ABTS•+, and OH radicals) during the first day of storage at 4°C (Figure 2A). Plain and probiotic yogurts were arranged in 2 separate clusters. Inclusion of probiotic cultures resulted in pronounced effects on the measured variables compared with the supplementation of inulin or PPP. Principal component analysis was also conducted and 2 components were identified based on Kaiser’s criterion of eigenvalues greater than 1.0 (Figure 2B). The first component explains higher variance (76.95%) than the second component (17.60%). Samples are distributed in all 4 quadrants of the score plot. Yogurts prepared with starter culture only are at the left quadrants, whereas yogurts prepared with both starter and probiotic cultures are at the right quadrants. Yogurt 6, designated in a separate quadrant, shows higher antioxidant and antimutagenic activities, indicating differences in properties compared with the other studied yogurts. Conclusions Improved growth and retention of viability of L. acidophilus (ATCC 4356), L. casei (ATCC 393), and L. paracasei spp. paracasei (ATCC BAA52) was observed during refrigerated storage at 4°C for 28 d in synbiotic yogurt formulations with added PPP. The proteolytic activities of cultures were enhanced substantially in the presence of prebiotic during storage. All WSPE prepared from yogurt samples possessed antimutagenic activity, which increased during storage. In addition, the WSPE exhibited excellent antioxidant properties measured through in vitro assays employing DPPH, ABTS•+, and OH free radicals. Interestingly, WSPE of the probiotic yogurt with PPP exhibited the most potent antimutagenic and antioxidant activities. In addition, incorporation of PPP and probiotics in food products would provide new opportunities in the design and preparation of novel functional foods. Establishing the stability of bioactive peptides during refrigerated storage would add in expanding the functional food market as well. However, these findings demand further investigations to identify and purify these bioactive peptides from WSPE and to understand the possible health benefits, as well as their effect on sensory attributes of food products enriched with protein hydrolyzates. Acknowledgments The authors are grateful to the Australian government for offering an Australia Awards Scholarships and Australia Awards Leadership Program place to B. N. P. Sah. References • Al-Sheraji S.H. • Ismail A. • Manap M.Y. • Mustafa S. • Yusof R.M. 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Antioxidant capacity of bovine milk as assayed by spectrophotometric and amperometric methods. Int. Dairy J. 2003; 13 (http://dx.doi.org/10.1016/S0958-6946(03)00139-0): 927-935 • Choi J. • Sabikhi L. • Hassan A. • Anand S. Bioactive peptides in dairy products. Int. J. Dairy Technol. 2012; 65 (http://dx.doi.org/10.1111/j.1471-0307.2011.00725.x): 1-12 • Coman M.M. • Verdenelli M.C. • Cecchini C. • Silvi S. • Vasile A. • Bahrim G.E. • Orpianesi C. • Cresci A. Effect of buckwheat flour and oat bran on growth and cell viability of the probiotic strains Lactobacillus rhamnosus IMC 501®, Lactobacillus paracasei IMC 502® and their combination SYNBIO®, in synbiotic fermented milk. Int. J. Food Microbiol. 2013; 167 (http://dx.doi.org/10.1016/j.ijfoodmicro.2013.09.015): 261-268 • Corona-Hernandez R.I. • Álvarez-Parrilla E. • Lizardi-Mendoza J. • Islas-Rubio A.R. • de la Rosa L.A. • Wall-Medrano A. Structural stability and viability of microencapsulated probiotic bacteria: A review. Compr. Rev. 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Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here! # One type of electromagnetic radiation has a frequency of 107.1 $\mathrm{MHz}$ , another type has a wavelength of $2.12 \times 10^{-10} \mathrm{m},$ and another type of electromagnetic radiation has photons with energy equal to $3.97 \times 10^{-19} \mathrm{J} / \mathrm{photon}$ . Identify each type of electromagnetic radiation and place them in order of increasing photon energy and increasing frequency. ## $1.4 \times 10^{18}3$$6.00 \times 10^{14} \mathrm{s}^{-1}$Least $v_{A}<v_{c}<v_{s}$ Greatest ### Discussion You must be signed in to discuss. ##### Stephanie C. University of Central Florida ##### Morgan S. University of Kentucky ##### Jake R. University of Toronto ### Video Transcript June problem 52 Triple 71 type electromagnetic radiation has a frequency of 100 and 7.1 megahertz and we can rewrite that as a hunt during 7.1 times 10 toothy six in vour seconds. The other is a electromagnetic. Radiation has 3.97 times 10 to the negative. 19 Jules perf Otan. But we can just imply that's per photon. And another type has 2.12 times 10 to the negative 10 meters. So the way we do that is so we want to comparatively. So the first thing we're gonna do is we're gonna convert all of these two to ah, frequency. Remember, this is energy. Energy equals H mu. Energy equals place. Constant times. I was frequency. So we're gonna divide this energy. Hey, but blanks concept to get a frequency content 6.63 I'm stand to gain negative 34 Jule seconds. So that means that the frequency in this case is going to be Hey, five 0.9 nine alone times, Then toward the 14 inverse seconds. However, this one lamb day ik equals Sorry, Mu equals Sea of Orlando. We're seeing speed of light. So we have three times 10 to D eight meters per second, which is the speed of light over over. Our wade left 2.12 times 10 to the negative 10 meter. There's some of you in this case that's going to be so one point for one times tend to be 18th 18th inverse seconds, so good. So identify each type of electromagnetic radiation and placed them in order of increasing photon energy and increasing frequency. So there are different types of Elektra Matic take, uh, radiation, and you could turn that based on the frequency. So So this is in the tented six power there and might say 10 to 6. I mean, tend to the positive six x So this is a This is going to be radio. This is indeed 10 to the 14th range. So this is going to be, uh, e. I would say yes, and this is actually high in the 10 to the into the 14th range. So I would say this. It's close enough that I would say this in the visible spectrum spectrum. I would not say I would not say it's in the UV or in front, I would say it's invisible and this one tend to the 18. This would be in the X. This would be in the X ray spectrum. Now, since energy is remember, energy is proportional to frequency. So this has it. So these are arranged in order of increasing frequency and consequently there also were arranged in order of increasing being energy energy. Manhattan College ##### Stephanie C. University of Central Florida ##### Morgan S. University of Kentucky ##### Jake R. University of Toronto	{}
mersenneforum.org > Data mersenne.ca Register FAQ Search Today's Posts Mark Forums Read 2021-09-03, 15:49 #683 James Heinrich "James Heinrich" May 2004 ex-Northern Ontario 3×11×107 Posts Quote: Originally Posted by axn Ok, a new "sorta" bug. The total line also has a link. I've "sorta" fixed it. 2021-09-03, 16:53 #684 axn Jun 2003 145116 Posts Quote: Originally Posted by James Heinrich I've "sorta" fixed it. Ok 2021-09-04, 16:46 #685 masser Jul 2003 Behind BB 23·13·17 Posts Quote: Originally Posted by James Heinrich I believe I have reset the TF level for the affected exponents as it should be. The status report will correct itself at the next overnight update (sometime after midnight UTC) Still have a fluke exponent at 2^70 here. I tried to find the offending exponent in the public-facing mersenne.org pages, but couldn't find anything. 2021-09-04, 17:56 #686 James Heinrich "James Heinrich" May 2004 ex-Northern Ontario 3×11×107 Posts Quote: Originally Posted by masser Still have a fluke exponent at 2^70 here. I tried to find the offending exponent in the public-facing mersenne.org pages, but couldn't find anything. Thanks, fixed that one. It's not impossible that one or more other outliers exist in other ranges, let me know if you see anything that seems out of place. 2021-09-04, 21:03 #687 petrw1 1976 Toyota Corona years forever! "Wayne" Nov 2006 3·1,619 Posts Quote: Originally Posted by James Heinrich Thanks, fixed that one. It's not impossible that one or more other outliers exist in other ranges, let me know if you see anything that seems out of place. These are definitely not all at TF67 https://www.mersenne.ca/status/tf/0/0/4/0 937.9 does not have 58 at TF71 https://www.mersenne.ca/status/tf/0/0/4/93700 2021-09-04, 21:27 #688 James Heinrich "James Heinrich" May 2004 ex-Northern Ontario 67138 Posts Quote: Originally Posted by petrw1 These are definitely not all at TF67 https://www.mersenne.ca/status/tf/0/0/4/0 I've been told to assume that all GIMPS-range exponents have been TF'd to at least 67 thanks to the efforts of TJAOI. Quote: Originally Posted by petrw1 937.9 does not have 58 at TF71 https://www.mersenne.ca/status/tf/0/0/4/93700 I fixed these, and a few other apparent outliers. There are still 26 exponents in the 204M range that are outliers at 72 -- they've had higher TF levels done (e.g. 74-75), but there are intermediate TF levels missing from the history. Exponents listed here in case someone wants to take them to 73: Code: 204437939 204438029 204438253 204438497 204438671 204438823 204438919 204438979 204439087 204439219 204439223 204440213 204440347 204440377 204440641 204440773 204441239 204441359 204441403 204441437 204441509 204441607 204441841 204442013 204442237 204442321 2021-09-21, 02:31 #689 SethTro "Seth" Apr 2019 18416 Posts The "Download all finalized archives (1997-2020, torrent)" torrent from the export page is not being seeded by anyone. (again thanks for the phenomenal site) Last fiddled with by James Heinrich on 2021-09-21 at 12:23 Reason: fix broken link to export page 2021-09-21, 03:00 #690 Zhangrc "University student" May 2021 Beijing, China 127 Posts Quote: Originally Posted by James Heinrich Code: 204437939 204438029 204438253 204438497 204438671 204438823 204438919 204438979 204439087 204439219 204439223 204440213 204440347 204440377 204440641 204440773 204441239 204441359 204441403 204441437 204441509 204441607 204441841 204442013 204442237 204442321 I'll do them all by TFing them to 74 bits. I'll finish them by tonight. Are there other exponents with higher TF done but intermediate levels missing in the 100-120M range? Last fiddled with by Zhangrc on 2021-09-21 at 03:08 2021-09-21, 12:22 #691 James Heinrich "James Heinrich" May 2004 ex-Northern Ontario DCB16 Posts Quote: Originally Posted by SethTro The "Download all finalized archives (1997-2020, torrent)" torrent from the export page is not being seeded by anyone. Sorry, should be seeding now. Last fiddled with by James Heinrich on 2021-09-21 at 12:23 2021-09-23, 07:25 #692 Happy5214 "Alexander" Nov 2008 The Alamo City 77610 Posts Quote: Originally Posted by James Heinrich Sorry, should be seeding now. I can help seed too, if I could get any of it to download. (I can't get any peers.) 2021-10-17, 03:48 #693 axn Jun 2003 145116 Posts Similar Threads Thread Thread Starter Forum Replies Last Post GP2 mersenne.ca 44 2016-06-19 19:29 LaurV mersenne.ca 8 2013-11-25 21:01 siegert81 Math 2 2011-09-19 17:36 optim PrimeNet 13 2004-07-09 13:51 All times are UTC. The time now is 17:32. Mon Dec 6 17:32:23 UTC 2021 up 136 days, 12:01, 0 users, load averages: 1.55, 1.70, 1.69	{}
Browse Questions # The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt 2$. Its equation is $\begin{array}{1 1}(A)\;x^2-y^2=32 \\(B)\;\frac{x^2}{4} -\frac{y^2}{9}=1 \\(C)\;2x^2-3y^2=7 \\(D)\; none\;of\;these \end{array}$ $2ae=16$ $ae=8\qquad e=\sqrt 2$ $a=4 \sqrt 2$ $a^2=32$ $b^2=a^2(e^2-1)$ $\qquad= 32 (2-1)$ $\qquad=32$ Hence the hyperbola is $x^2-y^2=32$ Hence A is the correct answer.	{}
# General context: particle physics The question is conceptually quite simple: what are we made of? Particle physics attemps to address this question by studying the elementary particle (ie. the one which are not made of something else) and how they interact together to form different composit system up to our human-scale matter that one can touch and see. At the very small scale, the particle are actually described by fields which obey to quantum dynamics (each particle appears to be a quantum of vibration of the corresponding field). In this mathematical description, symmetries are the corner stone of the structure of the theory. Indeed for both free particle and interacting particle, specific invariances strongly constrain the structure of field dynamics, thus particles behaviour. ### The Standard Model The Standard Model is the current mathematical framework which allows a quite extensive and precise prediction of plenty of observations at the sub-atomic scale. The basic ingredients are quantum fields and symmetries. In spite of very attractive features - such as understanding three of the fourth fundamental interactions as coming from a similar symmetry, and predictions - like the existence of neutral current discovered at CERN in 1984), few observational and conceptual questions remain open. To mentione just an example for both conceptual and observational aspects: the gravitation cannot be described in the Standard Model and we do know by experiment that neutrinos are massive (unlike in the Standard Model). ### How to improve our understanding of Nature? One of the most appealing feature of the Standard Model is to unify the electromagnetic force and the weak force into the so-called electro-weak interaction. This unification should be happening at high energy and that is why these two interactions look different at our scale. What makes the transition between the unified interaction and how they appear to us? The Higgs field, and its associated particle the Higgs boson, plays a crucial role in this question and its discovery and study is clearly a good area to look to push further our understanding. Another option (and it is cleary a non-exhaustive list) is study particles which seems a bit peculiar with respect to the others. In the Standard Model, the top quark is a bit appart because he is the heaviest matter particle, and by far! No one know why the top quark seems to be different and people think it might be because it is connected to a new set of not-yet-known particles and interactions. It represents then also a good area to search for laws beyond the Standard Model. ### Particle collider experiments In order to probe Nature at this scale, one needs to reach the highest energies since smaller distance always go with higher energy. A way to concentrate a lot of energy is to produce collisions between very energetic particles. Few facilities in the world allow to reach big enough collision energies to study physics at the electroweak scale. The last one was the Tevatron near Chicago (with proton-antiproton collision at 1.92 TeV) and the only currently working machine is the Large Hadron Collider (LHC) near Geneva (with proton-proton collisions at 13 TeV). In order to establish what happened during the collision at the microscopic level, it is necessary to measure all the out-going particles which are produced. This is done with very complex detectors measuring energy deposits and tracks in all directions of space. The hadronic calorimer of the ATLAS experiment is one the sub-dectector on which I work on, involved in these measurements. All together, every sub-detectors allow physicists to reconstruct the full collision by knowing, let say, if there were an electron in this direction or a muon in that direction. Sophisticated alogorithms have to be developped in order to recognize a give particle from all the energy deposit the detector sees: those are called particle reconstruction and identification. In particular, I worked on identification of tau lepton (a kind of very heavy electron). Once every particle is identified using the detector, physicist are actually able to select collision where a specific particle were produced, like the Higgs boson, that was the highest goal in particle physics during the last decades and that was discovered in 2012. I partipcated to his search at the Tevatron and to the evidence of one of his properties at the LHC. It is also very important to search for collisions which are not predicited by the Standard Model (or not with the same rate) to discover new physics laws. I am participating to search for new physics with the ATLAS detector in event with two lepton having the same electric charge. # Search for new physics in the top quark sector The top quark is the heaviest known particle in the Standard Model and deserve then a special attention. A search for new physics in the top quarks sector can be performed in collisions with two leptons with the same electric charge and some b jets. This final state is sensitive to many beyond the Standard Model signatures and was already studied at the LHC Run 1. This analysis is being performed for the LHC Run 2 data and a preliminary public result is available with a fraction of Run 2 data. Below, few details are given concerning the physics motivations and the Run 1 results as well as a Run 2 update. ### Top quark pair production LHC collisions produce many top-antitop pairs allowing to precisely measure top quark properties. Once produced, the top quark almost immediatly decay into a W boson and a b quark leading to 3 typical final states for top-antitop, depending on the W boson decay, with 0, 1 or 2 leptons. This kind of collision is now well known and any deviation from what we expect could be a sign for new phenomena. ### Four top quarks final state - Run 1 The question is how many top quark can we produce in a single collision? In fact, the top quark can be also singly produced (via electroweak interaction rather than strong interaction in case of pair production). Here, the appraoch to search for new physics is to look for four top quark production, since the rate predicted by the Standard Model is very low for this production, and its signature can be quite spectacular and then easy to extract from known processes. But how new particle or interaction could enhence four top quark production? In the Standard Model, there is no direct interaction between top quarks since they always interact via gluon exchange. Let's assume there is a new interaction between top quarks mediated by a very heavy particle (too heavy to be directly produced at the LHC). Then, this would appear at the LHC energy scale as a direct interaction between top quarks, leading to four top quarks production. The interesting feature in this approach is that the experimental signature doesn't depend on the structure of the new interaction. Selecting collisions with two leptons with the same electric charge allows to get four top quarks production. The number of Standard Model processes leading to such a signature is reasonably low so the dominant background for this analysis is mostly related to detector effects leading, for example, to the wrong charge assignment. In this case, processes creating two opposit sign leptons (like top-antitop pair production) could be wrongly identified as a same-sign dilepton collision. The data collected by ATLAS during the Run 1 were analyzed and an interesting excess of observed data were found with respect to the expected background. This result allowed to set limit on new physics models and was published in JHEP 10 (2015) 150. ### Four top quarks final state - Run 2 update In June 2016, we released a preliminary result based on Run 2 data collected in 2015, having a center-of-mass energy 60% higher, detailed in ATLAS-CONF-2016-032. The constraints are much more stringent (as shown in figures below), especially for processes requiring high energy such as four top quarks production, but the modest excess is not present anymore (with 3.2 fb-1 of data). This result has been presented in several international conferences such as LHCP2016 or ICHEP2016. An update with more data is now in preparation. References: JHEP 10 (2015) 150 (Run 1) , ATLAS-CONF-2016-032 (Run 2) # Detector R&D for the High Luminosity LHC The ATLAS hadronic calorimeter is a subdetector of ATLAS dedicated to energy measurement of hadrons, particles made of quarks (such as proton, neutron or pions), heavily produced in proton-proton collisions. This subdetector is made of an alternance of steel and scintillating tiles. Steel plates allow to interact with incoming hadrons during which many secondary particles are produced. Scintillating tiles produce light when one of these secondary particles go through. This light, reflecting the energy of the initial hadron, is collected via optical fiber and converted in electric signal by photomultipliers. ### High Luminosity LHC The high luminosity LHC is a project for 2023 aiming to upgrade the LHC to produce proton-proton collision with a much higher rate than today. This allows to capture possible rare phenonmenon and also to probe the Standard Model with a higher precision, especially the Higgs sector. Detector are also upgraded in order to cope with the much higher radiation exposure and the unprecedent collision rate. The HL-LHC project was approved by the CERN scientific Council in June 2016 and will then happen. The readout electronics is the last step before the signal treatment process. It reads the charge delivered by the photomultiplier, shape the pulse and digitize it so that it can be analyzed to extract the energy corresponding the collected light and the time at which it was deposited. The readout electronics must satisfy several specifications in term of precision, noise, response linearity but also in term of radiation hardness. I am involved in the design and the test of an Application-Specific Integrated Circuit (ASIC), called FATALIC, able to read photomultiplier signal fulfilling all specifications required by the physics. FATALIC is based on 130nm technology build in three different blocks: a current conveyer able to directly read a current (and not a voltage), a shaper and an Analog-Digital Converter (ADC). One feature of this readout system to have three different gains in order to have a very good precision over a large dynamic range. ### Testing prototype with particle beams ... There is a prototype module at CERN which is used to test the whole system (low/high voltage power supplies, data acquisition system, noise of the complete electronic chain, etc ...). In addition, this demonstrator allows to compare the various options of readout electronics, which are designed by different labs. In practice, a lot of tests can be done without real particles, for example using electronic charge injection system (needed for in-situe calibration). The communication between the different components of the readout chain (very front-end, frond-end, back-end) are also studied on this demonstrator. However, it is crucial at some point to see how the demonstrator reacts to real particles and this is done using particle beam provided by the CERN facilities. The demonstrator is then measuring electrons, muons and pions with various incidence angles and electronics readout can be tested in real conditions. # Higgs boson physics The Higgs boson was searched for many years before being discovered in 2012. The search strategy was quite wide in order to cover all possible experimental signatures of the Higgs boson. Indeed, its mass wasn't predicited and has a strong impact on the way the Higgs boson decays and thus on how the detector will "see" it. They are two types of decay mode which allow to access different properties of the Higgs sector: 1. fermionic decay modes reflect the Yukawa coupling terms, not constrained by any invariance, like in this search. 2. bosonic decay modes reflect the behaviour of the Higgs field under electro-weak symmetry, like in this search. ### Search and evidence for Higgs boson decaying into tau leptons in ATLAS Unlike couplings to electroweak bosons, couplings between the Higgs boson and fermions are not constrained by any symmetry. Thus, measuring the branching ratio of the Higgs boson decaying into a fermion pair is testing a part of the Standard Model, called Higgs Yukawa sector, which has no fundamental reason to be like we think it is. They are two fermions which provide a good opportunity to experimentally probe the Yukawa part of the Higgs sector: the b quarks and the tau lepton. Both benefits from a significant branching ratio but are extremely challenging from the experimental point of view. If we focus on the tau-tau final state, there are three actual different final states seen by the detector depending on how each tau lepton will decay (more details here) , namely tauleptaulep (dilepton final state), tauleptauhad (lepton plus a hadronically decaying tau lepton) and tauhadhadlep (two hadronically decaying tau lepton). Since tau leptons decays before reaching the detecor, they have to be identified from their decay products. Hadronically decaying tau lepton are particularly challenging to identify among jets, largely produced in hadronic collider (for more details, see my researches on tau lepton identification). On top of the instrumental background due to the hadronic final states, there is an irreducible background where a Z boson decays into tau-tau having the exact same final state of the collision we are looking for. I mostly participated to the tauleptauhad analysis exploiting a lot of properties to extract the signal from the fake tau lepton background and the Z boson production. For example, specifically looking a collisions where the (tau,tau) system is highly boosted allowed to significantly reduce the background while keeping a good fraction of the signal. Actually, one of the biggest achievement was to use a multivariate analysis to optimally exploit these numerous properties together and reach a good enough sensitivity to actually see the Higgs boson production in the tau-tau decay mode. At the end, the analysis was based on six different categories of events, classified depending on their topology. Each of them had a dedicated set of observables entering in the final MVA. This channel is particularly relevant because it allows to access to one of the finger print of the Higgs boson. Indeed, one of the specificity of the Higgs boson is that his coupling to other particle follow a very simple law: it is proportional to the masse of the particle. The coupling between electroweak boson and the Higgs boson are well measured (Higg boson decay into two vector bosons - W, Z, photon - were actually the golden channel to discover the Higgs boson), but what's about the fermions? The coupling to the top quark is well known but only accessible indirectly. The direct observation of couplings between the Higgs boson and a fermion is provided by the tau-tau channel. And once again, the observation is in good agreement with prediction from the Standard Model. For more details, see ATLAS-CONF-2015-044. References: JHEP 04 117, ATLAS page ### Search for Higgs boson decaying into WW in the mu+tau final state at DØ The Tevatron collider was mostly sensitive to the WW decay channel for a Higgs boson mass of mH=165 GeV. Indeed H almost always decays in this mode for mH ~ 2 mW. Experimentally, the e-e, e-mu and mu-mu final state are the easiest to identify and to reconstruct, in particular on a hadronic collider. However, before the LHC started, the Tevatron was the only collider able to constrain the Higgs sector and it was important to exploit every possible channels, including tau lepton channels. The strategy was then to search for this Higgs boson in mu+tau events, coming from WW decay of the Higgs boson where one W decays into a muon and a neutrino and the other decays into a tau and a neutrino: $$pp \to H \to W+W \to \mu\nu_{\mu} + \tau\nu_{\tau}$$ Since tau leptons decays before reaching the detecor, they have to be identified from their decay products. Hadronically decaying tau lepton are particularly challenging to identify among jets, largely produced in hadronic collider (for more details, see my researches on tau lepton identification). Therefore, the main background of this search is the production of W boson is association with jets, where the W boson decays into a muon and a jet is wrongly identified as hadronically decaying tau lepton. Even more important than reducing the background, is how to accurately estimate it. Indeed, this part is crucial since we search for an excess of collision with respect to a predicted number of background collisions. The number of background events coming from W+jets strongly depends on how frequently a jet can look like tauhad, which is quite difficult to model using simulation. That is why a dedicated method based on data was elaborated in order to correct the simulation. In order to separate collisions from background processes and collisions where H were produced, a multi-variate analysis (MVA) approach was adopted. Many observables allowing to help identifying signal collision were combined in a Neural Network (NN) in order to get an optimal discriminant exploiting all the physics behind the background and the Higgs boson production. Using this strategy, the experimental sensitvity to H production reached 8 times the SM predicted rate. # Tau lepton in hadrons collider ### Why tau lepton are challenging? The tau lepton is the heaviest known lepton and it is also the only one decaying after a very short distance (~100 micrometers). Thus, if one wants to probe particle decaying into tau leptons, one need to reconstruct them from they decay products. It also means that the reconstructed particles must be identified as a tau lepton decay, which might not always be trivial depending on the decay mode. They are two kinds of decay decay modes: 35% of leptonic decay (with one neutrino) and 65% of hadronic decay (with two neutrinos). The leptonic decay gives either an electron or a muon which are relatively easy to reconstruct (but impossible to distinguish from lepton originating from heavy boson). The hadronic decay mode is more tricky because it is subdivided in 3 main types of signature, depending on the hadrons produced in the decay. In addition, hadrons are heavily produced in hadronic collisions and will easily mimic tau leptons. Since tau lepton mostly decay into hadrons, it is quite important to be able to identify them among all hadrons produced in a collision, in particular, those from jets - experimental signature of a quark. This is done exploiting specific properties of the hadrons coming from the tau decay. For example, hadrons from a tau will be very collimated and not very numerous (mostly 1, 2 or 3). Looking around the most energetic hadron will also provide a good insight whether this is a hadronic tau - low activity, few tracks and energy deposit - or a jet - high activity, many tracks and energy deposits. The fraction of charged hadrons and neutral hadrons will also be a nice handle to distinguish between hadronic taus and jets. ### How to improve hadronic tau identification I worked on several aspects to better identify hadronics taus. First, I tried to explore detailed properties of the hadronic decay. For example, in some cases a particular hadron is produced a ρ, decaying into a neutral and a charged pions. It means that the two final pions will have a particular correlation when they come from a tau, which won't be the case if they are randomely produced in a jet. At DØ, I developped a dedicated algorithm to reconstruct close-by photons coming from neutral pion decay, using scintillating strip detector in DØ. Several other ideas were tested in order better reject fake taus, based on physics but also on sophisticated data analysis like a neutral network. The result was a significant improvement of tau identification efficiency, 20% on signal (true taus) over background (jets) ratio.	{}
# Acceleration due to gravity on Mars is about one—third that on Earth. Suppose you throw a ball upward with the same velocity on Mars as on Earth. How would the balls maximum height compare to that on Earth? Nov 10, 2015 The ball would travel 3 times as high. #### Explanation: The equation for finding the displacement (vertical height in this case) of an object with a constant acceleration is: ${v}^{2} = {u}^{2} - 2 a s$ where $s$ is the displacement , $u$ is the initial velocity , $v$ is the final velocity , and $a$ is the acceleration . We can rearrange this formula to give: $s = \frac{{v}^{2} - {u}^{2}}{2 a}$ since we want to know about displacement (height). When throwing a ball vertically, the velocity at the top of the throw (when the ball is highest) is 0 , so we can remove the ${v}^{2}$ part of the equation , as ${0}^{2} = 0$. Now we can look at this equation in terms of Earth. ${s}_{e a r t h} = \frac{- {u}^{2}}{2 a}$ The initial velocity $u$ remains constant in both cases, but the acceleration due to gravity on mars is one third the acceleration due to gravity on earth or $\frac{1}{3} a$, so our equation for Mars is: ${s}_{m a r s} = \frac{- {u}^{2}}{2 \times \frac{1}{3} a}$ This can be rearranged to produce: ${s}_{m a r s} = 3 \frac{- {u}^{2}}{2 a}$ From this we see that ${s}_{m a r s} = 3 \left({s}_{e a r t h}\right)$. Therefore the displacement or height ($s$) on mars is three times that on earth so the ball will travel 3 times as high .	{}
# Colección Música » Sleeping with Sirens » ## You Kill Me (In a Good Way) 255 scrobblings \| Ir a la página del tema Temas (255) Tema Álbum Duración Fecha You Kill Me (In a Good Way) 3:41 14 Jun 2011, 19:31 You Kill Me (In a Good Way) 3:41 14 Jun 2011, 19:31 You Kill Me (In a Good Way) 3:41 14 May 2011, 17:14 You Kill Me (In a Good Way) 3:41 14 May 2011, 17:14 You Kill Me (In a Good Way) 3:41 14 May 2011, 17:14 You Kill Me (In a Good Way) 3:41 10 May 2011, 13:39 You Kill Me (In a Good Way) 3:41 29 Abr 2011, 0:10 You Kill Me (In a Good Way) 3:41 10 Abr 2011, 2:39 You Kill Me (In a Good Way) 3:41 30 Mar 2011, 20:54 You Kill Me (In a Good Way) 3:41 30 Mar 2011, 20:54 You Kill Me (In a Good Way) 3:41 14 Feb 2011, 13:23 You Kill Me (In a Good Way) 3:41 14 Feb 2011, 13:23 You Kill Me (In a Good Way) 3:41 28 Dic 2010, 21:32 You Kill Me (In a Good Way) 3:41 28 Dic 2010, 21:32 You Kill Me (In a Good Way) 3:41 20 Dic 2010, 21:10 You Kill Me (In a Good Way) 3:41 23 Sep 2010, 16:26 You Kill Me 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12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You Kill Me (In a Good Way) 3:41 15 Sep 2010, 12:01 You 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a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 8 Sep 2010, 16:53 You Kill Me (In a Good Way) 3:41 3 Sep 2010, 20:55 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 2 Sep 2010, 16:41 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 29 Ago 2010, 20:15 You Kill Me (In a Good Way) 3:41 27 Ago 2010, 20:36	{}