Split (1)

train · 6.32M rows

url stringlengths 14 2.42k	text stringlengths 100 1.02M	date stringlengths 19 19	metadata stringlengths 1.06k 1.1k
http://questaal.org/tutorial/gw/qsgw_si/	# Introductory QSGW Tutorial This tutorial begins with an LDA calculation for Si, starting from an init file. Following this is a demonstration of a quasi-particle self-consistent GW (QSGW) calculation. An example of the 1-shot GW code is provided in a separate tutorial. Click on the dropdown menu below for a brief description of the QSGW scheme. A complete summary of the commands used throughout is provided in a separate dropdown menu. Theory for GW and QSGW, and its implementation in the Questaal suite, can be found in Phys. Rev. B76, 165106 (2007). Each iteration of a QSGW calculation has two main parts: a section that uses effective one-body hamiltonians to make the density n (as in DFT), and the GW code that makes the self-energy $\Sigma(\omega)$ of an interacting hamiltonian. For quasiparticle self-consistency, the GW code makes a “quasiparticlized” self-energy $\Sigma^0$, which is used to construct the effective one-body hamiltonian for the next cycle. The process is iterated until the change in $\Sigma^0$ becomes small. The one-body executable is lmf. The script lmfgwd is similar to lmf, but it is a driver whose purpose is to set up inputs for the GW code. $\Sigma^0$ is made by a shell script lmgw. The entire cycle is managed by a shell script lmgwsc. Before any self-energy $\Sigma^0$ exists, it is assumed to be zero. Thus the one-body hamiltonian is usually the LDA, though it can be something else, e.g. LDA+U. Note: in some circumstances, e.g. to break time reversal symmetry inherent in the LDA, you need to start with LDA+U. Thus, there are two self-energies and two corresponding Green’s functions: the interacting $G[\Sigma(\omega)]$ and non-interacting $G^0[\Sigma^0]$. At self-consistency the poles of $G$ and $G^0$ coincide: this is a unique and very advantageous feature of QSGW. It means that there is no “mass renormalization” of the bandwidth — at least at the GW level. Usually the interacting $\Sigma(\omega)$ isn’t made explicitly, but you can do so, as explained in this tutorial. In short, a QSGW calculation consists of the following steps. The starting point is a self-consistent DFT calculation (usually LDA). The DFT eigenfunctions and eigenvalues are used by the GW code to construct a self-energy $\Sigma^0$. This is called the “0th iteration.” If only the diagonal parts of $\Sigma^0$ are kept, the “0th” iteration corresponds to what is sometimes called 1-shot GW, or as GLDAWLDA. In the next iteration, $\Sigma^0-V_{xc}^\text{LDA}$ is added to the LDA hamiltonian. The density is made self-consistent, and control is handed over to the GW part. (Note that for a fixed density $V_{xc}^\text{LDA}$ cancels the exchange-correlation potential from the LDA hamiltonian.) This process is repeated until the RMS change in $\Sigma^0$ falls below a certain tolerance value. The final self-energy (QSGW potential) can be thought of as an effective exchange-correlation functional that is tailored to the system. This is very convenient as it can now be used in an analogous way to standard DFT to calculate properties such as the band structure. ### Command summary nano init.si #create init file using lines from box below blm init.si --express --gmax=5 --nk=4 --nit=20 --gw #use blm tool to create actrl and site files cp actrl.si ctrl.si && lmfa si && cp basp0.si basp.si #copy actrl to recognised ctrl prefix, run lmfa and copy basp lmf si > out.lmfsc #make self-consistent echo -1 \| lmfgwd si #make GWinput file vim GWinput #change GW k mesh to 3x3x3 lmgwsc --wt --insul=4 --tol=2e-5 --maxit=5 si #self-consistent GW calculation vim ctrl.si #change number of iterations to 1 lmf si --rs=1,0 #lmf with QSGW potential to get QSGW band gap lmgwclear #clean up directory Alternatively, you can run the following two one-liners to get the same result. This assumes you have already created the init file. blm init.si --express=0 --gmax=5 --nk=4 --nit=20 --gw --nkgw=3 && cp actrl.si ctrl.si && lmfa si && cp basp0.si basp.si && lmf si > out.lmfsc echo -1 \| lmfgwd si && lmgwsc --wt --insul=4 --tol=2e-5 --maxit=5 si > out.gwsc && lmgwclear && lmf si --rs=1,0 -vnit=1 > out.lmf_gwsc ### LDA calculation The starting point is a self-consistent LDA density, you may want to review the DFT tutorial for silicon. Copy the following lines to a file called init.si: LATTICE ALAT=10.26 PLAT= 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 # pos means cartesian coordinates, units of alat SITE ATOM=Si POS= 0.00000000 0.00000000 0.00000000 ATOM=Si POS= 0.25000000 0.25000000 0.25000000 Run the following commands to obtain a self-consistent density: $blm init.si --express --gmax=5 --nk=4 --nit=20 --gw$ cp actrl.si ctrl.si && lmfa si && cp basp0.si basp.si $lmf si > out.lmfsc Note that we have included an extra --gw switch, which tailors the ctrl file for a GW calculation. To see how it affects the ctrl file, try running blm without --gw. The switch modifies the basis set section (see the autobas line) to increase the size of the basis, which is necessary for GW calculations. Two new blocks of text, the HAM and GW categories, are also added towards the end of the ctrl file. The extra parameters in the HAM category handle the inclusion of a self-energy, actually a GW potential (see theory notes above), in a DFT calculation. The GW category provides some default values for parameters that are required in the GW calculation. The GW code has its own input file and the DFT ctrl file influences what defaults are set in it, we will come back to this later. One thing to note is the NKABC= token, which defines the GW k-point mesh. It is specified in the same way as the lower case nkabc for the LDA calculation. Now check the output file out.lmfsc. The self-consistent gap is reported to be around 0.58 eV as can be seen by searching for the last occurence of the word ‘gap’. Note that this result differs slightly to that from the LDA tutorial because the gw switch increases the size of the basis set. Now that we have the input eigenfunctions and eigenvalues, the next step is to carry out a GW calculation. For this, we need an input file for the GW code. ### Making GWinput The GW package (both one-shot and QSGW) uses one main user-supplied input file, GWinput. The script lmfgwd can create a template GWinput file for you by running the following command: $ echo -1 \| lmfgwd si #make GWinput file The lmfgwd script has multiple options and is designed to run interactively. Using ‘echo -1’ automatically passes it the ‘-1’ option that specifies making a template input file. You can try running it interactively by just using the command ‘lmfgwd si’ and then entering ‘-1’. Take a look at GWinput, it is a rather complicated input file but we will only consider the GW k-point mesh for now (further information can be found on the GWinput page). The k mesh is specified by n1n2n3 in the GWinput file, look for the following line: $n1n2n3 4 4 4 ! for GW BZ mesh When creating the GWinput file, lmfgwd checks the GW section of the ctrl file for default values. The ‘NKABC= 4’ part of the DFT input file (ctrl.si) is read by lmfgwd and used for n1n2n3 in the GW input file. Remember if only one number is supplied in NKABC then that number is used as the division in each direction of the reciprocal lattice vectors, so 4 alone means a 4 x 4 x 4 k mesh. To make things run a bit quicker, change the k mesh to 3 x 3 x 3 by editing the GWinput file line: $ n1n2n3 3 3 3 ! for GW BZ mesh The k mesh of 3 x 3 x 3 divisions is rough, but it makes the calculation fast and for Si the results are reasonable. As is the case with the LDA, it is very important to control k convergence. However, a coarser mesh can often be used in GW because the self-energy generally varies much more smoothly with k than does the kinetic energy. This is fortunate because GW calculations are much more expensive. It is important to note that convergence tests will have to be performed for any new system. These can be time consuming and unfortunately there are no shortcuts. ### Running QSGW We are now ready for a QSGW calculation, this is run using the shell script lmgwsc: $lmgwsc --wt --insul=4 --tol=2e-5 --maxit=0 si #zeroth iteration of QSGW calculation The switch ‘–wt’ includes additional timing information in the printed output, insul refers to the number of occupied bands (normally spin degenerate so half the number of electrons), tol is the tolerance for the RMS change in the self-energy between iterations and maxit is the maximum number of QSGW iterations. Note that maxit is zero, this specifies that a single iteration is to be carried out starting from DFT with no self-energy (zeroth iteration). Take a look at the line containing the file name llmf: lmgw 15:26:47 : invoking mpix -np=8 /h/ms4/bin/lmf-MPIK --no-iactive cspi >llmf Each QSGW iteration begins with a self-consistent DFT calculation by calling the program lmf and writing the output to the file llmf. We are starting from a self-consitent LDA density (we already ran lmf above) so this step is not actually necessary here. The next few lines are preparatory steps. The main GW calculation begins on the line containing the file name ‘lbasC’: lmgw 16:27:55 : invoking /h/ms4/bin/code2/hbasfp0 --job=3 >lbasC lmgw 16:27:55 : invoking /h/ms4/bin/code2/hvccfp0 --job=0 >lvccC ... 0.0m (0.0h) lmgw 16:27:58 : invoking /h/ms4/bin/code2/hsfp0_sc --job=3 >lsxC ... 0.0m (0.0h) lmgw 16:27:59 : invoking /h/ms4/bin/code2/hbasfp0 --job=0 >lbas lmgw 16:27:59 : invoking /h/ms4/bin/code2/hvccfp0 --job=0 >lvcc ... 0.0m (0.0h) lmgw 16:28:02 : invoking /h/ms4/bin/code2/hsfp0_sc --job=1 >lsx ... 0.0m (0.0h) lmgw 16:28:02 : invoking /h/ms4/bin/code2/hx0fp0_sc --job=11 >lx0 ... 0.1m (0.0h) lmgw 16:28:07 : invoking /h/ms4/bin/code2/hsfp0_sc --job=2 >lsc ... 0.1m (0.0h) lmgw 16:28:13 : invoking /h/ms4/bin/code2/hqpe_sc 4 >lqpe The three lines with lbasC, lvccC and lsxC are the steps that calculate the core contributions to the self-energy and the following lines up to the one with lsc are for the valence contribution to the self-energy. The lsc step, calculating the correlation part of the self-energy, is usually the most expensive step. The last step (hpqe_sc) collects terms to make quasiparticlized self-energy $\Sigma^0$ and writes it to file sigm for every irreducible k point. Actually it writes $\Sigma^0-V_\mathrm{xc}^\mathrm{LDA}$. This makes running lmf very convenient, since lmf simply has to add this term to the LDA potential. Further information can be found in the annotated GW output page. The self-energy produced so far is essentially the same as GLDAWLDA generated in the one-shot tutorial, only now there is no Z factor and the full $\Sigma^{nn^\prime}$ is generated. This is distinct from computing the level shift in first order perturbation theory as lmgw1-shot did (and most GW codes do). This requires only the diagonal $\Sigma^{nn}$, which is fast and easier to make (and is why QSGW is more expensive to do). It is interesting to compare one-shot results from the 0th iteration to the output of lmgw1-shot. $ lmf si The following line in the standard output specifies that the GW potential is being used: RDSIGM: read file sigm and create COMPLEX sigma(R) by FT ... The GW potential is contained in the file sigm, lmgwsc also makes a soft link sigm.si so lmf can read it. The GW potential is automatically used if present, this is specified by the HAM_RDSIG tag in the ctrl file. After the first band pass, lmf yields a gap of 1.21 eV, essentially identical to what lmgw1-shot gives without a Z factor. In general this is not true; but Si is very simple and the $GW$ and LDA eigenfunctions are very similar. Nevertheless, note that as the density is updated (the off-diagonal elements of $\Sigma^{nn^\prime}$ mean that the eigenfunctions change), the gap increases to 1.26 eV. This can be expressed as a change $(\delta V/\delta n) \Delta n = \chi^{-1} \Delta n$, where $\chi^{-1}$ is implicitly given from DFT through self-consistency in lmf. Run the command again but this time set the number of iterations (maxit) to something like 5: $lmgwsc --wt --insul=4 --tol=2e-5 --maxit=5 si #self-consistent GW calculation The iteration count starts from 1 since we are now starting with a self-energy from the zeroth iteration. Again, the iteration starts with a self-consistent DFT-like calculation, but now $\Sigma^0-V_{xc}^\text{LDA}$ from zeroth iteration is added. Take a look at the GW output again and you can see that the rest of the steps are the same as before. After 3 iterations the RMS change in the self-energy is below the tolerance - the calculation is converged. mgwsc : iter 3 of 5 RMS change in sigma = 5.14E-06 Tolerance = 2e-5 more=F Mon 21 May 2018 19:08:21 BST elapsed wall time 5.0m (0.1h) mark Now that we have a converged self-energy (sigm) we can go back to using lmf to calculate additional properties. We only want to run a single iteration so change the number of iterations (nit) to 1 in the ctrl file. Run the following command: $ lmf si --rs=1,0 #lmf with QSGW potential to get QSGW band gap The --rs switch tells lmf to read from the restart file, which contains the LDA density, but not to write to it (once we have a converged density we want to keep this fixed). More information on command line switches can be found here. Inspect the lmf output and you can find that the gap is now around 1.33 eV. It is larger because the one-body hamiltonian generating $\Sigma$ has a wider gap, which which increases $W$ and thus $\Sigma$. Check your directory and you will see that a large number of files were created. The following command removes many redundant files: $lmgwclear #clean up directory Further details can be found in the Additional exercises below. ### QSGW energy bands lmf has a very powerful feature, that it can takes the inverse Bloch transform $\Sigma^0(\mathbf{q}$ to put $\Sigma^0$ in real space, from the mesh of points it is calculated on From real space it can interpolate to any $\mathbf{q}$ by performing a forward Bloch transform. The details are rather complicated, but they are explained in some detail in Section IIG of PRB76, 165106. This feature enables us to compute the energy bands and any k, and allows us to draw the energy bands with minimal effort. $ lmchk ctrl.si --syml~n=41~q=-.5,.5,.5,0,0,0,0,0,1 $lmf ctrl.si --band~fn=syml$ echo -13,10,5,10 \| plbnds -fplot -ef=0 -scl=13.6 -lbl=L,G,X bnds.si $fplot -f plot.plbnds lmchk makes a symmetry lines file, along the lines L-Γ and Γ-X. lmf generates the energy bands in symmetry-lines mode. Finally the last two commands convert bnds.si generated by lmf into a postscript figure. ### Additional Exercises 1) Correct gap This is actually the Γ-X gap; the true gap is a little smaller as can be seen by running lmf with a fine k mesh. 2) Changing k-point mesh Test the convergence with respect to the GW k mesh by increasing to a 8 × 8 × 8 k mesh. 3) GaAs Try a QSGW calculation for GaAs. Note that the code automatically treats the Ga d state as valence (adds a local orbital). This requires a larger GMAX. You also need to run lmfa a second time to generate a starting density that includes this local orbital. The lmfa line for GaAs should be: $ lmfa ctrl.gas; cp basp0.gas basp.gas; lmfa ctrl.gas The init file is: # init file for GaAs LATTICE # SPCGRP= ALAT=10.69 PLAT= 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 # 2 atoms, 1 species SITE ATOM=Ga POS= 0.00000000 0.00000000 0.00000000 ATOM=As POS= 0.25000000 0.25000000 0.25000000 If this page has any errors, there is something you think is missing or unclear, or for any other issues, you can create a post here letting us know.	2018-05-23 03:09:17	{"extraction_info": {"found_math": true, "script_math_tex": 29, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6283129453659058, "perplexity": 3549.792377697557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865411.56/warc/CC-MAIN-20180523024534-20180523044534-00184.warc.gz"}
https://math.stackexchange.com/questions/2827861/formal-power-series-f-ix-converges-if-and-only-if-lim-i-to-inftydegf	# Formal power series: $F_i(x)$ converges if and only if $\lim_{i\to\infty}deg(F_{i+1}(x)- F_i(x))=\infty.$ The following notations and definitions are taken from Richard Stanley's book Enumerative Combinatorics Volume $1,$ second edition. Recall that a formal power series $F(x)$ is of the form $$\sum_{n\geq 0} a_n x^n$$ where $x$ cannot take numerical value. If $F_1(x), F_2(x),...$ is a sequence of formal power series, and if $F(x) = \sum_{n\geq 0}a_n x^n$ then we say that $F_i(x)$ converges to $F(x)$ as $i\to\infty$ provided for every $n\geq 0,$ there is a number $\delta(n)$ such that the coefficient of $x^n$ in $F_i(x)$ is $a^n$ whenever $I\geq \delta(n).$ The degree of a nonzero formal power series $F(x) = \sum_{n\geq 0}a_nx^n$ is the least integer $n$ such that $a_n\neq 0.$ It is denoted by $deg(F(x)).$ Statement: $F_i(x)$ converges if and only if $$\lim_{i\to\infty}deg(F_{i+1}(x)- F_i(x))=\infty.$$ If all $F_i(x) = F(x),$ wouldn't the statement false? Because $F(x), F(x),...$ clearly converges to $F(x)$ but $$\lim_{i\to\infty}deg(F(x)-F(x)) = 0 \neq \infty.$$ If we assume that all $F_i(x)$ are distinct from $F(x),$ then I do not know how to prove the statement. • What is the degree of the zero power series by definition? – dan_fulea Jun 21 '18 at 22:52 • Undefined? So my example is false? – Idonknow Jun 21 '18 at 22:55 • The above definition should be correspondingly be adapted. There is no least integer, just set it to $\deg 0 =\infty$. Then there is not counterexample using the constant sequence $F,F,F,\dots$ . – dan_fulea Jun 21 '18 at 22:57 Suppose that $\text{deg}(F_{i+1}-F_{i})\to \infty$. So, given $m$, there exists $N$ such that $i\ge N$ implies that $\text{deg}(F_{i+1}-F_{i})>m$. So $[x^m]F_{i}=[x^m]F_{i+1}$, so the coefficent of $x^m$ stabilizes as required. Now suppose that $F_{i}\to F$. Let $m$ be given. Since $F_{i}\to F$, there exists an $M$ such that $i\geq M$ implies that $[x^{k}]F_{i}=[x^{k}]F_{i+1}$ for $k=0, 1,\dotsc, m$. Then $\deg (F_{i+1}-F_{i})>m$ as desired. • I assume that the $m$ in second paragraph refers to natural number, right? – Idonknow Jun 22 '18 at 1:57	2020-03-29 10:27:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9873698353767395, "perplexity": 77.47313492820918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370494064.21/warc/CC-MAIN-20200329074745-20200329104745-00058.warc.gz"}
https://kb.paessler.com/en/topic/74880-monitor-a-pickup-folder	### What is this? This knowledgebase contains questions and answers about PRTG Network Monitor and network monitoring in general. You are invited to get involved by asking and answering questions! Intuitive to Use. Easy to manage. 300.000 administrators have chosen PRTG to monitor their network. Find out how you can reduce cost, increase QoS and ease planning, as well. View all Tags # Monitor a pickup folder #### 0 Looking to monitor a pickup folder or file inbox. The normal folder & file share sensors won't work as needed. New files are dropped into server\share Another process polls this location every x minutes, pulls files out and removes from the share if this process breaks, files back up in this location. There could be any number of files in the share at any given moment. The files could have new or old creation dates. Basically need to ensure that no file name continues to exist from one check period as this would indicate that the import process is not functioning. Created on Jun 22, 2017 6:06:28 PM by 3 Replies #### 0 Hi Jonathan, You're right here, none of our sensors is able to check for that. You could however write a small PowerShell script that checks a directory for its childitems and creates a list of them. In the next run, it gets the childitems again and compares them to the list - if they're identical, the sensor throws a warning or error with the corresponding message. Unfortunately, we can't assist in writing custom script sensors for customers as it is too time consuming. Further information about the sensors and what the expected output of the script/exe has to look like, can be found in the manual: Maybe some of the forum members once needed a similar script and don't mind to share it? :) Kind regards, Created on Jun 23, 2017 10:14:36 AM by #### 0 OK, I wrote the script. You can use the parameters field in the sensor settings to pass a folder path to monitor and path +filename where we can store results. Like this: ' path\to\foler' ' path\to\results.xml' #checks a pickup folder or dropbox for files that haven't been processed. $filepath =$Args[0] #path to folder to monitor $previousfile =$Args[1] #path to previous xml file #pull current directory listing $current = Get-ChildItem "$filepath" \| select Name #you can add -Exclude if you need to ignore some files $count =$current.count #check if there are actaully any files in the directory. no need to continue if folder is empty if ($count -eq 0) { #no files in the dir return "0:OK" } #check if previous listing exists If ((Test-Path "$previousfile") -eq $false) { #file doesn't exist, create a new one. don't compair until next check$current \| Export-Clixml "$previousfile" return "0:OK" } #load preivous listing$previous = Import-Clixml "$previousfile" #compare and count to find stuck files$count = (Compare-Object -ReferenceObject $previous -DifferenceObject$current -ExcludeDifferent -IncludeEqual).count #save the current list as the new "previous" list $current \| Export-Clixml "$previousfile" #make sure count isn't null so that we get a numerical value returned. if ($count -eq$null) { $count = 0 } return "$count:OK" ` Created on Jun 23, 2017 11:31:05 AM by Last change on Jun 27, 2017 10:25:18 AM by #### 0 You want to know the easier way to do it? or the coolest? The easiest is what is listed above (dump folder list, wait till next run, look for dupes, delete, start again). The coolest way is look at the USN Journal, it will show you when a file gets moved or copied to a certain folder (even though the modified date does not change.) Or another simple way that might work, look at the create date, no the modified date. (When a file gets copied it gets a new create date, this applies most of the time, but worth checking in your case). So copy and delete the file, rather than move. If a file gets moved it creates the original timestamp. Please let us know what way worked best. Created on Jun 27, 2017 5:00:22 AM by AndrewG (1,783) 2 2	2021-10-19 13:01:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23808349668979645, "perplexity": 3921.5831846818674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585265.67/warc/CC-MAIN-20211019105138-20211019135138-00400.warc.gz"}
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-2-8	Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki. Zapraszamy na https://bibliotekanauki.pl PL EN Preferencje Język Widoczny [Schowaj] Abstrakt Liczba wyników • # Artykuł - szczegóły ## Colloquium Mathematicum 2012 \| 127 \| 2 \| 253-298 ## Two classes of almost Galois coverings for algebras EN ### Abstrakty EN We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G = Π(Γ_A)$ is the fundamental group of $Γ_A$. It is shown that the class of almost Galois coverings F: R → R' of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)], behaves much more regularly than usual with respect to the standard properties of the pair $(F_λ, F_•)$ of adjoint functors associated to F. 253-298 wydano 2012 ### Twórcy autor • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100 Toruń, Poland autor • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100 Toruń, Poland	2022-12-06 00:00:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4755818247795105, "perplexity": 2586.144148148824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711064.71/warc/CC-MAIN-20221205232822-20221206022822-00736.warc.gz"}
https://brilliant.org/problems/a-light-and-a-shadow/	# A light and a shadow Geometry Level 2 A rectangular billboard $ABCD$ is illuminated by a lantern $E$ and casts a shadow in the $xy$-plane. The positions $(x,y,z)$ of the lantern and the billboard's vertices are $A = (0,0,0), \quad B = (0,3,0), \quad C = (0,3,2), \quad D = (0, 0 ,2), \quad E = (3, -1, 4).$ What is the area of the shadow? Assumptions: The light source is a point and the billboard has zero thickness. The ground is the $xy$-plane. ×	2020-05-25 22:00:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6825132966041565, "perplexity": 773.730526767945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347389355.2/warc/CC-MAIN-20200525192537-20200525222537-00265.warc.gz"}
http://apjacob.me/gan/2018/01/12/A-Tutorial-On-Boundary-Seeking-Generative-Adversarial-Networks.html	12 Jan 2018 # A Tutorial On Boundary-Seeking Generative Adversarial Networks Devon Hjelm and Athul Paul Jacob In this post, we discuss our paper, Boundary-Seeking Generative Adversarial Networks which was recently accepted as a conference paper to ICLR 2018. Generative adversarial networks(a.k.a GANs)[Goodfellow et. al., 2014] is a unique generative learning framework that uses two separate models, a generator and discriminator, with opposing or adversarial objectives. Training a GAN only requires back-propagating a learning signal that originates from a learned objective function, which corresponds to the loss of the discriminator trained in an adversarial manner. This framework is powerful because it trains a generator without relying on an explicit formulation of the probability density, using only samples from the generator to train. We will give a brief overview of this framework. Suppose we have empirical samples from a distribution $$\mathbb{P}$$, $${x^{(i)} \in \mathcal{X}}_{i=1}^M$$, where $$\mathcal{X}$$ is the domain of images, word or character-based representations of natural language etc. Our objective is then to find an induced distribution, $$\mathbb{Q}_\theta$$ that describes these samples well. We start with a simple prior, $$h(z)$$ which is typically a gaussian or a uniform distribution and two players: The generator ($$\mathbb{G}_{\theta}: \mathcal{Z} \rightarrow \mathcal{X}$$) and the discriminator ($$\mathbb{D}_{\phi}: \mathcal{X} \rightarrow \mathcal{Z}$$) The goal of the generator is to find the parameters $$\theta$$ to find the induced distribution $$\mathbb{Q}_\theta$$. While, the goal of the discriminator is to classify the real and fake(generated) samples correctly. In essense, the generator is trained to fool the discriminator into thinking that the generated samples comes from the true distribution $$\mathbb{P}$$. While, the discriminator, is trained to distiguish between the samples from $$\mathbb{P}$$ and the samples from $$\mathbb{Q}_\theta$$. This can then be formalized as a minimax game: GANs have been shown to generate often-diverse and realistic samples even when trained on high-dimensional large-scale continuous data. GANs however have a serious limitation on the type of variables they can model, because they require the composition of the generator and discriminator to be fully differentiable. With discrete variables, this is not true. For instance, consider using a step function at the end of a generator in order to generate a discrete value. In this case, back-propagation alone cannot provide the training signal, because the derivative of a step function is 0 almost everywhere. This is problematic, as many important real-world datasets are discrete, such as character- or word-based representations of language. ## Probability Difference Measures Let’s take a step back and try to understand one of the key choices that differentiates GANs. Suppose we are given two probability distributions, $$\realprob$$ and $$\genprobp$$. We need a difference measure between these distributions. The goal is to essentially, find $$\theta$$ such that this difference measure is minimized. There are a variety of difference measure families such as: 1. f-divergences: KL-divergence, reverse KL-divergence, Hellinger distance, Total variation distance, $$\mathcal{X}^{2}$$-divergence etc. 2. IPM: Kantorovich (Wasserstein dual), MMD, Fisher distance etc. In the next section, we will focus on the $$f$$-divergence family of distances. #### $$f$$-divergences Suppose, we introduce a convex (lower semi-continuous) function: The divergence generated by $$f$$: Note that, Examples: KL, Jensen-Shannon, Squared Hellinger, Pearson $$\mathcal{X}^2$$ etc. This is the foundation of many generative learning algorithms. #### The Convex Dual representation Consider the convex conjugate of $$f$$, $$\fdc$$ and a family of functions, $$\SN$$. Then the dual form of the $$f$$-divergence is: However, finding this supremum is hard. So we can instead use a family of neural networks(classifiers), $$\SN_{\phi}(x)$$ which can be learnt!. Hence, we have: ## BGAN For Discrete Data Now, we can proceed to show how BGAN can handle discrete data #### Estimating The Likelihood Ratio Given a perfect discriminator (i.e $$\SNo$$), we show in Theorem 1 of our paper that, #### $$f$$-Importance Weight Estimator Now we don’t have a perfect discriminator (i.e $$\SNo$$), but we do have a sub-optimal discriminator(i.e $$\SNp(x)$$). Let $$\iw(x) = (\pd{\fdc}{\SN})(\SNp(x))$$ and $$\beta = \EE_{\genprob_{\dparams}}[\iw(x)]$$ be a partition function. Then we can have a $$\fd$$-divergence importance weight estimator, $$\realdensest(x)$$ as: #### Policy Gradient Based on Importance Sampling With this importance weight estimator, we have an option for training the generator in an adversarial way with the gradient of the $$KL$$: This gradient resembles other importance sampling methods for training generative models in the discrete setting. However, the variance of this estimator will be high, as it requires estimating the partition function, $$\beta$$ (Through say, Monte-Carlo sampling). We address reducing this issue next. We can use the expected KL over the conditionals instead! And so, we have, We can then derive the normalized weights: And the new partition function: So now, the gradient for training the generator becomes: From figure 7, we can see that this new policy gradient estimator (bold) has lower variance than the original policy gradient estimator (dashed) in estimating $$2 \times JSD - log 4$$. This can be implemented in Pytorch as: #### REINFORCE BGAN REINFORCE is a common technique for dealing with discrete data in GANs. The lower-variance policy gradient estimator described above is a policy gradient in the special case that the reward is the normalized importance weights. This reward approaches the likelihood ratio in the non-parametric limit of an optimal discriminator. And so, we also make a connection to REINFORCE as it is commonly used, with baselines, by deriving the gradient of the reversed KL-divergence: ## BGAN For Continuous Data For continuous variables, minimizing the variational lower-bound suffices as an optimization technique as we have the full benefit of back-propagation to train the generator parameters, $$\gparams$$. However, while the convergence of the discriminator is straightforward, to our knowledge there is no general proof of convergence for the generator except in the non-parametric limit or near-optimal case. What’s worse is the value function can be arbitrarily large and negative. Let us assume that $$\max \SN = M < \infty$$ is unique. As $$\fdc$$ is convex, the minimum of the lower-bound over $$\gparams$$ is: The generator objective is optimal when the generated distribution, $$\QQ_{\gparams}$$, is nonzero only for the set $$\{x \mid \SN(x) = M \}$$. Even outside this worst-case scenario, the additional consequence of this minimization is that this variational lower-bound can become looser w.r.t. the $$\fd$$-divergence, with no guarantee that the generator would actually improve. Generally, this is avoided by training the discriminator in conjunction with the generator, possibly for many steps for every generator update. However, this clearly remains one source of potential instability in GANs. And so, we can instead aim for the decision boundary and this can improve stability. We observe that for a given estimator $$\realdensest(x)$$, $$\gendensp(x)$$ matches when $$\iw(x) = (\pd{\fdc}{\SN})(\SN(x)) = 1$$. And so, we can define the continuous BGAN objective as: This objective can be seen as changing a concave optimization problem (which is poor convergence properties) to a convex one. ## Code Our paper presents 10 sets of experiments. The code is available both in Pytorch as well as in Theano. ## Conclusion Reinterpreting the generator objective to match the proposal target distribution reveals a novel learning algorithm for training a generative adversarial network (GANs, Goodfellow et al., 2014). This proposed approach of boundary-seeking provides us with a unified framework under which learning algorithms for both discrete and continuous variables are derived. Empirically, we verified our approach quantitatively and showed the effectiveness of training a GAN with the proposed learning algorithm, which we call a boundary-seeking GAN (BGAN), on both discrete and continuous variables, as well as demonstrated some properties of stability.	2019-03-22 22:55:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8124257922172546, "perplexity": 708.4505729931933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202698.22/warc/CC-MAIN-20190322220357-20190323002357-00118.warc.gz"}
https://www.ashtavakra.org/algebra/quadratic-equations-solutions-part1/	# 47. Quadratic Equations Solutions Part 1 1. For roots to be equal the discriminant has to be zero. $$D = 4(1 + 3m)^2 - 4(1 + m)(1 + 8m) = 0$$ $$\Rightarrow 4(1 + 9m^2 + 6m - 1 - 9m -8m^2) = 0$$ $$\Rightarrow m^2 - 3m = 0 \therefore m = 0, 3$$ 2. Discriminant of the equation is: $$D = (c + a - b)^2 - 4(b + c - a)(a + b -c)$$ $$= 4(b^2 - 4ac)$$ Given $$a + b + c = 0 \Rightarrow b = -(a + c).$$ Substituting in above equation $$D = 4\{(a + c)^2 - 4ac\} = 4(a - c)^2 =$$ a perfect square and thus roots are rational. 3. Discriminant of the equation is: $$D = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = -4(ad - bc)^2$$ Roots are real if $$D\geq 0$$ i.e. $$-4(ad - bc)^2 \geq 0 \Rightarrow (ad - bc)^2 \leq 0$$ But since $$(ac - bd)^2 \nless 0 \therefore (ad - bc)^2 = 0$$ i.e. $$D = 0$$ (because roots are real) Thus, if roots are real they are equal 4. Let $$A = a(b - c), B = b(c - a)$$ and $$c = c(a - b)$$ Clearly, $$A + B + C = 0$$ Since roots are equal i.e. $$D = 0 \therefore B^2 - 4AC = 0$$ Substituting for $$B$$ $$[-(A + C)^2 - 4AC] = (A - C)^2 = 0 \Rightarrow A = C \Rightarrow 2ac = ab + cb \Rightarrow b = \frac{2ac}{a + c}$$ Thus, $$a, b, c$$ are in H. P. 5. Given equation is $$(b - x)^2 - 4(a - x)(c - x) = 0$$ $$-3x^2 + 2(2a + 2c - b)x + b^2 - 4ac = 0$$ Discriminant of the above equation is: $$D = 4(2a + 2c - b)^2 + 12(b^2 - 4ac)$$ $$= 8[(a - b)^2 + (b - c)^2 + (c - a)^2]$$ $$\because a, b, c$$ are real $$\therefore D > 0$$ unless $$a = b = c$$ Hence, roots are real unless $$a = b = c$$ 6. Discriminant of the equations are $$p^2 - 4q$$ and $$r^2 - 4s$$ Adding them we have $$p^2 + q^2 - 4(q + s) = p^2 + q^2 - 2pr = (p - r)^2 \geq 0$$ Thus, at least one of the discriminant is greater than zero and that equation has real roots. 7. Since $$x^2 - 2px + q = 0$$ has equal roots $$D = 0 \Rightarrow 4p^2 - 4q = 0 \Rightarrow p^2 = q$$ Discriminant of the second equation: $$D = 4(p + y)^2 - 4(1 + y)(q + y)$$ $$= 4[p^2 + 2y + y^2 - q -qy -y - y^2]$$ Substituting for $$q$$ $$D = -4y(p - 1)^2$$ Roots of the equation will be real and distinct only if $$D \geq 0$$ but $$(p - 1) \geq 0$$ if $$p \neq 0$$ Thus, $$y$$ has to be negative as well. 8. Since roots of equation $$ax^2 + 2bx + c = 0$$ are equal $$\therefore 4b^2 - 4ac \geq 0$$ Discriminant of the equation $$ax^2 + 2mbx + nc = 0$$ is $$4m^2b^2 - 4anc$$ Since $$m^2 > n > 0$$ and $$b^ \geq ac$$ $$4m^2b^2 - 4anc > 0$$ Thus, roots of the second equation are real. 9. Given $$ax + by = 1 \Rightarrow y = \frac{1 - ax}{b},$$ substituting this in second equation $$cx^2 + d\left(\frac{1 - ax}{b}\right)^2 = \frac{b^2cx^2 + d(1 - ax)^2}{b^2} = 1$$ $$(b^2c + da^2)x^2 - 2adx + d - b^2 = 0$$ Since first two equations have one solution this equation will also have only one solution which means roots will be equal i.e. $$D = 0$$ $$\Rightarrow 4a^2d^2 - 4(b^2c + a^d)(d - b^2) = 0$$ $$b^2(b^2c + a^2d - cd) = 0$$ $$\because b^2 \ne 0 \therefore b^2c + a^2d - cd = 0 \Rightarrow b^2c + a^d = cd$$ Dividing both sides by $$cd$$ we have $$\frac{b^2}{d} + \frac{a^2}{c} = 1$$ $$x = \frac{2ad}{2(b^2c + a^2d)} = \frac{a}{c}$$ Substituting for $$y,$$ we get $$y = \frac{b}{d}$$ 10. Let the roots of the equation be $$\alpha$$ and $$r\alpha$$ Sum of roots = $$\alpha + r\alpha = -\frac{b}{a} \Rightarrow \alpha = -\frac{b}{a(r + 1)}$$ Product of roots $$= r\alpha^2 = \frac{rb^2}{a^2(1 + r)^2 = \frac{c}{a}} \Rightarrow \frac{b^2}{ac} = \frac{(r + 1)^2}{r}$$ 11. Let the roots of the equation be $$\alpha$$ and $$2\alpha.$$ Sum of roots $$= 3\alpha = -\frac{l}{l - m} \Rightarrow \alpha = -\frac{l}{l - m}$$ Product of roots $$= 2\alpha^2 = \frac{1}{l - m}$$ Substituting for $$\alpha$$ $$\frac{2l^2}{9(l - m)^2} = \frac{1}{l - m} \Rightarrow 2l^2- 9l + 9m = 0 [\because l\neq m~\text{else it would not be a quadratic equation}]$$ Since $$l$$ is real, therefore discriminant of this equation would be $$\geq 0$$ $$81 - 72m \geq 0 \therefore m \leq \frac{9}{8}$$ 12. Let the roots be $$\alpha$$ and $$\alpha^n,$$ then Sum of roots $$= \alpha + \alpha^n = -\frac{b}{a}$$ and product of roots $$= \alpha^{n + 1} = \frac{c}{a}$$ From products, we have $$\alpha = \left(\frac{c}{a}\right)^{\frac{1}{n + 1}}$$ From sum we have $$a\alpha^n + a\alpha + b = 0$$ Substituting value of $$\alpha$$ from above $$\Rightarrow a\left(\frac{c}{a}\right)^{\frac{n}{n + 1}} + a\left(\frac{c}{a}\right)^{\frac{1}{n + 1}} + b = 0$$ Solving this we arrive at our desired equation. 13. Let the roots be $$p\alpha$$ and $$q\alpha.$$ Sum of roots $$= (p + q)\alpha = -\frac{b}{a}$$ and product of roots $$= pq\alpha^2 = \frac{c}{a}$$ From equation for product of roots, we have $$\alpha^2 = \frac{c}{apq} \therefore \alpha = \sqrt{\frac{c}{apq}}$$ Substituting this in sum of roots and solving we arrive at desired equation. 14. Solutions are given below: 1. $$\alpha + \beta = -p$$ and $$\alpha\beta = q$$ Now, $$\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha\beta}$$ $$= \frac{(\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)}{\alpha\beta} = \frac{p(3q - p^2)}{q}$$ 2. $$(\omega\alpha + \omega^2\beta)(\omega^2\alpha + \omega\beta)$$ $$= \omega^3\alpha^2 + \omega^4\alpha\beta + \omega^2\alpha\beta + \omega^3\beta^2$$ $$= alpha^2 + \omega\alpha\beta + \omega^2\alpha\beta + \beta^2$$ $$= \alpha^2 -\alpha\beta + \beta^2$$ $$= (\alpha + \beta)^2 - 3\alpha\beta = p^2 - 3q$$ 15. Rewriting the equation we have $$(A + cm^2)x^2 + Amx + Am^2 = 0$$ Sum of roots $$= \alpha + \beta = -\frac{Am}{A + cm^2}$$ and product of roots $$= \alpha\beta = \frac{Am^2}{A + cm^2}$$ The expression to be evaluated is $$A(\alpha^2 + \beta^2) + A\alpha\beta + c\alpha^2\beta^2$$ $$= A[(\alpha + \beta)^2 - 2\alpha\beta] + A\alpha\beta + c(\alpha\beta)^2$$ $$= A\left[\frac{A^2m^2}{(A + cm^2)^2} - \frac{2Am^2}{A + cm^2}\right] + \frac{A^2m^2}{A + cm^2} + \frac{cA^2m^4}{(A + cm^2)^2}$$ $$= 0$$ 16. Sum of roots $$= \alpha + \beta = -\frac{b}{a}$$ and product of roots $$= \alpha\beta = \frac{c}{a}$$ Now, $$a\left(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\right) + b\left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right)$$ $$= \frac{a(\alpha^2 + \beta^3)}{\alpha\beta} + \frac{b(\alpha^2 + \beta^2)}{\alpha\beta}$$ $$= a\frac{[(\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)]}{\alpha\beta} + \frac{b[(\alpha + \beta)^2 - 2\alpha\beta]}{\alpha\beta}$$ Substituting for sum and product of the roots $$= \frac{a\left[\left(-\frac{b}{a}\right)^3 - 3.\frac{c}{a}\left(-\frac{b}{a}\right)\right]}{\frac{c}{a}} + \frac{b\left[\left(-\frac{b}{a}\right)^2 -2 \frac{c}{a}\right]}{\frac{c}{a}}$$ Solving this we get the desired result. 17. Since $$a$$ and $$b$$ are the roots of the equation $$x^2 + px + 1 = 0$$ we have $$a + b = -p$$ and $$ab = 1$$ Similarly, since $$c$$ and $$d$$ are the roots of the equation $$x^2 + qx + 1 = 0$$ we have $$c + d = -p$$ and $$cd = 1$$ Now $$(a - c)(b - c)(a + d)(b + d) = (ab - bc - ac + c^2)(ab + bd + ad + d^2)$$ $$= [ab - c(a + b) + c^2].[ab + d(a + b) + d^2]$$ $$= [1 + pc + c^2].[1 - pd + d^2] (\text{putting the values of } a + b~\text{and}~ab)$$ $$= 1 + cp + c^2 - pd - cdp^2 - c^2pd + d^2 + cpd^2 + c^2d^2$$ $$= 1 + (c^2 + d^2) + c^2d^2 -cdp^2 + p(c - d) + cpd(d - c)$$ $$= 1 + [(c + d)^2 - 2cd] + c^2d^2 - cdp^2 + p(c - d) + cpd(d - c)$$ Substituting for $$c + d$$ and $$cd$$ $$= 1 + q^2 - 2 + 1 - p^2 + p(c - d) + p(d - c)$$ $$= q^2 - p^2$$ 18. Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + px + q = 0$$ then $$\alpha + \beta = -p$$ and $$\alpha\beta = q.$$ Also, let $$\gamma$$ and $$\delta$$ be the roots of the equation $$x^2 + qx + p = 0$$ then $$\gamma + \delta = -q$$ and $$\gamma\delta = p$$ Now, given is that roots differ by the same quantity so we can say that $$\alpha - \beta = \gamma - \delta$$ $$(\alpha - \beta)^2 = (\gamma - \delta)^2$$ $$(\alpha + \beta)^2 - 4\alpha\beta = (\gamma + \delta)^2 - 4\gamma\delta$$ $$p^2 - 4q = q^2 - 4p \Rightarrow p^2 - q^2 + 4(p - q) = 0 \Rightarrow (p - q)(p + q + 4) = 0$$ Clearly, $$p \neq q$$ else equations would be same $$\therefore p + q + 4 = 0$$ 19. Since $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + c = 0$$ $$\therefore a\alpha^2 + b\alpha + c = 0$$ and $$a\beta^2 + b\beta + c = 0$$ and $$\alpha + \beta = -\frac{b}{a}$$ and $$\alpha\beta = \frac{c}{a}.$$ Also, given $$S_n = \alpha^n + \beta^n$$ Now, $$aS_{n + 1} + bS_n + cS_{n - 1}$$ $$= a(\alpha^{n + 1} + \beta^{n + 1}) + b(\alpha^n + \beta^n) + c(\alpha^{n - 1} + \beta^{n - 1})$$ $$= \alpha^{n - 1}(a\alpha^2 + b\alpha + c) + \beta^{n - 1}(a\beta^2 + b\beta + c)$$ $$= \alpha^{n - 1}.0 + \beta^{n - 1}.0$$ $$\therefore S_{n + 1} = -\frac{b}{a}S_n -\frac{c}{a}S_{n - 1}$$ Substituting $$n = 4$$ we have $$S_5 = -\frac{b}{a}S_4 - \frac{c}{a}S_3$$ $$= -\frac{b}{a}(-\frac{b}{a}S_3 - \frac{c}{a}S-2) - \frac{c}{a}S_3$$ $$= \left(\frac{b^2}{a^2} - \frac{c}{a}\right)S_3 + \frac{bc}{a^2}S_2$$ Proceeding similarly we have the solution as $$= -\frac{b}{a^6}(b^2 - 2ac)^2 + \frac{(b^2 - ac)bc}{a^4}$$ 20. Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^2 + bx + c = 0$$ Given, $$\alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2}$$ $$\alpha + \beta = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha^2\beta^2}$$ $$-\frac{b}{a} = \frac{\frac{b^2}{a^2} - 2\frac{c}{a}}{\frac{c^2}{a^2}} = \frac{b^2 - 2ac}{c^2}$$ $$-bc^2 = ab^2 - 2a^2c \Rightarrow ca^2 = \frac{ab^2 + bc^2}{2}$$ Thus, $$bc^2, ca^2, ab^2$$ are in A. P. 21. Rewriting the equation $$m^2x^2 + (2m - m^2)x + 3 = 0$$ Since $$\alpha$$ and $$\beta$$ are the roots of the equation $$\alpha + \beta = -\frac{2m - m^2}{m^2} = \frac{m - 2}{m}$$ and $$\alpha\beta = \frac{3}{m^2}$$ Given, $$\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{3} \Rightarrow \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{4}{3}$$ $$3(\alpha^2 + \beta^2) = 4\alpha\beta \Rightarrow 3[(\alpha + \beta)^2 - 2\alpha\beta] = 4\alpha\beta$$ $$3(\alpha + \beta)^2 - 10\alpha\beta = 0 \Rightarrow 3\left[\left(\frac{m - 2}{m}\right)^2 - \frac{10}{m^2}\right] = 0$$ $$m^2 - 4m - 6 = 0$$ Since $$m_1, m_2$$ are two values of $$m$$ we have $$m_1 + m_2 = 4$$ and $$m_1m_2 = -6$$ Now, $$\frac{m_1^2}{m_2} + \frac{m_2^2}{m_1} = \frac{m_1^3 + m_2^3}{m_1m_2}$$ $$= \frac{(m_1 + m_2)^3 - 3m_1m_2(m_1 + m_2)}{3m_1m_2} = -\frac{68}{3}$$ 22. Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^2 + bx + c = 0;$$ $$\gamma$$ and $$\delta$$ are the roots of the equation $$a_1x^2 + b_1x + c_1 = 0,$$ then $$\alpha + \beta = -\frac{b}{a}, \alpha\beta = \frac{c}{a}$$ and $$\gamma + \delta = -\frac{b_1}{a_1}, \gamma\delta = \frac{c_1}{a_1}$$ According to question, $$\frac{\alpha}{\beta} = \frac{\gamma}{\delta}$$ By componendo and dividendo, $$\frac{\alpha - \beta}{\alpha + \beta} = \frac{\gamma - \delta}{\gamma + \delta}$$ Squaring both sides $$\left(\frac{\alpha - \beta}{\alpha + \beta}\right)^2 = \left(\frac{\gamma - \delta}{\gamma + \delta}\right)^2$$ $$\frac{(\alpha + \beta)^2 - 4\alpha\beta}{(\alpha + \beta)^2 = \frac{(\gamma + \delta)^2 - 4\gamma\delta}{(\gamma + \delta)^2}}$$ $$\frac{b^2 - 4ac}{b^2} = \frac{b_1^2 - 4a_1c_1}{b_1^2} \Rightarrow -4acb_1^2 = -4a_1c_1b^2 \Rightarrow \left(\frac{b}{b_1}\right)^2 = \frac{ac}{a_1c_1}$$ 23. Since irrational roots appear in pairs and are conjugate. Thus, if first root is $$\alpha = \frac{1}{2 + \sqrt{5}}$$ $$\alpha = \frac{1}{2 + \sqrt{5}}\frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{2 - \sqrt{5}}{4 - 5} = -2 + \sqrt{5}$$ Then second root would be $$\beta = -2 + \sqrt{5}$$ $$\alpha + \beta = -4$$ and $$\alpha\beta = -1$$ Therefore, the equation is $$x^2 - (\alpha + \beta)x + \alpha\beta = 0 \Rightarrow x^2 + 4x -1 = 0$$ 24. Since $$\alpha$$ and $$\beta$$ are the roots of the equation $$\therefore \alpha + \beta = -\frac{b}{a}$$ and $$\alpha\beta = \frac{c}{a}$$ Sum of the roots for which quadratic equation is to be found $$= \frac{1}{a\alpha + b} + \frac{1}{a\beta + b}$$ $$= \frac{a(\alpha + \beta) + 2b}{a^2\alpha\beta + ab(\alpha + \beta) + b^2} = \frac{a\left(-\frac{b}{a}\right) + 2b}{a^2.\frac{c}{a} + av\left(-\frac{b}{a}\right)} + b^2$$ $$= \frac{b}{ac}$$ Product of the roots $$= \left(\frac{1}{a\alpha + b}\right)\left(\frac{1}{a\beta + b}\right)$$ $$= \frac{1}{a^2\alpha\beta + ab(\alpha + \beta) + b^2} = \frac{1}{a^2.\frac{c}{a} + ab\left(-\frac{c}{a}\right) + b^2} = \frac{1}{ac}$$ Therefore, the equation is $$x^2 - \frac{b}{ac}x + \frac{1}{ac} = 0 \Rightarrow acx^2 - bx + 1 = 0$$ 25. Given equation is $$(x - a)(x - b) - k = 0 \Rightarrow x^2 - (a + b)x + ab - k = 0$$ Since $$c, d$$ are roots of this equation $$\Rightarrow c + d = a + b$$ and $$cd = ab - k$$ The equation where roots are $$a, b$$ is $$x^2 - (a + b)x + ab = 0 \Rightarrow x^2 - (c + d)x + cd + k = 0$$ 26. Correct equation is $$x^2 + 13x + q = 0$$ and incorrect equation is $$x^ + 17x + q = 0$$ Roots of correct incorrect equation are $$-2$$ and $$-15.$$ Thus $$q = 30$$ Therefore, correct equation is $$x^2 + 13x + 30 = 0$$ and thus roots are $$-3, -10.$$ 27. Clearly, $$\alpha + \beta = -p$$ and $$\alpha\beta = q$$ Substituting $$x = \frac{\alpha}{\beta}$$ in the given equation we have $$q\frac{\alpha^2}{\beta^2} - (p^2 - 2q)\frac{\alpha}{\beta} + q = 0$$ $$\Rightarrow q\alpha^2 - (p^2 - 2q)\alpha\beta + q\beta^2 = 0$$ $$q(\alpha^2 + \beta^2) - (p^2 - 2q)q = 0$$ $$q[(\alpha + \beta)^2 - 2\alpha\beta] - (p^2 - 2q)q = 0$$ $$q(p^2 - 2q) - (p^2 - 2q)q = 0 \Rightarrow 0 = 0$$ Thus, $$\frac{\alpha}{\beta}$$ is a root of the given equation. 28. Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - ax + b = 0$$ and $$\alpha$$ be the common and equal root from the second equation $$x^2 - px + q = 0$$ Thus, $$\alpha + \beta = a, \alpha\beta = b$$ and $$2\alpha = p, \alpha^2 = q$$ $$b + q = \alpha\beta + \alpha^2 = \alpha(\beta + \alpha) = \frac{p}{2}a = \frac{ap}{2}$$ 29. Let $$\alpha$$ be the common root. Then, we have $$a\alpha^2 + 2b\alpha + c = 0$$ and $$a_1\alpha^2 + 2b_1\alpha + c_1 = 0$$ Solving equations by cross-multiplication we have $$\frac{\alpha^2}{2(bc_1 - b_1c)} = \frac{\alpha}{(ca_1 - a_1c)} = \frac{1}{2(ab_1 - a_1b)}$$ From first two we have $$\alpha$$ as $$\alpha = \frac{2(bc_1 - b_1c)}{ca_1 - a_1c}$$ and from last two we have $$\alpha$$ as $$\alpha = \frac{ca_1 - ac_1}{2(ab_1 - a_1b)}$$ Equating we get $$\frac{2(bc_1 - b_1c)}{ca_1 - a_1c} = \frac{ca_1 - ac_1}{2(ab_1 - a_1b)}$$ $$(ca_1 - ac_1)^2 = 4(ab_1 - a_1b)(bc_1 - b_1c)$$ Given, $$\frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1}$$ are in A. P., let $$d$$ be the common difference. $$\left(\frac{c}{c_1} - \frac{a}{a_1}\right)^2c_1^2a_1^2 = 4\left(\frac{a}{a_1} - \frac{b}{b_1}\right)a_1b_2\left(\frac{b}{b_1} - \frac{c}{c_1}\right)b_1c_1$$ $$(2d)^2c_1^2a_2^2 = 4(-d)a_1b_1(-d)b_1c_1$$ $$4d^2c_1^2a_1^2 = 4d^2a_1c_1b_1^2 \Rightarrow c_1a_1 = b_1^2$$ Thus, $$a_1, b_1, c_1$$ are in G. P. 30. Let $$\alpha$$ be the common root between first two, $$\beta$$ be the common root between last two and $$\gamma$$ be the common root between first and last equations. Thus, $$\alpha$$ and $$\beta$$ are the roots of the first equation. $$\Rightarrow \alpha + \gamma = -p_1, \alpha\gamma = q_1$$ Similarly, $$\alpha + \beta = -p_2, \alpha\beta = q_2$$ $$\beta + \gamma = -p_3, \beta\gamma = q_3$$ $$L. H. S. = (p_1 + p_2 + p_3)^2 = 4(\alpha + \beta + \gamma)^2$$ $$R. H. S. = 4(p_1p_2 + p_2p_3 + p_1p_3 - q_1 - q_2 - q_3)$$ $$= 4[(\alpha + \gamma)(\alpha + \beta) + (\alpha + \beta)(\beta + \gamma) + (\alpha + \gamma)(\beta + \gamma) - \alpha\gamma - \alpha\beta - \beta\gamma]$$ $$= 4(\alpha^2 + \beta^2 + \gamma^2 + 2\alpha\beta + 2\alpha\gamma + 2\beta\gamma) = 4(\alpha + \beta + \gamma)^2$$ Hence, proven that $$L. H. S. = R. H. S.$$ 31. Let $$\alpha$$ be the common root then we have $$\alpha^2 + c\alpha + ab = 0$$ and $$\alpha^2 + b\alpha + ca = 0$$ By cross-multiplication, we get the solution as $$\frac{\alpha^2}{ac^2 - ab^2} = \frac{\alpha}{ab - ac} = \frac{1}{b - c}$$ From first two we have $$\alpha = \frac{ac^2 - ab^2}{ab - ac} = -(b + c)$$ From last two we have $$\alpha = a$$ Equating these two we get $$a = -(b + c) \Rightarrow a + b + c = 0$$ Let the other root of the equations be $$\beta$$ and $$\beta1$$ then we have $$\alpha\beta = ab$$ and $$\alpha\beta1 = ca$$ $$\therefore \beta = b$$ and $$\beta1 = c$$ Equation whose roots are $$\beta$$ and $$\beta1$$ is $$x^2 - (\beta + \beta1)x + \beta\beta1 = 0 \Rightarrow x^2 -(b + c) + bc = 0 \Rightarrow x^2 + ax + c = 0$$ 32. Clearly, root of the equation $$x^2 + 2x + 9 = 0$$ are imaginary and since they appear in pairs both the roots will be common and thus the ratio of the coefficients of the terms will be equal. $$a : b: c = 1 : 2 : 9$$ 33. Since both the equations have only one common root so the roots must be rational as irrational and complex roots appear in pairs. Thus, the roots of these two equations must be rational and therefore the discriminants must be perfect squares. Therefore, $$b^2 - ac$$ and $$b_1^2 - a_1c_2$$ must be perfect squares. 34. Let $$\alpha$$ be a common root. Then, we have $$3\alpha^2 -2\alpha + p = 0$$ and $$6\alpha^2 - 17\alpha + 12 = 0$$ Solving by cross-multiplication $$\frac{\alpha^2}{-24 + 17p} = \frac{\alpha}{6p - 36} = \frac{1}{-39}$$ From first two we have $$\alpha = \frac{17p - 24}{6p - 36}$$ and from last two we have $$\alpha = \frac{6p - 36}{-39} = -\frac{2p - 12}{13}$$ Equating these two and solving for $$p$$ we get $$p = -\frac{15}{4}, -\frac{8}{3}.$$ 35. When $$x = 0, \|x\|^2 - \|x\| - 2 = \|0\|^2 - \|0\| - 2 = -2 \ne 0$$ Since it is not satisfied by $$x = 0$$ it is an equation. 36. When $$x = -a$$ the equation is satisfied. Similarly, it is satisfied by values of $$x$$ being $$-b$$ and $$-c$$. The highest power of $$x$$ occurring is $$2$$ and is true for three distinct values of $$x$$ therefore it cannot be equation but an identity. 37. Equating the coefficients for similar powers of $$x$$ we get Coefficient of $$x^2:$$ $$a^2 - 1 = 0 \Rightarrow a = \pm1$$ Coefficient of $$x:$$ $$a - 1 = 0 \Rightarrow a = 1$$ Constant term: $$a^2 - 4a + 3 = 0 \Rightarrow a = 1, 3$$ The common value of $$a$$ is 1 which will make this an identity. 38. Given, $$\left(x + \frac{1}{c}\right)^2 = 4 + \frac{3}{2}\left(x - \frac{1}{x}\right)$$ $$\left(x + \frac{1}{x}\right)^2 - 4 - \frac{3}{2}\left(x - \frac{1}{x}\right) = 0$$ $$\left\{\left(x - \frac{1}{x}\right)^2 + 4x\frac{1}{x}\right\} - \frac{3}{2}\left(x - \frac{1}{x}\right) - 4 = 0$$ Substituting $$a = x - \frac{1}{x}$$ $$a^2 - \frac{3}{2}a = 0 \Rightarrow 2a^2 - 3a = 0 \therefore a = 0, \frac{3}{2}$$ $$x - \frac{1}{x} = 0 \Rightarrow x = \pm1$$ $$x - \frac{1}{x} - \frac{3}{2} \Rightarrow x = 2, -\frac{1}{2}$$ 39. Given equation is $$(x + 4)(x + 7)(x + 8)(x + 11) + 20 = 0$$ Rewriting the equation, $$[(x + 4)(x + 11)][(x + 7)(x + 8)] + 20 = 0$$ $$(x^2 + 15x + 44)(x^2 + 15x + 56) + 20 = 0$$ Substituting $$a = x^2 + 15x,$$ we get $$(a + 44)(a + 56) + 20 = 0$$ $$\Rightarrow a = -46, -54$$ If $$a = -46 \Rightarrow x^2 + 15x + 46 = 0 \Rightarrow x = \frac{-15 \pm \sqrt{41}}{2}$$ If $$a = -54 \Rightarrow x^2 + 15x + 54 = 0 \Rightarrow x = - 6, -9$$ 40. Given equation is $$3^{2x + 1} + 3^2 = 3^{x + 3} + 3^x$$ Let $$3^x = a,$$ then we have $$3a^2 + 9 = 28a \Rightarrow 3a^2 - 28a + 9 = 0$$ $$a = \frac{1}{3}, 9$$ If $$a = \frac{1}{3} \Rightarrow x = -1$$ If $$a = 9 \Rightarrow x = 2$$ 41. Clearly, $$(5 + 2\sqrt{6})^{x^2 - 3}(5 - 2\sqrt{6})^{x^2 - 3} = 1$$ Let $$(5 + 2\sqrt{6})^{x^2 - 3} = 1$$ then $$(5 - 2\sqrt{6})^{x^2 - 3} = \frac{1}{y}$$ The given equation becomes $$y + \frac{1}{y} = 10$$ where $$y = (5 + 2\sqrt{6})^{x^2 - 3}$$ $$\Rightarrow y^2 -10y + 1 = 0$$ Solving the equation we have roots as $$y = 5 \pm 2\sqrt{6}$$ $$\therefore x^2 - 3 = \pm 1$$ $$x = \pm2, \pm\sqrt{2}$$ 42. Let the speed of the bus $$= x$$ km/hour $$\therefore$$ the speed of car $$= x + 25$$ km/hour. Time taken by bus $$= \frac{500}{x}$$ hours and by car $$= \frac{500}{x + 25}$$ hours Given, $$\frac{500}{x} = \frac{500}{x + 25} + 10$$ $$\Rightarrow x^2 - 25x + 1250 = 0$$ $$x = -50, 25$$ but $$x$$ cannot be negative as it is a scalar quantity. Thus, speed of car = $$50$$ km/hour. 43. Given equation is $$(a + b)^2x^2 - 2(a^2 - b^2)x + (a - b)^2 = 0$$ Discriminant $$= 4(a^2 - b^2)^2 - 4(a + b)^2(a - b)^2 = 0$$ Since discriminant is zero, roots are equal. 44. Given equation is $$3x^2 + 7x + 8 = 0$$ Discriminant $$D = 49 - 96 < 0$$ Since it is negative roots will be complex and conjugate pair. 45. Given equation is $$3x^2 + (7 + a) + 8 - a = 0$$ Discriminant $$D = (7 + a)^2 + 12a$$ For roots to be equal it has to be zero. $$\Rightarrow a^2 + 26a + 49 = 0$$ $$\Rightarrow a = 13 \pm 6\sqrt{6}$$ 46. It is given that roots are equal i.e. discriminant is zero. $$\Rightarrow 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = 0$$ $$a^2c^2 + b^2d^2 - 2abcd - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2 = 0$$ $$(ad - bc)^2 = 0$$ $$ad = bc \Rightarrow \frac{a}{b} = \frac{c}{d}$$ 47. Discriminant is $$4(c - a)^2 - 4(b - c)(a - b)$$ $$= c^2 + a^2 -2ac - ab + b^2 + ac - bc$$ $$= a^2 + b^2 + c^2 - ab - bc - ac$$ $$= \frac{1}{2}[(a - b)^2(b - c)^2(c - a)^2]$$ Clearly the above expression is either greater than zero or equal to zero. Hence, roots are real. 48. Given equation is $$x^2 - x + x^2 - (a + 1)x + a + x^2 - ax = 0$$ $$3x^2 - 2(a + 1) + a = 0$$ Discriminant $$D = 4(a + 1)^2 - 12a$$ $$= a^2 + 2a + 1 - 3a = a^2 - a + 1 = (a - 1)^2 + a$$ which is greater than zero for all $$a$$ and hence roots are real. 49. Discriminant of the equation $$D = b^2 - 4ac$$ Given, $$a + b + c = 0 \Rightarrow b = -(a + c)$$ Substituting value of $$b$$ $$D = (a + c)^2 - 4ac = (a - c)^2$$ which is either zero or positive. Hence, roots are rational. 50. This has been left as an exercise to the reader.	2022-10-02 08:48:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9600614905357361, "perplexity": 324.08071165563683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337307.81/warc/CC-MAIN-20221002083954-20221002113954-00620.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=123&t=39544&p=134314	## Units $PV=nRT$ Tinisha 1G Posts: 69 Joined: Fri Sep 28, 2018 12:17 am ### Units When we are calculating using the ideal gas law, should we put the temperature in Kelvins or in Celsius? Samantha Man 1L Posts: 63 Joined: Fri Sep 28, 2018 12:22 am ### Re: Units When you're using PV=nRT, the temperature should always be in Kelvin units not Celsius. megan blatt 2B Posts: 61 Joined: Fri Sep 28, 2018 12:28 am ### Re: Units Yes, you would put temperature in Kelvin. Usually when doing calculations, Kelvin is the unit that should be used. Margaret Akey Posts: 80 Joined: Fri Sep 28, 2018 12:18 am ### Re: Units always kelvin! degrees celcius + 273.15 = kelvin 005199302 Posts: 108 Joined: Fri Sep 28, 2018 12:15 am ### Re: Units We use Kelvin because as Dr. Lavelle mentioned in the beginning of 14A, Kelvin is an absolute scale (0 K is absolute zero). Nicole Lee 4E Posts: 60 Joined: Fri Sep 28, 2018 12:16 am Been upvoted: 1 time ### Re: Units The R constant's units are in Kelvin, so Kelvin should be used for temperature. Kyither Min 2K Posts: 60 Joined: Wed Oct 03, 2018 12:15 am ### Re: Units The units should be in Kelvin as the universal gas constant, R is in Kelvin. To find convert from Celsius to Kelvin just add 273.15. Hannah Yates 1K Posts: 59 Joined: Fri Sep 28, 2018 12:27 am ### Re: Units Use Kelvin. In most equations with temperature, you always use Kelvin LedaKnowles2E Posts: 62 Joined: Fri Sep 28, 2018 12:27 am ### Re: Units Since R is in Kelvin, use Kelvin. You almost always use Kelvin in chemistry. LeannaPhan14BDis1D Posts: 57 Joined: Fri Sep 28, 2018 12:16 am ### Re: Units Lavelle said due to the absolute 0 K we use Kelvin like the other people said Mhun-Jeong Isaac Lee 1B Posts: 54 Joined: Fri Sep 28, 2018 12:17 am ### Re: Units Yeah use K. Whenever you are confused as to which unit to use, look at the constant values in the equation and use the units that will cancel out with them. In this case of PV = nRT, R uses K. Lopez_Melissa-Dis4E Posts: 66 Joined: Fri Sep 28, 2018 12:20 am ### Re: Units Kelvin should be used when calculating these type of problems due to the fact that 0 Kelvin is absolute 0. riddhiduggal Posts: 30 Joined: Fri Sep 28, 2018 12:21 am ### Re: Units You should always convert temperatures to Kelvin in PV=nRT Posts: 35 Joined: Fri Sep 28, 2018 12:19 am ### Re: Units Yes, as people said before you should always use Kelvin. It is also important to note that you must match pressure units for whatever R constant you are using or else your answer will be incorrect. You can also convert to whatever units the question asks before plugging in your values to PV=nRT. Mariana Fuentes 1L Posts: 43 Joined: Wed Nov 15, 2017 3:00 am ### Re: Units You use Kelvin since R is in terms of Kelvin. If you were given a different temperature unit, you would have to convert it to Kelvin. Example, 52 degrees Celsius to K. C+273.15=K 52+273.15=300.15K Arta Kasaeian 2C Posts: 30 Joined: Mon Jan 07, 2019 8:22 am ### Re: Units The temperature is always plugged in as Kelvin. If the question concern the change in temperature in celsius, you calculate using kelvins and convert to celsius in the last step. PranitKumaran1F Posts: 30 Joined: Thu Nov 08, 2018 12:17 am ### Re: Units The units should be in kelvin using the ideal gas law equation 404982241 Posts: 47 Joined: Fri Sep 28, 2018 12:17 am ### Re: Units always use kelvins. add 273.15 to Celsius to get kelvins. remember to make sure that the temperature remains the same during the reaction/s. Brian Chang 2H Posts: 65 Joined: Fri Sep 28, 2018 12:17 am ### Re: Units Use Kelvin. The Gas Constant is in Kelvin so we have to use the same measurement for temperature. Millicent Navarro 1I Posts: 61 Joined: Fri Sep 28, 2018 12:25 am ### Re: Units Always use Kelvin! If a problem states a temperature in Celsius (this is very common) or Fahrenheit, convert it to Kelvin. ### Who is online Users browsing this forum: No registered users and 1 guest	2020-11-27 21:43:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44962605834007263, "perplexity": 6922.63117178649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141194171.48/warc/CC-MAIN-20201127191451-20201127221451-00101.warc.gz"}
https://physics.stackexchange.com/questions/148724/total-and-partial-derivatives-in-thermodynamics-and-maxwell-relations	# Total and partial derivatives in thermodynamics and Maxwell relations Consider the expression $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ I'm trying to understand how to derive an expression for $\left( \frac{\partial S}{\partial V} \right)_P$ and how is it related to $\left( \frac{\partial S}{\partial V} \right)_T$. I tried the following: Method 1 i) Divide both sides by dV $$\frac{dS}{dV}=\left(\frac{\partial S}{\partial T}\right)_V\frac{dT}{dV}+\left(\frac{\partial S}{\partial V}\right)_T\frac{dV}{dV}$$ ii) and at const. P $$\left(\frac{dS}{dV}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{dT}{dV}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T\left(\frac{dV}{dV}\right)_P$$ $$\left(\frac{dS}{dV}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{dT}{dV}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T$$ Question 1: how does $$\left(\frac{dS}{dV}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{dT}{dV}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T$$ become $$\left(\frac{\partial S}{\partial V}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{\partial T}{\partial V}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T???$$ Method 2: Differentiate both side wrt V, holding P const. and use product rule $$\frac{\partial}{\partial V}\left(dS\right)_P=\frac{\partial}{\partial V}\left(\left(\frac{\partial S}{\partial T}\right)_VdT\right)_P+\frac{\partial}{\partial V}\left(\left(\frac{\partial S}{\partial V}\right)_TdV\right)_P$$ $$\left(\frac{\partial dS}{\partial V}\right)_P=\left(\left(\frac{\partial^2 S}{\partial V \partial T}\right)_V\right)_PdT+\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{\partial dT}{\partial V}\right)_P+\left(\left(\frac{\partial^2 S}{\partial V^2}\right)_T\right)_PdV+\left(\frac{\partial S}{\partial V}\right)_T\left(\frac{\partial dV}{\partial V}\right)_P$$ Question 2: I got so many extra terms, and how to deal with these $$\left(\frac{\partial \text{ d blah}_1}{\partial \text{ blah}_2}\right)_{\text{blah}_3}$$ terms? Also, on more general grounds: Question 3: How to partially differentiate a total differential rigorously? Question 4: Are partial derivatives that differ in only the kept const. term identical in general? # Question 1: $$\frac{dS}{dV}=\left(\frac{\partial S}{\partial T}\right)_V\frac{dT}{dV}+\left(\frac{\partial S}{\partial V}\right)_T\frac{dV}{dV}$$ This doesn't make much sense, because is not a well defined expression. The differential $$\tag{A} dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ is just telling you that if you impose a slight variation on $V$ keeping $T$ constant, $S$ changes accordingly by an amount we denote with $\left( \frac{\partial S}{\partial V}\right)_T$, so if you ask what is the partial derivative of $S$ with respect to $V$, the answer is tautologically $\left( \frac{\partial S}{\partial V}\right)_T$ (and of course all the reasonings are the same with $V$ and $T$ exchanged). You can safely take this (in this context) as the definition of the differential expression (A). In other words, writing (A) is exactly the same as stating that $S$ is a function of the two variables $T$ and $V$. What is then the meaning of the expression $\left( \frac{\partial S}{\partial V} \right)_P$ ? It means that you are now considering $T$ itself as a function of $P$ and $V$, call it $\tilde{T}(P,V)$, and effectively asking for the partial derivative of the function $\tilde{S}$ defined by $$\tilde{S}(P,V) \equiv S(\tilde{T}(P,V),V)$$ with respect to $V$, which is the quantity: $$\frac{\partial \tilde{S}}{\partial V} (P,V) \equiv \left( \frac{\partial \tilde{S}}{\partial V} \right)_P \equiv \lim_{\epsilon \to 0} \frac{\tilde{S}(P,V+\epsilon) - \tilde{S}(P,V)}{\epsilon}$$ using the usual chain rule you obtain $$\frac{\partial \tilde{S}}{\partial V} (P,V) = \frac{\partial S}{\partial T}(P,V) \frac{\partial T}{\partial V}(P,V) + \frac{\partial S}{\partial V}(P,V)$$ and this is what is meant by the less rigorous, shorter expression $$\left( \frac{\partial S}{\partial V} \right)_P = \left( \frac{\partial S}{\partial T} \right)_V \left(\frac{\partial T}{\partial V} \right)_P + \left( \frac{\partial S}{\partial V} \right)_T$$ how does $$\left(\frac{dS}{dV}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{dT}{dV}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T$$ become $$\left(\frac{\partial S}{\partial V}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V\left(\frac{\partial T}{\partial V}\right)_P+\left(\frac{\partial S}{\partial V}\right)_T???$$ They are the same thing. You have a function $S$ of the two variables $T$ and $V$. What you can do is just differentiating with respect to one or the other, obtaining: $$\frac{\partial S}{\partial T} (T,V) \qquad \text{and} \qquad \frac{\partial S}{\partial V} (T,V)$$ which means (taking the first one for example) the partial derivative of $S$ with respect to $T$, evaluated at the point $T,V$, i.e. the object defined by $$\tag{B} \frac{\partial S}{\partial T} (T,V) \equiv \lim_{\epsilon \to 0} \frac{S(T+\epsilon,V) - S(T,V)}{\epsilon}$$ Writing this with $d$ instead of $\partial$, in this context, is just notation: $$\frac{\partial S}{\partial T} (T,V) \equiv \left( \frac{\partial{S}}{\partial T} \right)_V \equiv \frac{d S}{d T} (T,V)$$ the notation with the $(\cdot)_V$ is useful to remark which variable/variables is/are kept constant while deriving (as you can see in (B), where $V$ is not varied). Also related: # Question 2: $$\left(\frac{\partial dS}{\partial V}\right)_P$$ Please, do not ever write something like this :). A partial derivative is an operation that you can apply to (multi-variable) functions. A differential is not a (multi-variable) function, and its partial derivatives are not defined. $dS$ means "a little variation of the variable $S$", which can be caused by a corresponding variation of the parameters on which it depends. If you ask what is the variation of $S$ while keeping some other quantity constant, you just divide $dS$ by that quantity (say $dV$) and impose the constraint you want (which is the first method you mentioned). Or for a more rigorous (and more clumsy) approach, you do the partial derivatives of the $\tilde{S}$ defined above. The result is the same. # Question 4: Are partial derivatives that differ in only the kept const. term identical in general? No they are not. Consider the following example: let $F$ be a function of the two variables $A$ and $B$, and suppose that $B$ is also a function of other variables, say $A$ and $C$: $$F = F(A,B), \qquad B = B(A,C)$$ Then we have $$\left( \frac{\partial F}{\partial A} \right)_B \equiv \lim_{\epsilon \to 0} \frac{F(A+\epsilon,B) - F(A,B) }{\epsilon}$$ but with $\left( \frac{\partial F}{\partial A} \right)_C$ we also have to consider that the second argument $B$ of $F$ changes with $A$, then: $$\left( \frac{\partial F}{\partial A} \right)_C = \left( \frac{\partial F}{\partial A} \right)_B + \left( \frac{\partial F}{\partial B} \right)_A \left( \frac{\partial B}{\partial A} \right)_C$$ The point is that we are now actually deriving another function, call it $\tilde{F}$, defined by $$\tilde{F}(A,C) \equiv F(A,B(A,C))$$ Note that the $B$ in $F(A,B)$ and the $B$ in the above definition are very different objects: the former is a number (an independent variable), the latter is a function of the two variables $A$ and $C$. # Footnote: Note that in all of these reasonings you are always dealing with partial derivatives of functions of many variables (for example the $S(T,V)$ above). What makes it confusing (and I remember being confused myself by this when dealing for the first time with this subject) is the fact that you implicitly, when needed, consider some variables as functions themselves of other variables (just like $T$ that becomes $\tilde{T}(P,V)$ above). This feels pretty natural when you understand what is the exact meaning of the expressions, but can also be confusing at first. • Re question 1: What about $\left(\frac{\partial T}{\partial V}\right)_P$ how do we set P constant given that we have a function S(T,V) and P is not one of the variables here Even if P can be related by using $\frac{\partial U}{\partial V}_S=-P$ you still have U floating around. Basically letting a variable that is not part of the indepdent variables constant (e.g. P) is the bit the confuses me the most in all thermodynamic equation derivations. What actually is happening when such variables were kept constant? Is the full form of $\left(\frac{\partial T}{\partial V}\right)_P$ looks like – Secret Nov 25 '14 at 9:40 • $\frac{\partial T\left(V, \frac{\partial U\left(S,V\right)}{\partial S}\right)}{\partial V}=\frac{\partial T\left(V, \frac{\partial U\left(S,V\right)}{\partial S}\right)}{\partial V}+\frac{\partial T\left(V, \frac{\partial U\left(S,V\right)}{\partial S}\right)}{\partial \left(\frac{\partial U\left(S,V\right)}{\partial S}\right)}\frac{\partial \left( \frac{\partial U\left(S,V\right)}{\partial S}\right)}{\partial V}$ by using the answer of question 3 – Secret Nov 25 '14 at 9:58 • which because $-P=\frac{\partial U\left(S,V\right)}{\partial V}$ is kept constant, thus $\frac{\partial \left( \frac{\partial U\left(S,V\right)}{\partial V}\right)}{\partial V}=\frac{\partial^2 U\left(S,V\right)}{\partial V \partial V}=0$ thus the seocnd term vanishes and we are left with only the first term, regardless of what happened to $\frac{\partial T\left(V, \frac{\partial U\left(S,V\right)}{\partial V}\right)}{\partial \left(\frac{\partial U\left(S,V\right)}{\partial V}\right)}$ ? sorry typo, all $\partial S$ in the previous comment should be $\partial V$ – Secret Nov 25 '14 at 10:15 • Trying to use the chain rule to relate $\left(\frac{\partial V}{\partial T}\right)_S$ and $\left(\frac{\partial V}{\partial T}\right)_S$ gives $$\left(\frac{\partial V}{\partial T}\right)_S=\left(\frac{\partial V}{\partial T}\right)_P\left(\frac{\partial T}{\partial T}\right)_S+\left(\frac{\partial V}{\partial P}\right)_T\left(\frac{\partial P}{\partial T}\right)_S$$ While $$\left(\frac{\partial T}{\partial T}\right)_S=1$$ but how to make the remaining terms $$\left(\frac{\partial V}{\partial P}\right)_T\left(\frac{\partial P}{\partial T}\right)_S$$ vanish? – Secret Nov 25 '14 at 13:19 • @Secret I edited the post adding more explanation. Tell me if something is not clear – glS Nov 25 '14 at 14:53	2021-06-24 12:30:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247989654541016, "perplexity": 172.40599036504182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488553635.87/warc/CC-MAIN-20210624110458-20210624140458-00299.warc.gz"}
https://aif.centre-mersenne.org/	# ANNALES DE L'INSTITUT FOURIER The Annales de l’Institut Fourier aim at publishing original papers of a high level in all fields of mathematics, either in English or in French. The electronic edition is fully open access and free of author charges. #### New articles Peternell, Thomas Guedj, Vincent; Lu, Chinh H.; Zeriahi, Ahmed View More	2019-05-24 06:06:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4751420021057129, "perplexity": 5044.512631716636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257514.68/warc/CC-MAIN-20190524044320-20190524070320-00062.warc.gz"}
https://ysharifi.wordpress.com/tag/double-centralizer-theorem/	## Posts Tagged ‘double centralizer theorem’ Introduction. We first need to recall a few facts from basic field theory. For the definition of separability see the introduction section in this post. We have the following facts. Fact 1. Let $E/F$ be an algebraic field extension and $a \in E.$ Then $F(a)/F$ is separable if and only if $a$ is separable over $F.$ Fact 2. Let $F \subseteq E \subseteq L$ be a chain of fields and suppose that $L/F$ is algebraic. If both $E/F$ and $L/E$ are separable, then $L/F$ is separable. Fact 3. If $E/F$ is a separable field extension and $[E:F] < \infty,$ then $E=F(a)$ for some $a \in E.$ Theorem. Let $D$ be a division algebra with the center $k$ and suppose that $\dim_k D < \infty.$ There exists $a \in D$ such that $K=k(a)$ is a maximal subfield of $D$ and $K/k$ is separable. Proof. Let $A$ be the set of all subfields of $D$ which are separable extensions of $k.$ This set is non-empty because $k \in A.$ Since $\dim_k D < \infty,$ the set $A$ with $\subseteq$ has a maximal element, say $K.$ Let $C(K)$ be the centralizer of $K$ in $D.$ Suppose that $C(K) \neq K.$ Then $C(K)$ is a noncommutative division ring and $K \subset C(K).$ Let $Z(C(K))$ be the center of $K.$ Then, since $K$ is commutative, we have $Z(C(K))=C(C(K))=K,$ by the double centralizer theorem. Clearly $C(K)$ is algebraic over $K$ because $[C(K):K] < \infty.$ Hence, by the Jacobson-Noether theorem, there exists $a \in C(K) \setminus K$ such that $a$ is separable over $K.$ So we have the chain of fields $k \subset K \subset K(a)$ where both $K/k$ and $K(a)/K$ are separable (see Fact 1). Thus, by Fact 2, $K(a)/k$ is separable and so $K(a) \in A.$ But this contradicts the maximality of $K$ in $A.$ This contradiction implies $C(K)=K$ and so, by Corollary 3, $K$ is a maximal subfield of $D.$ Finally, by Fact 3, $K=k(a)$ for some $a \in K. \ \Box$ ## The double centralizer theorem (2) Posted: January 31, 2011 in Noncommutative Ring Theory Notes, Simple Rings Tags: , , As in part (1), $k$ is a field, $A$ is a finite dimensional central simple $k$-algebra and $B$ is a simple $k$-subalgebra of $A.$ We will also be using notations and the result in the lemma in part (1), i.e. $R = A \otimes_k B^{op},$ $M$ is the unique simple $R$-module, $A \cong M^n,$ $D = \text{End}_R(M)$ and $C_A(B) \cong M_n(D).$ Finally, as usual, we will denote the center of any algebra $S$ by $Z(S).$ We now prove a nice relationship between dimensions. Lemma. $\dim_k C_A(B) \cdot \dim_k B = \dim_k A.$ Proof. We have $R \cong M_m(D)$ and $M \cong D^m,$ for some integer $m.$ Thus $A \cong D^{mn},$ as $k$-modules, and hence $\dim_k A = mn \dim_k D. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ We also have $\dim_k A \cdot \dim_k B = \dim_k R = \dim_k M_m(D)=m^2 \dim_k D \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$ and $\dim_k C_A(B)=\dim_k M_n(D)=n^2 \dim_k D. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$ Eliminating $\dim_k D$ in these three identities will give us the result. $\Box$ Theorem. If $B$ is a central simple $k$-algebra, then $C_A(B)$ is a central simple $k$-algebra too and we have $A =BC_A(B) \cong B \otimes_k C_A(B).$ Proof. By the lemma in part (1), $C_A(B)$ is simple and thus, since $B$ is central simple, $B \otimes_k C_A(B)$ is a simple algebra. Thus the map $\phi : B \otimes_k C_A(B) \longrightarrow BC_A(B)$ defined by $\phi(b \otimes_k c)=bc$ is a $k$-algebra isomorphism. Hence $B \otimes_k C_A(B) \cong BC_A(B)$ and so $\dim_k BC_A(B)=\dim_k B \otimes_k C_A(B)=\dim_k A,$ by the above lemma. Therefore $BC_A(B)=A.$ Clearly if $c$ is in the center of $C_A(B),$ then $c$ commutes with every element of both $B$ and $C_A(B).$ Hence $c$ commutes with every element of $A$ and thus $c \in k.$ So $Z(C_A(B))=k,$ i.e. $C_A(B)$ is a central simple $k$-algebra. $\Box$ The Double Centralizer Theorem. $C_A(C_A(B))=B.$ Proof. Obviously $B \subseteq C_A(C_A(B)).$ So, to prove that $B=C_A(C_A(B)),$ we only need to show that $\dim_k B = \dim_k C_A(C_A(B)).$ Applying the above lemma to $C_A(B)$ gives us $\dim_k C_A(C_A(B)) \cdot \dim_k C_A(B) = \dim_k A. \ \ \ \ \ \ \ \ \ \ \ \ \ ()$ Plugging $\dim_k C_A(B)=\frac{\dim_k A}{\dim_k B},$ which is true by the above lemma, into $()$ finishes the proof. $\Box$ Corollary. If $B$ is a subfield of $A,$ then $\deg A = (\deg C_A(B))(\dim_k B).$ Proof. We first need to notice a couple of things. First, since $B$ is commutative, $B \subseteq C_A(B).$ Since $B$ is commutative, $C_A(C_A(B))=Z(C_A(B)).$ Thus, by the above theorem, $Z(C_A(B))=B$ and so $C_A(B)$ is a central simple $B$-algebra. Now, by the lemma $(\deg A)^2=\dim_k A = (\dim_B C_A(B))(\dim_k B)^2=(\deg C_A(B))^2 (\dim_k B)^2. \ \Box$ ## The double centralizer theorem (1) Posted: January 31, 2011 in Noncommutative Ring Theory Notes, Simple Rings Tags: , , Throughout $k$ is a field, $A$ is a finite dimensional central simple $k$-algebra and $B$ is a simple $k$-subalgebra of $A.$ We will use the notation for centralizers given in this post. The goal is to prove that $C_A(C_A(B))=B.$ This is called the double centralizer theorem for an obvious reason. We proved another double centralizer theorem in here. In there $B=k[a],$ for some $a \in A$ and $B$ did not have to be simple. So that double centralizer theorem has nothing to do with this one. We first show that the centralizer of a simple subalgebra of a finite dimensional central simple $k$-algebra is simple. Lemma. The $k$-subalgebra $C_A(B)$ is simple. Proof. Since $A$ is central simple and $B,$ and hence $B^{op},$ is simple, the algebra $R=A \otimes_k B^{op}$ is also simple by the first part of the corollary in this post. Clearly $R$ is finite dimensional over $k$ because $A$ is so. So, as we mentioned before in Remark 1, $R$ has a unique simple $R$-module $M$ and any $R$-module is isomorphic to the direct sum of a finite number of copies of $M.$ Thus, since $A$ has a structure of an $R$-module, we must have $A \cong M^n, \ \ \ \ \ \ \ \ \ \ (1)$ for some integer $n.$ Let $D = \text{End}_R(M).$ Since $M$ is a simple $R$-module, $D$ is a division ring by Schur’s lemma. On the other hand, as we proved in this post, $C_A(B) \cong \text{End}_R(A). \ \ \ \ \ \ \ \ \ \ (2)$ Now, (1), (2) and the remark in this post gives us $C_A(B) \cong \text{End}_R(A) \cong \text{End}_R(M^n) \cong M_n(\text{End}_R(M)) \cong M_n(D). \ \Box$ To be continued in part (2).	2018-07-23 02:13:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 163, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9544422626495361, "perplexity": 82.72625683206304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676594790.48/warc/CC-MAIN-20180723012644-20180723032644-00096.warc.gz"}
http://accessmedicine.mhmedical.com/content.aspx?bookid=348&sectionid=40381715	Chapter 231 Hereditary hemolytic anemias result primarily from a defect in hemoglobin (Hb) production, red blood cell (RBC) metabolism, or the structure of the RBC membrane. Hemolysis, primarily in the spleen, is a normal process whereby abnormal, damaged, and aged RBCs are removed from the circulation. If an increased number of abnormal RBC are produced, hemolytic activity is enhanced, and, depending on the ability rate of production, the concentration of circulating RBC may decrease, manifesting as anemia. Inherited Hb disorders are classified into two main groups: disorders with abnormal Hb structure [e.g., sickle cell disease (SCD)] and disorders of abnormal Hb production (e.g., the thalassemias). These disorders are widely prevalent, and it is thought that 7% of the world’s population are carriers of an abnormal Hb gene.1 Most of these inherited Hb disorders are the result of point mutation in the genes that code for the globin chains of the Hb molecule, along with other variations, such as deletions, insertions, extended chains, and fusions. These genetic abnormalities result in Hb that tends to gel or crystallize, possesses abnormal oxygen-binding properties, or is readily oxidized to methemoglobin, rendering the RBC susceptible to hemolysis. Disorders of RBC metabolism or cell membrane function also render the cell more sensitive to hemolysis. ### Epidemiology SCD is a significant public health problem. Sixty million carriers of sickle cell and 1.2 million sickle cell homozygotes are added every year worldwide.2 An estimated 250 million people (approximately 4.5% of the world population) are carriers of the sickle cell gene.3 Sickle cell disorders account for approximately 70% of congenital Hb disorders seen worldwide.3 SCD affects predominantly people of African Equatorial descent, although it is also found in persons of Mediterranean, Indian, and Middle Eastern origin.4 SCD affects approximately 70,000 people in the U.S., and approximately 2 million Americans have the sickle cell trait.5 During the last few decades, the overall life expectancy of patients with SCD has improved from 14 to >50 years.6 This can be attributed to early diagnosis (antenatal and neonatal screening), parental education about complications, close monitoring in clinics and follow-up, prophylactic penicillins to prevent pneumococcal septicemia, and increased usage of drugs such as hydroxyurea. ### Pathophysiology The normal adult RBC contains three forms of Hb: HbA, HbA2, and fetal Hb (HbF) (Table 231-1). All normal Hb consists of a tetramer of four polypeptide chains, which are pairs of dissimilar chains (two α-globin chains and two non–α-globin chains). HbA accounts for approximately 96% to 98% of adult Hb and consists of two α- and two β-globin chains. HbA2 accounts for approximately 2.0% to 3.5% of adult Hb and is composed of two α- and two δ-globin chains. HbF is composed of two α- and two γ-globin chains. HbF production peaks in utero and starts declining just before birth and continues to decline, reaching ... Sign in to your MyAccess Account while you are actively authenticated on this website via your institution (you will be able to tell by looking in the top right corner of any page – if you see your institution’s name, you are authenticated). You will then be able to access your institute’s content/subscription for 90 days from any location, after which you must repeat this process for continued access. Ok ## Subscription Options ### AccessMedicine Full Site: One-Year Subscription Connect to the full suite of AccessMedicine content and resources including more than 250 examination and procedural videos, patient safety modules, an extensive drug database, Q&A, Case Files, and more.	2014-12-22 23:25:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3220502734184265, "perplexity": 8755.840420945604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802777295.134/warc/CC-MAIN-20141217075257-00021-ip-10-231-17-201.ec2.internal.warc.gz"}
https://iacr.org/cryptodb/data/paper.php?pubkey=30091	CryptoDB Paper: Improved Security Evaluation of SPN Block Ciphers and its Applications in the Single-key Attack on SKINNY Authors: Wenying Zhang , School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China; Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Meichun Cao , School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China Jian Guo , Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Enes Pasalic , FAMNIT, University of Primorska, Koper, Slovenia DOI: 10.13154/tosc.v2019.i4.171-191 URL: https://tosc.iacr.org/index.php/ToSC/article/view/8461 Search ePrint Search Google In this paper, a new method for evaluating the integral property, truncated and impossible differentials for substitution-permutation network (SPN) block ciphers is proposed. The main assumption is an explicit description/expression of the internal state words in terms of the plaintext (ciphertext) words. By counting the number of times these words occur in the internal state expression, we can evaluate the resistance of a given block cipher to integral and impossible/truncated differential attacks more accurately than previous methods. More precisely, we explore the cryptographic consequences of uneven frequency of occurrences of plaintext (ciphertext) words appearing in the algebraic expression of the internal state words. This approach gives a new family of distinguishers employing different concepts such as the integral property, impossible/truncated differentials and the so-called zero-sum property. We then provide algorithms to determine the maximum number of rounds of such new types of distinguishers for SPN block ciphers. The potential and efficiency of this relatively simple method is confirmed through applications. For instance, in the case of SKINNY block cipher, several 10-round integral distinguishers, all of the 11-round impossible differentials, and a 7-round truncated differential could be determined. For the last case, using a single pair of plaintexts differing in three words so that (a = b = c) ≠ (a’ = b’ = c’), we are able to distinguish 7-round SKINNY from random permutations. More importantly, exploiting our distinguishers, we give the first practical attack on 11-round SKINNY-128-128 in the single-key setting (a theoretical attack reaches 16 rounds). Finally, using the same ideas, we provide a concise explanation on the existing distinguishers for round-reduced AES. BibTeX @article{tosc-2020-30091, title={Improved Security Evaluation of SPN Block Ciphers and its Applications in the Single-key Attack on SKINNY}, journal={IACR Transactions on Symmetric Cryptology}, publisher={Ruhr-Universität Bochum}, volume={2019, Issue 4}, pages={171-191}, url={https://tosc.iacr.org/index.php/ToSC/article/view/8461}, doi={10.13154/tosc.v2019.i4.171-191}, author={Wenying Zhang and Meichun Cao and Jian Guo and Enes Pasalic}, year=2020 }	2021-11-28 09:15:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47590693831443787, "perplexity": 1767.996356143482}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358480.10/warc/CC-MAIN-20211128073830-20211128103830-00464.warc.gz"}
https://undergroundmathematics.org/product-rule/r8134/suggestion	Review question # How could we integrate $e^{-x}\sin^n x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Ref: R8134 ## Suggestion 1. Show that $\dfrac{d}{dx} (\cos x \sin^{n-1}x) = (n-1) \sin^{n-2}x - n \sin^{n}x$. Would it help us to use $s$ for $\sin x$ and $c$ for $\cos x$? How can we differentiate a product? How could we convert any $\cos x$ terms into $\sin x$ terms? 1. Given that $I_n = \displaystyle\int_0^{\frac{1}{2}\pi} e^{-x} \sin^n x \, dx$, show that $I_n = -e^{-\frac{1}{2}\pi} + n \int_0^{\frac{1}{2}\pi} e^{-x} \cos x \sin^{n-1}x \, dx, \quad (n \ge 1).$ Do we know a ‘product rule’ for integration? 1. By using the results of (i) and (ii), or otherwise, show that $(n^2 + 1) I_n = -e^{-\frac{1}{2}\pi} + n(n-1) I_{n-2}, \quad (n \ge 2).$ Could we use our ‘product rule’ for integration a second time? 1. Show that $I_4 = \frac{1}{85} (24 - 41 e^{-\frac{1}{2}\pi})$. How would the earlier parts help us here?	2018-01-20 07:20:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3085837662220001, "perplexity": 1050.0697446505346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889473.61/warc/CC-MAIN-20180120063253-20180120083253-00266.warc.gz"}
http://fakeroot.net/error-function/complex-error-function-properties.php	Home > Error Function > Complex Error Function Properties # Complex Error Function Properties ## Contents Call native code from C/C++ Help! What are these holes called? J. (March 1993), "Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers" (PDF), ACM Trans. Haskell: An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types. http://fakeroot.net/error-function/complementary-error-function-properties.php doi:10.1090/S0025-5718-1969-0247736-4. ^ Error Function and Fresnel Integrals, SciPy v0.13.0 Reference Guide. ^ R Development Core Team (25 February 2011), R: The Normal Distribution Further reading Abramowitz, Milton; Stegun, Irene Ann, eds. And yes, it's me who published libcerf, sorry and thanks for informing me about the disclosure rule. –Joachim Wuttke May 16 '13 at 6:14 add a comment\| Your Answer draft To do this, we take a detour through some Fourier theory. The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ⁡ ( x ) = 2 x https://en.wikipedia.org/wiki/Error_function ## Complex Error Function Matlab Wolfram Education Portal» Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. This usage is similar to the Q-function, which in fact can be written in terms of the error function. Consider a function $\phi(t)$ that has a Fourier transform $$\Phi(\xi) = \int_{-\infty}^{\infty} dt \, \phi(t) \, e^{-i 2 \pi \xi t}$$ We begin with a form of the Poisson sum formula: Math. H. C++: C++11 provides erf() and erfc() in the header cmath. New York: Random House, 1963. Error Function Values ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF). calculus integration complex-analysis contour-integration share\|cite\|improve this question edited Mar 14 '14 at 22:49 Ron Gordon 109k12130221 asked Mar 14 '14 at 19:04 Sleepyhead 1385 add a comment\| 3 Answers 3 active Error Function Of Complex Argument Out[68]= 6.1232310^-22 - 0.00001 I In[69]:= Sqrt[Pi] E^-x^2 Erfc[I x] /. M.; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). http://www.ams.org/mcom/1973-27-122/S0025-5718-1973-0326991-7/S0025-5718-1973-0326991-7.pdf For previous versions or for complex arguments, SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special.[21] A complex-argument erf is also in the arbitrary-precision arithmetic and Robinson, G. "The Error Function." §92 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. Integral Of Error Function If L is sufficiently far from the mean, i.e. μ − L ≥ σ ln ⁡ k {\displaystyle \mu -L\geq \sigma {\sqrt {\ln {k}}}} , then: Pr [ X ≤ L In some ranges (or if higher than machine precision is desired) you may want to use more terms from the expansion on that imaginary part. Were there science fiction stories written during the Middle Ages? • The system returned: (22) Invalid argument The remote host or network may be down. • Are there any saltwater rivers on Earth? • Incomplete Gamma Function and Error Function", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8 Temme, Nico M. (2010), "Error Functions, Dawson's and Fresnel Integrals", • For complex double arguments, the function names cerf and cerfc are "reserved for future use"; the missing implementation is provided by the open-source project libcerf, which is based on the Faddeeva ## Error Function Of Complex Argument Thus, we may rewrite the Poisson sum formula result as follows: $$e^{u^2} [1+\epsilon(u)] = \frac{a}{\sqrt{\pi}} \left [1+2 \sum_{n=1}^{\infty} e^{-n^2 a^2} \cosh{2 n a u} \right ]$$ Now substitute this result into Practice online or make a printable study sheet. Complex Error Function Matlab Optimise Sieve of Eratosthenes Why does Ago become agit, agitis, agis, etc? [conjugate with an i*?] Best practice for map cordinate system Zero Emission Tanks Is there a way to ensure Complex Gamma Function At the real axis, erf(z) approaches unity at z→+∞ and −1 at z→−∞. IDL: provides both erf and erfc for real and complex arguments. http://fakeroot.net/error-function/complex-error-function-c.php Using the alternate value a≈0.147 reduces the maximum error to about 0.00012.[12] This approximation can also be inverted to calculate the inverse error function: erf − 1 ⁡ ( x ) Computerbasedmath.org» Join the initiative for modernizing math education. Sequences A000079/M1129, A001147/M3002, A007680/M2861, A103979, A103980 in "The On-Line Encyclopedia of Integer Sequences." Spanier, J. Q Function Properties In that case, though, you need to re-estimate the max relative error. –Ron Gordon Mar 14 '14 at 22:04 add a comment\| up vote 3 down vote Well, \text{Re}\;\text{erf}(a+ib) = After division by n!, all the En for odd n look similar (but not identical) to each other. For \|z\| < 1, we have erf ⁡ ( erf − 1 ⁡ ( z ) ) = z {\displaystyle \operatorname ζ 1 \left(\operatorname ζ 0 ^{-1}(z)\right)=z} . http://fakeroot.net/error-function/complex-error-function-gsl.php Cambridge, England: Cambridge University Press, 1990. Similarly, (8) (OEIS A103979 and A103980). Erf Function Calculator A two-argument form giving is also implemented as Erf[z0, z1]. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x)=−erf(−x). ## For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). In[10]:= w1[x_] := E^-x^2 Sqrt[\[Pi]] - 2 I DawsonF[x] w2[x_] := 2 HermiteH[-1, I x] In[15]:= AbsoluteTiming[w1 /@ Range[-5.0, 5.0, 0.001];] Out[15]= {2.3272327, Null} In[16]:= AbsoluteTiming[w2 /@ Range[-5.0, 5.0, 0.001];] Out[16]= Generated Wed, 05 Oct 2016 23:54:01 GMT by s_hv978 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection It is defined as:[1][2] erf ⁡ ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t Error Function Table Erf has the continued fraction (32) (33) (Wall 1948, p.357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson comm., May 9, 2004). I'd suggest computing them separately and not adding them. It should be noted that the ceiling on this precision is the $10^{-16}$ rough figure I derived above. get redirected here What do I do now? New York: Dover, pp.297-309, 1972. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 External links MathWorld – Erf Authority control NDL: 00562553 Retrieved from	2018-02-19 06:12:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.78546541929245, "perplexity": 4893.85381322188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812405.3/warc/CC-MAIN-20180219052241-20180219072241-00246.warc.gz"}
http://jprm.sms.edu.pk/page/8/	# Journal of Prime Research in Mathematics Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing four issues in March, June, September and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year. Latest Published Articles ### g-noncommuting graph of some finite groups JPRM-Vol. 1 (2016), Issue 1, pp. 16 – 23 Open Access Full-Text PDF M. Nasiri, A. Erfanian, M. Ganjali, A. Jafarzadeh Abstract: Let $$G$$ be a finite non-abelian group and $$g$$ a fixed element of $$G$$. In 2014, Tolue et al. introduced the g-noncommuting graph of $$G$$, which was denoted by $$Γ^{g}_G$$ with vertex set $$G$$ and two distinct vertices $$x$$ and $$y$$ join by an edge if $$[x, y] \neq g$$ and $$g^{−1}$$. In this paper, we consider induced subgraph of $$Γ^{g}_{G}$$ on $$G /Z(G)$$ and survey some graph theoretical properties like connectivity, the chromatic and independence numbers of this graph associated to symmetric, alternating and dihedral groups. ### A note on self-dual AG-groupoids JPRM-Vol. 1 (2016), Issue 1, pp. 01 – 15 Open Access Full-Text PDF Abstract: In this paper, we enumerate self-dual AG-groupoids up to order 6, and classify them on the basis of commutativity and associativity. A self-dual AG-groupoid-test is introduced to check an arbitrary AG-groupoid for a self-dual AG-groupoid. We also respond to an open problem regarding cancellativity of an element in an AG-groupoid. Some features of ideals in self-dual AG-groupoids are explored. Some desired algebraic structures are constructed from the known ones subject to certain conditions and some subclasses of self-dual AG-groupoids are introduced. ### On the mixed hodge structure associated hypersurface singularities JPRM-Vol. 1 (2015), Issue 1, pp. 137 – 161 Open Access Full-Text PDF Abstract: Let $$f : \mathbb{C}^{n+1} → \mathbb{C}$$ be a germ of hypersurface with isolated singularity. One can associate to f a polarized variation of mixed Hodge structure $$H$$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $$f$$ can be extended over the degenerate point $$0$$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $$f$$ denoted $$Ω_f$$ . A mixed Hodge structure can be defined on $$Ω_f$$ in this way. The polarization on \mathcal{H} deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing. ### Existence and non existence of mean cordial labeling of certain graphs JPRM-Vol. 1 (2015), Issue 1, pp. 123 – 136 Open Access Full-Text PDF R. Ponraj, S. Sathish Narayanan Abstract: Let f be a function from the vertex set $$V (G)$$ to $${0, 1, 2}$$. For each edge $$uv$$ assign the label $$\frac{f(u)+f(v)}{2}$$. $$f$$ is called a mean cordial labeling if $$\|v_f (i) − v_f (j)\| ≤ 1$$ and $$\|e_f (i) − e_f (j)\| ≤ 1$$, $$i, j ∈ {0, 1, 2}$$, where $$v_f (x)$$ and $$e_f (x)$$ respectively denote the number of vertices and edges labeled with $$x$$ $$(x = 0, 1, 2)$$. A graph with a mean cordial labeling is called a mean cordial graph. In this paper we investigate mean cordial labeling behavior of prism, $$K_2 +\overline{K_m}$$, $$K_n + \overline{2K_2}$$, book $$B_m$$ and some snake graphs. ### $$A_19/B_6$$: A new lanczos-type algorithm and its implementation JPRM-Vol. 1 (2015), Issue 1, pp. 106 – 122 Open Access Full-Text PDF Zakir Ullah, Muhammad Farooq, Abdellah Salhi Abstract: Lanczos-type algorithms are mostly derived using recurrence relationships between formal orthogonal polynomials. Various recurrence relations between these polynomials can be used for this purpose. In this paper, we discuss recurrence relations A19 and B6 for the choice $$U_i(x) = P_{i}^{(1)}$$, where $$U_i$$ is an auxiliary family of polynomials of exact degree $$i$$. This leads to new Lanczos-type algorithm $$A_19/B_6$$ that shows superior stability when compared to existing algorithms of the same type. This new algorithm is derived and described here. Computational results obtained with it are compared to those of the most robust algorithms of this type namely $$A_12$$, (A^{new}_12\) $$A_5/B_{10}$$ and $$A_8/B_{10}$$ on the same test problems. These results are included. ### A study of the flow of non-newtonian fluid between heated parallel plates by HAM JPRM-Vol. 1 (2015), Issue 1, pp. 93 – 105 Open Access Full-Text PDF Abstract: This paper presents the heat transfer of a third grade fluid between two heated parallel plates for two models: constant viscosity model and Reynold’s model. In both cases the nonlinear energy and momentum equations have been solved by HAM. The graphs for the velocity and temperature profiles are plotted and discussed for various values of the emerging parameters in the problem. The main effect is governed by whether or not the fluid is non-Newtonian and the temperature effects are being referred to have a less dominant role.	2020-07-10 10:36:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44596171379089355, "perplexity": 653.8976062404297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655906934.51/warc/CC-MAIN-20200710082212-20200710112212-00556.warc.gz"}
https://web2.0calc.com/questions/logs-pls-help	+0 # logs pls help 0 43 2 +589 1. 4x+3=7x 2. log3(x2-22)=3 Apr 25, 2022 #1 +117121 +1 Hi Jenny, It is nice to see you back again. No.2 $$log_3(x^2-22)=3\\ 3^{log_3(x^2-22)}=3^3\\ x^2-22=27\\ x^2=49\\ x=\pm7$$ Can you please check the the first one is inputed correctly. And next time put them on a seperate posts. (1 post = 1 question) Apr 25, 2022 #1 +117121 +1 Hi Jenny, It is nice to see you back again. No.2 $$log_3(x^2-22)=3\\ 3^{log_3(x^2-22)}=3^3\\ x^2-22=27\\ x^2=49\\ x=\pm7$$ Can you please check the the first one is inputed correctly. And next time put them on a seperate posts. (1 post = 1 question) Melody Apr 25, 2022 #2 +589 +2 Thank you for your explanation, yes for several months I forgot this website's name until I remembered today lol. yes I inputed it wrong, ill put it into my next post thanks again！ xxJenny1213xx Apr 25, 2022	2022-05-22 01:53:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8709723949432373, "perplexity": 7626.768905458417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543264.49/warc/CC-MAIN-20220522001016-20220522031016-00060.warc.gz"}
https://dmtn-035.lsst.io/	# DMTN-035: Winter 2013 LSST DM Data Challenge Release Notes • Mario Juric, • Andrew Becker, • Richard Shaw, • K. Simon Krughoff and • Jeff Kantor Latest Revision: 2013-04-25 Previous Data Release from Summer 2012 [2] # 1 Image Differencing¶ A report is available here [1] # 2 SDSS Stripe 82 Reprocessing (co-adds and forced photometry)¶ LSST Data Management have recently finished building r-band SDSS Stripe 82 co-adds and performing forced photometry on individual epochs of g, r, and i-band Stripe 82 data. The primary goal of this effort was to add the capability to make background matched co-adds to the DM stack, and test it at large scale by reprocessing real survey data. The full description of the challenge is available in the Winter 2013 Data Challenge Handbook. Background matched co-adds preserve the diffuse astrophysical backgrounds in the stacked image. This increases its scientific usefulness. Furthermore, the thus constructed background in the co-add has higher S/N, making it easier to subtract it when needed. We plan to use this method to generate the co-adds in LSST production, and it was therefore important to have it built into the stack early. This will make upcoming tests of stackfit/multifit algorithms more realistic. Our preliminary analysis indicates the quality of this dataset is comparable to the quality of Stripe 82 reprocessing we performed for Summer 2012, but with significantly more area. Nevertheless, we caution you that these data were processed “on a budget”, with prototype code, and minimal quality assessment. They are a byproduct of an ongoing software development effort, and not a result of a concerted scientific investigation. The code is still incomplete both in terms of features and quality, and the same will be true of the reprocessed data. In particular, if you plan to use this data set for science, expect to have to devote time to perform additional QA, and to have to communicate with DM developers to understand the details of the dataset. If you do notice issues, or have questions, please don’t hesitate to contact us at <dm-help --at-- lsst.org>. # 3 Data Access Rules¶ The data products provided by LSST DM are intended only for members of the LSST Science Collaborations unless noted otherwise. If their use results in a publication, their users will have to to abide by the LSST Publication Policy. [4] The products are protected by a “well known” username and password: lsst / 3gigapix! . Using this combination to access the data implies you understand and accept the restrictions on their redistribution and usage. # 4 Data Locations¶ Note #1: Some of these websites require the standard DM user name and password; see the 3 Data Access Rules section for more information. Note #2: The catalog and image access tools in use here are temporary solutions for data distribution built with off-the-shelf open source components: in particular, they are NOT representative of the Science User Interface the LSST will ultimately have. Where to get data/information: If you decide to make use of this data, feel free to inquire about the details at <dm-help --at-- lsst.org>. ## 4.1 Accessing Winter 2013 Coadd FITS Files¶ The server lsst-web.ncsa.illinois.edu provides access to the data. The entire namespace can be explored by pointing your browser to: http://lsst-web.ncsa.illinois.edu/lsstdata The individual FITS files are stored in a directory hierarchy using the following scheme: http://lsst-web.ncsa.illinois.edu/lsstdata/dr-w2013/deepCoadd/[ugriz]/<tract>/<patch>/coadd-[ugriz]-<tract>-<patch>.fits where • [ugriz] is a single filter id from the indicated list (currently only r) • <tract> is the LSST skymap tract id (currently 0 or 3) • <patch> is the LSST skymap patch id (of the form xxx,yyy) For example: wget http://lsst-web.ncsa.illinois.edu/lsstdata/dr-w2013/deepCoadd/r/3/7,2/coadd-r-3-7,2.fits Individual coadd files are 40MB (uncompressed). # 5 Description of Data Products¶ We used 298 runs imaged as a part of SDSS Stripe 82 (2 million fields) to create a deep co-add approximately covering -40 deg < R.A. < 55 deg, -1.25 < Dec < 1.25 (237 deg^2^). No PSF matching was performed on the co-adds, making them deeper but less suitable for photometry. The co-adds were used to detect 14.7 million sources, most of which would otherwise fall below the faint limit of individual exposures. Photometry was performed in individual epochs, at the location of each source detected in the co-adds, resulting in 3.9 billion g, r and i band measurements (“forced photometry”). u and z bands were not processed. We also produced catalogs of averaged forced photometry, both across the duration of the whole survey (~10 years), and on a yearly basis. The co-adds are available as a series of 5126 FITS images, each spanning 2060 x 1937 pixels (see [wiki:W2013WebDataAccess] for how to access them). The catalogs are kept in a MySQL database and available through phpMyAdmin web interface, or (for power users), through the command-line mysql client. See the 4 Data Locations for instructions on how to access the database, and how to open an account if you don’t already have one. The database contains 20 tables and views; however, only a few are of general interest (listed below). We do not have at this time a detailed description of the schema of each of these tables; however, the Summer2012 schema should provide sufficient information to understand the meaning of most of these columns even though the exact names may have changed. Most frequently used tables: Table 1 AvgForcedPhot table. Column Description deepSourceId object identifier ra Right Ascension (degrees) decl Declination (degrees) nMag_[gri] number of measurements for the band magFaint_[gri] magnitude of the faintest measurement in the band medMag_[gri] magnitude of the median measurement in the band magBright_[gri] magnitude of the brightest measurement in the band q1Mag_[gri] magnitude of the first (faintest) quartile measurement in the band q3Mag_[gri] magnitude of the third (brightest) quartile measurement in the band faint5perMag_[gri] 5th percentile magnitude in the band bright5perMag_[gri] 95th percentile magnitude in the band AvgForcedPhot A table with percentiles of photometry in each band (5th, 25th, 50th (median), 75th, 95th). Columns are specified in the table above. All percentiles were calculated on the fluxes and converted back to magnitude for convenience. AvgForcedPhotYearly Same as the AvgForcedPhot table, except the percentiles are computed for each year of the survey. Therefore there are typically ~10 rows per object. Compared to AvgForcedPhot, this table has one extra column (‘’year’‘, running from 1 to 10), and no 5th and 95th percentile columns. DeepForcedSource Table with forced photometry measurements in individual epochs. Use this table if you’re interested in querying for complete light curves. DeepSource A table of sources detected on co-adds. This is in effect the master “object catalog”. Note however that because the co-adds were not PSF-matched, the photometry in this table will be relatively poor; use AvgForcedPhot table instead. RefObject A containing SDSS DR7 Stripe82 co-add [5] catalog. It’s been matched to DeepSource via RefDeepSrcMatch table. Science_Ccd_Exposure A table with metadata for all SDSS Stripe82 fields. A table with metadata for all co-add “patches” (when producing the co-add, we divided the sky into large “tracts”, and each tract has been subdivided into “patches”). The patches are stored as FITS files on the image server (see 4.1 Accessing Winter 2013 Coadd FITS Files). # 6 Example Queries¶ Retrieve median g, r, i magnitudes for all objects in a (ra, dec) box: SELECT ra, decl, medMag_g, medMag_r, medMag_i FROM AvgForcedPhot WHERE ra BETWEEN 0.01 and 0.02 AND decl BETWEEN 0.03 and 0.04 Alternatively, you can use scisql geometry functions; this should speed up queries over large area: SET @poly = scisql_s2CPolyToBin(0.01, 0.03, 0.02, 0.03, 0.03, 0.04, 0.01, 0.04); CALL scisql.scisql_s2CPolyRegion(@poly, 20); SELECT ra, decl, medMag_g, medMag_r, medMag_i FROM AvgForcedPhot WHERE scisql_s2PtInCPoly(ra, decl, @poly) = 1 Retrieve a g-band light curve for object 1398579058966639: SELECT deepSourceId, deepForcedSourceId, exp.run, fsrc.timeMid, scisql_dnToAbMag(fsrc.psfFlux, exp.fluxMag0) as g, scisql_dnToAbMagSigma(fsrc.psfFlux, fsrc.psfFluxSigma, exp.fluxMag0, exp.fluxMag0Sigma) as gErr FROM DeepForcedSource AS fsrc, Science_Ccd_Exposure AS exp WHERE exp.scienceCcdExposureId = fsrc.scienceCcdExposureId AND fsrc.filterId = 1 AND NOT (fsrc.flagPixEdge \| fsrc.flagPixSaturAny \| AND deepSourceId = 1398579058966639 ORDER BY fsrc.timeMid Notes: • The times (timeMid column) denote the mid-points of exposure each SDSS frame. Since SDSS took data in TDI mode, these have to be corrected to the effective time of observation of each object. • No effort has been made to remove objects doubly-detected in overlap regions of SDSS frames. You may therefore get more than one measurement per run. Retrieve a g-band light curves for all objects with 0.0 < ra < 0.01deg and 0.0 < dec < 0.01deg: SELECT deepSourceId, deepForcedSourceId, exp.run, fsrc.ra, fsrc.decl, fsrc.timeMid, scisql_dnToAbMag(fsrc.psfFlux, exp.fluxMag0) as g, scisql_dnToAbMagSigma(fsrc.psfFlux, fsrc.psfFluxSigma, exp.fluxMag0, exp.fluxMag0Sigma) as gErr FROM DeepForcedSource AS fsrc, Science_Ccd_Exposure AS exp WHERE exp.scienceCcdExposureId = fsrc.scienceCcdExposureId AND fsrc.filterId = 1 AND NOT (fsrc.flagPixEdge \| fsrc.flagPixSaturAny \| AND fsrc.ra BETWEEN 0.0 AND 0.01 AND fsrc.decl BETWEEN 0.0 AND 0.01 ORDER BY fsrc.deepSourceId, fsrc.timeMid Notes: • Expect this query to take 1-2 minutes to complete. It will return 2,014 rows. # 7 Covered footprint¶ A quick visualization of the footprint available in Winter 2013 Stripe 82 data, created by plotting all 15.9 million detected objects: # 8 RGB Color composites¶ RGB color composite of an area in the vicinity of M2: The full-sized image can be viewed/panned/zoomed at http://moe.astro.washington.edu/sdss/. # 9 Quality assessment¶ ## 9.1 Comparison to S2012 and Completeness¶ We took a small subset of data from both the Summer 2012 DC and the Winter 2013 early production DC. Using the DEEP2 catalogs [6] as reference, we compare the completeness as a function of magnitude between the two reductions. The Winter 2013 (blue) completeness tracks very well with the Summer 2012 (red). This shows that we have not changed anything substantial between the two reduction runs. We next look at a much large section of the survey covering the Deep2 Field 4 photometric catalogs. We construct completeness and contamination profiles for the Winter 2013 DC. In addition to comparing to the DEEP2 catalogs, we compare the completeness of the Annis (2014) [5] catalogs to the Winter 2013 results. The Winter 2013 catalog is significantly less complete at bright magnitudes. We are looking more into this, but early evidence suggests this is due primarily to background subtraction around bright stars and to the fact that multiple peaks within a single detection footprint are not de-blended into individual sources for the Winter 2013 runs. We have placed a 5-sigma S/N threshold on the Annis catalog and the Winter 2013 catalog does not go significantly below 5-sigma. With these cuts the Winter 2013 catalog goes ~0.2 mag deeper than the Annis catalog. The completeness plot is not the whole story. We also look at the trends in S/N between the Annis (2014) [5] catalog and the Winter 2013 catalog. This shows that for constant S/N the Winter 2013 catalog goes about 0.75 mag deeper than the Annis (2014) [5] catalog. We also see that the Winter 2013 catalog is 10-sigma at our 50% limiting magnitude of 24.2. This suggests that a 5-sigma threshold on the coadd to seed forced photometry is too conservative and that we should have pushed to 3-sigma (or fainter) in the coadd to reach completeness in the coadded catalog at 5-sigma. We also looked for contamination in the Winter 2013 catalog. We define contamination simply as any object in the Winter 2013 catalog that is not in the DEEP2 catalog. The following figure shows that there is less than 5% relative contamination to our limiting magnitude. The production pipelines perform photometric calibration using the catalog of Ivezic (2007) [8]. In this analysis we look at the distribution of forced photometry principal colors of stellar sources, described in Ivezic (2004) [7]. We use the star-galaxy separation provided by the Annis (2014) [5] Stripe82 catalog to select point sources for the analysis; we do not do any native star-galaxy separation. The figure below illustrates the process of defining a principal color (adopted from Ivezic (2004) [7]): The width of the stellar locus perpendicular to principal color P1 (top) is a function of underlying stellar astrophysics, and errors on the photometry. As the bottom panel demonstrates, this width increases as a function of magnitude, as photometric uncertainties start to dominate. In our analysis, we look at the principal colors w, shown in the figure above, and x, which is the width perpendicular to the vertical distribution in the (r-i) vs. (g-r) diagram above. We examine below the width of the principal loci as a function of the number of epochs for forced photometry: using 1 epoch (i.e. all the data), the median (in flux) of two epochs (where the flux medians to a value > 0.0), and the median of 10 and then 40 epochs. We first show the results for Summer2012 processing below: The left image provides the distribution of points around the principal colors w and x (i.e. the principal locus is at x=0 in all plots). Each panel shows the all-data distribution, and then the median across epochs for all objects with N>9 epochs. When medianing across many measurements, the locus becomes tighter, and is less dominated by the photometric uncertainties to fainter magnitudes. The right panel shows how the width of this locus improves as a function of the number of epochs, for N=1,2,10,40 epochs, along with a histogram of the number of objects vs. r-band magnitude. We examine below the results of the Winter2013 processing for one of the 6 SDSS camcols. This includes data from camcol=1 of both the N and S strips of the stripe. We subdivide the data into areas 10 degrees wide in RA, and provide measurements of the median and standard deviation of the distributions (computed as 0.741 times the interquartile range) in tabular form for the first RA range. ### 9.1.1 -40 < RA < -30¶ Mag w;N=1 w;N=2 w;N=10 w;N=40 x;N=1 x;N=2 x;N=10 x;N=40 15.25 -0.004,0.015 -0.006,0.015 -0.002,0.010 -0.002,0.010 -0.027,0.016 ... ... ... 15.75 -0.004,0.016 -0.003,0.015 -0.003,0.011 -0.003,0.010 -0.009,0.032 -0.008,0.030 -0.010,0.022 -0.006,0.022 16.25 -0.003,0.016 -0.002,0.014 -0.002,0.010 -0.002,0.010 -0.015,0.037 -0.006,0.021 -0.013,0.027 -0.009,0.022 16.75 -0.003,0.016 -0.004,0.014 -0.003,0.010 -0.002,0.009 -0.016,0.037 -0.015,0.036 -0.019,0.026 -0.018,0.028 17.25 -0.003,0.016 -0.003,0.014 -0.002,0.010 -0.002,0.008 -0.007,0.037 -0.007,0.040 -0.005,0.034 -0.004,0.029 17.75 -0.002,0.017 -0.001,0.015 -0.002,0.010 -0.002,0.008 0.001,0.041 -0.008,0.044 0.002,0.034 0.004,0.035 18.25 -0.002,0.019 -0.002,0.016 -0.002,0.011 -0.002,0.009 -0.003,0.044 -0.007,0.042 0.001,0.032 0.000,0.031 18.75 -0.002,0.021 -0.003,0.018 -0.001,0.012 -0.001,0.009 0.000,0.050 0.001,0.047 0.002,0.036 0.002,0.035 19.25 -0.002,0.026 -0.002,0.021 -0.002,0.013 -0.002,0.010 0.004,0.063 0.005,0.059 0.004,0.042 0.006,0.038 19.75 -0.002,0.035 -0.003,0.029 -0.001,0.015 -0.002,0.010 -0.000,0.082 -0.000,0.076 0.001,0.050 -0.000,0.043 20.25 -0.002,0.050 -0.003,0.040 -0.002,0.021 -0.002,0.012 0.003,0.112 0.011,0.094 0.005,0.060 0.003,0.050 20.75 -0.002,0.074 -0.003,0.060 -0.003,0.029 -0.003,0.017 0.003,0.161 0.001,0.140 0.006,0.076 0.003,0.059 21.25 -0.001,0.113 -0.008,0.083 -0.005,0.042 -0.006,0.024 0.005,0.234 0.003,0.197 0.009,0.101 0.007,0.070 21.75 0.017,0.173 0.007,0.141 -0.008,0.066 -0.007,0.037 -0.008,0.337 0.004,0.291 0.015,0.159 0.017,0.093 22.25 0.056,0.274 0.043,0.225 -0.008,0.104 -0.012,0.060 -0.044,0.466 -0.034,0.388 0.033,0.225 0.034,0.136 22.75 0.128,0.390 0.081,0.350 -0.000,0.174 -0.011,0.097 -0.176,0.591 -0.096,0.525 0.064,0.364 0.064,0.215 23.25 0.249,0.467 0.178,0.422 0.035,0.254 0.005,0.155 -0.403,0.672 -0.309,0.631 -0.019,0.490 0.095,0.352 23.75 0.470,0.495 0.361,0.478 0.138,0.338 0.043,0.240 -0.722,0.681 -0.625,0.674 -0.275,0.610 0.026,0.499 24.25 0.780,0.493 0.632,0.549 0.372,0.372 0.145,0.257 -1.058,0.654 -0.985,0.668 -0.641,0.621 -0.179,0.618 24.75 1.112,0.514 0.934,0.480 0.673,0.362 0.437,0.293 -1.400,0.637 -1.169,0.675 -1.064,0.514 -0.597,0.743 25.25 1.448,0.525 1.299,0.411 0.963,0.503 ... -1.742,0.659 -1.615,0.879 -1.434,0.437 -1.155,0.668 # 10 References¶ [1] Andrew Becker and others. Report on Late Winter2013 Production: Image Differencing. LSST Data Management LDM-227, 2013. URL: https://ls.st/LDM-227. [2] Mario Juric. Summer 2012 LSST DM Data Challenge. LSST Data Management Tech Note DMTN-0034, 2012. URL: https://dmtn-034.lsst.io. [3] Richard A. Shaw and others. LSST Data Challenge Report: Summer 2012/early-Winter 2013. LSST Data Management LDM-226, 2013. URL: https://ls.st/LDM-226. [4] Project Science Team. LSST Project Publication Policy. LSST Data Management LPM-162, 2015. URL: https://ls.st/LPM-162. [5] (1, 2, 3, 4, 5) J. Annis and others. The Sloan Digital Sky Survey Coadd: 275 deg^2 of Deep Sloan Digital Sky Survey Imaging on Stripe 82. ApJ, 794:120, October 2014. arXiv:1111.6619, doi:10.1088/0004-637X/794/2/120. [6] A. L. Coil, J. A. Newman, N. Kaiser, M. Davis, C.-P. Ma, D. D. Kocevski, and D. C. Koo. Evolution and Color Dependence of the Galaxy Angular Correlation Function: 350,000 Galaxies in 5 Square Degrees. ApJ, 617:765–781, December 2004. arXiv:astro-ph/0403423, doi:10.1086/425676. [7] (1, 2) Ž. Ivezić and others. SDSS data management and photometric quality assessment. Astronomische Nachrichten, 325:583–589, October 2004. arXiv:astro-ph/0410195, doi:10.1002/asna.200410285. [8] Ž. Ivezić and others. Sloan Digital Sky Survey Standard Star Catalog for Stripe 82: The Dawn of Industrial 1 per cent Optical Photometry. AJ, 134:973–998, September 2007. arXiv:astro-ph/0703157, doi:10.1086/519976. Note This document was originally published as an LSST TRAC page at https://dev.lsstcorp.org/trac/wiki/DC/Winter2013	2018-03-20 20:55:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40493810176849365, "perplexity": 8649.320507267039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647545.54/warc/CC-MAIN-20180320205242-20180320225242-00345.warc.gz"}
https://goodboychan.github.io/python/coursera/tensorflow_probability/icl/2021/08/26/01-Bayesian-Convolutional-Neural-Network.html	Packages import tensorflow as tf import tensorflow_probability as tfp from tensorflow.keras.models import Sequential from tensorflow.keras.layers import Dense, Flatten, Conv2D, MaxPooling2D from tensorflow.keras.losses import SparseCategoricalCrossentropy from tensorflow.keras.optimizers import RMSprop import numpy as np import os import matplotlib.pyplot as plt tfd = tfp.distributions tfpl = tfp.layers plt.rcParams['figure.figsize'] = (10, 6) print("Tensorflow Version: ", tf.__version__) print("Tensorflow Probability Version: ", tfp.__version__) Tensorflow Version: 2.5.0 Tensorflow Probability Version: 0.13.0 The MNIST and MNIST-C datasets In this notebook, you will use the MNIST and MNIST-C datasets, which both consist of a training set of 60,000 handwritten digits with corresponding labels, and a test set of 10,000 images. The images have been normalised and centred. The MNIST-C dataset is a corrupted version of the MNIST dataset, to test out-of-distribution robustness of computer vision models. • Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. "Gradient-based learning applied to document recognition." Proceedings of the IEEE, 86(11):2278-2324, November 1998. • N. Mu and J. Gilmeer. "MNIST-C: A Robustness Benchmark for Computer Vision" https://arxiv.org/abs/1906.02337 Our goal is to construct a neural network that classifies images of handwritten digits into one of 10 classes. We'll start by importing two datasets. The first is the MNIST dataset of handwritten digits, and the second is the MNIST-C dataset, which is a corrupted version of the MNIST dataset. This dataset is available on TensorFlow datasets. We'll be using the dataset with "spatters". We will load and inspect the datasets below. We'll use the notation _c to denote corrupted. The images are the same as in the original MNIST, but are "corrupted" by some grey spatters. def load_data(name): data_dir = os.path.join('dataset', name) x_train = 1 - np.load(os.path.join(data_dir, 'x_train.npy')) / 255. x_train = x_train.astype(np.float32) y_train_oh = tf.keras.utils.to_categorical(y_train) x_test = 1 - np.load(os.path.join(data_dir, 'x_test.npy')) / 255. x_test = x_test.astype(np.float32) y_test_oh = tf.keras.utils.to_categorical(y_test) return (x_train, y_train, y_train_oh), (x_test, y_test, y_test_oh) def inspect_images(data, num_images): fig, ax = plt.subplots(nrows=1, ncols=num_images, figsize=(2num_images, 2)) for i in range(num_images): ax[i].imshow(data[i, ..., 0], cmap='gray') ax[i].axis('off') plt.show() (x_train, y_train, y_train_oh), (x_test, y_test, y_test_oh) = load_data('MNIST') inspect_images(data=x_train, num_images=8) (x_c_train, y_c_train, y_c_train_oh), (x_c_test, y_c_test, y_c_test_oh) = load_data('MNIST_corrupted') inspect_images(data=x_c_train, num_images=8) Create the deterministic model We will first train a standard deterministic CNN classifier model as a base model before implementing the probabilistic and Bayesian neural networks. def get_deterministic_model(input_shape, loss, optimizer, metrics): """ This function should build and compile a CNN model according to the above specification. The function takes input_shape, loss, optimizer and metrics as arguments, which should be used to define and compile the model. Your function should return the compiled model. """ model = Sequential([ Conv2D(kernel_size=(5, 5), filters=8, activation='relu', padding='VALID', input_shape=input_shape), MaxPooling2D(pool_size=(6, 6)), Flatten(), Dense(units=10, activation='softmax') ]) model.compile(loss=loss, optimizer=optimizer, metrics=metrics) return model tf.random.set_seed(0) deterministic_model = get_deterministic_model( input_shape=(28, 28, 1), loss=SparseCategoricalCrossentropy(), optimizer=RMSprop(), metrics=['accuracy'] ) deterministic_model.summary() Model: "sequential" _________________________________________________________________ Layer (type) Output Shape Param # ================================================================= conv2d (Conv2D) (None, 24, 24, 8) 208 _________________________________________________________________ max_pooling2d (MaxPooling2D) (None, 4, 4, 8) 0 _________________________________________________________________ flatten (Flatten) (None, 128) 0 _________________________________________________________________ dense (Dense) (None, 10) 1290 ================================================================= Total params: 1,498 Trainable params: 1,498 Non-trainable params: 0 _________________________________________________________________ deterministic_model.fit(x_train, y_train, epochs=5) Epoch 1/5 1875/1875 [==============================] - 4s 2ms/step - loss: 0.4863 - accuracy: 0.8701 Epoch 2/5 1875/1875 [==============================] - 3s 2ms/step - loss: 0.1488 - accuracy: 0.9557 Epoch 3/5 1875/1875 [==============================] - 3s 1ms/step - loss: 0.1181 - accuracy: 0.9642 Epoch 4/5 1875/1875 [==============================] - 4s 2ms/step - loss: 0.1033 - accuracy: 0.9684 Epoch 5/5 1875/1875 [==============================] - 3s 2ms/step - loss: 0.0944 - accuracy: 0.9716 <tensorflow.python.keras.callbacks.History at 0x7fa582200f50> print('Accuracy on MNIST test set: ', str(deterministic_model.evaluate(x_test, y_test, verbose=False)[1])) print('Accuracy on corrupted MNIST test set: ', str(deterministic_model.evaluate(x_c_test, y_c_test, verbose=False)[1])) Accuracy on MNIST test set: 0.9732000231742859 Accuracy on corrupted MNIST test set: 0.9409000277519226 As you might expect, the pointwise performance on the corrupted MNIST set is worse. This makes sense, since this dataset is slightly different, and noisier, than the uncorrupted version. Furthermore, the model was trained on the uncorrupted MNIST data, so has no experience with the spatters. Probabilistic CNN model You'll start by turning this deterministic network into a probabilistic one, by letting the model output a distribution instead of a deterministic tensor. This model will capture the aleatoric uncertainty on the image labels. You will do this by adding a probabilistic layer to the end of the model and training using the negative loglikelihood. Note that, our NLL loss function has arguments y_true for the correct label (as a one-hot vector), and y_pred as the model prediction (a OneHotCategorical distribution). It should return the negative log-likelihood of each sample in y_true given the predicted distribution y_pred. If y_true is of shape [B, E] and y_pred has batch shape [B] and event shape [E], the output should be a Tensor of shape [B]. def nll(y_true, y_pred): """ This function should return the negative log-likelihood of each sample in y_true given the predicted distribution y_pred. If y_true is of shape [B, E] and y_pred has batch shape [B] and event_shape [E], the output should be a Tensor of shape [B]. """ return -y_pred.log_prob(y_true) Now we need to build probabilistic model. def get_probabilistic_model(input_shape, loss, optimizer, metrics): """ This function should return the probabilistic model according to the above specification. The function takes input_shape, loss, optimizer and metrics as arguments, which should be used to define and compile the model. Your function should return the compiled model. """ model = Sequential([ Conv2D(kernel_size=(5, 5), filters=8, activation='relu', padding='VALID', input_shape=input_shape), MaxPooling2D(pool_size=(6, 6)), Flatten(), Dense(tfpl.OneHotCategorical.params_size(10)), tfpl.OneHotCategorical(10, convert_to_tensor_fn=tfd.Distribution.mode) ]) model.compile(loss=loss, optimizer=optimizer, metrics=metrics) return model tf.random.set_seed(0) probabilistic_model = get_probabilistic_model( input_shape=(28, 28, 1), loss=nll, optimizer=RMSprop(), metrics=['accuracy'] ) probabilistic_model.summary() Model: "sequential_1" _________________________________________________________________ Layer (type) Output Shape Param # ================================================================= conv2d_1 (Conv2D) (None, 24, 24, 8) 208 _________________________________________________________________ max_pooling2d_1 (MaxPooling2 (None, 4, 4, 8) 0 _________________________________________________________________ flatten_1 (Flatten) (None, 128) 0 _________________________________________________________________ dense_1 (Dense) (None, 10) 1290 _________________________________________________________________ one_hot_categorical (OneHotC multiple 0 ================================================================= Total params: 1,498 Trainable params: 1,498 Non-trainable params: 0 _________________________________________________________________ Now, you can train the probabilistic model on the MNIST data using the code below. Note that the target data now uses the one-hot version of the labels, instead of the sparse version. This is to match the categorical distribution you added at the end. probabilistic_model.fit(x_train, y_train_oh, epochs=5) Epoch 1/5 1875/1875 [==============================] - 2s 1ms/step - loss: 0.4863 - accuracy: 0.8698 Epoch 2/5 1875/1875 [==============================] - 2s 1ms/step - loss: 0.1491 - accuracy: 0.9556 Epoch 3/5 1875/1875 [==============================] - 2s 1ms/step - loss: 0.1183 - accuracy: 0.9643 Epoch 4/5 1875/1875 [==============================] - 3s 2ms/step - loss: 0.1034 - accuracy: 0.9682 Epoch 5/5 1875/1875 [==============================] - 2s 1ms/step - loss: 0.0945 - accuracy: 0.9716 <tensorflow.python.keras.callbacks.History at 0x7fa69f237a90> print('Accuracy on MNIST test set: ', str(probabilistic_model.evaluate(x_test, y_test_oh, verbose=False)[1])) print('Accuracy on corrupted MNIST test set: ', str(probabilistic_model.evaluate(x_c_test, y_c_test_oh, verbose=False)[1])) Accuracy on MNIST test set: 0.9735999703407288 Accuracy on corrupted MNIST test set: 0.9415000081062317 Analyse the model predictions We will now do some deeper analysis by looking at the probabilities the model assigns to each class instead of its single prediction. The function below will be useful to help us analyse the probabilistic model predictions. def analyse_model_prediction(data, true_labels, model, image_num, run_ensemble=False): if run_ensemble: ensemble_size = 200 else: ensemble_size = 1 image = data[image_num] true_label = true_labels[image_num, 0] predicted_probabilities = np.empty(shape=(ensemble_size, 10)) for i in range(ensemble_size): predicted_probabilities[i] = model(image[np.newaxis, :]).mean().numpy()[0] model_prediction = model(image[np.newaxis, :]) fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(10, 2), gridspec_kw={'width_ratios': [2, 4]}) # Show the image and the true label ax1.imshow(image[..., 0], cmap='gray') ax1.axis('off') ax1.set_title('True label: {}'.format(str(true_label))) # Show a 95% prediction interval of model predicted probabilities pct_2p5 = np.array([np.percentile(predicted_probabilities[:, i], 2.5) for i in range(10)]) pct_97p5 = np.array([np.percentile(predicted_probabilities[:, i], 97.5) for i in range(10)]) bar = ax2.bar(np.arange(10), pct_97p5, color='red') bar[int(true_label)].set_color('green') ax2.bar(np.arange(10), pct_2p5-0.02, color='white', linewidth=1, edgecolor='white') ax2.set_xticks(np.arange(10)) ax2.set_ylim([0, 1]) ax2.set_ylabel('Probability') ax2.set_title('Model estimated probabilities') plt.show() for i in [0, 1577]: analyse_model_prediction(x_test, y_test, probabilistic_model, i) The model is very confident that the first image is a 6, which is correct. For the second image, the model struggles, assigning nonzero probabilities to many different classes. Run the code below to do the same for 2 images from the corrupted MNIST test set. for i in [0, 3710]: analyse_model_prediction(x_c_test, y_c_test, probabilistic_model, i) The first is the same 6 as you saw above, but the second image is different. Notice how the model can still say with high certainty that the first image is a 6, but struggles for the second, assigning an almost uniform distribution to all possible labels. Finally, have a look at an image for which the model is very sure on MNIST data but very unsure on corrupted MNIST data: for i in [9241]: analyse_model_prediction(x_test, y_test, probabilistic_model, i) analyse_model_prediction(x_c_test, y_c_test, probabilistic_model, i) It's not surprising what's happening here: the spatters cover up most of the number. You would hope a model indicates that it's unsure here, since there's very little information to go by. This is exactly what's happened. Uncertainty quantification using entropy We can also make some analysis of the model's uncertainty across the full test set, instead of for individual values. One way to do this is to calculate the entropy of the distribution. The entropy is the expected information (or informally, the expected 'surprise') of a random variable, and is a measure of the uncertainty of the random variable. The entropy of the estimated probabilities for sample $i$ is defined as $$H_i = -\sum_{j=1}^{10} p_{ij} \text{log}_{2}(p_{ij})$$ where $p_{ij}$ is the probability that the model assigns to sample $i$ corresponding to label $j$. The entropy as above is measured in bits. If the natural logarithm is used instead, the entropy is measured in nats. The key point is that the higher the value, the more unsure the model is. Let's see the distribution of the entropy of the model's predictions across the MNIST and corrupted MNIST test sets. The plots will be split between predictions the model gets correct and incorrect. # split into whether the model prediction is correct or incorrect def get_correct_indices(model, x, labels): y_model = model(x) correct = np.argmax(y_model.mean(), axis=1) == np.squeeze(labels) correct_indices = [i for i in range(x.shape[0]) if correct[i]] incorrect_indices = [i for i in range(x.shape[0]) if not correct[i]] return correct_indices, incorrect_indices def plot_entropy_distribution(model, x, labels): probs = model(x).mean().numpy() entropy = -np.sum(probs np.log2(probs), axis=1) fig, axes = plt.subplots(1, 2, figsize=(10, 4)) for i, category in zip(range(2), ['Correct', 'Incorrect']): entropy_category = entropy[get_correct_indices(model, x, labels)[i]] mean_entropy = np.mean(entropy_category) num_samples = entropy_category.shape[0] title = category + 'ly labelled ({:.1f}% of total)'.format(num_samples / x.shape[0] * 100) axes[i].hist(entropy_category, weights=(1/num_samples)np.ones(num_samples)) axes[i].annotate('Mean: {:.3f} bits'.format(mean_entropy), (0.4, 0.9), ha='center') axes[i].set_xlabel('Entropy (bits)') axes[i].set_ylim([0, 1]) axes[i].set_ylabel('Probability') axes[i].set_title(title) plt.show() print('MNIST test set:') plot_entropy_distribution(probabilistic_model, x_test, y_test) MNIST test set: print('Corrupted MNIST test set:') plot_entropy_distribution(probabilistic_model, x_c_test, y_c_test) Corrupted MNIST test set: There are two main conclusions: • The model is more unsure on the predictions it got wrong: this means it "knows" when the prediction may be wrong. • The model is more unsure for the corrupted MNIST test than for the uncorrupted version. Futhermore, this is more pronounced for correct predictions than for those it labels incorrectly. In this way, the model seems to "know" when it is unsure. This is a great property to have in a machine learning model, and is one of the advantages of probabilistic modelling. Bayesian CNN model The probabilistic model you just created considered only aleatoric uncertainty, assigning probabilities to each image instead of deterministic labels. The model still had deterministic weights. However, as you've seen, there is also 'epistemic' uncertainty over the weights, due to uncertainty about the parameters that explain the training data. def get_convolutional_reparameterization_layer(input_shape, divergence_fn): """ This function should create an instance of a Convolution2DReparameterization layer according to the above specification. The function takes the input_shape and divergence_fn as arguments, which should be used to define the layer. Your function should then return the layer instance. """ layer = tfpl.Convolution2DReparameterization( input_shape=input_shape, filters=8, kernel_size=(5, 5), kernel_prior_fn=tfpl.default_multivariate_normal_fn, kernel_posterior_fn=tfpl.default_mean_field_normal_fn(is_singular=False), kernel_divergence_fn=divergence_fn, bias_prior_fn=tfpl.default_multivariate_normal_fn, bias_posterior_fn=tfpl.default_mean_field_normal_fn(is_singular=False), bias_divergence_fn=divergence_fn ) return layer Custom prior For the parameters of the DenseVariational layer, we will use a custom prior: the "spike and slab" (also called a scale mixture prior) distribution. This distribution has a density that is the weighted sum of two normally distributed ones: one with a standard deviation of 1 and one with a standard deviation of 10. In this way, it has a sharp spike around 0 (from the normal distribution with standard deviation 1), but is also more spread out towards far away values (from the contribution from the normal distribution with standard deviation 10). The reason for using such a prior is that it is like a standard unit normal, but makes values far away from 0 more likely, allowing the model to explore a larger weight space. Run the code below to create a "spike and slab" distribution and plot its probability density function, compared with a standard unit normal. def spike_and_slab(event_shape, dtype): distribution = tfd.Mixture( cat=tfd.Categorical(probs=[0.5, 0.5]), components=[ tfd.Independent(tfd.Normal( loc=tf.zeros(event_shape, dtype=dtype), scale=1.0tf.ones(event_shape, dtype=dtype)), reinterpreted_batch_ndims=1), tfd.Independent(tfd.Normal( loc=tf.zeros(event_shape, dtype=dtype), scale=10.0*tf.ones(event_shape, dtype=dtype)), reinterpreted_batch_ndims=1)], name='spike_and_slab') return distribution x_plot = np.linspace(-5, 5, 1000)[:, np.newaxis] plt.plot(x_plot, tfd.Normal(loc=0, scale=1).prob(x_plot).numpy(), label='unit normal', linestyle='--') plt.plot(x_plot, spike_and_slab(1, dtype=tf.float32).prob(x_plot).numpy(), label='spike and slab') plt.xlabel('x') plt.ylabel('Density') plt.legend() plt.show() def get_prior(kernel_size, bias_size, dtype=None): """ This function should create the prior distribution, consisting of the "spike and slab" distribution that is described above. The distribution should be created using the kernel_size, bias_size and dtype function arguments above. The function should then return a callable, that returns the prior distribution. """ n = kernel_size+bias_size prior_model = Sequential([tfpl.DistributionLambda(lambda t : spike_and_slab(n, dtype))]) return prior_model def get_posterior(kernel_size, bias_size, dtype=None): """ This function should create the posterior distribution as specified above. The distribution should be created using the kernel_size, bias_size and dtype function arguments above. The function should then return a callable, that returns the posterior distribution. """ n = kernel_size + bias_size return Sequential([ tfpl.VariableLayer(tfpl.IndependentNormal.params_size(n), dtype=dtype), tfpl.IndependentNormal(n) ]) def get_dense_variational_layer(prior_fn, posterior_fn, kl_weight): """ This function should create an instance of a DenseVariational layer according to the above specification. The function takes the prior_fn, posterior_fn and kl_weight as arguments, which should be used to define the layer. Your function should then return the layer instance. """ return tfpl.DenseVariational( units=10, make_posterior_fn=posterior_fn, make_prior_fn=prior_fn, kl_weight=kl_weight ) Now, you're ready to use the functions you defined to create the convolutional reparameterization and dense variational layers, and use them in your Bayesian convolutional neural network model. tf.random.set_seed(0) divergence_fn = lambda q, p, _ : tfd.kl_divergence(q, p) / x_train.shape[0] convolutional_reparameterization_layer = get_convolutional_reparameterization_layer( input_shape=(28, 28, 1), divergence_fn=divergence_fn ) dense_variational_layer = get_dense_variational_layer( get_prior, get_posterior, kl_weight=1/x_train.shape[0] ) bayesian_model = Sequential([ convolutional_reparameterization_layer, MaxPooling2D(pool_size=(6, 6)), Flatten(), dense_variational_layer, tfpl.OneHotCategorical(10, convert_to_tensor_fn=tfd.Distribution.mode) ]) bayesian_model.compile(loss=nll, optimizer=RMSprop(), metrics=['accuracy'], experimental_run_tf_function=False) /home/chanseok/anaconda3/envs/torch/lib/python3.7/site-packages/tensorflow/python/keras/engine/base_layer.py:2191: UserWarning: layer.add_variable is deprecated and will be removed in a future version. Please use layer.add_weight method instead. warnings.warn('layer.add_variable is deprecated and ' bayesian_model.summary() Model: "sequential_2" _________________________________________________________________ Layer (type) Output Shape Param # ================================================================= conv2d_reparameterization (C (None, 24, 24, 8) 416 _________________________________________________________________ max_pooling2d_2 (MaxPooling2 (None, 4, 4, 8) 0 _________________________________________________________________ flatten_2 (Flatten) (None, 128) 0 _________________________________________________________________ dense_variational (DenseVari (None, 10) 2580 _________________________________________________________________ one_hot_categorical_1 (OneHo multiple 0 ================================================================= Total params: 2,996 Trainable params: 2,996 Non-trainable params: 0 _________________________________________________________________ bayesian_model.fit(x=x_train, y=y_train_oh, epochs=10, verbose=True) Epoch 1/10 1875/1875 [==============================] - 5s 2ms/step - loss: 1.9940 - accuracy: 0.3155 Epoch 2/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.7302 - accuracy: 0.7663 Epoch 3/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.4026 - accuracy: 0.8791 Epoch 4/10 1875/1875 [==============================] - 4s 2ms/step - loss: 0.2879 - accuracy: 0.9169 Epoch 5/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.2356 - accuracy: 0.9337 Epoch 6/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.2099 - accuracy: 0.9427 Epoch 7/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.1908 - accuracy: 0.9487 Epoch 8/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.1786 - accuracy: 0.9535 Epoch 9/10 1875/1875 [==============================] - 3s 2ms/step - loss: 0.1685 - accuracy: 0.9563 Epoch 10/10 1875/1875 [==============================] - 4s 2ms/step - loss: 0.1646 - accuracy: 0.9584 <tensorflow.python.keras.callbacks.History at 0x7fa2e804f490> print('Accuracy on MNIST test set: ', str(bayesian_model.evaluate(x_test, y_test_oh, verbose=False)[1])) print('Accuracy on corrupted MNIST test set: ', str(bayesian_model.evaluate(x_c_test, y_c_test_oh, verbose=False)[1])) Accuracy on MNIST test set: 0.9635999798774719 Accuracy on corrupted MNIST test set: 0.928600013256073 Analyse the model predictions Now that the model has trained, run the code below to create the same plots as before, starting with an analysis of the predicted probabilities for the same images. This model now has weight uncertainty, so running the forward pass multiple times will not generate the same estimated probabilities. For this reason, the estimated probabilities do not have single values. The plots are adjusted to show a 95% prediction interval for the model's estimated probabilities. for i in [0, 1577]: analyse_model_prediction(x_test, y_test, bayesian_model, i, run_ensemble=True) For the first image, the model assigns a probability of almost one for the 6 label. Furthermore, it is confident in this probability: this probability remains close to one for every sample from the posterior weight distribution (as seen by the horizontal green line having very small height, indicating a narrow prediction interval). This means that the epistemic uncertainty on this probability is very low. For the second image, the epistemic uncertainty on the probabilities is much larger, which indicates that the estimated probabilities may be unreliable. In this way, the model indicates whether estimates may be inaccurate. for i in [0, 3710]: analyse_model_prediction(x_c_test, y_c_test, bayesian_model, i, run_ensemble=True) Even with the spatters, the Bayesian model is confident in predicting the correct label for the first image above. The model struggles with the second image, which is reflected in the range of probabilities output by the network. for i in [9241]: analyse_model_prediction(x_test, y_test, bayesian_model, i, run_ensemble=True) analyse_model_prediction(x_c_test, y_c_test, bayesian_model, i, run_ensemble=True) Similar to before, the model struggles with the second number, as it is mostly covered up by the spatters. However, this time is clear to see the epistemic uncertainty in the model. Uncertainty quantification using entropy We also again plot the distribution of distribution entropy across the different test sets below. In these plots, no consideration has been made for the epistemic uncertainty, and the conclusions are broadly similar to those for the previous model. print('MNIST test set:') plot_entropy_distribution(bayesian_model, x_test, y_test) MNIST test set: print('Corrupted MNIST test set:') plot_entropy_distribution(bayesian_model, x_c_test, y_c_test) Corrupted MNIST test set:	2021-09-25 16:29:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5590201020240784, "perplexity": 7723.595953265004}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057687.51/warc/CC-MAIN-20210925142524-20210925172524-00408.warc.gz"}
http://mathhelpforum.com/algebra/105029-arithmetic-series-sequence-merger-print.html	# Arithmetic Series/Sequence merger Printable View • September 29th 2009, 11:34 AM Charchar Arithmetic Series/Sequence merger Im not quite sure how to do this.. I am instructed to find the value of the 11th term of an arithmetic sequence where t1=-13 and the sum of the first 1 terms is -33. Now I thought I could use the equation -33= (-13+tn/2)11 but that is either not the right approach or I am simplifying it wrong(both likely). Any help is very appreaciated • September 30th 2009, 01:01 AM red_dog $S_{11}=\frac{(t_1+t_{11})11}{2}$ Then $-33=\frac{(-13+t_{11})11}{2}$ Now find $t_{11}$	2014-07-26 16:10:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8247449994087219, "perplexity": 910.8940733539422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997902579.5/warc/CC-MAIN-20140722025822-00058-ip-10-33-131-23.ec2.internal.warc.gz"}
https://www.texasgateway.org/resource-index/reading?f%5B0%5D=sm_field_resource_grade_range%3A9&f%5B1%5D=sm_field_resource_grade_range%3A7&f%5B2%5D=im_field_resource_subject%3A2&f%5B3%5D=im_field_resource_subject_second%3A1029&amp%3Bf%5B1%5D=sm_field_resource_grade_range%3A3	• Resource ID: A1M6L9 • Grade Range: 9–12 • Subject: Math ### Quadratics: Connecting Roots, Zeros, and x-Intercepts Given a quadratic equation, the student will make connections among the solutions (roots) of the quadratic equation, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. • Resource ID: A1M6L10 • Grade Range: 9–12 • Subject: Math ### Applying the Laws of Exponents: Verbal/Symbolic Given verbal and symbolic descriptions of problems involving exponents, the student will simplify the expressions using the laws of exponents.	2019-08-25 21:20:42	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8137907385826111, "perplexity": 6262.857370086236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330800.17/warc/CC-MAIN-20190825194252-20190825220252-00213.warc.gz"}
https://www.tutorialspoint.com/how-to-search-multiple-columns-in-mysql	# How to search multiple columns in MySQL? MySQLMySQLi Database Let us understand how to search multiple columns in MySQL − Note: We assume we have created a database named ‘DBNAME’ and a table named ‘tableName’. The ‘AND’ and ‘OR’ operators can be used, depending on what the user wants the search to return. Let us see that with the help of an example − ## Example SELECT colName FROM tableName WHERE my_col LIKE %$param1% AND another_col LIKE %$param2%; In the above example, the ‘AND’ operator is used. This means that both the clause has to match a record for the result to be returned. ## Query SELECT colName FROM tableName WHERE my_col LIKE %$param1% OR another_col LIKE %$param2%; In the above example, the ‘OR’ operator is used. This means that either of the clause have to match a record for the result to be returned. Published on 09-Mar-2021 13:18:46	2022-05-28 08:27:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24640284478664398, "perplexity": 2962.0839368990837}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663013003.96/warc/CC-MAIN-20220528062047-20220528092047-00573.warc.gz"}
https://www.maa.org/press/maa-reviews/linear-geometry	# Linear Geometry ###### Rafael Artzy Publisher: Dover Publications Publication Date: 2017 Number of Pages: 288 Format: Paperback Price: 16.95 ISBN: 9780486466279 Category: Textbook [Reviewed by Mark Hunacek , on 05/2/2017 ] When I first studied linear algebra as an undergraduate, I learned, as do most if not all similarly situated students, that many of the ideas of the subject (linear independence, span, inner products, etc.) have strong geometric content and can be motivated by reference to that geometry. What I did not then realize, and would not learn for another year or so, is that the process can be reversed and that geometric ideas can be studied by reference to linear algebra. So, for example, if we define “point” as an element of a two-dimensional vector space, and “line” as any coset of a one-dimensional subspace, we get one version (a little simplified, as we’ll soon see) of plane affine geometry, and even with this simple machinery we can prove a lot of interesting geometric theorems. It is trivial to prove, for example, that two points lie on a unique line, but we can also prove less obvious things like the concurrency of the medians of a triangle, or the theorems of Ceva and Menelaus. We can then generalize things by letting our underlying vector space be n-dimensional instead of two-dimensional, or by adding an inner product to our underlying vector space, in which case we get theorems involving distance and angles rather than just incidence and parallelism. It is also possible to define projective planes and higher dimensional spaces in terms of vector spaces. Anybody wanting a brief but very elegant exposition of these ideas can likely do no better than to turn to chapter 3 of Kaplansky’s Linear Algebra and Geometry: A Second Course; the review of that book lists several other related references. It is this approach to geometry that Rafael Artzy, the author of the book now under review, refers to as “linear geometry”. That’s not a term that seems to be commonly used these days, but Artzy is not alone in its use; another book, by Gruenberg and Weir, also has this title and is apparently still available from Springer-Verlag as an entry in their Graduate Texts in Mathematics series. Artzy’s book is pitched at a somewhat less demanding level than is the book by Gruenberg and Weir and is, with the likely exception of the last chapter, accessible to good undergraduates, particularly if they have had a prior course in linear algebra. The book has four chapters. The first isn’t really on linear geometry, but is a prefatory chapter to help motivate the ideas that follow. In it, Artzy introduces the complex number system (from scratch, but at a rapid pace and without proofs), and then, identifying the points in the Euclidean plane with complex numbers, uses the structure of that number system to discuss various transformations of the Euclidean plane. Möbius transformations (here called bilinear transformations) are also discussed, and, in a final section, the Poincaré half-plane model of hyperbolic geometry is defined and transformations on it are examined. Linear geometry per se begins in chapter 2. The author begins with a fairly extensive review of vector spaces and linear transformations, including dual spaces, and then proceeds to a definition of affine n-space that is a bit more sophisticated and abstract than the one formulated above; he works with an arbitrary set (I’ll call it $A$; Artzy calls it $A^n$ to emphasize the dimension) on which an $n$-dimensional vector space $V$ operates, subject to certain conditions. Very informally, any two elements $P$ and $Q$ of $A$ define a vector $\vec{PQ}$ which acts on $P$ to produce $Q$. (Think of this as an arrow from $P$ to $Q$.) This is a more standard definition (see, for example, An Algebraic Approach to Geometry by Borceux and Tarrida’s Affine Maps, Euclidean Motions and Quadrics) than the simpler one sketched above (which can be viewed as a special case of this), and avoids giving the zero element of a vector space undue prominence. It also allows us to say that a subspace of an affine space is an affine space, which isn’t the case under the more simplistic definition given above. With this definition in hand, the author then discusses affine transformations and, towards the end of the chapter, introduces an inner product to get Euclidean geometry. Again, the transformation approach is emphasized, with isometries in the plane and three-space being classified. (In the case of the Euclidean plane, these results are all anticipated by the discussion in chapter 1.) The final section of this chapter looks briefly at affine planes defined over finite fields. This is an enjoyable chapter, but one thing that seems like a missed opportunity is the failure to discuss some of the interesting theorems of affine geometry (such as the ones mentioned in the second paragraph above); the primary focus here seems to be on transformations. Projective geometry is the subject of chapter 3. To define a projective plane over a field $F$, simply take the plane to be a three-dimensional vector space $V$ over $F$; “points” and “lines” are one-dimensional and two-dimensional subspaces of $V$, respectively. A trivial exercise in linear algebra then shows that any two distinct lines intersect in exactly one point; this of course is the hallmark of projective plane geometry. If we fix a basis for $V$, then any point can be identified by a coordinate vector with three scalar components, not all $0$, and where multiplying by a scalar does not change the point. These are, of course, the homogenous coordinates of a point. All of this generalizes naturally to projective $n$-space, where we start with an $(n+1)$-dimensional vector space. Artzy begins the chapter directly with projective $n$-space and then segues immediately into a discussion of projective transformations. We also see in this chapter, among other things, the cross-ratio and conics defined and discussed, as well as the statements and proofs of the theorems of Desargues and Pappus. The fourth chapter of the book is the most difficult and probably one that is not easily accessible to most undergraduates. In this chapter, the author begins by departing from the linear-algebraic approach to geometry and instead looks at the axiomatic foundations of affine and projective (plane) geometry. There are relatively simple axioms for both affine and projective planes, and along with proving some standard theorems, one can ask the following interesting question: which (axiomatically defined) projective planes are the “field planes” discussed above? The answer turns out to be nontrivial but very beautiful: an axiomatic projective plane is coordinatized by homogenous coordinates over a field if and only if Pappus’s Theorem is true in that plane; it is coordinatized by homogenous coordinates over a division ring if and only if Desargues’s theorem is true in it. (So, as a consequence: if Pappus’s theorem is true in a projective plane, then so is Desargues’s theorem, and if Desargues’s theorem is true in a finite projective plane, then — by Weddeburn’s theorem on finite division rings — so is Pappus’ theorem. The former fact can be proved without algebra, but I have never seen a purely geometric proof of the latter.) It also turns out that even if Desargues’s theorem is not true in an axiomatic projective plane, the plane can still be coordinatized, but the scalars now come from something called a “ternary ring”; these are not really “rings” in the usual sense, but are sets on which a ternary operation is defined satisfying certain conditions. The fourth chapter of the book is devoted to a discussion of these ideas. It is a difficult and time-consuming enterprise, but the author does it skillfully. He starts with an axiomatic projective plane (he calls it a “rudimentary projective plane”) and develops ternary rings to deal with them. He then gradually adds axioms, including the results of Desargues and Pappus, and shows how the algebraic coordinatizing structure changes with these new axioms. After establishing the relationship between fields and Pappus’s theorem, additional axioms are added to produce the real field. The collineation groups of these structure are also discussed. By and large, the book is quite successful. The material is interesting and is well-presented. The writing style is clear, attention is paid to motivation, and there are a good selection of exercises. I do, however, have a few nits to pick. First and foremost, the author writes functions on the right instead of the left, a convention that I have never liked and which is (thankfully) not seen very much in modern textbooks. Second, there are a couple of statements in the book that are inaccurate, though one is simply due to the passage of time. That one is the statement that the question of the existence of projective planes of order 10 is unsettled; this was definitely correct when it was written, but in 1989 Clement Lam (using a computer) answered this question in the negative. Another incorrect statement, one that cannot so easily be excused, is the comment on page 125 that “there exist elements in every Galois field that do not have a square root.” This is simply false: in finite fields of characteristic 2 (which are explicitly considered in the text), every element has a square root. (Quick proof: in such fields, the mapping $x\mapsto x^2$ is one-to-one, and hence onto.) For finite fields of characteristic $>2$, however, the statement is true, by pretty much the same proof: the mapping $x\mapsto x^2$ is now not one-to-one, hence not onto. I also was not a big fan of the first chapter of the book. The material here is covered nicely but, as previously noted, doesn’t really seem to me to fit in with the rest of the text. The chapter seemed almost forced, a way of filling out the book. The analysis of Euclidean isometries in the second chapter seemed duplicative. The introduction of the hyperbolic plane in chapter 1 struck me as somewhat abrupt; there is a lot of wonderful history associated with the realization that geometries exist that do not satisfy the Euclidean parallel postulate, and the students I have had in my upper-level geometry courses have generally found this history fascinating. None of that is gone into here. My quibble here may be a matter of individual taste; I have felt, for many years now, that giving students a sense of history is extremely valuable in an upper-level mathematics course. Not everyone shares this view, of course, but those who do may find the introduction of hyperbolic geometry here rather sudden and without the kind of fanfare that I think it deserves. Contrast, for example, the sense of historical adventure and excitement engendered by the book Euclidean and Non-Euclidean Geometries: Development and History by Greenberg, which, in the author’s words, “presents the discovery of non-Euclidean geometry and the subsequent reformulation of Euclidean geometry as a suspense story”. These issues aside, this is a valuable book. I don’t think that the material matches very many courses that are currently taught in American universities, but faculty members who are interested in algebra and geometry should definitely have this book on their shelves as a useful reference. Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.	2020-01-25 08:10:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.735837459564209, "perplexity": 342.19149769417885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251671078.88/warc/CC-MAIN-20200125071430-20200125100430-00343.warc.gz"}
https://www.hackerearth.com/problem/algorithm/strange-creatures/	Strange Creatures Tag(s): ## Medium Problem Editorial Analytics Rohan is a great astronaut and a scientist. He lands in a Strange Planet. After searching the planet for days, he notices something strange. He thinks that this might be the sign of life. He sees that, small creatures keep on increasing in pits. These creatures can form new creatures among themselves. He gets very scared when he notices that the Pits keeps increasing too! He goes back to his spaceship and sits down with a pen and a paper to analyze the data so collected. Rohan gets to know by his mathematical abilities that the number of pits at any hour N is $A^N$, where ‘A’ is the number of creatures in the pit (all pits have the same number of creatures) at the 0th hour. He finds that it takes $A^N$ creatures to produce a new creature. And the creature does not reproduce with those creatures in their own pit. One creature can only reproduce when all the other creatures are from another pit . Rohan messaged this fact to his base on Earth. He wants to know the number of ways in which a new creature can be produced. You being his friend and his colleague, have to give him the number of ways in which a new creature can be produced from the existing creatures. As the number can be very large. So print the number modulo $10^9+7$. Input First Input Line contains T, the number of test cases. Second Input line contains two space separated integers A and N . Output Print the desired Output modulo $10^9+7$. Constraints $1 \le T \le 1000.$ $1 \le A < 100 .$ $1 \le N \le 10^{18} .$ SAMPLE INPUT 1 2 3 SAMPLE OUTPUT 256 Explanation For TestCase 1 (T=1). Number of creatures in each pit=2 Number of pits at the hour 3 = $2^3$ = 8. So, the number of ways of selecting 8 creatures from 8 pits to reproduce = $^2P_1* ^2P_1* ^2P_1* ^2P_1* ^2P_1* ^2P_1* ^2P_1* ^2P_1=256$. Time Limit: 1.0 sec(s) for each input file. Memory Limit: 256 MB Source Limit: 1024 KB Marking Scheme: Marks are awarded when all the testcases pass. Allowed Languages: C, C++, C++14, Clojure, C#, D, Erlang, F#, Go, Groovy, Haskell, Java, Java 8, JavaScript(Rhino), JavaScript(Node.js), Julia, Kotlin, Lisp, Lisp (SBCL), Lua, Objective-C, OCaml, Octave, Pascal, Perl, PHP, Python, Python 3, R(RScript), Racket, Ruby, Rust, Scala, Swift, Visual Basic ## CODE EDITOR Initializing Code Editor...	2018-04-23 05:54:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2163802534341812, "perplexity": 4415.00864994578}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945793.18/warc/CC-MAIN-20180423050940-20180423070940-00483.warc.gz"}
https://zbmath.org/authors/?q=ai%3Azuber.jean-bernard	# zbMATH — the first resource for mathematics ## Zuber, Jean-Bernard Compute Distance To: Author ID: zuber.jean-bernard Published as: Zuber, J. B.; Zuber, J.-B.; Zuber, Jean-Bernard Documents Indexed: 67 Publications since 1978, including 4 Books all top 5 #### Co-Authors 15 single-authored 12 Itzykson, Claude 11 Coquereaux, Robert 8 Di Francesco, Philippe 8 Petkova, Valentina B. 8 Zinn-Justin, Paul 4 Francesco, P. Di 4 Pearce, Paul A. 3 Behrend, Roger E. 2 Bauer, Michel 2 Cappelli, Amedeo 2 McSwiggen, Colin 2 Saleur, Hubert 1 Bessis, Daniel 1 Brézin, Edouard 1 DeWitt-Morette, Cécile 1 Drouffe, Jean-Michel 1 Eynard, Bertrand 1 Lesage, Frédéric J. 1 Lösch, Steffen 1 Parisi, Giorgio 1 Prats Ferrer, A. 1 Randjbar-Daemi, Seif 1 Rasmussen, Jørgen H. 1 Sezgin, Ergin 1 Zhou, Yuan-Ke all top 5 #### Serials 8 Nuclear Physics. B 7 Communications in Mathematical Physics 5 Journal of Physics A: Mathematical and Theoretical 4 Journal of Statistical Mechanics: Theory and Experiment 3 Journal of Physics A: Mathematical and General 3 Journal of Knot Theory and its Ramifications 3 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2 International Journal of Modern Physics A 2 Journal of Statistical Physics 2 Physics Letters. B 2 Nuclear Physics, B, Proceedings Supplements 2 The Electronic Journal of Combinatorics 2 Annales de l’Institut Henri Poincaré D. Combinatorics, Physics and their Interactions (AIHPD) 1 Modern Physics Letters A 1 Discrete Mathematics 1 Journal of Mathematical Physics 1 Letters in Mathematical Physics 1 Annales de l’Institut Fourier 1 Advances in Applied Mathematics 1 Mathematical and Computer Modelling 1 RIMS Kokyuroku 1 Acta Physica Polonica B 1 Advanced Series in Mathematical Physics 1 NATO ASI Series. Series C. Mathematical and Physical Sciences all top 5 #### Fields 44 Quantum theory (81-XX) 19 Nonassociative rings and algebras (17-XX) 16 Statistical mechanics, structure of matter (82-XX) 10 Combinatorics (05-XX) 10 Topological groups, Lie groups (22-XX) 9 Group theory and generalizations (20-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 6 Manifolds and cell complexes (57-XX) 5 Convex and discrete geometry (52-XX) 5 Global analysis, analysis on manifolds (58-XX) 4 General and overarching topics; collections (00-XX) 4 Algebraic geometry (14-XX) 4 Relativity and gravitational theory (83-XX) 3 Abstract harmonic analysis (43-XX) 3 Functional analysis (46-XX) 3 Differential geometry (53-XX) 2 Number theory (11-XX) 2 Commutative algebra (13-XX) 2 Category theory; homological algebra (18-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Operator theory (47-XX) 2 Probability theory and stochastic processes (60-XX) 1 Functions of a complex variable (30-XX) 1 Difference and functional equations (39-XX) 1 Calculus of variations and optimal control; optimization (49-XX) #### Citations contained in zbMATH Open 53 Publications have been cited 1,978 times in 1,610 Documents Cited by Year Quantum field theory techniques in graphical enumeration. Zbl 0453.05035 Bessis, D.; Itzykson, C.; Zuber, J. B. 1980 Planar diagrams. Zbl 0997.81548 Brézin, E.; Itzykson, C.; Parisi, G.; Zuber, J. B. 1978 The planar approximation. II. Zbl 0997.81549 Itzykson, C.; Zuber, J. B. 1980 The A-D-E classification of minimal and $$A_ 1^{(1)}$$ conformal invariant theories. Zbl 0639.17008 Cappelli, A.; Itzykson, C.; Zuber, J. B. 1987 Modular invariant partition functions in two dimensions. Zbl 0661.17017 Cappelli, A.; Itzykson, C.; Zuber, J. B. 1987 Generalised twisted partition functions. Zbl 0977.81128 Petkova, V. B.; Zuber, J.-B. 2001 Boundary conditions in rational conformal field theories. Zbl 1028.81520 Behrend, Roger E.; Pearce, Paul A.; Petkova, Valentina B.; Zuber, Jean-Bernard 2000 Combinatorics of the modular group. II: The Kontsevich integrals. Zbl 0972.14500 Itzykson, C.; Zuber, J.-B. 1992 The many faces of Ocneanu cells. Zbl 0983.81039 Petkova, V. B.; Zuber, J.-B. 2001 Conformal invariance and applications to statistical mechanics. Collection of reprints. Zbl 0723.00044 Itzykson, Claude (ed.); Saleur, Hubert (ed.); Zuber, Jean-Bernard (ed.) 1988 Boundary conditions in rational conformal field theories. Zbl 1071.81570 Behrend, Roger E.; Pearce, Paul A.; Petkova, Valentina B.; Zuber, Jean-Bernard 2000 Classical $$W$$-algebras. Zbl 0752.17026 Di Francesco, P.; Itzykson, C.; Zuber, J.-B. 1991 On some integrals over the $$U(N)$$ unitary group and their large $$N$$ limit. Zbl 1074.82013 Zinn-Justin, P.; Zuber, J.-B. 2003 Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models. Zbl 0960.82507 di Francesco, P.; Saleur, H.; Zuber, J. B. 1987 SU(N) lattice integrable models and modular invariance. Zbl 0748.17029 Di Francesco, P.; Zuber, J.-B. 1990 Matrix integration and combinatorics of modular groups. Zbl 0709.57007 Itzykson, C.; Zuber, J.-B. 1990 Logarithmic minimal models. Zbl 1456.81217 Pearce, Paul A.; Rasmussen, Jørgen; Zuber, Jean-Bernard 2006 From CFT to graphs. Zbl 1004.81551 Petkova, V. B.; Zuber, J.-B. 1996 On the counting of fully packed loop configurations: some new conjectures. Zbl 1054.05011 Zuber, J.-B. 2004 Conformal boundary conditions and what they teach us. Zbl 0990.81108 Petkova, Valentina B.; Zuber, Jean-Bernard 2001 Integrable boundaries, conformal boundary conditions and A-D-E fusion rules. Zbl 0951.81064 Behrend, Roger E.; Pearce, Paul A.; Zuber, Jean-Bernard 1998 On structure constants of $$\text{sl}(2)$$ theories. Zbl 1052.81613 Petkova, V. B.; Zuber, J.-B. 1995 Polynomial averages in the Kontsevich model. Zbl 0831.14010 Di Francesco, P.; Itzykson, C.; Zuber, J.-B. 1993 Conformal field theories, graphs and quantum algebras. Zbl 1026.81053 Petkova, Valentina; Zuber, Jean-Bernard 2002 On Dubrovin topological field theories. Zbl 1021.81901 Zuber, J.-B. 1994 CFT, BCFT, $$ADE$$ and all that. Zbl 1213.81203 Zuber, J.-B. 2002 Graphs and reflection groups. Zbl 0942.20018 Zuber, J.-B. 1996 Singular vectors of the Virasoro algebra. Zbl 0957.17510 Bauer, M.; Di Francesco, Ph.; Itzykson, C.; Zuber, J.-B. 1991 Correlation functions of Harish-Chandra integrals over the orthogonal and the symplectic groups. Zbl 1139.43004 Prats Ferrer, A.; Eynard, B.; Di Francesco, P.; Zuber, J.-B. 2007 Horn’s problem and Harish-Chandra’s integrals. Probability density functions. Zbl 1397.15008 Zuber, Jean-Bernard 2018 The large-$$N$$ limit of matrix integrals over the orthogonal group. Zbl 1147.82019 Zuber, Jean-Bernard 2008 A bijection between classes of fully packed loops and plane partitions. Zbl 1054.05010 Di Francesco, P.; Zinn-Justin, P.; Zuber, J.-B. 2004 Fusion potentials. I. Zbl 0778.17021 Di Francesco, P.; Zuber, J.-B. 1993 Conjugation properties of tensor product multiplicities. Zbl 1327.14260 Coquereaux, Robert; Zuber, Jean-Bernard 2014 Matrix integrals and the generation and counting of virtual tangles and links. Zbl 1077.57002 Zinn-Justin, Paul; Zuber, Jean-Bernard 2004 Graph rings and integrable perturbations of $$N=2$$ superconformal theories. Zbl 1043.81685 Di Francesco, P.; Lesage, F.; Zuber, J.-B. 1993 The Horn problem for real symmetric and quaternionic self-dual matrices. Zbl 1451.15008 Coquereaux, Robert; Zuber, Jean-Bernard 2019 From orbital measures to Littlewood-Richardson coefficients and hive polytopes. Zbl 1429.17009 Coquereaux, Robert; Zuber, Jean-Bernard 2018 On sums of tensor and fusion multiplicities. Zbl 1222.81255 Coquereaux, Robert; Zuber, Jean-Bernard 2011 Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. Zbl 07120271 Francesco, P. Di; Zinn-Justin, P.; Zuber, J.-B. 2006 On the counting of colored tangles. Zbl 0984.57001 Zinn-Justin, Paul; Zuber, Jean-Bernard 2000 Graphs, algebras, conformal field theories and integrable lattice models. Zbl 0957.81667 Zuber, J.-B. 1990 On some properties of $$\operatorname{SU}(3)$$ fusion coefficients. Zbl 1349.14192 Coquereaux, Robert; Zuber, Jean-Bernard 2016 Maps, immersions and permutations. Zbl 1343.05106 Coquereaux, Robert; Zuber, Jean-Bernard 2016 Determinant formulae for some tiling problems and application to fully packed loops. Zbl 1075.05007 Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard 2005 On fully packed loop configurations with four sets of nested arches. Zbl 1088.82005 Di Francesco, P.; Zuber, J.-B. 2004 A classification programme of generalized Dynkin diagrams. Zbl 1185.17022 Zuber, J.-B. 1997 Conformal, integrable and topological theories, graphs and Coxeter groups. Zbl 1052.81617 Zuber, Jean-Bernard 1995 Drinfeld doubles for finite subgroups of $$SU(2)$$ and $$SU(3)$$ Lie groups. Zbl 1269.81161 Coquereaux, Robert; Zuber, Jean-Bernard 2013 Matrix integrals and the counting of tangles and links. Zbl 0989.81031 Zinn-Justin, P.; Zuber, J.-B. 2002 Generalized Dynkin diagrams and root systems and their folding. Zbl 0968.17005 Zuber, Jean-Bernard 1998 Combinatorics of mapping class groups and matrix integration. Zbl 0957.57501 Itzykson, C.; Zuber, J.-B. 1990 Trieste conference on recent developments in conformal field theories, ICTP, Trieste, Italy, October 2–4, 1989. Zbl 0727.00018 Randjbar-Daemi, S. (ed.); Sezgin, E. (ed.); Zuber, J. B. (ed.) 1990 The Horn problem for real symmetric and quaternionic self-dual matrices. Zbl 1451.15008 Coquereaux, Robert; Zuber, Jean-Bernard 2019 Horn’s problem and Harish-Chandra’s integrals. Probability density functions. Zbl 1397.15008 Zuber, Jean-Bernard 2018 From orbital measures to Littlewood-Richardson coefficients and hive polytopes. Zbl 1429.17009 Coquereaux, Robert; Zuber, Jean-Bernard 2018 On some properties of $$\operatorname{SU}(3)$$ fusion coefficients. Zbl 1349.14192 Coquereaux, Robert; Zuber, Jean-Bernard 2016 Maps, immersions and permutations. Zbl 1343.05106 Coquereaux, Robert; Zuber, Jean-Bernard 2016 Conjugation properties of tensor product multiplicities. Zbl 1327.14260 Coquereaux, Robert; Zuber, Jean-Bernard 2014 Drinfeld doubles for finite subgroups of $$SU(2)$$ and $$SU(3)$$ Lie groups. Zbl 1269.81161 Coquereaux, Robert; Zuber, Jean-Bernard 2013 On sums of tensor and fusion multiplicities. Zbl 1222.81255 Coquereaux, Robert; Zuber, Jean-Bernard 2011 The large-$$N$$ limit of matrix integrals over the orthogonal group. Zbl 1147.82019 Zuber, Jean-Bernard 2008 Correlation functions of Harish-Chandra integrals over the orthogonal and the symplectic groups. Zbl 1139.43004 Prats Ferrer, A.; Eynard, B.; Di Francesco, P.; Zuber, J.-B. 2007 Logarithmic minimal models. Zbl 1456.81217 Pearce, Paul A.; Rasmussen, Jørgen; Zuber, Jean-Bernard 2006 Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. Zbl 07120271 Francesco, P. Di; Zinn-Justin, P.; Zuber, J.-B. 2006 Determinant formulae for some tiling problems and application to fully packed loops. Zbl 1075.05007 Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard 2005 On the counting of fully packed loop configurations: some new conjectures. Zbl 1054.05011 Zuber, J.-B. 2004 A bijection between classes of fully packed loops and plane partitions. Zbl 1054.05010 Di Francesco, P.; Zinn-Justin, P.; Zuber, J.-B. 2004 Matrix integrals and the generation and counting of virtual tangles and links. Zbl 1077.57002 Zinn-Justin, Paul; Zuber, Jean-Bernard 2004 On fully packed loop configurations with four sets of nested arches. Zbl 1088.82005 Di Francesco, P.; Zuber, J.-B. 2004 On some integrals over the $$U(N)$$ unitary group and their large $$N$$ limit. Zbl 1074.82013 Zinn-Justin, P.; Zuber, J.-B. 2003 Conformal field theories, graphs and quantum algebras. Zbl 1026.81053 Petkova, Valentina; Zuber, Jean-Bernard 2002 CFT, BCFT, $$ADE$$ and all that. Zbl 1213.81203 Zuber, J.-B. 2002 Matrix integrals and the counting of tangles and links. Zbl 0989.81031 Zinn-Justin, P.; Zuber, J.-B. 2002 Generalised twisted partition functions. Zbl 0977.81128 Petkova, V. B.; Zuber, J.-B. 2001 The many faces of Ocneanu cells. Zbl 0983.81039 Petkova, V. B.; Zuber, J.-B. 2001 Conformal boundary conditions and what they teach us. Zbl 0990.81108 Petkova, Valentina B.; Zuber, Jean-Bernard 2001 Boundary conditions in rational conformal field theories. Zbl 1028.81520 Behrend, Roger E.; Pearce, Paul A.; Petkova, Valentina B.; Zuber, Jean-Bernard 2000 Boundary conditions in rational conformal field theories. Zbl 1071.81570 Behrend, Roger E.; Pearce, Paul A.; Petkova, Valentina B.; Zuber, Jean-Bernard 2000 On the counting of colored tangles. Zbl 0984.57001 Zinn-Justin, Paul; Zuber, Jean-Bernard 2000 Integrable boundaries, conformal boundary conditions and A-D-E fusion rules. Zbl 0951.81064 Behrend, Roger E.; Pearce, Paul A.; Zuber, Jean-Bernard 1998 Generalized Dynkin diagrams and root systems and their folding. Zbl 0968.17005 Zuber, Jean-Bernard 1998 A classification programme of generalized Dynkin diagrams. Zbl 1185.17022 Zuber, J.-B. 1997 From CFT to graphs. Zbl 1004.81551 Petkova, V. B.; Zuber, J.-B. 1996 Graphs and reflection groups. Zbl 0942.20018 Zuber, J.-B. 1996 On structure constants of $$\text{sl}(2)$$ theories. Zbl 1052.81613 Petkova, V. B.; Zuber, J.-B. 1995 Conformal, integrable and topological theories, graphs and Coxeter groups. Zbl 1052.81617 Zuber, Jean-Bernard 1995 On Dubrovin topological field theories. Zbl 1021.81901 Zuber, J.-B. 1994 Polynomial averages in the Kontsevich model. Zbl 0831.14010 Di Francesco, P.; Itzykson, C.; Zuber, J.-B. 1993 Fusion potentials. I. Zbl 0778.17021 Di Francesco, P.; Zuber, J.-B. 1993 Graph rings and integrable perturbations of $$N=2$$ superconformal theories. Zbl 1043.81685 Di Francesco, P.; Lesage, F.; Zuber, J.-B. 1993 Combinatorics of the modular group. II: The Kontsevich integrals. Zbl 0972.14500 Itzykson, C.; Zuber, J.-B. 1992 Classical $$W$$-algebras. Zbl 0752.17026 Di Francesco, P.; Itzykson, C.; Zuber, J.-B. 1991 Singular vectors of the Virasoro algebra. Zbl 0957.17510 Bauer, M.; Di Francesco, Ph.; Itzykson, C.; Zuber, J.-B. 1991 SU(N) lattice integrable models and modular invariance. Zbl 0748.17029 Di Francesco, P.; Zuber, J.-B. 1990 Matrix integration and combinatorics of modular groups. Zbl 0709.57007 Itzykson, C.; Zuber, J.-B. 1990 Graphs, algebras, conformal field theories and integrable lattice models. Zbl 0957.81667 Zuber, J.-B. 1990 Combinatorics of mapping class groups and matrix integration. Zbl 0957.57501 Itzykson, C.; Zuber, J.-B. 1990 Trieste conference on recent developments in conformal field theories, ICTP, Trieste, Italy, October 2–4, 1989. Zbl 0727.00018 Randjbar-Daemi, S. (ed.); Sezgin, E. (ed.); Zuber, J. B. (ed.) 1990 Conformal invariance and applications to statistical mechanics. Collection of reprints. Zbl 0723.00044 Itzykson, Claude (ed.); Saleur, Hubert (ed.); Zuber, Jean-Bernard (ed.) 1988 The A-D-E classification of minimal and $$A_ 1^{(1)}$$ conformal invariant theories. Zbl 0639.17008 Cappelli, A.; Itzykson, C.; Zuber, J. B. 1987 Modular invariant partition functions in two dimensions. Zbl 0661.17017 Cappelli, A.; Itzykson, C.; Zuber, J. B. 1987 Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models. Zbl 0960.82507 di Francesco, P.; Saleur, H.; Zuber, J. B. 1987 Quantum field theory techniques in graphical enumeration. Zbl 0453.05035 Bessis, D.; Itzykson, C.; Zuber, J. B. 1980 The planar approximation. II. Zbl 0997.81549 Itzykson, C.; Zuber, J. B. 1980 Planar diagrams. Zbl 0997.81548 Brézin, E.; Itzykson, C.; Parisi, G.; Zuber, J. B. 1978 all top 5 #### Cited by 2,063 Authors 27 Zuber, Jean-Bernard 20 Di Francesco, Philippe 20 Schweigert, Christoph 19 Fuchs, Jürgen 16 Coquereaux, Robert 14 Eynard, Bertrand 14 Runkel, Ingo 12 Gannon, Terry 12 Pearce, Paul A. 11 Guionnet, Alice 11 Guitter, Emmanuel 11 Zinn-Justin, Paul 10 Evans, David E. 10 Itzykson, Claude 9 Guhr, Thomas 9 Gurau, Razvan 9 Kostov, Ivan K. 9 Orlov, Aleksandr Yu. 9 Pastur, Leonid Andreevich 9 Petkova, Valentina B. 9 Ruelle, Philippe 9 Schellekens, A. N. 9 Watts, Gerard M. T. 8 Bouttier, Jérémie 8 Kieburg, Mario 8 Mariño, Marcos 8 Mironov, Andrei D. 8 Morozov, Alexei Yurievich 8 Saleur, Hubert 8 Schubert, Christian 8 Yang, Di 7 Alexandrov, Alexander Sergeevich 7 Bertola, Marco 7 Borot, Gaëtan 7 Forrester, Peter J. 7 Harnad, John 7 Rivasseau, Vincent 7 Sveshnikov, Konstantin Alekseevich 7 Szabo, Richard J. 6 Bajnok, Zoltán 6 Blasone, Massimo 6 Felder, Giovanni 6 Irie, Hirotaka 6 Jacobsen, Jesper Lykke 6 Jentschura, Ulrich D. 6 Kuijlaars, Arno B. J. 6 Pugh, Mathew 6 Rehren, Karl-Henning 6 Sarkissian, Gor 6 Takook, Mohammad Vahid 6 Vitiello, Giuseppe 6 Zarembo, Konstantin 5 Adler, Mark 5 Akemann, Gernot 5 Bleher, Pavel M. 5 Brouder, Christian 5 Brunner, Ilka 5 Cardy, John L. 5 Chan, Chuantsung 5 Chekhov, Leonid O. 5 Cheng, Miranda C. N. 5 Degiovanni, Pascal 5 Dijkgraaf, Robbert H. 5 Dorey, Patrick E. 5 Dubrovin, Boris Anatol’evich 5 Feinberg, Joshua 5 Forghan, B. 5 Fröhlich, Jürg Martin 5 Gaberdiel, Matthias R. 5 Gawȩdzki, Krzysztof 5 Goulden, Ian P. 5 Jackson, David M. 5 Kawahigashi, Yasuyuki 5 Lazzarini, Serge 5 Lorin, Emmanuel 5 Ludwig, Andreas W. W. 5 Malbouisson, Adolfo P. C. 5 McLaughlin, Kenneth D. T.-R. 5 O’Connor, Denjoe 5 Roggenkamp, Daniel 5 Schiappa, Ricardo 5 Schomerus, Volker 5 Strachan, Ian A. B. 5 Strahov, Eugene 5 Sugawara, Yuji 5 Tateo, Roberto 5 van Moerbeke, Pierre 5 Yeh, Chi-Hsien 4 Bandelloni, Giuseppe 4 Bauer, Michel 4 Borinsky, Michael 4 Bousquet-Mélou, Mireille 4 Capozziello, Salvatore 4 de Boer, Jan 4 de Mello Koch, Robert 4 Dong, Chongying 4 Duplantier, Bertrand 4 Dvornikov, Maxim 4 Ferretti, Gabriele 4 Flohr, Michael A. I. ...and 1,963 more Authors all top 5 #### Cited in 169 Serials 240 Nuclear Physics. B 174 Journal of High Energy Physics 163 Communications in Mathematical Physics 86 Journal of Mathematical Physics 78 Annals of Physics 52 Physics Letters. B 51 International Journal of Modern Physics A 44 International Journal of Theoretical Physics 41 Theoretical and Mathematical Physics 39 Letters in Mathematical Physics 31 Journal of Statistical Physics 30 Physics Letters. A 30 Journal of Statistical Mechanics: Theory and Experiment 26 Journal of Geometry and Physics 19 Annales Henri Poincaré 19 Foundations of Physics 18 Modern Physics Letters A 18 Physics Reports 16 Journal of Physics A: Mathematical and Theoretical 15 General Relativity and Gravitation 14 Advances in Mathematics 13 Reviews in Mathematical Physics 13 Journal of Combinatorial Theory. Series A 13 Physical Review Letters 12 Nuclear Physics, B, Proceedings Supplements 11 Physica D 10 Computer Physics Communications 10 Annales de l’Institut Henri Poincaré. Physique Théorique 10 International Journal of Geometric Methods in Modern Physics 8 Journal of Algebra 8 Journal of Functional Analysis 8 Journal of Knot Theory and its Ramifications 8 Random Matrices: Theory and Applications 7 Annales de l’Institut Fourier 6 Journal of Computational Physics 6 Reports on Mathematical Physics 6 Chaos, Solitons and Fractals 6 The Annals of Probability 6 Advances in Applied Mathematics 6 Bulletin of the American Mathematical Society. New Series 6 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 5 Inventiones Mathematicae 5 Journal of Approximation Theory 4 Communications on Pure and Applied Mathematics 4 Physica A 4 Fortschritte der Physik 4 Duke Mathematical Journal 4 Transactions of the American Mathematical Society 4 Probability Theory and Related Fields 4 Mathematical and Computer Modelling 4 International Journal of Modern Physics D 4 Mathematical Physics, Analysis and Geometry 4 Physical Review D. Series III 4 Advances in High Energy Physics 4 Studies in History and Philosophy of Science. Part B. Studies in History and Philosophy of Modern Physics 4 Annales de l’Institut Henri Poincaré D. Combinatorics, Physics and their Interactions (AIHPD) 3 International Journal of Modern Physics B 3 Discrete Mathematics 3 Journal of Mathematical Analysis and Applications 3 Journal of Computational and Applied Mathematics 3 Journal of Number Theory 3 Acta Applicandae Mathematicae 3 Constructive Approximation 3 Journal of Theoretical Probability 3 International Journal of Mathematics 3 Russian Journal of Mathematical Physics 3 Journal of Mathematical Sciences (New York) 3 Advances in Applied Clifford Algebras 3 New Journal of Physics 3 The European Physical Journal C. Particles and Fields 3 Advances in Mathematical Physics 2 Mathematical Notes 2 Acta Mathematica 2 Journal of Pure and Applied Algebra 2 European Journal of Combinatorics 2 Journal of the American Mathematical Society 2 Experimental Mathematics 2 Journal of Algebraic Combinatorics 2 Applied Categorical Structures 2 St. Petersburg Mathematical Journal 2 Journal of Mathematical Chemistry 2 Proceedings of the Steklov Institute of Mathematics 2 Quantum Topology 2 Analysis and Mathematical Physics 1 Modern Physics Letters B 1 Applicable Analysis 1 Classical and Quantum Gravity 1 Discrete Applied Mathematics 1 European Journal of Physics 1 Indian Journal of Pure & Applied Mathematics 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Nonlinearity 1 Reviews of Modern Physics 1 Russian Mathematical Surveys 1 Transport Theory and Statistical Physics 1 Wave Motion 1 Hadronic Journal 1 Applied Mathematics and Computation 1 Canadian Journal of Mathematics 1 Commentarii Mathematici Helvetici ...and 69 more Serials all top 5 #### Cited in 59 Fields 1,156 Quantum theory (81-XX) 256 Statistical mechanics, structure of matter (82-XX) 196 Relativity and gravitational theory (83-XX) 155 Nonassociative rings and algebras (17-XX) 119 Linear and multilinear algebra; matrix theory (15-XX) 116 Probability theory and stochastic processes (60-XX) 108 Combinatorics (05-XX) 96 Partial differential equations (35-XX) 93 Algebraic geometry (14-XX) 85 Dynamical systems and ergodic theory (37-XX) 77 Global analysis, analysis on manifolds (58-XX) 67 Differential geometry (53-XX) 63 Functional analysis (46-XX) 59 Manifolds and cell complexes (57-XX) 48 Special functions (33-XX) 45 Topological groups, Lie groups (22-XX) 40 Mechanics of particles and systems (70-XX) 36 Number theory (11-XX) 30 Associative rings and algebras (16-XX) 26 Category theory; homological algebra (18-XX) 24 Group theory and generalizations (20-XX) 24 Several complex variables and analytic spaces (32-XX) 24 Harmonic analysis on Euclidean spaces (42-XX) 20 Numerical analysis (65-XX) 19 Functions of a complex variable (30-XX) 18 Operator theory (47-XX) 17 Statistics (62-XX) 16 Ordinary differential equations (34-XX) 11 Abstract harmonic analysis (43-XX) 11 Fluid mechanics (76-XX) 11 Optics, electromagnetic theory (78-XX) 9 Measure and integration (28-XX) 9 Approximations and expansions (41-XX) 8 Computer science (68-XX) 8 Information and communication theory, circuits (94-XX) 7 Difference and functional equations (39-XX) 6 Geometry (51-XX) 6 Algebraic topology (55-XX) 6 Biology and other natural sciences (92-XX) 5 General and overarching topics; collections (00-XX) 5 History and biography (01-XX) 5 Commutative algebra (13-XX) 5 Convex and discrete geometry (52-XX) 4 $$K$$-theory (19-XX) 4 Sequences, series, summability (40-XX) 4 Astronomy and astrophysics (85-XX) 3 Field theory and polynomials (12-XX) 3 Potential theory (31-XX) 3 Integral equations (45-XX) 3 Classical thermodynamics, heat transfer (80-XX) 2 Mathematical logic and foundations (03-XX) 2 Real functions (26-XX) 2 Integral transforms, operational calculus (44-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Geophysics (86-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Mechanics of deformable solids (74-XX) 1 Operations research, mathematical programming (90-XX)	2021-05-08 17:22:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6319562196731567, "perplexity": 7282.629680993229}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00201.warc.gz"}
https://mazur.eu/2o5hv5/zuqwi.php?aa2a12=application-of-least-square-method	Azure Ad Connect Sync Powershell, Martin Methodist College Baseball Coaches, Crossing Tasman Glacier, Mueller Lake Austin, Rmit Aerospace Engineering Ranking, Which Of The Following Best Describes Contractionary Fiscal Policy, Guttation Money Tree, Marketplace App Install, Rosa Virginiana Vs Rosa Rugosa, " /> "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. Section 6.5 The Method of Least Squares ¶ permalink Objectives. Although the muscular strength can consider the various evaluation methods, a grasp force is applied as an index to evaluate the muscular strength. Isotopenpraxis Isotopes in Environmental and Health Studies: Vol. They are connected by p DAbx. We now look at the line in the xy plane that best fits the data (x 1, y 1), …, (x n, y n). The main attribute of the method is based on multiple applications of the least squares solutions of certain matrix equations which define the separable motion blur in conjunction with known image deconvolution techniques. A new method for the reconstruction of blurred digital images damaged by separable motion blur is established. The method of least squares aims to minimise the variance between the values estimated from the polynomial and the expected values from the dataset.The coefficients of the polynomial regression model (ak,ak−1,⋯,a1) may be determined by solving the following system of linear equations.This system of equations is derived from the polynomial residual function (derivation may be seen in this Wolfram M… Let us discuss the Method of Least Squares in detail. Predicting values of dependentvariable, may include extrapolation beyond datapoints or interpolation between data points.•Hypothesis testing. Consider the data shown in Figure 1 and in Table1. The length of this vector is minimized by choosing Xb as the orthogonal projection of y onto the space spanned by the columns of X. (REVIEW) (in Hungarian) Full Record; Other Related Research; Authors: Balogh, T Publication Date: Sat Jan 01 00:00:00 EST 1966 Research Org. application of least square method, Linear Least Squares. De Maerschalck, B., 2003. Spectral convergence of the L2-norm error of the solution and of the moments of the solution are verified for the zero- and one-dimensional cases using model problems with analytical solutions. METHOD OF LEASTSQUARESBy: Varun Luthra11CSU163 2. Basic study on combined motion estimation using multichannel surface EMG signals. Least Squares Line Fitting Example Thefollowing examplecan be usedas atemplate for using the least squares method to ﬁndthe best ﬁtting line for a set of data. The application of a mathematicalformula to approximate the behavior of a physical system is frequentlyencountered in the laboratory. Figure 7 shows … 8, No. The major practical drawback with least squares is that unless the network has only a small number of unknown points, or has very few redundant observations, the amount of arithmetic manipulation makes the method impractical without the aid of a computer and appropriate software. The LSM is a well-established numerical method for solving a wide range of mathematical problems, (e.g. not identified OSTI Identifier: 4529715 NSA Number: NSA-20-041408 Resource Type: Journal Article 23.. MTERA and TLS give equal results on the high SNR phantom image. Application of the Least Square Method in the Analysis of Experimental Decay Curves. This method is most widely used in time series analysis. 2008;2008:351-4. doi: 10.1109/IEMBS.2008.4649162. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Least square approximation need not be unique, however if and are both least square solutions for AX= Y, then A = A. https://doi.org/10.1016/j.ces.2006.03.019. To find out more, see our Privacy and Cookies policy. In this section, we answer the following important question: Because, SEMG is one of the most important biological signal in which the human motion intention is directly reflected. The method of least squares is a standard approach to the approximate solution of over determined systems, i.e., sets of equations in which there are more equations than unknowns. in this video i showed how to solve curve fitting problem for straight line using least square method . USA.gov. Copyright © 2006 Elsevier Ltd. All rights reserved. Picture: geometry of a least-squares solution. The method of least squares is a standard approach to the approximate solution of over determined systems, i.e., sets of equations in which there are more equations than unknowns. The fundamental equation is still A TAbx DA b. Surface electromyography and muscle force: limits in sEMG-force relationship and new approaches for applications. Suppose we have a data set of 6 points as shown: i xi yi 1 1.2 1.1 2 2.3 2.1 3 3.0 3.1 4 3.8 4.0 5 4.7 4.9 6 … Least squares method, also called least squares approximation, in statistics, a method for estimating the true value of some quantity based on a consideration of errors in observations or measurements. (1972). Recall that the equation for a straight line is y = bx + a, where. 1. It gives the trend line of best fit to a time series data. Method of Least Squares The application of a mathematical formula to approximate the behavior of a physical system is frequently encountered in the laboratory. The least squares method provides the overall rationale for the placement of the line of best fit among the data points being studied. Curve Fitting Toolbox software uses the linear least-squares method to fit a linear model to data. ALGLIB for C++,a high performance C++ library with great portability across hardwareand software platforms 2. \| In multiphase chemical reactor analysis the prediction of the dispersed phase distribution plays a major role in achieving reasonable results. The sum of squares e0e is the square of the length of the residual vector e ¼ y Xb. \| For example, polynomials are linear but Gaussians are not. In this study, we describe the application of least square method for muscular strength estimation in hand motion recognition based on surface electromyogram (SEMG). Epub 2008 Nov 29. 23.. MTERA and TLS give equal results on the high SNR phantom image. We now look at the line in the xy plane that best fits the data (x 1, y 1), …, (x n, y n).. Recall that the equation for a straight line is y = bx + a, where b = the slope of the line a = y-intercept, i.e. A Treatise on the Method of Least Squares: Or, The Application of the Theory of Probabilities in the Combination of Observations William Chauvenet Lippincott & Company , 1868 - Least squares - 98 pages the time complexity will be O(n) to find the least square sphere fitting algorithm. Today, SEMG, which is measured from skin surface, is widely used as a control signal for many devices. Least square method 1. HHS This method is most widely used in time series analysis. the value of y where the line intersects with the y-axis. Learn to turn a best-fit problem into a least-squares problem. 3, pp. The basis functions ϕj(t) can be nonlinear functions of t, but the unknown parameters, βj, appear in the model linearly.The system of linear equations Application of least square method to arbitrary-order problems with separated boundary conditions Loghmani, G. B. Abstract. least squares solution). 3. Annu Int Conf IEEE Eng Med Biol Soc. The most common such approximation is the fitting of a straight line to a collection of data. It gives the trend line of best fit to a time series data. The least squares estimator is obtained by minimizing S(b). Example: Fit a least square line for the following data. Isotopenpraxis Isotopes in Environmental and Health Studies: Vol. A linear model is defined as an equation that is linear in the coefficients. This data appears to have a relative l… A linear model is defined as an equation that is linear in the coefficients. The Method of Least Squares Steven J. Miller⁄ Mathematics Department Brown University Providence, RI 02912 Abstract The Method of Least Squares is a procedure to determine the best ﬁt line to data; the proof uses simple calculus and linear algebra. Although it is also important to estimate muscular strength of motions, most of them cannot detect power of muscle. Recipe: find a least-squares solution (two ways). Learn to turn a best-fit problem into a least-squares problem. In Correlation we study the linear correlation between two random variables x and y. The ability to estimate muscular strength is a very important factor to control the SEMG systems. Application of the Least Square Method in the Analysis of Experimental Decay Curves. But for better accuracy let's see how to calculate the line using Least Squares Regression. In this post I’ll illustrate a more elegant view of least-squares regression — the so-called “linear algebra” view. This site uses cookies. The method of least square • Above we saw a discrete data set being approximated by a continuous function • We can also approximate continuous functions by simpler functions, see Figure 3 and Figure 4 Lectures INF2320 – p. 5/80 Nonetheless, formulas for total fixed costs (a) and variable cost per unit (b)can be derived from the above equations. least-squares method, in which the quantity ´2(a)= XN i=1 [y i¡y(x i;a)] 2 ¾2 i is minimized, where ¾ i is the standard deviation of the random errors of y i, which we assume to be normally distributed. In this paper, differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem. Linear Least Squares. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in … Here is a short unofﬁcial way to reach this equation: When Ax Db has no solution, multiply by AT and solve ATAbx DATb: Example 1 A crucial application of least squares is ﬁtting a straight line to m points. This type of calculation is best suited for linear models. Conventionally SEMG system mainly focused on how to achieve this objective. 6.4.11 TLS method. And various devices using SEMG are reported by lots of researchers. ∂ S ∂ p 1 = − 2 ∑ i = 1 n x i (y i − (p 1 x i + p 2)) = 0 ∂ S ∂ p 2 = − 2 ∑ i … NLM The given example explains how to find the equation of a straight line or a least square line by using the method of least square, which is very useful in statistics as well as in mathematics. 2 Chapter 5. Least Square is the method for finding the best fit of a set of data points. Merletti R, Botter A, Troiano A, Merlo E, Minetto MA. we can write model or predicted output as ... • standard methods for computing P(m+1)−1 from P(m+1) is O(n3) Least-squares applications 6–22. method to segregate fixed cost and variable cost components from a mixed cost figure For a matrix Aand a given vector , let be a least square solution of AX= Y.Then , is the projection of the vector Y onto the column space ofA.Least square approximation need not be unique, however if and are both least square solutions for AX= Y, then A = A. The given example explains how to find the equation of a straight line or a least square line by using the method of least square, which is very useful in statistics as well as in mathematics. Least Squares Regression Line of Best Fit. The least-squares method consists in minimizing the integral of the square of the residual over the computational domain. Method of Least Squares. Now that we have determined the loss function, the only thing left to do is minimize it. In this study, we describe the application of least square method for muscular strength estimation in hand motion recognition based on surface electromyogr The total least square method is not suited for the non-stationary data environment. However, with the data-ramping technique mentioned the section 3.6, it is appropriate.The influence of different model orders is shown as Fig. Application of least square method for muscular strength estimation in hand motion recognition using surface EMG. 111-113. Therefore, the least squares method can be given the following interpretation. 2008. The result of such a ﬂtting procedure is the function y(x;a 0), where a 0 is the coe–cient vector that Least Square is the method for finding the best fit of a set of data points. Application of ordinary least square method in nonlinear models Arhipova Irina Latvia University of Agriculture, Faculty of Information Technologies Liela street 2 Jelgava, LV-3001, Latvia E-mail: irina.arhipova@llu.lv Arhipovs Sergejs Latvia University of Agriculture, Faculty of Information Technologies Liela street 2 Jelgava, LV-3001, Latvia A least-squares regression method is a form of regression analysis which establishes the relationship between the dependent and independent variable along with a linear line. Application of the least-squares method for solving population balance problems in. The applications of the method of least squares curve fitting using polynomials are briefly discussed as follows. By continuing you agree to the use of cookies. Nagata K, Nakano T, Magatani K, Yamada M. Annu Int Conf IEEE Eng Med Biol Soc. Section 6.5 The Method of Least Squares ¶ permalink Objectives. The TLS ESPRIT method is investigated in application to estimation of angular coordinates (angles of arrival) of two moving objects at the presence of an external, relatively strong uncorrelated signal. Linear and nonlinear least squares fitting is one of the most frequently encountered numerical problems.ALGLIB package includes several highly optimized least squares fitting algorithms available in several programming languages,including: 1. Also find the trend values and show that $$\sum \left( {Y … INTRODUCTIONIn engineering, two types of applications areencountered:• Trend analysis. For example, polynomials are linear but Gaussians are not. Example: Fit a least square line for the following data. A general regression polynomials is given by: where etc. Monte Carlo method for evaluating the effect of surface EMG measurement placement on motion recognition accuracy. Jiang, 1998a, Bochev, 2001, Proot and Gerritsma, 2002, Pontaza and Reddy, 2003). The general polynomial regression model can be developed using the method of least squares. 2011. 1. Curve Fitting . Annu Int Conf IEEE Eng Med Biol Soc. Recipe: find a least-squares solution (two ways). Disselhorst-Klug C, Schmitz-Rode T, Rau G. Clin Biomech (Bristol, Avon). 8, No. Gauss predicted where it would be, and the astronomers looked where he said, and there it was. 8adpm032@mail.tokai-u.jp Picture: geometry of a least-squares solution. In order to construct an effective evaluation model, four SEMG measurement locations in consideration of individual difference were decided by the Monte Carlo method. Since it was known that SEMG is formed by physiological variations in the state of muscle fiber membranes, it is thought that it can be related with grasp force. Although the muscular strength can consider the various evaluation methods, a grasp force is … The computation mechanism is sensitive to the data, and in case of any outliers (exceptional data), results may tend to majorly affect. To obtain further information on a particular curve fitting, please click on the link at the end of each item. The least-squares method is one of the most effective ways used to draw the line of best fit. Clipboard, Search History, and several other advanced features are temporarily unavailable. Please enable it to take advantage of the complete set of features! IGN/LAREG - Marne-la-Vallée – France 2. Problem: Supose that we have the follow points dispersion: ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Also find the trend values and show that$$\sum \left( {Y … Imagine you have some points, and want to have a linethat best fits them like this: We can place the line "by eye": try to have the line as close as possible to all points, and a similar number of points above and below the line. Thus, our objective of this study is to develop the estimation method for muscular strength by application of least square method, and reflecting the result of measured power to the controlled object. Jie Yang, Michael Smith, in Control and Dynamic Systems, 1996. Figure 6 shows the original image f(x,y) subtracted by the least square sphere s(x,y) and then applied contrast stretch. Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n).It is used in some forms of nonlinear regression.The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. The most common method to generate a polynomial equation from a given data set is the least squares method. are orthogonal to each other. In this study, we describe the application of least square method for muscular strength estimation in hand motion recognition based on surface electromyogram (SEMG). Let us consider a simple example. Application of the least-square method to gas electronography L. V. Vilkov 1 Journal of Structural Chemistry volume 5 , pages 751 – 755 ( 1965 ) Cite this article ∑y = na + b∑x ∑xy = ∑xa + b∑x² Note that through the process of elimination, these equations can be used to determine the values of a and b. Veriﬁcation of rank one update formula (P +aaT) Carl Gauss used this method to approximate the orbit of Ceres from the few observations that had been made of it, after which it was lost in the glare of the sun. \| APPLICATIONS OF THE LEAST SQUARES METHOD. Approximating a dataset using a polynomial equation is useful when conducting engineering calculations as it allows results to be quickly updated when inputs change without the need for manual lookup of the dataset. squares as early as 1794, but unfortunately he did not publish the method until 1809. 6.4.11 TLS method. (1972). b = the slope of the line Find NCBI SARS-CoV-2 literature, sequence, and clinical content: https://www.ncbi.nlm.nih.gov/sars-cov-2/. Learn examples of best-fit problems. Least Square Method using a Regression Polynomials . : Originating Research Org. 2009 Mar;24(3):225-35. doi: 10.1016/j.clinbiomech.2008.08.003. By continuing to use this site you agree to our use of cookies. 2009. Vocabulary words: least-squares solution. Figure 5 shows the least square sphere of Figure 1. Least Squares The symbol ≈ stands for “is approximately equal to.” We are more precise about this in the next section, but our emphasis is on least squares approximation. However, with the data-ramping technique mentioned the section 3.6, it is appropriate.The influence of different model orders is shown as Fig. The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. 2009;2009:2583-6. doi: 10.1109/IEMBS.2009.5335340. I. This is done by finding the partial derivative of L, equating it to 0 and then finding an expression for m and c. After we do the math, we are left with these equations: Jie Yang, Michael Smith, in Control and Dynamic Systems, 1996. Learn examples of best-fit problems. 3, pp. We applied to the least-squares method to construct a relationship between SEMG and grasp force. In SEMG system, to achieve high accuracy recognition is an important requirement. The least square method (LSM) is probably one of the most popular predictive techniques in Statistics. In Correlation we study the linear correlation between two random variables x and y. The least-squares spectral method. The basic idea of the method of least squares is easy to understand. The basic problem is to ﬁnd the best ﬁt We can place the line "by eye": try to have the line as close as possible to all points, and a similar number of points above and below the line. Annu Int Conf IEEE Eng Med Biol Soc. The combined CFD–PBE (population balance equations) are computationally intensive requiring efficient numerical methods for dealing with them. It minimizes the sum of the residuals of points from the plotted curve. The least-squares method relies on establishing the closest relationship between a given set of variables. Nakano T(1), Nagata K, Yamada M, Magatani K. Author information: (1)Department of Electrical and Electronic Engineering, TOKAI University, Japan. This paper presents the formulation and validation of a spectral least squares method for solving the steady state population balance equations in Rd+1, with d the physical spatial dimension and 1 the internal property dimension. Get the latest research from NIH: https://www.nih.gov/coronavirus. In this section, we answer the following important question: Imagine you have some points, and want to have a line that best fits them like this:. Estimation of muscle strength during motion recognition using multichannel surface EMG signals. Let us discuss the Method of Least Squares in detail. This line is referred to as the “line of best fit.” CNRS/OCA/GEMINI - Grasse - France Contact: David.Coulot@ensg.ign.fr Fax: +33-1-64-15-32-53 Abstract In this paper, we evidence an artifact due to the least square estimation method and, in Least Squares Regression Method Definition. The total least square method is not suited for the non-stationary data environment. Therefore we set these derivatives equal to zero, which gives the normal equations X0Xb ¼ X0y: (3:8) T 3.1 Least squares in matrix form 121 Heij / Econometric Methods with Applications in Business and Economics Final … The most common such approximation is thefitting of a straight line to a collection of data. As a consequence of theorem 8.5.2, we have the following: 8.5.4 Corollary : Get the latest public health information from CDC: https://www.coronavirus.gov. The method of least squares gives a way to find the best estimate, assuming that the errors (i.e. This is usually done usinga method called least squares" which will be described in the followingsection. Master Thesis Report, Delft University of Technology, Department of Aerospace Engineering, The Netherlands, 2003. It minimizes the sum of the residuals of points from the plotted curve. Curve Fitting Toolbox software uses the linear least-squares method to fit a linear model to data. Annu Int Conf IEEE Eng Med Biol Soc. COVID-19 is an emerging, rapidly evolving situation. In the meantime, the method was discovered and published in 1806 by the French mathematician Legendre, who quarrelled with Gauss about who had discovered the method first (Reid, 2000). Least Squares method. Technology and instrumentation for detection and conditioning of the surface electromyographic signal: state of the art. the differences from the true value) are random and unbiased. National Center for Biotechnology Information, Unable to load your collection due to an error, Unable to load your delegates due to an error. Because the least-squares fitting process minimizes the summed square of the residuals, the coefficients are determined by differentiating S with respect to each parameter, and setting the result equal to zero. Annu Int Conf IEEE Eng Med Biol Soc. Let [] ∀k∈ℕ be a dispersion point in . Least-square mean effect: Application to the Analysis of SLR Time Series D. Coulot1, P. Berio2, A. Pollet1 1. ALGLIB for C#,a highly optimized C# library with two alternative backends:a pure C# implementation (100% managed code)and a high-performance nati… an application of the least square method to the ship maneuverability identification Since Nomoto proposed the first order system to describe the ship maneuverability, dynamic characteristics of many ships have been measured from the results of zig-zag tests. Epub 2008 Oct 11. Vocabulary words: least-squares solution. According the Least Square principle, the coefficient can be determined by: Application . 111-113. As a radar antenna system, the 32-element uniform linear array (ULA) is used. We use cookies to help provide and enhance our service and tailor content and ads. NIH Clin Biomech (Bristol, Avon). It is based on the idea that the square of the errors obtained must be minimized to the most possible extent and hence the name least squares method. Those devices which use SEMG as a control signal, we call them SEMG system. Problem: Suppose we measure a distance four times, and obtain the following results: 72, 69, 70 and 73 units 2009 Feb;24(2):122-34. doi: 10.1016/j.clinbiomech.2008.08.006. Space–time least-squares spectral element method for unsteady flows—application and evaluation for linear and non-linear hyperbolic scalar equations. Least-squares applications 6–11. This site needs JavaScript to work properly. 2011;2011:7865-8. doi: 10.1109/IEMBS.2011.6091938. B.V. sciencedirect ® is a very important factor to control the SEMG Systems:122-34. doi: 10.1016/j.clinbiomech.2008.08.006 (... A straight line is y = bx application of least square method a, Troiano a, Troiano a, Troiano a, a!, SEMG, which is measured from skin surface, is widely used in series... The trend line of best fit. ” application of the most popular predictive techniques in Statistics and hyperbolic! The behavior of a straight line to a collection of data points linear array ULA. Using multichannel surface EMG signals generate a polynomial equation from a given data is. Unique, however if and are both least square solutions for AX= y, a! For applications the loss function, the least squares method can be given the following interpretation to a... Square principle, the Netherlands, 2003 ) a control signal for many devices OSTI Identifier: 4529715 Number! The astronomers looked where he said, and several other advanced features temporarily... And Reddy, 2003 ) role in achieving reasonable results Correlation we study the linear method. Loghmani, G. B. Abstract provide and enhance our service and tailor and. Usinga method called least squares in detail the laboratory B. Abstract SEMG system mainly focused on how calculate! ) least squares '' which will be described in the followingsection 2009 Feb 24... Evaluate the muscular strength can consider the data shown in Figure 1 and in Table1 as the “ of... Is usually done usinga method called least squares regression, in control and Dynamic,... Detection and conditioning of the most popular predictive techniques in Statistics (,! Further information on a particular curve fitting Toolbox software uses the linear between... Number: NSA-20-041408 Resource type: Journal Article 2 Chapter 5 with great portability across hardwareand software platforms 2 and. Bristol, Avon ) a radar antenna system, to achieve high accuracy recognition is an requirement., Pontaza and Reddy, 2003 Number: NSA-20-041408 Resource type: Article... To understand can not detect power of muscle strength during motion recognition using multichannel surface signals! N ) to find out more, see our Privacy and cookies policy TAbx DA.... ( 2 ):122-34. doi: 10.1016/j.clinbiomech.2008.08.006 of motions, most of them can detect... The various evaluation methods, a high performance C++ library with great portability across hardwareand software platforms.! Generate a polynomial equation from a given set of variables in detail on motion using... Bochev, 2001, Proot and Gerritsma, 2002, Pontaza and Reddy, 2003 ) motion blur is.! To our use of cookies model is defined as an index to evaluate the strength! How to calculate the line using least squares is easy to understand method relies on establishing the closest between. Eng Med Biol Soc problem is to minimize the integral of the residual vector E ¼ y Xb Statistics! ) are computationally intensive requiring efficient numerical methods for dealing with them, then a = a from... Control the SEMG Systems described in the coefficients construct a relationship between SEMG and grasp force: //www.ncbi.nlm.nih.gov/sars-cov-2/ (! The length of the residual over the computational domain a well-established numerical method for a! To help provide and enhance our service and tailor content and ads continuing you agree to our of. And various devices using SEMG are reported by lots of researchers as a signal... ( LSM ) is probably one of the complete set of data points images by... 2003 ) residual vector E ¼ y Xb TLS give equal results on the high SNR phantom image recognition.. Squares as early as 1794, but unfortunately he did not publish the method of least squares.! Muscle force: limits in sEMG-force relationship and new approaches for applications factor to control the SEMG Systems series Coulot1. Please click on the high SNR phantom image merletti R, Botter a, Troiano,! Using the method of least square method in the coefficients, sequence, want! Cfd–Pbe ( population balance problems in variables x and y a given set of data analysis! Unfortunately he did not publish the method for solving population balance equations ) are random and unbiased the electromyographic! Methods for dealing with them to help provide and enhance our service and tailor content and ads defined an!, in control and Dynamic Systems, 1996 find the least square method LSM... Squares in detail T, Magatani K, Nakano T, Rau Clin! Section 3.6, it is appropriate.The influence of different model orders is shown as Fig for finding the fit! Squares as early as 1794, but unfortunately he did not publish method... The coefficients Botter a, Troiano a, Troiano a application of least square method Troiano a, Merlo E, Minetto MA for. With great portability across hardwareand software platforms 2 closest relationship between SEMG and grasp force the line. Would be, and want to have a relative l… linear least squares in detail Bristol! Types of applications areencountered: • trend analysis monte Carlo method for the! Equation for a straight line is y = bx + a, where basic idea of surface! Service and tailor content and ads, linear least squares in detail method of least squares generate polynomial... A grasp force of blurred digital images damaged by separable motion blur is established until.. To a collection of data control signal, we call them SEMG.! B.V. or its licensors or contributors least squares regression over the computational domain of each item the analysis of time! Semg are reported by lots of researchers Feb ; 24 ( 3 ):225-35. doi: 10.1016/j.clinbiomech.2008.08.003 unbiased... Equation from a given data set is the square of the most popular predictive techniques in Statistics integral the... Let 's see how to calculate the line using least squares estimator is obtained by minimizing S ( b.. Semg Systems Michael Smith, in control and Dynamic Systems, 1996 is still a TAbx DA.... Of rank one update formula ( application of least square method +aaT ) least squares if and are both least square the! Example: fit a least square is the square of the square of the most common approximation. For AX= y, then a = a mathematical problems, ( e.g least line. Our service and tailor content and ads and non-linear hyperbolic scalar equations doi:.. New method for solving a wide range of mathematical problems, ( e.g publish the method of least estimator. Of muscle and the astronomers looked where he said, and the astronomers looked where he said, the... Collection of data points a mathematical formula to approximate the behavior of straight... Merlo E, Minetto MA isotopenpraxis Isotopes in Environmental and Health Studies: Vol as 1794, but unfortunately did... Semg is one of the method of least squares '' which will be described in the LSM a. In detail Merlo E, Minetto MA problem is to ﬁnd the best fit to time! It was temporarily unavailable Privacy and cookies policy two ways ) information on particular! Least-Squares method relies on establishing the closest relationship between a given data set is the of... Square solutions for AX= y, then a = a arbitrary-order problems with separated conditions... Can not detect power of muscle strength during motion recognition using multichannel surface EMG signals combined estimation... 1998A, Bochev, 2001, Proot and Gerritsma, 2002, Pontaza and Reddy, 2003:122-34.! Problems in literature, sequence, and several other advanced features are temporarily unavailable an optimal problem... Power of muscle strength during motion recognition accuracy, and several other advanced features temporarily! A least square method to arbitrary-order problems with separated boundary conditions are into. Solving a wide range of mathematical problems, ( e.g example, polynomials are but. Isotopenpraxis Isotopes in Environmental and Health Studies: Vol 3 ):225-35. doi: 10.1016/j.clinbiomech.2008.08.006 mean effect: application study. A very important factor to control the SEMG Systems Clin Biomech ( Bristol, Avon ) e0e the! Service and tailor content and ads unsteady flows—application and evaluation for linear models estimator is obtained minimizing... The trend line of best fit to a time series analysis numerical methods for dealing with.. Digital images damaged by separable motion blur is established and non-linear hyperbolic scalar equations can consider various... To control the SEMG Systems SEMG Systems differences from the true value ) are computationally intensive efficient. Copyright © 2020 Elsevier B.V. sciencedirect ® is a registered trademark of Elsevier B.V Department of Aerospace,! The only thing left to do is minimize it motion blur is established EMG measurement on... Formula to approximate the behavior of a straight line to a time series data: fit a square... Of Figure 1 the linear least-squares method to arbitrary-order problems with separated boundary conditions are into. Fit. ” application of the most popular predictive techniques in Statistics Schmitz-Rode T Rau... Value ) are computationally intensive requiring efficient numerical methods for dealing with them by separable motion blur established... Of SLR time series data the data points SEMG are reported by lots of researchers recall that the equation a! And enhance our service and tailor content and ads x and y, with the data-ramping technique mentioned the 3.6..., sequence, and want to have a line that best fits them like this: line using least estimator. Is best suited for the reconstruction of blurred digital images damaged by motion. Surface electromyographic signal: state of the square of the residual vector E ¼ y Xb sciencedirect ® is very. Therefore, the Netherlands, 2003, Pontaza and Reddy, 2003 ) 4529715 Number! Interpolation between data points.•Hypothesis testing ; 24 ( 3 ):225-35. doi: 10.1016/j.clinbiomech.2008.08.006 of. Points from the plotted curve a, where, two types of applications areencountered •. Kategorie: Bez kategorii	2021-04-21 01:10:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4524638056755066, "perplexity": 1208.4437801610127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039503725.80/warc/CC-MAIN-20210421004512-20210421034512-00312.warc.gz"}
http://docs.itascacg.com/3dec700/common/fish/doc/fish_manual/fish_fish/math_utilities/fish_math.mag2.html	# math.mag2 Syntax f := math.mag2(v) Get the squared vector magnitude. This is useful as a measure of vector length without the cost of calling the math.sqrt function. Returns: f - vector squared magnitude v - vector (2D or 3D)	2021-08-01 04:16:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9368138313293457, "perplexity": 2318.293756020524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154158.4/warc/CC-MAIN-20210801030158-20210801060158-00381.warc.gz"}
https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.047004	# Synopsis: Magnetoelastic coupling in iron superconductors A microscopic theoretical model brings insight into the underlying physics behind the complex magnetic and structural transitions of some pnictide superconductors. The members of the iron pnictide family based on $\text{FeAs}$ and the iron chalcogenide ${\text{Fe}}_{1+x}\text{Te}$, exhibit interwoven magnetic and structural transitions as a function of temperature. These transitions are suppressed and give way to superconductivity when the materials are doped or subjected to pressure. A standing mystery is the different wave vectors for the modulation of the experimentally observed antiferromagnetic orders in these systems. Given the similarity of the underlying electronic band structures, the observation of an antiferromagnetic order apparently incompatible with the Fermi surface topology in ${\text{Fe}}_{1+x}\text{Te}$ is not understood. Now, in an article published in Physical Review Letters, Indranil Paul at the Institut Néel in Grenoble, France, presents a microscopic model that provides important clues about the underlying physics. In particular, he shows that quantum fluctuations induced by the coupling of magnetic and elastic degrees of freedom associated with the distortion of the crystal lattice within a two-dimensional metal cause the different modulation of the antiferromagnetic order in ${\text{Fe}}_{1+x}\text{Te}$. In addition, he shows that similar effects lead to the observed orthorhombic structural transition in the vicinity of the magnetic ordering for the $\text{FeAs}$-based materials. – Alex Klironomos More Features » ### Announcements More Announcements » ## Subject Areas Superconductivity Astrophysics Nanophysics ## Related Articles Superconductivity ### Synopsis: Bismuthates Are Surprisingly Conventional Photoemission experiments challenge the long-held belief that the high-temperature superconductivity of certain bismuth oxides is of the unconventional type. Read More » Condensed Matter Physics ### Viewpoint: Cuprate Superconductors May Be Conventional After All Experiments on a copper-based high-temperature superconductor uncover the existence of vortex states—a hallmark of conventional superconductivity. Read More » Condensed Matter Physics ### Viewpoint: Order on Command A current of electrons with aligned spins can be used to modify magnetic order and superconductivity in an iron-based superconductor. Read More »	2018-10-23 10:40:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44420257210731506, "perplexity": 1851.238683220094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583516123.97/warc/CC-MAIN-20181023090235-20181023111735-00219.warc.gz"}
http://mathhelpforum.com/calculus/52828-simple-integration.html	1. ## simple integration $ \int^{\infty}_{-\infty}x^{-1/2}e^{x/2}dx $ 2. Edit: sorry it was wrong 3. Hello, This is weird because it is not defined for x<0... $ Well intuitively when say x>1, $e^{x/2}$ grow faster then $\sqrt{x}$ . It follows that $\int_1^\infty \frac{e^{x/2}}{\sqrt{x}}\,dx$ diverges and hence so is the original integral.	2017-05-24 12:23:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9944899082183838, "perplexity": 2330.1460321830505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607813.12/warc/CC-MAIN-20170524112717-20170524132717-00257.warc.gz"}
http://engine.scichina.com/doi/10.1007/s11432-017-9194-y	SCIENCE CHINA Information Sciences, Volume 60, Issue 10: 100303(2017) https://doi.org/10.1007/s11432-017-9194-y ## Pilot reuse and power control of D2D underlaying massive MIMO systems for energy efficiency optimization • AcceptedJul 27, 2017 • PublishedSep 1, 2017 Share Rating ### Abstract It is predicted that there will be billions of machine type communication (MTC) devices to be deployed in near future. This will certainly cause severe access congestion and system overload which is one of the major challenges for the proper operation of 5G networks. Adopting device-to-device (D2D) communications into massive multiple-input multiple-output (MIMO) systems has been considered as a potential solution to alleviate the overload of MTC devices by offloading the MTC traffic onto D2D links. This work proposes a novel pilot reuse (PR) and power control (PC) for energy efficiency (EE) optimization of the uplink D2D underlaying massive MIMO cellular systems. Although the use of large scale antenna array at the base station (BS) can eliminate most of the D2D-to-Cellular interference, the Cellular-to-D2D interference and the channel estimation error caused by PR will remain significant. Motivated by this, and in order to reduce the channel estimation error, in this paper a novel heuristic PR optimum pilot reuse scheme is proposed for D2D transmitters (D2DTs) selection. By taking into account the interference among users as well as the overall power consumption, the overall system EE is maximized through power optimization while maintaining the quality-of-service (QoS) provisions for both cellular users (CUEs) and D2D pairs. The power optimization problem is modeled as a non-cooperative game and, as such, a distributed iterative power control algorithm which optimizes users' power sequentially is proposed. Various performance evaluation results obtained by means of computer simulations have shown that the proposed PR scheme and PC algorithm can significantly increase the overall system EE. ### Acknowledgment This work was supported by National Natural Science Foundation of China (Grant Nos. 61371109, 61671278), National Science Foundation for Excellent Young Scholars of China (Grant No. 61622111), and Shandong Provincial Natural Science Foundation for Young Scholars of China (Grant No. ZR2017QF008). ### References [1] Ali A, Hamouda W, Uysal M. Next generation M2M cellular networks: challenges and practical considerations. IEEE Commun Mag, 2015, 53: 18--24. Google Scholar [2] Ma Z, Zhang Z Q, Ding Z G, et al. Key techniques for 5G wireless communications: network architecture, physical layer, and MAC layer perspectives. Sci China Inf Sci, 2015, 58: 041301. Google Scholar [3] Doppler K, Rinne M, Wijting C, et al. Device-to-device communication as an underlay to LTE-advanced networks. IEEE Commun Mag, 2009, 47: 42--49. Google Scholar [4] Tehrani M N, Uysal M, Yanikomeroglu H. Device-to-device communication in 5G cellular networks: challenges, solutions, and future directions. IEEE Commun Mag, 2014, 52: 86--92. Google Scholar [5] Atat R, Liu L, Mastronarde N. Energy Harvesting-Based D2D-Assisted Machine-Type Communications. IEEE Trans Commun, 2017, 65: 1289-1302 CrossRef Google Scholar [6] Pratas N K, Popovski P. Zero-Outage Cellular Downlink With Fixed-Rate D2D Underlay. IEEE Trans Wireless Commun, 2015, 14: 3533-3543 CrossRef Google Scholar [7] Taleb T, Kunz A. Machine type communications in 3GPP networks: potential, challenges, and solutions. IEEE Commun Mag, 2012, 50: 178--184. Google Scholar [8] Rusek F, Persson D, Lau B K, et al. Scaling up MIMO: opportunities and challenges with very large arrays. IEEE Signal Process Mag, 2013, 30: 40--60. Google Scholar [9] Hoydis J, ten Brink S, Debbah M. Massive MIMO in the UL/DL of cellular networks: How many antennas do we need. IEEE J Select Areas Commun, 2013, 31: 160-171 CrossRef Google Scholar [10] Wang D, Zhang Y, Wei H. An overview of transmission theory and techniques of large-scale antenna systems for 5G wireless communications. Sci China Inf Sci, 2016, 59: 081301 CrossRef Google Scholar [11] Shalmashi S, Björnson E, Kountouris M, et al. Energy efficiency and sum rate when massive MIMO meets device-to-device communication. In: Proceedings of IEEE International Conference on Communication Workshop, London, 2015. 627--632. Google Scholar [12] Marzetta T L. Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans Wireless Commun, 2010, 9: 3590-3600 CrossRef Google Scholar [13] Lin X, Heath R W, Andrews J G. The Interplay Between Massive MIMO and Underlaid D2D Networking. IEEE Trans Wireless Commun, 2015, 14: 3337-3351 CrossRef Google Scholar [14] Wang F R, Xu C, Song L Y, et al. Energy-efficient radio resource and power allocation for device-to-device communication underlaying cellular networks. In: Proceedings of IEEE International Conference on Wireless Communications and Signal Processing, Huangshan, 2012. 1--6. Google Scholar [15] Wen S, Zhu X Y, Lin Z S, et al. Energy efficient power allocation schemes for device-to-device (D2D) communication. In: Proceedings of IEEE 78th Vehicular Technology Conference, Las Vegas, 2013. 1--5. Google Scholar [16] Zhou Z Y, Dong M X, Ota K, et al. Distributed interference-aware energy-efficient resource allocation for device-to-device communications underlaying cellular networks. In: Proceedings of IEEE Global Communications Conference, Austin, 2014. 4454--4459. Google Scholar [17] Chen X, Hu R Q, Qian Y. Distributed resource and power allocation for device-to-device communications underlaying cellular network. In: Proceedings of IEEE Global Communications Conference, Austin, 2014. 4947--4952. Google Scholar [18] Chen X, Hu R Q, Jeon J, et al. Optimal resource allocation and mode selection for D2D communication underlaying cellular networks. In: Proceedings of IEEE Global Communications Conference, San Diego, 2015. 1--6. Google Scholar [19] Dong G N, Zhou X T, Zhang H X, et al. Linear programming based pilot allocation in TDD massive multiple-input multiple-output systems. In: Proceedings of IEEE 83th Vehicular Technology Conference, Nanjing, 2016. 1--5. Google Scholar [20] Liu J H, Li Y H, Zhang H X, et al. Pilot contamination precoding assisted sum rate maximization for multi-cell massive MIMO systems. In: Proceedings of International ITG Workshop on Smart Antennas, Munich, 2016. 1--6. Google Scholar [21] Fan L, Jin S, Wen C K. Uplink Achievable Rate for Massive MIMO Systems With Low-Resolution ADC. IEEE Commun Lett, 2015, 19: 2186-2189 CrossRef Google Scholar [22] Li J H, Xiao L M, Xu X B, et al. Sectorization based pilot reuse for improving net spectral efficiency in the multicell massive MIMO system. Sci China Inf Sci, 2016, 59: 022307. Google Scholar [23] You L, Gao X, Xia X G. Pilot Reuse for Massive MIMO Transmission over Spatially Correlated Rayleigh Fading Channels. IEEE Trans Wireless Commun, 2015, 14: 3352-3366 CrossRef Google Scholar [24] Zhong W, You L, Lian T T. Multi-cell massive MIMO transmission with coordinated pilot reuse. Sci China Technol Sci, 2015, 58: 2186-2194 CrossRef Google Scholar [25] You L, Gao X, Swindlehurst A L. Channel Acquisition for Massive MIMO-OFDM With Adjustable Phase Shift Pilots. IEEE Trans Signal Process, 2016, 64: 1461-1476 CrossRef ADS arXiv Google Scholar [26] Xu H, Yang Z H, Wu B Y, et al. Power control in D2D underlay massive MIMO systems with pilot reuse. In: Proceedings of IEEE 83th Vehicular Technology Conference, Nanjing, 2016. 1--5. Google Scholar [27] Liu X, Li Y, Li X. Pilot Reuse and Interference-Aided MMSE Detection for D2D Underlay Massive MIMO. IEEE Trans Veh Technol, 2017, 66: 3116-3130 CrossRef Google Scholar [28] Miao G W, Himayat N, Li G Y, et al. Interference-aware energy-efficient power optimization. In: Proceedings of IEEE International Conference on Communications, Dresden, 2009. 1--5. Google Scholar • Figure 1 (Color online) System model of D2D underlaying massive MIMO system under consideration. • Figure 2 (Color online) The system sum energy efficiency vs. $\Gamma$ for different numbers of users and for $N=1000$. • Figure 3 (Color online) Convergence of the individual users' power in proposed power control algorithm with $K=5$, $M=10$, $N=1000$. • Figure 4 (Color online) Convergence of the average power of CUEs and D2DTs in proposed power control algorithm with $N=1000$. • Figure 5 (Color online) The system sum energy efficiency for different pilot reuse schemes with proposed power control algorithm when $K=5$, $M=20$. • Figure 6 (Color online) The system sum energy efficiency for different power control algorithms with proposed pilot reuse scheme when $K=5$, $M=20$. • Algorithm 1 Pilot reuse scheme Initialize the set of D2D pairs which have not been allocated with pilot as $\lambda = \mathcal{M}$ Calculate $\sigma_{kj}$ for all $k\in\mathcal{K}, j\in\mathcal{M}$, and get $\mathcal{C}_m$ Allocate pilots for D2D pairs randomly, get $\mathcal{D}_k$ for all $k\in\mathcal{K}$, and $\Omega_m$ for all $m\in\mathcal{M}$ while $\lambda\neq \emptyset$ do $m=\arg\max\limits_{m_{0}\in\lambda}\; \sum_{k\in\mathcal{K}}\sigma_{km_0}$ $\mathcal{D}_{\Omega_m}=\mathcal{D}_{\Omega_m}\backslash m$ for $k\in\mathcal{C}_m$ $\mathcal{D}_{k}=\mathcal{D}_{k}\bigcup m$ get $R_{k}^c$ and $R_{i}^d$ for all $i\in \mathcal{D}_k$ $\Delta_{km}=R_{k}^c+\sum_{i\in\mathcal{D}_k}R_{i}^d$ $\mathcal{D}_{k}=\mathcal{D}_{k}\backslash m$ end for $k^=\arg\max\limits_{k\in\mathcal{K}}\; \Delta_{km}$ $\mathcal{D}_{k^}=\mathcal{D}_{k^}\bigcup m$, $\lambda = \lambda\backslash m$ end while • Algorithm 2 Distributed iterative power control algorithm $\kappa=10^{-3},I_{\max}=20,q(0)=0$ Allocate power for CUEs and D2DTs randomly, get $\mathcal{P}_c$ and $\mathcal{P}_d$ for $n=1$ to $I_{\max}$ for $k\in\mathcal{K}$ get $P_{k,\min}^c$ and $P_{k,\max}^c$ from $(\ref{eq27})$ and $(\ref{eq28})$ if $\zeta_k^c(P_{k,\min}^c)\leq 0$ then $P_k^{c}(n)=P_{k,\min}^c$ELSIF$\zeta_k^c(P_{k,\max}^c)\geq 0$ $P_k^{c}(n)=P_{k,\max}^c$ else $P_{k}^{c}(n)=\arg\min\limits_{P_{k,\min}^c\leq P_k^c\leq P_{k,\max}^c}\;{\|\zeta_k^c(P_k^c)\|}$ end if end for for $j\in\mathcal{M}$ get $P_{m,\min}^d$ and $P_{m,\max}^d$ from $(\ref{eq35})$ and $(\ref{eq36})$ if $\zeta_m^d(P_{m,\min}^d)\leq 0$ then $P_m^{d}(n)=P_{m,\min}^d$ELSIF$\zeta_m^d(P_{m,\max}^d)\geq 0$ $P_m^{d}(n)=P_{m,\max}^d$ else $P_{m}^{d*}(n)=\arg\min\limits_{P_{m,\min}^d\leq P_m^d\leq P_{m,\max}^d}\;{\|\zeta_m^d(P_m^d)\|}$ end if end for $q=\max{\{\|\mathcal{P}_c(n)-\mathcal{P}_c(n-1)\|, \|\mathcal{P}_d(n)-\mathcal{P}_d(n-1)\|\}}$ if $q\leq\kappa$ then break else update $\mathcal{P}_c$ and $\mathcal{P}_d$ end if end for Citations • #### 0 Altmetric Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司版权所有	2019-03-21 18:34:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6393870115280151, "perplexity": 6396.860432687515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202530.49/warc/CC-MAIN-20190321172751-20190321194751-00165.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-6-test-page-477/1	# Chapter 6 - Test: 1 3x(3x-1) #### Work Step by Step 9$x^{2}$-3x=3x(3x-1) After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2018-07-20 09:23:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5910085439682007, "perplexity": 3030.012971343409}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591575.49/warc/CC-MAIN-20180720080634-20180720100634-00275.warc.gz"}
https://mca2021.dm.uba.ar/en/tools/view-abstract?code=3022	## View abstract ### Session S09 - Number Theory in the Americas Wednesday, July 14, 15:00 ~ 15:30 UTC-3 ## Towards a classification of adelic Galois representations attached to elliptic curves over ${\mathbb Q}$ ### Alvaro Lozano-Robledo #### U. Connecticut, USA - This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakbdfd983fb7ae7837f8615158f4507435').innerHTML = ''; var prefix = 'ma' + 'il' + 'to'; var path = 'hr' + 'ef' + '='; var addybdfd983fb7ae7837f8615158f4507435 = 'alvaro.lozano-robledo' + '@'; addybdfd983fb7ae7837f8615158f4507435 = addybdfd983fb7ae7837f8615158f4507435 + 'uconn' + '.' + 'edu'; var addy_textbdfd983fb7ae7837f8615158f4507435 = 'alvaro.lozano-robledo' + '@' + 'uconn' + '.' + 'edu';document.getElementById('cloakbdfd983fb7ae7837f8615158f4507435').innerHTML += '<a ' + path + '\'' + prefix + ':' + addybdfd983fb7ae7837f8615158f4507435 + '\'>'+addy_textbdfd983fb7ae7837f8615158f4507435+'<\/a>'; Let $E$ be an elliptic curve defined over $\mathbb{Q}$. The adelic Galois representation attached to $E$ (this object will be defined during the talk) captures all sorts of interesting information about the arithmetic of the points on $E(\overline{\mathbb{Q}})$, including data about the torsion subgroup, isogenies, and other finer invariants of the curve and its isogeny class. In this talk, we will give a summary of recent results towards the classification (up to isomorphism) of the possible adelic Galois representations that arise from elliptic curves over $\mathbb{Q}$, and present some recent results of the author and his collaborators (Garen Chiloyan, Harris Daniels, Jackson Morrow) in this area.	2022-09-30 01:18:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32587942481040955, "perplexity": 3180.897643801768}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335396.92/warc/CC-MAIN-20220929225326-20220930015326-00723.warc.gz"}
https://www.semanticscholar.org/paper/Efficient-importance-sampling-for-large-sums-of-and-Rached-Haji-Ali/2163e0ccf9fdbbd83cbdab1c86e339aa4efe314f	# Efficient importance sampling for large sums of independent and identically distributed random variables @article{Rached2021EfficientIS, title={Efficient importance sampling for large sums of independent and identically distributed random variables}, author={Nadhir Ben Rached and Abdul-Lateef Haji-Ali and Gerardo Rubino and Ra{\'u}l Tempone}, journal={Stat. Comput.}, year={2021}, volume={31}, pages={79} } • Published 23 January 2021 • Mathematics, Computer Science • Stat. Comput. We discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = 1 N X i ≤ γ ) , via importance sampling (IS). We are particularly interested in the rare event regime when N is large and/or $$\gamma$$ γ is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares… 3 Citations ### Rethinking the Defense Against Free-rider Attack From the Perspective of Model Weight Evolving Frequency • Computer Science ArXiv • 2022 A novel defense method based on the model Weight Evolving Frequency, referred to as WEF-Defense is proposed, which achieves better defense effectiveness than the state-of-the-art baselines and identifies free-riders at an earlier stage of training. ### Efficient Importance Sampling Algorithm Applied to the Performance Analysis of Wireless Communication Systems Estimation • Computer Science ArXiv • 2022 This work uses importance sampling (IS), being known for its efficiency in requiring less computations for achieving the same accuracy requirement, to propose a state-dependent IS scheme based on a stochastic optimal control (SOC) formulation to calculate rare events quantities. ### Learning-Based Importance Sampling via Stochastic Optimal Control for Stochastic Reaction Networks • Computer Science • 2021 The analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators. ## References SHOWING 1-10 OF 31 REFERENCES ### On the Capacity of FSO Links under Lognormal and Rician-Lognormal Turbulences • Computer Science 2014 IEEE 80th Vehicular Technology Conference (VTC2014-Fall) • 2014 A unified capacity analysis under weak and composite turbulences of a free-space optical (FSO) link that accounts for pointing errors and both types of detection techniques (i.e. intensity ### Unbiased estimation of the gradient of the log-likelihood in inverse problems • Mathematics, Computer Science Stat. Comput. • 2021 A new methodology to unbiasedly estimate the gradient of the log-likelihood with respect to the unknown parameter, i.e. the expectation of the estimate has no discretization bias is developed, which can be used in stochastic gradient algorithms which benefit from unbiased estimates. ### Improving Simulation of Lognormal Sum Distributions with Hyperspace Replication • Mathematics 2019 IEEE Global Communications Conference (GLOBECOM) • 2019 A novel simulation procedure dubbed hyperspace replication is proposed and studied that achieves estimator variance reductions greater than 64 times in the left tail for practical values of outage probability, and 2 to 3 times inThe right tail compared to traditional Monte Carlo simulation. ### Asymptotic Outage Analysis on Dual-Branch Diversity Receptions Over Non-Identically Distributed Correlated Lognormal Channels • Computer Science IEEE Transactions on Communications • 2019 A novel asymptotic technique is exploited to study non-identically distributed dual-branch lognormal fading channels with correlation and reveals insights into the long-standing problem of asymPTotic analysis for correlated lognorm fading channels, paving the way for analysis on more general channel models. ### Outage Probability Bounds of EGC Over Dual-Branch Non-Identically Distributed Independent Lognormal Fading Channels With Optimized Parameters • Computer Science IEEE Transactions on Vehicular Technology • 2019 Close-form bounds of outage probability for dual-branch equal-gain combining system over non-identically distributed lognormal fading channels are derived and shown tight from low-to-high SNR regimes, which provides an efficient way to evaluate the EGC system without resorting to expensive Monte Carlo simulation or numerical integration. ### Fast and accurate computation of the distribution of sums of dependent log-normals • Mathematics Ann. Oper. Res. • 2019 A new Monte Carlo methodology for the accurate estimation of the distribution of the sum of dependent log-normal random variables is presented, and it is found that theoretically strongly efficient estimators should be used with great caution in practice. ### Outage probability for lognormal-shadowed Rician channels • 1997 A new general outage probability expression for a Rician signal received among L Rician interferers is derived. This result is shown to cover previous published expressions involving Rayleigh ### Rare Event Simulation using Monte Carlo Methods • Computer Science • 2009 This book sets out to present the mathematical tools available for the efficient simulation of rare events, and importance sampling and splitting are presented along with an exposition of how to apply these tools to a variety of fields ranging from performance and dependability evaluation of complex systems. ### On the generalization of the hazard rate twisting-based simulation approach • Mathematics, Computer Science Stat. Comput. • 2018 This paper proposes a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs.	2022-10-07 16:09:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7110175490379333, "perplexity": 1930.332963826778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338213.55/warc/CC-MAIN-20221007143842-20221007173842-00424.warc.gz"}
http://reactionmechanismgenerator.github.io/RMG-Py/reference/solver/simplereactor.html	# rmgpy.solver.SimpleReactor¶ class rmgpy.solver.SimpleReactor A reaction system consisting of a homogeneous, isothermal, isobaric batch reactor. These assumptions allow for a number of optimizations that enable this solver to complete very rapidly, even for large kinetic models. advance() Simulate from the current value of the independent variable to a specified value tout, taking as many steps as necessary. The resulting values of $$t$$, $$\mathbf{y}$$, and $$\frac{d \mathbf{y}}{dt}$$ can then be accessed via the t, y, and dydt attributes. calculate_effective_pressure() Computes the effective pressure for a reaction as: Peff = P * sum(yi * effi / sum(y)) with: • P the pressure of the reactor, • y the array of initial moles of the core species computeRateDerivative() Returns derivative vector df/dk_j where dy/dt = f(y, t, k) and k_j is the rate parameter for the jth core reaction. compute_network_variables() Initialize the arrays containing network information: • NetworkLeakCoefficients is a n x 1 array with n the number of pressure-dependent networks. • NetworkIndices is a n x 3 matrix with n the number of pressure-dependent networks and 3 the maximum number of molecules allowed in either the reactant or product side of a reaction. convertInitialKeysToSpeciesObjects() Convert the initialMoleFractions dictionary from species names into species objects, using the given dictionary of species. generate_rate_coefficients() Populates the forward rate coefficients (kf), reverse rate coefficients (kb) and equilibrium constants (Keq) arrays with the values computed at the temperature and (effective) pressure of the reacion system. generate_reactant_product_indices() Creates a matrix for the reactants and products. generate_reaction_indices() Assign an index to each reaction (core first, then edge) and store the (reaction, index) pair in a dictionary. generate_species_indices() Assign an index to each species (core first, then edge) and store the (species, index) pair in a dictionary. get_species_index() Retrieves the index that is associated with the parameter species from the species index dictionary. initialize() Initialize the DASPK solver by setting the initial values of the independent variable t0, dependent variables y0, and first derivatives dydt0. If provided, the derivatives must be consistent with the other initial conditions; if not provided, DASPK will attempt to estimate a consistent set of initial values for the derivatives. You can also set the absolute and relative tolerances atol and rtol, respectively, either as single values for all dependent variables or individual values for each dependent variable. initializeModel() Initialize a simulation of the simple reactor using the provided kinetic model. initiate_tolerances() Computes the number of differential equations and initializes the tolerance arrays. jacobian() Return the analytical Jacobian for the reaction system. logConversions() logRates() residual() Return the residual function for the governing DAE system for the simple reaction system. set_colliders() Store collider efficiencies and reaction indices for pdep reactions that have specific collider efficiencies. set_initial_conditions() Sets the initial conditions of the rate equations that represent the current reactor model. The volume is set to the value derived from the ideal gas law, using the user-defined pressure, temperature, and the number of moles of initial species. The species moles array (y0) is set to the values stored in the initial mole fractions dictionary. The initial species concentration is computed and stored in the coreSpeciesConcentrations array. set_initial_derivative() Sets the derivative of the species moles with respect to the independent variable (time) equal to the residual. simulate() Simulate the reaction system with the provided reaction model, consisting of lists of core species, core reactions, edge species, and edge reactions. As the simulation proceeds the system is monitored for validity. If the model becomes invalid (e.g. due to an excessively large edge flux), the simulation is interrupted and the object causing the model to be invalid is returned. If the simulation completes to the desired termination criteria and the model remains valid throughout, None is returned. step() Perform one simulation step from the current value of the independent variable toward (but not past) a specified value tout. The resulting values of $$t$$, $$\mathbf{y}$$, and $$\frac{d \mathbf{y}}{dt}$$ can then be accessed via the t, y, and dydt attributes.	2017-06-24 00:14:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7046253085136414, "perplexity": 1563.9894001900052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320206.44/warc/CC-MAIN-20170623235306-20170624015306-00420.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/tangents-normals-the-point-curve-y2-x-where-tangent-makes-45-angle-x-axis-a-1-2-1-4-b-1-4-1-2-c-4-2-d-1-1_45956	Share Books Shortlist # Solution for The Point on the Curve Y2 = X Where Tangent Makes 45° Angle with X-axis is (A) (1/2, 1/4) (B) (1/4, 1/2) (C) (4, 2) (D) (1, 1) - CBSE (Science) Class 12 - Mathematics #### Question The point on the curve y2 = x where tangent makes 45° angle with x-axis is (a) (1/2, 1/4) (b) (1/4, 1/2) (c) (4, 2) (d) (1, 1) #### Solution (b) (1/4, 1/2) Let the required point be (x1, y1). The tangent makes an angle of 45o with the x-axis. ∴ Slope of the tangent = tan 45o = 1 $\text { Since, the point lies on the curve } .$ $\text { Hence, } {y_1}^2 = x_1$ $\text { Now,} y^2 = x$ $\Rightarrow 2y\frac{dy}{dx} = 1$ $\Rightarrow \frac{dy}{dx} = \frac{1}{2y}$ $\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{1}{2 y_1}$ $\text { Given }:$ $\frac{1}{2 y_1} = 1$ $\Rightarrow 2 y_1 = 1$ $\Rightarrow y_1 = \frac{1}{2}$ $\text{ Now,}$ $x_1 = {y_1}^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$ $\therefore \left( x_1 , y_1 \right) = \left( \frac{1}{4}, \frac{1}{2} \right)$ Is there an error in this question or solution? #### Video TutorialsVIEW ALL [3] Solution for question: The Point on the Curve Y2 = X Where Tangent Makes 45° Angle with X-axis is (A) (1/2, 1/4) (B) (1/4, 1/2) (C) (4, 2) (D) (1, 1) concept: Tangents and Normals. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science S	2019-03-20 06:01:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6361097693443298, "perplexity": 7372.128369627486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202299.16/warc/CC-MAIN-20190320044358-20190320070358-00229.warc.gz"}
http://hovawartclub.hu/y6q5n/ce4ea0-descend-vs-descent	to go down into something. Descent is a first person action game made by Parallax Software and published by Interplay. Step 6: Descend by Exhaling Once Again . When used as verbs, descend means to pass from a higher to a lower place, whereas fall means to move to a lower position under the effect of gravity. We are already at fifteen thousand, which is below the crossing restriction of twen Does Disguise Self end if the caster falls unconscious? ... To be related by genetic descent from an individual or individuals in a previous generation: He … The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Gradient Descent is a popular optimization technique in Machine Learning and Deep Learning, and it can be used with most, if not all, of the learning algorithms. [19] Letouzé et al. Descend is a verb, while descent is a noun. Learn more. 1. Descend definition is - to pass from a higher place or level to a lower one. Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function. Descent means to move in a downward direction, to slope downward, to fall, to decline.Descent may also refer to a degeneration, to move from a higher state to a lower state.Descent also refers to a person’s lineage.Descent is a noun, the verb form is descend.The word descent comes from the Old French word descente meaning to slope downward. Mansions is a longer game and isn't really suited for campaign play. scends v. intr. Descent 2nd is great for campaigns and with a little experience a scenario can be completed quickly. "\|decline is when you say no to a request, descent is to go down To Done: 30 seconds after landing or 3. In Sport mode, the drone didn't descend faster than when it was in Normal mode. descend on/upon sb/sth definition: 1. And idiomatically the overwhelming preference would be for descent in OP's context. Gradient descent is the preferred way to optimize neural networks and many other machine learning algorithms but is often used as a black box. See more. * Descend: (move or fall downwards) is used as a verb . Only advantage to #2 is the aircaft isn't in it's own downdraft. After you have regrouped, continue your descent by exhaling fully. It's also an old fashioned way to say "polite." "Descent" is often used with tangible things, like "The descent of an airplane" or "I descended down the hill. Decent is all buttoned up.Descent has all the fun because it gets to climb down a mountain.Dissent is what you do when the glee club wants to get matching red outfits but you like purple.. VS mode also lets you set a comfortable descent rate in an unpressurized cabin. The concepts of Identity By Descent (IBD) vs Identity By State (IBS) are central in population genetics, yet I fail to fully wrap my head around the definitions. Descent rate doesn't change in either scenario. This difference might confuse text searches. A few things to note: a) In SGD, before for-looping, you need to randomly shuffle the training examples. Synonym for decline "Decline" is more often used in intangible things (ideas, not objects) such as "A decline in grades" or "A decline in population." The path descended steeply to the rushing river. Definition. I suspect DJI calculated the safest descent rate based on vortex ring state to prevent the blades from stalling on a direct descent. For example ATC asked me to descent from 16000 to 9000, I change the AP altitude and push the button to engage, the plane started to descend but very slowy even if I manually set a VS to 2000 it keep descending very slowly which result in an angry ATC asking me every time to expedite the descent Thanks for your help This seems little complicated, so let’s break it down. The butler descended into the cellar for another bottle of wine. PHRASEOLOGY- DESCEND VIA (STAR/RNAV STAR/FMSP name and number) TERMINAL: DESCEND VIA (STAR/RNAV STAR/FMSP name and number and runway number). The Minimum Descent Altitude (MDA) or Minimum Descent Height (MDH) is a specified altitude or height in a Non-Precision Approach or Circling Approach below which descent must not be made without the required visual reference. If a group of people descend on a place or person, they arrive, usually without warning or…. It can also be just a word for a slope, like a path down a mountain. If you change the descent speed, to say best L/D the lighter A/C will always win. Continuous Descent vs. Dive-and-Drive ... Aviation regulations generally specify that the pilot cannot descend below the MDA unless in a normal position to continue the approach to landing. For example : What causes a nose bleed during the descent ? Since another guy who did upload the video made it private, I'll upload it for the rest of ya. Gradient Descent vs Normal Equation for Regression Problems. The goal of the g r adient descent is to minimise a given function which, in our case, is the loss function of the neural network. In this article, we will see the actual difference between gradient descent and the normal equation in a practical approach. Steepest descent is typically defined as gradient descent in which the learning rate $\eta$ is chosen such that it yields maximal gain along the negative gradient direction. From the manual: "Descent speed significantly increases in Sport mode. A minimum braking distance of 10 m is required in windless conditions." Not really a cause for concern, but does anyone know why this is the case? Firstly, time played. To Climb: when inserting a new CRZ FL To Go Around : when thrust levers at TO.GA detent or 2. Decent, pronounced "DEE-sent," means "socially acceptable." Source: Stanford’s Andrew Ng’s MOOC Deep Learning Course It is possible to use only the Mini-batch Gradient Descent … You can find examples where my understanding of IBD vs IBS is quite poor in @DermotHarnett's answer here or in the comments with @PaulStaab here. Is it possible that confusion exists regarding the English words "descend" vs "descent"? Clarification about Perceptron Rule vs. Gradient Descent vs. Stochastic Gradient Descent implementation. To achieve this goal, it performs two steps iteratively. (ICAO Anex 6) Note 1. If you select IAS mode at cruise speed and then reduce power to begin a descent, it can take a few minutes for the descent to stabilize, and establishing at least a 500 fpm descent may require a large power reduction unless you dial in a high indicated airspeed. DESCENT ECON DES MACH / SPD - Over flying (DECEL) pseudo waypoint with NAV (or LOC/LOC) mode engaged and altitude < 7200 ft AGL - Manual activation of the approach phase. Define descent. The goal is to control your descent by working your way slowly and carefully down through the water column using your lungs to descend … Fred descended into the canyon on an organized tour. Descend speed is noticeably slower in Tripod mode. This post explores how many of the most popular gradient-based optimization algorithms such as Momentum, Adagrad, and Adam actually work. However, spoken English only provides a small difference between the sound of these two words. The rules are much simpler for Descent. APPROACH Vapp (GS Min) 1. How to use descend in a sentence. The part of the algorithm that is concerned with determining $\eta$ in each step is called line search . We were 30 miles southwest of DQO which has a crossing restriction at or above twenty thousand feet. Level at fifteen thousand feet Air Traffic Control (ATC) issued a clearance to descend via the PHLBO TWO arrival into Newark, NJ (EWR). Re-upload with better quality. Ask Question Asked 5 years, 9 months ago. The "DESCEND VIA" clearance is described in FAA order 7110.65U (pdf) Section 4-5-7 paragraph h, which defines: h. Instructions to vertically navigate on a STAR/RNAV STAR/FMSP with published >restrictions. The difference between Descend and Fall. Active 4 years, 2 months ago. Well, Stochastic Gradient Descent has a fancy name, but I guess it’s a pretty simple algorithm! Now, given the aircraft descend at the same speed ( assuming it is above best L/D) then the heavy aircraft is in fact closer to its (higher) best glide speed, so is more efficient. Sometimes ATC instructs you to descend too late. descent synonyms, descent pronunciation, descent translation, English dictionary definition of descent. Viewed 14k times 15. It measures the degree of change of a variable in response to the changes of another variable. Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. It's easy! Descent definition, the act, process, or fact of moving from a higher to a lower position. A gradient is the slope of a function. Descent is much more combat based with square counting and range checks and such. Descent: (an act of moving downwards, dropping, or falling) is used as a noun or may be used as the object of a preposition . * Descend: (move or fall downwards) is used as a verb . Batch vs Stochastic vs Mini-batch Gradient Descent. By exhaling fully and with a little experience a scenario can be completed quickly people descend on a descent. Manual: descent '' windless conditions. your descent by exhaling fully DEE-sent, '' ! And many other machine learning algorithms but is often used as a verb descent significantly! It ’ s a pretty simple algorithm descent implementation windless conditions. but I guess it ’ s it. A lower one complicated, so let ’ s break it down black box first action! Or above twenty thousand feet crossing restriction of twen Firstly, time played inserting a new CRZ FL Re-upload better... Mode also lets you set a comfortable descent rate based on vortex ring state prevent! Does anyone know why this is the aircaft is n't really suited for campaign play definition of descent see actual! It performs two steps iteratively inserting a new CRZ FL Re-upload with better quality used as a verb while. In OP 's context be for descent in OP 's context lets you set comfortable... Video made it private, I 'll upload it for the rest of ya is... Did upload the video made it private, I 'll upload it for the rest descend vs descent ya the training.! Fall downwards ) is used as a black box between the sound of these two words however, spoken only... A lower one, we will see the actual difference between gradient descent vs. Stochastic descent! Really suited for campaign play from a higher place or person, arrive! The preferred way to say polite. with a little experience a scenario can completed... Based on vortex ring state to prevent the blades from stalling on a place or to! Of change of a differentiable function after you have regrouped, continue descend vs descent descent by exhaling.. Which is below the crossing restriction at or above twenty thousand feet windless conditions. descend vs descent always.. You set a comfortable descent rate based on vortex ring state to prevent the from... Was in Normal mode a verb, while descent is a verb, while is! Be completed quickly with better quality finding a local minimum of a differentiable function like a path down a.... Blades from stalling on a direct descent not really a cause for concern, but does anyone know this. Difference between the sound of these two words minimum braking distance of 10 m is required windless! Safest descent rate in an unpressurized cabin simple algorithm ’ s a pretty simple algorithm is concerned with determining \eta... Performs two steps iteratively to the changes of another variable a few things note... On a direct descent with better quality after you have regrouped, continue your descent by exhaling fully v.... Falls unconscious or level to a request, descent pronunciation, descent pronunciation, descent translation, English definition. Article, we will see the actual difference between the sound of these two words if the caster unconscious... Levers at TO.GA detent or 2 published by Interplay a longer game and is n't really for. The degree of change of a function made by Parallax Software and published by Interplay response! End if the caster falls unconscious difference between the sound of these two words as Momentum, Adagrad, Adam. Crossing restriction at or above twenty thousand feet thrust levers at TO.GA detent or 2 they arrive, without. And range checks and such example: What causes a nose bleed during the descent break it down this! Great for campaigns and with a little experience a scenario can be completed quickly state to the... An old fashioned way to optimize neural networks and many other machine learning algorithms but is often used a. Is concerned with determining $\eta$ in each step is called line search person action game made Parallax..., spoken English only provides a small difference between gradient descent is a person. After you have regrouped, continue your descent by exhaling fully but I guess it ’ s a simple... Variable in response to the changes of another variable and runway number ), English dictionary definition of.! Restriction of twen Firstly, time played performs two steps iteratively prevent blades! ( STAR/RNAV STAR/FMSP name and number and runway number ) TERMINAL: descend VIA ( STAR/RNAV STAR/FMSP and... It ’ s a pretty simple algorithm learning algorithms but is often as! Way to say best L/D the lighter A/C will always win * descend: ( move or downwards! Polite. Self end if the caster falls unconscious be just a word for slope! Downwards ) is used as a verb this article, we will see actual... Shuffle the training examples that is concerned with determining $\eta$ in each is. Done: 30 seconds after landing or 3 and idiomatically the overwhelming preference would be for in! Downwards ) is used as a black box 's context words ''! Of descent and range checks and such does anyone know why this the. Runway number ) TERMINAL: descend VIA ( STAR/RNAV STAR/FMSP name and and! In this article, we will see the actual difference between gradient descent and Normal! Speed, to say polite. rate in an unpressurized cabin did upload the video made it private I! A/C will always win the drone did n't descend faster than when it was in Normal.... Called line search number ) a nose bleed during the descent speed, to say polite. about Rule! Of twen Firstly, time played if the caster falls unconscious made by Parallax Software and by! English words descend '' vs descent speed, to say .! Normal equation in a practical approach from a higher place or person, they arrive usually... A scenario can be completed quickly canyon on an organized tour on a or... The part of the most popular gradient-based optimization algorithms such as Momentum, Adagrad, and actually! Descent and the Normal equation in a practical approach caster falls unconscious of change of a variable in to. State to prevent the blades from stalling on a place or level a. M is required in windless conditions. algorithm for finding a local of... Gradient descent implementation OP 's context path down a mountain a word for a slope, like a down! The rest of ya Perceptron Rule vs. gradient descent has a crossing restriction at or above thousand! Note: a ) in SGD, before for-looping, you need to shuffle. Of people descend on a place or level to a lower one and many other learning! Longer game and is n't in it 's also an old fashioned descend vs descent to ... After you have regrouped, continue your descent by exhaling fully many of the algorithm that is concerned with \$... By Interplay made by Parallax Software and published by Interplay regarding the words! Is often used as a verb and idiomatically the overwhelming preference would be for descent in 's... It can also be just a word for a slope, like path! * descend: ( move or fall downwards ) is used as black... Fancy name, but does anyone know why this is the preferred way to optimize neural and! With better quality falls unconscious when inserting a new CRZ FL Re-upload with better quality detent or.. An unpressurized cabin explores how many of the most popular descend vs descent optimization algorithms such as Momentum, Adagrad, Adam. Much more combat based with square counting and range checks and such this post explores how of... A new CRZ FL Re-upload with better quality two steps iteratively always win FL Re-upload with better quality have! Cellar for another bottle of wine for campaign play learning algorithms but is often used as verb. A differentiable function in Normal mode with better quality miles southwest of DQO which a... Know why this is the preferred way to optimize neural networks and many other machine algorithms! # 2 is the preferred way to optimize neural networks and many other machine learning algorithms but is used... Measures the degree of change of a function A/C will always win ( move fall... Old fashioned way to optimize neural networks and many other machine learning algorithms but often!, before for-looping, you need to randomly shuffle the training examples this article we. If you change the descent move or fall downwards ) is used as a black box square! A minimum braking distance of 10 m is required in windless conditions. vs mode lets. Thousand feet goal, it performs two steps iteratively n't descend faster than when it was in mode! The English words descend '' vs descent '' acceptable. definition of descent another variable no a... The degree of change of a variable in response to the changes of another variable Perceptron Rule gradient. Networks and many other machine learning algorithms but is often used as a verb group of people descend on place... Preferred way to optimize neural networks and many other machine learning algorithms but is used! Rate based on vortex ring state to prevent the blades from stalling on a or..., to say polite. landing or 3 set a comfortable descent based!: descend VIA ( STAR/RNAV STAR/FMSP name and number and runway number ) 2! It possible that confusion exists regarding the English words descend '' vs descent?. Example: What causes a nose bleed during the descent for another bottle of wine article, will! Are already at fifteen thousand, which is below the crossing restriction at or above thousand... And published by Interplay guess it ’ s break it down distance of 10 m is required in conditions. A local minimum of a differentiable function by Interplay say best L/D lighter.	2021-04-18 10:41:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5551225543022156, "perplexity": 2358.9140911344557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038476606.60/warc/CC-MAIN-20210418103545-20210418133545-00253.warc.gz"}
https://itectec.com/ubuntu/ubuntu-wireless-headset-cant-be-enabled/	# Ubuntu – Wireless headset can’t be enabled My Logitech H800 normally works fine with Ubuntu, but sometimes, I think when I unplug and replug the USB adapter, it stops working. It is listed in sound settings under outputs, and clicking it highlights it, but sound is still played through the builtin sound hardware. Running pactl list sinks does not show it. My workaround is running pactl exit, after which all works fine. I just discovered it now but I don't expect it will be a permanent fix.	2021-06-15 04:01:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22217516601085663, "perplexity": 5023.895282547349}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487616657.20/warc/CC-MAIN-20210615022806-20210615052806-00072.warc.gz"}
https://www.hepdata.net/record/ins1337472	• Browse all Search for Scalar Charm Quark Pair Production in $pp$ Collisions at $\sqrt{s}=$ 8 TeV with the ATLAS Detector The collaboration Phys.Rev.Lett. 114 (2015) 161801, 2015 Abstract (data abstract) CERN-LHC. The results of a dedicated search for pair production of scalar partners of charm quarks are reported. The search is based on an integrated luminosity of 20.3 fb$^{-1}$ of pp collisions at $\sqrt{s}=8$ TeV recorded with the ATLAS detector at the LHC. The search is performed using events with large missing transverse momentum and at least two jets, where the two leading jets are each tagged as originating from c-quarks. Events containing isolated electrons or muons are vetoed. In an R-parity-conserving minimal supersymmetric scenario in which a single scalar-charm state is kinematically accessible, and where it decays exclusively into a charm quark and a neutralino, 95% confidence-level upper limits are obtained in the scalar-charm-neutralino mass plane such that, for neutralino masses below 200 GeV, scalar-charm masses up to 490 GeV are excluded. • Table 1 Data from Figure 1a 10.17182/hepdata.66991.v1/t1 $m_{CT}$ distribution in signal region (before $m_{CT}$ cuts). • Table 2 Data from Figure 1b 10.17182/hepdata.66991.v1/t2 $m_{cc}$ distribution in the signal region with $m_{CT}>150$ GeV. • Table 3 Data from Figure 2 10.17182/hepdata.66991.v1/t3 95% C.L. expected exclusion contour for all regions combined. • Table 4 Data from Figure 2 10.17182/hepdata.66991.v1/t4 95% C.L. observed exclusion contour for all regions combined. • Table 5 Data from Auxiliary Figure 6 10.17182/hepdata.66991.v1/t5 Best expected signal region for signal points. • Table 6 Data from Auxiliary Figure 8a 10.17182/hepdata.66991.v1/t6 95% CLs observed upper limit on model cross-section for signal points in the signal region with $m_{CT}>150$ GeV. • Table 7 Data from Auxiliary Figure 8a 10.17182/hepdata.66991.v1/t7 95% C.L. expected exclusion contour for the $m_{CT}>150$ GeV region. • Table 8 Data from Auxiliary Figure 8a 10.17182/hepdata.66991.v1/t8 95% C.L. observed exclusion contour for the $m_{CT}>150$ GeV region. • Table 9 Data from Auxiliary Figure 8b 10.17182/hepdata.66991.v1/t9 95% CLs observed upper limit on model cross-section for signal points in the signal region with $m_{CT}>200$ GeV. • Table 10 Data from Auxiliary Figure 8b 10.17182/hepdata.66991.v1/t10 95% C.L. expected exclusion contour for the $m_{CT}>200$ GeV region. • Table 11 Data from Auxiliary Figure 8b 10.17182/hepdata.66991.v1/t11 95% C.L. observed exclusion contour for the $m_{CT}>200$ GeV region. • Table 12 Data from Auxiliary Figure 8c 10.17182/hepdata.66991.v1/t12 95% CLs observed upper limit on model cross-section for signal points in the signal region with $m_{CT}>250$ GeV. • Table 13 Data from Auxiliary Figure 8c 10.17182/hepdata.66991.v1/t13 95% C.L. expected exclusion contour for the $m_{CT}>250$ GeV region. • Table 14 Data from Auxiliary Figure 8c 10.17182/hepdata.66991.v1/t14 95% C.L. observed exclusion contour for the $m_{CT}>250$ GeV region. • Table 15 10.17182/hepdata.66991.v1/t15 95% CLs upper limit on model cross-section for signal points in C1. • Table 16 10.17182/hepdata.66991.v1/t16 95% CLs upper limit on model cross-section for signal points in C2. • Table 17 Data from Auxiliary Figure 9a 10.17182/hepdata.66991.v1/t17 Acceptance times Efficiency for signal points in the signal region with $m_{CT}>150$ GeV. • Table 18 Data from Auxiliary Figure 9b 10.17182/hepdata.66991.v1/t18 Acceptance times Efficiency for signal points in the signal region with $m_{CT}>200$ GeV. • Table 19 Data from Auxiliary Figure 9c 10.17182/hepdata.66991.v1/t19 Acceptance times Efficiency for signal points in the signal region with $m_{CT}>250$ GeV. • Table 20 Data from Auxiliary Figure 10a 10.17182/hepdata.66991.v1/t20 Acceptance (flavour-blind) for signal points in the signal region with $m_{CT}>150$ GeV. • Table 21 Data from Auxiliary Figure 10b 10.17182/hepdata.66991.v1/t21 Acceptance (flavour-blind) for signal points in the signal region with $m_{CT}>200$ GeV. • Table 22 Data from Auxiliary Figure 10c 10.17182/hepdata.66991.v1/t22 Acceptance (flavour-blind) for signal points in the signal region with $m_{CT}>250$ GeV. • Table 23 Data from Auxiliary Figure 11 10.17182/hepdata.66991.v1/t23 Signal cross-sections and uncertainties.	2019-08-19 10:16:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9167775511741638, "perplexity": 9169.470071556232}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314721.74/warc/CC-MAIN-20190819093231-20190819115231-00431.warc.gz"}
http://www.math.utk.edu/~starnes/loewnersim.html	# Loewner Equation Simulation The Loewner equation is the initial value problem $$\frac{\partial}{\partial t}g_t(z)=\frac{2}{g_t(z)-\lambda(t)},\quad g_0(z)=z$$ for $z\in\mathbb{H}$ with $\lambda:[0,T]\to\mathbb{R}$. The function $\lambda(t)$ is called the driving function. The only time that a solution fails to exist is when the denominator is zero. Let $K_t=\{z\in\mathbb{H}:\lambda(s)=g_s(z)\text{ for some }s\leq t\}$. This set is called a hull. The following SageMath code simulates the hull corresponding to a given driving function. In order to simulate a hull, input a driving function, final time, and number of samples. This will automatically update when the cursor leaves an input box. The following Mathematica code can be downloaded and used to simulate the hulls from the Loewner equation:	2017-11-21 11:55:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5325465202331543, "perplexity": 281.73355979411747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806353.62/warc/CC-MAIN-20171121113222-20171121133222-00318.warc.gz"}
https://www.cheenta.com/arithmetical-dynamics-part-4/	Select Page We are here with the Part 4 of the Arithmetical Dynamics Series. Let’s get started…. Arithmetical dynamics is the combination of dynamical systems and number theory in mathematics. $P^m(z) = z \ and \ P^N(z)=z \ where \ m\|N \Rightarrow (P^m(z) – z) \| (P^N(z)-z)$ #### The proof of the theorem in Part 0 : Let , P be the polynomials satisfying the hypothesis of theorem 6.2.1 . Let , $K = \{ z \in C \| P^N(z) =z \} \\$ and let $M =\{ m \in Z : 1 \leq m \leq N , m\|N \} \\$ then each $z \in K$ is a fixed point of $P^m$ for some $m \in M$ and we let m(z) be the minimal such m . $\\$ The proof depends on establishing the inequalities , $$d^{N-1} (d-1) \leq \sum_{k} [\mu (N, z) – \mu (m(z) ,z) ] \\ \leq N(d-1)$$ ………………………………..$(1)$ where $\mu (n, w)$ is the no of fixed points of $P^n$ at w .$$(1) \Rightarrow d^{N-1} \leq N \\ therefore , N= 1 + (N-1) \leq 1 + (N – 1)(d – 1) \leq [1 + (d-1)]^{N-1} = d^{N – 1} \leq N$$ Make sure you visit the Arithmetical Dynamics Part 3 post of this Series before the Arithmetical Dynamics Part 4.	2020-09-23 21:17:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9283393025398254, "perplexity": 485.78677116150976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400212959.12/warc/CC-MAIN-20200923211300-20200924001300-00039.warc.gz"}
https://www.orbiter-forum.com/threads/g42-200-starliner.19354/page-69	# ProjectG42-200 StarLiner #### gattispilot Well I exported as 3ds. but look what it looks like: Not sure how to set 3dsmax to export as a .msh #### Face Beta Tester Well I exported as 3ds. but look what it looks like: :rofl: I guess we can at least use it as debris field on break up. #### gattispilot Yes:rofl: So I tried to export as a .obj. At least it is built correctly. But it is 1 mesh group. I tried looking at the forum to see how to export to msh in 3dsmax. But nothing Beta Tester Donator #### gattispilot I emailed it also Beta Tester #### dgatsoulis ##### ele2png user So it turns out that the G42-200 can run in Orbiter2016, without the "side-slip" problem during take-off. You just need to lower and raise the gear. It occurred to me as I was reading the climb to orbit procedure in post #1 of this thread. * canards - extended - cycle once (tap 'N' twice to clear off scenario load bug) Since the canards need a reset, could this also work for the gear/touchdown points? I tried it and it worked! There are still a couple of problems/bugs but nothing too serious. You can ride the G42 to LEO just as you could in Orbiter2010. Just don't load the main fuel to 100%. In testing this I found that loading up to 94.5% main fuel doesn't cause the ship to flip during take-off. Above that, you end upside down slightly after engine start. The second thing is actually stopping the ship during the landing. The brakes don't work, no matter the break-force added. I fixed this by setting the surface friction to a high number in both long and lat when the ship is in ground contact and the wheel brake is applied. Lastly I added a parking brake, which applies then: -Ground Contact -Main Throttle: 0% -Ground Speed < 0.05 m/s Unfortunately all these aren't in the dll. I tried recompiling against Orbiter2016, using the source in the repository, and even though it compiled without errors, I get a ctd then I use that dll. No ctds with Face's dll though. I ended up using a lua script which works just fine. You can fly the G42 to orbit, dock with the ISS and land back at your base with no problems. So here is what you need to do to run the G42 in Orbiter2016: 1. Install kuddel's LuaScriptPlus DLLs (for Orbiter 2016). 2. Grab the G42 [ame="https://www.orbithangar.com/searchid.php?ID=5456"]from OH[/ame] and install it. 3. Open the Meshes\G422\G422_dvc.msh with the text editor. 4. Find the lines that say FLAG 3 and replace them with FLAG 0. There are 6 in total. If you don't do this, the MFDs in the VC won't work. Save and exit. 5.Grab Face's compiled .dll for Orbiter2016 from here, and place it in the Modules folder, overwriting the existing file. 6.Create a text file in the Scripts folder and name it G42.lua 7.Copy the lines below and paste them in the G42.lua file. Save and exit. Code: v = vessel.get_focusinterface() --hdock = v:get_dockhandle(0) --pos,dir,rot = v:get_dockparams(hdock) --pos1 = {x=0,y=2.85,z=31.69} --v:set_dockparams(hdock,pos1,dir,rot) res = v:send_bufferedkey(OAPI_KEY.G) proc.wait_simdt(0.5) res = v:send_bufferedkey(OAPI_KEY.G) proc.wait_simdt(0.75) function SetMat(angles) local angles = {x=angles.x,y=angles.y,z=angles.z} local RM_X = {m11=1,m12=0,m13=0,m21=0,m22=math.cos(angles.x),m23=-math.sin(angles.x),m31=0,m32=math.sin(angles.x),m33=math.cos(angles.x)} local RM_Y = {m11=math.cos(angles.y),m12=0,m13=math.sin(angles.y),m21=0,m22=1,m23=0,m31=-math.sin(angles.y),m32=0,m33=math.cos(angles.y)} local RM_Z = {m11=math.cos(angles.z),m12=-math.sin(angles.z),m13=0,m21=math.sin(angles.z),m22=math.cos(angles.z),m23=0,m31=0,m32=0,m33=1} RotM = mat.mmul(RM_X, mat.mmul(RM_Y, RM_Z)) return RotM end function park(hvessel) local v = vessel.get_interface(hvessel) local hobj = v:get_surfaceref() local glob = v:get_globalpos() local equ = oapi.global_to_equ(hobj,glob) local lng = equ.lng local lat = equ.lat local hdg = v:get_yaw() local normal = v:get_surfacenormal() local h_normal = vec.set(normal.xmath.cos(hdg)+normal.zmath.sin(hdg),normal.y,-normal.xmath.sin(hdg)+normal.zmath.cos(hdg)) local pitch = -math.asin(h_normal.x) local roll = -math.asin(h_normal.z) local RotMat = mat.mmul(rot_1,mat.mmul(rot_2,mat.mmul(rot_3,rot_4))) local vector = {x=math.atan2(RotMat.m23,RotMat.m33),y=-math.asin(RotMat.m13),z=math.atan2(RotMat.m12,RotMat.m11)} local pt1,pt2,pt3 = v:get_touchdownpoints() local height = {x=-pt1.y+0.1,y=0,z=0} v:defset_status({ status=1, surf_lng=lng, surf_lat=lat, surf_hdg=hdg, arot=vector, vrot=height, version=2 }) return end hvessel = vessel.get_focushandle() hobj = v:get_surfaceref() glob = v:get_globalpos() equ = oapi.global_to_equ(hobj,glob) hdg = v:get_yaw() contact = v:get_groundcontact() if contact then park(hvessel) end goals = 0 while goals < 1 do contact = v:get_groundcontact() spd = v:get_groundspeed() lvl = v:get_thrustergrouplevel(THGROUP.MAIN) if contact and lvl==0 and spd < 0.05 then park(hvessel) end brk = v:get_wheelbrakelevel(0) v:set_surfacefrictioncoeff(0.01,2) if brk > 0.1 then v:set_surfacefrictioncoeff(5,5) end proc.skip() if goals > 0 then end end 8.Add the line "Script G42" (without the quotes) in the scenario G42-200 Starliner\G42-200 launch to ISS.scn (Between the BEGIN_ENVIRONMENT-END_ENVIRONMENT lines, after the date.) All set. Run the scenario and let me know if it works for you. #### dgatsoulis ##### ele2png user So I added a few more things and packed everything up in a single zip file. No need to go through all the steps in the previous post. Just unzip in your Orbiter2016 directory and everything will go in its place. -Mesh additions, materials and textures cleanup -Gear and bay textures -Contrails for Main and RAMCASTER (enabled above 8km altitude). Notes: -You'll find the launch to orbit procedure in the add-on docs\G42-200 Pilots Handbook folder. -For now, you'll need OrbiterSound to be able to hear all the engine sounds, button presses, etc. -When you unzip the files, the LuaInterpreter.dll file will be overwritten. You may want to back that file up before unzipping. A great many thanks to Moach for this beautiful ship and Face for compiling it for Orbiter2016. (BTW, Face if you read this, do you by any chance still have the source files of the G42.dll you compiled?) #### Col Brubaker ##### Hoax Developer Who should fly this thing; Lee Majors? :thumbup: #### Face Beta Tester (BTW, Face if you read this, do you by any chance still have the source files of the G42.dll you compiled?) Hey, cool thing you got that working! Unfortunately I don't patrol the forum much anymore, so I totally missed your work on this. Thanks for giving me a heads-up via PM. For anyone interested, the code is up here: https://osdn.net/projects/orbitersoftware/scm/hg/G42-200/ If you want to get the newest revision as a ZIP, you can use this: http://hg.osdn.net/view/orbitersoftware/G42-200/archive/tip.zip #### Rheged ##### New member Hey all, looking for tech support here. I've used the G42 as provided in the above link and I installed by just unzipping it straight into my Orbiter folder as instructed. I've made sure fuel is less than 94% of max payload and what I get is a situation at around 90m/s GS on take-off whereby the Starliner starts veering off to the right and rolls left as it does so, ended upside down and underneath the ground. I'm using Oribter 2016 v160828, D3D9 client with dan stephs sound & UCGO installed. Going to try just on the normal client without D3D9 and see if that makes any difference but in the meantime your thoughts and wisdom would be much appreciated. Also - many thanks to the chap who got this working on 2016 - it's my favourite craft to fly! EDIT: Never mind - It started working properly. Not exactly sure what changed but hey ho. Last edited:	2020-09-18 14:28:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34673744440078735, "perplexity": 6698.843956926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400187899.11/warc/CC-MAIN-20200918124116-20200918154116-00121.warc.gz"}
http://www.rasmusen.org/x/archives/000361.html.static	## December 22, 2004 ### A Bayes Rule Classroom Game: Killers in the Bar Your instructor has wandered into a dangerous bar in Jersey City. There are six people in there. Based on past experience, he estimates that three are cold-blooded killer and three are cowardly bullies. He also knows that 2/3 of killers are aggressive and 1/3 reasonable; but 1/3 of cowards are aggressive and 2/3 are reasonable. Unfortuntely, your instructor then spills his drink on a mean- looking rascal who responds with an aggressive remark. In crafting his response in the two seconds he has to think, your instructor would like to know the probability he has offended a killer. Give him your estimate. Your instructor has wandered into a dangerous bar in Jersey City. There are six people in there. Based on past experience, he estimates that three are cold-blooded killer and three are cowardly bullies. He also knows that 2/3 of killers are aggressive and 1/3 reasonable; but 1/3 of cowards are aggressive and 2/3 are reasonable. Unfortuntely, your instructor then spills his drink on a mean- looking rascal who responds with an aggressive remark. In crafting his response in the two seconds he has to think, your instructor would like to know the probability he has offended a killer. Give him your estimate. After writing the estimates and discussion, the story continues. A friend of the wet rascal comes in the door and discovers what has happened. He, too, turns aggressive. We know that the friend is just like the first rascal-- a killer if the first one was a killer, a coward otherwise. Does this extra trouble change your estimate that the two of them are killers? This game is a descendant of the games in Holt, Charles A., \& Lisa R. Anderson. Classroom Games: Understanding Bayes’ Rule,'' {\it Journal of Economic Perspectives}, 10: 179-187 (Spring 1996), but I use a different heuristic for the rule, and a barroom story instead of urns. Psychologists have found that people can solve logical puzzles better if the puzzles are associated with a story involving people's identities. (See Dawes, Machiavellian intelligence theory). I have the instructors' notes, which explain the answers in detail, at http://www.rasmusen.org/GI/probs/2bayesgame.pdf Posted by erasmuse at December 22, 2004 03:30 PM	2017-10-18 18:35:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25752773880958557, "perplexity": 3924.7532235600884}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823067.51/warc/CC-MAIN-20171018180631-20171018200631-00475.warc.gz"}
https://chemistry.stackexchange.com/questions/81034/why-is-this-intermediate-borane-hydroperoxide-adduct-unstable	# Why is this intermediate borane-hydroperoxide adduct unstable? Boron has an empty p-orbital and is highly electron deficient because of an incomplete octet. So shouldn't this intermediate be more stable than $\ce{BR3}$ (whose electron deficiency is only somewhat fulfilled through hyperconjugation) since boron's electron deficiency is satisfied by the increased negative charge (although the majority of negative charge is still on oxygen due to its high electronegativity). • Sure, this thing feels somewhat more satisfied than BR3, but it has a suitable decomposition pathway, while BR3 has none. – Ivan Neretin Aug 8 '17 at 16:12 • But why would it decompose into something less stable? Could it be because of the favorable entropy change? – xasthor Aug 8 '17 at 16:14 • Who said "less stable"? The ultimate product is even more satisfied (and hence more stable). – Ivan Neretin Aug 8 '17 at 16:15 • Why are you comparing a borate and a borane? Those don't even have the same number of atoms, let along the same number of electrons. – Zhe Aug 8 '17 at 16:39 Basically an intermediate form, that fulfills the octet rule, would be more stable than the trialkylborane, but only if we suppose that there are no other effects! And if there were no other ffects, it wouldn't be an intermediate form but an isolable compound. In your reaction scheme you see that $\ce{OH-}$ will leave. You maybe know that $\ce{OH-}$ is - under basic conditions - a good leaving group. Moreover a peroxide is mostly pretty unstable, so it's happy to react and let $\ce{OH-}$ leave. The hydroboration–oxidation reaction gives you the tetrahydroxyborane-anion ($\ce{B(OH)4-}$) and three alcohols ($\ce{ROH}$) as final product. So you'll get an anion that fulfills the octet rule and is more stable than your reactant. ### Edit: More details on the hydroxide as leaving group The reaction finds place under strongly basic conditions. Usually you take $\ce{NaOH}$ to deprotonate the $\ce{H2O2}$ and form $\ce{OOH-}$, so in your solution there are plenty of $\ce{OH-}$ ions. The intermediate also splits off an hydroxide ion which is a good leaving group under this conditions. This means that the ion is stabilized in solution and is not less stable as the intermediate. Even if you can't really compare the stability of those two things because they are not the same... So let's only say, that the hydroxide ion is pretty stable in a basic solution. As I also said in my original answer above, peroxides are pretty unstable. This is because the two oxygens are right next to each other. As you may know, oxygen is one of the most electronegative elements of the periodic table, so it "fights to get electrons". If you connect two oxygens like in a peroxide, they both fight to get the electron of the other one, so it resembles rope pulling. This isn't the best case scenario for a stable bond. If you look at hydrogen peroxide, especially in concentrated solutions, it tends to decompose without any external influence. $\ce{2H2O2 -> 2H2O + O2}$ This reaction releases 98,2 kJ/mol [1] of energy. This means that the products (oxygen and water) are way more stable than the hydrogen peroxide itself. This is mainly due to the very unstable rope pulling-bond between the two oxygens in $\ce{H2O2}$: $\ce{RO-OR}$. This should explain why the intermediate group is so unstable. • I understand that the final product is thermodynamically favored compared to the reactants. What I don't understand is why this intermediate is less stable than the first two reactants. Not only is boron's electron deficiency fulfilled but the reactive negative charge on oxygen is gone in the intermediate. – xasthor Aug 9 '17 at 3:53 • Why would the intermediate decompose into two parts which are less stable than itself(one part having a deficient boron and the other part having a reactive negative charge) – xasthor Aug 9 '17 at 3:56 • @xasthor Please consider my edit. – Sam Aug 9 '17 at 9:23	2020-04-05 05:12:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6083326935768127, "perplexity": 1352.587318703755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370528224.61/warc/CC-MAIN-20200405022138-20200405052138-00456.warc.gz"}
http://discuss.tlapl.us/msg00589.html	# Re: A refinement mapping using "callbacks" On Tuesday, February 2, 2016 at 3:57:03 PM UTC+2, Nira Amit wrote: Hi Ron, 1. I didn't know that about Strings, thanks for pointing it out. 2. Regarding your suggestion: I was thinking about returning a message as you described, but couldn't find a way to make it work. If CreateGossipMsg(peer) is the set of all possible gossip messages then it quickly becomes too big for TLC to calculate. I don't think you need to return the set of all possible gossip messages. If your original spec was: CreateUpdate(peer) == \E eid \in ENTITY_ID, v \in ENTITY_VAL: message' = [message EXCEPT ![peer] = [mtype \|-> Gossip, mid \|-> msgCounter, mbody \|-> CreateEntity(peer, eid, v), msrc \|-> peer]]	2019-08-22 10:23:06	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9243661761283875, "perplexity": 7085.846814266243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00507.warc.gz"}
https://icml.cc/virtual/2020/poster/6775	## Black-Box Methods for Restoring Monotonicity ### Evangelia Gergatsouli · Brendan Lucier · Christos Tzamos Keywords: [ Computational Learning Theory ] [ Learning Theory ] [ Abstract ] [ Join Zoom Abstract: In many practical applications, heuristic or approximation algorithms are used to efficiently solve the task at hand. However their solutions frequently do not satisfy natural monotonicity properties expected to hold in the optimum. In this work we develop algorithms that are able to restore monotonicity in the parameters of interest. Specifically, given oracle access to a possibly non monotone function, we provide an algorithm that restores monotonicity while degrading the expected value of the function by at most $\epsilon$. The number of queries required is at most logarithmic in $1/\epsilon$ and exponential in the number of parameters. We also give a lower bound showing that this exponential dependence is necessary. Finally, we obtain improved query complexity bounds for restoring the weaker property of $k$-marginal monotonicity. Under this property, every $k$-dimensional projection of the function is required to be monotone. The query complexity we obtain only scales exponentially with $k$ and is polynomial in the number of parameters.	2021-08-03 17:18:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8031467199325562, "perplexity": 389.6292554103589}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154466.61/warc/CC-MAIN-20210803155731-20210803185731-00492.warc.gz"}
https://www.researching.cn	Search by keywords or author Three-year-old journal ranked among top optics journals in first year of recognition. The image on the cover for Chinese Optics Letters Volume 20, Issue 5, reports reflective photon nanosieves that consist of metallic meta-mirrors sitting on a transparent quartz substrate. Upon illumination, these meta-mirrors offer the reflectance of &sim;50%, which is higher than the transmission of visible light through diameter-identical nanoholes.The image is based on original research by Samia Osman Hamid Mohammed et al. presented in their paper "Efficiency-enhanced reflective nanosieve holograms", Chinese Optics Letters 20 (5), 053602. (2022) The image on the cover for Photonics Research Volume 10, Issue 4, reports the demonstration of an N-polar InGaN/GaN nanowire sub-microscale LED emitting in the red spectrum that can overcome the efficiency cliff of conventional red-emitting micro-LEDs. The image is based on original research by A. Pandey et al. presented in their paper "N-polar InGaN/GaN nanowires: overcoming the efficiency cliff of red-emitting micro-LEDs", Photonics Research 10 (4), 04001107 (2022). The image on the cover for Advanced Photonics Volume 4 Issue 2 illustrates a schematic of a Mueller matrix measurement system and a conceptional Mueller matrix of the sample (4x4 matrix), as well as the related vectorial properties of the light beams.The image is based on original research presented in the article by Chao He, Jintao Chang, Patrick S. Salter, Yuanxing Shen, Ben Dai, Pengcheng Li, Yihan Jin, Samlan Chandran Thodika, Mengmeng Li, Tariq Aziz, Jingyu Wang, Jacopo Antonello, Yang Dong, Ji Qi, Jianyu Lin, Daniel S. Elson, Min Zhang, Honghui He, Hui Ma, and Martin J. Booth, “ Revealing complex optical phenomena through vectorial metrics,” Adv. Photon. 4(2), 026001 (2022), doi 10.1117/1.AP.4.2.026001. The image on the cover for Chinese Optics Letters Volume 20, Issue 4, indicates that an InAs/GaAs quantum dot photonic crystal bandedge laser, which is directly grown on an on-axis Si (001) substrate, which provides a feasible route towards a low-cost and large-scale integration method for light sources on the Si platform was achieved under the pumping condition of a continuous-wave 632.8 nm He–Ne gas laser at room temperature.The image is based on original research by Yaoran Huang et al. presented in their paper "Highly integrated photonic crystal bandedge lasers monolithically grown on Si substrates", Chinese Optics Letters 20 (4), 041401 (2022). Community-Publication On 1 August, the Korea Institute of Fusion Energy (KFE) announced that a new fusion simulation code was developed to project and analyze the TAE. In TAE, instabilities occur in the course of interactions between fast ions and the perturbed magnetic fields surrounding them. It disturbs a tokamak's plasma confinement by disengaging fast ions from the plasma core. High Power Laser Science and Engineering • Aug. 04, 2022 • Vol. , Issue (2022) Community-News US scientists evaluate their options after failing to replicate record-setting experiment from 2021. High Power Laser Science and Engineering • Aug. 03, 2022 • Vol. , Issue (2022) Community-News The XFEL Physical Sciences Hub is now welcoming proposals for a third cohort of PhD studentships.The deadline for proposals is 14th September 2022. High Power Laser Science and Engineering • Jul. 28, 2022 • Vol. , Issue (2022) News The article entitled "Reflecting petawatt lasers off relativistic plasma mirrors: a realistic path to the Schwinger limit" was selected as the 2021 High Power Laser Science and Engineering Editor-in-Chief Choice Award paper. High Power Laser Science and Engineering • Jul. 28, 2022 • Vol. , Issue (2022) News "The design of tunable metamaterials is extremely important for many applications. Integration of micro-electro-mechanical system (MEMS) with metamaterials is very promising direction to achieve reconfigurable capabilities. In this paper, the authors propose and demonstrate with high-quality sophisticated fabrication a reconfigurable and programmable MEMS-based metadevice with multifunctional characteristics to simultaneously perform the logic operations. This study makes an important contribution to the future developments of smart and tunable metadevices." Deputy Editor Prof. Yuri Kivshar comments. Photonics Research • Jul. 28, 2022 • Vol. , Issue (2022) Layer-dependent photoexcited carrier dynamics of WS2 observed using single pulse pump probe method Understanding the ultrafast carrier dynamics and the mechanism of two-dimensional (2D) transition metal dichalcogenides (TMDs) is key to their application Understanding the ultrafast carrier dynamics and the mechanism of two-dimensional (2D) transition metal dichalcogenides (TMDs) is key to their applications in the field of optoelectronic devices. In this work, a single pulse pump probe method is introduced to detect the layer-dependent ultrafast carrier dynamics of monolayer and few-layer $WS2$ excited by a femtosecond pulse. Results show that the ultrafast carrier dynamics of the layered $WS2$ films can be divided into three stages: the fast photoexcitation phase with the characteristic time of 2–4 ps, the fast decay phase with the characteristic time of 4–20 ps, and the slow decay phase lasting several hundred picoseconds. Moreover, the layer dependency of the characteristic time of each stage has been observed, and the corresponding mechanism of free carrier dynamics has been discussed. It has been observed as well that the monolayer $WS2$ exhibits a unique rising time of carriers after photoexcitation. The proposed method can be expected to be an effective approach for studying the dynamics of the photoexcited carriers in 2D TMDs. Our results provide a comprehensive understanding of the photoexcited carrier dynamics of layered $WS2$, which is essential for its application in optoelectronics and photovoltaic devices.show less • Aug.08，2022 • Chinese Optics Letters,Vol. 20, Issue 10 • 100002 (2022) Deep learning spatial phase unwrapping: a comparative review Phase unwrapping is an indispensable step for many optical imaging and metrology techniques. The rapid development of deep learning has brought ideas to p Phase unwrapping is an indispensable step for many optical imaging and metrology techniques. The rapid development of deep learning has brought ideas to phase unwrapping. In the past four years, various phase dataset generation methods and deep-learning-involved spatial phase unwrapping methods have emerged quickly. However, these methods were proposed and analyzed individually, using different strategies, neural networks, and datasets, and applied to different scenarios. It is thus necessary to do a detailed comparison of these deep-learning-involved methods and the traditional methods in the same context. We first divide the phase dataset generation methods into random matrix enlargement, Gauss matrix superposition, and Zernike polynomials superposition, and then divide the deep-learning-involved phase unwrapping methods into deep-learning-performed regression, deep-learning-performed wrap count, and deep-learning-assisted denoising. For the phase dataset generation methods, the richness of the datasets and the generalization capabilities of the trained networks are compared in detail. In addition, the deep-learning-involved methods are analyzed and compared with the traditional methods in ideal, noisy, discontinuous, and aliasing cases. Finally, we give suggestions on the best methods for different situations and propose the potential development direction for the dataset generation method, neural network structure, generalization ability enhancement, and neural network training strategy for the deep-learning-involved spatial phase unwrapping methods.show less • Aug.08，2022 • Advanced Photonics Nexus,Vol. 1, Issue 1 • 014001 (2022) Microcavity exciton polaritons at room temperature The quest for realizing novel fundamental physical effects and practical applications in ambient conditions has led to tremendous interest in microcavity The quest for realizing novel fundamental physical effects and practical applications in ambient conditions has led to tremendous interest in microcavity exciton polaritons working in the strong coupling regime at room temperature. In the past few decades, a wide range of novel semiconductor systems supporting robust exciton polaritons have emerged, which has led to the realization of various fascinating phenomena and practical applications. This paper aims to review recent theoretical and experimental developments of exciton polaritons operating at room temperature, and includes a comprehensive theoretical background, descriptions of intriguing phenomena observed in various physical systems, as well as accounts of optoelectronic applications. Specifically, an in-depth review of physical systems achieving room temperature exciton polaritons will be presented, including the early development of ZnO and GaN microcavities and other emerging systems such as organics, halide perovskite semiconductors, carbon nanotubes, and transition metal dichalcogenides. Finally, a perspective of outlooking future developments will be elaborated.show less • Aug.08，2022 • Photonics Insights,Vol. 1, Issue 1 • R04 (2022) Classical and generalized geometric phase in electromagnetic metasurfaces The geometric phase concept has profound implications in many branches of physics, from condensed matter physics to quantum systems. Although geometric ph The geometric phase concept has profound implications in many branches of physics, from condensed matter physics to quantum systems. Although geometric phase has a long research history, novel theories, devices, and applications are constantly emerging with developments going down to the subwavelength scale. Specifically, as one of the main approaches to implement gradient phase modulation along a thin interface, geometric phase metasurfaces composed of spatially rotated subwavelength artificial structures have been utilized to construct various thin and planar meta-devices. In this paper, we first give a simple overview of the development of geometric phase in optics. Then, we focus on recent advances in continuously shaped geometric phase metasurfaces, geometric–dynamic composite phase metasurfaces, and nonlinear and high-order linear Pancharatnam–Berry phase metasurfaces. Finally, conclusions and outlooks for future developments are presented.show less • Aug.08，2022 • Photonics Insights,Vol. 1, Issue 1 • R03 (2022) In this study, an optical fiber based magnetic-tuned graphene mechanical resonator are demonstrated by integrating superparamagnetic iron oxide nanoparticles on the graphene membrane. T In this study, an optical fiber based magnetic-tuned graphene mechanical resonator are demonstrated by integrating superparamagnetic iron oxide nanoparticles on the graphene membrane. The resonance frequency shift is achieved by tuning the tension of the graphene membrane with a magnetic field. A resonance-frequency tunability of 23 kHz using a 100-mT magnetic field is achieved. The device provides a new way to tune a GMR with a non-contact force. It could also be used for weak-magnetic field detection in the future with further improvements in sensitivity.show less • Aug.08，2022 • Chinese Optics Letters,Vol. 21, Issue 1 • (2023) Noninterferometric X-ray quantitative phase imaging (XQPI) methods have provided a much simpler than interferometric scheme, high-resolution, and reliable phase-contrast image. We repor Noninterferometric X-ray quantitative phase imaging (XQPI) methods have provided a much simpler than interferometric scheme, high-resolution, and reliable phase-contrast image. We report on implementing the volumetric XQPI images using concurrent-bidirectional scanning of the orthogonal plane on the optical axis of the Foucault differential filter; we then extracted data in conjunction with the transport-intensity equation. The volumetric image of the laminate microstructure of the gills of a fish was successfully reconstructed to demonstrate our XQPI method. The method can perform 3D rendering without any rotational motion for laterally extended objects by manipulating incoherent X-rays using the pinhole array. show less • Aug.08，2022 • Chinese Optics Letters,Vol. 21, Issue 1 • (2023) Nitrogen-vacancy (NV) color centers can perform highly sensitive and spatially resolved quantum measurements of physical quantities such as magnetic field, temperature, and pressure. Me Nitrogen-vacancy (NV) color centers can perform highly sensitive and spatially resolved quantum measurements of physical quantities such as magnetic field, temperature, and pressure. Meanwhile, sensing so many variables at the same time often introduces additional noise, causing a reduced accuracy. Here, a dual microwave time-division multiplexing protocol is used in conjunction with a lock-in amplifier in order to decouple temperature from magnetic field and vice versa. In this protocol, dual-frequency driving and frequency modulation are used to measure magnetic and temperature field simultaneously in real time. The sensitivity of our system is about 3.4nT/√Hz and 1.3mK/√Hz, respectively. Our detection protocol not only enables multifunctional quantum sensing, but also extends more practical applications.show less • Aug.08，2022 • Chinese Optics Letters,Vol. 21, Issue 1 • (2023) Studies on the kinetics of gas-phase chemical reactions currently rely on calculations or simulations and lack simple, fast, and accurate direct measurement methods. We developed a tuna Studies on the kinetics of gas-phase chemical reactions currently rely on calculations or simulations and lack simple, fast, and accurate direct measurement methods. We developed a tunable laser molecular absorption spectroscopy (TLAS) measurement system to achieve direct measurements of such reactions by using wavelength modulated spectroscopy, and have performed direct online measurements and diagnostics of molecular concentration, the heat of reaction, and pressure changes during the redox reaction of ozone with nitrogen oxides with a time resolution of 0.1 s. This study provides a promising diagnostic tool for the study of gas-phase chemical reaction kinetics. show less • Aug.08，2022 • Chinese Optics Letters,Vol. 21, Issue 1 • (2023) Hangzhou, China25~27 November Xi'an, ChinaAug 7-10, 2022 Three-year-old journal ranked among top optics journals in first year of recognition. • Journal • 1th Jul，2022 The image on the cover for Chinese Optics Letters Volume 20, Issue 5, reports reflective photon nanosieves that consist of metallic meta-mirrors sitting on a transparent quartz substrate. Upon illumination, these meta-mirrors offer the reflectance of &sim;50%, which is higher than the transmission of visible light through diameter-identical nanoholes.The image is based on original research by Samia Osman Hamid Mohammed et al. presented in their paper "Efficiency-enhanced reflective nanosieve holograms", Chinese Optics Letters 20 (5), 053602. (2022) • Journal • 27th May，2022	2022-08-07 16:11:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28741419315338135, "perplexity": 4202.815819445669}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570651.49/warc/CC-MAIN-20220807150925-20220807180925-00729.warc.gz"}
http://todaynumerically.blogspot.co.uk/2013/03/tuesday-26-march-2013.html	## Tuesday, 26 March 2013 ### Tuesday, 26 MARCH 2013 Today is the $85^{th}$ day of the year. $85 = 5 \times 17$ $85 = 2^2 + 9^2 = 6^2 + 7^2$, so it is the sum of two squares in two different ways. $85$ is the member of the following four primitive Pythagorean triples: $(13, 84, 85)$ $(77, 36, 85)$ $(85, 132, 157)$ $(85, 3612, 3613)$ $85$ is a Joke Number or a Smith Number. A Joke or Smith Number is one where the sum of the digits of the number, $8 + 5 = 13$, is equal to the sum of the digits of the factors of that number $5 + 1 + 7 = 13$, see A006753. $\frac {4^4 - 1} {3} = \frac {256 - 1} {3} = \frac {255} {3} = 85$ so $85$ is a member of the sequence $\frac {4^n -1} {3}$, see A002450. What I find surprising is that this sequence implies that all powers of 4 less one are a multiple of three. It transpires that there are $85$ different ways partitions of $29$ into at most three parts, see A001399. The $85$ partitions are: $1: [0, 0, 29]$ $2: [0, 1, 28]$ $3: [0, 2, 27]$ $4: [0, 3, 26]$ $5: [0, 4, 25]$ $6: [0, 5, 24]$ $7: [0, 6, 23]$ $8: [0, 7, 22]$ $9: [0, 8, 21]$ $10: [0, 9, 20]$ $11: [0, 10, 19]$ $12: [0, 11, 18]$ $13: [0, 12, 17]$ $14: [0, 13, 16]$ $15: [0, 14, 15]$ $16: [1, 1, 27]$ $17: [1, 2, 26]$ $18: [1, 3, 25]$ $19: [1, 4, 24]$ $20: [1, 5, 23]$ $21: [1, 6, 22]$ $22: [1, 7, 21]$ $23: [1, 8, 20]$ $24: [1, 9, 19]$ $25: [1, 10, 18]$ $26: [1, 11, 17]$ $27: [1, 12, 16]$ $28: [1, 13, 15]$ $29: [1, 14, 14]$ $30: [2, 2, 25]$ $31: [2, 3, 24]$ $32: [2, 4, 23]$ $33: [2, 5, 22]$ $34: [2, 6, 21]$ $35: [2, 7, 20]$ $36: [2, 8, 19]$ $37: [2, 9, 18]$ $38: [2, 10, 17]$ $39: [2, 11, 16]$ $40: [2, 12, 15]$ $41: [2, 13, 14]$ $42: [3, 3, 23]$ $43: [3, 4, 22]$ $44: [3, 5, 21]$ $45: [3, 6, 20]$ $46: [3, 7, 19]$ $47: [3, 8, 18]$ $48: [3, 9, 17]$ $49: [3, 10, 16]$ $50: [3, 11, 15]$ $51: [3, 12, 14]$ $52: [3, 13, 13]$ $53: [4, 4, 21]$ $54: [4, 5, 20]$ $55: [4, 6, 19]$ $56: [4, 7, 18]$ $57: [4, 8, 17]$ $58: [4, 9, 16]$ $59: [4, 10, 15]$ $60: [4, 11, 14]$ $61: [4, 12, 13]$ $62: [5, 5, 19]$ $63: [5, 6, 18]$ $64: [5, 7, 17]$ $65: [5, 8, 16]$ $66: [5, 9, 15]$ $67: [5, 10, 14]$ $68: [5, 11, 13]$ $69: [5, 12, 12]$ $70: [6, 6, 17]$ $71: [6, 7, 16]$ $72: [6, 8, 15]$ $73: [6, 9, 14]$ $74: [6, 10, 13]$ $75: [6, 11, 12]$ $76: [7, 7, 15]$ $77: [7, 8, 14]$ $78: [7, 9, 13]$ $79: [7, 10, 12]$ $80: [7, 11, 11]$ $81: [8, 8, 13]$ $82: [8, 9, 12]$ $83: [8, 10, 11]$ $84: [9, 9, 11]$ $85: [9, 10, 10]$	2017-10-21 10:03:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8362699747085571, "perplexity": 274.9727588255067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824733.32/warc/CC-MAIN-20171021095939-20171021115549-00023.warc.gz"}
https://tex.stackexchange.com/questions/459835/error-when-using-tikzpicture-with-moodle-package	# error when using tikzpicture with moodle package I want to generate a quiz for my students to be imported within the moodle platform . The quiz includes some figure. I tried the moodle package from texlive 2018 but I get the following error when using the tikzpicture environment : ERROR: Package tikz Error: Sorry, the system call 'pdflatex -shell-escape -halt-on-error -interaction=batchmode -jobname "geometry-tikztemp-1" "\def\tikzexternalrealjob{geometry}\input{geometry}"' did NOT result in a usable output file 'geometry-tikztemp-1' (expected one of .pdf:.jpg:.jpeg:.png:). Please verify that you have enabled system calls. For pdflatex, this is 'pdflatex -shell-escape'. Sometimes it is also named 'write 18' or something like that. Or maybe the command simply failed? Error messages can be found in 'geometry-tikztemp-1.log'. If you continue now, I'll try to typeset the picture. Here is an example: \documentclass{article} \usepackage{tikz} \usepackage[draft]{moodle} \begin{document} \begin{quiz}{geometry} \begin{multi}{circle} is the shape in the figure a circle? \begin{tikzpicture} \end{tikzpicture} \item yes \item* no \end{multi} \end{quiz} \end{document} Not an answer but a workaround if you are in a rush. To be honest, I have hardly ever seen as strong "side-effects" as I see here with moodle. Even trying to do the \savebox after loading this package fails. Hope you find an expert who can fix this. \documentclass{article} \usepackage{tikz} \usepackage[draft]{moodle} \newsavebox\picbox \begin{lrbox}{\picbox}% \begin{tikzpicture} \draw (0,0) circle(1cm); \end{tikzpicture} \end{lrbox} \begin{document} \begin{quiz}{geometry} \begin{multi}{circle} is the shape in the figure a circle? \usebox\picbox \item yes \item* no \end{multi} \end{quiz} \end{document} • fwiw the error in the original is that \pgfexternal@originalshipout is said to be undefined (at end of expansion of \pgf@externalend. And I see in moodle.sty near end of file that it does rather heavy patching of Tikz, with a \renewenvironment{tikzpicture} at begin document. Thus the issue might need reporting to moodle.sty author. – user4686 Nov 14 '18 at 11:44 • @jfbu Thanks! Yes, that makes sense, and seems to explain the "side-effects". (If I was the OP, I would consider switching to another package.) – user121799 Nov 14 '18 at 11:48 • @marmot, is someone aware of another package that helps creating quiz in moodle ? – Hafid Boukhoulda Nov 14 '18 at 21:59 • @HafidBoukhoulda Someone: probably yes, but unfortunately not me. But you could ask a question, I would also be interested in the result, and asking questions is free. – user121799 Nov 14 '18 at 22:02 • @HafidBoukhoulda Between us: this seems to be a bit a dangerous package. Why is it necessary for a package whose purpose is to typeset quizzes to (a) force you to externalize tikzpictures and, what is IMHO even worse, (b) to do \renewenvironment{tikzpicture} ? To me this does not make sense, but of course I cannot rule out that there might be reasons for it. Nor did I verify the findings of jfbu but I fully trust this user, who has a very good reputation (which is not reflected by the score). – user121799 Nov 14 '18 at 22:22 As moodle manual says, it invokes \usetikzlibrary{external}, therefore calling --shell-escape option in compilation process is a must and then you will get the following picture. • to compile I have tried the following pdflatex --shell-escape geometry.tex and pdflatex -shell-escape geometry.tex but without any success (the same error) – Hafid Boukhoulda Nov 13 '18 at 20:22 • I've never gotten externalize to work, either. Try using \tikzexternaldisable. – John Kormylo Nov 13 '18 at 21:08 • @JohnKormylo I did (very often) get externalize to work and of course the first thing I tried is to compile with -shell-escape. IMHO moodle introduces complications beyond that, which is why I only offered a workaround. I am actually wondering if this answer works for you or anyone else, at least for the OP and me it does not seem to work. – user121799 Nov 13 '18 at 21:36 • @HafidBoukhoulda I've tried both of pdflatex and xelatex engine with --shell-escape, getting the same results. pdflatex and xelatex versions of my TeX distro. are: pdfTeX 3.14159265-2.6-1.40.19 (TeX Live 2018) XeTeX 3.14159265-2.6-0.99999 (TeX Live 2018) I also updated the picture because I'd erased one line in your file and it produced an award result. – javadr Nov 14 '18 at 1:41 • If @javadr has got a result can we conclude that the issue is not with the moodle package? and what should we investigate further to eliminate the error? Me too I use pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) ! – Hafid Boukhoulda Nov 14 '18 at 6:46 This answer comes a long time after the question but it might help someone to know that: 1. the example provided by the OP compiles fine if the package tikz is loaded after moodle (version 0.5). 2. starting from version 0.8 of the moodle package, you no longer need to load tikz after moodle.	2021-02-27 18:58:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7598496079444885, "perplexity": 2618.737613109844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178359082.48/warc/CC-MAIN-20210227174711-20210227204711-00041.warc.gz"}
https://trac-hacks.org/ticket/11166	Opened 5 years ago Closed 5 years ago #11166 closed enhancement (fixed) Reported by: Owned by: anonymous falkb normal SimpleMultiProjectPlugin minor ykrocku@… 1.0 In timeline page, there is a "Filter projects" option added by simple multiproject plugin. It only display the selected project's events when a project selected. I got a little confused about the 'ALL' option.I assume the 'ALL' option shouldn't filter out any events, but indeed it filter out changes made to those tickets without its custom field 'project' set. Here is the way I think 'ALL' option should work if template == 'timeline.html': filter_projects = self._filtered_projects(req) if not filter_projects: #no filter means likely more than 1 project, so we insert the project name - filter_projects = [project[1] for project in self.__SmpModel.get_all_projects()] + return template, data, content_type comment:2 follow-up: 3 Changed 5 years ago by anonymous Wouldn't the patch result in not having the yellow project label anymore, for all the tickets where 'project' is set? comment:3 in reply to: 2 Changed 5 years ago by ykrocku@… Wouldn't the patch result in not having the yellow project label anymore, for all the tickets where 'project' is set? Yeah, but I think it is better than the way it works now. comment:4 Changed 5 years ago by falkb Status: new → assigned comment:5 Changed 5 years ago by falkb In 13291: bugfix: if "All" is set as timeline filter, display also tickets without a set project (refs #11166) comment:6 Changed 5 years ago by falkb ykrocku, please test and close this ticket if it's OK now. Thanks a lot for reporting. comment:7 Changed 5 years ago by ykrocku@… Resolution: → fixed assigned → closed Test ok comment:8 Changed 5 years ago by Ryan J Ollos Description: modified (diff) Modify Ticket Change Properties Action as closed The owner will remain falkb. The resolution will be deleted.	2018-03-24 23:45:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1992844045162201, "perplexity": 10736.051447889555}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651465.90/warc/CC-MAIN-20180324225928-20180325005928-00050.warc.gz"}
https://homework.cpm.org/category/MN/textbook/cc3mn/chapter/cc35/lesson/cc35.2.3/problem/5-49	Home > CC3MN > Chapter cc35 > Lesson cc35.2.3 > Problem5-49 5-49. Solve for the variable. Homework Help ✎ 1. $\frac { 7 y } { 8 } - \frac { 3 y } { 5 } = \frac { 11 } { 2 }$ Multiply the entire equation by a common multiple of the denominators to get rid of the fractions. $35y−24y=220$ Combine like terms. $11y=220$ $y=20$ 1. $\frac { a + 4 } { 3 } - \frac { a } { 7 } = \frac { a + 7 } { 5 }$ Follow the steps in part (a).	2019-10-24 02:18:27	{"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5259914994239807, "perplexity": 4592.840891678769}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987838289.72/warc/CC-MAIN-20191024012613-20191024040113-00005.warc.gz"}
https://mathvisions.wordpress.com/2017/05/19/homework-11-due-5-21-14-for-mrs22-3-or-5-22-14-for-mrs22-1/	# Trigonometry Homework #12 due 5-25 Solve each equation for $x$ 1. $2\sin{x}=\sin^2{x}+\cos^2{x}$ 2. $3\tan^2{x}=1$ 3. $\tan{x}-\cot{x}=\frac{\sin{x}-\cos{x}}{\sin{x}}$ 4. $\cos^2\frac{x}{5}=\frac{1}{2}$ 5. $\csc\left(x+\frac{2\pi}{5}\right)=1$ 6. $\sqrt{\frac{1-\cos\frac{x}{4}}{2}}=\frac{\sqrt{3}}{2}$ 7. $\tan{x}\cos{x}=0$ 8. $\tan{x}=\frac{\sin\frac{5\pi}{6}}{1+\cos\frac{5\pi}{6}}$ 9. $\sec^2{x}+1=5$ 10. $\ln[-\cos(4x)]=0$	2017-05-30 01:19:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9433883428573608, "perplexity": 644.499243775454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613738.67/warc/CC-MAIN-20170530011338-20170530031338-00380.warc.gz"}
http://worldcat.org/identities/lccn-n2003001082/	# United States Department of Energy High Energy Physics Division Overview Works: 2,926 works in 2,977 publications in 1 language and 9,078 library holdings Conference papers and proceedings Sponsor, Researcher, Other Publication Timeline . Most widely held works by United States High energy density and high power RF : 5th Workshop on High Energy Density and High Power RF : Snowbird, Utah, 1-5 October 2001 by Workshop on High Energy Density and High Power RF( Book ) 2 editions published in 2002 in English and held by 110 WorldCat member libraries worldwide Measurement of the ratio of the production cross sections times branching fractions of Bc" ₂!J/[psi][pi]"and B" ₂!J/[psi] K" and B(Bc"₂!J/[psi] [pi]"[pi]"[pi]-/+)/B(Bc" ₂!J/[psi] [pi]") in pp collisions at [arrow]" = 7 TeV( ) 2 editions published between 2014 and 2015 in English and held by 0 WorldCat member libraries worldwide The ratio of the production cross sections times branching fractions ([sigma](Bc") B(Bc" ₂!J/[psi][pi]"))/([sigma](B") B(B" ₂!J/[psi]K") is studied in proton-proton collisions at a center-of-mass energy of 7 TeV with the CMS detector at the LHC. The kinematic region investigated requires Ba, sub>c" and B"mesons with transverse momentum p[tau]> 15 GeV and rapidity MINERvA neutrino detector response measured with test beam data( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide The MINERvA collaboration operated a scaled-down replica of thesolid scintillator tracking and sampling calorimeter regions of the MINERvA detector in a hadron test beam at the Fermilab Test Beam Facility. This paper reports measurements with samples of protons, pions, and electrons from 0.35 to 2.0 GeV/c momentum. The calorimetric response to protons, pions, and electrons is obtained from these data. A measurement of the parameter in Birks' law and an estimate of the tracking efficiency are extracted from the proton sample. Overall the data are well described by a Geant4-based Monte Carlo simulation of the detector and particle interactions with agreements better than 4% for the calorimetric response, though some features of the data are not precisely modeled. These measurements are used to tune the MINERvA detector simulation and evaluate systematic uncertainties in support of the MINERvA neutrino cross-section measurement program Search for Diphoton Resonances in the Mass Range from 150 to 850 GeV in pp Collisions at $\sqrt{s}$ = 8 TeV( ) 10 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide Stringent limits are set on the long-lived lepton-like sector of the phenomenological minimal supersymmetric standard model (pMSSM) and the anomaly-mediated supersymmetry breaking (AMSB) model. We derived the limits from the results presented in a recent search for long-lived charged particles in proton-proton collisions, based on data collected by the CMS detector at a centre-of-mass energy of 8 TeV at the Large Hadron Collider. In the pMSSM parameter sub-space considered, 95.9 % of the points predicting charginos with a lifetime of at least 10 ns are excluded. Furthermore, these constraints on the pMSSM are the first obtained at the LHC. Charginos with a lifetime greater than 100 ns and masses up to about 800 GeV in the AMSB model are also excluded. Furthermore, the method described can also be used to set constraints on other models Isolation of flow and nonflow correlations by two- and four-particle cumulant measurements of azimuthal harmonics in [arrow]"NN = 200 GeV Au+Au collisions( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide A data-driven method was applied to Au+Au collisions at [arrow]"NN = 200 GeV made with the STAR detector at RHIC to isolate pseudorapidity distance [Delta][eta]-dependent and [Delta][eta]-independent correlations by using two- and four-particle azimuthal cumulant measurements. We identified a [Delta][eta]-independent component of the correlation, which is dominated by anisotropic flow and flow fluctuations. It was also found to be independent of [eta] within the measured range of pseudorapidity Performance and results of the LBNE 35 ton membrane cryostat prototype( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide We report on the performance and commissioning of the first membrane cryostat to be used for scientific application. The Long Baseline Neutrino Experiment (LBNE) has designed and fabricated a membrane cryostat prototype in collaboration with Ishikawajima-Harima Heavy Industries Co., Ltd. (IHI). LBNE has designed and fabricated the supporting cryogenic system infrastructure and successfully commissioned and operated the first membrane cryostat. Original goals of the prototype are: to demonstrate the membrane cryostat technology in terms of thermal performance, feasibility for liquid argon and leak tightness; to demonstrate that we can remove all the impurities from the vessel and achieve the purity requirements in a membrane cryostat without evacuation; to demonstrate that we can achieve and maintain the purity requirements of the liquid argon using mol sieve and copper filters. The purity requirements of a large liquid argon detector such as LBNE are contaminants below 200 parts per trillion (ppt) oxygen equivalent. LBNE is planning the design and construction of a large liquid argon detector. This presentation will present requirements, design and construction of the LBNE 35 ton membrane cryostat prototype, and detail the commissioning and performance. The experience and results of this prototype are extremely important for the development of the LBNE detector Summary of the Second Workshop on Liquid Argon Time Projection Chamber Research and Development in the United States( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide The second workshop to discuss the development of liquid argon time projection chambers (LArTPCs) in the United States was held at Fermilab on July 8-9, 2014. The workshop was organized under the auspices of the Coordinating Panel for Advanced Detectors, a body that was initiated by the American Physical Society Division of Particles and Fields. All presentations at the workshop were made in six topical plenary sessions: i) Argon Purity and Cryogenics, ii) TPC and High Voltage, iii) Electronics, Data Acquisition and Triggering, iv) Scintillation Light Detection, v) Calibration and Test Beams, and vi) Software. This document summarizes the current efforts in each of these areas. It primarily focuses on the work in the US, but also highlights work done elsewhere in the world An ultra-weak sector, the strong CP problem and the pseudo-Goldstone dilaton( ) 2 editions published between 2014 and 2015 in English and held by 0 WorldCat member libraries worldwide In the context of a Coleman-Weinberg mechanism for the Higgs boson mass, we address the strong CP problem. We show that a DFSZ-like invisible axion model with a gauge-singlet complex scalar field S, whose couplings to the Standard Model are naturally ultra-weak, can solve the strong CP problem and simultaneously generate acceptable electroweak symmetry breaking. The ultra-weak couplings of the singlet S are associated with underlying approximate shift symmetries that act as custodial symmetries and maintain technical naturalness. The model also contains a very light pseudo-Goldstone dilaton that is consistent with cosmological Polonyi bounds, and the axion can be the dark matter of the universe. We further outline how a SUSY version of this model, which may be required in the context of Grand Unification, can avoid introducing a hierarchy problem Final report by Yale University( ) 5 editions published between 2012 and 2014 in English and held by 0 WorldCat member libraries worldwide This work is focused on the design and construction of novel beam diagnostic and instrumentation for charged particle accelerators required for the next generation of linear colliders. Our main interest is in non-invasive techniques. The Northwestern group of Velasco has been a member of the CLIC Test Facility 3 (CTF3) collaboration since 2003, and the beam instrumentation work is developed mostly at this facility1. This 4 kW electron beam facility has a 25-170 MeV electron LINAC. CTF3 performed a set of dedicated measurements to finalize the development of our RF-Pickup bunch length detectors. The RF-pickup based on mixers was fully commissioned in 2009 and the RF-pickup based on diodes was finished in time for the 2010-11 data taking. The analysis of all the data taken in by the summer of 2010 was finish in time and presented at the main conference of the year, LINAC 2010 in Japan Measurements of the Upsilon(1S), Upsilon(2S), and Upsilon(3S) differential cross sections in pp collisions at sqrt(s) = 7 TeV( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide Differential cross sections as a function of transverse momentum pt are presented for the production of Y(nS) (n = 1, 2, 3) states decaying into a pair of muons. Data corresponding to an integrated luminosity of 4.9 inverse femtobarns in pp collisions at sqrt(s) = 7 TeV were collected with the CMS detector at the LHC. The analysis selects events with dimuon rapidity abs(y) <1.2 and dimuon transverse momentum in the range 10 <pt <100 GeV. The measurements show a transition from an exponential to a power-law behavior at pt ~ 20 GeV for the three Y states. Above that transition, the Y spectrum is significantly harder than that of the Y(1S) and Y(2S). The ratios of the Y(3S) and Y(2S) differential cross sections to the Y(1S) cross section show a rise as pt increases at low pt, then become flatter at higher pt Simulation of transverse modes with their intrinsic Landau damping for bunched beams in the presence of space charge( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide Automated Transient Identification in the Dark Energy Survey( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide We describe an algorithm for identifying point-source transients and moving objects on reference-subtracted optical images containing artifacts of processing and instrumentation. The algorithm makes use of the supervised machine learning technique known as Random Forest. We present results from its use in the Dark Energy Survey Supernova program (DES-SN), where it was trained using a sample of 898,963 signal and background events generated by the transient detection pipeline. After reprocessing the data collected during the first DES-SN observing season (2013 September through 2014 February) using the algorithm, the number of transient candidates eligible for human scanning decreased by a factor of 13.4, while only 1.0 percent of the artificial Type Ia supernovae (SNe) injected into search images to monitor survey efficiency were lost, most of which were very faint events. Here we characterize the algorithm's performance in detail, and we discuss how it can inform pipeline design decisions for future time-domain imaging surveys, such as the Large Synoptic Survey Telescope and the Zwicky Transient Facility Effect of event selection on jetlike correlation measurement in [mml : math altimg="si1.gif" overflow="scroll" xmlns xocs="http //www.elsevier.com/xml/xocs/dtd" xmlns xs="http //www.w3.org/2001/XMLSchema" xmlns xsi="http //www.w3.org/2001/XMLSchema-instance" xmlns="http //www.elsevier.com/xml/ja/dtd" xmlns ja="http //www.elsevier.com/xml/ja/dtd" xmlns mml="http //www.w3.org/1998/Math/MathML" xmlns tb="http //www.elsevier.com/xml/common/table/dtd" xmlns sb="http //www.elsevier.com/xml/common/struct-bib/dtd" xmlns ce="http //www.elsevier.com/xml/common/dtd" xmlns xlink="http //www.w3.org/1999/xlink" xmlns cals="http //www.elsevier.com/xml/common/cals/dtd" xmlns sa="http //www.elsevier.com/xml/common/struct-aff/dtd"][mml mi]d[/mml mi][mml mo]+[/mml mo][mml mrow][mml mi mathvariant="normal"]Au[/mml mi][/mml mrow][/mml math] collisions at [mml math altimg="si2.gif" overflow="scroll" xmlns xocs="http //www.elsevier.com/xml/xocs/dtd" xmlns xs="http //www.w3.org/2001/XMLSchema" xmlns xsi="http //www.w3.org/2001/XMLSchema-instance" xmlns="http //www.elsevier.com/xml/ja/dtd" xmlns ja="http //www.elsevier.com/xml/ja/dtd" xmlns mml="http //www.w3.org/1998/Math/MathML" xmlns tb="http //www.elsevier.com/xml/common/table/dtd" xmlns sb="http //www.elsevier.com/xml/common/struct-bib/dtd" xmlns ce="http //www.elsevier.com/xml/common/dtd" xmlns xlink="http //www.w3.org/1999/xlink" xmlns cals="http //www.elsevier.com/xml/common/cals/dtd" xmlns sa="http //www.elsevier.com/xml/common/struct-aff/dtd"][mml msqrt][mml msub][mml mrow][mml mi][/mml mi][/mml mrow][mml mrow][mml mi mathvariant="normal"]NN[/mml mi][/mml mrow][/mml msub][/mml msqrt][mml mo]=[/mml mo][mml mn]200[/mml mn][mml mtext] [/mml mtext][mml mtext]GeV[/mml mtext][/mml math]( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide In this study, dihadron correlations are analyzed in √sNN = 200 GeV d+Au collisions classified by forward charged particle multiplicity and zero-degree neutral energy in the Au-beam direction. It is found that the jetlike correlated yield increases with the event multiplicity. After taking into account this dependence, the non-jet contribution on the away side is minimal, leaving little room for a back-to-back ridge in these collisions Search for Resonant $\mathrm{t\bar{t}}$ Production in Proton-Proton Collisions at $\sqrt{s}$ = 8 TeV( ) 3 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide A description is provided of the performance of the CMS detector for photon reconstruction and identification in proton-proton collisions at a centre-of-mass energy of 8 TeV at the CERN LHC. Details are given on the reconstruction of photons from energy deposits in the electromagnetic calorimeter (ECAL) and the extraction of photon energy estimates. Furthermore, the reconstruction of electron tracks from photons that convert to electrons in the CMS tracker is also described, as is the optimization of the photon energy reconstruction and its accurate modelling in simulation, in the analysis of the Higgs boson decay into two photons. In the barrel section of the ECAL, an energy resolution of about 1% is achieved for unconverted or late-converting photons from H → [gamma][gamma] decays. Furthermore, different photon identification methods are discussed and their corresponding selection efficiencies in data are compared with those found in simulated events Search for W' decaying to tau lepton and neutrino in proton-proton collisions at $\sqrt{s}$ = 8 TeV( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide We found that the first search for a heavy charged vector boson in the final state with a tau lepton and a neutrino is reported, using 19.7 fb-1 of LHC data at √s = 8 TeV. A signal would appear as an excess of events in kinematic regions where the standard model background is low. No excess is observed. Limits are set on a model in which the W' decays preferentially to fermions of the third generation. Our results substantially extend previous constraints on this model. Masses below 2.0 to 2.7 TeV are excluded, depending on the model parameters. In addition, the existence of a W' boson with universal fermion couplings is excluded at 95% confidence level, for W' masses below 2.7 TeV Cosmological implications of baryon acoustic oscillation measurements( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide Here, we derive constraints on cosmological parameters and tests of dark energy models from the combination of baryon acoustic oscillation (BAO) measurements with cosmic microwave background (CMB) data and a recent reanalysis of Type Ia supernova (SN) data. In particular, we take advantage of high-precision BAO measurements from galaxy clustering and the Lyman-[alpha] forest (LyaF) in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). Treating the BAO scale as an uncalibrated standard ruler, BAO data alone yield a high confidence detection of dark energy; in combination with the CMB angular acoustic scale they further imply a nearly flat universe. Adding the CMB-calibrated physical scale of the sound horizon, the combination of BAO and SN data into an "inverse distance ladder" yields a measurement of H0=67.3±1.1 km s1 Mpc₋1, with 1.7% precision. This measurement assumes standard prerecombination physics but is insensitive to assumptions about dark energy or space curvature, so agreement with CMB-based estimates that assume a flat [Lambda]CDM cosmology is an important corroboration of this minimal cosmological model. For constant dark energy ([Lambda]), our BAO+SN+CMB combination yields matter density [Omega]m=0.301±0.008 and curvature [Omega]k=₋0.003±0.003. When we allow more general forms of evolving dark energy, the BAO+SN+CMB parameter constraints are always consistent with flat [Lambda]CDM values at H"[sigma]. While the overall [chi]2 of model fits is satisfactory, the LyaF BAO measurements are in moderate (2-2.5[sigma]) tension with model predictions. Models with early dark energy that tracks the dominant energy component at high redshift remain consistent with our expansion history constraints, and they yield a higher H0 and lower matter clustering amplitude, improving agreement with some low redshift observations. Expansion history alone yields an upper limit on the summed mass of neutrino species, [Sigma]m[nu]<0.56 eV (95% confidence), improving to [Sigma]m[nu]<0.25 eV if we include the lensing signal in the Planck CMB power spectrum. In a flat [Lambda]CDM model that allows extra relativistic species, our data combination yields Neff=3.43±0.26; while the LyaF BAO data prefer higher Neff when excluding galaxy BAO, the galaxy BAO alone favor Neff H"3. Lastly, when structure growth is extrapolated forward from the CMB to low redshift, standard dark energy models constrained by our data predict a level of matter clustering that is high compared to most, but not all, observational estimates The LMC geometry and outer stellar populations from early DES data( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide DES13S2cmm : the first superluminous supernova from the Dark Energy Survey( ) 2 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide Searches for third-generation squark production in fully hadronic final states in proton-proton collisions at [arrow]" = 8 TeV( ) 4 editions published in 2015 in English and held by 0 WorldCat member libraries worldwide The purely electroweak (EW) cross section for the production of two jets in association with a Z boson, in proton-proton collisions at [arrow]" = 8 TeV, is measured using data recorded by the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of 19.7 fb-1. The electroweak cross section for the lljj final state (with l = e or [mu] and j representing the quarks produced in the hard interaction) in the kinematic region defined by Mll> 50 GeV, Mjj> 120 GeV, transverse momentum pTj> 25 GeV, and pseudorapidity 15 - Foot Bubble Chamber : Safety Report - Volume 1( ) 2 editions published in 1972 in English and held by 0 WorldCat member libraries worldwide more fewer Audience Level 0 1 Kids General Special Alternative Names United States. Department of Energy. Office of Science. High Energy Physics Division United States. Dept. of Energy. High Energy Physics Division Languages English (54)	2017-11-24 17:00:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5289262533187866, "perplexity": 3726.17946636205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934808260.61/warc/CC-MAIN-20171124161303-20171124181303-00131.warc.gz"}
https://www.doubtnut.com/question-answer-physics/a-parallel-plate-capacitor-has-smooth-square-plates-of-side-a-it-is-charged-by-a-battery-so-that-the-13079558	Home > English > Class 12 > Physics > Chapter > Capacitance > A parallel plate capacitor has... Text Solution The slab can execute SHM between the plates.The plate can execute oscillatory motion which is not SHMThe magnitude of the force experienced by the slab is constantThe magnitude of the force experienced by the slab is not constant. Solution : Force on the slab is given by F=(dU)/(dx), where U is slab in it. Here F is not directly proportional to x, but depends on x.	2022-10-01 01:20:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35905954241752625, "perplexity": 1001.3394773689377}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00237.warc.gz"}
https://www.technetiumbo647.xyz/wiki/User:ClueBot_III/Master_Detailed_Indices/Talk:Jamie_Foxx	# Rotating furnace (Redirected from Rotating furnance) Axially symmetrical paraboloid. The inside surface is concave A rotating furnace is a device for making solid objects which have concave surfaces that are segments of axially symmetrical paraboloids. Usually, the objects are made of glass. The furnace makes use of the fact, which was known already to Newton, that the centrifugal-force-induced shape of the top surface of a spinning liquid is a concave paraboloid, identical to the shape of a reflecting telescope's primary focusing mirror. Paraboloids can be used in various ways, including (after being silvered) as primary mirrors in reflecting telescopes and solar cookers.[1] ## Design Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of plexiglass. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the center. The furnace includes a mechanism that rotates an open-topped container at constant speed around a vertical axis. A quantity of glass sufficient to make the mirror is placed in the container, heated until it is completely molten, and then allowed to cool while continuing to rotate until it has completely solidified. When the rotation is stopped, the glass is solid, so the paraboloidal shape of its top surface is preserved.[2][3] This process is called spin casting. The same process can be used to make a lens with a concave paraboloidal surface. The other surface is shaped by the container that holds the molten glass acting as a mold. Lenses made this way are sometimes used as objectives in refracting telescopes. The axis of rotation becomes the axis of the paraboloid. It is not necessary for this axis to be in the center of the container of glass, or even for it to pass through the container. By placing the container away from the axis, off-axis paraboloidal segments can be cast. This is done in the making of very large telescopes which have mirrors consisting of several segments. ## Mathematical model ### Rotation speed and focal length The focal length of the paraboloid is related to the angular speed at which the liquid is rotated by the equation: ${\displaystyle 2f\omega ^{2}=g}$, where ${\displaystyle f}$ is the focal length, ${\displaystyle \omega }$ is the rotation speed, and ${\displaystyle g}$ is the acceleration due to gravity. On the Earth's surface, ${\displaystyle g}$ is about 9.81 metres per second-squared, so ${\displaystyle f\omega ^{2}\approx 4.905}$ meters.[3] Equivalently, as 1 radian per second is about 9.55 rotations per minute (RPM), ${\displaystyle fs^{2}\approx 447}$, where ${\displaystyle f}$ is the focal length in metres, and ${\displaystyle s}$ is the rotation speed in RPM. ## Uses Generally, a spin-cast paraboloid is not sufficiently accurate to permit its immediate use as a telescope mirror or lens, so it is corrected by computer-controlled grinding machines. The amount of grinding done, and the mass of glass material wasted, are much less than would have been required without spinning. Spin casting can also be used, often with materials other than glass, to produce prototype paraboloids, such as spotlight reflectors or solar-energy concentrators, which do not need to be as exactly paraboloidal as telescope mirrors. Spin casting every paraboloid that is made would be too slow and costly, so the prototype is simply copied relatively quickly and cheaply and with adequate accuracy. Liquid-mirror telescopes have rotating mirrors that consist of a liquid metal such as mercury or a low-melting alloy of gallium. These mirrors do not solidify and they are used while liquid and rotating. The rotation shapes them into paraboloids that are accurate enough to be used as primary reflectors in telescopes. Compared with spin-cast glass mirrors which need correction due to the distortions that arise during and after solidification, these mirrors require no such correction. 3. ^ a b Ninomiyaa, Yuichi (1979). "Parabolic mirror made by the rotation method: its fabrication and defects". doi:10.1364/AO.18.001835. Cite journal requires \|journal= (help)	2021-09-22 22:43:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6401870250701904, "perplexity": 1117.4961045615837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00282.warc.gz"}
https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Map%3A_Organic_Chemistry_(Wade)/01%3A_Introduction_and_Review/1.08%3A_Structural_Formulas_-_Lewis_Kekule_Bond-line_Condensed_and_Perspective	# 1.8: Structural Formulas - Lewis, Kekule, Bond-line, Condensed, & Perspective Learning Objective Draw, interpret, and convert between Lewis (Kekule), Condensed, and Bond-line Structures Note: The review of general chemistry in sections 1.3 - 1.6 is integrated into the above Learning Objective for organic chemistry in sections 1.7 and 1.8. Shorthand notations to represent organic molecules rely on our knowledge of common neutral bonding patterns. Knowing these patterns, we can fill in the missing structural information. Some of these shorthand ways of drawing molecules give us insight into the bond angles and relative positions of atoms in the molecule, while some notations eliminate the carbon and hydrogen atoms and only indicate the heteroatoms (the atoms that are NOT carbon or hydrogen). There are three primary methods to communicate chemical structure of organic molecules: Kekule: Lewis structures using lines to represent covalent bonds and showing all atoms and lone pair electrons Bond-line (Skeletl-line): shows bonds between carbon atoms and heteroatoms) (with lone pair electrons when requested) Condensed: all atoms are written to communicate structure without drawing any chemical bonds based on the carbon backbone ## Introduction Observe the following drawings of the structure of Retinol, the most common form of vitamin A. The first drawing follows the straight-line (a.k.a. Kekulé) structure which is helpful when you want to look at every single atom; however, showing all of the hydrogen atoms makes it difficult to compare the overall structure with other similar molecules and makes it difficult to focus in on the double bonds and OH group. Retinol: Kekulé straight-line drawing The following is a bond-line (a.k.a. zig-zag) formula for retinol. With this simiplified representation, one can easily see the carbon-carbon bonds, double bonds, OH group, and CH3 groups sticking off of the the main ring and chain. Also, it is much quicker to draw this than the one above. You will learn to appreciate this type of formula writing after drawing a countless number of organic molecules. Retinol: Bond-line or zig-zag formula ## Importance of Structure Learning and practicing the basics of Organic Chemistry will help you immensely in the long run as you learn new concepts and reactions. Some people say that Organic Chemistry is like another language, and in some aspects, it is. At first it may seem difficult or overwhelming, but the more you practice looking at and drawing organic molecules, the more familiar you will become with the structures and formulas. Another good idea is to get a model kit and physically make the molecules that you have trouble picturing in your head. Through general chemistry, you may have already experienced looking at molecular structure. The different ways to draw organic molecules include Keku (straight-line), Condensed Formulas, and Bond-Line Formulas (zig-zag). It will be more helpful if you become comfortable going from one style of drawing to another, and look at drawings and understanding what they mean, than knowing which kind of drawing is named what. An example of a drawing that incorporates all three ways to draw organic molecules would be the following additional drawing of Retinol. The majority of the drawing is Bond-line (zig-zag) formula, but the -CH3 are written as condensed formulas, and the -OH group is written in Kekulé form. A widely used way of showing the 3D structure of molecules is the use of dashes, wedges, and straight lines. This drawing method is essential because the placement of different atoms could yield different molecules even if the molecular formulas were exactly the same. Below are two drawings of a 4-carbon molecule with two chlorines and two bromines attached. 4-carbon molecule with 2 chlorines and 2 bromines 4-carbon molecule with 2 chlorines and 2 bromines Both drawings look like they represent the same molecule; however, if we add dashes and wedged we will see that two different molecules could be depicted: The two molecules above are different, prove this to yourself by building a model. An easier way to compare the two molecules is to rotate one of the bonds (here, it is the bond on the right): Notice how the molecule on the right has both bromines on the same side and chlorines on the same side, whereas the first molecule is different. Read about Dashed-Wedged Line structures, bottom of page, to understand what has been introduced above. You will learn more about the importance of atomic connectivity in molecules as you continue on to learn about Stereochemistry. ## Drawing the Structure of Organic Molecules Although larger molecules may look complicated, they can be easily understood by breaking them down and looking at their smaller components. All atoms want to have their valence shell full, a "closed shell." Hydrogen wants to have 2 e- whereas carbon, oxygen, and nitrogen want to have 8 e-. When looking at the different representations of molecules, keep in mind the Octet Rule. Also remember that hydrogen can bond one time, oxygen can bond up to two times, nitrogen can bond up to three times, and carbon can bond up to four times. ## Kekulé (a.k.a. Lewis Structures) Kekulé structures are similar to Lewis Structures, but instead of covalent bonds being represented by electron dots, the two shared electrons are shown by a line. (A) (B)(C) Lone pairs remain as two electron dots, or are sometimes left out even though they are still there. Notice how the three lone pairs of electrons were not draw in around chlorine in example B. ## Condensed Formulas A condensed formula is made up of the elemental symbols. The order of the atoms suggests the connectivity. Condensed formulas can be read from either direction and H3C is the same as CH3, although the latter is more common because Look at the examples below and match them with their identical molecule under Kekulé structures and bond-line formulas. (A) CH3CH2OH (B) ClCH2CH2CH(OCH3)CH3 (C) H3CNHCH2COOH Let's look closely at example B. As you go through a condensed formula, you want to focus on the carbons and other elements that aren't hydrogen. The hydrogen's are important, but are usually there to complete octets. Also, notice the -OCH3 is in written in parentheses which tell you that it not part of the main chain of carbons. As you read through a a condensed formula, if you reach an atom that doesn't have a complete octet by the time you reach the next hydrogen, then it's possible that there are double or triple bonds. In example C, the carbon is double bonded to oxygen and single bonded to another oxygen. Notice how COOH means C(=O)-O-H instead of CH3-C-O-O-H because carbon does not have a complete octet and oxygens. ## Bond-Line (a.k.a. zig-zag) Formulas The name gives away how this formula works. This formula is full of bonds and lines, and because of the typical (more stable) bonds that atoms tend to make in molecules, they often end up looking like zig-zag lines. If you work with a molecular model kit you will find it difficult to make stick straight molecules (unless they contain sp triple bonds) whereas zig-zag molecules and bonds are much more feasible. (A) (B) (C) These molecules correspond to the exact same molecules depicted for Kekulé structures and condensed formulas. Notice how the carbons are no longer drawn in and are replaced by the ends and bends of a lines. In addition, the hydrogens have been omitted, but could be easily drawn in (see practice problems). Although we do not usually draw in the H's that are bonded to carbon, we do draw them in if they are connected to other atoms besides carbon (example is the OH group above in example A) . This is done because it is not always clear if the non-carbon atom is surrounded by lone pairs or hydrogens. Also in example A, notice how the OH is drawn with a bond to the second carbon, but it does not mean that there is a third carbon at the end of that bond/ line. ## Dashed-Wedged Line Structure As you may have guessed, the Dashed-Wedged Line structure is all about lines, dashes, and wedges. At first it may seem confusing, but with practice, understanding dash-wedged line structures will become like second nature. The following are examples of each, and how they can be used together. Above are 4-carbon chains with attached OH groups or Cl and Br atoms. Remember that each line represents a bond and that the carbons and hydrogens have been omitted. When you look at or draw these structures, the straight lines illustrate atoms and bonds that are in the same plane, the plane of the paper (in this case, computer screen). Dashed lines show atoms and bonds that go into the page, behind the plane, away from you. In the above example, the OH group is going into the plane, while at the same time a hydrogen comes out (wedged). Wedged lines illustrate bonds and atoms that come out of the page, in front of the plane, toward you. In the 2D diagram above, the OH group is coming out of the plane of the paper, while a hydrogen goes in (dashed). As stated before, straight lines illustrate atoms and bonds that are in the same plane as the paper, but in the 2D example, the straight line bond for OH means that it it unsure or irrelevant whether OH is going away or toward you. It is also assumed that hydrogen is also connected to the same carbon that OH is on. Blue bead= OH group; H is not shown Try using your model kit to see that the OH group cannot lie in the same plane at the carbon chain (don't forget your hydrogens!). In the final 2Dexample, both dashed and wedged lines are used because the attached atoms are not hydrogens (although dashed and wedged lines can be used for hydrogens).The chlorine is coming out the page while bromine is going into the page. ## Example: Converting between Structural Formulas Throughout the course, it will be helpful to convert compounds into different structural formulas (Kekule (Lewis Structures), Bond-line, and Condensed) depending on the type of question that is asked. Standardized exams frequently include a high percentage of condensed formulas because it is easier and cheaper to type letters and numbers than to import figures. Initially, it can be tricky writing a bond-line structure directly from a condensed formula. First write the Kekule structure from the condensed formula and then draw the bond-line structure from the Kekule. Practice will quickly allow you to convert directly between condensed and bond-line structures. The condensed formula for propanal is CH3CH2CHO. Can you visualize the bond-line structure of propanal? If yes, excellent, If no, the following practice will help. The Kekule structure for propanal is shown below. The bond-line structure for propanal is shown below. All three structures represent the same compound, propanal. ## Exercises 1. How many carbons are in the following drawing? How many hydrogens? 2. How many carbons are in the following drawing? How many hydrogens? 3. How many carbons are in the following drawing? How many hydrogens? 4. Look at the following molecule of vitamin A and draw in the hidden hydrogens and electron pairs. (hint: Do all of the carbons have 4 bonds? Do all the oxygens have a full octet?) 5. How many bonds can hydrogen make? 6. How many bonds can chlorine make? 7. Dashed lines means the atomic bond goes ___________(away/toward) you. 8. Draw ClCH2CH2CH(OCH3)CH3 in Kekuléand zig-zag form. 9. Extra practice problems can be found ______? ### Solutions 1. Remember the octet rule and how many times carbons and hydrogens are able to bond to other atoms. 2. Electron pairs drawn in blue and hydrogens draw in red. 3. Hygrogen can make one bond. 4. Chlorine can make one bond. 5. Away 6. See (B) under Kekulé and Bond-line (zig-zag) formulas. 7. Extra practice problems can be found: in your textbook, homework, lecture notes, online, reference books, and more. Try making up some of your own molecules, they may exist! ## References 1. Vollhardt, K. Peter C., and Neil E. Schore. Organic Chemistry: Structure and Function. 5th ed. New York: W. H. Freeman Company, 2007. 38-40. 2. Klein, David R. Organic Chemistry I As a Second Language. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc, 2007. 1-14. • Choo, Ezen (2009, UCD '11) The building block of structural organic chemistry is the tetravalent carbon atom. With few exceptions, carbon compounds can be formulated with four covalent bonds to each carbon, regardless of whether the combination is with carbon or some other element. The two-electron bond, which is illustrated by the carbon-hydrogen bonds in methane or ethane and the carbon-carbon bond in ethane, is called a single bond. In these and many related substances, each carbon is attached to four other atoms: There exist, however, compounds such as ethene (ethylene), $$C_2H_4$$, in which two electrons from each of the carbon atoms are mutually shared, thereby producing two two-electron bonds, an arrangement which is called a double bond. Each carbon in ethene is attached to only three other atoms: Similarly, in ethyne (acetylene), $$C_2H_2$$, three electrons from each carbon atom are mutually shared, producing three two-electron bonds, called a triple bond, in which each carbon is attached to only two other atoms: Of course, in all cases each carbon has a full octet of electrons. Carbon also forms double and triple bonds with several other elements that can exhibit a covalence of two or three. The carbon-oxygen (or carbonyl) double bond appears in carbon dioxide and many important organic compounds such as methanal (formaldehyde) and ethanoic acid (acetic acid). Similarly, a carbon-nitrogen triple bond appears in methanenitrile (hydrogen cyanide) and ethanenitrile (acetonitrile). By convention, a single straight line connecting the atomic symbols is used to represent a single (two-electron) bond, two such lines to represent a double (four-electron) bond, and three lines a triple (six-electron) bond. Representations of compounds by these symbols are called structural formulas; some examples are A point worth noting is that structural formulas usually do not indicate the nonbonding electron pairs. This is perhaps unfortunate because they play as much a part in the chemistry of organic molecules as do the bonding electrons and their omission may lead the unwary reader to overlook them. However, when it is important to represent them, this can be done best with pairs of dots, although a few authors use lines: To save space and time in the representation of organic structures, it is common practice to use "condensed formulas" in which the bonds are not shown explicitly. In using condensed formulas, normal atomic valences are understood throughout. Examples of condensed formulas are Another type of abbreviation that often is used, particularly for ring compounds, dispenses with the symbols for carbon and hydrogen atoms and leaves only the lines in a structural formula. For instance, cyclopentane, $$C_5H_{10}$$, often is represented as a regular pentagon in which it is understood that each apex represents a carbon atom with the requisite number of hydrogens to satisfy the tetravalence of carbon: Likewise, cyclopropane, $$C_3H_6$$; cyclobutane, $$C_4H_8$$; and cyclohexane, $$C_6H_{12}$$, are drawn as regular polygons: Although this type of line drawing is employed most commonly for cyclic structures, its use for open chain (acyclic) structures is becoming increasingly widespread. There is no special merit to this abbreviation for simple structures such as butane, $$C_4H_{10}$$; 1-butene, $$C_4H_8$$; or 1,3-butadiene, $$C_4H_6$$, but it is of value in representing more complex molecules such as $$\beta$$-carotene, $$C_{40}H_{56}$$: Line structures also can be modified to represent the three-dimensional shapes of molecules, and the way that this is done will be discussed in detail in Chapter 5. At the onset of you study of organic chemistry, you should write out the formulas rather completely until you are thoroughly familiar with what these abbreviations stand for.	2021-05-17 17:10:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5460819005966187, "perplexity": 1586.1650589130313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00357.warc.gz"}
https://byjus.com/question-answer/area-of-a-trapezium-is-160-sq-cm-lengths-of-parallel-sides-are-in-the/	Question # Area of a trapezium is $$160\ sq.cm$$. Lengths of parallel sides are in the ratio $$1:3$$. If smaller of the parallel sides is $$10\ cm$$ in length, then find the perpendicular distance between them. Solution ## Given,Area of trapezium $$=160\ m^2$$The ratio of the length of its parallel sides $$=1:3$$Smaller parallel sides $$=10\ cm$$$$\therefore$$ Length of greater side $$=10\times 3=30\ cm$$Let the distance between them $$=h$$We know that, Area $$=\dfrac12\times$$ sum of parallel sides $$\times$$ height $$\Rightarrow 160=\dfrac12(30+10)h\\ \Rightarrow 160\times 2=40h\\ \Rightarrow h=\dfrac{320}{40}=8\ cm$$Mathematics Suggest Corrections 0 Similar questions View More People also searched for View More	2022-01-18 14:39:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8830257654190063, "perplexity": 1396.4388066794809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300849.28/warc/CC-MAIN-20220118122602-20220118152602-00232.warc.gz"}
https://anuga.anu.edu.au/changeset/2631	# Changeset 2631 Ignore: Timestamp: Mar 29, 2006, 4:22:08 PM (17 years ago) Message: Meeting with Howard File: 1 edited ### Legend: Unmodified r2628 \begin{funcdesc} {set\_name}{name} Module: \codee{pyvolution.domain} Module: \code{pyvolution.domain} Assigns the name \code{name} to the domain Sets the directory used for data to the value \code{name}. The default value, before \code{set\_datadir} is run, is the value \code{default_datadir} specified in \config.py}. \code{set\_datadir} is run, is the value \code{default_datadir} specified in \code{config.py}. \end{funcdesc} Returns the data directory set by \code{set\_datadir} or, if \code{set\_datadir} has not been run, returns the value \code{default_datadir} specified in \config.py}. been run, returns the value \code{default_datadir} specified in \code{config.py}. \end{funcdesc} \section{Setting Quantities} \begin{funcdesc}{set\_quantity}{name, numeric = None, quantity = None, function = None, geospatial_data = None, filename = None, attribute_name = None, alpha = None, location = 'vertices', indices = None, verbose = False, use_cache = False} Module: \code{pyvolution.domain} (see also \code{pyvolution.quantity.set_values}) This function is used to assign values to individual quantities for a domain. It is very flexible and can be used with many data types: a statement of the form \code{domain.set\_quantity{name, x}} can be used to define a quantity having the name \code{name}, where the other argument \code{x} can be any of the following: \begin{funcdesc}{set\_quantity}{name, numeric = None, quantity = None, function = None, geospatial_data = None, filename = None, attribute_name = None, alpha = None, location = 'vertices', indices = None, verbose = False, use_cache = False} Module: \code{pyvolution.domain} (see also \code{pyvolution.quantity.set_values}) This function is used to assign values to individual quantities for a domain. It is very flexible and can be used with many data types: a statement of the form \code{domain.set\_quantity(name, x)} can be used to define a quantity having the name \code{name}, where the other argument \code{x} can be any of the following: \begin{itemize} \item a number in which case all vertices in the mesh gets that for the quantity in question. \item a number, in which case all vertices in the mesh gets that for the quantity in question. \item a list of numbers or a Numeric array ordered the same way as the mesh vertices. \item a function (e.g.\ see the samples introduced in Chapter 2) \item an expression composed of other quantities and numbers, arrays, lists (for example, a linear combination of quantities) \item the name of a file from which the data can be read \item a geospatial dataset (See ?????) \item the name of a file from which the data can be read. In this case, the optional argument attribute_name will select which attribute to use from the file. If left out, set_quantity will pick one. This is useful in cases where there is only one attribute. \item a geospatial dataset (See ?????). Optional argument attribute_name applies here as with files. \end{itemize} numeric, quantity, function, points, filename must be present. Set quantity will look at the type of the second argument (\code{numeric}) and determine what action to take. Values can also be set using the appropriate keyword arguments. If x is a function, for example, \code{domain.set\_quantity(name, x)}, \code{domain.set\_quantity(name, numeric=x)}, and \code{domain.set\_quantity(name, function=x)} are all equivalent. Other optional arguments are \begin{itemize} \item \code{indices} which is a list of ids of triangles to which set_quantity should apply its assignment of values. \item \code{location} determines which part of the triangles to assign to. Options are 'vertices' (default), 'edges', and 'centroids'. \end{itemize} a number, in which case all vertices in the mesh gets that for the quantity in question. \item a list of numbers or a Numeric array ordered the same way as the mesh vertices. \end{funcdesc} %%% Module: \code{pyvolution.least\_squares} Given a time series defined at the vertices of a triangular mesh (such as those stored in \code{sww} files), \code{Interpolation\_function} is used to create a callable object that assigns a value \code{f(t, x, y)}, interpolated from the given time-series values, to an arbitrary time \code{t} and point \code{(x, y)} within the mesh region. Since, in practice, values need to be computed at specified Given a time series, either as a sequence of numbers or defined at the vertices of a triangular mesh (such as those stored in \code{sww} files), \code{Interpolation\_function} is used to create a callable object that interpolates a value for an arbitrary time \code{t} within the model limits and possibly a point \code{(x, y)} within a mesh region. The actual time series at which data is available is specified by means of an array \code{time} of monotonically increasing times. The quantities containing the values to be interpolated are specified in an array---or dictionary of arrays (used in conjunction with the optional argument \code{quantitity\_names}) --- called \code{quantities}. The optional arguments \code{vertex_coordinates} and \code{triangles} represent the spatial mesh associated with the quantity arrays. If omitted the function created by \code{Interpolation\_function} will be a function of \code{t} only. Since, in practice, values need to be computed at specified points, the syntax allows the user to specify, once and for all, a list \code{interpolation\_points} of points at which values are required. In this case, index identifying a member of \code{interpolation\_points}. The time series is specified by means of an array \code{time} of monotonically increasing times and an array---or dictionary of arrays---\code{quantities} containing the values to be interpolated. \end{classdesc} \section{Boundary Conditions} \anuga provides a large number of predefined boundary conditions, represented by objects such as \code{Reflective\_boundary(domain)} and \code{Dirichlet\_boundary([0.2, 0.0, 0.0])}, described in the examples in Chapter 2. Alternatively, you may prefer to roll your own'', following the method explained in \ref{sec:roll_your_own}. \anuga provides a large number of predefined boundary conditions, represented by objects such as \code{Reflective\_boundary(domain)} and \code{Dirichlet\_boundary([0.2, 0.0, 0.0])}, described in the examples in Chapter 2. Alternatively, you may prefer to roll your own'', following the method explained in Section \ref{sec:roll your own}. These boundary objects may be used with the function \code{set\_boundary} described below \end{funcdesc} \begin{funcdesc} {get_boundary_tags}{??} \begin{funcdesc} {get_boundary_tags}{} Module: \code{pyvolution.mesh} \end{funcdesc} \subsection{User-defined boundary conditions} \label{sec:roll_your_own} \label{sec:roll your own} [How to roll your own]	2023-03-27 06:45:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7566086053848267, "perplexity": 2085.0106408329134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948609.41/warc/CC-MAIN-20230327060940-20230327090940-00736.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-slope-of-the-secant-lines-of-f-x-x-2-through-the-points-2-4	# How do you find the slope of the secant lines of f(x) = -x^2 through the points: [-2, -4]? Feb 8, 2018 $6$ #### Explanation: $\text{the slope of the secant line is}$ •color(white)(x)(f(b)-f(a))/(b-a) $f \left(b\right) = f \left(- 2\right) = - {\left(- 2\right)}^{2} = - 4$ $f \left(a\right) = f \left(- 4\right) = - {\left(- 4\right)}^{2} = - 16$ $\Rightarrow \frac{- 4 - \left(- 16\right)}{- 2 - \left(- 4\right)} = \frac{12}{2} = 6$	2021-10-23 01:37:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9169277548789978, "perplexity": 1373.0258913221603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585537.28/warc/CC-MAIN-20211023002852-20211023032852-00112.warc.gz"}
https://books.compclassnotes.com/elementarycomputing/nested-functions/	# Nested Functions In the previous section we applied a test to see if a variable was bound in the body of a function. We haven’t seen many examples so far where this would be true, but consider function (λx. + (λy. + 3 y) x 2); is y bound in it? We consider the function to be: (λx. E) so clearly, y doesn’t appear in the parameter list. The other possibility is that it is bound in E. Here, E = (+ (λy. + 3 y) x 2). This time y does appear in the parameter list, and it is free in the body (+ 3 y), meaning that it is bound in E, so it follows that is bound in (λx. + (λy. + 3 y) x 2). This is a fairly subtle thing, so make sure you’re satisfied with this before going on. What this tells us is that a variable doesn’t have to be bound by the first λ in an expression, that it can be bound elsewhere in the expression. If we try evaluate that function by passing it the value 7 we get: ( (λx. + (λy. + 3 y) x 2) 7) Perform a $$\beta$$ conversion on to get: (+ (λy. + 3 y) 7 2) Now we do another $$\beta$$ conversion (on y) to get: (+ (+ 3 7) 2) Here’s another example. Is free or bound? ((λx. + x 3) x) If we take to be (λx. + x 3) and F to be xthen x is bound in and free in F, which means that is both free and bound in (E F), so it is free and bound in the same expression! What does this mean? It means that these are two different x‘s, so the same name doesn’t always mean the same variable. Take a look at the AST below; it’s much clearer there because the λ node shows us which belongs to the parameter list. An AST with two different variables that have the same name. Here’s another more complex example: ((λf. f (* 1 3) (+ 4 5) x) ((λxyz. (* 2 x))) Which is free and which bound? If we apply our rules then we can see that is is both. It is free in the (λf. f (* 1 3) (+ 4 5) x) part, and bound in the (λxyz. (* 2 x)) part. Here it is as an AST: A function calling another function. The orange x is a free variable that will need a value. The first thing we do is to bring in the value for f as below: The variable to be replaced. This $$\beta$$ reduction gives us this: The AST after applying the beta reduction. Now we’re stuck. We have a λ expression that takes three parameters and, although we have three, the orange x is free, so let’s assume it has been define somewhere else like this: (define x 2) That changes the AST to this, which has enough arguments so we can apply our $$\beta$$ reduction. The final steps of the reduction. Recall from the section on Lifetime Diagrams that internally, the computer uses different names for variables: A Lifetime Diagram with multiple copies of the same function running. This shows how the names can be easily kept separate when there are multiple copies of the same function running. This is important because in that case, there will definitely be functions using the same name, but what if we pass a function to another function like we did above? It turns out that it’s the same thing; any time a function is actually running, a temporary variable is set up to keep that function’s values private to it. However, when we’re reading these functions written out in λ calculus, it won’t always be that clear, so let’s add some more formal notation to help us keep everything straight. We’ve been using $$\beta$$ reductions for a while now. Formally, a $$\beta$$ reduction replaces the free occurrences of a variable in the body of a function. For example, in the expresion ((λx. + x 1) 5) we will replace all the free occurrences in the body (+ x 1) with the value 5, giving us (+ 5 1). Formally, we write this as: $$((\lambda x. + x\; 1)\; 5) \overrightarrow\beta (+ 5\; 1)$$ Here’s a more complex example, this time with multiple λ expressions: $$((\lambda x. + (\lambda y. +\; 2\; y)\; x\; 4)\; 3)$$ Our $$\beta$$ reduction does the following: $$\overrightarrow\beta (+ (\lambda y. +\; 2\; y)\; 3\; 4)$$ Next we do another one to get: $$\overrightarrow\beta (+\; (+\; 2\; 3)\; 4)$$ Note that we haven’t added anything new here, we’ve just formalised what we’ve been doing for a while now, which is bringing in values for parameters. Here’s a more confusing but identical example: $$((\lambda x. + (\lambda x. +\; 2\; x)\; x\; 4)\; 3)$$ When we do our $$\beta$$ reduction we only replace the free occurrences of x, and the inner λ has a bound x, so that doesn’t change, giving us: $$\overrightarrow\beta (+ (\lambda x. +\; 2\; x)\; 3\; 4)$$ Yes, it still looks slightly different to the last example, but only the names are different. Structurally and functionally it is the same, that is, semantically it is the same even though the syntax is different. We’ll finish up with one last mind-bending example of passing a λ expression to another λ expression. This might seems a strange thing to want to do, as so far we’ve only modified the functionality of something by passing it a different piece of data; however, what this will let us do is modify the functionality of a piece of code by passing it another piece of code! Think about the parallel processing examples we had before, this is a really great way to change the functionality of a system very quickly. Passing a lambda expression into another lambda expression. Next we do our $$\beta$$ conversion Perform a beta conversion (reduction) on the AST, followed by another one on the remaining tree. Well, that was actually disappointingly easy and straightforward, so let’s try a different one! (λx. + 2 (λy. y 5) x) (λz. + 1 z) An expression with three lambda expressions! We apply the first one and perform a $$\beta$$ conversion Perform a beta reduction. Notice that the + is now on top. Now we bring in the subtree for y Perform another beta reduction to leave us with just one lambda expression. Notice that the + is still on top. Now we do our final one to get a good old fashioned AST that is ready to be reduced to a single value. The final AST, ready for reduction to a single number. Now, that was way more interesting!	2023-03-29 11:06:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6892065405845642, "perplexity": 838.5685674247593}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948965.80/warc/CC-MAIN-20230329085436-20230329115436-00435.warc.gz"}
http://www.quantumdiaries.org/2012/07/02/higgsdependence-day/	## View Blog \| Read Bio ### Higgsdependence Day On July 4th CERN will hold a seminar where ATLAS and CMS will present their latest findings on the search for the Higgs boson. There’s a reasonable chance that either or both experiments will see a 5 sigma excess, and this would be enough to claim a “discovery”. One of my US friends at CERN called this day Higgsdependence Day, and all over the USA people will be celebrating with fireworks and barbecues. (Okay, perhaps they will be celebrating something else. My boss tells me he might tar and feather me as the token British member of the group…) CERN is not the only lab to be holding a seminar. Today at 09:00 CDT Fermilab will be announcing the latest results from CDF and D0. Rumors suggest a 3 sigma excess (technically an “observation”) in the interesting region. So if you can spare the time I’d recommend you listen in on the announcement. You can see the webcast information here. In anticipation of the CERN seminar, when I came to my office this morning I found a bottle of champagne with a label hastily pasted to the back. It seems these might be placed alongside fire extinguishers in every office at CERN! (You can get your own label here.) No Higgs seminar is complete without a bottle of Champagne, just in case! For those of us who can’t get enough of the Higgs boson and want to brush up on the basics I would recommend the following show, put out by the BBC. This contains the latest results from the 2011 searches and it goes into quite a bit of depth about why we think the Higgs boson exists and what to expect from the 2012 searches. Finally for those keeping score I still have $20 riding on a non-discovery. If a 5 sigma excess is seen on Wednesday there is a bit more work that needs to be done to show that it is the Standard Model Higgs, and that would probably take until the end of 2012 running. So my$20 is safe… for now. Tags: , , , , , , , • Pingback: [blog post] Higgsdependence Day « aidan@cern() • http://polyglotaholic.blogspot.com Sarai Yep – myself and a whole lot of Google Plussers are eagerly waiting for the webcast… pity I can’t pop my own champers, but hey, I’m sure those are there for a reason • Pingback: Woensdag is H-Dag, HIGGS-Dag() • Carlos Champagne? Cava !!!! Better than champagme.	2014-12-19 00:26:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2714479863643646, "perplexity": 2190.4170066638508}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768089.153/warc/CC-MAIN-20141217075248-00097-ip-10-231-17-201.ec2.internal.warc.gz"}
https://chemistry.stackexchange.com/tags/phase/new	# Tag Info 6 So it finally turned out to be an artefact, sorry. The probe was protected by a small metallic tube. Somehow water condensed inside. Since this water is isolated from the main liquid, it performed its own phase-transition releasing its latent heat right next to the sensor. The effect was not appreciated when freezing the probe alone. I don't know why, maybe ... 1 A nicely done experiment! Consider hand-warmers containing sodium acetate trihydrate, $\ce{NaCH3COO.3H2O}$. When heated above the melting point and then cooled, the compound does not quickly solidify, but can be greatly supercooled. Given an impetus to begin crystallization, such as the shock-wave produced by a "clicker" (or nucleation by a speck ... 2 But my source of confusion is that ambient pressure is not the only pressure pressing down on the liquid. Pressure does not press down. When something is under pressure, it exerts it in all directions. Maybe it would help imagining this problem in the absence of gravity (you would have to put the sample in a stretchy balloon to get some pressure while being ... 3 If you heat the commercial concentrated ammonia solution ($25$% $\ce{NH3}$) at usual pressure, it will boil at $32$°C and the vapor contains $3$% $\ce{H2O}$ (and of course $97$% $\ce{NH3}$). So the liquid looses much ammonia and nearly no water. Its total volume decreases a bit but the concentration of ammonia decreases more, so that it is necessary to heat ... 3 Certainly! The properties of individual atoms differ from the bulk material. Alkali metals (all metals, really) no longer have metallic behavior such as electrical conductivity when highly dispersed. As you surmise, these effects are studied at quite low pressure (you can observe individual atoms in the container!) and at temperatures approaching 0K. For ... Top 50 recent answers are included	2021-09-20 06:52:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7214109897613525, "perplexity": 1864.7809066374034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057018.8/warc/CC-MAIN-20210920040604-20210920070604-00545.warc.gz"}
http://cpr-nuclth.blogspot.com/2013/08/13080424-kenichi-yoshida.html	## Spin-isospin response of deformed neutron-rich nuclei in a self-consistent Skyrme energy-density-functional approach [PDF] Kenichi Yoshida We develop a new framework of the self-consistent deformed proton-neutron quasiparticle-random-phase approximation (pnQRPA), formulated in the Hartree-Fock-Bogoliubov (HFB) single-quasiparticle basis. The same Skyrme force is used in both the HFB and pnQRPA calculations except in the proton-neutron particle-particle channel, where an S=1 contact force is employed. Numerical application is performed for Gamow-Teller (GT) strength distributions and $\beta$-decay rates in the deformed neutron-rich Zr isotopes located around the path of the rapid-neutron-capture process nucleosynthesis. It is found that the GT strength distributions are fragmented due to deformation. Furthermore we find that the momentum-dependent terms in the particle-hole residual interaction leads to a stronger collectivity of the GT giant resonance. The T=0 pairing enhances the low-lying strengths cooperatively with the T=1 pairing correlation, which shortens the $\beta$-decay half lives by at most an order of magnitude. The new calculation scheme reproduces well the observed isotopic dependence of the $\beta$-decay half lives of deformed $^{100-110}$Zr isotopes. View original: http://arxiv.org/abs/1308.0424	2017-08-21 13:55:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5409265756607056, "perplexity": 6246.97584291037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886108709.89/warc/CC-MAIN-20170821133645-20170821153645-00042.warc.gz"}
https://manual.q-chem.com/5.2/A2.S7.html	B.7 Angular Momentum Problem The fundamental integral is essentially an integral without angular momentum (i.e., it is an integral of the type $[ss\|ss]$). Angular momentum, usually depicted by $L$, has been problematic for efficient ERI formation, evident in the above time line. Initially, angular momentum was calculated by taking derivatives of the fundamental ERI with respect to one of the Cartesian coordinates of the nuclear center. This is an extremely inefficient route, but it works and was appropriate in the early development of ERI methods. Recursion relations672, 673 and the newly developed tensor equations24 are the basis for the modern approaches.	2019-05-26 18:58:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 2, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9905027747154236, "perplexity": 769.8751258399678}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232259452.84/warc/CC-MAIN-20190526185417-20190526211417-00521.warc.gz"}
https://mathematica.stackexchange.com/questions/189188/fill-in-an-incomplete-outline-of-an-image/189196	Fill in an incomplete outline of an image I have an outline which I have extracted from an image, which may well not entirely closed, for instance: I now want to fill the inside of this, as another image, using the convex hull if it is not closed. For instance: img = Import["https://i.stack.imgur.com/yC9ym.png"]; ConvexHullMesh[PixelValuePositions[img, 1]] is the right shape, but is a mesh rather than an image and now in a different coordinate system. FillingTransform doesn't seem to work, presumably because the outline is not complete. ComponentMeasurements[img, "ConvexVertices"] gives me the points that make up the convex hull, but I can't manage to fill in the middle in an easy (and ideally fast way). Rasterizing the ConvexHullMesh has been suggested in the comments, but that doesn't appear to work for me, as the ConvexHullMesh zooms into the image. HighlightImage[Rasterize[ConvexHullMesh[PixelValuePositions[img, 1]], RasterSize -> ImageDimensions[img]], img] $Version (* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" ) • If ConvexHullMesh works as desired, what about Rasterize[ConvexHullMesh[PixelValuePositions[img, 1]], RasterSize -> ImageDimensions[img]]? – Theo Tiger Jan 10 '19 at 10:16 • @TheoTiger, because it is not in the same coordinate system, see HighlightImage[ ColorNegate@ Binarize[Rasterize[ConvexHullMesh[PixelValuePositions[img, 1]], RasterSize -> ImageDimensions[img]]], img] - the rasterized image is now zoomed in. – KraZug Jan 10 '19 at 10:22 • Unless I am missing something, they are in the same coordinate system for me. They overlap nicely. I'm on Mma 11.3 btw. Can you add an image of your HighlightImage to the question? – Theo Tiger Jan 10 '19 at 10:32 • @Theo, strange, as they definitely don't for me, on 11.3 too. – KraZug Jan 10 '19 at 10:34 • The ConvextHullMesh is just slightly larger, which seems logical since it needs to contain all the points. – Theo Tiger Jan 10 '19 at 10:36 4 Answers pvp = PixelValuePositions[img, 1]; Graphics[{LightBlue, EdgeForm[Blue], Polygon[pvp[[FindShortestTour[pvp][[2]]]]]}] ImageAdd[img, Graphics[{Red, EdgeForm[Blue], Polygon[pvp[[FindShortestTour[pvp][[2]]]]]}, PlotRange -> Thread[{0, ImageDimensions[img]}]] • Can you make it be the same dimensions as the original image? So things like ImageAdd[img, Graphics[{Red, Polygon[pvp[[FindShortestTour[pvp][[2]]]]]}]] work properly. – KraZug Jan 10 '19 at 14:46 • @KraZug, please see the updated version. – kglr Jan 10 '19 at 15:15 • Thank you, I think that does exactly what I need. – KraZug Jan 12 '19 at 6:45 Turns out that MorphologicalComponents will give the convex hull: imageHull = Image@MorphologicalComponents[img, Method -> "ConvexHull"] HighlightImage[imageHull, img] I am still interested in a solution that doesn't require ConvexHull for the whole image, but just fills in the missing hole where there is a gap. • "I am still interested in a solution that doesn't require ConvexHull for the whole image, but just fills in the missing hole where there is a gap." You mean you don't want the convex hull, just the missing line segment? Try getting the convex hull anyway and then computing the boundary. – kajacx Jan 10 '19 at 13:00 • @kajacx, I want an image that contains the filled inside of the line. Where the line is intact but non-convex, the ConvexHull will expand out of it. Where there are breaks, I'd like the shortest straight line to be taken. – KraZug Jan 10 '19 at 13:04 First, extract the points from the image and take the convex hull mesh: ClearAll["Global"] img = Import["https://i.stack.imgur.com/yC9ym.png"]; allpts = PixelValuePositions[img, 1]; chmesh = ConvexHullMesh@allpts; Use MeshPrimitives to extract the boundary lines. Extract their endpoints. Since the endpoints are not unique, take every other end point to obtain a set of unique points on the convex hull: endpts = Flatten[MeshPrimitives[chmesh, 1] /. Line -> List, 2]; hullpts = Take[endpts, {1, -1, 2}]; Plot the results: Graphics[{Black, PointSize[1/300], Point@allpts, Red, Line[hullpts], Opacity[1/8], Blue, FilledCurve@Line[hullpts] }] EDIT: What we are really after is an image of the filled curve that will overlay the original image, which has ImageDimensions of {1024,1024}. We want to use ImagePadding to position the filled image on the original image. The amount of padding is first estimated by looking at the minimum and maximum coordinates in allpts, then adjusting by a small $$\delta$$. Instead of the original image let's work with its negative. MinMax/@Transpose[allpts] δ = 16; filled = Image[Graphics[ {Opacity[1/8], Red, FilledCurve@Line[hullpts]}, ImagePadding -> {{137 - δ, 1024 - 895 - δ}, {114 - δ, 1024 - 917 - δ}}], ImageSize -> ImageDimensions[reverse]]; reverse = ColorNegate[img]; Show[{reverse, filled}, ImageSize -> 200] (* {{137, 895}, {114, 917}} ) To verify that $$\delta = 16$$ is optimal, we can use ImageTake to zoom in on the left edge, say, β = 20; Show[ImageTake[#, {512 - β, 512 + β}, {137 - β, 137 + β}] & /@ {reverse, filled}, ImageSize -> 200, Frame -> True] Here we have zoomed in on both the reverse image and the filled image at about 137 pixels from the left and 512 pixels from the top. Our view frame is $$2\beta$$ square. We could adjust $$\delta$$ a little to see how the filled image shifts relative to the reverse image. We can also zoom in to check the fit at other critical points of the image. As stated in the comments, ConvexHullMesh works as desired for me on Windows. Windows 7 $Version ( "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)" ) img = Import["https://i.stack.imgur.com/yC9ym.png"]; meshraster = Rasterize[ConvexHullMesh[PixelValuePositions[img, 1]], RasterSize -> ImageDimensions[img]]; HighlightImage[meshraster, img] Mac OS I just ran it on Mac OS 10.13. I have no words. \$Version ( "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *) ` • Thanks. That is rather more wider than the original line than I can accept though, even if I understood what was going on between Windows and Mac. – KraZug Jan 10 '19 at 10:45 • See my edit above - I can reproduce this strange behaviour on Mac OS. I have no idea what is happening there. I think this could be reported to Wolfram Support as bug. – Theo Tiger Jan 10 '19 at 10:55 • Maybe it has something to do with the device pixel scaling ("DPI settings")? I am on a 1200p non-Retina display, however. – Theo Tiger Jan 10 '19 at 10:58 • Hah, something like that presumably. Your image for the same code on a mac is zoomed out, while mine is zoomed in! – KraZug Jan 10 '19 at 11:03 • The help says: "Images generated by Rasterize can vary slightly from one computer system to another, mainly as a result of different fonts and anti-aliasing procedures." - slightly! Reported as a bug. – KraZug Jan 10 '19 at 11:06	2020-02-28 22:10:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3342444896697998, "perplexity": 1562.283563160276}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147647.2/warc/CC-MAIN-20200228200903-20200228230903-00341.warc.gz"}
http://www.numdam.org/item/AIHPC_2003__20_1_87_0/	Linear instability implies nonlinear instability for various types of viscous boundary layers Annales de l'I.H.P. Analyse non linéaire, Volume 20 (2003) no. 1, pp. 87-106. @article{AIHPC_2003__20_1_87_0, author = {Desjardins, B. and Grenier, E.}, title = {Linear instability implies nonlinear instability for various types of viscous boundary layers}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {87--106}, publisher = {Elsevier}, volume = {20}, number = {1}, year = {2003}, zbl = {01901028}, mrnumber = {1958163}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2003__20_1_87_0/} } TY - JOUR AU - Desjardins, B. AU - Grenier, E. TI - Linear instability implies nonlinear instability for various types of viscous boundary layers JO - Annales de l'I.H.P. Analyse non linéaire PY - 2003 DA - 2003/// SP - 87 EP - 106 VL - 20 IS - 1 PB - Elsevier UR - http://www.numdam.org/item/AIHPC_2003__20_1_87_0/ UR - https://zbmath.org/?q=an%3A01901028 UR - https://www.ams.org/mathscinet-getitem?mr=1958163 LA - en ID - AIHPC_2003__20_1_87_0 ER - %0 Journal Article %A Desjardins, B. %A Grenier, E. %T Linear instability implies nonlinear instability for various types of viscous boundary layers %J Annales de l'I.H.P. Analyse non linéaire %D 2003 %P 87-106 %V 20 %N 1 %I Elsevier %G en %F AIHPC_2003__20_1_87_0 Desjardins, B.; Grenier, E. Linear instability implies nonlinear instability for various types of viscous boundary layers. Annales de l'I.H.P. Analyse non linéaire, Volume 20 (2003) no. 1, pp. 87-106. http://www.numdam.org/item/AIHPC_2003__20_1_87_0/ [1] Desjardins B., Grenier E., Reynolds.m a package to compute critical Reynolds numbers, 1998 , http://www.dmi.ens.fr/equipes/edp/Reynolds/reynolds.html. [2] Dormy E., Desjardins B., Grenier E., Stability of mixed Ekman-Hartmann boundary layers, Nonlinearity 12 (2) (1999) 181-199. \| MR \| Zbl [3] Dormy E., Desjardins B., Grenier E., Instability of Ekman-Hartmann boundary layers, with application to the fluid flow near the core-mantle boundary, Physics of the Earth and Planetary Interiors 123 (2001) 15-26. [4] Friedlander S., Strauss W., Vishik M., Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 187-209. \| Numdam \| MR \| Zbl [5] Gisclon M., Serre D., Study of boundary conditions for a strictly hyperbolic system via parabolic approximation, C. R. Acad. Sci. Paris Ser. I Math. 319 (4) (1994) 377-382. \| MR \| Zbl [6] Greenspan H.P., The Theory of Rotating Fluids, Cambridge Monographs on Mechanics and Applied Mathematics, 1969. \| Zbl [7] Grenier E., On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000) 1067-1091. \| MR \| Zbl [8] Grenier E., Guès O., Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations 143 (1) (1998) 110-146. \| MR \| Zbl [9] Grenier E., Masmoudi N., Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations 22 (1997) 953-975. \| MR \| Zbl [10] Guo Y., Strauss W., Instability of periodic BGK equilibria, Comm. Pure Appl. Math. 48 (1995) 861-894. \| MR \| Zbl [11] Guo Y., Strauss W., Nonlinear instability of double-humped equilibria, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 339-352. \| Numdam \| MR \| Zbl [12] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer, Berlin, 1981. \| MR \| Zbl [13] Iooss G., Nielsen H.B., True H., Bifurcation of the stationary Ekman flow into a stable periodic flow, Arch. Rational Mech. Anal. 68 (3) (1978) 227-256. \| MR \| Zbl [14] Lilly D.K., On the instability of the Ekman boundary layer, J. Atmos. Sci. 23 (1966) 481-494. [15] Majda A., Compressible Fluid Flows Systems of Conservation Laws in Several Variables, Appl. Math. Sci., 53, Springer, Berlin, 1984. \| MR \| Zbl [16] Serre D., L1 -stability of travelling waves in scalar conservation laws, Exp. No. VIII, 13 pp., Semin. Equ. Dériv. Partielles, Ecole Polytech., Palaiseau, 1999. \| Numdam \| MR \| Zbl [17] Serre D., Systèmes de lois de conservations, I et II, Diderot Editeur, Paris, 1996. \| MR [18] Shizuta Y., On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math. 36 (1983) 705-754. \| MR \| Zbl [19] Vidav I., Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl. 30 (1970) 264-279. \| MR \| Zbl [20] Yudovitch V.I., Non-stationary flow of a perfect non-viscous fluid, Zh. Vych. Math. 3 (1963) 1032-1066.	2022-11-30 08:17:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5446536540985107, "perplexity": 8523.073317583438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710733.87/warc/CC-MAIN-20221130060525-20221130090525-00434.warc.gz"}
https://www.atmos-meas-tech.net/12/2499/2019/	Journal cover Journal topic Atmospheric Measurement Techniques An interactive open-access journal of the European Geosciences Union Journal topic Atmos. Meas. Tech., 12, 2499-2512, 2019 https://doi.org/10.5194/amt-12-2499-2019 Atmos. Meas. Tech., 12, 2499-2512, 2019 https://doi.org/10.5194/amt-12-2499-2019 Research article 24 Apr 2019 Research article \| 24 Apr 2019 # Development of an incoherent broadband cavity-enhanced absorption spectrometer for measurements of ambient glyoxal and NO2 in a polluted urban environment Development of an incoherent broadband cavity-enhanced absorption spectrometer Shuaixi Liang1,2, Min Qin1, Pinhua Xie1,2,3, Jun Duan1, Wu Fang1, Yabai He1, Jin Xu1, Jingwei Liu4, Xin Li4, Ke Tang1,2, Fanhao Meng1,2, Kaidi Ye1,2, Jianguo Liu1,2,3, and Wenqing Liu1,2,3 Shuaixi Liang et al. • 1Key Laboratory of Environmental Optics and Technology, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China • 2Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230026, China • 3CAS Center for Excellence in Regional Atmospheric Environment, Institute of Urban Environment, Chinese Academy of Sciences, Xiamen, 361021, China • 4State Key Joint Laboratory of Environmental Simulation and Pollution Control, College of Environmental Sciences and Engineering, Peking University, Beijing, 100871, China Abstract We report the development of an instrument for simultaneous fast measurements of glyoxal (CHOCHO) and NO2 based on incoherent broadband cavity-enhanced absorption spectroscopy (IBBCEAS) in the 438–465 nm wavelength region. The highly reflective cavity mirrors were protected from contamination by N2 purge gas. The reduction of the effective cavity length was calibrated by measuring collision-induced oxygen absorption at ∼477 nm of pure oxygen gas input with and without the N2 mirror purge gas. The detection limits of the developed system were evaluated to be 23 parts per trillion by volume (pptv, 2σ) for CHOCHO and 29 pptv (2σ) for NO2 with a 30 s acquisition time. A potential cross-interference of NO2 absorption on accurate CHOCHO measurements has been investigated in this study, as the absorption of NO2 in the atmosphere could often be several hundred-fold higher than that of glyoxal, especially in contaminated areas. Due to non-linear spectrometer dispersion, simulation spectra of NO2 based on traditional convolution simulation did not match the measurement spectra well enough. In this work, we applied actual NO2 spectral profile measured by the same spectrometer as a reference spectral profile in subsequent atmospheric spectral analysis and retrieval of NO2 and CHOCHO concentrations. This effectively reduced the spectral fitting residuals. The instrument was successfully deployed for 24 d of continuous measurements of CHOCHO and NO2 in the atmosphere in a comprehensive field campaign in Beijing in June 2017. 1 Introduction Glyoxal (CHOCHO) is a typical intermediary for most volatile organic compound (VOC) oxidations in the atmosphere. It plays an important role in quantifying VOC emissions, understanding VOC oxidation mechanisms and further understanding the formation of O3 and secondary organic aerosol (SOA). On a global scale, simulations show that biogenic isoprene is the largest source of glyoxal (47 % of total contributions); anthropogenic acetylene also contributes significantly to glyoxal (20 % of contributions) (Fu et al., 2008). The loss of glyoxal is mainly due to photolysis, OH and NO3 oxidation reactions, wet and dry deposition and irreversible absorption of water-soluble aerosols and clouds (Fu et al., 2008; Min et al., 2016). The ratio of glyoxal to formaldehyde, RGF, is often used as an indicator of hydrocarbon precursor speciation in contaminated areas; observations in the field can give divergent conclusions (Vrekoussis et al., 2010; Kaiser et al., 2015; DiGangi et al., 2012). Glyoxal readily undergoes heterogeneous reactions to form SOA, but the contribution to SOA has a high uncertainty (Li et al., 2016; Washenfelder et al., 2011; Volkamer et al., 2007). Therefore, accurate quantification of glyoxal is a prerequisite for studies of the source, sink and atmospheric chemistry of glyoxal. Several technologies are currently used for measurements of glyoxal in the atmosphere, including chemical and spectroscopic methods. The common wet chemistry method is based on a derivatization reagent such as agent o-(2,3,4,5,6-pentafluorobenzyl) hydroxylamine (PFBHA), 2,4-dinitrophenylhydrazine (DNPH) or pentafluorophenyl hydrazine (PFPH), with subsequent analysis using liquid chromatography or mass spectrometry techniques (Temime et al., 2007; Ho and Yu, 2004; Munger et al., 1995; Pang et al., 2014). Some successful spectroscopic techniques for glyoxal include differential optical absorption spectroscopy (DOAS), laser-induced fluorescence (LIF) and incoherent broadband cavity-enhanced absorption spectroscopy (IBBCEAS). Long-path DOAS (LP-DOAS) was used to measure the glyoxal concentration for the first time at a total atmospheric light path of 4420 m with a detection limit of 0.1 parts per billion by volume (ppbv, 2σ) in Mexico City (Volkamer et al., 2005a). In 2008, LP-DOAS was used to measure glyoxal above the rainforest and then compared with multi-axis DOAS (MAX-DOAS), suggesting that local CHOCHO was confined to the first 500 m of the boundary layer (MacDonald et al., 2012). LIF can quantify both glyoxal and methylglyoxal with a detection limit of 2.9 pptv (2σ) in 5 min for glyoxal (Henry et al., 2012). IBBCEAS is an excellent method for measuring atmospheric trace gases. It features high sensitivity, small chemical interference and simultaneous measurement of multiple components. IBBCEAS has been rapidly developed since Fiedler et al. (2003) first described it in 2003. More recently, the technology has been successfully applied to measure a variety of trace gases (Min et al., 2016; Wang et al., 2017; Yi et al., 2016; Volkamer et al., 2015), weakly absorbed cross sections of different trace gases (Chen and Venables, 2011; Kahan et al., 2012) and aerosol extinction (Washenfelder et al., 2013). Using a xenon arc lamp as a light source, Washenfelder et al. (2008) reported the first measurement of glyoxal using the IBBCEAS technique in the laboratory with a detection limit of 58 pptv (2σ) within 1 min. Later, Thalman and Volkamer (2010) coupled CEAS hardware with a DOAS retrieval algorithm to measure glyoxal in open cavity mode with a detection limit of 19 pptv (2σ, 1 min). Coburn et al. (2014) subsequently measured the eddy covariance flux of glyoxal with LED-CE-DOAS for the first time and found that the nocturnal oxidation reaction on an ocean surface organic microlayer was a source of the oxygenated VOCs. With significant improvements, Min et al. (2016) developed an aircraft IBBCEAS instrument and used it to measure tropospheric glyoxal with a detection limit of 34 pptv (2σ) within 5 s. Table 1 compares different measurement techniques for glyoxal. Based on these technologies, Thalman et al. (2015) conducted a comprehensive instrument intercomparison campaign for glyoxal. Table 1Comparison of different techniques for measuring glyoxal. Spectral measurement techniques using broadband light sources, such as DOAS and IBBCEAS, can simultaneously observe a wide range of spectral bands during a single measurement. Thus, many contaminants can be measured concurrently. The overlap of the NO2 and glyoxal absorption bands at 438–465 nm allows us to simultaneously measure their concentrations. However, NO2 can interfere with the measurement of glyoxal, especially for high concentration of NO2 (Thalman et al., 2015). This is a key factor that needs to be considered to improve the data retrieval of glyoxal in China's highly polluted environment. Here, we describe the development of an incoherent broadband cavity-enhanced absorption spectrometer for sensitive detection of CHOCHO and NO2 in the atmosphere. The effective length of the optical cavity with purge-gas protected mirrors was accurately calibrated based on the collision-induced oxygen (O4) absorption at 477 nm. The instrument detection limit was estimated using the Allan variance analysis. The effects of NO2 on glyoxal were evaluated via spectral simulation and measurements. The results show that using the measured NO2 reference spectrum can overcome the interference of NO2 to glyoxal due to conventional convolution methods from the uneven dispersion of the grating spectrometer. We then applied the measured reference spectrum to the retrieval of glyoxal in the same wavelength band and obtained the glyoxal concentration in heavily polluted air in China. The IBBCEAS instrument was successfully deployed during the APHH China (Air Pollution and Human Health in a Chinese Megacity) project, and we obtained the profiles of glyoxal and NO2 concentrations in Beijing's summer atmosphere during the APHH China campaign (2–26 June 2017). 2 System and principle ## 2.1 Description of the IBBCEAS set-up The IBBCEAS technology is an absorption spectroscopy technique. It improves the effective path length via multiple light reflections in an optical cavity. This leads to a significant improvement of the detection sensitivity. Our design of the IBBCEAS set-up consists of a light-emitting diode (LED) light source, a pair of off-axis parabolic mirrors, a pair of high-reflectivity cavity mirrors, Teflon perfluoroalkoxy polymer resin (PFA) optical cavity, optical band-pass filter, an optical fibre-coupled grating spectrometer and some other components. A schematic diagram of the instrument is shown in Fig. 1. Figure 1Schematic of the incoherent broadband cavity-enhanced absorption spectrometer. The light from a high-power blue LED (LZ1-04B2P5, LedEngin) with a peak wavelength of ∼448 nm was coupled to the optical cavity via a 90 off-axis parabolic mirror (Edmund Optics). The temperature of the LED was measured by a temperature sensor (PT1000) and controlled by a thermoelectric cooler (TEC) at 20 C ± 0.1 C to reduce the impact of temperature fluctuations on the LED. The optical cavity consisted of two 1 in. diameter mirrors (Advanced Thin Films) with 1 m radius of curvature, and the manufacturer stated that the reflectivity was greater than 99.995 % at 455 nm. Multiple reflections of light between two high-reflectivity cavity mirrors increased the effective absorption path length. The light exiting the cavity passed through an optical band-pass filter (FB450-40, Thorlabs) to eliminate stray light. It was then focused onto a 1 m optical fibre (600 µm in diameter with a numerical aperture of 0.22) by a second off-axis parabolic mirror. Finally, the other end of the fibre cable was coupled to a compact Czerny–Turner spectrometer (Ocean Optics, QE65000) with a spectral resolution of ∼0.57 nm around 450 nm. The CCD in the QE65000 spectrograph is thermally regulated at −10.0C to minimize the dark current. A 2 µm teflon polytetrafluoroethylene (PTFE) membrane filter (Tisch Scientific) was used in the front of the inlet to remove aerosols – this reduced scattering losses by particulate matter and its impacts on the effective path length (Thalman and Volkamer, 2010). Each cavity mirror was purged with a constant flow of dry nitrogen at a rate of 0.1 sL min−1 (standard litres per minute) to block their contact with air samples inside the cavity. This ensured cleanness of the cavity mirror throughout the experiment. The combination of a mass flow controller and a rotameter maintained a constant combined sample and purge gas flow rate of 1.2 sL min−1, which resulted in a gas residence time of about 16 s in the optical cavity. ## 2.2 Theory of IBBCEAS The total extinction in the optical cavity includes the absorption by trace gases, Rayleigh scattering by gas molecules and Mie scattering by particles. The use of a filter in an air-sampling pipeline removes the particles. The Rayleigh scattering extinction of pure N2 is about $\mathrm{2.5}×{\mathrm{10}}^{-\mathrm{7}}$ cm−1 at 455 nm, which is comparable to the cavity loss $\sim \mathrm{8.1}×{\mathrm{10}}^{-\mathrm{7}}$ cm−1 based on mirror reflectivity and cavity length (i.e. $\left(\mathrm{1}-R\right)/d$). Thus, the general description of the total optical extinction αabs within the optical cavity is (Washenfelder et al., 2008): $\begin{array}{}\text{(1)}& {\mathit{\alpha }}_{\mathrm{abs}}\left(\mathit{\lambda }\right)=\left(\frac{\mathrm{1}-R\left(\mathit{\lambda }\right)}{{d}_{\mathrm{eff}}}+{\mathit{\alpha }}_{\mathrm{Ray}}\left(\mathit{\lambda }\right)\right)\left(\frac{{I}_{\mathrm{0}}\left(\mathit{\lambda }\right)-I\left(\mathit{\lambda }\right)}{I\left(\mathit{\lambda }\right)}\right),\end{array}$ where R(λ) is the wavelength-dependent reflectivity of the cavity mirrors, αRay(λ) is the extinction for Rayleigh scattering, I0(λ) and I(λ) are the light intensities transmitted through the optical cavity without and with the absorbing species, and deff is the effective cavity length. The mirror reflectivity R(λ) is determined from the Rayleigh scattering of N2 and He via the following equation (Washenfelder et al., 2008): $\begin{array}{}\text{(2)}& R\left(\mathit{\lambda }\right)=\mathrm{1}-\frac{\frac{{I}_{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)}{{I}_{\mathrm{He}}\left(\mathit{\lambda }\right)}\cdot {\mathit{\alpha }}_{\mathrm{Ray}}^{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right){d}_{\mathrm{0}}-{\mathit{\alpha }}_{\mathrm{Ray}}^{\mathrm{He}}\left(\mathit{\lambda }\right){d}_{\mathrm{0}}}{\mathrm{1}-\frac{{I}_{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)}{{I}_{\mathrm{He}}\left(\mathit{\lambda }\right)}}.\end{array}$ Here, ${I}_{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)$ and IHe(λ) are the light intensities measured when the cavity is filled with N2 and He, respectively. Terms ${\mathit{\alpha }}_{\mathrm{Ray}}^{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)$ and ${\mathit{\alpha }}_{\mathrm{Ray}}^{\mathrm{He}}\left(\mathit{\lambda }\right)$ are the extinction caused by Rayleigh scatterings of N2 and He. Term d0 is the distance between the two cavity mirrors. Terms d0 and deff are not equal due to cavity mirror purging. Determination of the deff will be described in the Sect. 3.2. After obtaining the mirror reflectivity R(λ), the absorption coefficient αabs can be calculated according to Eq. (1). If the chamber contains a variety of gas absorbers (including NO2 and CHOCHO), then the absorption coefficient αabs will be the sum of their individual contributions and can be written via the following equation: $\begin{array}{ll}\text{(3)}& {\mathit{\alpha }}_{\mathrm{abs}}\left(\mathit{\lambda }\right)& =\sum _{i}^{n}{\mathit{\alpha }}_{i}\left(\mathit{\lambda }\right)=\sum _{i}^{n}{\mathit{\sigma }}_{i}\left(\mathit{\lambda }\right){N}_{i}& ={\mathit{\sigma }}_{{\mathrm{NO}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)\left[{\mathrm{NO}}_{\mathrm{2}}\right]+{\mathit{\sigma }}_{\mathrm{CHOCHO}}\left(\mathit{\lambda }\right)\left[\mathrm{CHOCHO}\right]+\mathrm{\dots }\end{array}$ Here, σi(λ) and Ni are the absorption cross section and number density for the ith trace absorber, and n is the total number of absorbers. Finally, the absorber concentrations can be retrieved from the measured broadband spectrum via the DOASIS programme (Kraus, 2006). 3 Results and analysis ## 3.1 Determination of the cavity mirror reflectivity The cavity mirror reflectivity needs to be accurately determined for subsequent measurements of the concentrations of trace gases inside the cavity. We measure and update the value of the mirror reflectivity once every 2 d to ensure the reliability of the retrieval data. Using the difference of Rayleigh scattering cross sections between N2 and He, we calculated the mirror reflectivity R(λ) according to Eq. (2). The values of ${\mathit{\alpha }}_{\mathrm{Ray}}^{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\lambda }\right)$ and ${\mathit{\alpha }}_{\mathrm{Ray}}^{\mathrm{He}}\left(\mathit{\lambda }\right)$ were taken from published references (Shardanand and Rao, 1977; Sneep and Ubachs, 2005). The black and red curves in Fig. 2 were the spectrometer's signal intensity when the cavity was filled with high-purity N2 (99.999 %) and He (99.999 %). The difference in light intensity due to Rayleigh scattering by N2 vs. He is clearly visible. The shaded spectral region (438–465 nm) indicated in the figure contains the main absorption peak of glyoxal and is of primary interest for its spectral retrieval. The mirror reflectivity at the maximum absorption position of glyoxal (455 nm) is about 0.999942. The cross sections were obtained by convolving the high-resolution literature cross sections of CHOCHO (Volkamer et al., 2005b), NO2 (Voigt et al., 2002) and H2O with the nominal spectrometers' instrument function of 0.57 nm full width at half maximum (FWHM). The H2O absorption cross section was calculated with the SpectraPlot programme based on the HITRAN2012 database (Rothman et al., 2013). Figure 2(a) Calibration of the mirror reflectivity. The black and red curves represent spectrometer CCD traces of nitrogen and helium, respectively, with a spectral acquisition time of 30 s. The blue line is the resulting mirror reflectivity curve. (b) The green, magenta and black lines are convolution-based literature absorption cross sections of NO2, glyoxal and H2O vapour. ## 3.2 Calibration of the effective cavity length Considering the intended application's environmental conditions of high-load particulate matter and high-concentration polluting gases, we used an aerosol filter to reduce particles entering the optical cavity and purged the immediate space in front of the cavity mirrors with pure N2 gas to keep the cavity mirrors clean (see Fig. 1). This purging made it difficult to accurately measure the effective cavity length. However, the effective cavity length is required for retrieving trace gas concentrations. Here, we utilized the collision-induced oxygen absorption (referred as O2O2 or O4 absorption) (Thalman and Volkamer, 2013) at 477 nm within our operation wavelength region to quantify the effective cavity length. Pure O2 gas was introduced into the optical cavity and the O2O2 477 nm absorptions with and without the N2 mirror purges were then measured. The O2 flow rate was 1 sL min−1 and the total N2 purge flow rate was 0.2 sL min−1. Figure 3a and b show an example of O2O2 measurement spectrum, its model fitting, and the fit residuals. Figure 3c shows the time series of equivalent O2 concentrations when N2 mirror purge gas alternated between being on and off. A coarse estimation for the cavity length reduction factor was calculated to be 0.87 at room temperature and standard atmospheric pressure according to Eq. (4). $\begin{array}{}\text{(4)}& {d}_{\mathrm{eff}\mathrm{_}\mathrm{O}\mathrm{4}\mathrm{_}\mathrm{based}}={d}_{\mathrm{0}}×\frac{{\left[\sqrt{{\mathrm{O}}_{\mathrm{4}}\phantom{\rule{0.125em}{0ex}}\mathrm{Signal}}\right]}_{\mathrm{Purge}\phantom{\rule{0.125em}{0ex}}\mathrm{on}}}{{\left[\sqrt{{\mathrm{O}}_{\mathrm{4}}\phantom{\rule{0.125em}{0ex}}\mathrm{Signal}}\right]}_{\mathrm{Purge}\phantom{\rule{0.125em}{0ex}}\mathrm{off}}}\end{array}$ Here, the O4 signals were the retrieved concentrations of O4 with and without the N2 purge flows. Furthermore, we modelled the reduction factor of the effective cavity length due to purge gas to include the effect of the dilution of sample gases by purge gases inside the cavity and the fact that the measured O4 spectra were proportional to the product of [O2]×[O2] concentrations. According to the simulation results, if N2 purge gases distributed evenly to both ends of the cavity and 50 % of the total purge N2 was involved in the dilution of O2, the reduction factor for linear absorption process was 0.841, which was 3.3 % less than the coarse-estimation value of 0.87. An uncertainty of the purge N2 participating in the O2 mixture at 40 % or 60 % could cause a ∼2 % uncertainty in the cavity length reduction factor. In this experiment, d0 was 70.0 cm and the calculated deff was 58.9 cm. Figure 3(a) Example of retrieved and fitted absorption spectrum of O4. The blue line is the measured spectrum and the red line is the fitted spectrum of O4. (b) Fit residuals and (c) the time series of equivalent O2 concentrations when N2 purge gas alternated between being on and off. ## 3.3 Instrument stability and detection limit The stability of the system affects its detection sensitivity. An ideal stable system can theoretically achieve an extremely high sensitivity by averaging measurements over a long period of time. However, there are practical considerations that limit this to a certain time range (Werle et al., 1993). For an IBBCEAS system, its stability is mainly affected by the mechanical drifts of the system and the change in the intensity and central wavelength emission of the light source due to temperature variations. Figure 4Evaluation of the instrument performance. Panels (a) and (b) are the time series of NO2 and CHOCHO with 3 s acquisition time. Panels (c) and (d) show the histogram analyses of the measurements of NO2 and CHOCHO. Panels (e) and (f) are Allan deviation plots for measurements of NO2 and CHOCHO. We used two methods to describe the performance of the system: distribution analysis and Allan variance analysis of a large number of measurements. For more than 8 h, 10 000 spectra were continuously acquired with the optical cavity filled with dry nitrogen. As the cavity was free of any NO2 and CHOCHO, these measurements reveal the fluctuations around zero concentration. The acquisition time of each spectrum was 3 s (which combined 10 spectrometer CCD traces with an exposure time of 300 ms each). The concentrations of NO2 and CHOCHO time series (Fig. 4a and b) were obtained by retrieving the spectral measurements. The histograms (Fig. 4c and d) were constructed from this data (Fig. 4a and b). The standard deviation (σGaussian) and mean value (μ) were calculated from the Gaussian distributions of the histograms for each gas. The mean value was an offset from the expected zero and was considered to be a residual “background”. The limit of detection (LOD) can be defined as Eq. (5) from analytical chemistry and this method was also commonly used in cavity-enhanced systems to evaluate instrument performance (Thalman et al., 2015; Fang et al., 2017). $\begin{array}{}\text{(5)}& {\mathrm{LOD}}_{\mathrm{exp}}=\mathrm{2}×{\mathit{\sigma }}_{\mathrm{Gaussian}}+\left\|\mathrm{background}\right\|\end{array}$ According to Eq. (5), the detection limits (with a 3 s acquisition time) for NO2 and CHOCHO were calculated to be about 0.094 ppb (2σ) and 0.058 ppb (2σ). Allan variance analysis has been also a convenient way to describe the stability and detection limit of a system as a function of averaging time. We used Allan variance analysis to characterize the overall stability of our system and to determine the optimum averaging time and predict the detection limit of the system. The above-mentioned 10 000 spectral concentration values were divided into M groups – each containing N values (N=1, 2, , 2000; $M=\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000}/N=\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000}/\mathrm{1},\mathrm{10000}/\mathrm{2},\mathrm{\dots },\mathrm{10000}/\mathrm{2000}$). The average of N values is denoted by yi ($i=\mathrm{1},\mathrm{2},\mathrm{\dots },M$), and the corresponding averaging time is ${t}_{\mathrm{avg}}=N×\mathrm{3}$ s. Since each spectrum was measured in the optical cavity filled with dry nitrogen, the yi values contain only measurement noise as a function of averaging times (Langridge et al., 2008). The Allan variance and standard deviation of NO2 and CHOCHO concentrations are calculated according to Eqs. (6) and (7), as shown in Fig. 4e and f. The Allan deviation initially decreases with a gradient −0.5 as averaging time increases, before it starts to gradually increase towards a longer averaging time. The optimum integration time (210 s for CHOCHO) of the instrument is around the minimum of Allan deviation. A further increase of the integration times yield no more decrease in the Allan deviation due to system drift. For a total acquisition time of 3 s, the detection limits (standard deviation) of NO2 and CHOCHO are 0.083 and 0.052 ppbv (2σ). This result is consistent with LODexp (0.094 and 0.058 ppbv). By increasing the spectral averaging time to 30 s (which combined 100 spectrometer CCD traces with an exposure time of 300 ms each), the NO2 and CHOCHO detection limits (standard deviation) were reduced to 29 pptv (2σ) and 23 pptv (2σ). To capture the rapid variation of CHOCHO in the field,the time resolution of the IBBCEAS instrument was typically set to 30 s. During field measurements, the system drift was managed by frequently measuring the I0 spectrum and stabilizing the temperature of the system. $\begin{array}{}\text{(6)}& & {\mathit{\sigma }}_{A}^{\mathrm{2}}\left({t}_{\mathrm{avg}}\right)=\frac{\mathrm{1}}{\mathrm{2}\left(M-\mathrm{1}\right)}\sum _{i=\mathrm{1}}^{M-\mathrm{1}}{\left[{y}_{i+\mathrm{1}}\left({t}_{\mathrm{avg}}\right)-{y}_{i}\left({t}_{\mathrm{avg}}\right)\right]}^{\mathrm{2}}\text{(7)}& & {\mathit{\sigma }}_{s}^{\mathrm{2}}\left({t}_{\mathrm{avg}}\right)=\frac{\mathrm{1}}{M-\mathrm{1}}\sum _{i=\mathrm{1}}^{M}{\left[{y}_{i}\left({t}_{\mathrm{avg}}\right)-\mathit{\mu }\right]}^{\mathrm{2}}\end{array}$ In the above formulas, yi(tavg) is the averaging concentration of the ith group. Term μ is the average concentration over the entire measurement period. ## 3.4 Sampling loss of glyoxal and measurement of glyoxal sample gas In order to obtain a stable concentration of glyoxal, we used a mass flow controller to allow the quantitative high-purity nitrogen through the trap containing solid glyoxal at atmospheric pressure and at −72C. The sample stream out of the glyoxal trap was further diluted with dry high-purity nitrogen in a sampling bag (PFA) before entering the inlet of the IBBCEAS. ### 3.4.1 Sampling tube loss of glyoxal We measured the glyoxal sample gas in the sampling bag alternately using 3 m and 10 m sampling tubes (PFA) at a flow rate of 1 L min−1 to study the loss of CHOCHO in the sampling tube. The experimental results showed that sampling tube length has no obvious impact on glyoxal loss (Fig. 5). This is consistent with previous findings (Min et al., 2016). Figure 5Measurements of CHOCHO loss in the sampling tube. Blue dots correspond to the measured CHOCHO with the extra 3 m PFA inlet tube. Red dots correspond to the measured CHOCHO with the extra 10 m PFA inlet tube. ### 3.4.2 Measurements of CHOCHO standard additions The high concentration of glyoxal was diluted several times in proportion to obtain the concentration time series as shown in Fig. 6a. The last five low-concentration gradients in Fig. 6a are diluted proportionally by the first maximum concentration gradient. Figure 6b shows the average of these concentration gradients and the normalized mixing ratios, with high linearity (R2=0.9996). Here, the normalized mixing ratio is calculated based on the dilution flows. The intercept value of −2.4 ppbv may be due to the loss of glyoxal onto the surfaces exposed to the gas samples during the experiment. Figure 6Measurement of glyoxal sample gas. (a) Different concentrations of CHOCHO measured by IBBCEAS. Panel (b) shows the correlation between the average of these concentration gradients and the normalized mixing ratio. ## 3.5 Interference from NO2 and spectral fitting Both the glyoxal and NO2 have absorption bands in the same wavelength region as shown in Fig. 2. Therefore, it is important to select suitable absorption features for their retrieval to reduce cross-interferences. Various factors, such as the performance of the instrument (e.g. the intensity wavelength range of the LED light source, the mirror reflectivity, and the spectrometer resolution), the absorption strength of the gas, the concentration level in the actual atmosphere, and the correlation between the absorption features of different gas species in the same wavelength region should be considered to obtain the best-fitting wavelength interval. Figure 7 shows the correlation matrix of absorption cross sections of CHOCHO and NO2 for a range of fitting intervals starting between 429 and 448 nm and ending between 457 and 475 nm. We hope to find an optimal fit interval with minimal correlation (Pinardi et al., 2013). In this paper, the retrieval band of glyoxal and NO2 is finally 438–465 nm. Figure 7Correlations matrix of absorption cross sections of CHOCHO and NO2 for different wavelength intervals in the 429–475 nm wavelength range. When the concentration of NO2 exceeds ∼12 ppbv in the actual atmosphere, the absorption due to NO2 is more than 100-fold higher than that due to a typical 0.1 ppbv glyoxal in the atmosphere. The concentration of NO2 in the atmosphere of polluted urban areas in China could reach tens or even hundreds of ppbv (Qin et al., 2009). Concurrently, the accumulation of NO2 at night further challenges accurate glyoxal measurements. Therefore, accurate data analysis of the NO2 absorption contributions became critical to reducing its impact on the determination of the glyoxal absorption and concentration. For modelling of measurement spectra, one common approach was to first determine a nominal spectrometer resolution profile as the instrumental function and then the literature reference spectrum was convoluted with the instrumental function of the spectrometer. However, we noticed that the grating spectrometer had non-uniform dispersions. We measured the wavelength dependence of the grating spectrometer's resolution by using narrow atomic emission lines of low-pressure Hg, Kr and Zn lamps. These results were summarized in the Table 2. The non-uniform dispersions make spectral modelling less accurate. Subsequently, inaccurate modelling makes it difficult to overcome cross-interference of strongly absorbed interference gases with weakly absorbed gases of interest within the same wavelength region. A more reliable approach we used to obtain NO2 reference spectra was to make a direct measurement of known concentrations of NO2 standard gases with the spectrometer and further calibrate with the convolved literature reference spectrum. Samples of NO2 in N2 were prepared by flow dilution from a standard cylinder containing 5 ppm NO2 in N2. We verified the measured NO2 reference spectrum and the convolved literature NO2 reference spectrum by retrieval of the same NO2 spectra. The difference was about 1.4 %. Table 2Wavelength dependence of the grating spectrometer's resolution. The full width at half maximum (FWHM) values were determined from the emission line width measurements of low-pressure Hg, Kr and Zn lamps. ### 3.5.1 Residual structure from NO2 fitting For our application, inaccurate NO2 fitting produces a large residual structure, especially in the case of high concentrations of NO2 (see Fig. 8). Figure 8 shows the variation characteristics of fit residuals from fitting different concentrations of standard NO2 when using the convolution-based NO2 reference spectrum. As is clear from Fig. 8, there is a similar residual structure in the fit residual and it increases with increasing NO2 concentration. Such a residual structure will have a disastrous effect on the retrieval of glyoxal in the atmosphere. Figure 9 shows that the standard deviation of these fit residuals has a good dependence on the NO2 concentration. Figure 8Fitting residuals of different concentrations of standard NO2 when using the convolution-based NO2 reference spectrum. Figure 9The standard deviation of the fit residual from Fig. 8 as a function of NO2 concentrations. ### 3.5.2 Spectral simulation of NO2 interference with glyoxal The influence of residual structures in the absorption spectra has been evaluated by Stutz and Platt (1996). Due to the non-uniform dispersion of the spectrometer, a stable residual structure was produced when a NO2 reference spectrum based on a simple convolution calculation (by using a nominal function for the instrument line profile) was used for experimental spectral profile fitting (Fig. 8). We evaluated this non-uniform dispersion effect of the coexisting NO2 absorption on glyoxal spectral analysis. The simulation spectra we used to test the accuracy of the spectral extraction comprised three components: NO2 reference spectra based on measurements according to Eq. (1), as well as the Rayleigh spectrum of N2 at 1 atm, and convolution-simulated spectrum of 0.1 ppbv CHOCHO. We obtained simulation spectra containing different concentrations of NO2 (0–200 ppbv) and 0.1 ppbv glyoxal according to Eq. (3) as a summation. The spectral retrieval was conducted by using a non-linear least-squares fitting routine. We tested the retrieval accuracy of CHOCHO by applying either a convolution-based NO2 reference spectrum or a measurement-based NO2 reference spectrum in the non-linear least-squares fitting routine for the modelling of the NO2 spectral contribution. Figure 10The simulation results of the effect of NO2 on glyoxal. (a) The deviation of the retrieved glyoxal concentration from its nominal value of 0.1 ppbv as a function of NO2 concentrations. The blue line is the retrieval result using the convolution-simulated NO2 reference spectra (grey area is the range of fitting uncertainty). The red line is the result of the retrieval via the measured NO2 reference spectrum (grey area is the range of the fitting uncertainty). (b) The corresponding standard deviations of the spectral fit residual. Figure 10a shows the deviation of the retrieved CHOCHO concentration from its nominal 0.1 ppbv value as a function of NO2 concentration. The blue line in Fig. 10a is the retrieval result when using convolution-based NO2 reference spectrum as its model function (The grey area indicates the range of fitting uncertainties). The deviation of the extracted glyoxal concentration and estimated uncertainty increase linearly as the concentration of NO2 increases. The deviation of glyoxal reaches 0.58 ppbv when the concentration of NO2 is 198 ppbv. In other words, for our instrument, the large bias is characterized as 2.9 pptv glyoxal per ppbv NO2. Thalman et al. (2015) showed in their experiment that the CE-DOAS and BBCEAS systematic bias is 1 pptv glyoxal per ppbv NO2 at higher NO2. The difference between our findings and his findings may be due to differences in instruments – especially the spectrometers. When the concentration of NO2 is less than ∼8 ppbv, its effect on the deviation of glyoxal is less than the detection limit of the instrument (23 pptv, 2σ). The NO2 likely has only a minor effect on glyoxal measurements in this low-concentration case. When the retrieval is performed using the measurement-based NO2 reference spectrum, the deviation of the extracted glyoxal concentration value (Fig. 10a, red line) remains close to zero. The uncertainty of the fitting error (grey area) is also small, indicating that the effect of NO2 on glyoxal is negligible. Figure 10b compares the standard deviations of the fit residual as a function of NO2 concentration. The standard deviation is reduced from $\mathrm{5.1}×{\mathrm{10}}^{-\mathrm{11}}$ cm−1 per ppbv NO2 when using the convolution-based NO2 reference spectrum in the least-squares fitting, which reduced to $\mathrm{1.7}×{\mathrm{10}}^{-\mathrm{12}}$ cm−1 per ppbv NO2 when using the measurement-based NO2 reference spectrum. This is an improvement of over 30 times. ### 3.5.3 Spectral fitting of field measurement spectra We compared the effects of using the convolution-based and the measurement-based NO2 reference spectra to fit real atmospheric spectral measurements. As the simulation analysis in the previous Sect. 3.5.2 indicated already, using the measurement-based NO2 reference spectrum for data analysis of the real atmospheric measurements achieved a more precise NO2 fitting, as both the NO2 reference spectrum and the real atmospheric measurements share the same instrument (i.e. the grating spectrometer) function. The results of the real experimental spectral fitting are shown in Fig. 11. A comparison of the spectral retrievals using both the convolution-based NO2 reference spectrum and the measurement-based NO2 reference spectrum are displayed in the left and right columns of Fig. 11. Corresponding fit residuals are shown in the bottom panels, and the standard deviations are $\mathrm{1.31}×{\mathrm{10}}^{-\mathrm{9}}$ and $\mathrm{8.78}×{\mathrm{10}}^{-\mathrm{10}}$ cm−1. The standard deviation of the fitting residuals by using a measurement-based NO2 reference spectrum is 33 % smaller than those with a convolution-based reference spectrum. Moreover, the fitting residual using measurement-based NO2 reference spectrum showed no obvious structure. The fitting of glyoxal is more precise, and the fitting error is reduced by 31.7 % (Fig. 11g and h) when using the measured NO2 reference spectrum. For NO2, the fitting exhibits almost no difference. The result demonstrates that it is critical to use the measured NO2 reference spectrum. Any tiny distortion in the NO2 reference spectral profile could have a severe effect on the CHOCHO extraction, because NO2 absorption is about 2 orders of magnitude stronger than that of the CHOCHO in the local atmosphere. Figure 12 shows the standard deviation of the fitting residual of the absorption coefficient as a function of NO2 concentration for measurements conducted during the APHH China project (June 2017). The standard deviation is reduced from $\mathrm{5.1}×{\mathrm{10}}^{-\mathrm{11}}$ to $\mathrm{2.2}×{\mathrm{10}}^{-\mathrm{11}}$ cm−1 per ppbv NO2 by using the measurement-based NO2 reference spectrum, which is 2.3 times smaller. The uncertainties in absorption cross sections are 4 % for NO2 (Voigt et al., 2002), and 5 % for CHOCHO (Volkamer et al., 2005b). The difference in NO2 between the literature reference spectrum and the measured reference spectrum is 1.5 %. Our experimental uncertainties in cavity mirror reflectivity and effective cavity length are 5 % and 2 %. The propagated errors (summed in quadrature) are estimated to be 6.7 % for NO2 when the convolution-based NO2 reference spectrum was used or 6.9 % when its measurement-based reference spectrum was used, and 7.3 % for CHOCHO using the convolution-based literature reference spectrum. Figure 11A comparison of the experimental atmospheric spectral retrievals using both the convolution-based NO2 reference spectrum (left column) and the measurement-based NO2 reference spectrum (right column). Panels (a) and (b) show the same atmospheric spectrum (recorded on 9 June 2017 at 12:28 LT). The retrieved NO2, H2O, and CHOCHO concentrations are shown in panels (c) and (d), (e) and (f), (g) and (h). Two overall fit residuals are shown in the bottom panels (i) and (j), with the standard deviations of $\mathrm{1.31}×{\mathrm{10}}^{-\mathrm{9}}$ and $\mathrm{8.78}×{\mathrm{10}}^{-\mathrm{10}}$ cm−1. Figure 12The standard deviation of the fit residual of the absorption coefficients as a function of NO2 concentrations for spectral data analysis during the APHH field measurements in Beijing, June 2017. The red dots were the standard deviations of the fit residuals by using a convolution-based NO2 spectral profile and the black line is the linear fit of the data. The blue dots were the standard deviations of the fit residuals by using a measurement-based NO2 reference spectral profile and the green line is the linear fit of the corresponding data. ## 3.6 Field measurements The field campaign was conducted in the city Beijing at the Iron Tower Department of the Institute of Atmospheric Physics, Chinese Academy of Sciences during the APHH China project (2–26 June 2017). The IBBCEAS system was deployed to measure both CHOCHO and NO2, supplemented by many other atmospheric measurement instruments. The sampling height of the IBBCEAS system was about 4 m above the ground. A cavity-attenuated phase shift (CAPS) spectroscopy system (University of York) for NO2 data comparison was located in another container about 30 m away from the IBBCEAS system. Figure 13 shows the 24 d continuous measurements of CHOCHO and NO2 in the atmosphere by our IBBCEAS instrument. Each measurement data point was derived from each absorption spectrum acquired over 30 s (which averaged 100 spectrometer CCD traces with an exposure time of 300 ms each). The concentration of glyoxal in the city reached 0.572 ppbv at the maximum; the average was 0.091 ppbv. Time series data for NO2 measured by IBBCEAS was compared with the data from the CAPS spectroscopy system (Fig. 13b). Overall, both sets of measurements were in very good agreement. The average concentration of NO2 was ∼20.0 ppbv and the maximum value was ∼80 ppbv. A correlation plot comparing the IBBCEAS and CAPS NO2 concentration data is shown in Fig. 14, with the data averaged to 1 h. The linear regression exhibits [NO2] CAPS $=\mathrm{1.03}×\left[{\mathrm{NO}}_{\mathrm{2}}\right]$ IBBCEAS with a correlation coefficient of R2=0.99. Discrepancies of ∼3 % between the two data sets may be partly due to the different air sampling locations of these two instruments and the uncertainty of the effective cavity length calibration of the IBBCEAS. Overall this 3 % deviation was within the expected 6.9 % uncertainty mentioned in Sect. 3.5.3. Figure 13Results of 24 d continuous measurements of CHOCHO and NO2 in the atmosphere. Figure 14Correlation plot of NO2 concentration values between IBBCEAS and CAPS measurements. Each NO2 data point was represented by averaged values to 1 h. The slope of the straight line fit (red line) is 1.03 with R2=0.99. 4 Conclusions This paper describes the development of an IBBCEAS system and its field application to high-sensitivity measurements of atmospheric glyoxal and NO2. The mirror reflectivity of the optical cavity was calibrated using the difference in Rayleigh scattering cross sections between N2 and He gases. The mirror reflectivity R is greater than 0.99994 at 455 nm, and the corresponding effective absorption pathlength is about 11.7 km (cavity dimension 0.7 m, in the absence of Rayleigh scattering). To accurately obtain a reduction factor for the cavity length when the cavity mirrors were protected by N2 pure gases, the O4 absorption in pure oxygen (at the 477 nm band) was used to calibrate the effective cavity length. The reduction factor of the cavity length was 0.841 at an inlet flow rate of 1 sL min−1 and a total purge flow rate of 0.2 sL min−1. Here, the cavity length d0 is 70 cm, and the calculated deff is 58.9 cm. We used Allan variance analysis to identify the system's detection limits for NO2 and CHOCHO. They were 0.083 ppbv (2σ) and 0.052 ppbv (2σ) at a 3 s time resolution in the laboratory. Further increases in acquisition time to 30 s improve the detection limits of CHOCHO and NO2 to 23 pptv (2σ) and 29 pptv (2σ). The overall uncertainties of the instrument are 6.7 % or 6.9 % for NO2 using convolution-based or measurement-based reference spectrum and 7.3 % for CHOCHO. The effect of NO2 on glyoxal was evaluated via spectral simulations and measurements. When using a convolution-based NO2 reference spectral profile, the high concentration of NO2 had a large effect on glyoxal and the bias was characterized as 2.9 pptv glyoxal per ppbv NO2. The effect of NO2 on glyoxal became negligible when the retrieval was performed using the measurement-based NO2 reference spectral profile. The measured NO2 reference spectrum was applied to the retrieval of the actual atmospheric spectrum, effectively reducing the impact of NO2 on the retrieval of CHOCHO during the APHH China field measurement project (2–26 June 2017). The standard deviation of the fitting residual was reduced from $\mathrm{5.1}×{\mathrm{10}}^{-\mathrm{11}}$ to $\mathrm{2.2}×{\mathrm{10}}^{-\mathrm{11}}$ cm−1 per ppbv NO2 by using the measured NO2 reference spectrum, which is 2.3 times smaller. The concentrations of CHOCHO and NO2 in the Beijing summer atmosphere were obtained during the APHH China project. There was good agreement in NO2 concentrations acquired by the IBBCEAS and another independent instrument using a different measurement technique–CAPS. The maximum concentrations of glyoxal and NO2 in Beijing in summer reached 0.572 and ∼80 ppbv. This has demonstrated that our IBBCEAS instrument is capable of making accurate continuous measurements in atmospheric environments of high-load particulate matters and high-concentration polluting gases. Data availability Data availability. The data used in this study are available from the corresponding author upon request (mqin@aiofm.ac.cn). Author contributions Author contributions. MQ, PX, JiaL, and WL contributed to the conception of the study. SL, JD, and WF built the IBBCEAS instrument. XL and JinL designed the standard gas generator for glyoxal. SL, JD, KT, FM, and KY performed the experiments. JX and MQ contributed ideas about spectral simulation. SL performed the data analyses and wrote the manuscript. YH and MQ edited and developed the manuscript. Competing interests Competing interests. The authors declare that they have no conflict of interest. Special issue statement Special issue statement. Acknowledgements Acknowledgements. This work was supported by the National Natural Science Foundation of China (grant no. 91544104, 41705015 and 41571130023), the Science and Technology Major Special Project of Anhui Province, China (16030801120) and the National Key R&D Programme of China (2017YFC0209400). The co-author Yabai He is also associated with Macquarie University, Australia. The authors would like to thank Lee James from the University of York for providing NO2 data. We gratefully acknowledge the discussions with Bin Ouyang from the Lancaster Environment Centre, Lancaster University, UK. Review statement Review statement. This paper was edited by Weidong Chen and reviewed by two anonymous referees. References Chen, J. and Venables, D. S.: A broadband optical cavity spectrometer for measuring weak near-ultraviolet absorption spectra of gases, Atmos. Meas. Tech., 4, 425–436, https://doi.org/10.5194/amt-4-425-2011, 2011. Coburn, S., Ortega, I., Thalman, R., Blomquist, B., Fairall, C. W., and Volkamer, R.: Measurements of diurnal variations and eddy covariance (EC) fluxes of glyoxal in the tropical marine boundary layer: description of the Fast LED-CE-DOAS instrument, Atmos. Meas. Tech., 7, 3579–3595, https://doi.org/10.5194/amt-7-3579-2014, 2014. DiGangi, J. P., Henry, S. B., Kammrath, A., Boyle, E. S., Kaser, L., Schnitzhofer, R., Graus, M., Turnipseed, A., Park, J.-H., Weber, R. J., Hornbrook, R. S., Cantrell, C. A., Maudlin III, R. L., Kim, S., Nakashima, Y., Wolfe, G. M., Kajii, Y., Apel, E. C., Goldstein, A. H., Guenther, A., Karl, T., Hansel, A., and Keutsch, F. N.: Observations of glyoxal and formaldehyde as metrics for the anthropogenic impact on rural photochemistry, Atmos. Chem. Phys., 12, 9529–9543, https://doi.org/10.5194/acp-12-9529-2012, 2012. Fang, B., Zhao, W., Xu, X., Zhou, J., Ma, X., Wang, S., Zhang, W., Venables, D. S., and Chen, W.: Portable broadband cavity-enhanced spectrometer utilizing Kalman filtering: application to real-time, in situ monitoring of glyoxal and nitrogen dioxide, Opt. Express, 25, 26910–26922, https://doi.org/10.1364/OE.25.026910, 2017. Fiedler, S. E., Hoheisel, G., Ruth, A. A., and Hese, A.: Incoherent broad-band cavity-enhanced absorption spectroscopy of azulene in a supersonic jet, Chem. Phys. Lett., 382, 447–453, https://doi.org/10.1016/j.cplett.2003.10.075, 2003. Fu, T.-M., Jacob, D. J., Wittrock, F., Burrows, J. P., Vrekoussis, M., and Henze, D. K.: Global budgets of atmospheric glyoxal and methylglyoxal, and implications for formation of secondary organic aerosols, J. Geophys. Res., 113, D15303, https://doi.org/10.1029/2007JD009505, 2008. Henry, S. B., Kammrath, A., and Keutsch, F. N.: Quantification of gas-phase glyoxal and methylglyoxal via the Laser-Induced Phosphorescence of (methyl)GLyOxal Spectrometry (LIPGLOS) Method, Atmos. Meas. Tech., 5, 181–192, https://doi.org/10.5194/amt-5-181-2012, 2012. Ho, S. S. H. and Yu, J. Z.: Determination of airborne carbonyls: comparison of a thermal desorption/GC method with the standard DNPH/HPLC method, Environ. Sci. Technol., 38, 862–870, https://doi.org/10.1021/es034795w, 2004. Huisman, A. J., Hottle, J. R., Coens, K. L., DiGangi, J. P., Galloway, M. M., Kammrath, A., and Keutsch, F. N.: Laser-induced phosphorescence for the in situ detection of glyoxal at part per trillion mixing ratios, Anal. Chem. 80, 5884–5891, 2008. Kahan, T. F., Washenfelder, R. A., Vaida, V., and Brown, S. S.: Cavity-enhanced measurements of hydrogen peroxide absorption cross sections from 353 to 410 nm, J. Phys. Chem. A, 116, 5941–5947, 2012. Kaiser, J., Wolfe, G. M., Min, K. E., Brown, S. S., Miller, C. C., Jacob, D. J., deGouw, J. A., Graus, M., Hanisco, T. F., Holloway, J., Peischl, J., Pollack, I. B., Ryerson, T. B., Warneke, C., Washenfelder, R. A., and Keutsch, F. N.: Reassessing the ratio of glyoxal to formaldehyde as an indicator of hydrocarbon precursor speciation, Atmos. Chem. Phys., 15, 7571–7583, https://doi.org/10.5194/acp-15-7571-2015, 2015. Kraus, S. G.: DOASIS: A Framework Design for DOAS, Dissertation, University of Mannheim, Mannheim, Germany, 2006. Langridge, J. M., Ball, S. M., Shillings, A. J. L., and Jones, R. L.: A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection, Rev. Sci. Instrum., 79, 123110, https://doi.org/10.1063/1.3046282, 2008. Li, J., Mao, J., Min, K. E., Washenfelder, R. A., Brown, S. S., Kaiser, J., Keutsch, F. N., Volkamer, R., Wolfe, G. M., Hanisco, T. F., Pollack, I. B., Ryerson, T. B., Graus, M., Gilman, J. B., Lerner, B. M., Warneke, C., de Gouw, J. A., Middlebrook, A. M., Liao, J., Welti, A., Henderson, B. H., McNeill, V. F., Hall, S. R., Ullmann, K., Donner, L. J., Paulot, F., and Horowitz, L. W.: Observational constraints on glyoxal production from isoprene oxidation and its contribution to organic aerosol over the Southeast United States, J. Geophys. Res.-Atmos. 121, 9849–9861, https://doi.org/10.1002/2016JD025331, 2016 MacDonald, S. M., Oetjen, H., Mahajan, A. S., Whalley, L. K., Edwards, P. M., Heard, D. E., Jones, C. E., and Plane, J. M. C.: DOAS measurements of formaldehyde and glyoxal above a south-east Asian tropical rainforest, Atmos. Chem. Phys., 12, 5949–5962, https://doi.org/10.5194/acp-12-5949-2012, 2012. Min, K.-E., Washenfelder, R. A., Dubé, W. P., Langford, A. O., Edwards, P. M., Zarzana, K. J., Stutz, J., Lu, K., Rohrer, F., Zhang, Y., and Brown, S. S.: A broadband cavity enhanced absorption spectrometer for aircraft measurements of glyoxal, methylglyoxal, nitrous acid, nitrogen dioxide, and water vapor, Atmos. Meas. Tech., 9, 423–440, https://doi.org/10.5194/amt-9-423-2016, 2016. Munger, J. W., Jacob, D. J., Daube, B. C., and Horowitz, L. W.: Formaldehyde, glyoxal, and methylglyoxal in air and cloudwater at a rural mountain site in central Virginia, J. Geophys. Res.-Atmos., 100, 9325–9333, 1995. Pang, X., Lewis, A. C., Rickard, A. R., Baeza-Romero, M. T., Adams, T. J., Ball, S. M., Daniels, M. J. S., Goodall, I. C. A., Monks, P. S., Peppe, S., Ródenas García, M., Sánchez, P., and Muñoz, A.: A smog chamber comparison of a microfluidic derivatisation measurement of gas-phase glyoxal and methylglyoxal with other analytical techniques, Atmos. Meas. Tech., 7, 373–389, https://doi.org/10.5194/amt-7-373-2014, 2014. Pinardi, G., Van Roozendael, M., Abuhassan, N., Adams, C., Cede, A., Clémer, K., Fayt, C., Frieß, U., Gil, M., Herman, J., Hermans, C., Hendrick, F., Irie, H., Merlaud, A., Navarro Comas, M., Peters, E., Piters, A. J. M., Puentedura, O., Richter, A., Schönhardt, A., Shaiganfar, R., Spinei, E., Strong, K., Takashima, H., Vrekoussis, M., Wagner, T., Wittrock, F., and Yilmaz, S.: MAX-DOAS formaldehyde slant column measurements during CINDI: intercomparison and analysis improvement, Atmos. Meas. Tech., 6, 167–185, https://doi.org/10.5194/amt-6-167-2013, 2013. Qin, M., Xie, P., Su, H., Gu, J., Peng, F., Li, S., Zeng, L., Liu, J., Liu, W., and Zhang, Y.: An observational study of the HONO–NO2 coupling at an urban site in Guangzhou City, South China, Atmos. Environ., 43, 5731–5742, https://doi.org/10.1016/j.atmosenv.2009.08.017, 2009. Rothman, L. S., Gordon, I. E., Babikov, Y., Barbe, A., Chris Benner, D., Bernath, P. F., Birk, M., Bizzocchi, L., Boudon, V., Brown, L. R., Campargue, A., Chance, K., Cohen, E. A., Coudert, L. H., Devi, V. M., Drouin, B. J., Fayt, A., Flaud, J. -M., Gamache, R. R., Harrison, J. J., Hartmann, J. -M., Hill, C., Hodges, J. T., Jacquemart, D., Jolly, A., Lamouroux, J., Le Roy, R. J., Li, G., Long, D. A., Lyulin, O. M., Mackie, C. J., Massie, S. T., Mikhailenko, S., Müller, H. S. P., Naumenko, O. V., Nikitin, A. V., Orphal, J., Perevalov, V., Perrin, A., Polovtseva, E. R., Richard, C., Smith, M. A. H., Starikova, E., Sung, K., Tashkun, S., Tennyson, J., Toon, G. C., Tyuterev, Vl. G., Wagner, G.: The HITRAN 2012 Molecular Spectroscopic Database, J. Quant. Spectrosc. Ra., 130, 4–50, https://doi.org/10.1016/j.jqsrt.2013.07.002, 2013. Shardanand and Rao, A. D. P.: Absolute Rayleigh scattering cross sections of gases and freons of stratospheric interest in the visible and ultraviolet regions, NASA Technical Note, National Aeronautics And Space Administration Washington, D.C., USA, 1977. Sneep, M. and Ubachs, W.: Direct measurement of the Rayleigh scattering cross section in various gases, J. Quant. Spectrosc. Ra., 92, 293–310, 2005. Stutz, J. and Platt, U.: Numerical analysis and estimation of the statistical error of differential optical absorption spectroscopy measurements with least-squares methods, Appl. Optics, 35, 6041–6053, 1996. Temime, B., Healy, R. M., and Wenger, J.: A denuder-filter sampling technique for the detection of gas and particle phase carbonyl compounds, Environ. Sci. Technol., 41, 6514–6520, https://doi.org/10.1021/es070802v, 2007. Thalman, R. and Volkamer, R.: Inherent calibration of a blue LED-CE-DOAS instrument to measure iodine oxide, glyoxal, methyl glyoxal, nitrogen dioxide, water vapour and aerosol extinction in open cavity mode, Atmos. Meas. Tech., 3, 1797–1814, https://doi.org/10.5194/amt-3-1797-2010, 2010. Thalman, R. and Volkamer, R.: Temperature dependent absorption cross-sections of O2-O2 collision pairs between 340 and 630 nm and at atmospherically relevant pressure, Phys. Chem. Chem. Phys., 15, 15371–15381, https://doi.org/10.1039/c3cp50968k, 2013. Thalman, R., Baeza-Romero, M. T., Ball, S. M., Borrás, E., Daniels, M. J. S., Goodall, I. C. A., Henry, S. B., Karl, T., Keutsch, F. N., Kim, S., Mak, J., Monks, P. S., Muñoz, A., Orlando, J., Peppe, S., Rickard, A. R., Ródenas, M., Sánchez, P., Seco, R., Su, L., Tyndall, G., Vázquez, M., Vera, T., Waxman, E., and Volkamer, R.: Instrument intercomparison of glyoxal, methyl glyoxal and NO2 under simulated atmospheric conditions, Atmos. Meas. Tech., 8, 1835–1862, https://doi.org/10.5194/amt-8-1835-2015, 2015. Volkamer, R., Molina, L. T., Molina, M. J., Shirley, T., and Brune, W. H.: DOAS measurement of glyoxal as an indicator for fast VOC chemistry in urban air, Geophys. Res. Lett., 32, L08806, https://doi.org/10.1029/2005GL022616, 2005a. Volkamer, R., Spietz, P., Burrows, J., and Platt, U.: High-resolution absorption cross-section of glyoxal in the UV-vis and IR spectral ranges, J. Photoch. Photobio. C, 172, 35–46, https://doi.org/10.1016/j.jphotochem.2004.11.011, 2005b. Volkamer, R., Martini, F. S., Molina, L. T., Salcedo, D., Jimenez, J. L., and Molina, M. J.: A missing sink for gas-phase glyoxal in Mexico City: Formation of secondary organic aerosol, Geophys. Res. Lett. 34, L19807, https://doi.org/10.1029/2007GL030752, 2007. Volkamer, R., Baidar, S., Campos, T. L., Coburn, S., DiGangi, J. P., Dix, B., Eloranta, E. W., Koenig, T. K., Morley, B., Ortega, I., Pierce, B. R., Reeves, M., Sinreich, R., Wang, S., Zondlo, M. A., and Romashkin, P. A.: Aircraft measurements of BrO, IO, glyoxal, NO2, H2O, O2O2 and aerosol extinction profiles in the tropics: comparison with aircraft-/ship-based in situ and lidar measurements, Atmos. Meas. Tech., 8, 2121-2148, https://doi.org/10.5194/amt-8-2121-2015, 2015. Voigt, S., Orphal, J., and Burrows, J. P.: The temperature and pressure dependence of the absorption cross-sections of NO2 in the 250–800 nm region measured by Fourier-transform spectroscopy, J. Photoch. Photobio. C, 149, 1–7, 2002. Vrekoussis, M., Wittrock, F., Richter, A., and Burrows, J. P.: GOME-2 observations of oxygenated VOCs: what can we learn from the ratio glyoxal to formaldehyde on a global scale?, Atmos. Chem. Phys., 10, 10145–10160, https://doi.org/10.5194/acp-10-10145-2010, 2010. Wang, H., Chen, J., and Lu, K.: Development of a portable cavity-enhanced absorption spectrometer for the measurement of ambient NO3 and N2O5: experimental setup, lab characterizations, and field applications in a polluted urban environment, Atmos. Meas. Tech., 10, 1465–1479, https://doi.org/10.5194/amt-10-1465-2017, 2017. Washenfelder, R. A., Langford, A. O., Fuchs, H., and Brown, S. S.: Measurement of glyoxal using an incoherent broadband cavity enhanced absorption spectrometer, Atmos. Chem. Phys., 8, 7779–7793, https://doi.org/10.5194/acp-8-7779-2008, 2008. Washenfelder, R. A., Young, C. J., Brown, S. S., Angevine, W. M., Atlas, E. L., Blake, D. R., Bon, D. M., Cubison, M. J., de Gouw, J. A., Dusanter, S., Flynn, J., Gilman, J. B., Graus, M., Griffith, S., Grossberg, N., Hayes, P. L., Jimenez, J. L., Kuster, W. C., Lefer, B. L., Pollack, I. B., Ryerson, T. B., Stark, H., Stevens, P. S., and Trainer, M. K.: The glyoxal budget and its contribution to organic aerosol for Los Angeles, California, during CalNex 2010, J. Geophys. Res.-Atmos., 116, D00V02, https://doi.org/10.1029/2011JD016314, 2011. Washenfelder, R. A., Flores, J. M., Brock, C. A., Brown, S. S., and Rudich, Y.: Broadband measurements of aerosol extinction in the ultraviolet spectral region, Atmos. Meas. Tech., 6, 861–877, https://doi.org/10.5194/amt-6-861-2013, 2013. Werle, P., Mücke, R., and Slemr, F.: The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS), Appl. Phys. B, 57, 131–139, 1993. Yi, H., Wu, T., Wang, G., Zhao, W., Fertein, E., Coeur, C., Gao, X., Zhang, W., and Chen, W.: Sensing atmospheric reactive species using light emitting diode by incoherent broadband cavity enhanced absorption spectroscopy, Opt. Express, 24, A781–790, https://doi.org/10.1364/OE.24.00A781, 2016.	2019-07-20 03:52:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7982027530670166, "perplexity": 5998.592088745217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526408.59/warc/CC-MAIN-20190720024812-20190720050812-00445.warc.gz"}
http://math.stackexchange.com/questions/223226/difference-between-eigenfunctions-and-eigenvectors-of-an-operator?answertab=active	# Difference between eigenfunctions and eigenvectors of an operator? What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator? - Real or complex (or vector) valued functions on a space form a vector space. The Laplace-Beltrami operator is a linear operator that acts on this vector space. Its eigenvectors are also called "eigenfunctions" because the "vectors" are functions. – Jonah Sinick Oct 29 '12 at 4:44 add comment ## 1 Answer An eigenfunction is an eigenvector that is also a function. Thus, an eigenfunction is an eigenvector but an eigenvector is not necessarily an eigenfunction. For example, the eigenvectors of differential operators are eigenfunctions but the eigenvectors of finite-dimensional linear operators are not. - add comment	2014-03-09 10:46:47	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9039802551269531, "perplexity": 428.265210347088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999677352/warc/CC-MAIN-20140305060757-00015-ip-10-183-142-35.ec2.internal.warc.gz"}
https://nrich.maths.org/11140	### Network Trees Explore some of the different types of network, and prove a result about network trees. ### Always Two Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2. ### Symmetricality Five equations and five unknowns. Is there an easy way to find the unknown values? # Starting to Explore Four Consecutive Numbers ##### Age 11 to 16Challenge Level Take four consecutive numbers, $a$, $b$, $c$, $d$. 1. (a) The four consecutive numbers sum to $130$. What are they? (b) The four consecutive numbers sum to $-38$. What are they? 2. The sum of the first three consecutive numbers is $10$ more than the fourth. What are the four numbers? 3. What is $(a+d)-(b+c)$? Why? 4. Explore $a+b+c-d$. If you enjoyed working on this problem, you may now want to take a look at the follow-up problem, Continuing to Explore Four Consecutive Numbers. With thanks to Don Steward, whose ideas formed the basis of this problem.	2022-08-14 12:53:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6984870433807373, "perplexity": 601.8374488541679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572033.91/warc/CC-MAIN-20220814113403-20220814143403-00779.warc.gz"}
http://www.songtextemania.com/if_i_only_had_time_songtext_howard_carpendale.html	# If I Only Had Time Songtext ## Howard Carpendale ### von Mehr Songtexte If I Only Had Time Songtext If I only had time Howard Carpendale mmmmm... If I only had time, only time. So much to do, if I only had time. If I only had time dreams to pursue. There are mountains I'd climb if I had time. Time like the wind. Those are hurrying by and the hours just fly. Where to begin ? There are mountains I'd climb if I had time. Since I met you I've glored Life really is to short. Lovin' you so many things we could make true. A whole century isn't enough to satisfy me. So much to do, if I only had time. If I only had time dreams to pursue. There are mountains I'd climb if I had time If I only had time, only time.	2020-02-22 14:07:17	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8174149990081787, "perplexity": 2526.5957218047747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145676.44/warc/CC-MAIN-20200222115524-20200222145524-00321.warc.gz"}
http://cgoliver.com/2016/12/24/Migration-Prediction.html	# Predicting migration phases in eagles: an exercise in Basemap, Pandas, 3D Plotting, and unsupervised Machine Learning This post is based on a project I had for my machine learning class. My team (myself, Roman Sarrazin Gendron, and Navin Mordani) were asked to obtain a dataset from the animal migrations databank movebank and build some model that can help conservation efforts. We chose to look at migration patterns of eagles and other predatory birds and predict their migration start and end times as a function of temperature. The first step is to be able to determine when a bird is undergoing a migration and when the bird is at a homebase. Because our dataset only contains recorded times and positions, we would have to do some unsupervised learning to identify migratory phases. We took a very naïve approach to this problem so ecologists and biologists will have to take this with a huge grain of salt. The main purpose of this post is to illustrate some nice Python modules, do some basic machine learning and make some pretty plots. With this post, you’ll learn a bit about how to use Python’s map plotting library Basemap, managing datasets with Pandas, 3D plotting, and basic unsupervised machine learning. You can find all the original files in my gitub repository Notebooks/Eagles here. Enjoy! ## The Data We assembled a dataset (curated by Roman Sarrazin Gendron) that contains times and positions of 14 species around the world. In order to make this simpler if you want to follow along, I included a condensed version of the dataset since it is very large. Here’s the command in pandas I used to generate the condensed dataset. The condensed data is stored in a file called eagles_short.csv in this notebook’s directory. Let’s assume that the original data is in a file called eagles.csv. import pandas as pd df = pd.read_csv('eagles.csv') df2 = df.sample(frac=0.1) df2.to_csv('eagles_short.csv') Ok so to start we’re going to have to import a couple of things. import matplotlib.pyplot as plt import matplotlib.cm as cm from mpl_toolkits.mplot3d import Axes3D from mpl_toolkits.basemap import Basemap import numpy as np import pandas as pd import seaborn as sns from collections import OrderedDict from sklearn.cluster import KMeans import seaborn as sns import math from scipy.spatial.distance import mahalanobis %matplotlib inline sns.set_style(style="white") plt.rcParams['figure.figsize'] = 14, 12 Using Pandas we can easily load data from text files in various formats. In our case, the data is in csv format. Here, data is stored in rows with entries separated by columns. The first row is typically known as the header row and it contains labels for each of its corresponding columns. Pandas automatically reads these labels and lets you access the data by label when you load it. Let’s load the eagles_small.csv data into the main Pandas data structure known as a DataFrame. df = pd.read_csv('../Data/eagles_short.csv') #head() will print the first couple of lines of the DataFrame df.head() Unnamed: 0 timeofyear long lat species fulldate tag 0 455810 0.857534 21.654294 15.956181 Ciconia nigra 2015-11-09 13:50:09.000 2699 1 1314661 0.139726 90.367971 27.600898 Gyps himalayensis 2015-02-20 22:00:06.000 3932 2 764593 0.336986 87.284953 28.744269 Gyps himalayensis 2015-05-03 08:48:05.000 4003 3 1597617 0.418033 0.723650 42.564186 Aquila clanga 2016-06-01 11:40:19.000 15005 4 2239490 0.210959 -63.057830 -8.199500 Pandion haliaetus 2013-03-18 18:00:00.000 81057 ## Drawing the data on a world map Python has a very nice library called Basemap which is built on top of matplotlib and lets us do all sorts of geographical plotting. I am by no means an expert in this field so I am only introducing the most basic usage of the library. def draw_map(eagles): """ Draws a map of the world with points for each eagle measurment colored by species. """ #initialize a map eaglemap = Basemap(lat_0=0, lon_0=0) #draw the boundaries and coastlines eaglemap.drawmapboundary() eaglemap.drawcoastlines() #get the list of unique species names species = eagles.species.unique() #generate a color for each species colors = cm.gist_rainbow(np.linspace(0, 1, len(species))) #for each species, we will plot its locations on the map for i, s in enumerate(species): #extract a df of the species we want using the .loc operator from pandas #the arguments to loc are conditions for matches. in this case, we will extract any rows where the #value of 'species' matches the current species 's' we are plotting. species_df = eagles.loc[eagles['species'] == s] #convert the longitude and latitude from the DataFrame to map positions using the map object lo, la = eaglemap(species_df['long'], species_df['lat']) #we use the scatter function of our map object to plot each point. we assign a label, a point size, 's' #and a color from our list of colors eaglemap.scatter(lo, la, label=s, s=2, color=colors[i]) #we set a legend for our map. the frameon option draws a border around the legend. plt.legend(markerscale=2, frameon=True, loc="lower left", fontsize=12) plt.show() return eaglemap Now we can call the draw_map() function with our dataframe to draw the points on a world map. draw_map(df) <mpl_toolkits.basemap.Basemap at 0x111dd7cc0> ## Visualizing a trajectory in 3D To keep things simple, from now on we’ll work a single individual that has nice migration patterns, this is individual 208. The first thing we’re going to do is plot the trajectory through time which will result in a 3D plot with the dimensions being: longitude, latitude, time. The time dimension is normalized to be represented as a number between 0 and 1 where 0 is the first day of the year and 1 is the last. def plot3d_simple(df): #we get a figure object fig = plt.figure() #we add an Axes3D object to the plot ax = fig.add_subplot(111, projection='3d') #now we can do a scatter plot on the new axes. the c argument colors each point so let's color by time of year ax.scatter(df['timeofyear'], df['long'], df['lat'], c=df['timeofyear']) ax.set_xlabel('Time of year') ax.set_ylabel('Longitude') ax.set_zlabel('Latitude') plt.show() #you can play with the .loc function to plot different species and individuals d = df.loc[df['tag'] == '208'] plot3d_simple(d) ## Clustering We can see that this eagle has a very nice migration pattern. Towards the beginning of the year, the eagle is around -75 longitude and 38 latitude. Then around 0.5 of the year it flies north and returns to the same spot but further on in time. We can see that in longitude and latitude there are two concentrated areas which represent its two migration destinations. So let’s do something simple to pick these out and use an unsupervised learning technique known as K-means clustering. Since we know that we want two clusters, the K-means algorithm will identify two clusters of points which we can use as the ‘home bases’ for the eagle. Here’s a quick overview of how K-means clustering works: 1. Pick $k$ cluster centers $C_i$ at random 2. Assign each data point to the nearest center $C_i$. 3. For each $C_i$ compute a new $C_{i}^{‘}$ from the mean of all the points belonging to $C_i$ 4. Re-assign the points to the newly computed centers. 5. Repeat until convergence For more details, there is plenty of literature on this and other clustering algorithms with a quick google search. So now that we know how to cluster let’s try it on our dataset using scikit-learn’s KMeans implementation. def cluster(df, key, val): """ Takes a dataframe and computs clusters on longitude and latitude. Returns a tuple (df, centers) df with new column denoting the cluster ID of each point, and the cluster centers. """ #scikit takes as input a numpy matrix with numeric values so we need to convert our input dataframe accordingly #thankfully pandas has a nice function that lets us convert DataFrame columns to numpy matrices #get DF with the desired eagle(s) s = df.loc[df[key] == val] df_np = df.as_matrix(columns=['timeofyear', 'long', 'lat']) #now we can give this data to sickitlearn's KMeans() function. #this gives us an object that contains the clustering information such as where the centers are and which points #belong to which centers, i.e. labels kmeans = KMeans(n_clusters=2).fit(df_np) #store the labels in new DF column 'cluster' s['cluster'] = kmeans.labels_ return s #now we can cluster our 208 eagle df, centers = cluster(d, 'tag', '208') print(centers) [[ 0.47368786 -75.93400598 39.81164254] [ 0.52355137 -71.68675441 44.84591283]] To visualize the clustering let’s update our 3D plotting function to draw cluster centers and cluster labelling. def plot3d_cluster(df, labels=np.array([])): """ Now the function will accept two optional arguments: centers, and labels which we will use to visualize clustering. """ fig = plt.figure() ax = fig.add_subplot(111, projection='3d') if len(labels) > 0: ax.scatter(df['timeofyear'], df['long'], df['lat'], c=labels) else: ax.scatter(df['timeofyear'], df['long'], df['lat'], c=df['timeofyear']) ax.set_xlabel('Time of year') ax.set_ylabel('Longitude') ax.set_zlabel('Latitude') plt.show() plot3d_cluster(df, labels=df['cluster']) Okay so we see that K-means did a pretty nice job of separating what seem to be the two home bases. However, this clustering still doesn’t tell us which points are in migration and which are ‘stationary’. We need to find a way to separate those points around the middle that seem to be moving from those that are concentrated in one of the home bases. We need to find a way to separate these points away. ## Detecting migratory phases Let’s try something very naïve. Let’s assume that if eagles are at their home base, their position can be modelled with a Gaussian distribution where the center of the distribution is the cluster center we identified using K-means. Then, we compare the distance of each point to one of the two gaussian home bases and if an eagle is sufficiently far from either home base we say that it is migrating and ‘stationary’ otherwise. So if we want to model the position of an eagle at a homebase as a gaussian in 2D we will need two parameters: a mean vector ${\bf \mu }= (\mu_{timeofyear},\mu_{longitude}, \mu_{latitude})$ and a covariance matrix $\Sigma$. ${\bf \mu}$ we have from K-means, and $\Sigma$ we can estimate from the data as a matrix of sample variances. ### Mahalanobis Distance Now we need a measure of distance between a point and a distribution. For this, we use a metric known as the Mahalanobis Distance (Thank you to Vladimir Reinharz for the suggestion) which in general takes as input a vector to ${\bf x} = (x_1, x_2, .., x_n)^T$, distribution parameters ${\bf \mu} \in \mathbb{R}^n$, and $\Sigma \in \mathbb{R}^{nxn}$ and computes a distance $D_M({\bf x})$ as follows: Given the distribution parameters, we can apply this function to each of our points to get a measure of how far it is from the distribution center, or ‘home base’. Conveniently, the quantity $D_M$ is $\chi^2_{n}$ distributed so we can assign a probability to each point. def migration(df, key, val): #d is the dataframe containing the birds we want to work with d = df.loc[df[key] == val] #convert the data to a numpy matrix X = d.as_matrix(columns=['timeofyear', 'long', 'lat']) #do k-means clustering as before kmeans = KMeans(n_clusters=2).fit(X) #store labels and centers from clustering labels = kmeans.labels_ centers = kmeans.cluster_centers_ #print(centers) #as before, make new column with cluster ID for each point d['kmeans'] = labels #store parameters for gaussian models for each cluster in a dictionary gaussians = {} #get gaussian model parameters for i in range(len(centers)): #get datapoints belonging to a cluster dd = d.loc[d['kmeans'] == i][['timeofyear', 'long', 'lat']] #the mean vector is the same as the cluster center vector mu = centers[i] #compute the covariance matrix using the pandas function cov() that works on a DataFrame cov = dd.cov() #add the model parameters to the dictionary gaussians[i] = (mu, cov) #assign each point to migrating or not migrating, store 1 if migrating, 0 else mig = [] #go through each datapoint with the itertuples() function which gives us an iterator of the rows in the DF #the iterator yields a namedtuple object so we can access columns by name for i, eagle in enumerate(d.itertuples()): #get the keys of the gaussian dictionary models = gaussians.keys() #list to store the distance of each point to each cluster [dist_to_cluster_0, dist_to_cluster_1, ..] model_distances = [] for m in models: #get the model parameters for the current cluster model = gaussians[m] #position vector, will be input to mahalanobis distance pos_vec = np.array([eagle.timeofyear, eagle.long, eagle.lat]) #mean vector mu_vec = model[0] #inverse of covariance matrix which we convert to a numpy matrix S_inv = np.linalg.inv(model[1].as_matrix(columns=['timeofyear', 'long', 'lat'])) #scipy function for computing mahalanobis distance mala = mahalanobis(pos_vec, mu_vec, S_inv) #store the distance model_distances.append(mala) #check if distance above threshold of 1.5 which corresponds to a chi2 probability density of 0.68 #this means that points with scores greater than 2 to both clusters #will have a probability of 0.52 of belonging to #the cluster. we use a weak threshold because the data is fairly noisy. if model_distances[0] >= 2 and model_distances[1] >= 2: mig.append(1) else: mig.append(0) #store the migration status in a new column of the DataFrame d.loc[:,'migration'] = mig #make 3D plot using the migration status labels we just obtained plot3d_cluster(d, labels=d['migration']) return d mig = migration(df, 'tag', '208') Now you can see a nice separation between the points that appear to be in motion and the ones that are at their home base. And there you have it! Hope you enjoyed this notebook. Please feel free to send feedback :)	2018-04-21 05:59:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3496125638484955, "perplexity": 2207.6371518570577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945037.60/warc/CC-MAIN-20180421051736-20180421071736-00400.warc.gz"}
https://emeraldreverie.org/2022/06/07/matrix-adoption-so-far/	It’s been 9 months since the Ansible Steering Committee ratified the vote to adopt Matrix as part of the Ansible community ecosystem. As the main proponent of this, I promised regular updates, and while I’ve done two talks earlier this year (FOSDEM and Contributor Summit), I haven’t actually written anything for wider consumption. Time to rectify that. ## The State of the Chat I’ll start with the TL;DR; and those who want the supporting evidence can read further down: Matrix is promoting growth within the community, and is outpacing IRC as the choice of platform for users. Since this was an argument I made in my original pitch, I’m obviously happy to be right - but I’d better back it up, lest I be accused of cherry-picking! Let’s get to it! Note: All the data here concerns messages and reactions posted between Dec 1st 2021 and Jun 1st 2022. ## Unique Users per Day A natural place to start is our traffic levels, so let’s look at that: We can pick out few things here: - Chat in general is on a downward trend at the moment (but this may be seasonal). - Matrix is less down that IRC. - The huge spike on Matrix is Contributor Summit (Apr 12th). - no such spike occurs on the IRC side, suggesting new users are choosing Matrix. This gives us an idea of traffic levels, but it’s not a great view for comparison. Let’s do IRC & Matrix as a percentage of the total, for each day: Note these are mirror-images, every date adds up to 100%. You can see that Matrix occasionally hits 50% of the daily users now, but a value like 40% is probably more reasonable. Given that we have been cautious in our advertising of Matrix, that’s quite decent. ### The devil is in the detail There is, however, some nuance to this… While our documentation correctly lists both options for communication, I would guess there are plenty of places where IRC is still the only mention (no blame here, we have a lot of content!). In particular, the User Help room (#users:ansible.com / #ansible on IRC) is especially busy and especially IRC-centric. What happens if we differentiate between that room and all the rest? I kept only the Matrix fraction here, for brevity. You can see that while Matrix adoption in the Help room is low, the Matrix adoption in all the other rooms has been well over 50% for quite some time. I expect the slow adoption in the Help room is mostly down to the fact that culture moves more slowly in larger rooms - I expect it to catch up eventually, but it definitely confirms our caution in handling the rollout. As you can see, the data is quite volatile - we see drops every weekend (which we can account for), and seasonal variation too (which we can’t without more data). Dealing with what variation we can, it is possible to get an estimate of the changing trend for each platform: I’ve put these on a similar scale for Y (roughly 8 users/day) for ease of comparison. Two points here: - IRC has a similar shape to Matrix (both have declined recently) but it’s variation is 0.3; essentially, it’s a flatline. - Matrix, by comparison, has increased by 50% (12 -> 18) daily users over the last 6 months. So, when we look at our overall trend (the 3rd panel, on the right), all the growth in that curve comes from Matrix. And yes, most of the decline too, for what it is worth - but that is likely seasonal (we have seen similar dips earlier in the year that were quickly reversed after holidays, etc). ### Scenario modelling for trend One last game we can play with this data is to ask “what if?” - this is scenario forecasting. We build a model based on date vs number of unique users, and use percentage of users on Matrix as a predictor. We can then forecast the model, and we will look at 3 scenarios: • Flat: Status quo, Matrix remains at 40% • Rising: Wild increase, Matrix moves to 100% in the next 30 days • Drop: Stop using Matrix, drops to 0% in the next 30 days The red dots are the raw data, the dark-blue line is the model fit (just to show it’s a decent enough model), and then the non-seasonal component of each model is plotted on top (which is identical for the historical data, but then diverges as we move into the future). You can immediately see the model picking up on the recent dip - time-series models tend to treat recent data more strongly than older data, so this is expected… so don’t take the downward turn too badly! The big result, though, is looking at the three model lines, where you see a clear ranking - the higher the proportion of Matrix users, the higher the model forecasts the number of daily users. So again, we see support for my statement that Matrix is doing well - it’s growing, where IRC is not. ## Conclusion I’ve used users-per-day for this analysis, but the same results hold if you use messages-per-day. I suggest that adopting Matrix is making our community more accessible, and I think the trends we see here will continue. So, it’s not great leap that I also think we’d be wise to start basing more of our communcations policy on Matrix.	2022-06-28 09:51:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5749614834785461, "perplexity": 1469.5366798858515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103360935.27/warc/CC-MAIN-20220628081102-20220628111102-00657.warc.gz"}
https://forums.macrumors.com/threads/file-input-in-java.334146/	# MacFile Input in Java #### thefil ##### macrumors newbie Original poster I'm coming from a Windows environment and I have a bit of a file input problem... Code: ``glass = new Scanner( new File("\Users\filipkrynicki\Mydia 4.0\build\classes\mydia40\db.m4"));`` returns an error "illegal escape character" edit I fixed that problem by doubling the \\s, but now it can't find the file. Am I doing something wrong? #### itickings ##### macrumors 6502a My guess would be the use of \ instead of /. To avoid this kind of problem, consider using File.separator instead of assuming a specific separator.	2020-01-28 03:09:50	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.938301146030426, "perplexity": 4549.047125102868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251773463.72/warc/CC-MAIN-20200128030221-20200128060221-00001.warc.gz"}
http://www.wikihow.com/Play-Pool-Like-a-Mathematician	User Reviewed # wikiHow to Play Pool Like a Mathematician Billiard balls collide with nearly perfect elasticity. This means that the kinetic energy in their motion is almost completely preserved, and very little of it dissipates into heat or other energy sinks. This makes pool and billiards a great sport to analyze mathematically. If you have perfect control over how you strike the cue ball and where to aim it, you can always predict what will happen. ## Selected Takeaways • A ball that strikes a rail at angle X will bounce off it at angle X as well (if there is no spin). • If the cue ball and object ball are equidistant from a rail, you can strike the object ball by aiming at the point on the rail exactly between the two balls. • If the cue ball is X times as far from the rail as the object ball, imagine two perpendicular lines extending from the rail to the two balls. Aim for a point on the rail ${\displaystyle {\frac {X}{X+1}}}$ of the distance to the object ball's line. • Ghost ball method for angle shots: Draw a line from the pocket through the object ball. Imagine a ghost ball touching the object ball and sitting on this line. Aim for the center of the ghost ball. ### Part 1 Predicting the Angle a Ball will Bounce off a Rail 1. 1 Understand the law of reflection. Many pool players already know this simple mathematical lesson, since it comes up every time you carom the cue ball off a rail. This law tells you that the angle at which the ball strikes the rail is equal to the angle the ball bounces off at. In other words, if the ball approaches the rail at a 30º angle, it will bounce off at a 30º angle as well. • The law of reflection originally refers to the behavior of light. It's usually written "the angle of incidence is equal to the angle of reflection."[1] 2. 2 Set up the cue ball and object ball equidistant from the rail. In this scenario, the goal is to carom the cue ball off the rail, and have it return to strike the object ball. Now set up a basic geometry problem as possible: • Imagine a line from the cue ball to the rail, intersecting at right angles. • Now imagine the cue ball traveling to the rail. This path is the hypotenuse of a right triangle, formed by your first line and a section of the rail. • Now picture the cue ball bouncing off and hitting the object ball. Mentally draw a second right triangle pointing the opposite direction. 3. 3 Prove the two triangles are congruent. In this case, we can use the "Angle Angle Side" rule. If both triangles have two equal angles and one equal side (in the same configuration), the two triangles are congruent.[2] (In other words, they are the same shape and size). We can prove that these triangles meet these conditions: • The law of reflection tells us that the two angles between the hypotenuses and the rail are equal. • Both are right triangles, so they each have two 90º angles. • Since the two balls started equidistant from the rail, we know the two sides between the ball and the rail are equal. 4. 4 Aim at the midpoint of the rail section. Since the two triangles are congruent, the two sides that lie along the rail are also equal to each other. This means the point where the cue ball strikes the rail is equidistant from the two starting positions of the ball. Aim for this midpoint whenever the two balls are an equal distant from the rail. 5. 5 Use similar triangles if the balls are not equidistant from the rail. Let's say the cue ball is twice as far from the rail as the object ball. You can still picture two right triangles formed by the cue ball's ideal path, and use intuitive geometry to guide your aim:[3] • The two triangles still share the same angles, but not the same lengths. This makes them similar triangles: same shape, different sizes. • Since the cue ball is twice as far from the rail, the first triangle is twice as large as the second triangle. • This means the first triangle's "rail side" is twice as long as the second triangle's "rail side." • Aim for a point on the rail ⅔ of the way to the object ball, since ⅔ is twice as long as ⅓. ### Part 2 Calculating the Angle to Strike an Object Ball 1. 1 Learn the basics. Most shots in pocket billiards are angle shots or "cuts," meaning the cue ball does not strike the object ball dead on. The "thinner" (more glancing) the collision is, the greater the angle the object ball will travel at, relative to the trajectory of the cue ball. 2. 2 Estimate the fullness of the hit. An excellent way to estimate this effect is to sight along the planned trajectory of the ball. At the moment of collision, how much will the cue ball "overlap" the object ball from your perspective? The answer tells you how "full" the collision is: • A dead-on shots overlaps completely. You could say it has a "fullness" of 1. • If the cue ball covers ¾ of the object ball, the hit is ¾ full. 3. 3 Predict the angle based on the fullness. The graph of these two quantities is not quite linear, but it's close enough that you can estimate by adding 15º every time you subtract ¼ fullness. Alternatively, use these more accurate measurements:[4][5] • A direct hit (fullness 1) results in a cut angle of 0º. The object ball continues along the same path as the cue ball. • A ¾ shot sends the object ball out at 14.5º. • A ½ shot sends the object ball out at 30º. • A ¼ shot sends the object ball out at 48.6º. 4. 4 Use caution for very thin shots. Past ¼ fullness, it becomes difficult even to estimate how much of the ball is covered. More importantly, the cut angle rises more and more steeply, so tiny errors can have large effects. These glancing shots require plenty of practice and good technique even once you've figured out where to aim. If you can, look for another shot you can take. 5. 5 Aim with the ghost ball method instead. If the description of fullness doesn't help you, try the "ghost ball" approach:[6] • Imagine a straight line segment from the pocket to the center of the object ball. • Extend this line slightly past the object ball. Imagine a "ghost ball" at this spot, squarely on this line and touching the object ball. • To hit the object ball into the pocket, you should aim at the center of the "ghost ball." 6. 6 Follow the thirds rule for kiss shots. A kiss shot involves caroming the cue ball off ball A so it can strike ball B. If you're playing a game that allows kiss shots, remember this rule: if ball A is touching a rail, the desired cut angle is ⅓ of the angle formed by the three balls.[7] • For example, if the angle with ball A as the vertex is about 45º, the cut angle you want to achieve is about 15º. The fullness rule above tells us that a ¾ full collision should produce this angle. ### Part 3 Using English (Side Spin) 1. 1 Perfect your stroke first. Consistent stroke form and aim should be your first priorities when you start to take pool seriously. English is a very useful technique, but it has complex effects and you need consistency to practice it. • You'll have trouble narrowing down the effects of English (side spin) if you're not also controlling the amount of overspin and slipping. These effects are determined by the height you strike on the cue ball. Slipping is completely eliminated at 2/5 of the distance between the center and the top of the ball, but in practical terms 1/5 of this distance is often a better measure for optimal control and speed.[8][9] 2. 2 Avoid English when in danger of sinking the cue ball. As long as there is no English, the cue ball will come to a dead stop after a perfect head-on collision. Practice head-on collisions striking the ball with your cue at the midpoint of its horizontal axis. Once you can get the cue ball to stop dead every time, you have enough control to introduce English to your game. 3. 3 Practice different amounts of English. There are several types of English, but this article sticks to the most basic form. If your cue hits the cue ball left of center, the ball will spin along this axis — this is "left English." When this spinning ball strikes a surface, the spin will cause it to rebound further to the left than a ball with no English.[10] Similarly, striking the right side imparts "right English" and moves rebounds further to the right. The further from center you are, the more dramatic this effect:[11] • 100% English or maximum English means you strike halfway between the center and the edge of the ball. This is the farthest from the center you can strike and reliably avoid miscues. • 50% English means you strike halfway between the maximum point and the center (¼ of the way from the center to the edge of the ball). • You can use any other percentage of English by striking at different points between the center and the maximum point. 4. 4 Understand gearing. When two balls collide, the object ball starts rotating around a particular axis, determined by the angle and the amount of English. If you achieve "gearing," this rotation occurs along the axis of movement. In other words, the object ball's motion is not affected by spin. It will travel exactly along the "line of centers," or the line drawn between the centers of the two balls at the moment of impact.[12] • The term comes from the analogy of two gears meshing smoothly together, transferring the motion perfectly. 5. 5 Adjust your English to achieve gearing for any cut. Once you've aimed your angle shot using the fullness or "ghost ball" approaches from the last section, you'd like to ensure that the object ball doesn't pick up any funny spin and ruin your shot. Here's where a chart can save you a lot of trial and error. All numbers below are for "outside English," meaning you move the cue to the side of the cue ball farther from the object ball.[13][14] • If the cut angle is 15º, use slightly more than 20% English. (Remember, the cut angle is the angle between the cue ball's original path and the path of the object ball.) • If the cut angle is 30º, use 40% English. • If the cut angle is 45º, use about 55% English. • If the cut angle is 60º, use about 70% English. • As the cut angle approaches 90º, increase English to 80%. 6. 6 Know the effects of a collision without gearing. If you use less English than the "gearing" amount listed in the last step, the cue ball will slide forward during the collision, transferring side spin to the object ball. The object ball will move slightly to the right of the expected cut angle. If you use more English than the gearing amount, the object ball will move slightly to the left of the expected cut angle instead.[15] • This effect is called cut induced throw: the cut angle transferred a spin which threw the ball off the expected path. • You can use this to your advantage to make seemingly impossible shots. If your only clear shot would put the ball slightly too far to the right, increase the amount of outside English to throw the ball into the pocket. ## Community Q&A 200 characters left ## Tips • If the object ball is frozen to the rail and you need to slide it along the rail into a pocket, always strike the rail before the object ball. This way, the cue ball imparts momentum along the rail, instead of into it. (If the angle of collision is over 45º, you'll also need to use English.)[16] • The greater the angle of collision between two balls, the less momentum is transferred. This means you'll need a slightly stronger stroke for thin cuts (collisions at an extreme angle).[17] • After a collision, the angle between the cue ball's path and object ball's path will always equal 90º.[18] Use this knowledge to avoid sinking the cue ball. Note that extreme spin can break this rule, as can balls with unequal mass (as found on some coin-operated tables). ## Warnings • People and cue balls collide with spectacular inelasticity. Leave that experiment to the professionals. ## Article Info Featured Article Categories: Featured Articles \| Cue Sports In other languages: Español: jugar billar como un matemático, Português: Jogar Bilhar como um Matemático Thanks to all authors for creating a page that has been read 149,775 times.	2017-04-25 10:46:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4839401841163635, "perplexity": 1038.6652424159286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120338.97/warc/CC-MAIN-20170423031200-00291-ip-10-145-167-34.ec2.internal.warc.gz"}
http://en.wikibooks.org/wiki/Organic_Chemistry/Glossary	# Organic Chemistry/Glossary > Glossary ## A • Acetal - A molecule with two single bonded oxygens attached to the same carbon atom. • Acetyl - A functional group with chemical formula -COCH3. • Achiral - A group containing atleast two identical substituents. • Acid anhydride - Hydrocarbon containing two carbonyl groups.Acyl group attached with carboxylate group.eg- RCOOCOR' • Acid halide - Acyl group with any halogen attached with carbon of carbonyl group.eg.- RCO-X(X=F,Cl,Br,I). • Acidity constant Ka - • Activating group - Any group which activate any molecule by increasing positive or negative charge on carbon atom.Mainly towards neucleophilic or electrophilic substitution reactions. • Activation energy - The energy required to reactants to cross energy barrier to undergo any chemical change.denoted by Ea. • Acyl group - A group having alkyl or aryl group with a carbonyl group RCO- • Adam's catalyst - A catalyst for hydrogenation and hydrogenolysis in organic synthesis. Also known as platinum dioxide • Addition reaction - A reaction where a product is created from the coming together of 2 reactants. • Alcohol - A saturated hydrocarbon chain with an -OH functional group. • Aldehyde - A hydrocarbon containing atleast one carbonyl gp having one hydrogen attached to it.(>C=O) • Aldol reaction - When two similar aldehydes are reacted with each other,a product having both aldehyde(>C=O) and alcohol() group is formed.This reaction is called aldol reaction. • Aliphatic - A non-cyclic, non-aromatic, hydrocarbon chain (e.g. alkanes, alkenes, and alkynes) • Alkane - A hydrocarbon with all the carbon-carbon bonds are single bonds. • Alkene - A hydrocarbon with at least one carbon-carbon bond is a double-bond. • Alkoxide ion - The conjugate base of an alcohol without the terminal H atom. For any alcohol R-OH, the corresponding alkoxide form is R-O-. • Alkyl - A hydrocarbon having formula CnH2n+1 • Alkylation - Addition of alkyl group in a compound. • Alkyne - An unsaturated hydrocarbon containog triple bond.and having general formula CnH2n-2 • Allyl - An alkene hydrocarbon group with the formula H2C=CH-CH2- • α Position - Carbon attached to a functional group is called α-carbon and the position is known as α position. • α-carbon - Carbon attached to a functional group is called α-carbon • Amide - A hydrocarbon containing amnine group attached to acyl group. eg.- RCONH2 • Amine - A simple hydrocarbon containing atleast one -NH2 group. • Amino Acid - A fundamental unit of polypeptides or proteins.having general formula-COOHRCHNH2.eg.- glysine,alanine etc. • Anti conformation - • Anti periplaner - • Anti stereochemistry - • Anti bonding molecular orbital - Molecular orbitals having higher energy than bonding molecular orbitals after combination of atomic orbitals.denoted by an astric over Sigma or pi notations. • Arene - Another name for an aromatic hydrocarbon. • Aromacity - A chemical property in which a conjugated ring of unsaturated bonds, lone pairs, or empty orbitals exhibit a stabilization stronger than would be expected by the stabilization of conjugation alone. • Atomic mass - Total no of nucleon i.e. no. of proton and no. of neutrons.It is denoted by A. • Atomic number - Total no. of protons is called atomic no. • Axial bond - The bond parellel or anti parellel to axial coordinate passing center of gravity. • Azide synthesis - Dutt-Wormall reaction in which a diazonium salt reacts with a sulfonamide first to a diazoaminosulfinate and then on hydrolysis the azide and a sulfinic acid. • Azo compound - A compound containing -N=N group. ## B • Benzoyl Group - The acyl of benzoic acid, with structure C6H5CO- • Benzyl Group - The radical or ion formed from the removal of one of the methyl hydrogens of toluene (methylbenzene). • Benzylic - • β position - • β-carbon - • Bicylcoalkane - A compound containing two cyclic rings. • Bimolecular reaction - A second order reaction where the concentration of two compounds determine the reaction rate. • Boat cyclohexane - A less-stable conformation of cyclohexane that somewhat resembles a boat. • Bond - The attractive forces that create a link between atoms. Bonds may be covalent or ionic. • Bond angle - The angle formed between three atoms across at least two bonds. • Bond length - The average distance between the centers of two atoms bonded together in any given molecule. • Bond strength - The degree to which each atom linked to a central atom contributes to the valency of this central atom. • Bonding molecular orbital - • Bromonium ion - ## C • Cahn-Ingold-Prelog priorities - A rule for assigning priorities to substituents off of carbon in a double-bond or in a chiral center. • Carbocation - • Carbonyl group - A functional group composed of a carbon atom double-bonded to an oxygen atom: C=O. • Carboxylation - A chemical reaction in which a carboxylic acid group is introduced in a substrate. • Carboxylic acid - An organic acid characterized by the presence of a carboxyl group. • Chain reaction - A sequence of reactions where a reactive product or by-product causes additional reactions to take place. • Chair cyclohexane - • Chiral - A term chiral used to describe an object that is non-superposable on its mirror image • Chiral center - A carbon atom bonded to four different groups • Chromatography - The process of separating compounds such as a dye into its constituents • Cis-trans isomers - • Claisen condensation reaction - • Claisen rearrangement reaction - • Concerted - • Configuration - the permanent geometry of a molecule that results from the spatial arrangement of its bonds. • Conformation - • Conformer - • Conjugate acid - • Conjugate base - • Conjugation - A system of atoms covalently bonded with alternating single and multiple (e.g. double) bonds (e.g., C=C-C=C-C). • Covalent bond - A form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms. • Cracking - The process whereby complex organic molecules such as heavy hydrocarbons are broken down into simpler molecules (e.g. light hydrocarbons) by the breaking of carbon-carbon bonds. • Cycloalkane - An alkane that has one or more rings of carbon atoms in the chemical structure of its molecule. ## D • Debye - • Decarboxylation - • Delocalization - The ability of electrons to spread out among pi bonds to provide stabilization to electronically unstable areas of a molecule. • Dextrorotatory - • Diastereomers - Two or more isomers of a molecule which are not enantiomers of one another. • 1,3 Diaxial interaction - The steric intereaction between two methyl or larger groups attached at the 1 and 3 cis positions of cyclohexanes. The cyclohexane is in a higher energy state in the ring flip conformation that results in both 1 and 3 positions being axial due to steric strain between the 2 groups. This strain does not exist when hydrogens are bonded at these positions. • Diels-Alder reaction - • Dienophile - • Dipolar - • Dipole moment - • Disulfide - • Downfield - A term used to describe the left direction on NMR charts. A peak to the left of another peak is described as being downfield from the peak. ## E • E geometry - • E1 reaction - • E2 reaction - • Eclipsed conformation - • Eclipsing strain - • Electron - An elementary subatomic particle that carries a negative electrical charge and occupies an electron shell outside the atomic nucleus. • Electron configuration - The arrangement of electrons in an atom or molecule • Electron-dot structure - • Electron shell - The orbit followed by electrons around an atomic nucleus. The atom has a number of shells and they are normally labelled K, L, M, N, O, P, and Q. • Electronegativity - The ability of an atom to attract electrons towards itself in a covalent bond. • Electrophile - Literally, electron lover. A positively or neutrally charged reagent that forms bonds by accepting electrons from a nucleophile. Elecrophiles are Lewis Acids. • Electrophilic aromatic substitution - • Elimination reaction - A reaction where atoms and/or functional groups are removed from a reactant. • Endergonic - In an endergonic process, work is done on the system, and ΔG0 > 0, so the process is nonspontaneous. An exergonic process is the opposite: ΔG0 < 0, so the process is spontaneous. • Endothermic - An endothermic reaction is a chemical reaction that absorbs heat, and is the opposite of an exothermic reaction. • Enol - An alkene with a hydroxyl group affixed to one of the carbon atoms composing the double bond. • Enolate ion - • Entgegen - German word meaning "opposite". Represented by E in the E/Z naming system of alkenes. • Enthalpy - • Entropy - • Equatorial bond - • Ester - An inorganic or organic acid in which at least one -OH (hydroxyl) group is replaced by an -O-alkyl (alkoxy) group. • Ether - An organic compound which contains an ether group — an oxygen atom connected to two (substituted) alkyl or aryl groups — of general formula R–O–R'. • Exergonic - • Exothermic - An exothermic reaction is a chemical reaction that releases heat, and is the opposite of an endothermic reaction. ## F • Fingerprint region - • First order reaction - A reaction whose rate is determined by the concentration of only one of its reactants leading to a reaction rate equation of $Rate = k[X]$ • Fischer projection - • Formal Charge - • Friedel-Crafts reaction - • Functional group - This is a specific group of atoms within a molecule that is responsible for the characteristic chemical reactions of that molecule. The same functional group will undergo the same or similar chemical reaction(s) regardless of the size of the molecule it is a part of. ## G • Geminal - • Gibbs free energy - • Gilman reagent - • Glycol - A chemical compound containing two hydroxyl groups (-OH groups). Also known as a Diol. • Glycolysis - The metabolic pathway that converts glucose, C6H12O6, into pyruvate, C3H5O3 • Grignard reagent - • Ground state - ## H • Halohydrin formation - • Hammond postulate - • Hemiacetal - • Hemiaminal - • Heterocycle - A cyclic molecule with more than 2 types of atoms as part of the ring. (e.g. Furan, a 5-membered ring with four carbons and one oxygen, or a Pyran, a 6-membered ring with five carbons and one oxygen) • HOMO - Acronym for Highest Occupied Molecular Orbital. • Homolytic cleavage - Where bond breaks leaving each atom with one of the bonding electrons, producing two radicals. • Hybrid orbital - • Hydration - A chemical reaction in which a hydroxyl group (OH-) and a hydrogen cation (an acidic proton) are added to the two carbon atoms bonded together in the carbon-carbon double bond which makes up an alkene functional group. • Hybride shift - • Hydroboration - A reaction adding BH3 or B2H6 or an alkylborane to an alkene to produce intermediate products consisting of 3 alkyl groups attached to a boron atom. This molecule is then used in other reactions, for example, to create an alcohol by reacting it with H2O2 in a basic solution. • Hydrocarbon - A molecule consisting of hydrogens and carbons. • Hydrogen bond - • Hydrogenation - Addition of a hydrogen atoms to an alkene or alkane to produce a saturated product. • Hydrophilic - literally, "water loving". In chemistry, these are molecules that are soluble in water. • Hydrophobic - literally, "water fearing". In chemistry, molecules that aren't soluble in water. • Hydroxylation - A chemical process that introduces one or more hydroxyl groups (-OH) into a compound (or radical) thereby oxidizing it. • Hyperconjugation - ## I • Imide - • Imine - • Infrared spectroscopy - • Intermediate - • Isomer - Compounds with the same molecular formula but different structural formulae. There are two main forms of isomerism: structural isomerism and stereoisomerism. • Isotope - The different types of atoms of the same chemical element, each having a different atomic mass (mass number). Isotopes of an element have nuclei with the same number of protons (the same atomic number) but different numbers of neutrons. • IUPAC - Acronym for International Union of Pure and Applied Chemistry. • IUPAC Nomenclature - The international standard set of rules for naming molecules. ## K • Kekulé structure - • Keto-enol tautomerism - • Ketone - The functional group characterized by a carbonyl group (O=C) linked to two other carbon atoms, or a chemical compound that contains a carbonyl group ## L • Leaving group - • Levorotatory - • Lewis acid - A reagent that accepts a pair of electrons form a covalent bond. (see also Lewis Acids and Bases) • Lewis base - A reagent that forms covalent bonds by donating a pair of electrons. (see also Lewis Acids and Bases) • Lewis structure - • Lindlar catalyst - • Line-bond structure - • Lone pair electrons - • LUMO - Acronym for Lowest Unoccupied Molecular Orbital ## M • Markovnikov's rule - States that "when an unsymmetrical alkene reacts with a hydrogen halide to give an alkyl halide, the hydrogen adds to the carbon of the alkene that has the greater number of hydrogen substituents, and the halogen to the carbon of the alkene with the fewer number of hydrogen substituents." • Mass number - The total number of protons and neutrons (together known as nucleons) in an atomic nucleus • Mass spectrometry - • Mechanism - • Meso compound - • Meta - • Methylene group - • Molality - A measure of the concentration of a solute in a solvent given by moles of solute per kg of solvent. • Molarity - A measure of the concentration, given by moles of solute per liter of solution (solute and solvent mixed). • Mole - A measure of a substance that is approximately Avogadro's Number (6.022×1023) of molecules of the substance. More simply, calculate the molecule's atomic mass and that many grams of the substance is a mole. • Molecule - • Monomer - A small molecule that may become chemically bonded to other monomers to form a polymer. ## N • Nitrile - Any organic compound which has a -C≡N functional group. • NMR - See Nuclear magnetic resonance. • Non-bonding electrons - • Normality - • Nuclear magnetic resonance - • Nucleophile - Literally, nucleus lover. A negatively or neutrally charged reagent that forms a bond with an electrophile by dontating both bonding electrons. Nucleophiles are Lewis Bases. • Nucleophilic aromatic substitution reaction - • Nucleophilicity - ## O • Optical isomer - • Optical activity - • Orbital - • Ortho - • Oxidation - • Oxime - • Oxymercuration reduction reaction - ## P • Para - • Pauli exclusion principle - • Pericyclic reaction - • Periplanar - • Peroxide - • Peroxyacid - • Phenol - A toxic, colourless crystalline solid with the chemical formula C6H5OH and whose structure is that of a hydroxyl group (-OH) bonded to a phenyl ring. It is also known as carbolic acid, • Phenyl - A functional group with the formula -C6H5 • Pi bond - • Polar aprotic solvent - • Polar covalent bond - • Polar protic solvent - • Polar reaction - • Polarity - • Polarizability - • Polymer - A large molecule (macromolecule) composed of repeating structural units (monomers) typically connected by covalent chemical bonds. • Primary - • Prochiral - • Prochirality center - • Protic solvent - ## R • R group - • R,S convention - • Racemic mixture - • Rate constant - • Rate equation - • Rate-limiting step - • re face - • Reducation - • Regiochemistry - • Regioselectivity - • Resonance form - • Resonance hybrid - • Ring-flip - ## S • Saponification - The hydrolysis of an ester under basic conditions to form an alcohol and the salt of a carboxylic acid. • Saturated - A situation in which a compound has no double or triple bonds. Saturated can refer to the maximum amount of a solute being dissolved in a solution. Whether the context is chemical bonding or solutions will determine which meaning is appropriate. • Second order reaction - A reaction whose rate is dependent on the concentration of two reactants, leading to a reaction rate of $Rate = k[X][Y]$ • Secondary - • si face - • Side chain - • Sigma bond - • Simmons-Smith reaction - • SN1 reaction - • SN2 reaction - • Solvation - • Solvent - • sp orbital - • sp2 orbital - • sp3 orbital - • Spin-spin splitting - • Staggered conformation - • Stereochemistry - • Stereoisomer - • Steric hinderance - • Steric strain - • Substitution reaction - Reactions where one functional groups is replaced with another functional group. • Symmetry plane - • Syn periplanar - ## T • Tautomers - • Tertiary - • Thioester - • Thiol - A compound that contains the functional group composed of a sulfur atom and a hydrogen atom (-SH). • Thiolate ion - • Torisional strain - • Tosylate - • Transition state - • Twist-boat conformation - ## U • Ultraviolet spectroscopy - • Unsaturated - A situation in which a compound contains double or triple bonds. • Upfield - A term used to describe the right direction on NMR charts. A peak to the right of another peak is described as being upfield from the peak. ## V • Valence bond theory - • Valence electrons - • Valence shell - • Van der Waals forces - • Vicinal - • Vinyl - An organic compound that contains a vinyl group (also called ethenyl), −CH=CH2. • Vinylic - ## W, X, Y, Z • Zaitsev's rule - In elimination reactions, the major reaction product is the alkene with the more highly substituted double bond. This most-substituted alkene is also the most stable. • Zussamen - German word meaning "together". Represented by Z in the E/Z naming system of alkenes. Simple mnemonic, Z=Zame Zide (Same Side).	2014-10-30 19:01:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32055994868278503, "perplexity": 11144.738559245903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637898751.26/warc/CC-MAIN-20141030025818-00239-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.gmudatamining.com/lesson-12-lecture.html	In this lesson we learn about discriminant analysis, a popular classification algorithm for predicting response variables with two or more levels. We will also cover the K-Nearest Neighbor (KNN) algorithm, which can be used for regression or classification tasks. The KNN algorithm contains model hyperparameters - these are parameters of a model that cannot be estimated directly from our training data and must be estimated with a process known as hyperparameter tuning. The R tutorial in this lesson will introduce how to fit these models with tidymodels as well as how to perform hyperparameter tuning. # Next Steps Please head over to the R tutorial where you will learn how to fit discriminant analysis and KNN models with the tidymodels package.	2022-09-28 19:48:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40369969606399536, "perplexity": 819.8917977357916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335276.85/warc/CC-MAIN-20220928180732-20220928210732-00780.warc.gz"}
http://math.stackexchange.com/questions/506484/what-are-some-easily-stated-recently-proven-theorems	What are some easily-stated recently proven theorems? What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples: • The proof of Fermat's Last Theorem was completed in 1995, according to Wikipedia, by Wiles and others. • Green and Tao proved that for any $N$, there is an arithmetic sequence of primes of length at least $N$. • This year, Yitang Zhang made progress toward the twin prime conjecture - he found an integer $K$ such that there are infinitely many pairs of distinct primes that differ by less than $K$. I think that $K$ was in the neighborhood of 17 million or so but lower bounds were found within months. Sorry I don't have more specifics; see Yitang Zhang: Prime Gaps. • According to http://truthiscool.com/prime-numbers-the-271-year-old-puzzle-resolved, Helfgott has proved the weak Goldbach conjecture (any odd integer >5 is the sum of 3 primes (that is the wording from the article, I apologize if it is imprecise or wrong). The article states "Helfgott's preprint is endorsed and believed to be true by top mathematicians, Tao among them". The article is old (May 13, 2013) and I don't know if the result has been peer-reviewed and published in a journal. The conjecture is easy to state and if the proof is indeed valid it belongs on the list. Notice that all four theorems above are in number theory (the statements of the theorems, anyway. The proofs may have used stuff from other branches of mathematics, I don't know.) • Fairly recently, some young folks found a deterministic primality-testing algorithm that had polynomial computational complexity (in time). One might consider this more of a theoretical computer science result than a mathematics result. Again, sorry, I forgot the specifics. • I think Perelman's proof of the Poincare Conjecture almost qualifies. It is difficult to explain exactly what it means for a manifold to be orientable, even to most mathematicians, let alone laymen. - That result of Tao is actually a result of Green and Tao. – Isaac Solomon Sep 27 '13 at 2:24 It was Yitang Zhang who made the breakthrough on the twin primes problem. This sparked Tao's twin primes polymath project which improved on Zhang's result. – littleO Sep 27 '13 at 3:00 The weak Goldbach conjecture was apparently proved by H. Helfgott this year. – Jeppe Stig Nielsen Sep 27 '13 at 6:31 Dear Stefan, There is no reason to doubt Helfgott's result, but note that this result builds on the fundamental contributions of Vinagrodav from over 70 years ago; the difference with Helfgott's result is that Vinagradov proved ternary Goldbach for all odd numbers greater than an unspecified bound $N$, while Helfgott's work replaces $N$ by $5$. Regards, – Matt E Sep 27 '13 at 14:02 @StefanSmith: Dear Stefan, You might also be interested in Terry Tao's post where he explains his proof of a weaker result (sum of at most five primes, rather than three primes) but again, for all odd numbers. I think Helfgott's argument was inspired by Tao's. Regards, – Matt E Sep 27 '13 at 16:37 Catalan‘s conjecture (a.k.a. Mihăilescu’s theorem): http://en.wikipedia.org/wiki/Catalan%27s_conjecture - Well, I'm not sure if this has been confirmed yet, but apparently in March, Ciprian Manolescu claims to have refuted the Triangulation Conjecture in dimensions $\geq 5$. It's not the simplest result to state, but it's not terribly technical (unlike the proof, I imagine). The conjecture essentially states that "every compact topological manifold can be triangulated by a locally finite simplicial complex," in the language of the linked article. Put less rigorously, you can't necessarily take a nice, compact manifold of high dimension and cut it into triangles that fit together like puzzle pieces. - Thanks. I upvoted your answer. I looked at the link. The write-up was amusing and the result w as interesting. But any result with the word "homology" in the title can't fit on my list unless you are being very generous. – Stefan Smith Sep 28 '13 at 16:45 There are infinitely many pairs of primes $p_1 < p_2$ such that $$p_2 - p_1 < \text{(some specific large number)}.$$ - – lhf Sep 27 '13 at 3:14 Oh, I didn't even see that this was mentioned in the original question -- I just sort of skimmed it and saw something about FLT and Green-Tao. – Daniel McLaury Sep 27 '13 at 6:19	2015-05-26 08:35:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8796105980873108, "perplexity": 630.9246429009979}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928817.29/warc/CC-MAIN-20150521113208-00241-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.madrassah.mu/blog/index.php?entryid=1700	Blog entry by Andi Tabor CC3 Campaign Cartographer. 96. Click the Download button below. Watch the Video to the right. CC3 is now a fully supported product and has passed beta testing. Please click here to learn more about the available subscriptions.. Campaign Cartographer 3, the software which I've been using since version 3.3, is completely. on 9/1/2012. It's been a long journey since the very first version of. CC3 received an excellent review in RPG World (#3) for the. From the User's Manual: Model creation and city map creation. VBS Script, return if two letters are the same I'm very new to VBS scripting, and I am looking to create a script, which would check if 2 letters in an array are the same. if they are the same, it would return a value saying 'Error' (For example in this case a = 'Bm' and a = 'Bn' in this case the script would return 'Error') if a = 'Am' and a = 'An' the script would return 'Error' I've been trying to do this for a while now, and been searching the Internet but can't find how to do this. Any help would be much appreciated! A: You can loop through your array and then check: ' The variable Error refers to a global variable For Each Item In a If IsError(Error) Then Error = True ElseIf Item = Item.Substring(1) Then Error = True Else Error = False End If Next Or if you want to know only if the last two characters are equal, then you can: For Each Item In a If IsError(Error) Then Error = True ElseIf Item.Substring(0,2) = Item.Substring(0,2) Then Error = True Campaign Cartographer 3 Description. Please download the following files and place them in their respective folders:. /campaign_cartographer_3_rar_install (2.5 G).... 2.36 G FULL Version... 1.43 G EASE. You can use the settings to try a pilot version of the 2.3 version or the new 6. You do not need the installer to use the program... The text file is linked below with a torrent link.. I don't want the installer either, thanks. . maps and campaigns within. here for free.. cc3 1.7.3. 1154. campaign_cartographer_3_rar_install. 6.3.. I then went into CC3, saw the problem, and. Campaign Cartographer 3 is a fantastic tool that's hard to. Â . Oct 26, 2016. Campaign Cartographer 3 ($22) - the world's leading RPG map maker,. You'll have a view of the whole world. In CC3, the world is. In this Tutorial I'll show how to map your campaign. Below you can download a text file with 3 saves (this will give you.. The best EASY manner to erase Dungeondraft version 0. Oct 10, 2017. Campaign Cartographer 3 has a free trial that. Fight the Shadow, Diehard Gaming. Campaign Cartographer 3 (CC3) is a fantasy mapping software by. for the author and all backers, we offer a free copy of Campaign Cartographer 3 ($22). Cyberchase (reboot),. I think that is why there is no CC3 map for digital Dungeons and Dragons... I don't have much experience mapping maps for AdventureQuest or. What program do you use? This video contains original game footage and game data created by. Download CC3 at Â . Campaign Cartographer 3 is a fantasy map making software.. campaign_cartographer_3_rar_install: 6.3.. CC3 is a great tool for making. Campaign Cartographer 3: Battlezone Mania (CC3-BZ-M).	2022-11-27 09:15:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1846245378255844, "perplexity": 3920.5319763334023}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710218.49/warc/CC-MAIN-20221127073607-20221127103607-00462.warc.gz"}
https://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity?noredirect=1	# How exactly does curved space-time describe the force of gravity? I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or trampoline bending under an object, which causes other objects to come together, but I always thought this was a hopelessly circular explanation because the blanket only bends because of "real" gravity pulling the object down and then pulling the other objects down the sloped blanket. In other words, to me, it seems that curved space wouldn't have any actual effect on objects unless there's already another force present. So how is curved space-time itself actually capable of exerting a force (without some source of a fourth-dimensional force)? I apologize for my ignorance in advance, and a purely mathematical explanation will probably go over my head, but if it's required I'll do my best to understand. • In many "video" explanations of general relativity curvature of Time is omited, time is certainly not easy to graph with the blanket example, but sometimes it's not even mentioned, perhaps lack of self-questioning of the explainer, then it's a good question +1 – HDE May 16 '11 at 17:28 • I would modify this question as follows: If we could put a particle in orbit around a star with no other planets or satellites and then use a fictional device to cancel all the inertia of the particle, it is obvious that the curve of space-time is towards the star but what is not obvious is what would make the particle begin to move towards the star after all its momentum/inertia were canceled. Gravity is not a force so how would the particle 'know' that it needs to start accelerating towards the star? – Kelly S. French Sep 29 '15 at 20:39 • The blanket/trampoline isn't meant to explain anything in the sense of suggesting an underlying mechanism. It's a way of thinking about an esoteric subject far removed from ordinary experience in terms of something more familiar. "Vectors are like arrows" doesn't mean vectors are made of obsidian or fired from bows. In any case, the blanket/trampoline is entirely wrong as a model of curved space in general relativity, though it is a surprisingly accurate model of Newtonian gravity: see this answer. – benrg Feb 4 '19 at 8:37 • Nothing is as instructive as reading. Especially this: archive.org/details/TheClassicalTheoryOfFields – my2cts Apr 29 '19 at 18:27 Luboš's answer is of course perfectly correct. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve). Image a 2-sphere (a surface of a ball). If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. south poles). Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. E.g. for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. to the precession of the perihelion of the Mercury). So much for the explanation of how curved space-time (discussion above was just about space; if you introduce special relativity into the picture, you'll get also new effects of mixing of space and time as usual). But how does the space-time know it should be curved in the first place? Well, it's because it obeys Einstein's equations (why does it obey these equations is a separate question though). These equations describe precisely how matter affects space-time. They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. for a Sun you'll obtain that trampoline curvature and the planets (which will also produce little dents, catching moons, for example; but forget about those for a moment because they are not that important for the movement of the planet around the Sun) will follow straight lines, moving in ellipses (again, almost ellipses). • Thanks heaps guys, it's starting to make some sense. So that makes sense to me with moving objects, but I still don't quite understand how it causes objects to accelerate. For example, with your analogy, what if the ants were stationary on the ball? When we lift something off the ground and let go, it accelerates toward the earth. Is this just because that's the "straighest" line through the curved spacetime around the earth? Why must it always be "moving" through a straight line, and what does it mean in terms of curved spacetime for something to be stationary? – Zac Jan 16 '11 at 12:08 • Also @Zac, for something to be stationary in spaceTIME means that it only exists for a single instant in time! Even something that stays stationary in space for all time moves on a curve in spacetime. (think about what an x vs. t plot looks like for a stationary object) – wsc Jan 16 '11 at 18:33 • @wsc: it's really called perihelium in my language (Slovak) so I never imagined it might be something different in English. Anyway, thank you :) – Marek Jan 16 '11 at 19:34 • @Marek: nor did I realize it was different in Slovak; always nice to learn! Anyway, I was using that meaning of stationary since that was what @Zac was using: his question seemed to me to be: 'Sure you have geodesics on curved manifolds, but why do the ants have to move?' Which is a very good question, you just have to remember that time is a coordinate too. – wsc Jan 16 '11 at 20:07 • @AdamHughes: So the crucial difference is that spacetime includes curved time as well. No object can remain truly stationary in spacetime because that would require being stationary in time. An object that magically appears above Earth may begin stationary in the space dimension of spacetime, but it nevertheless continues to "move" through the time dimension. It must follow the geodesic (straightest/shortest possible path) from that point, and the geodesic from the point above Earth points through time but also towards Earth because of its effect on the curvature of spacetime. – Zac Aug 4 '17 at 16:06 There are actually two different parts of general relativity. They're often stated as 1. Spacetime tells matter how to move 2. Matter tells spacetime how to curve Point #1 is actually straightforward to explain: objects simply travel on the straightest possible paths through spacetime, called geodesics. The paths only seem curved because of the warping of spacetime. If you're a physicist, then you would want to know that that fact can be derived from the principle of extremal action (with all the requisite mathematical details), but if you don't want to wade through the math, hopefully it at least makes sense that objects move on "straight" lines. There is no actual force involved when a massive (or even a massless) object's trajectory curves in response to gravity, because it doesn't take any force to keep something moving on a straight line. (I can definitely expand on this point if you want) Now, I mentioned that spacetime needs to be warped in order for objects' trajectories to appear curved to us despite them actually being "straight." So the essence of point #2 is, why is spacetime warped in the first place? Physics doesn't have a good answer to that. Technically, we don't have an answer to point #1 either, but the "straight line" argument at least makes it seem plausible; unfortunately, there's no equivalent plausibility argument for why spacetime warps itself around matter. (Perhaps someday we will come up with one) All we can do right now is produce equations that describe how spacetime behaves around matter, namely the Einstein equations which can be written $G_{\mu\nu} = 8\pi T_{\mu\nu}$ among other ways. • I never got why Wheeler wasn't as famous as Feynman. He had that same magical way of reducing things down to really clear, simple statements that made complicated things seem obvious. – Jerry Schirmer Apr 17 '15 at 2:48 the trampoline analogy needs an extra source of gravity - because this is what the laymen, the recipients of the explanation, intuitively understand - but the real general relativity doesn't need any extra "external" gravity. Instead, general relativity says that the space is getting curved by Einstein's equations, $$G=T$$ where the left-hand side are numbers describing the curvature at a given point and the right hand side is the density of matter and momentum. I omit indices and constants haha. So general relativity says how the spacetime is curved under the influence of matter. The second part of the story is that general relativity also says how matter moves in external geometry. It moves along "geodesics", lines that are as straight as you can get. $$\delta S_{action\,ie\,proper\,length} = 0$$ This actually means that the objects move along the predicted, seemingly curved trajectories. These trajectories are actually as straight in the curved spacetime as you can get. Imagine that there is a hemisphere replacing a disk in the trampoline. So there exists a (nearly) straight line on the hemisphere - namely the equator near the junction with the rest of the trampoline. Note that the equator on the Earth is a maximum circle - so it is one of the straightest lines you can draw on the surface of Earth. The same is true for all actual trajectories that objects choose in spacetime of general relativity. So in the hemisphere-above-trampoline example, particles can orbit around the equator of the attached hemisphere, just like planets, because it is the straightest and most natural line they can choose. I don't use any external gravity to explain the real gravity; instead, I use the principle that particles choose the most natural - the straightest - line they can find in the curved spacetime. Best wishes Lubos The other answers are more or less correct, but perhaps I can say something more to the point of the question, *How is curved spacetime itself actually capable of exerting a force? No force whatsoever is needed. Gravity is not a force. What is a force, anyway? Newton clarified for almost the first time in Science what a force is: First I will say it, then explain it: A force is something which makes the motion of a body deviate from uniform straightline motion. Newton pointed out that bodies have a tendency, inertia, to continue in whatever direction they are already going, with whatever velocity they have at the moment. That means uniform, rectilineal motion: steady velocity, same direction. Newton actually knew this was what would be later called a geodesic, since « a straight line is the shortest distance between two points ». Newton then went on to say that to overcome inertia, to overcome this tendency, requires a force: force is what makes a body depart from the geodesic it is (even momentarily) headed on (its direction and speed). It was then Einstein (and partly Mach before him) who said this does not get to the essence of the question. For Einstein, any coordinate system had to be equally allowable, and in fact, space-time is curved (as already explained by other posters). A body or particle under the influence of gravity actually does travel in a geodesic....i.e., it does what a free particle does. I.e., it does what a particle not under the influence of any force does. So gravity is not a force. Newton did not realise that space-time could be curved and that then the geodesics would not appear to our sight to be straight lines when projected into space alone. That ellipse you see in pictures of planetary orbits? It is not really there of course since the planet only reaches different points of the ellipse at different times...that ellipse is not what the planet really traverses in space-time, it is the projection of the path of the planet onto a slice of space, it is really only the shadow of the true path of the planet, and seems much more curved than the true path really is. ( ¡ The curvature of space-time in the neighbourhood of the earth is really very small ! The path of the earth in space-time would even appear to be nearly straight to an imaginary Euclidean observer who, in a flat five-dimensional space larger than ours, was looking down on us in our slightly curved four dimensional space-time embedded in their world. It's $$ct$$, remember, so the curving around the ellipse gets distributed over an entire light-year, and appears to be nearly straight...and is straight when one takes into account the slight curvature of space-time.) Since every particle under the influence of gravity alone moves in a geodesic, it does not experience any force that would make it depart from its inertia and make it depart from this geodesic. So gravity is not a force, but electric forces still do exist. They could overcome the inertia of a charged body and make it deviate from the geodesic it is headed on: change its speed and direction (when speed and direction are measured in curved space-time). Einstein (and me too) did not want to change the definition of force in this new situation, since after all electric forces are known to exist and are still forces in GR. So the old notion of force still retains its usefulness for things other than gravity. To repeat: if a body is not moving in a geodesic in space-time, you go looking for a force that is overcoming its inertia....but since gravity and curvature of space-time do not make a body depart from a geodesic, neither of them is a force. See also http://www.einstein-online.info/elementary/generalRT/GeomGravity.html which avoids the trampoline fallacy and has a great image of the great circle. • Gravity is not a force in GR. Gravity was a force in classical mechanics. Gravity is ________ in quantum theories (Sorry, I don't know enough to fill in the blank.) My point is that all of these realms are models that predict the motion of terrestrial and astronomical objects. Some models (e.g., GR) make better predictions than others (e.g., classical), but do any of them tell us what gravity really is? – Solomon Slow Sep 21 '15 at 20:14 • @james large, the answer is no. There isn't a complete theory of gravity. No one knows what causes. – Ernesto Melo Jun 13 '17 at 16:26 As others mentioned, the main problem with the common visualization is, that it omits the time dimension. In the animation linked below the time-dimension is included to explain how General Relativity differs form Newton's model. It is straightforward to see how the geometry of spacetime describes the force of gravity -- you just need to understand the geodesic equation, which in general relativity describes the paths of things subject to gravity and nothing else. This is the "spacetime affects matter" side of the theory. To understand why curvature in particular, as a property of the geometry, is important, you need to understand the "matter affects spacetime" side of general relativity. The postulate is that the Gravitational Lagrangian of the theory is equal to the scalar curvature -- this is called the "Einstein-Hilbert Action" -- $$S=\int{\left( {\lambda R + {{\mathcal{L}}_M}} \right)\sqrt { - g}\, d{x^4}} {\text{ }}$$ You set the variation in the action to zero, as with any classical theory, and solve for the equations of motion. The conventional way to do this goes something like this -- $$\int{\left( {\frac{{\delta \left( {\left( {{{\mathcal{L}}_M} + \lambda R} \right)\sqrt { - g} } \right)}}{{\delta {g_{\mu \nu }}}}} \right)\delta {g_{\mu \nu }}\,d{x^4}} = 0$$ $$\sqrt { - g} \frac{{\delta {{\mathcal{L}}_M}}}{{\delta {g_{\mu \nu }}}} + \lambda \sqrt { - g} \frac{{\delta R}}{{\delta {g_{\mu \nu }}}} + \left( {{{\mathcal{L}}_M} + \lambda R} \right)\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} = 0$$ $$\frac{{\delta R}}{{\delta {g_{\mu \nu }}}} + \frac{R}{{\sqrt { - g} }}\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} = - \frac{1}{\lambda }\left( {\frac{1}{{\sqrt { - g} }}{{\mathcal{L}}_M}\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} + \frac{{\delta {{\mathcal{L}}_M}}}{{\delta {g_{\mu \nu }}}}} \right)$$ $${R_{\mu \nu }} - \frac{1}{2}R{g_{\mu \nu }} = \frac{1}{{2\lambda }}{T_{\mu \nu }}$$ To fix the value of $\kappa=1/{2\lambda}$, we impose Newtonian gravity at low energies, for which we only consider the time-time component, which Newtonian gravity describes (I'll use $C$ for the gravitational constant, reserving $G$ for the trace of the Einstein tensor) -- $$\begin{gathered} {G_{00}} = \kappa c^4\rho \\ {R_{00}} = {G_{00}} - \frac{1}{2}Gg_{00} \\ \Rightarrow {R_{00}} \approx \kappa \left( {c^4\rho - \frac{1}{2}\frac{1}{{c^2}}c^4\rho c^2} \right) \approx \frac{1}{2}\kappa c^4\rho \\ \end{gathered}$$ Imposing Poisson's law from Newtonian gravity with $\partial^2\Phi$ approximating $\Gamma _{00,\alpha }^\alpha$, $$4\pi C\rho \approx {\nabla ^2}\Phi \approx \Gamma _{00,\alpha }^\alpha \approx {R_{00}} \approx \frac{\kappa }{2}c^4\rho \\ \Rightarrow \kappa = \frac{{8\pi G}}{{c^4}} \\$$ (The fact that this is possible is fantastic -- it means that simply postulating that spacetime is curved in a certain sense produces a force that agrees with our observations regarding gravity at low energies.) Giving us the Einstein-Field Equation, $${G_{\mu \nu }} = \frac{{8\pi G}}{{c^4}}{T_{\mu \nu }}$$ • This is not an explanation in "layman's terms"... – Comp_Warrior Jun 22 '13 at 1:20 • I just think the average person interested in OP's question would not have knowledge of Lagrangians, Tensors etc. – Comp_Warrior Jun 22 '13 at 13:13 • @Comp_Warrior, the About says the site is for Academics, students, and researchers of physics and astronomy, so the average audience should not consist of laypeople and it is perfectly ok to give technical and advanced answers for the people who can stomach it. Even though it looks like this since quite some time, physics se is not meant to be a popular physics forum such as quora for example ... – Dilaton Jun 22 '13 at 14:46 • Btw the op says he does not mind technical answers, so why are you insisting on answers exclusively in layman terms? – Dilaton Jun 22 '13 at 15:00 • @Comp_Warrier what dimension10 says, and the SE system is exactly desined that the op can accept the answer he likes best, maybe a popular one, whereas there can legitimately be other more technical answers too, that are liked by other members of the community. Answers to a question are not only meant to serve the op, but the whole community. So there is absolutely nothing wrong with a question getting answers of different level. It would be nice if you stop discouraging good technical posts which are perfectly legitimate. – Dilaton Jun 22 '13 at 22:20 I think the problem for the layman is understanding why there is motion in spacetime and I think a sort of answer is that we already accept motion through time when we think of time and space as separate. Well we are in motion through spacetime where time and space are not separable and when we move through a region of spacetime that contains matter the shortest spacetime path between two events is the one that includes motion through the space bit as well as the time bit (ie not orthogonal to the space axes). That is experienced as falling under gravity. A complete replacement of the brief answer I wrote some time ago: More than one person has brought up the idea of a pair of ants walking on the surface of a sphere. Each ant is moving in what, for it, is a straight line, but the get closer together at an increasing rate until they collide. (Provided they're lined up right.) This is an excellent metaphor, but it can be confusing because each ant is propelling itself, so it could stop if it wanted, and also they do have to be lined up right when they start or they won't collide. If you hold a rock still and then let go, it starts to move, which seems different from the ant picture. All of these problems disappear if you realize that they don't call it spacetime for nothing. The surface of the balloon is two-dimensional in the ants-on-a-balloon analogy (and really the ants ought to be two-dimensional themselves, living embedded in the surface of the balloon just as we are embedded in spacetime). But it's wrong to think we are only throwing away one dimension to be able to visualize curved space. The right way to think of the balloon is that it has one dimension of space and one of time, so we're really throwing away two out of the four dimensions. Each ant is racing headlong into its own future, and it can't stop or even slow down. And the ants can't miss each other, because the paths they follow are really the histories of their lives. The paths are called world lines. Each point on a world line is a time and a place that the ant passed through. If two world lines cross, that means two ants were at the same place at the same time. This is still confusing, because the balloon is round. Which direction is time, and which direction is space? What happens when the ant goes all the way around the sphere? To make sense of these questions, you have to put a coordinate system on the sphere. For this toy universe, it actually makes sense to use latitude and longitude as coordinates. The south pole is kind of a big bang (take this with a lot of salt) and the north pole is the big crunch in the future (that definitely isn't going to happen in real life). The lines of latitude are the time coordinate, which means time progresses along the lines of longitude. A question was marked as a duplicate of a duplicate of this question, so I am posting my answer here. Gravity is due to the curvature of spacetime I believe it is true. That is what general relativity says, and general relativity has been confirmed in predictions ranging from the existence of black holes to the orbit of Mercury to the bending of light. Relation between spacetime, curvature, mass and gravity You say you are confused about how the curvature of spacetime and gravity are related. I am going to explain mainly that in my answer, starting with simpler examples, and moving to more complicated ones. Okay, let's say you have a sheet of rubber. This is the classic example of spacetime. Let's say you take a bowling ball, and set it on the taut sheet of rubber. It has a large mass (compared to what else we'll be putting on the sheet), therefore the sheet curves a lot for the bowling ball. We now have an image in our head like the one below: So mass leads to curvature. Then, take a baseball, say, and set it near the bowling ball. It rolls toward the bowling ball, right? This occurs because of the curvature of the sheet. So, then, curvature leads to gravity. So, if an object has large mass, it will curve spacetime dramatically, leading to strong gravity. This is, of course, an overly simplistic example. It is 2-d, and it doesn't take into account other factors. Let us move to 3-d (keeping in mind the universe is accepted to be at 4-d, ignoring the holographic principle). The mass of a bowling ball now sucks in space around it, sort of like in the picture below: And now, in this case, we can see (or understand) that more mass still leads to more curvature. The greater the mass, the more spacetime will "contract" around the object. So we still think that mass leads to curvature. Now, if we set an object near this massive object (like the moon next to Earth) it is "sucked in" sort of, by the curvature of spacetime, though of course the moon contracts spacetime around it as well. At this point, we can reasonably still conclude that in 3-d, mass leads to curvature which leads to gravity. But, as I said earlier, the universe is generally thought of as 4-d. What does our picture look like when we add time? Well, the time dimension is contracted around a massive object. So let us picture our previous example but that the fabric of spacetime has a few clocks embedded in it occasionally. As the space stretches and contracts, so will the clocks (the "time") and so the time on those clocks will be "wrong" - it'll differ from the other clocks. And in this case, as the Earth contracts space and time around it, it changes the time and space (it curves spacetime) and so when another object enters our region of spacetime, it is "sucked in" still, but so is it's time. This is, of course, a very extreme example, but I hope this shows that we can conclude that mass leads to curvature which leads to gravity. And a black hole, is simply so much mass that it leads to so much curvature that the gravity is so strong that light cannot escape. I hope this helps! • I like that you added the 3D image - the 2D one confuses many people : ) – BlackHoleSlice Feb 27 '20 at 20:56 • I see this explanation a lot, but I think it poses more questions than answers. The question that arises is why does a smaller ball fall into the pit? In a sheet analogy it is an xy-component of a sheet reaction on a z-force, but where does this z-force come from, as we're trying to explain it firsthand with this exact analogy? – user3125367 Jul 13 '20 at 3:40 • ... We can instead imagine that a ball on this sheet stands still, like it would do in zero-gravity setup. I think the issue with this explanation is that it doesn't take time into account, thus is self-referential. It is time-part of spacetime that is also bent, and through which our slow satellite ball "flies" at near the speed of light and that gradient makes it fall with time, even with no additional down-force analogies around. – user3125367 Jul 13 '20 at 3:40 What Einstein's equation tell us, at a basic level, is that the curvature of space-time and stress-energy are the same thing. In order for this law to be respected it is clear that the stress-energy of a test particle cannot be constant in a space-time with changing curvature. So, if you can choose a coordinate set in which the stress energy tensor is represented by the mass-energy of the particle, then the practical effect you can observe is changing energy and momenta of the test particle. When you therefore observe the test particle, you will see it as having changing energy and momenta, and therefore derive a force driving these changes. This is what we call gravity. However, general relativity gives a much deeper picture of gravity as a description of the curvature of space-time, so, in a way, gravity is an observed effect of the curvature of space-time, or, if you like, an observed effect of the distribution of mass and energy. • Part of this answer has been quoted in a new question. – Nat Sep 26 '18 at 10:01 Curvature affects motion by making the lines that are as straight as possible end up converging, just line how if you and your friends fly at constant altitude from the north pole, then no matter what directions you go (even if you and your friend head out in very different directions) then you start to converge on the south pole. This is a very good way to describe an effect that is determined by the path and not by the mass of the object taking the path. This is sometimes described as "spacetime tells matter how to move" but really this is just that the straightest possible lines converge when spacetime is curved the right way. But something not mentioned enough is that while mass, energy, momentum, stress, and pressure are sources of curvature, they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure. The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself). So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would. Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there. So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources. Having curved spacetime is something we observe. Having gravitational sources that can change the normal or usual way curvature evolves is something else entirely. We can make theories about how the sources evolve, and then the curvature is forced to co-evolve with it, and that's what gravity is about, about gravitational interactions (source and curvature together) changing how the curvature evolves changing the evolution that the curvature otherwise would have evolved a different way. So there is nothing circular, curvature is observed, and on its own it interacts and affects itself in a particular way (that is also observed), but gravitational sources get to change that and by interacting with the gravitational sources (which we can do) we can ourselves make the curvature change in different ways than it otherwise would! Here's a simple way to think about it: Newton's first law of motion says that in the absence of any force on a particle, the particle will move in a straight line. Hence, if we see a particle move in a curved path - that is of it deviates from a curved path - we can say that there is a force on it. Now, in GR, particles without any forces acting on it move on geodesics. This is the replacement for the notion of straight lines in a curved spacetime. Nevertheless we can detect the deviation from the usual notion of a straight line in flat space. This deviation will be correlated with the force of gravity as experienced by this object in its local frame.	2021-01-17 09:26:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6520724892616272, "perplexity": 442.7525851228464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703511903.11/warc/CC-MAIN-20210117081748-20210117111748-00176.warc.gz"}
https://dsp.stackexchange.com/questions/36786/analogue-filter-analysis-band-pass	# Analogue filter analysis (band-pass) The transfer-function: $$G(s) = \frac{\beta s}{s^2 + \beta s + \omega_0^2}$$ is to be used in an application that requires the magnitude of the frequency response to be of the band-pass form. • Show that this property is present by considering the cases when $\omega = 0, \omega = 1$ and show also that $\lvert G(j\omega_0)\rvert = 1$. • Determine also the equation that defines the frequencies where the filter magnitude is $−3\textrm{ dB}$. I use $G(s)=G(j\omega)\|_{s=j\omega}$, then $$\lvert G(j\omega_0)\rvert= \frac{j\beta \omega_0}{-\omega_0^2 + j\beta\omega_0 + \omega_0^2}=1$$ when $\omega=0$, the numerator equal to $0$ so $\omega=0$ is be filtered out. My questions are: • Could I use the $G(s)=\lvert G(j\omega)\rvert$ directly or $G(s)=G(j\omega)=\lvert G(j\omega)\rvert e^{-jk\omega}$? • I have no idea about the case when $\omega=+\infty$? • The last question, I use $-20\log\lvert G(j\omega)\rvert=-3\textrm{ dB}$, so we can solve the $\omega$ from this equation, is that correct? • Please add the homework tag. Where are you stuck ? – Gilles Jan 8 '17 at 12:03 • this is not a homework, should i add the tag? – Haoming Li Jan 8 '17 at 12:05 • Now we know the exercise that you need to solve, but we still don't know what your problem is with that exercise. – Matt L. Jan 8 '17 at 12:07 1. All questions refer to the filter's frequency response, which is simply obtained by evaluating $G(s)$ for $s=j\omega$. 2. The first question should read "Show that this property ... $\omega=0$, $\omega={\infty}$" (not $\omega=1$). 3. Your expression for $\|G(j\omega)\|$ is wrong; you wrote down $G(j\omega_0)$ (without magnitude). Of course, if $G(j\omega)=0$, or $G(j\omega)=1$, then also $\|G(j\omega)\|=0$ and $\|G(j\omega)\|=1$, respectively. 4. In order to see what happens for $\omega\rightarrow\infty$, write down $\|G(j\omega)\|^2$ (that's maybe easier than the magnitude), and take the limit. 5. For the 3 dB cut-off frequencies (there must be two positive solutions), use $\|G(j\omega)\|^2$ derived above and solve $\|G(j\omega_c)\|^2=\frac12$. Substitute $x=\omega^2$ and solve the resulting quadratic equation for $x$. From that solution, derive the two positive solutions for $\omega_c$.	2020-10-31 07:30:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9457468390464783, "perplexity": 411.0967720685889}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107916776.80/warc/CC-MAIN-20201031062721-20201031092721-00109.warc.gz"}
https://www.intel.com/content/www/us/en/docs/programmable/683283/18-1/enabling-multi-processor-compilation.html	ID 683283 Date 9/24/2018 Public ## 4.2.2. Enabling Multi-Processor Compilation The Compiler can detect and use multiple processors to reduce total compilation time. You specify the number of processors the Compiler uses. The Intel® Quartus® Prime software can use up to 16 processors to run algorithms in parallel. The Compiler uses parallel compilation by default. To reserve some processors for other tasks, specify a maximum number of processors that the software uses. This technique reduces the compilation time by up to 10% on systems with two processing cores, and by up to 20% on systems with four cores. When running timing analysis independently, two processors reduce the timing analysis time by an average of 10%. This reduction reaches an average of 15% when using four processors. The Intel® Quartus® Prime software does not necessarily use all the processors that you specify during a given compilation. Additionally, the software never uses more than the specified number of processors. This fact enables you to work on other tasks without slowing down your computer. The use of multiple processors does not affect the quality of the fit. For a given Fitter seed, and given Maximum processors allowed setting on a specific design, the fit is exactly the same and deterministic. This remains true, regardless of the target machine, and the number of available processors. Different Maximum processors allowed specifications produce different results of the same quality. The impact is similar to changing the Fitter seed setting. To enable multiprocessor compilation, follow these steps: 1. Open or create an Intel® Quartus® Prime project. 2. Click Assignments > Settings > Compilation Process Settings. 3. Under Parallel compilation, specify options for the number of processors the Compiler uses. 4. View detailed information about processor use in the Parallel Compilation report following compilation. To specify the number of processors for compilation at the command line, use the following Tcl command in your script: set_global_assignment -name NUM_PARALLEL_PROCESSORS <value> In this case, <value> is an integer from 1 to 16. If you want the Intel® Quartus® Prime software to detect the number of processors and use all the processors for the compilation, include the following Tcl command in your script: set_global_assignment -name NUM_PARALLEL_PROCESSORS ALL The actual reduction in compilation time when using incremental compilation partitions depends on your design and on the specific compilation settings. For example, compilations with multi-corner optimization enabled benefit more from using multiple processors than compilations without multi-corner optimization. The Fitter (quartus_fit) and the Intel® Quartus® Prime Timing Analyzer (quartus_sta) stages in the compilation can, in certain cases, benefit from the use of multiple processors. The Flow Elapsed Time report shows the average number of processors for these stages. The Parallel Compilation report shows a more detailed breakdown of processor usage. This report displays only if you enable parallel compilation. For designs with partitions, once you partition your design and enable partial compilation, the Intel® Quartus® Prime software can use different processors to compile those partitions simultaneously during Analysis & Synthesis. This can cause higher peak memory usage during Analysis & Synthesis. Note: The Compiler detects Intel® Hyper-Threading® Technology (Intel® HT Technology) as a single processor. If your system includes a single processor with Intel HT Technology, set the number of processors to one. Do not use the Intel® HT Technology for Intel® Quartus® Prime compilations. Did you find the information on this page useful? Characters remaining: Feedback Message	2023-04-01 02:36:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39505043625831604, "perplexity": 3404.180672313858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949694.55/warc/CC-MAIN-20230401001704-20230401031704-00505.warc.gz"}
https://stacks.math.columbia.edu/tag/032D	Lemma 10.160.11. Let $(R, \mathfrak m)$ be a Noetherian complete local domain. Then there exists a $R_0 \subset R$ with the following properties 1. $R_0$ is a regular complete local ring, 2. $R_0 \subset R$ is finite and induces an isomorphism on residue fields, 3. $R_0$ is either isomorphic to $k[[X_1, \ldots , X_ d]]$ where $k$ is a field or $\Lambda [[X_1, \ldots , X_ d]]$ where $\Lambda$ is a Cohen ring. Proof. Let $\Lambda$ be a coefficient ring of $R$. Since $R$ is a domain we see that either $\Lambda$ is a field or $\Lambda$ is a Cohen ring. Case I: $\Lambda = k$ is a field. Let $d = \dim (R)$. Choose $x_1, \ldots , x_ d \in \mathfrak m$ which generate an ideal of definition $I \subset R$. (See Section 10.60.) By Lemma 10.96.9 we see that $R$ is $I$-adically complete as well. Consider the map $R_0 = k[[X_1, \ldots , X_ d]] \to R$ which maps $X_ i$ to $x_ i$. Note that $R_0$ is complete with respect to the ideal $I_0 = (X_1, \ldots , X_ d)$, and that $R/I_0R \cong R/IR$ is finite over $k = R_0/I_0$ (because $\dim (R/I) = 0$, see Section 10.60.) Hence we conclude that $R_0 \to R$ is finite by Lemma 10.96.12. Since $\dim (R) = \dim (R_0)$ this implies that $R_0 \to R$ is injective (see Lemma 10.112.3), and the lemma is proved. Case II: $\Lambda$ is a Cohen ring. Let $d + 1 = \dim (R)$. Let $p > 0$ be the characteristic of the residue field $k$. As $R$ is a domain we see that $p$ is a nonzerodivisor in $R$. Hence $\dim (R/pR) = d$, see Lemma 10.60.13. Choose $x_1, \ldots , x_ d \in R$ which generate an ideal of definition in $R/pR$. Then $I = (p, x_1, \ldots , x_ d)$ is an ideal of definition of $R$. By Lemma 10.96.9 we see that $R$ is $I$-adically complete as well. Consider the map $R_0 = \Lambda [[X_1, \ldots , X_ d]] \to R$ which maps $X_ i$ to $x_ i$. Note that $R_0$ is complete with respect to the ideal $I_0 = (p, X_1, \ldots , X_ d)$, and that $R/I_0R \cong R/IR$ is finite over $k = R_0/I_0$ (because $\dim (R/I) = 0$, see Section 10.60.) Hence we conclude that $R_0 \to R$ is finite by Lemma 10.96.12. Since $\dim (R) = \dim (R_0)$ this implies that $R_0 \to R$ is injective (see Lemma 10.112.3), and the lemma is proved. $\square$ Post a comment Your email address will not be published. Required fields are marked. 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). Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 032D. Beware of the difference between the letter 'O' and the digit '0'.	2022-01-20 14:55:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9795234799385071, "perplexity": 165.0746989824559}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00252.warc.gz"}
https://or.stackexchange.com/questions/7754/confusion-between-different-types-of-optimization-problems	# Confusion Between Different Types of Optimization Problems I do not have a background in optimization and I am trying to teach myself more about this topic. I find myself having a lot of trouble understanding the different "types" of optimization problems that exist. For example, I understand the idea of optimizing continuous functions (e.g. $$y = x^2$$). For example, we could be interested in finding out the value of $$x$$ that results in the smallest value of $$y$$. I also understand that continuous functions can be optimized subject to some constraints. However, I find myself very confused when trying to sort through the following types of optimization problems: • Discrete Optimization • Integer Optimization • Mixed Integer Optimization • Combinatorial Optimization When I think of these problems, the first thing that comes to mind is that they are fundamentally different from optimizing continuous functions. For instance, the inputs of the above list of problems are usually "categorical" in nature. This is why I have heard that problems belong to the above list usually require "gradient free optimization methods" (e.g. evolutionary algorithms, branch and bound, etc.) , since it is impossible to take the derivatives of the objective functions corresponding to these problems. For example, if you take problems such as the "Traveling Salesman" or "Knapsack Problem" (note: I have heard that these problems belong on the above list, but I am not sure), I would visualize the objective function as something like this: This leads me to the following question: • Are 4 types of optimizations on the above list effectively the "same thing"? The way I see it, all 4 types of these problems have "discrete inputs" and in a mathematical sense, "integers" are always considered as "discrete". In all 4 types of problems, we are interested in finding out a "discrete combination" of inputs - i.e. "combinatorical". Thus, are 4 types of optimizations on the above list effectively the "same thing"? • I have heard the argument that "any optimization problem that can be formulated into a linear problem is always convex (because linear objective functions are always convex)". If we consider continuous optimization problems, we usually say that "convex optimization problems are easier than non-convex optimization problems" because non-convex functions can have "saddle points" that can result in the optimization algorithm getting stuck in these "saddle points". Using this logic, I have seen the objective function of the "Traveling Salesman Problem" being written as a linear function and thus the "Traveling Salesman Problem" being considered as a convex optimization problem. I have also heard the "Traveling Salesman Problem" is a very difficult problem to solve. If the "Traveling Salesman Problem" is convex and difficult to solve, does this imply that there are non-convex discrete/combinatorial problems that are even more difficult to solve? • I have heard the following argument: Discrete/Combinatorial Optimization Problems are more difficult to solve compared to Continuous Optimization Problems. This is apparently because discrete/combinatorial optimization problems involve "treating the problem as a continuous problem" to first come up with a solution, and then determine if the solution lies within the feasible region, thus effectively solving two optimization problems in one. Is this correct? • Finally, I have seen both the "Traveling Salesman" and the "Knapsack Problem" being formulated as a linear problem and therefore as convex. Are there any well known examples of non-convex discrete/combinatorial optimization problems? • I'm inclined to treat "discrete optimization" and "combinatorial" optimization as synonyms, but I'm not sure everyone does. • "Integer programing/optimization" is a specific approach to modeling and solving a discrete/combinatorial problem. It is not the only approach. For instance, constraint programming can be used productively to solve some combinatorial problems, and it is very different from integer programming both in terms of problem representation (modeling) and in terms of the algorithms used to solve the models. • "Mixed integer programming/optimization)" generalizes integer programming (slightly) by allowing non-integral real variables in the model. The most common algorithms for IPs make no distinction between IPs and MIPs. • When dealing with discrete optimization, the roles of linearity and convexity largely have to do with how easy (and perhaps how useful) it is to solve relaxations of the problem where the integrality restrictions are dropped. • Yes, nonconvex discrete problems can be bigger pains in the posterior to solve than discrete problems with convex relaxations are. Note that this is not guaranteed to be the case with every instance. I could probably conjure up a nonconvex discrete problem A and a convex discrete problem B where A was easier than B. Let's just say that when confronting a new problem I am always rooting (hard) for convexity and (fairly hard) for linearity. • Discrete problems tend to be harder than their continuous equivalents, in large part because the "move a small amount in this direction" logic of many continuous optimization algorithms is not applicable when the variables are discrete. A small step from an integer is not integer, hence not feasible. I would not, however, agree with characterization of solving the continuous relaxation to see if the solution is in the feasible region and calling that two problems. In branch and bound/branch and cut, for instance, you do solve the continuous relaxation, and assuming it is feasible with an objective value no worse than the current incumbent you do check whether the solution is integer-feasible ... but that check is trivial. (You just look at the values of the discrete variables and see if they are within rounding distance of integer.) The main reason for solving the relaxation, though, is not the hope of finding an integer solution but rather to get what you hope is a fairly tight bound for the objective value in that part of the solution space. ----If the "Traveling Salesman Problem" is convex and difficult to solve: TSP is not convex, it belongs to a class of hard problems, "NP-hard" to be more specific (for more details, see here: https://stackoverflow.com/a/49845003/9125267). ---- Discrete/Combinatorial Optimization Problems are more difficult to solve compared to Continuous Optimization Problems: I have to disagree with this statement since you can enforce the constraint x is binary just by adding the constraint x^2=x and leaving your formulation continuous. The real distinction should be convex vs non-convex. ----Are there any well known examples of non-convex discrete/combinatorial optimization problems?: Mixed-Integer Nonlinear Programs (MINLPs) are used to model non-convex discrete/continuous problems (they include mixed-integer linear and pure integer programs, see classification chart below). There are plenty of examples in the MINLP library: https://www.minlplib.org/applications.html. For a real-world example, have a look at the Grid Optimization competition: https://gocompetition.energy.gov/about-competition	2022-06-28 09:41:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.821094810962677, "perplexity": 460.40869376978543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103360935.27/warc/CC-MAIN-20220628081102-20220628111102-00692.warc.gz"}
https://stats.stackexchange.com/questions/430564/performing-and-interpreting-a-logistic-regression-using-ordered-variables-in-r	# Performing and interpreting a logistic regression using ordered variables in R I'm currently working on my first larger project with self-collected data and only few guidelines. My dataset contains 29 variables, all of which are categorical and most of which are ordered (with 2 to 6 levels). My aim is to perform several logistic regressions, and due to the ordered nature of (most of) my data I first stumbled across the polr function in the MASS package. However, I discovered that it can only be used if the dependent variable has at least 3 levels (is there a practical reason for this?), and so I hope to be able to use glm() in those cases where it has only 2. For now, I decided to work with a random sample (60% of my 4400 observations). For my first regression, the dependent variable has 2 levels (no/yes), while the five predictors I chose are all ordered and have 6, 3, 2 (no/yes), 4 and 5 levels respectively. Running summary(glm(P05 ~ P03 + P06 + P07 + P10 + P17, data = DataTrain, family = binomial("logit"))) produces the following output: Call: glm(formula = P05 ~ P03 + P06 + P07 + P10 + P17, family = binomial("logit"), data = DataTrain) Deviance Residuals: Min 1Q Median 3Q Max -3.2841 -0.1813 0.0906 0.3962 3.0173 Coefficients: Estimate Std. Error z value Pr(>\|z\|) (Intercept) 3.24548 71.40502 0.045 0.96375 P03.L 8.02192 256.03496 0.031 0.97501 P03.Q 4.18870 233.72680 0.018 0.98570 P03.C 3.59522 159.66645 0.023 0.98204 P03^4 2.17832 80.96710 0.027 0.97854 P03^5 1.30158 26.99216 0.048 0.96154 P06.L 3.15496 0.16009 19.707 < 2e-16 * P06.Q -1.21995 0.20944 -5.825 5.72e-09 * P07.L 0.31185 0.11018 2.830 0.00465 P10.L 0.98932 0.20832 4.749 2.04e-06 * P10.Q 0.26703 0.29000 0.921 0.35716 P10.C -0.08688 0.34696 -0.250 0.80227 P17.L 1.18209 0.38270 3.089 0.00201 ** P17.Q 0.14912 0.34263 0.435 0.66339 P17.C 0.23543 0.32997 0.713 0.47554 P17^4 -0.54607 0.26883 -2.031 0.04223 * --- Signif. codes: 0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 3197.3 on 2316 degrees of freedom Residual deviance: 1163.1 on 2301 degrees of freedom (310 observations deleted due to missingness) AIC: 1195.1 Number of Fisher Scoring iterations: 13 Three things about this are confusing me. First, the inclusion of the .Q, .C, ^4 and ^5 terms although I didn't specify this anywhere. In some tutorial I found (don't know if I'm allowed to link to other sites), they use one 4-level categorical (and two continuous) predictor, and instead of all those higher powers they receive three coefficients, one for level 2, 3 and 4 each. Second, they then proceed to use wald.test() to determine the overall significance of the categorial variable across all levels. While I can technically run the function using the different "power coefficients" I obtained instead of "level coefficients" like in the tutorial, I'm unsure if the interpretation remains similar, or if it even makes sense methodically. Third (I didn't want to work on this before the other two issues have been sorted out, so I haven't tried this myself yet), they use odd ratios as an intuitive way to interpret their results. I'm 99.9% certain that even if I were able to calculate them using my "power coefficients", this would be incorrect methodically.(?) So I assume the main problem are those unexpected "power coefficients". That's when I went through several other posts on here and found this one: Interpretation of .L & .Q output from a negative binomial GLM with categorical data The answer there says the "power coefficients" are a direct consequence of the variables being ordered, and while I COULD treat my variables as unordered, I wonder whether I'm actually allowed to do that, given that there IS indeed a natural order. If yes, does the interpretation of unordered variables have any less value? Is there any conclusion that I would be able to draw from ordered variables but not from unordered ones? Otherwise, what else am I supposed to do? I'd appreciate your help! Thanks!	2019-12-05 21:39:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7301321029663086, "perplexity": 1537.0618968977683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540482284.9/warc/CC-MAIN-20191205213531-20191206001531-00197.warc.gz"}
https://physics.stackexchange.com/questions/201461/is-there-any-reason-for-principle-of-least-action-to-be-true	# Is there any reason for principle of least action to be true? [duplicate] My question is not rigidly related to physics. The principle of least actions says that for any dynamical system there exists a function parameterized by $q$'s and $\dot{q}$'s such that the line integral of the function from state $A$ to state $B$ with respect to the parameter $s$ is stationary. Is there any rigorous mathematical proof of the principle? Can I apply the principle of least action to study the evolution of a dynamical system in phase Space?	2021-05-13 16:16:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9042878746986389, "perplexity": 91.84641336212776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989814.35/warc/CC-MAIN-20210513142421-20210513172421-00221.warc.gz"}
https://www.physicsforums.com/threads/crime-statistics-split.920996/	Crime Statistics - Split 1. Jul 22, 2017 Staff: Mentor Here is some data from the FBI, from 2013, with arrest percentages by race for a variety of crimes (https://ucr.fbi.gov/crime-in-the-u.s/2013/crime-in-the-u.s.-2013/tables/table-43). The data I've listed here comes from Table 43A. $\begin{array}{ccc} ~ & \text{White} & \text{Black/African American}\\ \text{Murder,nonnegligent manslaughter} & 45.3 & 52.2 \\ \text{Rape} & 45.3 & 52.2 \\ \text{Robbery} & 41.9 & 56.4 \\ \text{Aggravated assault} & 62.9 & 33.9 \\ \text{Burglary} & 67.5 & 30.4 \end{array}$ I did not include American Indians/Alaskan Natives or Asians, whose arrest percentages are almost all below 2% for all crimes listed, nor did I include Hawaiians and Pacific Islanders, whose arrest percentages are all 0.1% or below. Hispanics are not included by race - there is a separate section in this table for Hispanic vs. Non-hispanic. Focusing on black/African Americans, the arrest rates for violent crimes I listed above are disproportionately high for a group that makes up about 12% of the population. It is disingenuous, IMO, to say that a certain group is being incarcerated due to discriminatory policing. 2. Jul 22, 2017 UsableThought This is getting into race relations, about which a lot could be said. May I urge that the thread not get involved in a deep discussion of yet another absolutely enormous issue, especially given (a) it would grow contentious quite quickly as race is a hot topic, and (b) that it would be tangential to the OP's question? 3. Jul 22, 2017 Staff: Mentor My aim was not to steer the thread into a different direction, but only to rebut some earlier assertions that aren't borne out by the existing data. 4. Jul 22, 2017 StatGuy2000 We are getting off-topic, but let me respond accordingly. The FBI is reporting arrest percentages by race for a variety of crimes. What you fail to take into account is that decades of biased policing (based largely on the implicit bias that black people are inherently more likely to be violent, dangerous, or criminal) has led to black Americans being disproportionately being pulled over, questioned, and arrested for various crimes. Even if there isn't a deliberate attempt from police officers to be biased, there may be structural bias introduced by, say, patrolling differently in high crime neighbourhoods. See the following: http://www.slate.com/articles/healt..._20_states_suggests_evidence_of_racially.html http://www.rand.org/content/dam/rand/pubs/reprints/2011/RAND_RP1427.pdf All of these may well play into while you are seeing disproportionately high arrest rates for violent crimes for black/African Americans. Ditto for Hispanic Americans. 5. Jul 22, 2017 Staff: Mentor Nevertheless, anyone who watches the news or picks up a newspaper now and then can see the actual disparity in crime rates by counting the bodies. This should not be controversial. ...but it is off topic. I can split this to a different thread if people really want to discuss it more. I would also caution people against judging personal motives in statistics. Assume you are talking to someone reasonable until proven otherwise. 6. Jul 24, 2017 StatGuy2000 You are right that this is off topic and should belong in a different thread. That being said, your point about watching the news and picking the newspaper -- the TV or print news does not necessarily present a fully accurate picture of the true rate of crime due to the necessity to present the latest headlines to sell the news. That is a fact that is beyond dispute -- just look at how any mention of terrorism in the news today is almost invariably associated with Muslims, even though in the US today, right-wing extremist groups commit the most # of terrorist attacks or incidents (this is coming from a study commissioned by the Department of Homeland Security during the second Bush administration, so no liberal bias here). Here is a link to a PBS report summarizing the results. All of this has the (unintended) consequence of presenting a biased picture with respect to crime rates, and can often have a tendency to reinforce biases people have of various minority groups (e.g. African Americans, Hispanics, Muslims, etc.) Last edited: Jul 24, 2017 7. Jul 24, 2017 Staff: Mentor Agreed, but that was non-responsive to my statement. Please clarify: do you agree that blacks commit more murders in the USA than whites, per population, by several times (roughly a factor of 5)? Here's the data: https://www.usnews.com/news/articles/2016-09-29/race-and-homicide-in-america-by-the-numbers Please note, there is a difference between murder and drugs or other contraband found in a traffic stop: police can generate more of the latter by making more traffic stops, but they can't generate more of the former by making more dead bodies. So murder statistics are much less subject to such selection bias. This is true of most crimes not associated with traffic stops. Oy. Just awful. First off, please don't construe government funding of research with endorsement. If it were, we'd have NASA endorsing anti-gravity and reactionless propulsion (that's actually being researched directly by NASA, not just funded by them). Anyway, there are three major flaws in that research, two of which are provided for us right there in the title: 1. Notice the different terminology: "Deadly threat from far-right extremists" vs "Islamic terrorism". Why didn't it say "far-right extremist terrorism"? That isn't an accidental lack of clarity: it's an attempt to slip past you (successfully, since you misquoted it as "right-wing...terrorist attacks"!) the fact that they are comparing apples to oranges. They are comparing things like ordinary hate crimes and murders of police on one side to terrorism on the other. They aren't the same thing. The fact that they hide this bait-and switch in plain sight doesn't make it any less deceitful. 2. Why just far right? Doesn't the far left commit such crimes? (Since a large fraction are just run-of-the-mill hate crimes or murders of cops, yes, they do.) Indeed, if you look at the list of this group's published papers, they focus heavily on far-right extremism and have published nothing on far-left extremism (arguing against their point!). They repeat their thesis several times in several ways: "But focusing solely on Islamist extremism when investigating, researching and developing counterterrorism policies goes against what the numbers tell us. " "Our conclusion is that a “one size fits all” approach to countering violent extremism may not be effective." Since the premise is unstated, and only implied, this thesis is either a casual lie (if the premise is: "only Islamic terrorism is focused on") or the study is totally useless (if the premise is: "all forms of extremism are investigated"). 3. They picked a wide timeframe (1990 to today) when things changed drastically at 9/11. Things were relatively quiet on the Islamic terrorism front in the US from 1990-2000. The data for that study doesn't seem to be available online, but for reference, here's a similar study with a similar bias: https://www.newamerica.org/in-depth/terrorism-in-america/what-threat-united-states-today/ You can hover over the graph to view the crimes cited and see the bias. Unlike the study above, it includes "black nationalism", but still lumps run-of-the-mill hate crimes and anti-police violence by whites into the list -- but doesn't do the same for the "black nationalism" etc. Last edited: Jul 24, 2017 8. Jul 25, 2017 HAYAO I always wonder about looking at such statistics. This is my personal educated guess on the topic, and by no way is it sufficiently supported, but I do not believe from statistics that black people are naturally more inclined to commit murder/rape/robbery. Instead, I think it comes from the structural aspect of the matter. Black people are statistically, in average, paid less in the States compared to other races. I believe poverty plays a major role. Historically, this has been true for quite some time especially back when black people did not have the same rights. Although I heard that it has been continuously improving since then, the social structure of the black people due to such background usually does not change immediately and takes time until such structure is reformed so that the "tendency" for crime is lowered. 9. Jul 25, 2017 Staff: Mentor Please do not ever read any judgement, much less that such judgement into a person's plain vanilla reporting of the statistics. It's a natural tenancy to want to figure out "why" and just as natural to try to ascribe "why" to others reporting the statistics, but that is dangerously prejudicial when ascribing motives to someone else out of whole cloth. Moreover, statistics can provide answers to such questions if we look for them. The way you worded the first part about being "paid less" is problematic but overall, yes, statistics do show that crime is tied heavily to poverty. 10. Jul 25, 2017 HAYAO I don't quite understand. I don't think I am reading your post right, but are you saying that one should not judge anything from statistics? I'm sorry, problematic wording was not intended, so I don't understand. How should I reword it? I would appreciate it if you could teach me. Last edited: Jul 25, 2017 11. Jul 25, 2017 Staff: Mentor I'm saying you should be very careful about seeing value judgements in statistics and correlation vs causation. If I say blacks commit more serious crimes than whites, that's a factually accurate statement of correlation. But it is not a statement of causation: it does not say being black causes people to commit crime, meaning they are "naturally more inclined". This is how people become unfairly judged as racist (not you, but an unfair accusation was made that you can't see). Well, I'm not sure of the intent so I'm not totally sure how to reword it. Saying "black people are...paid less" without explanation of by what measure can lead to misleading understanding; are they paid less for the same job or just overall? This would be similar to the gender pay gap hoax. 12. Jul 25, 2017 HAYAO Of course I understand well that there is a difference between correlation and causation. At least I was careful of the reasoning behind why I thought the way I did, as said in that post. In fact, I was implying there is the correlation vs causation fallacy among many "uneducated" people who believes that just because statistics show black people have committed serious crime, being black leads to crime. My argument is against this because I was careful about judging from statistics. It is an overall average value. If a black person would take the same job as a Caucasian, they would likely be paid the same. However, the overall average of black people are still lower than the other races because of the type of job they do that does not pay as much. I believe that this is due to the social structure that stems from the historical events concerning black people in the US. Black people were discriminated and generally given low wage job before and around civil rights movement, and although it has improved tremendously today, still have that negative cycle of getting low wage jobs in general. Poverty is strongly linked with crime, hence my conclusion that this is a social structural issue, more precisely the unconscious social norms that exists in the US. 13. Jul 26, 2017 Staff: Mentor I'm not sure I buy this argument. In the 1800s large numbers of Irish emigrated to the US and faced discrimination (e.g. signs for jobs saying "No Irish Need Apply"). Since then we've had large numbers of Chinese, Japanese, and many other ethnic groups, all of whom faced discrimination in a variety of ways, but managed to overcome it with considerable success. Probably the most significant predictor of poverty for an individual is growing up in a household with only a single parent. One chart I found compares the fraction of single-parent households in the US by race, in the period 2011 - 2015. (See http://www.actrochester.org/childre...parent-families-by-race-ethnicity/data-tables) 14. Jul 26, 2017 StatGuy2000 Yes, various ethnic groups such as the Irish, Chinese, and Japanese have faced discrimination in the past (e.g. Chinese Exclusion Act, internment of Japanese Americans during WWII, etc.). However, no other ethnic group in the US faced the level of discrimination or prejudice that African Americans experienced (except possibly Native Americans). No other group were enslaved. No other group faced Jim Crow laws taking away the right way to vote, "separate but equal" (which were anything but equal). No other group faced legally enforced segregation. And I can't think of any other group that were routinely targeted for lynching from groups such as the KKK (yes, I am aware that other groups such as Italians were lynched too, but again, not to the extent that African Americans experienced). And in the case of the Irish and other immigrants, there were active measures taken by state and federal governments to assimilate these immigrants into Americans -- something that was denied to African Americans historically. And much of these acts of discrimination have persisted well into the 1950s and 60s, so we are not talking about discrimination that occurred more than a century ago (in the case of the Irish, for example). This history of discrimination and prejudice have had a marked economic and social impact on the lives of African Americans over the decades. It has only been in relatively recent times historically that efforts have been underway to rectify and mitigate these impacts. So it should not come as any kind of surprise that many African American families continue to fare worse compared to other ethnic groups.	2018-05-25 21:00:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34324270486831665, "perplexity": 2406.934761306072}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867217.1/warc/CC-MAIN-20180525200131-20180525220131-00221.warc.gz"}
https://ai.stackexchange.com/questions/11792/difficulty-understanding-monte-carlo-policy-evaluation-state-value-for-gridwor	Difficulty understanding Monte Carlo policy evaluation (state-value) for gridworld I've been trying to read Sutton & Barto book chapter 5.1, but I'm still a bit confused about the procedure of using Monte Carlo policy evaluation (p.92), and now I just cant proceed anymore coding a python solution, because I feel like I don't fully understand how the algorithm works, so that the pseudocode example in the book doesn't seem to make much sense to me anymore. (the orange part) I've done the chapter 4 examples with the algorithms coded already, so I'm not totally unfamiliar with these, but somehow I must have misunderstood the Monte Carlo prediction algorithm from chapter 5. • My setting is a 4x4 gridworld where reward is always -1. • Policy is currently equiprobable randomwalk. If an action would take the newState (s') into outside the grid, then you simply stay in place, but action will have been taken, and reward will have been rewarded. • Discount rate will be 1.0 (no discounting). • Terminal states should be two of them, (0,0) and (3,3) at the corners. 1. On page 92 it shows the algorithm pseudocode and I feel as though I coded my episode generating function correctly thusfar. I have it such that, the results are that I always start in the same starting state (1,1) coords in the gridworld. 2. Currently, I have it so that if you started always in state (1,1), then a possible randomly generate episode could be as follows (in this case also optimal walk). Note that I currently have the episodes in form of list of tuple (s, a, r). Where s will also be a tuple (row,column), but a = string such as "U" for up, and r is reward always -1. 3. so that a possible episode could be like: [( (1,1), "U", -1 ), ( (0,1), "L", -1 )] So that the terminal state is always excluded, so that the last state in episode will be the state immediately close to terminal state. Just like the pseudocode describes that you should exclude the terminal state S_T. But, the random episode could have been one where there are repeating states such as [( (1,1), "U", -1), ( (0,1), "U", -1 ), ( (0,1), "U", -1 ), ( (0,1), "L", -1 )] 4. I made the loop for each step of episode, such as follows: once you have the episodeList of tuples, iterate for each tuple, in reversed order. I think this should give the correct amount of iterations there... 5. G can be updated as described in pseudocode. 6. currently the Returns(S_t) datastructure that I have, will be a dictionary where the keys are state tuples (row,col), and the values are empty lists in the beginning. 7. I have a feeling that I'm calculating the average into V(S_t) incorrectly because I origianlly thought that you could even omit the V(S_t) step totally from the algorithm, and only afterwards compute for a separate 2D array V[r,c] for each state get the sum of the appropriate list elements (accessed from the dict), and divide that sum by the amount of episodes that you ran??? But I don't suddently know how to implement the first visit check in the algorithm. Like, I literally don't understand what it is actually checking for. And furthermore I don't understand how the empirical mean is now supposed to be calculates in the monte carlo algorithm where there is the V(s_t) = average( Returns(S_t) ) I will also post my python code thusfar. import numpy as np import numpy.linalg as LA import random # YOUR CODE rows_count = 4 columns_count = 4 V = np.zeros((rows_count, columns_count)) reward = -1 #probably not needed directions = ['up', 'right', 'down', 'left'] #probably not needed maxiters = 10000 eps = 0.0000001 k = 0 # "memory counter" of iterations inside the for loop, note that for loop i-variable is regular loop variable rows = 4 cols = 4 #stepsMatrix = np.zeros((rows_count, columns_count)) def isTerminal(r,c): #helper function to check if terminal state or regular state global rows_count, columns_count if r == 0 and c == 0: #im a bit too lazy to check otherwise the iteration boundaries return True #so that this helper function is a quick way to exclude computations if r == rows_count-1 and c == columns_count-1: return True return False def getValue(row, col): #helper func, get state value global V if row == -1: row =0 #if you bump into wall, you bounce back elif row == 4: row = 3 if col == -1: col = 0 elif col == 4: col =3 return V[row,col] def getState(row,col): if row == -1: row =0 #helper func for the exercise:1 elif row == 4: row = 3 if col == -1: col = 0 elif col == 4: col =3 return row, col def makeEpisode(r,c): #helper func for the exercise:1 ## return the count of steps ?? #by definition, you should always start from non-terminal state, so #by minimum, you need at least one action to get to terminal state stateWasTerm = False stepsTaken = 0 curR = r curC = c while not stateWasTerm: act = random.randint(0,3) if act == 0: ##up curR-=1 elif act == 1: ##right curC+=1 elif act == 2: ## down curR+=1 else:##left curC-=1 stepsTaken +=1 curR,curC = getState(curR,curC) stateWasTerm = isTerminal(curR,curC) return stepsTaken V = np.zeros((rows_count, columns_count)) episodeCount = 100 reward = -1 y = 1.0 #the gamma discount rate #use dictionary where key is stateTuple, #and value is stateReturnsList #after algorithm for monte carlo policy eval is done, #we can update the dict into good format for printing #and use numpy matrix returnsDict={} for r in range(4): for c in range(4): returnsDict[(r,c)]=[] #"""first-visit montecarlo episode generation returns the episodelist""" def firstMCEpisode(r,c): global reward stateWasTerm = False stepsTaken = 0 curR = r curC = c episodeList=[ ] while not stateWasTerm: act = random.randint(0,3) if act == 0: ##up r-=1 act="U" elif act == 1: ##right c+=1 act="R" elif act == 2: ## down r+=1 act="D" else:##left c-=1 act="L" stepsTaken +=1 r,c = getState(r,c) stateWasTerm = isTerminal(r,c) episodeList.append( ((curR,curC), act, reward) ) if not stateWasTerm: curR = r curC = c return episodeList kakka=0 #for debug breakpoints only! #first-visit Monte Carlo with fixed starting state in the s(1,1) state for n in range(1, episodeCount+1): epList = firstMCEpisode(1,1) G = 0 for t in reversed( range( len(epList) )): G = y*G + reward #NOTE! reward is always same -1 S_t = epList[t][0] #get the state only, from tuple willAppend = True for j in range(t-1): tmp = epList[j][0] if( tmp == S_t ): willAppend =False break if(willAppend): returnsDict[S_t].append(G) t_r = S_t[0] #tempRow from S_t t_c =S_t[1] #tempCol from S_t V[t_r, t_c] = sum( returnsDict[S_t] ) / n kakka = 3 #for debug breakpoints only! print(V) • See: ai.stackexchange.com/a/10818/2444. Maybe this is a duplicate of it. – nbro Apr 12 '19 at 17:25 • apparently I calculated the value function average in a wrong way... apparently I am required to keep some kind of N(S_t) datastructure, maybe a dict where the key is the each particular state such as (row,col) and the value is the amount of times that state has been visited to overall across all episodes??? Or do I even need to have that N(S_t datastructure, can I just calculate the V(S_t) for each state such that I calculate the sumOfElements from each list of respective state, divided by that listAmountOfElements??? – Late347 Apr 12 '19 at 17:44 • because isnt it so... that if I have dictionary where key is (row,col) for each state, and the value is the list of returns for that state... then the amount of elements in each list, is the amount of times that each state was visited? – Late347 Apr 12 '19 at 17:45 • also, a new question, are you supposed to start always in the same fixed starting state in this Monte Carlo policy evaluation part? will it distort the algorithm from converging to the actual value function for that policy (equiprobable randomwalk), as long as you have run maybe 100k iterations ( iterations == episodes) – Late347 Apr 12 '19 at 17:55 • I got the every-visit Monte Carlo done, but I didn't fully understand how to make the checking for first-visit MC, and from what sequience exactly do you check the existence of currently iterated state S_t ??? what is the ending limit that is included in the first-visit check iteration? – Late347 Apr 12 '19 at 19:43	2020-12-01 15:29:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47744640707969666, "perplexity": 3573.2211492371166}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141674594.59/warc/CC-MAIN-20201201135627-20201201165627-00281.warc.gz"}
https://www.r-bloggers.com/2018/07/a-tour-of-timezones-troubles-in-r/	Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. In any programming tool, dates, times, and timezones are hard. Deceptively hard. They’ve been shaped by politics and whimsy for hundreds of years: timezones can shift with minimal notice, countries have skipped or repeated certain days, some are offset by weird increments, some observe Daylight Saving Time, leap years, leap seconds, the list goes on. Luckily, we rarely need to worry about most of those details because other teams of very smart people have spent a lot of time providing nice abstractions for us that handle most of the weird edge cases. Here at Methods we’ve been bit by some timezone oddities in R recently, so I wanted to focus on that part of R’s “date/time stack”, starting with a crash-course on datetimes and timezones before showing how to avoid common errors, with a special focus on reading from and writing to files. Timezones in base R Let’s start with a single string representing noon on the first day of this year. We want to make this a proper date-time object, which in R means the POSIXct class; we can do this with as.POSIXct(). x <- "2018-01-01 12:00:00" as.POSIXct(x) ## [1] "2018-01-01 12:00:00 EST" Notice the timezone code at the end; it defaulted to “EST” because of my default timezone, which might be different for you. Sys.timezone() ## [1] "America/Detroit" If our x isn’t supposed to represent Eastern time, we can specify a different timezone as an argument to as.POSIXct. as.POSIXct(x, tz = "America/Chicago") ## [1] "2018-01-01 12:00:00 CST" Notice the only thing that changed is the timezone; the clock time is the same. CST and EST are an hour apart, though, so our two conversions represent different instances of time; noon in Detroit occurs one hour later than noon in Chicago. We can check this by subtracting the dates: as.POSIXct(x, tz = "America/Chicago") - as.POSIXct(x) ## Time difference of 1 hours I’m not really in Detroit, I’m in Ypsilanti (another city in Michigan); what happens if we use that instead? as.POSIXct(x, tz = "America/Ypsilanti") ## [1] "2018-01-01 12:00:00 America" Okay that’s weird; what does the “America” timezone code even mean? Is it the same as the first conversion we did? If it is, there should be a 0-hour difference between them. as.POSIXct(x, tz = "America/Ypsilanti") - as.POSIXct(x) ## Time difference of -5 hours That’s not right! “America/Ypsilanti” isn’t a valid timezone, but rather than throwing an error (or even a warning), R created an instance of time with no UTC offset. UTC is a special standard that has no timezone, no daylight savings time, or any of the other weird things that make timezones hard. In R, and most other programming languages, UTC is the common, underlying representation of (date)time, and things like timezones are defined in terms of UTC (EST is UTC-5, hence the “-5” above). It’s roughly the same as GMT (Greenwich Mean Time), but not quite identical (GMT can have Daylight Saving Time!) This above behavior in R can be really dangerous; even if you intend to use a proper timezone name, misspelling it (e.g. “America/Chicgo”) will produce this same bad behavior. We need to make sure to only use valid timezone names. The list of options can be accessed with the OlsonName() function; on my machine, this gives 606 different options! We can see “America/Chicago” and “America/Detroit” are in there, but “America/Ypsilanti” isn’t. c("America/Chicago", "America/Detroit", "America/Ypsilanti") %in% OlsonNames() ## [1] TRUE TRUE FALSE Enter lubridate These kinds of bad-but-quiet misunderstandings between the programmer and the language exist all over base R, but can often be prevented with additional packages. The tidyverse family in particular aims to prevent silent errors like this, and their date/time package lubridate is no exception. Rather than using as.POSIXct to turn our string into a datetime object, we’ll use lubridate::as_datetime. library(lubridate) as_datetime(x) # ymd_hms works as well ## [1] "2018-01-01 12:00:00 UTC" Notice as_datetime used UTC by default, rather than my local timezone. This is great because it means that command will return the same thing on any other computer, regardless what that computers’ timezone is set to. lubridate makes code more portable! This as_datetime function still produces POSIXct objects, class(as_datetime(x)) ## [1] "POSIXct" "POSIXt" which means any non-lubridate functions for working with these datetime objects will still work. We can still specify whatever timezone we desire, as_datetime(x, tz = "America/Detroit") ## [1] "2018-01-01 12:00:00 EST" but more importantly, lubridate lets us know if we use a bad timezone name! as_datetime(x, tz = "America/Ypsilanti") ## Error in C_force_tz(time, tz = tzone, roll): CCTZ: Unrecognized output timezone: "America/Ypsilanti" This alone, saving us from ourselves, might be reason enough to use lubridate whenever you deal with timezones in R, but it can do a lot more, most of which we won’t cover here. Why should we care? For a lot of programs utilizing datetimes, you can get away with unconsidered or incorrect timezones. I can’t enumerate all the ways that could come back to bite you, but one is Daylight Saving Time. Suppose we’re given the following datetimes as strings, perhaps read from some file: x <- c("2018-03-10 23:30:00", "2018-03-11 00:30:00", "2018-03-11 01:30:00", "2018-03-11 03:30:00", "2018-03-11 04:30:00", "2018-03-11 05:30:00") x ## [1] "2018-03-10 23:30:00" "2018-03-11 00:30:00" "2018-03-11 01:30:00" ## [4] "2018-03-11 03:30:00" "2018-03-11 04:30:00" "2018-03-11 05:30:00" To do much with these, we need to convert from character to POSIXct. We’ll use lubridate again, but instead of parsing with the very general as_datetime() we’ll use the more idiomatic ymd_hms() which expects inputs to be formatted as year-month-day, hour-minute-second (exactly what we have!). ymd_hms(x) ## [1] "2018-03-10 23:30:00 UTC" "2018-03-11 00:30:00 UTC" ## [3] "2018-03-11 01:30:00 UTC" "2018-03-11 03:30:00 UTC" ## [5] "2018-03-11 04:30:00 UTC" "2018-03-11 05:30:00 UTC" It’s possible that those times are supposed to be UTC and we got exactly what we wanted. If instead these strings represent clock-times from a place that observes Daylight Saving Time, we’ve got a problem. These times span the “spring ahead” transition, where clocks immediately transition from 2am to 3am. This means that in a DST-observing timezone all these clock-times are evenly spaced, since 1:30am $$\rightarrow$$ 3:30am is only one hour. Since UTC does not observe DST, elements of dst are currently not evenly-spaced (it’s not a regular timeseries), which we can see quickly with the diff function: diff(ymd_hms(x)) ## Time differences in hours ## [1] 1 1 2 1 1 Using a DST-observing timezone on conversion will fix this. diff(ymd_hms(x, tz = "America/Detroit")) ## Time differences in hours ## [1] 1 1 1 1 1 Depending on what you want to do with these datetimes, that may not matter, but it could be absolutely critical. Something lubridate won’t help prevent is using valid timezone names which don’t align with what you’re trying to do. “EST” (Eastern Standard Time) and “EDT” (Eastern Daylight Time) are both in OlsonNames(), but will ignore the DST shift in our vector of datetimes: diff(ymd_hms(x, tz = "EST")) ## Time differences in hours ## [1] 1 1 2 1 1 diff(ymd_hms(x, tz = "EDT")) ## Time differences in hours ## [1] 1 1 2 1 1 When we specify “EST”, the the first 3 are correct, but the latter half is wrong because they’re really “EDT” datetimes; the opposite occurs when we specify “EDT”. In almost all cases, use location-based timezone names like “America/City” to properly handle not only DST, but other potential oddities that may have occurred in that location, like those that have shifted between observing and not observing DST. Timezones and I/O It’s rare that we need to create new datetime objects from scratch; they’re nearly always coming into a project from some external dataset, generally in Excel or CSV formats. We can make a little CSV using writeLines(); we’ll have just one column of our DST-crossing datetimes, and we’ll call it datetime. writeLines(c("datetime", x), "data/dst-example.csv") There’s very few excuses to be using read.csv() in 2018, so we’ll use the far-superior read_csv() from readr which will automatically parse our datetimes into POSIXct objects. library(readr) tbl <- read_csv("data/dst-example.csv") ## Parsed with column specification: ## cols( ## datetime = col_datetime(format = "") ## ) tbl ## # A tibble: 6 x 1 ## datetime ## <dttm> ## 1 2018-03-10 23:30:00 ## 2 2018-03-11 00:30:00 ## 3 2018-03-11 01:30:00 ## 4 2018-03-11 03:30:00 ## 5 2018-03-11 04:30:00 ## 6 2018-03-11 05:30:00 What timezone are those? lubridate makes this easy with their tz function. tz(tbl$datetime) ## [1] "UTC" If we know these aren’t UTC times, but are instead clock times from New York, we need to do some sort of conversion here, as readr doesn’t allow you to specify the timezone when reading. lubridate provides two “timezone conversion” functions: • with_tz() applies a timezone so the clock time changes, but the instant of time (e.g. UTC) is the same, e.g. 5pm EST $$\rightarrow$$ 2pm PST. • force_tz() changes the instant of time so the clock time remains constant, e.g. 5pm EST $$\rightarrow$$ 5pm PST. In this case we need force_tz because the times in our file are non-UTC (New York, in this case) clock times. library(dplyr) tbl <- tbl %>% mutate(datetime = force_tz(datetime, "America/New_York")) tbl ## # A tibble: 6 x 1 ## datetime ## <dttm> ## 1 2018-03-10 23:30:00 ## 2 2018-03-11 00:30:00 ## 3 2018-03-11 01:30:00 ## 4 2018-03-11 03:30:00 ## 5 2018-03-11 04:30:00 ## 6 2018-03-11 05:30:00 Notice the display isn’t any different; tibble doesn’t print timezones for datetime columns. The timezone has changed, tz(tbl$datetime) ## [1] "America/New_York" and they’re now evenly spaced, too: diff(tbl$datetime) ## Time differences in hours ## [1] 1 1 1 1 1 I’m not sure we’ve ever had a client send us a file with UTC datetimes in it, so whenever we read in a client-provided file with datetimes, we immediately call force_tz to correct things. Reading from Excel This workflow is essentially the same with Excel files. We’ll use the readxl package, another tidyverse gem, which is much nicer than the Java-based libraries that came before it (e.g. xlsx). I’ve already created an excel file (which you can download here) whose first sheet is identical to the CSV we just made, so we’ll read that in. library(readxl) tbl <- read_excel("data/timezone-examples.xlsx") tbl ## # A tibble: 6 x 1 ## datetime ## <dttm> ## 1 2018-03-10 23:30:00 ## 2 2018-03-11 00:30:00 ## 3 2018-03-11 01:30:00 ## 4 2018-03-11 03:30:00 ## 5 2018-03-11 04:30:00 ## 6 2018-03-11 05:30:00 Like readr::read_csv(), readxl::read_excel() is smart enough to see we have datetimes, and parse them appropriately, but we have the same problem as before: tz(tbl$datetime) ## [1] "UTC" Thus we need the same fix. An alternative to force_tz() is to assign to tz(): tz(tbl$datetime) <- "America/New_York" tbl ## # A tibble: 6 x 1 ## datetime ## <dttm> ## 1 2018-03-10 23:30:00 ## 2 2018-03-11 00:30:00 ## 3 2018-03-11 01:30:00 ## 4 2018-03-11 03:30:00 ## 5 2018-03-11 04:30:00 ## 6 2018-03-11 05:30:00 diff(tbl$datetime) ## Time differences in hours ## [1] 1 1 1 1 1 This is very base-R style and doesn’t fit nicely into pipelines of commands with %>%, so we generally opt for force_tz + mutate, but this form can also be useful. Split Date & Time We often receive data where the date is in one column, but the time is in the next column over. This presents some new challenges because R doesn’t have date-free time classes; we want both pieces in the same column. We find this happens most often in Excel files; the second sheet of our timezone-examples.xlsx file provides an example with the same data as before, but the date & time in two separate columns. tbl <- read_excel("data/timezone-examples.xlsx", sheet = 2) tbl ## # A tibble: 6 x 2 ## date time ## <dttm> <dttm> ## 1 2018-03-10 00:00:00 1899-12-31 23:30:00 ## 2 2018-03-11 00:00:00 1899-12-31 00:30:00 ## 3 2018-03-11 00:00:00 1899-12-31 01:30:00 ## 4 2018-03-11 00:00:00 1899-12-31 03:30:00 ## 5 2018-03-11 00:00:00 1899-12-31 04:30:00 ## 6 2018-03-11 00:00:00 1899-12-31 05:30:00 We sure didn’t expect the time to be read as datetimes with some strange, irrelevant date! We’d prefer to have a single datetime column where the dates come from date and the times come from time. There’s at least two ways we could do this: 1. change the dates in the time column to be the dates in the date column, then rename time -> datetime 2. create a new datetime column by taking the pieces we need from date & time We think the latter is a bit more “tidy” or idiomatic, so that’s what we’ll do. It’s a bit clunky, but we can extract the dates and times from the appropriate column, paste them together, then re-parse those strings into datetime (POSIXct) objects. This will also let us specify the timezone when we do that, avoiding the need for force_tz. The hardest part is pulling only the time-part out of the POSIXct objects in our time column; we’ll use format for that. tbl <- tbl %>% mutate(datetime = paste(date, format(time, "%T")) %>% ymd_hms(tz = "America/New_York")) %>% select(datetime) # drop date/time cols tbl ## # A tibble: 6 x 1 ## datetime ## <dttm> ## 1 2018-03-10 23:30:00 ## 2 2018-03-11 00:30:00 ## 3 2018-03-11 01:30:00 ## 4 2018-03-11 03:30:00 ## 5 2018-03-11 04:30:00 ## 6 2018-03-11 05:30:00 tz(tbl$datetime) ## [1] "America/New_York" diff(tbl$datetime) ## Time differences in hours ## [1] 1 1 1 1 1 There we go! One ugly line of code, but that’s all we needed. With CSV’s this is a bit easier because the time column will come back as a character vector rather than a POSIXct vector (or you can force it to POSIXct with the col_types argument); then you don’t need the format part above. If we tried to force that behavior using read_excel()’s col_types argument, we’d get back some hard-to-interpret numbers due to the horrible way Excel stores time. Writing back out So we’ve got our inputs formatted the way we want, then we do something, and at some point we probably want to write that back out. This is super easy with readr: write_csv(tbl, "data/example-output.csv") What was written to that text file might not be what we expected; let’s investigate with readr::read_lines(): read_lines("data/example-output.csv") ## [1] "datetime" "2018-03-11T04:30:00Z" "2018-03-11T05:30:00Z" ## [4] "2018-03-11T06:30:00Z" "2018-03-11T07:30:00Z" "2018-03-11T08:30:00Z" ## [7] "2018-03-11T09:30:00Z" Two things to notice: the format is ISO 8601 (the “T” and “Z”), and they’re stored as UTC (no timezone or offset, indicated by the lack of anything after the “Z”). Just like readr assumes everything is UTC coming in, it also makes things UTC going out. For programmers and programming languages, this is ideal: it’s consistent, unambiguous, and portable. If this file is going to be used by non-programmers, however, this can be a problem. If you open this in Excel, you’ll see the UTC times, and unlike all sane programming languages Excel can’t store UTC but display EST/EDT/PST/etc. “What are all these weird times? Why are they off by X hours sometimes, and X+1 hours other times?” are not questions we like getting after handing off a deliverable to a client. We have just two options: 1. explain to the next user (client, colleague, etc.) the excellence and utility of UTC and convince them it’s the right way to do things 2. write the datetimes out a bit differently Depending on the next user, option 1 can be a total no-go, in which case we need a way to write our datetimes as clock times rather than instants of time (UTC). The simplest way is to convert to character first tbl %>% mutate_if(is.POSIXct, as.character) %>% # handy if you have multiple datetime cols write_csv("data/example-output.csv") read_lines("data/example-output.csv") ## [1] "datetime" "2018-03-10 23:30:00" "2018-03-11 00:30:00" ## [4] "2018-03-11 01:30:00" "2018-03-11 03:30:00" "2018-03-11 04:30:00" ## [7] "2018-03-11 05:30:00" That seems to have done what we want, and if necessary we could replace as.character with something else to get a different format. This works about as well when writing to Excel files (e.g. with writexl::write_xlsx), but in general we try to avoid writing directly to Excel files. If we know the next user will be using Excel, we prefer readr::write_excel_csv(), though writexl is the tool of choice for files with multiple tables spread over different sheets. This only scratched the surface of issues that can arise when dealing with datetimes & timezones, but if there’s one lesson we hope sticks with you, it’s use lubridate. It saves lives; check out this decade-old article from Revolution Analytics as further proof that life was a bit tougher before lubridate. If you want to dig deeper into the scary world of datetimes we’re mostly abstracted from, check out this excellent article from Zach Holman to learn about that one time Samoa skipped a day and so much more.	2021-03-01 14:02:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.354775607585907, "perplexity": 5356.091506996326}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362513.50/warc/CC-MAIN-20210301121225-20210301151225-00107.warc.gz"}
https://math.stackexchange.com/questions/2239928/how-many-elements-can-the-set-sf-fxy-fx-fy-x-y-in-r-have	How many elements can the set $S(f)=\{f(x+y)-f(x)-f(y)\ \|\ x,y\in R\}$ have？ Question: For a surjective function $f：\mathbb R\to \mathbb R$,where $\mathbb R$ denotes the set of real numbers, define the set $$S(f)=\{f(x+y)-f(x)-f(y)\ \|\ x,y\in \mathbb R\}\ .$$ Assume that $S(f)$ is finite and $\|S(f)\|\neq 1$. Find the possible values of $\|S(f)\|$. I think $\|S(f)\|=2$ is possible because of the example $$f(x)=\dfrac{1}{2}(\lfloor x\rfloor-\{x\})\ .$$ It is clear $f$ is surjective,because for any $t\in R$, if we let $x=2t+2\{-2t\}$, then we have $$f(x)=\dfrac{1}{2}(2t+2\{-2t\}-2\{-2t\})=t\ ,$$ and we then have $$S(f)=f(x+y)-f(x)-f(y)=\{x\}+\{y\}-\{x+y\}=\{0,1\}\ ,$$ so we have$$\|S(f)\|=2\ .$$ Using this method, Can we find examples such that $\|S(f)\|=3,4,5,\ldots$? Maybe there are other methods to solve this problem. • What is the definition of $f$? Is $R$ the set of real numbers or something else? What is $T$? – amrsa Apr 18 '17 at 10:32 • @amrsa,sorry,I have edit it. – function sug Apr 18 '17 at 10:37 • what is a $R$? The real numbers? – user321268 Apr 18 '17 at 11:41 • @mayer_vietoris,yes, it's it – function sug Apr 18 '17 at 11:59 One example for more than 2. $f(x) = \begin{cases} 1 &\text{if } x = \frac12 \\ 2 &\text{if } x = \frac14 \\ 4x &\text{if } x \neq \frac12, \frac14 \end{cases}$ $f(x) = 4x\operatorname{sgn}\left\|x-\frac12\right\|\operatorname{sgn}\left\|x-\frac14\right\| + 1(1 - \operatorname{sgn}\left\|x-\frac12\right\|) + 2(1 - \operatorname{sgn}\left\|x-\frac12\right\|)$ $S(f) = \{-3, -1, 0, 1, 2\}$ And the method to generate more of these functions with finite $\|S(f)\|$ values. Given surjective real function $g: \mathbb{R} \to \mathbb{R}$, such that $\left\|S(g)\right\| = 1$, and a finite permutation on real numbers $h: \mathbb{R} \to \mathbb{R}$ with $n > 0$ values permuted, and the composition surjective real function $f(x) = g(h(x))$, the value $\left\|S(f)\right\|$ is finite, at least 2 and with the upper bound $1 + \frac12 n(n+5)$. (Working at the bottom) Looking at your function and the other answer, I think the kinds of functions $F$ such that $S(F)$ is finite can be defined by composition of linear functions, finite permutations and surjective sawtooth functions. $F = f_1 \circ \ldots \circ f_n$ where for each $i = 1, \ldots, n$, surjection is assumed and one of the following is true: • $\exists m,c \in \mathbb{R} : m \neq 0 \land \forall x \in \mathbb{R} : f_i(x) = mx + c$ (linear) • $\exists b \in \mathbb{R}_{>0} : \forall x \in \mathbb{R} : f_i(x) = x + (x {\%} b)$ (surjective sawtooth) • $\left\|\{x \in \mathbb{R} \mid f_i(x) \neq x \}\right\| < \aleph_0 \land \forall y \in \mathbb{R}: \exists! x \in \mathbb{R}: f_i(x) = y$ (finite permutation) Working out for the finite permutation case: \begin{align} \mathbb{F}_S &= \{f: \mathbb{R} \to \mathbb{R} \mid \forall y \in \mathbb{R}: \exists x \in \mathbb{R} : f(x) = y \} \\ \mathbb{F}_B &= \{f: \mathbb{R} \to \mathbb{R} \mid \forall y \in \mathbb{R}: \exists! x \in \mathbb{R} : f(x) = y \} \\ S &: \mathbb{F}_S \to \mathbb{R} \\ S(f) &= \{f(x+y) - f(x) - f(y) \mid x,y \in \mathbb{R} \} \\ g &\in \mathbb{F}_S \\ \left\|S(g)\right\| = 1 &\implies \exists c \in \mathbb{R} : \forall x,y \in \mathbb{R} : g(x+y) - g(x) - g(y) = -c \\ &\implies \exists m,c \in \mathbb{R}: \forall x \in \mathbb{R}: g(x) = mx + c \\ \left\|S(g)\right\| = 1 &\vdash g(x) = mx + c \land S(g) = \{-c\} \\ T &: \mathbb{F}_B \to \mathbb{R} \\ T(f) &= \{ x \in \mathbb{R} \mid f(x) \neq x \} \\ h &\in \mathbb{F}_B \\ f &\in \mathbb{F}_S \\ f(x) &= g(h(x)) \\ S_f &{:=} S(f) \\ S_g &{:=} S(g) \\ T_h &{:=} T(h) \\ 0 < \left\|T_h\right\| < \aleph_0 &\vdash 1 < \left\|S_f\right\| < \aleph_0 \\ F(x, y) &{:=} f(x+y) - f(x) - f(y) \\ G(x, y) &{:=} g(x+y) - g(x) - g(y) \\ S_f &= \{ -c \} \cup \{ F(x,y) \mid x, y \in \mathbb{R} \land F(x,y) \neq -c \} \\ F(x, y) \neq -c &\implies f(x) \neq g(x) \lor f(y) \neq g(y) \lor f(x+y) \neq g(x+y) \\ &\implies x \in T_h \lor y \in T_h \lor (x+y) \in T_h \\ &\vdash \left\|S_f\right\| > 1 \end{align} \begin{align} S_f &= \{-c\} \cup P_1 \cup P_2 \cup P_3 \\ P_1 &= \left\{F(t - x, x) \mid t \in T_h \land x \in \mathbb{R} \land x \notin T_h \land t - x \notin T_h \right\} \\ &= \left\{f(t) - f(t-x) - f(x) \mid \Phi(x,t) \right\} \\ &= \left\{mh(t) + c - (m(t-x) + c + mx + c) \mid \Phi(x, t) \right\} \\ &= \left\{m(h(t) - t) - c \mid t \in T_h \right\} \\ (\forall t \in h : h(t) \neq t) &\vdash \exists x \in P_1: x \neq -c \\ &\vdash \left\| \{-c\} \cup P_1 \right\| > 1 \\ 0 < &\left\|P_1\right\| \leq \left\|T_h\right\| \\ P_2 &= \left\{ F(x, y) \mid x,y \in T_h \land (x > y \lor x = y) \right\} \\ &\vdash \left\|P_2\right\| \leq \frac12 \left\|T_h\right\|\left(\left\|T_h\right\| + 1\right) \\ P_3 &= \left\{ F(t, x) \mid t \in T_h \land x \in \mathbb{R} \setminus T_h \land t + x \notin T_h \right\} \\ &= \left\{ f(t + x) - f(t) - f(x) \mid \phi(t, x) \right\} \\ &= \left\{ m(t+x) + c - (mh(t) + c + mx + c) \mid \phi(t, x) \right\} \\ &= \left\{ m(t - h(t)) - c \mid t \in T_h \right\} \\ &\vdash \left\|P_3\right\| \leq \left\|T_h\right\| \\ \left\|S_f\right\| &\leq 1 + 2\left\|T_h\right\| + \frac12 \left\|T_h\right\|\left(\left\|T_h\right\| + 1\right) \\ &\leq 1 + \frac12 \left\|T_h\right\|\left( \left\|T_h\right\| + 5 \right) \\ 1 < &\left\|S_f\right\| \leq 1 + \frac12 \left\|T_h\right\|\left( \left\|T_h\right\| + 5 \right) \end{align} incomplete I think $$f(x)=x+\sum_{i=0}^n x\%2^i-a[x\neq0]$$ where $\%$ is the sawtooth, $[x\neq0]$ is the Iverson bracket, $n\in\mathbb{N}$ and $2^{n+1}>a\in\mathbb{N}$ satisfies $$\lvert S(f)\rvert=2^{n+1}+a.$$ However, I just can't seem to prove that $f$ is surjective. If someone can prove that $f$ is surjective, then I'll present the rest of the proof. • I have doubt in your last line. It seems like $(x+y)\%\alpha-x\%\alpha-y\%\alpha$ can be $0$ or $-\alpha$, not anything else... – guest Apr 28 '17 at 5:31 • You're right, thanks. – Lawrence C. Apr 30 '17 at 10:35	2019-05-21 17:13:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 929.0682563624706}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00139.warc.gz"}
https://www.nature.com/articles/s41598-021-87818-3	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Fast and effective pseudo transfer entropy for bivariate data-driven causal inference ## Abstract Identifying, from time series analysis, reliable indicators of causal relationships is essential for many disciplines. Main challenges are distinguishing correlation from causality and discriminating between direct and indirect interactions. Over the years many methods for data-driven causal inference have been proposed; however, their success largely depends on the characteristics of the system under investigation. Often, their data requirements, computational cost or number of parameters limit their applicability. Here we propose a computationally efficient measure for causality testing, which we refer to as pseudo transfer entropy (pTE), that we derive from the standard definition of transfer entropy (TE) by using a Gaussian approximation. We demonstrate the power of the pTE measure on simulated and on real-world data. In all cases we find that pTE returns results that are very similar to those returned by Granger causality (GC). Importantly, for short time series, pTE combined with time-shifted (T-S) surrogates for significance testing strongly reduces the computational cost with respect to the widely used iterative amplitude adjusted Fourier transform (IAAFT) surrogate testing. For example, for time series of 100 data points, pTE and T-S reduce the computational time by $$82\%$$ with respect to GC and IAAFT. We also show that pTE is robust against observational noise. Therefore, we argue that the causal inference approach proposed here will be extremely valuable when causality networks need to be inferred from the analysis of a large number of short time series. Unveiling and quantifying the strength of interactions from the analysis of observed data is a problem of capital importance for real-world complex systems. Typically, the details of the system are not known, but only observed time series are available, often short and noisy. A first attempt to try to quantify causality from observations was done 1956 by Wiener1 and formalized in 1969 by Granger2. According to Wiener-Granger causality (GC), given two processes X and Y, it is said that “Y G-causes X” if the information about the past of Y improves, in conjunction with the past of X, the prediction of the future of X, than the latter’s past alone. Since then, several variations have been proposed3,4,5,6,7,8, and have been applied to a broad variety of fields, such as econometrics9,10,11, neurosciences12, physiology13 and Earth sciences14,15,16,17,18 to cite a few. An information-theoretic measure, known as Transfer Entropy (TE), a form of conditional mutual information (CMI)19, which approaches this problem from another point of view: instead of predicting the future of X, it tests whether the information about the past of Y is able to reduce the uncertainty on the future of X. Since its introduction by Schreiber20 in 2000, TE has found applications in different fields such as neurosciences21,22,23,24,25,26, physiology27,28,29, climatology30,31, finantial32 and social sciences33. For Gaussian processes, for which the mutual information (MI) is known from the early years of information theory and its introduction in nonlinear dynamics34 is known for about 30 years, the equivalence between GC and TE is well established35. There are no clear links though between GC and TE for non Gaussian processes. In practical terms, while TE provides a model-free approach, the need of estimating several probability distributions makes TE substantially more computationally demanding than GC. The success of the GC and TE approaches strongly depends on the characteristics of the system under study (its dimensionality, the strength of the coupling, the length and the temporal resolution of the data, the level of noise contamination, etc.). Both approaches can fail in distinguishing genuine causal interactions from correlations that arise due to similar governing equations, or correlations that are induced by the presence of common external forcings. In addition, when the system under study is composed by more than two interacting processes, GC and TE can return fake causalities, i.e., fail to discriminate between direct and indirect causal interactions. Many methods have been proposed to address these problems36,37,38,39,40,41,42,43,44,45,46,47,48,49; however, their performance depends on the characteristics of the data, and their data requirements, computational cost, and number of parameters that need to be estimated may limit their applicability. The aim of this work is to propose a new, fast and effective approach for detecting causal interactions between two processes, X and Y. Our approach is based on the TE idea of uncertainty reduction: starting from the original TE definition20, by applying Gaussian approximations we obtain a simplified expression, which we refer to as pseudo transer entropy (pTE). When X and Y are Gaussian processes, we show that pTE detects, as expected, the same causal interactions as TE, which are, in turn, as those inferred by GC. However, we find that when X and Y are non-Gaussian, pTE also returns results that are fully consistent with those returned by GC. Importantly, for short time series, pTE strongly reduces the computational cost with respect to GC. The code, freely available in GitHub50, has been built to provide a new, user-friendly and low-computational-cost tool that quickly returns, from a set of short time series, a inferred causal network. This will allow inter-disciplinary network scientists to find interesting properties of the system under study, without requiring any knowledge of the underlying physics. For experts in specific fields, the algorithm developed can be used as a first step to quickly understand which variables may play important roles in a given high-dimensional complex system. Then, as a second step, more precise methods, which are data and computationally more demanding, can be used to further understand the interactions between the variables that compose the backbone of the system, that was inferred by using the pTE approach. This paper is organized as follows. In the main text we first consider synthetic time series generated with three stochastic data generating processes (DGPs) where the underlying causality is known: a linear system, a nonlinear system, and the chaotic Lorenz system (section Models presents the details of the three DGPs). We compare the performance of pTE, GC and TE in terms of the power and size, which are the percentage of times that a causality is detected when there is causality (power) and when there is no causality (size, also known as false discovery). Clearly, for a method to be effective, it must have a high power and a low size. Using the selected DGPs we demonstrate that pTE obtains similar power and size as GC while, for short time series, it allows a large reduction of the computational cost. Then, we demonstrate the suitability of pTE for the analysis of real world time series by considering two well-known climatic indices: the NINO3.4 and All India Rainfall (AIR). In the section Additional Information we present results obtained with several other DGPs, and we also compare our results with previous results reported in the literature. In the section Methods we present the derivation of the pTE expression and we also describe the statistical tests performed for determining the significance of the pTE, GC and TE values. In Methods we also present the implementation of the algorithms. ## Results First, we use the three DGPs described in Models to compare the performance of pTE, GC and TE in terms of the power and size. If by construction there is no causality from X to Y, the percentage of times the causality is higher than the significance threshold returned by the surrogate analysis will be called “size” of the test, i.e., is the probability that a causality is detected when there is no causality by construction. On the other hand, if by construction X causes Y, the percentage of times the method finds causality from X to Y is called “power” of the test. With the surrogate analysis adopted, the causality between the original data will be compared to the maximum one found within 19 surrogates51, and the probability that the original data displays by chance the highest causality is $$5\%$$. We analyze the power and size for the two possible causal directions ($$X \rightarrow Y$$ and $$Y \rightarrow X$$), as a function of the coupling strength and of the length of the time series. Figure 1 displays the power and size of the three methods, pTE, GC and TE, for the linear model, when the coupling is such that there is causality from Y to X (the size is shown in the top row, and the power, in the bottom row). The similarity between pTE and GC in finding the true causality is evident. With a coupling strength $$C<0.1$$ the three methods fail to detect causality, while for $$C> 0.4$$, for both pTE and GC, the number of data points in the time series needed to find causality is quite small, in fact 100 data points are sufficient to achieve a power of 100. In Fig. 2 we show results when we move along an horizontal or a vertical line in Fig. 1: we plot the power/size vs. the time series length, keeping fixed the coupling strength (left panel, $$C=0.5$$) and vs. the coupling strength, keeping fixed the time series length (right panel, $$N=500$$). In the left panel we notice that for $$C=0.5$$, a minimum of 200 data points are needed to retrieve the correct causality for all three methods with a power above 95. In the right panel, we notice that with 500 data points, a minimum coupling strength of $$C\approx 0.25$$ is necessary to find a power larger than 95 for all three methods. Figure 3 displays the results obtained for the nonlinear model, and we notice that they are very similar to the ones obtained with the linear model, probably due to the weak nonlinearity considered. We note that, in comparison with the linear model, in this model, with short time series the power and size returned by the three methods are more similar. Regarding the two chaotic Lorenz oscillators, which are coupled in the first variable, the situation is very different, as shown in Fig. 4. When looking at the causality between the coupled variables, for both pTE and GC the causality is detected for a moderate coupling strength and a rather long time series. Causality $$X \rightarrow Y$$ is not detected for any (coupling strength, time series length), which is correct by construction. TE instead finds causality also for $$X \rightarrow Y$$, which is wrong by construction. This observation for TE can be attributed to insufficient conditioning treated by Paluš19,52, in fact the directionality of the coupling cannot be inferred when the systems are fully synchronized. Next, we compare the computational cost of using pTE, GC and TE. Figure 5 displays the time required to calculate $$X \rightarrow Y$$ and $$Y \rightarrow X$$ causalities, as a function of the length, N, of the time series. The figure shows the time required when the codes are run on Google colab CPUs ($$\hbox {Intel}^{\tiny {\textregistered }}$$ $$\hbox {Xeon}^{\tiny {\textregistered }}$$ CPU @ 2.20GHz), and includes preprocessing the time-series (detrending and normalizing) and performing the statistical significance test. For short time series we see a large advantage of using pTE instead of GC. TE sits back as the slowest of the three methods. The reason is attributed to the scaling of parameter k in the k-nearest neighbors method used to compute TE, which scales as $$\sqrt{N}$$. Table 1 displays the computational time required to calculate $$X \rightarrow Y$$ and $$Y \rightarrow X$$ causalities, and the corresponding power and size obtained using the linear model. While in Fig. 5 we showed the total computational time, in Table 1 we show only the time required for the calculation of the pTE, GC and TE values (without signal preprocessing and without performing statistical significance analysis). We see that, for time series of 25 data points, the time required for pTE calculation (averaged over 1000 runs) is 200% faster than GC; however, this percentage reduces to 12% for time series of 500 data points. From these results, we argue on the value of using pTE to analyze a large number of short time series, which is often the case when causality methods are used to build complex networks from observed data. We remark that all the codes used to generate the results shown in this article are publicly available at GitHub50. The use of time-shifted (T-S) surrogates51,53 results in a substantial reduction of the computational time, in comparison to the widely used IAAFT surrogates, as seen in Fig. 5 and Table 2. The computational cost is reduced by approximately $$98\%$$, albeit displaying very similar results in terms of power and size. Clearly, T-S surrogates give a major boost in causality testing. As an example, for time series of length $$N=100$$, using pTE with T-S surrogates will reduce the computational cost by approximately $$82\%$$ with respect to GC with IAAFT surrogates, while a reduction of approximately $$77\%$$ is found with respect to GC with T-S surrogates. However, for causal inference T-S surrogates should be used with caution, because when there are time-delayed interactions, it can lead to fake conclusions. To study the resilience to observational noise, we add, to the time series generated with the DGPs, X and Y, a Gaussian noise $$\xi _{1,2}$$ of zero mean and unit variance, tuning its contribution with a parameter $$D\in [0,1]$$. In this way we generate and analyse the signals $$X^{'}$$ and $$Y^{'}$$ given by $$X^{'}_t = (1-D)X_t + D\xi _{1t}$$, $$Y^{'}_t = (1-D)Y_t + D\xi _{2t}$$. Figure 6 shows that pTE and GC perform very similarly (they are almost indistinguishable) and are quite resilient to noise. For the linear DGP, up to 40% of noise contribution can be present without a significant effect on the results, while for the nonlinear DGP, the methods start failing for a lower noise level. For the chaotic DGP the three methods are very resilient to noise. As previously noticed in Fig. 4, TE detects causality in both directions. Finally, moving beyond synthetic data, we apply the pTE measure to two well-known climatic indices, and compare the results with GC and TE. The time series analysed, the NINO3.4 index and All India Rainfall (AIR) index, shown in Fig. 7, represent the dynamics of two large-scale climatic phenomena, the El Niño–Southern Oscillation (ENSO) and the Indian Summer Monsoon (ISM), whose causal inter-relationship is represented by long-range links (tele-connections) between the Central Pacific and the Indian Ocean54. The time series were downloaded from55. The NINO3.4 index begins in 1854 while AIR index begins in 1813. Monthly-mean values are available, and their shared period is from 1854 to 2006 (153 years, 1836 months), Table 3 displays the results of the analysis of monthly-sampled data, and of yearly-sampled data. In the latter case we used the average of December, January and February (DJF) values, where the ENSO phenomenon peaks, and the average of June, July and August (JJA), where the monsoon peaks. Therefore, the length of the yearly-sample time series is 152 data points because for the last year the last data point, DJF, is not available. We used, for the yearly-sampled data, an autoregressive integrated moving average (ARIMA) model of order 4 (consistent with16) and, for the monthly-sampled data, of order 3. The order of the model was selected by using the Akaike information criterion (AIC). In Table 3 we see that for the yearly-sampled data, pTE and GC only detect the dominant causality (ENSO$$\rightarrow$$AIR), while TE detects both (in good agreement with16). We note similarities with the results presented in Fig. 4: while unidirectional causality is found with pTE and GC, TE causality is found in both directions. The computational times clearly show that pTE is faster than GC (and of course also faster than TE, which is the slowest method). In the monthly-sampled data we see an opposite direction of causality, a result that we interpret as due to different time scales in the mutual influence between ENSO and ISM: while ENSO effects on the Indian monsoon precipitations are pronounced on an annual time scale, the influence of the Indian monsoon on ENSO acts on a shorter, monthly time scale. To exclude the fact that this change in directionality is an artifact due to the different time series lengths, we analyzed the monthly-sampled time series using segments of 152 consecutive data points (which is the length of the annually-sampled data). In this case we did not find any significant causality, which suggests that the change in directionality when considering annually-sampled or monthly-sampled data is not an artifact but has a physical origin, that we interpret as due to different time scales in the mutual interaction and that 152 data points are not sufficient to find causality (in any direction) in the monthly-sampled data. Finally, we note that the computational times shown in Table 3 are higher than those that can be estimated from Fig. 5. In Fig. 5 we see that, for 150 datapoints, the time required for the GC calculation with T-S surrogate analysis is about 0.11 s while in Table 3 we see that the time required for GC and T-S calculation (two directions) is 0.36 s. The difference is due to the fact that in Fig. 5 a model of order 1 was used, while in Table 3, for the yearly-sampled data, a model of order 4 is used. The computational time increases with the order of the model, especially for GC, because the algorithm used (statsmodels grangercausalitytest) computes causality for all model orders up to the chosen one. For the NINO3.4 and AIR indices we also analysed the effect of varying the order of the model (from 1 to 10) and found either the same significant causal directionality (with stronger or weaker values), or we did not find any significant causality. ## Discussion We have proposed a new measure, pseudo transfer entropy (pTE), to infer causality in systems composed by two interacting processes. Using synthetic time series generated with processes where the underlying causality is known, and also, a real-world example of two well-known climatic indices, we have found a remarkable similarity between the results of pTE and Granger causality (GC), in terms of the power and size, and the robustness to noise, but pTE can be significantly faster, particularly for short time series. For example, for time series of 100 datapoints, while giving extremely similar results, pTE with time-shifted (T-S) surrogate testing reduces the computational time by approximatelly 92% with respect to GC with IAAFT surrogate testing, and by 48% with respect to GC with T-S surrogate testing (on Google colab CPU, the total computational time for pTE and T-S is 2.5 ms, while for GC and IAAFT is 32.5 ms, and for GC and T-S, 4.7 ms). Since the computational cost is of capital importance for the analysis of large datasets, the causality testing methodology proposed here will be extremely valuable for the analysis of short and noisy time series whose probability distributions are approximately Gaussian. We remark that many real-world signals follow distributions that are nearly normal. Although we do not claim that our method can be applied to any pair of signals, the information presented in the Additional information supports the method’s generic applicability. The algorithms are freely downloadable from GitHub50. ## Methods ### Derivation of the pseudo Transfer Entropy (pTE) Transfer entropy20 is a well-known measure that quantifies the directionality of information transfer between two processes. In the case of information transfer from process Y to X, it is defined as \begin{aligned} TE = \sum _{i,j} p\left( i_{n+1}, i_n^{(k)}, j_n^{(l)}\right) \log \left[ \frac{p\left( i_{n+1}\mid i_n^{(k)}, j_n^{(l)}\right) }{p\left( i_{n+1}\mid i_n^{(k)}\right) }\right] , \end{aligned} (1) where $$p(\cdot , \cdot , \cdot )$$ and $$p(\cdot \| \cdot )$$ are joint and conditional probability distributions that describe the processes, $$i_{n+1}$$ represents the state of process X at time step $$n+1$$, $$i_n^{(k)}$$ and $$j_n^{(k)}$$ are shorthand notations that represent the states of X and Y the previous k time steps, $$i_n^{(k)}=\{ i_n, \dots , i_{n-k+1}\}$$, $$j_n^{(k)}=\{ j_n, \dots , j_{n-k+1}\}$$. Equation (1) can be re-written as \begin{aligned} T_{Y\rightarrow X} = \sum _{i,j} p\left( i_{n+1}, i_n^{(k)}, j_n^{(l)}\right) \left\{ \log \left[ p\left( i_{n+1}\mid i_n^{(k)}, j_n^{(l)}\right) \right] - \log \left[ p\left( i_{n+1}\mid i_n^{(k)}\right) \right] \right\} , \end{aligned} (2) which, by using the definition of conditional probabilities and entropies, can be re-written as \begin{aligned} T_{Y\rightarrow X} = H\left( i_n^{(k)}, j_n^{(l)}\right) - H\left( i_{n+1}, i_n^{(k)}, j_{n}^{(l)}\right) + H\left( i_{n+1}, i_n^{(k)}\right) - H\left( i_n^{(k)}\right) . \end{aligned} (3) The computation of the TE with Eq. (1) is challenging because a good estimation of the probability distributions is often not available. Considering the processes X and Y to follow normal distributions i.e. $$X \sim {\mathscr {N}}(x\mid \mu _x, \Sigma _x)$$ and $$Y \sim {\mathscr {N}}(y\mid \mu _y, \Sigma _y)$$, the computation simplifies substantially, using in fact that the entropy of a p-variate normal variable x, is given by \begin{aligned} H_p\left( x\right) = \int _{-\infty }^{+\infty }{\mathscr {N}}(x\mid \mu _x, \Sigma _x) \log \left[ {\mathscr {N}}\left( x\mid \mu _x, \Sigma _x\right) \right] dx = -{\mathbb {E}}\left[ \log \left( {\mathscr {N}}(x\mid \mu _x, \Sigma _x)\right) \right] . \end{aligned} (4) By definition of the multivariate Gaussian, we can rewrite Eq. (4) as \begin{aligned} H_p\left( x\right) = -{\mathbb {E}}\left[ \log \left( (2\pi )^{-\frac{p}{2}}\mid \Sigma \mid ^{-\frac{1}{2}} e^{-\frac{1}{2}(x-\mu _x)^{T}\Sigma _x^{-1}(x-\mu _x)} \right) \right] , \end{aligned} (5) which, by the property of the logarithm of products becomes \begin{aligned} H_p\left( x\right) = \frac{p}{2}\log (2\pi ) +\frac{1}{2}\log (\mid \Sigma _x\mid ) + \frac{1}{2}{\mathbb {E}}\left[ (x-\mu _x)^T\Sigma ^{-1}(x-\mu _x)\right] . \end{aligned} (6) By noticing that $${\mathbb {E}}\left[ (x-\mu _x)^T\Sigma _x^{-1}(x-\mu _x)\right] = tr(\Sigma _x^{-1}\Sigma _x) = p$$, we obtain \begin{aligned} H_p(x) = \frac{1}{2}\left( p+p\log (2\pi ) + \log \|\Sigma _x\|\right) , \end{aligned} (7) where $$\|\Sigma \|$$ is the determinant of the $$p \times p$$ positive definite covariance matrix. By substituting Eq. (7) in Eq. (3), we can estimate the Transfer Entropy as follows: \begin{aligned} \begin{aligned} TE_{Y\rightarrow X}&= \frac{1}{2}\left[ k+l + (k+l)\log (2\pi ) + \log \left( \left\| \Sigma \left( {\mathbf {I}}^{(k)}_n\oplus {\mathbf {J}}^{(l)}_n \right) \right\| \right) \right] \\&\quad - \frac{1}{2}\left[ k+l+1+(k+l+1)\log (2\pi ) + \log \left( \left\| \Sigma \left( {\mathbf {i}}_{n+1}\oplus {\mathbf {I}}^{(k)}_n\oplus {\mathbf {J}}^{(l)}_n \right) \right\| \right) \right] \\&\quad + \frac{1}{2}\left[ k+1 + (k+1)\log (2\pi ) + \log \left( \left\| \Sigma \left( {\mathbf {i}}_{n+1}\oplus {\mathbf {I}}^{(k)}_n\right) \right\| \right) \right] \\&\quad -\frac{1}{2}\left[ k+ k \log (2\pi ) + \log \left( \left\| \Sigma \left( {\mathbf {I}}^{(k)}_n\right) \right\| \right) \right] ,\\ \end{aligned} \end{aligned} (8) which finally can be written as \begin{aligned} TE_{Y\rightarrow X} = \frac{1}{2} \log \left( \frac{\left\| \Sigma \left( {\mathbf {I}}^{(k)}_n\oplus {\mathbf {J}}^{(l)}_n\right) \right\| \cdot \left\| \Sigma \left( {\mathbf {i}}_{n+1}\oplus {\mathbf {I}}^{(k)}_n\right) \right\| }{\left\| \Sigma \left( {\mathbf {i}}_{n+1}\oplus {\mathbf {I}}^{(k)}_{n} \oplus {\mathbf {J}}^{(l)}_n\right) \right\| \cdot \left\| \Sigma \left( {\mathbf {I}}^{(k)}_n\right) \right\| }\right) , \end{aligned} (9) where $$\Sigma (A\oplus B)$$ is the covariance of the concatenation of matrices A and B, $${\mathbf {i}}_{n+1}$$ is the vector of the future values of X, $${\mathbf {I}}^{(k)}_n$$ and $${\mathbf {J}}^{(l)}_n$$ are the matrices containing the previous k and l values of processes X and Y respectively. Whenever X and Y are not Gaussian processes, we call the quantity in Eq. (9) pseudo Transfer entropy (pTE). For Gaussian variables pTE coincides with the Transfer Entropy and is equivalent to Granger causality35. The Gaussian form for CMI/TE for causality inference was also previously used56,57,58,59. ### Statistical significance We used surrogate data to test the significance of the pTE, TE and GC values. The number of surrogates needed depends on the characteristics of the data, the available computational resources and time limitations: given enough resources and time, one should use a large number of surrogates and select a confidence interval19; however, with limited time or computational resources, when the spread of surrogates data is not too large one can use an alternative strategy: analyze a small number of surrogates and, in the case of a one sided test, select as significance threshold the maximum or minimum value obtained with the surrogates. In this case, $$M = K/\alpha -$$1 surrogates should be generated, where K is a positive integer number and $$\alpha$$ is the probability of false rejection51. Therefore, a minimum of 19 surrogates ($$K=1$$) are required for a significance level of $$95\%$$. We used the algorithm developed by Schreiber and Schmitz60,61 known as iterative amplitude adjusted Fourier transform (IAAFT), which preserves both, the amplitude distribution and the power spectrum (for details, see Lancaster et al.51 and references therein). The python routine to compute the IAAFT surrogates is contained in the NoLiTSA package62. We also tested the time-shifted (T-S) surrogates51,53, which consist in randomly choosing a time shift independently for each surrogate and then shifting the signal in time, wrapping its end to the beginning. These surrogates are very fast to generate and they fully preserve all the properties of the original signal. Both surrogates test the null hypothesis of two processes with arbitrary linear or nonlinear structure but without linear or nonlinear inter-dependencies. ### Implementation To calculate pTE we developed an algorithm in python (available on GitHub50), while we used the statsmodels implementation of GC63 and the pyunicorn implementation of TE64. The code has been thought to be as user friendly as possible to be used to build networks. It takes as arguments all the time series of the studied system, the embedding parameter and the statistical significance test that the user decides to apply. As result it returns the matrix of pTE values computed from the original data, and the matrix of the maximum values obtained from the surrogates (i.e., the statistically significant thresholds). In the analysis of synthetic data generated with the DGPs the causality measures were run over 1000 realizations with different initial conditions and noise seeds. For each realization the first 100 data points were discarded. For the computation of GC and pTE we chose a lag equal to 1, which implies considering the models as auto-regressive processes of order 1, AR(1), since by the considered models construction, the dependent variable is influenced by the previous step of the independent one; for the computation of TE the k-nearest neighbors method is used, and we chose $$k=\sqrt{N}$$, where N is the number of data points in the time series65. In the analysis of the empirical data, from the physics of the problem, the choice of the order of the AR model used to represent the data is not trivial. We used an autoregressive integrated moving average (ARIMA) and the Akaike information criterion (AIC) to select order 4 for yearly-sampled data and order 3, for the monthly-sampled data. To calculate the causality between two time series, the time series were first linearly detrended and L2-normalized. The significance of the pTE, GC and TE values obtained were then tested against the values obtained from 19 couples of surrogates (as explained in the previous section, 19 surrogates is the minimum for achieving a significance level of $$95\%$$). Unless otherwise specifically stated, the results presented in the text were obtained by using IAAFT surrogates. ## Models In the main text three data generating processes (DGPs) were analyzed. For these DGPs the null hypothesis of non-causality is not satisfied for process Y to process X. Results obtained with other DGPs are presented in the Additional information. The first DGP is a linear model66 given by: \begin{aligned} X_t=0.6X_{t-1} + C\cdot Y_{t-1} +\epsilon _{1t}, \qquad Y_t = 0.6Y_{t-1} + \epsilon _{2t}, \end{aligned} (10) where $$\epsilon _{1t}$$ and $$\epsilon _{2t}$$ are white noises with zero mean and unit variance, and C is the coupling strength. The second DGP is a nonlinear model67 that reads: \begin{aligned} X_t = 0.5X_{t-1}+C\cdot Y_{t-1}^2 +\epsilon _{1t}, \qquad Y_t = 0.5Y_{t-1} + \epsilon _{2t}. \end{aligned} (11) The third DGP consists of two Lorenz chaotic systems, coupled on the first variable: \begin{aligned} \begin{array}{ll} {\dot{X}}_{1} = 10(-X_1+X_2)+ C\cdot (Y_1-X_1) &{}\quad {\dot{Y}}_{1} = 10(-Y_1+Y_2)\\ {\dot{X}}_{2} = 21.5X_{1} - X_{2} -X_1X_3 &{}\quad {\dot{Y}}_{2} = 20.5Y_{1} - Y_{2} -Y_1Y_3 \\ {\dot{X}}_{3} = X_{1}X_2 - \frac{8}{3}Y_3 &{}\quad {\dot{Y}}_{3} = Y_{1}Y_2 - \frac{8}{3}Y_3\\ \end{array} \end{aligned} (12) Examples of time series of these three DGPs, normalized to zero mean and unit variance, are displayed in Fig. 8. ### Comparison with literature The linear DGP was used by Diks and DeGoede66 to test nonlinear Granger causality. With a coupling strength of $$C=0.5$$ and a time series length of 100 points with a lag of 1, they obtained a power of 95.6 and a size of 3.0. Using pTE under the same conditions, we obtain a power of 99.8 and a size of 3.9. The nonlinear DGP was used by Taamouti et al.67 to quantify linear and nonlinear Granger causalities. With a coupling strength of $$C = 0.5$$, 200 data points, a pvalue of 5% and a resampling bandwidth k for the bootstrap as the integer part of $$2 \cdot 200^{1/2}$$, they obtained a power of 100 and a size of 4.4. Using pTE we obtained a power of 100 and a size of 3.3. The coupled Lorenz systems studied by Krakovská et al.68, are very similar to those studied here. By using three state-space based methods, including cross-mapping, they noticed that the highest directionality in the causality is for a coupling $$C \approx 4$$. From $$C > 4$$ synchronization is obtained, finding causality in both directions, using time series of 50000 data points. This observation is very similar to our results with TE, while for pTE and GC, once synchronization has been achieved, no causality is found. This supports their conclusion, warning the reader that the blind application of causality test can easily lead to incorrect conclusions. While GC and pTE can successfully be used to analyze AR processes and weakly nonlinear Gaussian-like processes, for more complex processes (high dimensional and/or highly nonlinear) advanced information-theoretic methods such as TE are needed. ## References 1. Wiener, N. Nonlinear prediction and dynamics. Proc. Third Berkeley Symp. Math. Stat. Probab. 3, 247–252 (1956). 2. Granger, C. W. J. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438 (1969). 3. Baccala, L. A. & Sameshima, K. Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84, 463–474. https://doi.org/10.1007/PL00007990 (2001). 4. Chen, Y., Rangarajan, G., Feng, J. & Ding, M. Analyzing multiple nonlinear time series with extended Granger causality. Phys. Lett. A 324, 26. https://doi.org/10.1016/j.physleta.2004.02.032 (2004). 5. Dhamala, M., Rangarajan, G. & Ding, M. Estimating Granger causality from Fourier and wavelet transforms of time series data. Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.100.018701 (2008). 6. Marinazzo, D., Pellicoro, M. & Stramaglia, S. Kernel method for nonlinear Granger causality. Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.100.144103 (2008). 7. Amblard, P. & Michel, O. The relation between Granger causality and directed information theory: a review. Entropy 15, 113–143. https://doi.org/10.3390/e15010113 (2013). 8. Barnett, L. & Seth, A. K. The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference. J. Neurosci. Methods 223, 50–68. https://doi.org/10.1016/j.jneumeth.2013.10.018 (2014). 9. Hiemstra, C. & Jones, J. D. Testing for linear and nonlinear Granger causality in the stock price-volume relation. J. Finance 49, 1639–1664 (1994). 10. Chiou-Wei, S. Z., Chen, C.-F. & Zhu, Z. Economic growth and energy consumption revisited: evidence from linear and nonlinear Granger causality. Energy Econ. 30, 3063–3076. https://doi.org/10.1016/j.eneco.2008.02.002 (2008). 11. Salahuddin, M. & Gow, J. The effects of internet usage, financial development and trade openness on economic growth in South Africa: a time series analysis. Telematics Inform. 33, 1141–1154. https://doi.org/10.1016/j.tele.2015.11.006 (2016). 12. Seth, A. K., Barrett, A. B. & Barnett, L. Granger causality analysis in neuroscience and neuroimaging. J. Neurosci. 35, 3293–3297. https://doi.org/10.1523/JNEUROSCI.4399-14.2015 (2015). 13. Porta, A. & Faes, L. Wiener–Granger causality in network physiology with applications to cardiovascular control and neuroscience. Proc. IEEE 104, 282–309. https://doi.org/10.1007/PL000079900 (2016). 14. Mosedale, T. J., Stephenson, D. B., Collins, M. & Mills, T. C. Granger causality of coupled climate processes: ocean feedback on the north Atlantic oscillation. J. Clim. 19, 1182–1194. https://doi.org/10.1007/PL000079901 (2006). 15. Tirabassi, G., Masoller, C. & Barreiro, M. A study of the air-sea interaction in the South Atlantic convergence zone through Granger causality. Int. J. Climatol. 35, 3440–3453. https://doi.org/10.1007/PL000079902 (2014). 16. Tirabassi, G., Sommerlade, L. & Masoller, C. Inferring directed climatic interactions with renormalized partial directed coherence and directed partial correlation. Chaos https://doi.org/10.1007/PL000079903 (2017). 17. McGraw, M. C. & Barnes, E. A. Memory matters: a case for Granger causality in climate variability studies. J. Clim. 31, 3289–3300. https://doi.org/10.1007/PL000079904 (2018). 18. Runge, J. et al. Inferring causation from time series in earth system sciences. Nat. Commun. 10, 2553. https://doi.org/10.1007/PL000079905 (2019). 19. Paluš, M. & Vejmelka, M. Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections. Phys. Rev. E https://doi.org/10.1007/PL000079906 (2007). 20. Schreiber, T. Measuring information transfer. Phys. Rev. Lett. 85, 461–464 (2000). 21. Pereda, E., Quiroga, R. Q. & Bhattacharya, J. Nonlinear multivariate analysis of neurophysiological signals. Prog. Neurobiol. 77, 1–37. https://doi.org/10.1007/PL000079907 (2005). 22. Staniek, M. & Lehnertz, K. Symbolic transfer entropy. Phys. Rev. Lett. https://doi.org/10.1007/PL000079908 (2008). 23. Lizier, J. T., Heinzle, J., Horstmann, A., Haynes, J.-D. & Prokopenko, M. Multivariate information-theoretic measures reveal directed information structure and task relevant changes in FMRI connectivity. J. Comput. Neurosci. 30, 85–107 (2011). 24. Vicente, R., Wibral, M., Lindner, M. & Pipa, G. Transfer entropy—a model-free measure of effective connectivity for the neurosciences. J. Comput. Neurosci. 30, 45–67 (2011). 25. Wibral, M. et al. Measuring information-transfer delays. PLoS ONE https://doi.org/10.1007/PL000079909 (2013). 26. Bielczyk, N. Z. et al. Disentangling causal webs in the brain using functional magnetic resonance imaging: a review of current approaches. Netw. Neurosci. 3, 1. https://doi.org/10.1016/j.physleta.2004.02.0320 (2019). 27. Faes, L., Nollo, G. & Porta, A. Information-based detection of nonlinear Granger causality in multivariate processes via a nonuniform embedding technique. Phys. Rev. E https://doi.org/10.1016/j.physleta.2004.02.0321 (2011). 28. Faes, L., Nollo, G. & Porta, A. Non-uniform multivariate embedding to assess the information transfer in cardiovascular and cardiorespiratory variability series. Comput. Biol. Med. 42, 290–297. https://doi.org/10.1016/j.physleta.2004.02.0322 (2013). 29. Mueller, A. et al. Causality in physiological signals. Physiol. Meas. 37, R46–R72. https://doi.org/10.1016/j.physleta.2004.02.0323 (2016). 30. Pompe, B. & Runge, J. Momentary information transfer as a coupling measure of time series. Phys. Rev. E https://doi.org/10.1016/j.physleta.2004.02.0324 (2011). 31. Deza, J. I., Barreiro, M. & Masoller, C. Assessing the direction of climate interactions by means of complex networks and information theoretic tools. Chaos https://doi.org/10.1016/j.physleta.2004.02.0325 (2015). 32. Sandoval, L. J. Structure of a global network of financial companies based on transfer entropy. Entropy 16, 4443. https://doi.org/10.1016/j.physleta.2004.02.0326 (2014). 33. Porfiri, M. et al. Media coverage and firearm acquisition in the aftermath of a mass shooting. Nat. Hum. Behav. 3, 913. https://doi.org/10.1016/j.physleta.2004.02.0327 (2019). 34. Paluš, M., Albrecht, V. & Dvořák, I. Information theoretic test for nonlinearity in time series. Phys. Lett. A 175, 203–209. https://doi.org/10.1016/j.physleta.2004.02.0328 (1993). 35. Barnett, L., Barrett, A. B. & Seth, A. K. Granger causality and transfer entropy are equivalent for gaussian variables. Phys. Rev. Lett. https://doi.org/10.1016/j.physleta.2004.02.0329 (2009). 36. Sugihara, G. et al. Detecting causality in complex ecosystems. Science 338, 496–500. https://doi.org/10.1103/PhysRevLett.100.0187010 (2012). 37. Kugiumtzis, D. Direct-coupling information measure from nonuniform embedding. Phys. Rev. E https://doi.org/10.1103/PhysRevLett.100.0187011 (2013). 38. Ma, H., Aihara, K. & Chen, L. Detecting causality from nonlinear dynamics with short-term time series. Sci. Rep. 4, 1–10. https://doi.org/10.1103/PhysRevLett.100.0187012 (2014). 39. Sun, J., Taylor, D. & Bollt, E. M. Causal network inference by optimal causation entropy. SIAM J. Appl. Dyn. Syst. 14, 73–106. https://doi.org/10.1103/PhysRevLett.100.0187013 (2015). 40. Jiang, J. J., Huang, Z. G., Huang, L., Liu, H. & Lai, Y. C. Directed dynamical influence is more detectable with noise. Sci. Rep. 6, 1–9. https://doi.org/10.1103/PhysRevLett.100.0187014 (2016). 41. Zhao, J., Zhou, Y., Zhang, X. & Chen, L. Part mutual information for quantifying direct associations in networks. Proc. Natl. Acad. Sci. U. S. A. 113, 5130–5135 (2016). 42. Hirata, Y. et al. Detecting causality by combined use of multiple methods: climate and brain examples. PLoS ONE https://doi.org/10.1103/PhysRevLett.100.0187015 (2016). 43. Ma, H. et al. Detection of time delays and directional interactions based on time series from complex dynamical systems. Phys. Rev. E 96, 1–8. https://doi.org/10.1103/PhysRevLett.100.0187016 (2017). 44. Harnack, D., Laminski, E., Schünemann, M. & Pawelzik, K. R. Topological causality in dynamical systems. Phys. Rev. Lett. 119, 1–5. https://doi.org/10.1103/PhysRevLett.100.0187017 (2017). 45. Vannitsem, S. & Ekelmans, P. Causal dependences between the coupled ocean-atmosphere dynamics over the tropical pacific, the north pacific and the north atlantic. Earth Syst. Dyn. 9, 1063–1083. https://doi.org/10.1103/PhysRevLett.100.0187018 (2018). 46. Runge, J., Nowack, P., Kretschmer, M., Flaxman, S. & Sejdinovic, D. Detecting and quantifying causal associations in large nonlinear time series datasets. Sci. Adv. 5, eaau4996. https://doi.org/10.1126/sciadv.aau4996 (2019). 47. Korenek, J. & Hlinka, J. Causal network discovery by iterative conditioning: comparison of algorithms. Chaos https://doi.org/10.1063/1.5115267 (2020). 48. Nowack, P., Runge, J., Eyring, V. & Haigh, J. D. Causal networks for climate model evaluation and constrained projections. Nat. Commun. 11, 1415. https://doi.org/10.1038/s41467-020-15195-y (2020). 49. Leng, S. et al. Partial cross mapping eliminates indirect causal influences. Nat. Commun. 11, 2632. https://doi.org/10.1038/s41467-020-16238-0 (2020). 50. Lancaster, G., Iatsenko, D., Pidde, A., Ticcinelli, V. & Stefanovska, A. Surrogate data for hypothesis testing of physical systems. Phys. Rep. 748, 1–60. https://doi.org/10.1016/j.physrep.2018.06.001 (2018). 51. Paluš, M., Komárek, V., Hrnčíř, Z. C. V. & Štěrbová, K. Synchronization as adjustment of information rates: detection from bivariate time series. Phys. Rev. E https://doi.org/10.1103/PhysRevE.63.046211 (2001). 52. Quiroga, R. Q., Kraskov, A., Kreuz, T. & Grassberger, P. Performance of different synchronization measures in real data: a case study on electroencephalographic signals. Phys. Rev. E https://doi.org/10.1103/PhysRevE.65.041903 (2002). 53. Dijkstra, H. A., Hernandez-Garcia, E., Masoller, C. & Barreiro, M. Networks in Climate (Cambridge University Press, Cambridge, 2019). 55. Molini, A., Katul, G. G. & Porporato, A. Causality across rainfall time scales revealed by continuous wavelet transforms. J. Geophys. Res. Atmos. https://doi.org/10.1029/2009JD013016 (2010). 56. Paluš, M. Multiscale atmospheric dynamics: cross-frequency phase-amplitude coupling in the air temperature. Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.112.078702 (2014). 57. Paluš, M. Cross-scale interactions and information transfer. Entropy 16, 5263–5289. https://doi.org/10.1126/sciadv.aau49963 (2014). 58. Cliff, O. M., Novelli, L., Fulcher, B. D., Shine, J. M. & Lizier, J. T. Assessing the significance of directed and multivariate measures of linear dependence between time series. Phys. Rev. Res. https://doi.org/10.1126/sciadv.aau49964 (2021). 59. Schreiber, T. & Schmitz, A. Improved surrogate data for nonlinearity tests. Phys. Rev. Lett. 77, 635–638. https://doi.org/10.1126/sciadv.aau49965 (1996). 60. Schreiber, T. & Schmitz, A. Surrogate time series. Physica D 142, 346–382. https://doi.org/10.1126/sciadv.aau49966 (2000). 62. Donges, J. et al. Unified functional network and nonlinear time series analysis for complex systems science: the pyunicorn package. Chaos https://doi.org/10.1126/sciadv.aau49969 (2015). 63. Lall, U. & Sharma, A. A nearest nighbor bootstrap for resampling hydrologic time seriess. Water Resour. Res. 32, 679–693. https://doi.org/10.1063/1.51152670 (1996). 64. Diks, C. G. H. & DeGoede, J. A General Nonparametric Bootstrap Test for Granger Causality (Institute of Physics, London, 2001). 65. Taamouti, A., Bouezmarni, T. & Ghouch, A. E. Nonparametric estimation and inference for conditional density based Granger causality measures. J. Econ. 180, 251–264. https://doi.org/10.1063/1.51152671 (2014). 66. Krakovská, A. et al. Comparison of six methods for the detection of causality in a bivariate time series. Phys. Rev. E https://doi.org/10.1063/1.51152672 (2018). 67. Péguin-Feissolle, A. & Teräsvirta, T. Causality tests in a nonlinear framework. Working paper, Stockholm School of Economics, Stockholm (2001). 68. Vilasuso, J. Causality tests and conditional heteroskedasticity: Monte Carlo evidence. J. Econ. 101, 25–35 (2001). 69. Tjostheim, T. Granger-causality in multiple time series. J. Econ. 17, 157–176 (1981). 70. He, F., Billings, S. A., Wei, H.-L. & Sarrigiannis, P. G. A nonlinear causality measure in the frequency domain: nonlinear partial directed coherence with applications to EEG. J. Neurosci. Methods 225, 71–80 (2014). 71. Rössler, O. E. An equation for continuous chaos. Phys. Lett. 57A, 397–398. https://doi.org/10.1063/1.51152673 (1976). 72. Aragoneses, A., Perrone, S., Sorrentino, T., Torrent, M. C. & Masoller, C. Unveiling the complex organization of recurrent patterns in spiking dynamical systems. Sci. Rep. 4, 1–6. https://doi.org/10.1063/1.51152674 (2014). ## Acknowledgements All of the computation of this article was done using free software and we are indebted to the developers and maintainers of the following packages: Google colab, TeXmaker, python, python-numpy, scikits.statsmodels to mention only a few. This work received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 8138444, Climate Advanced Forecasting of sub-seasonal Extremes (ITN CAFE). C.M. also acknowledges funding by the Spanish Ministerio de Ciencia, Innovacion y Universidades (PGC2018-099443-B-I00) and the ICREA ACADEMIA program of Generalitat de Catalunya. ## Author information Authors ### Contributions R.S. conducted the experiments, analyzed the results. C.M. supervised the study. Both authors wrote and reviewed the manuscript. ### Corresponding author Correspondence to Riccardo Silini. ## Ethics declarations ### Competing Interests The authors declare no competing interests. ### Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Silini, R., Masoller, C. Fast and effective pseudo transfer entropy for bivariate data-driven causal inference. Sci Rep 11, 8423 (2021). https://doi.org/10.1038/s41598-021-87818-3 • Accepted: • Published: • DOI: https://doi.org/10.1038/s41598-021-87818-3	2022-05-20 06:33:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8668285012245178, "perplexity": 2257.2386422480226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531762.30/warc/CC-MAIN-20220520061824-20220520091824-00355.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/electron-photon-wavelength-019-nm-isthe-momentum-electron-b-photon-energy-c-electron-d-pho-q393230	## Tricky problem that no one solve An electron and a photon each have a wavelength of 0.19 nm. What isthe momentum of the (a)electron and (b) photon?What is the energy of the (c) electron and (d) photon?	2013-05-23 08:30:43	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8683975338935852, "perplexity": 2722.634294364119}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703035278/warc/CC-MAIN-20130516111715-00000-ip-10-60-113-184.ec2.internal.warc.gz"}
http://viennafem.sourceforge.net/viennafem-example-weakform.html	Content # Example: Deduction of the Weak Form While partial differential equations are typically posed in the strong form, e.g. $- \Delta u = f \ ,$ the finite element method is based on the weak form, which is under the assumption of homogeneous Neumann boundary conditions in the above case given by $\int_\Omega \nabla u \cdot \nabla v \: dx = \int_\Omega fv \: dx \quad \forall v \in \mathcal{V} \ ,$ where $V$ is a suitable test space. By an extension of the functionality in ViennaMath, ViennaFEM can also deduce the weak form automatically: Deducing the weak form out of the strong form using namespace viennamath; function_symbol u; // instantiate the unknown // instantiate Poisson equation with f=1: equation poisson_eq = make_equation(laplace(u), -1.0); std::cout << "Strong form: " << poisson_eq << std::endl; std::cout << "Weak form: " << viennafem::make_weak_form(poisson_eq) << std::endl; // Inhomogeneous permittivity: div( eps * grad(u)) = -1.0: viennafem::cell_quan<CellType> eps; eps.wrap_constant( my_key() ); equation poisson_eq2 = make_equation( div( eps * grad(u)), -1); std::cout << "Strong form: " << poisson_eq2 << std::endl; std::cout << "Weak form: " << viennafem::make_weak_form(poisson_eq2) << std::endl; The second form of the Poisson equation includes a diffusion coefficient 'eps', which is constant within each ViennaGrid cell (of type 'CellType'), but may vary among the cells inside the mesh.	2017-07-26 10:29:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8951699137687683, "perplexity": 8072.142994380274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549426133.16/warc/CC-MAIN-20170726102230-20170726122230-00552.warc.gz"}
https://zbmath.org/?q=an:0958.60038&format=complete	# zbMATH — the first resource for mathematics Diffusion approximation of integral functionals in merging and averaging scheme. (English. Ukrainian original) Zbl 0958.60038 Theory Probab. Math. Stat. 59, 101-107 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 99-105 (1998). The authors consider an integral functional $$\zeta^{\varepsilon}(t)$$ with rapid Markov switchings $$x^{\varepsilon}(t)$$ in series scheme of the following form: $$\zeta^{\varepsilon}(t)=\int_{0}^{t}a(x^{\varepsilon}(s)) ds$$. Diffusion approximations of such integral functional in averaging and merging schemes are proposed. The diffusion approximation is constructed for the normalized and centered integral functionals. ##### MSC: 60G25 Prediction theory (aspects of stochastic processes) 60J75 Jump processes (MSC2010) 60F17 Functional limit theorems; invariance principles 60J25 Continuous-time Markov processes on general state spaces	2021-02-25 16:53:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9108541011810303, "perplexity": 3459.7803253943953}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178351374.10/warc/CC-MAIN-20210225153633-20210225183633-00109.warc.gz"}
http://www.nhehs.gdst.net/pi-day/	# Pi Day On 14th of March, or 3.14 in the US, and so otherwise known as Pi day, we endeavoured to calculate pi in two different ways as well as holding a pi recital competition. First we used the equation for the time period of pendulum swing to calculate pi. This involved Miss Croft dangling a weight from the second floor atrium and two helpers timing. Next up was the pi recital competition, which revealed some impressive memory skills, where Li An from Year 8 came out on top with a fantastic 109 decimal places of pi. The pi-nnacle of the pi day celebrations was when we tried to calculate pi using real pies. We did this by making a circle out of pies with a diameter of pies and dividing the number of pies in the circumference by the number of pies in the diameter. Pi came out to be a reasonably accurate 3.114, which we were very pleased with, and the spectators were equally pleased with the fact that they could now eat the pies used in the calculation. Back to all news	2017-10-20 07:01:39	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8734851479530334, "perplexity": 1744.5164659478824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823839.40/warc/CC-MAIN-20171020063725-20171020083725-00277.warc.gz"}
http://bappelitbangda.bekasikota.go.id/l8ovr/foil-math-equation-a853de	Global Panacea Spirits, The Water Is Wide Tab, 3-2-1 Ribs With Orange Juice, Vijayanagar 4th Stage 3rd Phase, Mysore, San Ysidro Ranch Montecito Wedding, " /> Global Panacea Spirits, The Water Is Wide Tab, 3-2-1 Ribs With Orange Juice, Vijayanagar 4th Stage 3rd Phase, Mysore, San Ysidro Ranch Montecito Wedding, " /> Global Panacea Spirits, The Water Is Wide Tab, 3-2-1 Ribs With Orange Juice, Vijayanagar 4th Stage 3rd Phase, Mysore, San Ysidro Ranch Montecito Wedding, "/> ## foil math equation The table equivalent to the FOIL rule looks like this: In the case that these are polynomials, (ax + b)(cx + d), the terms of a given degree are found by adding along the antidiagonals, so FOIL is primarily used with multiplication of these binomials. I struggle a lot with online calculator with foil method problems . In this Yay Math algebra playlist: equation solving, FOIL multiplying binomials, Solving Systems of Equations Elimination Method, absolute value inequalities Note that this process involves a total of three applications of the distributive property. is right after i've done FOIL, or after the FOIL and simplified? To find the factors, you must determine which of the sets of factors result in the polynomial when multiplied together. We have got a whole lot of great reference materials on subject areas ranging from a quadratic to solving systems of linear equations Blog. FOIL is a mnemonic for the standard method of multiplying two binomials, hence the method may be referred to as the FOIL method. In that sense, we are actually UN-FOILing, but students are likely to know what we mean. d More. The FOIL method of factoring calls for you to follow the steps required to FOIL binomials, only backward. So, we’re going to take the smaller of the two polynomials and distribute its terms into the larger using all of the same techniques… Solving quadratic equations by completing square. Foil Method Algebra plays a vital role not only in mathematics, but in our day-to-day life problems too.It is the branch of mathematics dealing with constants and variables.The combination of variables and constants is termed as an expression.. FOIL Method The FOIL method is an important algebra method that defines how two binomials are multiplied. Just repeat first, outside, inside, last and you'll remember it. Make a table with the terms of the first polynomial on the left edge and the terms of the second on the top edge, then fill in the table with products. FOIL is a mnemonic for the standard method of multiplying two binomials, hence the method may be referred to as the FOIL method. Here we can find plenty of resources to develop our Math skills. Remember that FOIL stands for first, outer, inner, and last. Applying this format to the given equation of , must equal 11 and must equal 24. x If either binomial involves subtraction, the corresponding terms must be negated. ACT Math : How to use FOIL with exponents Study concepts, example questions & explanations for ACT Math. Right from how to use foil in a math equation to multiplying and dividing rational expressions, we have all of it discussed. In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation that can be rearranged in standard form as ax²+bx+c=0 where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. You are supposed to multiply these pairs as shown below! Faire des équations de bases: addition, soustraction, … (00:36) Faire une division d’un groupe de termes (01:35) Changer de ligne (02:58) Faire des relations: plus petit que, égal, … (03:30) Faire une racine cubique (04:25) Faire une intégrale (05:10) Faire une matrice (06:02) Ajouter des symboles (07:08) Vidéos suivantes: LibreOffice Writer – Créer un document rapidement. There are other methods, although FOIL tends to be the most popular. If things go this way, I fear I will not be able to pass my math exam. BFF! b This is the order in which we multiple the binominals. Apply Foil rule to multiply the terms in a specific order that is First, Outside, Inside, Last. The letters in FOIL refer to two terms (one from each of two binomials) multiplied together in a certain order: First, Outer, Inner, and Last. NASA! Popular pages @ mathwarehouse.com . Math Study Guide School Study Tips 9th Grade Math Maths Ncert Solutions Algebra Worksheets Math Formulas Math Vocabulary Math Jokes Basic Math More information ... People also love these ideas Solving equations is the bedrock of Algebra. FOIL stands for first, outer, inner and last pairs. There are other methods, although FOIL tends to be the most popular. d You are supposed to multiply these pairs as shown below! c The FOIL rule converts a product of two binomials into a sum of four (or fewer, if like terms are then combined) monomials. f ( x) = x3. I’d like to get started today with a challenge. In National 5 Maths learn how to remove brackets and pairs of brackets using the distributive law and FOIL, then simplify by collecting like terms. The FOIL method won’t work for anything other than two binomials because there are more terms than the acronym FOIL allows, as Math is Fun accurately points out. Cubic Foot To Pint Calculator. $(2x + 4) (5x + 3) = 10x^{2} + 26x + 12$, Your email address will not be published. Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there! Binomials are defined as equations that contain two same terms. Foil Method - Sample Math Practice Problems The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. We provide a ton of really good reference tutorials on matters ranging from subtracting to basic concepts of mathematics In case you actually demand advice with math and in particular with math problem solver foil factoring or multiplying and dividing rational come pay a visit to us at Algbera.com. The word FOIL is an acronym for the four terms of the product: Note that a is both a "first" term and an "outer" term; b is both a "last" and "inner" term, and so forth. You may speak with a member of our customer support team by calling 1-800-876-1799. [6] The reverse process is called factoring or factorization. Apply the F O I L method in math to this problem: 2 x + 13 2 x - 17; Don't foil around with partial work; get answers down on paper before you check our answers. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. Example: x² - 7x + 12. Come to Emaths.net and master precalculus i, linear systems and plenty of other math … 2 In the event that you actually need to have assistance with math and in particular with algebra reverse foil calculator or radical expressions come pay a visit to us at Solve-variable.com. The method is most commonly used to multiply linear binomials. It should be noted that FOIL can only be used for binomial multiplication. Community. Inner means multiply the innermost two terms. Given a general quadratic equation of the form ax²+bx+c=0 with x representing an unknown, a, b and c representing constants with a ≠ 0, the quadratic formula is: where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. a Some of the worksheets for this concept are Multiplying binomials date period, Algebra 1 foil equations work, Lesson 26 quadratic equations, Algebra i work multiplying polynomials foil and the, Solving quadratic factoring, Factoring polynomials, Solving radical equations, Solve each equation with the quadratic. Meter To Inch Calculator. Answer Save. We carry a whole lot of quality reference material on topics varying from variables to intermediate algebra Solve your math problems using our free math solver with step-by-step solutions. Get up to 50% off. Unique Math Equation Stickers designed and sold by artists. For example: (x + 2) (2x + 4) FOIL stands for First, Outer, Inner, and Last. 2(a^2 + a)(3a^2 + 6a) I know i would follow the foil steps but at which point would i multiply the equation by 2? Solving one step equations. FOIL is a mnemonic for the standard method of multiplying two binomials, hence the method may be referred to as the FOIL method. It is a term used to help people remember how to multiply polynomials together. If perhaps you demand service with math and in particular with Foil Equation Solver Cubic or addition come pay a visit to us at Polymathlove.com. Foil Equations - Displaying top 8 worksheets found for this concept.. Some of the worksheets for this concept are Multiplying binomials date period, Algebra 1 foil equations work, Lesson 26 quadratic equations, Algebra i work multiplying polynomials foil and the, Solving quadratic factoring, Factoring polynomials, Solving radical equations, Solve each equation with the quadratic. d dx ( 3x + 9 2 − x ) … Should you actually have assistance with algebra and in particular with foil solver or the quadratic formula come visit us at Algebra-equation.com. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. ( 6 years ago. Foil Equations - Displaying top 8 worksheets found for this concept.. ( Foil Math Problems. a Split -7x into -3x and -4x: x² - 3x - 4x + 12. No matter how much I try, I just am not able to solve any problem in less than an hour. Trump mocks Biden for wearing a mask amid pandemic. In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials[1]—hence the method may be referred to as the FOIL method. The FOIL method won’t work for anything other than two binomials because there are more terms than the acronym FOIL allows, as Math is Fun accurately points out.. FOIL Method The FOIL Method allows us to multiply two Binomials together. Foil le numérateur. In this Yay Math algebra playlist: equation solving, FOIL multiplying binomials, Solving Systems of Equations Elimination Method, absolute value inequalities. It should be noted that FOIL can only be used for binomial multiplication. (x+1) (x+2) = (x. x) + (x.2) + (1.x) + (1.2) Free Online Calculators. Required fields are marked *. Foil Method For Quadratic Equation - Displaying top 8 worksheets found for this concept.. Hope that helps! CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, First (“first” terms of each binomial are multiplied together), Outer (“outside” terms are multiplied—that is, the first term of the first binomial and the second term of the second), Inner (“inside” terms are multiplied—second term of the first binomial and first term of the second), Last (“last” terms of each binomial are multiplied). . Multiply x by 2x = 2×2. ) Nature of the roots of a quadratic equations. In the first step, the (c + d) is distributed over the addition in first binomial. What Does FOIL in Math Mean? Popular pages @ mathwarehouse.com . Using the FOIL method, a set of factors with the form will result in . Show Instructions. What is the F O I L method in mathematics? FOIL is not the only method that can be used. 2015 - Cette minute de culture générale illustre l'épisode 14 de la saison Mathématiques 6ème sur les fractions. An acronym is a common word or abbreviation where all of the letters stand for another word. The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. Thus, Similarly, to multiply (ax2 + bx + c)(dx3 + ex2 + fx + g), one writes the same table. a References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. First, we multiple the first two terms of each binomial. CREATE AN ACCOUNT Create Tests & Flashcards. The FOIL rule cannot be directly applied to expanding products with more than two multiplicands or multiplicands with more than two summands. Last means multiply the terms which occur last in each binomial. If ever you will need assistance with math and in particular with foil math calculator or standards come pay a visit to us at Algebra1help.com. NFL! In contrast to the FOIL method, the method using distributive can be applied easily to products with more terms such as trinomials and higher. White or transparent. d If a = 0, then the equation is linear, not quadratic, as there is no ax² term. + Right from equation foiler to terms, we have every aspect discussed. The word FOIL is an acronym for the four terms of the product: $(2x + 4) (5x + 3) = (2x \times 5x + 2x \times 3) + (4 \times 5x + 4 \times 3)$ What Does FOIL in Math Mean? Free algebra fomulas, common denominator calculator, multiply exponents calculator, expanding the square root formula. ©F 32h041 l2 n sK fuFt EaX 2S6owfjtJw Kair iem 9L XLoC a.H l 8A ylLl b 1r IixgThht9s T YrNeUsie Lrjv le Md9.g S eM Caudze l hw Ti 7t 6hA 4Ien TfWirnji Ct4eL iA LlegvedbSr oam i1v. Home. This is my first post in this forum. Then you combine any like terms, which usually come from the multiplication of the outside and inside terms. Using the unFOIL method to factor quadratic equations into two binomials requires many steps. x2 – 7x + 12. when you have to add to get 7 and multiply to get 12, What is this process called? In general, you can skip parentheses, but be very careful: e^3x is e 3 x, and e^ (3x) is e 3 x. b In particular, if the proof above is read in reverse it illustrates the technique called factoring by grouping. Solving quadratic equations by factoring. Let’s see hwo this works with an example: ABC! FOIL is not the only method that can be used. End of Conversation. So we get laugh out loud. These are just a few popular acronyms that you may know. [3], Many students and educators in the United States now use the word "FOIL" as a verb meaning "to expand the product of two binomials".[4]. x Videos. Microsoft Math Solver. Then Outer means multiply the outermost terms in the product. Here is a place for all things Math. And try to put it in a form like. For example, The FOIL method is equivalent to a two-step process involving the distributive law:[5]. We carry a whole lot of really good reference tutorials on subjects starting from rational numbers to math review Come to Algebra-equation.com and master adding and subtracting rational expressions, scientific notation and a large number of other math … ... Once you finish this lesson you'll be able to factor quadratic equations by using the FOIL method of multiplying two binomials in reverse. We provide a large amount of great reference materials on subject areas ranging from course syllabus to exam review The word FOIL is an acronym for the four terms of the product: First (“first” terms of each binomial are multiplied together) I struggle a lot with online calculator with foil method problems . You will need to get assistance from your school if you are having problems entering the answers into your online assignment. Algebra 1 Foil Equations Solving video tutorial help. Decorate your laptops, water bottles, helmets, and cars. ) Once the process is completed, then simplify the algebraic expression. The order of the four terms in the sum is not important and need not match the order of the letters in the word FOIL. However, applying the associative law and recursive foiling allows one to expand such products. Any time you need support with math and in particular with math solver using the foil method or logarithmic functions come visit us at Mathfraction.com. So in LOL, the first 'L' stands for laugh, the 'O' stands for out, and the last 'L' stands for loud. 4 takeaways from the most juvenile debate in history. The FOIL method lets you multiply two binomials in a particular order. FOIL: First - Outer - Inner - Last. $prove\:\tan^2\left (x\right)-\sin^2\left (x\right)=\tan^2\left (x\right)\sin^2\left (x\right)$. Sum and product of the roots of a quadratic equations Algebraic identities Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there! We are here to assist you with your math questions. x + We provide a large amount of great reference materials on subject areas ranging from course syllabus to exam review It helps you remember which order to … A visual memory tool can replace the FOIL mnemonic for a pair of polynomials with any number of terms. The FOIL Method allows us to multiply two Binomials together. Alternate methods based on distributing forgo the use of the FOIL rule, but may be easier to remember and apply. Your email address will not be published. Welcome to the movement. Playlist for Algebra 1. Relevance. For example, This article is about a mnemonic. Really, FOIL refers to multiplying the terms in parentheses to get the quadratic form. Foil and simplified outermost terms in the binomial d ) is distributed over the foil math equation first. Of polynomials with any number of terms: how to use FOIL with exponents Study concepts, example &! Algebra, trigonometry, calculus and more fastest factoring method which helps you solve for quadratic equations algebraic identities math. Problems as they appear in the product two binomials, solving Systems of equations Elimination,! Juvenile debate in history got a lot of high-quality reference material on matters ranging from multiplying worksheet. It illustrates the technique called factoring foil math equation factorization réduisant à nouveau ( si nécessaire ) to show you picture! And cars binomials, hence the method is equivalent to a two-step process involving distributive! Learning algebra for the standard method of multiplying two binomials using the FOIL method problems and dividing rational,! To simplify each of the roots of a quadratic equations into two binomials using the FOIL method absolute. Si nécessaire ) algebra fomulas, common denominator calculator, expanding the square root formula standard method multiplying. Exponents Study concepts, example questions & explanations for act math: how to multiply binomials. Binomials requires many steps into your online assignment for multiplying algebraic expressions using the distributive law is used to each. Not the only method that can be used develop our math skills - 3x - 4x + 12 identities math! Example: ( x + 2 ) ( 2x + 4 ) FOIL for. + 12. when you have to multiply polynomials together [ 6 ] the reverse process is called by... Expanding the square root formula two parts of the roots of a quadratic equations into two binomials in particular... Sign, so 5 x is equivalent to 5 ⋅ x the polynomial when multiplied together is the in! Actually have assistance with algebra and in particular, if the proof above is read in it... Polynomials with any number of terms is primarily used with multiplication of these binomials 1-800-876-1799... Picture represents a challenge for binomial multiplication must be negated to guessing so... Than an hour stand for first, Outer, Inner, and I m! Plenty of resources to develop our math skills FOIL multiplying binomials, hence the method may be referred to the! Fomulas, common denominator calculator, expanding the square root formula combine any like terms, which helps you for. Exponents Study concepts, example questions & explanations foil math equation act math terms together is.! Letters FOIL stand for first, we have all of it discussed it! How to multiply the terms in a particular order x\right ) $, with steps shown expression! Parce que l'équation contient cos 2 2x, vous devez appliquer la formule de puissance réduisant à (., common denominator calculator, multiply exponents calculator, expanding the square root formula, when. These are just a few popular acronyms that you may speak with a challenge with step-by-step solutions,! Much I try, I just am not able to solve any problem less! And ( x+2 ) are the two binomials Mathématiques 6ème sur les fractions it discussed )... To the given equation of, must equal 11 and must equal 11 and must 24. Your online assignment to learn once you remember what it stands for I struggle a lot with calculator! Math equation Stickers designed and sold by artists the addition in first binomial in sense. Member of our customer support team by calling 1-800-876-1799 in mathematics the FOIL mnemonic for a pair of with. ] the reverse process is completed, then the equation is linear, not quadratic, there... Polynomial when multiplied together picture, and I ’ m going to show you picture... Are other methods, although FOIL tends to be the most juvenile debate in history many steps solely as mnemonic! Done FOIL, we have all of it discussed is linear, not,. Like terms, which usually come from the most juvenile debate in history your online.. A more general method for multiplying algebraic expressions using the unFOIL method to factor equations... Term used to multiply two binomial equations and not for polynomials trump mocks Biden for wearing a mask pandemic... We take a quadratic equations get 7 and multiply to get assistance from your if... Particular with FOIL method problems ’ m going to show you a picture, and cars member! Trump mocks Biden for wearing a mask amid pandemic called factoring or factorization for another word x 2. That FOIL stands foil math equation in math the use of the sets of factors with the form will result.... The two terms definition of FOIL method problems a challenge, a set of factors result in example (. Any like terms, which usually come from the multiplication of these binomials method in mathematics the... A picture, and cars … using the distributive law is used to simplify each of letters..., what is the order in which we multiple the binominals for you to follow the steps required FOIL... An acronym is a term used to multiply binomials by following the FOIL method your school you. Equations algebraic identities Hi math gurus mode refer to the overall difficulty of the form,. The main program the opposite of FOIL method lets you multiply the terms... Method for multiplying algebraic expressions using the distributive law is used to people... Online calculator with FOIL method is not the only method that can be.... Applying this format to the overall difficulty of the roots of a general. This concept with step-by-step solutions when you have to multiply these pairs as shown below process is completed, the... In math associative law and recursive foiling allows one to expand such products solve any problem in less than hour... 5 x is equivalent to 5 ⋅ x our math skills simplify the algebraic.! 2 2x, vous devez appliquer la formule de puissance pour réduire cosinus try to put in! Can find plenty of resources to develop our math Solver ( Free ) Free algebra Solver... type in! The distributive law I try, I just am not able to pass my math exam: equation,! )$ the outside and inside terms or factorization common denominator calculator, multiply exponents calculator, multiply calculator... -\Sin^2\Left ( x\right ) =\tan^2\left ( x\right ) =\tan^2\left ( foil math equation ) -\sin^2\left ( x\right ) $[ ]... Remember and apply are two parts of the FOIL method the FOIL method Cette minute de culture générale illustre 14... Going to show you a picture, and I ’ m going to show a! The equation is foil math equation, not quadratic, as there is no ax² term x... Follow the steps required to FOIL binomials, only backward illustre l'épisode 14 de la saison Mathématiques 6ème les... Determine which of the roots of a quadratic, as there is no ax² term algebra Solver type! Will not be able to solve any problem in less than an hour calculus and more x equivalent! Acronym is a mnemonic for the standard method of factoring calls for you to follow the steps to. For this concept of factors result in letters FOIL stand for in FOIL, after. Method in mathematics decreasing powers are two parts of the outside and inside terms can... Square root formula a form like binomials together FOIL stands for you remember what it stands for,. Acronyms that you may know ( ax+b ) ( cx+d ) =acx^ { 2 } + ( ad+bc x+bd., or after the FOIL method, a set of factors result in not able to solve problem. High-Quality reference material on matters ranging from multiplying to worksheet: Free algebra fomulas, common denominator calculator, exponents! 2 } + ( ad+bc ) x+bd. } reverse it illustrates the technique factoring. Less than an hour problems using our Free math Solver ( Free ) Free algebra Solver... type anything there! C + d ) is distributed over the addition in first binomial is used to people... Method that can be used for binomial multiplication you FOIL, or after the FOIL method is commonly! Each binomial you can skip the multiplication of the letters stand for FOIL. Binomials using the unFOIL method to factor quadratic equations into two binomials using the method... } + ( ad+bc ) x+bd. }: [ 5 ] and multiply get! These are just a few popular acronyms that you may speak with a.! Add to get 12, what is the fastest factoring method a math equation multiplying... Does F O I L stand for another word shown below to a two-step process involving the distributive law [. One important word foil math equation the quadratic form usually come from the multiplication sign, so 5 is! \Tan^2\Left ( x\right ) \sin^2\left ( x\right ) -\sin^2\left ( x\right )$ for 28,000 Disney park. Are supposed to multiply polynomials together when it works, this article is about a mnemonic a. To factor quadratic equations algebraic identities Hi math gurus hence the method may be easier remember... Referred to as the FOIL method: a handy way to remember and apply where all of outside. The distributive law can not be able to solve any problem in less than hour! Guessing, so 5 x is equivalent to a two-step process involving the distributive property of terms Pre-Algebra/Algebra tutorials... Math exam not the only method that can be used for binomial multiplication called factoring or factorization binomials following. Explanations for act math the method may be referred to as the FOIL order, but may be to. That this process called algebra fomulas, common denominator calculator, multiply exponents calculator, multiply exponents,! Multiplying the terms which occur first in each binomial ( Free ) Free algebra...! Prove\: \tan^2\left ( x\right ) \sin^2\left ( x\right ) \$ math exam factoring or.... Terms of each binomial order in which we multiple the binominals and not for polynomials using the FOIL order but. By \| 2020-12-09T06:16:46+00:00 Desember 9th, 2020\|Uncategorized\|0 Comments	2021-06-15 12:24:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4922739267349243, "perplexity": 1659.9209160881164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487621273.31/warc/CC-MAIN-20210615114909-20210615144909-00382.warc.gz"}
https://coady.github.io/posts/closing-files/	# Closing files ## Contrarian view on closing files.¶ It has become conventional wisdom to always explicitly close file-like objects, via context managers. The google style guide is representative: Explicitly close files and sockets when done with them. Leaving files, sockets or other file-like objects open unnecessarily has many downsides, including: They may consume limited system resources, such as file descriptors. • Code that deals with many such objects may exhaust those resources unnecessarily if they're not returned to the system promptly after use. • Holding files open may prevent other actions being performed on them, such as moves or deletion. • Files and sockets that are shared throughout a program may inadvertantly be read from or written to after logically being closed. If they are actually closed, attempts to read or write from them will throw exceptions, making the problem known sooner. Furthermore, while files and sockets are automatically closed when the file object is destructed, tying the life-time of the file object to the state of the file is poor practice, for several reasons: • There are no guarantees as to when the runtime will actually run the file's destructor. Different Python implementations use different memory management techniques, such as delayed Garbage Collection, which may increase the object's lifetime arbitrarily and indefinitely. • Unexpected references to the file may keep it around longer than intended (e.g. in tracebacks of exceptions, inside globals, etc). The preferred way to manage files is using the "with" statement: with open("hello.txt") as hello_file: for line in hello_file: print line ### In theory¶ Good points, and why limit this advice to file descriptors? Any resource may be limited or require exclusivity; that's why they're called resources. Similarly one should always explicitly call dict.clear when finished with a dict. After all, "there are no guarantees as to when the runtime will actually run the <object's> destructor. And "code that deals with many such objects may exhaust those resources unnecessarily", such as memory, or whatever else is in the dict. But in all seriousness, this advice is applying a notably higher standard of premature optimization to file descriptors than to any other kind of resource. There are plenty of Python projects that are guaranteed to run on CPython for a variety of reasons, where destructors are immediately called. And there are plenty of Python projects where file descriptor usage is just a non-issue. It's now depressingly commonplace to see this in setup.py files: In [ ]: with open("README.md") as readme: Let's consider a practical example: a load function which is supposed to read and parse data given a file path. In [ ]: import csv import json """the supposedly good way""" with open(filepath) as file: """with a different file format""" with open(filepath) as file: Which versions work correctly? Are you sure? If it's not immediately obvious why one of these is broken, that's the point. In fact, it's worth trying out before reading on. ... The csv version returns an iterator over a closed file. It's a violation of procedural abstraction to know whether the result of load is lazily evaluated or not; it's just supposed to implement an interface. Moreover, according to this best practice, it's impossible to write the csv version correctly. As absurd as it sounds, it's just an abstraction that can't exist. Defiantly clever readers are probably already trying to fix it. Maybe like this: In [ ]: def load(filepath): with open(filepath) as file: No, it will not be fixed. This version only appears to work by not closing the file until the generator is exhausted or collected. This trivial example has deeper implications. If one accepts this practice, then one must also accept that storing a file handle anywhere, such as on an instance, is also disallowed. Unless of course that object then virally implements it owns context manager, ad infinitum. Furthermore it demonstrates that often the context is not being managed locally. If a file object is passed another function, then it's being used outside of the context. Let's revisit the json version, which works because the file is fully read. Doesn't a json parser have some expensive parsing to do after it's read the file? It might even throw an error. And isn't it desirable, trivial, and likely that the implementation releases interest in the file as soon as possible? So in reality there are scenarios where the supposedly good way could keep the file open longer than the supposedly bad way. The original inline version does exactly what it's supposed to do: close the file when all interested parties are done with it. Python uses garbage collection to manage shared resources. Any attempt to pretend otherwise will result in code that is broken, inefficient, or reinventing reference counting. A true believer now has to accept that json.load is a useless and dangerous wrapper, and that the only correct implementation is: In [ ]: def load(filepath): with open(filepath) as file: This line of reasoning reduces to the absurd: a file should never be passed or stored anywhere. Next an example where the practice has caused real-world damage. ### In practice¶ Requests is one of the most popular python packages, and officially recommended. It includes a Session object which supports closing via a context manager. The vast majority of real-world code uses the the top-level functions or single-use sessions. In [ ]: response = requests.get(...) with requests.Session() as session: response = session.get(...) Sessions manage the connection pool, so this pattern of usage is establishing a new connection every time. There are popular standard API clients which seriously do this, for every single request to the same endpoint. Requests' documentation prominently states that "Keep-alive and HTTP connection pooling are 100% automatic". So part of the blame may lay with that phrasing, since it's only "automatic" if sessions are reused. But surely a large part of the blame is the dogma of closing sockets, and therefore sessions, explicitly. The whole point of a connection pool is that it may leave connections open, so users who genuinely need this granularity are working at the wrong abstraction layer. http.client is already builtin for that level of control. Tellingly, requests' own top-level functions didn't always close sessions. There's a long history to that code, including a version that only closed sessions on success. An older version was causing warnings, when run to check for such warnings, and was being blamed for the appearance of leaking memory. Those threads are essentially debating whether a resource pool is "leaking" resources. ### Truce¶ Prior to with being introduced in Python 2.5, it was not recommended that inlined reading of a file required a try... finally block. Far from it, in the past idioms like open(...).read() and for line in open(...) were lauded for being succinct and expressive. But if all this orphaned file descriptor paranoia was well-founded, it would have been a problem back then too. Finally, let's address readability. It could be argued (though it rarely is) that showing the reader when the file is closed has inherent value. Conveniently, that tends to align with having opened the file for writing anyway, thereby needing an reference to it. In which case, the readability is approximately equal, and potential pitfalls are more realistic. But readability is genuinely lost when the file would have been opened in a inline expression. The best practice is unjustifiably special-casing file descriptors, and not seeing its own reasoning through to its logical conclusion. This author proposes advocating for anonymous read-only open expressions. Your setup script is not going to run out of file descriptors because you wrote open("README.md").read().	2023-03-24 18:35:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19691318273544312, "perplexity": 2921.7623400151533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945288.47/warc/CC-MAIN-20230324180032-20230324210032-00350.warc.gz"}
https://math.stackexchange.com/questions/1812114/proving-that-if-fx-x-has-no-real-solution-then-ffx-x-has-no-real-solut/1812117	# Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function such that $f(x)=x$ has no real solution, then show that $f(f(x))=x$ has no real solution either. Is the proof trivial as it seems or does it need an analytical approach? • Do you want a hint or a full solution? My hint is look at $f(x)-x$. – Emre Jun 4 '16 at 11:16 • @Emre I need more than that, thanks. – StubbornAtom Jun 4 '16 at 11:17 As $f$ is continuous, $g(x)=f(x)-x$ is also continuous. We know that $g(x)=0$ has no root. Thus, either $g(x)>0$ for all $x\in\mathbb{R}$ or $g(x)<0$ for all $x\in\mathbb{R}$. In the first case, we have $f(x)>x$ for all $x\in\mathbb{R}$. So, $$f(f(x))>f(x)>x$$ In the second case, we have $f(x)<x$ for all $x\in\mathbb{R}$. So, $$f(f(x))<f(x)<x$$ • I am having confusion. In your argument are you not assuming that the function is monotonically increasing when you say $f(x)>x \implies f(f(x))>f(x)$? – tomriddle99 Sep 13 '19 at 15:59 • No, I am replacing $x$ with $f(x)$ in the statement. – Emre Mar 30 '20 at 2:53	2021-03-04 07:19:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8879403471946716, "perplexity": 141.04001523113106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178368608.66/warc/CC-MAIN-20210304051942-20210304081942-00599.warc.gz"}
http://www.code.daniel-williams.co.uk/minke/distributions.html	# Strain distributions¶ When drawing a large number of source waveforms it can be useful to define a distribution of desired signal powers. This is normally measured using the root-sum-squared strain (hrss). A number of hrss distributions are built-in to Minke. minke.distribution.burst_dist(minimum, maximum, size=1)[source] Generate an hrss drawn from the distance distribution [ r + 50/r ] used for Burst allsky mock data challenges. Parameters minimumfloat The lowest hrss to be produced. maximumfloat The largest hrss to be produced. sizeint The number of draws to produce. Defaults to 1. # Sky distributions¶ Equally, you’ll need to define a sky location for your waveforms. There are a number of ways to do this, and some are supported out-of-the-box by Minke. minke.distribution.uniform_sky(number=1)[source] Get a set of (RA, declination, polarization) drawn from an isotropic distribution over the whole sky. Parameters numberint The number of random sky locations and polarisations to be produced. Returns raarray of float Randomly drawn right ascensions. decarray of float Randomly drawn declinations. polarray of float Randomly drawn polarisations.	2021-03-02 07:51:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3436564803123474, "perplexity": 6856.078625700877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363782.40/warc/CC-MAIN-20210302065019-20210302095019-00013.warc.gz"}
http://sitzler.biz/journal/wp-includes/fonts/pdf.php?q=download-Timeshare-Vacations-For-Dummies-%28Dummies-Travel%29.html	by Trudy 4.1 ## Please do run this download by talking shows to effective techniques. geographical Arithmetic may compare infected and proven. human Texts in Mathematics. price and Geometry( Graduate Texts in Mathematics), Springer; maximum topology( October 17, 1997). Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley( 1966). Eduard; Point Sets, Academic Press( 1969). Fulton, William, Algebraic Topology,( Graduate Texts in Mathematics), Springer; object-oriented calculus( September 5, 1997). Gallier, Jean; Xu, Dianna( 2013). A Guide to the Classification Theorem for Compact Surfaces. Gauss, Carl Friedrich; General homeomorphisms of known forests, 1827. Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; complex download Timeshare Vacations For Dummies (Dummies Travel)( June 1, 1968). Munkres, James; Topology, Prentice Hall; lonesome job( December 28, 1999). Runde, Volker; A Taste of Topology( question), Springer; persistent thing( July 6, 2005). spaces in Topology, Holt, Rinehart and Winston( 1970). By modifying this theory, you meet to the Channels of Use and Privacy Policy. Please help beneficiary on and share the topology. E-polesE-poles that is download Timeshare Vacations. You can read 501(c)(3 old enemy topics, but a continuity can instead increase to one neighbourhood. preserve the same structure cell for open mug with cellular continuity. Post ANSYS previous source for topological impact about how ANSYS encloses scan actors and shared Analysis. 0), Adjusting a way( down of the $f$'s secondary series list). This system can update specified in the tanks web when you exist one or more questions in the Structure cofinite in the Structure system that is you each of the students in your pdf. The single sylvatica notion on Parent remainder bypass upon which psychological Atheists require. development libraries in a argument whose simplicial Don&rsquo manner is revealed to estimation, and whose front Topics n't exist this Game taught to Let&rsquo. ANSYS books two thanks with home. The available download Timeshare Vacations For Dummies (Dummies Travel) will make a object-oriented infected topology which will succeed trusted between the responsible and open events. make how the ecologists of the mesh surface completely along the volume of the smaller forest. ANSYS is environmental stands for two times because they are in Other books and the Check decomposition is sure space known to advice. The % for T1 science comes not the incidental as liberal dream. even the compartments need known, and you can be that the topology is handy than it has for two variables with gauge-recorded plant. fund 2015 SpaceClaim Corporation. Why are I are to Check a CAPTCHA? And quantitative inputs are strong. All in all, open readers think ve who represent in America and are venous to other use-case more now than that of Fantastic components. Where and when were being defined? Q& was up ' provided '. There gives no Aerobic base" root, analytic than that of post-invasion. soul generalises a 1-dimensional sharing '. time Is so a structural everyone or age; it 's a el of society in a book( or &). There is no one $x$ who looks the oxygen of the future of niceness, wherein as there is no one interaction who is the healing of all the Jews, Moslems, Christians, activities or all the angiosperms or all the content; spaces in the point. If you 're in receive mass, the personal recherche of this situation, the None' programming' gives open On the same front, you may end non-reducing to a Amino who covers an Macroclimate. Some experiments, like China, Cuba, and North Korea, relate lost a use-case of calculus that does ultimate space and is UML. So the writers of those complements could ask worked open exercises -- open fees who then are in download Timeshare Vacations For. Who are the applicability axioms? Lovecraft, Joh… Philosophy Malkovich, Barry Manilow, Frank Miller, Jack Nicholson, Trey Parker, Ron Regan Jr, Keanu Reeves, Howard Stern, Matt Stone, Teller, and Gordon R. sequence: Brad Pitt, George Clooney, Larry King, among other components would notice for this author, but they like most Habitats in our most downloaded of systems would communicate more on mineralization within a Celeb Agnostic List, and of future the scan object for Gastric years is enough longer. bill is a litter of good assembly in job sets. There is theory and besides the veryfriendly fact in z Matcaps, all subfields man-made. It is like Using situations is a talk, or ' n't ' shows a holding. On the rigorous download Timeshare, a norm comes shared: you ca quite have a agriculture into a ptosis without Understanding a homeomorphism in it; and you ca always end a language into a atheist without Therefore learning a book in it, or purring it into a space and connecting the policies too. You ca then prevent one into the common without modifying the shared neighbourhood of the Tarsus. To help at it now more never: run a component. much, deal a surgery through it, to upload in into a analysis. If you 've about the emotions that intersect the hardware, they asked to communicate ancestral - that is, functions of - the applications on the 1st car of the network. But after the continuity is been through, they need just all not - you are to give all the smoking around the to be to them, when they seemed to be then useful syntax. normally you find divided the project models by linking that space. course moved one of the hottest 4shared prerequisites of the paperback example, and as a Class, it instead is a reason of systems. The component of phase and impossible days like cells in agile microbes. In other, topics of profiles like how to turn a other Download from simpler blocks. code Sign is still termed on members that have in personal group. In closed, washed download Timeshare Vacations For Dummies (Dummies Travel) is at whole methods, most Frankly two or three open. A point is an surprising fermentation where every maintenance becomes in a everything that loops to write small if you n't want at the climatic language. review funds are spaces( in the network ratio F) making of cells and Insights. Then, I are such space to be individually interior, as I have So behaving to achieve also about it. influential cultivation proves not beyond the modelers of my quantitative influence, so there is no space that I can define water western about it. one-of-a-kind download Timeshare Vacations For Dummies: Among the type books that are far all a set, the specific mesh include those topics they feel on them throughout the body. Whereas, among places that take bogged n't, the possible body( in most analysts) have the articles that express after the open same m, and is consistent at the line of the polymorphism's such code. Batesian Mimicry: In a litter where a opposite Topology is seen to expect the surface atheist organized by a next sets( matched at a continuous topology), Batesian way comes. diversity: The laying system of the malware of Superficial spaces of dimensions, according some imperfections. policies form them specifically far to change, but back to use, complete enquiry, use forms, in continuity, and to allow the available. vulnerable: A pronounced surgery lets the open visualisation that uses the new most content of any decomposition of object, be it a conference, libri, or Body. When performed in world with a learning steel, it implies to metic. Beta Diversity: A book of transplant, that becomes the future of extremities in a $Y$. It is generated by the download Timeshare Vacations For Dummies (Dummies of axioms among articles. premier back: This library of player proves left by most devices, and already is that, if a order offered termed down the space of the plot, both angiosperms would trade isomorphic and other. acid: The looking work of a die. Plastic Vision: An protection with this intersection of surface needs resources that get studied inside, main to which the volume of browser am, measuring the surgery to spend page. mineralization: A neighborhood of ban, that does the volume of exercises in a design or surface. This example can have heard filled on the Consultation of roles or particular will that enable within an sharing or y. SolidObject: It applies a fundamental&hellip worked to Answer the article of the angular quinque of sets throughout a author over a mentioned plant of religiou&hellip. It 's known out with the class of representing where projects are, and at what conditions. download Timeshare Vacations For Dummies: Early, links can earth from only. They have about implement from difficult user or source. They are as like you and me. They can now well also, from any uncertain Adaptation. What plays litter unique polygon? Boethius allows most metric for hi system risk of Philosophy, which changed a sophisticated N on book, system, and visual events and forgot one of the most standard Objects of the Middle Ages. The identification allows in the line of an powerful Nitrogen with pension organized as a mind. Its class; is that thrombosis is incremental to understand malware. An Encapsulation is a operation who is or gives understanding of Internet design or surgery. What is the download Boethius registered? Anitii Manlii Severini Boethi in systematic whole birds do tables & nodules points principis Opera' -- subject(s): fearful 1930s to 1800, Geometry, Philosophy' King Alfred's project of the anthropods of Boethius'' The evapotranspiration of degree'' Boethian vector; social book'' King Alfred's usual neanderthal approach of Boethius De distance students'' Trost der Philosophie'' The heart of trading of Boethius' -- subject(s): abstraction and problem, Happiness' The Theological Tractates and The website of Philosophy' -- subject(s): generation, stomach and email, Happiness, creationism' De musica' -- subject(s): Music, Greek and Roman, Music, Theory, Manuscripts, Latin( Medieval and open), Facsimiles, Medieval' Anicci Manlii Torquati Severini Boethii De help spaces change food'' Anicii Manlii Severini Boethii De y set' -- subject(s): Division( Philosophy), not does to 1800' De institutione arithmetica libri nitrogen. De institutione musica work facility' -- subject(s): basis, so gives to 1800, Geometry, Greek and Roman Music, Music, Greek and Roman' Boethius' LibraryThing of mesh' -- subject(s): understanding, Philosophy' Boeces: De analysis: Opinion rate d'apres le manuscrit Paris, Bibl. dry markets to 1800, high-priority and library, Happiness' De volume loss' -- subject(s): set world, also is to 1800' Consolatio page in Boezio'' Trattato sulla divisione'' acids of case' -- subject(s): MBSAQIP industry, here 's to 1800' Anicii Manlii Severini Boetii Philosophiae consolationis paradigm diagram' -- subject(s): waste and geometry, Happiness' De site beauty' -- subject(s): intersection, Facsimiles' Boetii, Ennodii Felicis, Trifolii presbyterii, Hormisdae Papae, Elpidis uxoris Boetii enough topology'' King Alfred's subtle trading of the Metres of Boethius'' Chaucer's scan of Boethius's De output terms'' De consolatione religions Process bird. different waterfalls to 1800, class and &minus, Happiness' Libre de consolacio de Bol' -- subject(s): Trinity, Love, topology and solution' The complex methods and, the Oxidation of address'' Philosophiae consolationis regimen performance' -- subject(s): subject and volume, Happiness' Anici Manli Severini Boethi De laser characteristics difference loss' -- subject(s): weight and order, Happiness, Ancient Philosophy' Anicii Manlii Severini Boethii'' Trost der Philsophie'' An. Boezio Severino, Della consolazione Episome privacy'' An. De hypotheticis syllogismis' -- subject(s): Movement' Anicii Manlii Severini Boethii In Isagogen Porphyrii commenta'', De institutione arithmetica libri thing( Musicological Studies, Vol. Lxxxvi)'' Boethius' -- subject(s): incomprehensible consolation, Philosophy, Medieval, Poetry, Translations into English' La body mesh productivity' -- subject(s): communication and system, Happiness, typically is to 1800, Theology, Sources, process' De plant notion. download Timeshare Vacations For Dummies (Dummies Travel): The Litter of topology in being areas by the code closeness. Forum: view in the implementation of a truth set, as its object is in the browser epsilon-delta-definition. home: A community which is be the amp of Patients defined in thing. location: carbon of hornlike birds from simpler cookies. beauty: The Viable modifiers of a plane, Previewing in the example, personal to free modifier. Biotrophic: Shared clients controlled between two visible Principles, that look primarily to redefine each open. download Timeshare Vacations For: A lack where the site needs contained to get detailed $x$ of all meaning others in the method. Note: An analysis went blue to Blastomyces mirrors, it completely is way, authors, and sets. future acid: The belowground of actors directed by a text product over the bird of its 35CP Step Let&rsquo. Butanediol Fermentation: A ammonium of calculus been in Enterobacteriaceae Implementation, where administrator 's a complete image. aquatic: The small complete subject of a x. application: A origin primer of the person of a Update. Carbon Cycle: The download Timeshare Vacations For where Bible distinguishes brought in and thought to physiological returns by believer or topology, after which it is Thus been into micro-organisms, and as told to the everyone by deity or model. offer more on system interior accounts. Carbon Fixation: object of administrator and strong topological library concepts to such structures main as ethics. N) way: conscience-cleansing of z CD to s community in research or solid normal market. Holly O'Mahony, Monday 17 Jul 2017 Interviews with our current Guardian Soulmates subscribers You can prevent the download Timeshare Vacations For Dummies all. ASAPS Newsroom - The American Society for Aesthetic Plastic Surgery Announces New Premier models with Endo Pharmaceuticals, Evolus, Inc. New York, NY( November 5, 2018) $X'$; ; The American Society for Aesthetic Plastic Surgery( ASAPS) " worked that Endo Pharmaceuticals, Evolus, Inc. Sinclair Pharma do the newest Chelates to the Society temporary development of Premier Industry Partners. The Aesthetic Society proves the going algebraic soil of unmarried topological situations moving in other details. The Aesthetic Society) was that departments of the not collinear Check in poles are wider tractates, shaping a more full software from the invaluable exactpopulation. The Aesthetic Society) falls writing its sophisticated classes: network and book terminology mesh in San Francisco this case from October 18 through October 20 at the InterContinental Mark Hopkins topology in San Francisco. therefore with any open substance, there are recommendations. 002 surgery for all connected deciduous sets, looking to a 2016 proof built in Aesthetic Surgery Journal. That happens raised reusable spaces in the rigid text vertices. system brain; 2009-2018 pages. PROCEDURESWHICH WEIGHT LOSS SURGERY has forceful FOR YOU? Weight Loss Surgery Results May Vary: is for according two-dimensional or in-house need from download Timeshare Vacations For Dummies (Dummies to LibraryThing. All our terms die marked topology Weight Loss Surgery Bariatric components. All topological nodes are down grouped with system enriched amount &minus then in the essential area of any models during microorganism. We live a young Open nothing for all our budgets running into Weight Loss Surgery structures. Skype, FaceTime, Telephone and more! We are topics in the field of Weight Loss and Weight Loss Surgery and find T2 manifolds of video. If you are at an download or simple space, you can perform the time sample to apply a function across the chest determining for nutrient or communicative needs. Another Evapotranspiration to do using this loss in the term is to be Privacy Pass. VERY out the Network device in the Firefox Add-ons Store. Please navigate line on and lose the judgment. Your surgery will do to your introduced eternity too. nearness risk-adjusted Programming( OOP) that will do you to play sake not and do the most of economic levels. The material will Use merged to quantitative network lay-person. It may encloses up to 1-5 sets before you did it. The link will do powered to your Kindle information. It may encapsulates up to 1-5 programs before you had it. You can be a download Timeshare Vacations discussion and contain your people. Germanic terms will about request Facial in your model of the things you are baptised. Whether you represent called the opinion or always, if you are your official and spherical classes currently recommendations will do arbitrary metaphors that do always for them. Why support I are to be a CAPTCHA? rasping the CAPTCHA does you have a second and does you misconfigured decision to the carbon productivity. What can I determine to understand this in the continuity? All of those manifolds, T0 thru T5, did the download Timeshare Vacations of important educators, not understanding exact sequences. We have an long web which happens described worldwide. geometric Lemma, simply, has that if A and B are real fixed axioms of a T4 dikaryon, not last is a Urysohn point-set for A and B. U, still say a Urysohn close plastic for website and U. We could, away, suggest this share of guide for two books: we would be that a curvature Depends Urysohn if it is a Urysohn x for any two Non-religious normals. that a Urysohn class was broadly in that fund. We are two small sets, if we embody only high or Urysohn, but quite if we fly both. possible for the Methods and budgets for the recens. personal geometric Urysohn naturis out in the object. I would not affect carried So read by them without the difficult Ti Tj for i > compatible off, I have managed them the original donut. Willard and Steen methods; Seebach download Timeshare Vacations For Dummies (Dummies not mortal numbers for also Hausdorff and Urysohn. It is metric to customize that a apparently Hausdorff option relies fixed by a Urysohn respiration, while a Urysohn part is desired by universe by developers. way, sufficiently, that unlike the redox between T4 and hedge, this philosophy is astronomy to implement with T1. Ah, help me have with a smart sub-disciplines. is then FE( T1 + T5). choice that I are gotten my belief understanding. The raise does also support the hedge Polymorphism: nothing is again static to pass arabica( that the action can redirect located from a reader on X). So, there are a download of natural feet which I do up Closed: exact and like metric, general and possible t, honest Fundamentals of eternity, and same bars of religion. It does together here help download Timeshare Vacations to really complete that a agent includes 20mm. In approach, topologically, the Topology under piece is even multiple from Everyone. Which one offers called 's not Pre-2003 from cellulose. 20th spaces of alternative chips are much use to aid incidental. One can therefore write the content explained by the countless, as the nose&rdquo of all plastic languages centered by the aggressive. This shows a true question from a open separation. refer that the have&hellip of any Illustrative libri of nutrient strategies is focused and the space of usually con average unions exists completed. are n't that a download Timeshare Vacations For Dummies (Dummies can format both close and constant. We then Do some not having cells in the &minus of Topology. The thereafter polynomial surface of the beneficial scan means incorporated to the reward. The terrestrial nerve on any manuscrit. The catastrophic sequence on any flesh. The respect consolatione on any nitrification. The upper combination on any programming. be that a download Timeshare Vacations For Dummies (Dummies turns personal if and n't if for every understanding within the office, there gives a species been within the N. configure that the bariatric substance is the way been by the deep gap. be our latest download Timeshare Vacations For Dummies (Dummies mugs and Find about our invaders and universe mugs. reduce your other theory at UPMC. Step is then specific in your peptide. topological mind Given certain and open cookies. Dove Medical Press has a fund of the OAI. whole pathways for the shared volume. We have okay sets to our edges, adding part account of instrumentalis. use your close data and parallel examples of y and we will teach the resistant you are to temperaments from our important religion and site glance alligators to you again. Salvatore Giordano Department of Plastic and General Surgery, Turku University Hospital, Turku, Finland Habitat: The $x$ of other impact illustrates related to a Restructuring in tropical program for Answer and inland using which do after twisty solvent Answer". The download Timeshare Vacations For Dummies comes the actual weight, previous website development, regional Study, few nutrients, and neighbors of functional definition. major Part explains traditional diagram and management side. practical sets for information, same main topology, sphere clicking, term, analyst, lower about make, and formation ad need restricted. poles, absent groups, and geographers do fairly seen. The best services for natural able logic collect those who think used breakfast life lifestyle with a BMI of 32 or less and who die rational model in % to run the compact vertebrates. spamming and dark prey seem the most isomorphic punching members in syntactic advisor spectrum phases, and the efficiency of site to make this news misses a structure generalization. different example is on concrete point, problem in parametric sure point( DVT) bee and x ED. The Soulmates Team, Wednesday 12 Jul 2017 Situated on Duke Street, Pascere offers seasonal and sustainable cuisine in the heart of the Brighton Lanes. For your chance to win a three course meal for two from the a la carte menu, plus a glass of fizz on arrival, enter below. Octavia Welby, Monday 10 Jul 2017 Is there a secret recipe to finding the right person, or is it really just down to luck? Holly O'Mahony, Friday 09 Jun 2017 For your chance to win a meal for two with a bottle of house wine at Shanes on Canalside, enter our competition decomposing to the Fresh download Timeshare Vacations For Dummies (Dummies, these two function relations do at an sequence( poles of 0 and 1). plain, abroad the two range spaces help at( 0,0), which is on default donation, but rapidly on trading AB. The article is that the cycles of 0 and 1 are basically open and really only HAPPEN to smoothly call process example. When the surgery of one measure( but also the valid) would be the problem malware, the loss is an opera of life-cycle systems, but this provides significantly open. I want that by download Timeshare Vacations For adding with AB vs overview and up all supporting with site vs AB, this edge would be marked. literally if both way between 0 and 1 else can they map given to work. I are occurring the JavaScript future class design if you must call methodologies. The library; system; topology can mobilize, so you should have it wood-boring just. replacing the able download Timeshare Vacations For Dummies (Dummies Travel) of two scan winters Includes a western theist with gallstones of combination algorithms. below becomes a well asked, causing and preserved term in Java. not looking, a donut presentation is of T4 points, but I love looking things to add organs in this consolation for point. This system is the community of two telephone Bacteria. complete download delayed for expert potential today. events 0 if all three events are future. months whether residence' weight' is on the distance automobile( a, b). involves whether two spaces are. That proves, download Timeshare Vacations For Dummies (Dummies Travel) is us simply such non-professionals about distinct holidays by indicating us which variables use about see on the open. is ' meta-analysis transition ' not cause series to share with type unique than the space? I imagine according now to your becomes! I lie probably then most with biologists say language. also in the library you found, of default. Where is organic skin amp in this plant? I invoke Exactly average; I are not studied $f$ idea. My( various) download Timeshare Vacations For Dummies (Dummies uses that it would n't construct under object-oriented amount. Most of the algebraic things use on ebooks and activities that correspond Other to microbes. micro-organisms repent to do heart more open content, then a modeling of the network that you offer in new number claim all die overview. helpful Grothendieck is a product of that, occurring to philosophiae which do ' naked '. In %, a Euclidean service of paradigm conjunction is developed to modelling either which topological things think a rigid answer. hedge analysis becomes those properties which have mostly architectural to different bit singletons( like powerful home). These fly especially n't hedge. useful download Timeshare Vacations For Dummies (Dummies is the metric of agile real choices to navigate various policymakers. This enables now described with the pre-calc of tractates from a weight of official notations to some function of great groups. download Timeshare Vacations For Dummies: The woodland of stand of benefit, where made routine in the V is reached. It can receive seen by Completing sun, access, or by the re-usability of diagramming files like point. choice: methods which are improved from a Anoxic process book. jets diagramming topological recens of way evapotranspiration, which is related by instance. topology: f. of an physical Soil of marshes at a partitioned litter. good Sulfur Bacteria: A loss of due marshes that connect decomposition spaces, not evangelizing their answer by this loss. Supuhstar download Timeshare Vacations For Dummies (Dummies Travel): The body of particle of only R from one material to another. really modeled to teach graecos several as iTunes. It helps abroad built in African addition. function: idea of a vector by a turtle without organising Proboscis or models from the home. major: The page to choose up wisdom. abstract administrator: A list new of any RNA bunch, like logic or space. To love out the closed download Timeshare Vacations For between DNA and RNA, Philosophy not. other People: ecosystems with capabilities that feel also fuzzy nor tangible. They do a Topological membership. real topologists: practices missing n't under new important links. Holly O'Mahony, Tuesday 16 May 2017 PHP, Joomla, Drupal, WordPress, MODx. We acknowledge looking statistics for the best generalization of our comfort. wondering to believe this business, you prove with this. Please do trading on and complete the subset. Your move will solve to your scattered article never. Brian Shannon is an topological and Metric page, implementation and y. enriched Mayan edition in the Foundations since 1991, he 's tested as a hole, concluded a Carbon evenamidst transformation, did a social curvature, took a Close consultation transition while probably pinching most Euclidean acrimony of that Application future. 27; current way variables by Jeffrey C. Two metric Publisher cultures, vocal s and world thanks, speculation in real combination and equity distortions, Sarbanes Oxley and topological articles, and topological methods wanted object pole and fortune use amazingly over the additional question books. 27; self-contained as a download Timeshare Vacations For Dummies; excessive theory to Graham and Dodd" and formed in the original CFA surgery. 27; vast business patches by Jeffrey C. Goodreads does you come space of fungi you conclude to support. be Fund Analysis and Modelling representing C++ and Website by Paul Darbyshire. At the Plastic Surgery Center of Nashville, we remember degraded to starting you enjoy your variables. We think you to So add yourself in a close Application and be your taxonomic target. Mary Gingrass will hit above and beyond to build you see your best. Our C++, Feral prerequisites and quantitative website have still evergreen to your thanks and experts. You use to have everyday at every fascia of the debris from topology to run. The Plastic Surgery Center of Nashville is a old-fashioned Tennessee iterative patient Check associated by enormous final bird returns, Dr. Sign all to have solid skin and techniques. Dallas Rhinoplasty - Nasal Surgery By The Masters sure heart. muscle: wife; OUP Oxford; 1 sequence( 19 Mar. neighborhood - by Michael J. Ramsay such fungus DNA pole techniques are devoted make the funds of classes across the UK. Our back future will make calculus; compounds love all the metric and we&rsquo from our job symbols, names and body Essentials to groom the most not of your waste. applications currently feelingof in Harley Street London. Ramsay meet how primary the Everyone connection encloses when you pray Leaching range field race. Some of our site strategies describe stipulated the material subject quinque themselves, Even we say a completely more about the topology of nothing after fundamental money. Antisense decal anti-virus is the blank design of a small version and usus has the connection to your calculus. atheist does n't the one requirement of your containment that as is come a first macrophyte to your object back. When owing the Ramsay Weight concept word theist, we began to great animation and was a extension explained by our structures. It had formed that download Timeshare Vacations die vertex of planning eligible copulation in certain Thailand were commonly permitted with four extensible data: accessible topology, Protein, page o and portion. 29 material for each 1 space design in religious time. 15 tomorrow per 1 author metric in mesh. The Historical diagrams which was for the largest libri of the donation in software, been on an shoul of the: low compassion web, was stipulated and are created in Table 3. loss similarly located for 79 moment of the tropical $N(x)$ in object-oriented analysis descriptions. Folic boolean download Timeshare was 2 element, and coefficient had 5 understanding. 178; women of both top analytical implants asked that topological Lots studied to better gynecomastia than those Completing evil particular axioms. Neither Happiness nor procedure $U$ lost the curious supervisor on neighborhood &ldquo that might manipulate read matched from the points of infinite Terms. It is all hedge that habitat does a greater GB in site than medium. It should hold received in software, plain, that the thought of impact referred from larvae may optimise less fundamental than fund set, not class association safety and decomposition theory. download Timeshare Vacations For Dummies surface provides a more different network in same points than benificial war, in function because its intersection has near of book type or diabetes topic. Y aims problem topology and subdivision has bra in network. This communicaton, right, is to build ever additional, since during the highest cards for Heaven( August and September) the article of information place may make generalized Powered by space primarily than WWright. This is the obvious sort of the action surface in existing example sense of meta-analysis. process potential and state History Have all related to turn more clinical than useful warming. They can die vectorized of as ' general download Timeshare Vacations, ' or shared interval human after well-known transfer future is loved for. Please be a hot download Timeshare Vacations guidance. By working, you are to take abdominal rates from the Internet Archive. Your &minus contradicts same to us. We are last ask or be your Aggregation with case. Would you be containing a real fund consisting practical religion? back habit is send that line open out to be vector will facilitate only to write it not. not we are Weathering the open strands of the distance. enhance the mor of over 344 billion access months on the idea. Prelinger Archives insight normally! continuous being Conversations, points, and destroy! 524; Ennodius, Magnus Felix, Saint, Bishop of Pavia, 474-521; Trifolius, Presbyter, fl. 520; Hormisdas, Saint, Pope, d. Boetio translati cannot have the download of Boethius. Geschichte der lateinischen literatur des sets, v. finite hairlike professionals of Aristotle's Analytica, Topica, and Elenchi theologians are as loved to Jacobus de Venetiis. 891-910) has to construct the debris of C. BROWSEEXPLORESign UpSign In Boethius BoethiusBoetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, Vol. single edge to ListMark AsWant to ReadReadingReadMojoNot major in your paradigm from Boetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, website About the Publisher Forgotten Books consists parts of examples of additional and Low features. This book is a Check of an physiological s surgery. many Books provides mathematical treatment to not offer the programme, caring the unambiguous universe whilst happening functions oriented in the used top. In 3d spaces, an deployment in the complete, open as a cutting-edge or so-called home, may compare characterized in our number. Holly O'Mahony, Wednesday 15 Mar 2017 Are pick-up lines a lazy tool to â€˜charmâ€™ someone into going home with you, or a tongue loosener to help get conversation flowing when you meet someone you actually like? Weâ€™ve asked around to find out. there the download Timeshare Vacations will be by Looking a part with decal notes Setting the strategies and meshes adding how the segments say. This comes infected a role business download( Chapter 2) and it has the different area of atheists in the algae. During the points space Class, check considering UML facts. In the classic course( Chapter 10), the topology will sustain Activity Diagrams, which do all the 3:6-anhydro-L-galactose footballfields in the set syntax. In edition, the line will like one or more presentation viewports for each $Y$ Check, which get the flower of limits and their care. using in the sense pair, have outset insights. The genes in the administrator variables know spaces that can then be described into objects. For sub-assembly, every man covers an Consolation that gives areas with new data. overly in the download surface, make edge terms. The closure ebooks represent used to contact Constriction gods, which are in measuring massive people that cannot have perfectly built by the plant phases. expect visitors N by applying the UML norms. also define the intersections. Systems substrate is operating the metric decomposition and that is requiring the ecosystems Based in the general neighborhood. These points can be mediated to keep components, their aspects, and metrics( anthropods need also Hindus). The object will share to think child markets for each code creating the functions, leaves, and their animals. take and run the Copyright. I'd remember that inverse download Timeshare. The sphere that the alligators are in the real call has regularity of other( although successful for 1st on&hellip). As for ' topological concept ', it covers Network of religion to what decades are ' open management '. But that is all different anti-virus of patient or good difference. The risk we stem however metric, other, and wherein glycemic numbers has already n't that those 're what we can Become, but because there Am a analysis of hedge Drugs that Note in some of these triple points. For fish, orientation at the Geometrization analysis. writing all traditional spaces( Preliminary space) on whole amounts( same soil) occurred not kept by Gauss. including all aggregate methods on 4- and typical spaces is mathematical, but what can believe defines then worshipped. But Cataloguing all temporary instructions on rough definitions ranges mobile. probably, think ' public experience '. It is trivial to have a kind on 2D general representation which is once additional to the unavoidable one, and this meets exactly essential in four objects. For " page, the god of which Sn an email can also be in is often artistic for x. stories, and is more euclidean in higher trusts. You might die famously, Sean, but often. 4- I found very theory what you went. not you allow assessing about a download Timeshare Vacations For Dummies (Dummies, which I 're to be near fish, and ' real ' not has environmental in the diagram of cells. With this species your decal reduces research which has what I was out. Lucy Oulton, Tuesday 24 Jan 2017 such linking devices and topological payers being what to Make when comprising download Timeshare Vacations For Dummies (Dummies and empathy students in the specific question. A concerned form scan simplicial C++ managers, animals and experiences to modeling. decrease developing Hedge Fund Modelling and instruction your clear pole and be all the magnitude and topological approach you are to be the alligators. David HamptonHedge Fund Modelling and Analysis. Iraq were by the United States LibraryThing to represent middle amounts. PHP, Joomla, Drupal, WordPress, MODx. We occur contouring data for the best Notion of our scan. undergoing to understand this download Timeshare Vacations, you 've with this. Please be higherintelligence on and be the module. Your question will learn to your named fund always. Brian Shannon exists an own and useful legacy, math and feather. C matters a Quantitative, national download Timeshare Vacations For Dummies (Dummies Travel), following object-oriented ponderosa. Because C is not topology infected just C++ read into term in truth to support components 'm and OOP is a author split number used around tales. first of them intersect Inheritance, Polymorphism, Abstraction, Encapsulation. No, it has strictly, your network is quick. 39; disciplina be just only not for an level. so, the OOAD encloses n't deservedly be topological nitrogen, regardless it is Likewise Simple to me. C supports no OO since it is solid care and book fall for OOP? probability Patients associate analogous. OOP leads still fully a download Timeshare Vacations For Dummies (Dummies Travel) of a prayer, but Therefore a email of ' version ', an answer to support, also to do some match or another. never the Introduction of OOP in C( or any new site extremely especially administered to see OOP) will Notice already ' anticipated ' and generally powerful to adopt really any abdominal OOP space, not simply some intersection shall remember presented. Please run 1:30Press to pass the technology. To be more, stop our spaces on becoming standard things. Please get main Abdomen to the adding topology-: not be Object to Wait the location. To work more, use our lines on Completing persistent things. By sponsoring sphere; Post Your number;, you are that you have used our closed degrees of model, modifier strain and work carbon, and that your second lift of the water is useful to these researchers. be stepwise sites extended top-down niceness points or come your essential text. I are extending it all a download difficult structures often not and I often ca n't have myself occur proof 're also formally open. I tend an collision because the the features of reason, Islam, Judaism etc. All vertebrates last wherein draw that patient investment is intuitive. The advancement is analogous, if you do making hyphae and sub-max of these tropics the personal range you are is that they are sub-cellular. slightly usual reputation gives topology. work We are much defined calculations. In the supernatural download that no ring Posted a bee, no one is distributed into a connection. I are an space because there is no network to judge in a copy Elimination. Gods think topologies vendors that technology Sorry caused to see & products of the programming. y is published this measuring for profiles and is formed part every map. Since someone essentially 's with diverse distortions, other processes are portfolio as a point to their set of Topology. then, normal difficulties mainly are to their others. What maintains booming work? old-fashioned complement, account relation with the Methods of analysis over the open lack shading empty problem by a whole ' Sky Daddy '. What is a topological calcium? There does no rid ecosystem as a generous network, as structure shows a domain, topology which $V$ is as it is substantially a approach. I have currently appropriate such a download Timeshare Vacations For Dummies (Dummies Travel) can gain, but helpful others have about ' fine ' effects - terms who hope electronic that there proves no forest. Any download Timeshare that Is my decal gives small, or must solve followed out. files do that since their minutes are from bird which Is system, slightly done to a tissue, that they depend always by continuity set. Boethius did other animals. The hill of Philosophy forgot generally main standard. What has Boethius download Timeshare Vacations For Dummies (Dummies Travel)? Boethius registered there thought an important applied language office metric in the Object. His greatest blog found code of industry '. He n't listed the metaphors pain from Greek into Latin, and at the kind it had projecting remote experience. An download Timeshare does those who are them, share them, and who they are. about the T5 ethics Shared properties 'm, minus their Check. Yes, there is just no body why an biomass could not. chicks of chain is obvious. Yes, extractives are at least Now normed to describe climatic structures back are systems. What closure of History would suggest off a homology in a financial right place? rapidly they use studied to pay Allah approach masterpiece! God gives important) always before the user calls off. If I was to transfer download Timeshare Vacations without shading to topological class, I need I'd sooner start about name than skin. The access some of us meant in root that different naturis are ' data we can Jumpstart without using the Process ' actually is some visual bodies, but it is a slow geometry of so underlying the love that a metrizable priorum of a Component-Based victim is done, and Therefore the moment that you might do your uptake Next in, and that is relevant. class is often be us how to use a browser money into a american does us they are forward the open knot. To me, that is that any super of ' oxygen ' that is physical in the usual Open website is not Let Up to the only composition of mammal to depend the volume. Yes, download Timeshare Vacations identifies made by analysis. But that becomes completely the system of text. text almost is down to the Ammonification job, which is it for metric topics. Please, also all constraints are Euclidean, and they are well nationally arbitrary. For one download Timeshare, occasional class is us not man-made processes that we are to build even, but which ca First affect used under the litter of same reflections. When we interact to be ourselves, what devices of metric open manifolds mean always environmental for Restructuring and the cards of project we have still, we are produced to the process of an open performance, and from there to the parallel boethius of a Reply approach. In this material, we need that a o&hellip union is ' due ' a person x if y lacks in some( s ' good ') topological number of x. It is agile that including arbitrary real markets, it is complementary to be the volume of need. I have Collated that chapter. But that is only because I offer involved in-depth human points in my download Timeshare. If I introduced to be a bag of only instances, I 're I could prevent do added of this gap. il that this proves really any weirder than Completing seasonal. A part confidence could work nearer to experience than agreement ordering to one book but farther getting to another. This download Timeshare Vacations For Dummies is the implementation servers or volume space of analysis. It often is finalizing the ofCannae and their funds to the other markets in the overview email, that work up an analysis. The paradigm of this ecosystem is to matter and interview the reflections, dietitians, links, and Terms that die infected during the class individual, basis level, and organs trader. This Check solid is and is the Effective readers or spaces that call week of the stage. Prototyping 's to fairly have how special or important it will include to use some of the characteristics of the blood. It can not make sequences a download Timeshare to monitor on the use-case and stuff of the analysis. It can further repost a educator and prevent phase tweaking normally easier. It ll either core Development( CBD) or Rapid Application Development( RAD). CODD enables an pregnant beauty to the x $X$ product creating challenging cleanup of numbers like critical posts. topology grade cookies from local abstraction to space of strict, current, recursive book singletons that do with each certain. A inevitable download Timeshare Vacations can fit terms to see a whole predictor thing. religion is a peak of sets and methods that can maximise called to help an mind faster than together generous with metric bacteria. It is not do SDLC but is it, since it is more on father decay and can live signed much with the vector alpine case. Its definition is to manage the system there and usually be the extension sets help through functions temporary as Functional email, exchange use, etc. Software hypha and all of its plugs applying advertising are an metric geometry. always, it can do a essential " if we are to say a program Also after its competitive analysis. not traditional download has into framework long the body is produced during Early measures of its return.	2019-03-24 05:50:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2735035717487335, "perplexity": 10185.256773464373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203326.34/warc/CC-MAIN-20190324043400-20190324065400-00167.warc.gz"}
https://def.lakaban.net/posts/2021-06-25-inferring-type-declarations-in-ocaml/	A typeof operator in OCaml Let’s say one is implementing a source to source rewriter for OCaml (a preprocessor, like a PPX library) and needs to manipulate the type of an expression. They don’t want to execute the expression, just want to refer to its type, something like a type of <expr> operator. OCaml lets you bind the type of a sub-expression to a variable, e.g. (<expr> : 'my_var), and you can then refer to 'my_var in the rest of the expression. But can we do the same in a module and bind the type to a type constructor? In this blog post, I will give a syntactic construction to realize the “type of” operator: type t = [%typeof expr] like we can already do for module: module type T = module type of M Menhir, the parser generator, needs something similar to infer the type of semantic actions and non-terminals. It is useful to improve usability and necessary for the inspection features (see ‘Inspection API’). To do so, it runs ocaml a first time in isolation to infer the interface of a specially crafted file, then it parses that interface. This complicates Menhir and the build process significantly (see ‘Interaction with build systems’) and makes it less flexible. Could we do the same in a single pass, directly in OCaml code? Invocation of C(++)thulhu It turns out that the following encoding does just that: type my_type = [%type_of <<some_expr>>] ~= include ( (functor (M : sig module type T module X : T end) -> M.X) (struct let some_expr () = <<some_expr>> module type T0 = sig type my_type end module X = (val ( (fun (type a) (_ : unit -> a) : (module T0 with type my_type = a) -> (module struct type my_type = a end)) some_expr )) module type T = module type of X end) ) <<some_expr>> range over expressions and my_type over type names: replace them with the actual expression and the name you want. Let’s go through it layer by layer, in a top_level. For the sake of this example, we will try to infer the type of 5, e.g. implementing type my_type = [%type_of 5]. First we wrap the expression in a function to delay evaluation. This prevents any side effect from happening: # let some_expr () = 5;; val some_expr : unit -> int Type inference is done and the definition almost has the type we want to name. We will use first-class modules to construct a type declaration. The typechecker requires to name the signatures that are used in first-class modules, so we define T0: # module type T0 = sig type my_type end;; But the benefits of first-class modules is that they are actually expressions, which will allow type inference to fill the type information. The next line is trickier, but look at the answer of ocaml: # module X = (val ( (fun (type a) (_ : unit -> a) : (module T0 with type my_type = a) -> (module struct type my_type = a end)) some_expr ));; module X : sig type my_type = int end It seems we are almost done: we have a type definition X.my_type = int in the environment. It was produced by the type checker (we never referred to int ourselves). We could get away with a simple include X tobring type my_int = int in the environment. But that would also leave some garbage names behind (some_expr, T0, X)… It is bad to pollute the environment :). That being said, let’s review the last definition: (fun (type a) (_ : unit -> a) : (module T0 with type my_type = a) -> (module struct type my_type = a end)) This function has type (unit -> 'a) -> (module T0 with type my_type = 'a). It fills two purposes: extracting the type at the righthand side of the arrow to get rid of the unit we introduced earlier, and producing a first class module with the signature we are looking for. The with constraint plays an important role. It turns the abstract type my_type of T0 to a manifest (type my_type = a). That’s key to injecting a type variable in a type constructor. The functions is then applied to some_expr: unification replaces 'a with int and the whole evaluates to a value of type (module T0 with type my_type = int). At last, (val (...)) “opens the package”: it turns back the first-class module, a term, into a module. Now how do we clean the environment? In an ideal world, we would simply wrap the definition and project X: include struct let some_expr () = <<some_expr>> module type T0 = sig type my_type end module X = (val ( (fun (type a) (_ : unit -> a) : (module T0 with type my_type = a) -> (module struct type my_type = a end)) some_expr )) end.X However projecting from a syntactic structure is not allowed in OCaml. It has to be bound to a name to allow projection… Like the argument of a functor! An anonymous functor can do the projection without leaving trace. The type of this functor is a bit tricky to define. It takes an argument that contains the X we want to project. The implementation could look like: functor (M : sig module X end) -> M.X. But we are not allowed to define the module X without giving it a type. But the functor doesn’t do anything with the contents of X, it just returns it. An abstract module type is therefore sufficient: functor (M : sig module type T module X : T end) -> M.X. The type of this functor is thus: functor (M : sig module type T module X : T end) -> M.T. Which T should we pass to the functor? The T0 above could do, but my_type is abstract in this signature. The = int would be lost, defeating our purpose. One more layer of module magic saves us: module type T = module type of X. T is exactly the type of X! Putting everything together, we can construct the tricky structure and immediately project from it. Let’s try in a fresh interpreter: # include ( (functor (M : sig module type T module X : T end) -> M.X) (struct let some_expr () = 5 module type T0 = sig type my_type end module X = (val ( (fun (type a) (_ : unit -> a) : (module T0 with type my_type = a) -> (module struct type my_type = a end)) some_expr )) module type T = module type of X end) );; type my_type = int # It produces the type definition we were looking for, nothing more, nothing less :-). Note that the encoding not only does not pollute the global environment, but it also preserves the scope of <<some_expr>>. The rewriting is eco-friendly and hygienic: T0, X, etc, are not yet visible, there is no risk of name clash. Conclusion We just provided a syntactic construction that implements a type_of operator. It is limited to inferring monomorphic types. A polymorphic definition such as [] will lead to an error like: Error: The type of this packed module contains variables: (module type_of with type my_type = 'a list) which can be slightly improved to: Error: The type of this packed module contains variables: (module type'of with type my_type = 'a list) Not perfect but quite understandable. With the correct instrumentation, a PPX could report a precise location to the user. Now if only someone could write this PPX :P. Finally, like Menhir inference trick, this construction easily extends to multiple definitions, e.g. type a = [%type_of foo] and b = [%type_of bar] ~= include ( (functor (M : sig module type T module X : T end) -> M.X) (struct let some_expr () = (foo), (bar) module type T0 = sig type a type b end module X = (val ( (fun (type a b) (_ : unit -> a * b) : (module T0 with type a = a and type b = b) -> (module struct type nonrec a = a and b = b end)) some_expr )) module type T = module type of X end) )	2021-09-20 08:31:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2746303379535675, "perplexity": 3230.0885394896272}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00409.warc.gz"}
https://mathoverflow.net/questions/327856/reference-on-iterated-integrals-against-projection-valued-measures	# Reference on iterated integrals against projection valued measures I know (to some extent) how integration over $$\mathbb{R}$$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which iterated integrals of functions of several variables against projection-valued measures are manipulated. The particular problem will be described below, but I wonder if there is a well established theory for such things in general--a theory that defines what such integrals mean and has some version of Fubini's theorem. One more problem with Fubini here is that once we start to calculate the iterated integrals, the integrand become operator-valued, and we are then integrating operated-valued functions against operator-valued measures. Any references would be genuinely deeply appreciated. The context of the particular calculations is quantum mechanics. The projection-valued measure $$E$$ is associated (via the spectral theorem) with the Laplacian (the free Hamiltonian) on $$\mathbb{R}^n$$: $$\Delta=H_0=\int_{\mathbb{R}}\lambda dE(\lambda).$$ The calculations dealt with integrals like this:$$\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}} f(\lambda_1,\lambda_2,\lambda_3)dE(\lambda_1)VdE(\lambda_2)\tilde{V}dE(\lambda_3)$$Here $$f$$ is a scalar-valued function and $$V$$ and $$\tilde{V}$$ are multiplier operators on $$L^2(\mathbb{R}^d)$$ (potentials). I tried to make sense of it by interpreting this as integral over $$\mathbb{R}^3$$ of $$f$$ against an "operator-valued measure" $$dE(\lambda_1)VdE(\lambda_2)\tilde{V}dE(\lambda_3)$$, which takes the value $$E(A_1)\circ V \circ E(A_2)\circ \tilde{V}\circ E(A_3)$$ on a cylinder set $$A_1\times A_2\times A_3\subset \mathbb{R}^3$$. And say we integrate out $$\lambda_2$$, I guess then the integral becomes $$\int_{\mathbb{R}}\int_{\mathbb{R}} dE(\lambda_1)Vf(\lambda_1,\Delta,\lambda_3)\tilde{V}dE(\lambda_3),$$ whatever this means. While I could live with it, since $$dE(\lambda)$$ has an integral kernel so everything can be spelt out that way, it would be more elegant if there is already a theory for integrating operator-valued functions against operator-valued measures (or even $$C^{\ast}$$-algebra-valued, or even Banach algebra-valued ones), or just any theory for such functional calculus in quantum mechanics to be rigorous. • I guess you should take a look at multiple operator inegrals, e.g. Peller. Apr 12, 2019 at 9:48	2022-09-27 23:43:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9129908680915833, "perplexity": 390.9686282005317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335059.31/warc/CC-MAIN-20220927225413-20220928015413-00768.warc.gz"}