text:
bmc_jvenn.tex 30.3 KB
 Jerome Mariette committed Mar 03, 2014 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 %% BioMed_Central_Tex_Template_v1.06 %% % % bmc_article.tex ver: 1.06 % % % %%IMPORTANT: do not delete the first line of this template %%It must be present to enable the BMC Submission system to %%recognise this template!! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% LaTeX template for BioMed Central %% %% journal article submissions %% %% %% %% <8 June 2012> %% %% %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% For instructions on how to fill out this Tex template %% %% document please refer to Readme.html and the instructions for %% %% authors page on the biomed central website %% %% http://www.biomedcentral.com/info/authors/ %% %% %% %% Please do not use \input{...} to include other tex files. %% %% Submit your LaTeX manuscript as one .tex document. %% %% %% %% All additional figures and files should be attached %% %% separately and not embedded in the \TeX\ document itself. %% %% %% %% BioMed Central currently use the MikTex distribution of %% %% TeX for Windows) of TeX and LaTeX. This is available from %% %% http://www.miktex.org %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% additional documentclass options: % [doublespacing] % [linenumbers] - put the line numbers on margins %%% loading packages, author definitions %\documentclass[twocolumn]{bmcart}% uncomment this for twocolumn layout and comment line below \documentclass{bmcart} %%% Load packages  Jerome Mariette committed Mar 20, 2014 50 51 52 53 54 55 56 57 58 59 60 61 62 \usepackage{listings} \usepackage{color} \definecolor{gray}{rgb}{0.5,0.5,0.5} \lstset{ language=Java, showstringspaces=false, columns=flexible, basicstyle={\scriptsize \ttfamily}, numbers=none, stringstyle=\color{gray}, breaklines=true, breakatwhitespace=true }  Jerome Mariette committed Mar 03, 2014 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 %\usepackage{amsthm,amsmath} %\RequirePackage{natbib} %\RequirePackage{hyperref} \usepackage[utf8]{inputenc} %unicode support %\usepackage[applemac]{inputenc} %applemac support if unicode package fails %\usepackage[latin1]{inputenc} %UNIX support if unicode package fails %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% If you wish to display your graphics for %% %% your own use using includegraphic or %% %% includegraphics, then comment out the %% %% following two lines of code. %% %% NB: These line *must* be included when %% %% submitting to BMC. %% %% All figure files must be submitted as %% %% separate graphics through the BMC %% %% submission process, not included in the %% %% submitted article. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\includegraphic{} \def\includegraphics{} %%% Put your definitions there: \startlocaldefs \endlocaldefs %%% Begin ...  Jerome Mariette committed Aug 05, 2014 98 \begin{document}\sloppy  Jerome Mariette committed Mar 03, 2014 99 100 101 102 103  %%% Start of article front matter \begin{frontmatter} \begin{fmbox}  Jerome Mariette committed May 06, 2014 104 \dochead{Software}  Jerome Mariette committed Mar 03, 2014 105 106 107 108 109 110 111  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Enter the title of your article here %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Jerome Mariette committed May 05, 2014 112 \title{jvenn: an interactive Venn diagram viewer.}  Jerome Mariette committed Mar 03, 2014 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Enter the authors here %% %% %% %% Specify information, if available, %% %% in the form: %% %% ={,} %% %% = %% %% Comment or delete the keys which are %% %% not used. Repeat \author command as much %% %% as required. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \author[  Jerome Mariette committed Mar 11, 2014 129 130 131 132 133 134  addressref={aff2}, % id's of addresses, e.g. {aff1,aff2} noteref={n1}, % id's of article notes, if any email={Philippe.Bardou@toulouse.inra.fr} % email address ]{\inits{PB}\fnm{Philippe} \snm{Bardou}} \author[ addressref={aff1},  Jerome Mariette committed Mar 03, 2014 135 136  corref={aff1}, % id of corresponding address, if any noteref={n1}, % id's of article notes, if any  Jerome Mariette committed Mar 11, 2014 137 138  email={Jerome.Mariette@toulouse.inra.fr} ]{\inits{JM}\fnm{J\'{e}r\^{o}me} \snm{Mariette}}  Jerome Mariette committed May 22, 2014 139 140 141 \author[ addressref={aff1}, email={Frederic.Escudie@toulouse.inra.fr}  Jerome Mariette committed Jun 02, 2014 142 ]{\inits{FE}\fnm{Fr\'{e}d\'{e}ric} \snm{Escudi\'{e}}}  Jerome Mariette committed Mar 11, 2014 143 144 145 146 \author[ addressref={aff1}, email={Christophe.Djemiel@toulouse.inra.fr} ]{\inits{CD}\fnm{Christophe} \snm{Djemiel}}  Jerome Mariette committed Mar 03, 2014 147 148 \author[ addressref={aff1,aff2},  Jerome Mariette committed Mar 11, 2014 149 150  email={Christophe.Klopp@toulouse.inra.fr} ]{\inits{CK}\fnm{Christophe} \snm{Klopp}}  Jerome Mariette committed Mar 03, 2014 151 152 153 154 155 156 157 158 159 160 161  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Enter the authors' addresses here %% %% %% %% Repeat \address commands as much as %% %% required. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \address[id=aff1]{% % unique id  Jerome Mariette committed Mar 11, 2014 162 163 164 165 166  \orgname{Plate-forme bio-informatique Genotoul / MIA-T, INRA}, % university, etc \street{Borde Rouge}, % \postcode{31326} % post or zip code \city{Castanet-Tolosan}, % city \cny{France} % country  Jerome Mariette committed Mar 03, 2014 167 168 } \address[id=aff2]{%  Jerome Mariette committed Mar 11, 2014 169 170 171 172 173  \orgname{Plate-forme SIGENAE / GenPhySE, INRA}, % university, etc \street{Borde Rouge}, % \postcode{31326} % post or zip code \city{Castanet-Tolosan}, % city \cny{France} % country  Jerome Mariette committed Mar 03, 2014 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Enter short notes here %% %% %% %% Short notes will be after addresses %% %% on first page. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{artnotes} %\note{Sample of title note} % note to the article \note[id=n1]{Equal contributor} % note, connected to author \end{artnotes} \end{fmbox}% comment this for two column layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% The Abstract begins here %% %% %% %% Please refer to the Instructions for %% %% authors on http://www.biomedcentral.com %% %% and include the section headings %% %% accordingly for your article type. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstractbox} \begin{abstract} % abstract  Jerome Mariette committed Mar 03, 2014 206 \parttitle{Background} %if any  Jerome Mariette committed Aug 05, 2014 207 208 209 210 211 212 213 Venn diagrams are commonly used to display list comparison. In biology, they are widely used to show the differences between gene lists originating from different differential analyses, for instance. They thus allow the comparison between different experimental conditions or between different methods. However, when the number of input lists exceeds four, the diagram becomes difficult to read. Alternative layouts and dynamic display features can improve its use and its readability.  Jerome Mariette committed Mar 04, 2014 214   Jerome Mariette committed Mar 03, 2014 215 \parttitle{Results} %if any  Jerome Mariette committed Jun 11, 2014 216 217 jvenn is a new JavaScript library. It processes lists and produces Venn diagrams. It handles up to six input lists and presents results using classical or Edwards-Venn  Jerome Mariette committed Aug 05, 2014 218 219 layouts. User interactions can be controlled and customized. Finally, jvenn can easily be embeded in a web page, allowing to have dynamic Venn diagrams.  Jerome Mariette committed Mar 07, 2014 220   Jerome Mariette committed Mar 03, 2014 221 \parttitle{Conclusions} %if any  Jerome Mariette committed Aug 05, 2014 222 jvenn is an open source component for web environments helping  Jerome Mariette committed May 12, 2014 223 scientists to analyze their data. The library package, which comes with full  Jerome Mariette committed Jun 11, 2014 224 documentation and an example, is freely available at  Jerome Mariette committed May 06, 2014 225 http://bioinfo.genotoul.fr/jvenn.  Jerome Mariette committed Mar 20, 2014 226   Jerome Mariette committed Mar 03, 2014 227   Jerome Mariette committed Mar 03, 2014 228 229 230 231 232 233 234 235 236 237 238 \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% The keywords begin here %% %% %% %% Put each keyword in separate \kwd{}. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{keyword}  Jerome Mariette committed Mar 11, 2014 239 240 241 \kwd{Venn} \kwd{Edward-Venn} \kwd{vizualisation}  Jerome Mariette committed Mar 03, 2014 242 \kwd{jquery}  Jerome Mariette committed May 05, 2014 243 \kwd{JavaScript}  Jerome Mariette committed Mar 03, 2014 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 \end{keyword} % MSC classifications codes, if any %\begin{keyword}[class=AMS] %\kwd[Primary ]{} %\kwd{} %\kwd[; secondary ]{} %\end{keyword} \end{abstractbox} % %\end{fmbox}% uncomment this for twcolumn layout \end{frontmatter} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% The Main Body begins here %% %% %% %% Please refer to the instructions for %% %% authors on: %% %% http://www.biomedcentral.com/info/authors%% %% and include the section headings %% %% accordingly for your article type. %% %% %% %% See the Results and Discussion section %% %% for details on how to create sub-sections%% %% %% %% use \cite{...} to cite references %% %% \cite{koon} and %% %% \cite{oreg,khar,zvai,xjon,schn,pond} %% %% \nocite{smith,marg,hunn,advi,koha,mouse}%% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% start of article main body % %%%%%%%%%%%%%%%% %% Background %% %%  Jerome Mariette committed Mar 03, 2014 286 \section*{Background}  Jerome Mariette committed Mar 04, 2014 287   Jerome Mariette committed Aug 05, 2014 288 289 High-throughput biology has led to an increasing number of data, with more and more complex experimental designs. The analysis of these data often produces  Jerome Mariette committed Aug 06, 2014 290 291 292 293 294 biological identifier lists, including gene names or OTU (Operational Taxonomic Unit), obtained from different methods (for differential analysis) or from different experimental conditions. Venn diagrams \cite{Venn1880} are a common visualization chart, which allows to spot shared and unshared identifiers providing an insight on lists similarities.  Jerome Mariette committed Jul 23, 2014 295   Jerome Mariette committed Aug 05, 2014 296 In a Venn diagram, each list is presented by a transparent shape. Shape overlaps  Jerome Mariette committed Jun 11, 2014 297 contain the elements shared between lists or more often the corresponding counts.  Jerome Mariette committed Aug 05, 2014 298 299 300 301 302 303 304 In proportional Venn diagrams, the size of a shape is proportional to the number of elements of the corresponding list or of the corresponding lists intersection. Venn diagrams with up to four lists are easy to read and understand but Venn diagrams with more than four lists, are much harder to interpret. To solve this problem, the Edwards-Venn \cite{Edwards2004} representation introduces new shapes providing a clearer view, shown in the example of Fig.~\ref{fig::edwards}.  Jerome Mariette committed Jun 11, 2014 305 306  Many Venn diagram software packages are already available. The first six lines  Jerome Mariette committed Aug 06, 2014 307 of Table~\ref{table::features} present the main packages with their  Jerome Mariette committed Aug 05, 2014 308 309 main features (maximum number of input lists, input data formats, Venn diagram layouts, application types and output formats). The table gives insight on  Jerome Mariette committed Aug 06, 2014 310 311 312 313 314 315 316 317 318 several aspects of the Venn diagram production and highlights that, up to now, no web application handled up to six lists. VENNTURE \cite{Bronwen2012} is the only application able to produce such diagrams but it only implements Edwards layout and runs only under MS-Windows OS, producing static MS-PowerPoint and MS-Excel files. Proportional Venn diagrams can only display a very limited number of lists, three at maximum. The only feature available in other software which is not in jvenn is the proportional diagram. This is justified by the fact that jvenn was designed to display up to six lists and that proportional diagram is not suited to visualize more than three lists.  Jerome Mariette committed Jun 11, 2014 319 320 321  Hereafter we introduce jvenn, a JavaScript library, developed as a jQuery plug-in \cite{jquery}, including many features easing diagram production and  Jerome Mariette committed Aug 05, 2014 322 323 324 enhancing their readability. In particular, jvenn can handle up to 6 lists, is a dynamic tool and implements both proportional and Edwards layouts. The library has already been used and cited in two scientific publications  Jerome Mariette committed Aug 05, 2014 325 \cite{Bianchia2013, Aravindraja2013}. It is already embedded in different web  Jerome Mariette committed Aug 05, 2014 326 327 applications such as nG6 \cite{Mariette2012}, RNAbrowse \cite{Mariette} and WallProtDB \cite{SanClemente}.  Jerome Mariette committed Mar 03, 2014 328   Jerome Mariette committed Jul 15, 2014 329   Jerome Mariette committed Mar 05, 2014 330 \section*{Implementation}  Jerome Mariette committed Mar 04, 2014 331   Jerome Mariette committed Aug 05, 2014 332 333 334 335 336 This section presents the main features of the jvenn library, including the kind of inputs it accepts, the different types of charts it displays, the types of the outputs and how it can be integrated in websites or directly used on our example web page.  Jerome Mariette committed Jul 15, 2014 337 \subsection*{Inputs}  Jerome Mariette committed Jun 11, 2014 338   Jerome Mariette committed Aug 05, 2014 339 340 341 342 343 344 345 346 347 The jvenn library accepts three different input formats : Lists'', Intersection counts'' and Count lists''. Examples are presented in Table~\ref{table::format}, where the different lists are sample1'' and sample2'', the elements of the different lists are given in the fields data''. For Intersection counts'', the lists are given a label (A'' or B'') which is used to make the correspondence between the list and its count. Finally, Count lists'' provide a count number for each element of a list. Hence, with Count lists'' the figures presented in the diagram correspond to the sums of counts of all elements shared between lists. they can be  Jerome Mariette committed Aug 06, 2014 348 particularly useful to present OTU read counts \cite{Aravindraja2013}. For  Jerome Mariette committed Aug 05, 2014 349 350 Lists'' and Count lists'', jvenn computes the intersection counts and displays the chart. For intersection counts'', the intersection counts is  Jerome Mariette committed Aug 06, 2014 351 provided by the user.  Jerome Mariette committed Mar 07, 2014 352   Jerome Mariette committed Jul 15, 2014 353 \subsection*{Display features}  Jerome Mariette committed Mar 27, 2014 354   Jerome Mariette committed Aug 05, 2014 355 356 Venn diagrams are commonly used to present up to six lists but for six lists, the intersection areas obtained when using a proportional layout are  Jerome Mariette committed Aug 05, 2014 357 often too small to display the figures.  Jerome Mariette committed May 12, 2014 358   Jerome Mariette committed Aug 05, 2014 359 To display five or six lists diagrams, in a user-friendly manner, jvenn  Jerome Mariette committed Aug 05, 2014 360 implements several features.  Jerome Mariette committed Aug 05, 2014 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 First, the layout can be switched between the standard layout and the Edwards-Venn layout (Fig.~\ref{fig::edwards}) which gives a clearer graphical representation for six lists diagrams. To enhance the figure's readability for the classical six lists Venn chart, some count values are not shown and some are display outside the chart, using lines to line the count to its corresponding area. However, this is still not enough to show all figures. Therefore, a switch button panel (right side of Fig.~\ref{fig::features}) was added. It enables to switch on and off the different lists and to display the corresponding intersection counts. When the number of characters of the intersection count exceeds the available space to display it, the value is substituted by a question mark. When the mouse is mouved over this question mark, the value pops-up. To emphasize the list involved in an intersection area, jvenn highlights the intersection shapes when mouse is moved over, fading the others out. The extra charts presented under the Venn diagram ease the verification and comparison of multiple lists. The list size graph allows users to check the  Jerome Mariette committed Jun 11, 2014 378 379 380 homogeneity of the input list sizes. The intersection size graph can be used to compare the compactness of multiple Venn diagrams.  Jerome Mariette committed Jul 24, 2014 381 Scientists are usually interested in extracting identifier lists for some  Jerome Mariette committed Aug 05, 2014 382 383 384 385 intersections, therefore, jvenn implements a one-click function which retrieves the names of the corresponding sets and the identifiers. To find an identifier, one can use a dynamic search box. The shapes containing the matching identifiers are highlighted when using this tool.  Jerome Mariette committed Mar 27, 2014 386   Jerome Mariette committed Jul 15, 2014 387 388 \subsection*{Outputs}  Jerome Mariette committed Aug 05, 2014 389 390 391 392 jvenn display is based on a JavaScript canvas object that allows for PNG export. The intersection table can also be downloaded as a CSV file. This file contains a header line with the diagram area labels and, in column, the identifiers of the elements contained in the area.  Jerome Mariette committed Jul 25, 2014 393 394 395  \subsection*{Integration}  Jerome Mariette committed Aug 05, 2014 396 jvenn allows programmers having only moderate JavaScript experiences to embed  Jerome Mariette committed Aug 05, 2014 397 398 399 Venn diagrams in a web page without dependency. It has been designed following the examples of jbrowse \cite{Westesson01032013}, Cytoscape-Web \cite{Lopes2010}, and jHeatmap \cite{DeuPons2014}.  Jerome Mariette committed Jul 25, 2014 400 401 The integration documentation is included in the software package which can be downloaded from http://bioinfo.genotoul.fr/jvenn.  Jerome Mariette committed Jul 15, 2014 402 403 404  \subsection*{Web application}  Jerome Mariette committed Aug 05, 2014 405 jvenn can also be directly used as a web application, which is available at  Jerome Mariette committed Aug 05, 2014 406 http://bioinfo.genotoul.fr/jvenn/example.html (Fig.~\ref{fig::web}).  Jerome Mariette committed Aug 05, 2014 407 jvenn's web application performances depend on the client browser. Using the  Jerome Mariette committed Aug 05, 2014 408 409 current version on a standard Linux computer (one cpu, four GB of RAM), it displays a six lists diagram of 10,000 identifiers in two seconds.  Jerome Mariette committed Jul 23, 2014 410   Jerome Mariette committed Jul 15, 2014 411 412 413  \section*{Results}  Jerome Mariette committed Aug 05, 2014 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 M.A. Dillies and colleagues \cite{Dillies2012} have compared seven methods for normalization and search of differentially expressed genes in RNASeq data. This study is designed to provide a set of best practices to help biologists with their data processing. Table 2 of the cited article is the contingency table of the differentially expressed genes obtained from the seven methods, where counts in the table correspond to the intersection of two lists obtained from two different methods. The raw data table, kindly provided by the team, contains 5,277 lines and seven columns. The columns correspond to the different methods presented in the Differential expression analysis'' section of their article. The data in the table was filtered ($p < 0.05$) to retrieve the gene name lists corresponding to each method. As, jvenn handles only six list at most, six out of the seven lists were selected for further processing: we left out the median normalization method because, for one hand, this method is very similar to several other methods (as shown in the article) and, for the other hand, we believe that median is a poor estimate of the sequencing length, which is the bias that normalization methods try to correct. The lists were uploaded to the jvenn application and a Venn diagram was obtained, using an Edwards layout, which is shown in Fig.~\ref{fig::edwards}.  Jerome Mariette committed Jul 24, 2014 432   Jerome Mariette committed Jul 25, 2014 433 The same analysis was performed with VENNTURE, the only other tool able to  Jerome Mariette committed Jul 25, 2014 434 generate a six list Edwards Venn diagram. First, the software package was  Jerome Mariette committed Aug 05, 2014 435 436 437 438 installed on a computer running under MS-Windows OS. The six gene lists were loaded in an MS-Excel spreadsheet and VENNTURE was run using the spreadsheet as input generating a static MS-PowerPoint file containing the diagram and a MS-Excel file with all the intersection contents.  Jerome Mariette committed Jul 17, 2014 439   Jerome Mariette committed Jul 24, 2014 440 \section*{Discussion}  Jerome Mariette committed Jul 17, 2014 441   Jerome Mariette committed Aug 06, 2014 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 The lists overlaps, as produced by jvenn, are given in Fig.~\ref{fig::edwards} (Edwards layout) and \ref{fig::features} (standard layout). The highest counts are located in central areas of the graph, showing that the corresponding methods share large portions of gene lists. The jvenn statistics show that the different methods produce gene lists with very different sizes (minimum 417 - maximum 1,249) and that most of the genes are shared between methods: 1,069 genes out of 1,347 are common between at least four methods. In a very intuitive manner, the chart also points out that the results are strongly consensual since there are many zeros in the peripheral areas. Only a few genes (114) are specific to one list only (24 for FQ, 27 for UQ and 63 for DESeq, which appears to be the less restrictive method, as shown in the barplot below the Venn diagram, and also the most different from the others). Genes that are in two lists only are also very few (47: 13 for DESeq and TMM, 5 for UQ and FQ, 15 for TMM and UQ, 8 for FQ and DESeq and 6 for DESeq and UQ). Note that all these numbers are easily read from the chart and that the strong consensus between the lists is also clearly shown from the upper side figure Number of elements: specific or shared by several lists''). Such findings are not easily shown using only contingency tables. The largest count over all lists overlaps is found to be 484, which is the number of genes found to be differential by DESeq, TMM, UQ and FQ. As shown in Fig~\ref{fig::web}, this list is very easily retrieved from the web application in one click only, providing the biologist with a large list of very consensual list to study. On the other hand, if the biologist is interested in one specific gene, this gene can easily be tracked using the search box at the top side of Fig.~\ref{fig::web}. As no specific gene is of interest in the seminal work, we simply picked out one of the 5,277 genes randomly (G002562) and used it in the search box. It was found to be part of the five genes specific to FQ and UQ. Making the same analysis with VENNTURE is also possible but a bit harder: the 484 genes shared by DESeq, TMM, UQ and FQ can be found easily in the intersection spreadsheet outputed by VENNTURE but the diagram did not allow to search for gene G002562. Thus, this gene has to be found using MS-Excel text search in the intersection spreadsheet, which is less handy than a dynamic and interactive search. Moreover, the additional statistics are not provided by the tool.  Jerome Mariette committed Aug 05, 2014 480   Jerome Mariette committed Jul 17, 2014 481   Jerome Mariette committed Aug 05, 2014 482 \section*{Conclusion}  Jerome Mariette committed Mar 03, 2014 483   Jerome Mariette committed Aug 05, 2014 484 485 486 jvenn enables to compare up to six lists and updates the diagram automatically when modifying the lists content. Compared to VENNTURE it does not need any local installation of a new program and it gives access to a dynamic diagram  Jerome Mariette committed Aug 06, 2014 487 providing simple tools to extract gene lists and perform searches.  Jerome Mariette committed Aug 05, 2014 488 489 490 491 492 493 494 495 496 497 498 jvenn's statistics charts give a simple and quick overview of the sizes of the different lists and of their overlaps. It permits to compare different Venn diagrams. These features are not available in the VENNTURE software package. For biologists using different techniques in their experiment or in their statistical analysis, jvenn enables to quickly extract the shared identifiers. When comparing different methods applied to extract differentially expressed genes, these features ease the analysis. Thanks to its numerous features, dynamic behavior and graphical layout quality, jvenn can be efficiently used in many cases to compare different sets of results  Jerome Mariette committed Aug 06, 2014 499 500 and easily extract shared elements. Being a simple JavaScript plug-in allows developers to embed it in any web environment.  Jerome Mariette committed Jul 25, 2014 501   Jerome Mariette committed Mar 07, 2014 502   Jerome Mariette committed Mar 03, 2014 503 \section*{Availability and requirements}  Jerome Mariette committed Mar 03, 2014 504   Jerome Mariette committed May 12, 2014 505 jvenn is freely available under the GNU General Public License (GPL) and can be  Jerome Mariette committed Aug 06, 2014 506 downloaded with an example and its full documentation at  Jerome Mariette committed May 12, 2014 507 508 http://bioinfo.genotoul.fr/jvenn website. A running version is accessible at http://bioinfo.genotoul.fr/jvenn/example.html.  Jerome Mariette committed Mar 07, 2014 509   Jerome Mariette committed Mar 03, 2014 510 511 512 513 514 515 516 517 518 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Backmatter begins here %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{backmatter} \section*{Competing interests}  Jerome Mariette committed Mar 11, 2014 519 The authors declare that they have no competing interests.  Jerome Mariette committed Mar 03, 2014 520 521  \section*{Author's contributions}  Jerome Mariette committed May 22, 2014 522 JM conceived and designed the project. JM, PB, FE and CD implemented the project.  Jerome Mariette committed May 12, 2014 523 524 CK evaluated software capabilities, and provided feedback on implementation. JM and CK wrote the manuscript. All authors read and approved the final manuscript.  Jerome Mariette committed Mar 03, 2014 525 526  \section*{Acknowledgements}  Jerome Mariette committed Mar 03, 2014 527 We would like to acknowledge all our users for providing us useful feedback on  Jerome Mariette committed May 23, 2014 528 the system and for pointing out features worth developing. We thank the  Jerome Mariette committed Aug 05, 2014 529 530 531 reviewers and Nathalie Villa-Vialaneix for their insightful and constructive comments. We also thank Julie Aubert and the French StatOmique Consortium for providing us the data used in the Results'' section.  Jerome Mariette committed Mar 03, 2014 532   Jerome Mariette committed Mar 03, 2014 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The Bibliography %% %% %% %% Bmc_mathpys.bst will be used to %% %% create a .BBL file for submission. %% %% After submission of the .TEX file, %% %% you will be prompted to submit your .BBL file. %% %% %% %% %% %% Note that the displayed Bibliography will not %% %% necessarily be rendered by Latex exactly as specified %% %% in the online Instructions for Authors. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if your bibliography is in bibtex format, use those commands: \bibliographystyle{bmc-mathphys} % Style BST file  Jerome Mariette committed Mar 05, 2014 550 \bibliography{bmc_jvenn} % Bibliography file (usually '*.bib' )  Jerome Mariette committed Mar 03, 2014 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572  % or include bibliography directly: % \begin{thebibliography} % \bibitem{b1} % \end{thebibliography} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Figures %% %% %% %% NB: this is for captions and %% %% Titles. All graphics must be %% %% submitted separately and NOT %% %% included in the Tex document %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Do not use \listoffigures as most will included as separate files \section*{Figures} \begin{figure}[h!]  Jerome Mariette committed Mar 25, 2014 573  \caption{\csentence{A six lists classic Venn diagram.}  Jerome Mariette committed Aug 05, 2014 574 575 576 577 578  On mouse over a figure, the shape corresponding to the lists involved in the intersection are highlighted and the other ones faded out. In this example, the user pointed the intersection between DESeq, FQ, UQ and TMM which contains 484 different genes. }\label{fig::features}  Jerome Mariette committed Aug 05, 2014 579  \end{figure}  Jerome Mariette committed Mar 03, 2014 580   Jerome Mariette committed Aug 05, 2014 581  \begin{figure}[h!]  Jerome Mariette committed Mar 05, 2014 582  \caption{\csentence{A six lists Edwards-Venn diagram.}  Jerome Mariette committed Aug 05, 2014 583 584 585 586 587 588  This Venn diagram displays overlaps between six different biological samples. The icon, located on the top-right, allows users to download the diagram as a PNG file. The middle-right switch button panel allows to activate or dis-activate lists to access a specific intersection count. Charts showing the list size and intersection size repartition located underneath the diagram.}\label{fig::edwards}  Jerome Mariette committed Aug 05, 2014 589 590 591  \end{figure} \begin{figure}[h!]  Jerome Mariette committed Aug 05, 2014 592 593 594 595 596 597 598 599 600 601  \caption{\csentence{jvenn web application.} The running version of jvenn accessible at http://bioinfo.genotoul.fr/jvenn/example.html. This one allows the user to set all jvenn main features. The layout can be changed from classical to Edwards, a search box is accessible, the switch button panel and the exporting button are available and the statistical charts are displayed. On the right of the page, each of the six textareas can be filled with a list of elements (one per line). If the same item is given multiple times, this one will be considered as unique. The list labels can also be customized using the text field on the top of each textarea.}\label{fig::web}  Jerome Mariette committed Aug 05, 2014 602  \end{figure}  Jerome Mariette committed Mar 03, 2014 603   Jerome Mariette committed Mar 20, 2014 604 605 606 607 608 609 610 611 612 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Tables %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Use of \listoftables is discouraged. %% \section*{Tables}  Jerome Mariette committed Jun 10, 2014 613 614  \begin{table}[h!]  Jerome Mariette committed Aug 05, 2014 615 616 \caption{Features of a subset of already available software packages, and jvenn.}\label{table::features}  Jerome Mariette committed Jun 10, 2014 617 618 619 620 621 622 623  \begin{tabular}{c|cccccc} Application & Maximum & Layouts & Application & Proportionality & Input data & Output\\ & number of & & type & & formats & formats \\ & input lists & & & & & \\ \hline VENNTURE \cite{Bronwen2012} & 6 & Edwards & Stand-alone & No & Lists &  Jerome Mariette committed Jun 11, 2014 624 625  Powerpoint\\ & & & & & & and Excel \\ \hline  Jerome Mariette committed Jun 10, 2014 626 627  VennDiagram \cite{RVennDiagram} & 5 & Classical & R package & No & Lists  Jerome Mariette committed Jun 11, 2014 628 629  & R object \\ & & & & & & and TIFF \\ \hline  Jerome Mariette committed Jun 10, 2014 630   Jerome Mariette committed Aug 05, 2014 631  BioVenn \cite{Hulsen2008} & 3 & Classical & web application & Yes &  Jerome Mariette committed Jun 11, 2014 632  Lists & SVG and PNG \\ \hline  Jerome Mariette committed Jun 10, 2014 633   Jerome Mariette committed Aug 05, 2014 634  venny \cite{venny} & 4 & Classical & web application & No &  Jerome Mariette committed Jun 10, 2014 635 636 637  Lists & PNG \\ \hline Canvasxpress \cite{canvasxpress} & 4 & Classical & JavaScript library &  Jerome Mariette committed Jun 11, 2014 638 639  No & Intersection & JavaScript \\ & & & & & counts & canvas \\ \hline  Jerome Mariette committed Jun 10, 2014 640 641 642  Google & 3 & Classical & JavaScript library & Yes & Lists & PNG \\  Jerome Mariette committed Jun 11, 2014 643  Chart API \cite{googleAPI} & & & & & & \\ \hline \hline  Jerome Mariette committed Jun 10, 2014 644   Jerome Mariette committed Aug 05, 2014 645  jvenn & 6 & Classical & web application & No & Lists,  Jerome Mariette committed Jun 10, 2014 646 647  & Interactive \\  Jerome Mariette committed Jun 11, 2014 648 649  & & and Edwards & and JavaScript & & intersection & diagram, \\ & & & library & & counts and & PNG and CSV \\  Jerome Mariette committed Jul 24, 2014 650  & & & & & Count lists &  Jerome Mariette committed Jun 10, 2014 651 652 653 654  \end{tabular} \end{table}  Jerome Mariette committed Mar 20, 2014 655 \begin{table}[h!]  Jerome Mariette committed Aug 05, 2014 656 \caption{Available input formats.}\label{table::format}  Jerome Mariette committed Mar 20, 2014 657 658  \begin{tabular}{cccc} \hline  Jerome Mariette committed Jul 24, 2014 659 660  Format & Example\\ \hline Lists &  Jerome Mariette committed Mar 20, 2014 661 662 663 664 665 666 667 668 669 \begin{lstlisting} series: [{ name: 'sample1', data: ["Otu1", "Otu2", "Otu3", "Otu4", "Otu5", "Otu6", "Otu7"] }, { name: 'sample2', data: ["Otu1", "Otu2", "Otu5", "Otu7", "Otu8", "Otu9"] }] \end{lstlisting}\\ \hline  Jerome Mariette committed Jul 24, 2014 670  Intersection counts &  Jerome Mariette committed Mar 20, 2014 671 672 \begin{lstlisting} series: [{  Jerome Mariette committed Jun 05, 2014 673 674  name: {A: 'sample 1', B: 'sample 2'}, data: {A: ["Otu3", "Otu4", "Otu6"], B: ["Otu8", "Otu9"], AB: ["Otu1",  Jerome Mariette committed Jun 11, 2014 675  "Otu2", "Otu5", "Otu7"]} }],  Jerome Mariette committed Jun 05, 2014 676  values: {A: 3, B: 2, AB: 4}  Jerome Mariette committed Mar 20, 2014 677 \end{lstlisting}\\ \hline  Jerome Mariette committed Jul 24, 2014 678  Count lists &  Jerome Mariette committed Mar 20, 2014 679 680 681 682 683 684 685 686 687 688 689 690 691 692 \begin{lstlisting} series: [{ name: 'sample1', data: ["Otu1", "Otu2", "Otu3", "Otu4", "Otu5", "Otu6", "Otu7"], values: [5, 15, 250, 20, 23, 58, 89] }, { name: 'sample2', data: ["Otu1", "Otu2", "Otu5", "Otu7", "Otu8", "Otu9"], values: [90, 300, 10, 2, 45, 9] }] \end{lstlisting}\\ \hline \end{tabular} \end{table}  Jerome Mariette committed Mar 03, 2014 693 \end{backmatter}  Jerome Mariette committed Mar 24, 2014 694 \end{document}


text:
Eigen  3.4.90 (git rev a4098ac676528a83cfb73d4d26ce1b42ec05f47c)
Eigen::PastixLU< MatrixType_, IsStrSym > Class Template Reference

## Detailed Description

template<typename MatrixType_, bool IsStrSym>
class Eigen::PastixLU< MatrixType_, IsStrSym >

Interface to the PaStix solver.

Sparse direct LU solver based on PaStiX library.

This class is used to solve the linear systems A.X = B via the PaStix library. The matrix can be either real or complex, symmetric or not.

TutorialSparseDirectSolvers

This class is used to solve the linear systems A.X = B with a supernodal LU factorization in the PaStiX library. The matrix A should be squared and nonsingular PaStiX requires that the matrix A has a symmetric structural pattern. This interface can symmetrize the input matrix otherwise. The vectors or matrices X and B can be either dense or sparse.

Template Parameters
 MatrixType_ the type of the sparse matrix A, it must be a SparseMatrix<> IsStrSym Indicates if the input matrix has a symmetric pattern, default is false NOTE : Note that if the analysis and factorization phase are called separately, the input matrix will be symmetrized at each call, hence it is advised to symmetrize the matrix in a end-user program and set IsStrSym to true

This class follows the sparse solver concept .

Sparse solver concept, class SparseLU

Inherits Eigen::PastixBase< Derived >.

## Public Member Functions

void analyzePattern (const MatrixType &matrix)

void compute (const MatrixType &matrix)

void factorize (const MatrixType &matrix)

## ◆ analyzePattern()

template<typename MatrixType_ , bool IsStrSym>
 void Eigen::PastixLU< MatrixType_, IsStrSym >::analyzePattern ( const MatrixType & matrix )
inline

Compute the LU symbolic factorization of matrix using its sparsity pattern. Several ordering methods can be used at this step. See the PaStiX user's manual. The result of this operation can be used with successive matrices having the same pattern as matrix

factorize()

## ◆ compute()

template<typename MatrixType_ , bool IsStrSym>
 void Eigen::PastixLU< MatrixType_, IsStrSym >::compute ( const MatrixType & matrix )
inline

Compute the LU supernodal factorization of matrix. iparm and dparm can be used to tune the PaStiX parameters. see the PaStiX user's manual

analyzePattern() factorize()

## ◆ factorize()

template<typename MatrixType_ , bool IsStrSym>
 void Eigen::PastixLU< MatrixType_, IsStrSym >::factorize ( const MatrixType & matrix )
inline

Compute the LU supernodal factorization of matrix WARNING The matrix matrix should have the same structural pattern as the same used in the analysis phase.

format: plain


text:
# If E,F,G and $\mathrm{H}$ are respectively the mid-points of the sides of a parallelogram $\mathrm{ABCD}$, show that $\operatorname{ar}(\mathrm{EFGH})=\frac{1}{2} \operatorname{ar}(\mathrm{ABCD})$

AcademicMathematicsNCERTClass 9

#### Complete Python Prime Pack for 2023

9 Courses     2 eBooks

#### Artificial Intelligence & Machine Learning Prime Pack

6 Courses     1 eBooks

#### Java Prime Pack 2023

8 Courses     2 eBooks

Given:

$E,F,G$ and $\mathrm{H}$ are respectively the mid-points of the sides of a parallelogram $\mathrm{ABCD}$

To do:

We have to show that $\operatorname{ar}(\mathrm{EFGH})=\frac{1}{2} \operatorname{ar}(\mathrm{ABCD})$.

Solution:

Join $EF, FG, GH, HE$ and $FH$.

We know that,

Opposite sides of a parallelogram are equal and parallel.

This implies,

$A D \| B C$

$A D=B C$

$\Rightarrow \frac{1}{2} A D=\frac{1}{2}B C$

$\mathrm{AH} \| \mathrm{BF}$

$\mathrm{DH} \| \mathrm{CF}$

$\Rightarrow \mathrm{AH}=\mathrm{BF}$

$\mathrm{DH}=\mathrm{CF}$

$\mathrm{H}$ and $\mathrm{F}$ are mid points

Therefore, $\mathrm{ABFH}$ and $\mathrm{HFCD}$ are parallelograms.

$\triangle \mathrm{EFH}$ and parallelogram $\mathrm{ABFH}$ both lie on the same base $\mathrm{FH}$ and between the same parallels $\mathrm{AB}$ and $\mathrm{HF}$.

This implies,

Area of $\triangle \mathrm{EFH}=\frac{1}{2}\times$ area of  $\mathrm{ABFH}$.......(i)

Area of $\triangle \mathrm{GHF}=\frac{1}{2}\times$ area of $\mathrm{HFCD}$..........(ii)

Adding (i) and (ii), we get,

Area of $\triangle \mathrm{EFH}+$ area of $\triangle \mathrm{GHF}=\frac{1}{2}\times$ area of  $\mathrm{ABFH}+\frac{1}{2}\times$ area of  $\mathrm{HFCD}$

Area of $\mathrm{EFGH}=$ area of $\mathrm{ABFH}$

Therefore,

$\operatorname{ar}(\mathrm{EFGH})=\frac{1}{2} \operatorname{ar}(\mathrm{ABCD})$

Updated on 10-Oct-2022 13:41:36

Advertisements


text:
Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Algebra of Complex Numbers

Voice

Voice

Home

Complex Arithmetic as Numerical Expression

»  Real Number arithmetic

→  Addition with closure, commutative, associative, identity, inverse properties

→  Multiplication with closure, commutative, associative, distributive, identity, inverse properties

→  subtraction and division as inverse of addition and multiplication respectively

→  PEMDAS / BODMAS Precedence rule

»  Complex number arithmetic
→  extension of real arithmetic with additional number i
→  i is handled like a variable
→  i^2 = -1 maps i to a real number

### Understanding Complex Arithmetic

plain and simple summary

nub

plain and simple summary

nub

dummy

Complex arithmetic is the extension of Real numbers arithmetic.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

Support Nubtrek

You are learning the free content, however do shake hands with a coffee to show appreciation.
To stop this message from appearing, please choose an option and make a payment.

Complex number arithmetic like addition, multiplication, etc are nothing but extension of real number arithmetic.

Keep tapping on the content to continue learning.
Starting on learning "Understanding Complex Arithmetic". ;; Complex number arithmetic like addition, multiplication, etc are nothing but extension of real number arithmetic.

What is the complex number system?

• Real number system extended to include solutions to polynomials
• Real number system added with an additional number sqrt(-1)
• Model of amplitude and phase of sine waves and elements interacting with sine waves
• All the above

The answer is 'All the above'

For the complex number the following definitions and properties
•  multiplication and division
•  exponents and roots
•  properties like commutative, distributive, associative, etc.

Should one learn each of these afresh?

• No, just extend the definitions and properties of real numbers as though complex number is a numerical expression.
• Yes, sqrt(-1) is not a real number so it will not adhere to the definitions and properties of real numbers.

The answer is 'No, Just extend the definitions and properties of real numbers'

One person decides to write all real numbers in the form 3.1+7xx2.4 - that is, two real numbers with 7 appearing as in the expression. How would the representation affect the arithmetic operations, like addition, subtraction, associative, etc?

• All definitions and properties will hold true for the numerical expressions too
• The definitions and properties need to be figured out afresh.

The answer is 'All the definitions and properties will hold true' for the numerical expressions too, as long as the expression is kept in tact.

Consider the example of writing real numbers in the form a+7b:
•  Two identical numerical expressions may not be evident from the numbers in the expressions. eg: 3.1+7xx2.4 = 4.5+7xx2.2, though the expression looks different, they evaluate to the same.
•  There can be additional properties that is a combination of already existing properties. eg: To add 3.1+7xx2.4 + 1.1+7xx0.2, add the coefficients 3.1+1.1 and 2.4+0.2. This looks to be a new definitions for addition. But it is derived from the distributive and associative properties of addition and subtraction.

Complex number is a numerical expression with sqrt(-1) as an element. When learning rules and properties of complex numbers, visualize the complex number as a numerical expression of real numbers.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Complex Arithmetic Fundamentals: The definitions and properties of real number arithmetic is extended to include sqrt(-1) as a number that cannot be added or multiplied to other numbers.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

To understand complex numbers arithmetic operations and properties, which of the following number system is used?

• Integers
• Rational numbers
• Real numbers

Progress

Progress

What is the complex number system?
polynomials
Real number system extended to include solutions to polynomials
minus;1;square;root
Real number system added with an additional number square root minus 1
model;amplitude;phase
Model of amplitude and phase of sine waves and elements interacting with sine waves
all;above
All the above
The answer is 'All the above'
For the complex number the following definitions and properties ;; addition and subtraction ;; multiplication and division ;; exponents and roots ;; properties like commutative, distributive, associative, etc. ;; Should one learn each of these afresh?
No;just;extend
No, just extend the definitions and properties of real numbers as though complex number is a numerical expression.
yes;square;root;minus
Yes, square root of minus 1 is not a real number so it will not adhere to the definitions and properties of real numbers.
The answer is 'No, Just extend the definitions and properties of real numbers'
One person decides to write all real numbers in the form 3 point 1+7 multiplied by 2 point 4 ; - that is, two real numbers with 7 appearing as in the expression. How would the representation affect the arithmetic operations, like addition, subtraction, associative, etc?
will;hold;true
All definitions and properties will hold true for the numerical expressions too
need;figured;out;afresh
The definitions and properties need to be figured out afresh.
The answer is 'All the definitions and properties will hold true' for the numerical expressions too, as long as the expression is kept in tact.
Consider the example of writing real numbers in the form a+7b: ;; Two identical numerical expressions may not be evident from the numbers in the expressions. for example 3 point 1+7 into 2 point 4 = 4 point 5+7 into 2 point 2, though the expression looks different, they evaluate to the same.;; There can be additional properties that is a combination of already existing properties. eg: To add 3 point 1+7 into 2 point 4; + 1 point 1+7 into 0 point 2, add the coefficients 3 point 1+1 point 1 and 2 point 4+0 point 2. ;; This looks to be a new definitions for addition. But it is derived from the distributive and associative properties of addition and subtraction.
Complex number is a numerical expression with square root minus 1 as an element. When learning rules and properties of complex numbers, visualize the complex number as a numerical expression of real numbers.
Complex arithmetic is the extension of Real numbers arithmetic.
Complex Arithmetic Fundamentals: The definitions and properties of real number arithmetic is extended to include square root minus 1 as a number that cannot be added or multiplied to other numbers.
To understand complex numbers arithmetic operations and properties, which of the following number system is used?
integer;integers
Integers
rational
Rational numbers
real
Real numbers


text:
# Reducing Fractions

## Related Calculator: Fractions Calculator

Let's learn how to reduce fractions.

We already learned about equivalent fractions. There are many fractions that are equivalent to the given one, but there is one special fraction.

Fraction is irreducible if numerator and denominator have no common factors.

For example, 1/3 is irreducible, while 2/6 is not, because 2 and 6 have common factor 2.

Two ways to find irreducible fraction:

1. Try to find common factor of numerator and denominator. Divide them by that factor. Repeat this step until there are no common factors. This way is similar to finding prime factorization.
2. This method is essentially the same as first, but you need only one step. Find Greatest Common Divisor of numerator and denominator. Divide numerator and denominator by found number.

From second way it follows another definition of irreducible fraction.

Fraction a/b is irreducible if GCD(a,b)=1.

Indeed, if GCD(a,b)=1 then a and b have no common factors, so a/b is irreducible.

Example 1. Reduce fraction 18/24.

Both 18 and 24 are divisible by 2, so we divide them by 2: we get new equivalent fraction 9/12.

12 is divisible by 2, but 9 is not, so we try 3.

Both 9 and 12 are divisible by 3, so we divide them by 3: we get new equivalent fraction 3/4.

We are done because 3 and 4 have no common factors.

Answer: 3/4.

Now, let's try to use second way.

Example 2. Reduce fraction 40/60.

Find greatest common divisor: GCD(40,60)=20.

Divide both numerator and denominator by 20: (40/color(red)(20))/(60/color(red)(20))=2/3.

Answer: 2/3.

Next example.

Example 3. Reduce fraction 5/35.

Find greatest common divisor: GCD(5,35)=5.

Divide both numerator and denominator by 5: (5/color(red)(5))/(35/color(red)(5))=1/7.

Answer: 1/7.

Now, take pen and paper and do following exercises.

Exercise 1. Reduce fraction 15/25.

Answer: 3/5.

Next exercise.

Exercise 2. Reduce fraction 135/90.

Answer: 3/2.

Next exercise.

Exercise 3. Reduce fraction 2/9.

Answer: 2/9 (already irreducible).

Next exercise.

Exercise 4. Reduce fraction 60/12.

Answer: 5. Hint: 5/1=5.

Next exercise.

Exercise 5. Reduce fraction -18/18.


text:
Is there a natural (suitable) definition for functional derivative in Curved space time

If $$\delta S = \int \sqrt g F[\phi] \delta \phi\tag{1}$$

Then is it natural to define the functional derivative as follows,

$$\frac{\delta S}{\delta \phi} = F[\phi].\tag{2}$$

In particular does this definition satisfy the commutativity of the functional derivatives.

I understand that this is not the standard definition of a functional derivative, but if I define this way this makes a certain calculation I am doing much easier to control. So to Clarify what I want to know is that if I define the 'functional derivative' this way then if

$$S = \int \sqrt{g} L[\phi, g_{\mu \nu}]\tag{3}$$ then

is the following true?

$$\frac{\delta}{\delta\phi} \frac{\delta}{\delta g_{\mu \nu}} S = \frac{\delta}{\delta g_{\mu \nu}} \frac{\delta}{\delta\phi} S \tag{4}$$

This is my procedure for computing the second functional derivative suppose

$$\frac{\delta}{\delta \phi}S = E[\phi, g_{\mu \nu}]\tag{5}$$

and $$\frac{\delta}{\delta g_{\mu \nu}}S = E^{\mu \nu}[\phi, g_{\mu \nu}]\tag{6}$$

(equations of motion) Then to compute second functional derivative we write, eg.

$$E[\phi, g_{\mu \nu}](x) = \int \sqrt{g} d^4 y E(y) \hat \delta(x-y)\tag{7}$$

where $$\hat \delta (x-y) = \frac{\delta(x-y)}{\sqrt{g}}\tag{8}$$ is the generalised delta function. After this we can use the same definition to compute

$$\frac{\delta} {\delta g_{\mu \nu}} \int \sqrt{g} d^4 y E(y) \hat \delta(x-y)\tag{9} .$$ Of course now the result would involve delta functions.

• What you've written there is not a definition. Note that the general definition of the functional derivative does not presuppose a particular kind of integration measure, so I don't know what you are actually asking. – ACuriousMind Jun 9 '15 at 14:53

3 Answers

Comments to the question (v4):

1. First a disclaimer. Note that even for a smooth local functional, the existence of a functional/variational derivatives is not guaranteed, but depends on appropriately chosen boundary conditions.

2. Calculus of variations, functional/variational derivatives, Frechet derivatives, Gateaux derivatives, etc, is a huge mathematical topic. If OP is looking for rigor, then we suggest to post the question on Math.SE or Mathoverflow.SE. In this answer we will just make some heuristic comments and ignore boundary terms.

3. Let there be given an $n$-dimensional pseudo-Riemannian manifold $(M,g)$.

4. We can consider a Dirac delta distribution $$\tag{A} \delta_M(x,y)~=~\frac{\delta^n(x-y)}{\sqrt{|g(x)|}}, \qquad g(x)~:=~\det(g_{\mu\nu}(x)),$$ on the manifold, cf. OP's eq. (8). Here $x$ and $y$ denote two points in a coordinate neighborhood $U\subseteq M$, [and to simplify notation, they also denote the coordinates of the two points].

5. The Dirac delta distribution (A) is independent of the choice of coordinates, i.e. transforms as a scalar.

6. Next we would like to define the functional/variational derivative wrt. fields $\phi^{\alpha}(x)$ [which could include the metric field $g_{\mu\nu}(x)$].

7. The functional derivative $\frac{\delta_M S}{\delta\phi^{\alpha}(x)}$ should satisfy $$\tag{B} \delta S~=~ \int \! d\mu(x)~\frac{\delta_M S}{\delta\phi^{\alpha}(x)}\delta\phi^{\alpha}(x), \qquad d\mu(x)~:=~\sqrt{|g(x)|}d^nx,$$ for appropriate infinitesimal variations $\delta\phi^{\alpha}(x)$, cf. OP's eq. (1).

8. In particular, we have the rule $$\tag{C} \frac{\delta_M \phi^{\beta}(y)}{\delta\phi^{\alpha}(x)} ~=~\delta^{\beta}_{\alpha}~\delta_M(x,y).$$ When $\phi^{\alpha}$ is the metric field $g_{\mu\nu}$, note this subtlety.

9. When $\phi^{\alpha}$ and/or $\phi^{\beta}$ are the metric field $g_{\mu\nu}$, then commutativity of functional derivatives are generally not expected, since the manifold notion (B) of functional differentiation depends on the metric, cf OP's eq. (4). In fact, it is easy to find counterexamples.

10. For the rest of this answer we will assume that $\phi^{\alpha}$ and $\phi^{\beta}$ are not the metric field $g_{\mu\nu}$. It is then expected that the functional derivatives commute $$\tag{D} [\frac{\delta_M}{\delta\phi^{\alpha}}, \frac{\delta_M}{\delta\phi^{\beta}}]~=~0.$$ See also e.g. Ref. 1.

11. Example. If the functional is of the form $$\tag{E}S~=~\int \! d\mu(x)~L(\phi(x),\partial\phi(x),x),$$ and we define $$\tag{F}\nabla^{(x)}_{\mu} ~:=~ \frac{1}{\sqrt{|g(x)|}}d^{(x)}_{\mu}\sqrt{|g(x)|} ,\qquad d^{(x)}_{\mu}~:=~\frac{d}{dx^{\mu}},$$ then the 1st functional derivative reads $$\frac{\delta_M S}{\delta\phi^{\alpha}(x)} ~=~\frac{\partial L(x)}{\partial \phi^{\alpha}(x)} -\nabla^{(x)}_{\mu}\frac{\partial L(x)}{\partial \phi^{\alpha}_{,\mu}(x)}$$ $$\tag{G}~=~\int \! d\mu(y)\left[\frac{\partial L(y)}{\partial \phi^{\alpha}(y)}\delta_M(x,y) + \frac{\partial L(y)}{\partial \phi^{\alpha}_{,\mu}(y)}d^{(y)}_{\mu}\delta_M(x,y)\right],$$ so that the 2nd functional derivative becomes symmetric $$\frac{ \delta^2_M S}{\delta\phi^{\beta}(y)\delta\phi^{\alpha}(x)} ~=~\frac{\partial^2 L(y)}{\partial\phi^{\beta}(y)\partial \phi^{\alpha}(y)}\delta_M(x,y) -\nabla^{(y)}_{\nu}\left[\frac{\partial^2 L(y)}{\partial\phi^{\beta}_{,\nu}(y)\partial \phi^{\alpha}(y)}\delta_M(x,y)\right]$$ $$+\frac{\partial^2 L(y)}{\partial\phi^{\beta}(y)\partial \phi^{\alpha}_{,\mu}(y)}d^{(y)}_{\mu}\delta_M(x,y) -\nabla^{(y)}_{\nu}\left[\frac{\partial^2 L(y)}{\partial\phi^{\beta}_{,\nu}(y)\partial \phi^{\alpha}_{,\mu}(y)}d^{(y)}_{\mu}\delta_M(x,y)\right]$$ $$~=~\frac{\partial^2 L(y)}{\partial\phi^{\beta}(y)\partial \phi^{\alpha}(y)}\delta_M(x,y) -\nabla^{(y)}_{\nu}\left[\frac{\partial^2 L(y)}{\partial\phi^{\beta}_{,\nu}(y)\partial \phi^{\alpha}(y)}\delta_M(x,y)\right]$$ $$-\nabla^{(x)}_{\mu}\left[\frac{\partial^2 L(y)}{\partial\phi^{\beta}(y)\partial \phi^{\alpha}_{,\mu}(y)}\delta_M(x,y)\right] -\nabla^{(y)}_{\nu}\nabla^{(x)}_{\mu}\left[\frac{\partial^2 L(y)}{\partial\phi^{\beta}_{,\nu}(y)\partial \phi^{\alpha}_{,\mu}(y)}\delta_M(x,y)\right]$$ $$\tag{H}~=~\left[(x,\alpha) \leftrightarrow (y,\beta) \right].$$ Here we have used that $$\tag{I} [f(x)-f(y)]\delta_M(x,y)~=~0, \qquad [\nabla^{(x)}_{\mu}+d^{(y)}_{\mu}]\delta_M(x,y)~=~0.$$

References:

1. Bryce DeWitt, Supermanifolds, Cambridge Univ. Press, 1992; eq. (5.1.3).
• Thanks for your answer. May I ask you why you expect your points 9) , 10) to hold? That was the main point in my question. – vishmay Jun 15 '15 at 11:43
• I updated the answer. – Qmechanic Jun 20 '15 at 14:20
• Correction to the answer (v3): The last minus sign in eq. (H) should be a plus. – Qmechanic Jun 20 '15 at 20:14

When I'm confused about all the functional derivatives, I'm doing that:

$$S(\phi_1,\phi_2,\phi_3,g_1,g_2,g_3) =\sqrt{g_1}\,L(\phi_1,g_1) + \sqrt{g_2}\,L(\phi_2,g_2) + \sqrt{g_3}\,L(\phi_3,g_3)$$

I hope you'll recognize your $\int\sqrt{g}\, L[\phi,g]$ in the expression above.
The functional derivatives then become more familiar gradients:

$$\frac{\partial S}{\partial \phi_i} = \pmatrix{ \sqrt{g_1}\,\frac{dL}{d\phi}\left(\phi_1,g_1\right)\\ \sqrt{g_2}\,\frac{dL}{d\phi}\left(\phi_2,g_2\right)\\ \sqrt{g_2}\,\frac{dL}{d\phi}\left(\phi_3,g_3\right)\\ }\qquad \frac{\partial S}{\partial g_i} = \pmatrix{ \sqrt{g_1}\,\frac{dL}{dg}\left(\phi_1,g_1\right)\\ \sqrt{g_2}\,\frac{dL}{dg}\left(\phi_2,g_2\right)\\ \sqrt{g_2}\,\frac{dL}{dg}\left(\phi_3,g_3\right)\\ }+\pmatrix{ \frac{d\sqrt{\phantom{l}g_1}}{dg_1}\,L\left(\phi_1,g_1\right)\\ \frac{d\sqrt{\phantom{l}g_2}}{dg_2}\,L\left(\phi_2,g_2\right)\\ \frac{d\sqrt{\phantom{l}g_3}}{dg_3}\,L\left(\phi_3,g_3\right)\\ }$$

The mixed derivative will become a diagonal matrix: $$\frac{\partial^2 S}{\partial \phi_i\partial g_j} = \pmatrix{ \sqrt{g_1}\,\left.\frac{d^2L}{d\phi dg}\right|_{\phi_1g_1} + \frac{d\sqrt{\phantom{l}g_1}}{dg_1}\,\left.\frac{dL}{d\phi}\right|_{\phi_1g_1}&0&0\\ 0& \sqrt{g_2}\,\left.\frac{d^2L}{d\phi dg}\right|_{\phi_2g_2} + \frac{d\sqrt{\phantom{l}g_2}}{dg_2}\,\left.\frac{dL}{d\phi}\right|_{\phi_2g_2}&0\\ 0&0&...\\ }$$ From that point of view your "convenience definition" is (a) incomplete, (b) doesnt really look sensible and (c) doesn't look too convenient after all.

I mean -- are you going to divide the gradients componentwise (b) by the $\sqrt{g_i}$? Okay, but what about (a) the extra terms in the derivatives over $g$? Do you want to subtract the extra terms, or leave them? In any case that "new definitions" will definitely not commute (c).

I suggest that you make up your mind in those simple terms, and then generalize from 3D up to the functional spaces...

• Thanks for your answer. The extra terms 'derivative over g' , in the case of functional derivative by g_{\mu \nu} will give -1/2 \sqrt{g} g^{\mu \nu} \delta g_{\mu \nu} so you see I can again extract out the \sqrt{g}. Do you think this prescription have the chance of commutativity? – vishmay Jun 16 '15 at 9:58
• @Prag1 First of all -- are you sure about you derivative? The $\sqrt{g}$ means something more complex, isn't it? – Kostya Jun 16 '15 at 22:14

Assuming that g is constant, the only way your equations "hold," is if $F[\phi] = e^\phi$.

I hope this is helpful.

• this question is about functional derivatives – innisfree Jun 15 '15 at 21:51


text:
# Thread: Congruence proof, did I do it somewhat correct?

1. ## Congruence proof, did I do it somewhat correct?

I just want to see how well I did with this proof, and would like to know any corrections to make if I did it improperly, thanks!

The proof is Show if a ≡ b(mod m) and n ∈ Z+ where n | m, then a ≡ b(mod n)

suppose
a ≡ b(mod m). Then n ∈ Z+ , and a,b,m,k ∈ Z. Now suppose a=b+km, then a-b=km and km=n. Thus n|m. Then a ≡ b(mod n). Hence, if a ≡ b(mod m) then a ≡ b(mod n).

2. ## Re: Congruence proof, did I do it somewhat correct?

Originally Posted by michaelgg13
I just want to see how well I did with this proof, and would like to know any corrections to make if I did it improperly, thanks!

The proof is Show if a ≡ b(mod m) and n ∈ Z+ where n | m, then a ≡ b(mod n)

suppose
a ≡ b(mod m). Then n ∈ Z+ , and a,b,m,k ∈ Z. Now suppose a=b+km, then a-b=km and km=n. Thus n|m. Then a ≡ b(mod n). Hence, if a ≡ b(mod m) then a ≡ b(mod n).

your use of "thus n|m" is odd because you are given that n|m.

just do this

$a \equiv b (\bmod~ m) \Rightarrow \exists k \in \mathbb{Z} \ni a = km + b$

Now $n|m \Rightarrow \exists j \in \mathbb{Z} \ni m = jn$ so

$a \equiv b (\bmod~ m) \Rightarrow a = jkn + b = (jk)n + b$ and thus

$a \equiv b (\bmod ~n)$

3. ## Re: Congruence proof, did I do it somewhat correct?

Originally Posted by romsek
your use of "thus n|m" is odd because you are given that n|m.

just do this

$a \equiv b (\bmod~ m) \Rightarrow \exists k \in \mathbb{Z} \ni a = km + b$

Now $n|m \Rightarrow \exists j \in \mathbb{Z} \ni m = jn$ so

$a \equiv b (\bmod~ m) \Rightarrow a = jkn + b = (jk)n + b$ and thus

$a \equiv b (\bmod ~n)$
ahh ok, that makes more sense. thanks so much!

4. ## Re: Congruence proof, did I do it somewhat correct?

Originally Posted by michaelgg13
I just want to see how well I did with this proof, and would like to know any corrections to make if I did it improperly, thanks!

The proof is Show if a ≡ b(mod m) and n ∈ Z+ where n | m, then a ≡ b(mod n)
Here another way. You should know that if $m|n~\&~n|j$ then $m|j$.

We are given that $m|(a-b) ~\&~n|m$ so $n|(a-b)$. qed


text:
# Convergence of 1/(x_n) if x_n is convergentWatch

#1
Question: If is a convergent real sequence with limit , where . Show that is convergent with limit

Would it be valid to write the following?

For to be convergent with limit we must have that .

Suppose that we define then so .

Thus we have that .

As required.

I feel slightly uneasy about dividing by . Is this allowed?

Any help is appreciated
0
quote
1 year ago
#2
You write "Suppose that we define delta ..." but I'm at a loss to what you think this *does*? Certainly defining it isn't enough to decide that |x_n - L| is less than it.
0
quote
#3
(Original post by DFranklin)
You write "Suppose that we define delta ..." but I'm at a loss to what you think this *does*? Certainly defining it isn't enough to decide that |x_n - L| is less than it.
I have just realised this is terribly unclear. I don't really know what I meant by it. Here is hopefully a more clear (attempted) explanation.

For to be convergent with limit we must have that .

Suppose that then .

Thus we have that .
0
quote
1 year ago
#4
You can't suppose the thing your supposing. (I'm kind of puzzled how you thought you *could* suppose it).
0
quote
1 year ago
#5
Should there not be the requirement that .

isn't good enough.
quote
1 year ago
#6
(Original post by Cryptokyo)
Suppose that .
Under what premises are you certain that this is true?
0
quote
1 year ago
#7
(Original post by ghostwalker)
Should there not be the requirement that .

isn't good enough.
Good point.
0
quote
X

new posts

Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### University open days

• University of Lincoln
Mini Open Day at the Brayford Campus Undergraduate
Wed, 19 Dec '18
• University of East Anglia
Fri, 4 Jan '19
• Bournemouth University
Wed, 9 Jan '19

### Poll

Join the discussion

Yes (253)
27.23%
No (676)
72.77%


text:
# What is the second derivative of?

X^2 • lnX

I have to find the second derivative.

For the first derivative I have: 2x • 1/x

For the second Im stuck at: 2

Any help is much appreciated!

Relevance
• 8 years ago

1/x = x^(-1)

The derivative of that is -x^(-2)

But your first derivative is wrong.

Given f(x) = g(x) * h(x)

f'(x) = g(x) * h'(x) + g'(x) * h(x)

Here, g(x) = x^2

g'(x) = 2x

h(x) = ln(x)

h'(x) = 1/x

f'(x) = x^2 * 1/x + 2x * ln(x)

f'(x) = x + 2x * ln(x)

f''(x) = 1 + (2 * ln(x) + 2x * 1/x)

f''(x) = 1 + 2 * ln(x) + 2

f''(x) = 3 + 2 * ln(x)

• JoAn
Lv 4
8 years ago

y = x²(ln x)

using product rule

y' = (2x)(ln x) + x²/x

y' = (2x)(ln x) + x

y'' = 2(ln x) + 2x/x + 1

y'' = 2(ln x) + 2 + 1

y'' = 2(ln x) + 3


text:
## Thursday, August 11, 2011 ... /////

### CO₂ and ice-free Arctic Summer 2100

In recent 3 days, Andrew Revkin has pleased many climate skeptics by his opinions about the Arctic climate and ecosystems:

On Arctic Ice and Warmth, Past and Future

A Debate on Death Spirals and Arctic Resilience
Building on many recent papers that showed significant variations in the Arctic climate in the past (safely exceeding the recent changes), Revkin decided that there's no reason to be afraid of the Arctic ecosystems because they have survived similar or more dramatic changes in the past. I agree.

Swimming and getting some suntan in the warming Arctic

Nevertheless, I am probably going to repay debt to Revkin, shock some skeptics, and please by alarmists by this following opinion of mine:

I actually find it highly plausible that there will be an ice-free Arctic summer by 2100.

This opinion of mine stands on theoretical as well as empirical grounds. The observational support seems to be more robust among the two. At any rate, there seem to be many good reasons to believe that the Arctic climate is closely correlated with a very limited subset of forcings among which the carbon dioxide is an important one.

Let me start with some "theoretical" comments. First, there are differences between the polar regions on one side and the equatorial or moderate zones on the other side. Richard Lindzen believes that the equatorial temperatures have been constant - within a degree Celsius or so - for millions and maybe billions of years.

I am not sure whether such a bold statement is exactly right - or just an idealized slogan to promote the iris effect - but there surely exist many reasons to think that the variability of the temperature in the tropics is much smaller than it is in the extratropics. First of all, the recent observations show that the temperature variability near the equator is limited. Second of all, it's hotter over there and the same change of the temperature corresponds to a greater change in the energy fluxes. And the energy fluxes are somewhat likely to have the same variability over the globe which means that the temperature variability should be lower near the equator.

Also, there are lots of powerful negative feedbacks related to water vapor that are important exactly near the equator. And yes, the faint young Sun paradox also suggests that the Earth must be capable of regulating its equatorial temperatures much more than we would naively think. I could continue for a while but the main point I believe is that the variability of temperatures near the poles is higher than the variability of temperatures near the equator.

North vs South

Second, the North Pole seems to be different than the South Pole. Almost no temperature change in the recent 50 years has been detected around the Antarctica; on the other hand, the changes in the Arctic seem to be substantial. So it seems that only the Northern polar regions experience the variability we may clearly observe.

This difference between the North and the South impacts the global warming debate. It's perhaps unsurprising that it is always the North - and not the South - that the alarmists refer to. The Arctic just produces "more interesting data" for their case than the Antarctica.

However, it's not just the fearmongers' demagogy in which the North plays a more important role than the South. If you look at the details of the Milankovitch theory of glaciation cycles that was put on firm ground by Gerard Roe in 2006, you will see that it's the ice near the Arctic Circle that is being correlated, via the insolation near the Arctic circle, to the irregularities in the Earth's motion. Again, the North matters and the South doesn't. As far as I understand, if you calculated the Antarctic Circle's insolation, the Milankovitch theory wouldn't work too well.

This difference reduces to some circulations that exist near the North Pole - because of the free ocean over there - but not near the South Pole. Also, it may have something to do with the land near the Arctic Circle: there's a lot of continental ice that can grow and recede over there. On the other hand, almost all of Antarctica sits in the South from the Antarctic Circle (it is closer to the poles) so there's almost no land on the Antarctic Circle itself.

Antarctica is too cold - too close to the pole - to become ice-free; and the places which could grow large continental ice sheets aren't available in the South because there's no land over there. So for all these reasons, the fact that the interesting variability takes place in the North and not the South could be more than just a coincidence.

I almost forgot to say that the clouds in the Arctic are not "purely about cooling". In fact, they are more likely to have a warming effect. The fact that the solar radiation is arriving nearly horizontally most of the time is a part of the reason (especially in the winter): the clouds may reflect the solar radiation to the surface (down instead of up) and help to warm it up! So even if the cloud cover oscillates a lot, its average cooling effect on the surface is much smaller than it is in the mild zones. The same reasoning holds not only for clouds but even for aerosols and other players that are important away from the polar regions.

On the other hand, the greenhouse effect of $CO_2$ is always there in the Arctic because the $CO_2$ is always present and it always absorbs a part of the Earth's thermal radiation that is also there at all times. If you look at the situation, many effects "decouple" in the Arctic while the $CO_2$ greenhouse effect is among those that survive.

Also, the Arctic has some extra phenomena that are not important elsewhere - similar to the ice-albedo feedback. Those may amplify the initial perturbations (positive feedbacks) but on the other hand, they're really "close to being deterministic" (unlike things like the cloudiness): they don't carry "independent slow degrees of freedom". The shape of the sea ice may look complicated but it is almost exactly calculable from other things (air temperature above it) and it has a limited impact on the atmosphere above the ice. So the "turbulent phenomena" (hard to calculate ones) are suppressed near the poles. The polar environment is able to "cleanly amplify" a subset of forcings which is why the Milankovitch effects and even $CO_2$ may become visible in the Arctic temperature/ice data.

Empirical correlation

But let's look at the most important part of the story, the observed data. The graph below was created by Peter Hogarth for John Cook's blog.

Much of it takes the data from a paper by William Chapman and John Walsh. You may also download a gigabyte of bitmaps encoding historical data going back to 1870 from The Cryosphere Today. See the left upper corner of that page for many links.

Now, to be specific, look at the pink curve - the lowest one. It depicts the sea ice extent in millions of squared kilometers in July-August-September of various years; note that the minimum is reached around the mid September and this minimum is obviously even lower than the average of the adjacent 3-month period. (August-September-October would probably have even lower averages of the sea ice extent but it's not a conventional 3-month period.)

The pink curve does look like a hockey stick. More precisely, it is suggestively correlated with the carbon dioxide concentration. Let me be a bit more quantitative. I will assume the $CO_2$ concentrations to obey the "exponential business as usual". It is given by the following empirical formula of mine:
$C(y) = 280+22.3 \exp[(y-1920)/57]\,\,{\rm ppm}$ The formula says that the excess of the $CO_2$ concentration above 280 ppm is exponentially growing so that it reaches 280+22.3 in 1920, about 390 ppm in 2011, and the excess gets multiplied by $e\approx 2.718$ every 57 years so the annual increase of the $CO_2$ emissions is $100/57 = 1.75$ percent.

Of course, if the growth of the $CO_2$ emissions slows down in the future (which is pretty likely) - so that the multiplication of emissions (and excess carbon dioxide) by $e$ will take more than 57 years or we will even converge towards dropping emissions - all my estimates of the "speed of melting" below will become exaggerated (by a factor of two or so if you assume constant emissions from now).

Nevertheless, the sea ice area over the JAS 3-month period, as seen in the graph, may be approximated by
$A_{JAS,\,{\rm lin}} = 11.5 - [C(y)-280] / 27.$ Well, a more scientifically justifiable formula depends on the logarithm of the concentration (because the underlying theory is based on the nearly logarithmic greenhouse effect):
$A_{JAS,\,{\rm log}} = 11.5-11 \log [C(y)/280].$ In both cases, the units are millions of squared kilometers. The two formulae more or less agree before 2011 (and they agree with the pink graph) but the logarithmic formula predicts a less dramatic reduction of the sea ice extent in the future than the linear formula does.

The linear formula drops to zero around 2070 - we're talking about the ice-free average of July, August, and September in a given year - while the more realistic logarithmic formula drops to zero around 2100. So if you believe that the surprisingly convincing correlation between the $CO_2$ concentration and the sea ice extent that is seen on the graphs is more than a coincidence, it is very plausible that sometime in the 21st century, we will see an ice-free moment sometime in the Summer.

Just to be sure, it would take many, many centuries for the sea ice extent to drop to zero in the winter and I won't discuss those "more speculative" forecasts at all. Let me just cover them by the sentence that it is implausible that the Arctic will become ice-free throughout the year in any foreseeable future (even if we assume those persistently exponentially growing $CO_2$ emissions).

Fractal remainders of the temperature

People and Joe Romm make various bets about the ice-free Arctic. Yes, you're pretty safe if you bet that there won't be any ice-free moment in the Arctic in the next 20 years. I don't think that they will really see it in their "active lifetime", to enjoy the money they will win. But as I said, it's plausible that if they were living for 100+ years, they would see it. However, they would probably be too senile to be amazed by the observations (even more senile than Joe Romm is already today).

There is another subtlety that should be considered by all rational people who want to participate in similar bets: the behavior of the ice when its total area becomes very small. Imagine that the ice area drops to 0.5 million squared kilometers. Does it become easier to liquidate the remaining small amount of ice than to reduce ice from 6 to 5.5 million squared kilometers? Or does it become harder?

In other words, is the "most resilient ice" tougher and hard to be killed - which could mean that the Arctic will never be quite ice-free - or does the convergence towards $A=0$ accelerate when $A$ is very small?

I tend to think that the latter answer is correct. There are arguments in both directions and one must carefully compare them. First, there may be local variations of temperatures that make some "local islands of ice" very cold and almost indestructible. In the language of the normal distribution, 1/300 of the sea ice area has ice whose temperature is more than 3 sigma below what is needed for melting - and 3 sigma could be enough to beat the trend for many more decades or centuries.

On the other hand, when the sea ice area is very small, the remaining ice has a larger surface/volume ratio, so it's much easier to warm it up by its thermal contact with the warmer environment. I believe that this effect is probably more important than the previous, opposite one. So there probably won't be any important forces that will try to "preserve the last ice islands". Quite on the contrary, it seems plausible that the path towards "complete extinction" accelerates if the amount of sea ice becomes really small.

Of course, the tight observed correlation between the Arctic sea ice and the $CO_2$ concentration may be just a coincidence. If it's not, then the extrapolation is approximately legitimate. Within 50 years or so, there can be an ice-free moment sometimes in the Summer. Within 100 years or so, the whole 3-month period around the sea ice minimum may be ice-free, too.

Even if it is true, it will probably represent no threat for the ecosystems because similar things have happened in the past (Revkin mentions the end of the last ice age). The polar bear - a mammal, our relatively close cousin :-) - is indeed a rare exception of an animal who just pretends to prefer the life in the ocean but whose main bedroom is still on the land. ;-)

Just think about it: there is no "food" (carbon) richly available on the ice so there can't be any growing (reproducing) life forms living "on the ice" or "depending on the ice" except for life forms such as the polar bears that can quickly run over the ice (hundreds of kilometers), too. Ice is the ultimate lifeless zone (that's why we use fridges and freezers to protect meat against bacterial life) and polar bears are the exception among the "running animals" who boast that they can overcome this difficulty and use ice as a great ally that allows them to find some food beneath the ice. It makes no sense to dream about the "diversity of life" supported by ice itself.

Polar bears easily and logically move away from the pole when they find the sea ice around the poles too risky. If that's not the case, I believe that people in late August 2070 should spend a few million dollars and liberate the polar bears stuck on the shrinking (and, later, completely disappearing) last islands of ice near the pole. (I am less sure that today, we may effectively help the people in August 2070.)

While I view these considerations interesting from a scientific viewpoint - pure curiosity and speculations - and I think there's quite some evidence that the $CO_2$ could really be important for the Arctic climate, much like the Milankovitch (astrophysical) cycles, it's still important to realize that the Arctic is just a tiny (and for us nearly irrelevant, and too cold, anyway) piece of the globe and these insights obviously don't apply to warmer zones.

In the warmer zones, the internal variability of the atmosphere - and especially clouds (and everything that drives them internally or externally), water vapor, aerosols, and other things - are much more important for the climate and its evolution, so the warming trends are very noisy (a substantial portion of the Earth will be cooling) and their average may be between zero or slightly negative numbers (Lindzen's theory of the equator) and something comparable to the recently observed average trends (which are very small).

We shouldn't get carried away and of course, I am not suggesting that there's any climate threat. But still, the correlation between the Arctic ice and the $CO_2$ concentration looks much better than almost all other correlations that are claimed to be relevant.


text:
Negation of CH implied by lots of special subtrees?

In the following, I focus on trees of height $\omega_1$: if there exists a nonspecial tree any of whose $\aleph_1$-subtrees is special, must CH fail?

Some neither consistent nor coherent thoughts: Notice that the tree must have cardinality at least $\aleph_2$ and this is a fragment of $MA_{\aleph_1}$. It is known that it is possible to specialize Aronszajn trees with forcings that add no reals, hence obtaining a model of GCH along with all Aronszajn trees are special. But in this case, we are dealing with fat trees and maybe a possible attack is to think about forcings that specialize fat trees without adding reals.

Rado's conjecture is a combinatorial statement about instances of compactness in chromatic numbers of certain graphs, but Todorčević proved that it is equivalent to the statement that any tree, all of whose subsets of size $\aleph_1$ are special, is itself special.
So a counterexample to Rado's conjecture is a nonspecial tree, all of whose subsets of size $\aleph_1$ are special. In particular, all of its $\aleph_1$-subtrees are special.
In the same paper Todorčević also proves that Rado's conjecture implies that for every regular $\kappa>\omega_1$, every stationary subset of $\kappa\cap \operatorname{Cof}(\omega)$ reflects. Since the usual forcing to add a nonreflecting stationary set does not add reals, we can start in a model of CH and force the failure of Rado's conjecture, and consequently a tree as in your question, while preserving CH.


text:
# zbMATH — the first resource for mathematics

Intuitionistic fuzzy sets. (English) Zbl 0631.03040
By an intuitionistic fuzzy set (IFS) the author understands a pair of generalized characteristic functions and hence a generalization of a fuzzy set. These two functions for each point of the universe of discourse describe separately its degrees of membership and of nonmembership with respect to the IFS they characterize. These two degrees not necessarily add up to one. The motivation for introducing IFSs is explained with an example, set algebraic operations are introduced and some of their elementary properties given.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
Full Text:
##### References:
 [1] Atanassov, K., Intuitionistic fuzzy sets, () · Zbl 0597.03033 [2] Atanassov, K.; Stoeva, S., Intuitionistic fuzzy sets, (), 23-26 · Zbl 0597.03033 [3] Atanassov, K.; Stoeva, S., Intuitionstic L-fuzzy sets, (), 539-540 [4] Atanassov, K., Intuitionistic fuzzy relations, (), 56-57
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Image credits: One Nigeria

Recently, The president of the federal republic of Nigeria released the budget for 2019, going through the budget we’ll realise that there is nothing new expected to happen in the coming year.
Below is a quick analysis a friend on mine Mona Moxie gave me on whatsapp today about the budget, kindly follow through I assure you, you’d not be bored:

Here is a simple break down for the 2019 budget. I made it as simple as possible, no economic jargons so you understand those things you saw in the news and know how to plan your life.

The budget size is 8.83 trillion naira, this is the total amount the government is expected to spend in 2019.

A quarter (14) of this amount would be used to pay debt. The truth is Nigeria is owing a lot of debts, therefore around 2.14 trillion naira will be used to offset part of these debts. What would be left of the budget should be N8.83T - N2.14T = N6.69T.

The budget for capital expenditure is N2.031 trillion, capital expenditure means the money that will be used to fund projects such as the construction of roads, bridges, hospitals, etc. If Nigeria decides to construct any major project by next year, that’s the amount allocated for it. So what’s left will now be N6.69T -N2.031T = N4.65T.

Recurrent expenditure is 4.04 trillion naira. Unlike capital expenditures where you pay just once, this is allocated for stuff that may keep re-occurring and you have no choice but to settle, things like salaries, wages, service fees, etc. So what’s left will be N4.65T -N4.04 T = N619 billion.

Sinking fund is estimated at 120 billion naira. You know even after paying the debt earlier stated we will still be owing some money. See this money as something that is kept aside as savings to later pay back other entities we are currently in debt to. So what’s left will be N619B -N120B = N499B.

Last but not least is what they call statutory transfer, this is set at N492.36B.
This is the money that is paid to Senators and house of rep members. Even if Nigeria is burning, one naira out of this money must not be touched, they must be paid to the lawmakers and leaders. So what’s left will be N619 -N492B = N6.64B which will be kept in case of emergency.

So how and where does Nigeria get money to fund this budget? you may ask.
Usually, we fund budgets through many components but the most reliable means we have at the moment is via crude oil. As such, we plan most of our projects around the money we get from this sector.
Ceteris paribus (all things being equal), here is how Nigeria plans to get the money. Follow me, please.

Global oil prices are falling, some countries are leaving oil for renewable energy, the current price globally is around $45 per barrel but Nigeria is ‘hoping’ it will rise to$60 per barrel in 2019. How will this happen? We’re just hoping.
Now if oil prices do not rise as we hope then we are in big problem, we then have to go borrow else we won’t be able to fund the budget.

Secondly, we can only fund this budget if we keep producing 2.3 million barrels per day.
Currently, we can barely produce 1.9 million barrels per day.
But you know what, we are hoping if Jesus could raise dead Lazarus back to life, it won’t be hard for Him to help raise production limit from below 1.9 barrels per day to above 2.3 barrels per day.

Thirdly, we can only fund this budget without stress if the inflation rate is around 9.98%.
The CBN governor said we should expect the inflation rate to rise to 11.4% and the IMF even said it would rise to 13.5%.
We are hoping it won’t go that high, we are serving a living God and He will keep it at 9.98% for us.

Fourthly, we are hoping dollar will be roughly N305 per \$1 which may or may not be possible, but let’s hope it does.

Number 5: We hope the GDP growth rate will be 3.01% when the current reality is that GDP growth rate is barely up to 1.8%.
But we are hoping that it doubles. By the way, we are a prayerful nation, we will pray until our GDP rises from 1.8% to 3.01%.

So this is my opinion: 2019 doesn’t seem like the year where things will change for the economy of Nigeria and sorry to disappoint you, this post is not against the government of Presiden Buhari, he is trying his best no doubt but he does not have the answers. Atiku won’t be the answers either, our problem is beyond one man, it’s beyond PDP or APC.

By 31st of this month(Dec 2018), a lot of us will be gathered again in Church waiting to cross over to the new year, praying and kabashing for things to change, shaking and rolling, rolling and shaking.
Like our president, it’s all a series of hope and like our fathers before us, that’s how they prayed and hoped and here we are today.

But they forgot one thing:

Hope is not a Strategy

Hope deferred makes the heart sick, the times have changed my friend, I may not have the solution to all our problems in this country but I have a solution for you:

Fix Yourself.

Big shout out to Mona Moxie for his detailed analysis.


text:
# Is it helpful or harmful to create SEO filler pages?

I'm not sure exactly what these "filler pages" are called, so I couldn't do a proper search.

If I have a business, say a plumbing/heating company in NYC, and I created these HTML pages:

New-York-City-plumbing-company.html
Manhattan-plumbing-company.html
Brooklyn-plumbing-company.html

New-York-City-heating-service.html
Manhattan-heating-service.html
Brooklyn-heating-service.html

and the content for each page type was the same, except the name of the city/neighborhood changed, i.e.

[Neighborhood] Plumbing Company

description of plumbing service

[Neighborhood] Heating Service

description of heating service

Would this be helpful, harmful, or neutral in terms of SEO? I'm a little afraid Google might see that 95% of the content is the same, minus the name of the neighborhood, and punish my site for it.

-
That sounds like a really bad idea. This clearly is just for the sake of manipulating your search results. That's a big no-no. –  John Conde Mar 10 '14 at 19:17
@JohnConde what if the content for each page were different? Different images and text so that < 50% of the text was reused? –  andrewtweber Mar 10 '14 at 19:18
That's a grey area. I'd advise against it as it isn't worth risking a ban or penalty. –  John Conde Mar 10 '14 at 19:20
@JohnConde Thanks. Too bad, I thought I was being really clever! –  andrewtweber Mar 10 '14 at 19:28
@andrewtweber One of the great things about the search results from google now is that it displays local business from google maps. So in this case, if you have a physical location, creating a listing on google plus / maps would help you out greatly trying to achieve what you want. Search for "plumbing" and you'll see results from the maps. Create a listing here google.com/business/placesforbusiness see a –  guisasso Mar 10 '14 at 19:53

Google has a specific guideline against creating filler pages. They call it "doorway pages".

Doorway pages are typically large sets of poor-quality pages where each page is optimized for a specific keyword or phrase. In many cases, doorway pages are written to rank for a particular phrase and then funnel users to a single destination. Whether deployed across many domains or established within one domain, doorway pages tend to frustrate users.

Therefore, Google frowns on practices that are designed to manipulate search engines and deceive users by directing them to sites other than the one they selected, and that provide content solely for the benefit of search engines. Google may take action on doorway sites and other sites making use of these deceptive practices, including removing these sites from Google’s index.

Some examples of doorways include:

• Having multiple domain names targeted at specific regions or cities that funnel users to one page
• Multiple pages on your site with similar content designed to rank for specific queries like city or state names

Note that their last example specifically matches the example that you gave.

If you can tailor the site experience for users from these specific neighborhoods, then it doesn't sound like a bad idea to me. You would have to:

• Get content that would be interesting specifically to the targeted users.
• feature testimonials from customers in that neighborhood
• put information that is specific to that neighborhood such as any delivery charges
• give deals, discounts, or coupon codes that can be used by customers from that neighborhood
• Differentiate the page from other neighborhood pages
• Make sure that a significant portion of the content on each page is unique to that page (more than half)
• Make sure that the portion that is "above the fold" looks different from page to page
• Pictures specific to each page can really help
• Avoid funneling users to a single other page.
• Ensure that there are multiple links, forms, or calls to action on that page, similar to how your homepage is built.
-

1. Create your homepage with a <title> and a
 like Plumbing and heating on New York City (the best keywords for this page)


text:
interchange

0th

Percentile

Interchange algorithm for maximum entropy sampling

Starting point is a network $A[F]$ with $nf$ points. Now one has to select $ns$ points of a set of candidate sites to augment the existing network. The aim of maximum entropy sampling is to select a feasible D-optimal design that maximizes the logarithm of the determinant of all principal submatrices of $A$ arising by this expansion.

The interchange algorithm improves a feasible initial solution directly given or obtained by the greedy or dual greedy algorithm for maximum entropy sampling.

It is also possible to improve the initial solution for the construction of a completely new network, that means $nf=0$, but in this case the interchange algorithm fails for $ns=1$.

Keywords
spatial , design
Usage
interchange(A, nf, ns, S.start, etol=0, mattest = TRUE)
Arguments
A
Spatial covariance matrix $A$.
nf
Number of stations are forced into every feasible solution.
ns
Number of stations have to be added to the existing network.
S.start
Vector that gives the $ns$ indices contained in the initial solution of the dimension $dim(A)[1]-nf$ that should to be improved.
etol
Tolerance for checking positve definiteness (default 0)
mattest
Toggles testing matrix A for symmetry and positive definiteness (default T)
Details

$A[F]$ denotes the principal submatrix of $A$ having rows and columns indexed by $1..nf$.

Value

A object of class monet containing the following elements:
S.start
Vector containing the indices of the added sites in the initial solution or 0 for the other sites.
S
Vector containing the indices of the added sites in the solution or 0 for the other sites.
det
Determinant of the principal submatrix indexed by the initial solution.

References

Ko, Lee, Queyranne, An exact algorithm for maximum entropy sampling, Operations Research 43 (1995), 684-691.

Gebhardt, C.: Bayessche Methoden in der geostatistischen Versuchsplanung. PhD Thesis, Univ. Klagenfurt, Austria, 2003

O.P. Baume, A. Gebhardt, C. Gebhardt, G.B.M. Heuvelink and J. Pilz: Network optimization algorithms and scenarios in the context of automatic mapping. Computers & Geosciences 37 (2011) 3, 289-294

greedy, dualgreedy

• interchange
Examples
x <- c(0.97900601,0.82658702,0.53105628,0.91420190,0.35304969,
0.14768239,0.58000004,0.60690101,0.36289026,0.82022147,
0.95290664,0.07928365,0.04833764,0.55631735,0.06427738,
0.31216689,0.43851418,0.34433556,0.77699357,0.84097327)
y <- c(0.36545512,0.72144122,0.95688671,0.25422154,0.48199229,
0.43874199,0.90166634,0.60898628,0.82634713,0.29670695,
0.86879093,0.45277452,0.09386800,0.04788365,0.20557817,
0.61149264,0.94643855,0.78219937,0.53946353,0.70946842)
A <- outer(x, x, "-")^2 + outer(y, y, "-")^2
A <- (2 - A)/10
diag(A) <- 0
diag(A) <- 1/20 + apply(A, 2, sum)

S.c<-c(0,7,0,9,0,11,0,13,14,0,0,0,0,0,0)
interchange(A,5,5,S.c)
interchange(A,5,5,greedy(A,5,5)$S) interchange(A,5,5,dualgreedy(A,5,5)$S)

Documentation reproduced from package edesign, version 1.0-13, License: GPL (>= 2.0)

Community examples

Looks like there are no examples yet.