Consider the following equation:  \begin{align}  &amp; A x = b &amp; C x = d. \end{align}   Is there a possibility to count both the left equation and the right equation separately, so I can refer to them separately? I would prefer if this is done without subequation, so the count continues.
Having different horizontally  distributed equations inside an align environment, only lines are numbered, but not individual equations.    How can the left equations be numbered using subequations? See also the question   \documentclass{article} \usepackage{amsmath} \begin{document}   \begin{subequations}     \begin{align}        %\label{a}         a &amp;= a ,&amp;       \label{b}         b &amp;= b ,\\       %\label{c}         c &amp;= c ,&amp;       \label{d}       d &amp;= d .     \end{align}   \end{subequations} \end{document}     The solution should be able to either:    attach a label to the left equations, or modify the labels on the right to read (1a,b) and (1c,d)    I couldn't find anything in the documentation of amsmath or other packages.
Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then must there exist a homeomorphism from $A$ to $B$ ? I just installed Ubuntu 12.10 on an ASUS X401U, and after installation there is no list of wireless connections. I didn't have that problem with an installation on HP. What I can do to enable Wifi?  07:00.0 Network controller [0280]: Ralink corp. Device [1814:3290] Using WCS layer in QGIS for processing gives FAILURE: Missing destination I have a GeoServer serving raster dem data out via WCS and can add the layer easily to QGIS (3.2) but when I try to use it as the input for an operation (e.g. HillShade) I get an error.  Processing algorithm… Algorithm 'Hillshade' starting… Input parameters: { 'ALTITUDE' : 45, 'AZIMUTH' : 315, 'BAND' : 1, 'COMBINED' : False, 'COMPUTE_EDGES' : False, 'INPUT' : 'dpiMode=7&amp;identifier=training:panorama&amp;url=http://localhost:8080/geoserver/ows?version%3D1.1.1', 'MULTIDIRECTIONAL' : False, 'OPTIONS' : '', 'OUTPUT' : 'C:/Users/ishare/AppData/Local/Temp/2/processing_868cc25180f24d9f8e193bc0cfedb6d3/7cf48f64d8b54ee694c3ee93ed17e903/OUTPUT.tif', 'SCALE' : 1, 'ZEVENBERGEN' : False, 'Z_FACTOR' : 5 }  GDAL command: gdaldem hillshade dpiMode=7&amp;identifier=training:panorama&amp;url=http://localhost:8080/geoserver/ows?version%3D1.1.1 C:/Users/ishare/AppData/Local/Temp/2/processing_868cc25180f24d9f8e193bc0cfedb6d3/7cf48f64d8b54ee694c3ee93ed17e903/OUTPUT.tif -of GTiff -b 1 -z 5.0 -s 1.0 -az 315.0 -alt 45.0 GDAL command output:   FAILURE: Missing destination.   Clearly the missing destination is a red herring, and it is failing to understand the WCS layer as an input. But GDAL is built with WCS support included so shouldn't this work?  Do I need to do something else or start working out where to hack the QGIS code? Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$. Symmetric powers of modules What do we know about the creature whose skeleton we see in Star Wars? In Star Wars:  Episode IV - A New Hope, C-3PO passes by what looks like a dinosaur skeleton:      What is this creature?  Are they extinct?  How did such a large animal manage to survive in a barren desert? How do I get the local machine name in C#? Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$? If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$, then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$ hold? I know  it holds if we integrate over an interval of the form $[a,b]$ but not sure about improper integral! Unable to make recursive call in Javascript Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$?   Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using the normal core of $H$, one can find an $m\ge 1$ such that $x^m\in H$ for all $x\in G$. How can I monitor internet usage in my network? I need to know which computer is consuming my bandwidth. Is there a tool to do this? Which equals operator (== vs ===) should be used in JavaScript comparisons? The full question: suppose N is normal in G and $[G:N]=p$ is a prime. Prove that for any $K \leq G$, either (1) $K \leq N$ or (2)$KN=G$ and $[K:K \cap N]=p$.  Since $N,K \leq G$, it is of course possible that $K \leq N$ could happen.  but for (2), we can show $KN \subseteq G$  but how can we show $KN=G$?  And what information does "$[G:N]=p$" give us? Taking the * of these two large shapefiles   locally using ArcMap takes appr. 10 min.   Over the last few weeks, I've tried to obtain the same results using other, open-source approaches without success. I am curious why only ArcMap seems to generate results in a reasonable amount of time without errors.   The approaches I've tried so far:  QGIS  The most straight-forward alternative. There are two "union" functions available in QGIS. One default (part of vector overlay), one based on SAGA. Both functions create the desired results on smaller samples but fail to run (or take forever) on my shapefiles. I tried running union for 30 minutes after which the progress bar was at appr. 4%    PostGIS  I've ingested the shapefiles on an AWS RDS instance running postGreSQL with PostGIS enabled. The approach is not so straight forward and requires a hideous long SQL query to obtain the same result. Most importantly, the query did not create a meaningful result on the full datasets. It did run properly on a sample though.   See separate question   CREATE TABLE test.sqlislelijk AS  -- input data with polys1 AS (   SELECT      pfaf_id as df1,     geom as g   FROM hybas06_v04 ), polys2 AS (   SELECT      aqid as df2,     geom as g   FROM y2018m11d14_rh_whymap_to_rds_v01_v01 ), -- intersections intersections AS (   SELECT df1, df2, ST_INTERSECTION(a.g, b.g) i, a.g AS g1, b.g AS g2    FROM polys1 a, polys2 b WHERE ST_INTERSECTS(a.g, b.g) ), -- per-row union of intersections with this row diff1 AS (   SELECT df1, ST_UNION(i) i FROM intersections GROUP BY df1 ), diff2 AS (   SELECT df2, ST_UNION(i) i FROM intersections GROUP BY df2 ), -- various combinations of intersections pairs AS (   SELECT df1, df2, i AS g FROM intersections   UNION ALL   SELECT      p.df1,     NULL,     CASE       WHEN i IS NULL THEN g        ELSE ST_DIFFERENCE(g, i)     END   FROM polys1 p LEFT JOIN diff1 d ON p.df1 = d.df1   UNION ALL   SELECT     NULL,     p.df2,     CASE       WHEN i IS NULL THEN g       ELSE ST_DIFFERENCE(g, i)     END   FROM polys2 p LEFT JOIN diff2 d ON p.df2 = d.df2   ) SELECT    df1 as pfaf_id,   df2 as aqid,   g as geom FROM pairs WHERE NOT ST_IsEmpty(g);   Google BigQuery  Google's GIS extension for BQ looks promising but I experienced weird behavior and internal errors, probably due to mixing of GEOMETRY and GEOGRAPHY objects. Also, the syntax is similar to PostGIS and a command of 1 line in geopandas takes >20 lines in SQL.   GeoPandas  Probably my favorite approach. The out-of the box approach does't give any results after running it for several hours. My latest approach includes tiling the shapefiles, use multiprocessing on each tile and merge them. Not very straight forward and one 10x10 degree tile till takes 40+ minutes to calculate.   See   It is the first time in my career that I am impressed by the performance of ArcMap. It seems that ArcMap handles this use-case much, much better than anything else. The issue seems persistent over all solutions I've tried so far. Is there something wrong in the underlying libraries of the open-source/alternative solutions?    Note that Union in ArcMap means something different than in the Shapely / PostGIS world. Read binary file into a struct I'm trying to read binary data using C#. I have all the information about the layout of the data in the files I want to read. I'm able to read the data "chunk by chunk", i.e. getting the first 40 bytes of data converting it to a string, get the next 40 bytes.  Since there are at least three slightly different version of the data, I would like to read the data directly into a struct. It just feels so much more right than by reading it "line by line".  I have tried the following approach but to no avail:  StructType aStruct; int count = Marshal.SizeOf(typeof(StructType)); byte[] readBuffer = new byte[count]; BinaryReader reader = new BinaryReader(stream); readBuffer = reader.ReadBytes(count); GCHandle handle = GCHandle.Alloc(readBuffer, GCHandleType.Pinned); aStruct = (StructType) Marshal.PtrToStructure(handle.AddrOfPinnedObject(), typeof(StructType)); handle.Free();   The stream is an opened FileStream from which I have began to read from. I get an AccessViolationException when using Marshal.PtrToStructure.  The stream contains more information than I'm trying to read since I'm not interested in data at the end of the file.  The struct is defined like:  [StructLayout(LayoutKind.Explicit)] struct StructType {     [FieldOffset(0)]     public string FileDate;     [FieldOffset(8)]     public string FileTime;     [FieldOffset(16)]     public int Id1;     [FieldOffset(20)]     public string Id2; }   The examples code is changed from original to make this question shorter.  How would I read binary data from a file into a struct? Parse JSON in JavaScript? Can I cast a Warlock spell with another (higher) spell slot from a different multiclass? Reviewing an answer to a recent question I posted, I had a thought about Armor of Agathys. In the event of a Sorcerer 18 / Warlock 2 build, would the character be able to cast Armor of Agathys with a 9th level spell slot, thus getting 45 Temporary Hit Points and dealing 45 damage when hit?   Is the spell in any way limited to being cast using Warlock spell slots or to a maximum of a 5th level spell slot? What does !function ($) { $(function(){ }) }(window.jQuery) do? How do I calculate the date six months from the current date using the datetime Python module? Why does shrinkage work?