How did R-square get its name?
Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$
Function with limits only at irrational points This is the starred example 5.1.8 in Krantz' Real Analysis and Foundations.     Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x\rightarrow c} f(x)$ exists when $c$ is irrational but does not exist when $c$ is rational.   I'm aware of the Dirichlet and Thomae functions but this seems slightly different. I'm not even sure a solution exists. Thanks for any help! As my client software uses lat/lon coordinates when communicating with my (spherical mercator) postgis database i decided to ST_Transform every geometry to WGS-84.  However i noticed that ST_Distance for WGS-84 returns units as degrees (i need meters). So i decided to use ST_DistanceSpheroid(geom1, geom2, 'SPHEROID["WGS 84",6378137,298.257223563]') but this method seems to be extremely slow.  Therefor i switched back to using spherical mercator again and transforming my input and output to and from wgs-84 as that performs a lot better.  Am i using the correct method or is this a known issue? Countable sets have the same cardinality as the natural numbers (or a subset of the natural numbers, depending on who you ask).  Can one make the same claim about uncountable sets and real numbers?  In other words, do all uncountable sets have same cardinality as real numbers? What is recordSetVar and where is it used? I was looking at some code and in the 'apex:page' tag, an attribute called recordSetVar was mentioned. I'm new to VisualForce so could someone please tell me when and why is it used? Is it possible to enforce unique values in a column, excluding nulls? How do I kill the lockscreen after the Windows 10 anniversary update? Given that the majority of biologists do not currently consider viruses to be alive, a virus can never die.  It can, however, get destroyed by long exposures to soapy water, alcohol, and apparently certain frequencies of UV light.  Why is an individual SARS-CoV2 virion not infectious forever?  Or is it? How can I find the smallest set of groups of $n$ elements such that every element is in the same group as every other at least once? Background: I'm working on  for , and I've run into a problem with how I'm creating the individual matchups (groups of 4 entries).  Currently, I'm simply generating all combinations of 4 elements (in the combinatoric sense) of entries, but that gets really big really quickly:  $$x \choose 4$$    (from )  Therefore, my question is: How can I choose subsets (of length $n$) of a set such that each element of that set appears in the same subset as every other element at least once, while maintaining a minimal amount of subsets?  I can figure this out for $n = 2$, since the solution is quite intuitive: simply combine the first element with every other element, combine the second element with every other element that comes after it, etc. For a set of length $L$, this yields:  $$\sum_{x=1}^{L-1}(L - x)$$    (from )  which is much smaller and more manageable.  However, I still have not been able to figure out: How can I generalize this to any $n$? Definitions of "linearity" across branches of mathematics or levels of math education Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, operator, or whatsoever) is. These different definitions can lead to confusion, especially when people apply conflicting definitions of linearity to the same mathematical object (just  to figure out what I mean).  One of my favorite examples with potential for conflict is the natural logarithm $\ln : \mathbb{R}_{&gt;0} \to \mathbb{R}$, which the majority of people, as I could learn during hot disputations not only  but also with colleagues in my field, regard as clearly non-linear, even though it fully complies with the definition of a linear map from the vector space $\mathbb{R}_{&gt;0}$ to the vector space $\mathbb{R}$ (cf. Definition 4 below). A similar conflict arises analogously for the exponential function $\exp :\mathbb{R} \to \mathbb{R}_{&gt;0}$.  I do not want to initiate yet another discussion here, about what the right and what the wrong definition of linearity is! Rather, ...  ... I would like to compile a list of existing definitions, to get a better overview of the different uses of the term "linearity" across the different branches of mathematics and levels of education.    Let be begin with the following four, which are certainly far from constituting a complete list. What about bilinear or multilinear forms, linear morphisms and linear functors in category theory, linear metrics, or other concepts of linearity? I am looking forward to see your answers.  At school I was taught that     Definition 1:   A function $f$ is linear if it is of the form $f(x) = ax + b$.   Later I learned that $f(x) = ax + b$ is affine, rather than linear, because     Definition 2:   A function $f$ is linear if it is of the form $f(x) = ax$.   From real analysis and systems theory, I know the following     Definition 3:   A map $f$ is linear if         $f(x + y) = f(x) + f(y)$   $f(a \cdot x) = a \cdot f(x)$.      This definition also applies to the expected value in statistics, as        $\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)$   $\mathrm{E}(a \cdot X) = a \cdot \mathrm{E}(X)$.      In a linear algebra course, I learned for the first time that we must not forget to take into account that the vector-space operations of the domain and the codomain of a map $f$ may be dissimilar. In fact, if $(V, \oplus, \odot)$ and $(W, \boxplus, \boxdot)$ are two $\mathbb{K}$-vector spaces, then     Definition 4:   A map $f : V \to W$ is linear if         $f(x \oplus y) = f(x) \boxplus f(y)$   $f(a \odot x) = a \boxdot f(x)$.         Notice that I deliberately used distinct symbols for the operations in both vector spaces, to emphasize how this definition differs from Definition 3.   Correct me if I am wrong, but this definition should also apply to maps between two modules.    Updates: Reading order for Dragonlance? Find full path of the Python interpreter? What is the rule of thumb for mixing doughs? Without delving too deeply into specialty doughs, but addressing more than just  and , what concerns can I evaluate when calculating the technique for approaching a dough?  For instance, some recipes say not to mix too much, others would imply that beating every bit of air out of the dough is the best thing for it. What I am looking for is a set of criteria, maybe a flowchart, that outlines how to deal with having a well-reasoned technique (as opposed to just following a recipe).  Crumble, cream, or melt the fat first? Paddle or hook? Beat, fold or whisk? Knead once and rest or knead three times and throw? I would assume that to some extent getting air in or out, activating and mixing ingredients, distributing fats more or less evenly, and so forth play a factor in any decision. However, I have no idea what those different things do, or the food science behind them. What general principles would help me understand what each factor does, so that I can have more confidence when staring down the ingredients, whether for some cookies or pizza crusts or muffins? Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism Why is 24.0000 not equal to 24.0000 in MATLAB? I am writing a program where I need to delete duplicate points stored in a matrix. The problem is that when it comes to check whether those points are in the matrix, MATLAB can't recognize them in the matrix although they exist.  In the following code, intersections function gets the intersection points:  [points(:,1), points(:,2)] = intersections(...     obj.modifiedVGVertices(1,:), obj.modifiedVGVertices(2,:), ...     [vertex1(1) vertex2(1)], [vertex1(2) vertex2(2)]);   The result:  &gt;&gt; points points =    12.0000   15.0000    33.0000   24.0000    33.0000   24.0000  &gt;&gt; vertex1 vertex1 =     12     15  &gt;&gt; vertex2     vertex2 =     33     24   Two points (vertex1 and vertex2) should be eliminated from the result. It should be done by the below commands:  points = points((points(:,1) ~= vertex1(1)) | (points(:,2) ~= vertex1(2)), :); points = points((points(:,1) ~= vertex2(1)) | (points(:,2) ~= vertex2(2)), :);   After doing that, we have this unexpected outcome:  &gt;&gt; points points =    33.0000   24.0000   The outcome should be an empty matrix. As you can see, the first (or second?) pair of [33.0000 24.0000] has been eliminated, but not the second one.  Then I checked these two expressions:  &gt;&gt; points(1) ~= vertex2(1) ans =      0 &gt;&gt; points(2) ~= vertex2(2) ans =      1   % &lt;-- It means 24.0000 is not equal to 24.0000?   What is the problem?    More surprisingly, I made a new script that has only these commands:  points = [12.0000   15.0000           33.0000   24.0000           33.0000   24.0000];  vertex1 = [12 ;  15]; vertex2 = [33 ;  24];  points = points((points(:,1) ~= vertex1(1)) | (points(:,2) ~= vertex1(2)), :); points = points((points(:,1) ~= vertex2(1)) | (points(:,2) ~= vertex2(2)), :);   The result as expected:  &gt;&gt; points points =      Empty matrix: 0-by-2 We are developing a C# application for a web-service client. This will run on Windows XP PC's.  One of the fields returned by the web service is a DateTime field. The server returns a field in GMT format i.e. with a "Z" at the end.  However, we found that .NET seems to do some kind of implicit conversion and the time was always 12 hours out.  The following code sample resolves this to some extent in that the 12 hour difference has gone but it makes no allowance for NZ daylight saving.  CultureInfo ci = new CultureInfo("en-NZ"); string date = "Web service date".ToString("R", ci); DateTime convertedDate = DateTime.Parse(date);               As per :     UTC/GMT Offset       Standard time zone: UTC/GMT +12 hours   Daylight saving time: +1 hour   Current time zone offset: UTC/GMT +13 hours     How do we adjust for the extra hour? Can this be done programmatically or is this some kind of setting on the PC's? If Newton's third law is true, why can we sink in sand? Newton's third law of motion states that every action has an equal and opposite reaction. That is the reason we do not sink into the earth, because when our weight exerts a force on the earth it also exerts an equal and opposite force on us.  But when we stand on quicksand or on fluids we can sink in. How is this possible? Does it not exert an equal and opposite force on us? Or are Newton's laws different in the case of fluids and substances of low densities? I'm not sure how to remove the line that represents the 180th meridian (international dateline) on my polygon. I have the "Russia Country Boundary 2017" feature layer added from the Living Atlas. My Projected Coordinate System is set to "Asia North Albers Equal Area Conic". I've tried to google this question, but I haven't come across an answer I can fully understand. I'm a student and very new to all this!  The line is actually a gap between the polygons.   Using ArcGIS Pro 2.3.1  ANSWER:  It turns out I was not projecting correctly, which was creating issues down the line when editing polygons. Here's what I did...  1. Start a blank project in ArcGIS. Insert new map, go into its     properties, change the Coordinate System to "Asia North Albers Equal     Area Conic".  2. Add "World Countries (Generalized)" from the Living     Atlas. Select Russia and Export Features to new feature class layer.  3. Search and select the Project tool from Analysis>Geoprocessing>Tools.   4. Select the Russia layer for input. Select "Current Map" for Output Coordinate System. Run.  5.Now edit the polygons to close the gap and merge to remove any lines. All done!  The only thing that bothers me with this fix is the addition of shape area (484,914.49674 m²) and shape length (4.59 m) to the original Russia polygon. These numbers could be different depending on how you close the gap, and possibly affect data projections depending on your use of this map? How do I remove repeated elements from ArrayList? Is it possible to shutdown remote Computers without inputing the password? I used the following bash to shutdown remote computers,but for each computer,I must input the password for it.Is it possible to input the password Automatically?   #!/bin/bash  ssh -t root@192.168.1.11 "sudo poweroff" ssh -t root@192.168.1.12 "sudo poweroff" ssh -t root@192.168.1.13 "sudo poweroff" ssh -t root@192.168.1.14 "sudo poweroff" ssh -t root@192.168.1.15 "sudo poweroff" ssh -t root@192.168.1.16 "sudo poweroff" ssh -t root@192.168.1.17 "sudo poweroff" ssh -t root@192.168.1.18 "sudo poweroff" ssh -t root@192.168.1.19 "sudo poweroff" ssh -t root@192.168.1.20 "sudo poweroff" ssh -t root@192.168.1.21 "sudo poweroff" ssh -t root@192.168.1.22 "sudo poweroff" ssh -t root@192.168.1.23 "sudo poweroff" ssh -t root@192.168.1.24 "sudo poweroff" ssh -t root@192.168.1.25 "sudo poweroff" ssh -t root@192.168.1.26 "sudo poweroff" Why do we use Root Mean Square (RMS) values when talking about AC voltage What makes it a good idea to use RMS rather than peak values of current and voltage when we talk about or compute with AC signals.