How to get Heroes of Might and Magic 1 for Windows 95 work in Windows 7 64-bit
Installing a 16-bit Windows 95 game on 64-bit Windows 7 Home Premium
This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$ What are tag synonyms and merged tags? How do they work? My bibliography (made with biblatex) has all items close to each other. I would like to introduce some space between them, e.g. half a line.  How can I do this? It can be proved that there are two irrational numbers $a$ and $b$ such that $a^b$ is rational (see ) and that for each irrational number $c$ there exists another irrational number $d$ such that $c^d$ is rational (see ).  My question is: Is there an irrational number $a$ such that $a^a$ is rational (and how could you prove that)? Is it possible to see a list of all my edits using a query? Unable to cover get set in apex test class I am unable to cover the following class get set method. Need help in creating test class  public class Aresult{      public class Parent {         public String type {get; set;}         public String id {get; set;}         public String sequence_id {get; set;}         public String etag {get; set;}         public String name {get; set;}     } }   Please suggest  @isTest  public class CirriusGetItemsResultTest {     public static void myUnitTest(){         Aresult.parent a = new Aresult.parent();         a.type='hello';         a.id='hello';         a.sequence_id ='hello';         a.etag ='hello';         a.name ='hello';     } } Is there an optimal loss function for dealing with imbalanced classes? I'm aware that there are many ways of dealing with datasets where there is a strong class imbalance in the target variable: downsampling the more prevalent and less important class, over-weighting the valuable minority class, etc. Are there any loss functions that are ideally suited for these types of problems? I've heard of AUC, but AUC sounds like a means of evaluating a model once its training is complete (correct me if I'm wrong). Log loss and squared loss penalize a model's over confidence, but overconfidence and class imbalance seem unrelated. Is that true? Should we avoid negative words when writing error messages? Moving app to production mode in Symfony 2 Minecraft LAN "java.net.ConnectException: Connection refused: no further information:" I am trying to play a map on LAN. But every time I try to log on, I get:     java.net.ConnectException: Connection refused: no further information:   This used to work, but for some reason has stopped all of a sudden. I looked it up online, and everything I found said the firewall just wasn't allowing Java through. I followed tutorials to allow Java through the firewall, but even then it won't work. I know for a fact that I'm putting the IP in correctly, and everything seems to be correct, but yet I still get the same error message.  If it helps any, I'm using MC version 1.8.9 (the map requires it), and I don't have any mods. Why does Stack Exchange use Imgur over Amazon S3 or Azure? What benefit does  offer over , , etc. for image hosting?   Is it the simplicity of integration? The cost model? What's better about using a "gateway"? What's the easiest way to play a sound file (.wav) in Python? By easiest I mean both most platform independent and requiring the least dependencies. pygame is certainly an option, but it seems overkill for just sound. Young Adult Fantasy about children living with an elderly woman and learning magic related to their skills Stack Overflow in ES, PT, RU, JP have mixed-language descriptions Is the 2DS region locked? Will I be able to play NA games on a Japanese System? If I purchase a 2DS from Japan, will I be able to play American games on it? What are the differences between $$, \[, align, equation and displaymath? I once learnt, that it is uncool to use $$, but why is that? Why does \[ not act as a abbreviation to \begin{align}? I noticed, that there is a difference, since one cannot use &amp; and \\ inside a block started with \[. Connecting two Ethernet PHY without magnetics? In a (tentative) new design I want to connect two 100 Mbit/s Ethernet PHY similar to , separated by few inches on the same PCB, with same ground plane (but different power supplies). I have the choice of one of my PHY, but the other is embedded in an as-yet-unspecified PCIe to Ethernet IC (perhaps Gigabit but used in 100 Mbit/s mode), and it is critical that this PHY thinks there is a bona fide 100 Mbit/s Ethernet connection.  I could use the following         &ndash; Schematic created using   but I would like to achieve the same without the magnetics, in order to save on cost, purchasing headaches, and if at all possible power consumption.  I can imagine an RC network, perhaps as simple as in this , found thanks to a ; or maybe engineered with attenuation and lowpass, but that will not emulate the fact that with real magnetics, when there is a negative pulse on TX1P, there is a positive pulse on TX1M and RX2M. I am uncertain about if that would prevent some PHYs from working normally.  Anything to suggest that one is confident will work for practically any PHY around? Storing two kinds of point geometry in shapefile or creating multipoint shapefiles? Following , can I have a mixture of two kinds of geometry in a shapefile of type point. Just like I have about shapefiles of type polygon or I should always create a shapefile of type multipoint in order to support multipoints in my layer?    My problem is as you see at , I can't use ogrinfo in order to understand that the geometry type of a multipoint shapefile will be known as wkbMultipoint or wkbPointwhen I use this line of code:    OGRwkbGeometryType GeometryType = poLayer -&gt;GetGeomType();     for a layer of type multipoint before programming.     Will OGR know a layer of type multipoint as wkbpoint or wkbmultipoint?   When we create a shapefile in ArcCatalog, we have point and multipoint, but not multipolyline or multipolygon for polyline and polygon.Why? Wavy stream of liquid While pouring a liquid into a glass from a bottle, some streams have a wavy shape, like the one in the following photo:    What causes the stream to be of such a shape? Does a bad grade in one course (with otherwise OK record) affect graduate admissions?