Why is light, that was emitted right at the big bang (same time/place), arriving to us at different times
Did the Big Bang happen at a point?
How do I compare 2 rows from the same table (SQL Server)? I need to create a background job that processes a table looking for rows matching on a particular id with different statuses. It will store the row data in a string to compare the data against a row with a matching id. I know the syntax to get the row data, but I have never tried comparing 2 rows from the same table before. How is it done? Would I need to use variables to store the data from each? Or some other way? (Using SQL Server 2008) Any Set with Associativity, Left Identity, Left Inverse is a Group Related Link:    Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra     I will present my proof (distinct from those in the link) for critique and then ask my question.  $G$ is a set and $ \times $ is an associative binary operation.  Suppose that there exists a $e \in G$ such that, for all $a \in G$, $ea = a$ and $a^{-1}a = e$ for some $a^{-1} \in G$.  Show that for the same $e$, $ae = a$ and $aa^{-1} = e$.  $a^{-1}(aa^{-1})a = (a^{-1}a)(a^{-1}a) = ee = e = a^{-1}a = a^{-1}(a^{-1}a)a$  Since $a^{-1} \in G$, it has a left inverse; apply it to both ends, and we have $(aa^{-1})a = (a^{-1}a)a$.  As a result, $ae$ = $a(a^{-1}a) = (aa^{-1})a = (a^{-1}a)a = ea$.  For the right inverse, start with $aa^{-1} = a(a^{-1}a)a^{-1} = (aa^{-1})(aa^{-1})$.  Since $\times$ is a binary operation, $aa^{-1} \in G$ and has a left inverse; apply it to both ends, and we have $e = aa^{-1}$.  In second comment following the question in the link, Mr. Derek Holt pointed out that the requester did not word his/her question correctly.  Specifically, the identity in the second axiom is not well-defined.     Let $(G, *)$ be a semi-group. Suppose    1. $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;    2. $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group?    This formulation makes the same technical error as many textbooks. The $e$ in your second axiom is not well-defined. "But obviously it's intended to be the same $e$ as in the first axiom" you reply. But the first axiom does not necessarily specify a unique element $e$. So should we interpret the second axiom as meaning "for some $e$ as in 1" or "for all $e$ as in 1"? –  Derek Holt Sep 17 '11 at 15:31   Was he saying that if, in axiom 1, we have $ae_1 = a, ae_2 = a$, but $e_1 \neq e_2$,  when we get to axiom 2, do we have $aa^{-1} = e_1, aa^{-1} = e_2$, or two different inverses so that $aa_1^{-1} =  e_1, aa^{-1}_2 = e_2$?  I think my wording eliminated the ambiguity.  It does not imply that $e$ is unique, but if $e$ is a left identity and produces left inverses, then it is also a right identity and produces right inverses. I tried really hard on this one; please kindly point out my mistakes. In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$. Digitally selectable filter I have asked a  but I did not get an answer, as it was too specific. So I will build on it but make it a bit more broad spectrum.  I would like to make a digitally selectable BPF using transistors to select a component in the filter; resulting in the center frequency being changed.  I have used FilterPro to generate quick second order filters and get a rough filter response.  For all the designs, a Multiple-feedback was selected:  In the first picture, a center frequency at 108KHz with a bandwidth @ -3dB of 103K to 113KHz is given.    For the second, a center frequency at 117.5KHz with a bandwidth @ -3dB of 112.8K to 122.2KHz is given.    The only difference between these two circuits is R3.  If I were to implement a circuit as shown below, would it work and what transistor would be recommended? I have selected the  as the Opamp in the circuit design as that is what was used for the .    For the transistors, I was thinking of using the MOSFETs (2N7002) used in the ST7590 design as well. How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?  If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and $\varphi((0,1))=(0,1),(0,5),(2,1),(2,5)$.  So it seems to me that this groups contains 16 elements, am i right? And if I'm right there is a way to understand to which group of cardinality 16 this is isomorphic to?  EDIT: $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. And $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are characteristic (am I right?), so $\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\cong\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z})\times\mathrm{Aut}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}/8\mathbb{Z}^*\times\mathbb{Z}/3\mathbb{Z}^*$ and this group has cardinality 8, right? Where is my mistake? I have an AMD Radeon R7 M265, and when I force OPENCL with the batch file it only shows Iceland, Intel HD and the CPU.  I followed this instructions:   R, type CMD, press &#9166; Enter, go to your blender folder. ex:  cd C:\Program Files\Blender Foundation\Blender  Press &#9166; Enter and then type: set CYCLES_OPENCL_TEST=all Start Blender with blender.exe Inside blender go to File -> User preferences -> System Under compute device select OpenCL and then your card Close the preferences and go to the render panel and select device: GPU Compute.  I have the latest beta driver from AMD, but when I try to change the compute device it only shows this list:     Am I doing something wrong? Is my GPU is not supported? Or is it because it's in development and not every card is supported yet?  I've tried blender v2.74.5, v2.62, and v2.63. Every one gives the same list. Symfony plugin sfDoctrineActAsTaggablePlugin not working Windows 10 on asus x44h touchpad scolling not working I have just updated Windows 7 to Windows 10.  My touchpad scolling (right-side) is not working anymore.    I have no idea how to fix this in Windows 10.  I have hit the start button and typed "synaptic", but have found nothing useful.  I used to have an icon on the taskbar that showed where I was touching the touchpad.  I used to be able to right-click that icon to change settings, but Windows 10 has not brought that icon over.  Also, the asus website has no synaptic touchpad drivers available for Windows 10.  I have tried the enaltech drivers, but upon the installation atempt I am told that my hardware is not correct. Consider the following sentence:     The labour-intensive and time-intensive part starts tomorrow.   I want to write this without rewriting the word "intensive."  Is this the correct way to do it?     The labour- and time-intensive part starts tomorrow.   Note the hyphen hanging after the word labour. Or should there be no hyphen after labour? Can anyone point me to a reference that talks about this? Could "them" mean "those"? Filling a conical tank I have been working on this problem for about 2 hours and I can't seem to get it, here is exactly what the question reads. "Water is poured into the top of a conical tank at the constant rate of 1 cubic inch per second and flows out of an opening at the3 bottom at a rate of .5 cubic inches per second. The tank has a height of 4 inches, and a radius of 2 inches at the top. How fast is the water level changing when the water is 2 inches high?" Is there a systematic way of solving cubic equations? In my text book, to solve cubic equations, I need to    find by trial &amp; error what $f(a)$ will make the equation 0. The factor will be $(x-a)$ then the other factor will be $Ax^2+Bx+C$ then I can solve using compare coefficient   But for te first part, is there any other way apart from trail &amp; error? Or is there some technique to guess the factor of an equation?   The question I am currently doing happens to be $2x^3+3x+4=9$   UPDATE:  $2x^3+3x+4=9$ should be $2x^3+3x+4=9x^2$   UPDATE 2  So I am going to try substituting   $x=\pm{1}, \pm\frac{1}{2}, \pm{2}, \pm{4}$  When $x=-\frac{1}{2}$, one of the provided answers,   $2(−0.5)^3−9(−0.5)^2+3(−0.5)+4=-18.75\neq 0$   did I make a mistake? Bevel is not symmetrical Variable declaration placement in C In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized.  (Standardizing consists in subtracting the mean and dividing by the standard deviation.)  In which other cases do I need to standardize my data?  Are there cases in which I should only center my data (i.e., without dividing by standard deviation)? What is the safest way to move the iTunes Library from one hard drive to another? This may happen later with my iMac, but right now it is happening for my iTunes on PC.  The PC had 60GB of partition to install Windows 7, and right now it is down to about 2GB of free space.  All iTunes purchases, including Music and Movies, are downloaded to  C:\Users\mike\Music\iTunes\iTunes Media\Music  C:\Users\mike\Music\iTunes\iTunes Media\Movies   which will make the C: drive have less and less free space.  (and C: will be more healthy if it has at least a couple of GB of free space).  What is the safest, most proper way to migrate all the media to an external hard drive J: ?  On J:, the path might be:  J:\iTunes\iTunes Media\Music   or  J:\iTunes Media\Music   My guess is that we can change the folder in iTunes, but won't iTunes go to that folder or create it, and find nothing, and index everything wrong?  So, should an exact copy of C:\Users\mike\Music\iTunes\iTunes Media be made as J:\iTunes Media first (probably 30GB of files copying), and then start iTunes and change the preference for the folder to the J: one?  Is there a standard or official method to do this? How can I straighten ALL the vertical UV lines at once? Is there a hot keyshort cut? Or if I have to do them one by one .. is there at shortcut for that? Thank you! Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$? How can I multiply each item in an array easily with PHP? I have noticed that such terms as model hyperparameter and model parameter have been used interchangeably on the web without prior clarification. I think this is incorrect and needs explanation. Consider a machine learning model, an SVM/NN/NB based classificator or image recognizer, just anything that first springs to mind.   What are the hyperparameters and parameters of the model? Give your examples please.