[~/Discrete_Mathematics_and_Algorithm/Discrete_Mathematics_and_Its_Applications/miscs_snippets/stack_website/comment_crawler]$ scrapy crawl comment_crawler_main --nolog -O links.json
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<span class="comment-copy">Sorry, @Vass, it's not always easy to guess the background of others. $G$ is a group whose order $\varphi(n)$ is divisible by the prime $r$. Then by Cauchy's theorem <a href="http://en.wikipedia.org/wiki/Cauchy%27s_theorem_%28group_theory%29" rel="nofollow noreferrer">en.wikipedia.org/wiki/Cauchy%27s_theorem_%28group_theory%29</a> there is in $G$ an element $x$ of order $r$, so $x \ne 1$ and $x^{r} = 1$. Since $r$ also divides $e$, there is $k$ such that $e = r k$. Thus $x^{e} = x^{r k} = (x^{r})^{k} = 1^{k} = 1$.</span>
https://math.stackexchange.com/questions/285134/can-a-recurrence-relation-have-more-than-one-exact-solution#comment10285188_285134
<span class="comment-copy">Your question has some small problems with its description IMHO. 1. If caring about "more than one exact solution", then the answer is "Yes". Notice here the recurrence relation doesn't need to have initial terms as this <a href="https://redirect.cs.umbc.edu/~nikbar1/Lectures203/Lecture11_3-8" rel="nofollow noreferrer">lecture p2</a> says, so we can have multiple (exact) solutions. 2. If caring about "more than one general solution", then as Jonathan Christensen's answer says, it is "No".</span>
https://math.stackexchange.com/questions/116642/is-well-founded-the-same-as-well-ordered#comment271123_116642
<span class="comment-copy"><a href="http://en.wikipedia.org/wiki/Well-founded_relation" rel="nofollow noreferrer">The wikipedia page for well founded</a> also contains many examples of well founded relations which are not total orderings. (And hence they cannot be well orderings.)</span>
https://math.stackexchange.com/questions/3570899/every-not-empty-finite-subset-of-a-totally-ordered-set-has-a-maximum-and-minimum#comment10311443_3571343
<span class="comment-copy">Since it is one totally ordered set, every pair of elements are comparable. And "Every not empty finite subset" excludes the <a href="https://math.stackexchange.com/a/1258854/1059606">infinite case</a>. Then can this statement be stronger such that "Every not empty finite subset of a totally ordered set has <b>a greatest and least</b>"?</span>
https://math.stackexchange.com/questions/3201645/expanding-definition-of-simple-graph-isomorphism-to-include-multigraphs#comment6590030_3202725
<span class="comment-copy">P. J. Cameronin in <a href="http://vlsicad.eecs.umich.edu/BK/SAUCY/papers/Cameron2001.pdf" rel="nofollow noreferrer">"Automorphisms of Graphs"</a> says: This simple definition does not suffice for multigraphs; we need to specify a permutation of the edges as well as a permutation of the vertices, to ensure that the multiplicity of edges between two vertices is preserved. (Alternatively, a multigraph can be regarded as a weighted graph [...] This gives a slightly different description of automorphisms, but the action on the set of vertices is the same.) We will consider only simple graphs here.</span>
https://math.stackexchange.com/questions/881005/find-the-recurrence-relation-for-the-number-of-bit-strings-that-contain-the-stri#comment9151512_881005
<span class="comment-copy">Recurrences etc are provided in <a href="https://oeis.org/A000295" rel="nofollow noreferrer">oeis.org/A000295</a>, see in particular the comment by Curgus of 2013.</span>
https://tex.stackexchange.com/questions/36343/multiline-text-under-over-arrows#comment72468_36343
<span class="comment-copy">Have a look at <a href="http://tex.stackexchange.com/questions/30862/force-line-break-inside-a-lim-argument-in-align-environment">this post</a>, and the links within.</span>
https://tex.stackexchange.com/questions/14/how-to-look-up-a-symbol-or-identify-a-letter-from-a-math-alphabet-or-other-chara#comment1630644_14
<span class="comment-copy">Note: ■ deTeXify only helps with finding symbol, not determining the font. So it cannot e.g. finding characters in commercial font, e.g. <a href="https://tex.stackexchange.com/q/640360">Adobe Minion Pro</a>. See also: <a href="https://tex.stackexchange.com/q/45919">How do I find out what fonts are used in a document/picture?</a> ■ deTeXify appears to use DTW as the algorithm and unfortunately currently is sensitive to both order of strokes and direction of strokes. (try look up <code>⊛</code>. You need to draw it in order <code>○\/|</code> and the <code>○</code> needs to be drawn counter clockwise for the correct symbol to be shown on top.)</span>
https://math.stackexchange.com/questions/1418063/solution-of-recurrence-relation-for-roots-having-multiplicity-ge-1#comment10290196_1420007
<span class="comment-copy">Here I offer my understanding of this answer if someone is one newbie as me when I read this post. This post has one specific <a href="https://math.stackexchange.com/a/752184/1059606">application</a> with one specific example. 1. $\frac{a_m}{(z - \rho)^m} + \dotsb + \frac{a_1}{z - \rho}$ is due to: in the characteristic equation $r^k-c_1 r^{k-1}-\cdots-c_k=0$, divide both sides by $r^k$, then $z=r^{-1}$, so the repeated root $b$ in $(x-b)^t$ means $\rho=b^{-1}$. Then the above has $\rho^{-n}$ in its final equation of $a_n$.</span>
https://math.stackexchange.com/questions/2804893/show-that-the-composition-of-two-reflection-is-a-rotation#comment10305692_2804910
<span class="comment-copy">To help with the problems with "determinant $\pm 1$". 1. "orthogonal matrix with determinant 1" is not any "matrix with determinant 1". Counterexample is shown in <a href="https://en.wikipedia.org/wiki/Orthogonal_matrix#Matrix_properties" rel="nofollow noreferrer">wikipedia</a> 2. Then based on "orthogonal", we can ensure it must be rotation or reflection. It is <a href="https://en.wikipedia.org/wiki/Orthogonal_matrix#Lower_dimensions" rel="nofollow noreferrer">shown in wikipedia algebraically</a> assuming you know the <a href="https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions#Mathematical_expression" rel="nofollow noreferrer">patterns</a> of the rotation or reflection matrix.</span>
https://cs.stackexchange.com/questions/11438/why-does-dfs-only-yield-tree-and-back-edges-on-undirected-connected-graphs#comment316453_11552
<span class="comment-copy">This fills the missing pieces of the answer: <a href="https://stackoverflow.com/questions/28942262/dfs-in-an-undirected-graph-can-it-have-cross-edges" title="dfs in an undirected graph can it have cross edges">stackoverflow.com/questions/28942262/…</a></span>
https://math.stackexchange.com/questions/3677100/are-all-the-subgraphs-of-k5-planar#comment10333226_3677100
<span class="comment-copy">Based on this <a href="https://www.quora.com/Are-all-the-subgraphs-of-K5-planar-graph-theory-planar-graphs-and-math/answer/Michael-Harrison-482" rel="nofollow noreferrer">quora link</a>, you can also use the <a href="https://en.wikipedia.org/wiki/Kuratowski%27s_theorem#Statement" rel="nofollow noreferrer">Kuratowski's theorem</a>. Since the <b>proper</b> subgraphs of $K_5$ have at most 5 vertices (can't be homeomorphic to $K_{3,3}$) and can't be homeomorphic to $K_{5}$ by <a href="https://en.wikipedia.org/wiki/Homeomorphism_(graph_theory)#Subdivision_and_smoothing" rel="nofollow noreferrer">subdivision</a> which will add at least one more vertex. So they are planar.</span>
https://math.stackexchange.com/questions/1707407/what-is-the-correct-term-for-the-destination-tail-vertex-of-a-directed-edge#comment3484741_1707407
<span class="comment-copy">For a directed edge/arc/arrow $e = (u,v)$, which goes from $u$ to $v$, we call $u$ the tail and $v$ the head, exactly as you would draw an arrow from $u$ to $v$, i.e. $u \to v$. See also: <a href="https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Directed_graph" rel="nofollow noreferrer">Wikipedia</a> and <a href="http://mathworld.wolfram.com/DirectedGraph.html" rel="nofollow noreferrer">Wolfram</a>.</span>
https://math.stackexchange.com/questions/4143714/find-cofactor-matrix-of-a-certain-matrix#comment10346567_4143719
<span class="comment-copy">"The cofactor matrix of a laplacian matrix has all of its entries equal to each other". Proof of this is shown in the <a href="https://mathoverflow.net/q/172837">link</a> in this QA where 1. "Q has rank at most n - 2" because $n-1$ implies $\tau(X)\neq 0$. 2. $Q \;adj(Q)$ <a href="https://en.wikipedia.org/wiki/Adjugate_matrix" rel="nofollow noreferrer">equation</a> 3. "ker Q is spanned by I" is shown <a href="https://simonensemble.github.io/pluto_nbs/graph_connected_components.jl.html" rel="nofollow noreferrer">here "eigenvectors of the Laplacian matrix associated with eigenvalue zero"</a></span>
https://cs.stackexchange.com/questions/52949/understanding-proof-for-busy-beaver-being-uncomputable#comment345879_95517
<span class="comment-copy">2. "also upper-bounds the maximal number of steps run by the halting Turing machine with m states" because we runs $M$ on $U$ so the steps of $M$ is included in $UM$. (IMHO $M$ can be any machine as what the halting problem assumes.) 3. $BB(n)$ is <a href="https://en.wikipedia.org/wiki/Busy_beaver#cite_note-ligocki_bb6-14" rel="nofollow noreferrer">same</a> as $S(n)$. 4. "we can always determine corresponding value for n.": $n$ depends on how we "Construct a universal Turing machine U".</span>
https://math.stackexchange.com/questions/325279/proving-a-simple-connected-graph-is-a-tree-if-adding-an-edge-between-two-existin?rq=1#comment10355421_325279
<span class="comment-copy">I give one summary for the chat between OP and A.S which has <a href="https://chat.stackexchange.com/transcript/message/8438683#8438683">one valuable image</a> for explanation. If the original graph $G$ has one cycle $C$. Then take arbitrary 2 adjacent vertices $v,w$ (we can also take one pair not adjacent. For simplicity, I assume adjacent) inside this cycle. Then 2 cycles are created. One is the multiedge $(v,w)$, while the other is $C-\{v,w\}+\{v,w\}^{(*)}$ where the superscript $(*)$ shows whether the edge is in the original graph. This is contradiction. So $G$ is acyclic.</span>
https://cs.stackexchange.com/questions/18524/how-to-prove-that-a-language-is-context-free/26159#comment67981_18524
<span class="comment-copy">Until the material is moved here, note that Rick Decker and babou collected some typical context-free idioms <a href="http://cs.stackexchange.com/q/33228/98">at a duplicate question</a>.</span>
https://math.stackexchange.com/questions/3992275/if-a-winning-strategy-does-not-exist-for-player-2-does-it-exist-for-player-1#comment10457491_3993622
<span class="comment-copy">Here I give one sketch of the proof more briefly: By this useful <a href="https://oyc.yale.edu/sites/default/files/blackboard15_0_0.pdf" rel="nofollow noreferrer">Blackboard Note</a> of the above course link, we can just think the game as one <i>decision tree</i> which can be variant with different states for each node based on the context. Then we go from the leaf to the root of the whole tree in the induction process because the leaf state can be easily decided until the state of the root is decided.</span>
https://cs.stackexchange.com/questions/165101/how-can-the-approximation-algorithm-of-one-np-complete-problem-be-used-to-prove#comment342996_165101
<span class="comment-copy">I suggest you search out lecture notes on approximation algorithms and hardness of approximation, which should cover this topic. See also the first paragraph of <a href="https://en.wikipedia.org/wiki/Clique_problem#Hardness_of_approximation" rel="nofollow noreferrer">en.wikipedia.org/wiki/Clique_problem#Hardness_of_approximation</a> and <a href="https://sites.cs.ucsb.edu/~teo/papers/JACM-PCom.pdf" rel="nofollow noreferrer">P-Complete Approximation Problems</a> (Sahni & Gonzalez, JACM v23n3 1976).</span>
https://math.stackexchange.com/questions/2490502/counting-techniques-to-find-all-nonisomorphic-graphs-with-six-vertices-all-havi#comment9806829_2490933
<span class="comment-copy">This is the column k=2 in <a href="https://oeis.org/A167625" rel="nofollow noreferrer">oeis.org/A167625</a></span>
https://math.stackexchange.com/questions/1141412/mathbb-z-sqrt-5-is-not-a-ufd#comment10468076_1143135
<span class="comment-copy">For reference, '2 is not prime' is based on <a href="https://en.wikipedia.org/wiki/Irreducible_element#Example" rel="nofollow noreferrer">Euclid's lemma</a> where the reference is in p252 'Example 7' instead of p250.</span>
https://math.stackexchange.com/questions/182459/an-infinite-set-having-one-more-element-than-another-infinite-set#comment420603_182459
<span class="comment-copy">A simpler example: take the function $f:\mathbf N_{\geq 0}\to \mathbf N_{>0}$, $f(n)=n+1$. $f$ is a bijection. There are as many positive natural numbers as there are natural numbers. A sphere without one point has as many points as sphere. There were a lot of questions like yours here, for example <a href="http://math.stackexchange.com/questions/182171/are-all-infinities-equal">this one</a> just yesterday.</span>
https://cs.stackexchange.com/questions/160004/why-do-we-use-main-function-in-almost-all-the-programming-languages#comment334807_160008
<span class="comment-copy">If you use <code>gcc -c hello.c</code> / <code>ld -e main hello.o</code>, you won't get a working executable (even if you link <code>-lc</code>). See also <a href="https://stackoverflow.com/q/29694564">What is the use of _start() in C?</a> / <a href="https://stackoverflow.com/q/14558977">ld : _start not found defaulting to</a> , and <a href="https://stackoverflow.com/q/36861903">Assembling 32-bit binaries on a 64-bit system (GNU toolchain)</a> for some x86 Linux <code>main</code> vs. <code>_start</code> stuff.</span>
https://cs.stackexchange.com/questions/153049/prove-that-the-grammar-s-rightarrow-sss-epsilon-generates-precisely-all-we#comment346785_153049
<span class="comment-copy">1. For the future readers reading this question, OP may mean we should derive $x=[z]$ from the assumption of $z$ instead of the converse direction in the reference lecture p36 where <a href="https://www.abbreviations.com/serp.php?st=O%2FW" rel="nofollow noreferrer">'o/w' may mean otherwise</a>. 2. If someone is interested about this QA, <a href="https://stackoverflow.com/a/42565842/21294350">this similar SO QA</a> may be helpful which uses <i>one more simpler grammar</i>.</span>
https://unix.stackexchange.com/questions/127110/which-process-scheduler-is-my-linux-system-using#comment203380_127110
<span class="comment-copy">possible duplicate of <a href="http://unix.stackexchange.com/questions/36679/how-can-you-determine-which-process-scheduler-is-being-used">How can you determine which process scheduler is being used?</a></span>
https://dsp.stackexchange.com/questions/66513/q-format-doubts#comment134966_66513
<span class="comment-copy">Have you tried going through wikipedia page for Qm.n format. It's in pretty much detail : <a href="https://en.wikipedia.org/wiki/Q_(number_format)" rel="nofollow noreferrer">en.wikipedia.org/wiki/Q_(number_format)</a> , in Qm.n format n is used to denote fractional part and m to denote integral part. Go through this answer too : <a href="https://dsp.stackexchange.com/a/10707/49439">dsp.stackexchange.com/a/10707/49439</a></span>