Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com 2023-12-07T11:29:30Z https://math.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://math.stackexchange.com/q/4822015 0 Modulus of x raised to a DEEPAK KUMAR https://math.stackexchange.com/users/1183233 2023-12-07T11:28:49Z 2023-12-07T11:29:13Z <p>For what values of <span class="math-container">$a \epsilon R^+$</span> the function <span class="math-container">$$f(x) = |x|^a$$</span> is differentiable at <span class="math-container">$x=0$</span>.<br /> According to definition of differentiablity <span class="math-container">$$\lim_{x\to0} \frac{f(x)-f(0)}{x-0}$$</span> should exist &amp; equal for <span class="math-container">$\lim_{x\to0^+}$</span>and <span class="math-container">$\lim_{x\to0^-}$</span> . So according to this the limit for <span class="math-container">$a≤1$</span> right &amp; left limits are not equal &amp; for <span class="math-container">$a≥2$</span> they are equal but I have doubt for <span class="math-container">$a\epsilon(1,2)$</span> this formula gives limit = <span class="math-container">$0$</span> but when we plot the graph for <span class="math-container">$a\epsilon(1,2)$</span> <a href="https://i.stack.imgur.com/crrNf.png" rel="nofollow noreferrer">there is a steep change in slopes at <span class="math-container">$x=0$</span></a>. How to prove or disprove that it is differentiable at <span class="math-container">$0$</span> for <span class="math-container">$a\epsilon(1,2)$</span>?</p> https://math.stackexchange.com/q/4822014 -1 Confusion regarding the difference and thus the relation between Center of a group and Centraliser of an element. Struggling Mathematician https://math.stackexchange.com/users/1174843 2023-12-07T11:21:10Z 2023-12-07T11:27:06Z <p>First of all, I would like to apologise if this has been asked before but I have not found any answer to any question answer my doubt. This is embarassing considering I am beginning to study Sylow's Theorems yet can't distinguish between such concepts.</p> <p>My question: Center of a group is the set of elements of the group commuting with every element of the group meanwhile the Centraliser of an element is the set of all elements commuting with the element, then how come the center is a subset of the Centraliser?</p> https://math.stackexchange.com/q/4822011 0 Almost purity for perfectoid fields applied on a computation of Galois cohomology. Luc https://math.stackexchange.com/users/463468 2023-12-07T11:16:46Z 2023-12-07T11:16:46Z <p>Let <span class="math-container">$K=\mathbf{Q}_p(\zeta_{p^{\infty}})$</span> and <span class="math-container">$C:=\widehat{\overline{\mathbf{Q}_P}}$</span>. When I wrote an answer for a question <a href="https://math.stackexchange.com/a/4822001/463468">a question answer</a>, I have in my mind that</p> <blockquote> <p>&quot;<span class="math-container">$\widehat{K}$</span> is a perfectoid fields will imply for example <span class="math-container">$H^i(G_K, O_C)$</span> is almost zero for <span class="math-container">$i&gt;0$</span>&quot;</p> </blockquote> <p>as a result of some &quot;almost purity for for perfectoid fields&quot;.</p> <p>I then searched and find for example <a href="http://math.uchicago.edu/%7Eamathew/notes.pdf" rel="nofollow noreferrer">a lecture notes</a>, putting Corollary 185 as a consequence of Theorem 183. <strong>I am fine to use Corollary 185 to conclude</strong> (as in my answer linked). But I didn't see how can I directly use the fact &quot;<span class="math-container">$\widehat{K}$</span> is a perfectoid field + almost purity for for perfectoid fields&quot; to conclude. More precisely,</p> <blockquote> <p>Question: in <a href="http://math.uchicago.edu/%7Eamathew/notes.pdf" rel="nofollow noreferrer">a lecture notes</a>, how does Theorem 183 imply Corollary 185?</p> </blockquote> <p>Somehow Theorem 183 tells that &quot;<span class="math-container">$O_L/O_\widehat{K}$</span> is almost finite etale&quot;, but in Corollary 185 one needs somehow &quot;<span class="math-container">$O_L/O_K$</span> is almost finite etale.&quot;</p> <blockquote> <p>Or similar Question: Would &quot;<span class="math-container">$O_L/O_\widehat{K}$</span> being almost finite etale for any finite extension <span class="math-container">$L/\widehat{K}$</span>&quot; imply (removing the completion) &quot;<span class="math-container">$O_M/O_K$</span> is almost finite etale for any finite extension <span class="math-container">$M/K$</span>&quot;?</p> </blockquote> <p>Thank you for remarks, perhaps this is stupid and the answer is very direct.</p> https://math.stackexchange.com/q/4822010 0 Converting a number from base3 to base2 without going through base 10. a.moussa https://math.stackexchange.com/users/1263966 2023-12-07T11:08:29Z 2023-12-07T11:19:56Z <p>Could anyone guide me on how to directly convert a number from base 3 to base 2 without using base 10? For example, converting &quot;2101&quot; (base 3) directly to base 2. Any suggestions, method or algorithms would be greatly appreciated.</p> <p>Thanks in advance!</p> https://math.stackexchange.com/q/4822008 0 Semplification of a summation involving $\text{erf}(z)$ in LLoyd Algorithm Math Attack https://math.stackexchange.com/users/1173806 2023-12-07T11:04:19Z 2023-12-07T11:04:19Z <h2>Context and my work</h2> <p>I used the LLoyd Algorithm to evaluate the distorsion of <span class="math-container">$$f_{XY}(x,y)=\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$$</span> and I used the following thresholds and quantized values: <span class="math-container">$$(2k-N-1)h|_{k=1}^{N}\qquad (2k-N-2)h|_{k=1}^{N+1}$$</span> Where <span class="math-container">$N=3,...,31$</span> is odd and <span class="math-container">$h&gt;0$</span> and <span class="math-container">$h\propto\frac{1}{N}$</span></p> <p>So I wrote the following integral: <span class="math-container">$$\sum_{j=1}^{N+1}\sum_{k=1}^{N+1}\int_{\left(2j-N-3\right)h}^{\left(2j-N-1\right)h}\int_{\left(2k-N-3\right)h}^{\left(2k-N-1\right)h}\frac{1}{2\pi}e^{-\frac{x^{2}+y^{2}}{2}}\left(x-\left(2k-N-2\right)h\right)^{2}\left(y-\left(2j-N-2\right)h\right)^{2}dxdy=$$</span></p> <blockquote> <p><span class="math-container">$$\text{...Boring calculations not necessary for understanding the question...}$$</span></p> </blockquote> <p><span class="math-container">$${\frac{2h^{2}}{\pi}\left(\sqrt{\frac{\pi}{2}}\frac{N^{2}h^{2}+1}{h}\operatorname{erf}\left(\frac{N+1}{\sqrt{2}}h\right)+\left(N-1\right)e^{-\frac{h^{2}\left(N+1\right)^{2}}{2}}-2-4\sum_{k=1}^{\frac{N-1}{2}}\left(\sqrt{2\pi}hk\operatorname{erf}\left(\sqrt{2}kh\right)+e^{-2h^{2}k^{2}}\right)\right)^{2}}$$</span></p> <h2>Question</h2> <p>Is it possible to simplify the last summation somehow? I don't think there are any notable summations for Gaussians or erf functions, but is it possible for example through Taylor or some other method?</p> https://math.stackexchange.com/q/4822007 0 Conjecture about integer partitions Fabius Wiesner https://math.stackexchange.com/users/573047 2023-12-07T11:03:11Z 2023-12-07T11:17:25Z <p>I formulated this conjecture after reading <a href="https://math.stackexchange.com/q/4821904/573047">this related question</a>.</p> <p>Let <span class="math-container">$\mathcal{P}(n) = \{P_1(n), P_2(n), \ldots \}$</span> be the set of all integer partitions of a positive integer <span class="math-container">$n$</span>, and <span class="math-container">$p(n)=\vert \mathcal{P}(n) \vert$</span> the number of those partitions. Note that the <span class="math-container">$P_k(n)$</span> are multisets, i.e. elements can be repeated.</p> <p>Let <span class="math-container">$f(P_k(n))$</span> be the number of distinct permutations of the elements of <span class="math-container">$P_k(n)$</span>.</p> <p>Is it true that for <span class="math-container">$n \ge 2$</span>:</p> <p><span class="math-container">$$\sum_{k=1}^{p(n)} f(P_k(n))(-1)^{\vert P_k(n) \vert} = 0 \space ?$$</span></p> <p>I verified it for <span class="math-container">$2 \le n \le 6$</span>, while it evaluates to <span class="math-container">$-1$</span> for <span class="math-container">$n = 1$</span>.</p> <p>EDIT</p> <p>Searching the OEIS and precisely the comments section of <a href="https://oeis.org/A111786" rel="nofollow noreferrer">OEIS A111786</a> it seems this is a well known fact. I haven't found the explanation yet...</p> https://math.stackexchange.com/q/4822006 0 Error in Lemma 5.3.1 in (Qingliu) Algebraic Geometry and Arithmetic curve Z Wu https://math.stackexchange.com/users/502929 2023-12-07T11:01:42Z 2023-12-07T11:01:42Z <p>Here's the picture:</p> <p><a href="https://i.stack.imgur.com/krT8e.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/krT8e.png" alt="enter image description here" /></a></p> <p>This is alright when <span class="math-container">$d&gt;0$</span>. But when <span class="math-container">$d=0$</span> and <span class="math-container">$n&lt;0$</span>, we know <span class="math-container">$H^0(X,\mathcal{O}_X(n))=T_0^n\cdot A \neq 0$</span>.</p> <p>Similarly in Stacks Project tag 01XT,</p> <p><a href="https://i.stack.imgur.com/3Q8Ju.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3Q8Ju.png" alt="enter image description here" /></a></p> <p>things seem to be wrong when <span class="math-container">$q=n=0,d&lt;0$</span>, in this case <span class="math-container">$(R[T_0])_d=0\neq (\frac{1}{T_0}R[\frac{1}{T_0}])_d$</span>.</p> <p>Am I right? If I am right, how should we modify the statement so it is compatitable with this extreme case <span class="math-container">$\mathbb{P}^0$</span>.</p> https://math.stackexchange.com/q/4822003 0 Understanding how to approach a proof question that is related to using definitions. cosygod https://math.stackexchange.com/users/1235596 2023-12-07T10:55:16Z 2023-12-07T10:55:16Z <p>I struggle doing these type of questions, i have no idea how i should approach these type of question. Like for example, the question below:</p> <p>Suppose <span class="math-container">$f: A\to B$</span> is an injective function. Prove that <span class="math-container">$f^{-1}(f(C))=C$</span> for all <span class="math-container">$C \subseteq A$</span>.</p> <p>I now know that I should define everything that is in the question and then im not sure what do next.</p> <p>Please tell me how you would approach these type of question. So I can do the same approach for other questions.</p> https://math.stackexchange.com/q/4821979 1 If a deck of 54 cards (including 2 jokers) is evenly split into 3 groups of 18, what is the probability that any one group contains both jokers? Darrel https://math.stackexchange.com/users/1263900 2023-12-07T10:01:52Z 2023-12-07T10:48:58Z <p>I was just asked this interview question on combinatorics:</p> <blockquote> <p>A deck of 54 cards (includes 2 jokers) are split into 3 equal groups of 18. What is the probability of any single group having both jokers?</p> </blockquote> <p>I only had time to think of a solution post-interview, but please check whether my answer is correct:</p> <p><span class="math-container">$$P = \frac{\text{Ways to form group with 2 jokers} \times \text{Ways to form 1st non-joker group} \times \text{Ways to form 2nd non-joker group}}{\text{Ways to form 3 equal groups}}$$</span></p> <p><span class="math-container">$$P = \frac{{2 \choose 2}{52 \choose 16} \times {36 \choose 18} \times {18 \choose 18}}{{54 \choose 18}{36 \choose 18}{18 \choose 18} / 3!}$$</span></p> <p>I'm not sure whether I need to divide by <span class="math-container">$3!$</span> in the denominator, since it seems the numerator will also have this and it cancels out?</p> https://math.stackexchange.com/q/4821941 1 Is my approach correct to solve this group theory question The Riddler https://math.stackexchange.com/users/1084313 2023-12-07T08:27:22Z 2023-12-07T11:28:47Z <p>Let <span class="math-container">$A$</span> be an abelian group of order <span class="math-container">$2^{100}$</span>. Prove that <span class="math-container">$A$</span> is not a subgroup of <span class="math-container">$S_n$</span> for <span class="math-container">$n&lt;200$</span>.</p> <p>I was thinking of showing that there is no subgroup of order 2^200 by using an argument using lagrange to solve this question by I can't find a way to verify how many 2s are there in 200! and also this is a stronger claim so it seems unprobable. Does anybody have some other ideas?</p> <p>Edit: I realized we are dealine with 2^100 I made a mistake the question is about order 2^100</p> https://math.stackexchange.com/q/4821937 0 $\mu(E)=\mu(\theta^{-1}(E)) \forall E \in \mathbb{E} \Rightarrow \mu(A)=\mu(\theta^{-1}(A)) \forall A \in \mathbb{A}$ for generator E and measure \mu Dave https://math.stackexchange.com/users/1250828 2023-12-07T08:22:16Z 2023-12-07T10:52:33Z <p>I have to prove the following:</p> <p>Let <span class="math-container">$(S,\mathbb{A},\mu)$</span> be a finite measure space and <span class="math-container">$\mathbb{A}=\sigma(\mathbb{E})$</span> for an <span class="math-container">$\cap$</span>-stable generator <span class="math-container">$\mathbb{E}$</span>. Further let <span class="math-container">$\theta: S \to S$</span> be a function with <span class="math-container">$\theta^{-1}(A) \in \mathbb{A} \forall A \in \mathbb{A}$</span> . show that:</p> <p><span class="math-container">$\mu(E)=\mu(\theta^{-1}(E)) \forall E \in \mathbb{E} \Rightarrow \mu(A)=\mu(\theta^{-1}(A)) \forall A \in \mathbb{A}$</span></p> <hr /> <p>my ideas:</p> <p>I think i have to show that <span class="math-container">$D=( A\in \mathbb{A}: \mu(A)=\mu(\theta^{-1}(A)))$</span> is a Dynkin-System:</p> <p>But I'm already failing to show that <span class="math-container">$S\in D$</span></p> <p>Would need a little help here. Is my idea with the <span class="math-container">$\pi$</span>-System correct?</p> <p>Edit: Would I need the condition <span class="math-container">$S\in \mathbb{E}$</span>? I have the feeling that this is missing...</p> https://math.stackexchange.com/q/4821911 -4 the equality case for Hölder ineqality for series elif https://math.stackexchange.com/users/1142508 2023-12-07T07:35:29Z 2023-12-07T11:24:51Z <p><span class="math-container">$$\forall n\in N, \exists M&gt;0 : |u_n|^p=M|v_n|^q \iff(\sum_{i=1}^n |u_n|^p )^\frac{1}{p} (\sum_{i=1}^n |u_n|^q)^\frac{1}{q}$$</span> I proved if there is M&gt;0 a constant like that then equality is true but I couldn't prove the opposite.</p> https://math.stackexchange.com/q/4821885 0 Monotone Path: For any two points $x,y$, there must exists an increasing path $t$ with terminals at $x,y$ and $f(z)$ is monotonic along the path. High GPA https://math.stackexchange.com/users/438867 2023-12-07T06:38:00Z 2023-12-07T11:15:47Z <p><span class="math-container">$x,y\in\mathbb R^n$</span>. <span class="math-container">$f$</span> is a real function. We want a path inside <span class="math-container">$\mathbb R^2$</span> connecting <span class="math-container">$x,y$</span>. Now consider a new thing called &quot;increasing path&quot;.</p> <p>We say <span class="math-container">$t$</span> is an increasing path if <span class="math-container">$f(z)$</span> is monotonic when <span class="math-container">$z$</span> moving from <span class="math-container">$x$</span> to <span class="math-container">$y$</span> along <span class="math-container">$t$</span>.</p> <p><strong>Question:</strong> What is the assumption we need to put <span class="math-container">$f$</span> to ensure that every two points have an increasing path?</p> <p><strong>Motivation:</strong> this problem is related to the gradient descent method in optimization, in which you want your <span class="math-container">$f(z)$</span> be increasing in every step. If such path exists, then optimization method works. If not, then the method does not work.</p> <p>Ideas: for <span class="math-container">$n=1$</span>, the things are restricted, and we need <span class="math-container">$f$</span> to be monotonic (not necessarily continuous)</p> <p>For <span class="math-container">$n\geq2$</span>, however, things become more interesting. It seems like convexity or concavity is sufficient.</p> <p>Note: there is also similar idea in graph theory called &quot;<a href="https://www.tutorialspoint.com/monotonic-shortest-path-from-source-to-destination-in-directed-weighted-graph#:%7E:text=A%20path%20is%20monotonic%20if,strictly%20growing%20or%20strictly%20decreasing." rel="nofollow noreferrer">monotone path</a>&quot;</p> https://math.stackexchange.com/q/4821881 -1 Points H and I are the orthocenter and incenter of triangle ABC. If HBI=HCI then How much is the measurement of angle A? mathisdagoat https://math.stackexchange.com/users/1198331 2023-12-07T06:31:58Z 2023-12-07T11:18:03Z <p>I need some help with this question: Points H and I are the orthocenter and incenter of triangle ABC. If HBI=HCI (angles) then How much is the measurement of angle A? I don't know what the diagram will look like and I'm not quite sure what an incentre is and how to draw it with an orthocentre.</p> https://math.stackexchange.com/q/4821845 1 Subset of $\ell_2$ with the norm $\sup n|x_n|<\infty$ is a Banach space Captain Haddock https://math.stackexchange.com/users/940152 2023-12-07T04:38:41Z 2023-12-07T10:51:27Z <p>Consider the function <span class="math-container">$\lVert \cdot \rVert: \ell_2 \to [0,\infty]$</span> defined by <span class="math-container">$$\lVert x \rVert=\sup_{n\geq 1} n\lvert x_n\rvert$$</span> for every <span class="math-container">$x=\{x_n \}_{n=1}^{\infty}\in \ell_2$</span>. Let <span class="math-container">$$X=\{ x\in \ell_2 \mid \lVert x\rVert &lt;\infty \}$$</span> Show that <span class="math-container">$X$</span> with the said norm is a Banach space.</p> <p>I have shown that the function defined is indeed a norm. I can think of two ways to prove a space is Banach: one is to get a Cauchy sequence and show that it is convergent, and two is to show that every absolutely convergent series is convergent. None of them has led me far, but I think the first one is more useful if I can relate the norm with the usual <span class="math-container">$\ell_2$</span> norm <span class="math-container">$\lVert \cdot \rVert_2$</span> in some way. Let's say if we had <span class="math-container">$c \lVert x \rVert&lt;\lVert x \rVert_2 &lt; C\lVert x \rVert$</span>, then a Cauchy sequence in <span class="math-container">$X$</span> with the said norm yields a Cauchy sequence in <span class="math-container">$\ell_2$</span> by the right side of the inequality, and since <span class="math-container">$\ell_2$</span> is complete, the sequence will converge in <span class="math-container">$\ell_2$</span>, but again using the left side of the equality we can say that it converges in X as well.</p> https://math.stackexchange.com/q/4821751 0 How can I prove the existence of delta-incomplete / countable incomplete ultrafilters? David Gómez https://math.stackexchange.com/users/1124217 2023-12-07T01:16:22Z 2023-12-07T11:04:15Z <p>I am reading a W.A.J Luxemburg paper about nonstandard analysis (<a href="https://www.jstor.org/stable/3038221" rel="nofollow noreferrer">https://www.jstor.org/stable/3038221</a>). He presents the following definition</p> <p><a href="https://i.stack.imgur.com/waT2v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/waT2v.png" alt="enter image description here" /></a></p> <p>I am stuck trying to prove the existence of delta-incomplete ultrafilters. I suppose I have to construct a chain of filters over an infinite set with a certain condition such that, at the moment of using Zorn's lemma, the maximal satisfies the definition of delta-incomplete.</p> <p>I have this, and I am more confused as how I can proceed. Let <span class="math-container">$\mathcal{C}$</span> be a chain of filters over a set <span class="math-container">$I$</span> with a maximal <span class="math-container">$\mathscr{U}$</span> such that <span class="math-container">$\mathscr{U}$</span> is a <span class="math-container">$\delta$</span>-incomplete ultrafilter. That is, there exists <span class="math-container">$\{I_n\}_{n\in\mathbb{N}}$</span>, a countable partition of <span class="math-container">$I$</span> such that, for all <span class="math-container">$n$</span>, <span class="math-container">$I_n \not\in \mathscr{U}$</span>. So, we would have that same property for all <span class="math-container">$F \in\mathcal{C}$</span>. If some <span class="math-container">$I_n\in F$</span> then <span class="math-container">$I_n\in\mathscr{U}$</span>. that leads to <span class="math-container">$\mathcal{C} = \{\mathscr{U}\}$</span>.</p> https://math.stackexchange.com/q/4821277 5 What does "dx" mean in ∫𝑓(𝑥)𝑑𝑥? [duplicate] Zehan Li https://math.stackexchange.com/users/1216172 2023-12-06T10:20:02Z 2023-12-07T11:28:02Z <p>I have this confusion while studying indefinite integrals. Is dx a derivative of x or a notation?</p> <p><em><strong>If it is just a notation,how can we explain that the transformation from dx to du satisfies the operational law of differentiation?</strong></em></p> https://math.stackexchange.com/q/4821160 2 Are Distributions just functions with infinitesimal coefficients? Daniel Schwartz https://math.stackexchange.com/users/1122822 2023-12-06T04:57:35Z 2023-12-07T11:00:42Z <p>It is relatively well known that the Dirac delta, the classical example of a distribution, can be written as <span class="math-container">$$\frac{a}{π(a^2+x^2)}$$</span> where <span class="math-container">$a$</span> is an infinitesimal such as a hyperreal. This can be confirmed by simply integrating and evaluating the integral symmetrically about the y axis which is simply <span class="math-container">$\frac{\arctan(x/a)}{π}$</span> from <span class="math-container">$x=-b$</span> to <span class="math-container">$x=b$</span> with <span class="math-container">$b$</span> real which is of course equal to <span class="math-container">$1$</span>.</p> <p>My question is if this holds generally. Are there any distributions of note that cannot be written as straightforward functions in a setting equipped with infinitesimals?</p> https://math.stackexchange.com/q/4819446 0 Source of theorem regarding triangle presence in the graph Keithx https://math.stackexchange.com/users/364687 2023-12-03T16:55:14Z 2023-12-07T11:20:33Z <p>I know that there exists theorem that states the following:</p> <p>Graph <span class="math-container">$G$</span> contains triangle if and only if there are indices <span class="math-container">$i$</span> and <span class="math-container">$j$</span> existing so the both matrices <span class="math-container">$A_G$</span> and <span class="math-container">$A_G^2$</span> will have nonzero (<span class="math-container">$i,j$</span>).</p> <p>The problem is that I don't know the source of this theorem and I was unable to find it. Can you please give me a hint on where should I look for? :)</p> <p>Thank you.</p> https://math.stackexchange.com/q/4819330 0 Discrete change of an integral result Mr. Science https://math.stackexchange.com/users/1171986 2023-12-03T13:36:45Z 2023-12-07T10:54:49Z <p>So I was working on ΔS (it is called action in physics but the question I am about to present is purely mathematical).</p> <p><span class="math-container">$S=∫L dt =&gt; ΔS=Δ(∫(\frac{1}{2}mv^2 +V)dt)= Δ(∫\frac{1}{2}mv^2 dt)$</span></p> <p>The problem I face here is if we were to assume that t interval is small. Then could we say</p> <p><span class="math-container">$$ΔS= Δ(∫\frac{1}{2}mv^2 dt)=\frac{1}{2}mv^2Δt$$</span></p> <p>Is this acceptable?</p> <p>Note: I am reminded that this question may not be seen as &quot;purely mathematical&quot; and decided to wrote this note to avoid confusion regarding terms related to physics. However, this note will get a bit more mathematical towards the end. I asked this question here because I think my problem is more mathematical than physical. m is mass and v is velocity. It is assumed that they are constants. ΔS is &quot;change in action&quot; and is defined as ∫L dt with L being <span class="math-container">$L=\frac{1}{2}mv^2 +V$</span>. This &quot;change&quot; actually represents deviation from the action of the classical path a particle may follow.</p> <p>V (potential term) in my first definition of ΔS is not important. What I am trying to show here is that ΔS can be represented as <span class="math-container">$\frac{1}{2}mv^2Δt$</span>. Let's visualise the issue that I am having.</p> <p>When <span class="math-container">$Δt&lt;&lt;1$</span>, it is immediate to see that <span class="math-container">$∫\frac{1}{2}mv^2 dt=\frac{1}{2}mv^2Δt$</span></p> <p>So we have <span class="math-container">$Δ(\frac{1}{2}mv^2Δt)=\frac{1}{2}mv^2Δ(Δt)$</span>. This is a serious issue for me. I felt like we can overcome this by saying <span class="math-container">$Δt=t'$</span> and thus we have <span class="math-container">$$Δ(\frac{1}{2}mv^2Δt)=\frac{1}{2}mv^2Δ(Δt)=\frac{1}{2}mv^2Δt'=ΔS$$</span></p> <p>But I need <span class="math-container">$Δt'\equiv Δt$</span> and that feels wrong in a mathematical way. It works while making commentaries on a solution but I want to be able to show it mathematically.</p> <p>If <span class="math-container">$Δt'\equiv Δt$</span> somehow works than ΔS becomes <span class="math-container">$$ΔS=\frac{1}{2}mv^2Δt=\frac{1}{2}m(\frac{Δx}{Δt})^2Δt=\frac{1}{2}m\frac{(Δx)^2}{Δt}=\frac{1}{2}m\frac{\eta^2}{\epsilon}$$</span> Don't worry about <span class="math-container">$\eta$</span> and <span class="math-container">$\epsilon$</span>. They are just a symbol. They still show changes in the respective &quot;variables&quot;. That's not changed.</p> https://math.stackexchange.com/q/4818770 6 Approximation of the n'th prime Tommy R. Jensen https://math.stackexchange.com/users/160396 2023-12-02T15:29:34Z 2023-12-07T11:08:50Z <p>An old paper by Ernest Cesàro provides a suggested approximation of the n'th prime. The expression and the reference currently appears in the Wikipedia article on the Prime Number Theorem.</p> <p>It is inspired by an even earlier and difficult to locate article by one Monsieur Pervouchine aka Ivan Mikheevich Pervushin in a Russian Journal.</p> <p>Cesàro points out that his own original approximation that appeared in a paper in <em>Actes de l'Académie des Sciences de Naples</em> in 1893: <span class="math-container">$$\frac{p_n}{n} = \log p_n - 1 - \frac{1}{\log p_n} - \frac{3}{(\log p_n)^2} - \cdots$$</span> (presumably with the implicit understanding that all subsequent terms add up to <span class="math-container">$o(1/(\log p_n)^2)$</span>) implies Pervushin's approximation.</p> <p>What I would like to know is whether there exists a modern or at least better accessible account of this type of approximation. And I am curious why it seems much easier to get the attempted asymptotics right numerically when I change the &quot;3&quot; to a &quot;2&quot;?</p> https://math.stackexchange.com/q/4807218 -2 Showing equality of first norms Jeff https://math.stackexchange.com/users/1150130 2023-11-15T02:18:48Z 2023-12-07T11:21:19Z <p>We consider a function <span class="math-container">$f \in L_1[0,1]$</span>, and define <span class="math-container">$g : [0,\infty) \to [0,1]$</span> in terms of <span class="math-container">$f$</span>, where <span class="math-container">$g(y) = \lambda(|f|^{-1}(y,\infty])$</span> for <span class="math-container">$\lambda$</span> the Lebesgue measure, and <span class="math-container">$|f|^{-1}(y,\infty] = \{x \in [0,1] \mid |f(x)| &gt; y\}$</span>.</p> <p>We wish to show that <span class="math-container">$\|g\|_1 = \|f\|_1$</span>, what sort of methods would be appropriate to use here?</p> https://math.stackexchange.com/q/4794709 2 Method of Steepest Descent and Contour Deformation Leonardo https://math.stackexchange.com/users/714766 2023-10-26T18:57:05Z 2023-12-07T11:26:51Z <p>In the book <em>An Introduction to Quantum Field Theory</em> by Peskin and Schroeder, p. 14 in section 2.1, it is stated that, in looking at the asymptotic behavior for <span class="math-container">$x^{2} \gg t^{2}$</span> of the integral <span class="math-container">\begin{equation} \int_{-\infty}^{\infty} \mathrm{d}p \; p \, \mathrm{e}^{\phi(p)}, \qquad \phi(p) = i\left( p x - t \sqrt{p^{2} + m^{2}}\right),\tag{1} \end{equation}</span> one can implement the method of stationary phase.</p> <p>However, the stationary points of <span class="math-container">$\phi(p)$</span>, <span class="math-container">$$p_{\pm} = \pm imx/\sqrt{x^{2} - t^{2}},\tag{2}$$</span> are imaginary, which would seem to indicate to me that the method of stationary phase is not applicable as is. The book then mentions that one can &quot;freely push the contour upward&quot;, but exactly how and why is not clear to me.</p> <p>After looking around, it seems that the asymptotic behavior can be obtained by using the method of the steepest descent. However, I don't understand the following:</p> <ol> <li><p>How does one chooses a contour in the complex plane such that, through Cauchy's theorem I suppose, relates to the integral above, while at the same time passing through one of the stationary points.</p> </li> <li><p>Should one choose only one of the stationary points? If yes, why?</p> </li> </ol> <p>Any help is greatly appreciated.</p> https://math.stackexchange.com/q/4542814 7 Is it possible that an infinite group has exactly one infinite nontrivial proper subgroup that has a certain order? durianice https://math.stackexchange.com/users/806703 2022-10-01T07:08:02Z 2023-12-07T11:01:57Z <p>Can there be an infinite group <span class="math-container">$G$</span> and a nontrivial proper subgroup <span class="math-container">$1&lt;H&lt;G$</span> such that if <span class="math-container">$1&lt;K&lt;G$</span> and <span class="math-container">$|K|=|H|$</span>, then <span class="math-container">$K=H$</span>? Although I'm particularly interested in when <span class="math-container">$|H|$</span> is infinite, some examples for finite <span class="math-container">$H$</span> have been given in the comment:</p> <ol> <li><span class="math-container">$(G,H)=(\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z},\{0\}\times\mathbb{Z}/2\mathbb{Z})$</span></li> <li><span class="math-container">$(G,H)=(\mathbb{R}^*,\{1,-1\})$</span></li> <li><a href="https://en.wikipedia.org/wiki/Pr%C3%BCfer_group" rel="nofollow noreferrer">Prüfer groups</a> (I do not know about this)</li> </ol> <p>Note: Originally I allowed <span class="math-container">$K=G$</span>. I wasn't aware of this distinction but Qiaochu Yuan's answer shows that if <span class="math-container">$K=G$</span> is allowed, it is impossible to find such <span class="math-container">$(G,H)$</span> where <span class="math-container">$H$</span> is infinite.</p> https://math.stackexchange.com/q/4154051 2 Problem about the Definition of Large Deviation Principle MikeG https://math.stackexchange.com/users/579758 2021-05-28T13:35:14Z 2023-12-07T11:02:16Z <p>As in many classic textbooks, the definitions of Large Deviation Principle is as follows:</p> <p><strong><span class="math-container">$\{\mu_n\}$</span> has LDP with speed <span class="math-container">$a_n$</span> and rate <span class="math-container">$I(x)$</span> if the following holds for any measurable <span class="math-container">$A$</span>: <span class="math-container">$$\limsup\frac{1}{a_n}\log\mu_n(A)\leq -\inf_{x\in\bar{A}}I(x),$$</span> <span class="math-container">$$\liminf\frac{1}{a_n}\log\mu_n(A)\geq -\inf_{x\in A^{o}}I(x).$$</span></strong></p> <p>I was wondering why we need the closure and interior as upper and lower bounds here(resp.), instead of use the infimum on <span class="math-container">$A$</span> directly.</p> https://math.stackexchange.com/q/3462253 1 How to find the derivative of $x^2\sin(x)$ using only the limit definition of a derivative? spence https://math.stackexchange.com/users/644005 2019-12-04T03:23:50Z 2023-12-07T11:27:20Z <p>I’m trying to find the derivative of <span class="math-container">$x^2\sin x$</span> using only the limit definition of a derivative. I’ve tried two approaches, one using the difference quotient and another with the regular <span class="math-container">$x-a$</span> formula.</p> <p>I’m stumped on both approaches and not sure where to go. Maybe I’m on the wrong track completely. The difference quotient gets messy quickly and I can’t figure out how to factor out <span class="math-container">$h$</span> to get it into a definable form. So then I tried: <span class="math-container">$$\frac{x^2\sin(x) - a^2\sin(a)}{x-a}.$$</span></p> <p>Is it possible to apply the trig sum-to-product form to the numerator? I’m really just guessing playing around with identities trying to figure this out. Any tips would be appreciated!</p> https://math.stackexchange.com/q/2489541 4 Huge matrix multiplication Alessandro Pilleri https://math.stackexchange.com/users/495684 2017-10-25T18:24:36Z 2023-12-07T11:01:05Z <p>I have a sparse A matrix stored in column major order (it is intrisically column major) of ~80GB and another sparse matrix B relatively small (1GB) which can be loaded in row or column major with no particular effort. I need to compute a straight matrix product S = AB. My problem is that I have only 64 GB of RAM (I usually use the Eigen c++ library ) and i need to compute the product by blocks.</p> <p>I was thinking to re store the A matrix in row major (even it could imply a great increase in terms of storage) and later load the new matrix A by blocks, N rows at time, compute and store the various products and at the end assembly all the blocks together.</p> <p>Do you have any better ideas?</p> https://math.stackexchange.com/q/1680032 0 Why is the axiom of union harmless? asdfadsfasdfasdf https://math.stackexchange.com/users/319321 2016-03-02T15:00:34Z 2023-12-07T11:01:09Z <p>The axiom of union says:</p> <blockquote> <p>Let $M$ be a set of sets. Then $\bigcup M$ exists.</p> </blockquote> <p>The axiom of replacement says:</p> <blockquote> <p>Let $M$ be a set and replace every element $x\in M$ with another object $x'$. Then $\{x'\mid x\in M\}$ is a set.</p> </blockquote> <p>Most mathematicians believe that ZFC is consistent. But I wonder if it is possible to construct a set that is as big as the set of all sets or russel set by using axiom of replacement and axiom of union again and again. Why do mathematicians blindly believe that these constructions aren't contradictory?</p> <p>Is there an heuristic argument that the axiom of union and the axiom of replacement can't construct "sets that are too big to be sets" (that is "contradictory sets" or "proper classes") together?</p> <p>Note: If one uses an axiom system then one eventually thinks this axiom system isn't contradictory. So my question goes to everyone who uses ZFC: Why are you sure that ZFC isn't contradictory?</p> https://math.stackexchange.com/q/1668463 1 Geodesic curvature of parallels on two sheeted hyperboloid F. Chanal https://math.stackexchange.com/users/317054 2016-02-23T11:15:36Z 2023-12-07T11:24:01Z <p>I have troubles trying to imagine the geodesic curvature of curves on surfaces of positive gaussian curvature. Not being generally valid the "minimum distance" argument, I have difficulty to grasp the intuition of this quantity even in very simple cases. I am not interested in explicit calculations, I would like to know what could be some sensible ways I should look at two different curves on a "beautiful" surface with positive gaussian curvature and try to predict which is "further" from being geodesic (even if I don't know how all the geodesic of the surface look like - i.e. I don't have a comprehensive classification that allows me to make sensible comparisions)?</p> <p>More specifically, I want to know how to tell if the geodesic curvature of a parallel of a two-sheeted hyperboloid decreases as the parallel gets larger (i.e. as it gets further from the point of maximum gaussian curvature) without explicit calculation. Thinking of its definition as the norm of the covariant derivative doesn't seem to help much (maybe I'm just not confident enough with the covariant derivative?)...and using the fact that parallels through critical points of the curve whose rotation generates the surface are the only geodesic parallels doesn't seem to suggest the right answer (I think the geodesic curvature goes down to zero but the derivative of the generating curve does not...)? What goes wrong here? How should I go about it then?</p> <p>I hope my question is sensible enough, though I realize it is fairly vague and imprecise! And of course, I am just looking for ideas to make sensible expectations that could work in "beautiful" cases with lots of symmetries like the one presented, nothing too general. It's just that it doesn't feel like I've fully understood it until I can make sense of the simple cases (like this one) without uggly calculations. Thanks in advance so much for taking your time to answer me!</p> <p>P.S.: just so you know: I am un undergraduate student who has taken just one course of differential geometry.</p> https://math.stackexchange.com/q/4467 175 How to prove: if $a,b \in \mathbb N$, then $a^{\frac 1b}$ is an integer or an irrational number? anonymous https://math.stackexchange.com/users/0 2010-09-12T11:24:41Z 2023-12-07T11:22:48Z <p>It is well known that <span class="math-container">$\sqrt{2}$</span> is irrational, and by modifying the proof (replacing 'even' with 'divisible by <span class="math-container">$3$</span>'), one can prove that <span class="math-container">$\sqrt{3}$</span> is irrational, as well. On the other hand, clearly <span class="math-container">$\sqrt{n^2} = n$</span> for any positive integer <span class="math-container">$n$</span>. It seems that any positive integer has a square root that is either an integer or irrational number.</p> <blockquote> <ol> <li>How do we prove that if <span class="math-container">$a \in \mathbb N$</span>, then <span class="math-container">$\sqrt a$</span> is an integer or an irrational number?</li> </ol> </blockquote> <p>I also notice that I can modify the proof that <span class="math-container">$\sqrt{2}$</span> is irrational to prove that <span class="math-container">$\sqrt{2}, \sqrt{2}, \cdots$</span> are all irrational. This suggests we can extend the previous result to other radicals.</p> <blockquote> <ol start="2"> <li>Can we extend 1? That is, can we show that for any <span class="math-container">$a, b \in \mathbb{N}$</span>, <span class="math-container">$a^{\frac 1b}$</span> is either an integer or irrational?</li> </ol> </blockquote>