qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
738,940 | <p>Asked a question to evaluate the Gaussian Integral,</p>
<p>$$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$</p>
<p>using the the following approximation,</p>
<p>$J=\Bbb E[X^2] \sim J_N = 1/N \sum_1^NX_n^2 $</p>
<p>where $x_n \sim N(0, 1)$</p>
<p>Examin $J_N$ for different values of $N$ and... | mookid | 131,738 | <p><strong>Hint:</strong> integrate by parts:$$
x^2\exp (-x^2/2) = x\times x\exp (-x^2/2)
$$</p>
<p>details:
You get
$$
\int_{-\infty}^\infty x^2\exp (-x^2/2) dx =
[x\times -\exp (-x^2/2)]_{-\infty}^\infty
- \int_{-\infty}^\infty 1\times -\exp (-x^2/2) dx \\
= \int_{-\infty}^\infty \exp (-x^2/2) dx
=\sqrt{2\pi}
$$</p... |
244,046 | <p>Let $X$ bet a metric space with metric $d$.</p>
<p>Prove that $$p(x,y) = \frac{d(x,y)}{1 + d(x,y)}$$ defines a metric on $X$.</p>
<ol>
<li><p>Show $p(x,y)$ is is bounded by $1$. (i.e. $p(x,y) \le 1$ for all $x,y$).</p></li>
<li><p>Given $x \in X$, determine $B_1^p$.</p></li>
</ol>
<p>I can prove it is a metric, I... | Elchanan Solomon | 647 | <p>Note that</p>
<p>$$\frac{d(x,y)}{1 + d(x,y)} \leq 1 \Longleftrightarrow d(x,y) \leq 1 + d(x,y) \Longleftrightarrow 0 \leq 1$$</p>
<p>Note that $\frac{1}{n} \to 0$ but $\frac{1}{n} \not\leq 0$ for any $n$, so your proof does not suffice. </p>
|
4,226,115 | <p>I was going through some practice problems for an upcoming calculus exam and I came across the question: "Prove the following statement: if <span class="math-container">$\lim _{n\to \infty }a_n = L $</span> and <span class="math-container">$\lim _{n\to \infty }a_n = M$</span>, then <span class="math-container">... | Sebastiano | 705,338 | <p>I have thought immediately to the limit uniqueness theorem for sequences that asserts:</p>
<blockquote>
<p>A <span class="math-container">$\{a_n\}$</span> sequence of real numbers cannot have two distinct limit <span class="math-container">$L$</span> and <span class="math-container">$M$</span>. In fact if we suppose... |
1,331,975 | <p>What's the difference between logic (in a narrow sense, i.e. a logical system such as ZOL, FOL, etc.) and type system?</p>
<p>I will sketch my understanding of this -- please correct if I err. Under propositions-as-types, they coincide. In propositions-as-some-types, a logic(al system) is a proper subset of the typ... | Giorgio Mossa | 11,888 | <p>In my understanding formal logic-systems (like FOL, propositional logic, etc) and type theories (or type systems) are all families of derivation systems: i.e. systems having an underlying set of formulas (usually described by a formal grammar) and inference rules defined on these formulas (which can be thought as op... |
118,811 | <p>As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two intervals, etc. </p>
<p>The resulting set has Hausdorff dimension $s=\log 2/\log 3$, in view of Hutchinson's formula... | BSteinhurst | 11,332 | <p>Just a quick note before having to move on, but when thinking about IFSs which approached self-similarity the papers of Igudesman have been useful. He has a notion of lacunary self-similar sets which (without checking details) looks like your set up. The main paper I am thinking of is the one referred to <a href="ht... |
2,658,073 | <p>I've got a problem where I have to find the degree of the Maclaurin polynomial (meaning it's centered at $c = 0$, correct?) of $\ln(1.25)$ such that the error is less than 0.001.</p>
<p>I think I have a fairly good understanding of how you find the degree required to match that criteria for other functions, but I'm... | Claude Leibovici | 82,404 | <p><em>Just added for your curiosity since copper hat already gave the answer.</em></p>
<p>Say that you want the result to be in an error smaller than $\epsilon$, you need to find $n$ such that
$${x^{n+1} \over n+1} < \epsilon$$ Sooner or later, you will learn that
$${x^{n+1} \over n+1} = \epsilon\implies n=-1-\fr... |
1,980,012 | <p>I am facing issue to evaluate this particular type of definite integral,</p>
<p>$$I=\int_{0}^{\infty}\exp(-\alpha x^2)\cos(\beta x)dx$$</p>
<p>Please suggest a way to this.</p>
<p>Thanks</p>
| Mark Viola | 218,419 | <p>Let $I(\alpha,\beta)$ be given by the integral</p>
<p>$$I(\alpha,\beta)=\int_0^\infty e^{-\alpha x^2}\cos(\beta x)\,dx$$</p>
<p>Using Euler's Formula, $\cos(\beta x)=\text{Re}(e^{i\beta x})$. Then, we can write </p>
<p>$$\begin{align}
I(\alpha,\beta)&=\text{Re}\left(\int_0^\infty e^{-\alpha x^2}e^{i\beta x}\... |
1,980,012 | <p>I am facing issue to evaluate this particular type of definite integral,</p>
<p>$$I=\int_{0}^{\infty}\exp(-\alpha x^2)\cos(\beta x)dx$$</p>
<p>Please suggest a way to this.</p>
<p>Thanks</p>
| poweierstrass | 358,056 | <p>\begin{equation}
I = \int\limits_{0}^{\infty} \mathrm{e}^{-ax^{2}} \cos(bx) \mathrm{d}x
\tag{1}
\label{eq:1}
\end{equation}</p>
<p>Let
\begin{align}
I_{1} &= \int\limits_{0}^{\infty} \mathrm{e}^{-ax^{2}} \mathrm{e}^{ibx} \mathrm{d}x
= \int\limits_{0}^{\infty} \mathrm{e}^{-ax^{2}+ibx} \mathrm{d}x \\
&= \math... |
550,030 | <p>Hi I would like to prove this statement.</p>
<p>Show that there is no one-to-one correspondence from
the set of positive integers to the power set of the set of
positive integers.</p>
<p>[Hint: Assume that there is such a one-
to-one correspondence. Represent a subset of the set of
positive integers as an infinite b... | user85798 | 85,798 | <p>Let $S$ be the set of positive integers. Suppose there is a bijection $f : S \to P(S)$. Then <em>every</em> subset of $S$ is equal to $f(s)$ for some $s \in S$. For any $s\in S$, $f(s)$ is a subset of $S$, and it is certainly the case that either $s\in f(s)$ or $s\notin f(s)$. [For example, there exists $s_1$ such t... |
4,123,074 | <p>Let <span class="math-container">$\phi: X \rightarrow Y$</span> be a measurable function between measurable spaces. Let <span class="math-container">$\mu$</span> be a <span class="math-container">$\sigma$</span>- finite measure on <span class="math-container">$X$</span>. In general I have seen that the push forward ... | Ramiro | 190,563 | <p>Suppose</p>
<blockquote>
<p><span class="math-container">$\phi: X \rightarrow Y$</span> is a measurable function between measurable spaces, and <span class="math-container">$\mu$</span> is a <span class="math-container">$\sigma$</span>- finite measure on <span class="math-container">$X$</span>.</p>
</blockquote>
<p>... |
40,348 | <p>I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably simple, but it seems not so easy to find out whether one element is smaller than the other, let alone find the meet (... | Joel David Hamkins | 1,946 | <p>Since you seem at heart to be asking a question about how
to compute things with your partial order, let me offer
several observations from the perspective of computability
theory and computable model theory. I recognize, however,
that you may find this perspective unhelpful, and in this
case I apologize.</p>
<ul>
... |
51,895 | <p>I generated a spline function called <code>f</code> using the <code>BSplineFunction</code> with domain <code>{t, 0, 1}</code>.</p>
<pre><code>f = BSplineFunction[{{0., 5.81152}, {-0.909122, 5.73997}, {-1.86805, 5.5031},
{-1.79586, 5.52709}, {-2.40811, 5.28912}, {-3.4109, 4.70527}, {-4.68131, 3.501},
{-4.931... | Akku14 | 34,287 | <p>Define the spline for numerical values only and solve for x and y coordinate simultaneously.</p>
<pre><code>g[t_?NumericQ] = f[t]
fr = FindRoot[{First[k[0]], a} == g[t], {t, .7}, {a, 1}]
(* {t -> 0.674177, a -> 2.99958} *)
{f[t], k[0]} /. fr
(* {{-4.93131, 2.99958}, {-4.93131, 2.99958}} *)
</code><... |
154,726 | <p>The code is supposed to move the center of a circle of radius "d" incrementally <em>on</em> a function (the one in the first line), then solve for the two intercepts and repeat. I noticed that it gave two complex coordinates for every point as well (not really sure why, Id be glad if someone could explain), but I ig... | Lukas Lang | 36,508 | <p>It seems the general solution returned by <code>Solve</code> doesn't return all/any solutions when $y'(x)=0$. You can circumvent this by inserting the values first and solving afterwards, this ensures that <code>Solve</code> sees the special case and finds all solutions:</p>
<pre><code>y[x_] := (x Tan[Theta]) - ((1... |
996,432 | <p>The area of the unit <strong>circle</strong> is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as bounding the circumference by sequences of inscribed and outscribed polygons with successively higher ed... | Traklon | 103,134 | <p>You can mimic the way it is done in 2D, by randomly sampling points in the unit cube and calcultate the fraction that lie in the unit sphere inside. The theoretical result is $\dfrac{\frac43\pi}{8} = \dfrac{\pi}{6}$.</p>
<p>However, it don't see it being more efficient than the 2D version, because you need to make ... |
932,711 | <p>A simple question, but I like to be clinical with my choice of words:</p>
<p>I have a complex number, $z=-i$. </p>
<p>If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) \approx -\frac{π}{2}rad$.</p>
<p>I know that <em>something</em> is tending towards infinity. What is ... | Michael McGrady | 176,281 | <p>Four softwares with a bug: $\exists w \exists x \exists y \exists z (P(w) \wedge P(x) \wedge P(y) \wedge P(z) \wedge (\neg((w = x) \lor (w = y) \lor (w = z) \lor (x = y) \lor (x = z) \lor (y = z)))$</p>
|
3,557,252 | <p>By " same base, same exponent " rule I mean : </p>
<p><span class="math-container">$$b^x = b^n \iff x=n$$</span></p>
<p>Can the equation:
<span class="math-container">$$ 2^x+4^x = 6$$</span>
be solved by saying
<span class="math-container">$$ 2^x+4^x = 6 = 2+4= 2^1 + 4^1$$</span>
and consequently, that
<span c... | Taladris | 70,123 | <p>As a complement of the other answers:</p>
<p>consider the function <span class="math-container">$f(x)=a^x+b^x$</span> with <span class="math-container">$a>0$</span> and <span class="math-container">$b>0$</span>. Assume for simplicity that <span class="math-container">$a\neq 1$</span> and <span class="math-con... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Mariano Suárez-Álvarez | 1,409 | <p>A simple instance where infinity makes things simple is when proving that there exist transcendental numbers. Instead of having to come up with ways to tell a transcendental number from an algebraic one, you simply say «well, there are too many numbers for all of them to be algebraic, so there you have it».</p>
|
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Neel Krishnaswami | 1,610 | <p>One way of generating very simple examples of this kind is to define a family of predicates $F(n)$ by recursion over the natural numbers. Then, there can usually be no proof of $\forall n. F(n)$ in Peano arithmetic, since $F(n)$ won't usually be a schema in the sense the first-order Peano induction axiom requires -... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Michael Greinecker | 35,357 | <p>Every finite partial order has a maximal element. </p>
<p>One can prove that by induction and looking at cases, but I think the most natural way is to argue that if there is no maximal element, one must have a strictly increasing infinite sequence, which is impossible.</p>
|
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Andreas Blass | 6,794 | <p>Hindman's theorem is an example where (slightly?) higher infinity simplifies a proof greatly. The theorem asserts that, if you partition the set of positive integers into finitely many subsets, then there is an infinite set $H$ of positive integers such that all sums of finitely many members of $H$ lie in the same ... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Larry Rolen | 7,998 | <p>Using Eilenberg-Maclane spaces (namely $K(\mathbb{Z},2)=\mathbb{P}^{\infty}(\mathbb{C})$) and cellular approximation,one can show that any 3-fold with a positive Betti number has a map to the sphere $S^2$ which is not nullhomotopic. I am not sure how "down to earth" this example is, but it shows that even when study... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Mikhail Katz | 28,128 | <p>I once had a teaching assistant in calculus who admitted that he was unable to give an epsilon, delta proof that a the Heaviside stepfunction was not continuous. To take a slightly less trivial example, the proof that $y=x^2$ is not uniformly continuous using $\epsilon, \delta$ is somewhat involved for a freshman. ... |
239,004 | <p>From some geometric computations I get a lot of 3 dimensional points. From theory I know two facts: The points define a convex polyhedron, and this polyhedron has non-regular 3-, 4- and 5-gons as faces.</p>
<p>Here is a simple example:</p>
<pre><code>data = {
{0.21737620787507367`, 0.28204133994445385`, -0.638196... | Daniel Huber | 46,318 | <p>Maybe this will do what you want.</p>
<p>First we create use <code>ConvexHullMesh</code>to create a <code>BoundaryMeshRegion</code> from your <code>data</code>. From this we then extract the 2-dim primitives, that is the triangles.</p>
<pre><code>chm = ConvexHullMesh[data];
triangles = MeshPrimitives[chm, 2];
</code... |
239,004 | <p>From some geometric computations I get a lot of 3 dimensional points. From theory I know two facts: The points define a convex polyhedron, and this polyhedron has non-regular 3-, 4- and 5-gons as faces.</p>
<p>Here is a simple example:</p>
<pre><code>data = {
{0.21737620787507367`, 0.28204133994445385`, -0.638196... | kglr | 125 | <p>This works in version 12.2.0:</p>
<pre><code>MeshCellCount[ConvexHullMesh[data], 2]
</code></pre>
<blockquote>
<pre><code>30
</code></pre>
</blockquote>
<pre><code>ConvexHullMesh[data, MeshCellStyle -> { 1 -> Red}]
</code></pre>
<p><a href="https://i.stack.imgur.com/8eMv4.png" rel="nofollow noreferrer"><img... |
2,138,165 | <blockquote>
<p>Let <span class="math-container">$a,b,c,d$</span> be positive real numbers such that <span class="math-container">$a+b+c+d=1$</span>, then prove that <span class="math-container">$$S=\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}+\sqrt{4d+1} < 6$$</span></p>
<p>Source : <a href="http://imomath.com/pcpdf/f1/f41.... | Adren | 405,819 | <p>1) because it helps to prove the desired discontinuity</p>
<p>2) because both sequences are constant ones !</p>
|
2,308,060 | <p>For example I have $f(x)=x+1$ and $g(x)=x^2+2x+1$:</p>
<p>I don't have any problem when I calculate $g(x)/f(x)$, but I tried to calculate $f(x)/g(x)$ and I couldn't find the way to solve it.</p>
<ol>
<li><p>How should I do the polynomial long division for that situation?</p></li>
<li><p>If I have an expression $\f... | peterwhy | 89,922 | <p>Since $f(x)$ has a lower degree, the quotient will be $0$ and the remainder will be $f(x)$.</p>
<p>$$f(x) = 0g(x) + (x+1)$$</p>
<p>There is no way that the quotient $q(x)$ be something non-zero (constant or higher degree), that would make $q(x)g(x)$ something with degree 1 or less.</p>
|
357,501 | <p>This question is from Complex Variables and Applications by Brown & Churchill, 8ed. Section 62, #2. </p>
<p>Determine the Laurent Expansion of:</p>
<p>$$\frac{\exp(z)}{(z+1)^2}$$ </p>
<p>for the interval, $ 0 < |z + 1| < \infty$. How do I go about doing that? </p>
<p>Here's what I have so far. If I sub... | Mhenni Benghorbal | 35,472 | <p>Related problems <a href="https://math.stackexchange.com/questions/345139/how-to-construct-this-laurent-series/345160#345160">(I)</a>, <a href="https://math.stackexchange.com/questions/368734/find-the-laurent-series-at-the-given-point-and-its-residue/368862#368862">(II)</a> .</p>
<p><strong>Hint:</strong></p>
<p>... |
357,501 | <p>This question is from Complex Variables and Applications by Brown & Churchill, 8ed. Section 62, #2. </p>
<p>Determine the Laurent Expansion of:</p>
<p>$$\frac{\exp(z)}{(z+1)^2}$$ </p>
<p>for the interval, $ 0 < |z + 1| < \infty$. How do I go about doing that? </p>
<p>Here's what I have so far. If I sub... | M. Strochyk | 40,362 | <p>$$\frac{e^z}{(z+1)^2}=\frac{e^{z+1}}{e(z+1)^2}=\frac{1}{e(z+1)^2}\sum\limits_{n=0}^{\infty}\frac{(z+1)^n}{n!}=\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{(z+1)^{n-2}}{n!}$$</p>
|
2,662,509 | <p>The differential equation is:</p>
<p>${dP\over dT}=kP\Bigl(1-{P\over M}\Bigr)-\alpha P$</p>
<p>where $k > 0, \space M > 0$ and $\alpha > 0$ are constants.</p>
<p>How would I start an approach to solving this differential equation?</p>
| N. S. | 9,176 | <p>When you calculate P( there is a second game and Bo wins the first game in second round but lost the second game in second round), your calculation is multiplying the probability of there being a second round and the probability of Bo beating and then loosing to Al.</p>
<p>This is wrong, as the probability of there... |
414,642 | <p><strong>Background: Evaluation of</strong> <span class="math-container">$\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$</span></p>
<p>We can prove using the Beta-Function identity that</p>
<p><span class="math-container">$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac{1}{2} \ri... | Random Variable | 16,033 | <p>Another approach for evaluating the second integral using contour integration that avoids having to deform the contour around branch cuts is to consider $$ \displaystyle f(z) = \frac{\log(z+ e^{i \pi /4})}{(1+z^{2})^{2}}$$ and integrate around a contour that consists of the line segment $[-R,R]$ and the upper half o... |
2,714,660 | <p>I've got a rather elemental doubt about what "mapping a subspace homeomorphically onto its image" means, but I'd like to be sure what is being implied.</p>
<p>Let $f:X\to Y$ a continuous injective map between topological spaces. Let $U\subseteq X$ be an open subspace. Then we could say in some context that $f$ maps... | King Tut | 520,782 | <p>For a simple check, just differentiate it and try to integrate. That should give you a satisfactory answer:</p>
<p>$$\int_{-1}^{1} \frac{\mathrm d}{\mathrm dx}\arctan\left(\tfrac{1}{x}\right)\,\mathrm dx = -\int_{-1}^{1} \frac{1}{1+x^2}\,\mathrm dx = -\frac{\pi}{2}$$</p>
<p>I think <em>issue</em> is as pointed in ... |
1,868,458 | <p>Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$</p>
<p>Now let us define the Gaussian function with variance $\sigma>0$, $$p_\sigma:x \mapsto \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{x^2}{2... | Hans Engler | 9,787 | <p>Your guess in not correct. WLOG assume $x_0 = 0$. Now consider two functions $f, g$ as in your assumptions such that $f(0) = g(0)$, but $f \ge g$ everywhere and $f(x) = 1+g(x)$ on $[1,2]$. Then for all $\sigma$
$$
f\ast p_\sigma(0) - g \ast p_\sigma(0)= (f-g) \ast p_\sigma(0) \ge \int_1^2 p_\sigma(y) dy > 0 \, . ... |
540,636 | <p>I need to find the subgroup $H \leq A_4$ generated by $(123).$ I know that this subgroup will have order 1,2,3,4,6, or 12. </p>
| amWhy | 9,003 | <p>Hint: The order of a cycle is equal to its length. If the length of a cycle $\sigma$ is $k$, then the group it generates is given by $\{\sigma, \sigma^2, \ldots, \sigma^k\}$.</p>
|
1,225,277 | <p>I tried to prove that any subfield of $\mathbb{C}$ must contain every rational number by contradiction. </p>
<p>Proof:</p>
<p>Let $\mathbb{F}$ be any subfield of $\mathbb{C}$. Thus, $\mathbb{F}$ is itself a field under the usual operations of addition and multiplication of complex numbers. For $\mathb... | Crostul | 160,300 | <p>Your last argument is not correct: why do you say "Then $\mathbb{F}$ does not satisfy closure under addition and multiplication"? This argument is not explained.</p>
<p>What you should do is the following:</p>
<p>$1 \in\mathbb{F}$ so all elements of the form $1+ \cdots +1$ belong to $\mathbb{F}$. This means that $... |
1,825,016 | <p>I am trying to prove that $(p\rightarrow q)\wedge (p\rightarrow r) = p\rightarrow (q \wedge r)$. </p>
<p>This is my approach:</p>
<p>$(p\rightarrow q)\wedge(p\rightarrow r) = (-p \vee q) \wedge (-p \vee r)$</p>
<p>= ${[(-p \vee q) \wedge -p] \vee [(-p \vee q) \wedge r]}$</p>
<p>= ${[(-p \wedge -p) \vee (q \wedge... | Clive Newstead | 19,542 | <p>Unless the connective $\to$ is defined in terms of $\vee$ and $\neg$, there is no need to go about this proof using the equivalence $p \to q \equiv \neg p \vee q$ or using de Morgan's laws or any of that jazz. You can do it directly.</p>
<ul>
<li><p>($\to$) Assume $(p \to q) \wedge (p \to r)$. We'll prove $p \to (q... |
3,372,648 | <p><img src="https://i.stack.imgur.com/MRWJq.png" alt="image "></p>
<p>one could use linear approximation <span class="math-container">$g_{ab}=\eta_{ab}+h_{ab}$</span> to get linear ricci flow equation (2). How to do that? is any process ?
I am studying general relativity , i just use the linear approximation : <span... | Quarto Bendir | 680,680 | <p>You are correct, the paper is mistaken. In fact it's well known that the linearization of the Ricci flow is not strongly parabolic, so it is impossible that the linearization should be the standard heat equation.</p>
<p>It looks like that article ("Modified Ricci flow and asymptotically non-flat spaces") also has s... |
658,314 | <p>I was working on this question to find the following function's maximum value.Let $$y=f(x)={{(\sqrt{-3+4x-x^2}+4)}}^2 + (x-5)^2$$ where $$1 \le x \le 3$$.I have to find it's maximum value.
I tried by simple method of taking $$\frac{dy}{dx}=0$$ and got the answer.But this path was pretty lenghty and tiresome.Is there... | 5xum | 112,884 | <p>Since any method you use will end with you having solved the task, do not expect any easier way. After all, in the maximum value (if it is not achieved on the edges), the derivative IS zero, so whatever method you use, it will result in the value of $x$ where the derivative is zero.</p>
<p>Also, the method isn't th... |
3,617,619 | <p>Let <span class="math-container">$\textbf{P}_{2}(\textbf{R})$</span> be the real vector space of polynomials of degree less than or equal to <span class="math-container">$2$</span>.</p>
<p>Let <span class="math-container">$f:\textbf{P}_{2}(\textbf{R})\rightarrow \textbf{P}_{2}(\textbf{R})$</span> be the linear map ... | user0102 | 322,814 | <p>Since we are dealing with a linear operator, we can assume the same basis for the domain and the counterdomain.</p>
<p>Given the basis <span class="math-container">$\mathcal{B} = \{1,x+1,x^{2}+x\}$</span>, we have the following system of equations to solve
<span class="math-container">\begin{align*}
\begin{cases}
a... |
886 | <p>Okay so <a href="https://math.stackexchange.com/questions/835/optimal-strategy-for-deal-or-no-deal">this question</a> reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon <a href="http://en.wikipedia.org/wiki/Deal_or_no_deal" rel="noreferrer">Deal or no deal</a>.... | Community | -1 | <p>22 boxes. 2 at the final stage.</p>
<p>-Assume you will definately swap.
If you've picked 250k, which is 1/22 chance, you lose.
If you've not picked 250k, which is 21/22 chance, you win.
Every 22 times you play, you would expect to pick 250k once and expect to have 250k in the last 2 boxes twice (22 x 2 = 44).
Of t... |
4,088,439 | <p>First of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad.
<a href="https://math.stackexchange.com/questions/1896238/how-to-calculate-curvature-of-earth-per-surface-kilometer">How to calculate curvature of Earth per surface kilo... | lonza leggiera | 632,373 | <p>You're correct to suggest there is no answer of the form <span class="math-container">$x$</span> centimetres <em>per kilometre</em>. The formula <span class="math-container">$\ d=ks^2\ $</span>, where <span class="math-container">$\ d\ $</span> is the "drop" in centimetres, <span class="math-container">$\ ... |
4,529,275 | <p>Can we find a more <em>elegant</em> or <em>special</em> proof of <span class="math-container">$\frac {n!}{a^n}\to \infty, a>0$</span>?</p>
<p>For example I suggest this:</p>
<p><span class="math-container">$\forall a>0\forall n\in \Bbb N:\frac {n!}{a^n}=\int _{0}^{\infty}(\frac xa)^n e^{-x}dx=\int _0^a (\frac ... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$$m<a<m+1, m<n.$$</span>
Then <span class="math-container">$$\frac{a^n}{n!}=\frac{(a\cdot a\cdot a\cdot a\cdot a\cdot \ldots\cdot a)(a\cdot a\cdot a\cdot a\cdot \ldots\cdot a)}{[(1\cdot 2\cdot 3\cdot 4\cdot\ldots \cdot m)(m+1)(m+2)(m+3)\ldots n]}=\frac{a^m}{m!}\frac{a}{m+1}\... |
4,066,559 | <p>Question:</p>
<p>OA: y= 4x</p>
<p>AB: 3y+8x=400</p>
<p>OB: y=0</p>
<p>Find the Coordinates of A and B</p>
<p>I am really not sure how to do this problem.</p>
<p>I've tried using system of equations to see which numbers will have in common between O, A and B. But I was honestly guessing which method could work.
Could... | crankk | 202,579 | <p>Change the order of summation (overall there are only finitely many summands). For that note that</p>
<p><span class="math-container">\begin{align*}
M:&=\{ (i,j)\in\mathbb{N}^2~:~j=0,...,n~~i=j,...,n\}\\
&= \{(i,j)\in \mathbb{N}^2~:~i=0,...,n~~j=0,...,i\}.
\end{align*}</span></p>
<p>Therefore we can change ... |
2,864,538 | <p>Show that $$\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$$</p>
<p>I am studying for an exam and I bumped into this question. It's really bothering me because I really don't have any clue what to do. Does it have anything to do with the Cauchy condensation? Can somebody help me I would really appreciate it... | mengdie1982 | 560,634 | <h1>Solution</h1>
<p>Notice that
\begin{align*}
2^{2n}\sin^4\frac a{2^n}&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\sin^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\left(1-\cos^2\frac a{2^n}\right)\\&=2^{2n}\cdot\sin^2\frac a{2^n}-2^{2n}\cdot\sin^2\frac a{2^n}\cos^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\fra... |
52,607 | <p>If I am not mistaken there is a theorem that says any curve $C$ can be embedded in $\mathbf{P}^3$. What can be said about surfaces? Do we have a theorem like:</p>
<blockquote>
<p>All surfaces can be embedded in $\mathbf{P}^{N}$ for some fixed $N$. </p>
</blockquote>
<p>And if not what is the simplest counterexam... | J.C. Ottem | 3,996 | <p>All smooth projective surfaces can be embedded in $\mathbb{P}^5$ using a (linear) projection. This is a classical theorem in algebraic geometry. In general, it is known that any smooth projective variety of dimension $n$ can be embedded into $\mathbb{P}^{2n+1}$. </p>
<p>The proof runs as follows: The secant variety... |
1,397,400 | <p>Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product on $F_2^n$ in some papers, but I can only understand the general ideas. However, since it's possible that $v\cdot v =... | Sepideh Abadpour | 93,266 | <p>because the denumerator is $0$, the ambiguity should be of the type $\frac{0}{0}$ (because on any other case if the numerator is not zero the limit will be $\pm\infty$) so
$$\sqrt{a\times 0+b}-2=0\Rightarrow b=4$$<br>
now we can use hopital theorem:<br>
$$\lim_{x \to 0}\frac{\sqrt{ax+4}-2}{x}=\lim_{x\to 0}\frac{\fra... |
967,084 | <p>I am asked to find all equilibrium solutions to this system of differential equations:</p>
<p>$$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$</p>
<p>and to determine if they are stable, asymptotically stable or unstable.</p>
<p>I do not know how to proceed, I tried switching to polar coordinat... | Robert Lewis | 67,071 | <p><strong><em>This answer posted in response to the modified system, given in (1) and (2) below:</em></strong></p>
<p><strong><em>Oy Gevalt! NOW</em></strong> there is work!</p>
<p><strong><em>Where is Moshe now that we need him to lead us out from under Pharoah's toil?</em></strong></p>
<p>Well, I ain't no $Moshe... |
51,247 | <p>I have a matrix of 0's and 1's forming a number of disjoint paths:
<hr />
<img src="https://i.stack.imgur.com/pUlT7.jpg" alt="RandomPaths">
<hr />
I would like to find the lengths of the paths, and from that "spectrum,"
the longest length (in the above example: 27, starting at {1,14}).
(The shortest length po... | kglr | 125 | <p>Using a slightly modified version <a href="https://mathematica.stackexchange.com/a/50453/125">this answer</a> in a related Q/A:</p>
<pre><code>ClearAll[m, edges, v, vcs, g, cc, ap, ap2, pos];
m = Module[{i = 1}, mat /. 1 :> i++] (* where `mat` is the input matrix *);
v = ComponentMeasurements[m, "Label"][[All, 1... |
410,071 | <p>Let the function $f$ be increasing on the closed internal $[a,b]$. If $a \le f(a)$ and $f(b)\le b$, prove that:</p>
<blockquote>
<p>$\exists x_0\in [a,b]$, such that $f(x_0)=x_0$.</p>
</blockquote>
<p>Thanks for your help.</p>
<p><strong>Note that $f$ need not be continuous.</strong></p>
| Mher | 80,548 | <p>The solution of that question is almost the same as if we take that $f$ is countinious. By nested intervals' method we can choose such nested $[a_n,b_n]$, that
$$f(a_n)\geq a_n, f(b_n)\leq b_n.$$
Since the intervals are nested, there exists one $c\in[a_n,b_n]$ in all intervals.
Since
$$a_n\leq c \leq b_n,$$
then
$$a... |
4,031,286 | <p>I am doing a mathematical proof in which the answer from the solution is different from what I wrote. Is the one that I wrote also valid or is it wrong?<a href="https://i.stack.imgur.com/ITPRq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ITPRq.png" alt="enter image description here" /></a></p>
... | vertex | 797,931 | <p>Yes, your solution is correct. If you wanted to make it a little bit cleaner, you could do something similar to the following:</p>
<p>Let <span class="math-container">$y = x^2+5x-1,x\in\mathbb{Z}$</span></p>
<p><span class="math-container">$y$</span> is odd<span class="math-container">$\rightarrow y= $$ (2u+1) \lor ... |
3,151,564 | <p>I have been asked to calculate a general formula for <span class="math-container">$1 + b + 2b^2 + 3b^3 + ... + Nb^N$</span></p>
<p>I think that a part of this is a geometric sequence, and I have rewritten this as </p>
<p><span class="math-container">$f(x) = 1 + \sum_{i=1}^n i\cdot x^i$</span></p>
<p>(I couldn't f... | Henry | 6,460 | <p>Start with <span class="math-container">$$S=1 + b + 2b^2 + 3b^3 + \cdots + Nb^N$$</span></p>
<p>Multiply by <span class="math-container">$b$</span> and then subtract to give
<span class="math-container">$$bS=b + b^2 + 2b^3 + 3b^4 + \cdots +(N-1)b^N + Nb^{N+1}$$</span>
<span class="math-container">$$(b-1)S = -1 - b... |
4,457,926 | <p>I am practicing Calculus problems and the following one caught my attention:</p>
<p>Let <span class="math-container">$F:\mathbb{R}^3\to \mathbb{R}$</span> a <span class="math-container">$\cal{C}^1$</span> function, and let <span class="math-container">$P\in \mathbb{R}^3$</span> such that <span class="math-container"... | Umberto P. | 67,536 | <p>Apply the chain rule: since <span class="math-container">$F(x,y,z(x,y)) = 0$</span> you have
<span class="math-container">$$ F_1 + F_3 \frac{\partial z}{\partial x} = F_1 \frac{\partial x}{\partial x} + F_2 \frac{\partial y}{\partial x} + F_3 \frac{\partial z}{\partial x} = 0$$</span> where <span class="math-contain... |
316,046 | <p>What are some examples of (compact, say) manifolds <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> that are stably equivalent, i.e. <span class="math-container">$\Sigma^{\infty}X_+\simeq\Sigma^{\infty}Y_+$</span>, but are not homotopy equivalent?</p>
| mme | 40,804 | <p>Maybe it is worth adding some simply-connected examples.</p>
<p>Every simply connected closed 4-manifold may be described as <span class="math-container">$X = D^4 \cup_f (S^2 \vee \cdots \vee S^2)$</span>, where <span class="math-container">$f$</span> is a map <span class="math-container">$S^3 \to S^2 \vee \cdots \... |
2,027,453 | <p>I have a problem with an exercise.</p>
<p>The exercise is if given the usual topology on $\mathbb R.$</p>
<p>Suppose $ A $ is a closed subset of $\mathbb R$ containing all rational numbers.
Show that $A =\mathbb R.$</p>
<p>But I can't find a way to prove it</p>
| Bargabbiati | 352,078 | <p>$\mathbb Q$ is dense in $\mathbb R$, so $\forall B\subset\mathbb R \ open \ B\cap\mathbb Q\neq\emptyset$. Consider than $B=\mathbb R\smallsetminus A$...</p>
|
263,397 | <p>I'm having a trouble with this integral expression:</p>
<p>$$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$</p>
<p>I've done this substitution: $t= \tan(\theta/2)$</p>
<p>and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\fra... | Michael Hardy | 11,667 | <p>As $\theta$ goes from $0$ to $2\pi$, $\theta/2$ goes from $0$ to $\pi$, so $\tan(\theta/2)$ goes from $0$ up to $+\infty$ and then from $-\infty$ up to $0$. Add the two together and you get $\displaystyle\int_{-\infty}^\infty$.</p>
|
48,118 | <p>I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (<a href="https://doi.org/10.1007/BF02585431" rel="nofollow noreferrer">https://doi.org/10.1007/BF02585431</a>). I'm having a technical problem. Sorry for my ignorance, but I would like a referenc... | Jairo Bochi | 1,516 | <p>My "answer" is just a (still another) reformulation of Ian's.</p>
<p>Given two subspaces $V$, $W \subset \mathbb{R}^n$ with the same dimension, define their distance as:
$$
d(V,W):= \inf_J \| J - i_V\|
$$
where $i_V : V \to \mathbb{R}^n$ denotes the inclusion, $\|\mathord{\cdot}\|$ is the Euclidian operator norm, a... |
3,254,239 | <p>Given <span class="math-container">$f(x) = | \sin( | x | ) | $</span>, I am told to found the local and global minimum and maximum of <span class="math-container">$f$</span> (If they exist). </p>
<p>Simply from sketching the graph of <span class="math-container">$f$</span> I get that the function maximizes periodi... | FundThmCalculus | 153,550 | <p>Good insights to start off about the periodicity of the sine function.</p>
<p>Let's think about this in parts - dissecting the function from the outside inward.
Remember the definition of absolute value:
<span class="math-container">$$f(x) = | g(x) | = \left\{ \begin{array}{ll}
g(x) & g(x) \geq 0 \\
... |
77,527 | <p>I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.</p>
<p>Is there an infinite-dimensional Banach space (non-separable preferably) $E$ such that for some non-zero </p>
<p>$... | Uwe Stroinski | 17,879 | <p>For a given $\varepsilon>0$ the set never contains the operator $\varepsilon T$ unless $T=0$.</p>
|
1,195,337 | <p>If i suppose that $I$ is an interval but it is not connected,i.e there existe two open sets on $I$: $U,V$ such that $I=U\cup V$ et $U\cap V=\emptyset$ </p>
<p>Let $x\in U$ and $y\in V$ (for example $x<y$) and let $B_x=\{z\in U, [x,z]\subset U\}$ </p>
<p>Then $B_x$ is bounded by y .</p>
<p>My question is why $... | layman | 131,740 | <p>Suppose $B_{x}$ were <em>not</em> bounded by $y$, i.e., there is some $z$ such that $y \leq z$ and $[x,z] \subset U$. Then since $y \in [x,z]$, we have $y \in U$, which contradicts that $y \in V$ and $U \cap V = \emptyset$.</p>
|
1,650,371 | <p>What is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks</p>
| 5xum | 112,884 | <p>$f(x)=\sqrt{1-x^2}$ is a semicircle for $x\in [-1,1]$. Then simply shift this semicircle by $2$ to the right, and flip it over to get a second one, i.e.</p>
<p>$$f(x)=\begin{cases}\sqrt{1-x^2} & -1\leq x \leq 1\\
-\sqrt{1-(x-2)^2} & 1\leq x \leq 3\end{cases}$$</p>
|
2,496,950 | <p>This is probably a silly question for anyone "in the know", so please forgive me.</p>
<p>Consider the "group" given $G = \langle x,y : x^3=y^3=1,xy=yx^2\rangle$.</p>
<p>I got the elements to be $G = \{1,x,x^2,y,yx,yx^2,y^2,y^2x,y^2x^2\}$, meaning that $|G| = 9$.</p>
<p>I calculated the multiplication table (see b... | Angina Seng | 436,618 | <p>So $x=xy^3=yx^2y^2=y^2x^4y=y^3x^8=x^8$. As $x^3=1$ then $x=1$. The
relations then collapse to $y^3=1$, and $G$ has order $3$ generated by $y$.</p>
|
384,151 | <p>Let $ X$ be a Hilbert space and $(x_n)$ be a sequence in $ X$. If
$\|x_n\| \le 1$ and $x_n\to x$ weakly for some $x$ in $X$ with $\|x\| = 1$,
then show that $\|x_n- x\|^2 \to 0$.</p>
<hr>
<p>Here we just need to show that $\|x_n\| \to \|x\|$ using the fact $x_n\to x$ weakly for some $x$ in $X$ with $\|x\| = 1$, bu... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> Combine $\langle x_n,x\rangle\to\langle x,x\rangle=1$ and $\lvert\langle x_n,x\rangle\rvert\le\lVert x_n\rVert\cdot\lVert x\rVert=\lVert x_n\rVert$.</p>
<p>Then use $\lVert x_n-x\rVert^2=\lVert x_n\rVert^2-2\operatorname{Re}\langle x_n,x\rangle+\lVert x\rVert^2$.</p>
|
800,637 | <p>Buenos Dias, Ciao, Hello!</p>
<p>My fellow math stack users, I will try to solve this integral
$$
\int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx
$$
I did this $u=1-x$
$$
-\int_0^1 (u-1)u(u+1)e^{-u^2}\ln u \, du=\\
-\int_0^1 (u^2-u)(u+1)e^{-u^2}\ln u \, du=\\
-\int_0^1 (u^3-u)e^{-u^2}\ln u \, du=\\
-\int_0^1 u^3e^{-u... | Idris Addou | 192,045 | <p>The integral is of the form
\begin{equation*}
\int e^{-(1-x)^{2}}x(1-x)(2-x)\ln (1-x)dx=\int e^{g(x)}h(x)dx.
\end{equation*}
This form recalls the well-known formula
\begin{equation*}
\int e^{g(x)}\left( g^{\prime }(x)f(x)+f^{\prime }(x)\right)
dx=f(x)e^{g(x)}+C.
\end{equation*}
So we are done if we find a funct... |
3,338,312 | <p>In order to compute the derivative of the exponential function in an "early transcendentals" approach (where exponentials are defined not via power series or differential equations, but instead arithmetically via repeated multiplication, extended to reals), you first need to prove some fact like convexity of the exp... | Community | -1 | <p>Suppose the function <span class="math-container">$b^x$</span>, where <span class="math-container">$b>1$</span>, is <strong>not</strong> convex. Then there are <span class="math-container">$U=b^{x_1}$</span> and <span class="math-container">$V=b^{x_2}$</span>, with <span class="math-container">$x_1<x_2$</span>... |
81,173 | <blockquote>
<p>What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable first order theory with nice stability properties? </p>
</blockquote>
<p>Many standard examples, such as AC... | James Freitag | 6,789 | <p>I will refrain from discussing your general question, but since you specifically ask about differential fields, I can give the answer there. The theory of differentially closed fields of characteristic zero is decidable (lets say finitely many commuting derivations). There is in fact more that one can say. </p>
<p>... |
1,049,711 | <p>I'm having trouble finding a function to compare it to on [-1,0]...</p>
| Indika | 196,709 | <p>By the David Mitra's hint!</p>
<p>$$e^x\ge\dfrac{1}{e}\quad\text{ if}\quad x\ge-1$$ $$\int_{-1}^1\dfrac{e^x}{1+x}\ge\int_{-1}^1\dfrac{1}{e(1+x)}=\left.\dfrac{1}{e}\ln(1+x)\right|_{-1}^1$$ Clearly this is divergent.</p>
|
22,164 | <p>I have two vectors of matching lengths. They are readings from two different sensors (one is from a smartphone and the other is from a wiimote) of the same hand movement. I am trying to find the time offset between them to synchronise the readings for further processing. The readings I get are of the format (Time(ms... | Qiaochu Yuan | 232 | <p>Nope. Let $A = \mathbb{Z}_{(p)}, n = 1$. Then $\text{Spec } A[x]$ has a closed point given by the morphism $\phi : A[x] \to \mathbb{Q}$ sending $x$ to $\frac{1}{p}$, and this closed point maps to the generic point in $\text{Spec } A$, which is not closed. </p>
<p>$A$ is Noetherian and an integral domain, so neither... |
1,031,315 | <p>I don't know why $\sin x$ and $\cos x$ are lineary independent since if we take linear combination $a\cdot \sin x + b \cdot \cos x=0$ and for $a=\sqrt{3}$ and $b=1$ and $\displaystyle x=\frac{\pi}{6}$ we have that</p>
<p>$a\cdot \sin x + b \cdot \cos x=0$ and $a \neq 0 $ and $b \neq 0$ so it suggest they aren't ind... | Michael Hardy | 11,667 | <p>The problem is when you take $x=\pi/6$. It is true that the <em>values</em> of the function at $x=\pi/6$ are not linearly independent, and that's what you've shown here. They are in the space of all real numbers, and that is a one-dimensional space, so more than one element cannot be linearly independent.</p>
<p>... |
1,088 | <p>OK, so <a href="http://chat.stackexchange.com">chat</a> is now available... but; it has been suggested that for Mathematics we should have TeX support.</p>
<p>The current TeX processing has some non-trivial client impact. Before I even attempt trying to hack this in, is this something that the community would want ... | kahen | 1,269 | <p>With MathJax 2.x the way to load it dynamically has changed and <a href="http://docs.mathjax.org/en/latest/dynamic.html" rel="nofollow">the documentation</a> says that one should use a new method.</p>
<p>It seems to me that it ought to be possible to turn this into a <a href="https://en.wikipedia.org/wiki/Greasemon... |
2,587,395 | <blockquote>
<p>Find $\lim_{x \to 0} \frac{x^2e^{x^4}-\sin(x^2)}{1-\cos(x^3)}$</p>
</blockquote>
<p>By taylor polynomials we get: $e^{x^4}=1+x^4+\frac{x^8}{2}+\mathcal{O}(x^{12})$</p>
<p>$\sin(x^2)=x^2-\frac{x^6}{6}+\mathcal{O}(x^{10})$</p>
<p>$\cos(x^3)=1-\frac{x^6}{2}+\mathcal{O}(x^{12})$</p>
<p>so putting thes... | user | 505,767 | <p>From here you with some adjustment conclude:</p>
<p>$$...=\frac{\frac{7}{6}x^6+\frac{1}{2}x^{10}-\mathcal{O}(x^{10})+\mathcal{O}({x^{12}})}{\frac{1}{2}x^6-\mathcal{O}(x^{12})}=\frac{\frac{7}{6}x^6++\mathcal{O}(x^{10})}{\frac{1}{2}x^6+\mathcal{O}(x^{10})}=\frac73+\mathcal{O}(x^{4})\to\frac73$$</p>
|
3,646,072 | <p>I heard one can prove that GCH implies AC(the axiom of choice) in ZF. But I am confused with the meaning of GCH in ZF. Under AC, we can define cardinal exponentiation, so the cardinal 2 to the aleph is well defined. However, Without AC, How can we restate the GCH?</p>
| bof | 111,012 | <p>In Sierpiński's book <em>Cardinal and Ordinal Numbers</em>, second edition revised, 1965, on p. 88 he defines it as follows:</p>
<blockquote>
<p>The assumption that, no matter what an infinite set <span class="math-container">$A$</span> is like, there is no set which would be of greater power than <span class="ma... |
3,758,671 | <p>Let <span class="math-container">$p(\theta)$</span> be a real trigonometric polynomial of degree <span class="math-container">$N>0$</span>
<span class="math-container">$$
p(\theta) = \sum_{j=-N}^N a_{|j|}e^{\text ij\theta}
$$</span>
and for any <span class="math-container">$0\le s<N$</span> define the left tru... | Ivo Terek | 118,056 | <p>It's true. Take an orthonormal basis <span class="math-container">$(e_i)$</span> for <span class="math-container">$V$</span>. Then <span class="math-container">$(Te_i)$</span> is an orthonormal set, hence a basis. By orthonormal expansion, we have <span class="math-container">$$v = \sum_i \langle v,e_i\rangle e_i$$<... |
3,758,671 | <p>Let <span class="math-container">$p(\theta)$</span> be a real trigonometric polynomial of degree <span class="math-container">$N>0$</span>
<span class="math-container">$$
p(\theta) = \sum_{j=-N}^N a_{|j|}e^{\text ij\theta}
$$</span>
and for any <span class="math-container">$0\le s<N$</span> define the left tru... | Hypernova | 620,983 | <p>I found answer.
The point is using that <span class="math-container">$\langle x,x\rangle=0$</span> implies <span class="math-container">$x=0$</span>.
Consider <span class="math-container">$||T(u+v)-T(u)-T(v)||^2$</span> then the condition directly gives <span class="math-container">$||T(u+v)-T(u)-T(v)||^2=||u+v-u-v|... |
2,867,489 | <p>I want to rigorously verify that
$$\overline{\left(\frac{z_1}{z_2}\right)}={\left(\frac{\overline{z_1}}{\overline{z_2}}\right)}$$</p>
<p>Here's my attempt at thinking:</p>
<p>$$\overline{\left(\frac{z_1}{z_2}\right)}=\overline{\left({z_1}\cdot \frac{1}{z_2}\right)}={\left(\overline{z_1}\cdot \overline{\left(\frac{... | Quiver | 564,698 | <p>By multiplying numerator and denominator by $\bar{z}_2$, noting that $|z|^2=z\bar{z}$ we get
$$\left(\frac{z_1}{z_2}\right)=\left(\frac{z_1\bar{z}_2}{|z_2|^2}\right)$$ taking the conjugate on both sides $$\overline{\left(\frac{z_1}{z_2}\right)}=\overline{\left(\frac{z_1\bar{z}_2}{|z_2|^2}\right)} = \frac{\overline{z... |
3,623,803 | <p>I was reading a proof of the following proposition and had one small doubt about the proof:</p>
<blockquote>
<p><strong>Proposition:</strong> Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G,$</span> and <span class="math-container">$\mathcal{A}$</span> a <span class="m... | Cardioid_Ass_22 | 631,681 | <p>The goal seems to be to prove, for arbitrary <span class="math-container">$0\leq i< l$</span>, that the quotient <span class="math-container">$(A_{i+1}H/H)/(A_iH/H)$</span> is isomorphic to a quotient of <span class="math-container">$A_{i+1}/A_i$</span> (because that is the form of the factors of <span class="mat... |
263,727 | <p>Is there any relation between the limiting behaviour of $\Gamma({\epsilon})$ and $\Gamma(-1+{\epsilon})$?
I have seen the relation such as $\Gamma(-1+{\epsilon})$ $=$ $\Gamma({\epsilon})/(-1+{\epsilon})$. I think it is basically wrong? But does there exist such a similar relation?</p>
| akkkk | 28,826 | <p>Your relation /would/ hold <em>if</em> $\Gamma$ were continuous at $-1$. It is not, however: intuitively, we cannot take the factorial of negative integers.</p>
|
263,727 | <p>Is there any relation between the limiting behaviour of $\Gamma({\epsilon})$ and $\Gamma(-1+{\epsilon})$?
I have seen the relation such as $\Gamma(-1+{\epsilon})$ $=$ $\Gamma({\epsilon})/(-1+{\epsilon})$. I think it is basically wrong? But does there exist such a similar relation?</p>
| Old John | 32,441 | <p>The relation $\Gamma(-1+\epsilon) = \Gamma({\epsilon})/(-1+{\epsilon})$ is true so long as $\epsilon$ is not a negative integer (so that $-1+\epsilon$ will then also not be a negative integer) since the gamma function is extended to the complex plane minus the negative integers by using the relation $\Gamma(z)=\Gamm... |
18,753 | <p>Consider a compact manifold <em>M</em>. For a vector field <em>X</em> on <em>M</em>, let $\phi_X$ denote the diffeomorphism of <em>M</em> given by the time 1 flow of <em>X</em>.</p>
<p>If <em>X</em> and <em>Y</em> are two vector fields, is $\phi_X \circ \phi_Y$ necessarily of the form $\phi_Z$ for some vector field... | Theo Johnson-Freyd | 78 | <p>The BCH formula holds in a finite-dimensional Lie algebra for $X,Y$ in some open neighborhood of $0$. Therefore, the coefficients of BCH are bounded by $a^n$ for some $0&lt;a&lt;\infty$. So suppose that $X,Y$ are vector fields for which degree-$n$ bracket monomials (things like $[X,[X,Y]]$, with $n$ many $... |
693,761 | <p>Prove that the maximum connected components in a graph is $V+1-\left\lceil\frac{1+\sqrt{1+8E}}{2}\right\rceil$</p>
<p>I came up with this intuitively but was not able to prove it. The idea is that a graph with the maximum number of connected components would have individual nodes and 1 clique connected with at most... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> You may assume that each connected component is a complete graph. This only decreases your bound, hence it is still a maximum.</p>
<p><strong>Edit:</strong> Observe that if $E^* \geq E$, then $\lceil \frac{ 1 + \sqrt{1+8E^*}}{2} \rceil \geq \lceil \frac{ 1 + \sqrt{1+8E}}{2} \rceil$. Hence, if... |
17,257 | <p>What's the current state of Yang–Mills mass gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician? Thanks!</p>
| Q.Q.J. | 3,623 | <p>There's some guidelines about open problems in the FAQ that you might want to read, but there was a good article by Faddeev last November that you should know about:</p>
<p><a href="http://arxiv.org/abs/0911.1013">Mass in Quantum Yang-Mills Theory, by Faddeev</a></p>
|
17,257 | <p>What's the current state of Yang–Mills mass gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician? Thanks!</p>
| Tuna Yildirim | 56,336 | <p>For the 3D version of the proble, see</p>
<p>[1] Dimitra Karabali and V. P. Nair, Int. J. Mod. Phys. A 12, 1161 (1997)
<a href="http://arxiv.org/abs/hep-th/9610002" rel="nofollow noreferrer">http://arxiv.org/abs/hep-th/9610002</a></p>
<p>See eqn (20), (21) and the following paragraph.</p>
|
17,257 | <p>What's the current state of Yang–Mills mass gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician? Thanks!</p>
| Jon | 19,520 | <p>There is a rather strange situation here because a russian mathematician, Alexander Dynin, is claiming to have solved it. Please, refer to <a href="https://mathoverflow.net/questions/176029/some-explanation-about-dynins-formalism">this question</a> of mine or <a href="http://arxiv.org/abs/1005.3779" rel="nofollow no... |
1,585,960 | <p>How to prove the following?
$$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right)
$$
I obtained the right side with CAS. It seems like this function has many poles on the imaginary axis so a simple contour integral cannot be used. I also tried the following
... | Abdulhafeez Ayinde Abdulsalam | 527,979 | <p>This direct proof was discovered by Abdulhafeez Ayinde Abdulsalam (see <a href="https://arxiv.org/abs/2203.02675" rel="nofollow noreferrer">https://arxiv.org/abs/2203.02675</a>).</p>
<p>Let <span class="math-container">$\displaystyle \Delta\left(a\right) = \int_0^{\infty} \frac{\log\left(x^2 + a^2\right)}{\cosh\lef... |
1,830,594 | <p>What is the method for computing the inverse of this function?</p>
<p>I'm interested in asymptotic approximation. We can consider that $n \in \mathbb{R}$.</p>
<p>Thank you very much.</p>
| Dmytro Taranovsky | 384,034 | <p>I think the problem used "reasonable" to mean "somewhat reasonably close to optimal". I will give a more complete answer than what is probably expected in the problem, with an asymptotically optimal answer (up to a multiplicative error that approaches 1 in the limit) under different assumptions.... |
83,278 | <p>I am used to the following definition of a (proper) face of a polytope:</p>
<p>A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ imply $x, y \in F$. $F$ is a proper face of $C$ if $F$ is a face of $C$ and $F \neq C$... | Robert Israel | 8,508 | <p>In an infinite-dimensional real topological vector space $V$ with a discontinuous linear functional $f$, you could take $C$ to be the half-space $\{x \in V: f(x) \ge 0\}$, and $F = \{x \in V: f(x) = 0\}$ is a proper face according to the first definition but not according to the second.</p>
|
83,278 | <p>I am used to the following definition of a (proper) face of a polytope:</p>
<p>A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ imply $x, y \in F$. $F$ is a proper face of $C$ if $F$ is a face of $C$ and $F \neq C$... | Guillaume | 240,154 | <p>The general answer for convex sets is: no.</p>
<p>Your first definition corresponds to what is indeed called the face of a convex set.</p>
<p>The second definition (the one with hyperplanes) defines what is called the exposed faces of a convex set. As its name suggests, any exposed face is a face.</p>
<p>The fact... |
1,874,363 | <p>For example, how would you respond to this question?</p>
<blockquote>
<p>The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4, find an approximate value for the range.</p>
</blockquote>
| Mark Viola | 218,419 | <p>Note that $\max(x_1,x_2,\dots,x_n)=\max(x_1,\max(x_2,\dots,x_n))$. In addition, note that</p>
<p>$$\begin{align}
\int_0^1\int_0^1\max(x,y)\,dx\,dy&=\int_0^1\left(\int_0^y y\,dx+\int_y^1 x\,dx\right)\,dy\\\\
&=\int_0^1 \left(\frac12+\frac12 y^2\right)\,dy
\end{align}$$</p>
<p>Now, we can write</p>
<p>$$\b... |
137,306 | <p>Suppose I have a list that looks like the following (Town names and total distance to that town)</p>
<pre><code>resupply = {
{"Coleman", 0},
{"Highwood House", 106},
{"Canmore", 229},
{"Exshaw", 245},
{"Ghost Station", 288},
{"MountainAire", 370},
{"Nordegg", 552},
{"Robb", 672},
{"Hinton", 720}
... | ubpdqn | 1,997 | <pre><code>{place, dist} = Transpose[resupply];
With[{n = Length@resupply},
TableForm[
Partition[Abs[#1 - #2] & @@@ (dist[[#]] & /@ Tuples[Range[n], 2]),
n], TableHeadings -> {place, place}]]
</code></pre>
<p><a href="https://i.stack.imgur.com/6FIdr.png" rel="noreferrer"><img src="https://i.stack.im... |
2,005,847 | <blockquote>
<p>Consider the following figure (the length of the sides in the square is 1):</p>
<p><a href="https://i.stack.imgur.com/WLykc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WLykc.png" alt="enter image description here"></a></p>
<p>The unit square is transformed into the rec... | Rodrigo Dias | 375,952 | <p>Take the standard basis $\{e_1,e_2\}$ for $\Bbb{R}^2$. First, you have to rotate it by an angle of $\theta=\frac{3\pi}{4}\text{rad}$ (why?). So, you're mapping $e_1$ into $v_1$ and $e_2$ into $v_2$.</p>
<p>Then, you have to map $v_1$ into $w_1=\sqrt{2} v_1$ and $v_2$ into $w_2=2\sqrt{2}v_2$ (why?). And then, you're... |
251,301 | <p>$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$</p>
<p>Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$</p>
<p>I took the derivatives of f(x):
$$f(x) = 6/x$$
$$f'(x) = -6/x^2$$
$$ f''(x) = 12/x^3 $$
$$ f^{(3)}(x) = -36/x^4 $$
but I am unable to write out the series, please help! ... | Goonfiend | 17,338 | <p>Refer to the definition of Taylor Series. You will have to evaluate your derivatives at $a=-4$.</p>
|
251,301 | <p>$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$</p>
<p>Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$</p>
<p>I took the derivatives of f(x):
$$f(x) = 6/x$$
$$f'(x) = -6/x^2$$
$$ f''(x) = 12/x^3 $$
$$ f^{(3)}(x) = -36/x^4 $$
but I am unable to write out the series, please help! ... | 000 | 22,144 | <p>For a function $f$, the definition of the <em>Taylor Series</em> of $f$ is:
$$\text{Taylor}(f)=\sum_{n \ge 0}\frac{f^{(n)}(a)}{n!}(x-a)^n, $$
where $f^{(n)}$ indicates the $n$th derivative of $f$ with $f^{(0)}=f$.</p>
<p>The problem, then, reduces to the following: Is there an $n$th derivative of $f(x)=\frac{6}{x}$... |
835,806 | <p>Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1].</p>
<p>If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , $|f_n''(x)|\le 1$ then show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1].</p>
<p>D... | martini | 15,379 | <p>Yes, you can use Arzela-Ascoli's theorem. To prove boundedness of $B = \{f_n \mid n \in \mathbb N\}$ recall that
$$ f_n(x) = \int_0^x f_n'(\xi)\, d\xi = \int_0^x \int_0^\xi f_n''(\eta) \,d\eta\, d\xi $$
as $f_n(0) = f_n'(0)= 0$. For equicontinuity, prove that the $f_n$ are uniformly Lipschitz, using the mean value ... |
2,929,878 | <blockquote>
<p>Solve the Cauchy problem <span class="math-container">$$(x − y − 1)u_x + (y − x − u + 1)u_y = u,$$</span>
if <span class="math-container">$u=1$</span> on <span class="math-container">$x^2+(y+1)^2=1.$</span></p>
</blockquote>
<p><em>Attempt.</em> <span class="math-container">$$\frac{dx}{x-y-1}=\frac... | JJacquelin | 108,514 | <p><span class="math-container">$$(x − y − 1)u_x + (y − x − u + 1)u_y = u \tag 1$$</span>
<span class="math-container">$$\frac{dx}{x-y-1}=\frac{dy}{y-x-u+1}=\frac{du}{u}\quad\text{is correct}$$</span>
You rightly found a first characteristic equation :
<span class="math-container">$$x+y+u=c_1$$</span>
A second characte... |
1,824,046 | <p><a href="https://i.stack.imgur.com/AioYj.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AioYj.jpg" alt="enter image description here"></a></p>
<p>I'm pretty sure I haven't made any mistakes in part 1), but I have absolutely no idea how to do part 2). I don't even know what the question is asking... | mickep | 97,236 | <p>As mentioned in the comment, it should indeed be $2\,du=dx$.</p>
<p>For $x>2$ you can write the primitive as $\frac{1}{2}\text{arcsec}\,(x/2)+C$ (note the one-half, you probably lost the inner derivative. Try do differentiate your answer, and you'll see). For $x<-2$, there will typically be a problem with sig... |
4,320,591 | <p>Recently I came across a question <a href="https://gate.iitkgp.ac.in/documents/gatepapers/2016/ee_2016.pdf" rel="nofollow noreferrer">here</a> p.25 question 7. This is a question from the GATE EE 2016 exam. But I was having difficulty in analyzing the question as to whether it has a definite answer or not based on t... | Chickenmancer | 385,781 | <p>If <span class="math-container">$P$</span> is not the zero matrix, and as <code>zwim</code> noted, if <span class="math-container">$P^2 \ne P$</span> and <span class="math-container">$P^3=P,$</span> then we can use the Cayley-Hamilton theorem, since <span class="math-container">$P^3-P=0_{n\times n}$</span> this mean... |
2,835,660 | <p>Im practicing limits and integral transforms but I run into the following general problem </p>
<p>How to compute the following limit :</p>
<p>$$ \lim_{c \to a} \int_0^c f(x,c) \space dx $$</p>
<p>Ofcourse this reduces to a normal integral when the limit exists and $a$ is not $0$ nor $\infty$.
But when $a$ is one ... | tparker | 268,333 | <p>$x$ is just a dummy integration variable so the definite integral doesn't depend on $x$, and the limit doesn't make any sense.</p>
|
4,159,309 | <p>Let H is a subgroup of G. How we can show that <span class="math-container">$(1_{H^0})^G$</span> is reducible Brauer character of G. Where <span class="math-container">$1_{H^0}$</span> is principal Brauer character of H.</p>
| Jack Schmidt | 583 | <p>As Nate mentioned, this can be seen from Frobenius reciprocity since this module can be lifted to the integers. However, we can also directly find a nonzero, proper submodule.</p>
<p>Suppose <span class="math-container">$G$</span> is a finite group, <span class="math-container">$H$</span> is a subgroup of <span clas... |
44,200 | <p>Lets say I want to represent 20 Percent</p>
<p>What do you call/name the "0.2" and "20%" notations</p>
<p>I am calling them Fractional and WholeValue notations for the time being, but wondering if there was something better.</p>
| Zev Chonoles | 264 | <p>I would call "0.2" the decimal notation, and "20%" simply the percentage notation.</p>
|
2,987,238 | <p>I'm starting to study Hilbert Spaces for the very first time in my life and I had difficulty to understand one very simple proof:</p>
<p>Let <span class="math-container">$\{x_n:n=1,2,...\}$</span> be a sequence of vectors in the space; the sequence is said to converge to an element <span class="math-container">$x$<... | Bombadil | 611,234 | <p>In case you didn't notice I would remind you, result is direct consequence of the fact that linear functional <span class="math-container">$\phi(x) = \langle x, z \rangle$</span> defined for <span class="math-container">$x \in H$</span> is continuous. This of course follows from continuity of inner product which is ... |
3,638,741 | <p>Let <span class="math-container">$x_1, x_2, \ldots , x_n \in \{0, 1\}$</span>.</p>
<p>(a) Consider the equation <span class="math-container">$x_1 + x_2 + \ldots + x_n = 0 \pmod {2}$</span>. How many solutions does this equation have? Express your answer in terms of <span class="math-container">$n$</span>. For examp... | Sam | 584,704 | <p>If <span class="math-container">$x_1+\ldots +x_{10} = 0 \text{ mod}\ 2$</span>, then <span class="math-container">$x_{11}+\ldots+x_n = 0\text{ mod}\ 2$</span>, so each solution to the first equation can be paired to a solution of the second to obtain a solution to the original equation. Use that and the fact that th... |
42,814 | <h3><a href="http://en.wikipedia.org/wiki/Recursion#The_recursion_theorem" rel="noreferrer">The recursion theorem</a></h3>
<blockquote>
<p>In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set <span class="math-container">$X$</span>, <span class="math-container">$a \in X$</... | JDH | 413 | <p>The Recursion Theorem simply expresses the fact that definitions by recursion are mathematically valid, in other words, that we are indeed able correctly and successfully to define functions by recursion. Mathematicians implicitly use this fact whenever they define a function by recursion. </p>
<p>A more general ve... |
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