category stringclasses 107
values | title stringlengths 15 179 | question_link stringlengths 59 147 | question_body stringlengths 53 33.8k | answer_html stringlengths 0 28.8k | __index_level_0__ int64 0 1.58k |
|---|---|---|---|---|---|
analysis | Why can we assume that a piecewise continuous function has a maximum? | https://math.stackexchange.com/questions/2973787/why-can-we-assume-that-a-piecewise-continuous-function-has-a-maximum | <p>In a proof I found in a book regarding the ODE <span class="math-container">$\dot{x}(t) = A(t)x(t)$</span> where <span class="math-container">$A$</span> is piecewise continuous on a compact interval I, it is assumed that the maximum <span class="math-container">$K = \max_I ||A(t)||$</span> exists. In my opinion this... | <p>The reason for the discrepancy is that the book is
probably using a different definition of a piecewise continuous function which does not allow your example to hold.</p>
<p>Consider following definition of a piecewise
continuous function. Let <span class="math-container">$D_1, \ldots, D_m$</span> be
a finite par... | 600 |
analysis | Convergence in measure and convergence of norm implies convergence in L^p | https://math.stackexchange.com/questions/1162919/convergence-in-measure-and-convergence-of-norm-implies-convergence-in-lp | <p>I have been attacking this question but i got stuck. Please give me some hint.</p>
<p>Let $1\leq p<\infty$ and $\{f_n\}\subset L^p$. Suppose $f_n\to f$ in measure and $\|f_n\|_p\to\|f\|_p$, then $f_n\to f$ in $L^p$ norm.</p>
| <p>It's not easy. The key idea is to mimic the proof of Dominated Convergence Theorem. In order to do that, first establish the following inequality:
$$\lvert x+y\rvert^p\le\gamma_p(\lvert x\rvert^p+\lvert y\rvert)^p,\qquad\forall x,y\in\mathbb C$$
for some constant $\gamma_p$. Then apply <a href="http://en.wikipedia.o... | 601 |
analysis | Is $\mathbb{R}^{2}$ separable for the railmetric( or SNCF -metric or post-office metric)? | https://math.stackexchange.com/questions/1319739/is-mathbbr2-separable-for-the-railmetric-or-sncf-metric-or-post-office | <blockquote>
<p>It's the follow metric: <span class="math-container">$d(x,y)= ||x|| +||y||$</span> if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> don't lie on a line through the origin. And otherwise <span class="math-container">$d(x,y)= ||x-y||$</span>.</p>
<p>I think the answe... | <p>HINT: You're right: it's not. You can prove it by finding an uncountable family of pairwise disjoint, non-empty open sets. I've added a further hint in the spoiler-protected block below.</p>
<blockquote class="spoiler">
<p> Consider open rays leaving the origin.</p>
</blockquote>
| 602 |
analysis | A question on differentiability of a variant of Thomae function | https://math.stackexchange.com/questions/4178091/a-question-on-differentiability-of-a-variant-of-thomae-function | <p>Let <span class="math-container">$$f(x) =
\begin{cases}
\dfrac{1}{q^2}, & \text{if $x=\dfrac{p}{q} $ is rational and in lowest terms;} \\[2ex]
0, & \text{if $x$ is irrational}
\end{cases}$$</span>
Where is f continuous? Is f differentiable anywhere?</p>
<p>My attempt: I can prove that <span class="math-cont... | <p>Let's fix an irrational number <span class="math-container">$\gamma$</span> and a sequence of rationals <span class="math-container">$\frac{p_n}{q_n} \to \gamma$</span>. Due to <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem" rel="nofollow noreferrer">Dirichlet's theorem</a> we can also sa... | 603 |
analysis | Ask a step in Proof of L'Hospitals Rule | https://math.stackexchange.com/questions/4178423/ask-a-step-in-proof-of-lhospitals-rule | <p>Suppose that <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are differentiable functions on an open interval <span class="math-container">$I$</span> and that <span class="math-container">$p \in I$</span>. If <span class="math-container">$\lim_{x \to p} f(x) = \lim_{x \to p} g(x) ... | <p>You have <span class="math-container">$\lim_{x\to p}\big( \text{something}\big) = \ell.$</span></p>
<p>And you have <span class="math-container">$a>\ell.$</span></p>
<p>That implies that there is some open interval about <span class="math-container">$p$</span> for which, if <span class="math-container">$x$</span>... | 604 |
analysis | If $\{y_i\} \subset f([0,1])$ is increasing, there exists an monotonous $\{x_i\} \subset [0,1]$ such that $f(x_i)=y_i$? | https://math.stackexchange.com/questions/4181505/if-y-i-subset-f0-1-is-increasing-there-exists-an-monotonous-x-i | <p>Let <span class="math-container">$f \in C[0,1]$</span>. If <span class="math-container">$\{y_i\} \subset f([0,1])$</span> is increasing, there exists an monotonous <span class="math-container">$\{x_i\} \subset [0,1]$</span> such that <span class="math-container">$f(x_i)=y_i$</span>?</p>
<p>Intuitively, from the func... | 605 | |
analysis | Calculate $\lim _{x\to 0^+}\frac{\left(e^x-e^{-x}\right)}{\sin x}$ | https://math.stackexchange.com/questions/3982661/calculate-lim-x-to-0-frac-leftex-e-x-right-sin-x | <p>***How to find the this limit without using L'Hopital's Rule.</p>
<p><span class="math-container">$\lim _{x\to 0^+}\frac{\left(e^x-e^{-x}\right)}{\sin x}$</span></p>
| <p>Hint:</p>
<ul>
<li><p>Use maclaurin series for <span class="math-container">$e^x$</span> and <span class="math-container">$e^{-x}$</span></p>
</li>
<li><p>then apply <span class="math-container">$$\lim_{x\to 0} \dfrac{\sin x}{x}=1$$</span></p>
</li>
</ul>
| 606 |
analysis | Would it be possible to skip undergraduate college with self study/online courses | https://math.stackexchange.com/questions/4177654/would-it-be-possible-to-skip-undergraduate-college-with-self-study-online-course | <p>Hi I was wondering if it were possible to skip undergraduate and apply to a graduate school. I have been taking some math courses through CTY a somewhat accredited program offered by Johns Hopkins university. They offer up to real analysis 1 and complex analysis 1. If I take up to there, would it be possible to skip... | <p>If you plan to study further for Msc or Phd, you could and I think should just enroll in an integrated Bsc+Msc program. You can either</p>
<ul>
<li>request that you take an exam of the courses you feel like you have learned enough about without taking the course itself,</li>
<li>or take an otherwise extraordinary a... | 607 |
analysis | Why is $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n+1}\right)^{n+1}=e$ trivial? | https://math.stackexchange.com/questions/3115588/why-is-lim-limits-n-to-infty-left1-dfrac1n1-rightn1-e-trivial | <p><span class="math-container">$$\lim_{n\to\infty}\left(1+\dfrac{1}{n+1}\right) ^{n+1} =e$$</span></p>
<p>Could someone explain me why this is trivial ? Maybe subsitution such that <span class="math-container">$n+1=m$</span> which would give us the definition of <span class="math-container">$e$</span> ? </p>
| <p>Let
<span class="math-container">$$a_n = \left( 1 + \frac{1}{n}\right)^n$$</span></p>
<p>Since this sequence converges to <span class="math-container">$e$</span>, any subsequence of it also converges to <span class="math-container">$e$</span>.</p>
<p>Set <span class="math-container">$k_n=n+1$</span>. Then, the co... | 608 |
analysis | Resolution of a limit | https://math.stackexchange.com/questions/3705034/resolution-of-a-limit | <p>I want to formally prove that this limit is equal to <span class="math-container">$0$</span></p>
<p><span class="math-container">$\lim_{x \to 0} \frac{2}{x^{3}} e^{-\frac{1}{x^{2}}}$</span></p>
<p>can anybody help me?</p>
| <p>This is equivalent to</p>
<p><span class="math-container">$$\lim_{t\to\infty}t^{3/2}e^{-t}=0$$</span> as it is well-known that the exponential increases faster than any polynomial.</p>
| 609 |
analysis | A question on the Riemann integrable of Thomae function | https://math.stackexchange.com/questions/4189450/a-question-on-the-riemann-integrable-of-thomae-function | <p>The definition of Thomae function is
<span class="math-container">$$g(x) =
\begin{cases}
\dfrac{1}{q}, & \text{if $x=\dfrac{p}{q}$} \\[2ex]
0, & \text{if $x$ is irrational}
\end{cases}$$</span>
where <span class="math-container">$p\in \mathbb{Z}, q\in \mathbb{Z}^+$</span> and <span class="math-container">$\... | 610 | |
analysis | Explanation of composition of two onto functions? | https://math.stackexchange.com/questions/2913101/explanation-of-composition-of-two-onto-functions | <p>My book says that if functions $f$ and $g$ are both onto then $f\circ g$ and $g\circ f$ may or may not be onto. </p>
<p>Why is this so? Would someone please help me understand this, maybe with an example or diagrammatically? My book states that$ f\circ g$ and $g\circ f$ may or may not be onto. I think this might b... | <p>The composition of two surjective functions is always surjective: Let $f: X \to Y$ and $g: Y \to Z$ be functions and $z \in Z$. Then since g is surjective, there exists $y$ such that $g(y)=z$ and similarly there exists $x$ such that $f(x)=y$. Then $g(f(x))=z$ and your composition is surjective.</p>
<p>You can do th... | 611 |
analysis | Possible Properties of a Monotone Function | https://math.stackexchange.com/questions/2935065/possible-properties-of-a-monotone-function | <p>I'm working on the following question:</p>
<blockquote>
<p>Which of the following three properties imply <span class="math-container">$f$</span> is monotone (it maybe none or more than 1):</p>
<ol>
<li><span class="math-container">$f(x + y) = f(x)f(y)$</span></li>
<li><span class="math-container">$f(x - ... | <p>(1) does not imply monotonicity. Take <span class="math-container">$g$</span> to be a discontinuous (also non-integrable, non-monotone) solution to <a href="https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation" rel="nofollow noreferrer">Cauchy's Functional Equation</a>:
<span class="math-container">$$g(x + y... | 612 |
analysis | $\epsilon, \delta$ exercise | https://math.stackexchange.com/questions/2937282/epsilon-delta-exercise | <p>I'm working on the following problem:</p>
<blockquote>
<p>A real valued function <span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}$</span> has the property that <span class="math-container">$(\forall \epsilon>0)(\exists\delta>0)$</span> s.t. <span class="math-contai... | <p>Given <span class="math-container">$\epsilon >0$</span> there exists <span class="math-container">$\delta >0$</span> such that</p>
<p><span class="math-container">$$|x-1| \geq \delta \implies |f(x)-f(1)| \geq \epsilon.$$</span></p>
<p>In other words</p>
<p><span class="math-container">$$x\in (-\infty,1-\del... | 613 |
analysis | Cauchy Hadamard formula and starting index of power series | https://math.stackexchange.com/questions/2941938/cauchy-hadamard-formula-and-starting-index-of-power-series | <p>The radius of convergence <span class="math-container">$r$</span> can be calculated for every power series <span class="math-container">$\sum_{k=0}^\infty a_k z^k$</span> with <span class="math-container">$a_k\in \mathbb C$</span> and <span class="math-container">$z\in \mathbb C$</span> by using the Cauchy Hadamard ... | <p>Certainly. The two series converge for the same points <span class="math-container">$z$</span> and have the same radii of convergence. You can thing of the second as the first with <span class="math-container">$a_0=0$</span>. </p>
| 614 |
analysis | Uniform convergence of $\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$ | https://math.stackexchange.com/questions/2962462/uniform-convergence-of-sum-infty-n-1-x-e-nx-cosnx | <p>How can i prove uniform convergence on <span class="math-container">$E=[0, \frac{\pi}{2}]$</span> ?
<span class="math-container">$$\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$$</span></p>
| <p>Hint: For <span class="math-container">$N$</span> even, let</p>
<p><span class="math-container">$$T_N(x)=\sum_{n=N/2}^{N}xe^{-nx}\cos (nx).$$</span></p>
<p>If the convergence is uniform, then <span class="math-container">$T_N\to 0$</span> uniformly. Consider <span class="math-container">$T_N(1/N).$</span></p>
| 615 |
analysis | Countability of a set of subsequences . | https://math.stackexchange.com/questions/3035548/countability-of-a-set-of-subsequences | <p>Consider a sequence <span class="math-container">$x_n$</span> with positive values such that
<span class="math-container">$\sum _{n=1}^{\infty} x_n $</span> converges .
Is the set of the subsequences <span class="math-container">$ x_{k_n}$</span> of <span class="math-container">$x_n$</span> such that
<span class="... | 616 | |
analysis | Apostol - Mathematical Analysis - Theorem 1.1 | https://math.stackexchange.com/questions/3039044/apostol-mathematical-analysis-theorem-1-1 | <p>I realize there are several questions dealing with Theorem 1.1 from Apostol's "Mathematical Analysis", but those questions are mostly about how to prove the result or the geometrical intuition around it. My question is about the application of the theorem. The theorem states:</p>
<p>If <span class="math-container">... | <p>The premise says <strong>for all</strong> <span class="math-container">$\epsilon>0.$</span> You have only shown the inequality holds for <strong>one</strong> <span class="math-container">$\epsilon$</span> (namely <span class="math-container">$\epsilon=3$</span>). But it fails for <span class="math-container">$\ep... | 617 |
analysis | If at each point of a closed interval the $m$-th derivative of $f$ is $0$ for $m$ large enough, then $f$ is polynomial | https://math.stackexchange.com/questions/3042725/if-at-each-point-of-a-closed-interval-the-m-th-derivative-of-f-is-0-for-m | <p>I am working on exercice 9.5.2 of Analysis by Zorich and I am stuck at the question b.</p>
<blockquote>
<p>a) A set <span class="math-container">$E\subset X$</span> of a metric space <span class="math-container">$(X,d)$</span> is nowhere dense in X if it is not dense in any ball, that is, if for every ball <span cla... | <p>Here is a sketch of a possible approach, which may have some holes, but it is too long for a comment. Maybe someone can patch it up.Picking up on your idea, set </p>
<p><span class="math-container">$T = \{t\in [a,b]: \forall (c,d)\ni t: f\restriction_{(c,d)}$</span> is not a polynomial<span class="math-container">$... | 618 |
analysis | Proof that a sequence is Cauchy. | https://math.stackexchange.com/questions/3061670/proof-that-a-sequence-is-cauchy | <p>Show that <span class="math-container">$\left( x_{n}\right) $</span> is a Cauchy sequence, where
<span class="math-container">$$
x_{n}=\frac{\sin1}{2}+\frac{\sin2}{2^{2}}+\ldots+\frac{\sin n}{2^{n}}.
$$</span></p>
<p>We try to evaluate <span class="math-container">$\left\vert x_{n+p}-x_{n}\right\vert $</span> and... | <p>It is not sufficient that all <span class="math-container">$\left( \frac{1}{2^{n+j}}\right)$</span> converge to zero. A correct argument would be that
<span class="math-container">$$
\left\vert a_{n+p}-a_{n}\right\vert \le
\frac{1}{2^{n+1}}+\frac{1}{2^{n+2}}+\ldots+\frac{1}{2^{n+p}}
= \frac{1}{2^{n}} \left( \fr... | 619 |
analysis | Why is Completeness needed to demonstrate the Archimedean principle? | https://math.stackexchange.com/questions/3099533/why-is-completeness-needed-to-demonstrate-the-archimedean-principle | <p>Why is Completeness needed to demonstrate the Archimedean principle?</p>
<p>Could someone criticize the following proof.</p>
<p>Thanks in advance.</p>
<hr>
<p><strong>Proof</strong></p>
<p>Considering any <span class="math-container">$x \in \mathbb{R},$</span> <span class="math-container">$\lfloor{x}\rfloor \in... | <p>As for why completeness is needed? Here are two ordered fields that aren't complete, one with the Archimedean property and one without:</p>
<p><span class="math-container">$\mathbb{Q}$</span> is an ordered field with the property.</p>
<p>Construct an order on <span class="math-container">$\mathbb{R}(x)$</span> (th... | 620 |
analysis | $\frac{\int fg dx}{\int g dx}=f(0)$ | https://math.stackexchange.com/questions/3118986/frac-int-fg-dx-int-g-dx-f0 | <p>We already know that
<span class="math-container">$$\lim_{n \rightarrow +\infty} \int_{-1}^1 (1-x^2)^n dx = 0$$</span></p>
<p>If we have <span class="math-container">$f(x) \in C[-1,1]$</span> then prove
<span class="math-container">$$\lim_{n \rightarrow +\infty} \frac{\int_{-1}^1 f(x)(1-x^2)^n dx}{\int_{-1}^1 (1-x^... | <p>1) Observe that
<span class="math-container">$$
c_n := \int_{-1}^1 (1-x^2)^n\, dx \geq 2\int_0^{1/\sqrt{n}}(1-x^2)^n\, dx
\geq 2\int_0^{1/\sqrt{n}}(1- n x^2)\, dx = \frac{4}{3\sqrt{n}}\,.
$$</span></p>
<p>2) If <span class="math-container">$\delta\in (0,1)$</span>, then
<span class="math-container">$$
\frac{1}{c_n}... | 621 |
analysis | uniformly convergence of heat equation | https://math.stackexchange.com/questions/3155369/uniformly-convergence-of-heat-equation | <p><strong>Problem</strong></p>
<blockquote>
<p>Let <span class="math-container">$u(x,t) = \frac{e^\frac{-x^2}{4t}}{\sqrt{4 \pi t}} $</span> for <span class="math-container">$ t > 0,
> x \in \mathbb{R} $</span>. If a > 0, prove that <span class="math-container">$u(x,t) \rightarrow 0$</span> as <span class="mat... | <p>Showing that <span class="math-container">$u(a,t) \to 0$</span> is equivalent to showing that <span class="math-container">$u^{2}(a,t) \to 0$</span>. Make the substitution <span class="math-container">$s=\frac {a^{2}} {2t}$</span> and reduce to the proof to <span class="math-container">$se^{-s} \to 0$</span> as <spa... | 622 |
analysis | Shift of even periodic function | https://math.stackexchange.com/questions/3207241/shift-of-even-periodic-function | <p>When we shift <span class="math-container">$\sin(x)$</span> by <span class="math-container">$\pi$</span> (half the period) we get an odd function <span class="math-container">$-\sin(x)$</span>.
I was wondering if every periodic, even function can be made odd if we shift it by half it's period?</p>
<p>I guess this i... | <p>What you seem to be noticing is that <span class="math-container">$\sin(x)$</span> is odd, and
<span class="math-container">$\cos(x)$</span> is even and the second is a shifted version of the first by a quarter of a period. This generalizes. Suppose we have a function <span class="math-container">$f(x)$</span> defi... | 623 |
analysis | A problem from Zorich | https://math.stackexchange.com/questions/3230799/a-problem-from-zorich | <p>I am trying to read <em>Mathematical Analysis I</em> by Zorich on my own. Here is an exercise that I could not solve: </p>
<blockquote>
<p>For all <span class="math-container">$l \in \mathbb{R}$</span> that is not of the form <span class="math-container">$\frac{1}{n}$</span> for some <span class="math-container">... | 624 | |
analysis | Riemann integrability and lower and upper sums | https://math.stackexchange.com/questions/3275494/riemann-integrability-and-lower-and-upper-sums | <p>Let's say a function <span class="math-container">$f:[a,b] \to \mathbb{R}$</span> is bounded and Riemann integrable, then would there always exist a partition <span class="math-container">$P$</span> of <span class="math-container">$[a,b]$</span> such that the lower sum of <span class="math-container">$P$</span> equ... | <p>The lower and upper sums are sums of areas of rectangles. It should be clear from a diagram that unless <span class="math-container">$f$</span> is constant (or at any rate piecewise constant) we will always have
<span class="math-container">$$\{\hbox{lower sum}\}<\{\hbox{exact area}\}<\{\hbox{upper sum}\}\ .$... | 625 |
analysis | for real numbers w, x, y, and z. prove for x, z ≠ 0, (w/x)(y/z)=(wy)/(xz) and (w/x)+(y/z)=(wz+xy)/(xz) | https://math.stackexchange.com/questions/3355313/for-real-numbers-w-x-y-and-z-prove-for-x-z-%e2%89%a0-0-w-xy-z-wy-xz-and-w | <p>I'm lost with the assignment from my teacher.
For real numbers w, x, y, and z. Prove for x,z≠0, (w/x)(y/z)=(wy)/(xz) and (w/x)+(y/z)=(wz+xy)/(xz)</p>
| <p>You can write <span class="math-container">$$\frac{w}{x}+\frac{y}{z}=\frac{wz}{xz}+\frac{xy}{xz}=…$$</span>
<span class="math-container">$$\frac{w}{x}\cdot \frac{y}{z}=\frac{wy}{xz}$$</span></p>
| 626 |
analysis | Prove that $f(x)$ is uniformly continuous on $I$ if and only if the image of each Cauchy sequence under $f$ is also a Cauchy sequence. | https://math.stackexchange.com/questions/3381457/prove-that-fx-is-uniformly-continuous-on-i-if-and-only-if-the-image-of-eac | <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> defines on the bounded interval <span class="math-container">$I$</span>. Prove that <span class="math-container">$f(x)$</span> is uniformly continuous on <span class="math-container">$I$</span> if and only if the image of each Cauchy sequence under <s... | <p>Hint: The following can be used together with some basic theory of real analysis to complete the OP's analysis.</p>
<p>Given <span class="math-container">$a. b \in \Bbb R$</span> with <span class="math-container">$a \lt b$</span>.</p>
<p>Let <span class="math-container">$f: (a,b) \to \Bbb R$</span> be a continuou... | 627 |
analysis | Prove there are distinct $x_1,\,x_2,\cdots,\,x_n$ such that $ \sum_{i=1}^n\frac{p_i}{f'(x_i)}=\sum_{i=1}^n p_i. $ | https://math.stackexchange.com/questions/3383736/prove-there-are-distinct-x-1-x-2-cdots-x-n-such-that-sum-i-1n-frac | <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> is differentiable on <span class="math-container">$[0,\,1]$</span>, <span class="math-container">$f(0)=0$</span>, <span class="math-container">$f(1)=1$</span> and <span class="math-container">$p_1,\,p_2,\cdots,\,p_n$</span> are <span class="math-conta... | <p><strong>Proof</strong>. <span class="math-container">$\blacktriangleleft$</span> Assume <span class="math-container">$\sum p_j = 1$</span>, otherwise replace <span class="math-container">$p_j$</span> by <span class="math-container">$p_j/p$</span> for each <span class="math-container">$j$</span>. By continuity and th... | 628 |
analysis | Find the explicit form of $ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1} $. | https://math.stackexchange.com/questions/3386371/find-the-explicit-form-of-sum-n-1-infty-frac-1n-1nn2xn-1 | <p>Find the explicit form of
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}.
$$</span></p>
<p>Let <span class="math-container">$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$</span>. It has radius of convergence <span class="math-container">$1$</span>.</p>
<p>Let <span ... | <p><span class="math-container">$$\log(1+x)=\sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}~~~(1)$$</span>
Multipluy by <span class="math-container">$x$</span> on both sides and integrate w.r.t. <span class="math-container">$x$</span>
<span class="math-container">$$\int x \log(1+x) dx= \sum_{n=1}^{\infty} (-1)^{n-1}\int \... | 629 |
analysis | Evaluate $\lim_{x\to0+}\sum_{n=0}^{\infty}2^{-n}\frac{\sin (2^nx)}{x}$. | https://math.stackexchange.com/questions/3401237/evaluate-lim-x-to0-sum-n-0-infty2-n-frac-sin-2nxx | <blockquote>
<p>Suppose <span class="math-container">$\displaystyle f(x)=\sum_{n=0}^{\infty}2^{-n}\sin(2^nx)$</span>, evaluate
<span class="math-container">$$
\lim_{x\to0+}\frac{f(x)-f(0)}{x}.
$$</span></p>
</blockquote>
<p><span class="math-container">$f(x)$</span> converges uniformly on <span class="math-contain... | <p>After defining</p>
<p><span class="math-container">$$g_n(x) = \frac{\sin\left(2^nx\right)}{2^nx},$$</span></p>
<p>and
<span class="math-container">$$g(x) = \sum_{n=0}^{+\infty}g_n(x),$$</span></p>
<p>observe that in the interval
<span class="math-container">$$0<x<\frac{\pi}{2^N},$$</span></p>
<ol>
<li>For ... | 630 |
analysis | Is $\mathbb{N} \times\mathbb{R}$ closed, open or neither? | https://math.stackexchange.com/questions/3448363/is-mathbbn-times-mathbbr-closed-open-or-neither | <p>So I have to prove if is <span class="math-container">$\mathbb{N} \times \mathbb{R}$</span> closed, open or neither?</p>
<p>Any help?</p>
| <p>In what topological space? If <span class="math-container">$\mathbf{R}^2$</span>, closed. If <span class="math-container">$\mathbf{R}\times\mathbf{N}$</span> itself, both. In an arbitrary space <span class="math-container">$X$</span>, who’s to say?</p>
| 631 |
analysis | How to show $f\equiv 0$? | https://math.stackexchange.com/questions/3457881/how-to-show-f-equiv-0 | <p><span class="math-container">$f(x)$</span> is differentiable, and for any <span class="math-container">$x\in \mathbb R$</span>, <span class="math-container">$|f'(x)|\le \lambda |x|$</span>, then how to show <span class="math-container">$f\equiv 0$</span> ? </p>
<p>This is a question my student ask me, but I don'... | <p>This is not correct: <span class="math-container">$f(x)=x^{2}$</span>, <span class="math-container">$|f'(x)|=2|x|$</span>.</p>
<p>Even that <span class="math-container">$\lambda$</span> is required to be <span class="math-container">$\lambda\in(0,1)$</span> is still not correct: <span class="math-container">$f(x)=\... | 632 |
analysis | how to show $ \iint\limits_D(x\frac{\partial f }{\partial x}+ y \frac{\partial f}{\partial y}) \, dx \, dy= \frac{\pi}{2e} $? | https://math.stackexchange.com/questions/3465278/how-to-show-iint-limits-dx-frac-partial-f-partial-x-y-frac-partial-f | <p>Assume <span class="math-container">$f\in C^2(D)$</span>, where <span class="math-container">$D=\{(x,y)\in \mathbb R^2: x^2+ y^2 \le 1\}$</span>, if
<span class="math-container">$$
\frac{\partial^2 f }{\partial x^2} + \frac{\partial^2 f }{\partial y^2}
=e^{-x^2 -y^2}
$$</span>
how do I show
<span class="math-conta... | <p>Notice that the integral can be rewritten as</p>
<p><span class="math-container">$$\iint_D (x,y)\cdot \nabla f dA = \iint_D \nabla \left( \frac{x^2 + y^2}{2}\right) \cdot \nabla f dA$$</span></p>
<p>then use integration by parts (really the divergence theorem in higher dimensions) with <span class="math-container"... | 633 |
analysis | $g(x)=\lim _{t \to 0} \frac{f(x+t)-f(x-t)}{t}$,prove that there is a $c\in(a,b)$ ,such that $g(c)\geqslant0$ | https://math.stackexchange.com/questions/3544900/gx-lim-t-to-0-fracfxt-fx-tt-prove-that-there-is-a-c-ina-b | <p>Let <span class="math-container">$f(x)$</span> be a continuous function on <span class="math-container">$[a, b]$</span> (not necessarily derivable) and satisfy <span class="math-container">$f(a)<f(b)$</span>. At the same time, the limit <span class="math-container">$g(x)=\lim_{t \to 0} \frac{f(x+t)-f(x-t)}{t}$</... | 634 | |
analysis | Non-negative orthant: smoothing | https://math.stackexchange.com/questions/3606304/non-negative-orthant-smoothing | <p>Consider the non-negative orthant
<span class="math-container">$$
R\equiv \{(x_1,...,x_n): x_i\geq 0 \text{ }\forall i\}
$$</span>
The boundary of <span class="math-container">$R$</span> is not smooth.
I'm looking for a function which can smooth the boundary of <span class="math-container">$R$</span> and which depen... | <p>I guess the question is, why would you want to do that? Since your question is somewhat vague, I'm going to imagine that you are trying to solve an optimization problem like
<span class="math-container">$$
\max_{x \in \mathbb{R}^N} f(x)
$$</span>
subject to <span class="math-container">$x \ge 0$</span> and <span cl... | 635 |
analysis | Few questions (1.4, 1.5, 1.6) from Rudin 2e on roots of reals | https://math.stackexchange.com/questions/3647631/few-questions-1-4-1-5-1-6-from-rudin-2e-on-roots-of-reals | <p>I picked up a copy of Principles of Mathematical Analysis by Rudin. It just so happened to be the second edition which is drastically different from the third and I can't find solutions to a lot of exercises.</p>
<p>I'd like some hints on how to complete the following few problems.</p>
<ol>
<li>If <span class="mat... | <ol>
<li><p>I presume <span class="math-container">$\sqrt[n]x$</span> is defined as the unique solution of <span class="math-container">$y^n=x$</span> with <span class="math-container">$y>0$</span>
(and I presume that Rudin has proved there is a unique solution). Then to prove <span class="math-container">$\sqrt[n]{... | 636 |
analysis | Why can't we replace $\leq$ with $<$ in the limit location theorem? | https://math.stackexchange.com/questions/2784352/why-cant-we-replace-leq-with-in-the-limit-location-theorem | <p>The limit location theorem states for a sequence $\{a_n\}$, if for large enough $n$ we have $a_n \leq M$, then $\lim_{n \to \infty} a_n \leq M$ (the same holds if we replace $\leq$ with $\geq$). Why does the theorem not hold with strict inequalities?</p>
| <p>In general we have that even if $a_n < M$ then $\lim_{n \to \infty} a_n \leq M$, consider for example</p>
<p>$$\forall n>0 \quad a_n=\frac{n}{n+1}<1$$</p>
<p>but</p>
<p>$$\lim_{n \to \infty}a_n=\lim_{n \to \infty}\frac{n}{n+1}=1$$</p>
<p>thus the theorem you mentioned considers the more general case.</p... | 637 |
analysis | $|a_N(n)|\leq \frac{c}{n}$ | https://math.stackexchange.com/questions/2750867/a-nn-leq-fraccn | <p>For $N\geq 1$ the discrete Fourier Coefficient of $f$ are defined by
\begin{align}
a_N(n)=\frac{1}{N}\sum_{k=1}^{N}f(e^{2\pi i k/N})e^{-2\pi i k n/N}, \quad \forall n\in \mathbb{Z}
\end{align}
Suppose $f\in C^1$ function on circle, how to prove there exist $c>0$ such that $|a_N(n)|\leq \frac{c}{|n|}$ whenever $0&... | 638 | |
analysis | Determine the image of the map | https://math.stackexchange.com/questions/2822614/determine-the-image-of-the-map | <p>Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by \begin{equation*}f: \begin{pmatrix}x \\ y\end{pmatrix}\rightarrow \begin{pmatrix}u \\ v\end{pmatrix}=\begin{pmatrix}x(1-y) \\ xy\end{pmatrix}\end{equation*} </p>
<p>I want to determine th eimage $f(\mathbb{R}^2)$. </p>
<p>$$$$ </p>
<p>We have that
\begin{eq... | <p>You can notice that the map is invertible in certain conditions.</p>
<p>If $(u,v)\in\mathbb R^2$ then $f(x=u+v,y=\dfrac v{u+v})=(u,v)$ except for $v=-u$.</p>
<p>Indeed these points cannot be reached since $x(1-y)=-xy\implies x=0$ </p>
<p>And when $x=0$ then the whole axis is transformed to origin point : $f(0,y)=... | 639 |
analysis | Is this function a function of bounded variation? | https://math.stackexchange.com/questions/2823531/is-this-function-a-function-of-bounded-variation | <p>I am relevantly new to this concept of Bounded variation.
I want to know is the function $\sqrt {1-x^2} $, $x\in (-1,1) $ of bounded variation? How should I proceed. Is there any graphical interpretation of functions of bounded variation?</p>
| <p>As for the specific question, $f(x)=\sqrt{1-x^2}$ it is a function of bounded variation over the closed interval $[-1,1]$ (I don't know why the restriction here to the open interval, but if taken as a limit, the variation over $(-1,1)$ would also be finite, hence of bounded variation.</p>
<p>A simple proof relies o... | 640 |
analysis | $\{Q_i : i\in I\} \text{ is open} \rightarrow U_i:= f^{-1}(O_i)\text{ is open}$ | https://math.stackexchange.com/questions/2791505/q-i-i-in-i-text-is-open-rightarrow-u-i-f-1o-i-text-is-open | <p>$f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $K\subset\mathbb{R}$ is compact.<br>
$\{Q_i : i\in I\}$ is an open cover for $f(K)$.<br>
Does it follow that $U_i:= f^{-1}(O_i)$ is open for each $i \in I$? Is $\{U_i:i\in I \}$ an open cover for $K$?<br>
Intuitively, the answer to both of these questions seems ... | <p>Given a continuous function, the preimage of an open set is open.</p>
| 641 |
analysis | Trouble with intersection of infinitely many sets | https://math.stackexchange.com/questions/2824115/trouble-with-intersection-of-infinitely-many-sets | <p>So I have a problem that's asking me to compute the intersection of all sets $S(k)$ such that</p>
<p>$$ S(k) = \left(\frac{k-1}{k}, \frac{2k+1}{k}\right], k \ge 1 $$</p>
<p>My answer is $(1,2)$, but the book's answer is $(0,2)$. Maybe I'm just dumb, but I do not understand why the book's answer would be correct.</... | 642 | |
analysis | How do I calculate a set in a map ? | https://math.stackexchange.com/questions/1530276/how-do-i-calculate-a-set-in-a-map | <p>I've got a task, where a set A and a map g were given.
The task was to calculate g(A).
And I don't know what to do exactly, so I'd appreciate if s.o. could explain it with an example or give me a link, where it's explained.</p>
<p>For example let A = { x in IR | |x-3|= 2 } and g (x) = (x+2)/x.</p>
<p>So if I want ... | <p>You want to answer the question "what values are mapped to by $g$, from elements in $A$?". So, once you know what $A$ is, in the simplest case you just apply $g$ to the elements of $A$.</p>
<p>As you have already implicitly stated, after simplification we have $A = \{1,5\}$. Therefore, the set we seek is $g(A) = \{... | 643 |
analysis | For a bounded sequence $a_n$, $\lim\limits_{k\to\infty}(a_1+...+a_{n_k})/n_k=a$ implies $\lim_{n\to\infty}(a_1+...+a_n)/n=a$. | https://math.stackexchange.com/questions/1521861/for-a-bounded-sequence-a-n-lim-limits-k-to-inftya-1-a-n-k-n-k-a | <p>I am reading a paper on Markov chains and am trying to prove a lemma left to the reader that I need for the Ergodic Theorem for Markov Chains (though the lemma requires no knowledge of any of this). The statement of the lemma is as follows:</p>
<p>Let $\{a_n\}$ be a bounded sequence and suppose for $\{n_k\}$ a sequ... | <p>Let $C \le a_n \le D$ for all $n$. By using $a_n -C$ if necessary, we assume $a_n \ge 0$ (that is, $C=0$). For each $n\in \mathbb N$, there is $k$ so that $ n_k\le n <n_{k+1}$. Then </p>
<p>$$\begin{split}
\frac{a_1 +\cdots + a_n}{n} &\le \frac{a_1 + \cdots + a_n}{n_k} \\
&\le \frac{a_1+\cdots +a_{n_k}}{... | 644 |
analysis | transforming $\,x^3 + x^2 + 1 \implies x\left(x^2 + x + \dfrac{1}{x}\right) \implies\,$ div by $\,0\,$?! | https://math.stackexchange.com/questions/1536458/transforming-x3-x2-1-implies-x-leftx2-x-dfrac1x-right-imp | <p><strong>EDIT:</strong> Sorry. I basically was confused by that just valid mathematical transforming could lead into a undefined behavior.
I have to admit, my question has not much to do with the zero of a function, i've just used one method of the "zero calculating tool" set to reshape my function. (i explain that ... | <p>The function</p>
<p>$$f(x)= x^3 + x^2 + 1$$ </p>
<p>has domain $x \in \mathbb{R}$ when otherwise unspecified. It is not the same function as</p>
<p>$$g(x)= x \left(x^2 + x + \frac 1x\right) $$</p>
<p>whose domain is considered to be $x \in \mathbb{R}\setminus\{0\}$.</p>
<p>So $f(x) = g(x)$ if $x\ne0$.</p>
<p>I... | 645 |
analysis | If $i^2=-1$, at one point undefined, then why not define $\frac00$? | https://math.stackexchange.com/questions/1558737/if-i2-1-at-one-point-undefined-then-why-not-define-frac00 | <p>Would this even assist math in the way that $i$ did? Or is this just outright pointless and/or too exclusive to call for a definition?</p>
| <p>It would require us to give up many of the usual rules of algebra -- for example the cancellation rule which says that
$$ \frac{a\times b}{a} = b $$
whenever both sides exist.</p>
<p>However, taking $(a,b)$ first to be $(0,1)$ and $(0,2)$ we would get
$$ \frac00 = \frac{0\times 1}{0} = 1 \qquad\text{and}\qquad
\fra... | 646 |
analysis | Show the graph is Jordan region with volume 0 | https://math.stackexchange.com/questions/1374570/show-the-graph-is-jordan-region-with-volume-0 | <p>Let $f \colon [a, b] →\mathbb R$ be a continuous function. Then prove that the graph of $f$,
$$\operatorname{Graph}(f) := \{\,(y, x) \in \mathbb R^2\mid y = f(x), x \in [a, b]\,\}$$ is a Jordan region, and it has Jordan content $0$.</p>
<p>So I need to show that there exist an $\epsilon >0$ such that the volume... | <p>Note that $f$ is <em>uniformly</em> continuous. Thus given $\epsilon>0$, there exists $\delta>0$ such that yadda yadda.
With $n=\lfloor\frac{b-a}{2\delta}\rfloor+1$ we see that
the graph is contained in
$$\bigcup_{k=1}^n[x_k-\delta,x_k+\delta]\times [f(x_k)-\epsilon,f(x_k)+\epsilon] $$
where $x_k=a+\frac {2k-... | 647 |
analysis | Infinite closed subset of $S^1$ such that the squaring map is a bijection? | https://math.stackexchange.com/questions/1539215/infinite-closed-subset-of-s1-such-that-the-squaring-map-is-a-bijection | <p>Is there an infinite closed subset $X$ of the unit circle in $\mathbb C$ such that the squaring map induces a bijection from $X$ to itself?</p>
| <p>Let us think of $S^1$ as $\mathbb{R}/\mathbb{Z}$, so we want an infinite closed subset $X$ on which multiplication by $2$ is a bijection. Suppose you have such an $X$; write $T:X\to X$ for the multiplication by $2$ map. Each element $x\in X$ determines a biinfinite binary expansion $f_x:\mathbb{Z}\to\{0,1\}$, suc... | 648 |
analysis | Isolated points within a compact space. | https://math.stackexchange.com/questions/1596301/isolated-points-within-a-compact-space | <p>Here's an exercise in normed spaces that I can't get my head around.
It reads as follows:
"Let X be a compact space equipped with norm d. If X is countable, then the set of isolated points in X is both open and dense."
Just point me in the right direction.
Thanks a bunch!</p>
| <p>You've already figured out why the set of isolated points $I$ is open.</p>
<p>To show $I$ is dense: Because your space is countable, you can enumerate the <em>non</em>-isolated points $x_1, x_2,$ and so on. Define $U_0 = X$ and for all natural numbers $n$, let $U_{n+1} = U_n \setminus \{x_n\}$. Use induction to ... | 649 |
analysis | prove that a particular set is open | https://math.stackexchange.com/questions/1539580/prove-that-a-particular-set-is-open | <p>Let $S= \{(x,y): y<2x+1\}$.Then prove that $S$ is open in $\mathbb{R}\times\mathbb{R}$.</p>
<p>To prove that we must show that</p>
<blockquote>
<p>for every $(u,v)\in S $ there exists $r>0$ such that the ball $B[(u,v),r]$ is contained in $S$</p>
</blockquote>
<p>Or, equivalently,</p>
<blockquote>
<p>Fo... | <p>$f(x,y)=2x+1-y$ the set is $f^{-1}(]0,+\infty[)$ and is open since $f$ is continue and the inverse image of an open set by a continuous map is open.</p>
| 650 |
analysis | set operation for union and intersection | https://math.stackexchange.com/questions/1457643/set-operation-for-union-and-intersection | <p>How to write this in math notation
$$A=\{2,4,...,2n,...\}
\\B=\{3,6,9,...,3n,...\}$$
what is $A\cup B$ and $A\setminus B$</p>
<p>by inspection, I can see that $A\cup B=\{2,3,4,6,8,9...\}=\{2,4,6...3,6,9...\}$ that is even set or multiple of 3 but I do not know how to write this in math notation
$ A\setminus B=\{2,4... | <p>It depends on what set-builder notation you use, but something like this:
$\begin{align}A & = \{k:\Bbb N^+\mid \exists n\in\Bbb N^+\;k=2n\} & = \{2,4,6,\ldots\}\\ B & = \{k:\Bbb N^+\mid \exists n\in\Bbb N^+\;k=3n\} & = \{3,6,9,\ldots\}\\ A\cap B & = \{k:\Bbb N^+\mid \exists n\in\Bbb N^+\;(k=6n)\}... | 651 |
analysis | How to show $f(x)=x^2\sin^2(1/x)$ is Lipschitz continuous | https://math.stackexchange.com/questions/1598200/how-to-show-fx-x2-sin21-x-is-lipschitz-continuous | <p>$f(x)=x^2\sin^2(1/x)$ for $x\neq0$ and $f(0)=0$ for $x=0$</p>
<p>How to show $f(x)$ is Lipschitz continuous on $[-a,a]$.</p>
<p>I tried to use Mean Value Theorem, but couldn't finalized it.</p>
| <p>If you want to find an $K > 0$ such that $|f'(x)| \leq K$, then $K$ is not that difficult to calculate. In fact, $|f'(x)| = \left|2x\sin^2\left(\frac{1}{x}\right) - \sin\left(\frac{2}{x}\right)\right|\leq 2|x|+1 \leq 2a+1$. So $K = 2a+1$.</p>
| 652 |
analysis | Uniform convergence and maximum of an absolute difference | https://math.stackexchange.com/questions/1276263/uniform-convergence-and-maximum-of-an-absolute-difference | <p>I am trying to prove that: Consider the sequence $a_n = \sup_{x\in S}|f_n(x) - f(x)|$. Then $f_n$ converges to $f$ uniformly if and only if $a_n$ tends to $0$. But I can't prove that if $f_n$ converges to $f$ uniformly, then $a_n$ tends to $0$. I have not been able to go from $|f_n(x) - f(x)|<\epsilon... | <p>Suppose $f_n$ converges to $f$ uniformly. </p>
<p>Then for all $\epsilon>0$, there exists an $N$ such that for all $x \in S$ and all $n >N$, $|f_n(x) -f(x)|< \frac{\epsilon}{2}$.</p>
<p>Then $|f_n(x) -f(x)|$ is bounded above by $\frac{\epsilon}{2}$ for all $x$.</p>
<p>Suppose $c_n = \sup_\limits{x\in S... | 653 |
analysis | Is a nonzero linear functional is not constant on each ball? | https://math.stackexchange.com/questions/1564480/is-a-nonzero-linear-functional-is-not-constant-on-each-ball | <p>Let $T$ be a nonzero linear functional on a normed space $X$.
Is it then true that on each ball $B(x,r)$, $f$ is not constant?</p>
| <p>Sure, you can consider for the sake of clarity $B(0,r)$. Take $x_1$ and $x_2$ such that $||x_1||,||x_2|| < \frac{r}{4}$ and $T(x_1) \neq 0$ $T(x_2) \neq 0$ (these elements exist because $T$ is nonzero), then $||x_1+x_2||<r$, so $x_1+x_2 \in B(0,r)$.</p>
<p>Now apply $T$ to the new element:</p>
<p>$$T(x_1+x_2... | 654 |
analysis | Range of a function of two variable | https://math.stackexchange.com/questions/1609597/range-of-a-function-of-two-variable | <p>How to find $f(\mathbb{R}^2)$ if $f:\mathbb{R}^2 \to \mathbb{R}^2$</p>
<p>$$f(x,y)=(e^{x+y}+e^{x-y},e^{x+y}-e^{x-y} )$$</p>
| 655 | |
analysis | Analysis - Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0$ | https://math.stackexchange.com/questions/2155270/analysis-prove-that-there-exists-a-smallest-positive-number-p-such-that-co | <blockquote>
<p>Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0
$.</p>
</blockquote>
<p>I think I'm supposed to use either Rolle's theorem or the Mean Value Theorem but I'm not sure how, any help would be appreciated. Thanks!</p>
| <p>$\cos$ is a continous function, the set $\{0\}$ is closed, and therefore $\cos^{-1}(\{0\})$ is a closed set.</p>
<p>We conclude that $\cos^{-1}(\{0\})\cap[0,\infty)$ is closed. This set is equal to $\cos^{-1}(\{0\})\cap (0,\infty)$ (because $\cos(0)\neq 0$).</p>
<p>This set is bounded from below and must therefore... | 656 |
analysis | Let $z$ be an upper bound for $A \subset \Re$ such that for all $\epsilon > 0$ there is an $x \in A$ such that $x > z − \epsilon$. Then $z = \sup A$. | https://math.stackexchange.com/questions/4993999/let-z-be-an-upper-bound-for-a-subset-re-such-that-for-all-epsilon-0-t | <p>Let <span class="math-container">$z$</span> be an upper bound for <span class="math-container">$A \subset \Re$</span> such that for all <span class="math-container">$\epsilon > 0$</span> there is an <span class="math-container">$x \in A$</span> such that <span class="math-container">$x > z − \epsilon$</span>. ... | <p>Let <span class="math-container">$z$</span> be an upper bound for <span class="math-container">$A \subset \Re$</span> such that for all <span class="math-container">$\epsilon > 0$</span> there exists an <span class="math-container">$x \in A$</span> such that <span class="math-container">$x > z - \epsilon$</spa... | 657 |
analysis | continuous and strictly increasing implies differentiable | https://math.stackexchange.com/questions/130457/continuous-and-strictly-increasing-implies-differentiable | <p>I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable. </p>
<p>It may be because there are no bumps like in the absolute value. </p>
| <p>Not necessarily. Counterexample:
$$
f(x)=\begin{cases}
x & \text{if }x<0,\\
2x & \text{if }x\ge 0.\end{cases}
$$
Is continuous, strictly increasing but not differentiable at $x=0$.</p>
| 658 |
analysis | Divergence and curl both zero. What can we say about behavior at infinity? | https://math.stackexchange.com/questions/4428280/divergence-and-curl-both-zero-what-can-we-say-about-behavior-at-infinity | <p>All right, so my Professor claims that it is impossible to find a vector field that has both div and curl zero and yet vanishes at infinity (that is, becomes the zero vector).</p>
<p>I don't know if it is correct. I decide to trust my professor and check for myself.<br>
So I conclude that,<br> if the vector field is... | <p>First of all, if <span class="math-container">$\mathrm{curl}\;F = 0$</span> then there exists a function <span class="math-container">$U$</span> such that <span class="math-container">$F = \nabla U$</span>. For some reason I can't find a proper reference, but see e.g. <a href="http://persweb.wabash.edu/facstaff/foot... | 659 |
analysis | Divided differences and differentiability | https://math.stackexchange.com/questions/185848/divided-differences-and-differentiability | <p>Let $f: R \rightarrow R$, $y_0,y_1,y_2 \in R$. We define divided differences:
$$[y_0;f]=f(y_0),$$
$$[y_0,y_1;f]=\frac{f(y_1)-f(y_0)}{y_1-y_0},$$
$$[y_0,y_1,y_2;f]=\frac{[y_1,y_2;f]-[y_0,y_1;f]}{y_2-y_0}.$$</p>
<p>Assume that for each $x \in R$ and $\varepsilon>0$ there exists a $\delta>0$ such that $$|[y_... | <p>I'll try to provide a proof of the general case $[y_0,\ldots,y_n;f]$. This proof has many similarities to that of timur, but I'll formulate things more in terms of limits rather than using $\epsilon$s and $\delta$s. My other answer was wrong, but I leave it since it had some useful bits, one of which I'll redo here.... | 660 |
analysis | Is the set of linear functions from $[0,1]$ to $\mathbb{R}$ equicontinuous? | https://math.stackexchange.com/questions/4438313/is-the-set-of-linear-functions-from-0-1-to-mathbbr-equicontinuous | <p>By the set of linear functions I mean the functions of the form <span class="math-container">$$f(x)=\alpha x,$$</span> for <span class="math-container">$\alpha\in\mathbb{R}$</span>. We clearly have for any <span class="math-container">$x,y\in[0,1]$</span> <span class="math-container">$$|f(x)-f(y)|=|\alpha||x-y|.$$</... | <p>Here is an argument.</p>
<p>Suppose that your set of functions <em>is</em> equicontinuous on <span class="math-container">$[0,1]$</span>.
Then, given <span class="math-container">$\epsilon >0,$</span> there exists a <span class="math-container">$\delta >0$</span> such that
<span class="math-container">$|x-y| &... | 661 |
analysis | Doubly exponential sequence behaviour from inequality | https://math.stackexchange.com/questions/151811/doubly-exponential-sequence-behaviour-from-inequality | <p>I am investigating a strictly decreasing sequence $(a_i)_{i=0}^\infty$ in $(0, 1)$, with $\lim_{i\to\infty}a_i=0$, such that there exist constants $K>1$ and $m\in\mathbb{N}$ such that $$\frac{a_{i-1}^m}{K} \leq a_i \leq K a_{i-1}^m$$ for all $i$. Even though $K>1$, is it of the right lines to conclude that $a_... | <p>[Edit: Now that the question has been changed a day later, I removed analysis of the old version of the first inequality. Perhaps sometime I will update to fully answer the new question. The following still applies to the second inequality.]</p>
<p><s>On the other hand, f</s> For the second inequality, $$a_i\leq ... | 662 |
analysis | Preservation of compact sets | https://math.stackexchange.com/questions/1518100/preservation-of-compact-sets | <p>Decide whether the following statement is true or false, justify the conclusion.</p>
<p>If $f$ is defined on $\mathbb R$ and $f(K)$ is compact whenever $K$ is compact, then $f$ is continuous on $\mathbb R.$</p>
<p>Does this hold based on the Preservation of Compact Sets: Let $f : A → \mathbb R$ be continuous on $A... | <p>$$
f(x) = \begin{cases} 0 & \text{if } x<0, \\ 1 & \text{if } x\ge0. \end{cases}
$$</p>
<p>This is not continuous, but for all compact sets $K$ the set $\{f(x): x\in K\}$ is compact.</p>
| 663 |
analysis | For a compactly supported, continuously differentiable function $f$, do we have $\frac{f(x+h)-f(x)}{h}\to f'(x)$ uniformly? | https://math.stackexchange.com/questions/4455256/for-a-compactly-supported-continuously-differentiable-function-f-do-we-have | <p>For a compactly supported, continuously differentiable function <span class="math-container">$f$</span>, do we have <span class="math-container">$\frac{f(x+h)-f(x)}{h}\to f'(x)$</span> uniformly? (Euclidean space)</p>
<p>I think this is true, since compact supported-ness is pretty strong condition and this is geomet... | 664 | |
analysis | Prove that a unit circle contains its limit points (that it is closed under its analysis definition) | https://math.stackexchange.com/questions/4439601/prove-that-a-unit-circle-contains-its-limit-points-that-it-is-closed-under-its | <p>By a unit circle, I just mean <span class="math-container">$D = S^1$</span> i.e. a unit disk in <span class="math-container">$R^2$</span>.</p>
<p>My game is to use some arbitrary sequence <span class="math-container">$x_n = (a_n,b_n)$</span> such that <span class="math-container">$b_n = \sqrt{1-(a_n)^2}$</span> with... | <p>Suppose <span class="math-container">$(a,b)$</span> is a limit point. Then there are <span class="math-container">$(a_n,b_n) \in D$</span> such that <span class="math-container">$(a_n,b_n) \to (a,b)$</span>. Since <span class="math-container">$a_n^2+b_n^2 = 1$</span> for all <span class="math-container">$n$</span>, ... | 665 |
analysis | Uniform Lipschitz condition | https://math.stackexchange.com/questions/86111/uniform-lipschitz-condition | <blockquote>
<p>Does the function $f(x)=\sqrt{x}\sin(1/x),x\in(0,1],f(0)=0,$ satisfy the uniform Lipschitz condition $|f(x)-f(y)|<M|x-y|^{1/2},M>0$?</p>
</blockquote>
<p>Any help is appreciated.
Thanks</p>
| <p>no, take $x =\frac{1}{2n\pi +\frac{\pi}{2}}$ and $y =\frac{1}{2n\pi -\frac{\pi}{2}}$. Then $\vert f(x)-f(y)\vert$ behave like $\frac{1}{\sqrt{n}}$ and $\vert x-y \vert^\frac{1}{2}$ behaves like $\frac{1}{n}$.</p>
| 666 |
analysis | Rudin Chapter 1 Exercise 10 Hang-Ups? | https://math.stackexchange.com/questions/3499058/rudin-chapter-1-exercise-10-hang-ups | <blockquote>
<p>Suppose <span class="math-container">$z = a + bi$</span>, <span class="math-container">$w = u + iv$</span>, and
<span class="math-container">$$a = \left(\frac{|w| + u}{2}\right)^{1/2}, b = \left(\frac{|w| - u}{2}\right)^{1/2}$$</span>
Prove that <span class="math-container">$z^2 =w$</span> if <sp... | <p>1) This is a good point, but you can be sure that <span class="math-container">$|w| + u \ge 0$</span> and <span class="math-container">$|w|-u \ge 0$</span> because</p>
<p><span class="math-container">$(|w|+u)(|w|-u)=|w|^2-u^2=v^2 \ge 0$</span></p>
<p>If <span class="math-container">$v^2=0$</span> then <span class=... | 667 |
analysis | The "expected value" of a deterministic sequence | https://math.stackexchange.com/questions/21156/the-expected-value-of-a-deterministic-sequence | <p>Suppose I have a deterministic sequence $\{t_n\}$ that is uniformly distributed on $[0,1]$ (for example $t_n = \{ \pi n \}$, i.e the fractional part of $\pi n$) and a decreasing function $f : \mathbb{R} \rightarrow [0,1]$.</p>
<p>I think it is reasonable to expect that
$\#\{t_k < f(k) : k \leq n \} \approx \sum_... | <p>The problem with deterministic "random" sequences is that you can easily use them to provide an object (here a decreasing function) against which they won't be "random" at all :</p>
<p>If you pick $f(k) = \inf \{t_m, 1 \leq m \leq k\} /2$, you have that $\#\{t_k < f(k) : k \leq n \} = \#\{t_k = 0 : k \leq n \}$.... | 668 |
analysis | Uniform grid estimation for an integral by E.T. Jaynes | https://math.stackexchange.com/questions/1513568/uniform-grid-estimation-for-an-integral-by-e-t-jaynes | <p>E.T. Jaynes writes in "Probability Theory: The Logic of Science" (Chapter 17.7, page 531):</p>
<blockquote>
<p>Let a function y = f(x) have its domain of existence in the unit square <span class="math-container">$0 \leq x, y \leq 1$</span>; we wish to compute the integral
<span class="math-container">$$\th... | 669 | |
analysis | Rudin's theorem 7.29 | https://math.stackexchange.com/questions/1466885/rudins-theorem-7-29 | <p>In Rudin's POMA, the 7.29 theorem says:</p>
<blockquote>
<p>Let <span class="math-container">$\mathcal{B}$</span> be the uniform closure of an algebra <span class="math-container">$\mathcal{A}$</span> of bounded functions. Then <span class="math-container">$\mathcal{B}$</span> is a uniformly closed algebra.</p>
</bl... | 670 | |
analysis | Show that the real valued identity function is continuous | https://math.stackexchange.com/questions/2062471/show-that-the-real-valued-identity-function-is-continuous | <p>How do I show with the definition of continuity that <span class="math-container">$f : \mathbb{R} \to\mathbb{ R}$</span> defined by <span class="math-container">$f(x)= x$</span> for all <span class="math-container">$x\in\mathbb{R}$</span> is continuous?</p>
<p>I understand that this function is continuous, however I... | <p>Let $f$ be the identity function $\mathbb{R} \to \mathbb{R}$. Here are six weird reasons why $f$ is continuous (number 2 will make you cry, then number 3 will restore your faith in humanity!)</p>
<ul>
<li><p>The identity trivially satisfies that the preimage of any open set is open.</p></li>
<li><p>For every $\epsi... | 671 |
analysis | Basic Set Theory Proof Verification | https://math.stackexchange.com/questions/2256835/basic-set-theory-proof-verification | <blockquote>
<p><strong>Problem:</strong></p>
<p><span class="math-container">$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$</span></p>
</blockquote>
<p><strong>My proof:</strong></p>
<p>Let <span class="math-container">$x \in A \cap (B \cup C)$</span>. Then <span class="math-container">$x \in A$</span> and <span c... | <p>Your proof is good.</p>
<p>As for analysis tips: Here's a quote from my first analysis lecturer.</p>
<blockquote>
<p>"All I've ever done in my entire career is add zero, multiply by one, and use the triangle inequality."</p>
</blockquote>
<p>What he means is - when you're faced with something that looks a bit t... | 672 |
analysis | Minkowski-metric induct riamannian metric | https://math.stackexchange.com/questions/2286962/minkowski-metric-induct-riamannian-metric | <blockquote>
<p>Let the Minkowski-metric on <span class="math-container">$\mathbb{R}^3$</span> be (for a p<span class="math-container">$\in \mathbb{R}^3, v,w\in T_p\mathbb{R}^3$</span>) <span class="math-container">$<v,w>_p^{Mink}=-v_1w_1+v_2w_2+v_3w_3$</span>.</p>
<p>Show that:</p>
<p>i) <span class="math-contai... | 673 | |
analysis | Show that $F = \int_0^x f(t)dt $ is bijective function from $ [0,1]$ to $[0,A]$ | https://math.stackexchange.com/questions/2356269/show-that-f-int-0x-ftdt-is-bijective-function-from-0-1-to-0-a | <blockquote>
<p>Let <span class="math-container">$f: [0,1] \rightarrow \mathbb{R} $</span> a continuous function and <span class="math-container">$f > 0$</span>. Let <span class="math-container">$F(x) = \int_0^x f(t)dt $</span> and let <span class="math-container">$F(1)=A$</span>.</p>
<p><strong>1</strong>. Show tha... | <p>For (1), note that $F$ is continuous and moreover $F'(x)=f(x)>0$ so $F$ is also strictly monotone. Hence $F$ is injective and by the <a href="https://en.wikipedia.org/wiki/Intermediate_value_theorem" rel="nofollow noreferrer">Intermediate Value Theorem</a> $F$ attains all the values between $F(0)=0$ and $F(1)=A$.... | 674 |
analysis | Tangent plane equation in the point of intersection with $\space y$-axis | https://math.stackexchange.com/questions/3725351/tangent-plane-equation-in-the-point-of-intersection-with-space-y-axis | <p>I've encountered one tiny problem - I have to find tangent plane equation of the function <span class="math-container">$\space f(x,y)=y^3 - \sqrt{1-x^2y^2} \space$</span> in the point where it intersects with <span class="math-container">$\space y$</span>-axis. So <span class="math-container">$\space f(0,y) = y^3..... | <p><strong>Hints:</strong> Function <span class="math-container">$f(x,y)$</span> will have the intersection point with <span class="math-container">$y$</span>-axis in a point <span class="math-container">$P=(0, y_0, 0)$</span>. In your case it will be the point <span class="math-container">$P=(0, 1, 0)$</span>. Find no... | 675 |
analysis | Prove that the equation has no solution | https://math.stackexchange.com/questions/704990/prove-that-the-equation-has-no-solution | <p>Prove that there exist infinitely many positive real numbers <span class="math-container">$r$</span> such that the equation
<span class="math-container">$2^x +3^y + 5^z = r$</span> has no solution <span class="math-container">$(x, y, z) \subseteq \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$</span>.</p>
<p>First, ... | <p>Any number not in your set $S$ is a real number $r$ that you're looking for. Can you establish that there are infinitely many real numbers not in $S$, once you know that $S$ is countable?</p>
<p>In fact, every number in your set $S$ is algebraic, and the algebraic numbers are a countable subset of $\mathbb{R}$. Any... | 676 |
analysis | Why is the unit disk closed? | https://math.stackexchange.com/questions/57169/why-is-the-unit-disk-closed | <p>I read a question stating that if $z$ is complex, then $|z|\leq 1$ is a closed set. I think this is just saying that the unit disk is a closed set. Why is that so?</p>
| <p>Look at the complement of the set $S = \{ z \in \mathbb{C}: |z| \leq 1 \}$. The complement is given by $S^c = \{ z \in \mathbb{C}: |z| > 1 \}$. Consider a point $z_0 \in S^c$. $|z_0| > 1$ and hence let $|z_0| = 1 + r$ where $r > 0$. Consider the ball of radius $r$ centered at $z_0$ i.e. $B_r(z_0) = \{v \in ... | 677 |
analysis | Pointwise Upper Bound | https://math.stackexchange.com/questions/3250395/pointwise-upper-bound | <p>Consider <span class="math-container">$G\in C^{1}(\mathbb{R})$</span> such that <span class="math-container">$G(0)=0$</span>. Then there exists a constant <span class="math-container">$C$</span> such that <span class="math-container">$|G(s)|\leq C|s|$</span> for all <span class="math-container">$s\in [-M,M]$</span>,... | <p>Let <span class="math-container">$C$</span> be an upper bound for <span class="math-container">$G'$</span> on <span class="math-container">$[-M,M]$</span>, which is possible because <span class="math-container">$G'$</span> is continuous by hypothesis. By the mean value theorem, we have for every <span class="math-co... | 678 |
analysis | Prove or refute that $f$ is surjective (Are my thoughts correct ?) | https://math.stackexchange.com/questions/2348789/prove-or-refute-that-f-is-surjective-are-my-thoughts-correct | <p>Let $f : X \to Y$ and $g: Y \to Z$ be two functions, such that $(g \circ f) : X \to Z$ is bijective</p>
<p>Prove or refute that $f$ is surjective.</p>
<p>$\\$ </p>
<p>For $f$ to be surjective, $\forall b \in Y, \exists a \in X : f(a) = b$</p>
<p>Because $(g \circ f)$ is bijective, the sets X and Z have to have t... | <p>Your thoughts are not bad at all, but it seems that you think of $X$, $Y$ and $Z$ as finite sets. This is enough to give a counterexample. You realized that $Y$ has to have at least the same "number" of elements as $X$. But what if $Y$ has more elements? Can $f$ be surjective?<p></p>
<p>In general, you have to be c... | 679 |
analysis | $C^\infty$ version of Urysohn Lemma in $\Bbb R^n$ | https://math.stackexchange.com/questions/198748/c-infty-version-of-urysohn-lemma-in-bbb-rn | <p>I'm trying to solve an exercise which its conclusion seems to be the title of this post. The exercise is:</p>
<blockquote>
<ol>
<li>Show that the function $h:\Bbb R\to [0,1[$ given by
$$h(t)=\begin{cases}
e^{-1/t^2} &\text{if } t\neq 0\\
0 &\text{otherwise}
\end{cases}$$
is $C^\infty$.</li>... | <p>Since $K$ is compact and $U^C$ is closed, there is a positive distance, $\Delta$, from $K$ to $U^C$.</p>
<p>We can cover each point $k\in K$ with an open cube $Q_k(\Delta/\sqrt{n})$ centered at $k$ and side $\Delta/\sqrt{n}$. Note that the entire cube is within $\frac\Delta2$ of $k$. Since $K$ is compact, choose a ... | 680 |
analysis | $(a,b) \in \mathbb{R}^2$, $a + b = 2$ and $|a|<|b|$. Show that $1 \in ]|a|,|b|[ \iff ab \in ]-3,1[$ | https://math.stackexchange.com/questions/2243342/a-b-in-mathbbr2-a-b-2-and-ab-show-that-1-in-a-b | <p>We have: </p>
<p>$|a|<1<|b| \iff |a|+|b| < 1 + |b| < 2|b|$</p>
<p>$\iff (|a|+|b|)² < (1+|b|)² < 4|b|² $</p>
<p>$ \iff|a|² + 2|a||b| + |b|² < 1 + 2|b| + |b|² < 4|b|²$</p>
<p>$ \iff 2|a||b| < 1 + 2|b| - |a|² < 3|b|² - |a|² $</p>
<p>I don't see how to proceed to get $ab \in ]-3,1[$</p... | <p>We consider the function $f(a) = a(2 - a) = ab $</p>
<p>$\forall a \in (-1,1): f'(a) = 2(1 - a) > 0 $</p>
<p>$ f(-1) < f(a) < f(1) \iff -3 < ab < 1$ </p>
| 681 |
analysis | If $a < b$ then $((f \wedge b) -a)^-=(f-a)^-$ | https://math.stackexchange.com/questions/2124370/if-a-b-then-f-wedge-b-a-f-a | <p>If $a < b$ then
$$((f \wedge b) -a)^-=(f-a)^-$$
where $f$ is a real valued function. I can't figure out why this holds. I know that $f^-=-(f\wedge 0)$ so I tried showing this from definitions. I would greatly appreciate any help.</p>
| <p>It's easy checking cases…</p>
<p>For $f \ge b > a$ it holds: $$((f \wedge b) -a)^- = (b - a)^- = 0 = (f - a)^-$$</p>
<p>For $f < b$ it's $$((f \wedge b) -a)^- = (f - a)^-$$</p>
<p>Nothing wild…</p>
| 682 |
analysis | Properties of a subspace of sequence | https://math.stackexchange.com/questions/2227604/properties-of-a-subspace-of-sequence | <p>Let $p\in\left[ 1,\infty\right) $ and $\left(
X_{0},\left\Vert \cdot\right\Vert _{p}\right) $ a normed linear space, where
$$
X_{0}=\left\{ \left( x_{n}\right) \in
\mathbb{R}
^{
\mathbb{N}
}\mid\exists n_{0}\in%
\mathbb{N}
,\forall n\in
\mathbb{N}
:n_{0}\geq n\Rightarrow x_{n}=0\right\} .
$$</p>
<p>Let the s... | 683 | |
analysis | Proof Using Archimedean Property | https://math.stackexchange.com/questions/2125419/proof-using-archimedean-property | <p>An exercise from my book is as follows:</p>
<p>Assume that $x > 0$ for $x$ in $\mathbb{R}$ (Real numbers) then there is an $y$ in $\mathbb{N}$ (Natural Numbers) such that $1/y^3 < x$.</p>
<p>By the archimedean property, there exists an y in N such that $1/y < x$. How exactly would I continue on from here?... | <p>By Archimedean property there exists a natural number $y$ such that $xy >1$. Then $xy^3 >xy >1$. So $1/(y^3) <x$. </p>
| 684 |
analysis | Existance of global extremum | https://math.stackexchange.com/questions/2245651/existance-of-global-extremum | <p>I wonder if a matrix of second derivatives is positive definite and symmetric and the necessary condition of existance extremum is satisfy then exist exactly one extremum? </p>
| <p>Maybe if this log likelihood function is a concave function
($\frac{\partial^2 ln L(\theta)}{\partial \theta_i \partial \theta_j}= -n \frac{\partial^2 C(\theta)}{\partial \theta_i \partial \theta_j}$ with assumption is negative) and has a maximum then it must be absolute maximum?</p>
| 685 |
analysis | Get $x$ from exponential aquation | https://math.stackexchange.com/questions/2168267/get-x-from-exponential-aquation | <p>how can I get $x$ from equation</p>
<p>$ e^{(\frac{-a}{x})}\frac{x}{a} - \frac{x}{a} = b - 1 $</p>
<p>I tried to play with logaritms but I had no luck. Can I even get $a$ from this equation?</p>
| <p>The equation in unsolvable for $x$, the only way is a numerical approach</p>
| 686 |
analysis | How to find lim inf / lim sup | https://math.stackexchange.com/questions/2189040/how-to-find-lim-inf-lim-sup | <p>Let a$_n$ := $\frac {(n-1)(-1)^n}{n}$. Find lim sup a$_n$ and lim inf a$_n$.</p>
<p>I see that a$_n$ can either be $\frac {n-1}{n}$ when n is even or $\frac {-(n-1)}{n}$ when n is odd.</p>
<p>$\frac {n-1}{n}$ a$_2$ = $\frac {1}{2}$, a$_3$ = $\frac {2}{3}$, a$_4$ = $\frac {3}{4}$.</p>
<p>$\frac {-(n-1)}{n}$ a$_2$... | <p>$a_{2n}=\frac{2n-1}{2n} \to 1$ for $n \to \infty$</p>
<p>and</p>
<p>$a_{2n-1}=-\frac{2n-2}{2n-1} \to -1$ for $n \to \infty$.</p>
<p>Now suppose that $(a_{n_k})$ is a convergent subsequence of $(a_n)$ with limit $a$. Then $|a_{n_k}| \to |a|$.</p>
<p>But since $|a_n| \to 1$, we get $a=\pm 1$.</p>
<p>Consequence: ... | 687 |
analysis | Proving Convergent Subsequence | https://math.stackexchange.com/questions/2189158/proving-convergent-subsequence | <p>Let ${a_n}$ be a sequence and $b_n := \frac {2a_n + (-1)^n}{|a_n| +1}$ for $n$ in $\mathbb N$. Prove that ${b_n}$ has a convergent subsequence. </p>
<p>Attempt: </p>
<p>To show that the subsequence converges, I just have to show boundedness right? Is what I have here correct? Am I missing anything?</p>
<p>$|b_n| ... | 688 | |
analysis | Uniform Continuity of $f(x) = x+\frac1x$ on $(0,\infty)$ for any fixed $c>0$? | https://math.stackexchange.com/questions/1029490/uniform-continuity-of-fx-x-frac1x-on-0-infty-for-any-fixed-c0 | <p>Let $f(x) = x+\frac1x$. Show that $f(x)$ is uniformly continuous on $(0,\infty)$ for any fixed $c>0$?</p>
| <p>Hint:</p>
<p>I suppose you mean $x\in[c,\infty)$
$$|\frac{1}{x}+x-\frac{1}{y}-y|=|1-\frac{1}{xy}||x-y|\leq (1+\frac{1}{xy})|x-y|\leq(1+\frac{1}{c^2})|x-y|$$</p>
<p>Hence $f(x)$ is Lipschitz, hence uniformly continuous.</p>
| 689 |
analysis | Decomposition of a function as a sum | https://math.stackexchange.com/questions/1029606/decomposition-of-a-function-as-a-sum | <p>Let $f:I \times J\rightarrow \mathbb R$ be a function of class $C^2$ in the open rectangle $I \times J \subset \mathbb R^2$. If $\frac{\partial^2f}{\partial x\partial y}\equiv 0$, prove that there exist $\phi:I \rightarrow \mathbb R$ and $\chi:J \rightarrow \mathbb R$ of class $C^2$ such that $f(x,y)=\phi(x)+\chi(y)... | <p>Well, the first step is fine, but $f_y = a(y)$ for some $a$ rather than $f_y = f(x_0,y)$.
Otherwise, everything looks fine: you obtained the desired representation, the only thing you need to notice is that it concerns functions $\hat\phi(x) := \phi(x) - \phi(x_0) + f(x_0,y_0)$ and $\hat\psi(y) := \psi(y) - \psi(y_0... | 690 |
analysis | The strict convexity of the$ L^2$ norm | https://math.stackexchange.com/questions/1029825/the-strict-convexity-of-the-l2-norm | <p>I am sucked by a very simple question. I want to find an element which minimizes the$\|A\|$(in $L^2$-norm ),where A is a random variable.
Then the textbook says the uniqueness will follow from strict convexity of the $L^2$ norm.</p>
<p>What does this mean? What should I do if I want to prove the uniqueness? I thin... | 691 | |
analysis | Prove divergence of a series | https://math.stackexchange.com/questions/1026310/prove-divergence-of-a-series | <p>I have to prove that $\frac{e^x}{x^k} \to \infty$ for $x \to \infty$ with $k \in \mathbb N$</p>
<p>My idea is to calculate for $R \gt 0$ an $x_r$ so that for every $ x\gt x_r$ the inequation $\frac{e^x}{x^k} \gt R$ applies.
So, I struggle with solving the equation $\frac{e^{x_r}}{x_{r}^k} = R$ for $x_r$</p>
| <p>As $\displaystyle e^x=\sum_{r=0}^\infty\frac{x^r}{r!}$</p>
<p>For any finite $k\ge0,$
$$\lim_{x\to\infty}\frac{e^x}{x^k}\to\infty$$</p>
| 692 |
analysis | Existence of a function satisfying the above condition? | https://math.stackexchange.com/questions/2234149/existence-of-a-function-satisfying-the-above-condition | <p>Here is a question I asked to myself. Does there exist a function $f(z)$ satisfying the following conditions:</p>
<ul>
<li>$f(z)$ is continuous on the strip $0 \leq \text{Re}(z) \leq 1$</li>
<li>$f(z)$ is <strong>holomorphic</strong> on the strip $0 < \text{Re}(z) < 1$</li>
<li>$f(a)=0$ where $a$ is a fixed r... | <p>Assuming that you mean a real differentiable function, consider, for instance,
\begin{align*}
f(x+iy)=\frac{2(x-a)^2}{\min\{a^2,(1-a^2)\}}.
\end{align*}
This function is a nonnegative real polynomial (hence infinitely times real differentiable everywhere) with $f(a)=0$ and
\begin{align*}
f(it)&=f(0)=\frac{2a^2}{... | 693 |
analysis | The phrase "extension by $X$"? | https://math.stackexchange.com/questions/2234524/the-phrase-extension-by-x | <p>My professor used this term today, and I didn't get a chance to ask him what the heck it means. I can't find anything explicitly on the internet regarding the manner in which he used it.</p>
<p>for example, say we have a function $f(x)$ and a set $A$ where $f(x)$ is defined. If we say $f(x)$ is an extension by zero... | 694 | |
analysis | Help transforming a basic equation | https://math.stackexchange.com/questions/2192160/help-transforming-a-basic-equation | <p>Although this seems to be super basic, I cannot get my head around which steps I have to perform to transform $x^{2^{n+1}}\cdot x^{2^{n+1}}$ into $x^{2^{n+2}}$.</p>
<p>When I rewrite it as $x^{2^{{n+1}^2}}$ and the there are not braces, the power binds the strongest, that would lead to $x^{2^{n^2+2n+1}}$.</p>
<p>T... | <p>There's a basic fact that I'm assuming that you know:
$$
x^a \cdot x^b = x^ {a+b}.
$$
In your case, $a$ and $b$ are the same, and both are $2^{n+1}$. So all you need to know is what is
$$
a + b = 2^{n+1} + 2^{n+1}?
$$
Well, it's two copies of $2^{n+1}$, so it's
$$
2^1\cdot 2^{n+1} = 2^{n+2}
$$
by the same $a$-and-$b... | 695 |
analysis | Taylor series for $\frac{1}{1-x}$ in ascending powers of $(x-2)$ | https://math.stackexchange.com/questions/2194096/taylor-series-for-frac11-x-in-ascending-powers-of-x-2 | <p>I need to work out the Taylor series of $\frac{1}{1-x}$ in ascending powers of $(x-2)$ up to and including the term in $(x-2)^3$. </p>
<p>My method was to find all the derivatives (up to the third) and evaluate them at $x=0$ to give the standard geometric series formula</p>
<p>$\frac{1}{1-x} = 1 +x +x^2 + x^3 + ..... | <p>Observe
\begin{align}
\frac{1}{1-x} = \frac{-1}{1+(x-2)} = -\sum^\infty_{n=0}(-1)^n(x-2)^n.
\end{align}</p>
| 696 |
analysis | Is this proof of open set right? | https://math.stackexchange.com/questions/2194909/is-this-proof-of-open-set-right | <p>$\left( \mathsf{\mathbf{x}},\mathsf{\mathbf{y}}\right) $,
$_{[\mathsf{\mathbf{x}}=\left( x_{1},\cdots,x_{n}\right) \in R^{n}\text{
}\wedge\text{ }\mathsf{\mathbf{y}}=\left( y_{1},\cdots,y_{m}\right) \in
R^{m}]}$ $:=\left( x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right) \in R^{n+m}$</p>
<p>$W:=\left\{ \mathsf{... | <p>Yes, and no.</p>
<p>The proof looks formally correct, <strong>but</strong> still it looks a lot like smoke-and-mirror. Basically anyone that is able to actually follow the proof would probably find the statement quite obvious - which would make the proof so much easier.</p>
<p>Instead you should have written it in... | 697 |
analysis | Question about statement on p. 21 of Dieudonné's Infinitesimal Calculus | https://math.stackexchange.com/questions/2261139/question-about-statement-on-p-21-of-dieudonn%c3%a9s-infinitesimal-calculus | <p>On p. 21, Dieudonné states that: "... if $f$ is a continuous real function in a
bounded closed interval $I$, there is a smallest root and a largest root of the
equation $f(x)=0$ in $I$." </p>
<p>I haven't been able to convince myself of this basic
statement. I keep thinking of counterexamples that have only one roo... | <p>Let $R \subset I$ be the set of roots for $f$. Then because $f$ is continuous, $R$ is closed. Since $I$ bounded, $R$ is compact. In particular, it contains its infimum (which will be the smallest root of f in $I$) and supremum (largest root).</p>
<p>Is it clear?</p>
<p>If $f$ has no roots then $R = \emptyset$, but... | 698 |
analysis | Use induction to show that this improper integral exists | https://math.stackexchange.com/questions/2285577/use-induction-to-show-that-this-improper-integral-exists | <blockquote>
<p>Let $m,n \ge 0$ be positive integers. Using induction on $n$ or otherwise, show that the improper integral $$\int_0^1 x^m \left(\log x\right)^n dx$$ exists, and give a closed-form expression for it.</p>
</blockquote>
<p>I'm a bit confused by the concept of induction on an integral, any help with this... | <p><strong>Hint</strong></p>
<p>For $m,n>0$ and $0 <x <1$</p>
<p>$$|x^m\ln^n (x)|\leq -\ln (x) $$
and
$\int_0^1\ln (x)dx $ converges thus by comparison, the integral $I_{m,n} $ is absolutely convergent.</p>
<p>to find a recursive formula, use by parts integration.
it will be of the form</p>
<p>$$I_{m,n}=\... | 699 |
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