Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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How can I rewrite this sets using measure theory?

Let $(\Omega, F, \mu)$ be a measure space. Let $f_n:\Omega\rightarrow \Bbb{R}$ be measurable functions. Let $$B:=\{x\in \Omega: \lim_{n\rightarrow \infty} f_n(x)~~\text{does not exists}\}$$I need to ...
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Of a Controversy concerning the Order of Magnitude of Functions

Some time ago I started a chapter in my Calculus/Intro-Analysis textbook that had for its subject the notion of Order of Magnitude of functions tending to 0 or increasing beyond all bounds by absolute ...
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2 answers
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Using LMVT ( Lagrange's Mean Value Theorem) prove the following inequality

I want to prove the following inequality for $x > 0$ $$0 < \frac{1}{x} \log \frac{e^x - 1}{x} < 1$$ I think I can do this problem, but I just don't know what should be the function that is ...
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Does Caretheodory's extension theorem of premeasures on sigma-finite measure space give us a complete measure?

From Folland's analysis textbook, we constructed Lebesgue-Measure using the function $F(x)=x$ and defined a pre-measure on intervals of the form $(a,b]$ and extended using Caretheodory's extension ...
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Richardson's Extrapolation with a data tabel

I need to find the value of f'(1) using richardson's extrapolation formula and I have been given a set of data in a table to be used in the calculations. Based on the data, I assume that the step size ...
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2 answers
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Why is $\phi \circ \gamma: [0,1] \to U$ null-homotopic in $U$?

Let $R \subset \mathbb{R^2}$ be a rectangle and $\gamma:[0,1] \to R$ be it's boundary curve. Let $\phi: R \to U$, where $U \subset \mathbb{R^2}$ is open, be continuous. Why is then $\phi \circ \gamma: ...
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4 answers
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Prove that a half-open line is uncountable - corrected

I have to provide a proof that a halfopen line $L_a:={𝑥 ∈ ℝ:x > a}$ is uncountable. I have used Schröder-Bernstein-Cantor and tried to show that an injection exists between $L_a$ and $ℝ$ an ...
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156 views

Integration formula for cubic polynomial $\int_a^bq(x)dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a))$

Show that $\forall a,b\in \mathbb{R}$, with $a<b$, we have$$\int \limits _a^bq(x)\,dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a)),$$where $q\in \mathcal{P}_3$ is a cubic polynomial. I'...
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proving or disproving a certain inequality

In the attempt of proving a large deviations result, the following quantity pops up: $$H(\delta,h):= \frac{\frac{1}{2\alpha}(2\alpha-8\delta)+ \frac{4}{\alpha}\frac{1}{1-\gamma}\delta^{1-\gamma}}{h^{(\...
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3 answers
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Does the product of an exponential decrease with a polynomial with positive coefficients have a unique maximum?

Consider a function of the form $$f(x)=\exp(-x)p(x)$$ where $p$ is a polynomial with exclusively positive coefficients. Is it true that $f$ has a unique maximum? It seems like it when playing with ...
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Similar sums $\sum _{n=1}^\infty \frac{(-1)^{n-1}}{n^3}$ and $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)^3}=\frac{\pi ^3}{32}$

Is it possible to find $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3}$ if we know that $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)^3}=\frac{\pi^3}{32}$? Any help is welcome. ...
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Growth Rate of Supremum of Random Walk (with negative drift)

Let $(X_k)_{k=1}^\infty$ be a sequence of i.i.d. random variables, with $\mathbb{E}(X_i)$ finite and negative. Define $S_n := X_1 + ... + X_n$, and $M_n := \max(S_1,S_2,...,S_n)$. It follows from the ...
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river metric proof [closed]

I have river metric d(x,y)= |x2-y2| if x1 equals y1 and |x1-y1|+|x2|+|y2| if x1 doesn't equal x2. Proove that d(x,y)=<d(x,z)+d(z,y). I need the case if x1 doesn't equal x2 because I managed with ...
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3.1 Lemma of Four-manifolds with positive curvature operator

Picture below is from the Hamilton's Four-manifolds with positive curvature operator. I feel the red rectangle is not right. Maybe, it should be $$ \limsup_{h\searrow 0} \frac{f(c+h)- f(c)}{h} \le 0 $...
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one-dimensional Gelfand problem

For $t_0>0$, we define $$I(t_{0})=\int_{0}^{t_{0}} \frac{dt}{\sqrt {e^{t_{0}}-e^t}},$$ how to proof that $I(\cdot)$ takes values in some bounded interval and achieves its maximum at a unique point.
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4 answers
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Proof verification of the convergence question: If $\frac{1}{n}a_n\to0$, then $\frac{1}{n}\max\{a_1,\ldots,a_n\}\to0$.

I don't know how to use latex that well, so I'll upload the imagines of the question and the way I tried to prove it. I am unsure if my proof is correct. If it's not correct, then could you explain ...
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2 votes
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existence of a solution of a differential equation

we have a continuous function $\omega:\mathbb{R}_+\rightarrow\mathbb{R}_+ $ such that $$ p'(t)=\omega(p(t))\\ p(0)=0$$ has only the trivial solution $p=0$ on $J=[0,b]$. Now the author ...
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1 answer
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Cartan's proof that there is a smallest positive zero of cosine function

In section 3, paragraph 3 of Chapter 1 of his "Elementary theory of analytic functions of one or several complex variables", Cartan wants to prove that $\cos y$ vanishes for a certain value ...
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2 answers
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Are these two limits equal?

The problem basically goes as follows: Let $x_i$ be a sequence of real numbers. Let $f : \mathbf{N} \to \mathbf{N}$ be a bijection. And finally, let $h : \mathbf{R} \to \mathbf{R}$ be some function. I ...
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Some question of upper and lower limits in a paper.

I have read a paper and cofused some notions and some technical details. First, I state some definitions in this paper. Let $\mathcal{P}_\mathbb{N}$ and $\mathcal{P}_n$ be respectively the power sets ...
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Prove that $\sum\limits_{cyc}{} \frac{a^3}{b+c} \geq \sum_\limits{cyc}{} \frac{a^4+b^3c-b^2c^2+bc^3}{(a+b)(a+c)}$

If $ a,b,c>0 $, prove that :$$\sum_{cyc}{} \frac{a^3}{b+c} \geq \sum_{cyc}{} \frac{a^4+b^3c-b^2c^2+bc^3}{(a+b)(a+c)}.$$ my attempt: After uniting the denominator and dividing by$(a+b)(a+c)(b+c)$ $\...
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What is the error in my proof that every Cauchy sequence is bounded.

I am trying to find the error in my proof. I have used two separate cases of the triangle inequality to determine that $|s_n - s_m| < \epsilon \implies |s_n - L|< \epsilon$, for some $L\in\...
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How can you prove that $B(x,r)\subseteq B(0,s)$ where $0<r<s$?

I'm approaching the problem in the following way: If $B(x,r)=\{y\in X:|x-y|<r\}$ and $B(0,s)=\{y\in X:|y|<s\}$ then it must be the case that $y\in (x-r,x+r)$ for the first ball and $y\in (-s,s)$ ...
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On multiplication semigroups

I'm studying One-Parameter Semigroups for Linear Evolution Equations by Engel and Nagel. I have a question about multiplication semigroups. So we define $$T_q(t)f=e^{tq}f,$$ and we claim for $$\|T_q\|...
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How to calculate and show total derivative

$ℝ^2\to ℝ$ $f(x, y) = \frac{xy}{(x^2+y^2)}$ for $(x, y) ≠(0,0)$, and $f(x, y) = 0$ for $(x, y) =(0, 0)$ Calculate the partial derivative $\frac{df}{dx} (x, y)$ and $\frac{df}{dy} (...
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if a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$ [duplicate]

problem: a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$ my attempt: $LHS=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=(a+b+c) \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{a+b+...
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2 answers
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How to show that the integral is finite $\int (1+|x|^{2\delta}) e^{-B|x|^\gamma}dx<\infty$?

Suppose that $V$ is some continuously differentiable function on $\mathbb{R}^d$ and we have for some $0<\gamma \le 1$ such that for all $|x|\ge C_2$, $V(x)\ge C_1 |x|^\gamma$, for some constants $...
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-1 votes
0 answers
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$\lim _{n \rightarrow \infty} \int_{0}^{1} f\left(x, \sum_{k=1}^{n} a_{k} x^{k}\right) d x$ [closed]

Let $\left\{a_{k}\right\}_{k=1}^{\infty}$ be a sequence of numbers satisfying $\left|a_{k}\right| \leq k^{2} / 2^{k}$ for all $k$ and let $f:[0,1] \times \mathbf{R} \rightarrow \mathbf{R}$ be ...
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2 votes
0 answers
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Problem 18 chapter 4 from baby Rudin

Question: Every rational $x$ can be written in the form $x = m/n$, where $n > 0$, and $m$ and $n$ are integers without any common divisors. When $x=0$, we take $n=1$. Consider the function $f$ ...
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1 answer
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Show that $\lim f_n(x_n) = 0.$

Here is the question I want to solve $(b)$ in it: And here is a trial for the solution: But I did not get the idea for the answer of the first part, could someone explain this idea to me please? ...
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1 answer
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Trying to understand the statement of Rudin's Rank Theorem.

In Rudins's Principles of Mathematical Analysis, he states the (Constant) Rank Theorem like this: Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open ...
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1 vote
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Prove that if for a sequence $a_n$ we have that $a_{n+1} >a_{n}$, then for $m>n$, $a_m>a_n$.

The problem is as follows. Q: Let $(a_n)_{n=0}^\infty$ be a sequence such that $a_{n+1} >a_n$. Prove that for two natural numbers $m$ and $n$ such that $m>n$, $a_m>a_n$. I think I am ...
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Notation for dominating (or uniformly bounded) functions

While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function. A situation like this. For some true function $f:\mathbb{R} \to \...
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-1 votes
1 answer
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$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I want to prove this formula using Stirling's approximation or otherwise:$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ wher $s=\sigma+it$. Can someone please ...
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Duhamel's principle: show that this integral equation solves the inhomogeneous heat equation

Let $\Delta$ be the Laplacian on a bounded domain $\Omega \subseteq \mathbb{R}^{d}$. Let $s \geq 0$. Assume that $(x, t, s) \mapsto$ $v_{F}(x, t, s)$ solves the IVP of the heat equation $$ \begin{...
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0 votes
1 answer
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Finding an Upper bound for modulus of a function

Consider the integral $$f(s)=\int_{0}^{\infty}\frac{\left(\frac{1}{2}+ix\right)^{1-s}}{\cosh ^2(\pi x)} dx$$ where $s=\sigma+it$ and $\sigma,t\in\mathbb{R}$. Find a Big-O upper bound for the above ...
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4 votes
0 answers
36 views

Can differentiability classes be extended to negative values?

A function $f$ is said to be of differentiability class $C^k$ if its first $k$ derivatives are continuous. It has the property that $m>n \implies C^m \subset C^n$ and that all $C^k$s are algebras. ...
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2 votes
3 answers
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Calculate the area of the surface $x^2 + y^2 = 1 + z^2$ as $z \in [- \sqrt 3, \sqrt 3]$

The question states the following: Calculate the area of the surface $x^2 + y^2 = 1 + z^2$ as $z \in [- \sqrt 3, \sqrt 3]$ My attempt In order to solve this question, the first thing I think about is ...
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0 answers
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How do I show that show that lim (n^2/n!) = 0? [duplicate]

since this is the same as nn/(n-1)!n = n/(n-1)!, i can also prove that this goes to zero right? But I don't have any idea on how to write the prove.
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2 votes
0 answers
30 views

What is the equivalent condition for a smooth function on $\mathbb{R}^n$ to be well-defined as a smooth function on $S^{n}$?

The question is as in the title. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a smooth and bounded function. What would be the necessary and sufficient condition for this $f$ to be extended to a smooth ...
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1 vote
0 answers
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Changing index of a double summation [duplicate]

How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \sum_{\ell=1}^{P}\sum^{\ell-1}_{i=0} x(\ell,i,\ell-i) \quad = \quad \sum^{P}_{j=1}\sum^{P-...
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1 vote
0 answers
39 views

On Banach space $X$, does there exist a map similar to $1/z$ that maps elements with very large norm close to elements with norm $0$?

Basically, I want to ask is there a continuous mapping $F$ from the Banach space $X$ to its closed unit ball such that $||F(x_n)|| \to 0$ as $||x_n|| \to \infty$ kind of like the stereographic ...
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  • 3,191
0 votes
1 answer
78 views

Changing index of a summation [closed]

How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \sum_{\ell=1}^{P}\sum^{P-\ell}_{i=0} x(\ell,i,\ell-i) \quad = \quad \sum^{P}_{j=1}\sum^{P-...
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0 votes
1 answer
38 views

Questions regarding the Residue theorem (complex analysis)

I am trying to figure out how to calculate the residue of a function f(z) at z=$\infty$. My best guess is to define $g(z)=f(\frac{1}{z})$ then find the Laurent series for g(z) at z=0 and then find the ...
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2 votes
2 answers
136 views

Solution verification of a Double Integral

This question was left as an exercise in class of multivariable calculus of my brother and I am not sure about my solution of it when he asked me the same. Question: Compute the integral $\...
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2 votes
0 answers
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estimation of this formula, whether it is non-positive

I am reading a paper which gives $$-\frac{\cos\alpha}{2}\big [ \sum_{i=1}^N (\gamma_i^2+o(\gamma_i^4)\big ]+\sin\alpha\sum_{i=1}^No(\gamma_i^3)\leq 0$$ where $N$ in an integer that larger than one, $\...
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0 votes
0 answers
10 views

Why is the $D_{KL}(P||Q)\neq\infty$ as $P$ and $Q$ on the same support?

Define the Kullback-Lieibler (KL) divergence for two probability measures $P$ and $Q$ : $$ D_{KL}(P||Q):=\int p(x)\log (\frac{p(x)}{q(x)})dx $$ I try to convince myself that the KL divergence for the ...
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-5 votes
0 answers
43 views

Finding $\lim \limits _{(x,y)\to (0,0)}\frac{x^4-y^4}{x^2+y^2}$ [closed]

Determine whether the following limit exist, if so find the value:$$\lim \limits _{(x,y)\to (0,0)}\frac{x^4-y^4}{x^2+y^2}.$$
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3 votes
0 answers
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+100

$\lim_{r\to\infty} \int \cos^2(rz + t)\, d\pi_{\#}\mu(z) = \frac12$ for all $t\in \Bbb R$

This question stems from the identity listed as Equation $(6.6)$ in this paper on Pg. $11$. We want to show $$\color{blue}{\lim_{r\to\infty} \int \cos^2(rz + t)\, d\pi_{\#}\mu(z) = \frac12} \tag{6.6}$$...
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1 vote
0 answers
47 views

Proof in set theory with infinite indices

Prove that $\bigcap_{k=1}^{\infty}\{ \bigcup_{n=k}^{\infty} E_n\}=\{ x: x \in E_n$ for an infinite number of indices $n \}$ Can't solve that, just trying to begin with contradiction: $x \in E_n$ for ...
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