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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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9 views

Differntiable function prof

it is known that $f(0,0) = 0$ and $f$ is differentiable at $(0,0)$. It is also known that for every $t>0$ we have $f(\cos(t)/t , \sin(t)/t) > 0$. Show that necessarily $\operatorname{Gradient}(...
2
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1answer
31 views

Factorization for $e^{\lambda x}$

Let $\lambda, x$ be real numbers. Why can't we factorize $$e^{\lambda x}=f(\lambda)g(x)$$ for some functions $f$ and $g$?
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0answers
14 views

The convergence on an improper integral

Let us find all values of $\alpha$ such that the following improper integral converges: $$I(\alpha):=\int_{1}^{\infty} \frac{ e^{\dot{\imath}(x^2-2x)}}{x^{\alpha}}\,dx$$. When $\alpha>1$, $I(\...
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0answers
14 views

Property of supremum for any real exponent

For any $\alpha>0$ and $f:\Omega\to(0,\infty)$ bounded, is it true that $\sup f^{\alpha}\leq (\sup f)^\alpha$? Also what about $\alpha<0$. I know for $\alpha\in\mathbb{N}$, it holds. But for ...
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1answer
13 views

$\sigma$-finite measure $\mu$ so that $L^p(\mu) \subsetneq L^q(\mu)$ (proper subset)

I'm looking for a $\sigma$-finite measure $\mu$ and a measure space so that for $1 \le p <q \le \infty$ $$L^p(\mu) \subsetneq L^q(\mu)$$ I tried the following: Let $1 \le p <q \le \infty$ ...
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0answers
16 views

Estimate of the distance between a point and a closed real subset. [on hold]

Couldn't really find any idea for solving this... Do you have any? https://i.stack.imgur.com/Pr9Km.png
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3answers
23 views

Minimizing a function with a given inner product

I'm trying to solve a question but a not sure on how to approach. Here is the question Let $V$ be a finite dimensional space with a given inner product and $A \subset V$ be a convex set and $a \...
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0answers
22 views

A uniformly continuous function on a bounded set whose image is not bounded?

I'm trying to find an example of a uniformly continuous function $f:X\to Y$ such that $f(X)$ is not bounded, for some bounded metric space $X$ and a metric space $Y$. The classic exercise is to show ...
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0answers
28 views

Estimate on average with weight

Let $B_{2R}=B(x,2R)$ be the ball of radius $2R$ centered at $x$ and $v=log\,u$ for some positive function $u$ defined on $B_{2R}$. Denote by $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
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0answers
40 views

Find the general formula of the series

How to find the general formula: $${a_{n + 1}} = \frac{{a_n^2 + 3}}{{{a_n} + 1}},{a_1} = 1$$ and I have no ideas
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0answers
25 views

Exercise 4.6.11 (in Petrovic)

The question is given below: Could anyone give me a hint for its solution? or can I ask how is the proof different from the proof of the following theorem:
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2answers
39 views

A difficulty in understanding a proof for L'Hospital's rule (in Petrovic)

The theorem and its proof is given below: But I could not understand why $F$ & $G$ are defined as thought, could anyone explain this for me please?
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0answers
24 views

An inequality with constraints.

I came across a result in a control theory book (without proof), which states that: Given two variables $x,z \in \mathbb{R}$ and four parameters $c_{1}, c_{2}, k_{1}, k_{2}$ with $c_{1}, c_{2} > 0$...
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1answer
24 views

Is this correct? Trying to prove that the space $(C(K,\mathbb{R}^m), \| \cdot \|_{\infty})$ with $K\subseteq \mathbb{R}^n$ compact, is complete.

Now we did this proof in lecture and we had proved that if $(f_k)$ was a Cauchy Sequence in the space, then $f_k \to f$ (pointwise) for some function $f: K \rightarrow \mathbb{R}^{m}$. Then the ...
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2answers
25 views

Taylor series expansion for cos(z)/z about z=1

I feel like I am making this far too difficult for myself! The question states: Find $c_0, c_1, c_2, c_3$ from the Taylor Series expansion for ${cos(z)\over z}$ about $z=1$. I've tried rewriting ...
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1answer
29 views

How to show that f / f' has a removable singularity at 0

I'm new to complex analysis and am not sure where to start with this. The question states: Let the origin be a pole of order m > 0 of an otherwise analytic function f of a complex variable. Show ...
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1answer
33 views

Problem 4 Barry Simon a comprehensive course in analysis part 1.

(a) For any bounded Baire function, $f$, on a compact Hausdorff space, X, prove that $||f||_{\infty}:=\inf\left\{\sup_x |g(x)|: f-g=0\text{ for a.e. } x\right\}$ exists, defines a seminorm, and equals ...
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0answers
15 views

What does the word “analysis” mean when we say real/mathematical analysis or analytic geometry?

The choice of the word "analysis" has always confounded me. Is it because we're literally analyzing mathematical objects/relationships till we have a clear understanding of things? Is it quite ...
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1answer
26 views

Rudin 4.2 definition of a limit of a function

Baby Rudin theorem 4.2 presents an alternative definition of a limit: Suppose $X, Y$ are metric spaces, $E \subset X$, $f: X \rightarrow Y$, $p$ is a limit point of $E$. Then $\lim_{x \rightarrow p}...
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0answers
10 views

Stieltjes integral for monotonically increasing functions.

Let $$Z:[0,\infty)\rightarrow[0,\infty)$$ $$K:[0,\infty)\rightarrow[1,\infty)$$ monotonically increasing cadlag functions with $K(0)=1$ and $Z(\infty)<\infty$. I want to show that following ...
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1answer
16 views

Taylor's Polynomial with Lagrange's form of remainder.

Suppose our function be written as $$f(x)=P_n(x)+R_n(x)$$ where $P_n(x)$ is n-th Taylor's Polynomial about $x_0$ and $R_n(x)$ is associated remainder/error term both of which are given as $$P_n(x)= \...
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1answer
35 views

Prove these two functions are in $H^2(\mathbb{R}^n)$

I want to prove that the equation $$u -\Delta u=f$$ with $f \in L^2 (\mathbb{R}^n)$ admits a solution in $H^2 (\mathbb{R}^n)$ with $n=1,3$. Taking Fourier transforms and resolving, I get: $$u(x)=\...
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1answer
60 views

Show that $\overline{\mathbb{Q}\cap (0,1)}=[0,1]$.

Show that $\overline{\mathbb{Q}\cap (0,1)}=[0,1]$ My attemp: If $E=\mathbb{Q}\cap (0,1)$, then $$\overline{\mathbb{Q}\cap (0,1)}=\overline{\mathbb{Q}}\cap \overline{(0,1)}=\mathbb{R}\cap [0,1]=[0,1]$$...
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3answers
65 views

Is there a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{0}+\binom{4p}{1}+\cdots+\binom{4p}{p-1}}>\gamma^{p}$

In a proof of the Larman-Rogers conjecture (there is $\gamma>1$ such that $\chi(\mathbb{R}^{d})>\gamma^d) $ they used that there is a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{...
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2answers
66 views

Does the integral $\int_{0}^{\infty} \frac{x^3\,\cos{(x^2-x)}}{1+x^2}$ diverge

Does the integral $$J:=\int_{0}^{\infty} \frac{x^3\,\cos{(x^2-x)}}{1+x^2}dx $$ diverge ? If we integrate by parts we find $$J=\lim_{a\rightarrow +\infty} \frac{a^3}{(1+a^2)(2a-1)}\cos{(a^2-a)} -\\ \...
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2answers
59 views

Show that $\overline{E_{1}\cup E_{2}}=\overline{E_1}\cup\overline{E_{2}}$.

Show that $\overline{E_{1}\cup E_{2}}=\overline{E_1}\cup\overline{E_{2}}$. Hi! I know ther is a better proof of this problem, but I wrote my solution, and I need verify this is righ or not, thanks! ...
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1answer
27 views

How to prove a Convergent sequence with $\lim _{n→∞}$

Let $\{c_n\}$ be a convergent sequence with $\lim _{n→∞}c_n$ = 1. Show that there exists N ∈ $\mathbb N$ with n ≥ N we have $c_n$ ≥ 3/4.
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0answers
12 views

Problem with F be $C^1$ and determine its form.

Let $K \subset \mathbb{R^n}$ be a compact set,$U=K^o$, $a<b$.$f:K \times [a,b] \to \mathbb{R}^m$ continuous. Let $F: K \to \mathbb{R}^m$ defined by $$F(x)=\int_a^bf(x,t)dt=[\int_a^bf_1(x,t)dt,...,\...
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3answers
77 views

When does $\sum_{n=1} ^{\infty} (\sqrt [n]x -1)$ converge?

$A=\{ x| \sum_{n=1} ^{\infty} (\sqrt [n]x -1) \ \text{is convergent \}}$ . Now what is $A$ ? 1- $\{1\} $ 2- $(0, \infty)$ 3-$(0,1]$ 4-$(\frac{1}{e} , e) $ It is clear that $1 \in A$ and $\lim _{n ...
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1answer
30 views

If $f(\mathbb{R})$ is a subset of $\mathbb{Q}$, is it continuous or differentiable

I'm doing a practice paper before my exam and I came across the following question, but I don't have the memo. I know if a function is differentiable then it it must continuous, so that rules out ...
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0answers
6 views

Trivial Embeddings in Morrey & Campanato spaces

I'm taking a Nonlinear PDEs course this semester and the last time our professor introduced us to Morrey & Campanato Spaces. We have for $\lambda \gt 0$ that: The Morrey space $L^{2,\lambda}(\...
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2answers
18 views

Finding a limit of a function of two variables

Let $$u(x_1,x_2)=\frac{1}{n^2}\sin(nx_1)\sinh(nx_2)$$ with $(x_1,x_2) \in \mathbb{R}^2$. What happens to $u$ as $n \to +\infty.$ ? This is what I tried to do : $$-\frac{1}{n^2}\sinh(nx_2)\le u(x_1,...
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0answers
51 views

Ways to learn mathematical analysis. [on hold]

I'm studying mathematical analysis, more specifically differential calculus in $\mathbb{R}^n$, my main problems is that I didn't realize what is the best way to abbord some exercises, like what ...
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1answer
47 views

Prove that $\prod_{k=2}^{+\infty} (1+1/k^2) = \sinh(\pi)/(2 \pi)$.

My attempt 1: Let $x_n=\left(1+\frac{1}{2^2} + \cdots + \frac{1}{n^2} \right)$, and we have $x_{n+1}>x_n$. Since $$ 1+\frac{1}{n^2} \le 1+ \frac{1}{(n-1)(n+1)} = \frac{n}{n-1} \cdot \frac{n}{n+1}, $...
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0answers
29 views

Demicontinuity implies continuity

I want to check the continuity of the operator $A:X\to X^{*}$ defined by $$\langle A(u),v\rangle=\int_{\Omega}a(x,\nabla u)\cdot\nabla v\,dx-\int_{\Omega}g_k(u)v\,dx\quad\text{ for every }u,v\in X.$$ ...
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0answers
7 views

Prove Lemma 7.1.1 of the first competitiveness of competitive group testing

I got the problem of my proving lemma there is Lemma 7.1.1 of The first competitiveness of Competitive Group Testing. This is the lemma Lemma 7.1.1 For $0<d<ρn$, $M(d,n)≥d(\log n/d+\log(e\sqrt{...
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0answers
30 views

Prove that the series is Cauchy

Prove that $\ 2/5 +2/5^2 +2/5^3 +.....+ 2/5^n$ is Cauchy. Assuming $n$ is greater than or equal to $m$, it follows that $\ 2/5^{m+1} + 2/5^{m+2} + ...+2/{5^n}$ is less that or equal to 2/(m+1) + 2/(m+...
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1answer
26 views

Proving the integral of the cantor function

I'm trying to prove the integral of the cantor function on [0,1] is equal to 1/2. I'm thinking of using a symmetry argument by using the fact that the function is 1/2 on [1/3,2/3]. Then arguing that ...
3
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2answers
37 views

How to evaluate $\lim_{n\to\infty} \dfrac{1^p+2^p+…+n^p}{n^p}-\frac{n}{p+1}$? [duplicate]

$$\lim_{n\to\infty} \dfrac{1^p+2^p+...+n^p}{n^p}-\frac{n}{p+1}\text{ with } p\in\mathbb N$$ I don't really know how to start, I mean... I could try substituting by $\Big(\dfrac{n(n+1)}{2}\Big)^p$ ...
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1answer
21 views

Showing that a sequence of functions has no dominant (Dominated convergence theorem)

Define $u_n: \mathbb{R}^+ \to \mathbb{R}^+$ as $u_n(x) = n1_{(0,1/n)}$. I need to show that this has no dominant. A dominant is defined as an integrable function $w: X \to \mathbb{R}^+$ such that $\...
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1answer
17 views

How to find the supremum given a known infimum and supremum.

Let A ⊆ ${R}$ with supremum A = 3 and Infimum A = -1. Let B = {-a : a ∈ A}. Find the supremum of B. Justify your answer
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0answers
40 views

System of linear equations and asymptotic estimations: How to derive this solution? You

In the paper Slow Motion Manifolds, Dormant Instability, and Singular Perturbations by G. Fusco and J. K. Hale, the following system of equations is derived as part of a proof. Although I cannot ...
0
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1answer
20 views

$D=\{z\in\mathbb{C}:|z|<1\}$. Every compact subset $K$ of D, $K \subseteq D$ is in $rD$ for sufficiently large $0\le r<1$

$D=\{z\in\mathbb{C}:|z|<1\}$ How can I show that every compact subset $K$ of D, $K \subseteq D$ is in $rD$ for sufficiently large $0\le r<1$? When drawing a sketch of this problem it seems ...
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1answer
29 views

$f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$

I want to show: $f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$ Anyone got any hints how to prove ...
0
votes
0answers
21 views

Interverting things in analysis

I was wondering how exactly can you switch things in analysis, I can’t find a good summary on the net so Îm asking you. Here is my question: In which cases can you switch: _Derivatives and integrals/...
1
vote
1answer
20 views

Finding the series expansion of $\frac{1}{z-b}$ around $a$

I'm working through a question where I'm asked to find the series expansions of various functions around certain points. I've found the answer for most of the examples, but the last part of the ...
0
votes
1answer
28 views

On the intersection of Bouligand cones

Let $K, L$ be two closed convex subsets of the normed space $X$. If $$ 0 \in \operatorname{Int}(K-L) $$ Prove that $$ \forall x \in K \cap L \Rightarrow T_{L \cap K}(x)=T_{K}(x) \cap T_{L}(x) $$ In ...
0
votes
1answer
23 views

Showing $f$ is not differentiable at $(0,0)$ even though the partial derivatives exist.

$f\begin{pmatrix}x\\y\end{pmatrix}:=\begin{cases}\frac{x^2y}{x^6+2y^2}& (x,y)^T\neq(0,0)^T\\0&(x,y)^T=(0,0)^T\end{cases}$ I want to find out if $f$ is continuously partially differentiable ...
0
votes
0answers
23 views

Reciprocal of integrable function is integrable. [on hold]

Suppose $f$ is Riemann integrable on $[a,b]$ and $\frac{1}{f}$ is bounded on $[a,b]$. Prove that $\frac{1}{f}$ is Riemann integrable on $[a,b]$. I've stared at this for a while but I have no idea how ...
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votes
1answer
20 views

Product of integrable functions is integrable. [on hold]

Suppose that $f,g:[a,b]\to\mathbb{R}$ are bounded and integrable on $[a,b]$. Prove that $fg$ is integrable on $[a,b]$. I'm not sure how to go about solving this.