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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conversion of discrete to continuous

Given $N_{j+1}-N_j=kN_j$ How can I substitute some time variable in to make $delta(t)$ small? Meaning change in time. I need to show $N_j=e^{(j\ln(1+k))}$ How can I rewrite the given in terms of ...
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1answer
43 views

Reparametrization of an absolutely continuous curve

If $\alpha : [0,1] \rightarrow \mathbb{R^n} $ is $C^1$ and $\alpha'(t) \neq 0$ for all $t\in[0,1]$ then there always exists a reparametrization in which $\| \alpha'(s) \| = 1$. Is there an equivalent ...
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1answer
32 views

Asymptotic behaviour of a function of a bivariate normal vector

Let $(Z_1,Z_2)$ be a bivariate standard normal vector and $x\in\mathbb{R}$. We consider $$f(\sigma_l):=\left| \operatorname{E}[1\{Z_1\leq x/\sigma_l\}1\{Z_2\leq ...
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1answer
51 views

Is such a function of bounded variation?

Let $f:[a,b] \rightarrow \mathbb R$ and let $(x_n) $ be a given sequence of points such that: $$ a<x_{n+1} <x_n<b \textrm{ for } n\in \mathbb R, \atop x_n \rightarrow a. $$ Let's assume that ...
2
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0answers
45 views

Is $||f||_p$ continuous in $p$ [duplicate]

I just started learning about $L^p$ spaces today and I have this question: Let $(X,\scr{M},\mu)$ be a measure space. Let $f:X\rightarrow \mathbb{C}$ be measurable. Consider ...
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1answer
92 views

Integration and uniform norm

Suppose that $f$ is twice differentiable on $\mathbb{R}$ and $\|f\|_\infty = A$ and $\|f''\|_\infty = C$. Prove that $\|f'\|_\infty\leq \sqrt{2AC}$. Hint: $f'(x_0) = b > 0$. Show that $b-C|t| ...
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0answers
134 views

Are there bump functions which have infinitely many smooth integrals?

While it is well-known that bump functions are smooth yet non-analytic $C^\infty$ functions, I was wondering if like the latter bump functions do also possess infinitely many smooth integrals, i.e. ...
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1answer
15 views

Asymptotics of a real sequence

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence with $a_n\in O(n^d)$ $(d\in (-1,0))$. Now we consider the expression $$ b_n:=(1-\sqrt{1-a_n}).$$ Is $b_n\in O(\sqrt{n^d})$? Thanks!
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1answer
39 views

Is the set where $\mathrm{dist}(x,\{1,1/2,1/3,\ldots\})$ is not differentiable a closed set?

Suppose that $ A=\{1,1/2,1/3,...\}$ and $f: \mathbb{R}\to\mathbb{R}$ such that $f(x)=\inf \{|y-x|;y \in A\}$. Let $K$ be the set of points where $f$ is not differentiable. Is $K$ closed? Can $K$ be ...
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81 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
0
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1answer
75 views

Network throughput and message delay

I'm trying to figure out how to calculate the throughput. Throughput is defined as the rate (bits/sec) bits are transferred between a sender and receiver; also, . I have a source node and a ...
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1answer
220 views

Is $\mathbb R$ a normal topological space?

As in the title, in euclidean space is it always possible two find for two disjoint closed sets $A,B$ two open sets $U,V$ disjoint such that $A \subseteq U$ and $B \subseteq V$ (T4-property, normal)?
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1answer
34 views

Need help clarifying a proof ( limSn=SupS)

Let $S$ be a bounded nonempty subset of $R$ such that $Sup(S)$ is not in $S$. Prove $\exists$ a sequence $(S_n)$ of points that belong to $S$ such that $ limS_n=Sup(S)$. Let $t=Sup(S)$.then for ...
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1answer
104 views

differentiability of jump functions

In Stein's real analysis book, we consider a bounded increasing function $F$ on $[a,b].$ Consequently, we know that the set of discontinuities of $F$ on this interval is countable. Because $F$ is ...
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106 views

On the existence of $\sqrt{2}$ (guided)

Introduction: This is a homework assignment of mine, first I want to mention that I am aware of that there are many proofs all over the internet (including this site) about the existence of ...
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3answers
108 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
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1answer
125 views

prove Cauchy sequence

I have a problem in this exercise Suppose that ${(a_n)}$ is a sequence such that ${a_{2n}}$ ${}\le{}{}$ ${a_{2n+2}}$ ${}\le{}{}$ ${a_{2n+3}}$ ${}\le{}{}$ ${a_{2n+1}}$ for all n ${}\geq{}{}$ 0. Show ...
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2answers
106 views

Zeros of $C^\infty$ functions

If $f(x) \in C^\infty(\Bbb{R})$,and $f(a)=0$, do we have $$f(x)=(x-a)g(x)$$? where $g(x) \in C^\infty(\Bbb{R})$ and $g(a)=f'(a)$
2
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2answers
120 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
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1answer
39 views

Linear operators over rational coefficients

This is related to a question I answered earlier which raised a question in my mind. My question is the following, Suppose we have a vector space $\mathbb{V}$ with real coefficients. Let ...
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1answer
145 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
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0answers
69 views

How to prove that this integral converges absolutely?

$f:[a,{\infty}[\to\mathbb{R}\ $ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely ...
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1answer
169 views

Theorem 3.55 Rudin (rearrangement and convergence)

if $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement of $\sum a_n$ converges, and they all converge to the same sum. $\boldsymbol{proof:} let \sum a_n'$ ...
2
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1answer
652 views

Proving that the total variation of an absolutely continuous function is absolutely continuous

I am trying to prove this statement which appears in the real analysis text by Stein which he just passes as a remark. If $F$ is absolutely continuous on $[a,b]$, then the total variation of $F$ is ...
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1answer
216 views

Prove every open ball is both an open and closed set

I'm asked to prove that every open ball is both open an open and closed set So far, I've managed to show it's open: Given a ball $ B=B(x,r) $ I made a new ball, $ B=B(y,\delta)$ let z be an ...
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1answer
111 views

Smooth approximation with bounded derivatives

I want to approximate a periodic continuous function in $\mathbb{R}$ (for instance, the function $abs(x)$ defined in $[-1,1]$ and extended periodically) by a sequence of uniformly bounded period ...
0
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1answer
67 views

Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
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2answers
111 views

The complement of the closed subset of a closed set

Suppose that I had a closed set. Suppose that I made it so that there was nothing else outside such closed set. I will call this set "A". Now, suppose that I picked a closed subset from A. I will call ...
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1answer
96 views

Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.

Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x)  g(y)$ but under the polar coordinate system $F(x, y) = h(r)$. My ...
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1answer
93 views

Question on dense sets

We have a function $f(x)$, $x \in X$ where $X$ is a complete metric space and say $f()$ is continuous. Then say $f(y)=g(y)$, $y\in Y$, $Y\subset X$, $g()$ is continuous in $Y$, and $Y$ is dense. (a) ...
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1answer
96 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
4
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1answer
89 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
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2answers
42 views

$A \subseteq (X,d)$ is compact. Which metric $p$ makes $(A \times A,p)$ also compact and $d: (A \times A,p) \rightarrow [0,\infty)$ continuous?

$(X,d)$ is a metric space. And $A \subseteq X$ is a non-empty compact set in the metric space $(X,d)$. Then, does there exists a metrics $p$ and if so which metrics $p$ make $(A \times A,p)$ compact ...
0
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1answer
122 views

Positive real number has a finite number of binary when is in form $ m/2^n $

Prove that positive real number $ ( x \in \mathbb{R} \ x > 0) $ has a finite number of binary if and only if when is in form $ \frac{m}{2^n} $, where $ m, n \in \mathbb{N} $ I found this solution: ...
0
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1answer
74 views

Sequence with lower bound on gaps [closed]

Suppose a sequence satisfies $|a_i-a_j| \geq 1/j$ whenever $i<j$. Can such a sequence be bounded?
1
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1answer
53 views

How find the $\frac{1}{e}\le R\le 1$

Qustion: let $a_{n}> 0$,and such $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converge,let $$b_{m}=\sum_{n=1}^{\infty}\left(1+\dfrac{1}{n^m}\right)^na_{n}$$. show that $$\dfrac{1}{e}\le ...
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1answer
81 views

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$? Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two ...
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1answer
53 views

Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?

In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact ...
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1answer
29 views

How prove this $\overrightarrow{r_{i}}\cdot\overrightarrow{r_{j}}\ge\frac{1}{2}$

let $a,b,c$ are real numbers,and such $a+b+c=0,a^2+b^2+c^2=1$, we define: $\overrightarrow{r}=(x_{i},y_{i},z_{i})(i=1,2,3,4,5,6)$,where $\{x_{i},y_{i},z_{i}\}=\{a,b,c\}$, show that: there are exst ...
3
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1answer
156 views

$C^{2, \alpha}$ regularity for elliptic equations with Neumann boundary conditons

Say $\Omega\subseteq \mathbb{R}^n$ is a bounded open set and $0<\alpha<1$. I need some $C^{2, \alpha}(\overline\Omega)$ regularity result for elliptic equations with Neumann boundary conditions ...
2
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2answers
371 views

path connected subspaces of $\mathbb R^2$

I am trying to prove two statements that visually I think they are "obvious", but I am totally lost when it comes to do a formal proof. The statement of the exercise is: Decide whether $\mathbb R^2 ...
3
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1answer
326 views

Differentiating an integral using dominated convergence

Let's say we have $$F(x)=\int^b_af(x,t)\ \mathrm{dt}$$ And we want to calculate $F'(x)$. Then: ...
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1answer
69 views

Rolles Theorem Simple and multiple zeros

I have this problem with Legendre polinomials Use Rolle's Theorem to show that Pn cannot have multiple zeros in the open interval (-1, 1). In other words, any zeros of Pn which lie in (-1, 1) must be ...
2
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2answers
52 views

Positive semi/definite matrix claim.

If $A$, $B$ is positive semidefinite (PSD) and $C$ is positive definite (PD), all are Hermitian, complex valued. I want to claim that $$(B+C)^{-1/2}A(B+C)^{-1/2}$$ is PD. (I am sure it is PSD but ...
2
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0answers
350 views

Exercise 6.9 in Rudin's RCA (Real and Complex Analysis)

The following is an exercise 6.9 in Rudin's Real and Complex Analysis: Suppose that $\{ g_n \}$ is a sequence of positive continuous functions on $I=[0,1]$, that $\mu$ is a positive Borel measure on ...
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2answers
51 views

What is the answer of this problem?

Suppose that $f(x)$ is bounded on interval $[0,1]$, and for $0 < x < 1/a$, we have $f(ax)=bf(x)$. (Note that $a, b>1$). Please calculate $$\lim_{x\to 0^+} f(x) .$$
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1answer
209 views

definition of total variation of a complex measure does not depend on any algebra generating the sigma-algebra of that measure

While studying a course on "Vector Measures", I come to this problem: Let $\mu$ be a complex measure on a $\sigma$-algebra $\Sigma$, generated by an algebra $\mathcal{A}$. Its total variation $|\mu|$ ...
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1answer
70 views

Many points on hyperplane with probability zero

Let $m$ be a finite measure on $X \subseteq \mathbb{R}^n$, so that $m(\mathbb{R}^n) < \infty$. Define the hyperplanes on $\mathbb{R}^n$, parametrized by $A \in \mathbb{R}^{n \times n}$ and $b \in ...
1
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2answers
43 views

If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$

The problem as stated is Let $H$ be a Hilbert space. If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$. I ...
3
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1answer
130 views

How prove this analysis function $a\le\frac{1}{2}$

let $$f(x)=\begin{cases} x\sin{\dfrac{1}{x}}&x\neq 0\\ 0&x=0 \end{cases}$$ show that:there exsit $M>0,(x^2+y^2\neq 0)$ , $$F(x,y)=\dfrac{f(x)-f(y)}{|x-y|^{a}}|\le M ...