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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5 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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0answers
14 views

local inverse functions

consider $f(x,y)=(xsin(y),xcos(y)),\; (x,y)\in (0,\infty)\times (0,3\pi)=U$. f is locally invertible at every point in U, because $det(Df(x,y))\not= 0$ for all $(x,y)\in U$. I want to know : What are ...
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0answers
27 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
2
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4answers
91 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
2
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1answer
40 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
0
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1answer
40 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
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1answer
24 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
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0answers
52 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
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1answer
52 views

Zeros of an analytic function

How to prove zeros of a real analytic function (non-zero function) is always countable?
2
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1answer
60 views

Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$ [duplicate]

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
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1answer
23 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
1
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1answer
38 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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0answers
19 views

Searching for a condition on the derivative $f_u$

Please wht can be the condition on $f_u$ such that we obtain the following equality: $$\int_0^1 \int_0^1 G(t,s)f_u(s,0) v(s) w(t) \ ds\ dt=\int_0^1 \int_0^1 G(t,s)f_u(s,0) w(s) v(t) \ ds\ dt$$ ...
0
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0answers
9 views

Calculate factor for FWHM in a sech(x)-function

I have a $sech\left(\frac{\pi}{2}a\cdot x\right)$-function, and I want to calculate $a$ such that the FWHM of the function meets a specific width $\Delta x$. So I started with ...
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0answers
39 views

Research areas lying at the confluence of Analysis and Geometry

I wanted to get expert opinion on what are the areas of active research lying at the confluence of Analysis and Geometry. Two areas that I have heard about are : (1)Geometric Analysis and ...
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1answer
26 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
2
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2answers
62 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
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1answer
23 views

Nest of intervals explanation

I am currently reading Konrad Knopp book about infinite series, I just don't get the part where he mentions that the nest of intervals would determine or define as he said a rational number s if it ...
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0answers
24 views

Methods of Real Analysis solution book [on hold]

Where can I find a solutions book for Methods of Real Analysis by RIchard Goldberg?
3
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2answers
37 views

When is a continuous function piecewise monotone?

Given a continuous function $f:[a,b]\mapsto \mathbb{R}$, are there known additional conditions that ensure $f$ is piecewise monotone? Like this question, my motivation is to decompose the interval ...
0
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1answer
14 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
3
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2answers
37 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
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0answers
16 views

The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is positive. We can observe that the function $|f|^2$ has some nice property : ...
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0answers
22 views

How to find the domain of the support function

First the definition of the support function of set $S$: Let $S$ be a nonempty convex set. The support function $h$ of $S$ is the real-valued function defined by $$h(x)=\sup_{s\in S}\langle ...
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1answer
29 views

Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$

I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam. We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential ...
1
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3answers
59 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
2
votes
1answer
42 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
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1answer
31 views

Is the integral with respect to increasing continuous functions the limit of integrals with respect to $C^1$ functions?

if $\xi$ is continuous increasing can we find $\xi^n\in C^1$ such that $$\int_0^t f(u)\, d\xi = \lim_n\int_0^t f(u)\, d\xi^n$$ for every continuous $f$?
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1answer
29 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
1
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1answer
24 views

Complement of the union of finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , for $n>1$ , path connected? [on hold]

Let $D_1,D_2,...,D_k$ be finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , where $ n \ge 2$ . Then is $\mathbb R^n \setminus \cup_{i=1}^k D_i$ path connected ?
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2answers
44 views

set of all accumulation points of A is countable

How do I approach this question stating: Construct a compact set A of R such that the set of all accumulation points of A is countable. F compact means closed and bounded. Let $x_k$ element in it. ...
4
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1answer
38 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
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2answers
47 views

Is this sequence is dense?

Define $S _m, _n = $ n th smallest square number which is bigger or same than $10^ {m-1}$and smaller than $10^m$ Then is the sequence $ \frac{S_m,_n} {10^m}$ is dense in (0,1) or arbitary ...
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3answers
39 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...
0
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1answer
50 views

Does $\sum_{n=1}^{\infty}a_k^2<\infty$ imply $\sup_{n\in \mathbb{N}}na_n<\infty$? [on hold]

Let $a_k>0, k\in \mathbb{N}$. Suppose that $\sum_{n=1}^{\infty}a_n^2<\infty$. Does it implies $\sup_{n\in \mathbb{N}}na_n<\infty$? Thanks.
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0answers
43 views

Application of Arzela Ascoli theorem?

Let $\Omega$ an open bounded domain in $\mathbb R^n$ and consider $u_k: \Omega \to \mathbb R , k=1,2,...$ a sequence of functions. Suppose that $|u_k(x)| \leq 1$ for all $x \in \Omega$. Suppose that ...
0
votes
1answer
37 views

Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
2
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0answers
36 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
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0answers
36 views

Fundamental Lemma of the Calculus of Variations with higher derivatives

The fundamental lemma of the calculus of variations is often presented as: If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...
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1answer
38 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
0
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1answer
14 views

Jacobian of the Kelvin transform

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$ ...
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0answers
37 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
4
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2answers
43 views

If $\langle f'(x) \cdot v , v \rangle > 0$ then $f$ is injective

Question: Let $f: U \to \mathbb R^m$ differentiable at the convex set $U \subseteq \mathbb R^m$. If $$\langle f'(x) \cdot v , v \rangle > 0 , \,\,\, \forall\,\, x \in U, v \neq 0 \in \mathbb ...
2
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1answer
23 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
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0answers
40 views

Explicit expression of a sequence [on hold]

In my research, I am stack in finding the explicit expression of the sequence $(x_k)_{k \geq 1}$ defined as the following: Given a positive integer $r.$ The sequence is such that $x_k=\frac{1}{rk+1}$ ...
3
votes
1answer
30 views

Show the following are not connected in $\mathbb{R}^n$

Is the interior, boundary and closure of a connected set in $\mathbb{R}^n$ connected? I know the interior is not connected we can show it by a counterexample but I am not quite sure for the ...
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0answers
79 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
1
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0answers
18 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
0
votes
1answer
34 views

Existence of such a function

I am supposed to construct a function $f \in C_c^1((-\frac{3R}{4},\frac{3R}{4}))$ such that $f|_{(-\frac{R}{2},\frac{R}{2})}=1$ and $|f'(x)| \le \frac{4}{R}$ for almost all $x \in (-R,R)?$ I ...
2
votes
0answers
48 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...