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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1answer
350 views

Topological degree

I need help for this exercice 1)Let $\Omega$ be an open and bounded set from $\mathbb{R}^n$ and $f\in C(\overline{\Omega})$ ,we suppose that there exists $x_0 \in \Omega$ such that :if for $x\in ...
6
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1answer
4k views

square root of a sum? Bound?

Is there any bound for an expression like: $$\left( a_1 + a_2 + \cdots + a_n\right)^{1/2} \leq ?$$ I need it for $n=3$. I know Hardy's inequality but it is for exponent greater than 1. Is there ...
3
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4answers
3k views

composition of continuous functions

I was wondering if a function $f:[a,b]\rightarrow[c,d]$ is continuous, $g:[c,d]\rightarrow\mathbb{R}$ is continuous, does it necessarily imply that $g\circ f$ is continuous? Are there counterexamples? ...
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1answer
147 views

When is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$? [duplicate]

Given a measure space $(X,\Sigma,\mu)$, when is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$ for $p>r$, or for $p<r$ . Thanks.
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1answer
77 views

A function not in $L^2(\mathbb{R}^3)$

From this equation $$ (p^2-\alpha)\hat{f}(p)=\frac{e^{-ip\cdot y}}{p^2+\lambda}$$ where $\hat{f}$ is the Fourier transform, $\alpha,\lambda>0$ e $y$ a fixed point in $\mathbb{R}^3$ can I conclude ...
1
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1answer
43 views

Maximizing the Volume of Body under a Function

Given is the function $y = 1/2 (4-x)\sqrt{x} $ One has to calculate a) the volume of revolution between function and x-axis, restricted by the function's zeros. b) What is the biggest volume possible ...
4
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2answers
223 views

Second Countability of Euclidean Spaces

Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ...
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3answers
148 views

Non Uniformly Elliptic Equations page 117 [G-T]

Let $\Omega\subset\mathbb{R}^n$ be open and bounded. Suppose also that $\Omega$ satisfies the exterior sphere condition at $x_0$ and let $B=B_R(y)$ be a ball such that $B\cap\overline{\Omega}=x_0$. ...
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1answer
119 views

Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$

Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
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1answer
288 views

why $\exp({1\over-x^2})$ is not real analytic?

Real analytic function. Can someone explain why $\exp({1\over-x^2})$ is not real analytic? i have read a book talking about smoothness of functions and it talks about this function is not real ...
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1answer
93 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
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1answer
74 views

Equintinuity of bounded linear functions equivalent to uniform boundedness

The claim is the following: Every family of bounded linear functions is equicontinuous if and only it is uniformly bounded. Equicontinuity is defined here. Any suggestions about this? I only ...
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10answers
13k views

Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
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1answer
113 views

Mean Value Property From Brownian Motion

Let $B_t$ $(t \geq 0)$ be a Brownian motion on $\mathbb{R}^3$. That is, $B_t = (B_{t}^{(1)},B_{t}^{(2)},B_{t}^{(3)})$, where each $B_{t}^{(i)}$ is a Brownian motion on $\mathbb{R}$. Let $D$ be a open, ...
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2answers
302 views

$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$

Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$ Show that $$3\int_{0}^{1}(f'(x))^2 ...
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1answer
123 views

Regarding Hölder continuity

Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - ...
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2answers
392 views

Prove that $f$ is discontinuous at $(0,0)$

Let $f$ be defined by $$ f(x,y) = \begin{cases} \biggl\lvert \frac{y}{x^2} \biggr\rvert e^{-\bigl\lvert \frac{y}{x^2} \bigr\rvert} , \quad \text{ if $x \neq 0$} \\ 0, \qquad \qquad \quad \text{if $x ...
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2answers
143 views

Series of Fourier coefficients

Let $\mathscr B=\{\phi_n\}_{n\in\mathbb N}$ be an orthonormal basis of the real Hilbert space $L^2([0,1])$, and given $f,g \in L^2([0,1])$ let $\langle f,g\rangle$ denote the usual scalar product ...
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3answers
565 views

Weak convergence in reflexive Banach space

Let $X$ be a reflexive Banach space. Let $T: X \to Y$ a linear operator. I want to show that: $$T \in \mathcal{L}(X, Y) \iff ((x_n \stackrel{w}{\rightharpoonup} x) \implies (T(x_n) ...
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2answers
137 views

Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
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2answers
29 views

Problems with determining convergence of integral

It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent: $$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$ I get for all $\alpha \neq 1,2$ ...
2
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2answers
141 views

Stability analysis, or, Can we prove this limit to be zero?

Let's think about this ODE $$ \dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0, $$ where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes ...
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1answer
122 views

Laplacian(F) = (n-1/r)g'(r) + g''(r)

I got one more problem from my self reading of Methods of Advanced Calculus by Edwards, hints and solutions are equally appreciated: If f(x) = g(r), r= |x|, and n>=3, show that Laplace(f) = ...
5
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1answer
167 views

An Integral Involving Brownian Motion

Let $B_t$ $(t \geq 0)$ be a Brownian motion on $\mathbb{R}^3$. That is, $B_t = (B_{t}^{(1)},B_{t}^{(2)},B_{t}^{(3)})$, where each $B_{t}^{(i)}$ is a Brownian motion on $\mathbb{R}$. Let $Y$ be a Borel ...
3
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1answer
105 views

Is this function convex

Is function $$f(x, y) = \left(\frac{x}{y} - a\right)^2 \left(\frac{y}{x} - \frac{1}{a}\right)^2$$ convex on the domain $$\{(x,y): x, y \in \mathbb{R}, x >0, y >0 \}\quad?$$ Now I think that it ...
0
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1answer
85 views

Prove directly from the definition of mapping that h is differentiable

Hey guys I'm going through some problems on my own, currently going through both chapter 2 and chapter 3 of Advanced calculus of several variables by C.H. Edwards. Anyway I'm having problems with ...
2
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1answer
119 views

Self-adjoint operator and inner product

I am wondering whether there is a way to make sense of self adjointness of an operator on $C[0,1]$ without resorting to the inner product of $L^2[0,1]$. I am not referring to concrete alternative ...
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6answers
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Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$

Prove that : $$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$ the only thing I can think of is differentiating the RHS and trying to get : $$ -2 ...
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2answers
747 views

Exponential Function - proof of continuity at $x=0$

How do you prove that the exponential function is continuous at $x=0$ and how do you prove it is continuous for all real $x$?
9
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1answer
128 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
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2answers
117 views

Let $f \in C(\mathbb{R})$ and $A \subset \mathbb{R}$ open. Prove that $f^{−1}(A) := \{x \in \mathbb{R}:f(x) \in A\}$ is open.

Let $f$ be continuous on $\mathbb{R}$ and $A$, a subset of the reals, be open. Prove that $f^{-1}(A) := \{x \in \mathbb{R}:f(x) \in A\}$ is open.
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3answers
102 views

How to prove there is no positive and continuous function satisfying some conditions

Let $\alpha \in \mathbb R $. How could I prove there isn't any positive and continuous function $f$ such that the following conditions hold? $\int_{0}^1 f(x)dx=1$ $\int_{0}^1 xf(x)dx=\alpha $ ...
0
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1answer
114 views

Prove that $f$ is continuous if and only if $f(\cdot+t) \to f$ pointwise as $t \to 0$

Let $f:\mathbb{R}\to \mathbb{C}$ be a function. Prove that $f$ is uniformly continuous if and only if $f(•+t) \to f$ in $L_{∞}$ as $t\to0$ Prove that $f$ is continuous if and only if $f(•+t) \to f$ ...
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1answer
1k views

Working with the ~ (tilde) notation (asymptotic analysis)

For positive functions $f$ and $g$ on real domains, define $f(n) \sim g(n)$ to mean $\displaystyle\lim_{n\to\infty}\frac {f(n)}{g(n)}=1$. Given that $$\frac{n^{n+\frac12}}{e^{n-1}n!}\sim\frac ...
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2answers
282 views

$y''\pm e^ty=0 \implies \mid \cup x_i \mid =? s.t. y(x_i)=0 $

I have this question, and i don't know how to solve it: Show that the solutions of $y''+e^ty=0$ admit an infinite number of zeros. Also, how to prove that the solutions of $y''-e^ty=0$ admit not ...
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0answers
738 views

Green's function for the Laplace operator $\Delta$ in a rectangle (or square)?

What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm ...
1
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1answer
105 views

Picone formula (Sturm oscillation)

We consider the equations: $(p_1y')'+q_1y=0 ...(E_1)$ ,$(p_2y')'+q_2y=0 ...(E_2)$ $p_1,p_2 \in C^1([0,1],(0,\infty)) ; q_1,q_2 \in C([0,1],\mathbb{R})$ $y_1$and $y_2$ are respectivly solutions of ...
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4answers
88 views

$f(n)=2f(\lfloor{n\over2}\rfloor)+1$ . Prove that $f(n)=O(n)$.

$f(n)=2f(\lfloor{n\over2}\rfloor)+1$ where $n$ is positive integer and $f:Z^+\to Z^+$. Prove that $f(n)=O(n)$. Attempts: I have figured the case that for $n=2^k$, $f(n)=2n-1$ which can be obtainly by ...
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2answers
137 views

Cauchy sequence property

I do not see how this is even valid. Could someone point this out to me: Assume that $x_n$ is a cauchy sequence of rational numbers satisfying $|x_n| \geq r$ for all $n\in\mathbb{N}$. Show that there ...
2
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1answer
350 views

Limit of $\arctan z$

,$\displaystyle \lim_{z \rightarrow \infty} \arctan(z) = \frac{\pi}{2} $. One way to see this is to put $\displaystyle z = \frac{y}{x}$ and imagine $y$ and $x$ as the sides of a right triangle. Then ...
3
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2answers
146 views

Banach-algebra homeomorphism.

Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ ...
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2answers
117 views

Does the Intergral Test require the integrand to converge to zero to be applicable?

Is there an error in the following statement of the Integral Test (from David Brannan's Mathematical Analysis)? The author requires the integrand (equivalently, the general term of the series) to ...
3
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1answer
387 views

Cauchy but not fast cauchy

As the title indicates, I am trying to find a Cauchy sequence that is not fast (or rapidly) Cauchy. Could anyone suggest something? A sequence $\{a_n\}_{n \in \Bbb N}$is termed fast (or rapidly) ...
2
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1answer
301 views

Uniqueness of PDE BVP/IVP Modified Wave Equation

Let $U\subset\mathbb{R}^{n}$ be open, $q(x)\geq0$ continuous, and suppose $u\in\mathscr{C}^{2}(U\times[0,T])$ solves $$\left\{\begin{array}{rl} u_{tt}-\Delta u=q(x)u&\text{in}\;U_{T}\\ ...
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0answers
161 views

Change of variables formula for a general measure

In a paper, pp11, I read the equality ...
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0answers
15 views

Conservative Fields and Denseness

Let $\Omega\subset\mathbb{R}^3$ be a open domain and $F=(F_1,F_2,F_3):\Omega\rightarrow\mathbb{R}^3$ a continuous field. Suppose that does not exist $u:\Omega\rightarrow\mathbb{R}$ with $\nabla u=F$. ...
2
votes
2answers
54 views

Integrating factor of a differential arising from thermodynamics

Let $\delta E = (xy^2 + xye^x)dx + (2x^2y + xe^x)dy$ I now need to find the integrating factor $\mu (x,y)$ s.t. $dS = \mu (x,y) \delta E$ is a exact differential. Now as far as I know $\delta E$ is ...
0
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1answer
84 views

Aronszajn's criterion for Euclidean space

Referring to "Aronszajn's Criterion for Euclidean Space " by R.D. Arthan: Can any one help me to understand why in lemma one $\|(a+1)p+q\| = \|(a+1)p-q\|$ and again how $\|ap+(b+1)q\|=\|ap-(b+1)q\|$? ...
1
vote
1answer
529 views

Integration over inequality

What are the conditions for taking integral over an inequality? For example given the following inequality: $f(x)<g(x)$ when can we say $$\int_{a}^{b}f(x)dx<\int_{a}^{b}g(x)dx $$ holds? Please ...
2
votes
1answer
88 views

What is the limit of this series?

assume $$x_n=\frac{n+1}{2^{n+1}}\sum_{k=1}^n\frac{2^k}{k} ,n=1,2,.....$$ how compute $\lim_{n\to +\infty}x_n$? Thanks for any hint