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
215 views

Slightly confused about the definition of upper limits and lower limits.

I'm reading "The way of Analysis" by Strichartz, and the following is the definition of an upperlimit. The upper limit (limsup) of a sequence $\{x_j\}$ is the extended real number $$\limsup _{k ...
2
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1answer
84 views

Completeness for bimetric spaces

Let $(X,d)$ be a complete metric space. Is it possible to find a second metric $d'$ such that $d(x,y) \le d'(x,y)$, $\forall x,y\in X$ for which $(X, d')$ is not complete?
0
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1answer
114 views

prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N} B_j$

Prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N}B_j$. I have the solution, but it is too short and it is not enough to prove it, also it's too ...
3
votes
1answer
133 views

Differentiate a hypergeometric function expression

I have the following function $$f_\epsilon (p)=\frac{1}{2}(1-p)^\epsilon 2^\epsilon {_2}F_1(1-\epsilon,\epsilon;1+\epsilon;\frac{1-p}{2}),\qquad p\in(-1,1).$$ Here $F$ is the hypergeometric ...
2
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1answer
66 views

Making a function in $W^{1,2}$ continuous

Let $\Omega$ be an open domain in $\mathbb{R}^n$, $u\in W^{1,2}(\Omega)$ and assume that for any $y$ in $\Omega$ $$\lim_{\varrho \to 0} \operatorname{osc}(u,B(y,\varrho)) \rightarrow 0 , \varrho ...
1
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0answers
92 views

Finding a complex function $f$ and the residue of $f '(z)$ at $z=0$

Let $U,a$ real positive constants, $\varphi_1, \varphi_2$ $C^1$ functions on $[0,a]$ with $\varphi_1(0) = \varphi_2(0)$ and $\varphi_1(a) = \varphi_2(a)$. The problem is to find an analytic function ...
2
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2answers
231 views

How can I know the time difference between two cities almost at the same latitude?

Well I know that's the earth rotation speed is: $v=1669.756481\frac{km}{h}$ I have two cities New York, Madrid almost at the same latitude and the distance between them is: $d=5774.39$ $km$ ...
1
vote
1answer
191 views

Unexpected Probability Theory Uses

I am a french student in mathematical engineering. I had to go trough an intensive 3 year "preparation" to pass a "concours" to go to High School. In mathematics, I have been taught a lot of algebra, ...
1
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2answers
170 views

How can I know the time difference between two cities by knowing the distance between them and earth speed?

Well I know that's the earth speed is: $v=1669.756481\frac{km}{h}$ and I have two cities Moscow and NewYork the distance between them is: $d=7518.92$ $km$ Actually I know that's : ...
1
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0answers
31 views

Distributional convergence question for Feller processes

To briefly go over the setup, $S$ the state space is a separable locally compact metric space, and $C_0$ is the space of continuous functions on $S$ that vanish at infinity. $D$ is the space of ...
0
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1answer
94 views

Extension of Cauchy sequentially regular function

To prove: If $A$ is a subset of a metric space $(X,d)$ and there is a function $f$ from $A$ to a complete metric space $(Y,e)$ which maps Cauchy sequences to Cauchy. Then there exists a unique ...
8
votes
1answer
197 views

Algebraic Proof of Stone-Weierstrass

I have seen a few proofs of the Stone-Weierstrass theorem today for the first time. While some were completely analytic, the proof given in Rudin's Principles of Mathematical Analysis hinted at an ...
6
votes
1answer
169 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
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0answers
31 views

Multiplication $ H\times P(1/x) $ in sense of distributions

If $ P(1/x) $ means the principal value of $1/x$ and $ H(x) $ is the Heaviside step function is this then correct (regularization) ...
1
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2answers
88 views

To express a maximum in terms of the original function

Suppose that you have a continuous function $$ S \colon \mathbb{R}\times [0, 1]\to [0, \infty).$$ Define an auxiliary function $$S^\star(x)=\max_{t\in[0,1]}S(x, t).$$ Does there exist a continuous ...
0
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1answer
285 views

prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $V$ of $\Bbb R^m$

Suppose that $A$ is open in $\Bbb R^n$ and $f$ is a function from $A$ to $\Bbb R^m$. Prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $V$ of ...
7
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1answer
155 views

Prove the density of this SDE is not smooth in a parameter

Consider the following, 1-dimensional, equation $$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$ where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ...
1
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3answers
386 views

Can we call domain as inverse image of a function?

I was going through the definition of inverse image of a function http://www.northeastern.edu/suciu/U565/MATH4565-sp10-handout1.pdf, and I was wondering if inverse image of a function is the domain of ...
3
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1answer
1k views

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
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2answers
97 views

Compact sets in a metric space

On a non finite metric space, every compact can be finite? I guess it is false, but I don't know how can one prove it, I already know that at most the space has finite compact sets, but I have not a ...
5
votes
1answer
77 views

A basic question on Type and Cotype theory

I'm studying basic theory of type and cotype of banach spaces, and I have a simple question. I'm using the definition using averages. All Banach spaces have type 1, that was easy to prove, using the ...
1
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1answer
28 views

$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$ and $\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$ implies $x=0$?

Let $X,Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ ...
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0answers
69 views

Showing that $\hat f(\xi) = \int_{\mathbb R} f(x) \exp (-2 \pi i \xi x) dx$ is continuous

Assume that $f(x) \leq \frac A {1+x^2}$ on $\mathbb R$. I now want to show that $$ \hat f(\xi) = \int_{\mathbb R} f(x) \exp (-2 \pi i \xi x) dx $$ is continuous. By some simple calculations I get $$ ...
3
votes
2answers
69 views

Multiple integrals with $x_1 < x_2< \dots < x_n$

I was looking at an example with the following integral: $$\iiiint_{0 \le x \le y \le z \le t,\ 0 \le t \le \frac{1}{2}} 1 \,dx\,dy\,dz\,dt = \frac{1}{16}$$ Is it true in general that $$\int \dots ...
2
votes
2answers
740 views

How to measure the distance between two cities in the map by knowing latitude point and longitude point of them?

I want to measure the distance between two points in a map. For example between London and Moscow by knowing that the latitude point and longitude point of them. ...
0
votes
1answer
32 views

Inegrable functions $f_k$ with $\frac 1 {2 \pi} \int_0^{2 \pi} |f_k(x)|^2 dx \rightarrow 0$ where $\lim_{k \rightarrow \infty} f_k$ does not exist

I am searching for integrable functions $(f_k)_{k=0}^\infty$ on the circle with $$ \lim_{k \rightarrow \infty} \frac 1 {2\pi} \int_0^{2 \pi} |f_k(x)|^2 dx = 0 $$ and s.t. $\lim_{k \rightarrow \infty} ...
1
vote
1answer
74 views

Prove this inequality concerning integral average.

Let $f\in L^1([a,b])$, and extend $f$ to be $0$ outside $[a,b]$. Let $$ \phi(x)=\frac{1}{2h}\int_{x-h}^{x+h}f $$ How to prove $$ \int_a^b\left | \phi\right | \leqslant\int_a^b\left|f\right| $$ To ...
3
votes
1answer
81 views

convergence of $\prod_{m=2}^\infty \frac {1}{1-m^{-s}}$

Is the product $$\prod_{m=2}^\infty \frac {1}{1-m^{-s}}$$ convergent for all real $s>1$$\space$ ?
3
votes
1answer
163 views

Question on $\liminf$ and $\limsup$

Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow ...
2
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3answers
69 views

Evaluating $\lim\limits_{x\to 0^{+}} \frac{x}{\ln^2 x}$

How can I find: $$\lim_{x\to 0^+} \frac{x}{\ln^2 x} $$ I know that the limit is $0$. I tried sandwich theorem but I don't know what could be bigger. Thanks in advance.
1
vote
1answer
138 views

taylor series for a function of matrices

Say I have a function $(A+B)^{-1}$ where $A$, $B$ are matrix-valued functions of some vector $x$. Can I then expand this function around $x=0$ as: $$(A+B)^{-1} = (A[0]+B[0])^{-1} - (A[0]+B[0])^{-2} ...
4
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0answers
84 views

There exists unique $g$ s.t. $g(x) = f(x) + A\int_0^1\sin(x-y)g(y)dy$

I'm doing past papers for a first course on functional analysis. We are not allowed to assume any results from real analysis or topology, so I was surprised to find an exam question, where I couldn't ...
4
votes
1answer
133 views

How can I measure the distance between two cities in the map?

Well i know that the distance between Moscow and London by km it's about 2,519 km and the distance between Moscow and London in my map by cm it's about ...
1
vote
0answers
167 views

Compact integral operator

I have this exercise and I don't know how to solve the last question. In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by ...
2
votes
2answers
59 views

$\lim_{t \to 0}g(t)=10$ $\lim_{t \to 10}f(t)=100$ but $\lim_{t \to 0}f(g(t))$ does not exist

$\lim_{t \to 0}g(t)=10$ $\lim_{t \to 10}f(t)=100$ but $\lim_{t \to 0}f(g(t))$ does not exist. Can anyone suggest two possible functions for f(t) and g(t)? Both functions are defined on R
4
votes
4answers
692 views

Find nth derivative of $\frac{x^{n}}{(1-x)^{2}}$, please?

I need to find the nth derivative of $\frac{x^{n}}{(1-x)^{2}}$ for $0<x<1$ So far, I tried the same method used for $\frac{x^{n}}{1-x}$ and here's what I got: \begin{equation} ...
0
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2answers
2k views

Is a uniformly continuous functions bounded? [duplicate]

Let f be uniformly continuous on (a,b). How do you prove that it is bounded on (a,b)?
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2answers
145 views

Question about Green function

please how to find the Green function associated to the operator $\frac{-d^2}{dx^2}$ and to the boundary condition of Dirichlet $u(a)=u(b)=0$ Please help me Thank you
1
vote
1answer
108 views

torus filling curve

I'm trying solve this problem but I didn't many ideas how to do it. So, if someone can give me a hint or the step of a solution I would greatly appreciate it. This is the problem: "Let ...
1
vote
2answers
61 views

Brouwer fixed poin in 1-D

In $\mathbb R_{+}$ (non-negative) I realize that any nonnegative valued continuous function $f\colon X \rightarrow Y$ where $X, Y \subseteq \mathbb R_{+}$ with a negative finite gradient has a fixed ...
0
votes
1answer
31 views

Prove that the polynomial divided by a fraction of the power of n is equal to the sum of fractions of any constans and successive powers of

Let n≥1 and n is integer. P(x) - polynomial and $deg P(x)<n$. Prove if $ a \in \Bbb R/{0} $ then: $ \frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2}+...+\frac{c_n}{(ax+b)^n}$ for ...
8
votes
4answers
160 views

Why the members of $\mathbb R$ will be certain subsets of $\mathbb Q$?

$\mathbb{R}$ is real numbers set, $\mathbb{Q}$ denotes rational numbers set. This is quoted from Rudin's mathematical analysis book page 17 about Dedekind' s construction. Why the members of ...
3
votes
2answers
57 views

Can we always find an analytic function if we know countable points?

I hope to find an analytic function such that $f(n)=b_{n}$, $n\in \mathbb{N}$? Can we always take an analytic function $f$?
2
votes
2answers
250 views

Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally?

I think that the series $$\sum_{n=1}^{\infty} \dfrac{\sin(n)}{n}$$ converges conditionally, but I´m not able to prove it. Any suggestions ?
4
votes
1answer
157 views

showing $f(x_1,x_2)=\sqrt[3]{x_1x_2}$ is differentiable

Given $f\colon\mathbb R^{>0}\times\mathbb R^{>0}\rightarrow\mathbb R, (x_1,x_2)\mapsto \sqrt[3]{x_1\cdot x_2}$ I want to prove that $f$ is differentiable. I know $f$ is partial differentiable ...
11
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0answers
286 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
4
votes
2answers
607 views

Lipschitz continuity and sup of derivative norm

On this wikipedia page, it is stated: For a differentiable Lipschitz map $f : U \rightarrow R^m$ the inequality $\|Df\|_{\infty,U}\le K$ holds for the best Lipschitz constant of $f$, and it ...
1
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0answers
262 views

Monotone Decreasing Sequences of Functions: Is the Dominated Convergence Thm applicable?

Reading through Rudin's Real and Complex Analysis, I came across the following exercise: Suppose $(f_n: X \to [0,\infty])$ is a monotone decreasing sequence of measurable functions such that ...
1
vote
1answer
648 views

weak derivative of a vector valued function

Consider $T>0$ and $U$ a opensubset of $R^n$ ,bounded and with smooth boundary. Consider ${\Omega}_T = U \times (0,T]$. Let $u: {\Omega}_T \rightarrow R $ a smooth function. Define $h : [0,T] ...
18
votes
2answers
847 views

A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

I encountered a strange integral with a strange result. $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$ Believe it or not, the result agrees ...