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
56 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
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2answers
101 views

Proving $ \frac{x^3y^2}{x^4+y^4}$ is continuous.

The problem asks to show that $$f(x,y) = \left\{ \begin{align} \frac{x^3y^2}{x^4+y^4}, & (x,y) \neq (0,0), \\ 0, & (x,y) = (0,0), \end{align} \right.$$ is continuous at the origin, however it ...
4
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3answers
312 views

Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$

I'm trying to solve this limit $$\lim_{x\to 0} \frac{\sin x-x}{x^3}$$ Solving using L'hopital rule, we have: $$\lim_{x\to 0} \frac{\sin x-x}{x^3}= \lim_{x\to 0} \frac{\cos x-1}{3x^2}=\lim_{x\to ...
1
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1answer
85 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
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3answers
84 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
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1answer
47 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
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2answers
44 views

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ [duplicate]

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ Not sure how to go about this problem. I tried Fubini. But that ...
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2answers
75 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
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3answers
142 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
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1answer
62 views

For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + 1}{n}}f(x)dx$.

Let $f ∈ L_1(\mathbb{R}).$ For $n ∈ \mathbb{N}$ define the function $g_n :\mathbb{R}→\mathbb{R}$ as follows. For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + ...
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2answers
222 views

Show that $A = \{p\in\mathbb{ Q}^+\mid p^2<2\}$ contains no largest number and $B= \{p\in\mathbb {Q}^+\mid p^2>2\}$ contains no smallest number

I was reading Principles of Mathematical Analysis - Walter Rudin. In the start (pg-11), it is shown that the equation $p^2 = 2$ is not satisfied by any rational $p$ i.e. $\sqrt{2}$ is irrational. ...
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1answer
57 views

Bound the derivative norm of a convolution by the function norm

Is there a bound of the form $$ \|(f*\phi_\epsilon)'\|_{L^2}\leq C(\epsilon) \|f\|_{L^2}, $$ where $\{\phi_\epsilon\}$ are standard mollifiers, and $C(\epsilon)$ does not depend on $f$?
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1answer
116 views

How do you find the surface area of a boundary in $\mathbb{R}^3$?

I need to solve this problem: Let $D=\{(x,y,z):4(x-2+z)^2+4y^2\le(2-z)^2,0\le x-z\le1\}$ Calculate the area of $\partial D$ So how do you calculate the area of the boundary of a volume defined ...
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1answer
502 views

Space of continuous functions vanishing at infinity

Let's denote with $C_0(X)$ the space of continuous functions $f$ on $X$ such that for every $\epsilon>0$ there exists a compact set $K_\epsilon\subset X$ satisfying $sup_{x\notin K_\epsilon ...
4
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1answer
71 views

is it possible to find out a partition of $[a,b]$

Let, $f:[a,b]$$\rightarrow$$\mathbb{R}$ be a continuous function. Is it possible to find out a partition of $[a,b]$ such that $f$ is monotone there? I am stuck here. How to proceed from here?
4
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1answer
90 views

About p-adic numbers

I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, ...
2
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1answer
42 views

Explicit delta for polynomial limit

I'm looking for an explicit formulation for $\delta$ in the $\epsilon-\delta$ formulation of the limit for a polynomial $p(x) = \sum_{n=0}^N a_nx^n$. For example, in the the specific linear case ...
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2answers
175 views

$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that $$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
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1answer
52 views

Alternative definition of Euclidean operator norm

Given $A\in \mathbb{C}^{n\times n}$, $\|.\|$ the Euclidean operator norm, and $\rho(A)$ the spectral radius of A, how to show that $$ \|A\| = \sup\{\rho(AB):B\in \mathbb{C}^{n\times n}, \|B\|=1\} $$
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1answer
40 views

Cauchy's Integral Formula: conditions vs singularities

I'm sure this is a simple misunderstanding but it was annoying me. So using the version of Cauchy's Integral Formula given on Wikipedia http://en.wikipedia.org/wiki/Cauchy's_integral_formula, it is ...
2
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1answer
56 views

Letting $r \rightarrow 1$ in $\frac{1}{2\pi}\int_{0}^{2\pi}\log |f(re^{i\theta})|\, d\theta$

Suppose $f$ is continuous on $\{z: |z| \leq 1\}$, analytic on $\{z: |z| < 1\}$, and $f(0) \neq 0$. For $0 < r < 1$, consider the integral $$\frac{1}{2\pi}\int_{0}^{2\pi}\log ...
0
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1answer
50 views

Is $F(f)=\int_{a}^{b}\phi(f(t))dt$a differentiable function?

Let $E=C[a,b]$ the Banach space of functions which are continuous from $[a,b]$ to $\mathbb{R}$, with the norm of max (or sup). Let $\phi:\mathbb{R}\to \mathbb{R}$ twice continuously differentiable. ...
2
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1answer
80 views

A problem in the space $C[a,b]$

Let $E=C[a,b]$ provide with the $\max$ norm. Let $S\neq \emptyset$, let and $D(t,\lambda)$ be a continuous function (for each $\lambda\in S$), from $[a,b]$ to $\mathbb{R}$, such that $\displaystyle ...
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0answers
37 views

Proving that a function is a $C^\infty$ submanifold in $\Bbb{R}^2$ of dimension 1

We need to prove that for all $c\in\Bbb{R}$ the set $\{x\in\Bbb{R}\,\colon\, g(x)=c\, \}$ is a $C^\infty$ submanifold ($g\,\colon\,\Bbb{R}^2\rightarrow \Bbb{R};(x_1,x_2)\mapsto x_1^3-x_2^3$) in ...
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1answer
73 views

Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
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1answer
88 views

Any idea with this problem of distances???

Let $E$ a normed linear space and $H$ the closed hyperplane $H=\ker f$, where $f\in L(E,\mathbb{R})$, $f\not\equiv 0$. Show that if $a\in E$ then $$d(a,H)=\frac{|f(a)|}{||f||}$$ And the problem have a ...
2
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1answer
76 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
0
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2answers
46 views

Study of a function

I have this function $\displaystyle g(s)=\frac{s^{2-\sigma}}{1+s^2}, ~\text{for all} ~s\in \mathbb{R}$ , i need to find the interval of $\sigma$ and the maximum of the function $g$. I calculate the ...
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2answers
136 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
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1answer
247 views

What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
2
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1answer
89 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
2
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1answer
213 views

How to prove that $F(x,y)=(f(x)h(y),g(y))$ is a diffeomorphism?

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be given by $F(x,y)=(f(x)h(y),g(y))$, where $h:\mathbb{R}\to\mathbb{R}$ is a diferentiable function and $f,g:\mathbb{R}\to\mathbb{R}$ are diffeomorphisms. ...
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1answer
62 views

primitive function involving logarithm, square integrability

I want to ask if the following function, which is given by an integration $f(y):=\frac{1}{y}\int_0^y \frac{1}{x^{1/2}\log{x}}dx,$ is locally square integrable near $y=0$? Or equivalently, ...
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2answers
100 views

Prove: the equation $p^2=2$ is not satisfied by any rational $p$.

Rudin has gone through complex scenarios to prove this theorem, but isn't the following correct? By contradiction, let $\exists p\in \mathbb{Q}; p^2=2$. Take $p=a/b$ where $a,b\in \mathbb{Z}$ and ...
4
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1answer
85 views

How to show that $\varphi(x,y)=(x+f(y),f(x)+y)$ is bijective?

Let $f:\mathbb{R}\to\mathbb{R}$ be a $C^1$ function such that $|f'(t)|\leq k<1$ for all $t\in \mathbb{R}$. Let $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ be the function given by ...
3
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1answer
40 views

Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
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1answer
45 views

Given a histogram, programatically, how do I find the normal distributions that comprise it?

I will be getting data in at around 100 frames per second, and I need to compute the normal distributions that comprise a set of 48 data points. The distributions can partially overlap, but will ...
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2answers
165 views

Is true that $n=m$?

Let $O\subseteq \mathbb{R}^m$ and $U\subseteq\mathbb{R}^n$ and $f:O\to U$ a bijective function with $f$ and $f^{-1}$ differentiable in their domains. Is true that $n=m$? $O$ and $U$ are open. I ...
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1answer
41 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
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1answer
85 views

Two notions of total variation norms

I found these two definitions of the total variation norm for probability measures on $(X,\mathcal{F})$: $$ \left \|\mu- \nu \right \|_{TV} = \sup_{\text{$f:X \rightarrow [-1,1]$ measurable}} \left ...
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1answer
91 views

Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit.

Let $f : [0,1] → \mathbb{R}$ be absolutely continuous, satisfy $f(0) = 0$ and $f′ ∈ L_2([0,1]).$ Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit. From absolute ...
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2answers
212 views

Prove Jensen’s inequality: $F(\frac{1}{μ(X)}\int f \,dμ) ≤ \frac{1}{μ(X)} \int F(f)\,dμ.$ [closed]

Let $(X,A,μ)$ be a finite measure space, and let $F : \mathbb{R} → \mathbb{R}$ be a $C^2$ function with second derivative $F'' > 0$. Let $f \in L_1(\mu)$ be real-valued. Prove Jensen’s inequality: ...
1
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1answer
56 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
0
votes
1answer
511 views

Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy $$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
1
vote
1answer
95 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
3
votes
0answers
90 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
0
votes
1answer
45 views

For $a\in\Bbb{R\setminus Q}$, is it possible to make $a(2n+1)\pi$ “almost” an integer?

My textbook says the following: Let $a$ be any irrational number. Now consider $a(2n+1)\pi$ for various $n\in\Bbb{N}$. For any $\epsilon>0$, it is possible to choose an $n\in\Bbb{N}$ and a ...
1
vote
1answer
55 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
1
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0answers
66 views

Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
5
votes
3answers
110 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...