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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37 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
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1answer
25 views

Proving a particular subset of a Hilbert space is a subspace

I have a small question please, how to prove that this set: $F=\lbrace h\in H, \langle f''(u)h,h\rangle <0\rbrace$ is a sub space of the Hilbert space $H$, where $f''(u)$ is a self-adjoint ...
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0answers
34 views

Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
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0answers
19 views

Can I calculate the overall % each drug is prescribed if the list of drugs shown to each respondent varies?

I have a survey where the Q's are: Q1: Which of these drugs are you aware of - list of 10 drugs shown Q2: What % of your patients receive each drug (only drugs identified at Q1 shown). As the drugs ...
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4answers
171 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
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2answers
37 views

Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$

I have to prove the following by induction: $$(1+x)^n \le 1+(2^n-1)x$$ for $n \ge 1$ and $0 \le x \le 1$. I start by showing that it's true for $n=1$ and assume it is true for one $n$. ...
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0answers
58 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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1answer
91 views

Some properties of points with Lebesgue density equal to $1$.

I am studying Evans-Gariepy book and in corollary 1 of section 3.1.2, he prove that if $f:\mathbb{R}^N\to\mathbb{R}^M$ is locally Lipschitz and $$Z=\{x:\ f(x)=0\},$$ then $Df(x)=0$ a.e. $x\in Z$. He ...
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0answers
52 views

Clarification of a theorem from Chang's Methods in Nonlinear Analysis

The following theorem is taken from Chang's Methods in Nonlinear Analysis. It has a complete proof; however, I have some trouble understanding it (for example, I don't see what $K(f_{\sigma_i})$ ...
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0answers
46 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
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1answer
115 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
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1answer
43 views

Norm of the multiplication operator $f\mapsto (x\mapsto xf(x))$ on $L^2[a,b]$ [duplicate]

We have a linear operator $T : L^2[a,b] \rightarrow L^2[a,b]$ (with $|a| \le |b|$), $f \mapsto (x \mapsto xf(x))$ Now I shall determine what $\Vert T\Vert$ is. We clearly have $\Vert x \mapsto ...
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5answers
111 views

Find all real numbers $x$ such that $2+\sqrt{|x|}=x$

I'm revising for an exam and my lecture notes and online sheets etc. from my lecturers are very unhelpful, and I've completely forgotten how to do this type of question. So any help would be greatly ...
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0answers
29 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
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0answers
16 views

Lemma. $\exists N\in\mathbb{N}\forall n>N: x_n=0\Leftrightarrow x=\frac{q}{p^N}$ for some $q\in\mathbb{N}.$

https://proofwiki.org/wiki/User:J_D_Bowen/Math710_HW1 in the lemma of exercise 5 in direction $\Leftarrow$: Why $S_{N-1}=x$? I dont see! Thanks!
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2answers
109 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
0
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1answer
123 views

Few Questions about analysis in Rudins book

I have been looking at intro to real analysis. I am using the text book "Principals of Mathematical Analysis, third edition" by Walter Rudin. I have some questions about things I found confusing and ...
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2answers
49 views

Why is this a quotient map

Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...
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1answer
33 views

Bounding the $\ell^{1}$ norm given the $\ell^{2}$ norm

Suppose $x = (x_{1}, x_{2}, \ldots) \in \ell^{2}$. If $\sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1$, is it possible to bound $\sum_{n = 1}^{\infty}|x_{n}|$?
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3answers
130 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
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2answers
116 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
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0answers
213 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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2answers
34 views

$ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $

Let $ \Bbb B( \Bbb R^{n} ) $ denote a Borel algebra on $ \Bbb R^n $. Why is it true, that: $ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $ I think, that "$ \supset$" ...
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1answer
40 views

Lyapunov stability and finite time convergence

I have two questions: Problem 1: Let $V (x)$ be the Lyapunov function candidate with $x \in \mathbf{R}$, and the time derivative of $V(x)$ is given by $\dot{V} (x) \le - x ( x - \alpha (t) )$ where ...
2
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1answer
80 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
6
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1answer
85 views

Which negative integer powers of 2 belong to the Cantor Set?

Consider the Cantor set $C$, and negative integer powers $2^{-k}$. Clearly, for $k=1$, $2^{-1} \notin C$ since $1/2 \in (1/3, 2/3)$, the first deleted open interval. It is known that $1/4 = ...
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1answer
56 views

Which technique of integration should use to solve the question?

Quote from the paper I read: Given $F=(1-\lambda)f$ + $\lambda zf'$, we find that $f(z)= \tfrac{1}{\lambda} z^{1- \tfrac{1}{\lambda}}\int_0^zF(t)t^{\tfrac{1}{\lambda}-2}dt.$ My questions is Which ...
3
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3answers
103 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
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3answers
52 views

Length of Difference Curve

Let $\varphi : [a,b] \to \mathbb R^n$ be a curve, and for some partition $\pi = \{ t_0 = a, t_1, \ldots, t_m = b \}$ of $[a,b]$ set $$ l(\pi, \varphi) = \sum_{i=1}^m \| \varphi(t_i) - ...
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0answers
39 views

Rules to evaluate Suprema and Infima

Note: All functions considered are supposed to be bounded. Nowhere I found rules to evaluate suprema and infima. Obviously, $$ c \cdot \sup_{x\in \mathbb R} f(x) = \sup_{x\in\mathbb R} cf(x) ...
2
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1answer
54 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
3
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1answer
47 views

$1/r^2\int_{\mathbb{S}_r}u-u(x)$ converging to $\Delta u(x)$?a

When reading some papers on PDEs, the following shows up several times: For a $C^{\infty}$ function $u$, $\frac{\int_{\mathbb{S}_r(x)}u-u(x)}{r^2}$ converges to $1/2n\Delta u(x)$ uniformly on ...
0
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0answers
25 views

Integration of nonlinear and linear ODEs

\begin{equation} \frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber \end{equation} \begin{equation} ...
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0answers
45 views

Inequality among trigonometric sums of normal random variables

This is an inequality used in a proof which I do not know how to prove. $$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi ...
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1answer
89 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } ...
0
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1answer
30 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
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2answers
52 views

Limits in complex plane.

Let $z$ and $z_n$ be complex numbers and assume $z_n \rightarrow z$. It it true that $$\lim\limits_{n\to \infty}\left(1+\frac{z_n}{n}\right)^n= \lim\limits_{n\to \infty}\left(1+\frac{z}{n}\right)^n ...
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1answer
64 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
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1answer
31 views

Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
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1answer
26 views

Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
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1answer
46 views

Problem with constructing a smooth function with given properties

I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq ...
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1answer
34 views

Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
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0answers
35 views

Proving Equalities in Analysis

In measure theory, I saw that while proving some "equalities" - $``a=b"$ - (such as measure of any type of an interval is its length, ...), the argument goes as follows: We prove that $a\leq b$ ...
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1answer
44 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
0
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1answer
43 views

What is the necessary and sufficient condition of linear dependence of $n$ functions?

If $n$ functions are linear dependent, then the Wronskian determinent is zero, While that the Wronskian determinent is zero cannot imply $n$ functions are linear dependent. So what is the necessary ...
5
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1answer
64 views

Fubini's theorem applied to heaviside step functions

First of all, I should probably mention that I am a physicist, not a mathematician so I sincerely apologize for any lack of rigour in my explanation of my problem. Recently, I have been trying to ...
2
votes
1answer
37 views

Is set with this property is homeomorphic to Cantor set?

(1) $A$ is nonempty subset of $\mathbb{R}$. (2) For all $x<y \in A$ there is $z \notin A$ such that $x<z<y$. (3) $A$ is perfect. Then is there homeomorphism between $A$ and cantor set? ...
1
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1answer
67 views

Fourier transform of a compactly supported function.

Can someone help me the question below? Is there a positive-valued compactly supported function $f$ such that the Fourier transform ${{f}^{\operatorname{ft}}}\left( t \right)=\int_{-\infty }^{\infty ...
1
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4answers
136 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
0
votes
1answer
41 views

Partial derivatives of $xy^2/(x^2+y^2)$ at the origin

I noticed that this is a big black hole in my understanding of partial derivatives at the point. I don't know how to count it: $$ f(x,y) = \frac {xy^2}{x^2+y^2} $$ $$ \frac {df}{dx}(0,0)=\lim_{t\to ...