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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Let $f_n\in\mathcal C(0,1)$, $f_n\xrightarrow{\mathrm{unif}}f$ on every compact $K\subseteq(0,1)$. Is $f$ uniformly continuous on $(0,1)$?

This is part of a question on an old preliminary exam in Analysis at my institution. I think my answer is sufficient but I am not confident about it, and would appreciate feedback.
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2answers
67 views

If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?

Let $f:[0,\infty)\to [0,\infty)$ be a continuous function such that $\displaystyle \int_0^{\infty}f(t)\,dt<\infty$. Which of the following statements are true ? (A) The sequence $\{f(n)\}$ is ...
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0answers
47 views

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$. I have no idea to find this sum. Can anyone give me a hint? Thank you in advance !
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1answer
24 views

What is wrong with this line integral? (Line integral change of variables)

I have this line integral $$\oint_{\partial D}(f\nabla f)\cdot\hat{n}\,ds$$ which I would like to "change variables" so that the final result is in terms of a line integral of $g$ on $[0,2\pi]$ where ...
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1answer
37 views

Uniformly bound exists for a continuous function sequence in a neighbhorhood of a convergent point?

Assume that $$\lim_{n\to\infty}f_n(x_0) < \infty$$ and also that $\forall n\ge1$, $f_n(x)$ is continuous in a neighborhood of $x_0$, say $(x_0-\delta_1, x_0+\delta_1)$. Besides, for any fixed $x$ ...
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0answers
18 views

Intra Office Walking Competition [on hold]

My office is having a walking competition to get employees to be more active. However, there has been some debate about what measure we should use to determine how well each person has done. My office ...
2
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1answer
46 views

Norms on $\mathcal{P}_N$ Vector Space of Polynomials up to Order N

$\|p\|_\infty :=\sup_{x\in [0,1]}|p(x)|$ and $\|p\|_{L^1}:=\int_0^1 |p(x)| dx$. As the space of real-valued polynomials on $[0,1]$ up to order $N$ is a $N+1$ dimensional vector space and ...
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1answer
54 views

Definition of limit point: Is the superset necessary?

Consider: (A) The definition of a limit point from wiki: "Let S be a subset of a topological space X. A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S ...
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0answers
21 views

Is this correct? to prove the linearity of a function

Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be diferentiable in $U$ (open set), let $a \in U$ such that $[a,a+h] \in U$ and $|f'(x) - T| \leq M$ $\forall x \in (a,a+h)$ we know that ...
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1answer
391 views

Finding the tangent line through the origin

Find the tangent line to: $$f(x) = \sqrt{x-1}$$ that passes through the origin $(0, 0)$. $$f'(x) = \frac{1}{2\sqrt{x-1}}$$ The line will be tangent at $(a, b)$ so then: $$f'(a) = ...
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1answer
34 views

Showing a Lipschitz function is differentiable

Let $f : \mathbb{R} \to \mathbb{R}$ be a Lipschitz function. Suppose $$ \lim_{n\to \infty} n[f(x + \frac{1}{n}) - f(x)] = 0.$$ Prove that $f$ is differentiable. I am tempted to just let $n = ...
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1answer
11 views

How is it true that for large $t$, $(1+O(1/t))e^{-2\ln t O(1/t)}=1+O(\ln t/t)$?

The title pretty much says it all. At some point in large time analysis, the following claim popped out but I don't see how it is true: For sufficiently large $t>0$, $$ \frac{2\ln ...
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0answers
15 views

Limit superior of a function [on hold]

How to prove that if $f \colon [a,b] \to \mathbb{R}$ is continouos at $x \in [a,b]$ and $$\limsup_{y \to x} \frac{f(y)-f(x)}{y-x} = \inf_{\delta >0} \sup_{x \in (x- \delta , x + \delta)} ...
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0answers
9 views

Existence of partial derivatives of a homogeneous function

Here is an exercise I found in R.Creighton Buck's advanced calculus: A function f is said to be homoegeneous of degree $k$ in a neighborhood $N$ of the origin if $f(tx,ty)=t^k*f(x,y)$ for all points ...
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2answers
28 views

Winding number of closed curves

Let $c_1,c_2$ be closed curves in $\mathbb C^{\times}$ and we define $c(t):=\frac{c_1(t)}{c_2(t)}$. Proof the following for the winding number $win(c,0)=win(c_1,0)-win(c_2,0)$. I have no idea to ...
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1answer
20 views

Global Lipschitz implies bounded in coefficient

Consider $g:\mathbb{R}^2\to \mathbb{R}$ of the form $g(x,y)=p(x)q(y).$ Assume $g$ is uniformly Lipschitz in $x,y$ in the sense that there exists $K>0$ such that for any $(x_1,y_1),(x_2,y_2)\in ...
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0answers
14 views

troubles integrating by parts with $\int \sqrt{x^2+y^2}D_x\phi(x,y)$

I want to show that $f(x)=\|x\|^a$ is weakly differentiable in $B_1(0)\subset\mathbb R^2$ iff $a>-1$. Therefore I want to show that for all $\phi\in C_0^\infty$ we have $$ (*)\quad\int \phi D_i ...
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0answers
13 views

Cycle homology to closed curve

Let $c$ be a cycle in $\mathbb C^{\times}$ and $c_n:[0,1]\to \mathbb C^{\times},c_n(t)=e^{2\pi int}$. Show that $c$ and $c_n$ are homologous. They are homologous if the winding number ...
2
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2answers
52 views

The Dirac delta does not belong in L2

I need to prove that Dirac's delta does not belong in $L^2(\mathbb{R})$. First, I found the next definition of Dirac's delta $\delta :D(\mathbb R)\to \mathbb R$ is defined by: ...
2
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1answer
44 views

Show Lipschitz continuity of a function

I'm stuck trying to solve the following exercise: Let $f:\mathbb R^n \to \mathbb R^m$ a function with the property that, for all $v \in \mathbb R^n$, there is $L=L(v) > 0$ such that the ...
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0answers
21 views

For some $f:\Bbb{R}^n\to \Bbb{R}$, $A\subset \Bbb{R}^n$ and $B\subset A$, show that $\max_{A}(f)=\max_{B}(f)$

Let $A=\{(x_1,...,x_n)|{1\over n}(\sum_{i=1}^n{x_i})={1\over 3},{1\over n}(\sum_{i=1}^{n}{x_i^2}))=1\}\subset \Bbb{R}^n$, and let $B\subset A$ be a subsets of points from $A$ of the form ...
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0answers
38 views

Intersection of a smooth plane curve and a circle

Let $\gamma(t)=(x(t),y(t)):[0,2\pi] \rightarrow \mathbb{C}$ be a simple and closed $C^1$-curve. Prove that there is a small circle that intersects $\gamma$ only at two points?
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1answer
473 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point ...
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1answer
133 views

How to evaluate this limit about Bernoulli number?

First,we define $\displaystyle I_{1}\left ( x \right )=\frac{\sin x}{x}$, then $\displaystyle \lim_{x\rightarrow 0^+}I_{1}\left ( x \right )=1$, also we have \begin{align*} I_2\left ( x \right ...
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0answers
23 views

Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [on hold]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very ...
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1answer
14 views
+100

Find function $h$ so that $h(U,V)$ equals density of $f(a) da$ for $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be independant and uniform distributed on [0,1]. Now I want to find a function $h$ so that $h(U,V)$ is equal to the density ...
4
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1answer
117 views

A continuous onto function from $[0,1)$ to $(-1,1)$

How I can construct a continuous onto function from $[0,1)$ to $(-1,1)$ ? I know that such a function exists and also I have a function $\displaystyle f(x)=x^2\sin\frac{1}{1-x}$ which is ...
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1answer
27 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in ...
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4answers
94 views

Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} \frac{1}{a+x}$ = $\frac{1}{a+b}$.

I've been working on this epsilon delta proof for the longest time now, and I can't quite get it. Let $a>0$ and $b>0$. Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} ...
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0answers
27 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
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2answers
43 views

Are all solutions to the ODE $ay''(t) + by'(t) + cy(t) = 0$ of the form $y(t)= \alpha e^{(\beta + i\gamma)t}$?

Let $a$ $b$ and $c$ be complex numbers. Consider the complex solution of the ODE $$ay''(t) + by'(t) + cy(t) = 0.$$ If there exist solutions to this, are they necessarily of the form $$y(t)= \alpha ...
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Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $.

How does one prove the following limit? $$ \lim_{n \to \infty} \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$
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2answers
27 views

Ultrametric example

Can anybody give an example for ultrametric space? i.e., in the metric space definition, instead of triangle inequality, we have strong triangle inequality namely $d(x,y) \leq \max \{{d(x,z),d(z,y)}$} ...
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1answer
38 views

Differentiability of a function $f:\mathbb R^2\rightarrow \mathbb R$

I want to prove that the funktion $f:\mathbb R^2\rightarrow \mathbb R,\hspace{0.5cm} f(x,y)=\begin{cases} (x^2+y^2)\sin\big(\frac{1}{\sqrt{x^2+y^2}}\big),& \text{if } (x,y)\neq (0,0)\\ ...
1
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1answer
64 views

Harmonic function — Application of Divergence Theorem

Suppose $f$ is a harmonic function on $D=\{(x,y)\in\mathbb{R}^2: x^2+y^2<1\}$. Assume $f$ is twice continuously differentiable on $cl(D)=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 1\}$. How do we express ...
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1answer
53 views

Use Power Series to solve system of differential equations

Problem: Hello, I wonder how you would use a Power Series to solve a system of differential equations. Lets say I have the system $$\begin{cases}(1)\text{ }\text{ }x_1'=2x_1+4x_2 \\ (2)\text{ ...
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0answers
15 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=\theta(t)e^{-\sigma_0 t}\sin(\omega_0 t)$, where $\theta(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed. It has the ...
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1answer
15 views

Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all ...
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0answers
20 views

Every cycle in a domain $D$ is null homolog

Let $D$ be a domain, where every cycle is null homolog and $f$ be a biholomorphism. Proof that every cycle $c$ in $f(D)$ is nullhomolog. Let $c$ be a cylce in D, it is null homolog, if ...
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0answers
34 views

Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: ...
2
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1answer
455 views

Conservation of momentum for nonlinear Schrödinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
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2answers
51 views

Find a subset of A such that its boundary does not have measure zero

Question Find a subset $A$ of $[0,1]$ such that $A=cl(intA)$ and yet $bd(A)$ does not have measure $0$. I don't know how to construct it. I think it should be closed set, cannot be empty by ...
575
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16answers
53k views

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
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1answer
788 views

Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq C(b-a)mesh(P)$.

Suppose $f:[a,b]\to \mathbb{R}$ is Lipschitz, i.e $|f(x)-f(y)| \leq C|x-y|$ for all $x,y$ in $[a,b]$ and thus $f$ is continuous. Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq ...
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0answers
33 views

I want to draw one expression of fraction which have sin and cos. [on hold]

I want to know what shape of next expression. $a$ is a real number and $0<a<1$. $t$ moves $0<t<\frac{2\pi}{a}$. $x=\frac{\cos(a+1)t}{\cos(at)}$ $y=\frac{\cos(a+1)t}{\sin(at)}$
2
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2answers
227 views

Saturated measure defined as a supremum of a semifinite measure and countable unions

Here is what I am working on: Suppose that $\mu$ is semifinite. For E in $\overline{M}$, define $\underline{\mu}(E)=\sup\{\mu(A):A$ in $M$ and $A \subseteq E$$\}$. Then $\underline{\mu}$ is a ...
3
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0answers
11 views

Imposing boundary conditions AND self-similarity on a PDE

I have a PDE in the form $$u_t=F(u,u_x)$$ where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form $$u(x,t)=t\tilde ...
0
votes
1answer
31 views

Linear function: relation between linearity and continuity

Given a linear function $A$ between two normed Vectorspaces i have to show euquality of the follwing statements: $A$ is continuous There exists a point where $A$ is continuous $A$ is ...
1
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2answers
44 views

Prove or disprove continuity of two maps

Yet another time I need help to prove continuity of a certain map and don't know how to do it: Look at the vector space $$C_b^1(\mathbb R; \mathbb C) := \{f \in C^1(\mathbb R;\mathbb ...
0
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1answer
32 views

Simply Connected sets

In my textbook it states, that the Union of two open docs is simply connected but not connected Why is this. I know simply connected means any closed path or loop can be shrunk to a point ...