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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66 views

Bijections and disjoint sets by finite summation

Suppose $X$ is finite and $f:X\rightarrow\mathbb{R}$ a function. Let $g:Y\rightarrow X$ be a bijection. Then $$\sum_{x\in X} f(x) =\sum_{y\in Y} f(g(y))$$ That is what I first want to prove. The ...
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3answers
92 views

convergence of a sequence (partial fractions)

Let's define {${x_n}$} as $x_1=0.1, x_2=0.101, x_3=0.101001,....$ . Then we need to find out if the sequence convergences or not and the limit. This is how i proceeded. $x_1=\frac{1}{10}, ...
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1answer
35 views

$\lim \limits_{t \to a} \int_{c}^{d} f(t, s) ds= \int_{c}^{d} \lim \limits_{t \to a} f(t,s) ds$?

When is $\lim \limits_{t \to a} \int_{c}^{d} f(t, s) ds= \int_{c}^{d} \lim \limits_{t \to a} f(t,s) ds$. My guess is that $f$ uniformly continuous in a neighborhood of $(a,s)$ for $c\leq s \leq d$ is ...
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2answers
91 views

Properties of a surjective local diffeomorphism

Assume that $f:\mathbb{R}^N\to\mathbb{R}^N$ is a surjective function and in addition suppose that $f$ is a local diffeomorphism. Take two points in the image of $f$, let's say, $f(x),f(y)$ with ...
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1answer
13 views

Can I say $f$ is differentiable at $c$ if $D_u(c) = \nabla f(c) \cdot u$ for all unit vectors $u$?

Can I say $f$ is differentiable at $c$ if $D_u(c) = \nabla f(c) \cdot u$ for all unit vectors $u$? I think I can because it guarantees a tangent plane. But I don't know how to prove this precisely. ...
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2answers
240 views

Convex function problem and mean value theorem

I dont know how to solve this problem with not convex condition! Let $f \colon A \subseteq \mathbb R^n \to \mathbb R$ differentiable with $A$ convex and suppose $\|\mathop{\rm grad} f(x)\| \le M$ ...
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91 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial ...
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2answers
49 views

An induction problem.

I am trying to prove the following problem by induction on $n$. Let $T: (0,1]\rightarrow (0,1]$ be given by $T(x)=\left\{ \begin{array}{ll} 2x & \quad \text{if} \hspace{4mm} ...
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2answers
390 views

Show that for every real number x, there exists a natural number n such that $x < 2^n$

Let x be a positive real number. I want to prove that $\forall$ x, $\exists$ n $\in$ N such that x < $2^n$ . To me it seems that as x increases, I can just pick larger and larger values for n to ...
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2k views

Give an example of two $\sigma$ algebras whose union is not an algebra

Give an example of two $\sigma$ algebras in a set $X$ whose union is not an algebra. I've considered the sets $\{A|\text{A is countable or $A^c$ is countable}\}\subset2^\mathbb{R}$, which is a ...
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1answer
1k views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
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1answer
44 views

Prove $(X,d)$ is a metric. Where d is the number of entries a vector is not equal.

I am reading through my applied analysis book and trying to prove all of the things that have not been proven in order to prepare for an exam. Here is one of the examples: Let $X$ be the set of ...
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0answers
436 views

2- norm of a vector in spherical and cylindrical coordinates

I was wondering how the 2-norm of a vector in cylindrical and spherical coordinates looks like? Or more general, what is the idea to derive it? Does anybody have an idea?
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1answer
212 views

$\max, \min$ relation for probability LP

Suppose we have 2 LPs; $$\text{maximize } c^T\mathbf{x}$$$$\text{subject to} \,A \mathbf{x}\geq0 $$ $$\sum \mathbf{x}=1$$ $$\mathbf{x}\geq 0$$ and the other is, $$\text{minimize } ...
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2answers
198 views

How prove this $\int_{0}^{1}|f(x)|dx\le2$

let $f$ be Riemann integable in $[0,1]$, and for any $[a_{i},b_{i}]\subset [0,1]$,and $[a_{i},b_{i}]\bigcap [a_{j},b_{j}]= \emptyset,1\le i\neq j\le n$ then have ...
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2answers
2k views

Understanding the Frechet derivative

What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...
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0answers
48 views

question about hausdorff distance

Let $ 0 \leq u_n \leq 1$ a sequence of smooth functions defined in a bounded open domain $\Omega$. Supoose that $u_{n+1} \leq u_n$. Fix $l \in (0,1)$ and consider the level sets $A_n = \{ x \in ...
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Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
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3answers
111 views

Why does $\sum\limits_{k=n+1}^\infty\frac{r^{2k+1}}{(2k+1)!}$ converge to $0$?

I'm being told that because the following series is the tail end of a convergent series, it converges to zero as $n$ gets large: $$\sum\limits_{k=n+1}^\infty\frac{r^{2k+1}}{(2k+1)!}$$ The tail ...
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1answer
744 views

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
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1answer
81 views

the absolute value of a measurable set is measurable?

Suppose we have a set $A$ that is a measurable set, does it follow that $$E = \{ |x| : x \in A \} $$ is also measurable??
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1answer
511 views

Uniform convergence of $\sin x$ - what to choose for $N$?

I'm trying to show uniform convergence of the power series of $\sin x$ on $[0,r] (r>0)$ from first principles. So let $\epsilon>0$ and $n\geq N$. I'm stuck at this step: What do I choose for $N$ ...
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1answer
47 views

approaching the border

Let $U\subset \mathbb{R}^m$ $(U\neq \mathbb{R}^m)$ open and connected. Given $b \in \partial{U}$ there is some way $\varphi:[0,1]\rightarrow \mathbb{R}^m$ with finite length such that $\varphi(t) \in ...
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1answer
49 views

Complex function with two inverses?

I was computing the inverse of the complex function $$\xi(z) = z + \frac{1}{z} \quad \text{ where } z \ne 0$$ which lead me to a strange conclusion. If we set $$\xi(\xi^{-1}(z)) = z$$ and solve ...
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1answer
245 views

Proof of convergence of sequences

Problem: Show that if $x_n\to 5$ then $\dfrac{x_n +1}{\sqrt{x_n -1}} \to 3$. So I know $|x-5|<\epsilon$ and I need to show that $\left|\dfrac{x_n +1}{\sqrt{x_n -1}} -3\right| < \epsilon$. I ...
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2answers
136 views

Maximize the determinant

Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is $$\max_{A \in S} \det A = \det(I_n)$$ ...
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1answer
98 views

Estimate divergence by gradient in H1

I am currently trying to fully understand the stationary Stokes equations of incompressible fluid. In the mixed form (homogeneous boundary data), for $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, a ...
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1answer
85 views

Trigonometric inequality for the sum of sin and cos

I need a proof for the following trigonometric inequality $$\frac{|\sin x| + |\cos x|}{\sqrt2} \leq 1- \frac{\cos^2(2x)}{8}$$ Can someone please help me with this?
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1answer
91 views

Convergence of mean value of entire function bounded on the real line

Let $f(z)$ be entire analytic function which is bounded for $z \in \mathbb{R}$. Is it true that the following limit always exists $$\lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T f(t) \, d t\text{ ...
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1answer
223 views

Expressing intervals as a union or intersection of intervals of the form $(a,b]$

I want to express all intervals as countable union or intersection of intervals of the form $(a,b]$. I already know $$ (a,b) = \bigcup_{n} (a, b - \frac{1}{n} ]$$ $$ [a,b] = \bigcup_{n} (a + ...
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2answers
272 views

How prove this mathematical analysis by zorich from page 233

Let $f$ be twice differentiable on an interval $I$,Let $$M_{0}=\sup_{x\in I}{|f(x)|},M_{1}=\sup_{x\in I}{|f'(x)|},M_{2}=\sup_{x\in I}{|f''(x)|}$$ show that (a):$$M_{1}\le ...
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0answers
72 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
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0answers
151 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...
3
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2answers
468 views

Dido's problem with Euler equations

I'm considering Dido's problem: Consider 2 differentiable arcs $C$ and $C_0$ in $\mathbb{R}^2$ from the point $P$ to $Q$ and back. We keep $C_0,P,Q$ fixed, and want to choose the arc $C$ such that ...
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2answers
272 views

Why is the l.u.b. property equivalent to Cauchy-sequence convergence for $\mathbb{R}$?

Math people: I browsed some questions with similar titles and could not find a duplicate. I apologize if it is. If you read my question it will be obvious that I am not a logician, so please be ...
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1answer
431 views

A question about Hausdorff locally compact spaces

In one of the proofs of Rudin's real complex analysis, the following implication seems to be assumed: Let $X$ be a locally compact Hausdorff space. Let $V$ be an open subset of $X$. Let $x\in V$. ...
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1answer
36 views

What is this total length

What is the value of the total length of all the edges connecting the vertices of a regular $k$-gon that is inscribed on a unit circle?
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2answers
1k views

The local lipschitz condition implies differentiability?

I know differentiability implies the local lipschitz condition. however, I am not sure the converse. Actually, I think it might be. The definition of the local lipschitz condition is that for $$ ...
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1answer
148 views

Explanation on a proof of a property of mollifiers

Here are some definitions that was taken of PDE Evans book: Here is a proof of a property of mollifiers: My (elementary) question is: Why is the convergence uniform on $V$? Thanks.
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49 views

Does $|\textbf{x}-\textbf y|<\delta$ imply $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$

I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ ...
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0answers
74 views

The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
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1answer
41 views

Would this be bounded

Let $a_{m}$ be supremum of the minimum of the angle between the line segments between any $m$ points, in which the supremum is taken over all configurations of $m$ points. Is $\sqrt{m}a_{m}$ bounded ...
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1answer
159 views

A problem about limit

Problem: Suppose a sequence $\{x_n\}_{n\in\mathbb{N}}$ satisfies that $$\lim_{n\to\infty}\bigg(x_n\cdot\sum_{k=0}^nx_k^2\bigg)=1.$$ Prove that $\lim_{n\to\infty}\sqrt[3]{3n}x_n=1$. I do not have ...
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1answer
93 views

Coercivity and Compactness

My question is: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous on all of $\mathbb{R}^n$. Prove $f$ is coercive and and only if for every $\alpha\in\mathbb{R}$ the set $\{x|f(x)\leq\alpha\}$ ...
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1answer
91 views

Proving $ \lambda^{*}(A \cup B) = \lambda^{*}(A)+\lambda^{*}(B)$

Let $\lambda$ denote Lebesgue measure and $\lambda^{*}$ denote outer measure. Prove that if A and B are subsets of $R^n$ which are separated in the sense that there exists a measurable set C such that ...
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2answers
100 views

about periodic functions

Could anyone give some idea about the following problem? Many thanks! Suppose that $f,g: \mathbb{R}\to\mathbb{R}$ are two periodic functions such that $\lim_{x\to\infty}[f(x)-g(x)]=0$. Show that ...
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1answer
93 views

Is this bounded

Let $d_{k}$ be supremum of the minimum of the pairwise distances between any $k$ points in the unit square. Is $kd_{k}$ bounded as $k\rightarrow\infty$ ?
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1answer
56 views

Show that $\varphi_{j+1}(x)-C_j x \varphi_j (x) = \sum_{k=0}^j \alpha_{jk} \varphi_k (x)$ where $\{\varphi_j \}$ is a syst. of orth. polynom.

This is a homework exercise. I'm only asking for hints, please don't give a full solution. This is the exercise: This is my attempt to solve this problem: If $C_j$ is chosen to be equal to the ...
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1answer
274 views

Inequality remain true if we pass to the limit?

Say we have $$X \geq \sum_{k=1}^{n} F(k) $$ Does it follow that the inequality remain true if we pass to the limit $$X \geq \sum_{k=1}^{\infty} F(k) $$ Given that the $F$'s are all non negative?
2
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2answers
39 views

How can I show $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$ where $S(t)f(x)=e^{-t^2-2tx}f(x+t)$..

I need some help with the following: For every $t\in [0, \infty)$ let $S(t):C_0([0, \infty))\rightarrow C_0([0, \infty))$ be the bounded linear operator given by, $$S(t)f(x)=e^{-t^2-2tx}f(x+t)),$$ ...