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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2answers
83 views

Derivative of matrix inversion function?

Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?
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1answer
79 views

how to prove that $k+1 \ge (1+\frac{1}{k})^{k}$?

How to prove that $$k+1\ge \bigg(1+\frac{1}{k}\bigg)^{k} $$ when $k>2$
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1answer
36 views

Change of variable in differential equation legitimate?

Just a general question ( I don't want to solve this ODE, I just want to understand why this is legitimate to do or not): Assuming we have the ODE $$y'(x) - \cos(x) y(x)=0$$ on $[0,2\pi]$ Am I ...
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1answer
58 views

Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
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2answers
54 views

Absolute value inequality, difficult proof

Prove that $|x - y| \le |x| + |y|$ Let $x > y$ without the loss of generality, $x - y > 0 \implies |x - y| > 0$ $|x| > 0, |y| > 0 \implies |x| + |y| > 0$ But how can you ...
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2answers
287 views

Every infinite subset of E in R having a limit point in E implies E is closed and bounded

Every infinite subset of E in R having a limit point in E implies E is closed and bounded. Could you please help with a formal proof of this result ?
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2answers
195 views

Spivak “min” notation confusion

Spivak uses a notation: min$(1, \frac{\epsilon}{2|a| + 1})$ What does he mean by this notation? especially by "min"??
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1answer
72 views

How find this sum $I\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}=1-\frac{1}{2}\ln{(2\pi)}$

Question: show that $$I=\sum_{k=1}^{\infty}\dfrac{B_{2k}}{2k(2k-1)}=1-\dfrac{1}{2}\ln{(2\pi)}$$ where $B_{n}$ is Bernoulli number:Bernoulli number I think we can $$I=\sum_{k=1}^{\infty}\left(...
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0answers
38 views

Bound of the gradient of a $C^{m}$ function.

Let $f$ be a $C^{m}(\mathbb{R}^{n})$ function, that is a function which is $m$ times continuously differentiable. Is it true that $$ |\nabla f(x)| \leq \frac{C}{|x|^{n}}$$ for each $x \in \mathbb{R}^{...
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0answers
39 views

Finding strictly function that satisfies this limit

I am trying to find a strictly increasing function $\varphi: [a,b] \rightarrow \mathbb{R}$ such that $ \lim_{y \to x} \frac{2x - a - b}{\varphi(y) - \varphi(x)} = 0 $ and $ \lim_{y \to x} \frac{a + b -...
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0answers
36 views

Continuity of integral

How can I prove the continuity or discontinuity of the following $\mu$ as a function of $x$? $$\mu(x) = \int_0^x\exp\left\{ -\int_0^t\frac{d\alpha(s)}{\beta(s)+1} \right\}dt + \int_x^{+\infty}\exp\...
2
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4answers
281 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
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1answer
51 views

Exercise on discontinuous coefficients in first order pde

I'm trying to solve the following exercise on first order pde with discontinuous coefficients, which I've found online. It consists in giving an unambiguous meaning to the following equation $$u_t+\...
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1answer
86 views

Find the limit of $\sum\limits_{n=0}^\infty \frac{n}{3^n}$ [duplicate]

Hi all What would the best way/method be to approach this, any advice would be appreciated
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0answers
73 views

Rotational invariant curve definition

I have a question about the definition of rotational invariant curve. The definition I have is the following "By an invariant curve for a twist map $F$ we mean a simple closed curve that is invariant ...
2
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1answer
83 views

Distributions (Generalized Functions)

Why is a distribution defined in terms of the inequality $$ |\langle\Gamma, \psi\rangle| \leq C \sum_{|\alpha| \leq N} \sup_{x \in S} | \partial^\alpha \psi |$$ for all $\psi \in C^\infty_c (\Omega)$...
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2answers
32 views

Boundedness of continuous functions

So I am asked to give an example of a continuous function $f: [a,b] \rightarrow \mathbb{R}$ that has an image which is unbounded. I give the function $f(x)=1/x, (0,1)$. It is bounded below by $0$ ...
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2answers
61 views

Prove that $\lim_{n \rightarrow \infty} \frac{2^{n}}{n!} = 0$

Prove that $\lim_{n \rightarrow \infty} \frac{2^{n}}{n!} = 0$ using the hint that $0 < \frac{2^{n}}{n!} \leq 2 \, \left(\frac{2}{3}\right)^{n-2}$ for all $n \geq 3$? I know there is a thread ...
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2answers
120 views

How find this sum $\frac{1}{1^2}+\frac{2}{2^2}+\frac{2}{3^2}+\frac{3}{4^2}+\frac{2}{5^2}+\frac{4}{6^2}+\cdots+\frac{d(n)}{n^2}+\cdots$

Question: Find the value $$\dfrac{1}{1^2}+\dfrac{2}{2^2}+\dfrac{2}{3^2}+\dfrac{3}{4^2}+\dfrac{2}{5^2}+\dfrac{4}{6^2}+\cdots+\dfrac{d(n)}{n^2}+\cdots$$ where $d(n)$ is The total number of ...
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4answers
131 views

Spivak Sine confusion (possible error)

quote from Spivak: "Let us consider the function $f(x) = \sin(1/x)$." The goal is to show it is false that as $x \to 0$ that $f(x)\to 0$ He says we have to show "we simple have to find one $a > 0$...
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1answer
130 views

Homeomorphism between two locally compact spaces

Suppose $X_1$ and $X_2$ are two locally compact spaces. Define $\phi:X_1\to X_2$. Suppose $\phi$ is bijective and continuous. I know that if $X_1$ is compact, I can conclude $\phi$ is a homeomorphism. ...
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2answers
302 views

Misunderstanding about the definition of a limit (Spivak Calculus)

In Spivak's text, I quote: "In general, if $\epsilon > 0$ to ensure that $|x^2\sin(\frac{1}{x})| < \epsilon$ we need only require that $|x| < \epsilon$ and $x \ne 0$" This can easily be ...
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2answers
18 views

homogeous of degre zero function

let $f:\mathbb{R}^{n} \to \mathbb{R}$ be a smooth function on $\mathbb{R}^{n}-{0}$, a positive homogeneous of degree zero function and $\lim_{x \to o} f(x) $ be exit, then f is a constant function. ...
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1answer
74 views

Why is the derivative of $\ln |x|$ equal to $1/x$?

In a complex analysis book, I read that $f(z) = 1/z$ doesn't have a primitive on $\mathbb{C}\setminus\{0\}$. The reason given used the much stronger fact that if any $g : \mathbb{C} \to \mathbb{C}$...
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0answers
46 views

Example of a continuous bijection on $\mathbb R^n$ whose inverse is not continuous [duplicate]

For $n \ge 2$ give example of a bijective continuous map $f: \mathbb R^n \to \mathbb R^n$ whose inverse is not continuous ; example of such a function is also an example of Does there exist a ...
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0answers
49 views

Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on $\...
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1answer
251 views

Definition of convex combination with matrix-vector multiplication

Is there any similar definition to "convex combination of vectors", for the case of matrix coefficient not the scalars? E.g: w=(I-A)v+A u. What conditions A needs to satisfy to make it a "convex ...
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2answers
29 views

Show a certain sequence converges

Let sequence $(x_n)$ be recursively defined as: $x_1=0, x_2=1, x_{n+2}=\frac{1}{2}(x_{n+1}+x_n)$ What I need to do is show that $ (x_n) $ converges, not necessarily to show that it converges to $\...
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1answer
67 views

What is meant by“…on se ramène par régularisation…”?

I am currently attempting to translate the paper 'Sur l'équation de convolution $\mu = \mu \ast \sigma$' by Choquet and Deny. In the paper, a locally compact abelian group $G$ and a positive measure $...
4
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1answer
252 views

calculation of normal derivative

Suppose $\Omega$ is a bounded region in the plane $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Suppose $u$ is a smooth function in $\Omega$. I want to calculate $$\frac{\partial}{\partial\nu}|...
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1answer
21 views

Showing $\cos{\pi \frac{mx}{R}}\sin{ \pi\frac{nx}{R}}$ are orthogonal in $L^{2}([0,R])$?

I'm trying to show that $\sin \left(\frac{n\pi}{R}x\right)$ and $\cos \left(\frac{m\pi}{R}x\right)$ are orthogonal using the trigonometric identity that $2\cos{mx}\sin{nx} = \sin((m+n)x) + \sin((m-n)x)...
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2answers
710 views

When is the difference of two convex functions convex?

Assume that $X$ is a finite dimensional Banach space. I know that in general if two functions $f:X \mapsto \mathbb{R}$, $g:X \mapsto \mathbb{R}$ are convex then the function $(f-g):X \mapsto \mathbb{R}...
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1answer
419 views

References on integration: collections of fully worked problems (and explanations) of (1) advanced and (2) unusual techniques

I am searching for two kinds of books. (1) Comprehensive books that collect, explain, and provide many examples (that is, fully worked problems) of advanced integration techniques (that is, ...
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2answers
226 views

Expectation of Truncated Random Variables

Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $0<\delta<0.5$ and $\epsilon >0$ and define $c_n:=\...
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0answers
31 views

Help with the definition of weak solution

I've just begun studying PDE. In order to prove that $w \in C^1(D)$ is a weak solution of \begin{equation} \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; \partial ...
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1answer
51 views

Kolmogorov Exponential Bounds (Upper)

This is one version of Kolmogorov exponential bound from Allan Gut's Probability: A Graduate Course (2005, p385-386). Let $Y_k$ be an independent sequence of random variables with zero mean and finite ...
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7answers
358 views

If $\,f^{7} $ is holomorphic, then $f$ is also holomorphic. [closed]

I need some help with this problem: Let $ \Omega $ be a complex domain, i.e., a connected and open non-empty subset of $ \mathbb{C} $. If $ f: \Omega \to \mathbb{C} $ is a continuous function and $...
2
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3answers
120 views

What is the meaning of $dy=dx^2$?

When I read the mathematical analysis ,I think if the differential is $dy=Adx^2$ $A$ is a function about x, what will happen? Maybe, it is not proper defined ,but I think the "function" meet $dy=A(dx)^...
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1answer
35 views

Simple function sequence for measurable functions with range $[0,1]$.

Let $[0,1]$ have the usual topology. Consider $f: X \rightarrow [0,1]$ such that $f$ is measurable. Fix $n$. I want to show that there is a measurable simple function $\phi_{n}$ such that \begin{...
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1answer
53 views

Integral Inequality: Does multiplying integrand by $t$ maintain positivity?

Let $g: [0, 1]\to \mathbb R$ be a continuous function such that, for all $x\in [0, 1]$, $\displaystyle\int_x^1 g(t) \ dt\geq 0$. I'd like to show that $\displaystyle\int_0^1 tg(t)\ dt\geq 0$. Well, ...
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1answer
90 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
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1answer
66 views

Is the algebra generated by sine and cosine separating points on $[0, 2 \pi]$?

Consider an Algebra $\cal A$ generated by $\{\cos x, \sin x\}$. I was wondering does this algebra separate the points over $[0, 2\pi]$ ? I think it is. So I'm trying to show for any two distinct ...
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0answers
71 views

Can the Hamburger moment problem be solved for probability measures?

The hamburger moment problem states that given any real sequence $\{a_n\}$, there exists a positive Borel measure $\mu$ such that $$ a_n =\int_{\mathbb{R}} x^{n}\,d\mu. $$ In other words, the ...
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1answer
105 views

general mean value theorem

Can anyone give me the intuitive explanation of the general mean value theorem stated in my notes as under: Let $f:U\rightarrow \mathbb R$ and $U\subseteq \mathbb R^n$ and let $f$ is differentiable ...
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0answers
48 views

Is locally Lipshitz mapping on a compact set globally Lipschitz? [duplicate]

Let $f: K \rightarrow \mathbb R$ be a locally Lipschitz mapping on a compact subset $K$ of $\mathbb R^n$. Is it then $f$ Lipschitz?
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0answers
67 views

Exponential Function Limit Question

When I was first introduced to a derivation of the Taylor series representation of the exponential function here (pg 25): I noted the author, Dunham mentioning that the argument was non-rigorous. I ...
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4answers
160 views

Where is $\,\,f(x)=x^2\,\,$ a contraction mapping?

I can't understand this: My notes define a function $f$ to be a contraction if $$\lvert\, f(x)-f(y)\rvert\leq c\lvert x-y\rvert,$$ for some 0$\lt c\lt 1$. But then I have a question in my ...
0
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1answer
18 views

'O' order of a function

I'm having some trouble finding the 'O' orders of given functions. I have the following definitions. Let $\phi(x),\psi(x)$ be real or complex valued functions. Let $x_0$ be a limit point of a set $R$ ...
2
votes
1answer
98 views

Prove that $\mathbb{Q}\!\smallsetminus\!\mathbb{Z}$ is dense in $\mathbb{R}$

Can someone just tell me if this is a correct way to prove it. let $(a,b)$ be a nonempty open interval in $\mathbb{R}$. Then by density of $\mathbb{Q}$ in $\mathbb{R}$ there exists $q\in \mathbb{Q}$ ...
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0answers
54 views

Prove that $w$ is a weak solution, basic questions.

I have this problem: \begin{equation} \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; \partial D.\end{cases} \end{equation} The domain $D$ is made by two triangles ${...