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
27 views

Is it possible to find equation for ellipse when focus and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
0
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0answers
7 views

from each one-third part that eliminated in construting the Cantor set pick a point, what apout the resulting set?

During constructing the cantor set, pick up a point from the one-third that eliminated. if we call the set of this points A, then what is the internal of A? is the complement of A countable?
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1answer
15 views

Solving second order nonlinear ODE given boundary condition at infinity

I am trying to solve the following differential equation $$\frac{d^2 u}{dx^2} = - \frac{d V}{du} \; \; , \;\; where \;\; \; V = \frac{1}{2}u^2 - \frac{1}{4}u^4 $$ And the given boundary conditions are ...
-3
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0answers
21 views

analysis/number theory study group (online) [on hold]

I plan on studying analysis from landau, rudin probably others and am looking for people (hopefully more than 1) where we could solve theorems/problems and ask each other questions. Online ...
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0answers
19 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
1
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0answers
14 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative to not exist. Moreover, it's possible for all the ...
0
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0answers
50 views

A lower bound for $\log\left( \frac{a+x^2}{b+x^2}\right)$

I am looking for a tight lower bound for $$f(x)=\log\left( \frac{a+x^2}{b+x^2} \right)$$ $x>0$ and $1<b<<a$. I didn't check for convexity analytically, but I plotted this function ...
-1
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1answer
33 views

$ \exists c \in( a, b) \text{ such that } f(c)=\max\limits_{x \in [a, b]} f (x) $ [on hold]

I saw in a corrected. We have $ f $ continuous on $ [a, b] $ with $ f (a) = f (b) $ and $ f $ differentiable left and right at $ (a,b)$ then $$ \exists c \in (a,b) \text{ such that } ...
1
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1answer
15 views

Inequality connecting inf and liminf

Suppose $f(x,y)$ is a continuous, nonnegative-valued and bounded function on $\mathbb{R}^2$. Is the following correct? $$ \inf_{x,y} f(x,y)\le \liminf_{y\to\infty}\inf_{x}f(x,y) $$
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0answers
20 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
0
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4answers
57 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
0
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1answer
30 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
1
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0answers
22 views

Relationship between supremum of the partial derivative and the derivative of the supremum

To fully set up the problem: I have a function $$F : \mathbb R^n \times \mathbb R^+ \to \mathbb R$$ such that $F$ is positive and for any fixed $t^* > 0$, $$F(x,t^*) \in H^\infty(\mathbb R^n);$$ ...
0
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1answer
17 views

References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
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0answers
14 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
1
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1answer
23 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
0
votes
1answer
86 views

Theorem $4.3.12$ on ( Mathématiques en BCPST Tome 1 Pascal BEAUGENDRE )

Let $f\in \mathbb{R}^{I}$ and $x_0 \in \stackrel{\ \circ}{I} = \mathrm{int}(A)$ Show that $$\left.\begin{matrix} f \text{ is continuous at } x_0 \\ f \text{ is differentiable at all } x \in I ...
0
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0answers
16 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
1
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3answers
30 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
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1answer
27 views

Prove that the following statement is false [on hold]

If $s_n \le 1/2^n$ for all $n$, then the series $\sum_{n=1}^\infty s_n$ converges. Prove that this statement is false.
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1answer
34 views

Inequality between norms on $\mathbb{R}^n$

I am trying to show that $\lVert x\rVert_q\leq\lVert x\rVert_p\leq\lVert x\rVert_1$ for $1<p<q$ where $\lVert x\rVert_p=(\sum |x_i|^p)^{1/p}$ and $x\in\mathbb{R}^n$. This seems like it should be ...
1
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2answers
18 views

Explanation of uniqueness of square root

Let $c$ be a positive number. Then there is a unique positive number whose square is $c$. That is, $x^2=c$ Start: Suppose $a$ and $b$ are numbers whose square is $c$. then $a^2=c$ and $b^2=c$ ...
0
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0answers
22 views

Check Functional Analysis Proof

I seem to have proved something with elementary techniques even though the paper I found it in suggests it requires heavy tools. There could be a mistake but I can't find it if there is one. Theorem: ...
1
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0answers
50 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
1
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3answers
35 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
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0answers
15 views

Question about the Hessian matrix and its eigenvalues

Let $\Omega$ a bounded and connected domain in $R^n$ and $a_{ij} \in C(\Omega)$, $i,j=1,...,n$ such that $$\lambda |\theta|^2 \leq a_{ij}(x)\theta_i \theta_j \leq \gamma |\theta|^2, i,j=1,...,n$$ ...
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1answer
109 views
0
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0answers
30 views

calculate $\lim_{n \to \infty}(\int_{a}^{b} f^nx(x)g(x)dx)^{\frac{1}{n}}$ [on hold]

Suppose that $g,f:[a,b] \to (0,\infty) $ are two continous functions, calculate $$\lim_{n \to \infty}\Bigg(\int_{a}^{b} f^n\ x (x)\ g(x) \ dx \Bigg)^{\frac{1}{n}}$$ where $f^n (x)= f(x)^n =f(x)\ ...
1
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4answers
48 views

how to find the lim of $(1+\arcsin x) ^{\cot x} $ as $x$ goes to $0$?

$$\lim_{x\to 0} \ln (1+\arcsin x) ^{\cot x}=\lim_{x\to 0} \cot x \ln (1+\arcsin x)=\lim \frac{(1+\arcsin x)}{\tan x}$$ from l'hospital $$ \lim_{x\to 0}\frac{ \frac{1} {\sqrt{1-x^2}}} {\sec^2 x} $$ ...
2
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0answers
32 views

$\lim_{n \to \infty}\prod_{k=1}^n \cos(k\sqrt{\frac{3}{n^3}} t) = e^{- \frac{t^2}{2}}$ [duplicate]

Is it true that $$\lim_{n \to \infty}\prod_{k=1}^n \cos\left(k\sqrt{\frac{3}{n^3}} t\right) = e^{- \frac{t^2}{2}}$$ ? How to proceed ?
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3answers
50 views

“Standard” proof that open disks in $\mathbb{R}^2$ are connected?

Homework for a complex analysis course asks me to prove as homework that open disks are connected. I do know a way to do this: open disks are convex, and an old exercise in Rudin's "Principles of ...
1
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2answers
25 views

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$.

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$. What I have: Assume that $k<0$, need to show that this contradicts the ...
0
votes
1answer
26 views

What can we say about the definite integral and Riemann sums?

Consider $$f(x) = \int_{1}^{x} \frac{dt}{t}$$ and $$g(x) = \sum_{1}^{x} \frac{1}{t}.$$ I would like to say that $f(x) < g(x)$ for all natural numbers $x \ge 1$. Is there an easy way to derive ...
3
votes
1answer
102 views

Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
0
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0answers
20 views

Weak star convergence question

Let $C$ be a convex cone in $L^{\infty}$, that is if $x,y \in C$ and $\alpha, \beta > 0 $ then $\alpha x + \beta y \in C$. Let $U$ be the unit ball in $L^{\infty}$. Assume that for each sequence ...
1
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2answers
67 views

$ \int \frac{dx}{4x^2-12x+13}$

This is probably not too hard but i can't get it right: I am trying to calculate $$\displaystyle \int \frac{dx}{4x^2-12x+13}$$. The solution is $\displaystyle ...
1
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1answer
37 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
1
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1answer
43 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
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0answers
55 views
+50

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
0
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0answers
13 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
0
votes
1answer
46 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
1
vote
1answer
12 views

Proof via strong induction of a string output

I'm still new to the whole proof thing (first class of discrete mathematics and analysis right now). I could do general induction problems, but the fact that 'n' is the output here along with the ...
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0answers
32 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
1
vote
2answers
41 views

Jacod Protter “Probability Essentials” Problem 2.8

The question asks to show that a sigma-algebra $\mathcal A$ consisting of $A$ s.t. $A=f^{-1}(B)$, where $B$ is in $\mathcal B$ are Borel subsets of $R$ and $f$ is continuous, is contained in $\mathcal ...
1
vote
1answer
26 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...
0
votes
1answer
21 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
0
votes
1answer
17 views

$\lim_{n \to \infty} \int_{\Omega}X_n d \mu = +\infty$ under some conditions

Suppose $X_n$ are measurable functions in $L^1$ defined on the measure space $(\Omega, \mathfrak{F}, \mu)$. Suppose that $0 \leq X_n$ a.e. for all $n$ and $X_n \leq X_{n+1}$ a.e. for all $n$. Thus ...
0
votes
0answers
22 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
0
votes
0answers
25 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
3
votes
1answer
51 views

False equations with Euler's Identity [duplicate]

What's wrong with the following equations? $$1 = 1^{-i} = (e^{2πi})^{-i} = e^{-i2πi} = e^{2π}$$ My guess would be the third equation, but I can't really tell why... in the first equation, we use the ...