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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2
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1answer
16 views

Covering Lemma (Folland Lemma 3.15)

Lemma 3.15 from Folland's Real Analysis: Let $\mathcal{C}$ be a collection of open balls in $\mathbb{R}^n$, and let $U = \cup_{B \in \mathcal{C}}B$. If $c < m(U)$, there exist disjoint ...
1
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1answer
17 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
1
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2answers
15 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
1
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1answer
32 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
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0answers
13 views

find angular velocity for so that: $\exp(jt) = \exp( j(3t+\pi/3) )$ [on hold]

I have a fourier series in which there are two different arguments on the exponential function: $jt$ and $j(3t+\pi/3)$ and I have to "choose" a fitting angular velocity. It it probably easy yet it ...
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0answers
12 views

Cavalieri's principle and integrals

Let $f$ be a continuous function an $[a,b]$. Let $P\subset R^2$ be the figure under the graph of $f$ and $D \subset R^3$ a solid figure obtained by rotating a plane curve around $x$-axis. Using ...
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0answers
10 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
0
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1answer
13 views

Modify the Cantor pairing function

I have an infinite set of pairs $I:=\{(k,m) \mid k,m \in \mathbb{N},\quad m\geq 1, \quad 1\leq k\leq m\}$. I want to establish a bijective correspondence $\phi$ between $I$ and $\mathbb{N}$. I've ...
1
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0answers
20 views

Fourier transform all steps walkthrough for wave vector $k$ and $x$

Below is my walkthrough of a fourier transform. My problem is that I want to do all the similar steps for a fourier transform between position x and the wave vector k. That is working on a solution of ...
2
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1answer
55 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
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2answers
35 views

is this correct that if $\frac{\partial f}{\partial y}=0$ then $f$ is independent from $y$?

Suppose that $A=\{(x,y) \in \Bbb R^2 : x> 0 $ or $ y=0 \}$ and $f:A\to \Bbb R$ is an arbitary function. Prove that If $\frac{\partial f}{\partial x}=0$ then $f$ is independent from $x$ If ...
2
votes
1answer
35 views

What is the point of big Oh notation when it is used for estimation?

I'm reading a book on number theory at the moment that assumes familiarity with big Oh notation...and while I think I do understand the notation I cannot understand the point of it. For instance let ...
0
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1answer
18 views

Refreshing solving second order ODE

I have a boundary value problem for the following differential equation $$\frac{d^2 v}{d \chi^2} = q^2 \left( v - C \right), \; 0<\chi<S \; and \;\; v(0)=v(S)=0 $$ where $q$ and $C$ are certain ...
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2answers
21 views

Clarification from old post: Union of sigma-algebras is non sigma-algebra

I have been working on slightly different problem from one posted back in 2013 here. I followed closely the hints given by @martini there, but nevertheless I still got stuck. I am retyping the ...
1
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1answer
31 views

Conditions for a supremum of a set.

Suppose a function $f(x)$ is continuous on $[a, b]$ and there exists, $x_0 \in (a, b)$ such that $f(x_0) > 0$. And then define a set, $$A = \{ a \le x < x_0 \space | \space f(x) = 0 \}$$ We ...
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2answers
42 views

Is it possible to find equation for ellipse when focus, eccentricity 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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1answer
10 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
16 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 ...
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0answers
28 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
26 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 ...
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0answers
22 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 need not exist. Moreover, it's possible for all the ...
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0answers
51 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 ...
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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
17 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
60 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 ...
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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
votes
1answer
21 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
vote
1answer
25 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
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1answer
87 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 ...
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0answers
17 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) := ...
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3answers
31 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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votes
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.
0
votes
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 $b^2=c$ ...
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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: ...
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0answers
56 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
36 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
110 views
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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)\ ...
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4answers
49 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
51 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
2answers
116 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
votes
0answers
21 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 ...