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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24 views

Measure concentrated at a point

What does "a finite random measure $\nu$ is concentrated at a point" mean? And in this case, what is equal to $\int_{\Omega} x d\nu$ ? Thank you.
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2answers
35 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $(x_n)$ tending to $0$ such that $f(x_n)=0$ for all $n$, then $f(x)=0$ for all $x$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
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1answer
23 views

Which of two quantities is greater?

Let $x$ and $y$ be two positive real numbers such that $x>y$. Which of the quantities is bigger and when? $(x-y)\log\left(1-\frac{y}{x}\right)$ $x\log\left(1-\frac{y}{x+y}\right)$
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1answer
30 views

Show by substitution that: $\int^{xy}_{x}\frac{dt}{t}=\int^{y}_{1}\frac{dt}{t}$.

probably it is an easy one, but I can't get my head around it. Show by substitution that: $$\int^{xy}_{x}\frac{dt}{t}=\int^{y}_{1}\frac{dt}{t}$$ Any help would be greatly appreciated.
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0answers
26 views

Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, show $f(xy)=f(x)+f(y)$. [duplicate]

I would like to ask you for some help with the following problem. Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, and $f(1)=0$. Show, ...
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0answers
16 views

construction of some smooth function

I want to construct $\varphi \in C_c(\mathbb R)$ whose support is in $[\frac 1 2 ,2]$, satisfying $\displaystyle\sum_{n=-\infty}^\infty \varphi(2^{-n}t)=1$ for all $t>0$. I guess I should use ...
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1answer
21 views

Comparison of limits

I would like to obtain the values of n for which the following is satisfied: $$\lim_{x\to 0} x^n = \lim_{x\to 0} - (-x)^n$$ Upon observing I think it should be for all odd values of n. However the ...
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4answers
48 views

General expression for the k-th derivative of $(\cos(x))^n$

Is there a general expression for the higher-order derivatives of $(\cos(x))^n$ evaluated at the origin? The odd derivatives are zero due to the symmetry, but what about the even derivatives?
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0answers
12 views

Non-asymptotic error bound

I am looking for a sufficiently tight estimate of the following over the interval $[-T,T]$: $| \exp(t) - (1+t/n)^n|$. This is of course $o_n(1)$. What I am looking for is a non-asymptotic estimate ...
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0answers
21 views

Growing slower than exponential?

Consider $2^{cn}+a(n)$ with $c$ being an exponential growth rate. Am I right that $$ 2^{cn}+a(n)\sim 2^{cn} $$ only, if $a(n)$ grows slower than exponential? And if yes.. when does $a(n)$ grow ...
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1answer
14 views

Is this legitim to see polynomial growth?

Let $(a_n)$ be any monotoninally in creasing sequence where $a_n$ is a real number dependent on $n$. Of course it is $a_n\leq a_n^2$ for all $n\geq 1$. Does this means that any increasing sequence ...
0
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1answer
28 views

if the integrals of two function are equal then the functions are equal almost everywhere. true or false?

Actually I know that if the integral of a non negative function is equal to zero then that function is equal to zero almost everywhere. Can I use that to prove or is there a counter example for my ...
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2answers
36 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
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1answer
19 views

Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$

Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $. Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| ...
6
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1answer
45 views

Wick Rotation technique

I am trying to get my head around the Wick rotation technique. I have tried to play around with some elementary examples. Let us imagine I need to solve on the real line $$ y’ = \cos (x)$$ the prime ...
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0answers
31 views

integral inequalities and continuous functions

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
3
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2answers
25 views

How to prove this sequence converges

Here is a problem in analysis: Suppose $x_n\geq0$ and for all $n$, there is $$ x_{n+1}\leq x_n+\dfrac1{n^2} $$ Prove that $x_n$ converges. My approach: it is easy to prove $x_m-x_n\leq ...
2
votes
1answer
21 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
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0answers
31 views

Representing a function as a Poisson Integral.

This is a question I came across in Ahlfors' book Complex Analysis. It is found on page 171 of the 3rd Edition, Exercise 2. "Prove that a function $T(z)$ which is harmonic and bounded in the upper ...
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0answers
22 views

Indexed family of sets/power sets proof [on hold]

suppose there is an indexed family of sets $(Ai|\in I)$ where I is not an empty set. prove that the intersection of $A_i$ exists in the intersection of $P(A_i) ( \cap(i\in I) Ai \in \cap(i\in I) P(Ai) ...
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0answers
25 views

N-Functions (Nice Young functions)

A mapping $\Phi:[0,\infty)\to[0,\infty)$ is termed an N-function (nice Young function) if (i) $\Phi$ is continuous on $[0,\infty)$; (ii) $\Phi$ is convex on $[0,\infty)$; (iii) $\lim\limits_{t ...
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1answer
11 views

Limit superior and inferior of a sequence that satisfies the asymptotic formula $\sum\limits_{n\leq x} a_n \sim x $

Suppose $\{a_n\}$ is a sequence such that $\sum\limits_{n\leq x} a_n \sim x $. I have to show that: $$ \liminf a_n\leq 1 \leq \limsup a_n$$ I've no idea on how to approach this, in all honesty.
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0answers
10 views

Equivalence between $\limsup\frac{f(x)}{g(x)^{2-\epsilon}}=0$ and $\liminf \frac{|\log(f(x))|}{|\log(g(x)^2)|}\geq 1$

Suppose that $f,g\geq 0$ are positive functions on $(0,\infty)$, and assume that $g(x)\rightarrow 0$ as $x\rightarrow\infty$. I am trying to prove that the following two claims are equivalent. I have ...
3
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0answers
40 views

How to evalute: $\int_0^1 \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)) dx$ and $a, b >0$

How to evalute: $$\int_0^1 \left[ \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)\right) \right] dx$$ and $a, b >0$
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2answers
45 views

Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist.

I would like to ask you a question about the following question. Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow ...
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2answers
31 views

Inverse of $I +T^*T$

I am trying to show that the inverse of the operator $I +T^*T$ exists. What I have been trying to do is trial and error taking inverses of $T$ and $T^*$ but to no avail.
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1answer
48 views

Struggling with a basic math question in T. Tao's Analysis I

How to prove that a++ doesn't equal to a? In one example, the book showed that 0 not equal to 0++ by this: 0++ cannot be equal to 0 by the Axiom "n++ not equal to 0 for every natural number n" ...
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1answer
12 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ such that, ...
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0answers
30 views

Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete?

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
2
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1answer
35 views

IVT and fixed point theorem

Suppose that $f:[0,1]→[0,2]$ is continuous. Use the Intermediate Value Theorem to prove that there exists $c∈[0,1]$ such that $f(c)=2c^2$ The answer to this goes from the Fixed point theorem. But in ...
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1answer
15 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
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0answers
17 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
0
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1answer
31 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
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0answers
9 views

Harmonic functions locally null on connected open set

Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$. So, I tried to ...
3
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1answer
54 views

Let $f$ be a non-constant entire function. Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$

Let $f$ be a non-constant entire function satisfying the following conditions: $$f(0)=0$$ for each $M \gt 0$ the set $$\{z \mid \lvert f(z)\rvert \lt M\}$$ is connected. Prove that $f(z)=cz^n$ for ...
3
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3answers
70 views

How to prove the elementary inequality?

The inequality is the following: $$\frac{(1+x)^q-1}{x+x^q} \leq C(q),$$ where $q\in [1,+\infty)$ and $x > 0$, and the constant $C$ depends only on $q$. It's very nice if someone can provide the ...
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0answers
13 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
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0answers
26 views

How can I show the equivalent condition for $\mu^*$-measurability

This is Exercise 4.15 from "Real Analysis for Graduate Students": Let $X$ be a set and $A$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a pre-measure on $A$ such that ...
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0answers
12 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
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0answers
20 views

Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal

also compute $||f-p_2||_2$. Write $p_2$ with respect to $\{P_0,P_1,P_2\}$ and $\{1,x,x^2\}$ I know its helpful to show what I have so far but I really don't know where to start. I'm looking at ...
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2answers
43 views

Confused about proof that diameter of a closure of a set is the same as the diameter of the set.

Definition Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$, with $p \in E$ and $q \in E$. The supremum of $S$ is called the diameter ...
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4answers
35 views

$\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$

Why is $\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$ (here is $\mathbb{R}$ endowed with the standard topology? I could use the criterion: Is $(x_n)\subseteq M$ such that $x_n\to ...
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0answers
15 views

Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
2
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0answers
11 views

Series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$

As stated in the title, I'd like to compute the series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$ where $z$ is a (small enough) complex number. More generally, for any real value $\alpha$, is there a ...
1
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2answers
46 views

Clarification of notation $\|fw\|$

this is the question: Show that for each linear map $f:\mathbb R^d → \mathbb R^e$ there exists $a < \infty$ so that $\|fw\|< a\|w\|$ for each $w$ in $\mathbb R^d.$ And my problem is that $f$ ...
0
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0answers
13 views

Additive and multiplicative model

I'm new in study of survival analysis. There are some basic-questions that I wanna ask and it will be great if somebody could help me to aswer these following questions: What the differences between ...
0
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1answer
41 views

Compute the volume element in a differentiable manifold.

Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable manifold $ M = g^{-1}(0)$. The thing is that ...
2
votes
1answer
17 views

Question about disconnected metric spaces

The definition of disconnectedness that I've been taught is that a metric space $(X,d)$ is disconnected if there exists two non-empty disjoint open sets $A$ and $B$ such that $X=A\cup B$. My ...
1
vote
1answer
22 views

Incomplete beta integral

Let n be greater than one, and B be the beta integral, $$\sum _{j=0}^{\infty } C_j B_{\frac{1}{n}}(j+1,j+2)=\frac{1}{n}$$ Is it correct to call this an inversion formula? What possible ideas are ...
1
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1answer
33 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...