Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ...
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1answer
9 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
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0answers
16 views

cardinality of connected subset [on hold]

Let X be a connected subset of real numbers. if every element of x is irrational then the cardinality of x is (a) infinite (b) countably infinite (c)2 ...
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1answer
17 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by f(x) = ...
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2answers
55 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
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1answer
26 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
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13 views

Biorthogonal functionals continuous?

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
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12 views

On reconciling different definitions of the $\nabla$ operator in curvilinear coordinates

Note: This questions was originally asked in iMechanica. The main confusion appears to be on whether Christoffel symbols should appear in the divergence of a field expressed in curvilinear ...
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26 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
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1answer
24 views

when can you extend a map from a Hilbert basis?

Suppose $H$ and $K$ are Hilbert spaces and $H$ has Hilbert basis $h_i$. What is a necessary and sufficient condition for elements $k_i$ of K so that $h_i \mapsto k_i$ extends to a continuous linear ...
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2answers
37 views

Minimum value of a function

For $x \in [0, 5]$, let $$f(x) = \sum_{i = 1}^{5}\frac{1}{|x - i|}.$$ Why is $$f(x) \geq 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = f(0)?$$ This of course is true if one simply plots ...
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1answer
25 views

Where to go with continuity?

I've started to see continuity at university. The lecturer has given the $\delta-\epsilon$ definition of continuity, but we've seen applications of those only to silly examples of functions: ...
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1answer
11 views

how can I find the limit-pointwise convergence

Let $f_n(x)=\left\{\begin{matrix} 0, & x<\frac{1}{n+1} \text{ or } \frac{1}{n}<x\\ \sin^2(\frac{\pi}{x}),&\frac{1}{n+1} \leq x \leq \frac{1}{n} \end{matrix}\right.$ Show that $(f_n)$ ...
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0answers
22 views

Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
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21 views

question about a proof: sequence of picard iteration converges uniformly

Given $u'(t)=t\cdot u(t)+t^3$ with $u(0)=0$ I want to show that $$u_{n+1}(t)=u(t_{0})+\int_{t_0}^{t}f(x,u_{n}(x))dx$$ converges uniformly on $[-b;b]$ and solves the differential equation. After ...
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1answer
17 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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20 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
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19 views

find a minimum of a function

let $F(x)= \frac{2}{nx} + L^{2}e^{\frac{\alpha}{x^{r}}}$ for given $ n,L,\alpha$ show that $x^*=argminF(x) $ where $ x^*=\frac{\alpha^{\frac{1}{r}}}{(ln(n))^{\frac{1}{r}}}$ i know that F is convex ...
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1answer
33 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
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0answers
16 views

ratio test for convergence of series, different versions

In lecture we had the ratio test: Let $(a_k)$ be a sequence in $\mathbb{K} \in\{ \mathbb{R}, \mathbb{C}\}, a_k\not= 0$ for all $k \ge k_0$, where $k_0\in \mathbb{N}$. (I) If there is a $q\in (0,1)$, ...
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0answers
28 views

Radius of convergence and uniqueness

I was reviewing some fundamental concepts of real analysis, and the following exercise popped up: Let there be a power series: $\sum\limits_{k=0}^\infty a_k \mid x - 3 \mid $ Such that for $\ x = ...
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21 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
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22 views

If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

First I'd like to say that although this question was asked before (here) and is from the same text, the answer used methods that were not introduced in the text. Let $P_n(x)$ be a polynomial of ...
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1answer
24 views

Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$. Now it is obvious that the graph reaches a maximum at $x=e$ but how ...
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6 views

convergence double series in Hilbert spaces

Let $(a_{ij})$ be sequence in Hilbert spaces H. What is sufficient condition convergence $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$ ? Whether $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$ = ...
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1answer
110 views

How was this sequence discovered?

Let $N$ be a positive integer and consider the following rational sequence for $n \ge 0$: $$ a_{n+1} = \frac{N a_n + N}{a_n + N}, a_0 \in \Bbb{Q}. $$ If $-\sqrt{N} < a_0 < \sqrt{N}$, then ...
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1answer
28 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
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16 views

On the uniform convergence of generalized integral

Is the integral $$ \int_{1}^{\infty} e^{-yx^2}\sin{y}dx.$$ uniformly convergent in $y \in [0,\infty]$? Why or why not?
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24 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
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25 views

Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
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1answer
36 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
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1answer
14 views

Performance Index for sports [on hold]

I am trying to create a performance index weighted of strength and speed values. I have the equation working fine for the strength index. Ex. ...
1
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1answer
36 views

Uncountability of a nonmeasurable set

As per the Vitali's theorem, every measurable set of positive measure has a subset which is nonmeasurable. Which proceeds by defining a rational equivalence, followed by using the axiom of choice on ...
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2answers
72 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
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2answers
47 views

Constructing $\mathbb{R}$ from $\mathbb{Q}$ and showing $\mathbb{Q}$ is dense in $\mathbb{R}$

This is a very long, multi-part problem that we were told to figure out by any means possible. There are no limits on getting help or finding answers online. I haven't had much luck at all solving ...
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0answers
16 views

bounded and compact orthonormal basis for L^2(R) [on hold]

Whether there is a bounded and compact support orthonormal basis for $L^2(\mathbb{R})$ such that $(B_jK_j)$ is contained in $L^1(\mathbb{R})$ (where for every j, $B_j$ is $esssup e_j$ and $supp e_j$ ...
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1answer
22 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
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27 views

Is it a sufficient and necessary condition?

Let $f_n \to f$ uniformly in $[a,b]$. If each $f_n$ is integrable in $[a,b]$,then $f$ is integrable in $[a,b]$ and $\int_a^b f_n(x) dx \to \int_a^b f(x) dx$. If $\int_a^b f_n(x) dx \nrightarrow ...
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1answer
12 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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1answer
27 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
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0answers
32 views

Justification for exponents other than positive integers

Here's a question that's bothered me ever since highschool, and I've never heard a good answer. I know that mathematicians can define operators to mean whatever they want, as long as their system of ...
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3answers
45 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
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1answer
21 views

define convergence for a sequence of elements in a metric space [on hold]

Define the convergence for a sequence of real numbers. How will you modify the definition to define convergence for a sequence of elements in a metric space? Give examples of convergent and non ...
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0answers
31 views

how can I find the supremum??

We set $f_n(x)=(1+\frac{x}{n})^n, x \in \mathbb{R}$.Check the uniform convergence of $f_n$ at the intervals $(-\infty,a)$ and $(a,+\infty)$,where $a$ a random real number. $\lim_{x \to +\infty} ...
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2answers
77 views

If $f$ is twice differentiable then $f^{-1}$ is twice differentiable

$f:(a,b) \rightarrow (c,d)$ is a bijection and $f$ is differentibale with $f'(x) \neq 0$ for all $x \in (a,b)$, then $f^{-1}$ is also everywhere differentiable. Show that if $f$ is twice ...
2
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1answer
44 views

Does $\mathrm{Im}(f(z))$ bounded above $\implies$ $|f|$ is bounded, for analytic $f$?

If $f$ is analytic on $\Omega$ s.t. $\mathrm{Im}(f(z))$ is bounded from above, then does this imply that $|f|$ is itself bounded? I know that if $\Omega = \mathbb{C}$, then the result follows as a ...
2
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3answers
49 views

Why do we pick $n_0$ such that $\frac{1}{n_0}< \delta$?

Let $f: \mathbb{R} \to \mathbb{R}$ uniformly continuous.We set $f_n(x)=f(x+\frac{1}{n})$.Show that $f_n \to f \text{ uniformly }$. Let $\epsilon>0$. Since $f: \mathbb{R} \to \mathbb{R}$ uniformly ...
2
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3answers
842 views

“nice functions”

I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write "Morrey ...
3
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1answer
25 views

Showing that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from above

Problem: Suppose $f$ is analytic on the domain $\Omega$ except at the isolated singularity $a \in \Omega$. Show that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from ...
2
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2answers
40 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...