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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1answer
342 views

Integrability of a piecewise function on $[0,1]$

Consider the function $f(x)=\begin{cases} x & x\in \mathbb{Q}\cap [0,1] \\ -x & x\in [0,1]-\mathbb{Q} \\ \end{cases}$ I argue that this function is not integrable since ...
2
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1answer
112 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
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1answer
227 views

Limit of a sequence bounded below but has no cluster points

Question is this: Let $a_n$ be a sequence of real numbers. Prove that if $a_n$ is bounded below and has no cluster points then $a_n$ → ∞. I could not really find a way to prove it. Could you give me ...
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1answer
220 views

parametrizing quarter of a circle

I am given the circle whose equation is: $(x-\frac{1}{2})^{2}+(y+\frac{1}{2})^{2}=\frac{1}{2}$. So, the coordinates of the origin of the circle are: $(\frac{1}{2},-\frac{1}{2})$ and the radius of the ...
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1answer
73 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
2
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1answer
107 views

Convergence of a matrix

Let $A_j$ be a sequence in $\mathbb {C}^{n\times n}$. Show that $\displaystyle \sum_{j=0}^\infty A_j$ converges if $\displaystyle \sum_{j=0}^\infty ||A_j||$ does. Note that $\displaystyle \Vert ...
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3answers
1k views

Question regarding Nested Interval Theorem

Theorem Consider a family of closed intervals, $I_1 = [a_1, b_1], I_2 = [a_2, b_2], \ldots$. If $a_n \leq a_{n+1}$ and $b_{n+1} \leq b_n$ for all $n$ then there is an $x$ which is in every ...
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1answer
58 views

$a_n \downarrow a$, is $a$ the $\inf$ of the sequence?

My working to this question is implies that $a$ is indeed the inf, but I am not really sure whether $\{a_n\} \downarrow a$ implies that $a_n \ge a \quad \forall n$ since I am really not used to the ...
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0answers
47 views

Problem with a proof of one of characterization of manifold in $\mathbb R^n$

Let $M \subset \mathbb R^n$, $k \in \mathbb N$. Assume that $M$ is a $k$-dimensional manifold in $\mathbb R^n$, i.e. for each $x \in M$ there exists an open set $W \subset \mathbb R^k$ and a smooth ...
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0answers
113 views

Convergence of Lebesgue integrable functions in an arbitrary measure.

I'm a bit stuck on this problem, and I was hoping someone could point me in the right direction. Suppose $f, f_1, f_2,\ldots \in L^{1}(\Omega,A,\mu)$ , and further suppose that $\lim_{n \to \infty} ...
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3answers
161 views

Show that $\left(\frac{1}{2}\left(x+\frac{2}{x}\right)\right)^2 > 2$ if $x^2 > 2$

Okay, I'm really sick and tired of this problem. Have been at it for an hour now and we all know the drill: if you don't get to the solution of a simple problem, you won't, so ... I'm working on a ...
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1answer
33 views

Weird BigOmega statement - Totality

I've just encountered a weird statement regarding the BigOmega operator. I should prove that the BigOmega operator isn't totally ordered. As a prove hint, I should show that there are two functions, ...
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1answer
79 views

Subdifferential of a finite dimensional function.

I want to compute the subgradients of the absolute value function in $\mathbb{R}^n$. How do I do this?
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1answer
108 views

Equivalent definition of Riemann integrals

Given the following definition of the Riemann upper integral: $$\bar{\int_{a}^{b}}f =\inf \{ U(f;P): \text{ P is a partition of [a,b] } \}$$ Where $U(f;P)=\sum\limits_{i=1}^{n} ...
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1answer
166 views

Landau symbols and little o

I was wonderig if the following is true: $o(x^n+x^m)=o(x^n)+o(x^m)$ for $x\to 0$. I tried this way: suppose $m>n$ and let first $f=o(x^n+x^m)$. Then $$\frac{f}{x^n+x^m}=\frac{f}{x^n+o(x^n)} ...
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1answer
186 views

Analysis of the fundamental solution to the heat equation

It is well known that there is a heat kernel (or fundamental solution) of the Cauchy problem for the heat equation on $\mathbb{R}^{n}$. I have a simple question. How do I show that the fundamental ...
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4answers
61 views

Show convergence of a given series and find the limit.

Given the Series $$\sum_{k=1}^\infty \frac{1}{k(k+2)}$$ How exactly would I find out the limit is $\frac34$ as suggested by Wolframalpha? I already found out I can prove it actually converges by ...
3
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1answer
380 views

Show the usual Schwartz semi-norm is a norm on the Schwartz space

Let $f \in C^\infty(\mathbb R)$. Define the semi-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)| $$ where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$. Show ...
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1answer
333 views

Conditions for continuous extension of a function on an open set to its closure

Suppose $U$ is a open set in $\Bbb R^n$, and suppose $f\colon U\to \Bbb R$ is a continuous function. Suppose that $f$ is uniformly continuous on every bounded subset of $U$. Question: Can $f$ be ...
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1answer
116 views

Creating a 3D surface from 2D graphs

So I have two sets of equations: $\mathcal{A}$ = \begin{equation} \{ f(y_{0},x), \, f(y_{1},x) , \;... \;, f(y_{n},x) \} \end{equation} $\mathcal{B}$ = \begin{equation} \{ g(y,x_{0}), ...
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1answer
67 views

find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$

As an exercise for my analysis class, I have to find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$, where $f_0$ is differentiable ...
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1answer
343 views

Density of space in a Sobolev space

An exercise from Gilbarg-Trudinger Elliptic Partial Differential Equations states the following : "Using Lemma 9.12, show that for a $C^{1,1}$ domain $\Omega$ the subspace $$\{u \in ...
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1answer
95 views

How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $ \mathbb {R}_+$?

This is an excercise 2.2 from Hormander, vol. I: Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction $x \rightarrow e^{1/x}$ to $\mathbb{R}_+$? The answer, provided in the book, ...
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2answers
202 views

A polynomial can not have infinite number of constant values

This is my hypothesis and I am not sure if this is a valid one. I need a proof for this. My point is that infinite number of minimas are needed if this is valid, but I am lack of knowledge to prove ...
2
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1answer
631 views

Orders of Growth between Polynomial and Exponential

What is known in contemporary mathematics about orders of growth for functions that exceed any degree polynomial, but fall short of exponential? This is a subject for which I've found little ...
3
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2answers
50 views

Proving the existence of a point $a \in \mathbb{R}_+$ s.t. $\cos(a) < 0$

I am currently working on a challenge problem where I need to show that there is a point $x \in \mathbb{R_+}$ such that $\cos(x) = 0$ using only a few properties of the cosine function. In particular, ...
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2answers
230 views

Points of a Measure Zero Sets Covered by Intervals Infinitely Many Tmise

Given a measure zero set $E$, by definition we have forall $\varepsilon > 0$, a covering of $E$ by intervals whose lengths sum to $< \varepsilon$. I want to prove that we can cover $E$ in ...
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1answer
125 views

Need a good reference on Levy-Ito decomposition and martingale spaces

I am currently using Appelbaum but it does not go into too much detail how we deal with the part with small jumps. Can someone please recommend a good text book? I have Bertoin at my disposal, but I ...
3
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1answer
234 views

Schwartz space: semi norm estimate on translation

the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$: $$ \|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
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0answers
119 views

An integral inequality related to maximal function

Suppose that $f\in C^1([0,\infty),\mathbb{R})$,and $F(x)=\max_{x\leq y\leq 2x}|f(y)|$,then show that $$ \int_{0}^{\infty}F(x)dx\leq \int_{0}^{\infty}|f(x)|dx+\int_{0}^{\infty}x|f'(x)|dx $$ EDIT ...
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0answers
29 views

variation of a final state due to changes in period (where the period is a parameter)

I have a simple ordinary differential equation $\frac{dx}{dt}=f(x,t,p,T)$ $x(0) = x_0$, $x(T) = x_T$ where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks! NOTE: I ...
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1answer
249 views

Why Euler's formula is true? [duplicate]

Possible Duplicate: How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$? I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) ...
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0answers
89 views

A variation of fundamental lemma of variation of calculus .

I have a question on a variation of the fundamental lemma . If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $ then is it possible that there exist some ...
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4answers
586 views

Evaluate $\lim_{n\to\infty}n\int_0^1x^nf(x)dx$

Let $f:[0,1]\mapsto\mathbb{R}$ be a continuous function. Evaluate $\lim_{n\to\infty}n\int_0^1x^nf(x)dx$
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1answer
93 views

Show that $U \subset V \Leftrightarrow V^\bot \subset U^\bot$ for $U,V$ subspaces in a Hilbertspace

Let $(\mathcal H, \langle\cdot,\cdot\rangle)$ be a Hilbertspace, $U,V \subset \mathcal H$ are closed subspaces. I want to show $$U \subset V \Leftrightarrow V^\bot \subset U^\bot$$ $\Rightarrow$ is ...
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3answers
214 views

Showing that a set is not compact (not using the usual metric)

Consider $\mathbb{R}^2$ with the metric: $$d((x_1,y_1),(x_2,y_2))= \begin{cases} |y_1-y_2| \text{ if } x_1=x_2 \\ 1+|y_1-y_2| \text{ if } x_1 \neq x_2 \end{cases}$$ Show that $E= \{(x,0) : -1 \leq x ...
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3answers
76 views

For what values of $a$ will the following sequence converge?

$a \in \mathbb R$ has the decimal expansion $a = a_0.a_1a_2a_3 \ldots a_n \ldots$ Find all values for $a$ for which the sequence $\{a_n\}_{n=1}^{\infty}$ converges. I rule out irrationals first, ...
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1answer
100 views

If $B$ is a continuous bilinear function such that $B(h,k) = o(\lVert(h,k)\rVert^2)$, then $B=0$.

Suppose that $B: H \times K \Rightarrow F$ is a continuous bilinear function, where $H,K$ and $F$ are real normed spaces. I have to prove (not as homework) that if $B(h,k) = o(\lVert(h,k)\rVert^2)$, ...
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1answer
99 views

Prove the limit of a sequence

Prove that if $a_n$ is a nonnegative sequance and: $$\lim_{n\to \infty} a_n=a$$ then $$\lim_{n\to \infty} \sqrt[5]{a_n}=\sqrt[5]{a}$$ I tried to do this using the definition of the limit of a ...
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1answer
44 views

Prove the lower bound

Knowing That $ A \land B \subset \mathbb{R}$ and they both have a lower bound. Prove that (most likely using the definition of a bound): $$\inf (A \div B)=\min\{ \inf A,\ \inf B \}, $$ where $A \div ...
2
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1answer
92 views

Proof $a + \infty = \infty$

Given a convergent sequence $(a_n)$ with limit $a \in \mathbb{R}$ and a divergent sequence $(b_n)$ tending to infinity. I want to prove now using the boundedness of $(a_n)$: $\exists C \in ...
2
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0answers
143 views

$f$ mapping open sets to open sets

I know that the definition of a continuous mapping between two topologies is defined as: For $\mathcal{X}$ and $\mathcal{Y}$ and $f$ such that $f:\mathcal{X}\rightarrow \mathcal{Y}$ if $f^{-1}$ maps ...
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1answer
46 views

Does such a statement follow from compactness or connectedness of $[a,b]$

Assume that $I$ is a compact interval in $\mathbb R$. Does the following statement (I hope true) follows from compactness or from connectedness of $I$? For arbitrary family of open in $\mathbb R$ ...
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1answer
35 views

Problem with definition of $R$-analytic function

Let $f:[a,b]\rightarrow \mathbb R$. We say that $f$ is $\mathbb R$-analytic if for each $x_0 \in [a,b]$ there is $R(x_0)>0$ and power series $\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ convergent for ...
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1answer
75 views

Question on Gamma Function

In Gelfand and Shilov Vol I (of Generalized Function), on page 257, they write down the following equation that I don't know how to arrive at: $$\int_{0}^{1} (1-t)^{-\frac{n}{2}} t^{\frac{q-2}{2}}dt ...
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1answer
66 views

A question involving Poincaré inequality

I am struggling to prove the following. Let $\Omega$ be bounded set in $\bf{R}^n$ and $u$ a $C^2$ function on $\Omega$, such that $u=0$ in $\partial \Omega$. Prove that there is a constant $C$ ...
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1answer
46 views

Just would like a quick explanation regarding Lower Darboux Sums

On Example 1 in this following PDF: http://home.iitk.ac.in/~psraj/mth101/lecture_notes/lecture15-16.pdf Consider the function $f : [0, 1] \to \mathbb R$ defined by $f(1/2) = 1$ and $f(x) = 0$ ...
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0answers
125 views

Would this be bounded?

Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of $M$ is less than $1$. Let $I_{r}$ be an $m$ ...
3
votes
3answers
228 views

$L_p$ Spaces and limits of translated functions

If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$ where $g_{(t)}(x):=g(t+x)$. Any hints? Try to give me only ...
1
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1answer
64 views

simple calculus inquiry

Define $$f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0\\ 1 & \text{if } x = 0\\ \end{cases}$$ Show that $f$ is uniformly continuous (UC) on $\mathbb{R}$. My Approach: ...