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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413 views

What are the range and the norm of this bounded linear operator?

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
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1answer
115 views

A problem about $C^1$-convergence! (Elliptic theory)

Let a function $u:\overline\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies $$\Delta u+f(u)=0 \ \ \ \mbox{in} \ \ \Omega,$$ and consider ...
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1answer
43 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
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0answers
199 views

Disintegration of Measures

I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background. Let $(\Omega, ...
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2answers
123 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
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3answers
98 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
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1answer
172 views

How to find the range and inverse of this linear operator?

Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
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158 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
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0answers
72 views

Is the graph of every real function a null set? [duplicate]

This question popped to my mind during an analysis lecture: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a (general) function. Is there an $N\subset \mathbb{R}^2$ with $\lambda^2(N)=0$, such that ...
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1answer
117 views

$\exists a,b\in \mathbb R^+ $such that $|f(x)|\le a|x|+b$

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+~~~~\text{such that}~~~~|f(x)|\le a|x|+b.$$ Thanks in advance!
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67 views

Uniform best approximation in Chebyshev/Haar systems and the necessity of compactness of the function domain.

A great deal of Chebyshev/Haar systems are given for intervals $]-\infty,\infty[$, $[0,\infty[$ and other noncompact subsets of $\mathbb{R}$. Nonetheless, the theory of uniform best approximations in ...
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1answer
57 views

The area of the set in which a polynomial is “small”

Prove that there exist a constant $C$ such that for every monic polynomial $P$, the area of the set $A=\{x : |P(x)|<1\}$ is at most $C$. Remarks: This puzzle holds for both the real and the ...
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1answer
31 views

Help with limit $\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt$..

Let $D\subseteq \mathbb R^2$ be an open set and $P:D\rightarrow \mathbb R$ continuous. For $y$ fixed how to evaluate, $$\displaystyle\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt?$$ I know the ...
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1answer
409 views

About an extension of Riesz' Lemma for normed spaces

The Riesz' Lemma is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for ...
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1answer
217 views

Chebyshev rational approximations to $\cos x$

How can we construct all the Chebyshev rational approximations of degree $3$ for $f(x) = \cos(x)$. So, I note that we first get the Taylor series of $\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} ...
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1answer
212 views

How to define limsup of a function

Let $f: [0, \infty) \rightarrow \mathbb R$. What is the definition of $$ \limsup_{x \rightarrow a} f(x)$$ for $a \in [0, \infty)$?
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1answer
285 views

Unions of disjoint open sets.

Let $X$ be a compact metric space (hence separable) and $\mu$ a Borel probability measure. Given an open set $A$ and $r,\epsilon>0$ $\ $does there exist a finite set of disjoint open balls ...
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0answers
302 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
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2answers
343 views

characteristic curves for second-order equations

Reading about characteristic curves for second-order equations, in particular semi-linear equations of second order with two independent variables: ...
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92 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
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2answers
208 views

If the sum converges to zero, does that mean that each sequence converges to zero?

If we are given that the sequence sum of two sequences of positive real numbers converges to zero, does that mean that each sequence converges to zero? (by the squeeze theorem)
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1answer
46 views

Characterization homeomorphism between real intervals.

Suppose $I$ and $J$ are intervals in $\mathbb{R}$ and $f: I \rightarrow J$ is a continuous bijection. Can we state that $f$ is a homeomorphism?
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3answers
140 views

Linear instability implies nonlinear instability

I am trying to understand the following proof that linear instability implies nonlinear instability. Suppose we have the ODE, $ \frac{du}{dt}=A(u)$ for which $0$ is a solution. Suppose $L $ is the ...
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1answer
42 views

Question about moving planes method.

In the paper Inequalities for second-order elliptic equations with applications to unbounded domains I, by Berestycki, Caffarelli and Nirenberg (page $486$), they considered a set ...
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1answer
406 views

Is $\sum \sin^2(k)/k$ Convergent? [duplicate]

A student recently used the series $\displaystyle\sum_{k=1}^\infty\frac{\sin^2k}{k}$ as an example of a divergent series whose terms tend to $0$. However, I'm having trouble convincing myself that ...
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2answers
226 views

A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$

For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
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1answer
247 views

Automorphisms on Punctured Disc

I have to find the automorphism group of the punctured unit disc $D = \{|z| <1\}\setminus \{0\}$. I understand that if $f$ is an automorphism on $D$, then it will have either a (i) removable ...
0
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1answer
55 views

Failure of convergence to 0

Consider the interval $I = [0,1]$ and the sequence of functions: $$f_n(x) = (-1)^k \ \text{for} \displaystyle \frac{k}{2^n} \le x < \frac{k+1}{2^n} \ \text{where} \ 0 \le k < 2^n - 1$$ I want ...
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1answer
100 views

Total Ordering and relation

Consider the relation $<$ on $\mathbb{Q}$ defined by: $(m, n) < (j, k) \iff jn-mk \in \mathbb{N}.$ Where $m, j \in \mathbb{N}$ and $n, k\in\mathbb{Z}$ I want to show that $<$ is a total ...
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1answer
134 views

Weak convexity and continuity

For any open interval $(a, b)\subset {\mathbb R}\,$, define a weakly convex function $f:(a, b) \rightarrow {\mathbb R}$ as one for which $$f(q\;x_0 + (1 - q)\;x_1) \leq q\;f(x_0) + (1-q)\;f(x_1)$$ ...
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0answers
62 views

what are the borders of the convergence disks of series?

Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series: $$\sum_{k=1}^\infty{a_kz^k}$$ is convergent on ...
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2answers
833 views

The Least Upper Bound and The Greatest Lower Bound

I am taking math class, and I am not sure about LUB and GLB. I need someone to give this dummy a short explanation about them.... On interval (0,10), 0 is a lower bound and 10 is upper bound, but ...
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1answer
79 views

Conditions of parameter $\lambda$ ensuring integral is 0

Let $1 \le p \le \infty$. I am seeking to find the values of the parameter $\lambda$ such that: $$\displaystyle \lim_{\epsilon \to 0^+} \frac{1}{\epsilon^\lambda} \int_{0}^{\epsilon} f = 0 \ \ \forall ...
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3answers
332 views

Why does metric space which has the countable chain condition implies separable?

I've looked around but all I could find is that if X is separable then X has ccc. Can anybody give me some help?
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2answers
3k views

Monotonically increasing vs Non-decreasing [duplicate]

Is monotonically increasing is same as non-decreasing? Thank you for answer beforehand.
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2answers
51 views

Limit sequence sets

In my measure theory book I came across the following definition: Let $(A_n)_{n\ge1}$ be a sequence of subsets of some set $X$. Define: $\limsup_{n\to\infty} A_n:=\bigcap_{n\ge1}\bigcup_{k\ge n}A_k$ ...
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2answers
115 views

How do I prove $f:[0,1]^3\rightarrow\mathbb R$ has a minimum and maximum?

How do I prove continuous function $f:[0,1]^3\rightarrow\mathbb R$ has a minimum and maximum? Using compact, sequentially compact, continuous, or closed theorems.
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2answers
164 views

Supremum Axiom of $S = \{a+b\sqrt{2}: a,b \in Q\}$

Let: $$S = \{a+b\sqrt{2}: a,b \in Q\}.$$ It can be shown that $S$ ordered field. Dose the Supremum Axiom hold in $S$? Why? (with proof please)
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3answers
136 views

Question regarding definition of limit point and uniqueness

I found the following definition in a paper A sequence in a space $X$ is an ordered family $(x_j)_{j\in \mathbb{N}}$ where $x_j \in X$ but not necessarily $x_i \ne x_j$ for $i \ne j$, that is, ...
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1answer
46 views

Non-negative functions bounded from below.

Is there a result (or source) that says if I have a function $f$ defined on some set $S$ and that $f(y)>0~\forall y\in S$, then there exists a constant $K>0$ such that $f(y)\geq K~\forall y\in ...
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3answers
198 views

How many points of discontinuity?

I need to prove that any monotonic function whose domain is an interval $[a;b]$ can have only finite or countable number of discontinuity points... I don't seem to have any insightful ideas. It even ...
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1answer
134 views

d'alembert's formula

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} ...
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2answers
216 views

Taking supremum of inequality

If I have something like $$\int_0^T F(t,f) \leq \int_0^T G(t,f)$$ where f is a function, can I take supremums: $$\sup_{|f| = 1}\int_0^T F(t,f) \leq \sup_{|f| = 1}\int_0^T G(t,f)?$$ The integrands are ...
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1answer
32 views

Finding a minimal non-empty closed $G$-invariant set of a compact metric space

Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the ...
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1answer
78 views

Bounding $\liminf_{n} n |f^n(x)-x|$

I solved an exercise in which the first part asks to prove that for any measure preserving measurable transformation $f:[0,1]\rightarrow [0,1]$ we have $$\liminf_{n} n |f^n(x)-x| \leq 1, \ ...
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1answer
269 views

A Product of Connected Spaces Minus A Product of Proper Subsets of Both

Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, what can we say about the set $$(X \times Y) - (A \times B)$$ being connected or not?
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35 views

Is this function $(\partial / \partial x)f$ discontinuous at $(0,0)$?

Define $f$ on $\mathbb{R}^{2}$ by $f(x,y) = x^{3}\sin(1/x) + y^{2}$ if $x \neq 0$ and $f(0,y) = y^{2}$. I've got $D_{1} f(x,y) = 3x^{2}\sin(1/x) - x\cos(1/x)$ if $x \neq 0$ and $D_{1} f(0,y) =0$. ...
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1answer
298 views

Taylor expansion of an integral

I am interested in the Taylor series expansion around $t=0$ of the following expression: $$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$ Normally, I would proceed by ...
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1answer
142 views

I don't understand why the contrapositive part of the proof of continuity holds

So, assuming whoever can answer this knows the first part of this proof, that is, showing that for all$ \epsilon\gt0$ and for all $ p\in M$ there is a $\delta\gt0$. Also there is an $ x\in M$ such ...
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2answers
92 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...