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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4answers
64 views

Existence of a global minimum

Let $S = \{(x, y, z) \in \mathbb{R}^{3}: x > 0, y >0, z > 0\}$ and consider $f(x, y, z) = xyz + \frac{1}{xyz}$. Why must $f$ attain a global minimum at some $p \in S$?
4
votes
1answer
112 views

The Banach space $c_0$ is $C^{\infty}$-smooth.

In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces: A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
3
votes
0answers
68 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
4
votes
1answer
114 views

Question concerning mean value theorem

Here's the problem: Suppose the set $\{x\mid f(x)\not = 0,x\in[a,b]\}$ is not empty, and $f$ is differentiable on $[a,b]$, with $f(a)=f(b)=0$. Prove that $\exists c$, such that ...
0
votes
1answer
142 views

A question about sigma algebras and generators on the extended real line

We define the extended real line $\bar R = \mathbb{R} \cup \{-\infty\} \cup \{\infty\} $ The Borel sets $\beta(\bar R) = \sigma([-\infty, x]), x\in \mathbb{R}$. We want to show that $\beta(\bar R)$ ...
1
vote
3answers
94 views

A question on the upper bound of the limit supremum

My question is: Let $(x_n)$ be a bounded sequence of real numbers. Prove that for every $\epsilon > 0$ and every $N\in\mathbb{N}$ there are $n_1, n_2\geq N$ such that ...
1
vote
1answer
83 views

If $f:\mathbb{R}^2\to\mathbb{R}$ is continuous and $B\subset\mathbb{R}$ is open, is the intersection of all $f^{-1}(B,y)$, $y\in[a,b]$, open?

I've been reading a paper in which the authors seem to (implicitly) state the following: Suppose that $f:\mathbb{R}^2\to\mathbb{R}$ is continuous and $A\subseteq\mathbb{R}^2$, $B\subseteq\mathbb{R}$ ...
0
votes
1answer
79 views

Question about analysis

We know $x \sim y$ iff $y - x \in \theta \mathbb{Z}$ (mod 1) and $\theta$ irrational defines an equivalence relation on $[0,1]$ with equivalence classes $[x] = \{\{x + n \theta \}\} = Orb (x)$. My ...
0
votes
2answers
129 views

Interspersing of integers by rationals

I'm wondering if the next argument is sound or maybe need some adjustments; Proposition (Interspersing of integers by rationals): Let $x\in \mathbb{Q}$. Then there exists an integer such that $n\le ...
1
vote
1answer
49 views

Explore a function for extremums

I have an exam tomorrow and have to know how to explore for extremums a function of 3 variables. For example: Explore fo extremums the function: $$f(x,y,z)=x+ \frac{y^2}{4x} + \frac{z^2}{y} + ...
76
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
0
votes
0answers
62 views

Is it possible for all Cauchy-sum sequences to converge, but not to the same number?

First, let me define the Cauchy integral (which is just like the Riemann integral, except that we always pick the left endpoint of each subinterval when evaluating the function): (Partitions and ...
13
votes
4answers
391 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
0
votes
1answer
94 views

Help to understand a proof of Tao's book (analysis)

Hi I have troubles to understand a proof that is in the the notes and in the book of Terry Tao of Analysis. I Proposition in question is: The problems that I have it's to understand some tricky ...
5
votes
2answers
329 views

Convergence of $\sum_n a_nb_n$ for all $b_n\searrow 0$ implies convergence of $\sum_n a_n$

I need a hint for a practice problem: Let $a_n \geq 0$. Show that if $\displaystyle\sum_{n=1}^\infty a_nb_n$ converges for every monotonically decreasing sequence $b_n \to 0$, then ...
12
votes
2answers
326 views

Shift Operator has no “square root”?

Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb ...
0
votes
2answers
956 views

Riemann integral of $ f(x)=x^2$

Can someone suggest a method of finding the Riemann integral of $f(x)=x^2$ on [0,1]. I know that it is sufficient to show $U(P,f)≥1/3≥L(P,f)$ but could not do this. PLease help
0
votes
1answer
149 views

distance between a point and a set and open/closed sets

Let $(X, d)$ be a metric space and let $E$ be a subset of $X$ and $x$ be an element of $X$. Define the distance between $x$ and $E$ as $d(x, E) = \inf d(x, y)$, for all $y$ in $E$. Show that for a ...
2
votes
1answer
81 views

Please show error in my proof!

Theorem:(from Lecture Notes) Given a function $f: X \longrightarrow Y$ and sets $A_i \subset X$ and $B_i \subset Y$, $i\in \mathbb{N}$, we have: $$f(\cap_{i=1}^\infty A_i) \subseteq \cap_{i=1}^\infty ...
0
votes
1answer
41 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
2
votes
1answer
99 views

How can I construct envelop of unity?

Given a topological space $B$ and an open covering $\{ U_i \}_{i\in I}$ of $B$ with a partition of unity $\{ \varphi_i \}_{i \in I}$ such that $\operatorname{supp}(\varphi_i) \subset U_i$ $\forall i$, ...
2
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0answers
79 views

Solution of nonlinear waves( breathers)

The sine-Gordon equation is known as $$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0,$$ Can you please derive the equation which is known as breather equation ...
3
votes
1answer
147 views

calculate the volume formed by rotation of a given region

I have a question in elementary differential calculus: Let $S$ be a region given by $$S=\{(x,y): 0\leq x\leq 1,\ \ 3^x-x-1\leq y\leq x\}$$ then define $V$ the solid obtained by rotating $S$ around ...
1
vote
1answer
33 views

Differential Maping in Elementary Analysis

The problem I stuck was : Let $ f = R^{n} \rightarrow R^{m} $ and suppose there is a constant $M$ such that for $ x \in R^{n} $, $ || f(x) || \leq M || x ||^{2} $. Prove that $ f $ is differentable ...
2
votes
1answer
74 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
0
votes
2answers
901 views

Prove the convergence of the geometric series using $\epsilon$, N definition

Show $| \sum_{n=0}^{\infty}x^n - \frac{1}{1-x} | < \epsilon$ using the definition of convergence when |x|< 1.
5
votes
1answer
353 views

Convergence of sequences of inverse functions

Let $(X, \phi)$ and $(Y, \sigma)$ be metric spaces, and let $f, f_1, f_2, \ldots$ bijective function with inverse functions $g, g_1, g_2, \ldots$ $f_n \to f$ pointwise for $n \to \infty$. And all ...
3
votes
2answers
111 views

difference between $\mathbb{R}^2$ and $\mathbb{R} \times \mathbb{R}$

I was going through some of notes in regards to Fourier analysis and I noticed that in some cases when dealing with a 2 dimensional transform the function $f \in \mathbb{R}^2$ while other times $f \in ...
1
vote
1answer
173 views

How can I prove $f_M(x) = \min(f(x),M)$ is lower semi-continuous?

My question is: Suppose that $f:X\rightarrow \mathbb{R}$ is lower semicontinuous and M is a real number. Define $f_M:X\rightarrow\mathbb{R}$ by $$ f_M(x) = \min(f(x),M). $$ Prove that $f_M$ is lower ...
4
votes
2answers
335 views

fundamental lemma for variational calculus

Is it possible to use the fundamental lemma of calculus of variations in some way in the following case: $F(x,y)$ is a locally integrable function on $\mathbb{R}^n \times \mathbb{R}^n$. We know that ...
1
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0answers
98 views

Real and Rational Numbers

Intuitively, we often think of real numbers as existing in one-to-one correspondence with the points on a continuously drawn line, the real number line. One way of expressing the completeness of the ...
1
vote
1answer
72 views

limit of a recursive sequence:2

Let $$x_k = \frac{A}{1-C} x_{k-1} + \frac{B}{1-C}x_{k-2},$$ where $A, B, C$ are positive reals such that $A + B + C =1$. Let $$x_1 = 1$$ and $$x_2 = 1 + y,$$ with $y$ is positive. Which conditions ...
4
votes
1answer
162 views

The set of all compact non-empty subsets is perfect

Let $X$ be a perfect Polish space and let $H[X]$ be the set of all non-empty compact subsets of $X$. For $A,B \in H[X]$ define the so called Hausdorff-Distance $$ d_H(A,B) = \max \{ \sup_{x \in X} ...
3
votes
1answer
107 views

If a set could be represented as “arbitrary fine” finite union of open balls, then it is not closed

If $V$ is a subset of a metric space, such that for every $\varepsilon > 0$ there exists a finite number of open balls $B_{\varepsilon}(x_i)$ such that $$ V = \bigcup_{i = 1}^n ...
1
vote
1answer
45 views

$m^*(A) = m^*(A + t)$

Define $m^*(A) = \inf Z_A$ as the outer measure of $A \subseteq \mathbb{R}$ where $$Z_A = \left\{\sum_{n=1}^{\infty}|I_n| : I_n \text{ are intervals}, A \subseteq \bigcup_{n=1}^{\infty}I_n\right\} ...
1
vote
0answers
22 views

Angle between two centered noisy vectors

Let $\mathcal{H} = \lbrace u\in \mathbb{R}^n \mid \langle x, (1, 1, ...., 1) \rangle = 0 \rbrace $, the hyperplain where the avarage is zero i.e. $\frac{1}{n}\sum\limits_{i=1}^n x_i = 0$. Given two ...
1
vote
1answer
84 views

Null sets in $\mathbb{R}$

We know $A \subseteq \mathbb{R}$ is null if given $\epsilon > 0$, there exists intervals $\{I_n\}_{n \geq 1}$ such that $$ A \subseteq \bigcup_{n=1}^{\infty} I_n \text{ and } ...
0
votes
2answers
172 views

Proving $d$ is a metric of a power set

Let $E$ be a finite set. For every pair of subsets $A,\ B$ of $E$, define $d(A, B)$ as the number of elements $AΔB$, where $AΔB$ denotes the symmetric differencia of $A$ and $B$ defined by $(A\setminus ...
1
vote
2answers
285 views

Is there any “formula” that allows us make change of variables in surface integrals?

For example, here (wikipedia) there are some "formulas" (or better stated "theorems") that allows us make change of variables in some integrals. I need an analogous for surface integrals. Could ...
1
vote
1answer
89 views

What are the structure constants for the algebra of quaternions? Show this algebra is associative.

What are the structure constants for the algebra of quaternions? Show this algebra is associative. How can I find the structure constants? I know that for an algebra $\mathscr{A}$ and basis ...
0
votes
1answer
80 views

What is the sine of arcsine of $x$? Problem with using trigonometric substitution in integral.

I'm having problems with this $\int \sqrt{1-x^2}\,dx$. Now the text book (Spivak's Calculus) says we can replace $x$ by $\sin u$ ($u = \arcsin x$). Now my question is how can we replace $u$ by ...
0
votes
1answer
64 views

Question on weak topology

i need help to solve this exercise please "Let $E$ be a Banach space and let $K \subset E$ be a subset of $E$ that is compact in the strong topology. Let $(x_n)$ be a sequence in $K$ such that $x_n ...
0
votes
1answer
224 views

Asymptotic notation meaning in transitive relation

I'm attempting to prove the transitive relation on $\theta$ and I'm having trouble understanding the meaning of one of the symbols used. Here is the transitive relation: $f(n) = \theta(g(n)) ...
2
votes
2answers
156 views

Solving $v_{t}+v(x,t)v_{x}=0$ with initial condition

This problem comes from an undergraduate course in PDE. The first question of the problem was to solve the following PDE: $v_{t}+v(x,t)v_{x}=0$ with the following initial condition: $v(x,0)=5x$ ...
0
votes
0answers
81 views

All possible subsequences converging to same function $f$

Let $S = \{f_{n}\}$ be a sequence of continuous functions on a compact set $K \subset \mathbb{R}$. Furthermore suppose $S$ is a compact subset of $C(K)$ (the set of continuous functions on $K$ with ...
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0answers
84 views

Convergence of solutions in initial value problem

I am working on the following problem: Suppose $u_{n} : [-M, M] \rightarrow \mathbb{R}$ are differentiable and are such that $u_{n}'(x) = F(u_{n}(x), x)$ for $F$ continuous and bounded. Furthermore, ...
1
vote
2answers
38 views

Is the following information sufficient to guarantee a global maximum at the corner of some interval?

Suppose one has a continuous and twice differentiable function $h(x)$, defined on the interval $(\underline{x}(a), x^*]$, where $a \geq 0$ is a parameter. By definition, it holds that $h'(x^*) = 0$, ...
2
votes
2answers
106 views

Gauss hypergeometric function at z=-1

is there anything like a special value case of the hypergeometric function if $z=-1$ such that one can evaluate $_2F_1(\alpha,\beta; \gamma; -1)$? I mean there is a nice representation for the case ...
2
votes
3answers
378 views

union of infinitely many bounded sets is not bounded

Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?
1
vote
0answers
39 views

This limit of the hypergeometric function makes me stunning…

I am currently reading this paper Physics paper please have a look at the definition of (20) and then (36). In (36) they investigate the limit of the hypergeometric function ...