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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5
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1answer
336 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
162 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
320 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
95 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
156 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
105 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
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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
83 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
161 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
250 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
87 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
216 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
152 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
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0answers
80 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
82 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
105 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
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3answers
372 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?
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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 ...
2
votes
1answer
905 views

Modulus of continuity of a continuous function

Let $f : I\subset\mathbb{R} → \mathbb{C}$ be a continuous function on the closed interval $I$. A modulus of continuity of $f$ is any real-extended valued function $\omega: [0, ∞] → [0, ∞]$, vanishing ...
14
votes
3answers
341 views

How to prove :$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2…u_n}\right)^2=2011$

For sequence $u_n$ satisfing : $$\begin{cases} u_1=\sqrt{2015}\\ u_{n+1}=u_n^2-2\end{cases}$$ How to prove : $$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$$
1
vote
1answer
288 views

Continuity of the Lebesgue function

If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ ...
1
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0answers
148 views

total variation measure vs. total variation of its distribution function

Let's say that $f:[0, \infty)\rightarrow \mathbb{R}$ for simplicity, although the question is also intended for a variety of other types of domains. (Probably interval is all that is required.) ...
0
votes
1answer
40 views

I want to classify all these values by using an equivalence relation

We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $u<a$ and $f(u)=0$ to find a $c\in(u,a)$ such that: the expression ...
5
votes
3answers
792 views

Where is $\log(z+z^{-1} -2)$ analytic?

I need some help in determining where $\log(z+z^{-1} -2)$ is analytic, where $z$ is a complex number and $\log(z)=\ln|z|+\arg(z+2k\pi),k\in\mathbb{Z}$. Thank you in advanced.
1
vote
2answers
100 views

How to solve $f_{n}(x)=xf_{n-1}(x)-f_{n-2}(x)$?

How to solve the following recurrence relation $$f_{n}(x)=xf_{n-1}(x)-f_{n-2}(x)$$ with the initial conditions $f_{1}(x)=x, f_{2}(x)=x^2-1$? The answer is that ...
0
votes
1answer
77 views

Find the area/surface of a figure, specified by inequalities

I have some difficulties in solving this problem: Find the area of a figure, specified by the inequalities: $x^2 + y^2 \leq 2x$ and $x^2 + 2x + y^2 \leq 3$ I know that I have to use the formula ...
1
vote
2answers
76 views

When is this number irrational?

Lets say we have irrational numbers $\alpha _1, ..., \alpha _n$ in the interval $(0,1)$. Represent each $\alpha _i$ as a binary expansion $0.a_i^1 a_i^2 ...$ where each $a_i^j \in \{0,1\}$. Define ...
5
votes
1answer
606 views

Is this reading path recommended?

Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some ...
2
votes
2answers
161 views

Inequality involving partial derivatives

Suppose $f(x, y)$ is a twice continuously differentiable function with a unique minimum at $(0, 0)$. Why at $(0, 0)$ must we have $$\frac{\partial^{2}f}{\partial x^{2}}\frac{\partial^{2}f}{\partial ...
1
vote
1answer
45 views

choose relevent length of interval

i have question related to Riemann sum. Is then length of interval matters? Suppose we want to calculate Riemann net sum of function $f(x)=3-x/2$ in this interval $[2,14]$. I first take $n=2$, ...
2
votes
1answer
130 views

Convex set and weak topology

i have this question and i don't know how to answer it "Let $E$ be a Banach space. Let $A \subset E$ be a convex subset. Prove that the closure of $A$ in the strong topology and that in the weak ...
0
votes
1answer
39 views

surjectivity of a linear transformation and spanning

Okay, we have that $\{|a_i\rangle\}_{i=1}^n$ is a set of vectors spanning a vector space $V$. Also, $T\in L(V,W)$ is surjective, where $L(V,W)$ is the set of linear transformations (functionals) from ...
2
votes
1answer
35 views

The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$ \frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
7
votes
2answers
1k views

Must the (continuous) image of a null set be null?

Say $E \subset [0,1]$ is a null set. Let $f: [0,1] \rightarrow [0,1] $. Do you think $f(E)$ is a null set or not? Just being curious. (DEF): A set $A$ is null if given any $\epsilon > 0$, there ...
8
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3answers
697 views

Is the Riemann Integral good for anything?

Math people: I think it is a good idea to teach beginning calculus students the Riemann Integral (I refer to what calculus books call the "Riemann Integral" and ignore any controversy about whether ...
1
vote
1answer
233 views

Absolutely continuous functions and general absolute continuity

First, the definitions: $f$ is AC on $E$ if $$\forall \epsilon >0\ \exists \delta >0\ \forall \{[a_k,b_k]\}_{k=1}^N \mbox{ such that }a_k,b_k \in E,\ \Sigma(b_k - a_k) <\delta : \Sigma| ...
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2answers
143 views

If $f(x,y)=x^2+y$, what is the image of $K=\{(x,y):x^2+y^2\leq 1\}$?

Please disregard the first eight lines of the solution below (which I have provided for completeness; the referenced theorems simply state that continuous functions between metric spaces preserve ...
1
vote
2answers
218 views

least upper bound greatest lower bound theorem

I am trying to understand the following theorem: I can't understand how the author gets to the conclusion that $\alpha = \sup L$ is $\in L$ I'm ok until the "Our hypothesis about $S$ implies ...
3
votes
0answers
111 views

Upper semicontinuity of a probability measure

Let $m$ be an atomless probability measure on $\mathbb{R}^m$. Consider $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that for all $v \in \mathbb{R}^m$, $x \mapsto f(x,v)$ is ...
0
votes
1answer
207 views

Showing the complement of an open set is closed using sequences

Let $U \subseteq X \subseteq \mathbb{R^n}$. A set $U$ is open in $X$ if $X \setminus U$ is closed. Definitions: (1) A set $C\subseteq X \subseteq \mathbb{R}^n$ is closed in $X$ if sequences ...
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0answers
122 views

Continuous bijection between an annulus + a point and the open unit disk

The open annulus with a point I define as $\{ (r,\theta)\colon 1<r<2,0\leq \theta < 2\pi \}\bigcup \{\left(1,0\right)\}$. Call that $A$. Let $B=\{(r,\theta)\colon 0\leq r<1, 0\leq \theta ...
0
votes
1answer
59 views

Question about posets and maxima/minima

A thought just occurred to me, thinking about posets and maxima/minima... This is a "little" question just to make sure I am really grasping the definitions here: if $E$ is partially ordered by a ...
0
votes
1answer
61 views

Coercive problems

this is the complete problem and i have a problem that is : i dont understand step 2: step 1:"shows that $m>-\infty$ i dont understand how to prove it ? can someone help me please ? thank ...
3
votes
2answers
308 views

A pathological example of a differentiable function whose derivative is not integrable

First I'll make a definition: $$\operatorname{Loc-int}(g):=\left\lbrace x\in[0,1] : \exists \epsilon>0\text{ s.t. }\int_{(x-\epsilon,x+\epsilon)\cap[0,1]}|g|dm<\infty\right\rbrace,$$ where $m$ ...