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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1answer
19 views

Simple Question about Derivative property

Suppose $f:[-1,1] \to \mathbb{R}$ is twice differentiable and $f(-1) = f(1) = 0$ and $f(0) = 1$. Prove that there exists $x_0 \in (-1,1)$ with $f''(x_0) = -2$. I tried establishing this with ...
4
votes
1answer
69 views

How to integrate following indefinite integal?

The integral is $$ \int\frac{x-\sin x}{1-\cos x} \,dx $$ However, the only guess I have is that the denominator is the derivative of the numerator. Probably the integration by substitution will ...
1
vote
3answers
46 views

Finding partial derivative of $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$

I'm studying Mathematical Analysis II for a university course. There is a training exercise that asks me to: Find the partial derivatives at $(1,0)$ of $f(x,y)$, where: ...
1
vote
1answer
28 views

homeomorphism of a Banach space, constructed using a contraction

Let $X$ be a Banach space and $g: X \to X$ a contraction, meaning that for all $x, y \in X$, we have that $$||g(x) - g(y)|| ≤ L ||x - y||$$ for a constant $0 ≤ L < 1$. Now, consider the function ...
2
votes
4answers
32 views

Sum of two harmonic alternating series

I'm trying to solve the series $\sum_{n=1}^\infty (-1)^{n+1}\frac{2n+1}{n(n+1)}$ I've simplified it to the form $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n+1} + \sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ ...
1
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2answers
37 views

Evaluating a basic integral of the exponent.

Upon reading some mathematical literature, I have encountered the following computation: $x\in X$, a Banach space, $\alpha=\text{Re }(z)$ for $z\in\mathbb{C}$ and $\omega$ is the growth bound. ...
0
votes
1answer
25 views

Problem on Analysis (Functions)

Let f(x) be a function from reals to reals obeying the following: f(x) is continuous, f(0)=1, and f(m+n+1)=f(m)+f(n). Show that f(x)=1+x for all real numbers x. I am a bit confused on how to start ...
1
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0answers
18 views

Rayleigh quotient strictly increases

Consider the Rayleigh quotient $$\lambda_{L} := \max_{u \in H^{1}_{0}([0, L])}\frac{-\int_{0}^{L}u'^{2}\, dx}{\int_{0}^{L}u^{2}\, dx}.$$ Is $\lambda_{L}$ strictly increasing in $L$? Fix an $L_{1}, ...
2
votes
1answer
22 views

Reference for Gradient expression of a function on matricies

I'm looking for a reference (I suppose the statement is correct) for the following formula: $$ \langle\nabla f(\rho)^\dagger,V\rangle=\left.\frac d{dt} f(\rho+tV)\right|_{t=0} $$ for any direction ...
2
votes
2answers
40 views

$\{a_n\} \to a$ iff $\limsup_{n \to \infty} \{a_n\} = \liminf_{n \to \infty} \{a_n\}$

It is clear that if $$\limsup_{n \to \infty} \{a_n\} = \liminf_{\to \infty} \{a_n\},$$ then $\{a_n\} \to a$, since we can just squeeze the terms in the middle. I understand that to prove the ...
3
votes
1answer
63 views

Real Analysis, Cauchy but not null.

I came across this question in a book on p-adic numbers and thought it looked interesting. However, I am having trouble getting started with it. Any hints/suggestions is much welcomed Let $(a_n)$ be ...
1
vote
1answer
34 views

Algorithm and top-points.

Problem: For an array $A[1],\dots,A[n]$, with $n\geq 3$, it holds that $$A[i+1]>\frac{A[i]+A[i+2]}{2},\qquad i\in \{1,2,\dots, n-2\}$$ That is, it holds that $$A[2]>\frac{A[1]+A[3]}{2},\dots, ...
0
votes
1answer
19 views

Weak convergence of bounded nets

Let $(x_\alpha)_{\alpha\in A}$ be a bounded net in $c_0$. For all $\alpha\in A$, let $x_\alpha = (x_\alpha^n)\in c_0$; if, for every $n\in\mathbb{N}$, $(x_\alpha^n)_{\alpha\in A}$ is a net that ...
1
vote
1answer
27 views

Understanding this proof about the intersection of compact subsets

The following proof is theorem 2.36 from Rudin's Principles of Mathematical Analysis: Theorem: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection ...
1
vote
1answer
23 views

Justification for Interchange of integral and sum

Let $\mu$ be a probability measure and $t\in\mathbb{R}$. I would like to write this equality $$\int_{\mathbb{R}}e^{ixt}d\mu(x)=\sum_{n\geq0}\frac{(it)^{n}}{n!}\int_{\mathbb{R}}x^{n}d\mu(x).$$ This is ...
2
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1answer
12 views

Investigating uniform convergence of a sequence

I am trying to determine if the sequence $f_n$:= $\frac{x^{2n}}{1+x^{2n}}$ is uniformly convergent on $D_1:=[-q,q],0<q<1$, and $D_2:= (-\infty,r] \cup [r,\infty),r>1$. I have determined that ...
1
vote
1answer
31 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
1
vote
1answer
30 views

If the limit of power series exists, it converges.

Let $f(x) = \sum a_n x^n$ converges on $(-R, R)$. Does $\sum a_n R^n$ converge if $\lim _{x \to R-} f(x)$ exists?
2
votes
1answer
66 views

positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove$\displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent

When I do my homework ,I met this problem: If positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove:$ \displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent. I ...
3
votes
1answer
23 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
0
votes
1answer
19 views

If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
0
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0answers
53 views

rudin's definition of a compact set

Here are some definitions given in my book: Definition 2.31 By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha \}$ of open subsets of $X$ such that $E \subset ...
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votes
1answer
28 views

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$. Solution: Let $X_n=[x_0,x_1,\dots]$; define ...
0
votes
2answers
37 views

Let $X$ and $Y$ be finite sets. Then $X \cup Y$ is finite and $| X \cup Y| \leq |X| + |Y|$.

Let $X$ and $Y$ be finite sets. Let us assume that they are distinct at least, for otherwise $X \cup Y = X$ and $X$ is finite. Also let us assume that $X$ has cardinality $n$ and $Y$ has cardinality ...
1
vote
1answer
33 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
0
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0answers
27 views

Jacobian from $\mathbb{R}^n \to \mathbb{R}^m$

Consider a diffeomorphism $F: V \subset \mathbb{R}^N \to U \subset \mathbb{R}^n$, where $U$ and $V$ are open, $N>n$. Moreover, suppose that $f:=F|_M: M \subset V \to U' \subset U$ is also a ...
2
votes
0answers
20 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
0
votes
1answer
20 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
0
votes
0answers
12 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
0
votes
1answer
8 views

When we say f(x)->l as x->c then how c becomes a limit point of the domain of defination of f.

I think that if c be an isolated point of the domain of f then continuity of c does not imply existence of the limit of f at c.Is it the only cause?
1
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0answers
17 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
2
votes
3answers
53 views

Why is 1 + 1 = 0 when we make the addition table for F = {0, 1} (F = field)

In Analysis, I learned that any number system satisfying all the axioms (commutativity, associativity, identity elements, invertibility, distributivity) is called a field. Then the professor mentioned ...
5
votes
1answer
39 views

Showing that $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)$

Can anyone help me prove this: Let $X$ and $Y$ be nonempty sets and $f:X\times Y\to\Bbb R$ such that $f(X\times Y)$ is bounded. Prove the following statement: ...
1
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2answers
43 views

$y'=\frac{y^2}{2x(y-x)}$

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
1
vote
1answer
51 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
1
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2answers
42 views

Orthonormal basis of $L^2(T)$

Why is $\{e_n\mid n\in\mathbb{Z}\}$ an orthonormal basis of $L^2(T)$, where $T=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_T f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$? My try: If $n=m$, ...
1
vote
1answer
22 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
1
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0answers
63 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
2
votes
1answer
23 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
0
votes
0answers
25 views

In the geometrical interpretation for integration how lower and upper rectangular approximation are functions of natural number?

I've attempted to prove this in the following manner. Let Q be a subset of $P[a,b]$ which contains partitions of each order exactly once. Now, if we consider mappings $F:N \to Q$ defined by $F(n)=p$ ...
-1
votes
1answer
39 views

Reposting Question about Schroder-Bernstein

Assume there exists a $1$-$1$ function $f:X\to Y$ and another $1$-$1$ function $g:Y\to X$. Follow the steps to show that there exists a $1$-$1$, onto function $h:X\to Y$ and hence $X\sim Y$. a) The ...
0
votes
1answer
23 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
2
votes
1answer
60 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
-1
votes
0answers
36 views

On the continuity of Li's numbers. [on hold]

Consider Li's numbers defined by $\lambda_n = \sum_{\rho} \left(1-\left(1-\dfrac{1}{\rho}\right)^n\right)$ where $n$ is a nonnegative integer and the $\rho$ are the nontrivial zeros of the Riemann ...
0
votes
2answers
42 views

Decide what is the number of roots of the equation

Decide what is the number of roots of the equation $2^x=100x$. I know I can draw a sketch and then check but maybe there is a better method to do that? It's an exam question, thus it must require ...
1
vote
2answers
40 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
0
votes
1answer
13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
-1
votes
2answers
73 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
1
vote
0answers
16 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
1
vote
1answer
95 views
+300

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...