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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Proving least upper bound property implies greatest lower bound property

In Rudin 1.11 Theorem Proof he claims the following Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of ...
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0answers
10 views

Convergence of distances in metric space

If $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges?
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0answers
209 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasi-linear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
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4answers
51 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
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1answer
24 views

The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
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1answer
31 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
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5answers
92 views

Complex Analysis book including integration

FOR BEGINNERS: Currently, I am looking for a textbook on complex analysis, which covers complex analysis from the beginning, and majorly focuses on contour integration, and the residue theorem. On ...
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1answer
29 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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23 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
2
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2answers
30 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
2
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1answer
39 views

Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
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3answers
80 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
2
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3answers
151 views

a limit property at infinity

Let $k\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(x)=L$ then $\lim_{x\to\infty}f(kx)=L.$
2
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1answer
29 views

Predicting the increase/decrease of number

I have these entries in my database that looks like this: ...
2
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3answers
105 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
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1answer
27 views

A function differentiable only at $0$ and for $|z|=1$

I need to find a polynomial function that is differentiable at the origin where $f'(0)=1$ and at every point $|z|=1$ but at no other point in the complex plane. I just have no clue how to solve ...
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3answers
91 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
2
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1answer
25 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
2
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5answers
86 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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0answers
43 views

Properties of the Fourier transform

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$ \widehat{g}(kn)= \widehat{h}(k), \\ \widehat{g}(l)=0, l \not\equiv 0 \ \text{mod} \ n.$$ I ...
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0answers
37 views

Complex analysis (Analytic function, sharp upper bound)

I encouter complex analysis problems the I think it is quite to do. Could anyone please give a hint or guideline. Thank you very much in advanced. Let $D$ be an open unit disc $\{z \in \mathbb{C}| ...
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54 views

How to prove that an injection from a sphere into a Euclidean space is homotopic to a constant?

How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
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123 views

When does it make sense to say “the smallest measurable set containing $x\,$”?

We know for the Borel $\sigma$-algebra that each singleton set is measurable. I was working on the problem of proving that each infinite $\sigma$-algebra has uncountably many members. My solution went ...
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3answers
62 views

Showing instablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
2
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1answer
47 views

$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki

Paraphrased from Munkres' Topology: Lemma 9.2. Given a collection $\mathcal{A}$ of nonempty sets, there exists a choice function \begin{equation*} f: \mathcal{A} \to \bigcup\limits_{A \in ...
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1answer
77 views

Does $\int_{0}^{1}x^nf(x)\, dx=0$ imply that $f=0$ a.e. without assuming $f \in C[0,1]$?

Suppose that $f \in L^{1}[0,1]$ and $\int_{0}^{1}x^nf(x)\, dx=0$ for $n=0,1,2,\dots$ Does that imply that $f=0$ a.e.? I think that there will be a counterexample but it is hard to find out.
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4answers
237 views

To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.

Let $f$ be a compactly supported nonnegative $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2\leq C f(x) $ by showing that for every point ...
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3answers
719 views

Integration substitution: How does Wolfram Alpha come up with this step?

I have to integrate $$ \int \frac{1}{(\sin x) (\cos x)} \, dx $$ I looked at the Wolfram Alpha step by step solution to figure out how to do it. First, it rewrites the integral as: $$ \int (\csc ...
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1answer
25 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
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0answers
41 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
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106 views
+50

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
3
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4answers
71 views

How to prove that $\ln x\leq x-1 \forall x>0$?

I need to prove that $\ln x\leq x-1 \forall x>0$, using the Mean value theorem. For $x=1$, the equation is true. So, for starters I'll check for $x>1$. By applying the aforementioned ...
0
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1answer
29 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
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1answer
48 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
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2answers
57 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
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1answer
41 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
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0answers
29 views

Variational Inequalities and how they are used?

I am doing undergrad research in this field next semester and I have never heard of this topic before. I tried wikipedia and reddit for help but nothing seems to help. I just want to know what I'm up ...
3
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1answer
46 views

periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
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1answer
32 views

If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?

If $f \in L_2(a,b)$, then I want to show that the antiderivative $$ F(x) := \int_a^x f(y) d y $$ is in $L_2$ (I guess this is true). If $L_2(a,b)$ would be closed under pointwise product, i.e. if ...
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1answer
24 views

Limit of arc-length of a curve

Let $L(f)$ denote the length of a curve $f$, if $f = \lim\limits_{n\to\infty} f_n$ then do we necessarily have that $L(f) = \lim\limits_{n\to\infty} L(f_n)$? I assume that we will have some continuity ...
4
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3answers
266 views

Image of open set is not open?

I'm confused by the proof that $\epsilon$-$\delta$ continuity is equivalent to open-set continuity. One can prove that a function is $\epsilon$-$\delta$-continuous if and only if the preimage of any ...
0
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1answer
41 views

Hint on metric space [on hold]

I want show that $(a_n):=d(x_n,y_n)$ converges, if $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are cauchy sequences in $(X,d)$. Here is what i do; From the hypothesis, $(a_n)$ is bounded, because ...
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1answer
55 views

Question about milnor's proof of hairy ball theorem

Here is a link about the proof: http://people.ucsc.edu/~lewis/Math208/hairyball.pdf My question is: after lemma 2, Milnor takes the region A to be the region between two concentric spheres. Why can't ...
3
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1answer
42 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
4
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1answer
42 views

Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
2
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1answer
34 views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write ...
2
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2answers
55 views

There are $u$ in $W^{1,p}(D)$ and a subsequence $\left\{ u_{m_{k}}\right\} $ such that $\left\{ u_{m_{k}}\right\} $ weakly converges to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
1
vote
2answers
21 views

Proving Lipshitz continuous over a convex set with Projection Operator

Suppose a problem $$\min_{x \in \mathbb{R}^{n}} f(x)$$ subject to $x \in \Omega$ which is a closed and convex set. If $\nabla f(x)$ is Lipschitz continuous in $\Omega$, then prove that $$e(x) = x - ...
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0answers
25 views

statistical analysis [on hold]

Two independent random samples of annual starting salaries for individuals with masters and bachelors degrees in business were taken and the results are shown below ...
4
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1answer
30 views

Composition of relations. Both relations are functional and mutually inverse mappings. Zorich - MAI p22

$\def\R{\mathcal{R}}$ The composition $\mathcal{R}_2 \circ \mathcal{R}_1$ of the relations $\mathcal{R}_1$ and $\mathcal{R}_2$ is defined as follows: $$\mathcal{R}_2 \circ \mathcal{R}_1 := ...