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
36 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
votes
3answers
738 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
241 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| ...
1
vote
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
224 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
114 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
218 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 ...
1
vote
0answers
129 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
63 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
326 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 $m$ ...
0
votes
1answer
86 views

some confusions about extreme set

In Rudin's book "Functional Analysis",he defined an extreme set (called $S$) of $K$(in a topological vector space $X$) if no point of $S$ is an internal point of any lime interval whose end points are ...
2
votes
3answers
86 views

finding $\lim_{n \rightarrow +\infty}\frac{n}{2^n}= ?$

Finding the limit below..: $$\lim_{n \rightarrow +\infty}\frac{n}{2^n}= ?$$ I really think its 0. But intuitively, infinity over infinity. how can that be? indeterminate forms? Thanks
0
votes
1answer
96 views

Recursive formula for the integral

Derive a recursive formula for the integral $$ \int_0^\frac{\pi}{2} x^{2}\cos x $$ I tried to do it by parts but when I do it second time there is $ x^{n-2}$ I need some help... Is it a good way to ...
2
votes
3answers
154 views

Use of Newton method to find the value of $x$

A segment of a circle is the region enclosed by an arc and its chord (See figure below). If $r$ is the radius of the circle and $x$ the angle subtended at the center of the circle, find the value of ...
1
vote
1answer
155 views

Properties of $>$ for rational numbers

This question comes from an introductory undergraduate course in Analysis. We have just started from defining the set of rational numbers and then we will construct the set of real numbers. We define ...
2
votes
1answer
155 views

How to show that every bounded variation function on $[a,b]$ is differentiable a.e?

I'm struggling with this question for over a week now. I know the proposition is true, but haven't managed to prove it yet. any suggestions anyone? ($f$ is BV on $I$ if ...
0
votes
2answers
91 views

Smooth step between $-1$ and $1$

I am currently interpolating the step between two functions from $-1$ and $1$ smoothly therefore I used $\tanh$. Since I am quite confident with the result, but interested in further ways to do this, ...
0
votes
1answer
89 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
6
votes
2answers
140 views

Proving this function $f:[0,1]\to \mathbb{R}$ is continuous on $[0,1]$

I am looking for a hint or feedback on what I've already done, not a full solution. $f=t\sin{\left(\frac{1}{t}\right)}$ for $t\ne 0$, $f(0)=0$, My idea is that I only have to worry about the steep ...
2
votes
4answers
428 views

A sequence of functions $\{f_n(x)\}_{n=1}^{\infty} \subseteq C[0,1]$ that is pointwise bounded but not uniformly bounded.

We were talking about pointwise bounded vs. uniformly bounded in my analysis class, and this question came up. The problem is that we are working on a compact set, it would be much easier if the ...
2
votes
2answers
215 views

How to prove that $\|x\|_q\le \|x\|_p$ if $p \le q$

I was reading this: Proving an inequality with $\|x\|_p$ metrics? Since it is a really old post, and I'm not sure that I'll get an answer, I hope asking this way, as an independent post, doesn't ...
3
votes
3answers
123 views

Interesting question in analysis

I am trying to prove this : Consider $\Omega \subset R^n$ ( $n \geq 2$) a bounded and open set and $u $ a smooth function defined in $\overline{\Omega}$. Suppose that $u(y) = 0$ for $y \in \partial ...
0
votes
1answer
55 views

How to prove that $\|x\|_s\le n^{(r-s)/(rs)}\|x\|_r$ if $s \le r$

I have to prove that $\mathbf{(1)}$ $\|x\|_s\le n^{(r-s)/(rs)}\|x\|_r$ and $\mathbf{(2)}$ $\|x\|_s\le n^{1/s}\|x\|_{\infty}$ if $1\le s \le r\le \infty$. Unless I'm considering this wrong, the only ...
2
votes
1answer
69 views

Question on convex hull

Let $(x_n)_n$ be a countable subset of $C$ that is dense in $C$; For every $n$ let $C_n=conv\lbrace x_1, x_2, . . . , x_n \rbrace $ ($C\subset E$nonempty convex set, $E$ a finite-dimensional normed ...
2
votes
2answers
292 views

Almost sure convergence of a sum of random variables

Suppose $(X_i)_{i=1}^{\infty}$ is an i.i.d. sequence of rv's, where $X_i$ can take countably many values $\{x_1,x_2,\dots\}$ with probabilities $\{p_1,p_2\dots\}$, respectively. Let $p_{n,k}:= ...
5
votes
0answers
115 views

Do there exist solutions for this equation?

We know that solutions exist for equations of the following variety: $$ye^y=x \iff y=W(x)$$ Where W is the Lambert W function. We can augment the problem slightly, and ask if there exist solutions ...
1
vote
2answers
1k views

A set containing one element is an open set. Why?

I asked a question last night about proving that a discrete metric space is both open and closed. Once or twice it was mentioned that a set that contains only one element is open. I'd like to know: ...
2
votes
1answer
45 views

Simplify vector equation

I know that $div E=0$ and I know what $ curl E$ is. Further, I know what the vector laplacian of $E$ is. Now I want to simplify $\nabla \times (\nabla \times f(x,y,z) E(x,y,z))$, where ...
2
votes
3answers
220 views

How do I compute $\int_{-x^2}^{0}f(t)dt$ derivative ratio $x$?

If $$ f(t)=\begin{cases}\frac{\sin t}{t} & t\neq 0\\\\1 & t=0,\end{cases} $$ then, how do I compute the following derivative ratio $x$? The desired function is: ...
5
votes
1answer
120 views

Expected overlap

Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large ...
0
votes
2answers
70 views

Why commutatively convergent series iff summable?

maybe this is a stupid question, however I could not figure out its solution. I´am assuming that a summable series (convergent to $x$) is a series in a normed space $E$ indexed by $I$ such that, for ...
2
votes
1answer
122 views

Gaussian function

I want to scale the Gaussian function $\exp(-x^2)$ to the unit disc. In particular, I wish to represent $\int_0^\infty \exp(-x^2) dx$ as $\int_0^1 g(x) dx$, where $g$ should be the rescaled Gaussian ...
2
votes
0answers
37 views

Riemann integrable implies Stieltjes integrable?

Suppose $f:[a, b]\rightarrow \mathbb{R}$ is bounded and Riemann integrable. If $\alpha$ is a BV integrator that is, say, also, continuous, then is $f$ Riemann integrable ($\alpha$)? What if $f$ is ...
5
votes
2answers
318 views

If $f(x)$ is an integrable function, can we always find its anti-derivative using ordinary methods of integration?

Suppose that we have an integrable function $f(x)$ which is expressed in terms of elementary functions. By integrable, I mean that we can find its anti-derivative in terms of elementary functions, ...
1
vote
2answers
454 views

Proof that every subset of a discrete space is open and closed (visualizing 'Accumulation Points' definition)

I know this question is a duplicate, but I want to understand it in terms of accumulation points and internal points, etc. QUESTION Let X be any set and $d : X \times X \to \mathbf{R}$ be given by ...
1
vote
1answer
1k views

Proof that a discrete metric is indeed a metric space

QUESTION Let X be any set and $d : X \times X \to \mathbf{R}$ be given by $$ d(x,y) = \begin{cases} 0, & \text{if $x = y$} \\ 1, & \text{if $x \neq y$} \\ \end{cases}$$ Show that $d$ is a ...
1
vote
2answers
93 views

Partial fractions for inverse laplace transform

I have the following function for which I need to find the inverse laplace transform: $$\frac1{s(s^2+1)^2}$$ Am I correct in saying the partial fraction is: ...
1
vote
2answers
92 views

Proving a derivative exists

If f is differentiable at $x$ then for $\alpha\neq1$ $$f'(x)=\lim_{c\to 0} {{f(x+c)-f(x+\alpha c)}\over{c-\alpha c}}.$$ I am not really sure what it is I need to even show to prove the statement.
4
votes
2answers
450 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
2
votes
0answers
46 views

question on uniform convergence

Assume $p_0=0$ and for all $n\in\{0,1,2,....\}$ we define $P_{n+1}(x)=p_n(x)+\dfrac{x^m-p^m_n(x)}{2}$ then how should I find all natural numbers $m$ such that $\{p_n(x)\}$ be uniformly convergent ? ...
6
votes
1answer
183 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
2
votes
0answers
67 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
0
votes
1answer
79 views

Proving distance between a set and its supremum is zero

If $S$ is a bounded above set ant $t=SupS$ how do I prove that $dist(S,t)=0$ ?
3
votes
2answers
82 views

Help with a simple problem involving a functional inequality (trying to prove Gronwall's inequality)

So while trying to prove Grownwall's inequality, my proof led me to the following statement: $h'(x) \le h(x)g(x)$. Now when $h'(x)=h(x)g(x)$ the following holds: $h(x)=k \exp G(x)$, where $k$ is a ...
0
votes
1answer
42 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
1
vote
1answer
218 views

interchange summation and iterated integration

A math article makes use of the following equality, without proof. $$ \int_{0}^{1}\int_{0}^{1} \sum_{n\ge 0} (xy)^n dx\, dy = \sum_{n\ge 0} \int_{0}^{1}\int_{0}^{1}x^n y^n dx\, dy$$ But $s_k(x,y) = ...
2
votes
2answers
190 views

Vector of reduction

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a locally Lipschitz continuous and $\bar{x}\in\mathbb{R}^2$. Put $$ S(x):=\left\{x^*\in\mathbb{R}^2: \liminf_{u\rightarrow x}\frac{f(u)-f(x)-\langle ...
2
votes
1answer
250 views

Compactness of Hilbert-Schmidt Operator

I'm trying to show that a certain Hilbert-Schmidt operator is compact following some exercises in Rudin's Functional Analysis (exercise 15 on page 112): If $X, \mu$ is a finite measure space and $K ...