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).

learn more… | top users | synonyms (1)

4
votes
1answer
72 views

Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$

Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the ...
1
vote
3answers
712 views

How to determine whether an integral is convergent

I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent? For example: For which values of the parameters $p,q \in ...
2
votes
1answer
70 views

If $f_k \to 0$ a.e. and $\sum_n n 2^n \mu\{|f_k| \in (2^{n-1}, 2^n]\} \leq 1$ for all $k$, then $\int f_k \to 0$.

(Stanford Real Analysis Qualifying Exam: Spring 2012) (Ideal time: 18 minutes) (a) Let $\mu$ denote Lebesgue measure on $[0,1]$. Let $f_k\colon [0,1] \to \mathbb{R}$ be Lebesgue measurable ...
7
votes
4answers
435 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
1
vote
2answers
85 views

A question about metric spaces

Assume that we have a metric space $(S,d)$ and points $a,b,c \in S$ which statisfy the following conditions: for all $x \in S$, $d(a,x) \leq d(a,b)$, for all $y\in S$, $d(b,y) \leq d(b,c)$. Does ...
1
vote
1answer
61 views

Simpler way to check whether or not a sequence is uniformly convergent

Let $f_n(x)=(1+x^n)^\frac{1}{n}$ on $[0, \infty)$. I want to check if this is uniformly convergent. It's pointwise limit is $$f(x)=\begin{cases} x \text{ if } |x|\geq1\\ 1 \text{ if } |x|<1\\ ...
6
votes
1answer
189 views

Series counterexample

Give an example of series $$\sum_{n=1}^{\infty} a_n \quad \text{ and } \quad \sum_{n=1}^{\infty} b_n$$ such that both converge, and the series $$\sum_{n=1}^{\infty} c_n$$ defined by $$c_n = ...
4
votes
1answer
169 views

Analysis on Improper Integrals

This question is from Munkres' Analysis on Manifolds, section 15 question 1. Let $f: \mathbb{R} \to \mathbb{R}$ be the function $f(x) = x$. Show that, given $\lambda \in \mathbb{R}$, there exists a ...
0
votes
2answers
49 views

surface integral help

I tried to solve this test question. I dont know if I have to use arc length any help would be appreciate please A water fountain sprays water so that when it falls, its height above the water ...
1
vote
1answer
178 views

Extending a $C^2$-function from a $C^{1,1}$-curve to some neighbourhood

Suppose I have a simple, compact $C^{1,1}$-curve $L$ in $\mathbb{R}^3$ and a $C^2$-function $f$ on it ($C^2$ meaning with two continuous arclength derivatives). Can it be extended to a $C^2$-function ...
2
votes
1answer
37 views

$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous

Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define $$ f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$ Then is $f$ upper ...
0
votes
2answers
123 views

Is level set of sum of two continuous functions a closed set?

$f^i: R^{n}\to R^{n}$ is a continuous function for $i=1,2$. Let $$M=\{(x,y)\in R^{2n}~|~f^1(x)+f^2(y)=0\}$$ Is $M$ a closed set? If not, can you give a counter example.
0
votes
1answer
269 views

Weak Minimizer of a Functional

I showed that $u(x) = \frac{x^2}{2}$ is a potential minimizer for the functional $\int_0^2 \frac{n}{2}u'(x)^2-nu(x) \, dx$ in $C^2[0,2]$ with $u(0) = 0$ and $u(2)=2$ where $n$ is a positive constant ...
0
votes
1answer
59 views

Limit Involving Integral

I have been struggling with the following problem: $$\lim_{a\to\infty} \int_0^1 \frac {x^2e^x}{(2+ax)} dx .$$ My first reaction was to attempt to move the integral outside the limit, which would ...
1
vote
1answer
31 views

Interval of Uniform Convergence

If the sequence of functions $g_n$ converges to $g$ uniformly on the interval $[1/n, 1]$, where $n$ is a natural number, must it converge uniformly to $g$ on the interval $[0,1]$? I came across this ...
1
vote
1answer
54 views

Sequence of continued powers defined recursively

I have a series ${a_n}$ defined recursively by $a_1 = b$ and $a_{n+1} = b^{a_n}$, with $b \ge 1$ and $n \in \mathbb{Z} $. I am trying to show that ${a_n}$ is bounded above if $1 \le b < 3^{1/3}$, ...
2
votes
4answers
201 views

For $x > -1$ proof that $ \arctan x + \arctan\frac{1-x}{1+x} = \frac{\pi}{4} $

For $x > -1$ proof that $\arctan x + \arctan\dfrac{1-x}{1+x} = \dfrac{\pi}{4} $ I have no idea how to approach this, some kind of help would be greatly appreciated! edit: Thank you all!
1
vote
0answers
134 views

Rigorous hypothesis for Reynolds' transport theorem

I'm looking for rigorous hypothesis for the application of Reynolds' transport theorem : $$ \frac{\mathrm{d}}{\mathrm{d}t}\left[ \int_{\Omega(t)} \phi({\bf x},t) \mathrm{d}{\bf x} ...
4
votes
1answer
116 views

How much pure math should a physics/microelectronics person know [closed]

I do condensed matter physics modeling in my phd and I was struck up learning quite an amount of physics. But while having done lot of physics courses, I see that if I learn pure math I would ...
1
vote
1answer
94 views

Showing a function is discontinuous

I've used matlab to get some idea of how the following function behaves: $$g(\theta) = \frac{2}{\theta^3} - \frac{\pi\cos(\pi\theta)}{2\sin^3(\pi\theta)}.$$ It appears that it is discontinuous at ...
6
votes
1answer
481 views

Cauchy Integral Formula for Matrices

How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
7
votes
2answers
262 views

Prove the limit at infinity is 0

Suppose $f$ is continuous. For all $x>0$, the limit of $f(nx)$ when $n$ goes to infinity is $0$. Then please prove that the limit of $f(x)$ as $x$ goes to infinity is $0$. (I totally stuck at it) ...
1
vote
1answer
98 views

Correctness and help with Union and intersection proof of Open Sets

I need to prove the following: Let $A$ and $B$ be subsets of a metric space $(X, d)$ show that $A^o \cup B^o \subset (A \cup B)^o$. $(A \cap B)^o = A^o \cap B^o$ Here is my attempt: For 1) Let ...
0
votes
3answers
112 views

Uniform convergence of sequence of functions

Given that $$\gamma_n\rightarrow\gamma$$ uniformly, can we conclude that $$\int^b_a\|\gamma_n'\|\rightarrow\int^b_a\|\gamma'\|$$ uniformly? I know that we even do not have ...
2
votes
3answers
194 views

Some estimation a series by integral

Let $a \in (0,1)$. Does there exist a constant $C>0$ or function $C(a)>0$, which may be a function of $a$ but not $k$, such that $$ \sum_{n=1}^\infty a^n n^k \leq C(a) \int_0^\infty a^x x^k dx ...
2
votes
1answer
81 views

Is this function harmonic? [G-T] page 121

On page 121 of Gilbarg-Trudinger's book (Elliptic PDE of second order) they have the following Green's function in $\mathbb{R}^n (n\geq 3)$: \begin{equation} G(x, ...
2
votes
2answers
81 views

Radius of Convergence for$S=\sum^{\infty}_{0}\frac{2^n(x-2)^n}{(n+2)!}$

Given $$S=\sum^{\infty}_{0}\frac{2^n(x-2)^n}{(n+2)!}$$ After using root test, I got, $$-1\le\frac{2(x-2)}{(n+2)}\le 1$$ The n did not cancel. Now, How do I conclude about radius of convergence? ...
2
votes
1answer
155 views

Correctness of Converging sequence and Adherent Points

$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$ $B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$ ...
1
vote
1answer
331 views

Showing a sequence of functions converges uniformly on any bounded interval

Question: Let $\{f_n\}$ be a sequence of continuous functions on $\mathbb{R}$. Let $f_n \to f$ uniformly on $\mathbb{R}$. Let $g_n(x):=f_n(x+\frac{1}{n})$ for $n=1,2,3,....$ Then $g_n \to f$ ...
1
vote
1answer
57 views

how to prove this question about limit and derivative

Suppose $f:(a,b)\to\mathbb R$ that $ f $ satisfies: $$f\in C^1$$ $$\lim_{x\to a ^ +}f^2(x)=0$$ $$\lim_{x\to b ^ -}f^2(x)=e-1$$ if $\forall x \in(a,b) : 2f(x)f '(x)-f^2(x)\ge1 $, then how to ...
0
votes
1answer
125 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
17
votes
6answers
348 views

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-…$

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$ The number of signs increases by one in each "block". I have an idea. Group the series like ...
0
votes
1answer
193 views

How to come up with a function?

Say, I have a hypothetical value. And it increases at start and after some point it decreases (decays) at constant rate (average) and it tends to 0. It looks something like a Poisson distribution. ...
1
vote
0answers
52 views

Integration of $\int^\pi_0 \,dy \int^{\pi+y}_\pi \sin(x-y) \, dx$

What is the integration of $$\int^\pi_0 \,dy \int^{\pi+y}_\pi \sin(x-y) \, dx$$ Should use the change of variable technique?
3
votes
1answer
545 views

Set of Continuous Functions, Functionals, and Equicontinuity

Define the subset of $C^0[0,1]$ to be: $P = \{F(x) = \int_0^x f(t) \, dt : f \in C^0[0,1], \|f\|_\infty \le 1\}$ 1) Show that $P$ is not closed. 2) Show that $P$ is bounded and equicontinuous (using ...
0
votes
1answer
164 views

Abstract Algebra idempotent

An element $x$ in a ring $R$ is called an idempotent if $x^2 = x$. Prove that if $R$ has a unity, $e$, any idempotent $x$ (nonzero); $e$ is a zero divisor. So, since $e$ is a $0$ divisor, so every ...
3
votes
1answer
369 views

Examples of $f \in L^p$ iff $p_0 < p < p_1$, $p_0 \le p \le p_1$ or $p = p_0$

Hi how to show the following: Suppose $0 < p_0 < p_1 \leq \infty$. Find examples of functions $f$ on $(0,\infty)$ with Lebesgue measure such that $f \in L^p$ if and only if (a) $p_0 < p ...
2
votes
1answer
48 views

Lebesgue integrablity

For which $a$ and $b$ is the following function $f(x)$ on $(0,\infty)$ in $L^1$ $f(x) = x^{-a}|\log x|^b$ Thanks a lot!
10
votes
3answers
2k views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
0
votes
2answers
198 views

example of sub-harmonic function

A continuous function $u:\mathbb{R}^n\to\mathbb{R}$ is sub-harmonic if for every $x_0\in\mathbb{R}^n$ and $r>0$ $$u(x_0) \leq \frac{1}{|\partial B(x_0,r)|}\int_{\partial B(x_0,r)} \!\!u(x)\ ...
1
vote
1answer
351 views

Countable union of open balls in $\mathbb{R^n}$

Is it possible for any open set $U \in \mathbb{R^n}$ to be written as countable union of open balls? For example this is true when $n=1$. I would like to know if there is similar result in higher ...
2
votes
2answers
170 views

Does this integral go to zero?

I have $f_1,f_2$, $C^\infty$ functions with compact support and $f_3$ a smooth and bounded function; let $a\in\mathbb{R}^3$. I have to evaluate this limit ...
2
votes
1answer
230 views

Convergent sequence on a step function

I'm not quite sure how to go about proving this: Let $\phi:[a,b] \to \mathbb{R}$ be a step function and $s \in (a,b)$. Let $(x_n)$ be a sequence in $[a,b]$ with each $x_n > s$ and ...
1
vote
0answers
91 views

Inclusion maps on $L^p$

How to show for $1 \leq p < q < r$ the inclusion maps $$L^p \cap L^r \rightarrow L^q$$ $$L^q \rightarrow L^p + L^r$$ are continuous. where the norms are defined in the following: $L^p$ ...
1
vote
0answers
113 views

Question on measurability

I need help to prove that : If $(T,\mathcal{A})$ is measurable space, and $U$ a metric space , we say that $f:T\rightarrow U$ is (strongly) measurable if one of the following equivalent properties ...
2
votes
1answer
125 views

integer Random Walk with step size governed by a distribution.

This problem is for a final exam I am taking in a graduate probability class. Collaboration has been explicitly allowed, but with the remark that the professor felt he couldn't stop us even if he ...
1
vote
1answer
73 views

Limit of constant functions with countable discontinuities

Suppose we have a family of functions $\sum_{i=1}^n \alpha_i 1_{F_i}$ where $1_{F_i}$ is the characteristic function of $F_i \subset \mathbb{R}$, and $F_i$ is countable or $F_i^c$ is countable. This ...
3
votes
1answer
74 views

showing $W^{s,p}(\mathbb{R}^n) \subseteq W^{r,p}(\mathbb{R}^n)$ for $r < s$

I'm aware of a way of doing this using pseudo-differential operator theory. One can easily reduce to showing that $W^{t,p}(\mathbb{R}^n) \subseteq L^p$ for $t > 0$. This in turn follows because ...
1
vote
1answer
52 views

Show the monotonicity of the following expectation.

Now, there is an expectation as $$E\left[(1-a\cdot b^{X})^{m}\right]$$ where $m\in \mathbb{N}$ ,$a\in (0,1]$ and $b \in(0,1]$ are constants. $X\sim B(n-1,p)$ is a binomial random variable. I am not ...
1
vote
0answers
62 views

Bounded linear map bivariate

How to show the following: Let $X$, $Y$, $Z$ be Banach spaces. Let $T(x,y)=z$ be a bilinear map. Let $T(x,\cdot)$ and $T(\cdot,y)$ be linear bounded maps on each fixed $x$ and $y$ respectively. How ...