Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
19
votes
0answers
347 views
Ambiguous Curve: can you follow the bicycle?
Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
12
votes
0answers
806 views
When is the image of a null set also null?
It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ ...
9
votes
0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
9
votes
0answers
455 views
Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.
I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator
$$K(g) = \int_0^1 e^{xt} g(t) dt.$$
The sources I've checked* seem to say that the process is fairly ...
7
votes
0answers
313 views
Cauchy-Formula for Repeated Lebesgue-Integration
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given.
Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
7
votes
0answers
448 views
Errata for Dieudonné's Treatise on Analysis volume 2 second edition
I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
7
votes
0answers
183 views
Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?
When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
7
votes
0answers
156 views
Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
I'm wondering about a simple question that has multiple possible variants depending on a few parameters. The prototypical one would be:
Does there exist an infinite series such that any two ...
7
votes
0answers
247 views
How can we prove a simple case of the High Indices Theorem?
Let $(a_n)$ be a sequence of real numbers such that
$$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$
converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
6
votes
0answers
178 views
A curious theorem by Peano
Let $f$ be defined on $[a,b]$ and there differentiable.
Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
6
votes
0answers
138 views
Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
6
votes
0answers
264 views
On the weak and strong convergence of an iterative sequence
I have some difficulties in the following problem.
I would like to thank for all kind help and construction.
Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
0answers
104 views
Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$
I am trying to solve a qualifying exam problem.
I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
6
votes
0answers
162 views
Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$
Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$
For all $ n\in \mathbb{N} $ let
$$ F_n = \{ f: \exists ...
6
votes
0answers
470 views
Nasty Integral - Closed form solution?
Any suggestions on how to integrate this beast?:
$$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$
where:
$f = 2\pi+2\tan^{-1}(y,x)$
$y = ...
5
votes
0answers
122 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
votes
0answers
33 views
$M$ is compact, non-empty, perfect, and $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination?
Assume that $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination of ...
5
votes
0answers
64 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
5
votes
0answers
74 views
Can we do some scaling argument in the presence of inhomogeneous norms?
Notation:
$B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$.
$\hat{f}$ stands for the Fourier transform of $f$.
Question. The following inequality holds true for all $f\in ...
5
votes
0answers
130 views
Help me correct these properties of : $f_{n}(x)= nx(1-x)^{n}$? Is there maybe a typo in the sequence?
Examine the sequence of functions $(f_n)_{n\in \mathbb{N}}$ on $x\in[0,1]$:
$$f_{n}(x)= nx(1-x)^{n}$$
Does $(f_n)_{n\in \mathbb{N}}$ converge pointwise or uniform?
I will show that it does ...
5
votes
0answers
92 views
Linear algebra estimates
Here is something that has been troubleing me lately. I don't know if it is true of not. I suspect it is.
Suppose that $A,B$ are two $n \times n$ matrices with complex entries. $A^t = A$, $\bar B^t = ...
5
votes
0answers
57 views
derivative and cut off functions
Is there any way of constuct a cut off function which is $1$ in $B(0,\epsilon)$ and zero outside the ball $B(0,2\epsilon)$ and it's first and second derivative is smaller than $1/|x|^a$ , with ...
5
votes
0answers
82 views
Existence of a sequence which is good for mean convergence but not good for pointwise convergence
The ergodic theorems talk about the limit behaviour of the ergodic averages. For example, if $(X,\chi,\mu,T)$ is a measure preserving system, where $\mu$ is a probability measure on $X$, then we have ...
5
votes
0answers
133 views
Are these sets in $\mathbb{R}$ open and/or closed?
In $\mathbb{R}$, are these sets open? Are they closed?
$A = \{\frac{1}{n} : n \in \mathbb{N}\}$
$B = A \cup \{0\} $
$[0, 1)$
My thoughts:
$A$ is not open as if we have an open ball with $r > ...
5
votes
0answers
279 views
Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$
Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$
My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
5
votes
0answers
125 views
Convergence check (No steps/solutions/proofs please)
I just wish to check that I have got these right. Please just indicate whether the ans are right -- please don't show steps (I wish to figure those out myself).
Given $\phi_n (x)=n^k(1-x)x^n$ where ...
5
votes
0answers
153 views
Intuitive test of convergence
Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
5
votes
0answers
152 views
How does one determine $n$-spheres of curvature?
I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented ...
4
votes
0answers
88 views
Open Problem in Fixed Point Theory [Prize]
This open problem appeared on the bulletins of Evans Hall at Berkeley this week.
I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post ...
4
votes
0answers
34 views
Differential calculus on Banach space
I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly.
Problem
Given the ...
4
votes
0answers
47 views
When are two commuting linear operators functions of each other
I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up.
If we formally consider the integral operator $E ...
4
votes
0answers
114 views
Self-study Real analysis Tao or Rudin?
The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here come another one.
Recently I've began to learn real analysis ...
4
votes
0answers
98 views
The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.
The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric,
$$
d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \}
...
4
votes
0answers
62 views
Open map in Banach algebra
I'm having trouble showing a certian function is open and can be extended.
Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
4
votes
0answers
117 views
Disintegration of Measures
I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background.
Let $(\Omega, ...
4
votes
0answers
98 views
Prime numbers, analysis of polylogarithms
Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
4
votes
0answers
107 views
Integral of a gaussian function of trigonometric functions
I need help with the analytical solution of this integral:
...
4
votes
0answers
68 views
Mean value inequality and Fixed-point theorem
I need help to understand this question, it's not that clear for me:
Using the norm $|x|+|y|$ and the Mean value inequality, give a condition over the partial derivatives of $f(x,y)$ and $g(x,y)$ for ...
4
votes
0answers
77 views
Function that is discontinuous only for integer fractions
I have this question:
Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the
set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
4
votes
0answers
50 views
Quotient-lifting properties
I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies
Let $N\triangleleft G$. Then ...
4
votes
0answers
60 views
Spectrum of laplacian in a parallelogram
Is the spectrum of the laplacian on an arbitrary parallelogram with dirichlet boundary conditions known?
4
votes
0answers
51 views
Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
4
votes
0answers
47 views
Diffeomorphism on the Sphere
Suppose $f:S^2\rightarrow S^2$ is a diffeomorphism. Is true that $f$ must have a fixed point or a point $x\in S^2$ such that $f(f(x))=x$?
4
votes
0answers
70 views
Why Characterize Structure Preserving Functions in Terms of Pre-images?
In Analysis, functions are often characterized as "structure-preserving" if structures from codomains are preserved into the domain under the preimage operation. Specifically, if we let $f: (A, S_1) ...
4
votes
0answers
67 views
Prove that (M,+,*) is a field
Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*).
Group axioms:
1) ...
4
votes
0answers
73 views
How many points does one need for an epsilon-net
Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
4
votes
0answers
140 views
“Green Globs” question
When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
4
votes
0answers
168 views
Simple formulation, nontrivial problem
There's a problem from calculus I remember:
$$\forall x\ \exists n.\ f^{(n)}(x) = 0 \iff \exists n\ \forall x.\ f^{(n)}(x) = 0\,.$$
Function $f \in C^\infty(\mathbb{R})$, and the notation $f^{(n)}$ ...
4
votes
0answers
149 views
Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$
This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
...
4
votes
0answers
146 views
Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
Let $ 0 < r < 1$, fix $x > 1$ and consider the integral
$$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$
In the investigation of ...
