All Questions
5,674 questions
4
votes
2
answers
5k
views
Ratio of Sequences Sum Inequality
I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
- 1,182
2
votes
1
answer
255
views
Quotients of perfect powers separated by an integer
Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
- 3,595
6
votes
2
answers
2k
views
non-maximal prime ideal in the ring of continuous functions
Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\...
- 7,609
4
votes
2
answers
371
views
Heights of several interesting posets
Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).
Define several sets of total functions, in each ...
- 6,819
4
votes
1
answer
627
views
Does such a smooth function exist?
I am looking for a $C^\infty $ function $g:\mathbb{R}^3\to \mathbb{R}^3$ such that $g(x)=0$ for $|x|\le 1$ and $g(x)=x$ for $|x|\ge 2$. Certainly such $g$ can be constructed, but I also want it to ...
- 450
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
2
votes
1
answer
465
views
Showing the derivative of this function is equal to $0$ a.e [closed]
Define $f:[0,1]\to [0,1]$ by $f(0)=0$, and $$f(x)=\sum\limits_{r_n\le x} 2^{ -n }$$ with $0\lt x\le 1$ where $[r_n]_{n\in \mathbb{Z^+} } = \mathbb{ Q} \cap (0,1) $.
How to show that the derivative $...
- 133
17
votes
4
answers
2k
views
interlacing roots/eigenvalues results and modern analogues
Is there any relation between these theorems on interlacing roots?
The roots of $f(x), f'(x)$ interlace (if all the roots of $f(x)$ are real and have real coefficients).
The eigenvalues of an $n \...
- 22.8k
21
votes
3
answers
3k
views
Prime ideals in the ring of germs of continuous functions
We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.
Can anyone think of a single ...
- 13.5k
4
votes
2
answers
2k
views
mean value theorem for operators
This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \...
- 85
1
vote
2
answers
3k
views
Continuation of a smooth function
Setting
Suppose I have two bounded open domains $\Omega' \subset \Omega \subset \mathbb{R}^n$ (I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are $C^\...
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
2
votes
0
answers
494
views
Characterization of weak Lebesgue spaces [closed]
I would be interested to know whether the following is true:
Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a ...
- 373
11
votes
4
answers
5k
views
The metric space associated to a measure space
Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \...
- 3,830
8
votes
2
answers
753
views
Patching together homeomorphisms: how badly can it fail?
Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...
- 3,910
19
votes
2
answers
2k
views
Constants for Rolle's Theorem applied to polynomials
Rolle's Theorem states that $f(1/2)=f(-1/2)+f'(x)$ has a root in the open real
interval $(-1/2,1/2)$ if $f$ is continuous and differentiable. How large can the absolute value of such a root
$\xi$
be ...
- 17.9k
0
votes
1
answer
224
views
Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
- 1,271
8
votes
1
answer
2k
views
Does integrating with respect to a finitely additive measure respect addition?
Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we ...
- 3,830
1
vote
1
answer
6k
views
How to determine whether a multivariate function is bounded or not
Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function $f(x,y)=-x^2+...
- 141
10
votes
1
answer
607
views
Properties of a matrix-valued generalization of the $\Gamma$ function
I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$....
- 1,243
2
votes
2
answers
711
views
Power function inequality
Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...
- 373
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
- 2,270
9
votes
1
answer
10k
views
Can the supremum of continuous functions be discontinuous on a set of positive measure? [closed]
Given a sequence of continuous functions $f_n(x)$, all defined on a compact set $D$ and assuming $f_n(x)$ is uniformly bounded. Let $f(x) = sup_n f_n(x)$.
It is clear that $f(x)$ is not necessarily ...
- 93
3
votes
0
answers
211
views
Elementary analysis: reference request
Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...
- 2,895
2
votes
0
answers
224
views
Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental [closed]
I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts.
At page $6$, the ...
- 4,795
4
votes
2
answers
1k
views
$L^1$ norm of the Fourier transform of a truncated Gaussian
I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes:
Consider the Gaussian $G(x):=e^{-x^2}$ ...
- 852
21
votes
0
answers
1k
views
Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$
A while ago, I came across the following problem, which I was not able to resolve one way or the other.
Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
- 17.1k
5
votes
2
answers
560
views
implicit function theorem for algebraic sets
We know by the standard Implicit Function Theorem that
If $f:\mathbb R^4\rightarrow\mathbb
> R^2$ is a polynomial (or in fact any
continuously differentiable function),
then there is a ...
- 1,359
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199
6
votes
2
answers
929
views
reverse mathematics strength of "Lipschitz functions are somewhere differentiable"
What is the reverse mathematics strength of
"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
(...
user5810
0
votes
0
answers
165
views
minimizing the integral of a function over square sets.
Hi!
I'm interested in some problems, but to be honest i'm not sure of the field they belong to.
Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
- 1,352
2
votes
1
answer
942
views
A singular value inequality
Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
$$\left|s_{1}\...
- 173
67
votes
9
answers
7k
views
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
- 12.1k
3
votes
3
answers
522
views
Closure of singular points
Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular
form.
$$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y +
\frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + \frac{f_{40}}{...
- 3,245
9
votes
2
answers
2k
views
Does the Weierstrass function have a point of increase?
Problem
The Weierstrass function $W(x)$ is given by
$W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$
where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.
A function $f:\mathbb{R}\...
- 491
1
vote
1
answer
420
views
density of a set
let $S=\{\sin (n)|n \in N\}$. We can prove $S$ is dense in $[-1,1]$. So is the set $\{\sin( n^2)|n \in N\}$; but the set $\{\sin (n^3)| n \in N\}$ is not dense in $[-1,1]$. How to prove this?
- 11
-3
votes
1
answer
332
views
Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
- 1
6
votes
2
answers
2k
views
How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 71
11
votes
6
answers
18k
views
One-line proof of the Euler's reflection formula
A popular method of proving the formula is to use the infinite product representation of the gamma function. See ProofWiki for example.
However, I'm interested in down-to-earth proof; e.g. using the ...
- 111
4
votes
2
answers
323
views
Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?
Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that
$$\lim_{n} ~a_{nn} = L_1, \qquad
\lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$
exist. Is there a sufficient criteria to conclude ...
- 3,245
10
votes
1
answer
1k
views
Real analytic function, injective, non surjective and preserving the rationals ?
I'd like to prove the non-existence of a real analytic function, injective, non-surjective
that sends rationals to rationals.
Is it a classical result ? If not, any hints on how to prove it ?
Thanks ...
- 103
2
votes
3
answers
913
views
A definite integral
Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...
- 1,649
6
votes
2
answers
2k
views
Continuity of a convolution (Version 2)
Hello,
This problem bothers me for some time. Suppose that
$\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support);
$\psi$ is ...
2
votes
2
answers
2k
views
Does a bounded real function have an analytic continuation [closed]
Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where
$f$ is real-analytic on the open interval $(0,1)$
$f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-...
- 23
2
votes
1
answer
276
views
Conformal Extension from a closed set to open
Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
- 541
1
vote
2
answers
1k
views
An interesting doubly infinite series
Let $0<\mu<1$ and $\alpha:=1-\mu^2$. Consider the function
$$f(x):=x\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}x}-\frac{1}{x}\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}/x},$$
...
- 201
3
votes
0
answers
237
views
Monotonicity of a certain parametric integral
I would like to ask for some help (hints, ideas) in solving the following problem:
Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that
if we define for any $x\in\mathbb{R}...
- 696
1
vote
1
answer
528
views
Function space between uniform continuity and Hölder continuity
Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder ...
- 5,327
1
vote
0
answers
346
views
Gauge integral of the derivative of a function except on a set of measure 0.
For the entire question, the interval I am integrating over is $[0,1]$.
Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some ...
- 783
5
votes
1
answer
543
views
Acceleration via smoothing
Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature?
Let $f(t)=(\sum_{1 \leq n \leq ...
- 19.7k