All Questions
5,794 questions
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
2
votes
1
answer
2k
views
How to do (m)Gram-Schmidt orthogonalization with integers ? (real life problem) ("mathematicalized reformulation")
New edition of the question, "mathematicalized" (thanks to Gerhard).
Consider and integer valued n*n matrix M, with integers elements in the range -N < m < N.
I want to find integer-valued ...
- 24.9k
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
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
1
vote
2
answers
382
views
A question about zeros of Tate type integral
Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $...
- 2,709
8
votes
2
answers
471
views
Multiplying functions on the unit square as generalized matrices
Consider the $\mathbb{R}$-vector space of sufficiently nice real-valued functions on the unit square $I^2$, where "sufficiently nice" could be taken to mean any one of a number of things - say ...
- 3,910
4
votes
1
answer
561
views
Taylor Series Remainder
Suppose I have a $C^\infty$ smooth function $f$ defined on the reals.
I can apply Taylor's formula and get the local expression
$$
f(x) = \sum_{i=0}^l\frac{f^{(i)}(0)}{l!}x^i+ f^{(l+1)}(\xi(x))x^{l+...
- 979
1
vote
1
answer
3k
views
Is point to set distance continuous?
Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
- 27
4
votes
1
answer
1k
views
Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
- 3,630
13
votes
2
answers
1k
views
Seeking proof for linear algebra constraint problem.
Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
- 379
4
votes
1
answer
306
views
ordered fields with the bounded value property, without choice
In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ ...
- 19.7k
15
votes
1
answer
2k
views
Interpolating between piecewise linear functions, with a family of smooth functions
Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that:
there is a number $k\in\mathbb N-\{0\}$ ...
- 4,312
7
votes
1
answer
505
views
Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently
What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$?
Do we have to calculate $A^{-1}b$, or is this not necessary?
edit: I forgot to mention that A is symmetric and ...
- 493
5
votes
1
answer
878
views
Numerically finding a Mercer expansion for a given covariance kernel
Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise.
On ...
- 8,512
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 32.5k
4
votes
2
answers
2k
views
a different nested intervals theorem
Is there any literature on (and a standard name for) the proposition that for any arbitrary-cardinality collection of closed intervals in the reals that is nested (in the sense that, given any two of ...
- 19.7k
4
votes
1
answer
222
views
a closed-form for mean/integral, but weighting positive differences between values and "mean" differently from negative differences?
Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\...
- 41
4
votes
0
answers
382
views
Pseudoinverse of column submatrix, from pseudoinverse of entire matrix.
Hello,
I am working on a numerical method for the least-squares solution of a linear system. I know that I can approximate the solution to $Ax=b$ with $x=A^+b$, where $A^+$ is the Moore-Penrose ...
- 41