All Questions
5,657 questions
8
votes
3
answers
545
views
Approximation of pseudogeometric progression
Let $f_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( ...
- 728
8
votes
1
answer
783
views
Real-rooted polynomials
I proposed this question at MO which was resolved neatly by Gerald Edgar in the form
$$
u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k).$$
Now that we ...
- 43.2k
8
votes
1
answer
458
views
An interesting integration
For any positive integer $n$, let
$$A_n=\idotsint\limits_{\substack{x_1+\cdots+x_n+y_1+\cdots+y_n\leq1\\x_1,\cdots,x_n,y_1,\cdots,y_n\geq0}}\prod_{i,j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n.$$
It ...
- 1,997
8
votes
1
answer
1k
views
When can a sum be re-signed to converge to any limit?
Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold?
For any $S \in \mathbb R$ with $-\sum a_n \...
- 6,215
8
votes
1
answer
1k
views
Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?
Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
- 157
8
votes
2
answers
891
views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
- 205
8
votes
2
answers
2k
views
A generalized mean-value theorem
I'm pretty sure that if the function $f$ is continuous on $[x_1,x_3]$ and twice-differentiable on $(x_1,x_3)$, with $x_1 < x_2 < x_3$, then there must exist $x$ in $(x_1,x_3)$ for which $f''(x)$ ...
- 19.7k
8
votes
3
answers
2k
views
Does a weaker condition than vanishing derivative imply a function being constant?
I learned this question from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+...
- 183
8
votes
2
answers
3k
views
Finite measure on the power set
Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...
- 1,199
8
votes
2
answers
478
views
Whitney extension theorem preserving monotonicity
This question is related to Monotone version of one-dimensional Whitney extension theorem.
Let $m$ be a positive integer or $m=\infty$.
Suppose that $E\subset\mathbb{R}$ is a closed set and $f:E\to\...
- 28k
8
votes
3
answers
3k
views
Bounding supremum norm of Lipschitz function by L1 norm
Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ?
In ...
- 301
8
votes
3
answers
1k
views
On the set of divergence to infinity for sequences of positive continuous functions
Hi,
I have asked this question on math.stackexchange but it has not received much attention, so I ask it here.
This question is partly motivated by this one, which contains an example of a sequence $...
- 2,154
8
votes
3
answers
1k
views
Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 302
8
votes
4
answers
1k
views
For what sets does the Lebesgue Differentiation Theorem hold in one dimension?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
8
votes
3
answers
800
views
Continuous functions as uniformly continuous function
Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow (\Bbb{R},...
user38122
8
votes
1
answer
384
views
Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...
- 128k
8
votes
3
answers
786
views
truth vs. provability for ordered fields
In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
- 19.7k
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,145
8
votes
1
answer
1k
views
Off-diagonalize a matrix
Consider a self-adjoint matrix $M$ that has block form
$$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$
I am wondering if there exists any criterion to decide if ...
- 536
8
votes
2
answers
556
views
The limit of a function with derivative at least $1_\mathbb{Q}$
Let $f:\mathbb{R}\to \mathbb{R}$ be differentiable, such that $f'(x) \ge 1_{\mathbb{Q}}(x)$. Is it true that $\lim_{x\to\infty}f(x) = \infty$?
- 89
8
votes
3
answers
446
views
How to get this integral's asymptotics?
Consider the following integral
$$
\int_0^{\infty}\frac{e^{-x}-1}{x^{2+\frac{A}{\log b-5/6}}}\frac{1}{\log(b/x)-i\pi/2}\,dx
$$
where $A>0$ and $b>0$. I am interested in the small $b$ asymptotics ...
- 247
8
votes
2
answers
634
views
Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.?
Suppose $f \in L^\infty(\mathbb{R})$, $f_h(x) = f(x + h)$, and$$\lim_{h \to 0} \|f_h - f\|_\infty = 0.$$Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost ...
- 113
8
votes
1
answer
321
views
Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function.
If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
- 1,513
8
votes
1
answer
678
views
Inequality involving tensor product of orthonormal unit vectors
Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
- 1,495
8
votes
1
answer
617
views
Violating the Lebesgue density theorem
Can anyone exhibit a finite-dimensional metric space (preferably, $R^d$) equipped with a measure that does not satisfy the conclusions of the Lebesgue Density Theorem? Such examples exist in infinite-...
- 6,541
8
votes
1
answer
997
views
A seemingly trivial property of differentiable functions
NOTE. This is not really the question I wanted to ask. Somehow I forgot to mention that I am assuming $f$ is continuous. However, since Iosif's answer has been well-received I have left this question ...
- 1,149
8
votes
1
answer
574
views
On functions with strict Lipschitz constant
We say a measurable subset $S$ of $\mathbb R^n$ is measure dense if for every open set $U \subset \mathbb R^n$, $U \cap S$ is of positive Lebesgue measure.
Let $n \geq 2$, and let $f: \mathbb R^n \to \...
- 6,215
8
votes
1
answer
375
views
Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$?
In The Fourier Transform of the quartic Gaussian $\exp(-Ax^4)$: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to $\exp(-Ax^{2n})$, Boyd ...
- 860
8
votes
2
answers
773
views
Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,145
8
votes
2
answers
2k
views
Arzela-Ascoli for L_p-norm
Since I am from a different mathematical field and couldn't find it: Is there something which would be best called an Arzela-Ascoli version for the $L_p$-norm, namely:
Let $X,Y$ be two nice ...
- 255
8
votes
4
answers
338
views
Scaling a set of reals to be nearly integers
A version of this question was previously asked on MSE. I'll mention progress below.
A geometric construction I'm exploring
leads to a set $R$ of $n$ positive real numbers, for example:
$$
R = \{ \pi,...
- 151k
8
votes
2
answers
2k
views
Do proper Zariski closed sets of algebraic sets have measure zero
This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...
- 81
8
votes
1
answer
392
views
Malgrange preparation theorem with less regularity
(This question was previously posted on MSE
and I decided to post it here too.)
I am studying the proof of the Malgrange preparation theorem given in the book "Stable mappings and their ...
8
votes
1
answer
662
views
Can this inequality be proved using weighted maximal function estimates?
I am trying to understand the following fact:
Suppose $\{B_i\}_i$ are disjoint balls in $\mathbb R^n$, and $A_i \subset 100 B_i$ is a subset with $|A_i| \geq c |B_i|$. Then for any nonnegative $f$, ...
- 613
8
votes
1
answer
602
views
Example of a function with a curious property
Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$.
$\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that:
$\frac{F(x)}{x}\in ...
- 421
8
votes
1
answer
391
views
On the limit of partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n_k\}_{k=1}^\infty$ such that
$$ \lim_{k\to\infty}\frac{1}{n_k}\sum_{i=1}^{n_k}\sum_{j=1}^{n_k}...
- 171
8
votes
2
answers
979
views
Lebesgue outer measure
Denote the Lebesgue outer measure by $\mu^{\star}$. Is there a subset $X \subseteq [0, 1]$ such that $\mu^{\star}(X) > 0$ and $\mu^{\star} \upharpoonright \mathcal{P}(X)$ is a measure (countably ...
- 89
8
votes
1
answer
687
views
Measure without measurable sets
This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
- 3,574
8
votes
1
answer
461
views
On critical points of harmonic functions
Let $u \in C^{\infty}(\mathbb R^3)$ be harmonic. Suppose that $u$ has no critical points outside the unit ball but that it has at least one critical point inside the unit ball.
Does it follow that $u$ ...
- 4,145
8
votes
1
answer
759
views
Is there a real valued function whose limit exists only on irrational numbers?
I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational.
I tried variations ...
8
votes
4
answers
3k
views
Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
- 2,585
8
votes
3
answers
813
views
Strange real functions
I know there are a lot of strange functions $f~:~\mathbb R \to \mathbb R$.
I'm looking for an "elementary but complete" exposition of a result discovered by W. Sierpi\'nski and A. Zygmund in "Sur une ...
- 2,829
8
votes
1
answer
594
views
What is the minimum of this functional？
Recently I encountered an inequality from mathematical analysis.
Let $f(x)$ be twice continuously differentiable in $[0,1]$ with
$f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...
- 505
8
votes
2
answers
622
views
Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 903
8
votes
1
answer
460
views
Real-rootedness of some polynomials
Denote the unsigned Stirling numbers of the first kind by $s(n,j)$.
Question. Is it true that the polynomials
$$P_n(x)=\sum_{j\geq0}s(n,j)\binom{x}j$$
have only real roots?
Note. Obviously, the ...
- 43.2k
8
votes
3
answers
636
views
Method to compute fundamental solutions which are distributions
The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
8
votes
3
answers
540
views
Real varieties with enough algebraic loops
Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$).
We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...
- 43.2k
8
votes
1
answer
343
views
How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 6,215
8
votes
1
answer
454
views
Pseudo differentiable functions
Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. Let $\mathcal L$ be the set of linear functions $\mathbb R \to \mathbb R$.
Define the roughness $\mathcal Rf(x)$ of $f$ at $x \in \mathbb R^...
- 6,215
8
votes
2
answers
5k
views
Lipschitz constant of a function of matrix
The function is given by
$f(X) = (AX^{-1}A^\top + B)^{-1}$ where $X$, $A$, and $B$ are $n \times n$ positive definite matrices.
I'm trying to find the Lipschitz constant such that $\| f(X)-f(Y) \| \...
- 85