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31 votes
1 answer
2k views

Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$ I ...
user avatar
29 votes
6 answers
8k views

How to find a closest integer point to the intersection of two lines?

Here's a question that originates from StackOverflow. Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
P Shved's user avatar
  • 391
29 votes
3 answers
2k views

Wanted: Positivity certificate for the AM-GM inequality in low dimension

I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables $$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$ Can one write the LHS as a sum $\,\...
Hanno's user avatar
  • 489
28 votes
4 answers
3k views

Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$

Let $x>0$ and $n$ be a natural number. Prove that: $$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$ This question is very similar to many contests problems, but ...
Michael Rozenberg's user avatar
28 votes
4 answers
2k views

For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?

Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true : $$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
Aditya Guha Roy's user avatar
26 votes
2 answers
12k views

About the definition of Borel and Radon measures

I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature. More precisely, I have a doubt about the very definition of ...
Jeremy's user avatar
  • 281
26 votes
3 answers
16k views

the dual space of C(X) (X is noncompact metric space)

It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f: X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
yaoxiao's user avatar
  • 1,706
26 votes
4 answers
2k views

$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?

Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$. Then can we prove $f(x)$ is a convex ...
Anyu's user avatar
  • 271
26 votes
3 answers
3k views

Sum of Gaussian pdfs

I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum_{n\in\mathbb Z}f(x+n)$), the ...
Anthony Quas's user avatar
  • 23.2k
25 votes
2 answers
2k views

Writing a function on $\mathbb{R}$ as a sum of two injections

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
Burak's user avatar
  • 4,265
23 votes
4 answers
5k views

Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$. I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
Victor's user avatar
  • 1,437
23 votes
1 answer
706 views

Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...
Asaf Shachar's user avatar
  • 6,741
23 votes
1 answer
3k views

Does the average primeness of natural numbers tend to zero?

This question was posted in MSE. It got many upvotes but no answer hence posting it in MO. A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
Nilotpal Kanti Sinha's user avatar
22 votes
2 answers
652 views

Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
user521337's user avatar
  • 1,209
22 votes
5 answers
1k views

Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
BigbearZzz's user avatar
  • 1,245
21 votes
1 answer
564 views

Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series $\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely. Then there is a partition of the symmetric group ${\rm Sym}(\...
Stefan Kohl's user avatar
  • 19.6k
21 votes
2 answers
2k views

Boundedness of sum of sin(sin(n))

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$ is bounded. However, I did not succeed in proving this ...
Oleksandr Liubimov's user avatar
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,...
George Lowther's user avatar
21 votes
2 answers
2k views

Real rootedness of a polynomial

Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by: $$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$ I've found with Sage that for every $...
Luis Ferroni's user avatar
  • 1,889
20 votes
1 answer
685 views

Can all partial sums $\sum_{k=1}^n f(ka)$ where $f(x)=\log|2\sin(x/2)|$ be non-negative?

Let $f(x)=\log|2\sin(x/2)|$ (the normalizing factor $2$ is chosen to have the average over the period equal to $0$). Does there exist $a>0$ such that all sums $\sum_{k=1}^n f(ak)\ge 0$? The ...
fedja's user avatar
  • 61.9k
19 votes
3 answers
1k views

What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?

A colleague asked me the following question: "What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?" This ...
Ali Taghavi's user avatar
19 votes
4 answers
1k views

Applications of linear programming duality in combinatorics

So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
amakelov's user avatar
  • 997
19 votes
5 answers
1k views

Floors of powers of reals, how much do the first few determine the next?

Call an integer sequence $\mathbf{x}=\left( x_1,x_2,\cdots \right)$ feasible if it is $f(r)=\left(\lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, \ldots, \lfloor r^n \rfloor, \ldots \...
Aaron Meyerowitz's user avatar
19 votes
4 answers
3k views

Strange result about convexity

$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$. Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ? Source: AoPS
Dattier's user avatar
  • 4,074
18 votes
2 answers
2k views

Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$. Does any of the following generalizations Let $U\...
smyrlis's user avatar
  • 2,933
18 votes
3 answers
3k views

Deciding membership in a convex hull

Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$. This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
Mitch's user avatar
  • 667
18 votes
2 answers
1k views

Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
Noah Schweber's user avatar
17 votes
2 answers
2k views

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem: "Given a sequence of positive number $(x_n)$ such that for all $n$, $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$ Find the limit $\lim_{n \rightarrow \infty} \...
Paresseux Nguyen's user avatar
17 votes
2 answers
2k views

Explicit and fast error bounds for polynomial approximation

Main Question This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance. EDIT (Apr. 23): ...
Peter O.'s user avatar
  • 697
17 votes
3 answers
1k views

Decoupling a double integral

I came across this question while making some calculations. QUESTION. Can you find some transformation to "decouple" the double integral as follows? $$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
T. Amdeberhan's user avatar
17 votes
1 answer
985 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
Mohammad Golshani's user avatar
16 votes
2 answers
1k views

Is there always a way up?

I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P) For any $x, y \in U$ such that $f(...
Pluviophile's user avatar
  • 1,608
16 votes
1 answer
2k views

Are continuous functions almost completely determined by their modulus of continuity?

Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by $$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \...
James Baxter's user avatar
  • 2,069
16 votes
3 answers
1k views

A kernel 'more analytic' than $\exp(-x^2)$

I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \...
Ben Deitmar's user avatar
  • 1,295
16 votes
1 answer
661 views

Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
Dattier's user avatar
  • 4,074
16 votes
3 answers
1k views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
Bogdan's user avatar
  • 161
16 votes
3 answers
1k views

Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets. Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$. Can $\text{Proj}(P)$ have more than $f$ facets? ...
Pedro Ruiz's user avatar
15 votes
1 answer
904 views

Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?

Willie Wong asked here (MO) and here (MSE) very interesting question. As he phrased it: Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ ...
Right's user avatar
  • 225
15 votes
0 answers
749 views

Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$

I would like to prove that $$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$ for any $\omega > 0$ and $...
Tanya Vladi's user avatar
15 votes
3 answers
2k views

Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$

I've been trying to find an asymptotic expansion of the following series $$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$ and $$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
Trax's user avatar
  • 153
15 votes
3 answers
902 views

Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $

I am trying to prove or disprove $$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$ where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
Thomas Kojar's user avatar
  • 5,474
15 votes
2 answers
2k views

Where does the Lebesgue differentiation theorem fail?

The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
Vanessa's user avatar
  • 1,368
15 votes
1 answer
1k views

Characterizing $\mathbf{R}$ as an ordered group

A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ...
coudy's user avatar
  • 18.7k
15 votes
1 answer
2k views

Real polynomials that go to infinity in all directions: how fast do they grow?

Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
Stanley Yao Xiao's user avatar
14 votes
3 answers
2k views

How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$? A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
Maximilian Janisch's user avatar
14 votes
2 answers
807 views

Integral of power of binomials equal to sum of power of binomials?

Inspired by this MO question about integrating binomial coefficients and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And ...
Andreas Rüdinger's user avatar
14 votes
1 answer
899 views

“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
Mike Battaglia's user avatar
14 votes
0 answers
633 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
  • 24.7k
14 votes
2 answers
996 views

Does there exist some $C$ independent of $n$ and $f$ such that $ \|f''\|_p \geq Cn^2 \| f \|_p$, where $1 \leq p\leq \infty$?

Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. Does there exist some $C$ independent of $n$ and $f$ ...
user312503's user avatar
14 votes
1 answer
481 views

A question on a real sequence

Let $\{a_n\}_{n\ge1}$ be a real sequence that decays faster than any algebraic speed, that is, $\lim_{n\to \infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$\sum_{n\ge 1}(n+1)^kn^ka_n =...
Jacob Lu's user avatar
  • 903

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