All Questions
Tagged with real-analysis co.combinatorics
133 questions
4
votes
1
answer
481
views
Higher-order derivatives of $(e^x + e^{-x})^{-1}$
I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$
It is fairly straightforward to obtain
$$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...
- 749
4
votes
1
answer
215
views
On the number of repeated roots
Is there a number $c>0$ such that:
For any $n$ there is a polynomial $p(x) = a_nx^n +\cdots + a_0$ where the coefficients are $-1, 0$ or $1$ such that the number of repetition of the root $x=1$ ...
- 97
4
votes
1
answer
246
views
Is $C_n$ infinitely log-convex?
A sequence $a_n$ is called log-convex if $\mathcal{L}(a_n):=a_{n+1}a_{n-1}-a_n^2\geq0$ for all $n$; it is infinitely log-convex provided that all the iterates $\mathcal{L}^k(a_n)$ are still log-convex,...
- 43.2k
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
- 43.2k
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
4
votes
1
answer
119
views
Proving Equal Set Sizes in Sequential Point Selection on a Real Interval with Variable-Length Intervals
I'm here as an engineer working on a point sampling algorithm and I've noticed that when I perform the algorithm on an ordered set of points in one direction it selects the exact same number of points ...
- 151
4
votes
1
answer
95
views
Limiting values of particular functions
Let's define the functions
$$A_n(q)=\sum_{k=0}^n(-1)^k\cdot\frac{(1+q)q^k}{1+q^{2k+1}}\cdot\frac{2k+1}{n+k+1}\binom{2n}{n-k}.$$
I'm interested in the following:
QUESTION. Let $n\geq1$ be integers. ...
- 43.2k
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
0
answers
141
views
Level sets of function of inner products of vectors on hypercube
Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
- 1,127
4
votes
0
answers
188
views
Evaluate a multiple integral
I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
- 41
3
votes
3
answers
1k
views
Positivity of a one-variable rational function
Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$
Numerical evidence convinces me of the truth of the following.
QUESTION. Can you ...
- 43.2k
3
votes
1
answer
572
views
Remarkable limit involving $m_p=\log_p(p^{x_1} + \cdots + p^{x_n})-\log_p(n)$
It is easy to prove that
$\lim_{p\rightarrow 1} m_p = (x_1 + \cdots + x_n)/n$. The following fact about the derivative of $m_p$ with respect to $p$ is also elementary:
$$m'_p =\frac{dm_p}{dp}
=\frac{1}...
- 3,098
3
votes
3
answers
233
views
sequencial shift on families =flipped powers. How?
Consider the following family of functions
$$f_n(w):=\sum_{k=0}^{\infty}\frac{(-1)^{k-1}}{k!}(k+n)^{k-1}w^k.$$
QUESTION 1. Does the following hold?
$$f_n(w)=-\frac1{n(f_{-1}(w))^n}.$$
Deeper ...
- 43.2k
3
votes
2
answers
491
views
Unknown bias in a distribution related to prime numbers
If $n$ is composite then $\phi(n) < n-1$, hence there is at least one divisor $d$ of $n-1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trvially, if $n$ is prime then ...
- 5,470
3
votes
1
answer
222
views
Asymptotic for binomial sums
Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...
- 33
3
votes
1
answer
215
views
Laurent polynomials: what is the correspondence here?
Given a Laurent polynomial $f$, denote the number of terms by $\#f$ and let $\widehat{CT}(f)$ stand for the value of the constant term in $f$. For example, if $f(x,y)=2-\frac{y}x-\frac{x}y$ then $\#f=...
- 43.2k
3
votes
1
answer
147
views
Number and asymptotic for cyclic sequences
Cyclic sequence is equivalence class of cyclic shift action.
If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...
- 123
3
votes
1
answer
155
views
Smoothening a probability measure
Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define
$$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\},
\ z\in{\mathbb ...
- 23k
3
votes
1
answer
251
views
Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
3
votes
1
answer
449
views
Prove that when converge, the following expansions are equal
Prove $f_1(x)=f_2(x)=f_3(x)$ when converge.
$$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$
$$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^...
- 10.1k
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
3
votes
0
answers
155
views
asymptotics of the largest real root
Suppose you have a family of polynomials
$$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$
for $n=0,1,2,\dots$.
Further assumptions:
(1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)...
- 43.2k
2
votes
1
answer
377
views
Prove positivity of rational functions
We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative.
In this context, let
$$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - ...
- 43.2k
2
votes
2
answers
261
views
Prove a family of series having integer coefficients
I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true,...
- 43.2k
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
- 43.2k
2
votes
1
answer
228
views
Choosing finite subsets of natural numbers
Let $t>0$ and $\delta\in\big(0,\frac12\big)$ be fixed. For any $k\in\mathbb{N}$ let $I_k,J_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $|I_k|,|J_k|$, ...
- 375
2
votes
2
answers
269
views
Ratios of polynomials and derivatives under a certain functional
Let $p(x)$ be a polynomial of degree $n>2$, with roots $x_1,x_2,\dots,x_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V_m(p)=\sum_{1\leq i<j\...
- 43.2k
2
votes
2
answers
236
views
$q$-factorial coefficient asymptotics
Consider the $[n]!_q = \prod\limits_{k = 1}^{n} \frac{q^k - 1}{q - 1} = \sum\limits_{k = 0}^{\binom n 2} c_k q^k$ and let $\{f_n\}_{n \in \mathbb{N}}$ be the sequence of the functions on $[0; 1]$ ...
- 123
2
votes
1
answer
200
views
Subset in $[0,1]^k$ with positive density
Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?:
For any $A\subseteq\left[0,1\right]^k$ with the measure ...
- 349
2
votes
2
answers
539
views
Graph with complex eigenvalues
The question I am wondering about is:
Can the discrete Laplacian have complex eigenvalues on a graph?
Clearly, there are two cases where it is obvious that this is impossible.
1.) The graph is ...
user144765
2
votes
1
answer
249
views
linear recurrence inequality
Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does not imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the $...
- 145
2
votes
2
answers
277
views
Set of distinct real numbers such that all combination of sums are distinct
Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers.
For a given $p \in \mathbb N$, consider further the set
$$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, \...
user45183
2
votes
2
answers
128
views
Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
- 128k
2
votes
0
answers
231
views
Where does this trig. identity hold?
Fix an integer $n\geq1$.
QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that
$$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
- 43.2k
2
votes
0
answers
321
views
Distribution of $\frac{(\sin(n))^2}{2^n}$ in dyadic intervals?
Good morning all,
I was wondering what kind of methods could help in order to tackle the following problem :
Define the set $A = \left\{ \frac{(\sin(n))^2}{2^n}\right\}$ for $n$ integer. So A is a ...
- 125
2
votes
0
answers
58
views
Tail asymptotics of Durfee square identity
This post is related to the problem Asymptotics of a combinatorial series
According to the Durfee square identity:
$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$
where $(q;q)_k$ is ...
- 51
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
2
votes
0
answers
251
views
Volume of bounded regions in hyperplane arrangements
I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...
- 357
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
1
vote
1
answer
184
views
Meeting a set of spheres in $\mathbb{R}^n$
Let $n \geq 1$ be an integer. Let us call $S\subseteq \mathbb{R}^{n+1}$ an $n$-sphere if there is $x\in \mathbb{R}^{n+1}$ and $r\in \mathbb{R}$ with $r>0$ such that $$S = \{z\in \mathbb{R}^{n+1}: \|...
- 48.9k
1
vote
1
answer
603
views
The smallest altitude amongst the triangles formed by points in the unit circle
Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...
- 19
1
vote
1
answer
126
views
Evaluation of the multiple integral [closed]
Would you give me any suggestions or comments on evaluating the following $n$-dimensional
integral? $$ \int_{[0,t]^n} h(x) dx $$
where
$ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( \bar{...
- 245
1
vote
1
answer
117
views
Product/quotient of factorials beget dyadic powers
I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...
- 43.2k
1
vote
1
answer
474
views
Compare AM and GM
\begin{gather*}
M_g=(x_1\times x_2\times\dotsb\times x_n)^{1/n} \\
M_a=\frac1 n\times (x_1+x_2+\dotsb+x_n).
\end{gather*}
Is it true that $$\lvert M_g-M_a\rvert \leq (\max(x_i) /\min(x_i)) \times(\max(...
- 4,074
1
vote
1
answer
248
views
Expected value of maximal displacement in permutations of $\{1,\ldots,n\}$
For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S_n$ we let the maximal displacement be ...
- 48.9k
1
vote
2
answers
111
views
A two-parameter inequality on product of linear terms
I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
$$6\prod_{j=1}^k(...
- 43.2k
1
vote
1
answer
89
views
Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
- 573