All Questions
Tagged with real-analysis co.combinatorics
133 questions
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
1
vote
0
answers
79
views
Bounds on the inverse multivariate beta function
Can i find some constants $\mathbf a$ and $\mathbf b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function :
$$IB(\mathbf x) = \frac{\Gamma\left(\lvert \...
- 686
12
votes
1
answer
525
views
An inequality about unit vector orthogonal to $(1,1,...,1)$
Does there exist a constant $\alpha>0$ such that the following holds?
$$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
- 1,495
0
votes
1
answer
60
views
Bounding the ratio of the $\ell_1$-norms of two real-valued $n$-vectors as a linear combination of their $n$ element-wise ratios
Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a_i \le b_i\le 1$. Let the ratio $R$ defined as follows:
...
- 1,677
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
$$(\Delta^* u)(n)=...
- 192
4
votes
1
answer
481
views
Higher-order derivatives of $(e^x + e^{-x})^{-1}$
I am currently trying to build the derivatives of $$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
-1
votes
1
answer
155
views
Is this recurrent sequence decreasing?
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
1
vote
1
answer
181
views
Optimization problem with definite integral inequality constraints
Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$?
$$\int_{-1}^...
- 1,677
3
votes
1
answer
572
views
Remarkable limit involving $m_p=\log_p(p^{x_1} + \cdots + p^{x_n})-\log_p(n)$
It is easy to prove that
$\lim_{p\rightarrow 1} m_p = (x_1 + \cdots + x_n)/n$. The following fact about the derivative of $m_p$ with respect to $p$ is also elementary:
$$m'_p =\frac{dm_p}{dp}
=\frac{1}...
- 3,098
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
-1
votes
1
answer
103
views
Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]
I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$
$$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$
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 $...
- 1,889
6
votes
1
answer
274
views
Limits (growth rates) of power series coefficients
Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(...
- 43.2k
1
vote
2
answers
267
views
Is $ \{ \frac{1}{n} + \frac{1}{m} : n,m \in \mathbb{N} \}$ dense in some interval of $\mathbb{R}$? [closed]
Consider the set $\{\frac{1}{n} + \frac{1}{m}: n,m \in \mathbb{N} \}$. Is this set dense in some interval of $\mathbb{R}$?
More generally let $S_k= \{\sum_{i=1}^k \frac{1}{n_i}: n_i \in \mathbb{N} \} ...
- 93
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
2
votes
2
answers
539
views
Graph with complex eigenvalues
The question I am wondering about is:
Can the discrete Laplacian have complex eigenvalues on a graph?
Clearly, there are two cases where it is obvious that this is impossible.
1.) The graph is ...
user144765
6
votes
1
answer
290
views
What is the growth rate of the products of binomial coefficients?
Question 1: Are the following empirically observed relationships true
$$
{n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a}
\sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)
$$
...
- 5,470
1
vote
1
answer
457
views
A (surprising?) expression for $e$
I apologise if this is off topic.
Consider the quantity
$$
F(m,n,k)=\frac{(m)_k}{k!n^{k-1} }
$$
where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation
$$
\sum_{k=1}^{K} ...
- 10.4k
1
vote
0
answers
75
views
Generalization of Lagrange-Burmann to system of self-consistency equations
In my research, I have come across a system of probability generating functions of the following form:
$$H_1(x) = x A(H_1(x))B(H_2(x)) \text{,}$$
$$H_2(x) = x C(H_1(x))D(H_2(x)) \text{,}$$
and I am ...
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
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
7
votes
1
answer
321
views
Taylor's polynomials and loss of real roots
Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below.
Suppose the roots of a polynomial $p(x)$ are all real and ...
- 43.2k
1
vote
1
answer
72
views
Independent identical distribution sequence
given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
- 553
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
6
votes
3
answers
554
views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
- 61
0
votes
4
answers
570
views
How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$ [closed]
How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this ...
- 327
6
votes
1
answer
411
views
Interesting behaviour of binomial coefficients
Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...
- 63
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
8
votes
2
answers
323
views
Matrix rescaling increases lowest eigenvalue?
Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...
- 225
9
votes
0
answers
180
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 43.2k
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
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
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
- 111
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 $$S = \{z\in \mathbb{R}^{n+1}: \|...
- 48.9k
1
vote
1
answer
248
views
Expected value of maximal displacement in permutations of $\{1,\ldots,n\}$
For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S_n$ we let the maximal displacement be ...
- 48.9k
0
votes
1
answer
347
views
Asymptotic behaviour of fixed points in permutations
For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
- 48.9k
-3
votes
1
answer
227
views
Equation $\ \binom xn'\ =\ \log(n)$ [closed]
Problem: Solve equation
$$ \binom xn'\ =\ \log(n) $$
Here prime stands for the derivative with respect to $x$.
Observe that:
$\quad$ for integer $n$ large,
the approximate solution is $\ ...
- 4,786
5
votes
1
answer
333
views
Do monotone functions on the interval have an "Alexander duality" property?
Let $X,Y$ be two copies of the unit interval $[0,1]$. Consider functions $X\rightarrow Y$ and $Y\rightarrow X$ both as subsets of the cartesian product $X\times Y$. (More precisely: identify a ...
- 2,851
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 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
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?
$$6\prod_{j=1}^k(...
- 43.2k
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
- 43.2k
8
votes
2
answers
1k
views
does this sum have a limit?
Define the sequence given by the finite sum
$$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$
Questions.
(1) Is $0<a_n<1$?
(2) Does the limit $\lim_{n\rightarrow\infty}...
- 43.2k
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
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
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
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
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