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Vassilis G. Papanicolaou

Publications and source records attributed to Vassilis G. Papanicolaou.

At least 19 recordsLinked to original sources
arXiv2026

Factorization for the matrix-valued general Jacobi system on the full-line lattice

The Jacobi system with matrix-valued coefficients and with the spectral parameter depending on a matrix-valued weight factor is considered on the full-line lattice. The scattering from the full-line lattice is expressed in terms of the scattering from the fragments of the whole lattice by developing a factorization formula for the corresponding transition matrices. In particular, the matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments. Since the matrix-valued scattering coefficients are easier to determine for the fragments than for the full-line lattice, the factorization formula presented provides a method to determine the scattering coefficients for full-line lattices. The theory presented is illustrated with various explicit examples, including an example demonstrating that the matrix-valued left transmission coefficient in general is not equal to the matrix-valued right transmission coefficient for a lattice.

math-ph↗
arXiv2026

A Probabilistic Model for Forest Fires

We propose a discrete two-dimensional mathematical model for forest fires and we derive certain results describing its limiting behavior. We also pose a relevant open question.

math.PR↗
arXiv2025

Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval

Let $H = -d^2/dx^2 + q(x)$, $x \in \mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $σ(H)$ of $H$ is the positive semi-axis $[0, \infty)$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $σ(H) = [0, \infty)$. In this article we conjecture a characterization of all complex-valued entire potentials whose spectrum is $[0, \infty)$. We also present an analog of Borg's result for complex potentials.

math.SP↗
arXiv2024

Asymptotics Related to a Binary Search Scheme

Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated, independently of the other specimens. We assume group testing is applicable, namely one can take small portions from several specimens, mix them together, and test the mixture for contamination, so that if the test turns positive, then at least one of the samples in the mixture is contaminated. In this paper we derive asymptotics, as $N$ gets large, of the expectation and the variance of the number $T(N)$ of tests required in order to find all contaminated specimens, under the binary search scheme we introduced in \cite{P} (see, also, arXiv:2007.11910). In \cite{P} the probability $p$ was fixed, whereas in the present work we consider the case where $p \sim a / N^{-β}$, $a, β> 0$ with emphasis on the case $β= 1$, which turns out to be the most interesting.

math.PR↗
arXiv2024

A binary search scheme for determining all contaminated specimens

Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated (e.g., in the case of an infectious disease, $p$ is the prevalence rate), independently of the other specimens. In many cases group testing is applicable, namely one can take small portions from several specimens, mix them together and test the mixture for contamination, so that if the test turns positive, then at least one of the samples in the mixture is contaminated. In this paper we give a detailed probabilistic analysis of a binary search scheme, we propose, for determining all contaminated specimens. More precisely, we study the number $T(N)$ of tests required in order to find all the contaminated specimens, if this search scheme is applied. We derive recursive and, in some cases, explicit formulas for the expectation, the variance, and the characteristic function of $T(N)$. Also, we determine the asymptotic behavior of the moments of $T(N)$ as $N \to \infty$ and from that we obtain the limiting distribution of $T(N)$ (appropriately normalized), which turns out to be normal.

math.PR↗
arXiv2021

The Weierstrass $\wp$-function of the hexagonal lattice

We present some properties of the Weierstrass $\wp$-function associated to the hexagonal (or triangular) lattice. In particular, with the help of an old theorem of I.N. Baker \cite{B} on the characterization of meromorphic solutions of the equation $X^3 + Y^3 = 1$ we determine the zeros of the function $\wp'(z) \pm \sqrt{3}$.

math.CV↗
arXiv2020

On an Old Question of Erdős and Rényi Arising in the Delay Analysis of Broadcast Channels

Consider a broadcast channel with $n$ users, where different users receive different messages, and suppose that each user has to receive $m$ packets. A quantity of interest here, introduced by Sharif and Hassibi (2006-7) \cite{Sh}, \cite{S-H}, is the (\emph{packet}) \emph{delay} $D_{m,n}$, namely the number of channel uses required to guarantee that all users will receive $m$ packets. For the case of a \emph{homogeneous} network, where in each channel use the transmitter chooses a user at random, i.e. with probability $1/n$, and sends him$/$her a packet, the same quantity $D_{m,n}$ had already appeared in the \emph{coupon collector} context, in the works of Newman and Shepp (1960) \cite{N-S} and of Erdős and Rényi (1961) \cite{E-R}. A problem of particular interest in wireless communications, related to the delay $D_{m,n}$, is to determine its behavior as $n$ and $m$ grow large. Regarding this problem, Sharif and Hassibi \cite{Sh}, \cite{S-H} managed to calculated the asymptotics of the mean value $\mathbb{E}[D_{m,n}]$, as $n \to \infty$, for the cases (a) $m = \ln n$ and (b) $m = (\ln n)^ρ$, $ρ> 1$. It is remarkable that Erdős and Rényi \cite{E-R} had, also, raised the question of the determination of the asymptotic profile of $D_{m,n}$ for large $m$ and $n$ (in 1961). And in the 1970's the limiting distribution of $D_{m,n}$ for large $m$ and $n$ was determined (in great generality) by Ivchenko \cite{I1} and \cite{I2}. In this article we determine the asymptotics of the moments of $D_{m,n}$ for large $m$ and $n$. We also derive its limiting distribution in the "supercritical case" where $m$ grows faster than $\ln n$ and in the "critical case" $m \sim β\ln n$, by an approach which is different from the one used by Ivchenko.

math.PR↗
arXiv2020

The Coupon Collector's Brother

A popular variant of the collector's problem is the following: Assume there are $N$ different types of coupons with equal occurring probabilities. There is one main collector who collects coupons. When she gets a "double," she gives it to her younger brother. Hence, when the main collector completes her collection, her brother's album will still have, say, $U_N$ empty spaces. We show that, as $N \to \infty$, the limiting distribution of $U_N/\ln N$ is exponential with parameter $1$.

math.PR↗
arXiv2019

Interpolating the Derivatives of the Gamma Function

We consider a function $G(λ, z)$, entire in $λ$, which interpolates the derivatives of the Gamma function in the sense that $G(m, z) = Γ^{(m)}(z)$ for any integer $m \geq 0$ and we calculate the asymptotics of $G(λ, z)$ as $λ\to +\infty$.

math.CA↗
arXiv2019

Periodic Jacobi Operators with Complex Coefficients

We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill discriminant of finitely many discrete $N$-periodic Schrödinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem 5) which is the analog of Borg's Theorem for the non-self-adjoint Jacobi case.

math.SP↗
arXiv2019

Analytic Solutions of the Heat Equation

Motivated by the recent proof of Newman's conjecture \cite{R-T} we study certain properties of entire caloric functions, namely solutions of the heat equation $\partial_t F = \partial_z^2 F$ which are entire in $z$ and $t$. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function $f(z)$ to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the $z$-order and $z$-type of an entire caloric function $F(t, z)$, viewed as function of $z$, to its $t$-order and $t$-type respectively, if it is viewed as function of $t$. After that, we shift our attention to the zeros $z_k(t)$ of an entire caloric function $F(t, z)$, viewed as function of $z$. We show that the points $(t, z)$ at which $F(t, z) = \partial_z F(t, z) = 0$ form a discrete set in $\mathbb{C}^2$ and we derive the $t$-evolution equations of the zeros of $F(t, z)$. These are differential equations which hold for all but countably many $t \in \mathbb{C}$.

math.CV↗
arXiv2019

Uniform versus Zipf distribution in a mixing collection process

We consider the following variant of the classic collector's problem: The family of coupon probabilities is the mixing of two subfamilies one of which is the \textit{uniform} family, while the other belongs to the well known \textit{Zipf family}. We obtain asymptotics for the expectation, the second rising moment, and the variance of the random variable $T_N$, namely the number of trials needed for all the $N$ types of coupons to be collected (at least once, with replacement) as $N \rightarrow \infty$. It is interesting that the effect of the uniform subcollection on the asymptotics of the expectation of $T_N$ (at least up to the sixth term) appears only in the leading factor of the expectation of $T_N$. The limiting distribution of $T_N$ is derived as well. These results answer a question placed in a recent work of ours [\textit{Electron. J. Probab.} \textbf{18} (2012) 1--15].

math.PR↗
arXiv2018

Sampling from a mixture of different groups of coupons

A collector samples coupons with replacement from a pool containing $g$ \textit{uniform} groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur. For each $j = 1, \dots, g$ let $T_j$ be the number of trials needed to detect Group $j$, namely to collect all $M_j$ coupons belonging to it at least once. We derive an explicit formula for the probability that the $l$-th group is the first one to be detected (symbolically, $P\{T_l = \bigwedge_{j=1}^g T_j\}$). We also compute the asymptotics of this probability in the case $g=2$ as the number of coupons grows to infinity in a certain manner. Then, in the case of two groups we focus on $T := T_1 \vee T_2$, i.e. the number of trials needed to collect all coupons of the pool (at least once). We determine the asymptotics of $E[T]$ and $V[T]$, as well as the limiting distribution of $T$ (appropriately normalized) as the number of coupons becomes very large.

math.PR↗
arXiv2018

On the bound states of the discrete Schrödinger equation with compactly supported potentials

The discrete Schrödinger operator with the Dirichlet boundary condition is considered on the half-line lattice $n\in \{1,2,3,\dots\}.$ It is assumed that the potential belongs to class $\mathcal A_b,$ i.e. it is real valued, vanishes when $n>b$ with $b$ being a fixed positive integer, and is nonzero at $n=b.$ The proof is provided to show that the corresponding number of bound states, $N,$ must satisfy the inequality $0\le N\le b.$ It is shown that for each fixed nonnegative integer $k$ in the set $\{0,1,2,\dots,b\},$ there exist infinitely many potentials in class $\mathcal A_b$ for which the corresponding Schrödinger operator has exactly $k$ bound states. Some auxiliary results are presented to relate the number of bound states to the number of real resonances associated with the corresponding Schrödinger operator. The theory presented is illustrated with some explicit examples.

math-ph↗
arXiv2016

An Arctangent Law

Let $M_r$ be the maximum value of an one-dimensional Brownian motion on the (time) interval $[0, r]$. We derive an explicit formula for the distribution of the time required (after $r$) for the Brownian motion to exceed $M_r$.

math.PR↗
arXiv2015

The Coupon Collector's Problem Revisited: Generalizing the Double Dixie Cup Problem of Newman and Shepp

The "double Dixie cup problem" of D.J. Newman and L. Shepp (1960) is a well-known variant of the coupon collector's problem, where the object of study is the number $T_{m}(N)$ of coupons that a collector has to buy in order to complete $m$ sets of all $N$ existing different coupons. More precisely, the problem is to determine the asymptotics of the expectation (and the variance) of $T_{m}(N)$, as well as its limit distribution, as the number $N$ of different coupons becomes arbitrarily large. The classical case of the problem, namely the case of equal coupon probabilities, is here extended to the general case, where the probabilities of the selected coupons are unequal. In the beginning of the article we give a brief review of the formulas for the moments and the moment generating function of the random variable $T_{m}(N)$. Then, we develop techniques of computing the asymptotics of the first and the second moment of $T_{m}(N)$ (our techniques apply to the higher moments of $T_{m}(N)$ as well). From these asymptotic formulas we obtain the leading behavior of the variance $V[\,T_{m}(N)\,]$ as $N \to \infty$. Finally, based on the asymptotics of $E[\,T_{m}(N)\,]$ and $V[\,T_{m}(N)\,]$ we obtain the limit distribution of the random variable $T_{m}(N)$ for large classes of coupon probabilities. As it turns out, in many cases, albeit not always, $T_{m}(N)$ (appropriately normalized) converges in distribution to a Gumbel random variable. Our results on the limit distribution of $T_{m}(N)$ generalize a well-known result of P. Erdős and A. Rényi (1961) regarding the limit distribution of $T_{m}(N)$ for the case of equal coupon probabilities.

math.PR↗