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Vassilis G. Papanicolaou

Publications and source records attributed to Vassilis G. Papanicolaou.

29 records · Page 2Linked to original sources
arXiv2015

The logarithmic Zipf version of the coupon collector's problem

A collector wishes to collect $m$ complete sets of $N$ distinct coupons. The draws from the population are considered to be independent and identical distributed with replacement, and the probability that a type-$j$ coupon is drawn is noted as $p_{j}$. Let $T_{m}(N)$ the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of $T_{m}(N)$ (leading term), as well as its limit distribution as $N\rightarrow \infty$, when \begin{equation*} p_{j}=\frac{a_{j}}{\sum_{j=2}^{N+1} a_{j}}, \,\,\,\text{where}\,\,\, a_{j}=\left(\ln j\right)^{-p}, \,\,p>0. \end{equation*} These "log-Zipf" classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. Therefore, we enlarge the classes for which the collector's problem is solved (moments, variance, distribution).

math.PR↗
arXiv2015

Inverse problem with transmission eigenvalues for the discrete Schrödinger equation

The discrete Schrödinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the so-called transmission eigenvalues is analyzed. The Marchenko method and the Gel'fand-Levitan method are used to solve the inverse problem uniquely, except in one "unusual" case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential. It is shown that in the unusual case there may be a unique solution corresponding to certain sets of transmission eigenvalues, there may be a finite number of distinct potentials for some sets of transmission eigenvalues, or there may be infinitely many potentials for some sets of transmission eigenvalues. The theory presented is illustrated with several explicit examples.

math.SP↗
arXiv2015

Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator $\mathcal{L}$ with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.

math.SP↗
arXiv2014

The Siblings of the Coupon Collector

The following variant of the collector's problem has attracted considerable attention relatively recently (see, e.g., N. Pintacuda 1980, D. Foata H. Guo-Niu and B. Lass 2001, D. Foata and D. Zeilberger 2003, I. Adler, S. Oren and S. Ross 2003, and S. Ross 2010): There is one main collector who collects coupons. Assume there are $N$ different types of coupons with, in general, unequal occurring probabilities. When the main collector gets a "double", she gives it to her older brother; when this brother gets a "double", he gives it to the next brother, and so on. Hence, when the main collector completes her collection, the album of the $j$-th sibling, $j = 2, 3, \dots$, will still have $U_j^N$ empty spaces. In this article we develop techniques of computing asymptotics of the average $E[U_j^N]$ of $U_j^N$ as $N \rightarrow \infty,$ for a large class of families of coupon probabilities. We also give various illustrative examples.

math.PR↗
arXiv2014

Similarity Solutions of a Replicator Dynamics Equation Associated to a Continuum of Pure Strategies

We introduce a nonlinear degenerate parabolic equation containing a nonlocal term. The equation serves as a replicator dynamics model where the set of strategies is a continuum. In our model the payoff operator (which is the continuous analog of the payoff matrix) is nonsymmetric and, also, evolves with time. We are interested in solutions $u(t, x)$ of our equation which are positive and their integral (with respect to $x$) over the whole space is $1$, for any $t > 0$. These solutions, being probability densities, can serve as time-evolving mixed strategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approaching the Dirac delta function $δ(x)$ as $t \to 0^+$.

math.AP↗
arXiv2014

Transmission eigenvalues for the selfadjoint Schrödinger operator on the half line

The transmission eigenvalues corresponding to the half-line Schrödinger equation with the general selfadjoint boundary condition is analyzed when the potential is real valued, integrable, and compactly supported. It is shown that a transmission eigenvalue corresponds to the energy at which the scattering from the perturbed system agrees with the scattering from the unperturbed system. A corresponding inverse problem for the recovery of the potential from a set containing the boundary condition and the transmission eigenvalues is analyzed, and a unique reconstruction of the potential is given provided one additional constant is contained in the data set. The results are illustrated with various explicit examples.

math.SP↗
arXiv2013

Reconstruction of the wave speed from transmission eigenvalues for the spherically-symmetric variable-speed wave equation

The unique reconstruction of a spherically-symmetric wave speed $v$ is considered in a bounded spherical region of radius $b$ from the set of corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of $1/v$ on the interval $[0,b]$ is less than $b,$ assuming that there exists at least one $v$ corresponding to the data, $v$ is uniquely reconstructed from the data consisting of such transmission eigenvalues and their "multiplicities," where the multiplicity is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to $b,$ the unique reconstruction is presented when the data set contains one additional piece of information. Some similar results are presented for the unique reconstruction of the potential from the transmission eigenvalues with multiplicities for a related Schrödinger equation.

math.SP↗
arXiv2011

The uniqueness in the inverse problem for transmission eigenvalues for the spherically-symmetric variable-speed wave equation

The recovery of a spherically-symmetric wave speed $v$ is considered in a bounded spherical region of radius $b$ from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of $1/v$ on the interval $[0,b]$ is less than $b,$ assuming that there exists at least one $v$ corresponding to the data, it is shown that $v$ is uniquely determined by the data consisting of such transmission eigenvalues and their "multiplicities," where the "multiplicity" is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to $b,$ the unique recovery is obtained when the data contains one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with "multiplicities" for a related Schrödinger equation.

math-ph↗
arXiv2008

Time evolution of the scattering data for a fourth-order linear differential operator

The time evolution of the scattering and spectral data is obtained for the differential operator $\displaystyle\frac{d^4}{dx^4} +\displaystyle\frac{d}{dx} u(x,t)\displaystyle\frac{d}{dx}+v(x,t),$ where $u(x,t)$ and $v(x,t)$ are real-valued potentials decaying exponentially as $x\to\pm\infty$ at each fixed $t.$ The result is relevant in a crucial step of the inverse scattering transform method that is used in solving the initial-value problem for a pair of coupled nonlinear partial differential equations satisfied by $u(x,t)$ and $v(x,t).$

nlin.SI↗