Ehrhart Series,Unimodality, and Integrally Closed Reflexive Polytopes

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.     

Mean‐Field Evolution of Fermionic Mixed States

In this paper we study the dynamics of fermionic mixed states in the mean‐field regime. We consider initial states that are close to quasi‐free states and prove that, under suitable assumptions on the initial data and on the many‐body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi‐free state. In particular, we prove that the evolution of the reduced one‐particle density matrix converges, as the number of particles goes to infinity, to the solution of the time‐dependent Hartree‐Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many‐body dynamics towards the Hartree‐Fock evolution.© 2015 Wiley Periodicals, Inc.     

Sampling and recovery of multidimensional bandlimited functions via frames

In this paper, we investigate frames for L₂d[−π,π] consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L₂[−π,π], and a well-known theorem of Levinson is recovered as a corollary.     

Evolution of bank efficiency in Brazil: A DEA approach

This paper investigates cost, technical and allocative efficiencies for Brazilian banks in the recent period (2000–2007). We use Data Envelopment Analysis (DEA) to compute efficiency scores. Brazilian banks were found to have low levels of economic (cost) efficiency compared to banks in Europe and in the US. For the period with high macroeconomic volatility (2000–2002) the economic inefficiency in Brazilian banks can be attributed mainly to technical inefficiency rather than allocative inefficiency. State-owned banks are significantly more cost efficient than foreign, private domestic and private with foreign participation. There is no evidence of differences in economic efficiency due to type of activity and bank size. These results may provide some useful guidance for financial regulators and bank managers.     

Dynamics of Decision Making in Animal Group Motion

We analyze a continuous-time model of a multi-agent system motivated by simulation studies on dynamics of decision making in animal groups in motion. Each individual moves at constant speed in the plane and adjusts its heading in response to relative headings of others in the population. The population includes two subgroups that are “informed” such that individuals in each subgroup have a preferred direction of motion. The model exhibits fast and slow time scales allowing for a reduction in the dimension of the problem. The stable solutions for the reduced model correspond to compromise by individuals with conflicting preferences. We study the global phase space for the proposed reduced model by computing equilibria and exploring stability and bifurcations. B. Nabet’s and N.E. Leonard’s work is supported in part by AFOSR grant FA9550-07-1-0-0528 and ONR grants N00014-02-1-0826 and N00014-04-1-0534. I.D. Couzin’s work was supported by the Royal Society, Balliol College and EPSRC grants GR/S04765/01 and GR/T11234/01, a Searle Scholar Award and DARPA grant HR001-05-1-0057. S.A. Levin’s work was supported in part by DARPA grant HR0011-05-1-0057 and NSF grant EF-0434319.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ${?}^2\left({\mathbb{N}}^1\right)$. We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.     

Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G ⁿ , the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G ⁿ , generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre??s notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.     

Perfect state transfer in Laplacian quantum walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix \(U(t) = \exp (-itM)\), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time \(\tau \) if the (u, v)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if \(m \equiv 2\pmod {4}\). This was previously known for the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph G has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of G and \(\mu \) of H, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of \(P_{3}\) has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129–147, 2011).