Nonnegative Part Research Articles

Increases in risk aversion and the distribution of portfolio payoffs

Abstract Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff Aʼs payoff is always distributed as Bʼs payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

Stability of ZND detonations for Majda’s model

We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda’s model for reacting flow and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of Roquejoffre–Vila that viscous detonations of Majda’s model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.

KP solitons, total positivity, and cluster algebras

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili [Kadomtsev BB, Petviashvili VI (1970) Sov Phys Dokl 15:539-541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian.

An easy-to-compute factorization of rational generalized positive functions

We give a simple constructive proof of a factorization theorem for scalar rational functions with a non-negative real part on the imaginary axis. A Mathematica program, performing this factorization, is provided.

Cloaking of Small Objects by Anomalous Localized Resonance

We investigate solutions of ∇ · (a∇u) = 0 with various boundary conditions. The coefficient a is assumed to have a real part with changing sign and a small, non-negative imaginary part of order η. We investigate a two-dimensional ring geometry with unit inner radius and outer radius R. We use Fourier expansions in polar coordinates to analyze the qualitative behaviour of solutions when boundary conditions are imposed on a small circular inclusion, centred at x 0 . Our result is that u depends qualitatively on the position of the inclusion. If lx 0 l is larger than the cloaking radius R * = R 3/2 , then u behaves as if no ring were present. If, instead, lx 0 l < R*, then the small inclusion is invisible in the limit η→ 0.

Semiclassical Hypoelliptic Estimates for Non-Selfadjoint Operators with Double Characteristics

For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, with a leading symbol with a non-negative real part, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular spaces, we establish semiclassical hypoelliptic a priori estimates with a loss of the full power of the semiclassical parameter giving a localization for the low lying spectral values of the operator.

Geometric realizations of the multiplihedra

AbstractWe realize the multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.

Dynamic Properties of One‐Component Moderately Coupled Plasmas: The Mixed Löwner‐Nevanlinna‐Pick Approach

AbstractDynamic characteristics of strongly coupled one‐component plasmas are studied within the moment approach with local constraints, similar to that of Schur. The dynamic structure factor is modeled by the.rst three terms of its asymptotic decomposition at high frequencies and its values at a few interpolation points on the real axis. This makes the dynamic structure factor to be a non‐rational function whose extension onto the upper half‐plane of complex frequency is holomorphic with a nonnegative imaginary part and a continuous extension to the real axis. A free parameter of the non‐rational model function of the suggested Schur‐like algorithm is obtained from the Shannon entropy maximization procedure.The results automatically satisfy not only the sum rules and other exact relations but, also, interpolate between the interpolation frequencies. A quantitative agreement is obtained with simulation data on the plasma dynamic properties. The method permits to take into account the energy dissipation processes so that the results of alternative theoretical approaches are included into the moment scheme and are complemented. The algorithm tested herewith can also be used in a broader setting (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Increases in Risk Aversion and the Distribution of Portfolio Payoffs

Hart proved the difficulty of deriving general comparative statics in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff A’s payoff is always distributed as B’s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributionsover time (but not at every point in time).

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part ( G / P ) ⩾ 0 of an arbitrary (generalized, partial) flag variety G / P . He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that ( G / P ) ⩾ 0 – the closure of the top-dimensional cell – is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell ( G / P ) > 0 is homotopic to a sphere, is new for all G / P other than projective space.

Sums of Nevanlinna functions and differential equations on star-shaped graphs

Additive decompositions of a meromorphic function give rise to quotient representations of a particular form. We raise the question which quotient representations of a given function arise in this way. This question is answered by means of two characterizations via different terms. We pay particular attention to functions belonging to various subclasses of the Nevanlinna class of functions with nonnegative imaginary part throughout the upper half-plane. Our results lead to some direct and inverse spectral theorems for systems of strings or systems of Sturm-Liouville equations supported on a star-shaped graph. MSC 2000: Primary 34B45. Secondary 34B07, 30D35, 34B24.

Matching polytopes, toric geometry, and the totally non-negative Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr k,n )≥0. This is a cell complex whose cells Δ G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety X G . We use our technology to prove that the cell decomposition of (Gr k,n )≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr k,n )≥0 is 1.

The Totally Nonnegative Part of G/P is a CW Complex

The totally nonnegative part of a partial ag variety G/P has been shown in [18], [17] to be a union of semialgebraic cells. Moreover, the closure of a cell was shown in [19] to be a union of smaller cells. In this paper we provide glueing maps for each of the cells to prove that (G/P)⩾0 is a CW complex. This generalizes a result of Postnikov, Speyer and the second author [15] for Grassmannians.

Total positivity for cominuscule Grassmannians

In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively. Dans cet article nous schtroumpfons la combinatoire de la partie non-négative $(G/P)_{\geq 0}$ d'une Grassmannienne cominuscule. Pour chaque Grassmannienne de ce type nous définissons les Le-diagrammes ― certains remplissages de diagrammes de Young généralisés en bijection avec les cellules de $(G/P)_{\geq 0}$. Dans les cas classiques, nous décrivons les Le-diagrammes explicitement en termes d'évitement de certains motifs. Aussi nous définissons un jeu sur les diagrammes, avec lequel on peut réduire un diagramme arbitraire à un Le-diagramme. Nous donnons les résultats énumératifs et nous relions nos Le-diagrammes à d'autres objets combinatoires. Étonnamment, les cellules non-négatives dans la cellule de Schubert ouverte des Grassmanniennes orthogonales impaires et paires sont essentiellement en bijection avec les fonctions de préférence et les fonctions de préférence atomiques.

Quivers of semi-maximal rings

In this paper, the set of quivers of semi-maximal rings is investigated. It is proved that the elements of this set are formed by the elements of the set of quivers of tiled orders and that the set of quivers of tiled orders with n vertices is determined by the integer points of a convex polyhedral domain that lie in the nonnegative part of the space \(\mathbb{R}^{n^2 - n} \). It is also proved that the set of quivers of tiled orders with n vertices contains all simple, oriented, strongly connected graphs with n vertices and n loops, does not contain any graphs with n vertices and n − 1 loops, and contains only a part of the graphs with n vertices and m (m < n − 1) loops.

Irreducible modules for the quantum affine algebra [formula omitted] and its Borel subalgebra [formula omitted

Let g be an affine Kac–Moody Lie algebra, and let U q ( g ) be its quantized universal enveloping algebra. Let the Borel subalgebra U q ( g ) ⩾ 0 of U q ( g ) be the nonnegative part of U q ( g ) with respect to the standard triangular decomposition. Suppose ε ∈ { − 1 , 1 } n , where n is the number of simple roots of g . We construct a bijection between finite-dimensional irreducible U q ( g ) ⩾ 0 -modules of type ε and finite-dimensional irreducible U q ( g ) -modules of type ε. In particular: (i) Let V be a finite-dimensional irreducible U q ( g ) ⩾ 0 -module of type ε. Then the action of U q ( g ) ⩾ 0 on V extends uniquely to an action of U q ( g ) on V. The resulting U q ( g ) -module structure on V is irreducible and of type ε. (ii) Let V be a finite-dimensional irreducible U q ( g ) -module of type ε. When the U q ( g ) -action is restricted to U q ( g ) ⩾ 0 , the resulting U q ( g ) ⩾ 0 -module structure on V is irreducible and of type ε.

Hopf Algebras of Type A n , Twistings and the FRT-construction

We study pointed Hopf algebras of the form U(RQ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where RQ is the Yang–Baxter operator associated with the multiparameter deformation of GLn supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(RQ) is of type An in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(RQ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(RQ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(RQ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(RQ), U(RQ′) can be obtained from each other by twisting the comultiplication if and only if \(G(U_{\!Q})\cong G(U_{Q'}).\) In the last part we show that UQ is always a quotient of a double crossproduct.

Shelling totally nonnegative flag varieties

In this paper we study the partially ordered set Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety P^J_{\geq 0}. Our goal is to understand the geometry of P^J_{\geq 0}: Lusztig has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P^J_{\geq 0} is homeomorphic to a ball. The order complex |Q^J| is a simplicial complex which can be thought of as a combinatorial approximation of P^J_{\geq 0}. Using combinatorial tools such as Bjorner's EL-labellings and Dyer's reflection orders, we prove that Q^J is graded, thin and EL-shellable. As a corollary, we deduce that Q^J is Eulerian and that the Euler characteristic of the closure of each cell is 1. Additionally, our results imply that |Q^J| is homeomorphic to a ball, and moreover, that Q^J is the face poset of some regular CW complex homeomorphic to a ball.

Tableaux combinatorics for the asymmetric exclusion process

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability α and exit from the right with probability β. In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrímsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration τ is essentially the generating function for permutation tableaux of shape λ ( τ ) enumerated according to three statistics. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the matrix ansatz for the PASEP model. As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurrences of generalized patterns.