Generating functions for real character degree sums of finite general linear and unitary groups

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius–Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall–Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius–Schur indicator 1 or ?1.     

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].     

A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.     

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.     

A new generalization of the Pareto–geometric distribution

In this paper we introduce a new distribution called the beta Pareto–geometric. We provide a comprehensive treatment of the mathematical properties of the proposed distribution and derive expressions for its moment generating function and the rth generalized moment. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We also demonstrate its usefulness on a real data set.     

The minimal cone for conic linear programming

It is known that the minimal cone for the constraint system of a conic linear programming problem is a key component in obtaining strong duality without any constraint qualification. For problems in either primal or dual form, the minimal cone can be written down explicitly in terms of the problem data. However, due to possible lack of closure, explicit expressions for the dual cone of the minimal cone cannot be obtained in general. In the particular case of semidefinite programming, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. In this paper we develop a recursive procedure to obtain the minimal cone and its dual cone. In particular, for conic problems with so-called nice cones, we obtain explicit expressions for the cones involved in the dual recursive procedure. As an example of this approach, the well-known duals that satisfy strong duality for semidefinite programming problems are obtained. The relation between this approach and a facial reduction algorithm is also discussed.     

Conformal scalar curvature equations on complete manifolds

This note contains considerations on the existence and non-existence problem of conformal scalar curvature equations on some complete manifolds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radius. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conformally compact manifolds. We obtain non-existence results on both types of conditions. Then we study in more details the existence problem on manifolds with cones, manifolds with cusps and conformally flat manifolds of bounded positive scalar curvature.     

Eine kinematische Kennzeichnung der gleichseitigen Paraboloide

For a given quadric exist (in general) generating methods by means of cones. In the case of the equilateral hyperbolic paraboloid we know, due toH. Brauner, an explicit generation by an equilateral hyperbola. More general we show: The only quadrics which can be generated with cones by a cylindrical vibration are the equilateral paraboloids. The generating cones are equilateral hyperbolas or parabolas, resp. Some special properties are demonstrated.     

Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen–Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen–Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen–Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.     

On two-queue Markovian polling systems with exhaustive service

We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace–Stieltjes transforms of the waiting-time distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traffic behaviour of properly scaled versions of these distributions, which results in compact and closed-form expressions for the distribution functions themselves. The heavy-traffic behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution.     

Lagrangian duality for a class of infinite programming problems

First-order necessary conditions of the Kuhn-Tucker type and strong duality are established for a general class of continuous time programming problems. To obtain these results a generalized Farkas' Theorem, stated in terms of convex and dual cones, is implemented in conjunction with a constraint qualification analogous to that found in finite-dimensional programming. The assumptions imposed are weaker than those needed in previous approaches to duality for this type of problem.     

Belief structures,weight generating functions and decision-making

We describe the Dempster–Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.     

Risk aggregation in multivariate dependent Pareto distributions

In this paper we obtain closed expressions for the probability distribution function of aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with an individual risk model, where the probability density function corresponds to a second kind beta distribution, obtaining the VaR, TVaR and several other tail risk measures. Then, we consider a collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto–Poisson model, the probability density function is a function of the Kummer confluent hypergeometric function, and the density of the Pareto–negative binomial is a function of the Gauss hypergeometric function. Using data based on one-year vehicle insurance policies taken out in 2004–2005 (Jong and Heller, 2008) we conclude that our collective dependent models outperform other collective models considered in the actuarial literature in terms of AIC and CAIC statistics.     

Affine Processes on Symmetric Cones

We consider affine Markov processes taking values in convex cones. In particular, we characterize all affine processes taking values in irreducible symmetric cones in terms of certain Lévy–Khintchine triplets. This is the natural, coordinate-free formulation of the theory of Wishart processes on positive semidefinite matrices, as put forward by Bru (J Theor Probab 4(4):725–751, 1991) and Cuchiero et al. (Ann Appl Probab 21(2):397–463, 2011), in the more general context of symmetric cones, which also allows for simpler, alternative proofs.     

The k–Fibonacci difference sequences

In this paper we apply the concept of difference relation to the sequences of k–Fibonacci numbers. We will obtain general formulas to find any term of the ith k–Fibonacci difference sequence from the initial k–Fibonacci numbers. We also find formulas for the sum of the elements of these new sequences as well as their generating functions. Finally, we study the k–Fibonacci Newton polynomial interpolation.     

Inequalities Characterizing Coisotone Cones in Euclidean Spaces

The isotone projection cone, defined by G. Isac and A. B. Németh, is a closed pointed convex cone such that the order relation defined by the cone is preserved by the projection operator onto the cone. In this paper the coisotone cone will be defined as the polar of a generating isotone projection cone. Several equivalent inequality conditions for the coisotonicity of a cone in Euclidean spaces will be given. Thanks are due to A. B. Németh who draw the author’s attention on the relation of latticial cones generated by vectors with pairwise non-accute angles with the theory of isotone cones.     

On componentwise condition numbers for eigenvalue problems with structured matrices

We give explicit expressions for the componentwise condition number for eigenvalue problems with structured matrices. We will consider only linear structures and show a general result from which expressions for the condition numbers follow. We obtain explicit expressions for the following structures: Toeplitz and Hankel. Details for other linear structures should follow in a straightforward manner from our general result. Copyright © 2008 John Wiley & Sons, Ltd.     

Higher-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization with nonsolid ordering cones

In this paper, we consider higher-order Karush–Kuhn–Tucker optimality conditions in terms of radial derivatives for set-valued optimization with nonsolid ordering cones. First, we develop sum rules and chain rules in the form of equality for radial derivatives. Then, we investigate set-valued optimization including mixed constraints with both ordering cones in the objective and constraint spaces having possibly empty interior. We obtain necessary conditions for quasi-relative efficient solutions and sufficient conditions for Pareto efficient solutions. For the special case of weak efficient solutions, we receive even necessary and sufficient conditions. Our results are new or improve recent existing ones in the literature.     

Covariance of Lax Pairs and Integrability of the Compatibility Condition

We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DT). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DT via a version of non-Abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov–Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DT symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville–von Neumann equations for the density matrix are considered as illustrations.     

The Matsubara operator for the Fröhlich model

We use the generating functional method to consider the perturbation theory for the Matsubara operator for the Fröhlich model of an electron-phonon system. We obtain expressions determining the perturbed Hamiltonian of the electron-phonon system and the effective Hamiltonian of the electron system after averaging over phonon states.