Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.     

Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

Optimal production in a competitive market under yield uncertainty

Motivated by the U.S. influenza vaccine market, we study the impact of random yield and production capacity on the number of firms, total supply, consumer surplus and social welfare in a market with identical suppliers. We formulate a two-stage game with endogenous entry, where each entering firm aims to maximize its profit under yield uncertainty subject to a production capacity constraint. We show that if firms produce full capacity in the equilibrium, then there are fewer firms in the equilibrium compared to the social optimum even for small levels of yield uncertainty. Furthermore, we prove that if firms do not produce full capacity in the equilibrium, they will not produce full capacity in the social optimum.     

Local and Global Approximation for Certain (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Durrmeyer Type Operators

The present paper pertains to special approximation properties of certain (p, q)-Durrmeyer variant. The local and global approximation results are estimated along with comparison to the optimal convergence using the technique given by King. Graphically, we illustrate the convergence of these operators for different values of the two parameters p and q using MATLAB.     

Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.     

Antidirected Hamiltonian paths and directed cycles in tournaments

We prove that in a tournament of odd order $n\ge 5$, the number of antidirected Hamiltonian paths starting with a forward arc and the number of Hamiltonian circuits have the same parity.     

Bayesian Nonparametric Modeling for Multivariate Ordinal Regression

Univariate or multivariate ordinal responses are often assumed to arise from a latent continuous parametric distribution, with covariate effects that enter linearly. We introduce a Bayesian nonparametric modeling approach for univariate and multivariate ordinal regression, which is based on mixture modeling for the joint distribution of latent responses and covariates. The modeling framework enables highly flexible inference for ordinal regression relationships, avoiding assumptions of linearity or additivity in the covariate effects. In standard parametric ordinal regression models, computational challenges arise from identifiability constraints and estimation of parameters requiring nonstandard inferential techniques. A key feature of the nonparametric model is that it achieves inferential flexibility, while avoiding these difficulties. In particular, we establish full support of the nonparametric mixture model under fixed cut-off points that relate through discretization the latent continuous responses with the ordinal responses. The practical utility of the modeling approach is illustrated through application to two datasets from econometrics, an example involving regression relationships for ozone concentration, and a multirater agreement problem. Supplementary materials with technical details on theoretical results and on computation are available online.