To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.     

A Generalized Quasilinearization Method for Nonlinear Second-Order Impulsive Differential Equations Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

Using variational method and lower and upper solutions, we get a generalized quasilinearization method which construct an iterative scheme converging uniformly to a solution of a nonlinear second-order impulsive differential equations involving the p-Laplacian, and converging quadratically when p=2.     

AG codes from vector bundles

We discuss the properties of algebraic-geometric codes defined by a vector bundle on a curve X over a finite field. We investigate the correction capacities of the codes defined from vector bundles constructed by means of finite coverings of X.     

A new relaxation framework for quadratic assignment problems based on matrix splitting

Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to some strong bounds in the literature. Promising experimental results based on the new relaxations are reported.     

On <Emphasis Type="Italic">g</Emphasis>-measures in symbolic dynamics

Finitary Markov processes are described in G. Morvai and B. Weiss, Prediction for discrete time series, Probability Theory and Related Fields 132 (2005), 1–12. The transition functions of finitary Markov processes are residually locally constant g-functions that can be extended by continuity to their maximal domain of definition. The study of their associated symbolic dynamics leads one to the D-shifts as introduced in W. Krieger, On g-functions for subshifts, Institute of Mathematical Statistics Lecture Notes-Monograph Series, Vol. 48, Dynamics & Stochastics, arXiv:math.DS/0608259, (2006), 306–316, We study the phenomena that can arise in residually locally constant and locally constant maximally defined g-functions on D-shifts, Markov shifts and synchronizing systems with respect to future measures and g-measures     

Composition-diamond lemma for modules

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra sl ₂, the Verma module over a Kac-Moody algebra, the Verma module over the Lie algebra of coefficients of a free conformal algebra, and a universal enveloping module for a Sabinin algebra. As applications, we also obtain linear bases for the above modules.     

Simultaneous estimation and variable selection in median regression using Lasso-type penalty

We consider the median regression with a LASSO-type penalty term for variable selection. With the fixed number of variables in regression model, a two-stage method is proposed for simultaneous estimation and variable selection where the degree of penalty is adaptively chosen. A Bayesian information criterion type approach is proposed and used to obtain a data-driven procedure which is proved to automatically select asymptotically optimal tuning parameters. It is shown that the resultant estimator achieves the so-called oracle property. The combination of the median regression and LASSO penalty is computationally easy to implement via the standard linear programming. A random perturbation scheme can be made use of to get simple estimator of the standard error. Simulation studies are conducted to assess the finite-sample performance of the proposed method. We illustrate the methodology with a real example.     

Classification of Curves in 2D and 3D via Affine Integral Signatures

We propose new robust classification algorithms for planar and spatial curves subjected to affine transformations. Our motivation comes from the problems in computer image recognition. To each planar or spatial curve, we assign a planar signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures are based on integral invariants, which are significantly less sensitive to small perturbations of curves and noise than classically known differential invariants. Affine invariants are derived in terms of Euclidean invariants. We present two types of signatures: the global and the local signature. Both signatures are independent of curve parameterization. The global signature depends on a choice of the initial point and, therefore, cannot be used for local comparison. The local signature, albeit being slightly more sensitive to noise, is independent of the choice of the initial point and can be used to solve local equivalence problem. An experiment that illustrates robustness of the proposed signatures is presented.     

Energy Decay for the Strongly Damped Nonlinear Beam Equation and Its Applications in Moving Boundary

We study the existence and energy decay of solutions for the strongly damped nonlinear beam equation. We apply a method based on Nakao method to show that the solution decays exponentially, and to obtain precise estimates of the constants in the estimates. Finally, we discuss its applications in moving boundary.