Necessary and sufficient condition for uniqueness of solution to the first boundary value problem for the diffusion equation in unbounded domains

This paper introduces a notion of regularity of t=-∞$t=-\infty$ for the diffusion (or heat) equation and establishes a necessary and sufficient condition for the existence of a unique bounded solution to the first boundary value problem for the diffusion equation in a general domain Ω⊂R^N+1$\Omega \subset {\mathbb{R}}^{N+1}$ which extends up to t=-∞$t=-\infty$.     

Convexity and the difference property

We show that the class of all delta-convex selfmappings of R$\mathbb{R}$ (differences of two convex functions) enjoys the difference property in the sense of N.G. de Bruijn. The Q$\mathbb{Q}$-differentiability technique has been applied as a proof tool.     

Second-order multiobjective symmetric duality with cone constraints

In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under η$\eta$-bonvexity/η$\eta$-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric.     

Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

A novel method for mathematical modeling of a form-cutting-tool of the optimum design

A method that pertains to design of the optimal form-cutting-tool for machining of a given sculptured surface on a multi-axis NC   machine is discussed in the paper. The results reported in the paper are based in much on the author’s previous work in the field of R$\mathbb{R}$-mapping of a sculptured surface onto the machining surface of cutting tools. Mathematical foundations of a novel method of experimental modeling of the interaction of the form-cutting-tool and the work are disclosed. The last method is of critical importance for the experimental determination of the rate of conformity functions, those essential for consequent use of R$\mathbb{R}$-mapping of surfaces. The presented geometric criteria is helpful for designers since the maximal rate of conformity of the generating surface of the form-cutting-tool to the sculptured surface is a prerequisite for the development of extremely efficient machining operations, and in solving design problems. The results of the research reported in this paper can be considered as a portion of the DG/K-method of surface generation on a multi-axis NC machine earlier developed ¹ by the author. The method is based on the extensive use of reliable results worked out in classical differential geometry of surfaces. Topics covered in the paper enables one designing the form-cutting-tool for optimal machining of a given sculptured surface on a multi-axis NC machine. The usefulness of the approach is verified from two simple examples that are clear and easy for understanding.     

On symmetric and locally finite actions of groups on the quintic tree

In this paper, we determine all faithful, symmetric and locally finite actions of a group on the tree of valency five. As a corollary we complete the classification of the isomorphism types of vertex and edge stabilisers in a group acting symmetrically on a graph of valency five. This builds on work of Weiss and recent work of Feng, Zhou and Feng, Guo. Our approach is to classify the isomorphism types of finite, faithful amalgams of degree (5, 2).     

A combined scalarizing method for multiobjective programming problems

In this paper, a new general scalarization technique for solving multiobjective optimization problems is presented. After studying the properties of this formulation, two problems as special cases of this general formula are considered. It is shown that some well-known methods such as the weighted sum method, the ?$?$-constraint method, the Benson method, the hybrid method and the elastic ?$?$-constraint method can be subsumed under these two problems. Then, considering approximate solutions, some relationships between ε$\epsilon$-(weakly, properly) efficient points of a general (without any convexity assumption) multiobjective optimization problem and ?$?$-optimal solutions of the introduced scalarized problem are achieved.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.     

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space X$X$, our paper studies elements of vector analysis, _Lp$L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on X$X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.