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.     

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.     

Nonlinear maps preserving Jordan *-products

Let η$\eta$ be a non-zero scalar. In this paper, we investigate a bijective map ?$?$ between two von Neumann algebras, one of which has no central abelian projections, satisfying ?(AB+ηBA^∗)=?(A)?(B)+η?(B)?(A)^∗$?(AB+\eta B{A}^{\ast })=?\left(A\right)?\left(B\right)+\eta ?\left(B\right)?{\left(A\right)}^{\ast }$ for all A,B$A,B$ in the domain. It is showed that ?$?$ is a linear *-isomorphism if η$\eta$ is not real and ?$?$ is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η$\eta$ is real.     

A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k  -graph C^?$C^{?}$-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k  -graph C^?$C^{?}$-algebra, it is sufficient that the representation be injective on a distinguished abelian C^?$C^{?}$-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C^?$C^{?}$-algebra, each of which is the unique extension of a state on the distinguished abelian C^?$C^{?}$-subalgebra.     

p-embeddings

This is the sequel to Bhattacharjee et al. (in press) [3] where the notion of a p  -extension of commutative rings was investigated: a unital extension of commutative rings, say R?S$R?S$, is a p  -extension if for every s∈S$s\in S$ there is an r∈R$r\in R$ such that rS=sS$rS=sS$. In this article we apply the theory of p  -extensions to rings of continuous functions. We show that this concept lays between the concepts of C^?$C^{?}$-embeddings and z-embeddings.     

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.     

Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around _L2$L_2$-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the number of runs n$n$ is smaller than the number of all level-combinations m$m$, the construction problem can be transferred to a zero–one quadratic integer programming problem, and an efficient algorithm based on the simulated annealing is proposed. When n≥m$n\ge m$, another algorithm is proposed. Empirical study shows that when n$n$ is large, the proposed algorithms can generate designs with lower WD compared to many existing methods. Moreover, these algorithms are suitable for constructing both symmetrical and asymmetrical designs.     

Smale’s point estimate theory for Newton’s method on Lie groups

We introduce the notion of the (one-parameter subgroup) γ$\gamma$-condition for a map f$f$ from a Lie group to its Lie algebra and establish α$\alpha$-theory and γ$\gamma$-theory for Newton’s method for a map f$f$ satisfying this condition. Applications to analytic maps are provided, and Smale’s α$\alpha$-theory and γ$\gamma$-theory are extended and developed. Examples arising from initial value problems on Lie group are presented to illustrate applications of our results.     

Some decomposition results for a class of vacation queues

We analyze the MAP/PH/1 vacation system at arbitrary times using the matrix-analytic method, and obtain decomposition results for the R$R$ and G$G$ matrices. The decomposition results reduce the amount of computational effort needed to obtain these matrices. The results for the G$G$ matrix are extended to the BMAP/PH/1 system. We also show that in the case of the Geo/PH/1 and M/PH/1 systems with PH vacations both the G$G$ and R$R$ matrices can be obtained explicitly.     

Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities

For Ω$\Omega$, an open bounded subset of R^N${\mathbb{R}}^N$ with smooth boundary and 1<p<∞$1<p<\infty$, we establish W^1,p(Ω)$W^{1,p}\left(\Omega \right)$a priori bounds and prove the compactness of solution sets to differential inequalities of the form

|divA(x,∇u)|≤F(x,u,∇u),$\left|\text{div}A(x,\nabla u)\right|\le F(x,u,\nabla u)\text{,}$

which are bounded in L^∞(Ω)$L^{\infty }\left(\Omega \right)$. The main point in this work is that the nonlinear term F$F$ may depend on ∇u$\nabla u$ and may grow as fast as a power of order p$p$ in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.