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.     

Self-affine sets with positive Lebesgue measure

Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β$\beta$-expansion’ of different numbers in different bases.     

A proof for Antczak’s mean value theorem in invexity analysis

This paper provides another proof for Antczak’s mean value theorem in invexity analysis.     

A variant smoothing Newton method for P0-NCP based on a new smoothing function

In this paper, we present a new one-step smoothing Newton method proposed for solving the non-linear complementarity problem with _P0$P_0$-function based on a new smoothing NCP$NCP$-function. We adopt a variant merit function. Our algorithm needs only to solve one linear system of equations and perform one line search per iteration. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of _P0-NCP$P_0\text{-}NCP$. Furthermore, under the assumption that the solution set is non-empty and bounded, we can guarantee at least one accumulation point of the generated sequence. Numerical experiments show the feasibility and efficiency of the algorithm.     

Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

Rao’s statistic for constant and proportional hazard models

In this paper we introduce a new family of measures of divergence for the analysis of the degree of departure from a model with a constant hazard function and also for comparing if two models have proportional hazard rates. Our family of measures is based on the family of divergences introduced by Burbea and Rao (see [J. Burbea, C.R. Rao, On the convexity of higher order Jensen differences based on entropy functions, IEEE Transactions on Information Theory 28 (1982) 961–963]). Some well-known sets of data are reanalyzed using the new families of test statistics and confidence intervals introduced in this paper.     

A Bohr–Nikol'skii inequality

In this paper, we give a new inequality called Bohr–Nikol'skii inequality which combines the inequality of Bohr–Favard and the Nikol'skii idea of inequality for functions in different metrics.     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

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=-∞$t=-\infty$.     

Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

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.     

Some new inequalities in geometry and analysis

The paper announces some new inequalities that refer to broken lines, curves and real and complex functions. Their derivation is based on a new principle of angles and lengths for curves. The inequality in the complex analysis, called the principle of derivatives, is valid for analytic functions in arbitrary domains and extends to a broad class of sufficiently smooth complex functions. A new inequality follows for real functions of two variables concerning their level sets. For all cases mentioned above, both for curves and functions, we obtain some analogues of the second fundamental theorem in Nevanlinna theory of meromorphic functions. At the end we discuss a new point-domain inequality dealing with finite point sets in an arbitrary domain.     

Capability adjustment for gamma processes with mean shift consideration in implementing Six Sigma program

In the 1980s, Motorola, Inc. introduced its Six Sigma quality program to the world. Some quality practitioners questioned why the Six Sigma advocates claim it is necessary to add a 1.5σ shift to the process mean when estimating process capability. Bothe [Bothe, D.R., 2002. Statistical reason for the 1.5σ shift. Quality Engineering 14 (3), 479–487] provided a statistical reason for considering such a shift in the process mean for normal processes. In this paper, we consider gamma processes which cover a wide class of applications. For fixed sample size n, the detection power of the control chart can be computed. For small process mean shifts, it is beyond the control chart detection power, which results in overestimating process capability. To resolve the problem, we first examine Bothe’s approach and find the detection power is less than 0.5 when data comes from gamma distribution, showing that Bothe’s adjustments are inadequate when we have gamma processes. We then calculate adjustments under various sample sizes n and gamma parameter N, with power fixed to 0.5. At the end, we adjust the formula of process capability to accommodate those shifts which can not be detected. Consequently, our adjustments provide much more accurate capability calculation for gamma processes. For illustration purpose, an application example is presented.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

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.     

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 converse comparison theorem for anticipated BSDEs and related non-linear expectations

The converse comparison theorem has received much attention in the theory of backward stochastic differential equations (BSDEs). However, no such theorem has been proved for anticipated BSDEs. In this paper, we derive a converse comparison theorem by first giving an existence and uniqueness theorem for adapted solutions of anticipated BSDEs with a stopping time and then related to (f,δ)$(f,\delta )$-expectations induced by anticipated BSDEs.