Bishop–Phelps–Bollobás property for certain spaces of operators

We characterize the Banach spaces Y   for which certain subspaces of operators from _L1(μ)$L_1(\mu )$ into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.     

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.     

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.     

Solutions to degenerate complex Hessian equations

Let (X,ω)$(X,\omega )$ be an n-dimensional compact Kähler manifold and fix an integer m   such that 1?m?n$1?m?n$. We study degenerate complex Hessian equations of the form (ω+dd^cφ)^m∧ω^n−m=F(x,φ)ωⁿ${(\omega +d{d}^c\phi )}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$. Under some natural conditions on F, this equation has a unique continuous solution. When X is homogeneous and ω is invariant under the Lie group action, we further show that the solution is Hölder continuous.     

A perishable inventory system with service facility and finite source

In this article, we consider a continuous review (s,S)$(s\text{,}S)$ perishable inventory system with a service facility, wherein the demand of a customer is satisfied only after performing some service on the item which is assumed to be of random duration. We also assume that the demands are generated by a finite homogeneous population. The service time, the lead time are assumed to have Phase type distribution. The life time of the item is assumed to have exponential distributions. The joint distribution of the number of customers in the system and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time of the tagged customer is derived. Various system performance measures are derived and the total expected cost rate is computed under a suitable cost structure. The results are illustrated numerically.     

Global attraction to solitary waves for Klein–Gordon equation with mean field interaction

We consider a U(1)$\mathbf{U}(1)$-invariant nonlinear Klein–Gordon equation in dimension n?1$n?1$, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as t→±∞$t\to \pm \infty$ to the two-dimensional set of all “nonlinear eigenfunctions” of the form ?(x)e−iωt$?(x)e^{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.     

Labeling constructions using digraph products

In this paper we study the edge-magicness of graphs with equal size and order, and we use such graphs and digraph products in order to construct labelings of different classes and of different graphs. We also study super edge-magic labelings of 2$2$-regular graphs with exactly two components and their implications to other labelings.     

Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak^?${\mathrm{weak}}^{?}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L  -weakly compact sets coincide. In particular, in terms of almost Dunford–Pettis operators into _c0$c_0$, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E   is almost limited if and only if every continuous linear operator T:E→_c0$T:E\to c_0$ is an almost Dunford–Pettis operator.     

Generalized Zernike or disc polynomials: An application in quantum optics

An application of the “generalized Zernike or disc polynomials”, recently introduced in the literature, is shown, resorting to the Lie algebra based investigation of the dynamics of quantum systems driven by two-mode interaction Hamiltonians. Further properties of the associated “disc functions” are deduced. Also, a generalization of the disc polynomials towards the Hahn polynomials is suggested.     

Spaces of holomorphic functions on non-balanced domains

This paper studies the coincidence of the _τω${\tau }_{\omega }$ and _τδ${\tau }_{\delta }$ topologies on the space of holomorphic functions defined on an open subset U   of a Banach space. Dineen and Mujica proved that _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ when U   is a balanced open subset of a separable Banach space with the bounded approximation property. Here, we study the _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ problem for several types of non-balanced domains U.     

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley–Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation.