(h, Φ)-entropy differential metric

Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on (h, )-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic distances in testing statistical hypotheses is illustrated by an example within the Pareto family. We obtain the asymptotic distribution of the information matrices associated with the metric when the parameter is replaced by its maximum likelihood estimator. The relation between the information matrices and the Cramér-Rao inequality is also obtained.     

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.     

Infinite loop spaces,and coherence for symmetric monoidal bicategories

This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one.     

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.     

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.     

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.     

Reliability properties under a stopping time shift

Considering a series representation of a coherent system using a shift transform of the components lifetime T_i, at its critical level Y_i, we study two problems. First, under such a shift transform, we analyse the preservation properties of the non-parametric distribution classes and secondly the association preserving property of the components lifetime under such transformations.     

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.     

Avoiding σ-porous sets in Hilbert spaces

We give a constructive proof that any σ  -porous subset of a Hilbert space has Lebesgue measure zero on typical C¹$C^1$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-p   porous sets may meet many C¹$C^1$ curves in positive measure.     

Stability analysis of a complex standby system with constant waiting and different repairman criteria incorporating environmental failure

In this paper we investigate the solution of a complex standby system with constant waiting and different repairman criteria incorporating environmental failure. By using the method of functional analysis, especially, the _C0$C_0$ semigroup theory of bounded linear operators on Banach space, we prove the well-posedness and the existence of positive solution of the system. Under some appropriate hypothesis, we prove that the dynamic solution of the system converges its steady state in a stronger sense.     

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.     

On convergence rates in approximation theory for operator semigroups

We create a new, functional calculus, approach to approximation formulas for _C0$C_0$-semigroups on Banach spaces restricted to the domains of fractional powers of their generators. This approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense. In the case of analytic semigroups, we improve our general results obtaining better convergence rates which are optimal in that case too. The setting of analytic semigroups includes also the case of convergence on the whole space. As an illustration of our approach, we deduce optimal convergence rates in classical approximation formulas for _C0$C_0$-semigroups restricted to the domains of fractional powers of their generators.     

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.     

The existence of multiple equilibrium points in a global attractor for some p-Laplacian equation

In this paper, we are mainly concerned with some properties of the global attractor for some p-Laplacian equation with a Lyapunov function in a Banach space. Under some suitable assumptions, we prove the existence of multiple equilibrium points in a global attractor for some p-Laplacian equation.