On a theorem of Wald and Wolfowitz on randomization in statistics

In [9] it is proved that to each randomized decision function, there exists an equivalent randomized decision rule (cf. [3]), provided that the space of decisions is Polish. The object of the present paper is to show that the above theorem is a special case of risk-equivalence (cf. [5]) even if the space of decisions is analytic.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

New convergence results for continued fractions generated by four-term recurrence relations

In the study of simultaneous rational approximation of functions using rational functions with a common denominator (which can be viewed as the ‘German polynomial’ problem in simultaneous Pade´approximation, cf. [1]) the quest for convergence results lead to the study of generalized continued fractions, a type of Jacobi—Perron algorithm [2,3]. It then becomes important to exploit the connection between the convergence of the generalized continued fraction and the solutions of the associated difference equation (cf. [4,5]).     

Remarks on Steinhaus' property and ratio sets of sets of positive integers

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.     

Gorenstein gr-injective modules over graded isolated singularities

Graded isolated singularities appear very naturally in algebraic pro-jcctive geometry (cf. [18]) or invariant theory of the binary polyhedral groups (cf. [15], [17]). We define mock finitely generated modules in the category of graded modules and we use such graded modules which are also Gorenstein injective (cf. [1], [2]) in this category to characterize graded isolated Gorenstein singularities.     

On characterizations of atomistic lattices

In the present paper we characterize atomistic lattices. These characterizations are given in terms of concepts related to pure elements (cf. [4], [6]) and neat elements (cf. [3]).     

The propagation problem and far-field patterns in a stratified finite-depth ocean

In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves. A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].     

Filtered and stable boolean powers are relativized full boolean powers

The question whether the extended theorem of Feferman and Vaught (cf. Comer [7], Volger [13]) can be applied to filtered boolean powers with arbitrary base was raised by J. H. Schmerl [11]. It can be answered with the help of a relativized version of the theorem (cf. Comer [7], Volger [12], [14]) which seems to have escaped notice. Therefore we shall present in this note a precise formulation of the relativized version and apply it to filtered boolean powers with arbitrary base and to stable boolean powers with finite base and finite group. The former application was mentioned in Comer [7]. The latter application yields a decidability result which seems to be new. The question whether the second application can be extended to arbitrary bases remains open.Presented by E. Nelson.     

A misorientation dependent grain boundary yield criterion

An equivalent plastic strain gradient theory, cf. [1], is extended by a GB flow rule, that accounts for the misorientation of adjacent grains. In contrast to previous works, the grain boundary (GB) resistance against plastic flow is not treated as a constant parameter. Therefore, in the work at hand, the grain boundary dislocation density (GBD), cf. [2], is used as a measure for geometrical mismatch between all glide system orientations of two adjacent grains. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generic warped product submanifolds of locally conformal keahler manifolds

Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [9]). The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen's work on CR-warped product submanifolds of Kaehler manifolds (cf. [6], [7]). Later on, similar studies were carried out in the setting of l.c.K. manifolds and nearly Kaehler manifolds (cf. [3], [11]). In the present article, we investigate a larger class of warped product submanifolds of l.c.K. manifolds, ensure their existence by constructing an example of such manifolds and obtain some important properties of these submanifolds. With regard to the CR-warped product submanifold, a special case of generic warped product submanifolds, we obtain a characterization under which a CR-submanifold is reducesd to a CR-warped product submanifold.     

Robust cutpoints in the logical analysis of numerical data

Techniques for the logical analysis of binary data have successfully been applied to non-binary data which has been ‘binarized’ by means of cutpoints; see Boros et al. (1997, 2000) [7], [8]. In this paper, we analyze the predictive performance of such techniques and, in particular, we derive generalization error bounds that depend on how ‘robust’ the cutpoints are.     

Rings characterized by injectivity classes

Recently, there are a number of well known theorems which char­acterize rings in terms of injective modules (cf. [2], [3], [5] and [8]). The purpose of this paper is to give general theorems which show that some similar results remain true when many other classes of general injective modules are considered in place of injective modules. Our results encompass several well known results by Huynh and Smith, Liu, and Xue. In particular we answer negatively the question of Yue Chi Ming [11] and an example is given to show that E—injective modules form an intermediate class between continuous and quasi-continuous modules(cf. Example 3).     

Different Optimization Models for Crack-Free Laser Welding

Conventional laser beam welding of aluminum alloys often leads to hot cracking. This is caused by a complex process where thermo-mechanical and metallurgical aspects are involved; cf. [3], [2]. A possibility to prevent hot crack initiation yields the multi-beam welding technique (cf. [2]), where additional laser beams are led parallelly besides the main laser beam. There by optimal positions, sizes, and powers of the additional laser beams play an important role otherwise hot cracking can even be enhanced. In [1], [4], resp., a mechanical 1D and thermal 2D model of hot cracking was derived. It provides the basis for different formulations of constrained nonlinear programming problems to identify the optimal parameters of the additional laser beams. In the present paper a comparison between these formulations and between two different optimizers for the so far best formulation are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Théorie tannakienne non commutative

Inspired by a recent paper by Deligne [2], we extend the Tannaka-Krein duility results (over a field) to the non-commutative situation. To be precise, we establish a 1-1 corresponde:ice between ‘tensorial autonomous categories’ equipped with a ‘fibre functor’ (i. e. tannakian categories without the commutativity condition on the tensor product), and ‘quantum groupoids’ (as defined by Maltsiniotis, [9]) which are ‘transitive’ (7.1.). When the base field is perfect, a quantum groupoid over Spec B is transitive iff it is projective and faithfully fiat over B? _k B. Moreover, the fibre functor is unique up to ‘quantum isomorphism’ (7.6.). Actually, we show Tannaka-Krein duality results in the more general setting where there is no monoidal structure on the category (and the functor); the algebraic object corresponding to such a category is a ‘semi-transitive’ coalgebroid (5.2. and 5.8.).     

Total absolute curvature of polyhedral manifolds with boundary in E n

N. H. Kuiper has generalized the notion of total absolute curvature for compact polyhedra in euclidean space by considering the critical points of all height functions (cf. [12]). On the other hand in the case of compact smooth manifolds with boundary in E ⁿ there is a certain relation between the total absolute curvatures of the total space, the interior and the boundary (cf. [9]). In this note we show an analogous relation in the case of compact polyhedral manifolds with boundary leading to theorems of the Chern/Lashof type (cf. [3], [7]).     

Stochastic analysis of A 1-server n-unit system subject to inspection and several failure modes

The main intent of the paper is to investigate the stochastic behaviour of a single-server n-unit system subject to random inspection and several failure modes. The time between successive inspections is a random variable distributed exponentially. It is assumed that the life-time of a unit is arbitrarily distributed while all the other time distributicns involved are exponential. At any instant t, the system is characterized by the probabilities of its beinq in the ‘up’ or ‘down’ state . Integral equations are established for these probabilities by identifying the system at suitable regenerative epochs corresponding to different initial conditions. Various system parameters of significant importance, namely,

1. point-wise availability of the system at instant t,

2. steady-state availability of the system,

3. s-expected up-time of the system in [o, t],

4. s-expected inspection time of the server in [o, t],

5. s-expected repair time of type i (1 ≤ i ≤ r) in [o, t] and

6. s-expected net gain per unit time in [o, t], have been obtained.     

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood of a singular point

We study families of holomorphic vector fields, holomorphically depending on parameters, in a neighborhood of an isolated singular point. When the singular point is in the Poincaré domain for every vector field of the family we prove, through a modification of classical Sternberg's linearization argument, cf. Nelson (1969) [7] too, analytic dependence on parameters of the linearizing maps and geometric bounds on the linearization domain: each vector field of the family is linearizable inside the smallest Euclidean sphere which is not transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko (2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6], a version of Brjuno's Theorem in the case of linearization of families of vector fields near a singular point of Siegel type, and apply it to study some 1-parameter families of vector fields in two dimensions.     

Numerical Analysis of the Stochastic Moving Boundary Problem

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).     

A simple characterization of almost uniform convergence by stochastic convergence

Motivated by Egorov's theorem and the characterization of the equivalence of P-stochastic convergence and P-almost convergence by the property of the probability distribution P to be purely atomic and concentrated on a countable number of pairwise disjoint P-atoms (cf. [1], p. 68), it is proved that P-stochastic resp. P-almost convergence is equivalent to P-almost uniform convergence (cf. [2], p. 89/90) if and only if P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms. Furthermore, this property of P is equivalent to the condition that any P-stochastic convergent sequence admits a P-almost uniform convergent subsequence. Finally a proof is given that P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms if and only if there does not exist a purely finitely additive {0,1}-valued probability charge, which vanishes for all P-zero sets.