Study on statistically similar RVEs for real microstructures based on different statistical descriptors

A method for constructing statistically similar representative volume elements (SSRVEs) for a real dual-phase (DP) steel microstructure is presented in this contribution. The advantageous material properties of such kind of steels originate from the interaction of the microstructure constituents of the material on the microscale. In order to capture these effects directly in the material modeling, the FE² method is a suitable tool, where an RVE representing the microstructure of a material is used in the microscopic boundary value problem, which is solved at each Gauss-point of the macroscopic boundary value problem. However, RVEs based on real microstructures typically implicate high computational expenses due to the complexity of the underlying microstructure and its discretization. SSRVEs, which have a lower complexity than conventional RVEs but are still able to represent the material, can be used instead. Here, different statistical measures for the construction of SSRVEs and their comparison are the main focus. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some algorithms for the dynamic reconstruction of inputs

For the weight \(v_k \left( x \right) = \prod _{\alpha \in \mathbb{R}_ + } \left| {\left( {\alpha ,x} \right)} \right|^{2k\left( \alpha \right)}\) defined by a positive subsystem R ₊ of a finite root system R ? ? ^d and by a function k(α): R → ?₊ invariant under the reflection group generated by R, a sharp Jackson inequality in L ₂(? ^d ) is proved.     

Some basic principles for linear coupled dynamic thermopiezoelectricity

According to the basic idea of classical yin-yang complementarity and modern dualcomplementarity, in a simple and unified way some basic principles for linear coupled dynamic thermopiezoelectricity can be established systematically. An important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in linear coupled dynamic thermopiezoelectricity, but also to derive systematically the complementary functionals for eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. Project supported by the National Natural Science Foundation of China (Grant No. 19672074) and Research Grand Council of Hong Kong, No.RGC97/98, HKUST 6055/97E.     

Some further ideas concerning the interaction between insurance and investment risks

For many years it has been a frequently discussed question which is more important: insurance or investment risk. Based on Bühlmann’s (1995) method to separate these two risks with the help of conditional expectations, this paper presents a decomposition of the prospective portfolio loss into a sum of three addends that uniquely correspond to unsystematic insurance risk, systematic insurance risk, and investment risk. Calculating their variances for homogeneous portfolios of term life and pure endowment insurances shows that answering the initial question is more complex than frequently thought. In a second step, an extended duration concept is introduced, which allows one to analyze the non-diversifiable investment and systematic insurance in view of parameter changes at certain points in time.     

Some results on the spectral reconstruction problem

This paper is concerned with the spectral version of the reconstruction conjecture: Whether a graph with n>2 vertices is determined (up to isomorphism) by the collection of its spectrum and the spectrum of its vertex-deleted graphs? Some positive results as well as a method for constructing counterexamples to the problem are provided.     

Some properties of basic families of subsets

A basic system is a nonempty collection of finite incomparable subsets of a set such that for any two subsets or bases in the collection, any element of one basis can be replaced by some element of the other to give another basis in the collection. In a basic system, any subset of one basis can be bijectively exchanged for distinct elements of another; for a finite set, basis complements also have these properties; and certain conditions will guarantee that two such systems on the same set will contain a common basis. All proofs are new, elementary, and set-theoretic. In addition, they suggest efficient algorithmic procedures whose efficiencies are calculated.     

Some new basic results for singularly perturbed ordinary differential equations

Asymptotic results are obtained for an initial-value problem for singularly perturbed systems. Existence of bounded solutions to singularly perturbed systems is deduced from the results of a previous paper [9]. These results significantly enlarge the class of limiting asymptotic solutions of singularly perturbed systems inasmuch as the limiting solutions satisfy equations more general than the classical reduced system. These results generalize those of Tikhonov [3] for the initial value problem, Flatto and Levinson [6] for the existence of periodic solutions and Hale and Seifert [7] for the existence of almost-periodic solutions.     

Some basic properties of multiple Hamiltonian covers

The aim of this work is to introduce the concept of a multiple Hamiltonian cover(MHC). For the most part, attention is restricted to the class of cubic three-connected planar graphs. For those graphs having an MHC composed of three Hamiltonian cycles we are able to derive a Grinberg type result. On the other hand, for those graphs having an MHC consisting of six Hamiltonian cycles we find it convenient to impose the additional notion of balance, which then allows us to deduce some interesting consequences. We conclude with a problem from three-dimensional geometry. MHC's play a significant role in its solution.     

Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.     

Some basic observations on Kelly's conjecture for graphs

P.J. Kelly first mentioned the possibility of determining a graph from subgraphs obtained by deleting several points. While such problems have received a great deal of attention in the case of deletions of single points, the problem for several points is virtually untouched. This paper contains some basic results on that problem, including the negative observation that for every k, there exist two non-isomorphic graphs with the same collection of k-point subgraphs.     

Some fundamental issues of basic line search algorithm for linear programming problems

In this article we present the fundamental idea, concepts and theorems of a basic line search algorithm for solving linear programming problems which can be regarded as an extension of the simplex method. However, unlike the iteration of the simplex method from a basic point to an improved adjacent basic point via pivot operation, the basic line search algorithm, also by pivot operation, moves from a basic line which contains two basic feasible points to an improved basic line which also contains two basic feasible points whose objective values are no worse than that of the two basic feasible points on the previous basic line. The basic line search algorithm may skip some adjacent vertices so that it converges to an optimal solution faster than the simplex method. For example, for a 2-dimensional problem, the basic line search algorithm can find an optimal solution with only one iteration.     

Some basic properties of packing and covering constants

This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collectionK of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies fromK are used repeatedly, or one may require that these bodies should be used at most once. In each case one can define the packing and covering constants ofK as, respectively, the least upper bound and the greatest lower bound of the densities of all such packings and coverings. Three theorems are proved. First it is shown that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants. Then, a quantitative continuity theorem is proved which shows in particular that the packing and covering constants depend, in a certain sense, continuously onK. Finally, a kind of a transference theorem is proved, which enables one to evaluate the packing and covering constants when no repetitions are allowed from the case when repetitions are permitted. Furthermore, various consequences of these theorems are discussed.Supported by National Science Foundation Research Grant DMS 8300825.     

Some basic boundary value problems of the plane thermoelasticity with microtemperatures

The present paper is devoted to the two‐dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2? mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite-element heterogeneous multiscale method for the Helmholtz equation

We show that the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions to the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well-known results about FE-HMM and the notion of T-coercivity, we derive an a priori error bound. Numerical experiments corroborate the analytical findings.     

Some exact results basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

Six exact solutions (related to the conservation of number, energy and momentum) of the linearized Boltzmann equations for a binary mixture of rigid spheres, for the case of isotropic scattering in the center-of-mass system, are reported. The verification of the reported exact solutions (collisional invariants) is based on a recently reported explicit formulation of the linearized Boltzmann equation for a binary mixture of rigid spheres. Elementary analysis is used also to establish a basic flow condition.     

Some exact results basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

Six exact solutions (related to the conservation of number, energy and momentum) of the linearized Boltzmann equations for a binary mixture of rigid spheres, for the case of isotropic scattering in the center-of-mass system, are reported. The verification of the reported exact solutions (collisional invariants) is based on a recently reported explicit formulation of the linearized Boltzmann equation for a binary mixture of rigid spheres. Elementary analysis is used also to establish a basic flow condition.     

Analysis of FETI methods for multiscale PDEs

In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))² where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if varies only mildly in a layer Ω _i,η of width η near the boundary of each of the subdomains Ω _i , then , independent of the variation of α in the remainder Ω _i \Ω _i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests. C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306.