Homogenization in fractional elasticity – One spatial dimension

In this note we treat the equations of fractional elasticity in one spatial dimension. After establishing well-posedness, we use an abstract result in the theory of homogenization to derive effective equations in fractional elasticity with highly oscillating coefficients. The approach also permits the consideration of non-local operators (in time and space). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

Asymptotic expansions in limit theorems for stochastic processes. – III

For families of processes with independent increments (t), 0tT, with frequent small jumps, limit theorems for expectations of the functionals F([0, T]) are proved of the form where _diD, are positive numbers, A _di are linear integro-differential or differential operators acting on functionals, and some differentiability conditions are imposed on the functional F. The case of power tails of the jump distribution is considered.Mathematics Subject Classification (2000):60F17     

What does it mean for a student to understand the first-year calculus? Perspectives of 24 experts

This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of the first-year calculus, (b) construction of connections and relationships between and among concepts and skills, (c) the ability to use the ideas of the first-year calculus, and (d) a deep sense of the context and purpose of the calculus. Each end goal for student understanding is explored in detail and the potential for using the framework as an organizational tool is discussed.     

A Hybrid Micro­Macro­Model for the Description of Evolving Anisotropy in Finite Polycrystal Plasticity

The paper outlines recent developments of an efficient computational micro-macro modeling of evolving anisotropies in metallic polycrystals. Main focus is put onto large strain deformation processes where the anisotropy is caused by developments of crystallographic texture. We construct a hybrid micro-macro framework that mixes ingredients of a purely macroscopic modeling with scale bridging operations of selected micromechanisms. On the micromechanical side, we develop a new algorithmic procedure to capture the crystal reorientation for evolving fcc and bcc textures based on a parametrization of rotations in the Rodigues space. The computational model provides a fast and robust method for the estimation of evolving textures. This computational tool for texture estimation is incorporated in a modular format into a micro-macro-model, where it governs the evolution of macrostructural tensors due to texture development. The general framework for the hybrid embedding is a purely phenomenological setting of anisotropic finite plasticity in the logarithmic strain space. The model provides an efficient and computationally handable two-scale approach for the prediction of effects caused by complex microstructural changes in polycrystals. The capability of the proposed method is demonstrated by means of representative numerical examples. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

HOPS – Hamming-Oriented Partition Search for production planning in the spinning industry

In this paper, we investigate a two-stage lot-sizing and scheduling problem in a spinning industry. A new hybrid method called HOPS (Hamming-Oriented Partition Search), which is a branch-and-bound based procedure that incorporates a fix-and-optimize improvement method is proposed to solve the problem. An innovative partition choice for the fix-and-optimize is developed. The computational tests with generated instances based on real data show that HOPS is a good alternative for solving mixed integer problems with recognized partitions such as the lot-sizing and scheduling problem.     

Asymptotic sampling – a tool for efficient reliability computation in high dimensions

Monte Carlo methods are most versatile regarding applications to the reliability analysis of high-dimensional nonlinear structural systems. In addition to its versatility, the computational efficacy of Monte Carlo method is not adversely affected by the dimensionality of the problem. Crude Monte Carlo techniques, however, are very inefficient for extremely small failure probabilities such as typically required for sensitive structural systems. Therefore methods to increase the efficacy for small failure probability while keeping the adverse influence of dimensionality small are desirable. On such method is the asymptotic sampling method. Within the framework of this method, well-known asymptotic properties of the reliability index regarding the scaling of the basic variables are exploited to construct a regression model which allows to determine the reliability index for extremely small failure probabilities with high precision using a moderate number of Monte Carlo samples. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

NOTE – A Min-Max Relation for Monotone Path Systems in Simple Regions

xy -plane which bounds the simple polygonal (closed) region D. Let T and B be two finite, disjoint, equicardinal sets of points of D. We give a min-max relation for the maximum number of points of T and B which can be joined by a MPS in D, and a polytime algorithm for finding such a MPS. Received July 15, 1996/Revised October 7, 1999     

A derivative algorithm for inexact quadratic program – application to environmental decision-making under uncertainty

Inexact quadratic programming (IQP) is an extension of conventional quadratic programming for handling both nonlinearities in cost objectives and uncertainties with modeling parameters. It has been a useful tool for environmental systems analysis. However, inefficiency in its solution method has existed, leading to difficulties in its practical application. In this study, a derivative algorithm (DAM) is proposed for solving the IQP. It improves upon the existing method through provision of a quantitative expression for uncertain relationships between the quadratic objective function and the decision variables. The DAM requires much lower computational efforts than the existing algorithm, which is especially meaningful for the IQP's application to large-scale problems. The developed DAM is applied to a hypothetical problem of municipal solid waste management and planning. Detailed solution steps are provided to clearly demonstrate the method's advantages.     

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

Multisection in Interval Branch-and-Bound Methods for Global Optimization – I. Theoretical Results

We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The convergence properties of the multisplitting methods, an important class of multisection procedures are investigated in detail. We also studied theoretically the convergence improvements caused by multisection on algorithms which involve the accelerating tests (like e.g. the monotonicity test). The results are published in two papers, the second one contains the numerical test result.     

Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations

We describe conditions on non-gradient drift diffusion Fokker–Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.     

Regularity theory for the generalized Neumann problem for Yang-Mills Connections – Non-trivial examples in dimensions 3 and 4

We develop the existence and regularity theory for the generalized Neumann problem for Yang-Mills connections. This is the most general boundary value problem for connections on a compact manifold with smooth boundary, with geometric meaning. It is obtained by reflecting the base manifold across its boundary and lifting this action non-trivially to the bundle. The prescribed lifting corresponds to a geometric invariant, which is similar to the monopole number. When this invariant is non-zero, there exist non-trivial solutions of the generalized Neumann problem. We prove the existence of non-trivial solutions over the 3-dimensional disk and over the 4-dimensional manifold We outline the procedure for finding non-trivial examples of solutions over more general manifolds of dimension 3 and 4. Received: 20 June 1999     

Residual saturated porous media – from oscillating water blobs to waves in ground earth

A model for wave propagation in residual saturated porous media is presented distinguishing enclosed fluid clusters with respect to their eigenfrequency and damping properties. The additional micro-structure information of cluster specific damping is preserved during the formal upscaling process and allows a stronger coupling between micro- and macro-scale than characterisation via eigenfrequencies alone. A numerical example of sandstone filled with air and liquid clusters of different eigenfrequency and damping distributions is given. If energy dissipation due to viscous damping dominates energy storage due to cluster oscillations, the damping distribution is more influential than the eigenfrequency distribution and vice versa. Spreading the damping distribution around a constant mean value supported the effect of capillary forces and spreading the eigenfrequency distribution around a constant mean value supported the effect of viscous damping in the investigated samples. For a wide distribution of the liquid clusters' damping properties, two damping mechanisms of propagating waves occur at the same time: damping due to viscous effects (for highly damped clusters) and energy storage by cluster oscillations (for underdamped clusters). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extreme value theory for spatial random fields – with application to a Lévy-driven field

Extremes - First, we consider a stationary random field indexed by an increasing sequence of subsets of $mathbb {Z}^{d}$ . Under certain mixing and anti–clustering conditions combined with a...     

Analytic Wave Front Set for Solutions to Schrödinger Equations II – Long Range Perturbations

This paper is a continuation of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]], where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]] to long-range perturbations (in particular, we can allow potentials growing like ?x?^2?? at infinity). More precisely, we construct a modified quantum free evolution G ₀(?s, hD _z ) acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution e ^?itH u ₀ of the Schrödinger equation, in terms of the semiclassical exponential decay of G ₀(?th ^?1, hD _z )T u ₀, where T stands for the Bargmann-transform. The result is valid for t < 0 near the forward non trapping points, and for t > 0 near the backward non trapping points. It is an extension of [12 Nakamura , S. ( 2009 ). Semiclassical singularities propagation properties for the Schrödinger equations . J. Math. Soc. Japan 61 : 177 – 211 . [Google Scholar]] to the analytic framework.     

Lower bounds in cones for solutions to the schrödinger equation

We investigate the asymptotic behavior of square integrable solutions Ψ of the Schrödinger equation (?Δ+V)Ψ=EΨ,E<0, in an open cone Ω?R ⁿ . WhenV is a real function tending to zero at infinity in some specified sense, and when, in addition, the opening angle of Ω is greater than π/2, we prove precise lower bounds for Ψ in certain directions. In particluar, if Ω=R ⁿ andU?S ^n?1 is any nonempty open set, we show \( - \mathop {\lim }\limits_{r \to \infty } r^{ - 1} \ln \left( {\int\limits_U {\left| {\Psi \left( {r\omega } \right)} \right|^2 d\omega } } \right)^{1/2} = \sqrt { - E.} \)      

A constitutive model for the flow stress behavior and microstructure evolution in aluminum alloys under hot working conditions – with application to AA6099

A constitutive model for aluminum alloys under hot working conditions is proposed. The elastic-viscoplastic model is implemented in a finite strain continuum mechanical framework. The model accounts for the interplay between dynamic recovery and recrystallization during hot working of aluminum alloys and central aspects of microstructure evolution such as grain/subgrain size and dislocation density. The proposed model is generic in the sense that it can be used for arbitrary aluminum alloys, but in order to demonstrate its capabilities, the model is calibrated to a newly developed AA6099 alloy in the present study. The model is thoroughly discussed and details on the numerical implementation as well as on the calibration of the model against experimental data are provided.     

Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length

The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε ^q (ε is a small positive parameter, q = const > 0) and radius a _ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ²: x ₁ = 0} and their number is equal to N _ɛ= O(ɛ⁻¹) as ε → 0. Under the limit condition lim on the parameters characterizing the geometry of the domain, the weak H ¹-limit of the generalized solution of this problem is found. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005.