Modelling Ziegler-Natta polymerization in high-pressure reactors

We present a study of some mechanical aspects of the polymerization processes named after the names of the Nobel laureates (1963) Karl W. Ziegler and Giulio Natta. Using the ideas of [1] as a starting point, we formulate a mathematical model encompassing the mechanical behavior of the polymer while it grows around the nanometric catalytic particles. As a basis we use the theory of [5]. The full model is exposed in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

L形和十形截面杆的抗扭刚度

本文是作者前三篇论文的一个继续[1-3].利用调和函数延拓方法,L形和+形截面杆的抗扭刚度很容易求得.数值结果如表1-3所示.     

Inverse dynamics of underactuated multibody systems

The present work deals with controlled mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The governing equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. The rotationless formulation of multibody dynamics will be considered [1]. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [2]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Riemannian exponential and quantization

This article continues and completes the previous one [18]. First of all, we present two methods of quantization associated with a linear connection given on a differentiable manifold, one of them being the one presented in [18]. The two methods allow quantization of functions that come from covariant tensor fields. The equivalence of both is demonstrated as a consequence of a remarkable property of the Riemannian exponential (Theorem 5.1) that, as far as we know, is new to the literature. In addition, we provide a characterization of the Schrödinger operators as the only ones that by quantization correspond to classical mechanical systems. Finally, it is shown that the extension of the above quantization to functions of a very broad type can be carried out by generalizing the method of [18] in terms of fields of distributions.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

A thermodynamical consistent model for modeling of ionic electroactive polymers (EAPs) within the framework of the Theory of Porous Media

Ionic electroactive polymers are widely used in many engineering fields. These kind of materials can be stimulated to change their shape and size, see [1]. Since, the material under consideration has a complex multiphasic microstructure, such multiphasic materials are best described by a continuum mechanical approach. Thus, the presented model is based on the Theory of Porous Media (TPM), cf. [2]. In this contribution, we consider the Ionic Polymer Metal Composites (IPMCs). Stimulating by an electrical voltage, a structural deformation will be caused. Responsible for this deformation are the mobile ions. The focus of the presented model is to capture this material behavior, e.g. the distribution of the mobile cations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear congruence relations for 2-adic L-series at integers

In the paper we find a further generalization of congruences of the K. Hardy and K. S. Williams [5] type which seems to be a full generalization of congruences of G. Gras [4]. Moreover we extend results of [5], [7], [8], [9] and in part of [6]. We apply ideas and methods of [2], [7] and [9].     

Existence and multiplicity of solutions for elliptic problems with indefinite discontinuous nonlinearities

In this paper we study the critical points for a locally Lipschitz functional that in some sense will be solutions of an elliptic problem with indefinite discontinuous nonlinearities. We should mention that our results were inspired by the work of Ambrosetti-Badiale [3], Arcoya-Calahorrano [5], Alama-Tarantello [1] and Chang [8]. For the problem studied in [3] we introduce indefinite nonlinearities as in [1] and [6]. To obtain the existence and multiplicity of solutions we use the critical points theory developed by Chang. Applications for Plasma Physics are considered with nonlinearities that change sign. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

Fujimoto’s theorem - a further study

In the paper we improve Fujimoto’s sufficient condition for a finite set to be a Unique Range Set under relaxed sharing hypothesis. We introduce a new sharing notion which directly improves one result of the paper [3]. In particular, we rectify the Application Part of [3], and extend and improve all results of [3].     

Subdivisions of Toric Complexes

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner’s face ring for abstract simplicial complexes [20] and Stanley’s face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.     

Reelle algebraische Funktionen mit vorgegebenen Null- und Polstellen

This note continues the investigations of Knebusch on algebraic curves over real closed fields and was initiated by reading [3]. Especially we ask for the existence of real algebraic functions with given zeroes and poles, a question going back to Witt [4]. We study the real nature of coverings of real algebraic curves, and if the covering has degree two, we get algebraic proofs for results, which in the classical case have been obtained by topological methods in [2].     

On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1–3, 11, 12]. The issue of their computational complexity has received at-tention in the past, and was raised recently by E. Rassart in [11]. We prove that the problem of computing either quantity is #P-complete. Thus, unless P = NP, which is widely disbelieved, there do not exist efficient algorithms that compute these numbers.     

On the E-unitary covers for a Bruck semigroup

We give a characterization of theE-unitary covers for a Bruck semigroup, which is a generalization of Theorem 3 given in [1]. In a recent paper [1] we gave a characterization of theE-unitary covers for a Bruck semigroupB(T,α) whereT is a finite chain of groups, and now we give a characterization forB(T,α) whereT is any inverse semigroup. We use here the notations and the terminology of Petrich's book [2]. First we prove a Theorem which is more general than [[1], Theorem 2]. I wish to express my thanks to Dr. G. Pollák for his valuable advice.     

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

On General Transport Equations with Abstract Boundary Conditions. The Case of Divergence Free Force Field

This work is a continuation of our previous paper [8]. We investigate well-posedness (in the semigroup theory sense) of transport equations with general external fields and general measures associated to boundary conditions modeled by abstract boundary operators H. Fine properties of the traces are investigated, extending well-known results by M. Cessenat [15]. For dissipative boundary conditions, we revisit and generalize results from [12, 17] while, for multiplicative boundary conditions we extend techniques from [25]. Finally, we also investigate the case of boundary conditions associated to a boundary operator of norm one, extending the recent results of [6, 27] to more general fields and measures.     

<Emphasis Type="Italic">G</Emphasis>-projectable and Λ-projectable binary linear block codes

The binary [24,12,8] Golay code has projection O onto the quaternary [6,3,4] hexacode [9] and the [32,16,8] Reed-Muller code has projection E onto the quaternary self-dual [8,4,4] code [6]. Projection E was extended to projection G in [8]. In this paper we introduce a projection, to be called projection Λ, that covers projections O, E and G. We characterise G-projectable self-dual codes and Λ-projectable codes. Explicit methods for constructing codes having G and Λ projections are given and several so constructed codes that have best known optimal parameters are introduced.