Pathwidth of outerplanar graphs

We are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin [3], after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a constant c such that the pathwidth of every biconnected outerplanar graph is at most c plus the pathwidth of its dual. They also conjectured that this was actually true with c being one for every biconnected planar graph. Fomin [10] proved that the second conjecture is true for all planar triangulations. First, we construct for each p ≥ 1, a biconnected outerplanar graph of pathwidth 2p + 1 whose (geometric) dual has pathwidth p + 1, thereby disproving both conjectures. Next, we also disprove two other conjectures (one of Bodlaender and Fomin [3], implied by one of Fomin [10]. Finally we prove, in an algorithmic way, that the pathwidth of every biconnected outerplanar graph is at most twice the pathwidth of its (geometric) dual minus one. A tight interval for the studied relation is therefore obtained, and we show that all cases in the interval happen. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 27–41, 2007     

A Large-Deformation Phase-Field Approach for the Modeling of Magneto-Sensitive Elastomers

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magneto-mechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1, 7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Geometrically Exact Incremental Variational Formulation for Phase Field Models in Micromagnetics

Magnetic materials have been finding increasingly wide areas of application. We focus here on the continuum modeling of such materials and present an incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetically as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. Furthermore, the algorithmic preservation of the geometrical nature of the variables is an important challenge from the numerical perspective and to this end we present a novel FE discretization whereby the geometric property of the magnetization director is pointwise exactly preserved by nonlinear rotational updates at the nodes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet     

The geometry of Whitney’s critical sets

In this paper, the complete geometric characterizations, including decomposition and compression theorems, are obtained for a connected and compact set to be a critical set in Whitney’s sense, i.e., a set such that there exists a differentiable function critical but not constant on it. The problem to characterize these sets geometrically was posed by H. Whitney [21] in 1935. We also provide a complete geometrical characterization for monotone Whitney arc, i.e., there exists a differentiable function critical but also increasing along the arc. All examples appearing in the literature are monotone Whitney arcs, for example, the examples by Whitney [21] and Besicovitch [2], Norton’s t-quasi-arcs with Hausdorff dimension > t [14], and self-similar arcs [19]. Furthermore, after introducing the notion of homogeneous Moran arc, we can completely characterize all the monotone Whitney arcs of criticality > 1, which include t-quasi arcs and self-conformal arcs. Some applications to arcs which are attractors of Iterated Function Systems are discussed, including self-conformal arcs, self-similar arcs and self-affine arcs. Finally, we give an example of critical arc such that any of its subarcs fails to be a t-quasi-arc for any t, providing an affirmative answer to an open question by Norton.     

Quasi-separatrix layers and three-dimensional reconnection diagnostics for line-tied tearing modes

In three-dimensional magnetic configurations for a plasma in which no closed field line or magnetic null exists, no magnetic reconnection can occur, by the strictest definition of reconnection. A finitely long pinch with line-tied boundary conditions, in which all the magnetic field lines start at one end of the system and proceed to the opposite end, is an example of such a system. Nevertheless, for a long system of this type, the physical behavior in resistive magnetohydrodynamics (MHD) essentially involves reconnection. This has been explained in terms comparing the geometric and tearing widths [1] and [2]. The concept of a quasi-separatrix layer [3] and [4] was developed for such systems. In this paper we study a model for a line-tied system in which the corresponding periodic system has an unstable tearing mode. We analyze this system in terms of two magnetic field line diagnostics, the squashing factor [5], [6] and [7] and the electrostatic potential difference [8] and [9] which has been used in kinematic reconnection studies. We discuss the physical and geometric significance of these two diagnostics and compare them in the context of discerning tearing-like (reconnection-like) behavior in line-tied modes.     

Weak solutions of the Vlasov–Poisson initial boundary value problem

This paper deals with existence results for a Vlasov-Poisson system, equipped with an absorbing-type law for the Vlasov equation and a Dirichlet-type boundary condition for the Poisson part. Using the ideas of Lions and Perthame [21], we prove the existence of a weak solution having good L^p estimates for moment and electric field, by a good control on the higher moments of the initial data. As an application, we establish a homogenization result in the Hilbertian framework for this type of problem in non-homogeneous media, following the work by Alexandre and Hamdache [2] for general kinetic equations, and Cioranescu and Mural [11] for the Laplace problem.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

Forward and Inverse Viscoacoustic Modelling in a Tunnel Environment

In this work, the goal is to model forward acoustic waves in a tunnel environment with attenuation and to do full waveform inversion. In reality, there is no material without attenuation. Some materials, such as rocks, have so low attenuation that, in a small domain, the waves are almost not damped at all. At the same time, there are materials with high attenuation. In an environment with such materials, the attenuation has to be taken into account in order to model the waves properly. In this study, attenuation effect is integrated into acoustic equation by using Kolsky-Futterman model ( [1], [2]) which only replaces velocity field with a complex-valued field in frequency domain. Apart from attenuation, another objective is to consider an inhomogeneous density field. Mainly, acoustic equation with a constant density field is referred to in many studies. In many cases, it may suffice to model waves appropriately. However, in reality, the density field of ground can be highly inhomogeneous. The objective is to investigate the effect of the inhomogeneity in waves, and to search for density field ρ and attenuation parameter Q as well as pressure wave velocity c using full waveform inversion. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

Generalized Heteroclinic Cycles in Spherically Invariant Systems and Their Perturbations

Summary. In this paper we want to investigate the effects of forced symmetry-breaking perturbations—see Lauterbach & Roberts [29], as well as [28], [31]—on the heteroclinic cycle which was found in the l = 1 , l = 2 mode interaction by Armbruster and Chossat [1], [12] and generalized by Chossat and Guyard [25], [14]. We show that this cycle is embedded in a larger class of cycles, which we call a generalized heteroclinic cycle (GHC). After describing the structure of this set, we discuss its stability. Then the persistence under symmetry-breaking perturbations is investigated. We will discuss also the application to the spherical Bénard problem, which was the initial motivation for this work. Received March 11, 1997; first revision received October 10, 1997; second revision received April 13, 1998; accepted July 16, 1998     

可变形体的有限转动参数和回转磁效应

本文的第一部份将Synge[2]关于转动变换的推导用张量公式表达,进一步阐明作者在文[7]中所求得正交变换式的几何意义.文中并讨论转轴矢量的张量性质.文中后一部份应用拖带坐标系描述法讨论回转磁效应(Einstein-de Haas效应),建立一个求变形体中求磁化体力矩的简单公式.     

Geometric construction of the global base of the quantum modified algebra of\hat gl_N

A geometric construction of the modified quantum algebra ofgln was given in [BLM]. It was then observed independentely by Lusztig and Ginzburg-Vasserot (see [L1], [GV]) that this construction admits an affine analogue in terms of periodic flags of lattices. However the compatibility of the canonical base of the modified algebra and of the geometric base given by intersection cohomology sheaves on the affine flag variety was never proved. The aim of the paper is to prove this compatibility. As a consequence we prove a recent conjecture of Lusztig (see [L1]). Of course, our proof would work also in the finite type case.Partially supported by EEC grant no. ERB FMRX-CT97-0100.     

Hyperbolic unitals in the Hall planes

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.     

A family of cohen-macaulay modules over singularities of type xt + y3

Let K be a field with char K ≠ 3 and it two positive integers such that 1 ≤i <t/2,t ≠ 3i. The classification problem for maximal Cohen-Macaulay modules over K[[X,Y]]/(X^t+Y³ ) is complicated if t≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modules N, such that N/YN is a direct sum of copies of K[[X]]/(X ⁱ)K[[X]]/(X^t-i ).     

Generating Fractals Using Geometric Algebra

In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincaré disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.     

Quantization Formula for Symplectic Manifolds with Boundary

We extend our earlier work in [TiZ1], where an analytic approach to the Guillemin-Sternberg geometric quantization conjecture [GuSt] was developed, to the case of manifolds with boundary. We also give a general quantization formula that works for both regular and singular reductions. As simple applications, we prove an analytic analogue of the relative residue formula of Guillemin-Kalkman [GuK] and Martin [M], as well as a Guillemin-Sternberg type formula for singular reductions under circle actions. Submitted: February 1997, revised: January 1998 and July 1998, final version: March 1999.     

Values of rational functions on non-hilbertian fields and a question of Weissauer

We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions onK: we construct a non-hilbertian subfieldK of such that, given anyf ₁ ,…,f _h ∈K(x), each of degree ≥2, the union ∪ _z=1 ^h f _z(K) does not containK. See e.g. [FrJ], [L1], [L2], [Sch], [Se1], or [Se2] for the classical theory of hilbertian fields.