Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor-Coefficients G and Applications to Elasticity

Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

Determination of the derivative of the tangent stiffness matrix with respect to the load parameter

In order to solve the so-called consistently linearized eigenproblem in the frame of the Finite Element Method (FEM), the derivative of the tangent stiffness matrix with respect to the load parameter λ needs to be calculated. In this work, three schemes for calculation of are presented. The first scheme is based on an analytical expression for the first derivative of e the element tangent stiffness matrix with respect to λ for the special case of a co-rotational beam element. The second one is a finite difference approach for computation of . The third one is also a finite difference approach. However, it is based on a directional derivative of . An elastic beam, subjected to a compressive axial force and a small transverse uniform load, is chosen as a numerical example. The effectiveness and the accuracy of the three schemes are compared. The third scheme is found to be not only very practical but also more effective than the two competing schemes. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

Hamilton decompositions of complete multipartite graphs with any 2-factor leave

For m ≥ 1 and p ≥ 2, given a set of integers s₁,…,s_q with for and , necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph , where U is a 2-factor of consisting of q cycles, the jth cycle having length s_j. This result is then used to completely solve the problem when p = 3, removing the condition that . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 208–214, 2003     

Convergence Laws for Very Sparse Random Structures with Generalized Quantifiers

We prove convergence laws for logics of the form , where is a properly chosen collection of generalized quantifiers, on very sparse finite random structures. We also study probabilistic collapsing of the logics , where is a collection of generalized quantifiers and k ∈ ℕ₊, under arbitrary probability measures of finite structures.     

A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space where equals ℝ × N equipped with a warped product metric and show that a submanifold inN converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in converges to a minimal submanifold under a rescaled mean curvature flow in . We then define a notion of stability for conformal solitons and obtain L^p estimates as well as pointwise estimates for the curvature of stable solitons.     

Assessment of structural sensitivity on the basis of artificial neural networks

In engineering practice, the assessment of sensitivity is utilized to detect influential parameters in order to facilitate subsequent numerical simulation techniques. As sensitivity analyses are preprocessing methods for sophisticated numerical simulation techniques, e.g. reliability based optimization procedures, their application is always linked to an increase of the computational expense. In result, it is reasonable to couple sensitivity analysis and artificial neural networks (ANN). Therefore, multi-faceted global sensitivity measures (GSM) may be formulated, taking advantage of different characteristics of the ANNs. Additionally, to take into account nonlinearities of the response surface, a new approach of sectional global n sensitivity measures is introduced. Generally, the sensitivity can be determined with . Thereby, denotes the sensitivity of interest and a characteristic of the function under investigation. This can be either the response itself or the first partial derivative thereof . (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coloring edges of embedded graphs

In this paper, we prove that any graph G with maximum degree , which is embeddable in a surface Σ of characteristic χ(Σ) ≤ 1 and satisfies , is class one. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 197–205, 2000     

On the Eigenvalues of Half‐Linear Boundary Value Problems

We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .     

Comparing Packing Measures to Hausdorff Measures on the Line

For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures.     

A Mixed Finite Element Formulation for Piezoelectric Materials

A geometrically nonlinear finite element formulation to analyze multi-field problems as they arise e.g. in piezoelectric or magnetostrictive materials is presented. Here we focus on piezoelectric problems. The formulation is based on a Hu-Washizu functional considering six independent fields. These are displacements u , electric potential ϕ, strains E , stresses S , electric field , and the electric flux density . The finite element approximation leads to an 8-node hexahedral element with u and ϕ as nodal degrees of freedom. The fields E , S , , and are interpolated on element level by employing some internal degrees of freedom. These fields do not require continuity across the element boundaries, thus the internal degree of freedoms are eliminated on element level by a static condensation. The geometrically non-linear theory allows large deformations and accounts for stability problems. To fulfill the charge conservation law in bending dominated situations exactly a quadratic approximation of the electric potential is necessary. This leads in general to additional nodal degrees of freedom, which is circumvented by the presented formulation by employing appropriate interpolations of and . Numerical examples show that the locking effect which arise in low order elements are significantly reduced and that the element provides good accuracy with respect to experimental data. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Aspects For Bifurcation Problems in Continuous Mechanics

Consider a semilinear eigenvalue problem where λ ∈ R , the linear operator is defined in a real Hilbert space H and : H → H is generaly a nonlinear perturbation. We can define a coincidence degree of the pair ( ) under some conditions weaker than the ones when the classical coincidence degree was defined. Our final purpose is to extend the results to the case of the operators from the Banach space X into its dual X*, using the representation theorem due to Browder and Ton. We use these results to study resonance problems in mechanics of continua, such as the buckling in finite elastostatics and the steady state flow of incompressible fluids. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Data Based Regularization Matrices for the Tikhonov-Phillips Regularization

In Tikhonov-Phillips regularization of general form the given ill-posed linear system is replaced by a Least Squares problem including a minimization of the solution vector x, relative to a seminorm with some regularization matrix L. Based on the finite difference matrix L_k, given by a discretization of the first or second derivative, we introduce the seminorm where the diagonal matrix and is the best available approximate solution to x. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

Implicit linear time‐dependent differential‐difference equations and applications

The paper deals with the differential‐difference equation in a Banach space. The operator coefficient of the delay‐free derivative is allowed to be degenerate. Existence and uniqueness theorems are proved under the main assumption that for every the point is a polar singularity of the resolvent . The results are applied to evolution problems of microwave circuits. Copyright © 2000 John Wiley & Sons, Ltd.     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003