ANISOTROPIC POLARIZATION TENSORS FOR ELLIPSES AND ELLIPSOIDS

In this paper we present a systematic way of computing the polarization tensors, anisotropic as well as isotropic, based on the boundary integral method. We then use this method to compute the anisotropic polarization tensor for ellipses and ellipsoids. The computation reveals the pair of anisotropy and ellipses which produce the same polarization tensors.     

Proprietà di 20 ordine di un riferimento isotropo

We extend to the isotropic case, by adapted non-holonomic techniques [1], 2^° order properties of a relativistic frame of reference, generalized in the polar sense [2,3]: Riemann and gravitational tensors decomposition, Lie derivative of the Ricci rotation coefficients, commutation formulae and finally Bianchi identity. All the decomposition tensors are formally invariant (as regards the standard case), by means of the longitudinal derivative extension. Received: June 14, 2000?Published online: October 2, 2001     

Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmüller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.     

L2-Estimates and Existence Theorems for Generalized Analytic Functions in Several Variables

We consider the generalized Cauchy–Riemann system with nonlinear terms in an arbitrary domain of the complex space. Under some natural conditions on the coefficients and compatibility conditions, we prove solvability of this system in the space of locally square integrable functions.     

The forms of the relation between the stress and strain tensors in a non-linearly elastic material

New relations for the stress and strain tensors, which comprise energy pairs, are obtained for a non-linearly elastic material using a similar method to that employed by Novozhilov, based on a trigonometric representation of the tensors. Shear strain and stress tensors, not used previously, are introduced in a natural way. It is established that the unit tensor, the deviator and the shear tensor form an orthogonal tensor basis. The stress tensor can be expanded in a strain-tensor basis and vice versa. By using this expansion, the non-linear law of elasticity can be written in a compact and physically clear form. It is shown that in the frame of the principal axes the stresses are expressed in terms of the strains and vice versa using linear relations, while the non-linearity is contained in the coefficients, which are functions of mixed invariants of the tensors, introduced by Novozhilov, the generalized moduli of bulk compression and shear and the phase of similitude of the deviators. Relations for different energy pairs of tensors are considered, including for tensors of the true stresses and strains, where the generalized moduli of elasticity have a physical meaning for large strains.     

A neutral multi-coated sphere under non-uniform electric field in conductivity

We consider a neutral spherical inclusion for three-dimensional conductivity. Such an inclusion when inserted into a matrix does not disturb an arbitrary non-uniform electric field outside the inclusion. Our design is realized by means of multi-coating which cancels the lower order, generalized polarization tensors. Analytical results for a doubly coated sphere and detailed numerical results for a sphere with three or more coatings are presented to demonstrate the theory.     

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.     

A phase decomposition approach and the Riemann problem for a model of two-phase flows

We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data.     

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

Representations of Meromorphic Functions in Terms of Laurent Expansions and Partial Boundary Values

For any analytic function f on a Riemann surface S with an isolated singularity, we give a "good" formula representing f on a large domain containing any given point on which f is regular in terms of the coefficients of Laurent's expansion around any fixed isolated singular point using a conformal mapping of a doubly-connected domain. Representations of meromorphic functions in terms of partial boundary values will also be referred, clearly.     

Zeta functions interpolating the convolution of the Bernoulli polynomials

We prove several relations on multiple Hurwitz–Riemann zeta functions. Using analytic continuation of these multiple Hurwitz–Riemann zeta functions, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers. As an application, we give a general convolution identity for Bernoulli numbers.     

Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces

We investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors (GPTs), extending the previous terminology for inclusions with perfect interfaces. We derive explicit matrix expressions for the GPTs in terms of the incident field, material parameters, and geometry of the inclusion. As an application, we construct GPT-vanishing structures of general shape that result in negligible perturbations for all uniform incident fields. The structure consists of a simply connected core with an imperfect interface. We provide numerical examples of GPT-vanishing structures obtained by our proposed scheme.     

Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

Partitioning Infinite-Dimensional Spaces for Generalized Riemann Integration

To form Riemann sums for generalized Riemann integrals, thedomain of integration must be partitioned in a suitable manner.The existence of the required partitions is usually proved bya simple method of repeated bisection of the domain of integration.However, when the domain is the Cartesian product of infinitelymany copies of the set of real numbers, this simple method ofproof has frequently failed. A proof which works for infinite-dimensionalspaces is provided here. 2000 Mathematics Subject Classification28C20.     

Adaptive Smoothing Method,Deterministically Computable Generalized Jacobians,and the Newton Method

In this note, we show that a well-known integral method, which was used by Mayne and Polak to compute an -subgradient, can be exploited to compute deterministically an element of the plenary hull of the Clarke generalized Jacobian of a locally Lipschitz mapping regardless of its structure. In particular, we show that, when a locally Lipschitz mapping is piecewise smooth, we are able to compute deterministically an element of the Clarke generalized Jacobian by the adaptive smoothing method. Consequently, we show that the Newton method based on the plenary hull of the Clarke generalized Jacobian can be implemented in a deterministic way for solving Lipschitz nonsmooth equations.     

Exact solutions of a two-fluid model of two-phase compressible flows with gravity

We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave.     

Further results on the Drazin inverse of even‐order tensors

The notion of the Drazin inverse of an even‐order tensors with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402‐3413]. In this article, we further elaborate this theory by establishing a few characterizations of the Drazin inverse and $\mathrm{??}$‐weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution of multilinear systems by using the Drazin inverse and iterative (higher order Gauss‐Seidel) method of tensors. Besides these, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.     

An improved Riemann mapping theorem and complexity in potential theory

We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another “double quadrature domain,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann mapping theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain is close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains.     

Critical oscillation constant for half-linear differential equations with periodic coefficients

We compute explicitly the oscillation constant for certain half-linear second-order differential equations involving periodic coefficients. If these periodic functions are constants, our results reduce to the well-known oscillation constants for half-linear Euler and Riemann–Weber differential equations.     

On a Non-symmetric Problem in Electrochemical Machining

We consider a free boundary value problem arising from a non-symmetric problem of electrochemical machining (ECM). After a conformal mapping of the unknown domain the problem is transformed to a non-smooth non-linear Riemann–Hilbert problem for holomorphic functions in the complex unit disk. A special technique allows to remove the singularities in the boundary condition. Utilizing existence results for smooth non-orientable Riemann–Hilbert problems, existence and uniqueness of a solution are shown. Finally, we propose an iterative method of Newton type for the effective numerical computation of the free boundary of the anode and present some test results. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.