Symmetric Argument Research Articles

Multidimensional Frank–Laptev–Weidl improvement of the hardy inequality

AbstractWe establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.

A reflection formula for the Gaussian hypergeometric function of matrix argument

We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to generalizations of the Gaussian hypergeometric function defined in terms of series of Jack polynomials. Finally, we obtain a quadratic transformation formula for the Gaussian hypergeometric function of Hermitian 2×2 matrix argument.

Stable Difference Schemes with Interpolation for Delayed One-Dimensional Transport Equation

In this article, we consider the one-dimensional transport equation with delay and advanced arguments. A maximum principle is proven for the problem considered. As an application of the maximum principle, the stability of the solution is established. It is also proven that the solution’s discontinuity propagates. Finite difference methods with linear interpolation that are conditionally stable and unconditionally stable are presented. This paper presents applications of unconditionally stable numerical methods to symmetric delay arguments and differential equations with variable delays. As a consequence, the matrices of the difference schemes are asymmetric. An illustration of the unconditional stable method is provided with numerical examples. Solution graphs are drawn for all the problems.

Quasilinear Schrödinger equations with singular and vanishing potentials involving exponential critical growth in [formula omitted

We study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations: −Δu+V(|x|)u−[Δ(u2)]u=Q(|x|)h(u),x∈R2,u(x)→0as|x|→∞,where V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity and the nonlinearity h(s) is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. By combining variational methods in a suitable weighted Orlicz space with a version of the Trudinger–Moser inequality for this space, we obtain the existence of a nonnegative ground state solution. For this, we have used some regularity results and we need to explore a symmetric criticality type argument. Moreover, under some conditions on V, Q and h(s), we prove that this problem does not have positive radial solution. Schrödinger equations of this type have been studied as models of several physical phenomena.

Recent Developments in Macro-Econometric Modeling: Theory and Applications

Developments in macro-econometrics have been evolving since the aftermath of the Second World War.[...]

Do Seasonal Adjustments Induce Noncausal Dynamics in Inflation Rates?

This paper investigates the effect of seasonal adjustment filters on the identification of mixed causal-noncausal autoregressive models. By means of Monte Carlo simulations, we find that standard seasonal filters induce spurious autoregressive dynamics on white noise series, a phenomenon already documented in the literature. Using a symmetric argument, we show that those filters also generate a spurious noncausal component in the seasonally adjusted series, but preserve (although amplify) the existence of causal and noncausal relationships. This result has has important implications for modelling economic time series driven by expectation relationships. We consider inflation data on the G7 countries to illustrate these results.

Derivatives of compound matrix valued functions

Primary matrix functions and spectral functions are two classes of orthogonally invariant functions on a symmetric matrix argument. Many of their properties have been investigated thoroughly and find numerous applications both theoretical and applied in areas ranging from engineering, image processing, optimization and physics. We propose a family of maps that provide a natural connection and generalization of these two classes of functions. The family of maps also contains the well-known multiplicative and additive compound matrices. We explain when each member of this family is a differentiable function and exhibit a formula for its derivative.

On the spectrum of a Baouendi–Grushin type operator: an Orlicz–Sobolev space setting approach

We show that the spectrum of a nonhomogeneous Baouendi–Grushin type operator subject with a homogeneous Dirichlet boundary condition is exactly the interval \({(0,\infty)}\) . This is in sharp contrast with the situation when we deal with the “classical” Baouendi–Grushin operator (i.e., an operator of type \({-\Delta_x-|x|^{\xi}\Delta_y}\)) when the spectrum is an increasing and unbounded sequence of positive real numbers. Our proofs rely on a symmetric mountain-pass argument due to Kajikiya. In addition, we can show that for each eigenvalue there exists a sequence of eigenfunctions converging to zero.

Application of a coordinate‐free tensor formalism to the numerical implementation of a material model

AbstractWe analyze a coordinate‐free tensor setting in ℝ3 within the context of the classical tensor analysis. To this end, we formulate in a basis‐free manner the notions of second‐ and fourth‐rank tensors in ℝ3, and corresponding operations on tensors. Among the large number of different approaches to the tensor setting, we give the preference to the convenient ones, concerning the specific needs of computational solid mechanics. We use the well‐known Fréchet derivative to define the derivative of a function with respect to its tensor argument in a natural way. Furthermore, such aspects as the derivative with respect to a symmetric tensor argument and its uniqueness are covered in this paper. For the sake of completeness we present the coordinate representation of tensors and tensor operations. This representation is obtained in a straight‐forward manner from the coordinate‐free one. In particular, we elaborate the computation of the inverse of a fourth‐rank tensor and the inverse of a linear transformation on the space of symmetric second‐rank tensors. The tensor formalism is applied to the analysis of a nonlinear system of differential and algebraic equations governing visoplastic material response. An implicit time‐stepping algorithm is formulated and the numerical treatment of the algorithm is discussed.

Generalized Hadamard Product and the Derivatives of Spectral Functions

Real valued functions, $F(X)$, on a symmetric matrix argument are called spectral if $F(U^TXU) = F(X)$ for every orthogonal matrix $U$ and $X \in \mathrm{dom\,} F$. We are interested in a description of the higher order derivatives (when they exist) of $F$ with respect to $X$. Formulae for the gradient and the Hessian of $F$ are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576-588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368-386]. In this work we present common features of these two formulae that are preserved in the higher order derivatives.

Total Positivity Properties of Generalized Hypergeometric Functions of Matrix Argument

In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized (0F1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F1 functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( rFs) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP2 properties; the proofs of these results are based on Sylvester’s formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

A return map algorithm for general isotropic elasto/visco‐plastic materials in principal space

AbstractWe describe a methodology for solving the constitutive problem and evaluating the consistent tangent operator for isotropic elasto/visco‐plastic models whose yield function incorporates the third stress invariant . The developments presented are based upon original results, proved in the paper, concerning the derivatives of eigenvalues and eigenprojectors of symmetric second‐order tensors with respect to the tensor itself and upon an original algebra of fourth‐order tensors obtained as second derivatives of isotropic scalar functions of a symmetric tensor argument . The analysis, initially referred to the small‐strain case, is then extended to a formulation for the large deformation regime; for both cases we provide a derivation of the consistent tangent tensor which shows the analogy between the two formulations and the close relationship with the tangent tensors of the Lagrangian description of large‐strain elastoplasticity. Copyright © 2004 John Wiley & Sons, Ltd.

On the spatial formulation of anisotropic multiplicative elasto-plasticity

This contribution aims at a thermodynamically consistent and modular formulation of anisotropic multiplicative elasto-plasticity. Embedded in the well-established framework of non-standard dissipative materials, we adopt as a key ingredient the introduction of additional symmetric arguments––typically structural tensors––into the relevant scalar-valued isotropic tensor functions. On this basis, the fundamental covariance relation leads us to a formulation which allows itself to be set up directly in terms of spatial arguments and thus guarantees a convenient implementation within a finite element setting.

Doubly-focused echos from spheres unfold into a hyperbolic umbilic diffraction catastrophe

An underwater spherical target resides in an Airy field formed by reflection off a curved surface representing the sea floor or sea surface. In prior work [B. Dzikowicz and P. L. Marston, J. Acoust. Soc Am. 110, 2778 (2001)] direct returns of a tone burst from the surface reflection focused toward the target were shown to have a dependence on the target position described by an Airy function. The return echo can also be focused again by the surface onto the source and receive transducer. This gives the square of an Airy function for the case of a point target. With a finite sized target (as in the experiment) this goes over to a hyperbolic umbilic catastrophe with symmetric arguments. The arguments of the hyperbolic umbilic function are derived from only the relative return times of a transient pulse. Experiment confirms the predicted merging of transient echoes in the time domain, as well as the hyperbolic umbilic diffraction integral amplitudes for a tone burst. This method would allow for the observation of a target at a greater distance in the presence of a focusing surface. [Research supported by ONR.]

A closed-form representation for the derivative of non-symmetric tensor power series

In the present paper a closed-form representation for the derivative of non-symmetric tensor power series is proposed. Particular attention is focused on the special case of repeated eigenvalues. In this case, a non-symmetric tensor can possess no spectral decomposition (in diagonal form) such that the well-known solutions in terms of eigenprojections as well as basis-free representations for isotropic functions of symmetric tensor arguments cannot be used. Thus, our representation seems to be the only possibility to calculate the derivative of non-symmetric tensor power series in a closed form. Finally, this closed formula is illustrated by an example being of special importance in large strain anisotropic elasto-plasticity. As such, we consider the exponential function of the velocity gradient under simple shear. Right in this loading case the velocity gradient has a triple defective eigenvalue excluding the application of any other solutions based on the spectral decomposition.

Recurrence relations of coefficients of the generalized hypergeometric function in multivariate analysis

James(1960) defined the zonal polynomials and used it to represent the joint distributions of latent roots of VVisfiart matrix. The zonal polviionnals played an important role to define the generalized hypergeometric function of symmetric matrix argument Since then, many density functions and moments based on Wishart matrix have been expressed in terms of the generalized hy¬pergeometric Function. The purpose of this paper is to get the recurrence relations for the coefficients of it. In Section 1 we derive a partial differen¬tial equations having the generalized hypergeometric function as the unique solution. Then we ubtain the recurrence relations until order 7 in Section 2.

Dual differential geometry associated with the Kullbaek-Leibler information on the Gaussian distributions and its 2-parameter deformations

Amari showed that the geometry of a family of probability distributions is characterized by a dual differential geometry determined by a couple of affine connections and a divergence associated with a couple of dual potential functions. In this paper, a 2-parameter class of dual differential geometries is constructed on the manifold of the family of multivariate Gaussian distributions with nonzero means, as well as a new class of divergences. This class of geometry includes the Riemannian geometry studied by Skovgaard and the geometry associated with the Kullbaek-Leibler information. The specific dually flat charts for the latter geometry is given in conjunction with a detailed analysis of the associated connections. In order to facilitate the various calculations of differential geometric quantities in the analysis of our geometries, we introduce a new coordinate free differential calculus of a function of a symmetric matrix argument, based on a specific bilinear form defined on the domain of the function and its dual space. This calculus enables us to obtain a parallel formalism of the Legendre transformation in convex analysis even for a function of a matrix argument.

Comparison of two algorithms for the computation of fourth-order isotropic tensor functions

This paper compares two possible representations of a certain class of fourth-order isotropic tensor functions. It focusses on the derivatives of symmetric second-order isotropic tensor functions by a symmetric second-order tensor argument. This tensor argument is assumed to be given in spectral form. The first representation explicitly needs the eigenvectors of the argument tensor and is well-known in the literature. The second representation proposed here is based on the knowledge of second-order eigenvalue bases and avoids the computation of eigenvectors. The evaluation of a typical model problem shows that the second representation needs less computer time than the first one. The representations discussed here are of high importance for numerical solvers of nonlinear initial-boundary-value problems in large-strain elasticity and elastoplasticity.

Density estimation on the spaces of symmetric and rectangular matrices

This paper develops the theory of density estimation on the space S m of all m × m symmetric matrices and on the space R m, p of all m × p rectangular matrices. We consider the two methods, kernel density estimation and density estimation by orthogonal series, and illustrate them with examples for the normal kernel density functions and for the Hermite orthonormal bases with symmetric and rectangular matrix arguments on S m and R m, p , respectively. The problem of estimating unknown joint density functions of multiple random matrices is also discussed.

Generic existence of competitive equilibria when the asset market is incomplete: A symmetric argument

Generic existence of competitive equilibria when the asset market is incomplete: A symmetric argument