Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX +XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V , we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0 -property for La and show that La has the R0 -property if and only if a is invertible. Finally, we provide La with some characterizations of the E0 -property and the nondegeneracy property.     

A Note on the Uniqueness of Koebe–Andreev–Thurston Theorem

In this paper, we present a new proof of the uniqueness of Koebe-Andreev-Thurston theorem. Our method is based on the argument principle in complex analysis and reviews the connection between the circle packing theorem and complex analysis.     

On Connes’ trace formula for the Hankel transformation of order-1/2

A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes’ corresponding trace formulas in 1999.     

Clifford分析中Cauchy型奇异积分算子的一些性质

First,we give a module estimation of the singular integral with a differential element.Then by proving the existences of Cauchy principal value we obtain the transformation formula of the Cauchy-type singular integrals with a parameter.     

Solitons for the cubic-quintic nonlinear Schrdinger equation with varying coefficients

In this paper,by means of similarity transfomations,we obtain explicit solutions to the cubic-quintic nonlinear Schrdinger equation with varying coefficients,which involve four free functions of space.Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

A new reduction-based LQ control for dynamic systems with a slowly time-varying delay

Time delays in the feedback control often dete- riorate the control performance or even cause the instability of a dynamic system. This paper presents a control strategy for the dynamic system with a constant or a slowly time-varying input delay based on a transformation, which sire-plifies the time-delay system the relation is discussed for into a delay-free one. Firstly, two existing reduction-based linear quadratic controls. One is continuous and the other is discrete. By extending the relation, a new reduction-based control is then developed with a numerical algorithm presented for practical control implementation. The controller suggested by the proposed method has such a promising property that it can be used for the cases of different values of an input time delay without redesign of controller. This property provides the potential for stabilizing the dynamic system with a time-varying input delay. Consequently, the application of the proposed method to the dynamic system with a slowly time-varying delay is discussed. Finally, numerical simulations are given to show the efficacy and the applicability of the method.     

A continuum thermodynamics formulation for micro-magneto-mechanics with applications to ferromagnetic shape memory alloys

A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the evolution of magnetic domain and martensite twin structures in ferromagnetic shape memory alloys. The theory falls into the class of phase-field or diffuse-interface modeling approaches. In addition to the standard mechanical and magnetic balance laws, two sets of micro-forces and their associated balance laws are postulated; one set for the magnetization order parameter and one set for the martensite order parameter. Next, the second law of thermodynamics is analyzed to identify the appropriate material constitutive relationships. The proposed formulation does not constrain the magnitude of the magnetization to be constant, allowing for spontaneous magnetization changes associated with strain and temperature. The equations governing the evolution of the magnetization are shown to reduce to the commonly accepted Landau-Lifshitz-Gilbert equations for the case where the magnetization magnitude is constant. Furthermore, the analysis demonstrates that under certain limiting conditions, the equations governing the evolution of the martensite-free strain are shown to be equivalent to a hyperelastic strain gradient theory. Finally, numerical solutions are presented to investigate the fundamental interactions between the magnetic domain wall and the martensite twin boundary in ferromagnetic shape memory alloys. These calculations determine under what conditions the magnetic domain wall and the martensite twin boundary can be dissociated, resulting in a limit to the actuating strength of the material.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

Large deformation of anisotropic austenitic stainless steel sheets at room temperature: Multi-axial experiments and phenomenological modeling

A newly developed multi-axial testing technique for sheet materials is employed to investigate the inelastic response of a temper-rolled stainless steel 301LN under isothermal quasi-static loading conditions at room temperature. The experimental technique consists of a flat sheet specimen, which is subject to combinations of shear and normal loading using a custom-made dual-actuator system. The large deformation behavior under monotonic loading is determined along more than 20 distinct radial paths in the stress space. The experimental results indicate that Hill's quadratic yield function along with an associated flow rule provides a good approximation of the initial yield behavior of this anisotropic two-phase FCC/BCC sheet material. Based on the experimental data for radial monotonic loading, it is concluded that conventional isotropic-kinematic hardening models cannot successfully describe the strain hardening of this austenitic steel. Instead, a non-associated anisotropic hardening model is proposed that relates the increase in yield strength to an isothermal martensitic transformation kinetics law. The comparison of the model predictions with the experimental results shows very good agreement for all biaxial and uniaxial experiments.