Hyperbolic equations and inequalities in octants with the lateral Cauchy data at coordinate planes are consdered. Lipschitz stability estimate is established in the case when both the inhomogeneous right hand side and (unknown) initial conditions at {t = 0} have a finite support. This is the first stability estimate for such a Cauchy problem in an infinite domain. Refocusing of time reversal fi...

We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of the domain, ...

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bDq). Using this operator, we obtain extensi...

We study the Cauchy problem for the Aw-Rascle-Zhang model for traffic flow with a flux constraint at x = 0. More precisely we consider the Riemann solver, conserving the number of cars at x = 0 but not the generalized momentum, introduced in [9] for the problem with flux constrained. For such a Riemann solver, we prove existence of a solution for the Cauchy problem. The proof is based on the wa...

In this paper, we introduce the notion order quasi-Cauchy se-quences, downward and upward quasi-cauchy sequences, half Cauchy sequences. Next consider associated idea of continuity namely, ward continuous functions [2] investigated certain intersting results. The entire investigation is perform in (l)-group setting.

We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice p...

We study certain generalized Cauchy integral formulas for gradients of solutions to second order divergence form elliptic systems, which appeared in recent work by P. Auscher and A. Rosén. These are constructed through functional calculus and are in general beyond the scope of singular integrals. More precisely, we establish such Cauchy formulas for solutions u with gradient in weighted L2(R 1+...

— A meshfree computational scheme is developed to study the compressive deformation of single-walled carbon nanotubes (SWCNTs). The continuum constitutive relationship is based on an extended Cauchy-Born rule, the higher order Cauchy-Born rule, in which the atomic-scale deformed lattice vectors are calculated with both the first and second order deformation gradients. The compression of CNTs is...