A Sine function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval (-1, 1). It is shown that the quadrature error is of order 0(e~ ), where N is the number of nodes used, and w...

Semantic completeness and syntax completeness are important characters of the general formal systems, two the completeness have different meaning for a formal system. This paper analyze the relation between semantic completeness of a formal system S and syntax completeness under certain condition and the relation between the syntax completeness of S and the syntax completeness of consistent ext...

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H2 we get a formula that appears to be new. In the projective model of H2 we have P = (1/2) ∫ w dφ. Here w is the Euclidean length of the projection of K from the ideal boundary point R = (cosφ, sinφ) onto...

shows that a Carathéodory solution may not exist, even locally in time. Various assumptions on the discontinuous vector field f , which guarantee the existence or the uniqueness of local solutions, have been studied in the literature [1,2,4,5,8]. In essence, for the well-posedness of the Cauchy problem one needs that the vector field f should point transversally to the surfaces where discontinu...

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems (NCP ) { u̇(t) = A(t)u(t) u(s) = x ∈ X on a Banach space X by the existence of certain evolution semigroups. Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so called “...

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent t...

in this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete riemannian manifold floating in space rn+1and is tangent to the space rn at ∂h0(t); (2...

For many fixed-parameter problems that are trivially solvable in polynomial-time, such as kDOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed k the problem is solvable in time bounded by a polynomial of degree c, where c is a constant...

In this paper we introduce the uniqueness and completeness problems of array comprehensions. An array comprehension has the uniqueness property if it deenes each array element at most once. Uniqueness is a necessary condition for correctness in single assignment languages such as Haskell, Id, and Sisal. The uniqueness problem can be stated as a data dependence problem, which in itself can be re...

In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. Let G1 be a group and let G2 be a metric group with metric ρ ·, · . Given > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ρ f xy , f x f y < δ for all x, y ∈ ...