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Abstract: Let R be a local ring essentially of finite type over a field k of characteristic . In the paper [M. Furuya, H. Niitsuma, Regularity criterion of Noetherian local rings of prime characteristic, J. Algebra 247 (2002) 219–230], we constructed some regularity criterion for such a local ring R in terms of the higher differential algebra and the -basis. In this paper, we introduce the concept of a reduced index of a Noetherian ring and we show the sharpened result of the above criterion and further we also give a geometric regularity criterion in terms of the higher differential algebra and the -basis. The latter criterion yield the sharpened result of a part of Orbanz''s theorem [U. Orbanz, Höhere Derivationen und Regularität, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2]. [Copyright &y& Elsevier]

Abstract: In electrical impedance tomography, one tries to recover the conductivity inside a body from boundary measurements of current and voltage. In many practically important situations, the object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. It has been shown that the inhomogeneities can be characterized by the factorization technique if the conductivity coefficient jumps to a higher or lower value on the boundaries of the inclusions. In this paper, we extend the results to the case of weaker inclusions: If the inhomogeneities inside the body are more (or less) conductive than the known background, if the conductivity coefficient and its lowest normal derivatives are continuous over the inclusion boundaries, and if the mth normal derivative of the conductivity jumps on the inclusion boundaries, then the factorization method provides an explicit characterization of the inclusions. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established. [Copyright &y& Elsevier]

In this paper, by means of a new twin fixed-point theorem in a cone, the existence of at least two positive solutions of m-point boundary value problem for second order dynamic equations on time scales is considered. [Copyright &y& Elsevier]

In this paper, Schaefer''s fixed point theorem and nonlinear alternative of Leray–Schauder type are used to investigate the existence of solutions for second order boundary value problem for impulsive dynamic equations on time scales. [Copyright &y& Elsevier]

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions. [Copyright &y& Elsevier]

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L2(0,∞) and L2(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established. [Copyright &y& Elsevier]

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem y(2n)(t)=ft,y(t),y″(t),…,y(2(n−1))(t), 0&les;t&les;1,y(2i)(0)=0, y(2i)(1)=∑lower limit j=1, upper limit m−2 kijy(2i)(ξj), 0&les;i&les;n−1, where kij>0 (i=0,1,…,n−1; j=1,2,…,m−2), 0<ξ1<ξ2<⋯<ξm−2<1. The associated Green''s function for the higher order m-point boundary value problem is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models ut=[D(u)ux]x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c&ast; and we show the appearance of a sharp-type profile when c=c&ast;. These results solve recent conjectures formulated by Sa´nchez-Gardun˜o and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). [Copyright &y& Elsevier]