Nonradial Large Solutions of Sublinear Elliptic Equations

We show that the equation Δu = p(x)f(u) has a positive solution on R ^N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup _{s ≥ 1} f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial.     

The Generalized Cauchy-Riemann-Fueter Equation and Handedness

A generalization of the Cauchy-Riemann condition in complex analysis is described for complex numbers, quaternions and complex quaternions. The generalization called here generalized Cauchy-Riemann-Fueter analycity encompasses not just the left and right-handed versions of quaternion analysis but also generates other variants for complex quaternions. These multiple variants are shown to satisfy an analogue of Cauchy's Theorem and to have similarities with the generalized Cauchy-Riemann conditions that define monogenic functions on R ^{n + 1}; they are also similar to Fueter-type operators and the Moisil-Theodoresco operator. The multiple variants are shown to have an interpretation that unifies analycity into a single definition. Thus left and right-handedness in quaternions are shown to be two sides of the same concept, and likewise for complex quaternions. This is then shown to have possible physical interpretation for example in understanding the nature of chirality and the 'arrow of time'.     

Problems of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equation

By means of continuation theorem of coincidence degree theory, existence, nonexistence and unique existence criteria are established for the periodic solution of a kind of second order neutral functional differential equation.     

Extracting the Convex Hull of an Unknown Inclusion in the Multilayered Material

We consider the problem of extracting information about an unknown inclusion embedded in a background layered material that has different constant conductivities across finitely many parallel planes, from the Dirichlet-to-Neumann map. For this purpose we construct the exponentially growing solutions of the governing equation for the background material. The leading coefficient of the equation has discontinuity across finitely many planes that are parallel to each other. Using the property of those solutions, we give an extraction formula of the support function of the inclusion from the Dirichlet-to-Neumann map.     

Existence Theorem of Global Solutions for Monge-Ampere Equation

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Boundedness of Solutions of Dirichlet Problem for a Class of Nonlinear Elliptic Equations with Weights

In this article, under some Hypotheses on weights and on coefficients, using a modification of Moser's method, we establish the boundedness of solutions of Dirichlet Problem for a class of nonlinear elliptic equations.     

Some Results on the Linearized Oscillation of the Odd-order Neutral Difference Equation

In this article, we obtain some results on the linearized oscillation of the odd-order neutral difference equation <artwork name="GAPA31013eu1"> where Δ is the forward difference operator, m is odd, {p_n },{q_n } are sequences of nonnegative real numbers, k, l are nonnegative integers, g(x),h(x)∈ C(R,R) with xg(x) > 0 for x ≠0.     

Generalized Solutions of Semilinear Parabolic Equations

It is well-known that a semilinear parabolic equation has no unique solution in the classical sense. We study such equations from the viewpoint of generalized functions. By using approximations for generalized functions, we obtain results on existence and uniqueness of generalized solutions. Furthermore, we establish the relationship between generalized solutions and classical solutions. Current address: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan     

Computational Results for Seamount Imaging in a Water Waveguide

This article presents our computational results for imaging a seamount in a waveguide. We assume that there is no knowledge of what kind of boundary condition on the unknown seamount. The incident waves are sent from point sources along a straight line and the corresponding scattered fields are measured from a line above the unknown seamount. Analysis and numerical results are presented.     

Global Solutions of an Obstacle-Problem-Like Equation with Two Phases

Concerning the obstacle-problem-like equation , where ₊ > 0 and _– > 0, we give a complete characterization of all global two-phase solutions with quadratic growth both at 0 and infinity.     

Locally Symmetric Contact Metric Manifolds

We show that a locally symmetric contact metric space is either Sasakian and of constant curvature 1 or locally isometric to the unit tangent sphere bundle (with its standard contact metric structure) of a Euclidean space. The second author is corresponding author     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.     

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.     

一类微分差分方程的周期解

本文研究微分差分方程ｘ＇（ｔ）＝－ｆ（ｘ（ｔ），ｘ（ｔ－τ１））－ｆ（ｘ（ｔ），ｘ（ｔ－τ２））－．．．－ｆ（ｘ（ｔ），ｘ（ｔ－τｎ））非平凡周期解的存在性问题，得到了一些判别准则，推广和改进了文［１－４］的工作。     

Solutions to the Equation of Non－Newtonian Polytropic Filtration Under Optimal Condition On Initial Values

Ｓｏｌｕｔｉｏｎｓ　ｔｏ　ｔｈｅ　Ｅｑｕａｔｉｏｎ　ｏｆ　Ｎｏｎ－Ｎｅｗｔｏｎｉａｎ　Ｐｏｌｙｔｒｏｐｉｃ　Ｆｉｌｔｒａｔｉｏｎ　Ｕｎｄｅｒ　Ｏｐｔｉｍａｌ　Ｃｏｎｄｉｔｉｏｎ　Ｏｎ　Ｉｎｉｔｉａｌ　ＶａｌｕｅｓＹｕａｎＨｏｎｇｊｕｎ（袁洪君）（Ｄｅ...     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

A result on large surfaces of prescribed mean curvature in a Riemannian manifold

Plateau's problem (PP) is studied for surfaces of prescribed mean curvature spanned by a given contour in a 3-d Riemannian manifold. We consider the local situation where a neighborhood of a given point on the manifold is described by a single normal chart. Under certain conditions on and the contour, existence of a small -surface to (PP) is guaranteed by [HK]. The purpose of this paper is the investigation of large -surfaces. Our result states: For sufficiently large (constant) mean curvature and a sufficiently small contour depending on the local geometry of the manifold, (PP) has at least two solutions, a small one and a large one. The proof is based on mountain pass arguments and uses – in contrast to results in the 3-d Euclidean space and in order to derive conformality directly – also a deformation constructed by variations of the independent variable. Received November 8, 1995 / Accepted April 29, 1996     

Multiple Positive Radial Solutions of Elliptic Equations in an Exterior Domain

Several existence theorems on multiple positive radial solutions of the elliptic boundary value problem in an exterior domain are obtained by using the fixed point index theory. Our conclusions are essential improvements of the results in [7], [10] and [13].     

Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.