Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.

     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

Ausstrahlungsproblem des Laplace-Operators für eine Klasse der Gebiete mit unbeschränktem Rand

In this paper we prove the existence and uniqueness of a solution u satisfying the equation- u – k² y = f (k , k 0), homogeneous Dirichlet data on the boundary and a radiation condition at infinity. We consider this problem in some unbounded region with an infinite boundary for which the assumption (x) · (x) 0 holds; here denotes the exterior normal and a given field.     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

Large automorphism groups of 8-dimensional projective planes are Lie groups

We prove the following theorem: LetP be an 8-dimensional compact topological projective plane. If the connected component of its automorphism group has dimension at least 12, then is a Lie group.     

Асимптоический Анализ Распределения Статистики Диксона

. u 60-u u     

Infinite highly arc transitive digraphs and universal covering digraphs

A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL() is constructed and is shown to be a covering digraph for every digraph in a certain classD() of connected digraphs. Moreover, if is locally finite, thenDL() is a universal covering digraph forD(). Further constructions of infinite highly arc transitive digraphs are given.The second author wishes to acknowledge the hospitality of the Mathematical Institute of the University of Oxford, and the University of Auckland, during the period when the research for this paper was doneResearch supported by the Australian Research Council     

On the stability of the transport equation

We consider boundary value problems associated with the equation T=–A in a Hilbert Space, where T and A are bounded, self adjoint, injective, and A has a bounded inverse. We discuss the stability of the solution when A is perturbed by a self adjoint operator.     

Die Dupinschen Zykliden im pseudoeuklidischen Raum

In the pseudoeuclidean space¹ E ³ a surface of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus such that a (pseudoeuclidean) spherical inversion maps upon . If the axis of is respectively space-like, isotropic or time-like is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide is the envelope of two families of spheres, the caustic surfaces of degenerate in caustic curves and the lines of curvature of constitute two (orthogonal) families of circles.     

Parabelscharen und grenzkreise in der isotropen ebene

It is known by H. Sachs [5] that the classical curve theorem of ABRAMESCU also holds in isotropic geometry. Generalising an idea due to O. Röschel [2] we regard all inscribed parabolas (s, t) of a triangle (t). This triangle is formed by the tangents of three neighbouring points of a C -curve k(t) in an isotropic plane. Let U((t)) be the circumcircle of (t) and I((t)) the incircle of the triangle (t) whose midpoints of the sides are the vertices of (t). The circle U((t)) is the locus of the isotropic focal points of (s, t) and the incircle I((T)) the envelope of the isotropic axes of (s, t). We prove that the ABRAMESU-circle — lim U((t)) — is identical with the locus of the focal points of lim (s, t) and the circle lim I((t)) with the envelope of the axes of lim (s, t). The characteristic points, different from k(t), of the circles lim U((t)) and lim I((t)) determine the direction of the affine-normal of k(t).Herrn Professor Helmut Mäurer zum 60. Geburtstag gewidmet     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Calculi with monotone deductions and their economic interpretation

Let the possible ways of development of some system from the initial state X₀ be given by the deductive system <;X₀> (X₀ is an axiom, the algorithm defines the relation of deducibility in one step). Let Y₁,..., Y_e be all states directly derivable from X [i.e., (X)={y₁,..., Y_e]. Let be an algorithm assigning for each X transition probabilities p₁,..., p _l , where is the transition probability to the special state STOP. defines a probability measure on the set of all deductions. We define the information in the pair < ;X₀> by the forla: where px is the probability of being in X directly before STOP. We consider , assigning a fixed p for each X and satisfying the condition p₁= ...=p _l . Then the information in < ; X₀> becomes a function <;X₀> of one p. The essential characteristic of the system <;X > is given by the asymptotic behavior of <;X₀> as p 0. This characteristic corresponds well with the intuitive notion of the relative power of calculi. Now we consider <,X>(p) as a function of X. For many types of systems there is a useful strategy for maximizing this function (the strategy of increasing freedom of choice); we consider in this connection the simplest systems of economic character. Let X, Y, Z be n-dimensional vectors with nonnegative components (the components are interpreted as resources and products of a certain economic system, gives the technological possibility of transformations of the resources). Let Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 90–104, 1979.     

A modification of Newton's method to solve the Dirichlet problem for the equation Δu=f(x,u)

The Dirichlet problem for a weakly nonlinear equation u=f(x, u) is investigated. We use successive approximations constructed by modified Newton's scheme and apply the extremal properties of the solutions of the elliptic equation of the form u – c(x)u=F(x), where c(x) 0. Numerical solution of the resulting sequence of linear boundary-value problems is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 27–30, 1987.     

Hyperaffine planes over hyperfields

Summary In this paper we solve the problem of finding suitable conditions for an hyperaffine plane (,) under which a planar hyperfield K exists such that (,) is isomorphic to the hyperaffine plane (K²,').Research partially supported by C.N.R.-G.N.S.A.G.A.     

The thinnest holding-lattice of a set

In the present note we are going to study the following question proposed by L. Fejes Toth. In the Euclidean plane a lattice is called a holding-lattice of a planar setS if any set congruent toS contains at least one lattice point of . The density of equals (2)^–1, where denotes the area of a fundamental triangle of . A holding-lattice ofS of least possible density is said to be the thinnest holding-lattice ofS. The problem: Find the sets whose thinnest holding-lattice is a regular trianglelattice.To Professor L. Fejes Toth on his 70th Birthday     

Perturbation of the Laplacian by the Coulomb potential and a point interaction in L p (ℝ)3

Perturbations of -+/|x| (with >0) by a point interaction centered at zero are defined in L ^p(³). This is done for 3/20 (³{0}), such that the extension is the negative generator of an analytic semigroup on L ^p(³).