A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

On the uniqueness of a local martingale with a given absolute value

Summary D. Gilat has shown that any non-negative submartingale (X, .) is equal in law to the absolute value of a martingale (M, .). This result may be strenthened so that the pairs (X,.) and (¦M¦,.) are synonomous. In this paper the question of uniqueness of M is considered. Conditions on a local martingale (M, .) are found that lead to an explicit formula for the finite-dimensional distributions of M in terms of the Doob-Meyer decomposition of the local martingale X. In many cases of interest the conditions on M are unnecessary. For example, if X is the pth power of an Itô integral it is shown that (M) is unique if p> 1 but not in general if p=1.     

The extreme points of the unit ball of certain spaces of operators

In this note we discuss the set of extreme points of the unit ball of certain spaces of mappings. We prove that a mapping T: E F is an extreme point of the unit ball of the space I (E, F) of integral mappings, if and only if it has the formTx= ₀ >b ₀ , wherea, extS (E) andb ₀extS (F).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 521–527, October, 1976.     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

Superconnection currents and complex immersions

Summary Let i: M M be an immersion of complex manifolds, and let (, ) be a complex of holomorphic Hermitian vector bundles on M which provides a projective resolution of the sheaf of sections of a holomorphic vector bundle on M. We study the convergence as u + of a class of superconnection currents _u introduced by Quillen, and we calculate the limit. Microlocal estimates of the speed of convergence are also given.     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

Hasse-Witt matrices of Fermat curves

Let m be an integer with m3. Let K and K be perfect fields of characteristic p and p such that (p,m)=1 and (p,m)=1, respectively. Moreover let A and A be algebraic function fields over K and K defined by x^m+y^m=a(0, ak) and x^m+y^m=a(a0 ak), respectively. Put g=(m–1)(m–2)/2. Denote by M(K,p,a) and M(K,p,a) the Hasse-Witt matrices of A and A with respect to the canonical bases of holomorphic differentials. Then we show that if p+p0(mod.m) then rank M(K,p,a)+rank M(K,p,a)=g and if pp1 (mod.m) then rank M(K,p,a)=rank M(K,p,a).     

Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

On Localization and Stabilization for Factorization Systems

If (, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:AB is in if each of its pullbacks lies in (that is, if it is stably in ), and is in M ^* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (, M ^*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M ^*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Field extensions and isotropic subspaces in symplectic geometry

Let L/k be a finite field extension and let (V, B) be a finite dimensional symplectic space over L. We examine the action of the symplectic group Sp _L (V) on the set of B-isotropic k-subspaces of V, where B=°B is the k-symplectic form induced by a trace map :Lk. The orbits are completely classified in the case of a quadratic extension and for maximal B-isotropic subspaces in the case of a cubic extension; the number of orbits of maximal B-isotropic subspaces is shown to be infinite if the degree of the extension is at least 4.     

Famiglie di dischi analitici con bordo su sottovarietàCR non generiche

Summary In this work the authors prove that all sufficiently small analytic discs in the tangent space of a non generic embedded CR manifold MC^N (M at least of class C⁴) can be lifted uniquely to analytic discs in C^N with boundaries on M Moreover if M is real analytic or C then real analytic or C discs lift without loss of derivatives. If M is of class C^K then there is a 1+ loss of derivatives in the lifting. A stability result is also proven.     

Über die relativistische Elektronenoptik elektrostatis cher Beschleuniger

Summary Relativistic and nonrelativistic data for the coaxial two-cylinder electrostatic electron-lens are compared. Values of the potential along the axis were measured on a resistance network analogue, whilst trajectories were calculated on a high speed digital computer.IndexR indicating relativistic values, andN non-relativistic ones, quotientsF _R/F_N of the focal point coordinates (image sided),f _R/f_N of the focal length (image sided) and differences of the principal plane coordinates (H _R–H _N)/R (object sided) and (H _R–H _N)/R (image sided) are represented graphically as functions of the imagespace potential andU. The gap between the cylinders was in all cases 0,8R (R radius of the cylinders).U being the potential on the object side, andU on the image side, =U/U was varied from 0,1 to 0,966 andU from 0 to 2,000 kV; in the relativistic case a maximal departure of about 20% from the nonrelativistic values was found for focal length. A ten-lens accelerator of 1,5 MeV overall tension has been calculated both ways.     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.