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.     

Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.     

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.     

Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

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).     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

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).

     

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     

Support functions of convex compacta

The properties of the space (X_x) of all sublinear functionals, defined on a space X' (topologically adjoint to a Hausdorff locally convex barrelled space X) and continuous in the Arens topology × (X, X), equipped with topology of uniform convergence on bounded subsets of X are studied. It is shown that completeness and separability of a space X are hereditary for (X_x). Criteria for the compactness of subsets of (X_x) and conditions for the metrizability of compacta in (X_x) are given. The topological isomorphism between (X_x) and the space of all nonempty convex compacta in X with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.The author thanks S. S. Kutateladze for useful discussions regarding this article.     

On the definition of isomorphisms of linear spaces

Let (P, ) and (P, ) be linear spaces satisfying the exchange axiom with dim P=dim P . Then a bijection :PP which maps collinear points onto collinear points is an isomorphism. Also a surjection :PP which maps any three non-collinear points to non-collinear points is an isomorphism. This assertion is not true if dim P is not finite.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

Spline finite difference methods for singular two point boundary value problems

Summary In this paper we discuss the construction of a spline function for a class of singular two-point boundary value problemx ^–(x u)=f (x, u),u(0)=A,u(1)=B, 0<<1 or =1,2. The boundary conditions may also be of the formu(0)=0,u(1)=B. Three point finite difference methods, using the above splines, are obtained for the solution of the boundary value problem. These methods are of second order and are illustrated by four numerical examples.     

On the nuclearity of a dual space with the convergence in probability topology

Summary Let be a probability measure on a separable locally convex Fréchet space E and let s denote the topology on E of the convergence in . Then (E, s ) is nuclear iff ((E', s ))=1.     

Automorphism groups of posets and lattices with a given subset of fixed points

For a posetP, let Aut (P) denote the automorphism group ofP and let Fp (P) be the subposet of all fixed points of Aut (P). It is shown that for every posetP and every nontrivial groupG the posetsP satisfying Aut (P)G and Fp(P)=P form a proper class.Similarly, for a latticeL, let Aut (L) denote the automorphism group and Fp(L) the sublattice of fixed points. It is shown that ifL has more than one element andG is a nontrivial group then the latticesL for which Aut (L)G and Fp(L)=L also form a proper class. Moreover, if card (L)1 then this is still the case providingG is an infinite group. Since card (L)2 when Aut (L) is finite, this is the best possible result.With 3 FiguresThe support of the National Research Council of Canada is gratefully acknowledged.     

Présentations des groupes de Fischer (II)

In this paper we give Coxeter presentation (X, ) for the three Fischer groupsG=Fi₂₂, Fi₂₃, Fi₂₄; we apply methods exposed in the first part. Each of these groups is generated by a class of 3-transpositions (named here a Fischer class) in which elements ofX are chosen. A subset of is the set of all the relations (xy) ^m(x,y)=1, wherex andy are inX and wherem(x,y) means the order ofxy inG. We obtainG as a specified quotient of the Coxeter group (X, ) with the appropriate diagram .     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

On a conjecture of Siegel

Letk be an algebraically closed field of characteristic 0 and letf(x, y)k[t][x,y] be a polynomial in two variables with coefficients ink[t]. One is interested in solving the equationf(x,y)=0 with polynomialsx,yk[t]. Two solutions(x,y), (x, y) areproportional ifx/x andy/y are non-zero constants ink and a solution(x,y) isprimitive if the polynomialsx andy have no common root. The main result of this paper is that for a certain class of polynomialsf, which includes Thue equations with sufficiently lacunary exponents, the number of non-proportional, primitive solutions is bounded solely in terms of the number of monomials appearing in the polynomialf(x,y). This verifies the analogue of a conjecture of Siegel for this class of polynomials. The proof is an application of theabc-theorem in function fields to certain determinantal varieties arising from the elimination of the coefficients of the polynomialf(x,y), together with an inductive argument on the numberr of monomials inf(x,y).     

Necessary and sufficient condition of the existence of global Hopf bifurcation for Hamiltonian systems

ForH C ² (,R) where 0 R ²ⁿ ,H (0)=0 and detH(0)0, the paper proves that there is a global Hopf bifurcation fromx=0 for Hamiltonian systemx=JH(x) iffJH(0)possesses purely imaginary eigenvalues. The work improves the corresponding result of J.C.Alexander and J. Yorke (Amer. J. Math., 100 (1978), 263–292).