Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS ₅ ⁻ , one of the two nontrivial central extensions ofS ₅ byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA ₅ as a composition factor, isQ-admissible. This paper is part of a M.Sc. thesis written at the Technion — Israel Institute of Technology, under the supervision of Professor J. Sonn, whom the author wishes to thank for his valuable guidance.     

Sets of degrees of computable fields

Given a Σ₂ (resp. Σ₁) degree of recursive unsolvability a, a computable field (resp. a computable field with a splitting algorithm)F is constructed in any given characteristic, such that the set of dimensions of all finite extensions ofF has degree a. This is part of an M.Sc. Thesis presented at The Hebrew University. The author wishes to express his indebtedness to his Supervisor Professor H. Gaifman for his constant guidance and encouragement. Thanks are also due to Professor G. Sabbagh (University of Paris VII) for several useful suggestions.     

On the projection and macphail constants ofl n p spaces

We prove that the projection and Macphail constants ofl _n ^p (1≦p≦2) are asymptotically equivalent ton ^1/2 andn ^−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces. This note is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss, to whom the author wishes to express his thanks and appreciation.     

Markov processes on a locally compact space

Given a positive contraction,P, on C(X) we define the conservative and dissipative parts ofP and establish some properties which are analogous to known ones from measure theory (see [3]). We also prove a ratio limit theorem for certain processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor S.R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.     

Spectral synthesis in regular Banach algebras

We prove that every locally compact non-discrete abelian groupG contains a compact subsetE such thatA(E) — the restriction algebra ofA(G) toE — admits spectral synthesis, although it contains a closed, regular, self-adjoint subalgebra which is isomorphic to an algebra of infinitely differentiable functions on [−1, 1]. We also give some general results concerning the failure of spectral synthesis in regular Banach algebras. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor Y. Katznelson, to whom the author wishes to express his gratitude for his valuable remarks, and the interest he showed in the preparation of this paper.     

Zeros of abelian integrals for the reversible codimension four quadratic centersQ3r ∩Q4

We study the number of zeros of Abelian integrals for the reversible codimension four quadratic centersQ ₃ ^R ∩Q ₄, when we perturb such systems inside the class of all polynomial systems of degreen. Supported by grants from NSF of China (No. 10101031), Guangdong Natural Science Foundation (No. 001289), and NSF of Sun Yat-sen University for younger teachers. The author wishes to thank Prof. R. Conti and Prof. G. Villari for their discussions, and Dipartimento di Matematica “U. Dini”, Università Degli Studi di Firenze for its support and hospitality during the period when this paper was elaborated. Y. Zhao is grateful to the referee for helpful suggestions.     

Existence of identities inA⊗B

The following theorem is proved: ifA andB are twoPI-algebras over a fieldF, thenA⊗ _F B is aPI-algebra. This paper was written while the author was doing his Ph.D. thesis at the Hebrew University of Jerusalem under the supervision of Professor A. S. Amitsur, to whom the author wishes to express his warm thanks.     

Locally connected simplicial maps

In Propositions 1.6 and 7.6 of his paper onp-group complexes of finite groups [5], Quillen establishes fundamental results comparing the homology and the fundamental group of the order complexes of posetsP, Q admitting a mapf :P →Q of posets with good local behavior. We prove the analogue of Quillen’s results for mapsf :K→L of simplicial complexesK andL in a more general setup. This work was partially supported by BSF 88-00164. The first author is partially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.     

Onσ-finite invariant measures for Markov processes

Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem uuder the supervision of Professor S. R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

Examples of proper,closed, weakly dense subspaces in non locally convexF-spaces

It is shown that anyF-space havingl ^p as a continuous linear image for some 0<p<1 must contain proper closed subspaces which are dense in the weak topology. In particular, when 0<p<1, such subspaces are present inl ^p, and the same is shown to hold in certainF-spaces of analytic functions which nevertheless have enough continuous linear functionals to separate points. This work represents part of the author’s dissertation at the University of Michigan. The author wishes to thank Professor Allen Shields for his guidance and encouragement.     

L ∞ — Limit theorems for Markov processes

A necessary and sufficient condition for convergence of Markov processesL _∞ is given. As a consequence we get a theorem concerning the convergence of Harris processes. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

Groups with trivial virtual automorphism group

We answer a question of A. Lubotzky and A. Mann by constructing examples of infinite groupsG such that every isomorphismα:H →K between subgroupsH andK having finite index inG coincides with the identity on some subgroup of finite index. The structure of such a group is very restricted;G must be virtually a 2-group with finite central derived subgroup andG/G′ elementary abelian. This work was begun while the second author was a visitor at the University of Padova. He wishes to thank the Mathematics Department for its hospitality and the C.N.R. for its financial support.     

A homological characterization of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite. Dedicated to Otto Kerner on the occasion of his 65th birthday. The second author is supported by a grant from NSA. These results were obtained during a visit of the first author at Syracuse University. He would like to thank the second author for the hospitality during his stay.     

Preservation of the injectivity of the Gauss map for general projections

Let X be a smooth irreducible quasi-projective variety of dimension n in P ^N with N ≥ 2n + 2. Let γ be its Gauss map, let be the embedding obtained from the general projection in P ^N and let γ′ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ(Q) ≠ γ(Q′) implies γ′(Q) ≠ γ′ (Q′). We prove that this property holds in the following cases: N≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n−1 and X does contain a linear (n−1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map. The author is affiliated with the University at Leuven as a research fellow. This paper belongs to the FWO-project G.0318.06.     

Tactical problems involving uncertain actions

The purpose of this paper is to solve anm-silent versusn-silent duel with arbitrary accuracy functionsP andQ which are continuously differentiable in [0, 1] with positive derivatives in (0, 1) and such thatP(0)=Q(0)=0,P(1)(0, 1),Q(1)(0, 1). The game can be interpreted as a game in which the players know only that the numbers of their actions have binomial distribution.Part of this paper was written during the author's stay at the Institute of Statistics and Mathematical Economics, University of Karlsruhe, FRG. The author wishes to thank Prof. S. Trybua, Technical University of Wrocaw, Poland, for helpful discussion in preparing the paper and Prof. D. Pallaschke, University of Karlsruhe, FRG, for help in preparing the numerical example.     

Non-singly-generated closed ideals in group algebras

We investigate the problem whether every closed ideal in the group algebrasA(G) is generated by a single function. For the algebrasA _r(R ⁿ),n≧6, we give a negative solution. We also obtain some negative results for general locally compact abelian groups. This paper is a part of the author’s Ph. D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor Y. Katznelson, to whom the author wishes to express his thanks for his helpful guidance, and valuable remarks.     

Uniform embeddings of Banach spaces

It is proved that for 1<-p≤2,L _p(0,1) andl _p are uniformly equivalent to bounded subsets of themselves. It is also shown that for 1<=p<=2, 1≦q<∞,L _p is uniformly equivalent to a subset ofl _q. This is a part of the author’s Ph. D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. The author wishes to thank Professor Lindenstrauss for his guidance.     

Existence of Positive Solutions to Semipositone Singular Dirichlet Boundary Value Problems

The paper presents the conditions which guarantee that for some positive value of μ there are positive solutions of the differential equation （Ф（x＇））＇＋μQ（t, x, x＇） = 0 satisfying the Dirichlet boundary conditions x（0） = x（T） = 0. Here Q is a continuous function on the set [0, T] × （0, ∞） ~ （R ／ {0}） of the semipositone type and Q is singular at the value zero of its phase variables.