Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

Reduced products which are not saturated

Summary IfT is a complete theory of Boolean algebra, then we writeA ⊲_T B to denote that for every cardinal κ and every κ-regular filter over a setI such that the Boolean algebra 2 _F ^I of all subsets ofI reduced byF is a model ofT, the reduced powerA _F ^I isK ⁺-saturated wheneverB _F ^I isK ⁺-saturated. The relation ⊲_T generalizes the relation ◃ introduced by Keisler. As in the case of Keisler's ◃ it happens that ⊲_T’s are relations between complete theories, i.e. ifA≡B thenA ⊲_T B andB ⊲_T A. In this paper some examples of theories which are maximal (minimal) with respect to ⊲_T’s are provided and the relations ⊲_T are compared with each other. Presented by J. Mycielski     

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = b^σ and to be inverse-fused if a =(b^-1)^σ for some σ ? Aut(G). The fusion class of a ? G is the set {a^σ | σ ? Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:

(i) G has at most two fusion classes of order i for every i (23 examples); and

(ii) any two elements of G of the same order are fused or inversenfused.

The examples in case (ii) are: A₅, A₆,L₂(7),L₂(8), L₃(4), Sz(8), M₁₁ and M₂₃An application is given concerning isomorphisms of Cay ley graphs.     

On the Convexity of the Heinz Means

Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm | | | ·| | | ,{\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert ,}

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

Linear equations and sets of integers

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then |A|\leqq 2^-ck/(logk)2N|A|\leqq 2^{-ck/{(\log k)}^{2}}N, for some absolute constant c>0, provided that N is large enough.     

Finite groups with some given systems of Xm‐semipermutable subgroups

Let X be a nonempty subset of a group G. We call a subgroup A of G an X_m‐semipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T₁ of T there exists an element x ∈ X such that AM^x = M^xA. In this paper, we study the structure of finite groups with some given systems of X_m‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Sumsets being squares

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?[0,N] with |B|?(logN)^15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?[0,N] with |B|?(logN)^9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.     

An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S^′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS^′ denotes the subsequence of S obtained by deleting all terms in S^′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A₁,…,A_n, such that

then there exists a subsequence S^′ of S, with length |S^′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||A_i|−|A_j||≤1 for all i and j, or if |A_i|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S^′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge.     

Compact complex homogeneous manifolds with large automorphism groups

Let X be a compact complex homogeneous manifold and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. It is well-known that the dimension of Aut(X) is bounded by an integer that depends only on n=dim X. Moreover, if X is K?hler then dimAut (X)≤n(n+2) with equality only when X is complex projective space. In this article examples of non-K?hler compact complex homogeneous manifolds X are given that demonstrate dimAut(X) can depend exponentially on n. Let X be a connected compact complex manifold of dimension n. The group of holomorphic automorphisms of X, Aut(X), is a complex Lie group [3]. For a fixed n>1, the dimension of Aut(X) can be arbitrarily large compared to n. Simple examples are provided by the Hirzebruch surfaces F _m , m∈N, for which dimAut(F _m )=m+5, see, e.g. [2, Example 2.4.2]. If X is homogeneous, that is, any point of X can be mapped to any other point of X under a holomorphic automorphism, then the dimension of the automorphism group of X is bounded by an integer that depends only on n, see [1, 2, 6]. The estimate given in [2, Theorem 3.8.2] is roughly dimAut(X)≤(n+2) ⁿ . For many classes of manifolds, however, the dimension of the automorphism group never exceeds n(n+2). For example, it follows directly from the classification given by Borel and Remmert [4], that if X is a compact homogeneous K?hler manifold, then dimAut(X)≤n(n+2) with equality only when X is complex projective space P ⁿ . It is an old question raised by Remmert, see [2, p. 99], [6], whether this same bound applies to all compact complex homogeneous manifolds. In this note we show that this is not the case by constructing non-K?hler compact complex homogeneous manifolds whose automorphism group has a dimension that depends exponentially on n. The simplest case among these examples has n=3m+1 and dimAut(X)=3m+3 ^m , so the above conjectured bound is exceeded when n≥19. These manifolds have the structure of non-trivial fiber bundles over products of flag manifolds with parallelizable fibers given as the quotient of a solvable group by a discrete subgroup. They are constructed using the original ideas of Otte [6, 7] and are surprisingly similar to examples found there. Generally, a product of manifolds does not result in an automorphism group with a large dimension relative to n. Nevertheless, products are used in an essential way in the construction given here, and it is perhaps this feature that caused such examples to be previously overlooked. Oblatum 13-X-97 & 24-X-1997     

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on     

On Vertices of outdegree n in minimally n‐connected digraphs

Let |D| and |D|⁺_n denote the number of vertices of D and the number of vertices of outdegree n in the digraph D, respectively. It is proved that every minimally n‐connected, finite digraph D has |D|⁺_n ≥ n + 1 and that for n ≥ 2, there is a c_n > 0 such that for all minimally n‐connected, finite digraphs D. Furthermore, case n = 2 of the following conjecture is settled which says that every minimally n‐connected, finite digraph has a vertex of indegree and outdegree equal to n. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 129–144, 2002     

Local and Global LIPSCHITZian Mappings on Ordered Metric Spaces

We consider the set ?? of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces E_n, n ? T , with fixed local characteristics. For T = N we show that ?? has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z , ?? has at most N_-∞ · N_∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document}. We give examples, that for T = N for any number k, 1 ≦ k ≦ N_∞, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ l ≦ N_-∞, 1 ≦ k ≦ N_∞ occurs. We describe the inner structure of ??, the behaviour at infinity and the connection between the one-sided and the two-sided tail-fields. Simple examples of Markov fields which are no Markov processes are given.