Compact domination for groups definable in linear o-minimal structures

We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G ⁰⁰, m, π), where m is the Haar measure on G/G ⁰⁰ and π : G → G/G ⁰⁰ is the canonical group homomorphism.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

Linear Groups Definable in o-Minimal Structures

We study subgroups G of GL(n, R) definable in o-minimal expansions M = (R, +, · ,…) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, power functions, or an exponential function definable in M. (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser–Tits conjecture for real closed fields.     

Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

Colorings at minimum cost

We define by min_c∑_{u,v}∈E(G)|c(u)−c(v)| the min-costMC(G) of a graph G, where the minimum is taken over all proper colorings c. The min-cost-chromatic numberχ_M(G) is then defined to be the (smallest) number of colors k for which there exists a proper k-coloring c attaining MC(G). We give constructions of graphs G where χ(G) is arbitrarily smaller than χ_M(G). On the other hand, we prove that for every 3-regular graph G^′, χ_M(G^′)≤4 and for every 4-regular line graph G^″, χ_M(G^″)≤5. Moreover, we show that the decision problem whether χ_M(G)=k is -hard for k≥3.     

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group G, a parabolic subgroup P _J , and its unipotent radical U _J , we determine the ring structure of the cohomology ring H^?((U _J )₁, k). We also obtain new results on computing H^?((P _J )₁, L(??)) as an L _J -module where L(??) is a simple G-module with highest weight ?? in the closure of the bottom p-alcove. Finally, we provide generalizations of all our results to small quantum groups at a root of unity.     

M0(G)-boundaries are M(G)-boundaries

Let M(G) denote the convolution algebra of finite regular complex-valued Borel measures on a locally compact abelian group G, and let M₀(G) be the ideal consisting of those measures whose Fourier-Stieltjes transforms vanish at infinity. Then there is a natural inclusion of the maximal ideal space Δ₀ of M₀(G) in the maximal ideal space of M(G). The main result states that any subset of Δ₀ which is a boundary for M₀(G) is a boundary for M(G). An immediate corollary is that the ?ilov boundary of M₀(G) is dense in the ?ilov boundary of M(G).     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Erd?s-Zaks all divisor sets

Let ? _n be the finite cyclic group of order n and S ? ? _n . We examine the factorization properties of the Block Monoid B(? _n , S) when S is constructed using a method inspired by a 1990 paper of Erd?s and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {M _i } _i=1 ^n?1 which contains all the non-primary irreducible Blocks (or atoms) of B(? _n , S). This construction yields a weakly half-factorial Block Monoid (see [9]). After developing some basic properties of the set {M _i } _i=1 ^n?1 , we examine in Section 3 the connection between these irreducible blocks and the Erd?s-Zaks notion of ??splittable sets.?? In particular, the Erd?s-Zaks notion of ??irreducible?? does not match the classic notion of ??irreducible?? for the commutative cancellative monoids B(? _n , S). We close in Sections 4 and 5 with a detailed discussion of the special properties of the blocks M1 with an emphasis on the case where the exponents of M ₁ take on extreme values. The work of Section 5 allows us to offer alternate arguments for two of the main results of the original paper by Erd?s and Zaks.     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$ with 1 < n ₁ | ?? | n _r be a finite abelian group, d*(G) = n ₁ +??+n _r ?r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ?? d*(G), and the standing conjecture is that equality holds for G = C _n ^r . We show that equality does not hold for C ₂ ?? C _2n ^r , where n ?? 3 is odd and r ?? 4. This gives new information on the structure of extremal zero-sum free sequences over C _2n ^r .     

The evolution of the min-min random graph process

We study the following min-min random graph process G=(G₀,G₁,…): the initial state G₀ is an empty graph on n vertices (n even). Further, G_M+1 is obtained from G_M by choosing a pair {v,w} of distinct vertices of minimum degree uniformly at random among all such pairs in G_M and adding the edge {v,w}. The process may produce multiple edges. We show that G_M is asymptotically almost surely disconnected if M≤n, and that for M=(1+t)n, constant, the probability that G_M is connected increases from 0 to 1. Furthermore, we investigate the number X of vertices outside the giant component of G_M for M=(1+t)n. For constant we derive the precise limiting distribution of X. In addition, for n⁻¹ln⁴n≤t=o(1) we show that tX converges to a gamma distribution.     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

An extension of a theorem of Alan Camina on conjugacy class sizes

Let G be a finite group. Let n be a positive integer and p a prime coprime to n. In this paper we prove that if the set of conjugacy class sizes of primary and biprimary elements of group G is {1,p ^a , p ^a n}, then G ≌ G ₀ × H, where H is abelian and G ₀ contains a normal subgroup M × P ₀ of index p. Moreover, M × P ₀ is the set of all elements of G ₀ of conjugacy class sizes p ^a or 1, where M is an abelian π(n)-subgroup of G ₀ and P ₀ is an abelian p-subgroup of G ₀, neither being central in G. Finally, p ^a = p and P/P ₀ acts fixed-point-freely on M and ?(P) ≤ Z(P). This is an extension of Alan Camina’s theorems on the structure of groups whose set of conjugacy class size is {1,p ^a , p ^a q ^b }, where p and q are two distinct primes.     

Automorphism groups of finite groupoids

Given a finite set M of size n and a subgroup G of Sym(M), G is pertinent iff it is the automorphism group of some groupoid ??M; *??. We examine when subgroups of Sym(M) are and are not pertinent. For instance, A _n , the alternating group on M, is not pertinent for n > 4. We close by indicating a natural extension of our ideas, which relates to a question of M. Gould.     

Complexity of a 3-dimensional assignment problem

We show that a certain 3-dimensional assignment problem is NP-complete. To do this we show that the following problem is NP-complete: given bipartite graphs G₁, G₂ with the same sets of vertices, do there exist perfect matchings M₁, M₂ of G₁, G₂ respectively such that M₁ ∩ M₂ =Ø?     

The basis monomial ring of a matroid

We define the basis monomial ring M_G of a matroid G and prove that it is Cohen-Macaulay for finite G. We then compute the Krull dimension of M_G, which is the rank over Q of the basis-point incidence matrix of G, and prove that dim B_G ≥ dim M_G under a certain hypothesis on coordinatizability of G, where B_G is the bracket ring of G.