Two-parameter family of infinite-dimensional diffusions on the Kingman simplex

We construct a two-parameter family of diffusion processes X _α,θ on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers. For α = 0, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981) in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present paper, we extend Ethier and Kurtz’s main results to the two-parameter case. Namely, we show that the (two-parameter) Poisson-Dirichlet distribution PD(α,θ) is the unique stationary distribution for the process X _α,θ and that the process is ergodic and reversible with respect to PD(α, θ). We also compute the spectrum of the generator of X _α,θ . The Wright-Fisher diffusions on finite-dimensional simplices turn out to be special cases of X _α,θ for certain degenerate parameter values.     

Two generator subalgebras of Lie algebras

In [Thompson, J., 1968, Non-solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383–437.], Thompson showed that a finite group G is solvable if and only if every two-generated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [Grunewald et al., 2000, Two-variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161–176; 2003. Journal of Mathematical Sciences (New York), 116, 2972–2981.] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability.     

The Menger property for infinite ordered sets

It is shown that, if an ordered set P contains at most k pairwise disjoint maximal chains, where k is finite, then every finite family of maximal chains in P has a cutset of size at most k. As a corollary of this, we obtain the following Menger-type result that, if in addition, P contains k pairwise disjoint complete maximal chains, then the whole family, M (P), of maximal chains in P has a cutset of size k. We also give a direct proof of this result. We give an example of an ordered set P in which every maximal chain is complete, P does not contain infinitely many pairwise disjoint maximal chains (but arbitrarily large finite families of pairwise disjoint maximal chains), and yet M (P) does not have a cutset of size <x, where x is any given (infinite) cardinal. This shows that the finiteness of k in the above corollary is essential and disproves a conjecture of Zaguia.     

Structure of Quantum Doubles of Finite Clifford Monoids

Abstract

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

Cayley Digraphs of Finite Simple Groups with Small Out-Valency

Abstract

For a group G and a subset S of G which does not contain the identity of G, the Cayley digraph Cay(G, S) is called normal if R(G) is normal in Aut(Γ). In this paper, we investigate the normality of Cayley digraphs of finite simple groups with out-valency 2 and 3. We give several sufficient conditions for such Cayley digraphs to be normal. By using this result, we consider the digraphical regular representations of finite simple groups.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

Deciding finiteness for matrix groups over function fields

LetF be a field andt an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n,F(t)) is finite. WhenF is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n,F), for which the results of [1] can be applied. WhenF is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more efficient randomized algorithms. Supported in part by NSF DMS Awards 9404275 and Presidential Faculty Fellowship.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

On Finite Quasi-ℱ-Groups

Let G be a finite group and ? a saturated formation of finite groups. We say that G is quasi-?-group if for every ?-eccentric chief factor H/K of G and every x ∈ G, x induces an inner automorphism on H/K. In particular, we have the concepts of quasisoluble, quasisupersoluble, quasimetanilpotent groups, and so on. In this article, we obtain some results about the quasi-?-groups and use them to give the conditions under which a group is quasisoluble, quasimetanilpotent, or quasisupersoluble.     

Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems

Let a linear regression model be given with an experimental region [a, b] R and regression functions f ₁, ..., f _d+1 : [a, b] R. In practice it is an important question whether a certain regression function f _d+1, say, does or does not belong to the model. Therefore, we investigate the test problem H ₀ : "f _d+1 does not belong to the model" against K : "f _d+1 belong to the model" based on the least-squares residuals of the observations made at design points of the experimental region [a, b]. By a new functional central limit theorem given in Bischoff (1998, Ann. Statist. 26, 1398–1410), we are able to determine optimal tests in an asymptotic way. Moreover, we introduce the problem of experimental design for the optimal test statistics. Further, we compare the asymptotically optimal test with the likelihood ratio test (F-test) under the assumption that the error is normally distributed. Finally, we consider real change-point problems as examples and investigate by simulations the behavior of the asymptotic test for finite sample sizes. We determine optimal designs for these examples.     

Bayesian linear prediction in finite populations

In this paper, Bayesian linear prediction of the total of a finite population is considered in situations where the observation error variance is parameter dependent. Connections with least squares prediction (Royall (1976, J. Amer. Statist. Assoc., 71, 657–664)) in mixed linear models (Theil (1971, Principles of Econometrics, Wiley, New York)), are established. Extensions to the case of dynamic (state dependent) superpopulation models are also proposed.     

Consistency and Asymptotic Normality of Estimators in a Proportional Hazards Model with Interval Censoring and Left Truncation

Satten et al. (1998, J. Amer. Statist. Assoc., 93, 318–327) proposed an approach to the proportional hazards model for interval censored data in which parameter estimates are obtained by solving estimating equations which are the score equations for the full data proportional hazards model, averaged over all rankings of imputed failure times consistent with the observed censoring intervals. In this paper, we extend this approach to incorporate data that are left-truncated and right censored (dynamic cohort data). Consistency and asymptotic normality of the estimators obtained in this way are established.     

Confidence bands in nonparametric regression with length biased data

In this paper we deduce a confidence bands construction for the nonparametric estimation of a regression curve from length biased data, where a result from Bickel and Rosenblatt (1973,The Annals of Statistics,1, 1071–1095) is adapted to this new situation. The construction also involves the estimation of the variance of the local linear estimator of the regression, where we use a finite sample modification in order to improve the performance of these confidence bands in the case of finite samples.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Two-Vector Spaces and Groupoids

This paper describes a relationship between essentially finite groupoids and two-vector spaces. In particular, we show to construct two-vector spaces of Vect-valued presheaves on such groupoids. We define two-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation—a weak functor—from Span(FinGpd) (the bicategory of essentially finite groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.