Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.?We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal. Received February 8, 2000; accepted in final form December 23, 2000.     

COMPARATIVE PROBABILITY ON THE C* ALGEBRA OF COMPACT OPERATORS

Abstract

Let A be a von Neumann algebra on a Hilbert space H and let P(A) denote the projections of A. A comparative probability (CP) on A (or more correctly on P(A)) is a preorder ? on P(A) satisfying:

0 ? P ? P ε P(A) with Q ≠ 0 for some Q ε P(A).

If P, Q ε P(A) then either P ? Q or Q ? P.

If P, Q and R are all in P(A) and P⊥R, Q⊥R, then P ? Q ? P + R ? Q + R.

Let τ be any of the usual locally convex topologies on A. We say ? is τ continuous if the interval topology induced on P(A) by ? is weaker than the τ topology on P(A). If μ an additive (completely additive) measure on P(A) then μ induces a uniformly (weakly) continuous CP ?μ on P(A) given by P ?_μ Q if μ(P) ? μ(Q). We show that if A is the C* algebra C(H) of compact operators on an infinite dimensional Hilbert space H, the converse is true under an extra boundedness condition on the CP which is automatically satisfied whenever the identity is present in A = P(C(H)).     

Ramified partition algebras

For each natural number n, poset T, and |T|–tuple of scalars Q, we introduce the ramified partition algebra P _n ^(T) (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T|=1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representation theory in case ${{T=\underline{{2}}:=({1,2},\leq)}}$. We show that ${{P_n^{(\underline{{2}})$ (Q) is quasi–hereditary over field k when Q ₁ Q ₂ is invertible in k and k is such that certain finite group algebras over k are semisimple (e.g. when k is algebraically closed, characteristic zero). Under these conditions we determine an index set for simple modules of ${{P_n^{(\underline{{2}})$ (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that ${{P_n^{(\underline{{2}})$ (Q) is not semisimple for sufficiently large n, but that it is generically semisimple for all n. We construct tensor space representations of certain non–semisimple specializations of ${{P_n^{(\underline{{2}})$ (Q), and show how to use these to build clock model transfer matrices [24] in arbitrary physical dimensions. Sadly Ahmed died before this work was completed. His memory lives on.     

Square-free Lucas <Emphasis Type="Italic">d</Emphasis>-pseudoprimes and Carmichael-Lucas numbers

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P, Q) such that the rank of N in U(P, Q) is exactly (N − ε(N))/d, where ε is the signature of U(P, Q). We prove here that all but a finite number of Lucas d-pseudoprimes are square free. We also prove that all but a finite number of Lucas d-pseudoprimes are Carmichael-Lucas numbers.     

Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|² < |G|) or G is a direct product of 2-transitive Frobenius groups.     

Characterization of 3 D 4(q)

The author defined the concept order components in [2] and gave a new characterization of sporadic simple groups by their order components in [7]. Afterwards the following groups were characterized by the author: G₂(q), q = 0 (mod 3)^[8]; E₈(q)^[9]; Suzuki-Ree groups^[10]; PSL₂(q)^[11]. Here the author will continue such kind of characterization and prove that:Theorem 1. Let G be a finite group, M = ³D₄(q). If G and M has the same order components, then G M.And the following theorems follows from Theorem 1.Theorem 2. (Thompsons Conjecture) Let G be a finite group, Z(G) = 1,M = ³D₄(q). If N(G) = N(M), then G M. (ref. [6])Theorem 3. (Wujie Shi) Let G be a finite group, M = ³D₄(q). If|G| = |M|, _e(G) = _e(M), then G M. (ref. [15])All notations are the same as in [2]. According to the classification theorem of finite simple groups, [12] and [13], we can list the order components of finite simple groups with nonconnected prime graphs in Tables 1-4 (ref. [5]).American Mathematics Society Classification 20D05 20D60The author is indebted to Fred and Barbara Kort Sino-Israel Postdoctoral Programme for supporting my post-doctoral position (1999.10-2000.10) at Bar-Ilan University, also to Emmy Noether Mathematics Institute and NSFC for partially financial support.     

The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence D, satisfying certain conditions. Molloy and Reed defined a parameter Q = Q(D) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R( \begin{align*}\mathcal {D}\end{align*}) and prove that if |Q| = O(n^‐1/3R^2/3) then, with high probability, the size of the largest component of the random graph will be of order Θ(n^2/3R^‐1/3). If |Q| is asymptotically larger than n^‐1/3R^2/3 then the size of the largest component is asymptotically smaller or larger than n^2/3R^‐1/3. Thus, we establish that the scaling window is |Q| = O(n^‐1/3R^2/3). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A characterization of the n-cube by convex subgraphs

We prove that if Γ is a finite connected graph having the same convex subgraphs as the graph of the n-dimensional cube Q_n (n?3), then |V(Γ)|?|V(Q_n)|. Moreover, if |V(Γ)|=|V(Q_n)|, Γ is isomorphic to Q_n.     

Polynomial Time Uniform Word Problems

We have two polynomial time results for the uniform word problem for a quasivariety Q: (a) The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. (b) Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S(Q*) and the weak embedding closure S?(Q*) of Q* is finitely axiomatizable then the uniform word problem for Q is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length y ? \Bbb Q(b) y \notin {\Bbb Q}(\beta) . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.     

On Connected Transversals to Dihedral Subgroups of Order 2p n

Let G be a group with a dihedral subgroup H of order 2pⁿ, where p is an odd prime. We show that if there exist H-connected transversals in G, then G is a solvable group. We apply this result to the loop theory and show that if the inner mapping group of a finite loop Q is dihedral of order 2pⁿ, then Q is a solvable loop.1991 Mathematics Subject Classification: 20D10, 20N05     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

Characterization of finite simple group <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2)

Let G be a finite group, and let π _e (G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M), where M = D _n (2) and n is even.     

FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract

This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor¹ (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.     

On multiplication groups of relatively free quasigroups isotopic to Abelian groups

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group Mlt Q is a Frobenius group. Conversely, if Mlt Q is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an Abelian group. If Q is, in addition, finite, then it must be a central quasigroup (a T-quasigroup).This work was supported by institutional grant MSM 113200007, and by Grant Agency of Czech Republic, grant number 201/99/0263.     

Realisierbarkeit von darstellungen endlicher gruppen in einheitswurzelkörpern

Zusammenfassung LetD:G→GL(n,C) be an irreducible linear representation of a finite groupG with the characterX. IfD is realizible in Q(ξ _m ) and Q(ξ _m′ ) we give a condition for then realizability ofD in Q(ξ_(m′)). If the degreen is a prime ≠ 2, we show thatD realizible in Q(ξ _f ), wheref is the conductor of the abelian extensionQ(X)/Q.     

On finite groups whose derived subgroup has bounded rank

Let G be a finite group with derived subgroup of rank r. We prove that |G: Z ₂(G)| ≤ |G′|^2r . Motivated by the results of I. M. Isaacs in [5] we show that if G is capable then |G: Z(G)| ≤ |G′|^4r . This answers a question of L. Pyber. We prove that if G is a capable p-group then the rank of G/Z(G) is bounded above in terms of the rank of G′.     

SUBDIRECTLY IRREDUCIBLE LEFT SELF DISTRIBUTIVELY GENERATED ALGEBRAS

Abstract

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R _Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R _Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R _Δ(F) is solvable.     

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.