Equivaleces between D-categories of Inverse Monoids

There exist two equivalent small categories associated to an inverse monoid which reflect its divisorial structure, called its D-categories. It is well known the relationship between the Green's relation D and the structure, of certain subsemigroups, of an inverse monoid. In this work, an analogous result is established between the Green's relation J and the D-categories of those subsemigroups. They are also given equivalent conditions for the equivalence of the D-categories of two inverse monoids and likewise equivalent conditions for the isomorphism of two inverse monoids in terms of its D-categories. It is proved that for many important classes of inverse monoids the multiplicative structure is determined by the associated category. On the contrary, a sufficient condition to obtain families of counterexamples to the above is provided and three examples are explicitely exhibited.     

The third fundamental form of minimal surfaces in a sphere

We study minimal surfaces in a sphere Sⁿ with regard to the following question: to what extent minimal surfaces in Sⁿ are determined by restrictions on the Gaussian curvature of the Gaussian image in the sense of Obata?     

Permutability of subgroups of G×H that are direct products of subgroups of the direct factors

If M and S are two subgroups of a group G, M and S permute if MS = SM. Furthermore, M is a permutable subgroup of G if M permutes with every subgroup of G. We give necessary and sufficient conditions for M, a subgroup of G, to permute with a subgroup of G 2 H given that G and H are finite groups. The main part of the paper involves the development of a characterization of permutable subgroups of G 2 H that are direct products of subgroups of the direct factors; that is, subgroups that are equal to A 2 B where A \leqq \leqq G and B \leqq \leqq H.     

Eigenvalues of Random Power law Graphs

Many graphs arising in various information networks exhibit the "power law" behavior -the number of vertices of degree k is proportional to k^-# for some positive #. We show that if # > 2.5, the largest eigenvalue of a random power law graph is almost surely(1+ o(1))?m (1+ o(1))\sqrt{m} where m is the maximum degree. Moreover, the klargest eigenvalues of a random power law graph with exponent # have power law distribution with exponent 2# if the maximum degree is sufficiently large, where k is a function depending on #, mand d, the average degree. When 2<#< 2.5, the largest eigenvalue is heavily concentrated at cm^3-# for some constant c depending on # and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and [(d)\tilde] \tilde{d} , the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely (1+ o(1))?{m_k} (1+ o(1))\sqrt{m_k} where m_k is the k-th largest expected degree provided m_k is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.     

Integral Lie powers of a lattice for the cyclic group of order 2

Let L_n denote the n-th homogeneous component of the free Lie ring L(W) on a given \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice W. This paper gives explicit formulae for the multiplicities of the three indecomposable \Bbb ZC₂{{\Bbb Z}}C_{2}-lattices in a Krull-Schmidt decomposition of L_n. In the case where W is a free \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice, L_n is shown to have no non-zero direct summand on which C₂ acts trivially - this extends a result of R. M. Bryant for the special case where W is the regular \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice. As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of C₂ is determined.     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.     

The VT-operator Semigroup for Two Kinds of Regular Semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. For every rho element of Con S there exists a greatest congruence rhoV [smallest congruence rhov] on S such that the idempotent (rhoV/rho)-classes [(rho/rhov)-classes] are rectangular bands, and a greatest congruence rhoT [smallest congruence rhot] on S such that the idempotent (rhoT/rho-classes [(rho/rhot-classes] are groups. The subsemigroup of the transformation semigroup on Con S generated by the transformation rho → rhoV, rho → rhov, rho → rhoT, and rho → rhot, rho element of Con S, is investigated for orthodox semigroups and cryptogroups. It is shown that in this case this so-called Vt-operator semigroup Omega(S) contains 17 elements at most. A 17-element Vt-operator semigroup Omega(F) is realized for some regular orthogroup F.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

The finiteness of the reward function and the optimal value function in Markov decision processes

This paper studies the discrete time Markov decision processes (MDP) with expected discounted total reward, where the state space is countable, the action space is measurable, the reward function is extended real-valued, and the discount rate may be any real number. Two conditions (GC) and (C) are presented, which are weaker than that presented in literature. By eliminating some worst actions, the state space S can be partitioned into sets S_X, S_mX, S₀, on which the optimal value function equals +X , mX or is finite, respectively. Furthermore, the validity of the optimality equation is shown when its right hand side is well defined, especially, when it is restricted to the subset S₀. The reward function r (i, a) is finite and bounded above in a for each i]S₀. Finally, some sufficient conditions for (GC) and (C) are given.     

Topological algebras with ascending or descending chain condition

We examine some topological algebras with ascending or descending chain condition. We prove that a commutative noetherian F-algebra is necessarily a Q-algebra. We characterize noetherian F-algebras which are Q-algebras among those whose left ideals are closed. We show that any commutative artinian m-convex algebra is finite dimensional.     

The Fredholm determinant of an almost periodic arithmetical function

For functions f in the space of 2-limit-periodic functions on \Bbb Z{\Bbb Z}, the Fredholm determinant D(f; z) is defined as an everywhere convergent power series whose coefficients are mean values of certain determinants involving the values of f. A Weierstraß product representation is given for D(f; z) which shows in particular that the zeros of D(f; z) are the reciprocals of the nonvanishing Fourier coefficients of f.     

Glatte p-Gruppen

A p-group G is called smooth if there is a smooth chief series in G, that is a chief series in which any two factor groups of the same order are isomorphic. It is shown that in most cases non-abelian smooth p-groups are of nilpotency class two and that a smooth p-group is not much larger than the largest abelian group in its smooth chief series.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

On f-injective modules

In this paper, the notions of f-injective and f^*-injective modules are introduced. Elementary properties of these modules are given. For instance, a ring R is coherent iff any ultraproduct of f-injective modules is absolutely pure. We prove that the class S^* \Sigma^* of f^*-injective modules is closed under ultraproducts. On the other hand, S^* \Sigma^* is not axiomatisable. For coherent rings R, S^* \Sigma^* is axiomatisable iff every c₀ \chi_0 -injective module is f^*-injective. Further, it is shown that the class S \Sigma of f-injective modules is axiomatisable iff R is coherent and every c₀ \chi_0 -injective module is f-injective. Finally, an f-injective module H, such that every module embeds in an ultraprower of H, is given.     

Pressure in Classical Statistical Mechanics and Interacting Brownian Particles in Multi-dimensions

. Let P(u) denote the pressure at the density u defined in the Gibbs statistical mechanics determined by a 2 body potential U (_qi - _qj). The function U(x) is supposed rotationally invariant and of finite range but may be unbounded about the origin. We establish a representation of P(u) by means of the law of large numbers for the virial ?_i,j q_i ·? U(q_i-q_j)\sum_{i,j} q_i \cdot {\nabla} U(q_i-q_j), whether or not there occur phase transitions. This result on P(u) is motivated by a study of the hydrodynamic behavior of a system of a large number of interacting Brownian particles moving on a d-dimensional torus (d = 1, 2, ...) in which the interaction is given by binary potential forces of potential U. Employing our representation of P(u), we also show that in the hydrodynamic limit of such a system there arises a non linear evolution equation of the form u_t = ¹/₂ DP(u)u_t = {1\over2} \Delta P(u) under a certain hypothetical postulate concerning concentration of particles.     

Adapting Some Ideas from Stochastic Control Theory to Studying the Heat Equation in Closed Smooth Domains

Sufficient conditions on bounded domains D ² R ^d={(t,x)} of class C⁴ are given under which solutions of the heat equation u_t=j u+f in D have continuous second-order derivatives with respect to (t,x) in D^- . The equation is supplemented with C⁴ boundary data and it is assumed that f] C² .     

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(_{x, t}) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x₀, t) with non-zero derivative at a single sensor location x = x₀ is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x, t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.     

More on quotient polytopes

Summary. In an earlier paper, it was shown that every abstract polytope is a quotient Q = M(W)/N {\cal Q} = {\cal M}(W)/N of some regular polytope M(W) {\cal M}(W) whose automorphism group is W, by a subgroup N of W. In this paper, attention is focussed on the quotient Q {\cal Q} , and various important structures relating to polytopes are described in terms of N ', the stabilizer of a flag of the quotient under an action of W (the 'flag action'). It is pointed out how N ' may be assumed without loss of generality to equal N. The paper also shows what properties of N ' yield polytopes which are regular, section regular, chiral, locally regular, or locally universal. The aim is to make it more practical to study non-regular polytopes in terms of group theory.