On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

Transitivity and full transitivity for p-local modules

A p-local module M is called (fully) transitive if for all x,y ? Mx,y\in M with U_M(x) = U_M(y) ( U_M(x)\leqq U_M(y)U_M(x)\leqq U_M(y)) there exists an automorphism (endomorphism) of M which maps x onto y. In this paper we examine the relationship of these two notions in the case of p-local modules. We show that a module M is fully transitive if and only if M?MM\oplus M is transitive in the case where the divisible part of M/tMM/tM has rank at most one. Moreover, we show that for the same class of modules transitivity implies full transitivity if p > 2. This extends theorems of Files, Goldsmith and of Kaplansky for torsion p-local modules.     

A note on operator Banach spaces containing a complemented copy of c0

Let X and Y Banach spaces. Two new properties of operator Banach spaces are introduced. We call these properties "boundedly closed" and "d-boundedly closed". Among other results, we prove the following one. Let U(X, Y){\cal U}(X, Y) an operator Banach space containing a complemented copy of c₀. Then we have: 1) If U(X, Y){\cal U}(X, Y) is boundedly closed then Y contains a copy of c₀. 2) If U(X, Y){\cal U}(X, Y) is d-boundedly closed, then X^* or Y contains a copy of c₀.     

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.     

Uncorrelatedness and correlatedness of powers of random variables

Let x₁, ?, x_n \xi_1, \ldots, \xi_n be random variables and U be a subset of the Cartesian product \mathbbZ₊ⁿ, \mathbbZ₊ \mathbb{Z}_+^n, \mathbb{Z}_+ being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if¶¶E(x₁^j₁ ?x_n^j_n) = E(x₁^j₁) ?E(x_n^j_n) ? (j₁, ... ,j_n) ? U. \textbf {E}\big(\xi_1^{j_1} \cdots \xi_n^{j_n}\big) = \textbf {E}\big(\xi_1^{j_1}\big) \cdots \textbf {E}\big(\xi_n^{j_n}\big) \iff (j_1, \dots ,j_n) \in U. ¶It is proved that for an arbitrary subset U \subseteqq \mathbbZ₊ⁿ U \subseteqq \mathbb{Z}_+^n containing all points with 0 or 1 non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.     

Graphs whose strong endomorphism monoids are regular

Let X be a graph, S End X be its strong endomorphism monoid. It is proved that S End X is a regular monoid if and only if the canonical strong factor graph U of X contains no proper subgraph which is isomorphic to U. The result generalizes that of U. Knauer about the regularity of strong endomorphism monoids of graphs.     

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

Linear groups over a skew field of quaternions containing the group T_{n}(A, \Phi) over a subsfield

We study the subgroups of GL_n(D) (n \geqq 3) GL_{n}(D) (n \geqq 3) over a skew field of quaternions D that comprise the subgroup of the unitary group U_n(A, F) U_{n}(A, \Phi) over a subsfield A \subseteqq D A \subseteqq D generated by all transvections in U_n(A, F) U_{n}(A, \Phi) .     

Surfaces with affine Gauss-Kronecker curvature zero

We classify surfaces with affine Gauss-Kronecker curvature zero in locally and globally. If x:M ? A³x:M \rightarrow A^3 is Euclidean complete and its affine conormal surface U:M ? \Bbb R³U:M \rightarrow \Bbb R^3 is complete, thenx(M) is an elliptic paraboloid or unimodular affine equivalent to a surface determined by equation x²₁-x²₂+C² x^2_1-x^2_2+C^2\,cos h²x₃=0\,\hbox {h}^2x_3=0 where x₂ \leqq Cx_2 \leqq C and C is a positive constant.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

Wilson's functional equation on P3-groups

Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy^-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P₃-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P₃-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x^-1y^-1xy, has the order one or two.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.     

On a Conjecture of Kemnitz

A classic theorem of Erdis, Ginzburg and Ziv states that in a sequence of 2n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_nZ_n stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from Z_n^dZ_n^d (the direct sum of d copies of Z_nZ_n) there is a subsequence of length n whose sum is 0 in Z_n^dZ_n^d. Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that s(p,2) ￡ 4p - 2s(p,2) \le 4p - 2 holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that s(n, 2) ￡ (41/10)ns(n, 2) \le (41/10)n. The proof uses an algebraic approach.     

Total Dual Integrality of Matching Forest Constraints

Let G=(V, E, A) be a mixed graph. That is, (V, E) is an undirected graph and (V, A) is a directed graph. A matching forest (introduced by R. Giles) is a subset F of EèAE\cup A such that F contains no circuit (in the underlying undirected graph) and such that for each v ? Vv\in V there is at most one e ? Fe\in F such that v is head of e. (For an undirected edge e, both ends of e are called head of e.) Giles gave a polynomial-time algorithm to find a maximum-weight matching forest, yielding as a by-product a characterization of the inequalities determining the convex hull of the incidence vectors of the matching forests. We prove that these inequalities form a totally dual integral system. It is equivalent to an ``all-integer' min-max relation for the maximum weight of a matching forest. Our proof is based on an exchange property for matching forests, and implies Giles' characterization.     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each i 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)E \neq E(k), then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n^th cosyzygy has its Tor-depth equal to the depth of the ring whenever n 3 dn \geq d.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

How often can a given distance occur in a finite set of integers?

Let n, a, d be natural numbers and A a set of integers of the closed interval [0, n] with | A | = a. Then we establish sharp lower and upper bounds for the number of pairs (x,y) ? A×A(x,y)\in A\times A for which y - x = d. Roughly spoken, we investigate how often a distance d can occur in A.     

A characterization of infinite 3-abelian groups

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)³ = a³b³ for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)³ = x³y³.