A collection of n + 1 convex subsets of a Euclidean space E will be called an n-set in E provided each n of the sets have a common interior point although the intersection of all n + 1 interiors is empty. It is well-known that if {C0,C1} is a 1-set, then C0 and C1 can be separated by a hyperplane.

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,

for a connected graph G of n vertices and m edges, where

and ﹛x﹜ denotes the smallest integer ≧ x.

We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.

Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

A net {aα} of positive, norm one elements of a C*-algebra A excises a state f of A if

This notion has been used explicitly by the second author [4, 5, 6] for pure states, but the present paper will explore it more fully. The name is motivated by the following example. Let K be the unit disk in the complex plane, A = C(K) and f(a) = a(0). Define an(reiθ) = ϕn(r), where

Note that the sets {t ∊ K:an(t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence {an} “cuts out” the state f and, in the limit, isolates it from all other states.

A model subspace ${{K}_{\Theta }}$ of the Hardy space ${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane ${{\mathbb{C}}_{+}}$ is ${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where $\Theta $ is an inner function in ${{\mathbb{C}}_{+}}$. A function $\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for ${{K}_{\Theta }}$ if there exists an $f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta $'s some parts of Adm $\Theta $ (the set of all admissible majorants for ${{K}_{\Theta }}$) are explicitly described. These descriptions depend on the rate of growth of arg $\Theta $ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.

These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .

Let K be a (commutative) field and n be a positive integer. Consider the K-algebra E = Mat (n, K) of all n X n matrices over K, and the corresponding general linear group GL(n, K). We shall define the set R of rigid mappings of E to consist of all a in GLK(E) which can be written in one of two possible forms: either xσ= axb for all x ϵ E or xσ = ax'b for all x ϵ E (where a and b are fixed elements of GL(n, K) and x’ denotes the transpose of x).

Let E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p ≤ x such that E has supersingular reduction mod p is greater than for any positive δ and x sufficiently large. Here logkx is defined recursively as log(logk-1 x) and log1x = logx. We also establish several results related to the Lang-Trotter conjecture.

In this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar´e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Let be a category with nice factorization-properties. If a functor G: —> which has a left-adjoint behaves nice with respect to factorizations then it can be shown quite easily that G behaves well in many other respects, especially that it lifts nice properties from into .

In this paper we study generalized homomorphisms between two algebras, namely the binary relations whose graphs are subalgebras of the direct product of the given algebras. In 1897 Goursat proved that every subgroup of the direct product of two groups is determined by an isomorphism between factor groups of subgroups of the given groups (10, §§11, 12; 25, pp. 15, 16). A like result is here shown for a general class of algebras, including loops and quasigroups, by a method due to Riguet (22). This result is used to obtain general forms of the Zassenhaus lemma and the Jordan- Hälder-Schreier theorem for normal series (26, §9).

We shall consider here two generalizations to rings of the concept of induced representation as it occurs in the representation theory of finite groups (6).

If A is a ring, S a subring of A, we shall associate with each S-module M an induced pair (I(M), K) consisting of an A -modulo I(M) and an S-homomorphism K: M → I(M).

A graph G = G(n, e) consists of a set of n nodes e pairs of which are joined by a single edge; we assume that no edge joins a node to itself. A graph with modes is called a complete -graph if each pair of its nodes is joined by an edge. The graphs belonging to some collection of graphs are independent if no two of them have a node in common. The maximum number of independent complete -graphs contained in a given graph G will be denoted by Ik(G).

A general theory for Grôbner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras.

In this paper we examine the class G of simple undirected, connected graphs of diameter d > 1, girth 2d, and for any g ɛ G, if a pair of nodes are at distance d from each other, then that pair of nodes is connected by t distinct paths of length d, t > 1. (The girth of g is the length of the smallest circuit in g.)

We establish, in § 2, that for all g ɛ G, g is regular.

We establish necessary conditions for the existence of elements of G. If g ɛ G, we adopt the notation g = g(d, t, v, n), where v is the valence of g and n is the number of nodes. It is of course possible for g, h ɛ G, g ≠ h, and for given d, t, v, n to have both g(d, t, v, n) and h(d, t, v, n).

In particular, we show that if d = 2, t ≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular given t value.

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

For a given elliptic curve $\mathbf{E}$, we obtain an upper bound on the discrepancy of sets of multiples ${{z}_{s}}G$ where ${{z}_{s}}$ runs through a sequence $Z\,=\,\left( {{z}_{1}},\ldots ,{{z}_{T}} \right)$ such that $k{{z}_{1}},\ldots ,k{{z}_{T}}$ is a permutation of ${{z}_{1}},\ldots ,{{z}_{T}}$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$.

We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

To shed light on the following unsolved problem, several authors have considered related problems. The problem is that of finding commuting approximants to pairs of asymptotically commuting self-adjoint matrices:

Suppose that Hn and Kn are self-adjoint matrices of dimension m(n), with ║Hn║, ║Kn║ ≦ 1, which commute asymptotically in the sense that

Must there then exist commuting self-adjoint matrices H′n and K′n for which

One may alter the conditions imposed on Hn and Kn, for example, by requiring Hn to be normal and Kn to be self-adjoint, and ask whether commuting approximants H′n and K′n can be found satisfying the same conditions. Some of these related problems have been solved. This paper will examine their solutions from a K-theoretic point of view, illustrating the difficulty inherent in modifying them to work for the original problem.