A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M [ratio ] x < a} is strictly less than κ. In this paper we shall give constructions of κ-like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M [models ] Π2−Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ-like; also hierarchical variants of this result for Πn−Th(PA); and (2) for every n [ges ] 1, every singular κ and every M [models ] B[sum ]n+ exp+¬ I[sum ]n there is a κ-like model K elementarily equivalent to M.

We are concerned in this paper with the ideal structure of group rings of infinite simple locally finite groups over fields of characteristic zero, and its relation with certain subgroups of the groups, called confined subgroups. The systematic study of the ideals in these group rings was initiated by the second author in [15], although some results had been obtained previously (see [3, 1]). Let G be an infinite simple locally finite group and K a field of characteristic zero. It is expected that in most cases, the group ring KG will have the smallest possible number of ideals, namely three, (KG itself, {0} and the augmentation ideal), and this has been verified in some cases. In some interesting cases, however, the situation is different, and there are more ideals. We mention in particular the infinite alternating groups [3] and the stable special linear groups [9], in which the ideal lattice has been completely determined. The second author has conjectured that the presence of ideals in KG, other than the three unavoidable ones, is synonymous with the presence in the group of proper confined subgroups. Here a subgroup H of a locally finite group G is called confined, if there exists a finite subgroup F of G such that Hg∩F≠1 for all g∈G. This amounts to saying that F has no regular orbit in the permutation representation of G on the cosets of H.

Given a family of sets/vectors of the same cardinality/dimension you get the shadow by deleting one element/coordinate from a set/vector in all possible ways. You find the family with the smallest shadow by ordering all sets/vectors. The set case was solved by Kruskal (1963), and Katona (1966), and has many applications. We study two orderings which solve the 0, 1 vector case, and give the shadow size.

Let R be a ring (always understood to be associative with a unit element 1). It is well known that an R-module is Noetherian if and only if all its submodules are finitely generated and that it has a finite composition series if and only if it is Noetherian and Artinian. This raises the question whether every finitely generated Artinian module is Noetherian; here it is enough to consider cyclic Artinian modules, by an induction on the length. This question has been answered (negatively) by Brian Hartley [5], who gives a construction of an Artinian uniserial module of uncountable composition-length over the group algebra of a free group of countable rank. If we are just interested in finding cyclic modules that are Artinian but not Noetherian, there is a very simple construction based on the fact that over a free algebra every countably generated Artinian module can be embedded in a cyclic module which is again Artinian. This is described in §2 below.

In this paper we consider a generalization to algebraic number fields of the classical Hermite constant γn. For this constant we extend the well-known Minkowski bound and study the notion of extreme forms in this setting.

Let t(G, q) denote the smallest integer t such that the vertex set V of the graph G can be partitioned into t classes V(G)=[Cup ]ti=1Vi such that the number of edges in the induced subgraph 〈Vi〉 for 1[les ]i[les ]t, is divisible by q. Using an algebraic theorem due to Baker and Schmidt we prove that if q is a prime power then t(G, q) can be computed and a corresponding partition can be presented in a polynomial time.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.

Let d be a fixed positive integer, and ε>0. We show that there is a natural number M such that if G is any graph with m[ges ]M edges and maximum degree at most d, then the harmonious chromatic number h(G) satisfies

formula here

The mod two cohomology of the three connective covering of S3 has the form

formula here

where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.

Let G be a connected semisimple group over an algebraically closed field K of characteristic p>0, and [gfr ]=Lie (G). Fix a linear function χ∈[gfr ]* and let [zfr ][gfr ](χ) denote the stabilizer of χ in [gfr ]. Set [Nscr ]p([gfr ]) ={x∈[gfr ][mid ]x[p]=0}. Let [Cscr ]χ([gfr ]) denote the category of finite-dimensional [gfr ]-modules with p-character χ. In [7], Friedlander and Parshall attached to each M∈Ob([Cscr ]χ([gfr ])) a Zariski closed, conical subset [Vscr ][gfr ](M)⊂[Nscr ]p([gfr ]) called the support variety of M. Suppose that G is simply connected and p is not special for G, that is, p≠2 if G has a component of type Bn, Cn or F4, and p≠3 if G has a component of type G2. It is proved in this paper that, for any nonzero M∈Ob([Cscr ]χ([gfr ])), the support variety [Vscr ][gfr ](M) is contained in [Nscr ]p([gfr ])∩[zfr ][gfr ](χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [18].

Let R be a fully bounded Noetherian ring of finite global dimension. Then we prove that K dim (R) [les ] gldim (R). If, in addition, R is local, in the sense that R/J(R) is simple Artinian, then we prove that R is Auslander-regular and satisfies a version of the Cohen–Macaulay property. As a consequence, we show that a local fully bounded Noetherian ring of finite global dimension is isomorphic to a matrix ring over a local domain, and a maximal order in its simple Artinian quotient ring.

We introduce two related families of polynomials, easily computable by simple recursions into which any Kazhdan–Lusztig (and inverse Kazhdan–Lusztig) polynomial of any Coxeter group can be expanded linearly, and we give combinatorial interpretations to the coefficients in these expansions. This yields a combinatorial rule for computing the Kazhdan–Lusztig polynomials in terms of paths in a directed graph, and a completely combinatorial reformulation of the nonnegativity conjecture [15, p. 166].

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [8].

It is proved that quadratic systems of autonomous differential equations in the plane admit forty-five topologically distinct phase configurations at infinity. The related conditions on the coefficients are given in affine invariant form.

Let [Gscr ] be a Burnside 3-group constructed by Gupta-Sidki. The group algebra ℤ3[[Gscr ]] has a proper ring-quotient which embeds [Gscr ] and is just-infinite and primitive.

A negative answer to the Kuroš–Černikov Question 21 in [7], whether a group satisfying the normalizer condition is hypercentral, was given by Heineken and Mohamed in 1968 [6]. They constructed groups G satisfying:

(i) G is a locally finite p-group for a prime p,

(ii) G/G′≅Cp∞ and G′ is countable elementary abelian,

(iii) every proper subgroup of G is subnormal and nilpotent,

(iv) Z(G)={1},

(v) the set of normal subgroups of G contained in G′ is linearly ordered by set inclusion, see [3, p. 334],

(vi) KG′ is a proper subgroup in G for every proper subgroup K of G, see [6, Lemma 1(a)].

Let I be the graded ring of homogeneous rational polynomials in n-variables which are numerical over ℤ. Then I is a subring of Γ, the divided polynomial algebra over ℤ in the n-variables. A consequence of the main theorem is that for any positive integer k, the image of the induced homomorphism I[otimes ](ℤ/kℤ) → Γ [otimes ] (ℤ/kℤ) is a finite graded ring.

Let Fq be a finite field of characteristic r and order q=ra, V=Fnq the vector space of finite dimension n[ges ]1 over Fq, and Mat (n, q) the matrix algebra on V. We denote by Hn(q) (or Hn if q is clear from the context) one of the following classical groups: Sp (n, q) (with n even), the symplectic group on V; O(n, q) (with q odd), one of the full orthogonal groups on V in odd characteristic; U(n, q) (with q a square), the full unitary group on V.

Let f be analytic and univalent in Δ={z:[mid ]z[mid ]<1}. We write

formula here

for 0[les ]r<1 and θ∈[0, 2π], and remark that L(r, θ) is the length of the image in f(Δ) of the ray joining 0 to reiθ in Δ. We shall be concerned with the growth of L(r, θ), as a function of r, for univalent functions in Δ and, more particularly, for functions in the subclass of close-to-convex univalent functions in Δ.

For a large class of Banach algebras we solve the Cohomology Comparison Problem for the second cohomology groups, in that we prove that for all amenable strongly regular Banach function algebras A and all Banach A-bimodules E the second comparison map between continuous and algebraic cohomology is an embedding. Accepting the continuum hypothesis we show that for algebras of this kind there exist Banach A-bimodules E such that i2 is not onto.

Let G be a locally finite group of cardinality ℵn where n is a natural number. Let π(G) be the set of primes p for which G has an element of order p. In [5], Holt conjectures that if k is a finite field with char k ∉ π(G) then

(1) G has cohomological dimension n+1 over k;

(2) Hn+1(G, kG) has cardinality 2ℵn;

(3) Hi(G, kG) = 0 for 0 [les ] i [les ] n.