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Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matricesare such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number. In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links., 11 pages, 7 figures, correction to one of the skein relations and addition to attributions in introduction

Davis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$., Comment: This paper has been withdrawn by the authors, since it has already been published

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

In a fundamental paper in 1984, Vaughan Jones developed his new polynomial invariant of knots using a Markov trace on the Temperley–Lieb algebra. Subsequently, Lickorish used the associated bilinear pairing to provided an alternative proof for the existence of the 3-manifold invariants of Witten, Reshetinkin and Turaev. A key property of this form is the non-degeneracy of this form except at the parameter values ±2cos(π/(n+1)) [7]. Ko and Smolinsky derived a recursive formula for the determinants of specific minors of Markov's form, establishing the needed non-degeneracy [6]. In this paper, we define a triangular change of basis in which the form is diagonal and explicitly compute the diagonal entries of this matrix as products of quotients of Chebyshev polynomials, corroborating the determinant computation of Ko and Smolinsky. The method of proof employs a recursive method for defining the required orthogonal basis elements in the Temperley–Lieb algebra, similar in spirit to Jones' and Wenzl's recursive formula for a family of projectors in the Temperley–Lieb algebra. We define a partial order on the non-crossing chord diagram basis and give an explicit formula for a recursive construction of an orthogonal basis, via a recursion over this partial order. Finally we relate this orthogonal basis to bases constructed using the calculus of trivalent graphs developed by Kauffman and Lins [5].

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Q p generated by all p -power roots of unity. Let formula here where μ p n +1 denote the p n +1 th roots of 1. Let [pscr ] n be the maximal ideal of the ring of integers of Φ n and let U n be the units congruent to 1 modulo [pscr ] n . Let ζ n be a fixed primitive p n +1 th root of unity such that ζ p n = ζ n − 1 , ∀ n [ges ] 1. Put π n = ζ n − 1. Thus π n is a local parameter for Φ n . Let formula here Kummer already exploited the obvious fact that every u 0 ∈ U 0 can be written in the form formula here where f 0 ( T ) is some power series in Z p [[ T ]]. Here of course, the power series f 0 ( T ) is not uniquely determined. Let formula here the inverse limit being with respect to norm maps. Coates and Wiles (see [ 3 ]) discovered that any unit u = ( u n ) ∈ U ∞ has a unique power series f u ( T ) in Z p [[ T ]] with f u (π n ) = u n . The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Z p . Almost immediately, Coleman [ 4 ] gave a totally different proof of the existence of the f u ( T ), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [ 3 ]. We first prove the following result which is stronger than the original one in [ 3 ].

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that ct > 1/2. We have the elementary bound 0 < ct < 1; however, no bound of the form 0 < α ≤ ct or ct ≤ β < 1, valid for all t, is known. In this paper, we prove that ct > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).

Throughout this paper, let M be a finitely generated module over a Noether local ring ( A , [mfr ]) with dim M = d . Let x = ( x 1 , …, x d ) be a system of parameters of M and n = ( n 1 , …, n d ) ∈ ℕ d a d-tuple of positive integers. This paper is concerned with the following two points of view. First, it is well-known that, the difference between lengths and multiplicities formula here considered as a function in n , gives a lot of information on the structure of M . This function in general is not a polynomial in n for all n 1 , …, n d large enough ( n [Gt ] 0 for short). But, it was shown in [ C3 ] that the least degree of all polynomials in n bounding above I M ( n , x ) is independent of the choice of x . This numerical invariant is denoted by p ( M ) and called the polynomial type of the module M . By [ C2 ] and [ C3 ] this polynomial type does not change by the [mfr ]-adic completion M ˆ of M and p ( M ) is just equal to the dimension of the non-Cohen–Macaulay locus of M ˆ (see [ C2 , C3 , C4 , CM ] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p ( M ) = − ∞ or p ( M ) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p ( M ) > 0. Second, following Sharp and Hamieh ([ SH ]), we consider the difference formula here as a function in n , where formula here is the cyclic submodule of the module of generalized fractions U ( M ) − d −1 d +1 M defined in [ SZ1 ].

In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

There is an open-book decomposition of the 3-sphere which has open discs as pages and an unknotted circle as the binding. We can think of the 3-sphere as (R U {oo} and of the circle as the z-axis U {oo}. The pages are then half-planes Hg at angle 8 when the x-y plane has polar coordinates. In their investigation of the braid index of satellite links, Birman and Menasco[B-M] embed the companion knot in finitely many such half-planes so that the knot meets each half-plane in a single simple arc, and therefore meets the axis in a finite number of points. At the end of their paper they mention that the minimum number of planes required to present a given knot in this manner is a knot invariant and that it seems to have escaped attention. (Josef Przytycki has since pointed out to us that the phenomenon is evident in a onehundred-year-old paper by H. Brunn[Br].) We call this invariant the arc index of a link and denote it by a(L). The fundamental properties of this new presentation of knots and links were discussed in [Cr]. In this paper we investigate further the arc index of links and describe an algorithm for drawing a binding circle on a diagram of a knot or link. This works in most cases and gives an upper bound on the arc index. We conjecture that this bound applies to all knots and links, and that for alternating knots and links it achieves the lowest possible index. Our main result is the

Let (fn) and (gn) be two sequences of random variables adapted to an increasing sequence of σ-algebras (ℱn) such that the conditional distributions of fn and gn given ℱn coincide. Suppose further that the sequence (gn) is conditionally independent. Then it is known that where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free groupwhere Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the normW(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of in W(X, Y))

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

In this paper we concern ourselves with the Q-matrices of time-substituted Markov chains where the time-substitutions are based on fluctuating clocks with noise. The results of this paper follow on naturally from the analysis in Kennedy and Williams [3], in which the problems of characterization and of calculation of these Q-matrices were addressed. Here, we study the Q-matrices as functions of the variance σ2 of the noise.

In this paper we mainly consider local rings admitting dualizing complexes. It is well-known that if a Noetherian local ring A admits a dualizing complex, then the non-Cohen–Macaulay (abbreviated CM) locus of A is closed in the Zariski topology (cf. [8, 10]). If the dimension of this locus is zero and A is equidimensional, i.e. the punctured spectrum of A is locally CM and dim(A/P) = dim (A) for all minimal prime ideals P ∈ Ass (A), then A is a generalized CM ring and its structure is well-understood (see [2, 12]). For instance, one of the characterizations of generalized CM rings is the conditions that for any parameter ideal q contained in a large power of the maximal ideal m of A, the difference between length and multiplicityis independent of the choice of q. However, if the dimension of the non-CM locus is larger than zero, little is known about how this dimension is related to the structure of the local ring A. The purpose of this paper is to show that if M is a finitely generated A-module, then there exist systems of parameters x = (x1, …, xd) (where d = dim M) such that the differenceis a polynomial in n1, …, nd for all positive integers n1, …, nd and the degree of IM(n1, …, nd;x) is independent of the choice of x. We shall also give various characterizations of this degree by using the notion of reducing systems of parameters of Auslander and Buchsbaum[l]. In particular, if the module M is equidimensional we shall show that the degree of IM(n1, …, nd;x) is equal to the dimension of the non-CM locus of M.

In the paper [3] we obtained a lower bound for the Hausdorff dimension of certain sets of singular n-tuples in ℝn. In this paper we will obtain an upper bound for the dimension of these sets. These two bounds are not sharp enough to yield the exact dimension of all the sets considered but they do yield the dimension of the set of ‘highly singular’ (in a sense to be made precise below) n-tuples.

RésuméWe study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complexL-series for the family of quadratic twists of all optimal elliptic curves over${\mathbb Q}$.

In this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

Let $S(t) = \tfrac{1}{\pi} \arg \zeta (\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,{\rm d}\tau\, + \delta_n\,, \end{equation*} where $\delta_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that $S_n(t) = O(\log t/ (\log \log t)^{n+1})$. The order of magnitude of this estimate was never improved up to this date. The best bounds for $S(t)$ and $S_1(t)$ are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate \begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*} for all $n\geq 2$, with the constants $C_n^{\pm}$ decaying exponentially fast as $n \to \infty$. This improves (for all $n \geq 2$) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when $n \to \infty$. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when $n$ is odd, and an optimized interpolation argument for the cases when $n$ is even. In the final section we extend these results to a general class of $L$-functions., Comment: 23 pages. To appear in Math. Proc. Cambridge Philos. Soc

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

Let E/Q be an elliptic curve, p a prime and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has supersingular reduction at p.

In this paper, we prove a Tb theorem on product spaces $\mathbb{R}$n × $\mathbb{R}$m, where b(x1, x2) = b1(x1)b2(x2), b1 and b2 are para-accretive functions on $\mathbb{R}$n and $\mathbb{R}$m, respectively.

In this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ⩾ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erd\H{o}s., Comment: 14 pages

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic elements., Comment: 9 pages, 7 figures

In this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space. A well-known result of Falconer showed that under mild assumptions the Hausdorff dimension of typical self-affine sets is equal to its Singularity dimension. Heuter and Lalley subsequently presented a smaller open family of non-trivial examples for which there is an equality of these two dimensions. In this article we analyse the size of this family and present an efficient algorithm for estimating the dimension.

In this paper we prove the generalized version of the same n-type conjecture posed by McGibbon and Møller in [15, page 287]: it is shown that the set of all the same homotopy n-types of the suspension of the smash products of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type.

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those ofPSL(5,q) for some oddq.This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that ifq≡ – 1 (mod 4), the Sylow 2-subgroup ofPSL(5,q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization ofPSL(5,q) andPSU(5,q) (for all oddq) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.

Many of the basic concepts in this paper are defined in (2), and it will be useful to the reader if he is familiar with that paper. In it the concept of a loop transversal to a subgroup S of a group G is defined and discussed, where if G is a transitive permutation group and no explicit S is mentioned it is assumed that S is a point stabilizer. In (3) there appeared the following problem (P):Let G be a transitive permutation group which is 2-closed.(i) Is there a fixed-point-free transversal in G (i.e. to a point stabilizer)?(ii) Is there a loop transversal in G?Unfortunately a note was added in which it was falsely stated that the answer to both (i) and (ii) is yes.

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

Throughout this paper F denotes a (commutative) field. Let XF denote the class of all groups G such that every irreducible FC-module has finite dimension over F. In (1) P. Hall showed that if F is not locally finite and if G is polycyclic, then G∈XF if and only if G is abelian-by-finite. Also in (1), if F is locally finite he proved that every finitely generated nilpotent group is in XF and he conjectured that XF should contain every polycyclic group. This turned out to be very difficult, but a positive solution was eventually found by Roseblade, see (8). Meanwhile Levič in (4) had started a systematic investigation of the classes XF. Although his paper contains a number of errors, obscurities and omissions, it remains an interesting work, and it, or more accurately its recent translation, stimulated this present paper.

Knuth (12) seems to have been the first to carry out non-trivial matroid operations on a computer. However, as he remarks, there are considerable computational difficulties. In this paper we examine in detail the computational complexity of some fundamental matroid properties. The model of computation is the Hausmann–Korte oracle machine introduced in (10), (11) to deal with properties of independence spaces and fixed point problems. It was these papers (10), (11) and (12) which motivated the present work. Basically, the problems we will be discussing are matroid analogues of low-level complexity problems for graphs. For an excellent survey of this we refer to the recent monograph of Bollobés ((3), chapter 8). We shall use the approach of (3), and Bollobaé;s and Eldridge(4) rather than the more general treatment of Hausmann and Korte (11).

The purpose of this paper is to give natural characterizations of the countable graphs that admit tree-decompositions or simplicial tree-decompositions into primes. Tree-decompositions were recently introduced by Robertson and Seymour in their series of papers on graph minors [7]. Simplicial tree-decompositions were first considered by Halin[6], being the most typical kind of ‘simplicial decomposition’ as introduced by Halin[5] in 1964. The problem of determining which infinite graphs admit a simplicial decomposition into primes has stood unresolved since then; a first solution for simplicial tree-decompositions was given in [2].

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually proveThis may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

SummaryThe object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

An n-block is a sequence b1 … bn where bi ε {0,1} for 1 ≤ i ≤ n, and an n-block map is a function from the set of n-blocks to the set {0,1}. Block maps can be used to study endomorphisms of the shift dynamical system [7] and shift register sequences [4]. Composition of endomorphisms and cascading of shift registers [6] can be studied via composition of block maps. If fog is a given block map which is linear in the first variable and if g is also given, then f is unique and can be determined. If fog and f are given then g is unique and can be determined, at least up to a constant [10]. However, little is known about the level of computational complexity of the general task of searching for factors of a given block map. In this paper I identify a substantial class of maps which are linear in the first variable but not linear, for which there are effective methods of searching for factors. The methods are based on the presentation of block maps via successive principal parts, introduced in [9]. There I define the principal vector of a block map. One of the key results gives conditions under which there is a particularly close relationship between the principal vectors of two block maps f and g and their composition fog. In the second section of this paper I show that the same relationship holds under weaker conditions.

How must n equal non-overlapping circles be packed on a sphere so that the angular diameter of the circles will be as great as possible? In the paper, the conjectured solutions of this problem for n = 18, 27, 34, 35, 40 are improved on the basis of an idea of Danzer. Using the theory of bar structures it is ascertained that, in these cases, the edge-length of the graphs of the circle-packings can be increased till, in the graphs, additional edges appear which prevent further motions apart from rigid motions. The cases of n = 17 and 32 are also dealt with and there are references to the possibilities of further applications of the method applied in this paper (n = 59, 80, 110, 122).

All sets considered in this paper will be subsets of n-dimensional Eucidean space En. In this paper, we shall consider the ‘total edge-lengths’ of polytopes which are inscribed in a given sphere and which contain its centre. We first, however, mention some related problems considered by various authors which may be of interest to the reader.

A random submatroid ωrof the projective geometryPG(r− 1,q) is obtained fromPG(r− 1,q) by deleting elements so that each element has, independently of all other elements, probability 1 −pof being deleted and probabilitypof being retained. The properties of such random structures were studied in [5] and [6]. In the first of these papers,pwas kept fixed, while in the second, motivated by Erdös and Rényi's work ([3), [4]) on random graphs,pwas taken to be a function ofr. A recent paper of Bollobás[2] strengthens and extends a number of the results of Erdös and Rényi. In this paper we prove matroid analogues of several of Bollobàs's results thereby extending some of the results of [6].

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph X does not tend to infinity as the number of vertices of X tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author. However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when X is either a Cayley graph or an arc-transitive graph., 18 pages, 1 figure