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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.

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.

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

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

Recently Wang (5) has given some results concerning characterizations of Poisson, normal, gamma, binomial and negative binomial distributions. We shall show here that these results are incorrect and also attempt to rectify them.

Since the above-named paper was published my attention has been called, by Mr A. F. Crossley, to the fact that a solution for Rutherford's ‘Case 1’ had been given by H. Jeffreys in his Operational methods in mathematical physics (Cambridge Tracts in Mathematics), p. 35. This solution is very neat and direct. As, however, the method depends on the use of inverse operational factors, such as where it is less elementary in character than that followed in my paper.

Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.

Professor Oystein Ore of Yale University has kindly called my attention to the close relation between my paper and some earlier researches of Dedekind. I should like to correlate, as far as possible, my results with his.

Let Φ be a root system and let Γ ⊆ Φ. In this short paper we prove that Γ contains a ${\mathbb Z}$-basis of the lattice that it generates.

In a short paper recently published in these Proceedings, E. J. F. Primrose determines amongst other things a projective invariant N of a space cubic K and a linear complex L, N being of the third degree in the coefficients cik of the complex L. The purpose of the following remarks is to establish the connexion between this invariant N and the well-known projective comitants of the cubic K.

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.

Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

In this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

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

The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

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.

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

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].

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz systemx˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

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).

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic four manifolds., Comment: typos fixed, Rewrite Section 4 to reduce technicality

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 ].

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.

In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

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.)

The purpose of this paper is to determine the optimal order of decrease of the L 2 norm for the following degenerate and singular oscillatory integral operators defined by formula here for f ∈ C ∞ 0 (ℝ), where ψ( x , y ) is a smooth compactly supported function on ℝ 2 , S ( x , y ) is a homogeneous polynomial of degree n formula here and K ( x , y ) is a distribution kernel which satisfies the following hypotheses: K is a C 2 function away from the diagonal and the estimates formula here hold for 0 i =1, or 2. This paper basically uses the methods of [ 6 ], which are very general and can be widely used. The main idea of these methods is that the sizes of operator norms after the decomposition are estimated in different ways and the decomposition is then summed back by balancing the estimates of two types. The first type estimate takes into account the support sizes of the integrand after the decomposition. The second type estimate is delicate, depending on how to keep track of the distance to the singular varieties of phase function. In our case, we have to include the singular variety arising from distribution kernel as well. The analytic tool of the second type estimate is the operator version of the Van der Corput lemma [ 7 ]. We notice that the crux of matters is to locate the singular varieties of phase function and kernel. For higher dimensional situations, the problem is very difficult. One expects that the powerful theory of algebraic geometry may play a significant role. Here we would like to remark that Mather's results [ 2 ] may be useful although the geometrical shapes of singular varieties are basically undecidable.

In this paper we will study the Brill–Noether theory of vector bundles on a smooth projective curve X. As usual in papers on this topic we are mainly interested in stable or at least semistable bundles. Let Wkr, d(X) be the scheme of all stable vector bundles E on X with rank (E)=r, deg (E)=d and h0(X, E)[ges ]k+1. For a survey of the main known results, see the introduction of [6]. The referee has pointed out that the results in [6] were improved by V. Mercat in [14]; he proved that Wkr, d(X) is non-empty for d

The main aim of this paper is to provide two new proofs of Ramanujan's cubic transformation formula for 2F1(1/3, 2/3; 1; z) (see (1·8) below). For our first proof, we have to develop Ramanujan's elliptic functions in the theory of signature 3 using a different approach from that given in a recent paper by Berndt, Bhargava and Garvan. For our second proof, we use two of Goursat's formulas.

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

Let E be an elliptic curve with complex multiplication by the ring of integers [Ofr ] of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]; we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.

In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and ∞.

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case, orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather, they may equidistribute on a fractal subset. Understanding the behaviour of these orbits near the boundary is central to Patterson-Sullivan theory and much further work. Our theorem characterizes the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish an formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which was not known previously even in the lattice case., 33 pages, 1 figure; Accepted for Publication in Mathematical Proceedings of the Cambridge Philosophical Society

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

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

This paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Following his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in is a consequence of Bar-Natan's 4T relation.The last part of this paper consists of a proof that the space of weight systems is a sub-Hopf algebra of the space *, by means of the canonical projection.

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.

Tricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.