Let b ≥ 2 be an integer and let E/ be a fixed elliptic curve. In this paper, we estimate the number of primes p ≤ x such that the number of points nE(p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of nE(p).

Let andWe prove that there is an absolute constant c1 > 0 such thatfor every s ∈ (0, ∞) and n ≥ 9, where the supremum is taken for all f ∈ Gn withThis is what we call (an essentially sharp) Remez-type inequality for the class Gn. We also prove the right higher dimensional analog of the above result.

One of the fundamental problems in the study of moduli spaces is to give an intrinsic characterisation of representable functors of schemes, or of functors that are quotients of representable ones of some sort. Such questions are in general hard, leading naturally to geometry of algebraic stacks and spaces (see [1, 4]).

The relationship between posets that are cycle-free and graphs that have more than one end is considered. We show that any locally finite bipartite graph corresponding to a cycle-free partial order has more than one end, by showing a correspondence between the ends of the graph and those of the Hasse graph of its Dedekind–MacNeille completion. If, in addition, the cycle-free partial order is k-CS-transitive for some k ≥ 3 we show that the associated graph is end-transitive. Using this approach we go on to prove that, for infinite locally finite 3-CS-transitive posets with maximal chains of height 2, the properties of being crown-free and being cycle-free are equivalent. In contrast to this we show that the non-locally finite bipartite graphs arising from skeletal cycle-free partial orders each have only one end. We include a corrected proof of a result from an earlier paper on the axiomatizability of the class of cycle-free partial orders.

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

We prove a theorem that implies:

Let G1=〈G1, ⋅, ≤〉 andG2=〈G2, ⋅, ≤〉 be ordered groups with subgroupsH1andH2, respectively. If ϕ : H1 ≅ H2is an order-preserving isomorphism, then the free product ofG1andG2withH1andH2amalgamated via ϕ is right orderable.

This solves Problem 15⋅34 from the Kourovka Notebook.

We extend this result to an arbitrary family of ordered groups with order-preserving-isomorphic subgroups amalgamated.

The following theorem is proved. For any positive integer n there exists t depending only on n such that if the word w is a multilinear commutator, then the class of all groups G in which w(G) is locally finite and the product of any t values of w has order dividing n is a variety.

In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ∪S1, W1 for M1 and V2, ∪S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) − g(F) holds, and another is that under some circumstances we have a better lower bound.

We investigate the homology of finite index subgroups Gi of a given finitely presented group G. We fix a prime p, denote the field of order p by p and define dp(Gi) to be the dimension of H1(Gi; p). We will be interested in the situation where dp(Gi) grows fast as a function of the index [G : Gi]. Specifically, we say that a collection of finite index subgroups {Gi} has linear growth of mod p homology if infidp(Gi)/[G : Gi] is positive. This is a natural and interesting condition that arises in several different contexts. For example, the main theorem of [9] states that when G is a lattice in PSL(2, ℂ) with non-trivial torsion (equivalently, G is the fundamental group of a finite-volume hyperbolic 3-orbifold with non-empty singular locus), then G has such a sequence of subgroups. Another major class of groups G having such a collection of subgroups are those that are large. By definition, this means that G has a finite index subgroup that admits a surjective homomorphism onto a free non-abelian group. Large groups have many nice properties, for example super-exponential subgroup growth and infinite virtual first Betti number. One might wonder whether largeness is equivalent to the existence of some nested sequence of finite index subgroups {Gi} with linear growth of mod p homology for some prime p. If so, this would establish that lattices in PSL(2, ℂ) with non-trivial torsion are large, which would be a major breakthrough in low-dimensional topology.

Let β be a braid on n strands, with exponent sum w. Let Δ be the Garside half-twist braid. We prove that the coefficient of vw−n+1 in the Homfly polynomial of the closure of β agrees with (−1)n−1 times the coefficient of vw+n2−1 in the Homfly polynomial of the closure of βΔ2. This coincidence implies that the lower Morton–Franks–Williams estimate for the v–degree of the Homfly polynomial of is sharp if and only if the upper MFW estimate is sharp for the v–degree of the Homfly polynomial of .

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Veŕtesi's result [11].

The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact equivalent for all finitely presented groups.

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. A pair (E, φ) over X consists of an algebraic vector bundle E over X and a section φ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank n ≥ 2 are polarised pure Hodge structures, and they are isomorphic to H1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank n ≥ 2 and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.

We show how the techniques introduced in [1] and [2] can be employed to derive endpoint Lp → Lq bounds for the X-ray transform associated to the line complex generated by the curve t → (t, t2, . . ., td−1). Almost-sharp Lorentz space estimates are produced as well.

In this paper, we discuss the recognition problem for Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work “the geometry of fronts” for surfaces. The crucial point to prove our criteria for Ak-singularities is to introduce a suitable parametrization of the singularities called the “kth KRSUY-coordinates” (see Section 3). Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in Rn+1 (resp. Cn+1) is differentiably (resp. holomorphically) right-left equivalent to the Ak+1-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane Rn (resp. Cn) is right-left equivalent to the Ak-type singular point in Rn (resp. Cn). Moreover, we show that the restriction of a C∞-map f: Rn → Rn to its Morin singular set gives a wave front consisting of only Ak-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y′)′ + q(t)y = w(t)f(y) on [a, b] and a general separated boundary condition (BC). By comparing it with a linear Sturm--Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn, n = 0, 1, 2, . . ., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a, b) if λn is in the interior of the range of f(y)/y; and does not have any solution with exactly n zeros in (a, b) if λn is outside this range. These conditions become necessary and sufficient when f(y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.