A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z ∈ : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.

This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p>2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized étale cohomology theory called étale BP2.

Let Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.

We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

We study the distribution of a family of generalized Euler constants arising from integers sieved by finite sets of primes . For , the set of the first r primes, as r → ∞. Calculations suggest that is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of and the Riemann hypothesis.

We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and also show that each provides a necessary and sufficient set of conditions for the embeddability of a category in a groupoid. We show that all such conditions, and more besides, may be derived in a uniform way from a particular class of directed graphs which we call quadrangle clubs, and we prove a number of results (extending those of Mal'cev, Lambek, Bush and Krstić) on which families of quadrangle clubs provide sufficient conditions for embeddability.

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp2n) = n + 1 for n odd.

Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

A base for a commutative semigroup (S, +) is an indexed set 〈xt〉t∈A in S such that each element x ∈ S is uniquely representable as Σt∈Fxt where F is a finite subset of A and, if S has an identity 0, then 0 = Σn∈Øxt. We investigate those commutative semigroups or groups which have a base. We obtain the surprising result that has a base. More generally, we show that an abelian group has a base if and only if it has no elements of odd finite order.

We prove that if G is a centre-by-metabelian pro-p group of type FPm over , so every homomorphic image of G is of type FPm over .

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G2-structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G2-manifold is exactly G2 if and only if the fundamental group π1(M) is finite and neither M nor any double cover of M is homeomorphic to a cylinder.

Let P(n) = [x1, . . ., xn] = ⊕d≥0Pd(n) be the polynomial algebra viewed as a graded left module over the Steenrod algebra at the prime 2. The grading is by the degree of the homogeneous polynomials Pd(n) of degree d in the n variables x1, . . ., xn. The algebra P(n) realizes the cohomology of the product of n copies of infinite real projective space. We recall that a homogeneous element f of grading d in a graded left -module M is hit if there is a finite sum f = ΣiSqi(hi), called a hit equation, where the pre-images hi ∈ M have grading strictly less than d and the Sqi, called the Steenrod squares, generate . One of the important parts of the hit problem is to check whether a given polynomial in M is hit or not. In this article we study this problem in the 3-variable case.

Let (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

We prove the result stated in the title, which answers the equicharacteristic case of a question of Vasconcelos.

The LMO invariant is a very strong invariant such that it is expected to classify integral homology 3-spheres. In this paper we identify the set of the degree ≤ 6 parts of the logarithm of the LMO invariant for integral homology 3-spheres. As an application, we obtain a complete set of relations which characterize the set of Ohtsuki's invariants {λi(M)} for i ≤ 6. For any simple Lie algebra , we also obtain a complete set of relations which characterize the set of perturbative P invariants {(M)} for i ≤ 3.

For a locally compact group G, let B(G) denote its Fourier–Stieltjes algebra. Any continuous, piecewise affine map α: Y ⊂ H → G induces a completely bounded algebra homomorphism jα: B(G) → B(H) [14, 15] and we prove that jα is w* – w* continuous if and only if α is an open map. This extends one of the main results in [3], due to M.B. Bekka, E. Kaniuth, A.T. Lau and G. Schlichting. Several classical theorems regarding isomorphisms and extensions of homomorphisms on group algebras of abelian groups are extended to the setting of Fourier–Stieltjes algebras of amenable groups. As applications, when G is amenable we provide complete characterizations of those maps between Fourier–Stieltjes algebras that are either associated to a piecewise affine mapping, or are completely bounded and w* – w* continuous.

We use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta functionto , to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of .