The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerning

where T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet series

defines an entire function of s and satisfies the functional equation

The most penetrating statements that have been made on T(n) and LΔ(s)are:

Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

§ 1. Introductory. In § 3 a generalisation of the formula [MacRobert, Phil. Mag., Ser. 7, XXXI, p. 258]

where αp+1 = ½m + ½n, αp+2 = ½m - ½n, R(m ± n) > 0, and x is real and positive, will be established. In the course of the proof Hardy's formula [Mess, of Maths., LVI, (1927), p. 190],

where R(b)>0, will be required. This was originally proved by an application of Mellin's Inversion Formula. An alternative proof is given in § 2, and some related formulae are deduced.

Let A be a noetherian local ring, x a non-unit element of A, B=A/(x). Let E be the Koszul complex associated to an arbitrary set of generators of the ideal (x) of A. Assume that H1(E) is a free B-module. Then A is Gorenstein if and only if B is also.

1991 Mathematics Subject Classification 13H10, 13D03.

The Kadets path distance between Banach spaces X and Y is defined to be the infimum of the lengths with respect to the Kadets distance of all curves joining X and Y. If there is no curve joining X and Y, the Kadets path distance between X and Y is defined to be infty .

Some approaches to estimates of the Kadets path distance from above and from below are developed. In particular, the Kadets path distances between the spaces l_p^n,\ p\in [1,+\rm (inf)ty ], n\in {\b (N)} are estimated.

A sequence of solutions to the Galerkin approximation of a nonstationary magnetohydrodynamic system is proved to converge to a measure-valued solution, in the sense of R. J. DiPerna–A. J. Majda, to the three-dimensional stationary Euler equations.

We construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a non-singular variety. The result answers a question of Behrend–Fantechi. We correct here an error in our paper (D. Maulik, R Pandharipande and R. P. Thomas, Curves on K3 surfaces and modular forms, J. Topol.3 (2010) 937–996. arXiv:1001.2719v3), where an incorrect construction with the same claimed properties was proposed.

We prove the Löwner-Heinz inequality, via the Cordes inequality, for elements a,b>0 of a unital hermitian Banach *-algebra A. Letting p be a real number in the interval (0,1], the former asserts that a^p \le b^p if a \le b, a^p < b^p ifa<b , provided that the involution of A is continuous, and the latter that s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an element x.

Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.

We give a complete description of quadratic twisted potential algebras on three generators as well as cubic twisted potential algebras on two generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic other than 2 or 3.

Let w0 be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type An. We compare Lusztig's and Kashiwara's (string) parametrizations on canonical basis associated with w0. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show that this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov (O. Gleizer and A Postnikov, Littlewood–Richardson coefficients via Yang–Baxter equation, IMRN14 (2000) 741–774). We use combinatorics of the Auslander–Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities defining the string cone based on hammocks.

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.

Let γn = [x1,…,xn] be the nth lower central word. Denote by Xn the set of γn -values in a group G and suppose that there is a number m such that $|{g^{{X_n}}}| \le m$ for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete group G. We show that in two of the models we can also consider actions of any simplicial group G.

For a Banach space $X$ we consider three ways in which a subspace of $X^*$ can represent locally the whole dual space $X^*$. We obtain characterizations in terms of ultrapowers and we study the relationship between the subspaces of $X^*$ and the subspaces of the dual of an ultrapower of $X$.

We consider a class of deformations of Laplace-Beltrami operators on flat and constant curvature spaces, which possess a family of commuting operators. These are built as deformations of the symmetries of the underlying geometric space. In flat spaces it is also possible to extend some symmetries into ladder operators. In all cases it is possible to choose sub-classes which are super-integrable.

In recent years, certain varieties of semigroups with unary operations (of “inversion”) have received considerable attention. Generally speaking, these have been contained in one or other of the two classes of completely regular semigroups (that is, semigroups that are unions of groups) and inverse semigroups. For instances of the former see [1], [2], [3], [6], [10], [14] and [15], and for instances of the latter see [7], [8], [12] and [13].

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].

For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

The theory of isoparametric functions and a family of isoparametric hypersurfaces began essentially with E. Cartan in 1930's. He defined a real valued function V defined on a Riemannian space form to be isoparametric if ∥grad υ∥2=TV and ΔV = SV for some real valued functions S, T. Then a family of hypersurfaces Mt, is called isoparametric if Mt,=V-1 (t) where t is a regular value of V.

Let R be a commutative, semi-local ring. Let On,n be the group of linear automorphisms of R2n which preserve the bilinear form . The main result of this paper is the following theorem.

Theorem A. The natural inclusion of Onn into On,n into On+1,n+1induces an isomorphism on the ith homology group if only n is large enough with respect to i.