• Volume 104, Issue 1

February 1994,   pages  1-304

• On the equationx(x +d1)…(x + (k - 1)d1) =y(y +d2)…(y + (mk - 1)d2)

For given positive integersm ≥ 2,d1 andd2, we consider the equation of the title in positive integersx, y andk ≥ 2. We show that the equation implies thatk is bounded. For a fixedk, we give conditions under which the equation implies that max(x, y) is bounded.

• The density of rational points on non-singular hypersurfaces

LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤn;F(x)=0, |x|⩽X}, where$$\left| x \right| = \mathop {max}\limits_{1 \leqslant r \leqslant n} \left| {x_r } \right|$$. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪Xn−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.

• Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {NKm;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given.

• Non-surjectivity of the Clifford invariant map

The question whether there exists a commutative ringA for which there is an element in the 2-torsion of the Brauer group not represented by a Clifford algebra was raised by Alex Hahn. Such an example is constructed in this paper and is arrived at using certain results of Parimala-Sridharan and Parimala-Scharlau which are also reviewed here.

• Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ2 + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree).

• On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

In this article we establish the analogue of a theorem of Kuznetsov (theorem 6 of [3]) in the case of 3-dimensional hyperbolic space. We also consider a generalization of this result for higher dimensional hyperbolic spaces and discuss the relevant ingredients of a proof.

• On Zagier’s cusp form and the Ramanujan τ function

Zagier constructed a cusp form for each weightk of the full modular group. We use this construction to estimate Fourier coefficients of cusp forms. In particular, we get a non-trivial estimate, by elementary methods and indicate a relationship with the Lindelof hypothesis for classical Dirichlet L-functions.

• Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above.

In this paper, we generalize the theory to p.v.’s with symmetric structure ofKε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups.

• On a problem of G Fejes Toth

A solution is given for the following Problem of G Fejes Toth: In 3-space find the thinnest lattice of balls such that every straight line meets one of the balls.

• The number of ideals in a quadratic field

LetK be a quadratic Geld, and letR(N) be the number of integer ideals inK with norm at most AT. Letx with conductork be the quadratic character associated withK. Then |R(N)−NL(1,x)|⩽Bk50/73N23/73(logN)461/146 forNAk, whereA andB are constants. ForNAkc,C sufficiently large, the factork50/73 may be replaced by (d(k))4/73k46/73.

• On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples.

Theorem A.Let 0&lt;θ&lt;1/2and let {an}be a sequence of complex numbers satisfying the inequality$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1}$$for N = 1,2,3,…,also for n = 1,2,3,…let αnbe real and ¦αn¦ ≤ C(θ)where C(θ) &gt; 0is a certain (small)constant depending only on θ. Then the number of zeros of the function$$\sum\limits_{n = 1}^N {a_n \left( {n + \alpha _n } \right)^{ - s} } = \zeta \left( s \right) + \sum\limits_{n = 1}^\infty {\left( {a_n \left( {n + \alpha _n } \right)^{ - s} - n^{ - s} } \right)}$$in the rectangle (1/2-δ⩽σ⩽1/2+δ,Tt⩽2T) (where 0&lt;δ&lt;1/2)isC(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided TT0(θ,δ)a large positive constant.

Theorem B.In the above theorem we can relax the condition on an to$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1} N^0$$ and ¦aN¦ ≤ (1/2-θ)-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,Tt⩽2T)is &gt; C(θ,δ) Tlog T(log logT)-1.The upper bound for the number of zeros in σ⩾1/3+δ,Tt⩽2T) isO(T)provided$$\sum\limits_{n \leqslant x} {a_n } = x + O_s \left( {x^2 } \right)$$for every ε &gt; 0.

• Local zeta functions of general quadratic polynomials

This paper is concerned with the kind of local zeta functions now often called Igusa’s local zeta functions: A simple closed form of such a zeta function for an arbitrary quadratic form, its variants, and an application are given.

• Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:ZX which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH1(Z, O) →H1(X, EndE) is surjective.

• Finite arithmetic subgroups ofGLn, III

LetG be an algebraic group inGLn(C) defined over Q, andK an algebraic number field with the maximal orderOk. If the groupG(Ok) of rational points ofG inMn(Ok) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(Ok) is generated by roots of unity inK andG(Z).

• Reduction theory over global fields

The paper contains an exposition of the basic results on reduction theory in reductive groups over global fields, in the adelic language. The treatment is uniform: number fields and function fields are on an equal footing.

• Symplectic structures on locally compact abelian groups and polarizations

LetX be a locally compact abelian group and ω(.,.) a symplectic structure on it. A polarization for (X, ω) is a pair of totally isotropic closed subgroupsG, G* ofX such thatX =G.G* and ω(.,.) defines a dual pairing ofG andG*. In this paper we describe a class of such groups which always admit a polarization and also discuss their structure.

• Modular equations and Ramanujan’s Chapter 16, Entry 29

In this paper we illustrate how some of the classical modular equations can be proved by using only Ramanujan’s summation (see (1.1)) and dispensing completely with the Schröter-type methods.

• Gaussian quadrature in Ramanujan’s Second Notebook

Ramanujan’s notebooks contain many approximations, usually without explanations. Some of his approximations to series are explained as quadrature formulas, usually of Gaussian type.

• Two remarkable doubly exponential series transformations of Ramanujan

The purpose of this note is to prove two doubly exponential series transformations found in Ramanujan’s second notebook.

• Kolmogorov’s existence theorem for Markov processes inC* algebras

Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes.

• Iterations of random and deterministic functions

Letf be a probability generating function on [0, 1]. The convergence of its iteratesfn to fixed points is studied in this paper. Results include rates forf andf-1. Also iterates of independent identically distributed stable processes are studied and a trichotomy based on the order of the stability is established.

• Existence theory for linearly elastic shells

We review existence and uniqueness results, recently obtained for three of the most important linear two-dimensional shell models: Koiter’s model, the bending model and the membrane model. They rely on a crucial lemma of J L Lions, used in an essential way for establishing in each case a generalized Korn’s inequality, which is then combined with a generalized rigid displacement lemma of a geometrical nature.

• Absolutely expedient algorithms for learning Nash equilibria

This paper considers a multi-person discrete game with random payoffs. The distribution of the random payoff is unknown to the players and further none of the players know the strategies or the actual moves of other players. A class of absolutely expedient learning algorithms for the game based on a decentralised team of Learning Automata is presented. These algorithms correspond, in some sense, to rational behaviour on the part of the players. All stable stationary points of the algorithm are shown to be Nash equilibria for the game. It is also shown that under some additional constraints on the game, the team will always converge to a Nash equilibrium.

• Hierarchic control

Distributed control is applied to a system modelled by a parabolic evolution equation. One considers situations where there are two cost (objective) functions. One possible way is to cut the control into 2 parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.

• # Proceedings – Mathematical Sciences

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