A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over k. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over k, which is a consequence of the first one.     

The mean value theorem and analytic functions of a complex variable

The mean value theorem for real-valued differentiable functions defined on an interval is one of the most fundamental results in Analysis. When it comes to complex-valued functions the theorem fails even if the function is differentiable throughout the complex plane. we illustrate this by means of examples and also present three results of a positive nature.     

On the Shintani Zeta Function for the Space of Pairs of Binary Hermitian Forms

In this paper we determine the principal part of the adjusted zeta function for the space of pairs of binary Hermitian forms.     

Generalized bent functions and class group of imaginary quadratic fields

Several new results on non-existence of generalized bent functions are presented. The results are related to the class number of imaginary quadratic fields.     

The moments and statistical distribution of class numbers of quadratic fields with prime discriminant

We establish asymptotic formulas for all the moments of class numbers of quadratic fields with prime discriminant. Further, we study the statistical distribution of these class numbers, introducing a certain random Euler product.     

Parametrization of the quadratic function fields whose divisor class numbers are divisible by three

A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.     

Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY ²¹ =bX ²¹ +cZ ²¹ defined over finite fields F_q such thatq = p ^α? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_q. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_q, as rational functions in the variablet, for distinct cases ofa, b, andc, in F _q ^* . Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_q. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).     

On well-rounded sublattices of the hexagonal lattice

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.     

Fourier series, measures and divided power series in the theory of function fields

Much of classical number theory is based on Fourier series. Such series play a vital role in the study of characteristic-0 zeta-functions: In the complex theory one has theta-series and Tate's thesis. In the p-adic theory one has Mahler's theorem on binomial coefficients which is used to déscribe the ring of p-adic measures. In this paper, we discuss a version of binomial coefficients for function fields due to L. Carlitz. We will show how these functions arise naturally out of gamma functions for function fields. We will also use some work of C. Wagner to establish that the ring of -adic measures is canonically isomorphic to the ring of divided power-series. The computation of these power-series in specific instances is now an important problem in the theory. Finally, we show the existence of many Fourier transforms in the -adic theory. The explicit computation of these would also be very interesting.Partially supported by NSF grant DMS-8521678. Current address: Department of Mathematics, UMBC, MD 21228, U.S.A.Dedicated to L. Carlitz     

Solvability of a class of boundary value problems in the space of convergent sequences

We consider two types of infinite systems of second-order differential equations with boundary conditions in the Banach sequence space c. For each system, sufficient conditions are provided for the existence of solutions. Our approach is based on the use of measure of noncompactness concept and the application of Darbo’s theorem for condensing operators. Some examples are presented to illustrate the obtained results.     

Congruences for the class numbers of real cyclic sextic number fields

LetK ₆ be a real cyclic sextic number field, andK ₂,K ₃ its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh ^- =h (K ₆)/(h(K ₂)h(K ₃)) are obtained. In particular, when the conductorf ₆ ofK ₆ is a primep, , whereC is an explicitly given constant, andB _n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields. Project supported by the National Natural Science Foundation of China (Grant No. 19771052).     

The conjugacy problem of the modular group and the class number of real quadratic number fields

We shall discuss the conjugacy problem of the modular group, and show how its solution, in conjunction with a theorem of Olga Taussky can be used to compute the class number of certain real quadratic number fields.     

Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

On a class of nonlocal bifurcation concerning the Lorenz attractor

In this paper, we consider a two-parameter family of systemsE in whichE ₀ has a contour consisting of a saddle point and two hyperbolic periodic orbits, i.e., the situation is similar to that described by the Lorenz equations for parametersb= 8/3,=10,r=r ₁ 24.06. For the generic unfoldingE ofE ₀, we find three kinds of infinitely many bifurcation curves and establish the correspondence of the trajectories which stay forever in a sufficiently small neighborhood of the contour with symbolic systems of finite or countably infinite symbols; these results can be used to explain the turbulence behaviors appearing at the critical valuer=r ₁ 24.06 observed on computer for Lorenz equations in a precise mathematical way.     

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

The number of steps in the euclidean algorithm over complex quadratic fields

We obtain upper and lower bounds for the number of divisions in the Euclidean algorithm, for almost all pairs of algebraic integers lying in the complex quadratic fields (–m), form=1, 2, 3, 7 and 11. In addition, the order of the average length for almost all such pairs is deduced.