Representation of Integers in Subsequences of the Positive Integers by Binary Quadratic Forms

We consider positive-definite primitive binary quadratic forms of fundamental discriminant d < 0; R is the genus and C is the class of such forms. We obtain asymptotics for the sum of absolute values of the Fourier coefficients for the Hecke eigenforms of weight 1 and of dihedral type. In an earlier paper (Zap. Nauchn. Semin. POMI, 226 (1996)), the author showed that if C R, then almost all R-representable positive integers are C-representable. We extend this result to certain subsequences of such as {a _n = p _n + l}, {a _n = n(n + 1)}, etc. Finally, for certain genera R with class number greater than one, we prove an asymptotics (x ) for the sum

Representation of Binary Quadratic Forms by a Quaternary Form

The paper presents formulas for finding the number of representations r(A; Q) of a form A with odd square-free level by the form Q defined by the identity matrix of order 4. The case where the difference of the dimensions n — m is even (the codimension is even), which was not considered previously, is analyzed. Bibliography: 8 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 68–80.     

Representation of Integers by Ternary Quadric Forms

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number. Supported by NSFC and RFDP.     

Exceptional Integers for Binary Quadratic Forms

An integer m is said to be exceptional for a binary quadratic form if m is representable by a form from the genus of but not by the form itself. An asymptotic distribution law for exceptional integers is proved. Bibliography: 3 titles.     

Modular Properties of Theta-Functions and Representation of Numbers by Positive Quadratic Forms

By means of the theory of modular forms the formulas for a number of representations of positive integers by two positive quaternary quadratic forms of steps 36 and 60 and by all positive diagonal quadratic forms with seven variables of step 8 are obtain.     

On the Representation of Integers by the Lorentzian Quadratic Form

In this paper, we establish an asymptotic formula, for large radiusr, for the number of representations of a nonzero integerkby the Lorentzian quadratic formx²₁+x²₂+…+x²_n−x²_n+1that are contained in the ball of radiusrcentered at the origin in Euclidean (n+1)-space.     

Representations of Positive Integers by a Direct Sum of Quadratic forms

The number of representation of positive integers by quadratic forms $ F_{1}=x_{1}^{2}+3x_{1}x_{2}+8x_{2}^{2} $ and $ G_{1}=2x_{1}^{2}+3x_{1}x_{2}+4x_{2}^{2} $ of discriminant —23 are given. Moreover, a basis for the cusp form space S ₄(Γ₀(23), 1) are constructed. Furthermore, formulas for the representation of positive integers by direct sum of copies of F ₁ and G ₁, i.e. formulas for $ r(n; F_{4}), r(n; G_{4}), r(n; F_{3} \oplus G_{1}), r(n; F_{2} \oplus G_{2}), {\rm and}\ r(n; F_{1} \oplus G_{3}) $ , are derived using the elements of the space S ₄(Γ(23), 1).     

On the Representation of Numbers by Positive Diagonal Quadratic Forms with Five Variables of Level 16

A general formula is derived for the number of representations r(n; f) of a natural number n by diagonal quadratic forms f with five variables of level 16. For f belonging to one-class series, r(n; f) coincides with the sum of a singular series, while in the case of a many-class series an additional term is required, for which the generalized theta-function introduced by T. V. Vepkhvadze [4] is used.     

On the Number of Representations of Integers by Quadratic Forms in Twelve Variables

A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.     

On the Determinantal Representation of Quaternary Forms

We prove that a general polynomial form of degree d in 4 variables, over the complex field, can be written as the sum of two determinants of 2 × 2 matrices of forms, with given degree matrix (a _ij ), for any choice of non-negative integers a _ij  ≤ d with a ₁₁ + a ₂₂ = a ₁₂ + a ₂₁ = d.     

Positive Quadratic Forms on Intersections of Quadrics

The positive definiteness of a quadratic form on the cone defined as the set of points at which given quadratic forms vanish or are nonnegative is studied. Necessary and sufficient conditions for the positivity of a quadratic form on this cone are obtained. The results are applied to study second-order sufficient conditions for abnormal extremal problems.     

Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax²+bxy+cy² with discriminant d=b²-4ac. The behavior of h(5p²), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p² with large class numbers. In fact, one has the estimate

2-Universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

整数集的唯一加权表示基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. In this paper, we prove that there is a set A■Z such that every integer can be represented uniquely in the form n = k1a1 + k2a2, a1, a2 ∈ A.     

On an Asymptotics of the Number of Representations of a Pair of Integers by Quadratic and Linear Forms with Congruential Condition

Mathematical Notes - Asymptotic formulas with remainder for the number of representations of a pair of integers by quadratic and linear forms with a congruential condition are proved.     

A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors

Let the column vectors of X:: M×N, M<N, be distributed as independent complex normal vectors with the same covariance matrix Σ. Then the usual quadratic form in the complex normal vectors is denoted by Z=XLX^H where L: N×N is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distribution of the generalized variance |Z|.