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Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of $\mathcal{S}$-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among $\mathbb{Q}(\sqrt{u(n)})$ for n ∈ $\{M+1,\dots,M+N\}$. Fields of this type include the Shanks fields and their generalizations.

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

In this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

In this paper we consider the problem of an upper bound estimate for the least quadratic non-residue modulo prime number on special arithmetic sequences such as f(n) = [αn] and f(n) = [nc].

We classify the Morse indices for rank-convex quadratic forms defined on the space of linear elastic strains in two- and three-dimensional linear elasticity. For the higher-dimensional case n > 3, we give a universal lower bound of the largest possible Morse index and various upper bound of this index. We show in the three-dimensional case that the Morse index is at most 1, and in this case the nullity cannot exceed 2. Examples are given that show that the estimates can be reached. We apply the results to study the critical points for smooth rank-one convex functions defined on the space of linear strains. We also examine an example and construct a quasiconvex function that vanishes in a finite set in the direct sum of the null subspace and the negative subspace of the rank-one quadratic form.

We prove an explicit Hecke's bound for the Fourier coefficients of holomorphic cusp forms for SL2(Z) and apply it to derive effectively computable constants c (m) for each dimension m, divisible by 8, for which every even integer is always represented by every even unimodular form of m variables.

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

For a fieldkof characteristic not two, it is known thatkis algebraically closed in the function field of any (non-degenerate) quadratic form in three or more variables. In this note we consider fields of characteristic two and decide whenkis algebraically closed in a function field of a quadratick-form. For quadratic forms in three variables this has recently been done by Ohm.

We obtain simple and generally applicable conditions for the existence of mixed moments E([X′ AX]″/[X′BX]U) of the ratio of quadratic forms T = X′ AX/X′BX where A and B are n × n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.If u ≤ v, the precise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T; if u > v, a prerequisite for existence of the (u, v)th mixed moment is the existence of the 2(u − v)th moment of X When Xis not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2v.

Let denote the kth successive inhomogeneous minima for positive values of real indefinite ternary quadratic forms of type (2, 1). Here it is proved that for the class of zero forms, All the critical forms have also been obtained. is already known. For non-zero forms it is proved that .

The purpose of this paper is to prove functional equations of L-functions associated with the vector space of binary quadratic forms and determine their poles and residues. For a commutative ring K, let V(K) be the set of all symmetric matrices of degree 2 with coefficients in K. In V(C), we consider the inner productwhere for .

Recently, P. Kaplan and K. S. Williams [10] considered (as an example) the representation of primes by binary quadratic forms of discriminant –768. These forms fall into 4 genera, each consisting of two classes. In particular, they considered the formsF=3X2+642 and G = 12X2+12XY+19Y2.It follows from genus theory (as explained in [10]) that every prime p ≡ 19 mod 24 is represented by exactly one of the forms F and G. Based on numerical data, they conjectured that a prime p ≡ 19 mod 24 is represented bywhereVo = 2, V1 = -4, Vn+2=-4Vn+1 -Vn (n∨0).

Serre [6] has recently created a theory of some generality in response to a query from Manin about the size of the number N(x) of (indefinite) ternary quadratic forms AX2 + BY2 + CZ2 that represent zero and have coefficients of magnitudes not exceeding x.

Let s be a natural number, s ≥ 2. We seek a positive number λ(s) with the following property:Let ε > 0. Let Q1(x1, …, xs), Q2(x1, …, xs) be real quadratic forms, then for N > C1(s, ε) we havefor some integers n1, …, ns

Let X be the space of nondegenerate rational symmetric matrices of size 2 and putThe group G acts on X byWe are interested in the space (Γ\X) of Γ-invariant C-valued functions on X and its subspace &(Γ\X) of functions whose supports consist of a finite number of Γ-orbits. The Hecke algebra ℋ(G, Γ) of G with respect to Γ acts naturally on these spaces.

G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:Theorem.It is necessary and sufficient for the class number of an imaginary quadratic number fieldD= 1 — 4m, m> 0,to be one that x2—x+m is prime for any integer x such that1 ≤x≤m— 2.

Let ‖x‖ denote the distance of x from the nearest integer. In 1948 H. Heilbronn proved [5] that for ε>0 and N>c1(ε) the inequalityholds for any real α. This result has since been generalised in many different directions, and we consider here extensions of the type: For ε>0, N>c2{ε, s) and a quadratic formQ(x1,…, xs) there exist integersn1,…,nsnot all zero with |n1|,…,|ns≦N and with

The conventional linear/quadratic/Gaussian assumptions are modified in that minimisation of the expectation of cost G defined by (2) is replaced by minimisation of the criterion function (5). The scalar –θ is a measure of risk-aversion. It is shown that modified versions of certainty equivalence and the separation theorem still hold, that optimal control is still linear Markov, and state estimate generated by a version of the Kalman filter. There are also various new features, remarked upon in Sections 5 and 7. The paper generalises earlier work of Jacobson.

SynopsisThe groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to Fuchsian groups and which, for zero forms, are commensurable with the classical modular group. This is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms. The signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms. This last result should be compared with the corresponding result by Mennicke on non-zero forms.

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integert≥ 1 and a cusp formof weightk/2 (kodd, ≥ 3), levelN(divisible by 4) and characterϰ, he associated a certain functionf(t)(z) (Ft(z) in Shimura’s notation).

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

In this paper we shall define a generalised A-quadratic form and prove that in some way this form and an A-bilinear form are equivalent to each other. Our result characterises that of Vukman in the sense that we use any n vectors for a fixed n ≥ 2, instead of any two vectors. Consequently, a new generalisation of an inner product space among vector spaces is obtained. This also leads to a new relationship between a 2-inner product space and a 2-normed space.

Let L, M be regular quadratic lattices over Zp. The local density αp(L, M) is an important invariant in the theory of representation of quadratic forms and they appear naturally in Fourier coefficients of Eisenstein series. In spite of the importance we knew little except the case when either L = M or rk L = 1 and M is unimodular. Evaluating them is a laborious task.

Quadratic forms associated with graphs were introduced over a century ago by Jordan [4]. We are concerned with the optimisation of such quadratic forms, following Motzkin and Straus [5], and we use the setting of categories and functors to express the nice interplay between the algebra and the graph theory. Applications to interchange graphs are also obtained.

Our aim is to prove THEOREM. Let L be a positive lattice of E-type such that [L: L] and L is indecomposable . (i) If L ≅ L 1 ⊗ L 2 for positive lattices L 1 , L 2 , then L 1 , L 2 are of E-type and [L 1 : L 1 ], [L 2 :L 2 ] and L 1 , L 2 are indecomposable . (ii) If L is indecomposable with respect to tensor product, then for each indecomposable positive lattice X we have (1) L ⊗ X ≅ L ⊗ Y implies X ≅ Y for a positive lattice Y , (2) If X= ⊗ t L ⊗ X′ where X′ is not divided by L, then O(L ⊗ X) is generated by O(L), O(X′) and interchanges of L’s, and (3) L ⊗ X is indecomposable .

A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

In the projective plane, if H is a harmonic homology (linear transformation with H2 = I), and G a general inversion (quadratic transformation projectively equivalent to an inversion), then under a certain condition there is a pencil of cubics each of which is invariant under G, H separately. These are related to transformations discovered by Mandel, Todd and Lyness. As a near converse, we find that, given a Pascal configuration, there is a quadratic Cremona transformation under which each cubic passing through the vertices of the configuration is invariant. As a by-product, parametric expressions are found for elliptic functions of a fifth of a period.

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

The requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

A theory of quadratic forms is developed overR(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

A well-known theorem of Minkowski on the product of two linear forms states that ifare two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.

We use the notations and definitions of the introduction to the previous paper ((1)), and say that 2 lattices K and L on a quadratic space over R(t) with its Dedekind set C are in the same genus if, for all , the lattices and are in the same class. We mean to count the number of classes of lattices in a given genus over a binary space: we shall find that it is either 1 or .

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

§ 1. The groundwork of the present paper is a general transformation of Maxwell's equations. As relativist speculation had its origin in a simple transformation, it was to be expected that something more would result when the general transformation was explored: the determining influence has proved stronger than I had anticipated.

An ancipital form is a form [a, b, c] in which b= 0 or b= a; these fall into pairs of associates: [a, 0, c] and [c, 0, a] (type 1), and [a, a, c] and [4c–a, 4c–a, c] (type 2). The set of discriminantal divisors of discriminant d is formed by choosing, from each pair of primitive associate ancipital forms of discriminant d, exactly one of the two leading coefficients. In this article we study representations of discriminantal divisors of a given discriminant by binary quadratic forms of that discriminant, previously studied by the author and by G. Pall. We are concerned here with discriminants d= 4kpq, where k ≥ 1, p = 1, q = 3 (mod 4) are primes, and d = 4kp, where k ≥ 1 and p is an odd prime. This investigation arose in connection with the search for integral solutions of x2 –Dy2 = – 1.

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

Letbe a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfyingwhere F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimumm+(f) of the non-negative values1of an indefinite quadratic formfinnvariables, of given determinant det(f) ≠ 0 and of signatures. In particular it was announced (and later proved — see [2]) thatm+(f) ≦ (16/5)+for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is thatm+(f) ≦ (256/135)+for quaternary quadratic forms of determinant — 1 and signature — 2.

§ 1. In the transformation corresponding to constant acceleration, and in the resulting quadratic form, there is an arbitrary constant. In Einstein's formula for central acceleration a mass m defining the acceleration appears, but no arbitrary constant. In seeking to account for, or to repair, the deficiency I retained all constants naturally arising in the integration of the differential equations, and it then appeared that the result could be obtained by transformation. It will be convenient to state the transformation at the outset, and then to examine the differential equations with a view to a clear understanding of the assumptions made in reaching the more general result.

Let f = f (x1, x2,… xn) be an indefinite n-ary quadratic form of signature s and let m+(f), m−(f) denote the infimum of the non-negative values taken by f and —f respectively for integral (x1, x2,…, xn) ≠ (0, 0,…, 0). Furthermore let f satisfy the condition m+ (f) ≠ 0 and let for some integer k. Then Segré [3] has shown that, for n = 2, f must have determinant det (f) satisfying with equality if and only if f is equivalent under an integral unimodular transformation (denoted ˜) to a multiple of the form f1(x, y) = x2−kxy−ky2, while Oppenheim [2] has shown that, for n ≧ 3, is of the order of k2n−2

Let f = f(x) = f(x1, x2,…, xn) be an indefinite n-ary quadratic form of determinant det (f); that is, f(x) = x' Ax where A is a real symmetric matrix with determinant det (f). Such a form is said to take the value v if there exists integral x ≠ 0 such that f(x) = v.

LetQbe the rational number field, and letK=(D> 0 a rational integer) be a real quadratic field. Then, throughout this paper, we shall understand by the fundamental unitεDofthe normalized fundamental unitεD> 1.

SummaryLet represent the deviations from expectation of a set of multinomial or independent Poisson variables, and Η be a positive definite matrix. A lower bound is obtained for Pr in terms of Pr, Where is a vector of normal variables with the same mean and covariance matrix as Δ.