Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

The representation of quadratic forms by almost universal forms of higher rank

 In this article, we prove that there are only finitely many positive definite integral quadratic forms of rank n+3(n≥2) that represent all positive definite integral quadratic forms of rank n but finitely many exceptions. Furthermore we determine all diagonal quadratic forms having such property and its exceptions remaining four as candidates. Received: 29 November 2000 ; in final form: 8 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 11E12, 11E20.     

On the dimension of some spaces of generalized theta-series

We improve the upper bound of the dimension of the space of theta-series with respect to positive definite quadratic forms for some diagonal quadratic forms.     

Universal binary Hermitian forms

We will determine (up to equivalence) all of the integral positive definite Hermitian lattices in imaginary quadratic fields of class number 1 that represent all positive integers.

     

On primitively universal quadratic forms

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.     

Binary quadratic forms represented by sums of squares in an essentially unique way

The Ramanujan Journal - In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for...     

Positive definite quadratic forms representing integers of the form <Emphasis Type="Italic">an</Emphasis><Superscript>2</Superscript>+<Emphasis Type="Italic">b</Emphasis>

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S _a ={an ²∣n≥2} or S=S _a,b ={an ²+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S _a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S ₁-universal quadratic forms not representing 1.     

On the Construction of Nondecomposable Definite Hermitian Forms

51.Introducti0nLetF=Q(In)beanimaginaryquadraticfieldandD.theringofintegersofF.wheremisasquare-freepositiveinteger.ForapositivedefiniteD.-latticeLthesetSesq>.(L)ofpositivesemidefinitesesquilinearformsoverD.,0requivalently,thesetofpositivesemidefiniteHermitianmatrices0verD.isasemigroupunderadd1tlon.AformoverD.whichcanbewrittenasthesumoftwon0n-zer0elementsinSesq>,(L)iscalledadditivelydecomposable,otherwlseitiscallednondecomp0sab1e.IftheD.-lattice(L,9)istheorthogonalsumoftwonon-zerosublatti…     

Positive definite <Emphasis Type="Italic">n</Emphasis>-regular quadratic forms

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.     

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.     

Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms

In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over . Received: 13 November 1997 / Revised version: 17 November 1998     

Representations of finite number of quadratic forms with same rank

The Ramanujan Journal - Let m,&nbsp;n be positive integers with $$m\le n$$ . Let $$\kappa (m,n)$$ be the largest integer k such that for any (positive definite and integral) quadratic forms...     

Existence of Lyapunov functions of variable sign for linear extensions of dynamical systems on a torus

The problem of the existence of quadratic forms that have a positive definite derivative along the solutions of linear extensions of dynamical systems on a torus is considered. Assuming the existence of quadratic forms whose derivative is positive definite only with respect to part of the variables, conditions ensuring the existence of a quadratic form whose derivative is already positive definite with respect to all variables are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1713–1717, December, 1990.     

CONSTRUCTION OF INDECOMPOSABLEDEFINITE HERMITIAN FORMS

ＣＯＮＳＴＲＵＣＴＩＯＮＯＦＩＮＤＥＣＯＭＰＯＳＡＢＬＥＤＥＦＩＮＩＴＥＨＥＲＭＩＴＩＡＮＦＯＲＭＳ￥ＺＨＵＦＵＺＵ（ＤｅｐａｒｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙｔＳｈａｎｇｈａｉ２０００６２，...     

Manin’s conjecture for a class of singular cubic hypersurfaces

Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u³ = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.     

On locally primitively universal quadratic forms

The Ramanujan Journal - A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In...     

Quadratic forms that represent almost the same primes

Jagy and Kaplansky exhibited a table of pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that their list is complete outside of ``trivial' pairs. In this article, we confirm their conjecture, and in fact find all pairs of such forms that represent the same primes outside of a finite set.

     

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

We study eigenvalues of positive definite kernels of L² integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.     

Multiplicative factorizations of integral representations by quadratic forms

Multiplicative dependence of integral representations of integers by positive definite quadratic forms in an odd number of variables on square factors of the represented numbers is studied. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 15–55. Translated by A. S. Goloubeva.     

On embedding numbers into quaternion orders

A generalization of the chevalley-Hasse-Noether theorem from maximal orders to arbitrary Eichler orders in quaternion algebras is given. A stability property for the numbers of orbits for unit groups in quaternion orders acting on optimal embeddings of quadratic orders is proved. The results are applied to Siegel's meanvalue of integral representations by genera of integral definite ternary quadratic forms.