An asymmetric inequality for non-homogeneous ternary quadratic forms

LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and \(f(t) = \frac{4}{{(1 + t)^2 (1 + 5t)}}\) . Then given any real numbersx ₀,y ₀,z ₀ there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) \leqslant (f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained.     

A note on moment inequality for quadratic forms

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented.     

An inequality for means with applications

We show that an almost trivial inequality between the first and second moment and the maximal value of a random variable can be used to slightly improve deep theorems. Received: 25 November 2006     

A canonical decomposition for quadratic forms with applications to monotone convergence theorems

We prove monotone convergence theorems for quadratic forms on a Hilbert space which improve existing results. The main tool is a canonical decomposition for any positive quadratic form h = h_r + h_s where h_r is characterized as the largest closable form smaller than h. There is also a systematic discussion of nondensely defined forms.     

An inequality for multiple convolutions with respect to Dirichlet probability measure

A sharp multiple convolution inequality with respect to Dirichlet probability measure on the standard simplex is presented. Its discrete version in terms of the negative binomial coefficients is proved as well. The new bounds for the Dirichlet distribution and iterated convolutions are obtained as the consequences of the main result. Also some binomial, exponential, and generalized hypergeometric applications are discussed.     

Reduction theory of positive quadratic forms

The theory of Hermite and Minkowski reduction of positive n-ary quadratic forms is studied. The question of the equivalence of reduced forms is investigated. It is shown that for n3 the partition of the cone of positivity into precise domains of Hermite-Minkowski reduction is not normal (some domains intersect along pieces of the faces). For n6: 1) contiguous vectors of the Dirichlet domain D(f) are found, where f is a form which is Hermite-Minkowski-reduced; 2) the classical and precise domains of Hermite-Minkowski reduction are computed; 3) a convenient algorithm is developed for the verification of the equivalence of reduced forms. On the basis of this algorithm a precise fundamental domain is constructed for n=3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 6–96, 1975.     

A sum formula related to ellipsoids with applications to lattice point theory

In this article we consider finite and infinitep-dimensional sums over functionsf, where the argument off is represented by a positive definite quadratic form. We develop a sum formula like theEuler-Maclaurin orPoisson sum formula. Applications to exponential sums and lattice point problems are given.     

An inequality for a simplex and its applications

In this paper, we improve some inequalities for ann-dimensional simplex in Euclidean spaceE ⁿ and give some applications.     

Bilinear forms over a finite field,with applications to coding theory

Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF(q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given $\text{X ? Ω}$; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems $\text{X ? Ω}$, which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.     

Weighted sum of the extensions of the representations of quadratic forms

Let L, N and M be positive definite integral \({\mathbb{Z}}\) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of \(\tilde M_{\tilde \sigma}\) in the gen(M _σ) via N _η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.     

Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints

A symmetric matrix C is called copositive if the quadratic form x′Cx is nonnegative for all nonnegative values of the variables (x₁,x₂,…,x_n)=x′. A known sufficient condition for a quadratic form x′Qx to be positive unless x=0, subject to the linear inequality constraints Ax?0, is that there should exist a copositive matrix C such that Q?A′CA is positive definite. The main result of this paper establishes the necessity of this condition. For x'Qx to be merely nonnegative subject to Ax ? 0, the situation is less straightforward. The necessity of the existence of a copositive matrix C such that Q?A′CA is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of A, and counterexamples are given to show that, in general, no such matrix may exist, even when Slater's constraint qualification holds. Our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x′Qx subject to Ax?0, in which specific symmetric matrices, constructed by rational operations from A and Q and depending upon a single real parameter v, must be tested for positive definiteness or strict copositivity for large values of v. This technique is illustrated by several examples.     

An asymmetric Orlicz centroid inequality for probability measures

Using M-addition,an asymmetric Orlicz centroid inequality for absolutely continuous probability measures is established corresponding to Paouris and Pivovarov’s recent result on the symmetric case.As an application,we extend Haberl and Schuster’s asymmetric Lp centroid inequality from star bodies to compact sets.     

A martingale decomposition for quadratic forms of Markov chains (with applications)

We develop a martingale-based decomposition for a general class of quadratic forms of Markov chains, which resembles the well-known Hoeffding decomposition of U$U$-statistics of i.i.d. data up to a reminder term. To illustrate the applicability of our results, we discuss how this decomposition may be used to studying the large-sample properties of certain statistics in two problems: (i) we examine the asymptotic behavior of lag-window estimators in time series, and (ii) we derive an asymptotic linear representation and limiting distribution of U$U$-statistics with varying kernels in time series. We also discuss simplified examples of interest in statistics and econometrics.     

A Buzano type inequality for two Hermitian forms and applications

A Buzano type inequality for two nonnegative Hermitian forms is obtained. Applications to inequalities for norm and numerical radius of bounded linear operators in complex Hilbert spaces are given.