Semidefinite Research Articles

Fourier Optimization and Quadratic Forms

AbstractWe prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

Minimal universality criterion sets on the representations of binary quadratic forms

For a set S of (positive definite and integral) quadratic forms with bounded rank, a quadratic form f is called S-universal if it represents all quadratic forms in S. A subset S0 of S is called an S-universality criterion set if any S0-universal quadratic form is S-universal. We say S0 is minimal if there does not exist a proper subset of S0 that is an S-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for ‘most’ binary quadratic forms f, minimal S-universality criterion sets are unique in the case when S is the set of all subforms of the binary form f.

On Elliptic Curves and Binary Quartic Forms

Abstract A Dirichlet series is defined whose coefficients are determined by counting certain integral points on the quadratic twists of an elliptic curve. The function defined by this series has a meromorphic continuation with at most a simple pole at a certain distinguished point. In certain cases, the residue there gives a class number formula for positive definite binary quartic forms.

Positive definite matrix and its proof method

Advanced algebra is a required course of undergraduate mathematics, which plays a fundamental role in completing the study of other professional courses for students. Matrix theory is an important branch of mathematics, it is not only a basic subject, but also themost practical value. Widely used in mathematical theory, matrix is an important basic concept in matrix theory, is a major study of algebra, Positive definite matrix is a kind of important matrix, occupies an important position in “higher algebra”, its theory has rich content, positive definite quadratic form and Euclidean space, physics, probability and optimization control theory are related to it.

The number of representations of squares by integral quaternary quadratic forms

Let f be a positive definite (non-classic) integral quaternary quadratic form. We say f is strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly 34 strongly s-regular diagonal quaternary quadratic forms representing 1 (see Table 1). In particular, we use eta-quotients to prove the strong s-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number 2 (see Lemma 4.5 and Proposition 4.6).

Graph matching based on fast normalized cut and multiplicative update mapping

Point correspondence is a fundamental problem in pattern recognition and computer vision, which can be tackled by graph matching. Since graph matching is basically an NP-complete problem, some approximate methods are proposed to solve it. Continuous relaxation offers an effective approximate method for graph matching problem. However, the discrete constraint is not taken into consideration in the optimization step. In this paper, a fast normalized cut based graph matching method is proposed, where the discrete constraint is introduced into the optimization step. Specifically, first a semidefinite positive affinity matrix based form objective function is constructed by introducing a regularization term which is related to the discrete constraint. Then the fast normalized cut algorithm is utilized to find the continuous solution. Last, the discrete solution of graph matching is obtained by a multiplicative update algorithm. Experiments on both synthetic points and real-world images validate the effectiveness of the proposed method by comparing it with the state-of-the-art methods.

State Space Modeling with Non-Negativity Constraints Using Quadratic Forms

State space model representation is widely used for the estimation of nonobservable (hidden) random variables when noisy observations of the associated stochastic process are available. In case the state vector is subject to constraints, the standard Kalman filtering algorithm can no longer be used in the estimation procedure, since it assumes the linearity of the model. This kind of issue is considered in what follows for the case of hidden variables that have to be non-negative. This restriction, which is common in many real applications, can be faced by describing the dynamic system of the hidden variables through non-negative definite quadratic forms. Such a model could describe any process where a positive component represents “gain”, while the negative one represents “loss”; the observation is derived from the difference between the two components, which stands for the “surplus”. Here, a thorough analysis of the conditions that have to be satisfied regarding the existence of non-negative estimations of the hidden variables is presented via the use of the Karush–Kuhn–Tucker conditions.

A plural indefinite quantifier on the Romance-Slavic border

This study investigates the plural form uni/une deriving from the numeral ‘one’ in the Istriot dialect of Sissano. Sissano is located in the Istrian peninsula, an area characterized by high intensity of linguistic contact. We argue that the rise of such a peculiar form is indeed induced by contact with Croatian and that uni/une is unique in the Italo-Romance domain since, generally, the plural indefinite forms derived from the Latin numeral ‘one’ are pronouns and never occur in attributive position. The use of uni/ une is not attested in the few grammars of Istriot varieties because it is recent and still undergoing a process of grammaticalization. Therefore, we conducted interviews to verify how and to what extent contact with Croatian affects the meaning and the use of uni/une in Sissano. We found that this form is mostly used as a quantifier, bearing mainly the meaning ‘a pair of’, ‘one group of’, in the context of pluralia tantum and plural dominant nouns. We further observe that this quantifier has achieved a more advanced stage of grammaticalization in the younger generation of speakers than in the older ones. We discuss the role played by pluralia tantum as well as by the growing prestige of Croatian in triggering this borrowing and in fostering the grammaticalization process of uni/une on its way to become a marker of indefiniteness.

Siegel theta series for indefinite quadratic forms

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

Geometry on the variety of subspaces with positive definite form: geometry of the Grassmann spaces over generalized hyperbolic spaces

We consider Grassmann structures defined on the family consisting of subspaces on which a given nondegenerate bilinear form defined on a real vector space is positive definite. One may call such structures Grassmann spaces over generalized hyperbolic spaces. We show that the underlying (generalized) hyperbolic space can be recovered in terms of its Grassmannian, and the underlying projective space (equipped with respective associated polarity) can be recovered in terms of the generalized hyperbolic space defined over it.

Coupling of quantum gravitational field with Riemann and Ricci curvature tensors

The theoretical problem of establishing the coupling properties existing between the classical and quantum gravitational field with the Ricci and Riemann curvature tensors of General Relativity is addressed. The mathematical framework is provided by synchronous Hamilton variational principles and the validity of classical and quantum canonical Hamiltonian structures for the gravitational field dynamics. It is shown that, for the classical variational theory, manifestly-covariant Hamiltonian functions expressed by either the Ricci or Riemann tensors are both admitted, which yield the correct form of Einstein field equations. On the other hand, the corresponding realization of manifestly-covariant quantum gravity theories is not equivalent. The requirement imposed is that the Hamiltonian potential should represent a positive-definite quadratic form when performing a quadratic expansion around the equilibrium solution. This condition in fact warrants the existence of positive eigenvalues of the quantum Hamiltonian in the harmonic-oscillator representation, to be related to the graviton mass. Accordingly, it is shown that in the background of the deSitter space-time, only the Ricci tensor coupling is physically admitted. In contrast, the coupling of quantum gravitational field with the Riemann tensor generally prevents the possibility of achieving a Hamiltonian potential appropriate for the implementation of the quantum harmonic-oscillator solution.

On a class number formula of Hurwitz

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.

Krein reproducing kernel modules in Clifford analysis

Classic hypercomplex analysis is intimately linked with elliptic operators, such as the Laplacian or the Dirac operator, and positive quadratic forms. But there are many applications like the crystallographic X-ray transform or the ultrahyperbolic Dirac operator which are closely connected with indefinite quadratic forms. Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry. In this paper we are going to show that Clifford-Krein modules are naturally appearing in this context. Even taking into account the difficulties, e.g., the existence of different inner products for duality and topology, we are going to demonstrate how one can work with them Taking into account possible applications and the nature of hypercomplex analysis, special attention will be given to the study of Clifford-Krein modules with reproducing kernels. In the end we will discuss the interpolation problem in Clifford-Krein modules with reproducing kernel.

Weyl invariant Jacobi forms: A new approach

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type E8. This approach is also applicable to E8. Even if the algebra of E8 Jacobi forms is known to be non-free, we still derive a new structure result.

Ein korpuslinguistischer Beitrag zu Herkunft und Entwicklung des negativen Indefinitumskein

AbstractThis article discusses the origin and historical development of the German n-indefinitekein, which is an unusual negator because it does not share the initialnthat marks virtually all other negatives in German. Despite the discussion about its origins going back to the nineteenth century, it is still unclear howkeinfirst emerged and out of which other forms it developed. In this paper, new light is shed on an old controversy using new data and modern corpus-linguistic tools, in this case theReferenzkorpus Mittelhochdeutsch(ReM). The article first summarises the current state of research before presenting and analysing the data. In combination with additional evidence, the results show that certain hypotheses that have to this day been treated as accurate are in fact not viable. Subsequently, a solution that combines some of the existing theses and is compatible with the data is presented: Morphological reanalysis and the ensuing back-formation ofkein’s predecessornehein – in combination with a phonologically conditioned sound substitution triggered by a shift of the syllable boundary – in the context of negative concord seems the most likely candidate for an accurate explanation of the emergence and early usage patterns ofkeinin Middle High German. Incongruent evidence from Swiss German, however, suggests that partially convergent developments ensuing from different indefinite forms have taken place in that variety.

CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

PurposeThe author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.Design/methodology/approachThe author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.FindingsThe author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).Originality/valueTo the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

On exact Reznick, Hilbert-Artin and Putinar's representations

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers.We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and singly exponential in the number of variables. Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.

Event-triggered adaptive control of uncertain non-linear systems under input delay and limited resources

In this paper, both input delay and limited bandwidth challenges in uncertain networked control systems have been tackled. For this purpose, an event-triggered adaptive controller based on backstepping technique is designed for a general class of non-linear uncertain MIMO systems. The event-triggered mechanism is placed in the sensor-to-controller channel to reduce communication burden and save computational resources. Rather than predefined fixed or simple relative triggering thresholds, the proposed triggering condition is derived based on the negative semi-definiteness of derivative of Lyapunov function. To handle input delay problem, an auxiliary compensation system based on integral of input signal is introduced in the controller design. System stability are guaranteed under proposed scheme with no Zeno behavior. Simulation results on three-link cylindrical robotic arm show that the proposed control scheme successfully compensates for input delays and ensures a substantial saving in computational and network resources characterized by number of required control updates and signal transmissions over the network.

On the representation numbers satisfying partially multiplicative relations

Let rQ(n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2k variables. Let N be the level of Q. For a fixed prime p∤N, we assume that rQ(n) satisfies the relation rQ(1)rQ(p2n)=rQ(p2)rQ(n) for all positive integers n with p∤n. We actually show that this relation holds for any prime p∤N when (−1)kN is a fundamental discriminant.