Semidefinite Research Articles

Repeated integration and explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\)

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

Weyl roots and equivalences of integral quadratic forms

We study integral quadratic forms in the sense of Roiter, that is, quadratic forms whose integer coefficients satisfy certain divisibility condition assuring that the associated Weyl group is integral. Such forms are known to be useful for characterizing classes of finite-dimensional algebras and Lie algebras. We present a solution of the problem considered by Roiter in 1978 of providing a characterization of Weyl roots of positive definite forms which is independent on an explicit action of the Weyl group. We also prove some new facts on the Weyl group and the Weyl roots in general. Moreover, we provide a detailed comparison of several distinct equivalence relations of integral forms appearing in the literature. In particular, we discuss how these relations behave with respect to the Weyl roots and some properties of integral forms, and we supplement some missing proofs related to the classification of positive definite and principal forms.

Improved multiple Lyapunov functions of input–output-to-state stability for nonlinear switched systems

This study is concerned with the Lyapunov characterization of input–output-to-state stability (IOSS) for time-varying nonlinear switched systems. Considering the restriction in current methods that each subsystem must be IOSS, and the Lyapunov function of each subsystem is strictly descended in the definition domain, in this work, by extending multiple Lyapunov functions to an indefinite form, the restriction on the negativity of the time derivative of Lyapunov functions is relaxed; additionally, all subsystems are permitted to not be IOSS in the state space. The results obtained are then extended to verifying the integral input–output-to-state stability (iIOSS) and output-to-state stability (OSS); and a state-norm estimator is proposed based on the indefinite Lyapunov function as a by-product. Finally, a numerical example illustrates the effectiveness of the results.

On prescribing total orders and preorders to pairwise distances of points in Euclidean space

We show that any total preorder on a set of the form (X2) where X has n elements coincides with the order on pairwise distances of some point collection of size n in Rn−1. For total orders, a collection of n points in Rn−2 suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

On the number of integral binary n$n$‐ic forms having bounded Julia invariant

In 1848, Hermite introduced a reduction theory for binary forms of degree n $n$ which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational SL 2 ${\rm SL}_2$ -invariant of binary n $n$ -ic forms defined over R $\mathbb {R}$ , which is now known as the Julia invariant. In this paper, for each n $n$ and k $k$ with n + k ⩾ 3 $n+k\geqslant 3$ , we determine the asymptotic behavior of the number of SL 2 ( Z ) ${\rm SL}_2(\mathbb {Z})$ -equivalence classes of binary n $n$ -ic forms, with k $k$ pairs of complex roots, having bounded Julia invariant. Specializing to ( n , k ) = ( 2 , 1 ) $(n,k)=(2,1)$ and (3,0), respectively, recovers the asymptotic results of Gauss and Davenport on positive definite binary quadratic forms and positive discriminant binary cubic forms, respectively.

Ternary quadratic forms representing the same integers

In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this paper, we prove the existence of pairs of ternary quadratic forms representing the same integers which are not contained in Kaplansky’s list.

The rank of new regular quadratic forms

A (positive definite and integral) quadratic form f is called regular if it represents all integers that are locally represented. It is known that there are only finitely many regular ternary quadratic forms up to isometry. However, there are infinitely many equivalence classes of regular quadratic forms of rank n for any integer n greater than or equal to 4. A regular quadratic form f is called new if there does not exist a proper subform g of f such that the set of integers that are represented by g is equal to the set of integers that are represented by f. In this article, we prove that the rank of any new regular quadratic form is bounded by an absolute constant.

SOME PROPERTIES OF (n, m)-POWER-HYPONORMAL IN SEMI HILBERTIAN SPACE

In [1] (Kikete Wabuya, Luketero Wanyonyi, and Justus Mile) show that if an operator (n,m) hyponormal is isometrically equivalent to an operator S, then S is also (n ,m) hyponormal operator. In this paper, we prove results in the same spirit but in a semi Hilbertian space, i.e., spaces generated by positive semi-definite sesquilinear forms. This kind of spaces appears in many problems concerning linear and bounded operators on Hilbert spaces and is intensively studied in the present; some of the basic properties of this class are studied. Moreover, the product, tensor product and the sum of finite numbers of this type are discussed.

Improved Multiple Lyapunov Functions of input-to-output Stability and input-to-state Stability for Switched Systems

This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally, subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method, which extends multiple Lyapunov functions (MLFs) to an indefinite form. Indefinite multiple Lyapunov functions (iMLFs) are proposed on the basis of an inequality to estimate the upper boundary of the system state. Moreover, we apply the iMLFs method to verify the extended notions of IOS in a switched system. More relaxed sufficient conditions for ISS and IOS are proposed, a simulation experiment is carried out on a numerical example of the system, which demonstrates their effectiveness and superiority.

The Mellin Transform to Manage Quadratic Forms in Normal Random Variables

The problem of computing the distribution of quadratic forms in normal variables has a long tradition in the statistical literature. Well-established numerical algorithms that deal with this task rely on the inversion of Fourier transforms or series representations. In this article, the Mellin transform is proposed as a tool to compute both the density and the cumulative distribution functions of a positive definite quadratic form: an outline of the numerical algorithm is presented, providing details on the error analysis. The algorithm’s characteristics allow us to propose an efficient way to compute the random variables’ quantiles. From the theoretical point of view, the analytic properties of the Mellin transform are exploited to provide a novel representation of the distribution of the ratio of independent quadratic forms as a mixture of beta random variables of the second kind. Moreover, algorithms are proposed for computations related to ratios of both independent and dependent quadratic forms. The methods are tested and compared to popular numerical algorithms in terms of computational times and accuracy. The R package QF implementing all the proposed algorithms is also made available. Supplementary materials for this article are available online.

Finite dimensional systems of free fermions and diffusion processes on spin groups

In this article, we are concerned with “finite dimensional fermions,” by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra so(2n+1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator.

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

Let \(k \ge 2\) and N be positive integers and let \(\chi \) be a Dirichlet character modulo N. Let f(z) be a modular form in \(M_k(\Gamma _0(N),\chi )\). Then we have a unique decomposition \(f(z)=E_f(z)+S_f(z)\), where \(E_f(z) \in E_k(\Gamma _0(N),\chi )\) and \(S_f(z) \in S_k(\Gamma _0(N),\chi )\). In this paper, we give an explicit formula for \(E_f(z)\) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel’s formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities.

A proof of the mod 4 unimodal sequence conjectures and related mock theta functions

In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4 congruences for the coefficients of these generating functions. In 2016 Kim, Lim and Lovejoy obtained similar results for odd-balanced unimodal sequences and made similar mod 4 conjectures. We prove all of these mod 4 conjectures and similar congruences for the Andrews spt-function and related mock theta functions. Our method of proof involves new Hecke-Rogers type identities for indefinite binary quadratic forms and the Hurwitz class number.

On small values of indefinite diagonal quadratic forms at integer points in at least five variables

For anyε>0\varepsilon > 0we derive effective estimates for the size of a non-zero integral pointm∈Zd∖{0}m \in \mathbb {Z}^d \setminus \{0\}solving the Diophantine inequality|Q[m]|>ε\lvert Q[m] \rvert > \varepsilon, whereQ[m]=q1m12+…+qdmd2Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2denotes a non-singular indefinite diagonal quadratic form ind≥5d \geq 5variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport to higher dimensions combined with a theorem of Schlickewei. The result obtained is an optimal extension of Schlickewei’s result, giving bounds on small zeros of integral quadratic forms depending on the signature(r,s)(r,s), to diagonal forms up to a negligible growth factor.

Distribution of values of quadratic forms at integral points

The number of lattice points in d-dimensional hyperbolic or elliptic shells {m : a<Q[m]<b}, which are restricted to rescaled and growing domains r,Omega , is approximated by the volume. An effective error bound of order o(r^{d-2}) for this approximation is proved based on Diophantine approximation properties of the quadratic form Q. These results allow to show effective variants of previous non-effective results in the quantitative Oppenheim problem and extend known effective results in dimension d ge 9 to dimension d ge 5. They apply to wide shells when b-a is growing with r and to positive definite forms Q. For indefinite forms they provide explicit bounds (depending on the signature or Diophantine properties of Q) for the size of non-zero integral points m in dimension dge 5 solving the Diophantine inequality |Q[m] |< varepsilon and provide error bounds comparable with those for positive forms up to powers of log r.

Headlines of media texts: hidden contradictions

This inquiry aims to determine the ways and means through which the accuracy category manifests itself in the news genre and to identify internal contradictions formed by the interaction between logical and semantic components of the expression. The objectives were as follows: to determine how the accuracy category is presented through the categories of subject, time, and place of action, what are the transformative possibilities of these categories in the field of adequate transmission of meaning, which linguistic means actualize the accuracy category or, on the contrary, “blur” it. Continuous sampling, analysis and synthesis, transformational, and descriptive methods were used in the research process. We confirm that the subject in the headline is delivered explicitly and implicitly in the conclusions presenting the research results. “Nonsubjectivity” is realized using indefinite (non-personal) forms of the predicate or can be determined through a locus. For news headlines, the locus is an essential component, its place in the sentence can adjust the semantic accents of the statement. The time category in analyzed titles is almost always explicit. The accuracy category is most clearly manifested in direct indications of time and place and in two-syllable sentences where the subject of the action is specified.

Arithmetic properties of Fourier coefficients of meromorphic modular forms

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.

Statistical Beamforming Techniques for Power Domain NOMA System

Power-domain non-orthogonal multiple access (NOMA) assigns different power levels for near and far users in order to discriminate their signals by employing successive interference cancellation (SIC) at the near user. In this context, multiple-input-single-output NOMA (MISO-NOMA), where the base station (BS) is equipped with multiple antennas while each mobile user has a single antenna receiver, is shown to have a better overall performance by using the knowledge of instantaneous channel state information (CSI). However, this requires prior estimation of CSI using pilot transmission, which increases the transmission overhead. Moreover, its performance is severely degraded in the presence of CSI estimation errors. In this work, we provide statistical beamforming solutions for downlink power-domain NOMA that utilize only knowledge of statistical CSI, thus reducing the transmission overhead significantly. First, we derive the outage probabilities for both near and far users in the multi-user NOMA system without imposing strong assumptions, such as Gaussian or Chi-square distribution. This is done by employing the exact characterization of the ratio of indefinite quadratic form (IQF). Second, this work proposes two techniques to obtain the optimal solution for beam vectors which rely on the derived outage probabilities. Specifically, these two methods are based on (1) minimization of total beam power while constraining the outage probabilities to the QoS threshold, and (2) minimization of outage probabilities while constraining the total beam power. These proposed methods are non-convex function of beam vectors and, hence, are solved using numerical optimization via sequential quadratic programming (SQP). Since the proposed methods do not require pilot transmission for channel estimation, they inherit better spectral efficiency. Our results validate the theoretical findings and prove the supremacy of the proposed method.

A Krein space-based approach to event-triggered [formula omitted] filtering for linear discrete time-varying systems

This paper is concerned with the problem of event-triggered H∞ filtering for linear discrete time-varying (LDTV) systems. Using the lifting technique, we firstly establish an equivalent relationship with a certain equivalent minimum problem of indefinite quadratic form subject to LDTV systems with non-uniform sampling periods. Then, based on Krein space projection and innovation analysis, sufficient and necessary conditions for the existence of desired filter are derived and a feasible solution is obtained in terms of Riccati recursions. Thus, an algorithm based on the time-update and event-update recursions is given for the implementation of event-triggered H∞ filtering. Different from some existing results, a new event-triggered H∞ filtering scheme is provided so that the estimation error can be completely decoupled from the event-triggered transmission error. Moreover, the new proposed Krein space approach is less conservative and more computational attractive than the existing methods based on recursive linear inequality matrix. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Curl-free positive definite form of time-harmonic Maxwell’s equations well-suitable for iterative numerical solving

A new form of time-harmonic Maxwell’s equations is developed on the base of the standard ones and proposed for numerical modeling. It is written for the magnetic field strength H, electric displacement D, vector potential A and the scalar potential Φ. There are several attractive features of this form. The 1st one is that the differential operator acting on these quantities is positive. The 2nd is absence of curl operators among the leading order differential operators. The Laplacian stands for leading order operator in the equations for H, A and Φ, while the gradient of divergence stands for D. The 3rd feature is absence of space varied coefficients in the leading order differential operators that provides diagonal domination of the resulting matrix of the discretized equations. A simple example is given to demonstrate the applicability of this new form of time-harmonic Maxwell’s equations.