Semidefinite Research Articles

Real fibered morphisms and Ulrich sheaves

In this paper, we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces inPd\mathbb {P}^dand other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.

SINR and Multiuser Efficiency Gap Between MIMO Linear Receivers

Due to their low complexity, Minimum Mean Squared Error (MMSE) and Zero-Forcing (ZF) emerge as two appealing MIMO receivers. Although they provide asymptotically the same achievable rate as the signal-to-noise ratio (SNR) grows large, a non-vanishing gap between the signal to interference and noise ratio (SINR) obtained through the two receivers exists, affecting the error and outage probability, and the multiuser efficiency. Interestingly, both the SINR and the multiuser efficiency gaps can be compactly expressed as quadratic forms of random matrices, with a kernel that depends solely on the statistics of the interfering streams. By leveraging, we derive the closed-form distribution of such indefinite quadratic forms with random kernel matrix, which turns out to be proportional to the determinant of a matrix containing the system parameters. Then, specializing our result to different fading conditions, we obtain the closed-form statistics of both the SINR gap and the multiuser efficiency gap. Although the focus of this work is on the finite-size statistics, for completeness we also provide some results on the doubly-massive MIMO case. We validate all our derivations through extensive Monte Carlo simulations.

Intersection numbers of modular correspondences for genus zero modular curves

In this paper, we introduce modular polynomials for the congruence subgroup Γ0(M) when X0(M) has genus zero and therefore the polynomials are defined by a Hauptmodul of X0(M). We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of SL2(Z)-equivalence classes of positive definite binary quadratic forms over Z. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to Sp2(Z).

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

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 each integer $$k\ge 4$$.

Distributed Krein space-based attack detection over sensor networks under deception attacks

This paper is concerned with distributed attack detection for a discrete time-varying system monitored by a sensor network. An adversary simultaneously launches distinct data deception attacks on both the system dynamics and sensor intercommunication links so as to intentionally falsify the system state and the exchanged sensor measurement outputs. First, delicate distributed estimators are constructed, aiming to provide resilient local estimates for unavailable system state and appropriate residuals for attack detection. Second, an auxiliary Krein space state–space model as well as innovation analysis and a projection technique is skillfully employed to cast the finite horizon distributed estimator design problem into a minimization problem of a certain indefinite quadratic form. A necessary and sufficient condition on the existence of the minimum is derived. Third, a computational efficient recursive algorithm is developed to design desired distributed estimators such that the local estimates and residuals can be both determined at each time step. Furthermore, a two-stage distributed detection mechanism is proposed for each estimator to alert the attack occurrence. Specifically, it is shown that by properly choosing a weighting matrix parameter, each estimator can respectively detect the deception attacks launched on the system layer and sensor intercommunication links. Finally, the effectiveness of the proposed approach is demonstrated through numerical verification.

Palindromic Sequences of the Markov Spectrum

We study the periods of Markov sequences, which are derived from the continued fraction expression of elements in the Markov spectrum. This spectrum is the set of minimal values of indefinite binary quadratic forms that are specially normalised. We show that the periods of these sequences are palindromic after a number of circular shifts, the number of shifts being given by Stern's diatomic sequence.

Constructing Patterns of (Many) ESSs Under Support Size Control

As is well known, equilibrium analysis of evolutionary partnership games can be done by studying a so-called standard quadratic optimization problem, where a possibly indefinite quadratic form is maximized over the standard (probability) simplex. Despite the mathematical simplicity of this model, the nonconvex instances in this problem class allow for remarkably rich patterns of coexisting (strict) local solutions, which correspond to evolutionarily stable states (ESSs) in the game; seen from a dynamic perspective, ESSs form the asymptotically stable fixed points under the continuous-time replicator dynamics. In this study, we develop perturbation methods to enrich existing ESS patterns by a new technique, continuing the research strategy started by Chris Cannings and coworkers in the last quarter of the past century.

Age-related differences in muscle synergy organization during step ascent across step heights and directions

Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi-definite bilinear forms. As an application, the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections.

Ternary quadratic forms and the class numbers of imaginary quadratic fields

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then(ii) for , if m is a square-free positive integer with , then

A unified and improved Chebotarev density theorem

We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors. We also present a new application of our main result that exhibits considerable gains over earlier versions of the Chebotarev density theorem. If $f$ is a positive definite primitive binary quadratic form then we count lattice points $(u,v) \in \mathbb{Z}^2$ such that $f(u,v)$ is prime and $u, v$ have no prime factors $\leq z$ with uniformity in $z$ and the discriminant of $f$.

Vision‐based garbage dumping action detection for real‐world surveillance platform

In this paper, we propose a new framework for detecting the unauthorized dumping of garbage in real‐world surveillance camera. Although several action/behavior recognition methods have been investigated, these studies are hardly applicable to real‐world scenarios because they are mainly focused on well‐refined datasets. Because the dumping actions in the real‐world take a variety of forms, building a new method to disclose the actions instead of exploiting previous approaches is a better strategy. We detected the dumping action by the change in relation between a person and the object being held by them. To find the person‐held object of indefinite form, we used a background subtraction algorithm and human joint estimation. The person‐held object was then tracked and the relation model between the joints and objects was built. Finally, the dumping action was detected through the voting‐based decision module. In the experiments, we show the effectiveness of the proposed method by testing on real‐world videos containing various dumping actions. In addition, the proposed framework is implemented in a real‐time monitoring system through a fast online algorithm.

Superresolution 2D DOA Estimation for a Rectangular Array via Reweighted Decoupled Atomic Norm Minimization

This paper proposes a superresolution two-dimensional (2D) direction of arrival (DOA) estimation algorithm for a rectangular array based on the optimization of the atomic l0 norm and a series of relaxation formulations. The atomic l0 norm of the array response describes the minimum number of sources, which is derived from the atomic norm minimization (ANM) problem. However, the resolution is restricted and high computational complexity is incurred by using ANM for 2D angle estimation. Although an improved algorithm named decoupled atomic norm minimization (DAM) has a reduced computational burden, the resolution is still relatively low in terms of angle estimation. To overcome these limitations, we propose the direct minimization of the atomic l0 norm, which is demonstrated to be equivalent to a decoupled rank optimization problem in the positive semidefinite (PSD) form. Our goal is to solve this rank minimization problem and recover two decoupled Toeplitz matrices in which the azimuth-elevation angles of interest are encoded. Since rank minimization is an NP-hard problem, a novel sparse surrogate function is further proposed to effectively approximate the two decoupled rank functions. Then, the new optimization problem obtained through the above relaxation can be implemented via the majorization-minimization (MM) method. The proposed algorithm offers greatly improved resolution while maintaining the same computational complexity as the DAM algorithm. Moreover, it is possible to use a single snapshot for angle estimation without prior information on the number of sources, and the algorithm is robust to noise due to its iterative nature. In addition, the proposed surrogate function can achieve local convergence faster than existing functions.

Strict Lyapunov Functions for Model Reference Adaptive Control: Application to Lagrangian Systems

Model reference adaptive control is a well-understood and popular method used in the case when the plant's constant parameters are unknown and must be identified. The related literature is very rich and there exist many proofs of stability and convergence. Lyapunov functions for such systems, having the property that the total derivative is negative definite, are, however, very scarce. In this note, we use the Mazenc construction to design a simple strict Lyapunov function in a rather intuitive manner, based on a first-choice function whose derivative is negative semidefinite. Furthermore, we provide, for the first time in the literature, a Lyapunov function for a popular passivity-based adaptive controller for Lagrangian systems.

On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials

The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and play an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.

Root systems and inflations of non-negative quasi-Cartan matrices

Cartan matrices, quasi-Cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like Lie theory, representation theory and algebraic graph theory. We study quasi-Cartan matrices by means of the inflation algorithm, an idea used in Ovsienko's classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results. We prove that it is enough to perform linear number of inflations to reduce positive or non-negative principal quasi-Cartan matrix to its canonical form, i.e., to the Cartan matrix of a Dynkin or Euclidean diagram, respectively. Moreover, we show that in the positive case the length of every sequence of inflations has quadratic bound, and that the only other “natural” non-negative class having such universal bound is the class of so-called pos-sincerely principal quasi-Cartan matrices (and in this case the bound is linear). We obtain such low bounds by applying, among others, some new observations on the properties of the reduced root systems (in the sense of Bourbaki) associated with positive quasi-Cartan matrices.

On the structure of loop-free non-negative edge-bipartite graphs

We continue the study of a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) [33] and continued in Gąsiorek et al. (2016) [14] and Simson-Zając (2017) [38]. In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥2 vertices of corank r≥1, that is, bigraphs with the symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ. One of the main results of this paper are Theorems 2.1 and 2.17 asserting that every such a bigraph Δ with n+r≥2 vertices can be obtained from a corank zero (i.e. positive) connected loop-free bigraph Δ′ with n≥1 vertices by an r-point extension procedure (Δ′,u(1),…,u(r))↦Δ:=Δ′[[u(1),…,u(r)]] along the r≥1 roots u(1),…,u(r)∈Zn of the positive definite Gram form qΔ′:Zn→Z, v↦vGˇΔ′vtr. It is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop-free non-negative corank r≥1 bigraphs with n+r≥2 vertices, from the corank-zero bigraphs Δ′ with n≥1 vertices.

New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms

This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. The novel approximation strategy is based on the analysis of a suitably defined sequence of auxiliary variables which converges in distribution to the target variable. By leveraging such convergence, simple approximations for both the CDF and PDF of the target variable are given in terms of the derivatives of its moment generating function (MGF). In contrast to classical approximation methods based on truncated series of moments or cumulants, our approximations always represent a valid distribution and the relative error between variables is independent of the variable under analysis. The derived results are then used to approximate the statistics of positive-definite real Gaussian quadratic forms, comparing our proposed approach with other existing approximations in the literature.

Geometrically nonlinear Hessian eigenmode decomposition for local stability analysis of thin-walled structures

A nonlinear geometric formulation based on position and unconstrained vector is originally proposed to evaluate the local stability of thin-walled members. The Hessian is decomposed into linear and geometric equivalent installments. As the Hessian matrix exhibits a definite positive quadratic form, stability condition for large displacements requires that the Hessian matrix positivity be verified at each load small steps. This condition must be established by evaluating the smallest eigenvalue signal at the critical point imminence. The positional finite element method is formulated from a total Lagrangian reference. The mechanical system equilibrium is guaranteed by the total potential energy stationary principle. The constitutive law of Saint–Venant–Kirchhoff is obtained from the linear relationship between the second Piola–Kirchhoff stress tensor and Green–Lagrange deformation tensor. Some examples demonstrate the applicability of the methods.

Stability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach

In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the “current” state of a system, what does not impose restrictions on the approach.

Mean value results and $\Omega$-results for the hyperbolic lattice point problem in conjugacy classes

For \Gamma a cofinite Fuchsian group, we study the lattice point problem in conjugacy classes on the Riemann surface \Gamma\backslash\mathbb H . Let \mathcal{H} be a hyperbolic conjugacy class in \Gamma and \ell the \mathcal{H} -invariant closed geodesic on the surface. The main asymptotic for the counting function of the orbit \mathcal{H} \cdot z inside a circle of radius t centered at z grows like c_{\mathcal{H}} e^{t/2} . This problem is also related with counting distances of the orbit of z from the geodesic \ell . For X \sim e^{t/2} we study mean value and \Omega -results for the error term e(\mathcal{H}, X ;z) of the counting function. We prove that a normalized version of the error e(\mathcal{H}, X ;z) has finite mean value in the parameter t . Further, we prove that if \Gamma is cocompact then \begin{align*} \int_{\ell} e(\mathcal{H}, X;z)\, d s(z) = \Omega \big( X^{1/2} \log \log \log X \big). \end{align*} For \Gamma = {\hbox{PSL}_2( {\mathbb Z})} we prove the same \Omega -result, using a subconvexity bound for the Epstein zeta function associated to an indefinite quadratic form in two variables. We also study pointwise \Omega_{\pm} -results for the error term. Our results extend the work of Phillips and Rudnick for the classical lattice problem to the conjugacy class problem.