Semidefinite Research Articles

On Dwell Time Minimization for Switched Delay Systems: Time-Scheduled Lyapunov Functions

In the present paper, dwell time stability conditions of the switched delay systems are derived using scheduled Lyapunov-Krasovskii functions. The derivative of the Lyapunov functions are guaranteed to be negative semidefinite using free weighting matrices method. After representing the dwell time in terms of linear matrix inequalities, the upper bound of the dwell time is minimized using a bisection algorithm. Some numerical examples are given to illustrate effectiveness of the proposed method, and its performance is compared with the existing approaches. The yielding values of dwell time via the proposed technique show that the novel approach outperforms the previous ones.

On some new Bailey pairs and new expansions for some mock theta functions

In this paper we offer some new identities associated with mock theta functions and establish new Bailey pairs related to indefinite quadratic forms. We believe our proof is instructive use of changing base of Bailey pairs, and offers new information on some Bailey pairs that have proven important in the study of Mock theta functions.

On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables

In this work, we propose a unified approach to evaluating the CDF and PDF of indefinite quadratic forms in Gaussian random variables. Such a quantity appears in many applications in communications, signal processing, information theory, and adaptive filtering. For example, this quantity appears in the mean-square-error (MSE) analysis of the normalized least-mean-square (NLMS) adaptive algorithm, and SINR associated with each beam in beam forming applications. The trick of the proposed approach is to replace inequalities that appear in the CDF calculation with unit step functions and to use complex integral representation of the the unit step function. Complex integration allows us then to evaluate the CDF in closed form for the zero mean case and as a single dimensional integral for the non-zero mean case. Utilizing the saddle point technique allows us to closely approximate such integrals in non zero mean case. We demonstrate how our approach can be extended to other scenarios such as the joint distribution of quadratic forms and ratios of such forms, and to characterize quadratic forms in isotropic distributed random variables. We also evaluate the outage probability in multiuser beamforming using our approach to provide an application of indefinite forms in communications.

Trying to Explicit Proofs of Some Vey’s Theorems in Linear Connections

Let Χ a diferentiable paracompact manifold. Under the hypothesis of a linear connection r with free torsion Τ on Χ, we are going to give more explicit the proofs done by Vey for constructing a Riemannian structure. We proposed three ways to reach our object. First, we give a sufficient and necessary condition on all of holonomy groups of the connection ∇ to obtain Riemannian structure. Next, in the analytic case of Χ, the existence of a quadratic positive definite form g on the tangent bundle ΤΧ such that it was invariant in the infinitesimal sense by the linear operators ∇k R, where R is the curvature of ∇, shows that the connection ∇ comes from a Riemannian structure. At last, for a simply connected manifold Χ, we give some conditions on the linear envelope of the curvature R to have a Riemannian structure

Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D

In this paper, we study the extension of anisotropic metric-based mesh adaptation to the case of very high-order solutions in 3D. This work is based on an extension of the continuous mesh framework and multi-scale mesh adaptation [10] where the optimal metric is derived through a calculus of variation. Based on classical high order a priori error estimates [4], the point-wise leading term of the local error is a homogeneous polynomial of order k + 1. To derive the leading anisotropic direction and orientations, this polynomial is approximated by a quadratic positive definite form, taken to the power ▪. From a geometric point of view, this problem is equivalent to finding a maximal volume ellipsoid included in the level set one of the absolute value of the polynomial. This optimization problem is strongly non-linear both for the functional and the constraints. We first recast the continuous problem in a discrete setting in the metric-logarithm space. With this approximation, this problem becomes linear and is solved with the simplex algorithm [5]. This optimal quadratic form in the Euclidean space is then found by iteratively solving a sequence of such log-simplex problems. From the field of the local quadratic forms that representing the high-order error, a calculus of variation is used to globally control the error in Lp norm. A closed form of the optimal metric is then found. Anisotropic meshes are then generated with this metric based on the unit mesh concept [8]. For the numerical experiments, we consider several analytical functions in 3D. Convergence rate and optimality of the meshes are then discussed for interpolation of orders 1 to 5.

L'Ernesto, ovvero Il prigione di Umberto Saba

L'Ernesto, ovvero Il prigione di Umberto Saba

Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method

We consider differential equations with quadratic right-hand sides that admit two quadratic first integrals, one of which is a positive- definite quadratic form. We indicate conditions of general nature under which a linear change of variables reduces this system to a certain ` canonical' form. Under these conditions, the system turns out to be divergenceless and can be reduced to a Hamiltonian form, but the corresponding linear Lie-Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case, the equations can be reduced to the classical equations of the Euler top, and in four-dimensional space, the system turns out to be superintegrable and coincides with the Euler-Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplying by which the Poisson bracket satisfies the Jacobi identity. In the general case for we prove the absence of a reducing multiplier. As an example we consider a system of Lotka-Volterra type with quadratic right-hand sides that was studied by Kovalevskaya from the viewpoint of conditions of uniqueness of its solutions as functions of complex time. Bibliography: 38 titles.

Extremal entanglement witnesses

We present a study of extremal entanglement witnesses on a bipartite composite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus [Formula: see text] when [Formula: see text] is a witness and [Formula: see text] is a pure product state, [Formula: see text] with [Formula: see text]. The set of witnesses of unit trace is a compact convex set, uniquely defined by its extremal points. The expectation value [Formula: see text] as a function of vectors [Formula: see text] and [Formula: see text] is a positive semidefinite biquadratic form. Every zero of [Formula: see text] imposes strong real-linear constraints on f and [Formula: see text]. The real and symmetric Hessian matrix at the zero must be positive semidefinite. Its eigenvectors with zero eigenvalue, if such exist, we call Hessian zeros. A zero of [Formula: see text] is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic if it has at least one quartic zero. A main result we prove is that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, then the constraints defined by its zeros and Hessian zeros determine all directions in which we may search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions, by numerical methods, leads to an extremal witness which is nearly always quadratic and has the minimum number of zeros. We discuss briefly some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the so-called structural physical approximation (SPA) of an optimal witness. Finally, we discuss how to treat the entanglement witnesses on a complex Hilbert space as a subset of the witnesses on a real Hilbert space.

On Fedorov's parallelohedra - a review and new results

After a short introduction to the topic of Fedorov's parallelohedra, the fundamentals of this topic are reviewed in sections 2 and 3. The cone of positive-definite quadratic forms is described in section 4 where also new results on its shape are given. In the following sections aggregates of parallelohedra having certain properties are investigated and their connected domains in the cone of quadratic forms are determined. Results on combinatorial types of parallelohedra and their refined classification into contraction and similarity types are summarized in section 5. New results on Σ- and Ω-domains are discussed in sections 6 and 7, respectively.

Design methodology for the realization of zero-waste fashion design - Focused on the typology of ZWPM -

Zero-waste is sustainable development for ensuring continuous interactions with the environment as well as for the next generations, while expanding across industries. Zero-waste fashion design does not necessarily mean that we should stop making clothes in order to reduce waste, but we consider the social values of sustainability regarding the environment, humans, and profit. In particular, in the pre-use stage of zero-waste fashion design, fashion designers play critical roles. The purpose of this study is to develop a methodology for the realization of zero-waste fashion design through establishing the typology of zero-waste pattern making (ZWPM) as well as exploring the practical implications of zero-waste fashion design. For the realization of zero-waste fashion design that draws from pattern-making principals, this study categorizes zero-waste fashion design into zero-waste pattern cutting (ZWPC) and non-pattern cutting (NPC). ZWPC is based on drafting patterns on a piece of fabric, which can enable the sharing of patterns and processes, while NPC requires little- or non-cutting/sewing in optimizing a piece of fabric, bringing the possibility of creating indefinite forms. ZWPC is sub-categorized into tailored and non-tailored, and NCP into draped and folded. Then, by implementing the typology in undergraduate design programs, this study tests and completes the design methodology for the realization of zero-waste fashion design.

Polynomial spaces reproduced by elliptic scaling functions

In this paper, we consider the so-called elliptic scaling functions [V. G. Zakharov, Elliptic scaling functions as compactly supported multivariate analogs of the B-splines, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014) 1450018]. Any elliptic scaling function satisfies the refinement relation with a real isotropic dilation matrix; and, in the paper, we prove that any real isotropic matrix is similar to an orthogonal matrix and the similarity transformation matrix determines a positive-definite quadratic form. We prove that the polynomial space reproduced by integer shifts of a compactly supported function can be usually considered as a polynomial solution to a system of constant coefficient PDE’s. We show that the algebraic polynomials reproduced by a compactly supported elliptic scaling function belong to the kernel of a homogeneous elliptic differential operator that the differential operator corresponds to the quadratic form; and thus any elliptic scaling function reproduces only affinely-invariant polynomial spaces. However, in the paper, we present nonstationary elliptic scaling functions such that the scaling functions can reproduce no scale-invariant (only shift-invariant) polynomial spaces.

Testing multivariate economic restrictions using quantiles: The example of Slutsky negative semidefiniteness

This paper is concerned with testing a core economic restriction, negative semidefiniteness of the Slutsky matrix. We consider a system of nonseparable structural equations with infinite dimensional unobservables, and employ quantile regression methods because they allow us to utilize the entire distribution of the data. Difficulties arise because the restriction involves several equations, while the quantile is a univariate concept. We establish that we may use quantiles of linear combinations of the dependent variable, develop a new empirical process based test that applies kernel quantile estimators, and investigate its finite and large sample behavior. Finally, we apply all concepts to Canadian microdata.

The t-analog of string functions for $$A_1^{(1)}$$ A 1 ( 1 ) and Hecke indefinite modular forms

We study the generating functions for Lusztig’s t-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra \(A_1^{(1)}\). These generating series, termed t-string functions, are t-analogs of the string functions of \(A_1^{(1)}\). String functions are well studied for all affine Lie algebras and have important modularity properties. However, they are completely known in closed form only for the Lie algebra \(A_1^{(1)}\); in this case, Kac and Peterson showed that the string functions can be obtained in terms of certain indefinite modular forms of Hecke. In this paper, we generalize this description to the t-string functions of \(A_1^{(1)}\).

Commutative n-ary superalgebras with an invariant skew-symmetric form

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty}$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras.

Forms representing forms: the definite case

Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is everywhere locally representable and the number of variables of $F$ is large enough. In the quadratic case this supersedes a recent result due to Dietmann and Harvey. Another application addresses the number of primitive linear spaces contained in a hypersurface.

Harnack’s inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form div(A(x,u,∇u))=B(x,u,∇u)for x∈Ω as considered in our paper Monticelli et al. (2012). There we proved only local boundedness of weak solutions. Here we derive a version of Harnack’s inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin (1964) and N. Trudinger (1967) for quasilinear equations, as well as ones for subelliptic linear equations obtained in Sawyer and Wheeden (2006, 2010).

A Class of Explicitly Solvable Vehicle Motion Problems

A small but interesting result of Brockett is extended to the Euclidean group SE(3) and is illustrated by several examples. The result concerns the explicit solution of an optimal control problem on Lie groups, where the control belongs to a Lie triple system in the Lie algebra. The extension allows for an objective function based on an indefinite quadratic form. Applying the result requires explicit knowledge of the Lie triple systems of the Lie algebra se(3). Hence, a complete classification of the Lie triple systems of this Lie algebra is derived. Examples are considered for optimal trajectories in three cases. The first case concerns cars moving in the plane. The second looks at motions that rigidly follow the Bishop frame to a space curve. The final example does not have a particular name as it does not seem to have been studied before. The appendix gives a brief introduction to Screw theory. This is essentially the study of the Lie algebra se(3).

On the Interference Tolerance of the Primary System in Cognitive Radio Networks

In this letter, we consider underlay spectrum access in a cognitive radio network, where two coexisting cellular systems, i.e., primary and cognitive, share the same frequency band. The throughput performance of this system is directly linked to the tolerated interference level at the primary receiver. We investigate the fundamental trade-off between the interference tolerance in the primary system and the achievable throughput of the cognitive network. We formulate the network performance as a multi-objective optimization problem with a set of objectives including minimizing the interference imposed on each primary user, and maximizing the intended signal received at every secondary user. We then derive an equivalent standard semi-definite programming form and prove that the derived problem always yields rank-one solutions. The obtained solution set provides the best achievable throughput (characterized by the maximum intended signal power at each cognitive user) for a given level of interference tolerance in the primary system. It also gives the achievable gain on the system throughput if the primary system is able to compromise and increase its interference tolerance, such framework could be a base for negotiations between the operators. An improvement of 0.38 bits/s/channel-use is observed on the achievable throughput for 1 dB higher interference tolerance at the primary receivers.

Dominated splitting for exterior powers and singular hyperbolicity

We show that an invariant splitting for the tangent map to a smooth flow over a compact invariant subset is dominated if, and only if, the exterior power of the tangent map admits an invariant dominated splitting. For a C1 vector field X on a 3-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow Xt and its derivative DXt. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields.

The representation of integers by positive ternary quadratic polynomials

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the p-adic integers for every prime p. It is called complete if it is of the form Q(x+v), where Q is an integral quadratic form in the variables x=(x1,…,xn) and v is a vector in Qn. Its conductor is defined to be the smallest positive integer c such that cv∈Zn. We prove that for a fixed positive integer c, there are only finitely many equivalence classes of positive primitive ternary regular complete quadratic polynomials with conductor c. This generalizes the analogous finiteness results for positive definite regular ternary quadratic forms by Watson [18,19] and for ternary triangular forms by Chan and Oh [8].