Positive Polynomials Research Articles

The Myth of the Decline Effect in Psi Research: The Empirical Evidence

The decline effect (DE) has been discussed in “psi” research since the early times of experimental investigations, and many causes have been advanced from: individual psychology, social attitudes, electromagnetic fields, experimental artifacts, or physical properties often related to quantum physics. Bierman (2001) found small and statistically significant decline slopes in all experimental protocols, but the mind-matter interaction with random number generators, where he found a positive quadratic polynomial slope. This study aimed to update Bierman’s results, taking into account all studies completed up to 2023 and analyzed in different meta-analyses. Five experimental protocols were analyzed: including anomalous perception in a Ganzfeld condition, remote viewing, forced-choice design in extra-sensory perception, predictive physiological anticipation, and dream extra-sensory perception studies. The results showed that only one slope coefficient out of the five examined was statistically significant, indicating that there was no evidence of a general DE across the different experimental protocols.

A Novel Convexification Method for Control Synthesis Analysis of Continuous-Time Saturated Positive Polynomial Fuzzy Systems Under Imperfect Premise Matching

A Novel Convexification Method for Control Synthesis Analysis of Continuous-Time Saturated Positive Polynomial Fuzzy Systems Under Imperfect Premise Matching

Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.

A note on the Archimedean Positivstellensatz in real algebraic geometry

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic sets by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.

On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g=(g1,…,gr) defining S. They are polynomial in c and ε−1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

On the p-irreducibility of quintic positive polynomials

Abstract In this paper we will discuss the p-reducibility/irreducibility of positive polynomials, and we will give some sufficient conditions for quintic polynomials to be p-reducible/irreducible. This research is closely related to some problems in bio-chemistry, especially to the cooperativity in bio-systems. We will study some applications of our results to these problems.

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

Multi-antenna dual-blind deconvolution for joint radar-communications via SoMAN minimization

In joint radar-communications (JRC) applications such as secure military receivers, often the radar and communications signals are overlaid in the received signal. In these passive listening outposts, the signals and channels of both radar and communications are unknown to the receiver. The ill-posed problem of recovering all signal and channel parameters from the overlaid signal is termed as dual-blind deconvolution (DBD). In this work, we investigate DBD for a multi-antenna receiver. We model the radar and communications channels with a few (sparse) continuous-valued parameters such as time delays, Doppler velocities, and directions-of-arrival (DoAs). To solve this highly ill-posed DBD, we propose to minimize the sum of multivariate atomic norms (SoMAN) that depend on unknown parameters. To this end, we devise an exact semidefinite program using theories of positive hyperoctant trigonometric polynomials (PhTP). Our theoretical analyses show that the minimum number of samples and antennas required for perfect recovery is logarithmically dependent on the maximum of the number of radar targets and communications paths rather than their sum. We show that our approach is easily generalized to include several practical issues such as gain/phase errors and additive noise. Numerical experiments show the exact parameter recovery for different JRC scenarios.

Extremal structure of cones of positive homogeneous polynomials

It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.

Stability Analysis and Event-Triggered Control for IT2 Discrete-Time Positive Polynomial Fuzzy Networked Control Systems With Time Delay

Stability Analysis and Event-Triggered Control for IT2 Discrete-Time Positive Polynomial Fuzzy Networked Control Systems With Time Delay

Sum of squares decomposition of positive polynomials with rational coefficients

We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over R but not over Q. This answers an open question by C. Scheiderer posed as the second question in [14, Section 5.1]. We verify that the example we construct defines a nonsingular projective hypersurface, giving also a positive answer to the third question posed in that section.

Representation of positive polynomials on a generalized strip and its application to polynomial optimization

Representation of positive polynomials on a generalized strip and its application to polynomial optimization

Real factorization of positive semidefinite matrix polynomials

Suppose Q(x) is a real n×n regular symmetric positive semidefinite matrix polynomial. Then it can be factored asQ(x)=G(x)TG(x), where G(x) is a real n×n matrix polynomial with degree half that of Q(x) if and only if det⁡(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equationXSX−XR+RTX+P=0, where P,R,S are real n×n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization.

Event-Triggered Positive Filter Design for Positive Polynomial Fuzzy Systems via Premise Integration and Membership-Function-Dependent Methods

This paper focuses on the design of the positive fuzzy filter with event-triggered mechanism for positive polynomial fuzzy-model-based (PPFMB) system. The main objective is to design a positive and stable fuzzy filter which makes the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index of the estimated output signal as good as possible with limited networked bandwidth resources. Since the positive constraints exist in the positive systems, while the disturbance signal of the output signal and the transmission delay in the event-triggered mechanism are taken into account in this research, the augmented system is rebuilt by introducing the weight coefficient of disturbance signal, and a novel linear copositive Lyapunov function (LCLF) is proposed. In order to adapt to the analysis framework based on linear Lyapunov function, a novel linear event-triggered condition is proposed. Furthermore, to handle the mismatched premise variables caused by the event-triggered mechanism, the premise variable associated with event-triggered instants are integrated into the interval of continuous premise variable with the help of the novel event-triggered condition, which means that the original multi-dimensional type-1 membership functions (MFs) are transformed into single-dimensional interval type-2 MFs. Then, an interval type-2 membership-function-dependent (IT2MFD) method is employed to introduce MFs information into the resultant conditions, so that more optimized <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index is obtained. Finally, an example with simulation results is given to verify the effectiveness of all the fuzzy filter design strategies proposed in this paper for optimizing performance index.

An Optimal Strategy for Submodule Capacitance Sizing of Cascaded H-Bridge-Based Active Power Filter

This paper presents capacitor dimensioning to increase a system’s power density while the converter performance for the delta-connected cascaded H-bridge (CHB) active power filter (APF) is not impaired. After comparing aluminum electrolytic capacitors with film capacitors, this paper proposes capacitance design requirements for film capacitors. With the help of trigonometric transformation for periodic time-varying variables, including capacitor voltages and branch voltages, the capacitance design requirements can be represented as positive univariate polynomials, the coefficients of which are with regard to the capacitance value. The optimization solver SeDuMi is then applied to determine whether a capacitance value is feasible by checking the positiveness of the polynomials. This research has very appealing theoretical and practical properties: the time-domain analysis capability for periodic variables is improved, therefore, enabling a fast computation and a broad application. Both the simulations and experimental results validate the proposed strategy. Compared with the previous work, the proposed method can result in a lower capacitance value without degrading the performance, which is beneficial for a low-cost and low-volume system.

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

Constructing positive definite polynomials

We study positive definite functions on real Hilbert spaces and show that for a given polynomial p p , there exists a nontrivial polynomial q q such that p q pq is positive definite on every real Hilbert space if and only if p p has no positive roots. We use a generalized Fibonacci sequence to find an explicit construction of q q , in which the degree of q q depends on the location of the roots of p p that lie in the first quadrant of the complex plane.

A refinement of Reznick&amp;apos;s Positivstellensatz with applications to quantum information theory

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an N-th power of the squared norm of the variables and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Controllability Properties of Solar Sails

Trajectory design and stationkeeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when reemitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, the force generated by the solar radiation pressure is contained in a pointed convex cone of revolution with the axis in the sun–satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective because it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.