Central Polynomial Research Articles

Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment

Let (Zn) be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator Mn=n−1∑k=0n−1(Zk+1/Zk) introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized Mn is given.

D- and A-optimal designs for multi-response mixture experiments with qualitative factors

Mixture experiments are widely used in industry, agriculture, food manufacturing, medicine, and other fields, especially in the pharmaceutical field, which has played an indispensable role in recent years. Most predecessors focused on analyzing the single response problem for discussing mixture experiments, and the research of mixture experiments with qualitative factors is even less. Unlike previous studies, this article considers the optimal design problem of a class of multi-response mixture models with qualitative factors. It gives the general expression of the information matrix, D- and A-criterion functions of the multi-response second-order Scheffé central polynomial model with different mixture components. The variance function is derived under the corresponding optimality criterion, and finally, we conduct a comparative analysis of the relative efficiency of two optimal designs.

A new approach to the Lvov-Kaplansky conjecture through gradings

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture.

The Waring problem for matrix algebras

If a noncommutative polynomial f is neither an identity nor a central polynomial of $${\cal A} = {M_n}\left(\mathbb{C} \right)$$ , then every trace zero matrix in $${\cal A}$$ can be written as a sum of two matrices from $$f\left({\cal A} \right) - f\left({\cal A} \right)$$ . Moreover, “two” cannot be replaced by “one”.

General Odd and Even Central Factorial Polynomial Sequences

The δ2(·) operator, where δ(·) is the known central difference operator, is considered. The associated odd and even polynomial sequences are determined and their generalizations studied. Particularly, matrix and determinant forms, recurrence formulas, generating functions and an algorithm for effective calculation are provided. An interesting property of biorthogonality is also demonstrated. New examples of odd and even central polynomial sequences are given.

On central polynomials and codimension growth

Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of $A.$ In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.

Commutators and images of noncommutative polynomials

Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between [A,A], the linear span of commutators in A, and spanf(A), the linear span of the image of f in A. In particular, we show that [A,A]=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.

A new encryption scheme for multivariate quadratic systems

It is regarded as a difficult task to design a secure MPKC fundamental schemes such as an encryption scheme. In this paper we introduce a new central trapdoor for multivariate quadratic (MQ) public-key cryptosystems that allows for encryption, in contrast to time-tested MQ primitives such as Unbalanced Oil and Vinegar or Rainbow which only allow for signatures. The same as UOV or Rainbow, our construction is single field scheme where the central polynomial system is chosen to have a particular structure that enables efficient inversion. After applying this transformation, the plaintext can be recovered by solving a linear system. Our new central trapdoor can use to replace the broken extension field calculation trapdoor and simple matrix encryption trapdoor, thereafter, we use the minus and plus modifiers to inoculate our scheme against known attacks. It is highlight that our encryption scheme is a good explore in the area of multivariate cryptography. Finally, a straightforward Magma implementation confirms the efficient operation of the public key algorithms.

Extended the linear measurement range of four-quadrant detector by using modified polynomial fitting algorithm in micro-displacement measuring system

In this paper, we propose new method to improve the linear measurement range of four-quadrant detector (4QD) in micro-displacement measuring system. Based combines the central approximation method and polynomial fitting algorithm due to their opposite error characteristics of the spot position measurements, the voltage-displacement curves of four spot models, circular spot with flat energy distribution, circular spot with Gaussian energy, a novel method is designed to analyze the detecting the linear measurement range on a 4QD. The experimental results show that the proposed algorithm improves the detecting sensitivity and effective measure ranges of the 4QD, which the measuring range of 4QD is extend from 20 μm up to 60 μm, and the resolution is 20 nm. Finally, the proposed method was designed to ensure the most efficient use of the detecting sensitivity and effective measure ranges of the 4QD.

Dual-axis control of flexure-based motion system for optical fibre transceiver assembly using fixed-order controller

High-frequency resonant modes appearing in the flexure-based motion systems are the key factors that limit the motion/positioning performance. This paper presents an approach to design the fixed-order (low-order) controller for the flexure-based motion system, which is used for industrial optical fibre transceiver alignment and assembly. The uniqueness of the proposed algorithm is that one single controller can simultaneously stabilize the uncertain high-frequency resonant modes for both x and y-axis motion. Especially, the resulted controller can significantly enhance the tracking and disturbance rejection ability in low frequency. The novelty of the proposed method is mainly twofold. First, the finite-frequency specifications are introduced to robust control of the polytopic systems. As the entire-frequency specifications have the trend to place the dominant poles towards the central polynomial, the finite-frequency specifications largely relax the conservatism for the performance optimization. Second, a new and practical method is proposed to choose the central polynomial by designing a nominal controller, where two different central polynomials are utilized for x and y-axis separately for the practical applications, so that the stability and performance are guaranteed but the conservatism is reduced. Experiments are evaluated on the 2DOF flexure-based motion system, and the results show that the tracking error using designed fixed-order controller has a more than 50% reduction compared with the same-order notch filter based nominal controller, with respect to both the sinusoidal and triangular references.

Fixed-order ℋ∞ controller design for MIMO systems via polynomial approach

This paper presents a fixed-order [Formula: see text]∞ controller design based on linear matrix inequalities for multi-input–multi-output systems. The main difficulty in the development of a fixed-order controller design is that the associated solution set of the problem is defined in a non-convex cluster, and that makes the problem computationally intractable. The convex inner approximation is used to deal with this non-convexity. The proposed controller design approach is applied to some elegant numerical problems taken from various previous works. To show the effectiveness of the proposed method, the full-order [Formula: see text]∞ controller and fixed-order controllers are constructed for these models using the traditional method and popular toolboxes, respectively. Furthermore, in this paper, some strategies for choosing the central polynomial, which is the main conservatism of the proposed method, are discussed.

Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6

We study the centre of a relatively free associative algebra with the identity of Lie nilpotency of degree over a field of characteristic 0. It is proved that the core of the algebra (the sum of all ideals of contained in its centre) is generated as a -ideal by the weak Hall polynomial . It is also proved that every proper central polynomial of is contained in the sum of and the -space generated by and the commutator of degree 4. This implies that the centre of is contained in the -ideal generated by the commutator of degree 4. Similar results are obtained for ; in particular, it is proved that the core is generated as a -ideal by the commutator of degree 5. Bibliography: 15 titles.

On the primality property of central polynomials of prime varieties of associative algebras

In the paper, it is proved that, if f(x1,..., xn)g(y1,..., ym) is a multilinear central polynomial for a verbally prime T-ideal Γ over a field of arbitrary characteristic, then both polynomials f(x1,..., xn) and g(y1,..., ym) are central for Γ.

Quasi-identities on matrices and the Cayley–Hamilton polynomial

We consider certain functional identities on the matrix algebra Mn that are defined similarly as the trace identities, except that the “coefficients” are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley–Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity P and every central polynomial c with zero constant term there exists m∈N such that the affirmative answer holds for cmP. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley–Hamilton identity, and give a complete description of a certain family of such identities.

Robust Optimal Regional Closed-loop Pole Assignment over Positivity Conditions and Differential Evolution

The presented paper describes an optimal closed-loop pole assignment for a vast control feedback framework and structure. The main aim of this paper is a regional selection of the closed-loop central polynomial with additional flexibility and leeway for the following optimization procedure. The optimality of the approach is provided through the positivity condition of the quasi-convex spectral polynomial (QSP). The spectral polynomial presents the uncertainty and closed-loop characteristic of the feedback system. The derivation of the QSP is based on the characteristics of the uncertainty models and metrics with norm H∞. The structure of the controller is freely chosen and is also the subject of the optimization procedure. An objective function for the optimization algorithm Differential Evolution (DE) is presented in multi-criteria form and is directly composed of various QSP's. Such optimization of the objective function improves the transparence, computation expense and the regularity of the optimal/suboptimal solution.

A primeness property for central polynomials of verbally prime P.I. algebras

Let and be two non-commutative polynomials in disjoint sets of variables. An algebra is verbally prime if whenever is an identity for then either or is also an identity. As an analogue of this property, Regev proved that the verbally prime algebra of matrices over an infinite field has the following primeness property for central polynomials: whenever the product is a central polynomial for then both and are central polynomials. In this paper, we prove that over a field of characteristic zero, Regev’ s result holds for the verbally prime algebras and , where is the infinite-dimensional Grassmann algebra.

Central polynomials for matrices over finite fields

Let c(x 1, … , x d ) be a multihomogeneous central polynomial for the n × n matrix algebra M n (K) over an infinite field K of positive characteristic p. We show that there exists a multihomogeneous polynomial c 0(x 1, … , x d ) of the same degree and with coefficients in the prime field 𝔽 p which is central for the algebra M n (F) for any (possibly finite) field F of characteristic p. The proof is elementary and uses standard combinatorial techniques only.

A non-oscillatory and conservative semi-Lagrangian scheme with fourth-degree polynomial interpolation for solving the Vlasov equation

A conservative semi-Lagrangian scheme for the numerical solution of the Vlasov equation is developed based on the fourth-degree polynomial interpolation. Then, a numerical filter is implemented that preserves positivity and non-oscillatory. The numerical results of both one-dimensional linear advection and two-dimensional Vlasov–Poisson simulations show that the numerical diffusion with the fourth-degree polynomial interpolation is suppressed more than with the cubic polynomial interpolation. It is also found that inherent conservation properties of the Vlasov equation can be improved by combining numerical fluxes of the upwind-biased and central fourth-degree polynomial interpolations.

Trace-positive polynomials

In this paper positivity of polynomials in free noncommuting variables in a dimension-dependent setting is considered. That is, the images of a polynomial under finite-dimensional representations of a fixed dimension are investigated. It is shown that unlike in the dimension-free case, every trace-positive polynomial is (after multiplication with a suitable denominator—a a Hermitian square of a central polynomial) a sum of a positive semidefinite polynomial and commutators. Together with our previous results this yields the following Positivstellensatz: every trace-positive polynomial is modulo sums of commutators and polynomial identities a sum of Hermitian squares with weights and denominators. Understanding trace-positive polynomials is one of the approaches to Connes' embedding conjecture.

MIMO Controller Synthesis for LTI and LPV Systems Using Input-Output Models

AbstractThis paper considers the problem of designing MIMO fixed-structure controllers based on a polynomial framework for both Linear Time-Invariant (LTI) and Linear Parameter-Varying (LPV) systems having input-output models. For SISO systems, a polynomial method has been proposed recently to solve such problems, where the a priori selection of a central polynomial is carried out heuristically, and for LPV systems the synthesis problem is solved using a sum-of-squares relaxation approach. To reduce the complexity of the design procedure, we propose a combined gradient and LMI-based algorithm for optimizing the choice of the central polynomial matrix in terms of the closed-loop performance. Moreover, for LPV systems, the computational complexity of the sum-of-squares approach is reduced by introducing polytopic input-output LPV representations. Quadratic stability and performance is guaranteed in the admissible scheduling range. The proposed methods are illustrated with simulation examples.