Properties Of Polynomials Research Articles

Reduced submodules of finite dimensional polynomial modules

Reduced submodules of finite dimensional polynomial modules

New Multivariate Dimension Polynomials of Inversive Difference Field Extensions

New Multivariate Dimension Polynomials of Inversive Difference Field Extensions

New Difference Gröbner Bases and Bivariate Difference Dimension Polynomials

New Difference Gröbner Bases and Bivariate Difference Dimension Polynomials

Generalized fractional Levy-Meixner Appell polynomials in infinite dimensions

Generalized fractional Levy-Meixner Appell polynomials in infinite dimensions

TWO CLASSES OF TWO DIMENSIONAL MIXED GEGENBAUER-LEGENDRE POLYNOMIALS TO APPLY IN COMPUTATION OF THE REGION OF CONVERGENCE OF ARBITRARY FUNCTION CONTAINING THEM

In this article we introduce interesting two dimensional formulae of Legendre and Gegenbauer polynomials and then study their analytic and algebraic properties to derive some known and unknown results. Again we define two classes of two dimensional Gegenbauer-Legendre polynomials to obtain their series formulae. Finally, we use these results in the computation of the region of convergence of arbitrary function consisting of Gegenbauer-Legendre mixed polynomials.

Toroidal seawater conductivity sensors and instrumentation for anti-submarine warfare

This paper describes the development of toroid-based seawater conductivity sensors and instrumentation for 0 to 60 mS/cm conductivity range and 50 bar hydrostatic pressure operational capability which is equivalent to 500 metres of ocean depth. Double core toroidal transformer concept was used and investigated for Mn-Zn and Fe based nanocrystalline cores, turns ratio and temperature dependence. Different excitation and sense coil turn numbers were fabricated by keeping the turn’s ratio 1:2, 1:4, 1:6 and 1:8. Optimum frequency of Mn-Zn based conductivity sensors with lower and higher turn numbers were observed for turn ratio 1:8 and 1:4 respectively at 120 kHz and obtained voltage responses of 952 and 1567 mV. Fe based Nano crystalline core with turn ratio 1:8 shows an optimum frequency of 70 kHz and giving a voltage difference of 830 mV. Two dimensional polynomials of order 2 were presented for temperature dependence of conductivity for Mn-Zn core based sensors. Sensor responses of Fe based nanocrystalline core sensors in ‘lower turn number’ configuration with same turn ratio is presented and observed linear temperature dependence compared to Mn-Zn based sensors. The mechanical design and frequency of operation of developed conductivity sensors gives inherent immunity to ASW band of frequency interference. The efficacy of conductivity sensor with measuring electronics and commercial standard sensors profiling was proved experimentally compared up to 500 metres in sea profiling and obtained a root mean square error of 0.42 for profile data.

Shift modules, strongly stable ideals, and their dualities

We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSIs. The recently introduced duality on SSIs is given an effective conceptual and computational setting. We study SSIs in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSIs is introduced. This is the projective resolution in the category of shift modules.

On the partial differential Lüroth's theorem

We study the Lüroth problem for partial differential fields. The main result is the following partial differential analog of generalized Lüroth's theorem: Let F be a differential field of characteristic 0 with m commuting derivation operators, u=u1,…,un a set of differential indeterminates over F. We prove that an intermediate differential field G between F and F〈u〉 is a simple differential extension of F if and only if the differential dimension polynomial of u over G is of the form ωu/G(t)=n(t+mm)−(t+m−sm) for some s∈N. This result generalizes the classical differential Lüroth's theorem proved by Ritt and Kolchin in the case m=n=1. We then present an algorithm to decide whether a given finitely generated differential extension field of F contained in F〈u〉 is a simple extension, and in the affirmative case, to compute a Lüroth generator. As an application, we solve the proper re-parameterization problem for unirational differential curves.

Some Properties of Coefficients of the Kolchin Dimension Polynomial

This paper presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we obtain that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal differential dimension polynomial $$ {\omega}_{\mathcal{G}/\mathcal{F}} $$ (this concept was introduced by W. Sitt) we will prove a criterion for Macaulay constants to be equal. In this case, as our example shows, there are no bounds from above to the Macaulay constants of the polynomial $$ {\omega}_{\xi /\mathcal{F}} $$ for $$ \mathcal{G}=\mathcal{F}\left\langle \xi \right\rangle . $$

Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Let $ G $ be a Lie group, and let $ \Gamma $ be a lattice in $ G $. We introduce the notion of locally unipotent invariant measures on $ G/\Gamma $. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $ G/\Gamma $ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in [15].The proof relies on Ratner's measure classification theorem, a linearization technique for polynomial trajectories near singular sets, and a twisting technique of Shah.

Neural tensor contractions and the expressive power of deep neural quantum states

We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.

Sharp growth of the Ornstein—Uhlenbeck operator on Gaussian tail spaces

Let X be a standard Gaussian random variable. For any p ∈ (1, ∞), we prove the existence of a universal constant Cp > 0 such that the inequality $${\mathbb{E}\left({\left| {{h^\prime }{{\left(X \right)}^p}} \right|} \right)^{1/p}} \ge {C_p}\sqrt d {\left({\mathbb{E}{{\left| {h\left(X \right)} \right|}^p}} \right)^{1/p}}$$ holds for all d ≥ 1 and all polynomials h: ℝ → ℂ whose spectrum is supported on frequencies at least d, that is, \(\mathbb{E}h\left(X \right){X^k} = 0\) for all k = 0, 1, …, d − 1. As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of Mendel and Naor (2014) concerning the growth of the Ornstein—Uhlenbeck operator on tail spaces of the real line. We also show the same bound for the gradient of analytic polynomials in an arbitrary dimension.

A New Type of Difference Dimension Polynomials

A New Type of Difference Dimension Polynomials

On the recursive equivalence to Smith form of multivariate polynomial matrices

Abstract The Smith form of an $n$-$D$ polynomial matrix plays an important role in many areas of mathematics and engineering. In this paper, we investigate the recursive equivalence problem of $n$-dimensional polynomial matrices, i.e. if diag$(1,B)$ is equivalent to diag$(1,1,C)$, is B equivalent to diag$(1,C)$? We give a negative answer to this question by explicitly constructing a four-dimensional polynomial matrix which is not equivalent to its Smith form.

Positive dimensional parametric polynomial systems, connectivity queries and applications in robotics

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space.We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for “solving” positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots.Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight.

Optimisation based 3-dimensional polynomial regression to represent lithium-ion battery's open circuit voltage as function of state of charge and temperature

An accurate representation of Open Circuit Voltage (OCV) as a function of State of Charge (SoC) and temperature is a vital component of Battery Management System (BMS). The conventional representation of OCV through either look up table or temperature dependent equations requires enormous memory. In this article a novel approach of representing the OCV with a three dimensional polynomial equation through optimization based regression analysis is presented. The OCV of the cell is obtained from the incremental OCV test data performed at three different temperatures 0 °C, 25 °C and 45 °C. These experimental data of OCV is subjected to optimization based regression algorithms. The present study compares various optimization algorithms for three dimensional regressions through statistical parameters such as RMSE, MAE and Variance. Differential Evolution provides the least RMSE of 1.48 × 10−2, 1.34 × 10−2 and 0.37 × 10−2 for Charging, Discharging and Average OCV with a reduced MAE and variance which is in order of 10−9 and 10−5. It also provides optimised output in only 178 iterations which is lower than other algorithms considered in the article. The memory requirement of the proposed methodology of representation is of 30 bytes irrespective of number of data points. The computational time is reduced by ‘3P + 1’ cycles where P is the time taken for computing a power term in an ECU. Therefore, the comparison of the proposed methodology for three dimensional OCV representation with aforementioned methods reveals reduction of memory requirement and computational complexity in BMS.

Classification of noncommutative conics associated to symmetric regular superpotentials

Let [Formula: see text] be a [Formula: see text]-dimensional quantum polynomial algebra, and [Formula: see text] a central regular element. The quotient algebra [Formula: see text] is called a noncommutative conic. For a noncommutative conic [Formula: see text], there is a finite-dimensional algebra [Formula: see text] which determines the singularity of [Formula: see text]. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, [Formula: see text] is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs [Formula: see text], and calculate the algebras [Formula: see text].

All-in-one robust estimator of the Gaussian mean

The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time which is at most polynomial in dimension, sample size and the logarithm of the inverse of the contamination rate. Second, it is equivariant by translations, uniform scaling and orthogonal transformations. Third, it has a high breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown point approximately equal to $0.28$. Fourth, it is minimax rate optimal, up to a logarithmic factor, when data consists of independent observations corrupted by adversarially chosen outliers. Fifth, it is asymptotically efficient when the rate of contamination tends to zero. The estimator is obtained by an iterative reweighting approach. Each sample point is assigned a weight that is iteratively updated by solving a convex optimization problem. We also establish a dimension-free non-asymptotic risk bound for the expected error of the proposed estimator. It is the first result of this kind in the literature and involves only the effective rank of the covariance matrix. Finally, we show that the obtained results can be extended to sub-Gaussian distributions, as well as to the cases of unknown rate of contamination or unknown covariance matrix.

Applications of an improved Dixon elimination method for the inverse kinematics of 6R manipulators

Variable elimination of high dimensional polynomials is an essential problem for robot inverse kinematics, but it is hard to approach both high efficiency and precision simultaneously. This study proposes a unique elimination method for vector polynomial systems established for kinematic modelling of six-revolute serial manipulators, and then applies it to derive analytical inverse kinematic solutions in real time with high absolute precision. To begin, the kinematic modelling equations are parameterized into a non-redundant vector polynomial system, so as to make full use of vector properties in elimination processing. After that, an improved elimination method based on the Dixon resultant is proposed as a solution to the polynomial computation issue. Since there exist some duplicate terms in the Dixon polynomial, a linear constrained Dixon matrix is created by extracting linear terms from variable sequences. Furthermore, the inverse kinematic problem for six-revolute manipulators is addressed based on the elimination method, wherein the configurations of general decoupling manipulators are designed considering both precise machining and fast solutions. The proposed formulations are ported into the core algorithm of professional software, demonstrating that they are highly accurate and real-time required for precise manipulators of automatic use.

Transmission Line Fault Localization With Mesh and Surface Analysis Using PCA Features

— The proposed work investigates the application of principal component analysis (PCA) in localizing fault in a 150 km transmission line. Quarter-cycle pre-fault and half-cycle post-fault sending end noisy line currents are analyzed for faults conducted at intermediate locations with varying fault resistance. Principal component scores, so found, are further analyzed to form a planar mesh structure and smooth fit surface model, from which the distance of the test signal is predicted. Finally, the two models have been validated. The two dimension mesh and the three dimension polynomial surface model produce an average percentage error of 0.2923% and 0.2619%, respectively. The maximum Percentage Error obtained using the two methods are 2.167% and 2.033%, respectively. The high accuracy and low computational burden are achieved in presence of power system noise as well as variation of fault location and fault resistance, which indicates the robustness of the proposed localizer models. Besides, the development of the proposed algorithm using PCA only reduces the memory requirement by reducing dimensionality of a data set, enabling faster computation, especially compared to more extensively used supervised intricate analysis like neural networks, support vector machine or complex mathematical approaches like wavelet transform, fuzzy inference schemes.