Polynomial Modules Research Articles

Reduced submodules of finite dimensional polynomial modules

Reduced submodules of finite dimensional polynomial modules

Attached prime ideals over skew Ore polynomials

In this paper, we investigate the attached prime ideals of inverse polynomial modules over skew Ore polynomials.

Tensor product of generalized polynomial modules for the Virasoro algebra

In this paper, we construct a class of modules T over the Virasoro algebra by taking tensor products of the modules N(Ω(λ,b),V) defined in [24,19] and the irreducible modules defined in [31]. This provides a unified description of many known examples of Virasoro modules, for example, in [28,34,35,12,20,5]. We obtain the necessary and sufficient conditions for T to be irreducible and study their submodule structure when they are reducible. Then we also determine the conditions for two such modules to be isomorphic. In the last part of the paper, we compare the tensor product modules with other known Virasoro modules, concluding that they provide new simple modules in general, while admit some interesting isomorphisms with modules defined in [40] and [33].

A Class of Polynomial Modules over Map Lie Algebras

A Class of Polynomial Modules over Map Lie Algebras

Lithium-Ion Batteries SOH Estimation With Multimodal Multilinear Feature Fusion

As a key status for battery energy storage systems, state of health (SOH) can provide the fundamental information for lifespan management of the battery pack in electric vehicles (EV). Traditionally, features are extracted from various signals to establish a powerful data-driven model for the lithium-ion battery (Li-ion) SOH estimation. One issue left is how to utilize the existing features from diverse modalities of measurement signals for a superior battery aging information capture. The available options are selecting the features by analyzing their correlations with SOH. This paper aims to investigate the intercorrelations between various features through the multimodal multilinear fusion mechanism, which enables to utilize the multimodal multilinear features (MMF) and their interaction characteristics. A high-order polynomial module is designed to fuse the MMF from various sources. To improve the efficiency and performance of the SOH estimator, a 2D convolutional neural network (CNN) network is chosen to use the proposed MMF. The performance of the proposed method is validated on two independent datasets, which obtains the lowest mean absolute error (MAE) of 0.37% and the lowest root mean square error (RMSE) is 0.45%.

Fast Kötter–Nielsen–Høholdt interpolation over skew polynomial rings and its application in coding theory

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most O~sωM(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}\\left( s^\\omega {\\mathfrak {M}}(n)\\right) $$\\end{document} operations in Fqm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_{q^m}$$\\end{document}, where n is the length of the code, s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s$$\\end{document} the interleaving order, M(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {M}}(n)$$\\end{document} the complexity for multiplying two skew polynomials of degree at most n, ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} the matrix multiplication exponent and O~·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}\\left( \\cdot \\right) $$\\end{document} the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.

Minimal Realizations of Nonlinear Time-Delay Systems

The problem of finding a minimal realization of a single-input single-output nonlinear retarded time-delay input-output equation with constant commensurable delays is addressed. First, an algorithm is given to find an observable realization of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -th order input-output equation. Then, this algorithm is modified under two technical assumptions to find a realization of lower dimension, which is observable and accessible. It is shown that under the same technical assumptions an observable and accessible realization has minimal possible state-space dimension among all possible realizations. An algebraic approach based on polynomial tools and modules of differential 1-forms is used to derive the results.

The power-generalized k-Horadam sequences in different modules and a new cryptographic method with these sequences

In this study, unlike other power sequences, we define new ones in the integer and integer-valued polynomial module. We call these sequences the power-generalized [Formula: see text]-Horadam sequence in the integer module and the power-generalized [Formula: see text]-Horadam sequence in the integer-valued polynomial module. Using these sequences, we rebuild the ElGamal cryptosystem and then obtain a new cryptographic method by combining this and symmetric systems. We provide an encryption example where our method is applied. Finally, we see the advantages of the new cryptographic method when we compare the obtained new cryptographic method with some of the asymmetric cryptographic systems such as RSA and ElGamal. And we achieve that new cryptographic method that has more advantageous.

Polynomial extensions of Krull modules

Let M be a torsion-free module over an integral domain D with quotient field K. In this paper, we show that M is a Krull module over D in the sense of Costa-Johnson (Inert extensions of Krull domains, 1976) if and only if the polynomial module is a Krull module over in the sense of Costa-Johnson.

Cauchy transform and uniform approximation by polynomial modules

For a compact subset K of the complex plane C, let C(K) denote the algebra of continuous functions on K. For an open subset U⊂K, let A(K,U)⊂C(K) be the algebra of functions that are analytic in U. We show that there exists ϕ∈A(K,U) so that each f∈A(K,U) can uniformly be approximated by {pn+qnϕ} on K, where pn and qn are analytic polynomials in z. In particular, ϕ can be chosen as a Cauchy transform of a finite positive measure η compactly supported in C∖U. Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

New irreducible non-weight Virasoro modules from tensor products

The tensor product non-weight Virasoro modules are studied in this paper, where and (with and V being an irreducible module over a certain finite dimensional Lie algebra related to the Virasoro algebra) are irreducible Virasoro modules, respectively, defined in Liu and Zhao [Generalized polynomial modules over the Virasoro algebra. Proc Amer Math Soc. 2016;144:5103–5112] and Lü and Zhao [Irreducible Virasoro modules from irreducible Weyl modules. J Algebra. 2014;414:271–287]. The necessary and sufficient condition for to be irreducible is obtained. Then we determine the necessary and sufficient condition for two such irreducible modules to be isomorphic. At last, we show that the irreducible -modules are new for m>1.

The Restriction of Polynomial Modules for the Virasoro Algebra to sl2(C)

The Lie algebra [Formula: see text] may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to [Formula: see text] of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules.

Canonical bilinear form and Euler characters

An explicit formula for the canonical bilinear form on the Grothendieck ring of the Lie supergroup GL(n,m) is given. As an application we get an algorithm for the decomposition Euler characters in terms of characters of irreducible modules in the category of partially polynomial modules.

Rank one polynomial modules over the quantum group of type A 1

In this article, we classify all rank one polynomial modules over the quantum groups and for generic q. We also provide the equivalent conditions for the simplicity of these modules.

Some questions on biquadratic Pólya fields

A number field is called a Pólya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let L be a field which is a compositum of two quadratic Pólya fields. Some questions were raised on Pólyaness of L in [7]. Part was solved in [3] and [8]. Here we develop a general strategy allowing us to treat the remaining cases but also to find all these previous results.

A new offer of NTRU cryptosystem with two new key pairs

AbstractIn this study, it is aimed to contribute to the NTRU cryptologic system to increase the number of operations of taking module by taking a polynomial module, to increase the number of secret key by adding an isomorphism and also to increase the number of public key by choosing an irreducible polynomial. Since it is calculated module by suitably partitioning a chosen message polynomial, and since it is used properties of module being an algebraic structure in a phase of an operation and a decryption by adding to the system prime number module and prime polynomial module, this title is thought fit.

Continuous dependence of linear differential systems on polynomial modules

Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian.

On Syzygy Modules over Laurent Polynomial Rings

In this paper, we present a dynamical method for computing the syzygy module of multivariate Laurent polynomials with coefficients in a Dedekind ring (with zero divisors) by reducing the computation over Laurent polynomial rings to calculations over a polynomial ring via an appropriate isomorphism.

Algorithms for computing bases for 3-variable homogeneous integer-valued polynomials

A polynomial f in Q[x,y,z] is integer-valued if f(x,y,z)∈Z, whenever x,y,z are integers. This work will look at the case where f is homogeneous and will present algorithms for constructing polynomials fpk with f∈Z[x,y,z] such that k is as large as possible for a given degree. From this we will find bases for the modules of homogeneous integer-valued polynomials (IVPs) in a range of degrees. IVPs have been studied for their topological applications, including the homogeneous ones. We explain the connection between 3-variable homogeneous IVPs of degree m and 3-variable IVPs of degree m, as well as with 2-variable IVPs of degree m evaluated at odd values only, then use linear algebra to calculate bases for both cases in a range of degrees.