Search Results (9)

The skew generalized power series ring $R\left[\right[S,\omega \left]\right]$ is a ring construction involving a ring R, a strictly ordered monoid $(S,\le )$, and a monoid homomorphism $\omega :S\to End\left(R\right)$. The ring $R\left[\right[S,\omega \left]\right]$ is a common generalization of ring extensions such as (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) Mal’cev–Neumann series rings, and (skew) monoid rings. In this paper, we study the nilpotent elements of skew generalized power series rings and the relationships between the properties of the rings R and $R\left[\right[S,\omega \left]\right]$ expressed in terms of annihilators, such as weak symmetry, weak zip, and the nil-Armendariz and McCoy properties. We obtain results on transferring the weak symmetry and weak zip properties between the rings R and $R\left[\right[S,\omega \left]\right]$, as well as sufficient and necessary conditions for a ring R to be $(S,\omega )$-nil-Armendariz or linearly $(S,\omega )$-McCoy. Full article

The rational field $\mathbb{Q}$ is highly desired in many applications. Algorithms using the rational number field $\mathbb{Q}$ algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in $\mathbb{Q}$ or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of $2\times 2$ matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield $\mathbb{F}$ of $\mathbb{C}$, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. Full article

In 1974, the author proved that the codimension of the ideal $(g_1,g_2,\dots ,g_d)$ generated in the group algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]$ over a field $\mathbb{K}$ of characteristic 0 by generic Laurent polynomials having the same Newton polytope $\mathsf{\Gamma }$ is equal to $d!\times Volume\left(\mathsf{\Gamma }\right)$. Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set ${\mathbb{B}}^{sh}$ of monomials of cardinality $d!\times Volume\left(\mathsf{\Gamma }\right)$, whose equivalence classes form a basis of the quotient algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]/(g_1,g_2,\dots ,g_d)$. The set ${\mathbb{B}}^{sh}$ is constructed inductively from any shelling $sh$ of the polytope $\mathsf{\Gamma }$. Using the ${\mathbb{B}}^{sh}$ structure, we prove that the associated graded $\mathbb{K}$ -algebra $g{r}^{\mathsf{\Gamma }}\left(\mathbb{K}\left[{\mathbb{Z}}^d\right]\right)$ constructed from the Arnold–Newton filtration of $\mathbb{K}$ -algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]$ has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials $(f_1,f_2,\dots ,f_d)$ with the same Newton polytope $\mathsf{\Gamma }$, the set ${\mathbb{B}}^{sh}$ defines a monomial basis of the quotient algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]/(g_1,g_2,\dots ,g_d)$. Full article

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison where we check the theoretical properties. Full article

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series. Full article

In this paper, we propose and analyze a tensor product of nine-tic B-spline subdivision scheme (SS) to reduce the execution time needed to compute the subdivision process of quad meshes. We discuss some essential features of the proposed SS such as continuity, polynomial generation, joint spectral radius, holder regularity and limit stencil. Some results of the SS using surface modeling with the help of computer programming are shown. Full article

For a given pair of s-dimensional real Laurent polynomials $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ , which has a certain type of symmetry and satisfies the dual condition ${\overrightarrow{b}\left(z\right)}^T\overrightarrow{a}\left(z\right)=1$ , an $s\times s$ Laurent polynomial matrix $A\left(z\right)$ (together with its inverse $A^{-1}\left(z\right)$ ) is called a symmetric Laurent polynomial matrix extension of the dual pair $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ if $A\left(z\right)$ has similar symmetry, the inverse $A^{-1}\left(Z\right)$ also is a Laurent polynomial matrix, the first column of $A\left(z\right)$ is $\overrightarrow{a}\left(z\right)$ and the first row of $A^{-1}\left(z\right)$ is ${\left(\overrightarrow{b}\left(z\right)\right)}^T$ . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks. Full article

Let \(\mathbf{H}\) be the quaternion algebra. Let \(\mathfrak{g}\) be a complex Lie algebra and let \(U(\mathfrak{g})\) be the enveloping algebra of \(\mathfrak{g}\). The quaternification \(\mathfrak{g}^{\mathbf{H}}=\)\(\,(\,\mathbf{H}\otimes U(\mathfrak{g}),\,[\quad,\quad]_{\mathfrak{g}^{\mathbf{H}}}\,)\) of \(\mathfrak{g}\) is defined by the bracket \( \big[\,\mathbf{z}\otimes X\,,\,\mathbf{w}\otimes Y\,\big]_{\mathfrak{g}^{\mathbf{H}}}\,=\)\(\,(\mathbf{z}\cdot \mathbf{w})\otimes\,(XY)\,- \)\(\, (\mathbf{w}\cdot\mathbf{z})\otimes (YX)\,,\nonumber \) for \(\mathbf{z},\,\mathbf{w}\in \mathbf{H}\) and {the basis vectors \(X\) and \(Y\) of \(U(\mathfrak{g})\).} Let \(S^3\mathbf{H}\) be the ( non-commutative) algebra of \(\mathbf{H}\)-valued smooth mappings over \(S^3\) and let \(S^3\mathfrak{g}^{\mathbf{H}}=S^3\mathbf{H}\otimes U(\mathfrak{g})\). The Lie algebra structure on \(S^3\mathfrak{g}^{\mathbf{H}}\) is induced naturally from that of \(\mathfrak{g}^{\mathbf{H}}\). We introduce a 2-cocycle on \(S^3\mathfrak{g}^{\mathbf{H}}\) by the aid of a tangential vector field on \(S^3\subset \mathbf{C}^2\) and have the corresponding central extension \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). As a subalgebra of \(S^3\mathbf{H}\) we have the algebra of Laurent polynomial spinors \(\mathbf{C}[\phi^{\pm}]\) spanned by a complete orthogonal system of eigen spinors \(\{\phi^{\pm(m,l,k)}\}_{m,l,k}\) of the tangential Dirac operator on \(S^3\). Then \(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\) is a Lie subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}}\). We have the central extension \(\widehat{\mathfrak{g}}(a)= (\,\mathbf{C}[\phi^{\pm}] \otimes U(\mathfrak{g}) \,) \oplus(\mathbf{C}a)\) as a Lie-subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). Finally we have a Lie algebra \(\widehat{\mathfrak{g}}\) which is obtained by adding to \(\widehat{\mathfrak{g}}(a)\) a derivation \(d\) which acts on \(\widehat{\mathfrak{g}}(a)\) by the Euler vector field \(d_0\). That is the \(\mathbf{C}\)-vector space \(\widehat{\mathfrak{g}}=\left(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\right)\oplus(\mathbf{C}a)\oplus (\mathbf{C}d)\) endowed with the bracket \( \bigl[\,\phi_1\otimes X_1+ \lambda_1 a + \mu_1d\,,\phi_2\otimes X_2 + \lambda_2 a + \mu_2d\,\,\bigr]_{\widehat{\mathfrak{g}}} \, =\)\( (\phi_1\phi_2)\otimes (X_1\,X_2) \, -\,(\phi_2\phi_1)\otimes (X_2X_1)+\mu_1d_0\phi_2\otimes X_2- \) \(\mu_2d_0\phi_1\otimes X_1 + \) \( (X_1\vert X_2)c(\phi_1,\phi_2)a\,. \) When \(\mathfrak{g}\) is a simple Lie algebra with its Cartan subalgebra \(\mathfrak{h}\) we shall investigate the weight space decomposition of \(\widehat{\mathfrak{g}}\) with respect to the subalgebra \(\widehat{\mathfrak{h}}= (\phi^{+(0,0,1)}\otimes \mathfrak{h} )\oplus(\mathbf{C}a) \oplus(\mathbf{C}d)\). Full article

We consider a weighted family of \(n\) generic parallelly translated hyperplanes in \(\mathbb{C}^k\) and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plücker coordinates of the associated point in the Grassmannian Gr\((k,n)\). The Laurent polynomials are in involution. Full article