For a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, theCastelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understoodwhen $V(f)$ is smooth, as well as when $V(f)$ ...has isolated singularities. Westudy the regularity of $M(f)$ when $V(f)$ has a positive dimensional singularlocus. In certain situations, we prove that the regularity is bounded by$(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However,this is not always the case, and we prove that in $\mathbb{P}^n$ the regularityof the Milnor algebra can grow quadratically in $d$.
Let UJ2 be the Jordan algebra of 2×2 upper triangular matrices. This paper is devoted to continue the description given by recent works about the gradings and graded polynomial identities on UJ2(K) ...when K is an infinite field of characteristic 2. Due to the definition of Jordan algebras in terms of the commutative and Jordan identities being unsuitable in characteristic 2, we decided to study the gradings of the non-associative commutative algebra of 2×2 upper triangular matrices UT2=(UT2(K),∘) with the product x∘y=xy+yx. More precisely, fixed K a field of characteristic 2, we classify the gradings of (UT2(K),∘) and also, given an arbitrary grading, we calculate the generators of the ideals of graded identities and give a positive answer to the Specht property for the variety of commutative algebras generated by (UT2(K),∘) in each grading when K is infinite.
On Refined Neutrosophic Canonical Hypergroups Ibrahim, Muritala Abiodun; Agboola, Abdul Akeem Adesina; Hassan-Ibrahim, Zulaihat ...
Neutrosophic sets and systems,
05/2021, Volume:
45
Journal Article
Peer reviewed
Open access
Refinement of neutrosophic algebraic structure or hyperstructure allows for the splitting of the indeterminate factor into different sub-indeterminate and gives a detailed information about the ...neutrosophic structure/hyperstructure considered. This paper is concerned with the development of a refined neutrosophic canonical hypergroup from a canonical hypergroup R and sub-indeterminate I 1 and I 2. Several interesting results and examples are presented. The paper also studies refined neutrosophic homomorphisms and establishes the existence of a good homomorphism between a refined neutrosophic canonical hypergroup R(I 1 , I 2) and a neutrosophic canonical hypergroup R(I).
On Refined Neutrosophic Hyperrings Ibrahim, Muritala Abiodun; Agboola, Abdul Akeem Adesina; Hassan-Ibrahim, Zulaihat ...
Neutrosophic sets and systems,
05/2021, Volume:
45
Journal Article
Peer reviewed
Open access
This paper presents the refinement of a type of neutrosophic hyperring in which + and • are hyperopraetions and studied some of its properties. Several interesting results and examples are presented.
We analyze the space of differentiable functions on a quad-mesh $\cM$, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across ...shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of $G^{1}$ spline functions of bi-degree $\le (k,k)$ on each quadrangular face of $\cM$. We determine the dimension of this space of $G^{1}$ spline functions for $k$ big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of $G^{1}$ splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.
Wiles Defect for Modules and Criteria for Freeness Brochard, Sylvain; Iyengar, Srikanth B; Khare, Chandrashekhar B
International mathematics research notices,
04/2023, Volume:
2023, Issue:
8
Journal Article
Peer reviewed
Open access
Abstract
Diamond proved a numerical criterion for modules over local rings to be free modules over complete intersection rings. We formulate a refinement of these results using the notion of Wiles ...defect. A key step in the proof is a formula that expresses the Wiles defect of a module in terms of the Wiles defect of the underlying ring.
Let $k$ be a finite field of characteristic $p>0$. We construct a theory of weights for overholonomic complexes of arithmetic ${\cal D}$-modules with Frobenius structure on varieties over $k$. The ...notion of weight behave like Deligne's one in the $\ell$-adic framework: first, the six operations preserve weights, and secondly, the intermediate extension of an immersion preserves pure complexes and weights.
Degenerations of nilpotent algebras Fernández Ouaridi, Amir; Kaygorodov, Ivan; Khrypchenko, Mykola ...
Journal of pure and applied algebra,
March 2022, 2022-03-00, Volume:
226, Issue:
3
Journal Article
Peer reviewed
Open access
We give a complete description of the primary degenerations and non-degenerations between the 3-dimensional nilpotent algebras, the 4-dimensional nilpotent commutative algebras and the 5-dimensional ...nilpotent anticommutative algebras over C. It follows that all the varieties under consideration are irreducible and determined by the rigid algebra N2, the family of algebras C19(α) and the rigid algebra A11, respectively. In particular, as an auxiliary new result, we obtain an algebraic classification of the 4-dimensional complex nilpotent commutative algebras.
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