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We consider how a standard theorem in algebraic geometry relating properties of a curve with a (Z/2Z)²-action to the properties of its quotients generalizes to results about sets and graphs that admit (Z/2Z)²-actions. [ABSTRACT FROM AUTHOR]

In this note we improve a result of Steffens [7] on the lower bound for Seshadri constants in very general points of a surface with 1-dimensional Néron–Severi space. We also show a multi-point counterpart of such a lower bound. [ABSTRACT FROM PUBLISHER]

In this article, we commence to study the real (simple) left-symmetric algebras. From the known classification of certain complex (semi)simple left-symmetric algebras, we classify their corresponding real forms. We not only obtain the classification of real simple left-symmetric algebras in low dimensions, but also find certain examples of real simple left-symmetric algebras in higher dimensions. In particular, there exists a complex simple left-symmetric algebra without any real form. We also give a geometric construction for a class of real simple left-symmetric algebras. At last, we apply the classification results to study some structures related to geometry. [ABSTRACT FROM PUBLISHER]

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E 1 → E 0 → E 0 → E 1 → … of FP-injective left R-modules with M = ker(E 0 → E 1) such that Hom R (P, −) leaves the sequence exact whenever P is a finitely presented left R-module with pd R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules. [ABSTRACT FROM PUBLISHER]

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ℙ m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal. [ABSTRACT FROM PUBLISHER]

It is well known that convergence of the fictitious domain formulation with boundary Lagrange multipliers is slow due to the lower global regularity of its solution. This article presents a smoothed variant of this approach which is based on a formulation in the form of a state constraint optimal control problem. The convergence rate is increased as seen from a model example. [ABSTRACT FROM AUTHOR]

We show that if |Aut(G)| is odd, then without loss of generality we may assume the only primes dividing |G| are those dividing |Aut(G)|. We also classify finite abelian groups A such that |Aut(A)| = |A| and finite abelian groups A such that |Aut(A)| p = |A| p , where p is a prime dividing |A|. [ABSTRACT FROM PUBLISHER]

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo p for differential equations. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H G /H G is contained in the ℱ-hypercenter of G/H G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ. [ABSTRACT FROM PUBLISHER]

Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G0 is a torus and p does not divide the index (G : G0). [ABSTRACT FROM AUTHOR]

The reduced Burau representation is a one-parameter representation of Bn, the braid group on n strings. Specializing the parameter to a non-zero complex number x gives a representation βn(x) : Bn → GL(n-1) which is either irreducible or has an irreducible composition factor [image omitted]. It was proved in a previous work that the tensor product of an irreducible βn(x) or [image omitted] with an irreducible βn(y) or [image omitted] is irreducible unless x = y±1. In our work, we will consider three (irreducible) specializations of the Burau representations and we will find conditions on the non-zero complex numbers x, y and z that guarantee the irreducibility of the tensor product of the three linear representations βn(x) or [image omitted], βn(y) or [image omitted] and βn(z) or [image omitted]. [ABSTRACT FROM AUTHOR]

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ℓ-groups, we describe all varieties of symmetric top abelian unital ℓ-groups that cover the variety Auℓ of abelian unital ℓ-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, MV, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover MV. [ABSTRACT FROM AUTHOR]

A quiz concerning algebraic geometry is presented.

Given two affine algebraic varieties X and Y such that X ×  ≅ Y × , it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1. [ABSTRACT FROM AUTHOR]

The article provides information on the inequalities of lattice polygons in relation to Scott's inequality and the equation 2i + 7. It states that algebraic geometry is used to study lattice equivalent polygons, since toric geometry is a powerful link that connects discrete and algebraic geometry. Moreover, the proposition b ⩽ 2i + 7 is proven using Pick's Theorem with three equivalent inequalities. Furthermore, Scott's proof indicates that (x,y) = (3,3) is the only point where both upper bounds reach zero and the only case where equality can hold in Onion-Skin Theorem.

We show that there exists an essentially finitely indecomposable abelian p-group which is not a thick group, thus resolving once again in the negative an Irwin's conjecture which was already settled by Dugas and Irwin (1992) and Danchev (2008). Some other related questions are studied as well. [ABSTRACT FROM AUTHOR]

We prove that a finite-dimensional algebra over an algebraically closed field is of derived dimension 0 if and only if it is an iterated tilted algebra of Dynkin type. [ABSTRACT FROM AUTHOR]

We show that instanton bundles of rank r ≤ 2n - 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford-Takemoto. Furthermore, we show that rank r ≤ n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp. [ABSTRACT FROM AUTHOR]

This article presents a study regarding multivariate calculus, focusing on saddle points and inflection points. It discusses the similarities between saddle points on surfaces and inflection points on curves. It explores the direct connection between saddle points on surfaces and inflection points on curves. It presents the theorem which provides a simple constructive method to obtain continuous functions to connect the saddle points of functions of two variables with the inflection points of functions of one variable.

For a prime p, we denote by Bn the cyclic group of order pn. Let φ be a faithful irreducible character of Bn, where p is an odd prime. We study the p-group G containing Bn such that the induced character φG is also irreducible. The purpose of this article is to determine the subgroup NG(NG(Bn)) of G under the hypothesis [NG(Bn):Bn]4 ≦ pn. [ABSTRACT FROM AUTHOR]

We give a presentation for the Chow ring of the moduli space of degree 2 stable maps from 2-pointed rational curves to the projective line. Also, integrals of all degree 4 monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree 2, 2-pointed gravitational correlators of the projective line. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

As an outgrowth of the study of algebraic geometry over groups, discriminating groups were introduced. Many important universal type groups such as Higman's universal group and Thompson's group F were shown to be discriminating. Squarelike groups were then introduced to better capture axiomatic properties of discrimination. In the present article squarelike groups are reinterpreted in terms of discrimination of quasi varieties, and the relationship with an older version of discrimination, termed varietal discrimination here, is studied. [ABSTRACT FROM AUTHOR]

We consider a Birman, Ko, and Lee (BKL) presentation (defined by Vershinin, 2003) for the semigroup of singular braids SB n . We prove the embedding property for the monoid of positive singular braids and give a solution to the word and conjugacy problems in BKL generators. [ABSTRACT FROM AUTHOR]

This note takes up the issue of parallel curves while illustrating the utility of Mathematica in computations. This work complements results presented earlier. The presented treatment, considering the more general case of parametric curves, provides an analysis of the appearance of cusp singularities, and emphasizes the utility of symbolic algebra systems. [ABSTRACT FROM AUTHOR]

Let X = Spec ( R ) be a reduced equidimensional algebraic variety over an algebraically closed field k . Let Y = Spec ( R /??) be a codimension one ordinary multiple subvariety, where ?? is a prime ideal of height 1 of R . If U is a nonempty open subset of Y and ?? a closed point of U , we denote by A ? R ?? its local ring in X , by ?? the extension of ?? in A , and by K the algebraic closure of the residue field k (??). Then there exists a bijection ? ?? : Proj ( G ?? ( A ) ? A /?? k ) ? Proj ( G ( A ?? ) ? k (?? ) K ) such that for every subset S of Proj ( G ?? ( A ) ? A /?? k ), the Hilbert function of S coincides with the Hilbert function of ? ?? (S). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety. [ABSTRACT FROM AUTHOR]

Discusses various mathematical analysis. Concept of the superposition problem presented by Hilbert; Feasibility of representing any function on a tree in Euclidean space; Evaluations of topological complexity in real algebraic geometry.

Letzj = xj + iyj ∈ ℂ (j = 1, 2) and let Ω = {(z1, z2) ∈ ℂ2 : |x1|2 + |x2|2 < 1, |y1|2 + |y2|2 < 1} be a real bi-disc in ℂ2. In this article we will find the sharp lower bound forRsuch that Ω admits a symplectic embedding intoD(R) × ℂ, the complex cylinder with base radiusR [ABSTRACT FROM AUTHOR]

The article traces the contributions of Pierre Hérigone in making mathematics algebraic rather than geometric. It discusses the algebraic concepts and notation of Hérigone which can be found in his volume on algebra. It introduces his universal symbolic language applicable to all branches of mathematics. It presents his algebraic procedures and subsections on identities, equations, geometric constructions and ambiguous equations. His influence on later mathematicians including Isaac Barrow and Gottfried Wilhelm von Leibniz is also discussed.

The article reviews several articles, including "The Hidden Math behind Alice in Wonderland," by Keith Devlin in the March 2010 issue of MAA Online, "Alice's Secrets in Wonderland," by Melanie Bayley in the December 2009 issue of "New Scientist," and "Algebra in Wonderland," by Melanie Bayley in the March 7, 2010 issue of "New York Times."