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We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower. [ABSTRACT FROM AUTHOR]

It is proved that the problem of determining whether a given morphic sequence is almost periodic is decidable. [ABSTRACT FROM AUTHOR]

It is shown that if 햵 is a local monoidal pseudovariety of semigroups, then 햪 ⓜ 햵, 햣 ⓜ 햵 and are local. Other operators of the form 햹 ⓜ(_) are considered. In the process, results about the interplay between operators 햹 ⓜ(_) and (_) * 햣k are obtained. [ABSTRACT FROM AUTHOR]

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Abstract: Cox rings of normal prevarieties are factorially graded, i.e. homogeneous elements allow a unique decomposition into homogeneous factors. We study this property from an algebraic point of view and give a criterion which in a sense reduces it to factoriality. This will allow us to detect and construct Cox rings in a purely algebraic manner. [Copyright &y& Elsevier]

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines. [ABSTRACT FROM AUTHOR]

Abstract: We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph Γ and any field K of characteristic not 2, we consider an algebraic variety X over K whose K-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the nonedges of Γ. We show that, for all connected undirected finite graphs on at most 5 vertices, X is a finite-dimensional affine space. Furthermore, we show that for maximal-dimensional Lie algebras generated by 5 extremal elements, X is a single point. The latter result implies that the bilinear map describing extremality must be identically zero, so that all extremal elements are sandwich elements and the only Lie algebra of this dimension that occurs is nilpotent. These results were obtained by extensive computations with the Magma computational algebra system. The algorithms developed can be applied to arbitrary Γ (i.e., without restriction on the number of vertices), and may be of independent interest. [Copyright &y& Elsevier]

Let l1, … , ll be l lines in P² such that no three lines meet in a point. Let X(l) be the set of points {li ∩lj ∣ 1 ≤ i < j ≤ l} ⊆ P². We call X(l) a star configuration. We describe all pairs (d, l) such that the generic degree d curve in P² contains an X(l). Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems. [ABSTRACT FROM AUTHOR]

Abstract: We generalize Barr’s embedding theorem for regular categories to the context of enriched categories. [Copyright &y& Elsevier]

Abstract: A construction of Kleisli objects in 2-categories of noncartesian internal categories or categories internal to monoidal categories is presented. [Copyright &y& Elsevier]

Abstract: We prove the existence of an -cluster tilting object in a generalized -cluster category which is -Calabi–Yau and Hom-finite, arising from an -Calabi–Yau dg algebra. This is a generalization of the result for the case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension. [Copyright &y& Elsevier]

Abstract: In this note, we give a bound for the Castelnuovo–Mumford regularity of a homogeneous ideal in terms of the degrees of its generators. We assume that defines a local complete intersection with log canonical singularities. [Copyright &y& Elsevier]

We show that good quotients of algebraic varieties with finitely generated Cox ring have again finitely generated Cox ring. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety , the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any have truly topological interpretations. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra whose underlying algebra belongs to . Furthermore, each of the operations of coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in particular, the variety is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general. [Copyright &y& Elsevier]

Abstract: The intersection between a nilpotent orbit and the Lie algebra of a Borel subgroup is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties. [Copyright &y& Elsevier]

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom(G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), ..., N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim(RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, ..., xn〉, and letG = 〈x1, ..., xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, ..., xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, ..., xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, ..., xn〉, and A = PSL(2, ℂ), then Dim(RA(G)) = Max{3n, Dim(RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, ..., xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)). [ABSTRACT FROM AUTHOR]

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]

flat unar is a semilattice of height one with an additional unary operation. We characterise the primal, semi-primal and dualisable finite flat unars. Along the way, we give a useful characterisation of primality in general and use an array of techniques from the theory of natural dualities. [ABSTRACT FROM AUTHOR]

Abstract: Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced. [Copyright &y& Elsevier]

The parameter space $ \mathcal{S}_p$ is a smooth affine algebraic curve whose genus increases rapidly with $ p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ [ABSTRACT FROM AUTHOR]

In this paper, we present a new proof of the H-coloring dichotomy, which was first proved by Hell and Nešetřil in 1990, and then was reproved by Bulatov in 2005. Our proof is much shorter than the original proof and avoids the algebraic machinery of Bulatov's proof. [ABSTRACT FROM AUTHOR]

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]

Abstract  Spectral properties of normal (2k+1)-banded Toeplitz matrices of order n, with k≤⌊ n/2⌋, are described. Formulas for the distance of (2k+1)-banded Toeplitz matrices to the algebraic variety of similarly structured normal matrices are presented. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, and such that and for all . [Copyright &y& Elsevier]

Abstract: We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne–Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne–Mumford stacks. [Copyright &y& Elsevier]

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.

Abstract: In this article we show that the maximum size for the intersection of two distinct ovoids of , q even, is . This is less than the old bound of given by Segre. [Copyright &y& Elsevier]

We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of the Rubin-Stark Conjecture and the Burns Conjecture. [ABSTRACT FROM AUTHOR]

In is proved that, for a regular family of algebraic curves in E n , there exists a sequence of points at which all but the last curvature of the curves simultaneously tend to zero. [ABSTRACT FROM AUTHOR]

In is proved that, for a regular family of algebraic curves in E n , there exists a sequence of points at which all but the last curvature of the curves simultaneously tend to zero. [ABSTRACT FROM AUTHOR]

Abstract: We study the relation between certain increasing and decreasing subsequences occurring in the row form of certain elements in the symmetric group, following Schensted [C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961) 179–191] and Greene [C. Greene, An extension of Schensted''s theorem, Adv. Math. 14 (1974) 254–265], and the Kazhdan–Lusztig cells [D.A. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of the symmetric group to which they belong. We show that, in the two-sided cell corresponding to a partition λ, there is an explicitly defined element , each of whose prefixes can be used to obtain a left cell by multiplying the cell containing the longest element of the parabolic subgroup associated with λ on the right. Furthermore, we show that the elements of these left cells are those which possess increasing and decreasing subsequences of certain types. The results in this paper lead to efficient algorithms for the explicit descriptions of many left cells inside a given two-sided cell, and the authors have implemented these algorithms in GAP. [Copyright &y& Elsevier]

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]

The purpose of the paper is three-fold: (i) to review previous analyses of ambiguity and point out their merits and problems, (ii) to devise a formalized system of 'Integrated Ambiguity Analysis Model (IAAM),' which accounts for ambiguity detection, representation and optimal meaning selection, and (iii) to apply the IAAM to examples of English jokes, slogans, and literary works as well as Chinese/Japanese Zen köan samples in order to show the model's practicality and general applicability. Unlike the previous analyses, the IAAM accounts for intended ambiguities and produces the following three major results. First, the Ambiguity Detection together with the Shared-Knowledge Parameter accounts for the detection procedure of ambiguous expressions. Second, the Integrated Ambiguity Representation provides a unified phonetic/phonological, syntactic, semantic and pragmatic representation of ambiguous expressions. Finally, the Optimal Meaning Detector together with the Shared-Knowledge Parameter establishes a systematic optimal meaning selection procedure and its application to concrete examples in English and Zen koans, thereby not only contributing towards opening the door to the integrated linguistic analysis of intended ambiguity but also helping enhance a study of the linguistics-literature interface. [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.

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]

We show that given a collection of A lines in ℝn, n ≥ 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n-1)). An analogous result for smooth algebraic curves is also proven. [ABSTRACT FROM AUTHOR]

We show that given a collection of A lines in ℝn, n ≥ 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n-1)). An analogous result for smooth algebraic curves is also proven. [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.