On Positive and Weakly Positive Implicative BCI-Algebras

Prove that the notion of positive implicative BCI-algebras coincides with that of weakly positive implicative BCI-algebras, thus the whole results in the latter are still true in the former, in particular, one of these results answers definitely the first half of J. Meng and X.L. Xin’s open problem: Does the class of positive implicative BCI-algebras form a variety? The second half of the same problem is: What properties will the ideals of such an algebra have? Here, some further properties are obtained.     

Classification of two-orbit varieties

We obtain the classification of two-orbit varieties, i.e. the normal complete complex algebraic varieties on which a reductive complex algebraic group acts with two orbits. We prove also Luna's conjecture saying that these varieties are spherical, i.e. admit a dense orbit of a Borel subgroup. Received: September 1, 2000     

Matrix Lax Polynomials, Geometry of Prym Varieties and Systems of Hess–Appel’rot Type

We consider a four-dimensional generalization of Hess–Appel’rot system and costruct its Lax pair. Both classical and algebro-geometric integration procedure are proceeded. The algebro-geometric integration is based on deep facts from geometry of Prym varieties such as the Mumford relation and Mumford-Dalalyan theory. The integration is similar to the integration of Lagrange bitop which has recetly been performed by the authors.     

Laurent polynomials and Eulerian numbers

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels poses two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.     

Ratio of Protein Fractions in Cereal Grains Cultivated in Uzbekistan

The glutelin fraction was used as the food value index of local corn. The base-soluble fraction was used as a test of the glassiness of local strains of rice grain. The content of gliadins and glutelins did not change over 24 h after wheat grain cultivated in Uzbekistan was moistened.     

On polynomial digraphs

Let Φ(x,y) be a bivariate polynomial with complex coefficients. The zeroes of Φ(x,y) are given a combinatorial structure by considering them as arcs of a directed graph G(Φ). This paper studies some relationship between the polynomial Φ(x,y) and the structure of G(Φ).     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.     

Products of Jacobians as Prym-Tyurin varieties

Let X ₁, ..., X _m denote smooth projective curves of genus g _i ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to . We show that the product JX ₁ × ... × JX _m of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n ^m-1. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.      

On the crystalline cohomology of Deligne–Lusztig varieties

Let $X\to Y^0$ be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic $p>0$. Let Y be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural $W[F]$-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, $Y^0$ and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if $\mathbb{G}$ is a connected reductive algebraic group defined over k and $\mathbb{L}$ a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology (ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.