Potts models with magnetic field: Arithmetic,geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.     

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach that avoids geometry, this paper studies the structure of K_T(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis (the double Grothendieck polynomials), where is the structure sheaf of the Schubert variety X_w. For rank two cases we compute the corresponding structure constants of the ring K_T(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for H_T^*(G/B)). Let [X^λ]K_T(G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove “Pieri–Chevalley formulas” for the products , , , and , where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), H_T^*(G/B) and H^*(G/B).     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

On the variety of triangles for a hyper-Kähler fourfold constructed by Debarre and Voisin

We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [3]. We exhibit an analog of the notion of “triangle” for these varieties and prove that the 6-dimensional variety of “triangles” is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold.     

Antioxidant potential in the leaves of grape varieties (Vitis vinifera L.) grown in different soil compositions

Grape plant (Vitis vinifera L.) is a species of the family Vitaceae. It is an economically important fruit plant abundantly grown all over the world. Vitaceae comprised of 900 species distributed across tropical, sub-tropical and partially temperate sorts of climatic zones. In the current study we aimed to evaluate the antioxidant activities of grape leaves extract from different Pakistani varieties.We have analyzed the crude methanolic extract from the leaves of Pakistani grape varieties (i.e. NARC Black, Sundar Khani, Shogran-1, Shogran-2, Haita, Sheghali, Perlette, and Sultania-C) cultivated in various soil compositions. Total phenolic contents (TPC) of the extract were determind by using Folin-ciocalteu reagent method. Total flavonoid contents (TFC) were determind by the Aluminum Chloride Calorimetric method. Antioxidant activities of the extracts of different concentration were evaluated using Phosphomolybdenum method, Reducing power and DPPH (2, 2-disphenyl-1-picryl-hydrazyl) assays. Assays performed for grape varieties showed great antioxidant potential. All varieties significantly depicted with variation in polyphenolic contents and antioxidant potential with changes in the soil conditions. The highest TPC and TFC were quantified in Shogran-1 97.58 ± 0.09 (GAE, g⁻¹) and Sultania-C 97.47 ± 0.12 (µg QE/g DE) varieties respectively which were grown in (T6) and (T5), while lower were recorded in Sundar khani 5.16 ± 0.23 in (T6). Likewise, Shogran-1 128.26 ± 0.23 (µg AAE/g extract) variety grown in (T6) depicted highest total antioxidant capacity, NARC black 182.06 ± 0.24 (µg AAE/g extract) variety grown in (T6) resulted in maximum reducing power while DPPH free radical scavenging potential were determinded in Sultania-C 80.91 ± 0.10 (T4) equivalent µg/g. In the terms of bioassays, the promising results were showed by compost + sand + clay (T4), compost + clay + dung (T5) and sand + clay + dung (T6). Therefore, we conclude that variation in soil composition has a significant contribution in the quantity of phytochemicals and antioxidants expression. Our study further suggest that use of organic fertilizer (Compost) and loamy sort of soil enhance functioning of most of the varieties in terms of important phytochemicals.