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.     

Influence of ultrasound on the functional characteristics of indigenous varieties of mango (Mangifera indica L.)

The present study was conducted to evaluate the effect of ultrasonic (US) treatment on chemical characteristics and antioxidant potential of pulps obtained from eight mango varieties indigenous to Pakistan. There was a significant (p < 0.05) effect of varieties and US treatment on chemical characteristics i.e. pH, acidity, TSS, vitamin C contents, total sugars (%), reducing sugars (%) and non-reducing sugars (%). Microstructure evaluation of pulp from all mango varieties showed deshaped middle lamella and cell wall of cells after 8 min of US treatment. At 4 min of US treatment as per shaped cell wall and middle lamella, the chemical characteristics and antioxidant potential were higher. The total phenolics (TP), flavonoids (TF) and total antioxidant activity (TAA) of pulp from most varieties increased significantly (p < 0.05) after US treatment for 4 min but decreased successively after each treatment i.e. 8 and 12 min. The maximum value (314.17 μg AAE/mL pulp) of DPPH was shown by pulp from Dosehri and the minimum (158.67 μg AAE/mL pulp) was found in pulp from Langra before US treatment. The DPPH values of pulp from most of the varieties increased significantly (p < 0.05) after US treatment for 4 min but decreased successively after each treatment but pulp from Langra showed increasing trend after 8 min of US treatment which decreased after 12 min of treatment. The total anthocyanin (TA) values of pulp from Chaunsa, Dosehri, Sindhri, Gulab Khas and Langra increased abruptly after US treatment for 4 min but decreased successively after subsequent treatment. The pulp from Desi, Anwar Ratol, Gulab Khas and Langra showed an abrupt decrease in TA after 8 min of US treatment. An increasing trend of values of total carotenoids (TC) was shown by pulp from all mango varieties after 4 min of US treatment but decreasing trend was observed with subsequent increase in time of US treatment.     

Bioactive constituents of Iris hybrida (Iridaceae): Processing effect

Iris genus plants are a valuable source of bioactive compounds, which are an important component for pharmaceutical development. The present article shows the potential for mineral nutrition with application of magnesium sulfate, iron chelates and potassium oxide affecting the phenolic compound contents in Iris hybrida ‘Tsikavynka’, I. hybrida ‘Tambo’ and I. hybridа ‘Widecombe Fire’. The effect of mineral processing was specific to plant organs and varied in the component composition. The Iris rhizomes had an increased total phenolic compound content after treatment (up to 10% of the total isoflavonoid content, up to 8% of phenolic acids, up to 5% of γ-pyrones and up to 13% of flavonoids), determined using UV–vis spectroscopy. A positive effect of nutrition on the biosynthesis and content of individual isoflavonoids (tectoridin, nigricin d -glucoside, genistin, iristectorigenin B, nigricin, irigenin and irisolidone) and xanthone mangiferin in Iris rhizomes by HPLC was established. In addition, an increase in the chlorogenic acid amount in Iris leaves was noted. The results demonstrate the sensitivity of Iris phenylpropanoid metabolism to mineral nutrition and can be used to predict medical plant cultivation with increased content of bioactive constituents.     

The Bertini irreducibility theorem for higher codimensional slices

In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.     

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.     

Hyperplane sections of determinantal varieties over finite fields and linear codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated.