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.     

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.     

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.     

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R⁴. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.     

Hilbert function spaces and the Nevanlinna-Pick problem on the polydisc

In [4], Agler and McCarthy (2005) used Hilbert function spaces to study the uniqueness properties of the Nevanlinna-Pick problem on the bidisc. In this work we give a geometric procedure for constructing a Nevanlinna-Pick problem on Dn with a specified set of uniqueness. On the way to establishing this procedure, we prove a result about Hilbert function spaces and partially answer a question posed by Agler and McCarthy.     

On the universal Gröbner bases of toric ideals of graphs

The universal Gröbner basis of an ideal is a Gröbner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.