New C15 Acetogenins from Two Species of Laurencia from the Aegean Sea

The chemical diversity of the approximately 1,200 natural products isolated from red algae of the genus Laurencia, in combination with the wide range of their biological activities, have placed species of Laurencia in the spotlight of marine chemists’ attention for over 60 years. The chemical investigation of the organic (CH₂Cl₂/MeOH) extracts of Laurencia microcladia and Laurencia obtusa, both collected off the coasts of Tinos island in the Aegean Sea, resulted in the isolation of 32 secondary metabolites, including 23 C₁₅ acetogenins (1–23), 7 sesquiterpenes (24–30) and 2 diterpenes (31 and 32). Among them, six new C₁₅ acetogenins, namely 10-acetyl-sagonenyne (2), cis-sagonenyne (3), trans-thuwalenyne C (4), tinosallene A (11), tinosallene B (12) and obtusallene XI (17), were identified and their structures were elucidated by extensive analysis of their spectroscopic data. Compounds 1–3, 5–11, 13 and 15–32 were evaluated for their antibacterial activity against Staphylococcus aureus and Escherichia coli.     

Anthropogenic changes of nitrogen loads in a small river: external nutrient sources vs. internal turnover processes

Anthropogenic nutrient inputs increase the N-load in many aquatic systems, leading to eutrophication and potential changes of biological N-retention capacity. In this study, nitrate inputs in a small river were investigated along a gradient of anthropogenic influence. We aimed to determine changes in nitrate load and isotope signatures in the water column and to identify the anthropogenic influence on biological nitrogen assimilation and nitrification or denitrification in sediments. In seasonal sampling campaigns, we analysed dissolved inorganic nitrogen concentrations, and stable isotopes of nitrate. To differentiate rates of nitrate production and consumption in the pristine vs. agricultural river section, intact sediment cores were incubated with ¹⁵N-labelled nitrate. δ¹⁵N values of nitrate in the pristine river section were low, reflecting natural sources, but, as expected, increased with nitrate concentration in all seasons along the gradient. In general, nitrate retention and consumption were higher in the anthropogenically impacted than in the pristine river section, and nitrate consumption exceeded production. In addition to our measurements, modelled results also show that even in a small river, the anthropogenically enhanced consumption capacity is overwhelmed by surplus N-inputs, and nitrate consumption cannot increase in turn with external loads.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.     

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ₁, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.     

Authentication schemes from actions on graphs, groups, or rings

We propose a couple of general ways of constructing authentication schemes from actions of a semigroup on a set, without exploiting any specific algebraic properties of the set acted upon. Then we give several concrete realizations of this general idea, and in particular, we describe several authentication schemes with long-term private keys where forgery (a.k.a. impersonation) is NP-hard. Computationally hard problems that can be employed in these realizations include the Graph Colorability problem, the Diophantine problem, and many others.     

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

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The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by , where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.     

Monodromy for principal bundles

Given a strongly semistable principal bundle E_G over a curve, in Biswas et al. (2006) [4], a group-scheme for it was constructed, which was named as the monodromy group-scheme. Here we extend the construction of the monodromy group-scheme to principal bundles over higher dimensional varieties.     

Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.