The combinatorics and extreme value statistics of protein threading

In protein threading, one is given a protein sequence, together with a database of protein core structures that may contain the natural structure of the sequence. The object of protein threading is to correctly identify the structure(s) corresponding to the sequence. Since the core structures are already associated with specific biological functions, threading has the potential to provide biologists with useful insights about the function of a newly discovered protein sequence. Statistical tests for threading results based on the theory of extreme values suggest several combinatorial problems. For example, what is the number of waysm′=# _t {L _i >x _i } _i ⁼⁰ⁿ of choosing a sequence {X _i } _i ⁼¹ⁿ from the set {1, 2, ...,t}, subject to the difference constraints {L _i =X _i+1?X _i >x _i } _i ⁼⁰ⁿ , whereX ₀=0,X _n+1=t+1, and {x _i } _i ⁼⁰ⁿ is an arbitrary sequence of integers? The quantitym′ has many attractive combinatorial interpretations and reduces in special continuous limits to a probabilistic formula discovered by the Finetti. Just as many important probabilities can be derived from de Finetti's formula, many interesting combinatorial quantities can be derived fromm′. Empirical results presented here show that the combinatorial approach to threading statistics appears promising, but that structural periodicities in proteins and energetically unimportant structure elements probably introduce statistical correlations that must be better understood.     

Models of expansions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} with no end extensions

We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of ${\mathbb N}$ such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of ${\mathbb N}$ such that expanding ${\mathbb N}$ by any uncountably many of them suffice. Also we find arithmetically closed ${\mathcal A}$ with no ultrafilter on it with suitable definability demand (related to being Ramsey). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Extended Deligne–Lusztig varieties for general and special linear groups

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group.Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL₂(Fq???/(?²)), with odd q, the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.     

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Passivity-based control for Hopfield neural networks using convex representation

This paper considers the problem of passivity-based controller design for Hopfield neural networks. By making use of a convex representation of nonlinearities, a feedback control scheme based on passivity and Lyapunov theory is presented. A criterion for existence of the controller is given in terms of linear matrix inequality (LMI), which can be easily solved by a convex optimization problem. An example and its numerical simulation are given to show the effectiveness of the proposed method.     

On coderivatives and Lipschitzian properties of the dual pair in optimization

In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

C imaging by double resonance scalar-coupling editing

The two-dimensional FT Imaging of ¹³C-glucose obtained with twin spin-echo double resonance sequence is presented. The images have been obtained by a doubly tuned surface coil on samples containing water and ¹³C-enriched glucose in water. It is shown, both theoretically and experimentally, that the whole editing capability of the twin spin-echo double resonance imaging sequence is also preserved in the presence of the radiofrequency field inhomogeneity produced by the surface coil. As in an efficient selective irradiation method, the enhancement in the signal-to-noise ratio with respect to the direct ¹³C detection, depends on the number of protons J-coupled to ¹³C.