Global energy minimisation and cotranslational protein folding of HP models

The globally minimum energy configurations of simple HP lattice models (which use only two amino acid types, positioned on the vertices of a square lattice) of proteins have been established for short sequences. Here we investigate the folding of such proteins to this globally minimum energy configuration, both cotranslationally (as they are manufactured, sequentially, in the ribosome) and starting from a fully extended state. In order to do this we model the folding process and develop a heuristic method for finding local energy minima. Two main results emerge. First, some sequences do fold better cotranslationally than from a fully extended state and second, this can be due to cotranslational folding leading to an initial local energy minimum from which movement to the global minimum is efficient. Sequences for which this is true tend to have a higher density of hydrophobic residues at the start than at the finish. Structural properties of sequences that fold better cotranslationally than from a fully extended state are also identified.     

Low energy properties of the random displacement model

We study low-energy properties of the random displacement model, a random Schrödinger operator describing an electron in a randomly deformed lattice. All periodic displacement configurations which minimize the bottom of the spectrum are characterized. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states of the one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it has a 1/log²-singularity at the bottom of the spectrum. In particular, it is not Hölder-continuous.     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations. Received: February 24, 2005     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations.     

The Number of Lattice Rules of Specified Upper Class and Rank

The upper class of a lattice rule is a convenient entity for classification and other purposes. The rank of a lattice rule is a basic characteristic, also used for classification. By introducing a rank proportionality factor and obtaining certain recurrence relations, we show how many lattice rules of each rank exist in any prime upper class. The Sylow p-decomposition may be used to obtain corresponding results for any upper class.     

Maximal Condorcet Domains

A Condorcet domain is a subset of the set of linear orders on a finite set of candidates (alternatives to vote), such that if voters preferences are linear orders belonging to this subset, then the simple majority rule does not yield cycles. It is well-known that the set of linear orders is the Bruhat lattice. We prove that a maximal Condorcet domain is a distributive sublattice in the Bruhat lattice. An explicit lattice formula for the simple majority rule is given. We introduce the notion of a symmetric Condorcet domain and characterize symmetric Condorcet domains of maximal size.     

Résolution mathématique d'équations décrivant l'évolution d'une surface d'un matériau contraint

We consider a cristal structure, constituted by an elastic substrate and a film with a small thickness. The lattice parameters between the film and the substrate are not the same; consequently, a strain appears in the structure. This strain generates morphologies(see [1,2]).The difficulty consists in finding the profile of the film-vapor surface at any time, which depends on the elastic displacement of the structure. To this end, a physical model, detailed in [2], consists in solving a coupled system of partial derivative equations. The unknowns are the elastic displacement of the structure and the profile of the evolution surface. The elastic displacement solves the linearized elasticity equations posed over the domain occupied by the structure. The boundary of this domain depends on the evolution surface. The second equation is the evolution equation, depending on the elastic displacement by a term of the surface energy. This model is greatly simplified in order to obtain a decoupled two-dimensional model: the map of the film-vapor surface solves a non-linear partial derivatives equation, which is independent of the displacement of the structure.In this Note, we give some results of the existence and uniqueness of a solution for this model under some assumptions about the first derivative of the map.     

Arnol′d Diffusion in a Pendulum Lattice

The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way. © 2014 Wiley Periodicals, Inc.     

The lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of a positive integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an encoding of the obtained lattice which makes it possible to enumerate easily and efficiently all the partitions of a given integer. This approach also gives a recursive formula for the number of partitions of an integer, and some informations on special sets of partitions, such as length bounded partitions.     

The ultratriangular form for prime-power lattice rules

A recent development in the theory of lattice rules has been the introduction of the unique ultratriangular D-Z form for prime-power rules. It is known that any lattice rule may be decomposed into its Sylow p-components. These components are prime-power rules, each of which has a unique ultratriangular form. By reassembling these ultratriangular forms in a defined way, it is possible to obtain a canonical form for any lattice rule. A special case occurs when the ultratriangular forms of each of the Sylow p-components have a consistent set of column indices. In this case, it is possible to obtain a unique canonical D-Z form. Given the column indices and the invariants for an ultratriangular form, we may obtain a formula for the number of ultratriangular forms, and hence the number of prime-power lattice rules, having these column indices and invariants.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

A Constructive Approach to Lattice Rule Canonical Forms

The rank and invariants of a general lattice rule are conventionally defined in terms of the group-theoretic properties of the rule. Here we give a constructive definition of the rank and invariants using integer matrices. This underpins a nonabstract algorithm set in matrix algebra for obtaining the Sylow p-decomposition of a lattice rule. This approach is particularly useful when it is not known whether the form in which the lattice rule is specified is canonical or even repetitive. A new set of necessary and sufficient conditions for recognizing a canonical form is given.     

A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM

The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value prob-lem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L2-norms are independent of the Lame parameterλ.     

Lattice BCK logics with Modus Ponens as unique rule

Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic with modus ponens as unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule.     

Propagating belief functions in qualitative Markov trees

This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempster's rule.The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables.We show that efficient implementation of Dempster's rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempster's rule can be implemented by propagating belief functions through the tree.     

On a cancellation rule for subdirect products of lattice ordered groups and of <Emphasis Type="Italic">GMV</Emphasis>-algebras

The notion of internal subdirect decomposition can be defined in each variety of algebras. In the present note we prove the validity of a cancellation rule concerning such decompositions for lattice ordered groups and for GMV-algebras. For the case of groups, this cancellation rule fails to be valid. This work was partially supported by the Slovak Academy of Sciences via the project Center of Excellence — Physics of Information, Grant I/2/2005.     

Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation

One of the problems that focus the research in the linguistic fuzzy modeling area is the trade-off between interpretability and accuracy. To deal with this problem, different approaches can be found in the literature. Recently, a new linguistic rule representation model was presented to perform a genetic lateral tuning of membership functions. It is based on the linguistic 2-tuples representation that allows the lateral displacement of a label considering an unique parameter. This way to work involves a reduction of the search space that eases the derivation of optimal models and therefore, improves the mentioned trade-off.Based on the 2-tuples rule representation, this work proposes a new method to obtain linguistic fuzzy systems by means of an evolutionary learning of the data base a priori (number of labels and lateral displacements) and a simple rule generation method to quickly learn the associated rule base. Since this rule generation method is run from each data base definition generated by the evolutionary algorithm, its selection is an important aspect. In this work, we also propose two new ad hoc data-driven rule generation methods, analyzing the influence of them and other rule generation methods in the proposed learning approach. The developed algorithms will be tested considering two different real-world problems.     

Determination of the rank of an integration lattice

The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations. In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ^⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B. AMS subject classification (2000) 65D30