mean field theory for the q state potts glass neural network with biased patterns a systematic study of the q state potts model of neural networks extended to include biased patterns is made for extensive loading alpha mean field equations are written down within the replica symmetric approximation for general q and arbitrary temperature t for the q 3 model and two classes of representative bias parameters the storage capacity and retrieval quality at zero temperature are discussed as functions of the bias taking into account the mattis retrieval state and the lowest symmetric states the t alpha diagram is obtained and the stability properties of the retrieval state are analysed at finite temperatures a comparison is made with the biased hopfield model

categorization ability in a biologically motivated neural network abstract in this work we study the categorization properties of an extreme and asymmetrically diluted version of the hopfield model for associative memory when the effect of the refractory period of the neurons is taken into account in the dynamics of the system the simplest way of modeling these refractory periods is by means of a time dependent threshold that acts only on those neurons that emit a signal and favors them to be at rest during a given time interval the dynamic equations are derived in the limit of extreme dilution using an approach that explicitly preserves the dependence of the system on its whole history the categorization error is analyzed for different values of the parameters in particular we confront our analytical results with numerical simulations for the noiseless case t 0 when the number of examples or their correlations increases the system always categorizes independently of the amplitude of the potential that mimics the effect of the refractory period

a new linear time algorithm to compute the genomic distance via the double cut and join distance the genomic distance problem in the hannenhalli pevzner hp theory is the following given two genomes whose chromosomes are linear calculate the minimum number of translocations fusions fissions and inversions that transform one genome into the other this paper presents a new distance formula based on a simple tree structure that captures all the delicate features of this problem in a unifying way and a linear time algorithm for computing this distance

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

grassmann extensions of yang baxter maps in this paper we show that there are explicit yang baxter yb maps with darboux lax representation between grassman extensions of algebraic varieties motivated by some recent results on noncommutative extensions of darboux transformations we first derive a darboux matrix associated with the grassmann extended derivative nonlinear schrodinger dnls equation and then we deduce novel endomorphisms of grassmann varieties which possess the yb property in particular we present ten dimensional maps which can be restricted to eight dimensional yb maps on invariant leaves related to the grassmann extended nls and dnls equations we consider their vector generalisations

bayesian phylogenetic inference from animal mitochondrial genome arrangements summary the determination of evolutionary relationships is a fundamental problem in evolutionary biology genome arrangement data are potentially more informative than deoxyribonucleic acid sequence data for inferring evolutionary relationships between distantly related taxa we describe a bayesian framework for phylogenetic inference from mitochondrial genome arrangement data using markov chain monte carlo methods we apply the method to assess evolutionary relationships between eight animal phyla

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

3d compatible ternary systems and yang baxter maps according to shibukawa ternary systems defined on quasi groups and satisfying certain conditions provide a way of constructing dynamical yang baxter yb maps after noting that these conditions can be interpreted as three dimensional compatibility of equations on quad graphs we investigate when the associated dynamical yb maps are in fact parametric yb maps in some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints conversely parametric yb maps on quasi groups with an invariance condition give rise to three dimensional compatible systems the application of this method on spaces with certain quasi group structures provides new examples of multi parametric yb maps and three dimensional compatible systems

convergence time and finite size effects in neural networks the time needed for convergence to a stable state starting from an initial state having macroscopic overlap with a stored state is studied numerically for networks up to n 105 neurons although such systems are much larger than those traditionally used in studying neural networks they are still too small to extract reliable information about the asymptotic behaviour of the convergence time

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

simulating highly diluted neural networks the asymmetric diluted hopfield model is studied through numerical simulations and reliable results are obtained even when the network sizes are not exponentially large the mean final overlap the mean convergence time and a quantity that gives information on the relative importance of spurious states are measured as a function of the load capacity alpha

integrable two component systems of difference equations we present two lists of two component systems of integrable difference equations defined on the edges of the z2 graph the integrability of these systems is manifested by their lax formulation whic

neural networks with high order connections we present results for two different kinds of high order connections between neurons acting as corrections to the hopfield model equilibrium properties are analyzed using the replica mean field theory and compared with numerical simulations an optimal learning algorithm for fourth order connections is given that improves the storage capacity without increasing the weight of the higher order term while the behavior of one of the models qualitatively resembles the original hopfield one the other presents a new and very rich behavior depending on the strength of the fourth order connections and the temperature the system presents two distinct retrieval regions separated by a gap as well as several phase transitions also the spin glass states seems to disppear above a certain value of the load parameter ensuremath alpha mathrm ensuremath alpha mathit g

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

sorting by reversals and the theory of 4 regular graphs abstract we show that the theory of sorting by reversals fits into the well established theory of circuit partitions of 4 regular multigraphs which also involves the combinatorial structures of circle graphs and delta matroids in this way we expose strong connections between the two theories that have not been fully appreciated before we also discuss a generalization of sorting by reversals involving the double cut and join dcj operation finally we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates

a 2 approximation algorithm for genome rearrangements by reversals and transpositions recently a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of dna sequences was proposed sankoff et al 1992 the approach is based on the global rearrangements e g inversions and transpositions of fragments analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments we study sorting of signed permutations by reversals and transpositions a problem which adequately models genome rearrangements as the genes in dna are oriented we establish a lower bound and give a 2 approximation algorithm for the problem

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

on the multi neuron interaction model without truncating the interaction a replica symmetric mean field theory approach is presented to the multi neuron interaction model introduced by de almeida and iglesias 1990 phys lett 146a 239 fixed point equations are derived for the relevant order parameters of the model extended to include biased patterns without truncating the interaction the capacity bias and the temperature capacity phase diagrams are discussed compared with the truncated version of the model it is found that the capacity at zero temperature is infinite and that the retrieval states satisfy the de almeida thouless stability condition

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

yang baxter maps and symmetries of integrable equations on quad graphs a connection between the yang baxter relation for maps and the multidimensional consistency property of integrable equations on quad graphs is investigated the approach is based on the symmetry analysis of the corresponding equations it is shown that the yang baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations we use the classification results by adler bobenko and suris to demonstrate this method some new examples of yang baxter maps are derived in this way from multifield integrable equations

reversal and transposition medians in determining phylogenetic trees using gene order information medians provide a powerful alternative to pairwise distances on the other hand both breakpoint and reversal medians are np hard to compute and the use of medians has been limited to relatively closely related genomes in this paper we show that in spite of the greater non uniqueness of reversal medians compared to breakpoint medians medians of moderately distant genomes are often widely spread this means that regardless of which approximation algorithms one may devise for computing reversal medians the genomes need to be closely related for phylogenetic tree computations to be successful to show this we use results on transposition medians which behave similarly and also support our claims with simulations and a real data example with widely spread medians

higher order neural networks polya polynomials and fermi cluster diagrams the problem of controlling higher order interactions in neural networks is addressed with techniques commonly applied in the cluster analysis of quantum many particle systems for multineuron synaptic weights chosen according to a straightforward extension of the standard hebbian learning rule we show that higher order contributions to the stimulus felt by a given neuron can be readily evaluated via polya s combinatoric group theoretical approach or equivalently by exploiting a precise formal analogy with fermion diagrammatics

yang baxter maps and multi field integrable lattice equations a variety of yang baxter maps are obtained from integrable multi field equations on quad graphs a systematic framework for investigating this connection relies on the symmetry groups of the equations the method is applied to lattice equations introduced by adler and yamilov which are related to the nonlinear superposition formulae for the backlund transformations of the nonlinear schrodinger system and specific ferromagnetic models

chaotic dynamics of high order neural networks the dynamics of an extremely diluted neural network with high order synapses acting as corrections to the hopfield model is investigated the learning rules for the high order connections contain mixing of memories different from all the previous generalizations of the hopfield model the dynamics may display fixed points or periodic and chaotic orbits depending on the weight of the high order connections e the noise levelt and the network load defined as the ratio between the number of stored patterns and the mean connectivity per neuron p c as in the related fully connected case there is an optimal value of the weight e that improves the storage capacity of the system the capacity diverges

exact solutions of quantum mappings from the lattice kdv as multi dimensional operator difference equations the lattice kdv equation is defined on an associative algebra and a new initial value problem on the lattice kdv is considered this leads to two new families of iterative mappings discrete equations of motion which can be solved for all discrete times imposing a certain commutation relation on the algebra makes it possible to prove using a yang baxter structure that these mappings are quantum integrable maps in the classical limit the maps become a new class of integrable mappings in the sense of liouville arnold veselov lav

entwining yang baxter maps related to nls type equations we construct birational maps that satisfy the parametric set theoretical yang baxter equation and its entwining generalisation for this purpose we employ darboux transformations related to integrable nonlinear schrodinger type equations and study the refactorisation problems of the product of their associated darboux matrices additionally we study various algebraic properties of the derived maps such as invariants and associated symplectic or poisson structures and we prove their complete integrability in the liouville sense

the transposition median problem is np complete during recent years the genomes of more and more species have been sequenced providing data for phylogenetic reconstruction based on genome rearrangement measures where the most important distance measures are the reversal distance and the transposition distance the two main tasks in all phylogenetic reconstruction algorithms are to calculate pairwise distances and to solve the median of three problem while the reversal distance problem can be solved in linear time the reversal median problem has been proven to be np complete the status of the transposition distance problem is still open but it is conjectured to be more difficult than the reversal problem this in turn also suggests that the transposition median problem is np complete however this conjecture could not yet be proven we have now succeeded in giving a non trivial proof for the np completeness of the transposition median problem

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

integrable maps which preserve functions with symmetries we consider maps which preserve functions which are built out of the invariants of some simple vector fields we give a reduction procedure which can be used to derive commuting maps of the plane which preserve the same symplectic form and first integral we show how our method can be applied to some maps which have recently appeared in the context of yang baxter maps

integrable lattice equations with vertex and bond variables we present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables in some of the models the vertex dynamics is independent of the evolution of the bond variables and one can write the equations as non autonomous yang baxter maps we also present a model in which the vertex and bond variables are fully coupled integrability is tested with algebraic entropy as well as multidimensional consistency

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

an efficient algorithm for one sided block ordering problem under block interchange distance abstract in this work we study the one sided block ordering problem under block interchange distance given two signed permutations and of size n where represents a partially assembled genome consisting of several blocks i e contigs and represents a completely assembled genome the one sided block ordering problem under block interchange distance is to order i e assemble the blocks of such that the block interchange distance between the assembly of and is minimized the one sided block ordering problem is useful in genome resequencing because its algorithms can be used to assemble the contigs of partially assembled resequencing genomes based on their completely assembled genomes by using permutation groups in algebra we design an efficient algorithm to solve the one sided block ordering problem under block interchange distance in o n log n time moreover we show that the assembly of can be done in o n time and its block interchange distance from can also be calculated in advance in o n time

whole genome duplications and contracted breakpoint graphs the genome halving problem motivated by the whole genome duplication events in molecular evolution was solved by el mabrouk and sankoff in the pioneering paper siam j comput 32 2003 pp 754 792 the el mabrouk sankoff algorithm is rather complex inspiring a quest for a simpler solution an alternative approach to the genome halving problem based on the notion of the contracted breakpoint graph was recently proposed in m a alekseyev and p a pevzner ieee acm trans comput biol bioinformatics 4 2007 pp 98 107 this new technique reveals that while the el mabrouk sankoff result is correct in most cases it does not hold in the case of unichromosomal genomes this raises a problem of correcting a flaw in the el mabrouk sankoff analysis and devising an algorithm that deals adequately with all genomes in this paper we efficiently classify all genomes into two classes and show that while the el mabrouk sankoff theorem holds for the first class it is incorrect for the second class the crux of our analysis is a new combinatorial invariant defined on duplicated permutations using this invariant we were able to come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome halving problem

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

partially connected models of neural networks a partially connected hopfield neural network model is studied under the restriction that w the ratio of connections per site to the size of the system remains finite as the size n to infinity with the connection structure at each site being the same the replica symmetric mean field theory equations for the order parameters are derived the zero temperature forms of these equations are then solved numerically for a few different local connectivity architectures showing phase transitions at different critical storage ratios alpha c where the states which the authors are trying to store in the network become discontinuously unstable they show that the information capacity per connection improves for partially connected systems

1 ɛ approximation of sorting by reversals and transpositions gu et al gave a 2 approximation for computing the minimal number of inversions and transpositions needed to sort a permutation there is evidence that from the point of view of computational molecular biology a more adequate objective function is obtained if transpositions are given double weight we present a 1 e approximation for this problem based on the exact algorithm of hannenhalli and pevzner for sorting by reversals only