the generalization probability of a perceptron using the delta rule compares the generalization probability of a perceptron using the delta rule with other learning algorithms the numerical results indicate that the delta rule gives results which are superior to hebb s rule and comparable with the maximal stability perceptron rule minover both for randomly generated linear separable example sets and for optimal queries

time dependent fiber bundles with local load sharing fiber bundle models where fibers have random lifetimes depending on their load histories are useful tools in explaining time dependent failure in heterogeneous materials such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load sharing mechanisms but numerical verification has been hampered by severe computational demands in larger bundles to gain insight at large size scales interest has returned to idealized fiber bundle models in 1d such simplified models typically assume either equal load sharing els among survivors or local load sharing lls where a failed fiber redistributes its load onto its two nearest flanking survivors such models can often be solved exactly or asymptotically in increasing bundle size n yet still capture the essence of failure in real materials the present work focuses on 1d bundles under lls as in previous works a fiber has failure rate following a power law in its load level with breakdown exponent ensuremath rho surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed we develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications e g one million fibers in thousands of realizations in particular we develop an algorithm that adapts several concepts and methods that are well known among computer scientists but relatively unknown among physicists to dramatically increase the computational speed with no attendant loss of accuracy we consider various regimes of ensuremath rho that yield drastically different behavior as n increases for 1 2l ensuremath rho l 1 els and lls have remarkably similar behavior they have identical lifetime distributions at ensuremath rho 1 with approximate gaussian bundle lifetime statistics and a finite limiting mean for ensuremath rho g1 this gaussian behavior also applies to els whereas lls behavior diverges sharply showing brittle weakest volume behavior in terms of characteristic elements derived from critical cluster formation for 0l ensuremath rho l1 2 els and lls again behave similarly but the bundle lifetimes are dominated by a few long lived fibers and show characteristics of strongest link extreme value distributions which apply exactly to ensuremath rho 0

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

block transformations of one dimensional deterministic cellular automaton rules given a one dimensional cellular automaton rule f a block transform of f is a rule tbf such that there exists between the limit sets of both rules a bijection that replaces each site value x in a configuration belonging to the limit set of f with a string xb xx x of length b in the corresponding configuration belonging to the limit set of tbf if f is totalistic there exists a unique totalistic block transform and a large number of nontotalistic block transforms tbf if f is not totalistic there are no totalistic block transforms but there still exists a large number of non totalistic block transforms their number increases very rapidly with the block size b and the range of f the range of tbf may be any integer greater than or equal to rb many block transforms are studied the evolution according to rule tbf towards its limit set is discussed in terms of the annihilation of defects these defects are often simply related to the defects characterizing the evolution according to rule f

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

coupling coefficients of so n and integrals involving jacobi and gegenbauer polynomials the expressions for the coupling coefficients 3j symbols for most degenerate symmetric representations of orthogonal groups so n in a canonical basis with so n restricted to so n 1 and different semicanonical or tree bases with so n restricted to so n so n n n n are considered respectively in context of integrals involving triplets of the gegenbauer and the jacobi polynomials since the directly derived triple hypergeometric series do not reveal the apparent triangle conditions of the 3j symbols they are rearranged using their relation with semistretched isofactors of the second kind for the complementary chain sp 4 su 2 su 2 and analogy with the stretched 9j coefficients of su 2 into formulae with more rich limits for summation intervals and obvious triangle conditions the isofactors of class one representations of orthogonal groups or class two representations of unitary groups and of course the related integrals involving triplets of the gegenbauer and the jacobi polynomials turn into double sums in the cases of canonical so n so n 1 or u n u n 1 and semicanonical so n so n 2 so 2 chains as well as into the 4f3 1 series under more specific conditions some ambiguities of the phase choice of the complementary group approach are adjusted as well as problems with an alternating sign parameter of so 2 representations in the so 3 so 2 and so n so n 2 so 2 chains

missing label operators in the reduction o p o p 2 o 2 i consider the missing label problem for basis vectors of an o p representation corresponding to a group reduction chain with links o p o p 2 o 2 a chain with these links is required if the basis vectors are to be of definite weight i obtain two different sets of missing label operators which together with the casimir operators of group and subgroups from a complete set of labeling operators whose eigenvectors provide a canonical basis in the o p representation space the problem is solved for both the even and odd dimensional orthogonal groups

multistate perceptrons learning rule and perceptron of maximal stability a new perceptron learning rule which works with multilayer neural networks made of multistate units is obtained and the corresponding convergence theorem is proved the definition of a perceptron of maximal stability is enlarged in order to include these new multistate perceptrons and a proof of the existence and uniqueness of such optimal solutions is outlined

6j symbols for symmetric representations of so n as the double series the corrected triple sum expression of alisauskas 1987 j phys a math gen 20 35 for the recoupling racah coefficients 6j symbols of the symmetric most degenerate representations of the orthogonal groups so n previously derived from the fourfold sum expression of alisauskas also related to the result of hormes and junker 1999 j phys a math gen 32 4249 is rearranged into three new different double sum expressions related to the hypergeometric kampe de feriet type series and a new triple sum expression with preferable summation condition the regge type symmetry of special 6j symbols of the orthogonal groups so n in terms of special kampe de feriet f1 41 3 series is revealed the recoupling coefficients for antisymmetric representations of symplectic group sp 2n are derived using their relation with the recoupling coefficients of the formal orthogonal group so 2n

period doubling route to chaos for a global variable of a probabilistic automata network as a function of a parameter characterizing the degree of mixing of site values the density of nonzero sites of some one dimensional cellular automata is shown to exhibit a sequence of period doubling bifurcations and to behave chaotically when the degree of mixing is sufficiently large the automata network rules which are considered appear to be useful to model complex systems as in epidemiology in which the motion of the individuals is believed to play an important role

some 6 j symbols for symplectic and orthogonal groups by cerkaski s method the method devised by cerkaski 1987 has been used for the group sp 2n to find some 6 j symbols for which a single multiplicity index is required this approach possesses some advantages over that based on racah s techniques for so 3 the formulae for the 6 j symbols are comparatively simple and they automatically exhibit the symmetries of jucys 1973 elementary means are used to show that the eigenvalues of casimir s operator c2 for sp 2n are identical to those of c2 for o 2d if the authors take d n and replace the irreps of sp 2n by their transposed counterparts for o 2d where d is integral or half integral this enables the formulae for sp 2n to be used to generate expressions for some 6 j symbols for the orthogonal groups o 2d and so 2d illustrations are provided by so 2l 1 and o 4

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

solutions of the nonadditive ybe by spectral parameter factorization and re yang baxterization r matrix solutions for the nonadditive yang baxter equation are obtained in arbitrary dimension using a factorization ansatz in spectral parameters a symmetry transformation determined for the particle conserving r matrix is used to formulate a re yang baxterization scheme with multicomponent spectral parameters extending the notion of conventional yang baxterization the possible scope for such a transformation which has a striking resemblance to reshetikhin s deformation of r matrices but extended now to include spectral parameters is discussed

scaling properties of a transformation defined on cellular automaton rules a one parameter family of transformations defined on the set of all one dimensional cellular automata is studied the class of a given cellular automaton is unchanged under the set of these transformations for class 3 cellular automata transformed statistical quantities satisfy simple scaling properties

upper bound on the products of particle interactions in cellular automata abstract particle like objects are observed to propagate and interact in many spatially extended dynamical systems for one of the simplest classes of such systems one dimensional cellular automata we establish a rigorous upper bound on the number of distinct products that these interactions can generate the upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatio temporal information that a particle stores along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata thereby elucidating why that approach is of general utility the upper bound is tested against several relatively complex domain particle cellular automata and found to be tight

on a class of 6j coefficients with one multiplicity index for groups sp 2n so 2n and so 2n 1 an important class of 6j symbols for the groups sp 2n so 2n and so 2n 1 with one nontrivial multiplicity index is investigated an appropriate choice of a basis in the multiplicity space is made and the so called canonical form for 6j symbols is obtained their expressions depending on the roots of an nth order equation and explicit expressions for some simple class of representations are obtained

the isoscalar factors of o6 o6 using the tensor basis and utilising the wigner eckart theorem the authors obtain the matrix representations 00 and 10 of o6 st o3 s o3 t the isoscalar factors of o6 st o6 st o6 st o3 s o3 t are also calculated

learning and generalization with minimerror a temperature dependent learning algorithm we study the numerical performances of minimerror a recently introduced learning algorithm for the perceptron that has analytically been shown to be optimal both on learning linearly and nonlinearly separable functions we present its implementation on learning linearly separable boolean functions numerical results are in excellent agreement with the theoretical predictions

defect particle kinematics in one dimensional cellular automata let a z be the cantor space of bi infinite sequences in a finite alphabet a and let s be the shift map on a z a cellular automaton is a continuous s commuting self map f of a z and a f invariant subshift is a closed f s invariant subset s a z suppose a a z is s admissible everywhere except for some small region we call a defect it has been empirically observed that such defects persist under iteration of f and often propagate like particles we characterize the motion of these particles and show that it falls into several regimes ranging from simple deterministic motion to generalized random walks to complex motion emulating turing machines or pushdown automata one consequence is that some questions about defect behaviour are formally undecidable

learning and generalization in multineuron interacting feed forward neural networks we consider learning and generalization in the multi interacting feed forward network model recently proposed by h o carmesin 1994 with an a priori definition of the net architecture based on symmetries presented by the function to be learnt we define a generalized hebb rule extend the maximum stability learning algorithm to multi interactions and obtain training and generalization curves for rules where different orders of synapses are not correlated the results obtained for the simple perceptron concerning the hebb rule and through replica calculations in the space of couplings may be straightforwardly adapted to multi interactions through a simple renormalization of the total number of independent couplings analytical and numerical simulation results are compared and show excellent agreement

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

transformations of one dimensional cellular automation rules by translation invariant local surjective mappings abstract if m is a noninvertible translation invariant local surjective mapping it is shown that some local one dimensional deterministic cellular automaton rules f have a transform by m defined by m m f when it exists is local and its wolfram s class is the same as f the evolution of a cellular automaton according to rule is simply related to the evolution according to rule f in the case of class 3 rules the evolution to the attractor may often be viewed as particle like structures evolving in a regular background if the structure of these particles and their interactions for a rule f are known then the structure and interactions of the transformed particles for rule are also known if m is a nontrivial invertible translation invariant local surjective mapping always exists but it is in general site and time dependent

application of poisson maps on coadjoint orbits of sp 6 r group to many body dynamics the canonical transformation approach generated by the semisimple subgroup gcm 3 sp 6 r is applied to reduction of the lie poisson bracket on coadjoint orbits of the sp 6 r group and the poisson coalgebra spo 6 is determined investigating the construction of the n particle phase induced by this reduction we identify the poisson coalgebra spo 6 as the algebra of quadratic o k k n 1 invariant forms on symplectic 6 k 12 dimensional phase space t o k 3 o k k 3 the general classification scheme of poisson orbits for spo 6 is found and applied to the classification of coadjoint orbits of the sp 6 r group occurring in the decomposition of n particle phase spaces we show that the spo k k 4 6 poisson action on some class of surfaces determined by casimir invariants is not transitive the poisson maps for all classes of orbits spo 4 and spo 6 are found the quantum unitary irreducible representations of spo 4 are obtained

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance