energy landscape of neural networks storing spatially correlated patterns hopfield like neural networks with spatially organized data are studied by a mean field theory the internal structure of the data is described by a matrix c whose elements cij are equal to the correlation between two pixels i and j of any input pattern the model considered here is described by the matrix c in which the pixel pixel correlation is the same for all pairs of pixels and is equal to lambda n the statistical properties of the model depend on three parameters the reduced number of the stored patterns alpha the temperature t and the reduced number of strength correlations of the pixels lambda the phase diagram in the space of parameters lambda and alpha at temperature t 0 is obtained the network can retrieve patterns at t 0 for alpha alpha c where alpha c approximately 0 14 as for the usual hopfield neural network but there is a new transition line above which a new local minimum of the free energy arises this minimum corresponds to a ferromagnetic ordering of the neurons there is another additional minimum between the next two lines that corresponds to mixed ordering we also find the region where the ferromagnetic state becomes the ground state of the system

on mordell weil groups of isotrivial abelian varieties over function fields we show that the mordell weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in cm class introduced here this class of singularities includes all unibranched plane curves singularities as a corollary we describe a family of simple jacobians over the field of rational functions in two variables for which the mordell weil rank is arbitrarily large

interpolations of entanglement breaking channels and equivalent conditions for completely positive maps abstract we first investigate the interpolation problem between the strong entanglement breaking maps and countably decomposable separable states also we study the interpolation relation between a b and a b where is a completely positive trace preserving map a and b are self adjoint trace class operators at last we mainly give two results for completely positive maps to extend the choi theorem in the infinite dimensional hilbert space

properties of neural networks storing spatially correlated patterns the author studies the behaviour of a feedforward neural network supplied with spatially organized data this inner structure is taken into account by a matrix cij whose coefficients equal the average correlation between two pixels i and j of the input patterns the storage capacity alpha is computed as a function of the required stability and of the eigenvalues of c the author proposes a geometrical transformation allowing an intuitive interpretation of these results numerical simulations using real and binary patterns show a very good agreement with the theory finally the author analyses the synaptic couplings correlations resulting from the training of the network with structured patterns focusing on exponentially decreasing correlations one and two dimensions the author finds that they exhibit a mexican hat profile the excitatory centre size of which depends on alpha

noncommutative poisson boundaries of unital quantum operations in this paper poisson boundaries of unital quantum operations also called markov operators are investigated in the case of unital quantum channels compact operators belonging to poisson boundaries are characterized using the characterization of amenable groups by the injectivity of their von neumann algebras we will answer negatively some conjectures appearing in the work of arias et al fixed points of quantum operations j math phys 43 5872 2002 about injectivity of the commuting algebra of the kraus operators of unital quantum operations and their injective envelopes

fixed points of quantum operations quantum operations frequently occur in quantum measurement theory quantum probability quantum computation and quantum information theory if an operator a is invariant under a quantum operation we call a a fixed point physically the fixed points are the operators that are not disturbed by the action of our main purpose is to answer the following question if a is a fixed point is a compatible with the operation elements of we shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von neumann algebras

effective invariants of braid monodromy in this paper we construct new invariants of algebraic curves based on not necessarily generic braid monodromies such invariants are effective in the sense that their computation allows for the study of zariski pairs of plane curves moreover the zariski pairs found in this work correspond to curves having conjugate equations in a number field and hence are not distinguishable by means of computing algebraic coverings we prove that the embeddings of the curves in the plane are not homeomorphic we also apply these results to the classification problem of elliptic surfaces

fixed points of dual quantum operations abstract let a be a normal completely positive map on b h with kraus operators a i i 1 denote m the subset of normal completely positive maps by m a i 1 a i a i i i 1 a i a i i and a is normal in this note the relations between the fixed points of a and a are investigated we obtain that b k h a b b b k h a b b where k h is the set of all compact operators on h and a is the dual of a m in addition we show that the map a a is a bijection on m

the range of generalized quantum operations abstract a completely positive map is called a generalized quantum operation if tr a tr a for all positive operators a in this note we mainly characterize the sets of b is a generalized quantum operation b is a generalized quantum operation and b and are generalized quantum operations where is a dual of and b is a self adjoint operator in particular the third set is relevant to uhlmann s theorem in quantum information

some characterizations of quantum channel in infinite hilbert spaces we first show that for any quantum states on h and on k there exists a quantum channel such that where h and k are finite or infinite dimensional hilbert spaces then we consider some conclusions for the quantum channel such that and ih exists or ih ik

motivic infinite cyclic covers abstract we associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi projective manifold under certain finiteness conditions an element in the grothendieck ring k 0 var c which we call motivic infinite cyclic cover and show its birational invariance our construction provides a unifying approach for the denef loeser motivic milnor fiber of a complex hypersurface singularity germ and the motivic milnor fiber of a rational function respectively

characterizations of fixed points of quantum operations let a be a general quantum operation an operator b is said to be a fixed point of a if a b b in this note we shall show conditions under which b a fixed point a implies that b is compatible with the operation element of a in particular we offer an extension of the generalized luders theorem

characteristic varieties of hypersurface complements abstract we give divisibility results for the global characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface as an application we recast old and obtain new finiteness and divisibility results for the classical infinite cyclic alexander modules of complex hypersurface complements moreover for the special case of hyperplane arrangements we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties

normality and fixed points associated to commutative row contractions abstract let a a k k 1 n n is a positive integer or be a commutative row contraction on a complex hilbert space h and a the normal completely positive map associated with a we give some characterizations for a to be a normal sequence in the case that a is unital we show a is normal if either a is contained in a finite von neumann algebra or the set k h of all compact operators or k 1 n a k a k i moreover the fixed point set b h a of a is considered when a j i is convergent to a projection in strong operator topology

nearby cycles and alexander modules of hypersurface complements abstract let f c n 1 c be a polynomial map which is transversal at infinity using sabbah s specialization complex we give a new description of the alexander modules of the hypersurface complement c n 1 f 1 0 and obtain a general divisibility result for the associated alexander polynomials as a byproduct we prove a conjecture of maxim on the decomposition of the cappell shaneson peripheral complex of the hypersurface moreover as an application we use nearby cycles to recover the mixed hodge structure on the torsion alexander modules as defined by dimca and libgober we also explore the relation between the generic fibre of f and the nearby cycles

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

applications of the generalized luders theorem in this paper we prove that the main results in a more recent paper s gudder duality quantum computers and quantum operations int j theor phys in press also hold on a separable complex hilbert space the proofs in the present paper have been completed by using the block operator technique which are different from that in gudder s paper on the unitary freedom theorem of the finite dimensional setting

invariants of plane algebraic curves via representations of the braid groups recently new remarkable polynomial invariants of knots in 3 sphere were introduced using some representations of the braid groups which were shown to be effective in distinguishing knots j fyhlmo the purpose of this note is to introduce using representation of the braid groups an invariant of continuous equisingular families of plane algebraic curves a pleasant feature is that this invariant is defined for any representation of the braid group no coherence of representations for different number of strings is required though for many representations this invariant is trivial in the case of the reduced burau representation this invariant essentially coincides with the alexander polynomial of plane curves introduced in l 1 this fact is the counterpart of the well known relation between the alexander polynomial of closed braids and the reduced burau representation the definition of this invariant depends on braid monodromy associated with the curve m in particular we obtain a direct not involving calculations of the fundamental group though computationally cumbersome method of computation of the alexander polynomial via braid monodromy in the next section we shall recall the background on continuous equisingular families of plane algebraic curves and their braid monodromies in the sect 3 we introduce the invariant and in the last section we consider the case of reduced burau representations

braid monodromy and topology of plane curves in this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve