critical storage capacity of the j 1 neural network for neural networks in which the coupling jij are allowed to take on the values jij 1 or jij 1 the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability they use an exact enumeration scheme based on the gray code and a continuous distribution for the patterns to control finite size effects results are presented for n 25 they indicate an optimal storage capacity of c 0 82 n

on bipartite pure state entanglement structure in terms of disentanglement schrodinger s disentanglement e schrodinger proc cambridge philos soc 31 555 1935 i e remote state decomposition as a physical way to study entanglement is carried one step further with respect to previous work in investigating the qualitative side of entanglement in any bipartite state vector remote measurement or equivalently remote orthogonal state decomposition from previous work is generalized to remote linearly independent complete state decomposition both in the nonselective and the selective versions the results are displayed in terms of commutative square diagrams which show the power and beauty of the physical meaning of the antiunitary correlation operator inherent in the given bipartite state vector this operator together with the subsystem states reduced density operators constitutes the so called correlated subsystem picture it is the central part of the antilinear representation of a bipartite state vector and it is a kind of core of its entanglement structure

the binary perceptron and general aspects of non self averaged quantities abstract the possibility of a finite width distribution for the maximal capacity of the binary perceptron in the thermodynamic limit is discussed analytically and supported by a careful analysis of numerical simulations the results also indicate that the description of quenched random systems could take into account the possibility that in addition to non self averaged quantities other quantities such as the transition temperature might also be sample dependent

the norm of a composition operator with linear symbol acting on the dirichlet space abstract we obtain a representation for the norm of a composition operator on the dirichlet space induced by a map of the form z a z b we compare this result to an upper bound for c that is valid whenever is univalent our work relies heavily on an adjoint formula recently discovered by gallardo gutierrez and montes rodriguez

approximating composition operator norms on the dirichlet space abstract let be any univalent self map of the unit disk d whose image d is compactly contained in d we provide a method for approximating the norm of the composition operator c on the dirichlet space to any desired degree of accuracy the approximation uses a special basis which is orthogonal in both the bergman space on the disk and the bergman space on

the role of coherence entropy of physical twin observables in entanglement the concept of physical twin observables pto for bipartite quantum states introduced and proved relevant for quantum information theory in recent work is substantially simplified the relation of observable and state is studied in detail from the point of view of coherence entropy properties of this quantity are further explored it is shown that besides for pure states also for a class of mixed states quantum discord measure of entanglement can be expressed through the coherence entropy of a pto complete in relation to the state

finite size effects and bounds for perceptron models ablncl in lhis paper we consider two main aspsln of the binary perccptron problem the maximal capacity when random paltem are stored model a and its generalhation ability model b we have extended previous numerical estimates of critical capacilies and studied thermal properties of systems of small sizes to lest recent replica predictions we have also considered some simpler versions of thest models the discrete spherical vemions can be salved exactly using gardner s replica calculation for the spherical model and are shown to give a rigorous upper bound and lower bound on the capacities of models a and b respectively iby vemions of models a and b are soivcd in detail and provide information which is useful for interpreting the bnile size eeeciects present in the numerical studies of models a and b

derivation of the quantum probability law from minimal non demolition measurement one more derivation of the quantum probability rule is presented in order to shed more light on the versatile aspects of this fundamental law it is shown that the change of state in minimal quantum non demolition measurement also known as ideal measurement implies the probability law in a simple way namely the very requirement of minimal change of state put in proper mathematical form gives the well known luders formula which contains the probability rule

norm attaining composition operators on the bloch spaces abstract we prove that every composition operator c on the bloch space modulo constant functions attains its norm and characterize the norm attaining composition operators on the little bloch space modulo constant functions we also identify the extremal functions for c in both cases

norms of linear fractional composition operators we obtain a representation for the norm of the composition operator c on the hardy space h 2 whenever is a linear fractional mapping of the form z b cz d the representation shows that for such mappings the norm of c always exceeds the essential norm of c moreover it shows that a formula obtained by cowen for the norms of composition operators induced by mappings of the form z sz t has no natural generalization that would yield the norms of all linear fractional composition operators for rational numbers s and t cowen s formula yields an algebraic number as the norm we show e g that the norm of c 1 2 z is a transcendental number our principal results are based on a process that allows us to associate with each non compact linear fractional composition operator c for which c c e an equation whose maximum real solution is c 2 our work answers a number of questions in the literature for example we settle an issue raised by cowen and maccluer concerning co hyponormality of a certain family of composition operators

quantum probability law from environment assisted invariance in terms of pure state twin unitaries zurek s derivation of the born rule from envariance environment assisted invariance is retraced in an attempt to present a more detailed derivation which is extended to encompass the trace rule form of the quantum probability law in its full generality the investigation begins by a review presentation of schmidt decomposition and a detailed and complete theory of twin unitaries which are the other face of envariance and which stand in close connection with twin observables the trace rule tr e with e being an event projector and a quantum state density operator is derived in five stages the first three achieve the same as zurek s derivation of born s rule but in a different way using the full power of twin unitaries stage 4 extends the zurek like derivation to states ray projectors that are not eigenstates of the density operator finally stage 5 utilizes the assumption of continuity of the probability law in the density operators to extend the trace rule to isolated not correlated systems

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

which linear fractional composition operators are essentially normal abstract we characterize the essentially normal composition operators induced on the hardy space h2 by linear fractional maps they are either compact normal or the nontrivial case induced by parabolic nonautomorphisms these parabolic maps induce the first known examples of nontrivially essentially normal composition operators in addition we characterize those linear fractionally induced composition operators on h2 that are essentially self adjoint and present a number of results for composition operators induced by maps that are not linear fractional

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

how to define systematically all possible two particle state vectors in terms of conditional probabilities all possible two particle state vectors 12 can be generated in a systematic way from first particle reduced statistical operators 1 and anti unitary correlation operators it is pointed out that this procedure is easily completed so that 12 is defined in terms of 1 and conditional statistical operators for the second particle

three unfinished works on the optimal storage capacity of networks the optimal storage properties of three different neural network models are studied for two of these models the architecture of the network is a perceptron with or j interactions whereas for the third model the output can be an arbitrary function of the inputs analytic bounds and numerical estimates of the optimal capacities and of the minimal fraction of errors are obtained for the first two models the third model can be solved exactly and the exact solution is compared to the bounds and to the results of numerical simulations used for the two other models

adjoints of linear fractional composition operators on the dirichlet space the adjoint of a linear fractional composition operator acting on the classical dirichlet space is expressed as another linear fractional composition operator plus a two rank operator the key point is that in the dirichlet space modulo constant functions many linear fractional composition operators are similar to multiplication operators and thus normal as a particular application we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces even the norm of each linear fractional composition operator is computed on the dirichlet space modulo constant functions it is also shown that all this work can be carried out in the hardy space of the upper half plane

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

norms and spectral radii of composition operators acting on the dirichlet space we obtain new upper bounds on the norms of univalently induced composition operators acting on the dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero as an application we show that the spectral radius of every univalently induced composition operator on the dirichlet space is equal to one

adaptive genetic algorithm for the binary perceptron problem for neural networks with j couplings the perceptron problem for random unbiased patterns is considered an algorithm that uses concepts of the continuous perceptron problem as well as ideas of biological optimization is proposed and investigated the distribution of local stabilities and the critical storage capacity alpha c are determined while for n less than 50 the value of alpha c is approximately 0 83 the storage capacity goes down to alpha c 0 7 for n 255

on norms of composition operators acting on bergman spaces for arbitrary composition operators acting on a general bergman space we improve the known lower bound for the norm and also generalize a related recent theorem of d g pokorny and j e shapiro next we obtain a geometric formula for the norms of composition operators with linear fractional symbols thus extending a result of c cowen and p hurst and revealing the meaning of their computation finally we obtain a lower bound for essential norm of an arbitrary composition operator related to the well known criterion of b maccluer and j h shapiro as a corollary norms and essential norms are obtained for certain univalently induced noncompact composition operators in terms of the minimum of the angular derivative of the symbol

norm attaining integral operators on analytic function spaces abstract if f and g are analytic functions in the unit disc d we define s g f z 0 z f w g w d w z d if g is bounded then the integral operator s g is bounded on the bloch space on the dirichlet space and on b m o a we show that s g is norm attaining on the bloch space and on b m o a for any bounded analytic function g but does not attain its norm on the dirichlet space for non constant g some results are also obtained for s g on the little bloch space and for another integral operator t g from the dirichlet space to the bergman space