topological entanglement in polymers the brereton shah problem of two polymer loops topologically linked is studied by field theoretical methods in the limit that one polymer is allowed to fill a macroscopic volume at finite density rho for l 1 it is shown that typically for winding numbers m2 lalpha rho alpha 2 epsilon nu universality is observed with the critical exponents of the unconstrained system despite the presence of a dangerous renormalisation group instability here l is the length of the smaller polymer loop whose mean square size r2 approximately l2 nu nu 1 2 for random flight statistics or nu approximately 3 5 for the swelled chain case

a quantum computing primer for operator theorists this is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory after a brief introduction we discuss quantum algorithms we outline basic properties of quantum channels or equivalently completely positive trace preserving maps the main theorems for quantum error detection and correction are presented and we conclude with a description of a particular passive method of quantum error correction

path integral calculation of the writhe for circular semiflexible polymers earlier obtained numerical and analytical non path integral results for the writhe of a circular polymer are reproduced exactly employing the available path integral models for the semiflexible chains it is confirmed analytically that the average writhe scales with the root of the chain length we also briefly discuss related topological problems e g knotting of circular polymers which could be formulated and solved with the use of path integrals

crossover between random and self avoiding behaviour in a ring polymer the crossover between the random walk and the self avoiding walk limits of a large ring polymer is described using a direct renormalisation theory the methods of field theory renormalisation are applied directly to the ring polymer system to calculate the mean square radius of gyration rg2 ring through first order in epsilon 4 dimension of space the expansion factor alpha 2 for rg2 ring is not a universal function of the natural scaling variable x that characterises the strength of the self repelling excluded volume interaction the crossover exhibited by the radius of gyration ratio r x rg2 ring rg2 line is described by a universal scaling function of x and has a range space 1 2 or r x or 1 2 1 13 96 epsilon that is bounded below and above by finite universal numbers representing the random and self avoiding limits respectively

unital affine semigroups abstract affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable they were introduced by cohen and collins in 1959 we look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups most notably we improve a result of wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups

higher rank numerical ranges and compression problems we consider higher rank versions of the standard numerical range for matrices a central motivation for this investigation comes from quantum error correction we develop the basic structure theory for the higher rank numerical ranges and give a complete description in the hermitian case we also consider associated projection compression problems

loose flat knots in collapsed polymers we consider single ring polymers which are confined on a plane but maintain a fixed three dimensional knotted topology the equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by n step polygons with a number of self intersections restricted to the minimum compatible with the topology this allows to define the size s of the flat knots and to study their localization properties due to the presence of both excluded volume and attractive interactions the model undergoes a theta transition accurate monte carlo results show that while in the high temperature swollen regime both prime and composite knot components are localized s n n t with t 0 in the low temperature compact phase they are fully delocalized t 1 right at the theta transition weak localization prevails t 0 44 0 02 part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations and by applying the scaling theory of polymer networks of fixed topology in particular t 3 7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point

detecting knots in self avoiding walks in this comment the authors discuss problems in the method recently proposed by windwer ibid vol 22 p l605 1989 for computing knot probability in self avoiding walks on the simple cubic lattice

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

on the conditional distributions of low dimensional projections from high dimensional data we study the conditional distribution of low dimensional projections from high dimensional data where the conditioning is on other low dimensional projections to fix ideas consider a random d vector z that has a lebesgue density and that is standardized so that ez 0 and ezz id moreover consider two projections defined by unit vectors and namely a response y z and an explanatory variable x z it has long been known that the conditional mean of y given x is approximately linear in x under some regularity conditions cf hall and li 1993 however a corresponding result for the conditional variance has not been available so far we here show that the conditional variance of y given x is approximately constant in x again under some regularity conditions these results hold uniformly in and for most s provided only that the dimension of z is large in that sense we see that most linear submodels of a high dimensional overall model are approximately correct our findings provide new insights in a variety of modeling scenarios we discuss several examples including sparse linear modeling generalized linear models under potential link violation sliced inverse regression sliced average variance estimation and kernel learning machines

time dependent nonlinear growth theory of linear chains abstract a theory was developed for the nonlinear growth of linear chains by incorporating the chain dynamics into the rate equation this was achieved by using a modified boltzmann equation the chains were first divided into two groups as reacting and unreacting chains in any time interval the contribution of reacting chains to the chain growth was accounted by using a time propagator the nonlinearity in growth introduced by reacting chains was expressed in velocity space as a kind of enhanced diffusion the space independent and the space dependent solutions of chain growth were obtained by using an almost unchanging quantity the fractional change of length per collision the statistical mechanics of chain growth was studied and the chain growth distance was determined

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

cones with a mapping cone symmetry in the finite dimensional case abstract in the finite dimensional case we present a new approach to the theory of cones with a mapping cone symmetry first introduced by stormer our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the jamiolkowski choi isomorphism is an isometry we consider a slightly modified class of cones although not substantially different from the original mapping cones by stormer using the new approach several known results are proved faster and often in more generality than before for example the dual of a mapping cone turns out to be a mapping cone as well without any additional assumptions the main result of the paper is a characterization of cones with a mapping cone symmetry saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive a similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone our result is proved without the additional assumptions of symmetry and equality between the domain and the target space we show how it gives a number of older results as a corollary including an exemplary application

harmonic analysis of iterated function systems with overlap an iterated function system ifs is a system of contractive mappings i y y i 1 n finite where y is a complete metric space every such ifs has a unique up to scale equilibrium measure also called the hutchinson measure and we study the hilbert space l2 in this paper we extend previous work on ifss without overlap our method involves systems of operators generalizing the more familiar cuntz relations from operator algebra theory and from subband filter operators in signal processing these cuntz like operator systems were used in recent papers on wavelet analysis by baggett jorgensen merrill and packer contemp math 345 11 25 2004 where they serve as a first step to generating wavelet bases of parseval type alias normalized tight frames i e wavelet bases with redundancy similarly it was shown in work by dutkay and jorgensen rev mat iberoam 22 131 180 2006 that the iterative operator approach works well for generating wavelets on fractals from ifss without overla

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

statistical topology and knotting of fluctuating filaments abstract the aim of these notes is to provide an introduction to the topic of statistical topology with this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments some questions that we are going to address are the following i what is a knot and how can we identify it ii which is the probability of finding a random curve that is knotted iii how complex are these knots iv how big are they to try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some state of the art theoretical and numerical techniques exercises and lists of open problems will be provided too

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

the measure of a measurement we identify a fractal scale s in a family of borel probability measures on the unit interval which arises independently in quantum information theory and in wavelet analysis the scales s we find satisfy s r and s 1 some s 1 we identify these scales s by considering the asymptotic properties of u j j s where j are dyadic subintervals and j 0

what is the length of a knot in a polymer we give statistical definitions of the length l of a loose prime knot tied into a long fluctuating ring macromolecule monte carlo results for the equilibrium good solvent regime show that l nt where n is the ring length and t 0 75 is independent of the knot type in the collapsed regime below the theta temperature length determinations based on the entropic competition of different knots within the same ring show knot delocalization t 1

fixed points of normal completely positive maps on b h abstract given a sequence of bounded operators a j on a hilbert space h with j 1 a j a j 1 j 1 a j a j we study the map defined on b h by x j 1 a j x a j and its restriction to the hilbert schmidt class c 2 h in the case when the sum j 1 a j a j is norm convergent we show in particular that the operator 1 is not invertible if and only if the c algebra a generated by a j j 1 has an amenable trace this is used to show that may have fixed points in b h which are not in the commutant a of a even in the case when the weak closure of a is injective however if a is abelian then all fixed points of are in a even if the operators a j are not positive

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

entropy encoding hilbert space and karhunen loeve transforms by introducing hilbert space and operators we show how probabilities approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates our main results yield orthogonal bases which optimize distinct measures of data encoding

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

dimension reduction for non elliptically distributed predictors second order methods summary many classical dimension reduction methods especially those based on inverse conditional moments require the predictors to have elliptical distributions or at least to satisfy a linearity condition such conditions however are too strong for some applications li and dong 2009 introduced the notion of the central solution space and used it to modify first order methods such as sliced inverse regression so that they no longer rely on these conditions in this paper we generalize this idea to second order methods such as sliced average variance estimation and directional regression in doing so we demonstrate that the central solution space is a versatile framework we can use it to modify essentially all inverse conditional moment based methods to relax the distributional assumption on the predictors simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts

thermodynamics of relation based systems with applications in econophysics sociophysics and music abstract a methodology was developed to analyze relation based systems evolving in time by using the fundamental concepts of thermodynamics the behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors or pathways connecting subsequent values which characterize an event such as the index values in stock markets a system behaves in a rigid elastic way to an external effect and resists permanent deformation or it behaves in a viscous or soft way and deforms in an irreversible way it was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation based systems gunduz 2009 5 gunduz and gunduz 2010 6 in this research the work associated with elastic component and heat associated with viscous component were discussed and elaborated in a simple two subsequent pathway system in a scattering diagram the first vector represents the cause and the second the effect by using work and heat energy relations that involve force and also storage and loss modulus terms respectively one can calculate the energy involved in relation based systems the modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector once work like and heat like terms were determined the internal energy is also easily found from their summation the parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system three cases i the behavior of the nasdaq 100 index ii a social revolt and iii the structure of a melody were analyzed for their work like heat like and torque like energies in the course of their evolution nasdaq 100 exhibits highly dissipative behavior and its work terms are very small but heat terms are of large magnitude its internal energy highly fluctuates in time in the social revolt studied work and heat terms are of comparable magnitude the melody depicts highly organized structure and usually has larger work terms than heat terms but at some intervals heat terms burst out and attain very large magnitudes torque terms reach high values when the system is recovering from a minimum value

universal collective rotation channels and quantum error correction we present and investigate a new class of quantum channels what we call universal collective rotation channels that includes the class of collective rotation channels as a special case the fixed point set and noise commutant coincide for a channel in this class computing the precise structure of this c algebra is a core problem in a particular noiseless subsystem method of quantum error correction we prove that there is an abundance of noiseless subsystems for every channel in this class and that the young tableaux combinatorial machine may be used to explicitly compute these subsystems

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

resilient quantum computation error models and thresholds recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms these problems include factoring integers and certain physics simulations practical quantum computation requires overcoming the problems of environmental noise and operational errors problems which appear to be much more severe than in classical computation due to the inherent fragility of quantum superpositions involving many degrees of freedom here we show that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value the result is obtained by combining quantum error correction fault tolerant state recovery fault tolerant encoding of operations and concatenation it holds under physically realistic assumptions on the errors

row stochastic matrices with a common left fixed vector abstract we consider the convex polytope s n x that consist of those n n row stochastic matrices having a common nonnegative left fixed vector xt we examine the 1 skeleton of s n x and show how to construct all extreme points adjacent to a given one as vertices of the 1 skeleton connections with transportation polytopes are discussed further we give a formula for the degree of an extreme point in the 1 skeleton of s n x find its maximum and minimum values and determine when all degrees are equal an explicit description of the 1 skeleton is given for n 3

a geometric study of cores of idempotent stochastic matrices abstract for each idempotent e in the compact affine semigroup st n of n n stochastic matrices the core of e denoted by q e is defined as the maximal subsemigroup of st n with zero element e in this paper we study the algebraic and geometric structures of cores and describe how they are put together inside of st n in the process lattice structures of the idempotents and their cores inside a larger core are revealed and then employed to initiate a possible classification

a geometric description of the maximal monoids of some matrix semigroups abstract the maximal monoids of the form f s f are studied where f is a nonnegative idempotent matrix and s is one of the following matrix semigroups n n the nonnegative square matrices st n the stochastic matrices and d n the doubly stochastic matrices for the cases of n n and st n it is shown that these maximal monoids are affinely isomorphic to the full semigroups of lower order and for f d n f that it is a compact affine semigroup with zero here called the core of a primitive idempotent

empirical bayes estimates for large scale prediction problems classical prediction methods such as fisher s linear discriminant function were designed for small scale problems in which the number of predictors n is much smaller than the number of observations n modern scientific devices often reverse this situation a microarray analysis for example might include n 100 subjects measured on n 10 000 genes each of which is a potential predictor this article proposes an empirical bayes approach to large scale prediction where the optimum bayes prediction rule is estimated employing the data from all of the predictors microarray examples are used to illustrate the method the results demonstrate a close connection with the shrunken centroids algorithm of tibshirani et al 2002 a frequentist regularization approach to large scale prediction and also with false discovery rate theory

when is there a representer theorem vector versus matrix regularizers we consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements it is well known that if the regularizer is a nondecreasing function of the l2 norm then the learned vector is a linear combination of the input data this result known as the representer theorem lies at the basis of kernel based methods in machine learning in this paper we prove the necessity of the above condition in the case of differentiable regularizers we further extend our analysis to regularization methods which learn a matrix a problem which is motivated by the application to multi task learning in this context we study a more general representer theorem which holds for a larger class of regularizers we provide a necessary and sufficient condition characterizing this class of matrix regularizers and we highlight some concrete examples of practical importance our analysis uses basic principles from matrix theory especially the useful notion of matrix nondecreasing functions

the spectrum of kernel random matrices we place ourselves in the setting of high dimensional statistical inference where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n we consider the spectrum of certain kernel random matrices in particular n x n matrices whose i j th entry is f x i x j p or f x i x j 2 p where p is the dimension of the data and x i are independent data vectors here f is assumed to be a locally smooth function the study is motivated by questions arising in statistics and computer science where these matrices are used to perform among other things nonlinear versions of principal component analysis surprisingly we show that in high dimensions and for the models we analyze the problem becomes essentially linear which is at odds with heuristics sometimes used to justify the usage of these methods the analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high dimensional data encountered in practice

choi s proof as a recipe for quantum process tomography quantum process tomography is a procedure by which an unknown quantum operation can be fully experimentally characterized we reinterpret choi s proof linear algebr appl 10 285 1975 of the fact that any completely positive linear map has a kraus representation as a method for quantum process tomography the analysis for obtaining the kraus operators is extremely simple we discuss the systems in which this tomography method is particularly suitable

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

opacity and entanglement of polymer chains we argue that the mean crossing number of a random polymer configuration is simply a measure of opacity without being closely related to entanglement as claimed by several authors we present an easy way of estimating its asymptotic behaviour numerically these estimates agree for random walks theta polymers for self avoiding walks and for compact globules with analytic estimates giving log n a b n 2 1 and n 1 3 respectively for the average number of crossings per monomer in the limit n while the result for compact globules agrees with a rigorous previous estimate the result for self avoiding random walks disagrees with previous numerical estimates

knotted polygons with curvature in the knot probability of semiflexible polygons on the cubic lattice is investigated the degree of stiffness of the polygon is mimicked by introducing a bending fugacity conjugate to the curvature of the polygon by generalizing kesten s pattern theorem to semiflexible walks we show that for any finite value of the bending fugacity all except exponentially few sufficiently long polygons are knotted

rapid solution of problems by quantum computation a class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method the quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation

topological properties of a two dimensional polymer chain in the lattice of obstacles a new method of calculating the topological properties of a polymer chain in the lattice of obstacles on the plane is considered in detail it is shown that by applying the method of conformal transformations it is possible to reduce the problem of calculation of the partition function of the polymer chain in a given topological state with respect to the lattice of obstacles to a much more simple problem of the calculation of the green function of a free random walk without any topological restrictions on a riemann surface

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

coordinate independent sparse sufficient dimension reduction and variable selection sufficient dimension reduction sdr in regression which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information is very helpful when the number of predictors is large the standard sdr methods suffer because the estimated linear combinations usually consist of all original predictors making it difficult to interpret in this paper we propose a unified method coordinate independent sparse estimation cise that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently cise is subspace oriented in the sense that it incorporates a coordinate independent penalty term with a broad series of model based and model free sdr approaches this results in a grassmann manifold optimization problem and a fast algorithm is suggested under mild conditions based on manifold theories and techniques it can be shown that cise would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property simulation studies and a real data example demonstrate the effectiveness and efficiency of the proposed approach

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat