on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

generating equilateral random polygons in confinement in this paper we continue an earlier study diao et al 2011 j phys a math theor 44 405202 on the generation algorithms of random equilateral polygons confined in a sphere here the equilateral random polygons are rooted at the center of the confining sphere and the confining sphere behaves like an absorbing boundary one way to generate such a random polygon is the accept reject method in which an unconditioned equilateral random polygon rooted at origin is generated the polygon is accepted if it is within the confining sphere otherwise it is rejected and the process is repeated the algorithm proposed in this paper offers an alternative to the accept reject method yielding a faster generation process when the confining sphere is small in order to use this algorithm effectively a large reusable data set needs to be pre computed only once we derive the theoretical distribution of the given random polygon model and demonstrate with strong numerical evidence that our implementation of the algorithm follows this distribution a run time analysis and a numerical error estimate are given at the end of the paper

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

the abundance of unknots in various models of polymer loops a veritable zoo of different knots is seen in the ensemble of looped polymer chains whether created computationally or observed in vitro at short loop lengths the spectrum of knots is dominated by the trivial knot unknot the fractional abundance of this topological state in the ensemble of all conformations of the loop of n segments follows a decaying exponential form exp n n0 where n0 marks the crossover from a mostly unknotted i e topologically simple to a mostly knotted i e topologically complex ensemble in the present work we use computational simulation to look closer into the variation of n0 for a variety of polymer models among models examined n0 is smallest about 240 for the model with all segments of the same length it is somewhat larger 305 for gaussian distributed segments and can be very large up to many thousands when the segment length distribution has a fat power law tail

higher order models for fuzzy random variables a fuzzy random variable is viewed as the imprecise observation of the outcomes in a random experiment since randomness and vagueness coexist in the same framework it seems reasonable to integrate fuzzy random variables into imprecise probabilities theory nevertheless fuzzy random variables are commonly presented in the literature as classical measurable functions associated to a classical probability measure we present here a higher order possibility model that represents the imprecise information provided by a fuzzy random variable we compare it with previous classical models in the literature first some aspects about the acceptability function associated to a fuzzy random variable are investigated secondly we present three different higher order possibility models all of them arising in a natural way we investigate their similarities and differences and observe that the first one the fuzzy probability envelope is the most informative finally we compare the fuzzy probability envelope with the classical probability measure induced by the fuzzy random variable we conclude that the classical probability measure does not always contain all relevant information provided by a fuzzy random variable

knot complexity and the probability of random knotting the probability of a random polygon or a ring polymer having a knot type k should depend on the complexity of the knot k through computer simulation using knot invariants we show that the knotting probability decreases exponentially with respect to knot complexity here we assume that some aspects of knot complexity are expressed by the minimal crossing number c and the rope length of k which is defined by the smallest length of rope with unit diameter that can be tied to make the knot k

distribution of the distance between opposite nodes of random polygons with a fixed knot we examine numerically the distribution function fk r of the distance r between opposite polygonal nodes for random polygons of n nodes with a fixed knot type k here we consider three knots such as 31 and 3131 in a wide range of r the shape of fk r is well fitted by the scaling form 1 of self avoiding walks the fit yields the gaussian exponents and k 1 furthermore if we re scale the intersegment distance r by the average size rk of random polygons of knot k the distribution function of the variable r rk should become the same gaussian distribution for any large value of n and any knot k we also introduce a fitting formula to the distribution gk r of gyration radius r for random polygons under some topological constraint k

knot probability for lattice polygons in confined geometries we study the knot probability of polygons confined to slabs or prisms considered as subsets of the simple cubic lattice we show rigorously that almost all sufficiently long polygons in a slab are knotted and we use monte carlo methods to investigate the behaviour of the knot probability as a function of the width of the slab or prism and the number of edges in the polygon in addition we consider the effect of solvent quality on the knot probability in these confined geometries

on the variability of the concept of variance for fuzzy random variables fuzzy random variables possess several interpretations historically they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables accordingly there are two different approaches to the definition of the variance of a fuzzy random variable in the first one the variance of the fuzzy random variable is defined as a crisp number which makes it easier to handle in further processing in the second case the variance is defined as a fuzzy interval thus offering a gradual description of our incomplete knowledge about the variance of an underlying imprecisely observed classical random variable in this paper we also discuss another view of fuzzy random variables which comes down to a set of random variables induced by a fuzzy relation describing an ill known conditional probability this view leads to yet another definition of the variance of a fuzzy random variable in the context of the theory of imprecise probabilities the new variance is a real interval which achieves a compromise between both previous definitions in terms of representation simplicity our main objective is to demonstrate with the help of simple examples the practical significance of these definitions of variance induced by various existing views of fuzzy random variables

shapes of tight composite knots we present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings expanding our dataset to 943 knots and links we use the new data set to analyze two outstanding conjectures about tight knots namely that the ropelengths of composite knots are at least 4 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture we also present data on the number of configurations having straight segments and highly curved kinked regions

rational realizations of the minimum rank of a sign pattern matrix abstract a sign pattern matrix is a matrix whose entries are from the set 0 the minimum rank of a sign pattern matrix a is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of a it is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix the equivalence of this conjecture to several seemingly unrelated statements are established for some special cases such as when a is entrywise nonzero or the minimum rank of a is at most 2 or the minimum rank of a is at least n 1 where a is m n the conjecture is shown to hold connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either 1 or 1 are investigated

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

linking of uniform random polygons in confined spaces in this paper we study the topological entanglement of uniform random polygons in a confined space we derive the formula for the mean squared linking number of such polygons for a fixed simple closed curve in the confined space we rigorously show that the linking probability between this curve and a uniform random polygon of n vertices is at least our numerical study also indicates that the linking probability between two uniform random polygons in a confined space of m and n vertices respectively is bounded below by in particular the linking probability between two uniform random polygons both of n vertices is bounded below by

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

the average crossing number of equilateral random polygons in this paper we study the average crossing number of equilateral random walks and polygons we show that the mean average crossing number acn of all equilateral random walks of length n is of the form a similar result holds for equilateral random polygons these results are confirmed by our numerical studies furthermore our numerical studies indicate that when random polygons of length n are divided into individual knot types the for each knot type can be described by a function of the form where a b and c are constants depending on and n0 is the minimal number of segments required to form the profiles diverge from each other with more complex knots showing higher than less complex knots moreover the profiles intersect with the acn profile of all closed walks these points of intersection define the equilibrium length of i e the chain length at which a statistical ensemble of configurations with given knot type upon cutting equilibration and reclosure to a new knot type does not show a tendency to increase or decrease this concept of equilibrium length seems to be universal and applies also to other length dependent observables for random knots such as the mean radius of gyration rg

crossings and writhe of flexible and ideal knots the data of ideal knots nature 384 142 1996 are reanalyzed and the average crossing number of the ideal knots x mathrm ideal shows a nonlinear behavior with the essential crossing number c supplemented with our monte carlo simulations using the bond fluctuation model on flexible knotted polymers our analysis indicates that x mathrm ideal varies nonlinearly with both c and the corresponding average crossing number of the flexible knot which is contrary to previous claims our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of c furthermore our data on the average writhe number mathrm wr indicate that various knots are classified into holonomous groups and mathrm wr has a quantized linear increment with c in all four knot groups in our study

approximations of upper and lower probabilities by measurable selections a random set can be regarded as the result of the imprecise observation of a random variable following this interpretation we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable we link this problem to the existence of selectors of a multi valued mapping and with the inner approximations of the upper probability and prove that under fairly general conditions although not in all cases the upper and lower probabilities are an adequate tool for modelling the available information in doing this we generalise a number of results from the literature finally we study the particular case of consonant random sets and we also derive a relationship between aumann and choquet integrals

average size of random polygons with fixed knot topology we have evaluated by numerical simulation the average size r k of random polygons of fixed knot topology k ensuremath emptyset 3 1 3 1 ensuremath sharp 4 1 and we have confirmed the scaling law r k 2 ensuremath sim n 2 ensuremath nu k for the number n of polygonal nodes in a wide range n 100 char21 2200 the best fit gives 2 ensuremath nu k ensuremath simeq 1 11 char21 1 16 with good fitting curves in the whole range of n the estimate of 2 ensuremath nu k is consistent with the exponent of self avoiding polygons in a limited range of n n ensuremath gtrsim 600 however we have another fit with 2 ensuremath nu k ensuremath simeq 1 01 char21 1 07 which is close to the exponent of random polygons

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

algebraic invariants of knots and disordered potts model the authors propose the reformulation of the kauffman bracket invariant of the knot in terms of statistical mechanics of the 2d disordered potts model this allows one to put the question of the determination of knot entropy or of the probability of an arbitrary knot formation in terms of usual statistical mechanics to demonstrate the possibilities of their approach they give a constructive estimation for the trivial knot formation probability for a long strongly contracted closed random path confined in a thin slit they use the mean field approximation for the free energy of the potts system at the point of the transition from the paramagnetic phase to the spin glass one

patterns of alternating sign matrices abstract we initiate a study of the zero nonzero patterns of n n alternating sign matrices we characterize the row column sum vectors of these patterns and determine their minimum term rank in the case of connected alternating sign matrices we find the minimum number of nonzero entries and characterize the case of equality we also study symmetric alternating sign matrices in particular those with only zeros on the main diagonal these give rise to alternating signed graphs without loops and we determine the maximum number of edges in such graphs we also consider n n alternating sign matrices whose patterns are maximal within the class of all n n alternating sign matrices

the growth of the mean average crossing number of equilateral polygons in confinement the physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers such as dna in all organisms are known to be determined at least in part by their topological and geometrical properties with this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used however very few analytical results are known in this paper we investigate the effect of volume confinement on the mean average crossing number acn of equilateral random polygons the mean acn of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense for an equilateral random polygon of n segments without any volume confinement constrain it is known that its mean acn acn is of the order here we model the confining volume as a simple sphere of radius r we provide an analytical argument which shows that acn of an equilateral random polygon of n segments under extreme confinement meaning r n grows as o n2 we propose to model the growth of acn as a r n2 b r nln n under a less extreme confinement condition where a r and b r are functions of r with r being the radius of the confining sphere computer simulations performed show a fairly good fit using this model

sampling large random knots in a confined space dna knots formed under extreme conditions of condensation as in bacteriophage p4 are difficult to analyze experimentally and theoretically in this paper we propose to use the uniform random polygon model as a supplementary method to the existing methods for generating random knots in confinement the uniform random polygon model allows us to sample knots with large crossing numbers and also to generate large diagrammatically prime knot diagrams we show numerically that uniform random polygons sample knots with large minimum crossing numbers and certain complicated knot invariants as those observed experimentally we do this in terms of the knot determinants or colorings our numerical results suggest that the average determinant of a uniform random polygon of n vertices grows faster than we also investigate the complexity of prime knot diagrams we show rigorously that the probability that a randomly selected 2d uniform random polygon of n vertices is almost diagrammatically prime goes to 1 as n goes to infinity furthermore the average number of crossings in such a diagram is at the order of o n2 therefore the two dimensional uniform random polygons offer an effective way in sampling large prime knots which can be useful in various applications

anomalous finite size effects for the mean squared gyration radius of gaussian random knots anomalously strong finite size effects have been observed for the mean square radius of gyration r2k of gaussian random polygons with a fixed knot k as a function of the number n of polygonal nodes through computer simulations with n 2000 we find that the gyration radius r2k can be approximated by a power law r2k n2 keff for several knots where the effective exponents keff are larger than 0 5 and less than 0 6 furthermore a crossover occurs for the gyration radius of the trivial knot when n is roughly equal to the characteristic length nc of random knotting assuming an asymptotic fitting formula we also discuss possible asymptotic behaviours for rk2 of gaussian random polygons

exact and efficient sampling of conditioned walks a computationally challenging and open problem is how to efficiently generate equilibrated samples of conditioned walks we present here a general stochastic approach that allows one to produce these samples with their correct statistical weight and without rejections the method is illustrated for a jump process conditioned to evolve within a cylindrical channel and forced to reach one of its ends we obtain analytically the exact probability density function of the jumps and offer a direct method for gathering equilibrated samples of a random walk conditioned to stay in a channel with suitable boundary conditions unbiased walks of arbitrary length can thus be generated with linear computational complexity even when the channel width is much smaller than the typical bond length of the unconditioned walk by profiling the metric properties of the generated walks for various bond lengths we characterize the crossover between weak and strong confinement regimes with great detail

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

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

the dimensions of knotted polygons the authors study the dimensions mean square radius of gyration and mean span of self avoiding polygons on the simple cubic lattice with fixed knot type the approach used is a monte carlo algorithm which is a combination of the bfacf algorithm and the pivot algorithm so that the polygons are studied in the grand canonical ensemble but the autocorrelation time is not too large they show that although the dimensions of polygons are sensitive to knot type the critical exponent v and the leading amplitude are independent of the knot type of the polygon the knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour

on the variance of random sets on considere la notion de variance d un ensemble aleatoire on donne un algorithme de determination de la variance des ensembles aleatoires prenant leurs valeurs dans des intervalles compacts non vides sur r

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

dense alternating sign matrices and extensions abstract in this paper some connections of dense alternating sign matrices with total unimodularity combined matrices and generalized complementary basic matrices are explored

the l a mean squared dispersion associated with a fuzzy random variable in this paper we introduce a parameterized real valued measure of the mean dispersion of a fuzzy random variable with respect to an arbitrary fuzzy number this measure extends the second moment of a classical random variable and is based on a parameterized distance between fuzzy numbers properties of the measure presented are analyzed and the extension of the variance of a classical random variable which particularizes the mean squared dispersion is also examined some examples illustrating the computation and possible applications of the measure are included finally a brief discussion about the interest of using a parameterized distance and about some future directions of this study is developed

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

the average inter crossing number of equilateral random walks and polygons in this paper we study the average inter crossing number between two random walks and two random polygons in the three dimensional space the random walks and polygons in this paper are the so called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length we show that the mean average inter crossing number icn between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term which is in the case of two random polygons of length n the mean average inter crossing number icn is also linear but the prefactor of the linear term is different from that of the random walks these approximations apply when the starting points of the random walks and polygons are of a distance apart and is small compared to n we propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average icn for large values of our simulation result shows that the model in fact works very well for the entire range of we also study the mean icn between two equilateral random walks and polygons of different lengths an interesting result is that even if one random walk polygon has a fixed length the mean average icn between the two random walks polygons would still approach infinity if the length of the other random walk polygon approached infinity the data provided by our simulations match our theoretical predictions very well

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

characteristics of shape and knotting in ideal rings we present two descriptions of the local scaling and shape of ideal rings primarily featuring subsegments our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance defined to be the average squared distance between vertices k edges apart we calculate the exact averages of these values over the space of all such ideal rings not just a calculation of the order of these averages and compare these to the equivalent values in open chains this comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths these results will be corroborated by numerical experiments they will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape

alternating sign matrices related 0 1 matrices and the smith normal form abstract we investigate the smith normal form snf of alternating sign matrices asms and related matrices of 0 s and 1 s 0 1 matrices we identify certain classes of asms and 0 1 matrices whose snfs are 0 1 matrices we relate some of our work to various ranks in particular the 0 1 rank of a 0 1 matrix that is the bipartite partition number of a bipartite graph

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

natural classification of knots the principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types here we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages in addition we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot these numbers include crossing number average three dimensional writhe number of topological domains and the average relaxation value

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

advocating the use of imprecisely observed data in genetic fuzzy systems in our opinion and in accordance with current literature the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet in particular we question the existence of a set of problems for which the use of fuzzy rules in combination with genetic algorithms produces more robust models or classifiers that are inherently better than those arising from the bayesian point of view we will show that this set of problems actually exists and comprises interval and fuzzy valued datasets but it is not being exploited current genetic fuzzy classifiers deal with crisp classification problems where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions with a convenient linguistic interpretation provided that the customary use of fuzzy sets in statistics is vague data we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems the same can be said about genetic fuzzy models the use of a scalar fitness function assumes crisp data where fuzzy models a priori do not have advantages over statistical regression

a fast direct sampling algorithm for equilateral closed polygons sampling equilateral closed polygons is of interest in the statistical study of ring polymers over the past 30 years previous authors have proposed a variety of simple markov chain algorithms but have not been able to show that they converge to the correct probability distribution and complicated direct samplers which require extended precision arithmetic to evaluate numerically unstable polynomials we present a simple direct sampler which is fast and numerically stable and analyze its runtime using a new formula for the volume of equilateral polygon space as a dirichlet type integral

limits of analogy between self avoidance and topology driven swelling of polymer loops the work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length where the effects of knotting are small a simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data contrast between this expression and the well known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots which may be useful for understanding where the analogy between the two physical forms is valid the work also includes description of a new method for the computational generation of polymer loops via conditional probability although computationally intensive this method generates loops without statistical bias and thus is preferable to other loop generation routines in the region of interest

probability theory of random polygons from the quaternionic viewpoint we build a new probability measure on closed space and plane polygons the key construction is a map given by hausmann and knutson using the hopf map on quaternions from the complex stiefel manifold of 2 frames in n space to the space of closed n gons in 3 space of total length 2 our probability measure on polygon space is defined by pushing forward haar measure on the stiefel manifold by this map a similar construction yields a probability measure on plane polygons that comes from a real stiefel manifold the edgelengths of polygons sampled according to our measures obey beta distributions this makes our polygon measures different from those usually studied which have gaussian or fixed edgelengths one advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration another is that direct sampling according to our measures is fast linear in the number of edges and easy to code some of our methods will be of independent interest in studying other probability measures on polygon spaces we define an edge set ensemble ese to be the set of polygons created by rearranging a given set of n edges a key theorem gives a formula for the average over an ese of the squared lengths of chords skipping k vertices in terms of k n and the edgelengths of the ensemble this allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges 2014 wiley periodicals inc

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

scaling behavior of random knots using numerical simulations we investigate how overall dimensions of random knots scale with their length we demonstrate that when closed non self avoiding random trajectories are divided into groups consisting of individual knot types then each such group shows the scaling exponent of 0 588 that is typical for self avoiding walks however when all generated knots are grouped together their scaling exponent becomes equal to 0 5 as in non self avoiding random walks we explain here this apparent paradox we introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots we also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots

note on the spectral radius of alternating sign matrices abstract we show that the n n so called diamond alternating sign matrix d n is the unique n n alternating sign matrix with maximum spectral radius n and that lim n n n 2

alternating signed bipartite graphs and difference 1 colourings abstract we investigate a class of 2 edge coloured bipartite graphs known as alternating signed bipartite graphs asbgs that encode the information in alternating sign matrices the central question is when a given bipartite graph admits an asbg colouring a 2 edge colouring such that the resulting graph is an asbg we introduce the concept of a difference 1 colouring a relaxation of the concept of an asbg colouring and present a set of necessary and sufficient conditions for when a graph admits a difference 1 colouring the relationship between distinct difference 1 colourings of a particular graph is characterised and some classes of graphs for which all difference 1 colourings are asbg colourings are identified one key step is theorem 3 4 6 which generalises hall s matching theorem by describing a necessary and sufficient condition for the existence of a subgraph h of a bipartite graph in which each vertex v of h has some prescribed degree r v

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

scattering functions of knotted ring polymers we discuss the scattering function of a gaussian random polygon with n nodes under a given topological constraint through simulation we evaluate the form factor pk q of a gaussian polygon of n 200 having a fixed knot k for some different knots such as the trivial trefoil and figure eight knots here the gaussian polygons with different knots k have distinct values of the mean square radius of gyration r2 g k we obtain the kratky plots of the form factors i e the plots of qr g k 2pk q versus qr g k for the different topological constraints and discuss nontrivial large q behavior as well as small q behavior for the scattering functions we also find that the distinct values of r2 g k play an important role in the large q and small q properties of the kratky plots

imprecise distribution function associated to a random set some different extensions to random sets of the most common parameters of a random variable share a common rationale a random set represents the imprecise observation of a random variable hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable following the same principles in this paper it is proposed a new definition of the distribution function of a random set this definition is simpler in its formulation and it can be used in more general cases than previous proposals the properties of the distribution function defined here are discussed some issues about continuity convergence of the images of the distribution function monotonocity and measurability are studied it is also stated that not all the information conveyed by the random set about the original probability measure the probability measure induced by the imprecisely observed random variable is kept by its distribution function

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

curvature of random walks and random polygons in confinement the purpose of this paper is to study the curvature of equilateral random walks and polygons that are confined in a sphere curvature is one of several basic geometric properties that can be used to describe random walks and polygons we show that confinement affects curvature quite strongly and in the limit case where the confinement diameter equals the edge length the unconfined expected curvature value doubles from 2 to to study curvature a simple model of an equilateral random walk in spherical confinement in dimensions 2 and 3 is introduced for this simple model we derive explicit integral expressions for the expected value of the total curvature in both dimensions these expressions are functions that depend only on the radius r of the confinement sphere we then show that the values obtained by numeric integration of these expressions agrees with numerical average curvature estimates obtained from simulations of random walks finally we compare the confinement effect on curvature of random walks with random polygons

a topological invariant to predict the three dimensional writhe of ideal configurations of knots and links we present herein a topological invariant of oriented alternating knots and links that predicts the three dimensional 3d writhe of the ideal geometrical configuration of the considered knot link the fact that we can correlate a geometrical property of a given configuration with a topological invariant supports the notion that the ideal configuration contains important information about knots and links the importance of the concept of ideal configuration was already suggested by the good correlation between the 3d writhe of ideal knot configurations and the ensemble average of the 3d writhe of random configurations of the considered knots the values of the new invariant are quantized multiples of 4 7 for links with an odd number of components including knots and 2 7 plus multiples of 4 7 for links with an even number of components

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

a fast ergodic algorithm for generating ensembles of equilateral random polygons knotted structures are commonly found in circular dna and along the backbone of certain proteins in order to properly estimate properties of these three dimensional structures it is often necessary to generate large ensembles of simulated closed chains i e polygons of equal edge lengths such polygons are called equilateral random polygons however finding efficient algorithms that properly sample the space of equilateral random polygons is a difficult problem currently there are no proven algorithms that generate equilateral random polygons with its theoretical distribution in this paper we propose a method that generates equilateral random polygons in a step wise uniform way we prove that this method is ergodic in the sense that any given equilateral random polygon can be generated by this method and we show that the time needed to generate an equilateral random polygon of length n is linear in terms of n these two properties make this algorithm a big improvement over the existing generating methods detailed numerical comparisons of our algorithm with other widely used algorithms are provided

exponents of intrachain correlation for self avoiding walks and knotted self avoiding polygons we show numerically that critical exponents for two point intrachain correlation of an infinite chain characterize those of finite chains in self avoiding walk saw and self avoiding polygon sap under a topological constraint we evaluate short distance exponents i j through the probability distribution functions of the distance between the ith and jth vertices of n step saw or sap with a knot for all pairs 1 i j n we construct the contour plot of i j and express it as a function of i and j we suggest that it has quite a simple structure here exponents i j generalize des cloizeaux s three critical exponents for short distance intrachain correlation of saw and we show the crossover among them we also evaluate the diffusion coefficient of knotted sap for a few knot types which can be calculated with the probability distribution functions of the distance between two nodes

the generation of random equilateral polygons freely jointed random equilateral polygons serve as a common model for polymer rings reflecting their statistical properties under theta conditions to generate equilateral polygons researchers employ many procedures that have been proved or at least are believed to be random with respect to the natural measure on the space of polygonal knots as a result the random selection of equilateral polygons as well as the statistical robustness of this selection is of particular interest in this research we study the key features of four popular methods the polygonal folding the crankshaft rotation the hedgehog and the triangle methods in particular we compare the implementation and efficacy of these procedures especially in regards to the population distribution of polygons in the space of polygonal knots the distribution of edge vectors the local curvature and the local torsion in addition we give a rigorous proof that the crankshaft rotation method is ergodic

ranks of dense alternating sign matrices and their sign patterns abstract in this paper an explicit formula for the ranks of dense alternating sign matrices is obtained the minimum rank and the maximum rank of the sign pattern of a dense alternating sign matrix are determined some related results and examples are also provided

the mean squared writhe of alternating random knot diagrams the writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram it plays an important role not only in knot theory itself but also in various applications of knot theory to fields such as molecular biology and polymer physics the mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one half however such a diagram is usually not minimal if we restrict ourselves to a minimal knot diagram then the choice of which strand is the over or under strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams this paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced alternating knot diagrams we demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates in particular this is true when the sample space contains only diagrams of a special structure when the sample space is large enough to contain not only diagrams of a special type then the mean squared writhe for n crossing diagrams tends to grow linearly with n but at a faster rate than n indicating an intrinsic property of alternating knot diagrams studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used which is an important area of study in the applications of random knot theory