analysis of d2d underlaid cellular networks sir meta distribution and mean local delay we study the performance of device to device d2d communication underlaying cellular wireless network in terms of the meta distribution of the signal to interference ratio sir which is the distribution of the conditional sir distribution given the locations of the wireless nodes modeling d2d transmitters and base stations as poisson point processes ppps moments of the conditional sir distribution are derived in order to calculate analytical expressions for the meta distribution and the mean local delay of the typical d2d receiver and cellular downlink user it turns out that for d2d users the total interference from the d2d interferers and base stations is equal in distribution to that of a single ppp while for downlink users the effect of the interference from the d2d network is more complicated we also derive the region of transmit probabilities for the d2d users and base stations that result in a finite mean local delay and give a simple inner bound on that region finally the impact of increasing the base station density on the mean local delay the meta distribution and the density of users reliably served is investigated with numerical results

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

model evaluation for robust tracking under unknown upper bounds on perturbations and measurement noise this technical note addresses a problem of model evaluation in the framework of robust steady state tracking in the l1 setup the upper bounds on measurement noise exogenous disturbance and coprime factor perturbations in a discrete time linear time invariant siso system are assumed to be unknown for a given nominal model and a given tracking controller three different model validation tests are considered and the complexity of computing upper bounds on the worst case steady state tracking error consistent with these tests is analysed

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

knottedness in self avoiding walks in this letter the author explores the knottedness of self avoiding walks he was not able to represent the data as a power law

a fine grained analysis of millimeter wave device to device networks enabling device to device d2d communications in millimeter wave mm wave networks is of critical importance for the next generation mobile networks to support very high data rates multi gigabits per second for mobile devices in this paper we provide a fine grained performance analysis of the mm wave d2d communication networks specifically we first establish a general and tractable framework to investigate the performance of mm wave d2d networks using the poisson bipolar model integrated with several features of the mm wave band to show what fraction of users in the network achieve target reliability if the required signal to interference plus noise ratio sinr or qos requirement is given we derive the meta distributions of the sinr and the data rate interestingly in mm wave d2d networks the standard beta approximation for the meta distribution does not work very well when highly directional antenna arrays are used or the node density is small to resolve this issue we provide a modified approximation by using higher moments of the conditional sinr distribution which is shown to be closer to the exact result on this basis we also derive the mean local delay and spatial outage capacity to provide a comprehensive investigation on the impact of mm wave features on the performance of d2d communication

poland scheraga models and the dna denaturation transition poland scheraga models were introduced to describe the dna denaturation transition we give a rigorous and refined discussion of a family of these models we derive possible scaling functions in the neighborhood of the phase transition point and review common examples we introduce a self avoiding poland scheraga model displaying a first order phase transition in two and three dimensions we also discuss exactly solvable directed examples this complements recent suggestions as to how the poland scheraga class might be extended in order to display a first order transition which is observed experimentally

knot localization in adsorbing polymer rings we study by monte carlo simulations a model of a knotted polymer ring adsorbing onto an impenetrable attractive wall the polymer is described by a self avoiding polygon on the cubic lattice we find that the adsorption transition temperature the crossover exponent ensuremath phi and the metric exponent ensuremath nu are the same as in the model where the topology of the ring is unrestricted by measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized this knot localization transition is triggered by the adsorption transition but is accompanied by a less sharp variation of the exponent related to the degree of localization indeed for a whole interval below the adsorption transition one can not exclude a contiuous variation with temperature of this exponent deep into the adsorbed phase we are able to verify that knot localization is strong and well described in terms of the flat knot model

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

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

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

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

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

lattice stick number of knots the minimal number of straight line segments required to construct a polygonal presentation of the knot k in the cubic lattice is called the lattice stick number of the knot k denoted by it is known that if the crossing number of k satisfies and the main result of this paper is to improve this to if furthermore we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers

torsion of polygons in the torsion of polygons and self avoiding walks in the cubic lattice is a measure of the self entanglement of these objects we consider several definitions of torsion in polygons and introduce a fugacity conjugate to the torsion in our models we study the thermodynamic behaviour of these models using probabilistic methods and rigorous methods from statistical mechanics in particular we prove that at least one of our models has a non analyticity in its free energy corresponding to a transition between phases with high and low torsion

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

robust steady state tracking this paper addresses the robust performance problem when the performance measure is the steady state value of an error signal necessary and sufficient conditions are derived for robust steady state tracking of fixed inputs in the presence of structured time varying uncertainty are derived these conditions are easily computable and fit well with existing conditions on stability robustness and performance robustness when the performance measure is the level of disturbance rejection using these conditions it is shown that time varying perturbations of a nominal linear time invariant plant can result in large steady state tracking errors to fixed inputs even if the nominal plant and controller give zero steady state tracking errors the derived expressions for the worst case steady state tracking error give insight into how the time variation in the plant affects tracking errors and suggest that certain transfer function norms should be minimized to reduce the effect of these perturbations on the steady state value of error signals

new pattern theorems for square lattice self avoiding walks and self avoiding polygons a general pattern theorem for weighted self avoiding polygons saps and self avoiding walks saws in is obtained the pattern theorem for saps fits into the general framework of the pattern theorem for lattice clusters introduced by madras 1999 ann comb 3 357 84 note that unlike other pattern theorems proved for saps this pattern theorem does not rely on first establishing a relationship between saps and saws these results are applied to obtain pattern theorems for self interacting saps and self interacting saws

simple approximations of the sir meta distribution in general cellular networks compared to the standard success coverage probability the meta distribution of the signal to interference ratio sir provides much more fine grained information about the network performance we consider general heterogeneous cellular networks hcns with base station tiers modeled by arbitrary stationary and ergodic non poisson point processes the exact analysis of non poisson network models is notoriously difficult even in terms of the standard success probability let alone the meta distribution hence we propose a simple approach to approximate the sir meta distribution for non poisson networks based on the asappp approximate sir analysis based on the poisson point process method we prove that the asymptotic horizontal gap g 0 between its standard success probability and that for the poisson point process exactly characterizes the gap between the b th moment of the conditional success probability as the sir threshold goes to 0 the gap g 0 allows two simple approximations of the meta distribution for general hcns 1 the per tier approximation by applying the shift g 0 to each tier and 2 the effective gain approximation by directly shifting the meta distribution for the homogeneous independent poisson network given the generality of the model considered and the fine grained nature of the meta distribution these approximations work surprisingly well

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

on distances in uniformly random networks the distribution of euclidean distances in poisson point processes is determined the main result is the density function of the distance to the n nearest neighbor of a homogeneous process in ropfm which is shown to be governed by a generalized gamma distribution the result has many implications for large wireless networks of randomly distributed nodes

l1 robust performance of discrete time systems with structured uncertainty abstract this paper addresses the robust performance problem when a linear time invariant system is subjected to both norm bounded time varying uncertainty and worst case external bounded input the tightest upper bounds for steady state robust performance measures over classes of finite memory and fading memory perturbations are computed in the regulation and tracking problems since the classes of finite memory and fading memory perturbations are inappropriate to the purposes of model validation and adaptive control approximating subclasses of bounded memory perturbations and perturbations with exponentially decreasing impulse responses are considered it is shown that the robust performance bounds obtained are nonconservative in the case of bounded memory perturbations the worst case steady state robust performance measures for siso system under coprime factor perturbations are computed for illustration

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 number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

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

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

entropic exponents of lattice polygons with specified knot type ring polymers in three dimensions can be knotted and the dependence of their critical behaviour on knot type is an open question we study this problem for polygons on the simple cubic lattice using a novel grand canonical monte carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon we conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

suboptimal robust synthesis for mimo plant under coprime factor perturbations abstract suboptimal robust synthesis for mimo nominal system under coprime factor perturbations is considered in classical and non classical statements in the classical statement weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer suboptimal synthesis within e tolerance is reduced to the solution of log 2 1 e standard mixed sensitivity problems of l 1 optimization in the non classical statement the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1 e mixed sensitivity problems

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

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

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

adaptive robust control for siso systems abstract this paper addresses the problem of nonconservative adaptive robust regulating a siso system allowing for model perturbations in the coprime factor representation of the system as well as external additive bounded disturbance signals the perturbations and disturbances are assumed to belong to a particular parameterized class and are norm bounded the norm bounds are not available for control design but the class knowledge is available the performance measure is the worst possible steady state absolute value of the controlled output over the class of allowable disturbances and model perturbations an identification based adaptive control method is proposed where the identification criterion coincides with the control objective the proposed adaptive scheme achieves near optimal performance simulations comparing the new scheme with an adaptive controller based on classical recursive least squares identification are presented

bounds for the minimum step number of knots confined to slabs in the simple cubic lattice volume confinement is a key determinant of the topology and geometry of a polymer however the direct relationship between the two is not fully understood for instance recent experimental studies have constructed p4 cosmids i e p4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their dna knot distribution differs from that of wild type bacteriophage p4 in particular it was observed that the complexity of the knots decreases sharply with the length of the packed genome this problem is the motivation of this paper here a polymer is modeled as a self avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes i e bounded within a slab we estimate the minimum length required for such a polygon to realize a knot type our numerical simulations show that in order to realize a prime knot with up to ten crossings in a 1 slab i e a slab of height 1 one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition in the case of the trefoil knot we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1 slab is 26 which is greater than 24 the known minimum length required to tie a trefoil without a confinement condition additionally we find that in the 1 slab not all geometrical realizations of a given knot type are equivalent under bfacf moves this suggests that in certain confined volumes knowing the topology of a polymer is not enough to describe all its states

control oriented model validation and errors quantification in the spl lscr sub 1 setup a priori information required for robust synthesis includes a nominal model and a model of uncertainty the latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds if these bounds are unknown or too conservative they have to be estimated from measurement data in this paper the problem of errors quantification is considered in the framework of the spl lscr sub 1 optimal robust control theory associated with the spl lscr sub spl infin signal space the optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion for model with unstructured uncertainty entering the system in a linear fractional manner the optimal errors quantification is reduced to quadratic fractional programming for system under coprime factor perturbations the optimal errors quantification is reduced to linear fractional programming

upper bounds on the minimum length of cubic lattice knots knots have been considered to be useful models for simulating molecular chains such as dna and proteins one quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot in this paper we consider every knot in the cubic lattice in particular the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice diao introduced this term he used minimum edge number instead and proved that the minimum length of the trefoil knot 31 is 24 also the minimum lengths of the knots 41 and 51 are known to be 30 and 34 respectively in this paper we find a general upper bound of the minimum length of a nontrivial knot k except the trefoil knot in terms of the minimum crossing number c k the upper bound is moreover if k is a non alternating prime knot then the upper bound is our work can be considered a direct consequence of the results obtained by the authors in hong et al 2012 arxiv 1209 0048 furthermore if k is n 1 n torus knot then the upper bound is

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

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

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

monte carlo results for projected self avoiding polygons a two dimensional model for knotted polymers we introduce a two dimensional lattice model for the description of knotted polymer rings a polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice with possible crossings describing pairs of strands of polymer passing on top of each other each polygon configuration can be viewed as the two dimensional projection of a particular knot we study numerically the statistics of large polygons with a fixed knot type using a generalization of the bfacf algorithm for self avoiding walks this new algorithm incorporates both the displacement of crossings and the three types of reidemeister transformations preserving the knot topology its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof assuming this ergodicity we obtain numerically the following results for the statistics of knotted polygons in the limit of a low crossing fugacity we find a localization along the polygon of all the primary factors forming the knot increasing the crossing fugacity gives rise to a transition from a self avoiding walk to a branched polymer behaviour

reply to detecting knots in self avoiding walks the author replies to a comment by sumners and whittington ibid vol 23 p 1471 1990 on the detection of knots in self avoiding walks on a simple cubic lattice

analysis and design for robust performance with structured uncertainty abstract necessary and sufficient conditions for stability and performance robustness of discrete time systems are provided in terms of the spectral radius of a certain nonnegative matrix the conditions are easily computable and provide a simple and efficient method for computation of the robustness conditions for siso as well mimo perturbations the problem of robust controller synthesis is explored and an iteration scheme for controller synthesis is introduced

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

the statistics of collapsing square lattice trails with a fixed number of vertices of degree 4 a trail on the square lattice with a fixed number k of vertices of degree 4 is called a k trail we model polymer collapse using k trails by incorporating an interaction energy which is proportional to the number of nearest neighbour contact edges of the trail it is known that the number of square lattice n edge closed open k trails can be bounded above and below to o nk by the number of n step self avoiding circuits walks this along with pattern theorems for self interacting self avoiding circuits and walks are used herein to establish upper and lower bounds to o nk for the collapsing free energy of k trails in terms of self avoiding circuits or walks as appropriate we also use pattern theorems to obtain bounds on the limiting nearest neighbour contact density for collapsing k trails finally we investigate k trails with a fixed density of nearest neighbour contacts and show that their limiting entropy per monomer is independent of k

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

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

adaptive suboptimal control of a linear system with bounded disturbance abstract an adaptive suboptimal control of a linear discrete system with unknown parameters is proposed an additive disturbance v t acting on the system is supposed to be uniformly bounded the criterion is sup v t i y 1 u 1 where y t is the output u t is the control the adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters

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

sir meta distribution of k tier downlink heterogeneous cellular networks with cell range expansion heterogeneous cellular networks hcns constitute a necessary step in the evolution of cellular networks in this paper we apply the signal to interference ratio sir meta distribution framework for a refined sir performance analysis of hcns focusing on k tier heterogeneous cellular networks based on the homogeneous independent poisson point process ppp model with range expansion bias offloading bias in each tier expressions for the b th moment of the conditional success probability for both the entire network and each tier are derived based on which the exact meta distributions and the beta approximations are evaluated and compared key performance metrics including the mean success probability the variance of the conditional success probability the mean local delay and the asymptotic sir gains of each tier are obtained the results show that the biases are detrimental to the overall mean success probability of the whole network and that the b th moment curve of the conditional success probability of each tier can be tightly approximated by the horizontal shifted versions of the first moment curve of the single tier ppp network we also provide lower bounds for the region of the active probabilities of the base stations to keep the mean local delay of each tier finite

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

adaptive suboptimal tracking for the first order plant with lipschitz uncertainty a tracking problem for the first order discrete time plant with time invariant uncertainty of the lipschitz type and bounded exogenous disturbance is considered in the adaptive setting and two different adaptive laws are proposed the first law is based on a natural estimation of the unknown linear part of the plant and introduces some conservatism in stability margin and tracking performance the second adaptive law is suboptimal and includes additionally the estimation of the unknown lipshitz constant and the unknown norm of exogenous disturbance the suboptimality is achieved by exploiting a cone estimation algorithm and the linearity of the control criterion with respect to unknown parameters both adaptive laws utilize the nonlinear feedback by xie and guo based on nonparametric estimation of the uncertainty