constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

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

an algorithm for total symmetric obdd detection this paper presents an elegant and efficient algorithm for detection of symmetry in boolean functions represented by ordered binary decision diagrams obdds

parallel binary adders with a minimum number of connections an n bit parallel binary adder consisting of nor gates only in single rail input logic is proved to require at least 17n 1 connections for any value of n such an adder is proved to require at least 7n 2 gates an adder that attains these minimal values is shown also it is concluded that some of the parallel adders with the minimum number of nor gates derived by lai and muroga have the minimum number of connections as well as the minimum number of gates except for the two modules for the two least significant bit positions

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

input variable assignment and output phase optimization of pla s a pla minimization system having the following features is presented 1 minimization of both two level pla s and pla s with two bit decoders 2 optimal input variable assignment to the decoders 3 optimal output phase assignment and 4 essential prime implicants detection without generating all the prime implicants

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

symmetric minimal covering problem and minimal pla s with symmetric variables this paper first discusses the symmetric property of the minimal covering problem in terms of group theory and then the use of this property in the branch and bound method for solving the problem computational results on solving sample problems are shown to demonstrate the improvement of computational efficiency of the branch and bound method by the utilization of the symmetric property the sample problems contain the minimal sum derivation of symmetric switching functions as an application to the design of minimal programmable logic arrays pla s it is shown that when the minimal covering problem derived for the given switching function has symmetric permutations the given switching function may not be symmetric although when the given switching function is symmetric the minimum covering problem obtained from the function is symmetric the branch and bound method based on the symmetric property can be applied not only to symmetric switching functions but also to switching functions which are not symmetric if the corresponding minimal covering problems have symmetric permutations the method can be also applied to minimal covering problems which arise outside the minimal sum derivation problems for switching functions

logic networks with a minimum number of nor nand gates for parity functions of n variables design of logic networks in single rail input logic with a minimum number of nor gates for parity functions of an arbitrary number of variables is described this is partly based on minimum networks for parity functions of a small number of variables which are designed by the integer programming logic design method although it is generally difficult to design minimum networks for functions of an arbitrarily large number of variables we have previously designed minimum networks for adders of an arbitrary number of variables the minimum networks for parity functions of an arbitrary number of variables discussed in this paper is another case many unique properties of minimum nor networks for parity functions are shown minimum networks with nand gates for parity functions can be easily obtained from those with nor gates because of duality relationship between nand and nor

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

the complexity of the parity function in unbounded fan in unbounded depth circuits abstract almost everything is known on the complexity of the parity function in fan in 2 circuits over various bases also the minimal depth of polynomial size unbounded fan in dl circuits for the parity function has been studied here the complexity without any depth restriction is considered for the basis dl almost optimal bounds and for the basis of nor gates and the basis of all threshold functions optimal bounds on the number of gates are obtained for the basis dl the minimal number of wires is determined for threshold circuits an exponential gap between synchronous and asynchronous circuits is proved the results not only answer open questions in complexity theory but also have implications for the real life circuit design

edge disjoint packing of stars and cycles abstract we study the parameterized complexity of two graph packing problems edge disjoint k packing of s stars and edge disjoint k packing of s cycles with respect to the choice of parameters we show that although the two problems are fpt with both k and s as parameters they are unlikely to be fixed parameter tractable when parameterized by only k or only s in terms of kernelization complexity we show that edge disjoint k packing of s stars has a kernel with size polynomial in both k and s but in contrast unless np conp poly edge disjoint k packing of s cycles does not have a kernel with size polynomial in both k and s and moreover does not have a kernel with size polynomial in s for any fixed k we also show that edge disjoint k packing of 4 cycles admits a 96 k 2 kernel in general graphs and a 96 k kernel in planar graphs

on a new boolean function with applications consider a hypercube of 2 sup n points described by n boolean variables and a subcube of 2 sup m points m spl les n as is well known the boolean function with value 1 in the points of the subcube can be expressed as the product and of n m variables the standard synthesis of arbitrary functions exploits this property we extend the concept of subcube to the more powerful pseudocube the basic set is still composed of 2 sup m points but has a more general form the function with value 1 in a pseudocube called pseudoproduct is expressed as the and of n m exor factors each containing at most m 1 variables subcubes are special cases of pseudocubes and their corresponding pseudoproducts reduce to standard products an arbitrary boolean function can be expressed as a sum of pseudoproducts spp this expression is in general much shorter than the standard sum of products as demonstrated on some known benchmarks the logical network of an n bit adder is designed in spp as a relevant example of application of this new technique a class of symmetric functions is also defined particularly suitable for spp representation

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 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

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

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

boolean functions classification via fixed polarity reed muller forms in this paper we present a new method to characterize completely specified boolean functions the central theme of the classification is the functional equivalence a k a boolean matching two boolean functions are equivalent if there exists input permutation input negation or output negation that can transform one function to the other we have derived a method that can efficiently identify equivalence classes of boolean functions the well known canonical fixed polarity reed muller fprm forms are used as a powerful analysis tool the necessary transformations to derive one function from the other are inherent in the fprm representations to identify uniquely each equivalence class a set of well known characteristics of boolean functions and their variables including linearity symmetry total symmetry self complement and self duality are employed it is shown that all the equivalence classes of four variable functions are uniquely identified where majority of the classes have a single fprm form as their representative the boolean matching has applications in technology mapping and in design of standard cell libraries

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

approximating threshold circuits by rational functions motivated by the problem of understanding the limitations of threshold networks for representing boolean functions we consider size depth trade offs for threshold circuits that compute the parity function using a fundamental result in the theory of rational approximation we show how to approximate small threshold circuits by rational functions of low degree we apply this result to establish an almost optimal lower bound of n2 ln2n on the number of edges of any depth 2 threshold circuit with polynomially bounded weights that computes the parity function we also prove that any depth 3 threshold circuit with polynomially bounded weights requires n1 2 ln5 3n edges to compute parity on the other hand we give a construction of a depth d threshold circuit that computes parity with n1 1 d edges where 1 5 2 is the golden ratio we conjecture that there are no linear size bounded depth threshold circuits for computing parity

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

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

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

parameterized counting matching and packing a family of hard problems that admit fptras abstract in the field of parameterized counting complexity the problems that are w 1 hard and admit fixed parameter tractable randomized approximation scheme fptras have attracted much attention in recent years in this paper we focus on the problems on parameterized counting matching and packing these problems include counting set packing counting matching and counting subgraph packing including both vertex disjoint and edge disjoint versions we study the parameterized complexity on these problems on the basis of some results for counting graph matchings we show that a series of problems are w 1 hard furthermore by extending the previous algorithm for counting 3 d matching we obtain fptras for each considered problem respectively our results indicate that the problems on parameterized counting matching and packing form a large family of problems that are w 1 hard and admit fptras

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

a fast serial parallel binary multiplier a fast serial parallel fsp multiplier design is derived from the carry save add shift csas multiplier structure the csas technique accepts multiplier bits serially lsb first and produces outputs serially lsb first multiplication of two n bit unsigned numbers requires 2n clock cycles to complete the process out of which n clocks are used for n row carry save additions and the other n clocks are utilized only to propagate the remaining carries this csas structure is modified so that it operates as a csas unit for the first n clocks and reconfigures itself as an n bit ripple carry parallel adder at the n 1 st clock thus allowing the carries to ripple through eliminating the delay due to storage elements during the last n clocks it is shown that this modification results in an about one third increase in speed for an approximately one third increase in hardware the technique is extended to signed numbers represented in 2 s complement form also it is shown how these implementations can be modularized

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

linking numbers contacts and mutual inductances of a random set of closed curves we establish here a new general result of integral geometry concerning closed rigid curves of arbitrary shapes ine3 and their linking numbersi it generalizes by a different method the interesting integral property ofi2 found recently by pohl and extended by des cloizeaux and ball for two curves we considern closed curves linked successively to each other and forming a ring the cyclic product of their linking numbers is integrated over the group of rigid motions of the curves this integral is shown to factorize over a special algebra of characteristic functions each curve possesses two such intrinsic functions the same algebra is shown to describe a larger class of integral geometry properties a new theorem is established for a family of displacement integrals involving linking numbers contact angles and mutual inductances of the set ofn curves

generalized reed muller forms as a tool to detect symmetries in this paper we present a new method for detecting groups of symmetric variables of completely specified boolean functions the canonical generalized reed muller grm forms are used as a powerful analysis tool to reduce the search space we have developed a set of signatures that allow us to identify quickly sets of potentially symmetric variables our approach allows for detecting symmetries of any number of inputs simultaneously totally symmetric functions can be detected very quickly the traditional definitions of symmetry have also been extended to include more types this extension has the advantage of grouping input variables into more classes experiments have been performed on mcnc benchmark cases and the results verify the efficiency of our method

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

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

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

size energy tradeoffs for unate circuits computing symmetric boolean functions a unate gate is a logical gate computing a unate boolean function which is monotone in each variable examples of unate gates are and gates or gates not gates threshold gates etc a unate circuit c is a combinatorial logic circuit consisting of unate gates let f be a symmetric boolean function of n variables such as the parity function mod function and majority function let m 0 and m 1 be the maximum numbers of consecutive 0 s and consecutive 1 s in the value vector of f respectively and let l min m 0 m 1 and m max m 0 m 1 let c be a unate circuit computing f let s be the size of the circuit c that is c consists of s unate gates let e be the energy of c that is e is the maximum number of gates outputting 1 over all inputs to c in this paper we show that there is a tradeoff between the size s and the energy e of c more precisely we show that n 1 l m s e we also present lower bounds on the size s of c represented in terms of n l and m our tradeoff immediately implies that logn elogs for every unate circuit c computing the parity function of n variables

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

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

the parameterized complexity of counting problems we develop a parameterized complexity theory for counting problems as the basis of this theory we introduce a hierarchy of parameterized counting complexity classes w t for t ge 1 that corresponds to downey and fellows s w hierarchy r g downey and m r fellows parameterized complexity springer verlag new york 1999 and we show that a few central w completeness results for decision problems translate to w completeness results for the corresponding counting problems counting complexity gets interesting with problems whose decision version is tractable but whose counting version is hard our main result states that counting cycles and paths of length k in both directed and undirected graphs parameterized by k is w 1 complete this makes it highly unlikely that these problems are fixed parameter tractable even though their decision versions are fixed parameter tractable more explicitly our result shows that most likely there is no f k cdot n c algorithm for counting cycles or paths of length k in a graph of size n for any computable function f mathbb n to mathbb n and constant c even though there is a 2 o k cdot n 2 376 algorithm for finding a cycle or path of length k n alon r yuster and u zwick j acm 42 1995 pp 844 856

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

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

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

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

matching and weighted p2 packing algorithms and kernels parameterized algorithms and kernelization algorithms are presented for the weighted p 2 packing problem which is a generalization of the famous graph matching problem the parameterized algorithms are based on the following new techniques and observations 1 new study on structure relationship between graph matchings in general graphs and p 2 packings in bipartite graphs 2 an effective graph bi partitioning algorithm and 3 a polynomial time algorithm for a constrained weighted p 2 packing problem in bipartite graphs the kernelization algorithms are based on the following new techniques 1 the application of graph matching in kernelization 2 a crown reduction structure for weighted problems these techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions for the kernelization algorithm by using a weighted version of crown reduction a kernel of size o k 2 is presented where k is the given parameter of the problem

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

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

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

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

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

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

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

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