on the number of different permanents of some sparse 0 1 circulant matrices abstract starting from known results about the number of possible values for the permanents of 0 1 circulant matrices with three nonzero entries per row and whose dimension n is prime we prove corresponding results for n power of a prime n product of two distinct primes and n 2 3 h supported by some experimental results we also conjecture that the number of different permanents of n n 0 1 circulant matrices with k nonzero per row is asymptotically equal to n k 2 k o n k 3

on the permanent of certain 0 1 toeplitz matrices abstract we obtain convenient expressions and or efficient algorithms for the permanent of certain very sparse 0 1 toeplitz matrices the classes of matrices considered here include some nontrivial examples of circulants to which none of the previous approaches could be successfully applied

on prime factors of determinants of circulant matrices abstract we find the probability that the determinant of an integer circulant n n matrix is divisible by the prime p where p does not divide n

critical exponents for the self avoiding random walk in three dimensions we compute by direct monte carlo simulation the main critical exponents 4 andv and the effective coordination number for the self avoiding random walk in three dimensions on a cubic lattice we find both hyperscaling relationsdv 2 anddv 2 4 0 satisfied ind 3

a graph theoretic approach to statistical data security in this paper we study the problem of protecting sensitive data in an n by n two dimensional table of statistics when the nonsensitive data are made public along with the row and column sums for the table a sensitive cell is considered unprotected if its exact value can be deduced from the nonsensitive cell values and the row and column sums we give an efficient algorithm to identify all unprotected cells in a table the algorithm runs in linear time if the sensitive values are known and in o n 3 time if they are not known we then consider the problem of suppressing the fewest additional cell values to protect all the sensitive cells when some cells are initially unprotected we give a linear time algorithm for this problem in the useful special case that all cell values are strictly positive we next consider the problem of computing the tightest upper and lower bounds on the values of sensitive cells we show that each cell bound can be computed in o n 3 time but all theta n 2 value

one rooted trees attached to a triangular lattice surface the number of distinct weakly embedded n vertex one rooted trees attached to the surface of the triangular lattice at the root have been enumerated for trees with n or 12 writing the generating function g p sigma n ncnpn cn approximately n to infinity lambda nn theta indicates theta 2 008 or 0 010

tests of a method for estimating corrections to scaling from series the author studies a recently developed pade type method of series analysis which allows for the leading confluent singularity a number of properties are described and by applying the method to various test series some pitfalls of the technique are revealed

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

self avoiding random loops a random loop or polygon is a simple random walk whose trajectory is a simple jordan curve the study of random loops is extended in two ways first the probability p sub n x y that a random n step loop contains a point x y in the interior of the loop is studied and 1 2 1 2 is shown to be 1 2 1 n it is plausible that p sub n x y tends toward 1 2 for all x y but this is not proved even for x y 3 2 1 2 a way is offered to simulate random n step self avoiding loops numerical evidence obtained with this simulation procedure suggests that the probability p sub n 3 2 1 2 approximately 1 2 c n for some fixed c

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

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

non integrability of measure preserving maps via lie symmetries we consider the problem of characterizing for certain natural number m the local c m non integrability near elliptic fixed points of smooth planar measure preserving maps our criterion relates this non integrability with the existence of some lie symmetries associated to the maps together with the study of the finiteness of its periodic points one of the steps in the proof uses the regularity of the period function on the whole period annulus for non degenerate centers question that we believe that is interesting by itself the obtained criterion can be applied to prove the local non integrability of the cohen map and of several rational maps coming from second order difference equations

an analytical approach to the inference of summary data of additive type summary data take the form of a triple that consists of a summary attribute a category and a numeric value the inference problem of summary data consists in deciding whether or not a summary data of interest is evaluable i e can be computed from a given set of summary data we address the special case of the inference problem with homogeneous summary data i e summary data with the same summary attribute where the summary attribute is of additive nature owing to additivity one can model the information content of the given summary data by a linear equation system whose variables are constrained to take their values from the domain of the summary attribute we state two evaluability criteria one for a real or integral summary attribute and the other for a nonnegative real or nonnegative integral summary attribute using the two evaluability criteria we show that our inference problem can be solved in strongly polynomial time for a real or integral or nonnegative real summary attribute and is co i np complete for a nonnegative integral summary attribute moreover we prove that given a summary data of interest that is not evaluable even in the simplest case that the summary attribute is of a real type finding an evaluable summary data whose category is maximally contained in or minimally contains the category of the summary data of interest is an np hard problem

branched polymers with a prescribed number of cycles the authors consider the number of lattice animals with exactly c cycles for weakly embeddable bond clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c they use series analysis methods to estimate the critical exponents for c 1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles evidence from series analysis results and from expansions in inverse powers of the dimension suggests that in the case of strong embeddings i e site clusters the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings i e bond clusters they discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity

computation of sparse circulant permanents via determinants we consider the problem of computing the permanent of circulant matrices we apply some recent results on the computation of the number of perfect matchings of small genus graphs in order to show that the permanent of a circulant matrix with three nonzero entries per row is the linear combination of just four determinants of circulant matrices with the same structure as the original matrix we also show that the same result holds true for a class of circulant matrices with four nonzero entries per row related to the dimer problem with periodic boundary conditions conversely we give hints at the fact that more general circulants do not share similar properties and thus should be dealt with by means of radically different approaches

total protection of analytic invariant information in cross tabulated tables to protect sensitive information in a cross tabulated table it is a common practice to suppress some of the cells in the table an analytic invariant is a power series in terms of the suppressed cells that has a unique feasible value and a convergence radius equal to infty intuitively the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed this paper gives an optimal linear time algorithm for testing whether there exist nontrivial analytic invariants in terms of the suppressed cells in a given set of suppressed cells this paper also presents np completeness results and an almost linear time algorithm for the problem of suppressing the minimum number of cells in addition to the sensitive ones so that the resulting table does not leak analytic invariant information about a given set of suppressed cells

a linear algorithm for finding the invariant edges of an edge weighted graph given an edge weighted graph where all weights are nonnegative reals an edge reweighting is an assignment of nonnegative reals to edges such that for each vertex the sums of given and new weights assigned to the edges incident on the vertex do coincide an edge is then said to be invariant if its weight is the same for any edge reweighting we show that the set of invariant edges of an arbitrary edge weighted graph can be determined in time linear in the size of the underlying graph moreover an application to the security of statistical data is discussed

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

some new matroids on graphs cut sets and the max cut problem we define some new matroids on the edge set of a graph we show that minimal disconnecting edge sets and minimal edge sets which cover all the odd cycles are cocircuits of a binary matroid we give a polynomial algorithm for finding a minimum weight circuit in our matroids a necessary and sufficient condition for an edge set to be the minimum weight odd cycle cover is derived in terms of the family of cut sets of the graph

how fast can one compute the permanent of circulant matrices abstract in this paper we address the problem of computing the permanent of 0 1 circulant matrices we investigate structural properties of circulant matrices showing that i if they are dense enough then they contain large arbitrary submatrices and ii if they are very sparse then they are not too far from convertible matrices building upon ii we then develop an efficient algorithm which allows us to compute permanents of very sparse circulants of size up to 200

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

on the values of permanents of 0 1 circulant matrices with three ones per row abstract starting from previous works concerning permanents of 0 1 circulant matrices with three nonzero entries per row and just using algebraic methods we find some congruences satisfied by the values of such permanents along with simple generalizations to circulants with k ones per row k n k 3 we also give an alternative proof for known relations of fibonacci numbers with some of these values of permanents

on the parity of permanents of circulant matrices abstract starting from previous results concerning determinants and permanents of 0 1 circulant matrices and using theory on finite fields with characteristic 2 we first see denoting by a the n n 0 1 circulant matrix whose 1 s in the first row are exactly in positions i 1 i 2 i k that per a is even iff gcd x i 1 x i 2 x i k 1 x n 1 in z 2 x we then derive some consequences for specific classes of primes n in particular when n is a prime such that the group z n is generated by the residue class 2 we see that for k n and k odd the value per a is always odd similarly for n 7 and n prime with the group of squares in z n generated by the class 2 we see that for k 3 the value per a is always odd contrary to such cases we find that when n is a mersenne prime greater than 3 there are a significant number of circulants a of the above form with k odd and per a even

self avoiding walks attached to triangular and face centred cubic lattice surfaces extended exact enumeration data the analysis of exact enumeration data of self avoiding walks attached to the surfaces of triangular and face centred cubic lattices is consistent with gamma 1 0 718 or 0 008 136 delta pc where delta pc pc 0 0995 in three dimensions and 0 9530 or gamma 1 or 0 9548 in two dimensions where pc for the triangular lattice is assumed to lie in the range 0 240 915 or pc or 0 240 93

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

on the correction to scaling exponent of linear polymers in two dimensions the authors present new results which indicate that the leading correction to scaling exponent in the mean squared end to end distance in two dimensions is the analytic term a potential source of the various correction to scaling terms reported in the literature is pointed out

a new algorithm to enumerate the self avoiding random walk presents a new algorithm for enumerating the self avoiding walk and extend the existing series of 19 terms by one more term on a triangular lattice the leading correction to scaling exponent delta 1 for the mean square end to end distance of the self avoiding walk is found to be less than one

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

minimal invariant sets in a vertex weighted graph a weighting of vertices of a graph is admissible if there exists an edge weighting such that the weight of each vertex equals the sum of weights of its incident edges given an admissible vertex weighting of a graph an invariant set is an edge set such that the sum of the weights of its edges is the same for every edge weighting and a nonempty invariant set is minimal if none of its nonempty proper subsets is an invariant set it is easily seen that every nonempty invariant set is a disjoint union of minimal invariant sets a graphical characterisation of minimal invariant sets in a bipartite graph is known both in the case the vertex weights are reals and in the case the vertex weights are nonnegative reals we shall state a graphical characterisation of minimal invariant sets in an arbitrary vertex weighted graph moreover we give a linear algorithm to test an invariant set for minimality finally we state a complete axiomatisation of invariant sets and give a polynomial algorithm to find a set of minimal invariant sets that completely characterise the set of all invariant sets

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases