generalized domino shuffling the problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting perfect matchings of some subgraph of an aztec diamond graph an or more generally calculating the sum of the weights of all the matchings where the weight of a matching is equal to the product of the pre assigned weights of the constituent edges assumed to be non negative this article presents efficient algorithms that work in this context to solve three problems finding the sum of the weights of the matchings of a weighted aztec diamond graph an computing the probability that a randomly chosen matching of an will include a particular edge where the probability of a matching is proportional to its weight and generating a matching of an at random the first of these algorithms is equivalent to a special case of mihai ciucu s cellular complementation algorithm j combin theory ser a 81 1998 34 and can be used to solve many of the same problems the second of the three algorithms is a generalization of not yet published work of alexandru ionescu and can be employed to prove an identity governing a three variable generating function whose coefficients are all the edge inclusion probabilities this formula has been used duke math j 85 1996 117 as the basis for asymptotic formulas for these probabilities but a proof of the generating function identity has not hitherto been published the third of the three algorithms is a generalization of the domino shuffling algorithm presented in j algebraic combin 1 1992 111 it enables one to generate random diabolo tilings of fortresses and thereby to make intriguing inferences about their asymptotic behavior

a numerical analysis of a model of growth tumor in this paper we study a free boundary problem modeling the growth of tumors the model uses the conventional ideas of nutrient diffusion and consumption by the cells we consider the radially symmetric case of this free boundary problem we apply a spectral numerical method to the system of equations

a quadratic identity for the number of perfect matchings of plane graphs we present a quadratic identity on the number of perfect matchings of plane graphs by the method of graphical condensation which generalizes the results found by propp j propp generalized domino shuffling theoret comput sci 303 2003 267 301 kuo e h kuo applications of graphical condensation for enumerating matchings and tilings theoret comput sci 319 2004 29 57 and yan yeh and zhang w g yan y n yeh f j zhang graphical condensation of plane graphs a combinatorial approach theoret comput sci 349 2005 452 461

bounds on treatment effects from studies with imperfect compliance abstract this article establishes nonparametric formulas that can be used to bound the average treatment effect in experimental studies in which treatment assignment is random but subject compliance is imperfect the bounds provided are the tightest possible given the distribution of assignments treatments and responses the formulas show that even with high rates of noncompliance experimental data can yield useful and sometimes accurate information on the average effect of a treatment on the population

graphical condensation of plane graphs a combinatorial approach the method of graphical vertex condensation for enumerating perfect matchings of plane bipartite graph was found by propp generalized domino shuffling theoret comput sci 303 2003 267 301 and was generalized by kuo applications of graphical condensation for enumerating matchings and tilings theoret comput sci 319 2004 29 57 and yah and zhang graphical condensation for enumerating perfect matchings j combin theory ser a 110 2005 113 125 in this paper by a purely combinatorial method some explicit identities on graphical vertex condensation for enumerating perfect matchings of plane graphs which do not need to be bipartite are obtained as applications of our results some results on graphical edge condensation for enumerating perfect matchings are proved and we count the sum of weights of perfect matchings of weighted aztec diamond

dimers belonging to three orientations on plane honeycomb lattices it is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice and in each type all dimers are mutually parallel based on a previous result we can compute the partition function of the dimer problem of the plane free boundary honeycomb lattices with three different activities by using the number of its pure dimer coverings perfect matchings the explicit expression of the partition function and free energy per dimer for many types of plane honeycomb lattices with fixed shape of boundaries is obtained in this way for a shape of plane honeycomb lattices the procedure that the size goes to infinite corresponds to a way that the honeycomb lattice goes to infinite from these results an interesting phenomena is observed in the case of the regions of the plane honeycomb lattice has zero entropy per dimer when its size goes to infinite though in the thermodynamic limit there is no freedom in placing a dimer at all but if we distinguish three types of dimers with nonzero activities then its free energy per dimer is nonzero furthermore a sufficient condition for the plane honeycomb lattice with zero entropy per dimer when the three activities are equal to 1 is obtained finally the difference between the plane honeycomb lattices and the plane quadratic lattices is discussed and a related problem is proposed

empirical and counterfactual conditions for sufficient cause interactions summary sufficient component causes are discussed within the deterministic potential outcomes framework so as to formalize notions of sufficient causes synergism and sufficient cause in teractions doing so allows for the derivation of counterfactual and empirical conditions for detecting the presence of sufficient cause interactions the conditions are novel in that unlike other conditions in the literature they make no assumption about monotonicity the conditions can also be generalized and the conditions for three way sufficient cause interactions are given explicitly the statistical tests derived for sufficient cause interactions are compared with and contrasted to interaction terms in standard statistical models

maximum matchings and minimum dominating sets in apollonian networks and extended tower of hanoi graphs abstract the apollonian networks display the remarkable power law and small world properties as observed in most realistic networked systems their dual graphs are extended tower of hanoi graphs which are obtained from the tower of hanoi graphs by adding a special vertex linked to all its three extreme vertices in this paper we study analytically maximum matchings and minimum dominating sets in apollonian networks and their dual graphs both of which have found vast applications in various fields e g structural controllability of complex networks for both networks we determine their matching number domination number the number of maximum matchings as well as the number of minimum dominating sets

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

likelihood based inference for bounds of causal parameters it is a common causal inference problem that even with theoretically infinite samples we might be able to only provide bounds for the parameters of interest this problem occurs naturally for example in estimating causal interaction between two risk factors and in estimating the average causal effect using the instrumental variable or mendelian randomization method current procedures include linear programming to get the estimated bounds plus bootstrapping to get confidence intervals we describe a likelihood based procedure that automatically yields the interval estimate from the flat likelihood region and show some theory that allows us to construct confidence intervals from this non regular likelihood finally we illustrate the procedure with examples from the estimation of causal interaction between two risk factors and the treatment effect under partial compliance

cubic spline for a class of non linear singular boundary value problems arising in physiology abstract cubic spline method to analyze a class of non linear singular boundary value problems defined by y x m x y f x y y 0 0 y 1 y 1 is presented the quesilinearization technique is used to reduce the given non linear problem to a sequence of linear problems the resulting set of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution the numerical method is tested for its efficiency by considering two examples from physiology

applications of graphical condensation for enumerating matchings and tilings a technique called graphical condensation is used to prove various combinatorial identities among numbers of perfect matchings of planar bipartite graphs and tilings of regions graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph and then re partitioning the united matching actually a multigraph into matchings of two other subgraphs in one of two possible ways this technique can be used to enumerate perfect matchings of a wide variety of planar bipartite graphs applications include domino tilings of aztec diamonds and rectangles diabolo tilings of fortresses plane partitions and transpose complement plane partitions

bounds on causal interactions for binary outcomes summary a common goal of epidemiologic research is to study how two exposures interact in causing a binary outcome causal interaction is defined as the presence of subjects for which the causal effect of one exposure depends on the level of the other exposure for binary exposures it has previously been shown that the presence of causal interaction is testable through additive statistical interaction however it has also been shown that the magnitude of causal interaction defined as the proportion of subjects for which there is causal interaction is generally not identifiable in this article we derive bounds on causal interactions which are applicable to binary outcomes and categorical exposures with arbitrarily many levels these bounds can be used to assess the magnitude of causal interaction and serve as an important complement to the statistical test that is frequently employed the bounds are derived both without and with an assumption about monotone exposure effects we present an application of the bounds to a study of gene gene interaction in rheumatoid arthritis

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

symmetry classes of alternating sign matrices under one roof in a previous article math co 9712207 we derived the alternating sign matrix asm theorem from the izergin korepin determinant for a partition function for square ice with domain wall boundary here we show that the same argument enumerates three other symmetry classes of alternating sign matrices vsasms vertically symmetric asms even htsasms half turn symmetric asms and even qtsasms quarter turn symmetric asms the vsasm enumeration was conjectured by mills the others by robbins math co 0008045 we introduce several new types of asms uasms asms with a u turn side uuasms two u turn sides osasms off diagonally symmetric asms oosasms off diagonally off antidiagonally symmetric and uosasms off diagonally symmetric with u turn sides uasms generalize vsasms while uuasms generalize vhsasms vertically and horizontally symmetric asms and another new class vhpasms vertically and horizontally perverse osasms oosasms and uosasms are related to the remaining symmetry classes of asms namely dsasms diagonally symmetric dasasms diagonally anti diagonally symmetric and tsasms totally symmetric asms we enumerate several of these new classes and we provide several 2 enumerations and 3 enumerations our main technical tool is a set of multi parameter determinant and pfaffian formulas generalizing the izergin korepin determinant for asms and the tsuchiya determinant for uasms solv int 9804010 we evaluate specializations of the determinants and pfaffians using the factor exhaustion method

accuracy and speed in computing the chebyshev collocation derivative we study several algorithms for computing the chebyshev spectral derivative and compare their roundoff error for a large number of collocation points the elements of the chebyshev differentiation matrix if constructed in the usual way are not computed accurately a subtle cause is found to account for the poor accuracy when computing the derivative by the matrix vector multiplication method methods for accurately computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately the roundoff error of the matrix vector multiplication is as small as that of the transform recursion algorithm furthermore results of the cpu time usage are shown for several different algorithms for computing the derivative by the chebyshev collocation method for a wide variety of two dimensional grid sizes on both an ibm mainframe and a cray 2 computer we find that which algorithm is fastest on a particular machine depends not only on the grid size but also on small

a simple proof of an identity concerning pfaffians of skew symmetric matrices abstract in the present paper we prove an identity concerning pfaffians similar to the well known grassmann plucker relations for determinants using tools from multilinear algebra more precisely we shall derive the identity as a corollary to an equation concerning skew symmetric bilinear forms and operators acting on the exterior algebra m of an r module m over some fixed commutative ring r with 1 r

a linearisation method for non linear singular boundary value problems this paper presents a novel numerical approach for the solution of a class of nonlinear singular boundary value problems arising in physiology the approach is based on a new application of the successive linearisation method slm three illustrative examples are presented to demonstrate the effectiveness of the proposed method the new approach is found to give accurate results comparable to results in the literature found using existing numerical methods

maximum matchings in scale free networks with identical degree distribution the size and number of maximum matchings in a network have found a large variety of applications in many fields as a ubiquitous property of diverse real systems power law degree distribution was shown to have a profound influence on size of maximum matchings in scale free networks where the size of maximum matchings is small and a perfect matching often does not exist in this paper we study analytically the maximum matchings in two scale free networks with identical degree sequence and show that the first network has no perfect matchings while the second one has many for the first network we determine explicitly the size of maximum matchings and provide an exact recursive solution for the number of maximum matchings for the second one we design an orientation and prove that it is pfaffian based on which we derive a closed form expression for the number of perfect matchings moreover we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended sierpiski graph with the same average degree as both studied scale free networks our results indicate that power law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale free networks

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method