swapping labeled tokens on graphs consider a puzzle consisting of n tokens on an n vertex graph where each token has a distinct starting vertex and a distinct target vertex it wants to reach and the only allowed transformation is to swap the tokens on adjacent vertices we prove that every such puzzle is solvable in o n 2 token swaps and thus focus on the problem of minimizing the number of token swaps to reach the target token placement we give a polynomial time 2 approximation algorithm for trees and using this obtain a polynomial time 2 approximation algorithm for graphs whose tree spanners can be computed in polynomial time finally we show that the problem can be solved exactly in polynomial time on complete bipartite graphs

superinduction for pattern groups abstract it is well known that the representation theory of the finite group of unipotent upper triangular matrices u n over a finite field is a wild problem by instead considering approximately irreducible representations supercharacters one obtains a rich combinatorial theory analogous to that of the symmetric group where we replace partition combinatorics with set partitions this paper studies diaconis isaacs concept of superinduction in pattern groups while superinduction shares many desirable properties with usual induction it no longer takes characters to characters we begin by finding sufficient conditions guaranteeing that superinduction is in fact induction it turns out for two natural embeddings of u m in u n superinduction is induction we conclude with an explicit combinatorial algorithm for computing this induction analogous to the pieri formulas for the symmetric group

multiwavelets in the generalized sobolev space hwω rn a class of r regular multiwavelets is introduced with appropriate notation and definitions in h w w r n the oscillation properity of the orthonormal system is obtained a multiresolution analysis for multiwavelets is defined in h w w r n orthonormality conditions for multiscaling functions and multiwavelets are obtained a multiwavelet expansion formula of riesz potentials is given

a construction of multiwavelets a class of r regular multiwavelets depending on the smoothness of the multiwavelet functions is introduced with the appropriate notation and definitions oscillation properties of orthonormal systems are obtained in lemma 1 and corollary 1 without assuming any vanishing moments for the scaling functions and in theorem 1 the existence of r regular multiwavelets in l2 rn is established in theorem 2 a particular r regular multiresolution analysis for multiwavelets is obtained from an r regular multiresolution analysis for uniwavelets in theorem 3 an r regular multiresolution analysis of split type multiwavelets which are perhaps the simplest multiwavelets is easily obtained by using an r regular multiresolution analysis for uniwavelets and a 2n 1 fold regular multiresolution analysis for uniwavelets for some split type multiwavelets the support or width of the wavelets is shorter than the support or width of the scaling functions without loss of regularity nor of vanishing moments examples of split type multiwavelets in l2 r are constructed and illustrated by means of figures symmetry and antisymmetry are preserved in the case of infinite support

dynamic recomputation cannot extend the optimality range of priority indices sequencing rules that rely on priority indices are particularly attractive because they are simple to calculate and easy to implement it has been established that there are exactly two classes of delay cost functions for which policies that are determined by time invariant priority indices are capable of producing optimal sequences linear delay costs and discounted linear delay costs we consider index based policies that allow dynamic recalculation of priority indices and show that this class does not enlarge the set of delay cost functions for which index based rules can produce optimal policies our analysis relies on the argument that index based rules must induce a transitive ranking of the tasks we show that the classes of linear and discounted linear functions are the only ones that can be associated with such rankings

dominance and decomposition heuristics for single machine scheduling new heuristic dominance rules and a flexible decomposition heuristic are developed for the problem of minimizing weighted tardiness on a single processor extensive computational experience demonstrates that when our new heuristic dominance rules were incorporated into an optimal algorithm optimal or nearly optimal solutions were obtained quickly in fact solution times were orders of magnitude faster than those using the optimal algorithm alone on larger problems our decomposition heuristic obtained better solutions than previous heuristics furthermore on 50 job problems our decomposition heuristic obtained an optimal solution over ten times more often on the average than the best competing heuristic 22 versus 2 of the time since both our approaches are basically relaxations of optimal solution algorithms they could easily be adapted for use in the solution of other scheduling problems

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

random generators of the symmetric group diameter mixing time and spectral gap abstract let g h be a random pair of generators of g sym n or g alt n we show that with probability tending to 1 as n a the diameter of g with respect to s g h g 1 h 1 is at most o n 2 log n c and b the mixing time of g with respect to s is at most o n 3 log n c both c and the implied constants are absolute these bounds are far lower than the strongest worst case bounds known in helfgott seress 2013 they roughly match the worst known examples we also give an improved though still non constant bound on the spectral gap our results rest on a combination of the algorithm in babai beals seress 2004 and the fact that the action of a pair of random permutations is almost certain to act as an expander on l tuples where l is an arbitrary constant friedman et al 1998

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

cyclic group blocking polyhedra gomory linear algebra appl 2 451 558 1969 gave a subadditive characterization of the facets of the group polyhedron although there are exponentially many facets see gomory and johnson in math program 3 359 389 1972 example 4 6 and exponentially many vertices for the group polyhedron for the master cyclic group problem gomory s characterization of the non trivial facets has polynomially many subadditive inequalities in fact of order g 2 for a finite abelian group g we reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non trivial facets the system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed our minimal representation leads to a characterization of the vertices of length three gomory et al math program 96 321 339 2003 introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets we develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples hunsaker 2003 gave a knapsack shooting experiment which we use to find a simple pattern for the most hit knapsack facets

quantum data hiding we expand on our work on quantum data hiding hiding classical data among parties who are restricted to performing only local quantum operations and classical communication locc we review our scheme that hides one bit between two parties using bell 1964 states and we derive upper and lower bounds on the secrecy of the hiding scheme we provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme we give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement a candidate data hiding scheme that does not use entanglement is presented we show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme

random walks on the groups of upper triangular matrices this paper gives sharp bounds on the eigenvalues of a natural random walk on the group of upper triangular n times n matrices over the field of characteristic p an odd prime with 1 s on the diagonal in particular this includes the finite heisenberg groups as a special case as a consequence we get bounds on the time required to achieve randomness for these walks some of the steps are done using the geometric bounds on the eigenvalues of diaconis and stroock however the crucial step is done using more subtle and idiosyncratic techniques we bound the eigenvalues inductively over a sequence of subspaces

divisibility and laws in finite simple groups we provide new bounds for the divisibility function of the free group mathbf f 2 and construct short laws for the symmetric groups mathrm sym n the construction is random and relies on the classification of the finite simple groups we also give bounds on the length of laws for finite simple groups of lie type

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

super character theory and comparison arguments for a random walk on the upper triangular matrices abstract consider the random walk on the n n upper triangular matrices with ones on the diagonal and elements over f p where we pick a row at random and either add it or subtract it from the row directly above it the main result of this paper is to prove that the dependency of the mixing time on p is p 2 this is proven by combining super character theory and comparison theory arguments

on the cost of interchange rearrangement in strings consider the following optimization problem given two strings over the same alphabet transform one into another by a succession of interchanges of two elements in each interchange the two participating elements exchange positions an interchange is given a weight that depends on the distance in the string between the two exchanged elements the object is to minimize the total weight of the interchanges this problem is a generalization of a classical problem on permutations where every element appears once the generalization considers general strings with possibly repeating elements and a function assigning weights to the interchanges the generalization to general strings with unit weights was mentioned by cayley in the 19th century and its complexity has been an open question since we solve this open problem and consider various weight functions as well

stein s method and plancherel measure of the symmetric group we initiate a stein s method approach to the study of the plancherel measure of the symmetric group a new proof of kerov s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained the proof gives an error term the construction of an exchangeable pair needed for applying stein s method arises from the theory of harmonic functions on bratelli diagrams we also find the spectrum of the markov chain on partitions underlying the construction of the exchangeable pair this yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group

sorting of permutations by cost constrained transpositions the problem of finding a minimum decomposition of a permutation in terms of transpositions with predetermined non uniform and non negative costs is addressed alternatively computing the transposition distance between two permutations where transpositions are endowed with arbitrary non negative costs is studied for such cost functions polynomial time constant approximation decomposition algorithms are described for metric path costs exact polynomial time decomposition algorithms are presented the algorithms in this paper represent a combination of viterbi type algorithms and graph search techniques for minimizing the cost of individual transpositions and dynamic programing algorithms for finding minimum cost decompositions of cycles the presented algorithms have a myriad of applications in information theory bioinformatics and algebra

nilprogressions and groups with moderate growth abstract we show that doubling at some large scale in a cayley graph implies uniform doubling at all subsequent scales the proof is based on the structure theorem for approximate subgroups proved by green tao and the first author we also give a number of applications to the geometry and spectrum of finite cayley graphs for example we show that a finite group has moderate growth in the sense of diaconis and saloff coste if and only if its diameter is larger than a fixed power of the cardinality of the group we call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter we also give bounds on the cheeger constant first eigenvalue of the laplacian and mixing time this can be seen as a finite group version of gromov s theorem on groups with polynomial growth it also improves on a result of lackenby regarding property in towers of coverings another consequence is a universal upper bound on the diameter of all finite simple groups independent of the cfsg

applications of complex valued wavelet transforms to subband decomposition a method for constructing complex valued linear phase fir conjugate quadrature filters and associated wavelet bases is described each filter is derived by replating certain zeros of a real valued fir conjugate quadrature filter by their reciprocal conjugates the derived filters have the same frequency response magnitudes as the original filters and their linear phase property permits the use of symmetrization in subband decomposition to avoid border discontinuities that result from signal periodization subband decomposition and reconstruction using both a length 6 filter associated with a daubechies 1988 wavelet bases and a related length 6 complex valued linear phase filter are compared to illustrate the reduced border effects

geometric analysis for the metropolis algorithm on lipschitz domains this paper gives geometric tools comparison nash and sobolev inequalities for pieces of the relevant markov operators that give useful bounds on rates of convergence for the metropolis algorithm as an example we treat the random placement of n hard discs in the unit square the original application of the metropolis algorithm

computing short generator sequences abstract the diameter of a group is the length of the longest product of generators required to reach a group element we show that the diameter of a permutation group of degree n generated by cycles of bounded degree is o n 2 this bound is the best possible additionally such short products can be found in polynomial time the techniques presented here may be applied to many permutation group puzzles such as alexander s star the hungarian rings rubik s cube and some of their generalizations

the random k cycle walk on the symmetric group we study the random walk on the symmetric group s n generated by the conjugacy class of cycles of length k we show that the convergence to uniform measure of this walk has a cut off in total variation distance after frac n k log n steps uniformly in k o n as n rightarrow infty the analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

new results for single machine scheduling with past sequence dependent setup times and due date related objectives abstract we show that the single machine scheduling problem with past sequence dependent p s d setup times and either the minimum maximum lateness or the minimum maximum tardiness objective is solvable in o n2 time by an index priority rule followed by backward insertions of certain qualifying jobs we also show that the problem with the minimum number of tardy jobs objective is solvable in o n2 time by a variant of moore s algorithm we then show how to modify a general purpose dynamic programming algorithm to solve the problem with other due date related objectives such as the total weighted tardiness the weighted number of tardy jobs and in the case of an unrestricted common due date the total weighted earliness tardiness we also present heuristic decomposition algorithms for the np hard scheduling problems with p s d setup times and the objectives of total weighted tardiness weighted number of tardy jobs and total weighted earliness tardiness around an unrestricted common due date these algorithms decompose the respective problem heuristically into smaller sub problems which are then solved optimally by dynamic programming we show experimentally that the heuristic performs well with extended job due dates close to the makespan because in those cases the heuristic is capable of locating the optimal solution with either zero total weighted tardiness or with only a few tardy jobs in the remaining cases the quality of the ds solutions is superior with normally distributed data compared to the corresponding solutions with uniformly distributed data this is because sampling from the normal distribution reduces the variability among processing times and weights which in turn decreases the hardness of the problem finally we show that the heuristic s performance deteriorates as the problem size increases

a decomposition algorithm for sequencing with general precedence constraints many important unconstrained job sequencing problems for which optimal algorithms exist satisfy the adjacent job interchange property in recent years a more general property the adjacent sequence interchange property has been identified and shown to be a sufficient condition for the existence of efficient algorithms for sequencing problems with series parallel precedence constraints in this paper additional properties are defined which are sufficient to prove the existence of decomposition algorithms for general precedence structures these results are applicable to several problems including the two machine maximum flow time problem the maximum cumulative cost problem the least cost fault detection problem and the total weighted exponential completion time problem and represent an extension of sidney s results on the weighted completion time problem

multiple genome rearrangement by swaps and by element duplications we consider the swap distance and the element duplication distance we show that the swap centre permutation problem is np complete we show that the element duplication centre problem is np complete

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

universality of cutoff for the ising model on any locally finite geometry the stochastic ising model is known to be contractive when the inverse temperature beta is small enough via classical results of dobrushin and of holley in the 1970s by a general principle proposed by peres the dynamics is then expected to exhibit cutoff however so far cutoff for the ising model has been confirmed mainly for lattices heavily relying on amenability and log sobolev inequalities without these cutoff was unknown at any fixed beta 0 no matter how small even in basic examples such as the ising model on a binary tree or a random regular graph we use the new framework of information percolation to show that in any geometry there is cutoff for the ising model at high enough temperatures precisely on any sequence of graphs with maximum degree d the ising model has cutoff provided that beta kappa d for some absolute constant kappa a result which up to the value of kappa is best possible moreover the cutoff location is established as the time at which the sum of squared magnetizations drops to 1 and the cutoff window is o 1 just as when beta 0 finally the mixing time from almost every initial state is not more than a factor of 1 varepsilon beta faster then the worst one with varepsilon beta to0 as beta to0 whereas the uniform starting state is at least 2 varepsilon beta times faster

a computer search for simple multi dimensional wavelets the theory of multi dimensional wavelets is described in general terms then all the simple wavelets for a given three dimensional multiresolution analysis with a real valued function m0 multiplied by complex exponentials are exhibited by means of a computer search it is also proved by an exhaustive computer search that no such simple wavelets exist in dimension four and by an inductive argument nor in higher dimensions

tensor product markov chains abstract we analyze families of markov chains that arise from decomposing tensor products of irreducible representations this illuminates the burnside brauer theorem for building irreducible representations the mckay correspondence and pitman s 2 m x theorem the chains are explicitly diagonalizable and we use the eigenvalues eigenvectors to give sharp rates of convergence for the associated random walks for modular representations the chains are not reversible and the analytical details are surprisingly intricate in the quantum group case the chains fail to be diagonalizable but a novel analysis using generalized eigenvectors proves successful

analysis of the temporal decomposition procedure for scheduling with release and due dates this paper first presents an improved method of temporal decomposition for minimising the searching space of scheduling problems next the effects of the temporal decomposition procedure in scheduling problems are analysed it is theoretically shown that the complexity of a scheduling algorithm using decomposed subsets varies inversely with that of the decomposition procedure therefore the efficiency of the overall scheduling algorithm is strongly related to the decomposability of the set of operations to be processed on each machine this decomposability is evaluated using a probabilistic approach where the probability distributions of the scheduling parameters are obtained from historical workshop data the average number of decomposed subsets and the average size of these subsets are estimated both theoretical analysis and simulation results have revealed that the decomposition procedure leads to optimal effects when some conditions on scheduling parameters are met

walks on generating sets of groups we study a markov chain on generating n tuples of a fixed group which arises in algorithms for manipulating finite groups the main tools are comparison of two markov chains on different but related state spaces and combinatorics of random paths the results involve group theoretical parameters such as the size of minimal generating sets the number of distinct generating k tuples for different k s and the maximal diameter of the group

convergence properties of the gibbs sampler for perturbations of gaussians 1993 and smith and roberts 1993 a host of other applications of these methods in the context of hierarchical bayesian models is described in all these examples the question arises as to the rate of convergence of the markov chain there is no unified theory to answer this question especially in the case of continuous densities on r n and currently many different approaches are being attempted for the different types of simulation methods one direction of research uses doeblin type bounds on bounded sets together with geometric drift conditions or some form of lyapounov functions to obtain rather general results on exponential convergence for continuous n densities on r see mengersen and tweedie 1996 for the metropolis algorithm and rosenthal 1995 for the gibbs sampler these results are then applied to certain special cases and yield bounds on the rate of convergence in variational norm distance another approach analyzes a discretized version of the problem using general discrete state space markov chain techniques see frieze kannan and polson 1994 finally there are attempts to use spectral analysis to obtain rates of convergence in the l sense where the markov chain operator is considered 2 as a bounded operator in the hilbert space of square integrable functions

microlocal filtering with multiwavelets abstract hyperfunctions in r n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in c n orthonormal multiwavelets which are a generalization of orthonormal single wavelets generate a multiresolution analysis by means of several scaling functions microlocal analysis is briefly reviewed and a multiwavelet system adapted to microlocal filtering is proposed a rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions a fast algorithm for multiwavelet microlocal filtering is presented and several numerical examples are considered

sorting permutations with transpositions in o n3 amortized time abstract the problem of sorting a permutation over alphabet 1 2 3 n with the minimum number of transpositions is known to be intractable that is the computation of transposition distance i e d t is hard such sorting is equivalent to determining the distance between any of the symmetric group sn over under the transposition operation it has applications in computer interconnection networks where a node is denoted by a permutation and genetics where a genome is modeled by a permutation the maximum distance between any pair of permutations in sn is the diameter i e diam s n of the corresponding cayley graph s n it corresponds to the maximum latency in the computer interconnection network modeled by s n and the maximum genetic dissimilarity between any pair in sn we call transpositions prefix transpositions suffix transpositions and prefix suffix transpositions as block moves we design an algorithm to compute d t in sn and compute diam s n with transpositions in o n n 3 time and o n n 2 space at an amortized time of o n 3 the same can be computed for the remaining block moves in o n 2 log n amortized time and o n n space

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

mcclellan transformation and the construction of biorthogonal wavelet bases of l2r2 bidimensional wavelet bases are constructed by means of mcclellan s transformation applied to a pair of one dimensional biorthogonal wavelet filters it is shown that under some conditions on the transfer function f 1 2 associated to the mcclellan transformation and on the dilation matrix d it is possible to construct symmetric compactly supported biorthogonal wavelet bases of l2 r2 finally the construction method is illustrated by means of numerical examples

short wavelets and matrix dilation equations scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter in a two band orthogonal filter bank for multifilters those coefficients are matrices this gives a new block structure for the filter bank and leads to multiple scaling functions and wavelets geronimo hardin and massopust see j approx theory vol 78 p 373 401 1994 constructed two scaling functions that have extra properties not previously achieved the functions spl phi sub 1 and spl phi sub 2 are symmetric linear phase and they have short support two intervals or less while their translates form an orthogonal family for any single function spl phi apart from haar s piecewise constants those extra properties are known to be impossible the novelty is to introduce 2 spl times 2 matrix coefficients while retaining orthogonality of the multiwavelets this note derives the properties of spl phi sub 1 and spl phi sub 2 from the matrix dilation equation that they satisfy then our main step is to construct associated wavelets two wavelets for two scaling functions the properties were derived by geronimo hardin and massopust from the iterated interpolation that led to spl phi 1 and spl phi sub 2 one pair of wavelets was found earlier by direct solution of the orthogonality conditions using mathematica our construction is in parallel with recent progress by hardin and geronimo to develop the underlying algebra from the matrix coefficients in the dilation equation in another language to build the 4 spl times 4 paraunitary polyphase matrix in the filter bank the short support opens new possibilities for applications of filters and wavelets near boundaries

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc