faster calculation of the percolation correlation length on spatial networks the divergence of the correlation length at criticality is an important phenomenon of percolation in two dimensional systems substantial speed ups to the calculation of the percolation threshold and component distribution have been achieved by utilizing disjoint sets but existing algorithms of this sort cannot measure the correlation length here we utilize the parallel axis theorem to track the correlation length as nodes are added to the system allowing us to utilize disjoint sets to measure for the entire percolation process with arbitrary precision in a single sweep this algorithm enables direct measurement of the correlation length in lattices as well as spatial network topologies and provides an important tool for understanding critical phenomena in spatial systems

the application of the enhanced hoshen kopelman algorithm for processing unbounded images the enhanced hoshen kopelman ehk algorithm for the analysis of connected components in images that are unbounded in one of their dimensions is introduced the algorithm characterizes in a single pass the shapes of all the connected components in a multiple class image by computing the spatial moments the area the boundary and the bounding boxes of the connected components the algorithm is applied to a real time surface defect simulation and to a landsat image analysis the algorithm s performance is compared to the performances of related algorithms

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

completely reducible operatorname sl 2 homomorphisms keywords semisimple groups group scheme homomorphisms frobenius twists optimal homomorphisms geometrically completely reducible images twisted product homomorphisms cocharacters reference ctg article 2007 002doi 10 1090 s0002 9947 07 04289 4 record created on 2008 12 16 modified on 2017 05 12

an algorithm for detecting percolating structures in periodic systems application to polymer networks we consider the problem of detecting a percolating structure in an off lattice model polymer system when periodic boundary conditions are used physically with increasing polymer density the point at which this first occurs is the gel point a connected structure spans all space and the system becomes solid like this problem is similar to that of finding connected paths in lattice bond percolation and site percolation models we show that algorithms that detect these structures in finite systems will not always yield the correct answer for a system that is periodically repeated in space uses periodic boundary conditions here we describe an algorithm that detects clusters that connect to themselves over an arbitrary number of periodic replicas because of the periodic replication this means that they are connected over all space for system sizes that are typically tractable in simulations we find a relatively minor proportion of configurations are mis classified for a lattice model however the fraction that is mis classified is significant for the polymer system mis classified configurations when included in the calculation of ensemble averages will include configurations with spurious physical properties our algorithm allows this to be corrected for

double centralizers of unipotent elements in simple algebraic groups of type g2 f4 and e6 let u be a unipotent element of a simple algebraic group g over a field k of characteristic p we develop a method for computing the connected component of z c g u in the cases where p is positive and both it and the rank of g are small enough the method is then carried out for g of type g 2 f 4 and e 6 in bad characteristic c 2013 elsevier inc all rights reserved

on exponentiation and infinitesimal one parameter subgroups of reductive groups abstract let g be a reductive algebraic group over an algebraically closed field k of characteristic p 0 and assume p is good for g let p be a parabolic subgroup with unipotent radical u for r 1 denote by g a r the r th frobenius kernel of g a we prove that if the nilpotence class of u is less than p then any embedding of g a r in u lies inside a one parameter subgroup of u and there is a canonical way in which to choose such a subgroup applying this result we prove that if p is at least as big as the coxeter number of g then the cohomological variety of g r is homeomorphic to the variety of r tuples of commuting elements in n 1 g the p nilpotent cone of lie g

extension of hoshen kopelman algorithm to non lattice environments abstract we extend the hoshen kopelman hk algorithm for cluster labeling to non lattice environments where sites are placed at random at non lattice points this extension is useful for continuum systems and disordered networks our extension of the hk algorithm relies on several data structures that describe network connectivity regardless of its dimensionality just as for the classic hk algorithm on lattices our extension is completed in a single pass through the sites of the network and cluster relabeling operates on a vector whose size is much smaller than the size of the network our extension of the hk algorithm works for any environment lattice or non lattice of any dimensionality type sites bonds or both and with arbitrary connectivity between the sites the proposed extension is illustrated through a simple network consisting of 16 sites and 24 bonds and applied to a complex network extracted from a 3d micro focused x ray ct image of bentheimer sandstone consisting of 3677 sites and 8952 bonds

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

unipotent centralizers in algebraic groups let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 where p is a good prime for g and let u g be a unipotent element we study the embedding of u in abelian unipotent subgroups and certain reductive subgroups of g and use this information to obtain information on cg u of particular interest are z cg u and the reductive part of cg u the process of embedding a unipotent element u g in a connected abelian unipotent group is sometimes called saturation in previous work saturation results were achieved when u p where it was shown that there is a unique 1dimensional unipotent group the saturation of u which contains u and which is contained in a restricted sometimes called good a1 subgroup of g here an a1 subgroup of g is restricted if all weights on the adjoint module of g are at most 2p 2 it was shown in that these a1 subgroups provide the basis for a variant of the steinberg tensor product theorem hence the name it was also shown in that cg u can be factored as a product of the unipotent radical and the centralizer of a restricted a1 subgroup containing u in this paper we extend these results to cover unipotent elements of arbitrary order we describe a certain 1 dimensional torus which normalizes cg u and then decompose z cg u into indecomposable summands each invariant under the torus and such that u is contained in one of the summands say w if u p p we show that the saturation of up described above coincides with the subgroup of elements of order at most p in w we next establish the existence of a pair of reductive groups j r such that each is the centralizer of the other u is a semiregular unipotent element of j and r is the reductive part of cg u these subgroups provide insight into a number of issues surrounding unipotent elements for instance the reductive part of cg u appears as the centralizer of a reductive group and this is helpful

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

unipotent elements tilting modules and saturation the results in this paper were motivated by work of serre 19 where semisimplicity results for representations of arbitrary groups in positive characteristic were established a key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1 dimensional unipotent subgroup one of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups as a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type a1 and that the lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup the tilting decompositions are applied to show that the centralizers of the unipotent element a corresponding 1 dimensional unipotent group and its lie algebra all coincide and are closely related to the centralizer of the a1 subgroup we also establish a convenient factorization of the centralizer of the unipotent element results for finite groups of lie type are also obtained

exact solution of site and bond percolation on small world networks we study percolation on small world networks which has been proposed as a simple model of the propagation of disease the occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the disease and the transmissibility of the disease respectively we give an exact solution of the model for both site and bond percolation including the position of the percolation transition at which epidemic behavior sets in the values of the critical exponents governing this transition the mean and variance of the distribution of cluster sizes disease outbreaks below the transition and the size of the giant component epidemic above the transition

invaded cluster algorithm for potts models the invaded cluster algorithm a method for simulating phase transitions is described in detail theoretical albeit nonrigorous justification of the method is presented and the algorithm is applied to potts models in two and three dimensions the algorithm is shown to be useful for both first order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities the dynamic properties of the invaded cluster algorithm are studied numerical evidence suggests that the algorithm has no critical slowing for ising models s1063 651x 96 09208 2

critical percolation on the torus we compute the various crossing probabilities defined by r langlands p pouliot and y saint aubin for the critical percolation on the torus

percolation and cluster structure parameters the radius of gyration the squared radius of gyration of percolation clusters are determined in terms of the clusters second spatial moments using the enhanced hoshen kopelman algorithm for square and triangular lattices containing sites correlation length exponents and related exponents as well as their corresponding amplitudes are calculated in terms of the squared radius of gyration above and below the percolation threshold a coefficient for cluster compactness that is based on the squared radius of gyration of a cluster is introduced that coefficient is compared with a similar coefficient of compactness that is based on the cluster cyclomatic number

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

percolation as a dynamical phenomenon abstract we consider a percolation process where the probability p of having one site or bond occupied increases linearly with time we study the total number of clusters as a function of time or p the statistical distribution of jumps in the size of the major cluster as well as the frequency of these jumps we find that both distributions are power laws with different exponents below and above percolation threshold and we discuss these results

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

geometric invariants of representations of finite groups abstract for a finite group and a field k of characteristic p dividing the order of we construct a map p u p y of varieties over k with unions of affine spaces as fibers and with p y p isogenous to proj h k on u we construct a universal p nilpotent operator p which leads to the construction of equivariant coherent sheaves associated to finite dimensional k modules m these coherent sheaves are algebraic vector bundles on u if m has constant jordan type for any finite dimensional k module m these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of m for a linear algebraic group g we compare these constructions for the finite group g f p to previous constructions for the infinitesimal group scheme g r our comparison of bundle invariants for a rational g module m of exponential type p r upon restriction to g f p and to g r is a strengthened form of earlier comparisons of support varieties for m our constructions extend to semi direct products g of an infinitesimal group scheme g and a finite group thus to all finite group schemes if k is algebraically closed

involutions of reductive lie algebras in positive characteristic abstract let g be a reductive group over a field k of characteristic 2 let g lie g let be an involutive automorphism of g and let g k p be the associated symmetric space decomposition for the case of a ground field of characteristic zero the action of the isotropy group g on p is well understood since the well known paper of kostant and rallis b kostant s rallis orbits and representations associated with symmetric spaces amer j math 93 1971 753 809 such a theory in positive characteristic has proved more difficult to develop here we use an approach based on some tools from geometric invariant theory to establish corresponding results in good positive characteristic among other results we prove that the variety n of nilpotent elements of p has a dense open orbit and that the same is true for every fibre of the quotient map p p g however we show that the corresponding statement for g conjectured by richardson is not true we provide a new mostly calculation free proof of the number of irreducible components of n extending a result of sekiguchi for k c finally we apply a theorem of skryabin to describe the infinitesimal invariants k p k

subgroups of type a1 containing a fixed unipotent element in an algebraic group keywords simple algebraic groups unipotent elements subgroups of type a 1 frobenius morphisms graph morphisms finite groups of lie type reference ctg article 2000 003doi 10 1006 jabr 2000 8353 record created on 2008 12 16 modified on 2017 05 12

weil restriction and support varieties this paper extends the comparison results for cohomological support varieties for chevalley groups of the form g fp established by j carlson z lin and d nakano to chevalley groups of the form g fpd other topics considered are complexity and non maximal support varieties for modules techniques developed by j pevtsova and the author together with the use of weil restriction functors are employed

percolation thresholds for discrete continuous models with nonuniform probabilities of bond formation this work has been partially supported by the iuventus plus programme founded by the polish ministry of science and higher education ip2014 024373 m l acknowledges spanish mineco project foqus fis2013 46768 erc adg osyris eu ip siqs eu strepequam and eu fetproact quic j w has been partially funded by nsf grant dms 131271

cluster diversity and entropy on the percolation model the lattice animal identification algorithm we present an algorithm to identify and count different lattice animals la s in the site percolation model this algorithm allows a definition of clusters based on the distinction of cluster shapes in contrast with the well known hoshen kopelman algorithm in which the clusters are differentiated by their sizes it consists in coding each unit cell of a cluster according to the nearest neighbors nn and ordering the codes in a proper sequence in this manner a la is represented by a specific code sequence in addition with some modification the algorithm is capable of differentiating between fixed and free la s the enhanced hoshen kopelman algorithm j hoshen m w berry and k s minser phys rev e 56 1455 1997 is used to compose the set of nn code sequences of each cluster using monte carlo simulations on planar square lattices up to 2000x2000 we apply this algorithm to the percolation model we calculate the cluster diversity and cluster entropy of the system which leads to the determination of probabilities associated with the maximum of these functions we show that these critical probabilities are associated with the percolation transition and with the complexity of the system

scaling and critical probabilities for cluster size and la diversity on randomly occupied square lattices using monte carlo simulations we report the behaviour of the total number of clusters the cluster size diversity and the lattice animals la diversity on randomly occupied square lattices the critical probability associated with the maximum of these variables is determined in comparison with the percolation probability pc our simulations indicate that pc and the critical probability of the maximum cluster size diversity pc d s max occur at the same point as indicated in a previous paper tsang i r and tsang i j 1997 j phys a math gen 30 l239 the probability for the maximum number of clusters is obtained at a lower value pc n max 0 27 0 01 we describe the cluster identification algorithm used to count the different la and determined the critical probability for the maximum la diversity pc d f max 0 45 0 02 we derive the exponents characterizing the relation between d s max d f max and n max and several scaling relations between the variables measured the lattice size and the probability of occupation p in addition we show the scaling behaviour of la diversity versus cluster size diversity for each value of p

complete reducibility in good characteristic let g be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p 0 a closed subgroup h of g is called g completely reducible g cr if whenever h is contained in a parabolic subgroup p of g it is contained in a levi subgroup of p in this paper we determine the g conjugacy classes of non g cr simple connected subgroups of g when p is good for g for each such subgroup x we determine the action of x on the adjoint module l g and the connected centraliser of x in g as a consequence we classify all non g cr reductive subgroups of g and we additionally show the existence of and classify the following restricted non g cr semisimple subgroups non g cr reductive subgroups with trivial connected centraliser and subgroups which are maximal among reductive subgroups of g

winding clusters in percolation on the torus and the mobius strip using a simulation technique introduced recently we study winding clusters in percolation on the torus and the mobius strip for different aspect ratios the asynchronous parallelization of the simulation makes very large system and sample sizes possible our high accuracy results are fully consistent with predictions from conformal field theory the numerical results for the mobius strip and the number distribution of winding clusters on the torus await theoretical explanation to our knowledge this study is the first of its kind

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

logistic equation of arbitrary order abstract the paper is concerned with the new logistic equation of arbitrary order which describes the performance of complex executive systems x vs number of tasks n operating at limited resources k at non extensive heterogeneous self organization processes characterized by parameter f in contrast to the classical logistic equation which exclusively relates to the special case of sub extensive homogeneous self organization processes at f 1 the proposed model concerns both homogeneous and heterogeneous processes in sub extensive and super extensive areas the parameter of arbitrary order f where f depends on both the coefficient of external resource utilization u n k where 0 u 1 and the internal microscopic character of realized processes related to the depth of feedback the coefficient directly influences self organization of processes by the change of microscopic parameters v i s i i and z where v i is the number of references visit to the i th component of the system during the service of each task s i is the time of serving the task by the i th component and z is the think time of a given process in the general case of complex system parameters v i s i i and z can have values in the range from 0 to in this way the new equation includes all possible cases of a complex executive system s operation furthermore it allows us to define the optimal matching point between x and n with f as the parameter it also helps to balance the load in complex systems and to equip artificial systems with self optimization mechanisms similar to those observed in natural systems

variations on a theme of steinberg abstract the steinberg tensor product theorem is a fundamental tool for studying irreducible representations of simple algebraic groups over fields of positive characteristic this paper is concerned with extending the result replacing the target group sl v by an arbitrary simple algebraic group

percolation and cluster structure parameters the enhanced hoshen kopelman algorithm an enhanced hoshen kopelman ehk algorithm for calculating cluster structure parameters is presented for the site percolation problem the ehk algorithm enables efficient calculation of cluster spatial moments perimeters and bounding boxes for very large monte carlo simulated lattices the algorithm is used to compute the squared radius of gyration and internal perimeters of clusters in 3000 ifmmode times else texttimes fi 3000 square simulated lattices the squared radius of gyration was used to calculate the value of the correlation length exponent

a1 type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups keywords semisimple algebraic groups closed connected subgroups finite groups fixed point subgroups reference ctg article 1995 001doi 10 1006 jabr 1995 1285 record created on 2008 12 16 modified on 2017 05 12

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

carter subgroups in finite groups abstract let f be the class of finite groups which contain nonconjugate carter subgroups and assume that f is nonempty then it has been shown that a member of f of minimal order should be an almost simple group a in this paper using cfsg we restrict the possibilities for such a in particular we show that a cannot be simple except possibly when it is a unitary group

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

alternative techniques for cluster labelling on percolation theory we propose a new cluster labelling algorithm as a tool for computer aided simulation in the field of percolation theory due to the use of recursivity the basic labelling algorithm only needs a few lines of code and performs at least as well as the hoshen kopelman algorithm with the small lattices that can be stored in the computer memory additionally it can be extended to label the clusters compute cluster parameters and check for percolation in a single pass over the lattice we also detail how to deal with lattice dimensions higher than two or with lattices without complete connectivity huge lattices which cannot be stored as a whole in the computer s memory require a slight modification that permits the labelling by sublattices in this case a cluster association technique similar to that proposed by hoshen and kopelman is necessary nevertheless the proposed algorithm is capable of labelling lattices of virtually any size no matter what the memory capacity of the machine because it does not require an entire line or hyperplane of the whole lattice to be in memory it is only limited by the storage capacity of the devices attached to the machine if what is needed is just percolation checking the storage requirements are very low and processing times decrease using the percolation finders suggested listings of c programs are available in the online edition

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough