dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

hall polynomials for tame type abstract in the present paper we prove that hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line x over finite fields these polynomials are then used to define the generic ringel hall algebra of x as well as its drinfeld double combining this construction with a result of cramer we show that hall polynomials exist for tame quivers which not only refines a result of hubery but also confirms a conjecture of berenstein and greenstein

quantum groups via hall algebras of complexes we describe quantum enveloping algebras of symmetric kac moody lie algebras via a nite eld hall algebra construction involving z2 graded complexes of quiver representations

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

partial state reference model adaptive control of multivariable systems abstract in this paper the authors aim at designing a partial state reference model adaptive controller for multivariable systems the partial state reference model concept is particularly used to incorporate an appropriate tracking capability into the linear quadratic control design simulation studies involving a binary distillation column model are carried out to emphasize the performances of the proposed adaptive controller

the bridgeland s ringel hall algebra associated to an algebra with global dimension at most two abstract for any finite dimensional associative algebra with global dimension 2 we show that there is an embedding from the twisted ringel hall algebra to the bridgeland s ringel hall algebra in particular this result is true for tilted algebras and canonical algebras

the hall algebra approach to drinfeld s presentation of quantum loop algebras abstract the quantum loop algebra u v l g was defined as a generalization of the drinfeld s new realization of the quantum affine algebra to the loop algebra of any kac moody algebra g it has been shown by schiffmann that the hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra u v l g for some g with a star shaped dynkin diagram in this paper we study drinfeld s presentation of u v l g in the double hall algebra setting based on schiffmann s work we explicitly find out a collection of generators of the double composition algebra dc coh x and verify that they satisfy all the drinfeld relations

summability of the perturbative expansion for a zero dimensional disordered spin model we show analytically that the perturbative expansion for the free energy of the zero dimensional quenched disordered ising model is borel summable in a certain range of parameters provided that the summation is carried out in two steps first in the strength of the original coupling of the ising model and subsequently in the variance of the quenched disorder this result is illustrated by some high precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method

the hall algebra of the category of coherent sheaves on the projective line to an abelian category a of homological dimension one satisfying certain finiteness conditions one can associate an algebra called the hall algebra kapranov studied this algebra when a is the category of coherent sheaves over a smooth projective curve defined over a finite field and observed analogies with quantum affine algebras we recover here in an elementary way his results in the case when the curve is the projective line

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

on the hall algebra of an elliptic curve i these rings admit numerous algebraic and geometric realizations but one of the historically first constructions dating to the work of steinitz in 1900 completed later by hall was given in terms of what is now called the classical hall algebra h see ma chapter ii this algebra has a basis consisting of isomorphism classes of abelian q groups where q is a fixed prime power and the structure constants are defined by counting extensions between such abelian groups in fact these structure constants are polynomials in q and we can therefore consider h as a c q 1 algebra a theorem of steinitz and hall provides an isomorphism h q c q x1 x2 under this isomorphism the natural basis of h resp the natural scalar product is mapped to the basis of hall littlewood polynomials resp the hall littlewood scalar product in addition zelevinsky z endowed q with a structure of a cocommutative hopf algebra and the whole algebra q c q 1

modified ringel hall algebras green s formula and derived hall algebras abstract in this paper we define the modified ringel hall algebra m h a of a hereditary abelian category a from the category c b a of bounded z graded complexes two main results are obtained one is to give a new proof of green s formula on ringel hall numbers by using the associative multiplication of the modified ringel hall algebra the other is to show that in certain twisted cases the derived hall algebra can be embedded in the modified ringel hall algebra as a consequence of the second result we get that in certain twisted cases the modified ringel hall algebra is isomorphic to the tensor algebra of the derived hall algebra and the torus of acyclic complexes and therefore the modified ringel hall algebra is invariant under derived equivalences

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

root categories and simple lie algebras abstract by using the t 2 orbit category of the derived category of a hereditary algebra which is proved to be a triangulated category too we give a complete realization of a simple lie algebra this is a global version of ringel s work for the positive parts of simple lie algebras in this realization the grothendieck group provides the cartan subalgebra the indecomposable objects correspond to the chevalley basis and the structure constants are given by the evaluations of hall polynomials

three dimensional randomly dilute ising model monte carlo results we perform a high statistics simulation of the three dimensional randomly dilute ising model on cubic lattices l 3 with l le 256 we choose a particular value of the density x 0 8 for which the leading scaling corrections are suppressed we determine the critical exponents obtaining nu 0 683 3 eta 0 035 2 beta 0 3535 17 and alpha 0 049 9 in agreement with previous numerical simulations we also estimate numerically the fixed point values of the four point zero momentum couplings that are used in field theoretical fixed dimension studies although these results somewhat differ from those obtained using perturbative field theory the field theoretical estimates of the critical exponents do not change significantly if the monte carlo result for the fixed point is used finally we determine the six point zero momentum couplings relevant for the small magnetization expansion of the equation of state and the invariant amplitude ratio r xi that expresses the universality of the free energy density per correlation volume we find r xi 0 2885 15

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

universality in the off equilibrium critical dynamics of the three dimensional diluted ising model we study the off equilibrium critical dynamics of the three dimensional diluted ising model we compute the dynamical critical exponent z and we show that it is independent of the dilution only when we take into account the scaling corrections to the dynamics finally we will compare our results with the experimental data

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

determination of the dynamic and static critical exponents of the two dimensional three state potts model using linearly varying temperature we employ a successive monte carlo renormalization group procedure in the presence of a linearly varying temperature to study the three state q 3 potts model on square lattices by matching correlation functions of different lattice sizes at different renormalization levels a rate exponent r associated with the temperature sweep rate r and the correlation length exponent ensuremath nu are obtained the dynamic critical exponent z is then obtained by the scaling law z r ensuremath 1 ensuremath nu derived in our method the results of z 2 171 62 for q 3 in this work and the previously obtained z 2 15 13 for q 2 seem to support the extension of the weak universality hypothesis to dynamic critical behavior with these calculated exponents the dynamic scaling forms of both specific heat and order parameter at various r are presented and all the other static critical exponents are determined which verify our method discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

star graph expansions for bond diluted potts models we derive high temperature series expansions for the free energy and the susceptibility of random bond q state potts models on hypercubic lattices using a star graph expansion technique this method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions moreover we can keep the disorder strength p as well as the dimension d as symbolic parameters by applying several series analysis techniques to the new series expansions one can scan large regions of the p d parameter space for any value of q for the bond diluted four state potts model in three dimensions which exhibits a rather strong first order phase transition in the undiluted case we present results for the transition temperature and the effective critical exponent gamma as a function of p as obtained from the analysis of susceptibility series up to order 18 a comparison with recent monte carlo data chatelain et al phys rev e 64 036120 2001 shows signals for the softening to a second order transition at finite disorder strength

double hall algebras and derived equivalences abstract we show that the reduced drinfeld double of the ringel hall algebra of a hereditary category is invariant under derived equivalences by associating an explicit isomorphism to a given derived equivalence we also extend the results of burban and schiffmann 5 6 sevenhant and van den bergh 1999 25 and xiao and yang 2001 31

self averaging distribution of pseudocritical temperatures and finite size scaling in critical disordered systems we evaluate by monte carlo simulations various singular thermodynamic quantities x for ensembles of quenched random ising and ashkin teller models the measurements are taken at t c and we study how the distributions p x and in particular their relative squared width r x over the ensemble depend on the system size l the ashkin teller model was studied in the regime where bond randomness is irrelevant and we found weak self averaging r x ensuremath sim l ensuremath alpha ensuremath nu ensuremath rightarrow 0 where ensuremath alpha l0 and ensuremath nu are the exponents of the pure model fixed point governing the transition for the site dilute ising model on a cubic lattice known to be governed by a random fixed point we find that r x tends to a constant as predicted by aharony and harris we tested whether this constant is universal however this constant is different for canonical and grand canonical disorder we identify the pseudocritical temperature of each sample i t c i l as the temperature at which the susceptibility reaches its maximal value ensuremath chi mathrm max the distribution of these sample dependent t c i l was investigated we found that its variance scales as ensuremath delta t c l 2 ensuremath sim l ensuremath 2 ensuremath nu our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold we did observe deviations from a single function which imply that sample dependent scaling functions are needed these deviations are however relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ensuremath chi mathrm max than the commonly used ensuremath chi t c ensuremath infty

a note on bridgeland s hall algebra of two periodic complexes we show that the hall algebra of two periodic complexes which is recently introduced by t bridgeland coincides with the drinfeld double of the ordinary hall bialgebra

universal aging properties at a disordered critical point we investigate analytically near the dimension d uc 4 and numerically in d 3 the nonequilibrium relaxational dynamics of the randomly diluted ising model at criticality using the exact renormalization group method to one loop we compute the two times t phantom rule 0 2em 0ex t w correlation function and fluctuation dissipation ratio fdr for any fourier mode of the order parameter of finite wave vector q in the large time separation limit the fdr is found to reach a nontrivial value x ensuremath infty independently of small q and coincide with the fdr associated to the total magnetization obtained previously explicit calculations in real space show that the fdr associated to the local magnetization converges in the asymptotic limit to this same value x ensuremath infty through a monte carlo simulation we compute the autocorrelation function in three dimensions for different values of the dilution fraction p at t c p taking properly into account the corrections to scaling we find according to the renormalization group predictions that the autocorrelation exponent ensuremath lambda c is independent of p the analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state

noncommutative projective curves and quantum loop algebras we show that the hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the quantized enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding kac moody algebra in particular this yields a geometric realization of the quantized enveloping algebra of elliptic or 2 toroidal algebras of types d 4 1 1 e 1 1 6 e 1 1 7 and e 8 1 1 in terms of coherent sheaves on weighted projective lines of genus one or equivalently in terms of equivariant coherent sheaves on elliptic curves

hall algebras associated to triangulated categories by counting triangles and applying the octahedral axiom we find a direct way to prove the formula of toen in t section 5 1 for a triangulated category satisfying the left homological finiteness condition

derived hall algebras the purpose of this work is to define a derived hall algebra mathcal dh t associated to any dg category t under some finiteness conditions our main theorem states that mathcal dh t is associative and unital it is shown that mathcal dh t contains the usual hall algebra mathcal h t when t is an abelian category we will also prove an explicit formula for the derived hall numbers purely in terms of invariants of the triangulated category associated to t as an example we describe the derived hall algebra of an hereditary abelian category