ladders and simplicity of derived module categories abstract recollements of derived module categories are investigated using a new technique ladders of recollements which are maximal mutation sequences the position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category ladders also turn out to control derived simplicity on all levels an algebra is derived simple if its derived category cannot be deconstructed that is if it is not the middle term of a non trivial recollement whose outer terms are again derived categories of algebras derived simplicity on each level is characterised in terms of heights of ladders these results are complemented by providing new classes of examples of derived simple rings in particular indecomposable commutative rings as well as by a finite dimensional counterexample to the jordan holder problem for derived module categories moreover recollements are used to compute homological and k theoretic invariants

variational approximations and mean field scaling theory it is shown that mean field finite size scaling theory can be adjusted and applied to sequences of approximations that do not obey the well known power laws of finite size scaling the proposed modification of the theory seems to be well directed to the sequence of baxter s variational approximations this sequence of approximations shows a novel feature according to which exponential laws appear in places where power laws should be expected but the modified scaling theory still yields relations from which non classical critical exponents can be estimated thus two techniques for the estimation of the critical exponent beta are suggested from the finite order variational approximations these techniques are applied to the zero field ising model on the square lattice using the systematic series of the baxter tsang systems and excellent estimates of beta are obtained correct up to 13 significant figures

a homological characterization of piecewise hereditary algebras let be a finite dimensional algebra over a field k we will show here that is piecewise hereditary if and only if its strong global dimension is finite

critical finite size scaling of magnetization distribution function for baxter wu model abstract the distribution function pl m of the order parameter for the baxter wu model is studied using blocks of linear dimension l of a larger triangular lattice at a given temperature we use the metropolis algorithm for the generation of a sample of configurations on the triangular lattice the similarities and differences of this distribution with the usual cases of ising lattices are investigated we conclude that the present model obeys at the critical temperature a finite scaling law with the known critical exponents as expected however our numerical data strongly indicate that the analytic form of the scaling function does not conform to the corresponding function for the usual ising model an analytic expression that gives a good fit is presented

blocks of group algebras are derived simple a derived version of maschke s theorem for finite groups is proved the derived categories bounded or unbounded of all blocks of the group algebra of a finite group are simple in the sense that they admit no nontrivial recollements this result is independent of the characteristic of the base field

finite size scaling analysis of the critical behavior of the baxter wu model abstract we use the recently developed critical minimum energy subspace crmes approximation scheme to study the critical behavior of the baxter wu model this scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort the density of states is constructed by a multi range wang landau sampling and from this we obtain the critical properties of specific heat and of the reduced binder cummulant the good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique

epidemics and immunization on euclidean distance preferred small world networks abstract in this paper we propose a novel euclidean distance preferred model yielding small world networks in the euclidean space on which the epidemics dynamics is investigated a novel local region immunization strategy is proposed to suppress the epidemic prevalence on the euclidean small world networks and the critical immunization radius is extensively investigated which could be helpful in the practical applications of immunization strategies in the real world

on the uniqueness of stratifications of derived module categories abstract recollements of triangulated categories may be seen as exact sequences of such categories iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic objects we discuss the question of uniqueness of such a stratification up to ordering and derived equivalence for derived module categories the main result is a positive answer in the form of a jordan holder theorem for derived module categories of hereditary artin algebras we also provide examples of derived simple rings

concealed canonical algebras and separating tubular families we characterise those concealed canonical algebras which arise as endomorphism rings of tilting modules all of whose indecomposable summands have strictly positive rank as those artin algebras whose module categories have a separating exact subcategory that is a separating tubular family of standard tubes this paper develops further the technique of shift automorphisms which arises from the tubular structure it is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

scaling properties of the baxter wu model abstract we use a recently developed cluster algorithm for the baxter wu model to study characteristic features of its behavior magnetic scaling properties of the model are investigated for second order phase transitions we improve significantly the accuracy of monte carlo simulation results and we provide new results for the behavior of the binder cumulants for temperatures below above and at the critical point

phase diagrams of the triangular ising antiferromagnet variational approximations derives an appropriate generalisation of baxter s variational method and define a sequence of variational approximations for antiferromagnetic models on the triangular lattice expressions for the sublattice magnetisations are derived from a variational principle for the partition function per site he applies the method to the triangular antiferromagnet and obtains approximate phase diagrams in the temperature field and temperature magnetisation density planes

controlling the spreading in small world evolving networks stability oscillation and topology the spreading of viruses diseases and even disasters such as power blackouts and financial crises in many large scale and small world networks is one of the mostly concerned issues today in this note we study general spreading dynamical behaviors in small world evolving networks when control strategies are applied to suppress the propagation of diseases viruses and disasters after proposing a novel watts strogatz w s spreading model to capture the general spreading mechanism in small world networks we investigate the stability and hopf bifurcations of delay controlled spreading models with linear and nonlinear feedback controllers where parameters of small world rewiring probability feedback control gain and time delay are analyzed for the oscillating behaviors we conclude that the oscillatory spreading phenomena in delay controlled small world networks are topologically inherent

jordan holder theorems for derived module categories of piecewise hereditary algebras abstract a jordan holder theorem is established for derived module categories of piecewise hereditary algebras in particular for representations of quivers and for hereditary abelian categories of a geometric nature the resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories and also of the choice of finitely generated or arbitrary modules

epidemic processes on complex networks abstract we analyse the dynamics of the sis and reed frost models on complex networks in contrast to most of the previous analytical work which relied on mean field approximations and simulations we present a probabilistic approach whereby general results are derived and illustrated on various families of graphs

finite size scaling at first and second order phase transitions of baxter wu model abstract using a finite size phenomenological theory we investigate the behavior of the baxter wu model for both first and second order transitions in order to distinguish between the two kinds of transition we study the finite size scaling behavior of the order parameter and the susceptibility of the model at the critical temperature the known critical exponents of the system govern the scaling behavior while below the critical temperature the scaling exponents are the dimensionality of the system the presented scaling theory for the baxter wu model is an appropriate generalization of the corresponding theory of binder and landau for the ising model binder and landau phys rev b 30 1984 1477 we perform numerical simulations using standard monte carlo techniques and we find that our results are in good agreement with theoretical predictions

the telescope conjecture for hereditary rings via ext orthogonal pairs abstract for the module category of a hereditary ring the ext orthogonal pairs of subcategories are studied for each ext orthogonal pair that is generated by a single module a 5 term exact sequence is constructed the pairs of finite type are characterized and two consequences for the class of hereditary rings are established homological epimorphisms and universal localizations coincide and the telescope conjecture for the derived category holds true however we present examples showing that neither of these two statements is true in general for rings of global dimension 2

pseudospectral method for the kardar parisi zhang equation we discuss a numerical scheme to solve the continuum kardar parisi zhang equation in generic spatial dimensions it is based on a momentum space discretization of the continuum equation and on a pseudospectral approximation of the nonlinear term the method is tested in 1 11 and 211 dimensions where it is shown to reproduce the current most reliable estimates of the critical exponents based on restricted solid on solid simulations in particular it allows the computations of various correlation and structure functions with high degree of numerical accuracy some deficiencies that are common to all previously used finite difference schemes are pointed out and the usefulness of the present approach in this respect is discussed