grade dominant dimension and gorenstein algebras abstract we first give precise connections between auslander bridger s grade double centraliser properties and dominant dimension and apply these to homological conjectures then we introduce gendo d gorenstein algebras as correspondents of gorenstein algebras under a morita tachikawa correspondence we characterise these algebras by homological properties and derive several of their properties including higher auslander correspondence

fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

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

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

a time domain numerical modeling of two dimensional wave propagation in porous media with frequency dependent dynamic permeability an explicit finite difference scheme is presented for solving the two dimensional biot equations of poroelasticity across the full range of frequencies the key difficulty is to discretize the johnson koplik dashen jkd model which describes the viscous dissipations in the pores indeed the time domain version of biot jkd model involves order 1 2 fractional derivatives which amount to a time convolution product to avoid storing the past values of the solution a diffusive representation of fractional derivatives is used the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations the coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation then various methods of scientific computing are applied the propagative part of the equations is discretized using a fourth order finite difference scheme whereas the diffusive part is solved exactly an immersed interface method is implemented to discretize the geometry on a cartesian grid and also to discretize the jump conditions at interfaces numerical experiments are proposed in various realistic configurations

collective synchronization in spatially extended systems of coupled oscillators with random frequencies we study collective behavior of locally coupled limit cycle oscillators with random intrinsic frequencies spatially extended over d dimensional hypercubic lattices phase synchronization as well as frequency entrainment are explored analytically in the linear strong coupling regime and numerically in the nonlinear weak coupling regime our analysis shows that the oscillator phases are always desynchronized up to d 4 which implies the lower critical dimension d l p 4 for phase synchronization on the other hand the oscillators behave collectively in frequency phase velocity even in three dimensions d 3 indicating that the lower critical dimension for frequency entrainment is d l f 2 nonlinear effects due to the periodic nature of limit cycle oscillators are found to become significant in the weak coupling regime so called runaway oscillators destroy the synchronized ordered phase and there emerges a fully random disordered phase critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation collective behavior of globally coupled oscillators is also examined and compared with that of locally coupled oscillators

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

jordan holder theorems for derived categories of derived discrete algebras abstract for any positive integer n n derived simple derived discrete algebras are classified up to derived equivalence furthermore the jordan holder theorems for all kinds of derived categories of derived discrete algebras are obtained

eventually homological isomorphisms in recollements of derived categories abstract eventually homological isomorphisms are introduced between derived categories of algebras and these lead to a special recollement situation that is d b d a d c with the functor j d a d c being an eventually homological isomorphism this recollement turns out to be a very good context to compare the algebras a and c with respect to gorensteinness singularity categories and the finite generation condition fg for the hochschild cohomology the results are applied to stratifying ideals triangular matrix algebras and derived discrete algebras

interface conditions for biot s equations of poroelasticity interface conditions at a boundary between two porous media are derived directly from biot s equations of poroelasticity by replacing the discontinuity surface with a thin transition layer in which the properties of the medium change rapidly yet continuously and then taking the limit as the thickness of the transition layer approaches zero the interface conditions obtained in this way the well known open pore conditions are shown to be the only ones that are fully consistent with the validity of biot s equations throughout the poroelastic continuum including surfaces across which the medium properties are discontinuous but partially blocked or completely impermeable interfaces exist these may be looked upon as the case of a thin layer with its permeability taken to be proportional to the layer thickness again in the limit as layer thickness approaches zero this approach can serve as a simple recipe for modeling such an interface in any heterogeneous numerical scheme for poroelastic media

support varieties and hochschild cohomology rings we define a support variety for a finitely generated module over an artin algebra lambda over a commutative artinian ring k with lambda flat as a module over k in terms of the maximal ideal spectrum of the algebra rm hh lambda of lambda this is modelled on what is done in modular representation theory and the varieties defined in this way are shown to have many of the same properties as those for group rings in fact the notions of a variety in our sense and those for principal and non principal blocks are related by a finite surjective map of varieties for a finite dimensional self injective algebra over a field the variety is shown to be an invariant of the stable component of the auslander reiten quiver moreover we give information on nilpotent elements in rm hh lambda give a thorough discussion of the ring rm hh lambda on a class of nakayama algebras give a brief discussion of a possible notion of complexity and make a comparison with support varieties for complete intersections

numerical modeling of elastic waves across imperfect contacts a numerical method is described for studying how elastic waves interact with imperfect contacts such as fractures or glue layers existing between elastic solids these contacts have been classicaly modeled by interfaces using a simple rheological model consisting of a combination of normal and tangential linear springs and masses the jump conditions satisfied by the elastic fields along the interfaces are called the spring mass conditions by tuning the stiffness and mass values it is possible to model various degrees of contact from perfect bonding to stress free surfaces the conservation laws satisfied outside the interfaces are integrated using classical finite difference schemes the key problem arising here is how to discretize the spring mass conditions and how to insert them into a finite difference scheme this was the aim of the present paper for this purpose we adapted an interface method previously developed for use with perfect contacts j comput phys 195 2004 pp 90 116 this numerical method also describes closely the geometry of arbitrarily shaped interfaces on a uniform cartesian grid at negligible extra computational cost comparisons with original analytical solutions show the efficiency of this approach

wave field simulation for heterogeneous porous media with singular memory drag force the objective of this paper is to use biot s theory and the johnson koplik dashen dynamic permeability model in wave field simulation of a heterogeneous porous medium the johnson koplik dashen dynamic permeability model was originally formulated in the frequency domain in this paper the time domain drag force expression of the model is expressed in terms of the shifted fractional derivative of the relative fluid velocity in contrast to the exponential type viscous relaxation models the convolution operator in the johnson koplik dashen dynamic permeability model cannot be replaced by memory variables satisfying first order relaxation differential equations a new method for calculating the shifted fractional derivative without storing and integrating the entire velocity histories is developed using the new method to calculate the fractional derivative the governing equations for the two dimensional porous medium are reduced to a system of first order differential equations for velocities stresses pore pressure and the quadrature variables associated with the drag forces spatial derivatives involved in the first order differential equations system are calculated by fourier pseudospectral method while the time derivative of the system is discretized by a predictor corrector method for the demonstration of our method some numerical results are presented

on the finitistic dimension conjecture of artin algebras abstract let a be an artin algebra and e be an idempotent element of a we prove that if a has representation dimension at most three then the finitistic dimension of eae is finite and deduce that if quasi hereditary algebras have representation dimensions at most three then the finitistic dimension conjecture holds

homological theory of idempotent ideals let be an artin algebra u and a two sided ideal of then u is the trace of a projective module p in we study how the homological properties of the categories of finitely generated modules over the three rings u and the endomorphism ring of p are related we give some applications of the ideas developed in the paper to the study of quasi hereditary algebras

a discontinuous galerkin method for wave propagation in orthotropic poroelastic media with memory terms abstract poroelastic materials play an important role in biomechanical and geophysical research in this paper we investigate wave propagation in orthotropic poroelastic media by studying the time domain poroelastic wave equations both the low frequency biot s lf biot equations and the biot johnson koplik dashen biot jkd model are considered in lf biot equations the dissipation terms are proportional to the relative velocity between the fluid and the solid by a constant contrast to this the dissipation terms in the biot jkd model are in the form of time convolution memory as a result of the frequency dependence of fluid solid interaction at the underlying microscopic scale in the frequency domain the dynamic tortuosity and permeability described by darcy s law are two crucial factors in this problem and highly linked to the viscous force in the biot model the key difficulty is to handle the viscous term when the pore fluid is viscous in the biot jkd model the convolution operator involves order 1 2 shifted fractional derivatives in the time domain which is challenging to discretize in this work a new method of the multipoint pade or rational approximation for stieltjes function is applied to approximate the jkd dynamic tortuosity and then an augmented system of biot jkd model is obtained where the kernel of the memory term is replaced by the finite auxiliary variables satisfying a local system of ordinary differential equations the runge kutta discontinuous galerkin rkdg method with the un splitting method is used to compute the numerical solution and numerical examples are presented to demonstrate the high order accuracy and stability of the method compared with the existing approaches for solving the biot jkd equations the augmented system presented here require neither the storage of solution history nor the computation of the flux of the auxiliary variables

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

homological theory of recollements of abelian categories abstract we investigate several homological aspects of recollements of abelian categories in particular we study how various homological invariants and dimensions of the categories involved in a recollement situation are related and when recollements of abelian categories induce recollements at the level of the bounded derived categories finally we give applications to global finitistic and representation dimension of rings and artin algebras and to rouquier s dimension of triangulated categories

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

singular equivalence and the fg condition abstract we show that singular equivalences of morita type with level between finite dimensional gorenstein algebras over a field preserve the fg condition

stability of interfaces with mesh refinement a study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite difference approximations first a root condition which implies stability for the initial boundary value problem for this type of interface is derived from the root condition the stability of several interface equations is proved using the maximum principle in some cases the final verification steps can be done analytically in other cases a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle using this method stability for lax wendroff with all the interface conditions considered and for leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved

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

localization of triangulated categories and derived categories the notion of quotient and localization of abelian categories by dense subcategories i e serre classes was introduced by gabriel and plays an important role in ring theory 6 131 the notion of triangulated categories was introduced by grothendieck and developed by verdier 9 16 and is recently useful in representation theory 8 4 143 the quotient of triangulated categories by epaisse subcategories is constructed in 16 both quotients were indicated by grothendieck and they resemble each other in this paper we will consider triangulated categories and derived categories from the point of view of localization of abelian categories verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint and considered a relation between tpaisse subcategories 16 we show that localization of triangulated categories is similarly defined and have a relation between localizations and cpaisse subcategories beilinson bernstein and deligne introduced the notion of t structure which is similar to torsion theory in abelian categories 2 we in particular consider a stable t structure which is an epaisse subcategory and deal with a correspondence between localizations of triangulated categories and stable r structures and then recollement in the sense of 2 is equivalent to bilocalization next we show that quotient and localization of abelian categories induce quotient and localization of its derived categories in section 1 we recall standard notations and terminologies of quotient and localization of abelian categories in section 2 we define localization of triangulated categories and consider a relation between localizations and stable r structures theorem 2 6 in section 3 we show that if a a c is a quotient of abelian categories then d a d a c is a quotient of triangulated categories where or b theorem 3 2 moreover

death of period doublings locating the homoclinic doubling cascade abstract this paper studies a natural mechanism called a homoclinic doubling cascade for the disappearance of period doubling cascades in vector fields simply put an entire period doubling cascade collides with a saddle type equilibrium homoclinic doubling cascades are known to have self similar structure in contrast to the well known feigenbaum constant the scaling constants for homoclinic doubling depend on the eigenvalues of the saddle equilibrium specifically we present here for the first time a detailed study of homoclinic doubling cascades in a smooth vector field namely a three dimensional polynomial model proposed by sandstede a numerical algorithm is presented for computing homoclinic doubling cascades in general vector fields which makes use of the program auto homcont this allows us to compute two types of homoclinic doubling cascades one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation our results bring out the scaling constants in good agreement with analytical estimates obtained from one dimensional maps

derived equivalences via hrs tilting abstract let a be an abelian category and b be the happel reiten smalo tilt of a with respect to a torsion pair we give necessary and sufficient conditions for the existence of a derived equivalence between a and b which is compatible with the inclusion of b into the derived category of a as applications we obtain new derived equivalences related to splitting torsion pairs ttf triples and two term silting subcategories respectively we prove that for the realization functor of any bounded t structure its denseness implies its fully faithfulness

an upper bound for the dimension of bounded derived categories abstract let be an artin algebra we give an upper bound for the dimension of the bounded derived category of the category mod of finitely generated right modules in terms of the projective and injective dimensions of certain class of simple right modules as well as the radical layer length of in addition we give an upper bound for the dimension of the singularity category of mod in terms of the radical layer length of

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

an unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media the perfectly matched layer pml absorbing technique has become popular in numerical modeling in elastic or poroelastic media because of its efficiency in absorbing waves at nongrazing incidence however after numerical discretization at grazing incidence large spurious oscillations are sent back from the pml into the main domain the pml then becomes less efficient when sources are located close to the edge of the truncated physical domain under study for thin slices or for receivers located at a large offset we develop a pml improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first order system in velocity and stress we show its efficiency for both nondissipative and dissipative biot porous models based on a fourth order staggered finite difference method used in a thin mesh slice the results obtained are improved significantly compared with those obtained with the classical pml

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

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

recollements and hochschild theory it is shown that a recollement of derived categories of algebras induces those of tensor product algebras and opposite algebras respectively which is applied to clarify the relations between recollements of derived categories of algebras and smoothness and hochschild cohomology of algebras

stability of finite difference models containing two boundaries or interfaces it is known that the stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves here the waves point of view is applied to models containing two boundaries or interfaces where repeated reflection of trapped packets is a potential new source of instability our analysis accounts for various known instability phenomena in a unified way and leads to several new results three of which are as follows 1 dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface 2 algebraic gks instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present 3 gks stability and p stability can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one

some grid refinement schemes for hyperbolic equations with piecewise constant coefficients discontinuities in the coefficients of hyperbolic equations occur both naturally and artificially and must be treated in numerical schemes schemes for handling these discontinuities are derived an interesting stability result is derived and the schemes are shown to be exact under certain restric tions

discontinuous galerkin methods for wave propagation in poroelastic media we have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three dimensional domains this method is a discontinuous galerkin method that achieves arbitrary high order accuracy on unstructured tetrahedral meshes for the low frequency range and the inviscid case by using biot s equations and darcy s dynamic laws we have built a scheme that can successfully model wave propagation in fluid saturated porous media when anisotropy of the pore structure is allowed zero inflow fluxes are used as absorbing boundary conditions a continuous arbitrary high order derivatives time integration is used for the high frequency inviscid case whereas a space time discontinuous scheme is applied for the low frequency case we conducted a numerical convergence test of the proposed methods we used a series of examples to quantify the quality of our numerical results comparing them to analytic solutions as well as numerical solutions obtained by other methodologies in partic

finite difference modeling of biot s poroelastic equations across all frequencies an explicit time stepping finite difference scheme is presented for solving biot s equations of poroelasticity across the entire band of frequencies in the general case for which viscous boundary layers in the pores must be accounted for the time domain version of darcy s law contains a convolution integral it is shown how to efficiently and directly perform the convolution so that the darcy velocity can be properly updated at each time step at frequencies that are low enough compared to the onset of viscous boundary layers no memory terms are required at higher frequencies the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used in practice we never use more than 20 memory terms and often considerably fewer allowing for the convolution makes the scheme even more stable even larger time steps might be used than it is when the convolution is entirely neglected the accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results

homological embeddings for preprojective algebras for a fixed finite dimensional algebra a we study representation embeddings of the form mod b rightarrow mod a such an embedding is called homological if it induces an isomorphism on all ext groups and weakly homological if only ext 1 is preserved in case a is a preprojective algebra of dynkin type we give an explicit classification of all weakly homological and homological embeddings furthermore we show that for self injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the tachikawa conjecture

an error analysis of conservative space time mesh refinement methods for the one dimensional wave equation we study two space time mesh refinement methods as the one introduced in f collino t fouquet and p joly numer math 95 2003 pp 197 221 the stability of such methods is guaranteed by construction through the conservation of a discrete energy in this paper we show the l2 convergence of these schemes and provide optimal error estimates the proof is based on energy techniques and bootstrap arguments our results are validated with numerical simulations and compared with results from plane wave analysis f collino t fouquet and p joly numer math 95 2003 pp 223 251

gorenstein defect categories of triangular matrix algebras abstract we apply the technique of recollement to study the gorenstein defect categories of triangular matrix algebras first we construct a left recollement of gorenstein defect categories for a triangular matrix algebra under some conditions using it we give a categorical interpretation of the gorenstein properties of the triangular matrix algebras obtained by x w chen b l xiong and p zhang respectively second under some additional conditions a recollement of gorenstein defect categories for a triangular matrix algebra is constructed as an application for a special kind of triangular matrix algebras which are called simple gluing algebras we describe their singularity categories and gorenstein defect categories

numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain homogenization of a thin micro structure yields effective jump conditions that incorporate the geometrical features of the scatterers these jump conditions apply across a thin but nonzero thickness interface whose interior is disregarded this paper aims i to propose a numerical method able to handle the jump conditions in order to simulate the homogenized problem in the time domain ii to inspect the validity of the homogenized problem when compared to the real one for this purpose we adapt an immersed interface method originally developed for standard jump conditions across a zero thickness interface doing so allows us to handle arbitrary shaped interfaces on a cartesian grid with the same efficiency and accuracy of the numerical scheme than those obtained in an homogeneous medium numerical experiments are performed to test the properties of the numerical method and to inspect the validity of the homogenization problem

efficient solution of a wave equation with fractional order dissipative terms we consider a wave equation with fractional order dissipative terms modeling visco thermal losses on the lateral walls of a duct namely the webster lokshin model diffusive representations of fractional derivatives are used first to prove existence and uniqueness results then to design a numerical scheme which avoids the storage of the entire history of past data two schemes are proposed depending on the choice of a quadrature rule in the laplace domain the first one mimics the continuous energy balance but suffers from a loss of accuracy in long time simulation the second one provides uniform control of the accuracy however even though the latter is more efficient and numerically stable under the standard cfl condition no discrete energy balance has been yet found for it numerical results of comparisons with a closed form solution are provided

biot jkd model simulation of 1d transient poroelastic waves with fractional derivatives a time domain numerical modeling of biot poroelastic waves is presented the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce order 1 2 shifted fractional derivatives involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations thanks to the dispersion relation the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest a splitting strategy is then applied numerically the propagative part of biot jkd equations is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly comparisons with analytical solutions show the efficiency and the accuracy of this approach

wave simulation in 2d heterogeneous transversely isotropic porous media with fractional attenuation a cartesian grid approach a time domain numerical modeling of transversely isotropic biot poroelastic waves is proposed in two dimensions the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce shifted fractional derivatives of order 1 2 involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations resulting in the biot da diffusive approximation model the properties of both the biot jkd and the biot da models are analyzed hyperbolicity decrease of energy dispersion to determine the coefficients of the diffusive approximation two approaches are analyzed gaussian quadratures and optimization methods in the frequency range of interest the nonlinear optimization is shown to be the better way of determination a splitting strategy is then applied to approximate numerically the biot da equations the propagative part is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly an immersed interface method is implemented to take into account heterogeneous media on a cartesian grid and to discretize the jump conditions at interfaces numerical experiments are presented comparisons with analytical solutions show the efficiency and the accuracy of the approach and some numerical experiments are performed to investigate wave phenomena in complex media such as multiple scattering across a set of random scatterers

homological epimorphisms recollements and hochschild cohomology with a conjecture by snashall solberg in view abstract we show that recollements of module categories give rise to homomorphisms between the associated hochschild cohomology algebras which preserve the strict gerstenhaber structure i e the cup product the graded lie bracket and the squaring map we review various long exact sequences in hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict gerstenhaber structure as well as a byproduct we generalise a known long exact cohomology sequence of koenig nagase to arbitrary surjective homological epimorphisms we use our observations to motivate and formulate a variation of the finite generation conjecture by snashall solberg

abelian quotients of categories of short exact sequences abstract we mainly investigate abelian quotients of categories of short exact sequences the natural framework to consider the question is via identifying quotients of morphism categories as module categories these ideas not only can be used to recover the abelian quotients produced by cluster tilting subcategories of both exact categories and triangulated categories but also can be used to reach our goal let c e be an exact category we denote by e c the category of bounded complexes whose objects are given by short exact sequences in e and by s e c the full subcategory formed by split short exact sequences in general e c is just an exact category but the quotient e c s e c turns out to be abelian in particular if c e is frobenius we present three equivalent abelian quotients of e c and point out that the equivalences are actually given by left and right rotations the abelian quotient e c s e c admits some nice properties we explicitly describe the abelian structure projective objects injective objects and simple objects which provide a new viewpoint to understanding hilton rees theorem and auslander reiten theory furthermore we present some analogous results for triangulated versions

stability of a chain of phase oscillators we study a chain of n 1 phase oscillators with asymmetric but uniform coupling this type of chain possesses 2 n ways to synchronize in so called traveling wave states i e states where the phases of the single oscillators are in relative equilibrium we show that the number of unstable dimensions of a traveling wave equals the number of oscillators with relative phase close to ensuremath pi this implies that only the relative equilibrium corresponding to approximate in phase synchronization is locally stable despite the presence of a lyapunov type functional periodic or chaotic phase slipping occurs for chains of lengths 3 and 4 we locate the region in parameter space where rotations corresponding to phase slipping are present

types of serre subcategories of grothendieck categories abstract every serre subcategory s of an abelian category a is assigned a unique type m n where m resp n counts how many times one can form left resp right adjoints starting from i and q where i s a is the inclusion and q is the quotient functor the main result gives a complete list of all the types of serre subcategories of grothendieck categories 0 0 0 1 1 1 0 2 1 2 2 1 two observations are technically crucial in proving the main result the exactness of all the functors in a recollement of abelian categories forces the recollement to split and any left recollement of a grothendieck category can be extended to a recollement

collective synchronisation in lattices of nonlinear oscillators with randomness the authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies it is shown rigorously that the model exhibits no long range order in one dimension and that in higher dimensional lattices large clusters of synchronised oscillators necessarily have a sponge like structure surprisingly the phase locking behaviour of the mean field model is completely different from that of any finite dimensional lattice indicating that d infinity is the upper critical dimension for phase locking

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras