stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

proof of a conjecture on the spectral radius of c4 free graphs abstract let g be a graph with n vertices and be the spectral radius of its adjacency matrix recently nikiforov showed that if g has no 4 cycle then 2 n 1 0 with equality if and only if g is the friendship graph however this bound is not attainable when n is even he conjectured that if g is a c 4 free graph with even number of vertices then 3 2 n 1 1 0 with equality if and only if g is a star of order n with n 2 1 disjoint additional edges we prove the conjecture in this paper

affine grassmannians and the geometric satake in mixed characteristic we endow the set of lattices in q n p with a reasonable algebro geometric structure as a result we prove the representability of affine grassmannians and establish the geometric satake equivalence in mixed characteristic we also give an application of our theory to the study of rapoport zink spaces

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

on the principal eigenvectors of uniform hypergraphs abstract let a h be the adjacency tensor of r uniform hypergraph h if h is connected the unique positive eigenvector x x 1 x 2 x n t with x r 1 corresponding to spectral radius h is called the principal eigenvector of h the maximum and minimum entries of x are denoted by x max and x min respectively in this paper we investigate the bounds of x max and x min in the principal eigenvector of h meanwhile we also obtain some bounds of the ratio x i x j for i j n as well as the principal ratio h x max x min of h as an application of these results we finally give an estimate of the gap of spectral radii between h and its proper sub hypergraph h

foliations in moduli spaces of abelian varieties in this paper we study abelian varieties and p divisible groups in characteristic p even though a non trivial deformation of an abelian variety can produce a non trivial galois representation say on the tate group of the generic fiber in any characteristic the geometric generic fiber has a constant tate group we en counter the same phenomenon for the p structure in positive characteristic any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p divisible groups however for non ordinary abelian varieties this seems to break down the fas cinating structure which comes out of this is that the maximal locus where a given geometric isomorphism class of a p divisible group is realized e g in a family of abelian varieties is a locally closed set the locus defined by the geometric iso morphism type of a p divisible group will be called a central leaf see 2 1 and 3 4 this gives rise to a foliation of the open stratum attached to a newton polygon the dimension of any leaf in the same newton polygon stratum solely depends on in extreme cases either the leaf is the whole stratum as in the ordinary case or in the almost ordinary case the p rank equals g 1 or a leaf is zero dimensional as in the supersingular case in intermediate cases a leaf can be a proper subset and still be positive dimensional we have worked out the example of g 4 in 8 1

on torsion in the cohomology of locally symmetric varieties the main result of this paper is the existence of galois representations associated with the mod p or mod p m cohomology of the locally symmetric spaces for gln over a totally real or cm eld proving conjectures of ash and others following an old suggestion of clozel recently realized by harris lan taylor thorne for characteristic 0 cohomology classes one realizes the cohomology of the locally symmetric spaces for gln as a boundary contribution of the cohomology of symplectic or unitary shimura varieties so that the key problem is to understand torsion in the cohomology of shimura varieties thus we prove new results on the p adic geometry of shimura varieties of hodge type namely the shimura varieties become perfectoid when passing to the inverse limit over all levels at p and a new period map towards the ag variety exists on them called the hodge tate period map it is roughly analogous to the embedding of the hermitian symmetric domain which is roughly the inverse limit over all levels of the complex points of the shimura variety into its compact dual the hodge tate period map has several favorable properties the most important being that it commutes with the hecke operators away from p for the trivial action of these hecke operators on the ag variety and that automorphic vector bundles come via pullback from the ag variety

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

mod points on shimura varieties of abelian type we show that the mod p points on a shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of kottwitz and langlands rapoport along the way we show that the isogeny class of a mod p point contains the reduction of a special point

on the generic part of the cohomology of compact unitary shimura varieties the goal of this paper is to show that the cohomology of compact unitary shimura varieties is concentrated in the middle degree and torsion free after localizing at a maximal ideal of the hecke algebra satisfying a suitable genericity assumption along the way we establish various foundational results on the geometry of the hodge tate period map in particular we compare the fibres of the hodge tate period map with igusa varieties

convergence of lattice boltzmann methods for stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method m junk z yang asymptotic analysis of lattice boltzmann boundary conditions j stat phys 121 2005 3 35 with the stability estimate presented in m junk w a yong weighted l 2 stability of the lattice boltzmann equation preprint we are able to prove some strict convergence results the proof applies to the lattice boltzmann method with linear collision operator both in the case of periodic domains and bounded domains if the dirichlet boundary condition is realized with the bounce back rule

asymptotic analysis of lattice boltzmann boundary conditions in this article we use a general method for the analysis of finite difference schemes to investigate lattice boltzmann algorithms for navier stokes problems with dirichlet boundary conditions several link based boundary conditions for commonly used lattice boltzmann bgk models are considered with our method the accuracy of the algorithms can be exactly predicted moreover the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity the analysis is applicable to general geometries and instationary flows

the almost product structure of newton strata in the deformation space of a barsotti tate group with crystalline tate tensors in this paper we construct the almost product structure of the minimal newton stratum in deformation spaces of barsotti tate groups with crystalline tate tensors similar to oort s and mantovan s construction for shimura varieties of pel type it allows us to describe the geometry of the newton stratum in terms of the geometry of two simpler objects the central leaf and the isogeny leaf this yields the dimension and the closure relations of the newton strata in the deformation space in particular their nonemptiness shows that a generalisation of grothendieck s conjecture of deformations of barsotti tate groups with given newton polygon holds as an application we determine analogous geometric properties of the newton stratification of shimura varieties of hodge type and prove the equidimensionality of rapoport zink spaces of hodge type

the spectral turan problem about graphs with no 6 cycle abstract a spectral version of extremal graph theory problem is as follows what is the maximum spectral radius of an h free graph of order n nikiforov posed a conjecture on spectral extremal value of cycles in the past decade much attention has been paid to this conjecture and other questions on spectral extremal graphs in order to approach to nikiforov s conjecture we provide a new conjecture on spectral extremal value involving fans and wheels moreover we show it is true for k 2

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

on the search of more stable second order lattice boltzmann schemes in confined flows the von neumann linear analysis restricted by a heuristic selection of wave number vectors was applied to the search of explicit lattice boltzmann schemes which exhibit more stability than existing methods the relative stability of the family members of quasi incompressible collision kernels for the navier stokes equations in confined flows was analyzed the linear stability analysis was simplified by assuming a uniform velocity level over the whole domain where only the wave numbers of the first harmonic normal to the flow direction were permitted a singular equilibrium function that maximizes the critical velocity level was identified which was afterwards tested in particular cases of confined flows of interest validating the resulting procedure

on the cohomology of certain pel type shimura varieties in this article we study the local geometry at a prime p of pel type shimura varieties for which there is a hyperspecial level subgroup we consider the newton polygon stratification of the special fiber at p of shimura varieties and show that each newton polygon stratum can be described in terms of the products of the reduced fibers of the corresponding pel type rapoport zink spaces with certain smooth varieties which we call igusa varieties and of the action on them of a p adic group that depends on the stratum we then extend our construction to characteristic zero and in the case of bad reduction at p use it to compare the vanishing cycle sheaves of the shimura varieties to those of the rapoport zink spaces as a result of this analysis in the case of proper shimura varieties we obtain a description of the l adic cohomology of the shimura varieties in terms of the l adic cohomology with compact supports of the igusa varieties and of the rapoport zink spaces for any prime l p

a stability notion for lattice boltzmann equations lattice boltzmann equations are usually constructed to satisfy some physical requirements such as galilean invariance and isotropy to possess a velocity independent pressure and no compressible effects and so on in this paper a stability requirement is introduced as a new criterion for the constructions with this requirement we derive some relations of parameters for several lattice boltzmann models interestingly these relations are satisfied by many choices of parameters used in the literature

the maximum spectral radius of c4 free graphs of given order and size suppose that g is a graph with n vertices and m edges and let be the spectral radius of its adjacency matrix recently we showed that if g has no 4 cycle then 2 n 1 with equality if and only if g is the friendship graph here we prove that if m 9 and g has no 4 cycle then 2 m with equality if g is a star for 4 m 8 this assertion fails

weighted mathbb l 2 stability of the lattice boltzmann method this article is concerned with the linearized stability of the lattice boltzmann method both on periodic domains and on bounded domains with the bounce back rule used at the boundaries under a structural hypothesis we prove that a weighted mathbb l 2 norm of the solutions to the linearized lattice boltzmann method is decreasing with time moreover we show that the structural hypothesis holds true for many lattice boltzmann models

on central leaves of hodge type shimura varieties with parahoric level structure kisin and pappas publ math inst hautes etudes sci 2018 constructed integral models of hodge type shimura varieties with parahoric level structure at p 2 such that the formal neighbourhood of a mod p point can be interpreted as a deformation space of p divisible group with some tate cycles generalising faltings construction in this paper we study the central leaf and the closed newton stratum in the formal neighbourhoods of mod p points of kisin pappas integral models with parahoric level structure namely we obtain the dimension of central leaves and the almost product structure of newton strata in the case of hyperspecial level structure i e in the good reduction case our main results were already obtained by hamacher math z 287 3 4 1255 1277 2017 and we show that the result of this paper holds for ramified groups as well

on maximal entries in the principal eigenvector of graphs abstract for an undirected connected graph it is well known that an eigenvector belonging to the principal eigenvalue of g can be given such that all entries are positive we ask whether this vector carries information on the structure of the graph and approach this question by investigating the values that can occur in its maximal entry

the spectral radius of graphs without paths and cycles of specified length abstract let g be a graph with n vertices and g be the largest eigenvalue of the adjacency matrix of g we study how large g can be when g does not contain cycles and paths of specified order in particular we determine the maximum spectral radius of graphs without paths of given length and give tight bounds on the spectral radius of graphs without given even cycles we also raise a number of open problems

on the stability structure for lattice boltzmann schemes the stability structure for lattice boltzmann schemes has been introduced in banda et al 2006 16 junk and yong 2007 14 to analyze the stability of numerical algorithms the first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis its second goal is to illustrate and apply the results to different classes of lattice boltzmann collision operators in particular we formulate an equivalence condition just recently also reported in yong 2008 18 that guarantees the existence of a pre stability structure it is then illustrated by several examples how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators finally we point out some shortcomings of the stability structure approach arising in certain cases

projectivity of the witt vector affine grassmannian we prove that the witt vector affine grassmannian which parametrizes w k lattices in w k frac 1 p n for a perfect field k of characteristic p is representable by an ind perfect scheme over k this improves on previous results of zhu by constructing a natural ample line bundle along the way we establish various foundational results on perfect schemes notably h descent results for vector bundles

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

analytic methods for uniform hypergraphs abstract this paper presents some analytic methods for studying uniform hypergraphs its starting point is the spectral theory of 2 graphs in particular the largest and the smallest eigenvalues and min of 2 graphs first these two parameters are extended to weighted uniform hypergraphs second the eigenvalues numbers and min are extended to eigenvalues functions p and min p which also encompass other graph parameters like the lagrangian and the number of edges in this way the functions p and min p seamlessly join spectral and traditional results in hypergraphs in particular this new viewpoint helps to show that spectral extremal and edge extremal problems are asymptotically equivalent naturally all results about p and min p also extend spectral hypergraph theory but delve into deeper problems than before in fact the resulting theory is new even for 2 graphs where some well settled topics become research challenges again the paper covers a multitude of topics with more than a hundred concrete statements to underpin an analytic theory for hypergraphs essential among these topics are a perron frobenius type theory and methods for extremal hypergraph problems many open problems are raised and directions for possible further research are outlined

a spectral condition for odd cycles in graphs abstract let g be a graph of sufficiently large order n and let the largest eigenvalue g of its adjacency matrix satisfies g n 2 4 then g contains a cycle of length t for every t n 320 this condition is sharp the complete bipartite graph t 2 n with parts of size n 2 and n 2 contains no odd cycles and its largest eigenvalue is equal to n 2 4 this condition is stable if g is close to n 2 4 and g fails to contain a cycle of length t for some t n 321 then g resembles t 2 n

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

a structured approach to the construction of stable linear lattice boltzmann collision operator abstract we introduce a structured approach to the construction of linear bgk type collision operators ensuring that the resulting lattice boltzmann methods are stable with respect to a weighted l 2 norm the results hold for particular boundary conditions including periodic bounce back and bounce back with flipping of sign boundary conditions this construction uses the equivalent moment space definition of bgk type collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency the presented structured approach is then applied to the 3d isothermal linearized euler equations with non vanishing background velocity finally convergence results in the strong discrete l norm highlight the suitability of the structured approach introduced in this manuscript

eigenvalues and forbidden subgraphs i abstract suppose a graph g have n vertices m edges and t triangles letting n g be the largest eigenvalue of the laplacian of g and n g be the smallest eigenvalue of its adjacency matrix we prove that n g 2 m 2 3 nt m n 2 2 m n n g 3 n 3 t 4 m 3 nm n 2 2 m with equality if and only if g is a regular complete multipartite graph moreover if g is k r 1 free then n g 2 mn r 1 n 2 2 m with equality if and only if g is a regular complete r partite graph

spectral conditions for the existence of specified paths and cycles in graphs abstract let g be a graph with n vertices and n g be the least eigenvalue of its adjacency matrix of g in this paper we give sharp bounds on the least eigenvalue of graphs without given paths or cycles and determine the extremal graphs this result gives spectral conditions for the existence of specified paths and cycles in graphs

l adic etale cohomology of shimura varieties of hodge type with non trivial coefficients let mathsf g mathsf x be a shimura datum of hodge type let p be an odd prime such that mathsf g mathbb q p splits after a tamely ramified extension and p not mid pi 1 mathsf g mathrm der under some mild additional assumptions that are satisfied if the associated shimura variety is proper and mathsf g mathbb q p is either unramified or residually split we prove the generalisation of mantovan s formula for the l adic cohomology of the associated shimura variety on the way we derive some new results about the geometry of the newton stratification of the reduction modulo p of the kisin pappas integral model

bounds on graph eigenvalues i abstract we prove three results about the spectral radius g of a graph g a let t r n be the r partite turan graph of order n if g is a k r 1 free graph of order n then g t r n unless g t r n b for most irregular graphs g of order n and size m g 2 m n 1 2 m 2 n c let 0 k l if g is a graph of order n with no k 2 k k 1 and no k 2 l 1 then g min g k l 1 k l 1 2 4 l n 1 2