complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

a local level set method for paraxial geometrical optics we propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two dimensional 2 d acoustic wave equations the geometrical optics term consists of two multivalued functions a travel time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally the multivalued travel times are obtained by solving a level set equation and a travel time equation with a forcing term the multivalued amplitudes are computed by a neweulerian formula based on the gradients of travel times and takeoff angles as a byproduct the method is also able to capture the caustic locations the proposed eulerian method has complexity of o n 2 rm log n rather than o n 4 as typically seen in the lagrangian ray tracing method several examples including the well known marmousi synthetic model illustrate the accuracy and efficiency of the eulerian method

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

on the choice of a conformal gauss gauge near the cylinder representing spatial infinity a convenient approach to analyze spatial infinity is to use a cylinder representation i and impose a gauge based on a congruence of conformal geodesics this so called conformal gauss gauge comes along with the freedom to specify initial data for the conformal geodesics such a gauge has been constructed from an ordinary cauchy surface and from past null infinity i respectively the purpose of this note is to compare these gauges near the critical set i where i touches i as it turns out that they are related in a somewhat unexpected intricate way a convenient approach to analyze spatial infinity is to use a cylinder representation i and impose a gauge based on a congruence of conformal geodesics this so called conformal gauss gauge comes along with the freedom to specify initial data for the conformal geodesics such a gauge has been constructed from an ordinary cauchy surface and from past null infinity i respectively the purpose of this note is to compare these gauges near the critical set i where i touches i as it turns out that they are related in a somewhat unexpected intricate way

on the symplectic structure of general relativity the relation between the symplectic structures on the canonical and radiative phase spaces of general relativity is exhibited

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

kleiner s theorem for unitary representations of posets abstract a subspace representation of a poset s s 1 s t is given by a system v v 1 v t consisting of a vector space v and its subspaces v i such that v i v j if s i s j for each real valued vector 1 t with positive components we define a unitary representation of s as a system u u 1 u t that consists of a unitary space u and its subspaces u i such that u i u j if s i s j and satisfies 1 p 1 t p t 1 in which p i is the orthogonal projection onto u i we prove that s has a finite number of unitarily nonequivalent indecomposable representations for each weight if and only if s has a finite number of nonequivalent indecomposable subspace representations that is if and only if s contains any of kleiner s critical posets

on the regular and the asymptotic characteristic initial value problem for einstein s vacuum field equations the regular characteristic initial value problem for einstein s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations this is achieved by making use of the spin frame formalism instead of the harmonic gauge condition the method is applied to the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on part of past null infinity and on an incoming null hypersurface a uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

cauchy problems for the conformal vacuum field equations in general relativity cauchy problems for einstein s conformal vacuum field equations are reduced to cauchy problems for first order quasilinear symmetric hyperbolic systems the hyperboloidal initial value problem where cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two surface is discussed and translated into the conformally related picture it is shown that for conformal hyperboloidal initial data of classhs s 4 there is a unique up to questions of extensibility development which is a solution of the conformal vacuum field equations of classhs it provides a solution of einstein s vacuum field equations which has a smooth structure at past null infinity

on the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes we analyze the appearance of logarithmic terms at the critical sets of friedrich s cylinder representation of spatial infinity it is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions conversely it is proved that under the additional hypothesis that the spacetime has constant adm mass aspect and vanishing dual adm mass aspect by which we mean the limit of the bondi dual mass aspect on to the critical set this condition is also necessary for a spacetime to admit a smooth conformal representation at the critical sets

a rigidity property of asymptotically simple spacetimes arising from conformally flat data given a time symmetric initial data set for the vacuum einstein field equations which is conformally flat it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data coincides with schwarzschild data

on the existence of analytic null asymptotically flat solutions of einstein s vacuum field equations this paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for einstein s field equations for analytic data on past null infinity and on an incoming null hypersurface

estimates for the characteristic problem of the first order reduction of the wave equation we calculate certain estimates for the solution of the characteristic problem of the wave equation reduced to first order in terms of the free data prescribed on two transverse surfaces one of which is characteristic estimates of such kind ensure the stability of the solutions under small variations of the data similar estimates exist for the derivatives of the solution as well

exotic hyperbolic gluings we carry out exotic gluings a la carlotto schoen for asymptotically hyperbolic general relativistic initial data sets in particular we obtain a direct construction of non trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity preprint uwthph 2015 31

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

on the existence of c infinity solutions to the asymptotic characteristic initial value problem in general relativity the asymptotic characteristic initial value problem for einstein s vacuum field equations is treated it is shown that c infty solutions exist and are unique for c infty initial values the proof is based on friedrich s regular conformal vacuum field equations and rendall s method of reducing the characteristic to an ordinary initial value problem

on the hyperbolicity of einstein s and other gauge field equations it is shown that einstein s vacuum field equations respectively the conformal vacuum field equations in a frame formalism imply a symmetric hyperbolic system of reduced propagation equations for any choice of coordinate system and frame field and conformal factor certain freely specifiable gauge source functions occurring in the reduced equations reflect the choice of gauge together with the initial data they determine the gauge uniquely their choice does not affect the isometry class conformal class of a solution of an initial value problem by the same method symmetric hyperbolic propagation equations are obtained from other gauge field equations irrespective of the gauge using the concept of source functions one finds that einstein s field equation considered as second order equations for the metric coefficients are of wave equation type in any coordinate system

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

a new class of obstructions to the smoothness of null infinity expansions of the gravitational field arising from the development of asymptotically euclidean time symmetric conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6 to this end a certain representation of spatial infinity as a cylinder is used this setup is based on the properties of conformal geodesics it is found that these expansions suggest that null infinity has to be non smooth unless the newman penrose constants of the spacetime and some other higher order quantities of the spacetime vanish as a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders furthermore the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity then the initial data must in fact be schwarzschildean around spatial infinity

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

numerical relativity as a tool for computational astrophysics abstract the astrophysics of compact objects which requires einstein s theory of general relativity for understanding phenomena such as black holes and neutron stars is attracting increasing attention in general relativity gravity is governed by an extremely complex set of coupled nonlinear hyperbolic elliptic partial differential equations the largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of einstein s equations for the first time since they were written over 80 years ago allowing one to attempt full 3d simulations of such exciting events as colliding black holes and neutron stars in this paper we review the computational effort in this direction and discuss a new 3d multi purpose parallel code called cactus for general relativistic astrophysics directions for further work are indicated where appropriate

the asymptotic characteristic initial value problem for einstein s vacuum field equations as an initial value problem for a first order quasilinear symmetric hyperbolic system the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown it is delineated how the same method can be applied to the standard cauchy problems for einstein s vacuum and conformal vacuum equations