exotic elliptic algebras of dimension 4 abstract let e be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3 let be a translation automorphism of e that is not of order 2 in a previous paper we studied an algebra a a e that depends on this data a e s e m 2 k where s e is the 4 dimensional sklyanin algebra associated to e m 2 k is the ring of 2 2 matrices over k and is z 2 z 2 acting in a particular way as automorphisms of s and m 2 k the action of on s is compatible with the translation action of the 2 torsion subgroup e 2 on e following the ideas and results in papers of artin tate van den bergh smith stafford and levasseur smith this paper examines the line modules point modules and fat point modules over a and their incidence relations the right context for the results is non commutative algebraic geometry we view a as a homogeneous coordinate ring of a non commutative analogue of p 3 that we denote by proj n c a point modules and fat point modules determine points in proj n c a line modules determine lines in proj n c a line modules for a are in bijection with certain lines in p a 1 p 3 and therefore correspond to the closed points of a certain subscheme l of the grassmannian g 1 3 shelton vancliff call l the line scheme for a we show that l is the union of 7 reduced and irreducible components 3 quartic elliptic space curves and 4 plane conics in the ambient plucker p 5 and that deg l 20 the union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in p a 1 thus the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for a

numerical simulations for radiation hydrodynamics a finite difference scheme is proposed for two dimensional radiation hydrodynamical equations in the transport limit the scheme is of godunov type in which the set of time averaged flux needed in the scheme is calculated through riemann problems solved in the scheme flow signals are explicitly treated while radiation signals are implicitly treated flow fields and radiation fields are updated simultaneously an iterative approach is proposed to solve the set of nonlinear algebraic equations arising from the implicitness of the scheme the sweeping method used in the scheme significantly reduces the number of iterations or computer cpu time needed a new approach to further accelerate the convergence is proposed which further reduces the number of iterations needed by more than one order no matter how many cells radiation signals propagate in one time step only an extremely small number of iterations are needed in the scheme and each iteration costs only about 0 8 percent of computer cpu time which is needed for one time step of a second order accurate and fully explicit scheme two dimensional problems are treated through a dimensionally split technique therefore iterations for solving the set of algebraic equations are carried out only in each one dimensional sweep through numerical examples it is shown that the scheme keeps the principle advantages of godunov schemes for flow motion in the time scale of flow motion numerical results are the same as those obtained from a second order accurate and fully explicit scheme the acceleration of the convergence proposed in this paper may be directly applied to other hyperbolic systems

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

spaces of matrices of constant rank and uniform vector bundles abstract we consider the problem of determining l r a the maximal dimension of a subspace of a a matrices of rank r we first review in the language of vector bundles the known results then using known facts on uniform bundles we prove some new results and make a conjecture finally we determine l r a for every r 1 r a when a 10 showing that our conjecture holds true in this range

central singularities of quantum spaces abstract let a be a positively graded connected affine c algebra generated in degree one which is a finite module over a central subring r let a be the structure sheaf over proj r and spec z the central proj if injdim a spec z has singularities unless the ramification locus of a is pure of codimension one if gldim a a and the singular locus of spec z coincide

efficient solution algorithms for the riemann problem for real gases a procedure for constructing solutions to the riemann problem for gas dynamics with a general convex equation of state is given approximate procedures involving a local parametrization of the equation of state are introduced in order to calculate numerical fluxes in conservative finite difference schemes this leads to difference schemes which are as accurate and almost as fast as the analogous schemes for polytropic gases numerical results in one and two space variables are presented

galois extensions and subspaces of alternating bilinear forms with special rank properties abstract let k be a field admitting a cyclic galois extension of degree n the main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over k we show that this space of forms is the direct sum of n 1 2 subspaces each of dimension n and the non zero elements in each subspace have constant rank defined in terms of the orders of the galois automorphisms furthermore if ordered correctly for each integer k lying between 1 and n 1 2 the rank of any non zero element in the sum of the first k subspaces is at most n 2 k 1 slightly less sharp similar results hold for cyclic extensions of even degree

upwind differencing and lu factorization for chemical non equilibrium navier stokes equations abstract an efficient and robust upwind method for solving the chemical non equilibrium navier stokes equations has been developed the method uses either the roe or van leer flux splitting for inviscid terms and central differencing for viscous terms in the explicit operator residual and the steger warming sw splitting and lower upper lu approximate factorization for the implicit operator this approach is efficient since the sw lu combination requires the inversion of only block diagonal matrices as opposed to the block tridiagonal inversion of the widely used adi method and is fully vectorizable the lu method is particularly advantageous for systems with large number of equations such as for chemical and thermal nonequilibrium flow formulas of the numerical method are presented for the finite volume discretization of the navier stokes equations in general coordinates numerical tests in hypersonic blunt body ramped duct shock wave boundary layer interaction and divergent nozzle flows demonstrate the efficiency and robustness of the present method

space marching method for calculating steady supersonic flows on a grid adapted to the solution a noniterative implicit space marching method based on an adaptive grid approach is developed for solving the 2d steady state euler equations describing supersonic gas flows without streamwise separation a grid adapted to the solution is generated by using a variational method optimizing such important properties of a grid as smoothness orthogonality and grid cell volume variation simultaneously the weight function in the integral of adaptation is constructed so that the grid lines are accumulated near strong gradients and curvatures of the solution curve a special treatment of the boundary points is proposed to determine the location of nodes near slope discontinuities of the boundary mesh line the noniterative implicit space marching method of y c vigneronet al is applied to solve the euler equations written in an arbitrary curvilinear system of coordinates the streamwise and transverse derivatives in the governing equations are approximated by the second order upwind richardson and second order symmetric tvd schemes respectively implicit boundary condition procedure based on the theory of characteristics for hyperbolic systems of equations is employed numerical calculations show that the resolution of strong gradient flow fields can significantly be improved by using the grid adapted to the solution while the number of grid nodes is the same

on the dimension of spaces of linear transformations satisfying rank conditions abstract let l c n c m be the space from c n to c m let l k n m be the maximal dimension of a linear subspace v of l c n c m such that every nonzero map in v has rank k in this paper we develop some techniques for estimating l from above

classifying quadratic quantum p2s by using graded skew clifford algebras abstract we prove that quadratic regular algebras of global dimension three on degree one generators are related to graded skew clifford algebras in particular we prove that almost all such algebras may be constructed as a twist of either a regular graded skew clifford algebra or of an ore extension of a regular graded skew clifford algebra of global dimension two in so doing we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in p 2

constructing clifford quantum p3s with finitely many points abstract we present an algebro geometric technique for constructing regular clifford algebras a of global dimension four with associated point scheme consisting of a prespecified finite number of points in particular if a has more than one point in its point scheme then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors these cubic divisors correspond to regular clifford subalgebras of a of global dimension three if a has exactly a finite number n of distinct points in its point scheme then n 1 3 4 5 13 14 16 18 20 and all these possibilities occur we also prove that if a regular clifford algebra r of global dimension d 2 has exactly a finite number n of distinct isomorphism classes of point modules then n is odd if and only if r is an ore extension of a regular clifford subalgebra of r of global dimension d 1

on two extremal matrix problems abstract let w be a linear subspace of symmetric n n matrices whose rank is at most t it is shown that if the underlying field has more than n elements then let l be an affine space of n n matrices all having rank at least k it is shown that if the underlying field is algebraically closed then dim l n 2 k 1 2 the last result is applied to a problem of valiant concerning permanents and determinants

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

embedding a quantum nonsingular quadric in a quantum p3 a definition of regularity has been given for non commutative graded algebras and results of artin schelter tate and van den bergh classify the regular algebras of global dimension three that are generated by degree one elements our purpose is to classify a certain class of quadratic regular algebras of global dimension four letsbe a twisted homogeneous coordinate ring of a nonsingular quadricq p3 our interest is in algebrasrsuch that there is an embedding proj s proj r in this paper we classify all the quadratic regular algebrasrof global dimension four which have the same hilbert series as that of the polynomial ring on four variables and which map ontosvia a graded degree zero homomorphism our approach makes use of the point modules ofrand their associated geometric data we classify the algebrasraccording to their point scheme pand corresponding automorphism aut p those algebrasrwhich are determined by p belong to at most a five parameter family but those which are not determined by p belong to at most a four parameter family in the first case pis either p3or consists ofqtogether with a linel while in the second casep q it is also proved that under certain sufficient conditions the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism

star formation with 3 d adaptive mesh refinement the collapse and fragmentation of molecular clouds we describe a powerful methodology for numerical solution of 3 d self gravitational hydrodynamics problems with unprecendented resolution this code utilizes the technique of local adaptive mesh refinement amr employing multiple grids at multiple levels of resolution these grids are automatically and dynamically added and removed as necessary to maintain adequate resolution this technology allows solution of problems that would be prohibitively expensive with methods using fixed resolution and it is more versatile and efficient than competing methods of achieving variable resolution the application of this technique to simulate the collapse and fragmentation of a molecular cloud a key step in star formation is discussed the simulation involves many orders of magnitude of variation in length scale as fragments form in this paper we describe the methodology and present illustrative applications for both isothermal and nonisothermal cloud collapse we describe the numerical jeans condition a new criterion for stability of self gravitational gas dynamic problems we find that the uniformly rotating spherical clouds treated here first collapse to disks in the equatorial plane and then in the presence of applied perturbations form filamentary singularities that do not fragment while isothermal as the collapse enters the non isothermal phase we show the evolutionary sequence that leads to the formation of a binary system consisting of protostellar cores surrounded by distinct protostellar disks the scale of the disks of order 100 au is consistent with observations of gaseous disks surrounding single t tauri stars and debris disks surrounding systems such as beta pictoris

a high order streamline godunov scheme for steady supersonic flow computation abstract a space marching godunov method using control volumes consisted of streamlines and the coordinate lines in the marching direction is described for the computation of two dimensional and axisymmetric steady supersonic flows these streamline meshes are solution generated they follow the fluid particles closely and also simplify the implementation of the godunov method a second order extension to the godunov scheme using essentially nonoscillatory interpolation is also presented computations of various supersonic and hypersonic flow problems indicate that the method accurately represents smooth flow and crisply resolves all flow discontinuities such as shocks and slip lines even at high mach numbers

some finite quantum p3s that are infinite modules over their centers abstract a result of m artin j tate and m van den bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order we prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic noetherian regular algebras a of global dimension four such that i a is an infinite module over its center ii a has a finite point scheme which is the graph of an automorphism of finite order and iii a has a one parameter family of line modules such algebras are candidates for generic regular algebras of global dimension four the methods used to construct the algebras provide new techniques for creating other potential candidates

on koszul algebras and a new construction of artin schelter regular algebras abstract we prove the simple fact that the factor ring of a koszul algebra by a regular normal quadratic element is a koszul algebra this fact leads to a new construction of quadratic artin schelter regular algebras this construction generalizes the construction of artin schelter regular clifford algebras

on second fundamental forms of projective varieties the projective second fundamental form at a generic smooth pointx of a subvarietyx n of projective space n a may be considered as a linear system of quadratic forms ii x on the tangent spacet x x we prove this system is subject to certain restrictions 4 1 including a bound on the dimension of the singular locus of any quadric in the system ii x the only previously known restriction was that ifx is smooth the singular locus of the entire system must be empty one consequence of 4 1 is that smooth subvarieties with 2 a 1 n are such that their third and all higher fundamental forms are zero 4 14 this says that the infinitesimal invariants of such varieties are of the same nature as the invariants of hypersurfaces giving further evidence towards the principle e g h that smooth subvarieties of small codimension should behave like hypersurfaces

linear spaces of matrices of constant rank and instanton bundles abstract we present a new method to study 4 dimensional linear spaces of skew symmetric matrices of constant co rank 2 based on rank 2 vector bundles on p 3 and derived category tools the method allows one to prove the existence of new examples of size 10 10 and 14 14 via instanton bundles of charge 2 and 4 respectively and it provides an explanation for what used to be the only known example westwick 1996 25 we also give an algorithm to construct explicitly a matrix of size 14 of this type

vector spaces of skew symmetric matrices of constant rank abstract we study the orbits of vector spaces of skew symmetric matrices of constant rank 2 r and type n 1 n 1 under the natural action of sl n 1 over an algebraically closed field of characteristic zero we give a complete description of the orbits for vector spaces of dimension 2 relating them to some 1 generic matrices of linear forms we also show that for each rank two vector bundle on p 2 defining a triple veronese embedding of p 2 in g 1 7 there exists a vector space of 8 8 skew symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle

a new lagrangian method for steady supersonic flow computation i godunov scheme abstract this paper studies the problem of steady two dimensional supersonic flow of an inviscid compressible fluid using the new lagrangian formulation of hui and van roessel in which the stream function and the lagrangian time are used as independent variables a shock capturing method is developed by applying the first order godunov scheme to the conservation form equations of this formulation the method is fast and robust furthermore extensive comparisons with exact solutions and with the second order godunov scheme of glaz and wardlaw based on the eulerian formulation show that the first order lagrangian method generally attains the same level of accuracy as the second order eulerian method and is even better in resolving slip line discontinuities

a high order godunov scheme for steady supersonic gas dynamics abstract a high order extension of godunov s method is developed for the equations of steady supersonic flow sample calculations show that the method resolves smooth flow accurately and represents strong shock waves with very little diffusion and no post shock oscillation

generalized laurent polynomial rings as quantum projective 3 spaces abstract given a ring r we introduce the notion of a generalized laurent polynomial ring over r this class includes the generalized weyl algebras we show that these rings inherit many properties from the ground ring r this construction is then used to create two new families of quadratic global dimension four artin schelter regular algebras we show that in most cases the second family has a finite point scheme and a defining automorphism of finite order nonetheless a generic algebra in this family is not finite over its center

a higher order godunov method for the hyperbolic equations modelling shock dynamics a numerical method of solution is developed for steady state hyperbolic equations in conservation form an upwind direction is identified and a smooth mapping is introduced to align the equations with that direction a flux f u in the upwind direction is regarded as the state variable and advanced numerically on a grid according to a second order extension of godunov method the solution u is determined from the computed flux and a constraint on the mapping is that the jacobian det f u is of one sign the method is applied to the equations modelling shock wave propagation according to whitham theory of geometrical shock dynamics in this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times shock propagation in both uniform and non uniform gases is considered and several example calculations are presented a further application of the numerical method is inviscid supersonic flow as governed by the steady euler equations the steady supersonic flow in a converging diverging duct is computed to illustrate the numerical method for this latter application

vector spaces of matrices of low rank in this paper we study vector spaces of matrices all of whose elements have rank at most a given number the problem of classifying such spaces is roughly equivalent to the problem of classifying certain torsion free sheaves on projective spaces we solve this problem in case the sheaf in question has first chern class equal to 1 the characterization of the vector spaces of matrices of rank d 3 due to m d atkinson j ausfral math sot 34 1983 306 315 follows we speculate on the situation for higher rank and chern class free resolutions are used to verify some properties of the low rank examples and to produce an abundance of examples of somewhat higher rank connected with the theory of curves 0 1988 academic

a numerical study of oblique shock wave reflections with experimental comparisons a direct comparison is made for several occurrences of oblique shock wave reflections between interferometric results obtained at the university of toronto institute for aerospace studies utias 10 cm x 18 cm hypervelocity shock tube and numerical results obtained by using a current computational method for solving the euler equations very good qualitative agreement is obtained for equilibrium and frozen flow fields except in small regions where the experiments were dominated by viscous flow the quantitative agreement is very close in some cases but can be out by 10 15 in cases with non equilibrium flow or viscous structures or both additional parametrized sequences of calculations are presented to assess the utility of the present numerical method in constructing the various reflection transition lines for perfect inviscid flows in the shock wave mach number wedge angle m s theta w plane and the validity of the boundary layer defect theory

on two parameter families of symmetric matrices abstract lef f be a homogeneous polynomial mapping of degree d from c 2 into the space of complex n n symmetric matrices it is shown that if rank f x k for all 0 x c 2 then either k is even or d is even

counterexamples to additivity of minimum output p renyi entropy for p close to 0 complementing recent progress on the additivity conjecture of quantum information theory showing that the minimum output p renyi entropies of channels are not generally additive for p 1 we demonstrate here by a careful random selection argument that also at p 0 and consequently for sufficiently small p there exist counterexamples

on the dimension of some real spaces of bounded rank matrices abstract given n n let x be either the set of hermitian or real n n matrices of rank at least n 1 if n is even we give a sharp estimate on the maximal dimension of a real vector space v x 0 the results are obtained via k theory by studying a bundle map induced by the adjunction of matrices

on the dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

representation theory of two families of quantum projective 3 spaces abstract stephenson and vancliff recently introduced two families of quantum projective 3 spaces quadratic and artin schelter regular algebras of global dimension 4 which have the property that the associated automorphism of the scheme of point modules is finite order and yet the algebra is not finite over its center this is in stark contrast to theorems of artin tate and van den bergh in global dimension 3 we analyze the representation theory of these algebras we classify all of the finite dimensional simple modules and describe some zero dimensional elements of proj i e so called fat point modules in particular we observe that the shift functor on zero dimensional elements of proj which is closely related to the above automorphism actually has infinite order

point modules over regular graded skew clifford algebras abstract results of vancliff van rompay and willaert in 1998 8 prove that point modules over a regular graded clifford algebra gca are determined by commutative quadrics of rank at most two that belong to the quadric system associated to the gca in 2010 in 4 cassidy and vancliff generalized the notion of a gca to that of a graded skew clifford algebra gsca the results in this article show that the results of 8 may be extended with suitable modification to gscas in particular using the notion of rank introduced recently by the authors in 9 the point modules over a regular gsca are determined by noncommutative quadrics of rank at most two that belong to the noncommutative quadric system associated to the gsca

a split matrix runge kutta type space marching procedure abstract a new space marching method for the integration of the euler equations is developed it is robust and can deal with more complicated configurations the euler equations are formulated quasi conservatively and a split matrix method is used in order to apply a proper upwind discretisation the eigenvalues and the eigenvectors of the rather complicated hyperbolic system in space are calculated the bow shock is fitted using the rankine hugoniot equations together with the suitable characteristic compatibility equation the calculation of the flow variables at the wall is carried out by using a modification of the post correction technique which is explained in detail in this paper the finite difference equations have been solved by applying a three step runge kutta scheme which was also successfully used for the solution of the time dependent euler equations the validation of the method is done by a careful discussion and a detailed comparison of the results with the ones of other methods some three dimensional applications will be given

the connectedness of symmetric and skew symmetric degeneracy loci even ranks a degeneracy locus is the set of points where a vector bundle map has rank at most a given integer such a set is symmetric or skew symmetric according as whether the vector bundle map is symmetric or skew symmetric we prove a connectedness result first conjectured by fulton and lazarsfeld for skew symmetric degeneracy loci and for symmetric degeneracy loci of even ranks

a note on spaces of symmetric matrices we calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank generalizing a well known result of adams et al

flux coordinate splitting technique for the navier stokes equations the method of fractional steps is used to split the unsteady two dimensional navier stokes equations into a set of euler equations and a diffusion type equation a flux coordinate splitting technique is used to solve the euler equations so that shocks and contact discontinuities are resolved sharply in gas dynamic flows the technique involves solving the euler equations everywhere in the flowfield using a flux vector splitting technique except in small regions surrounding the contact discontinuities