resolutions of fat point ideals involving eight general points of p2 abstract the main result theorem 1 provides an algorithm for determining the minimal free resolution of fat point subschemes of p 2 involving up to eight general points of arbitrary multiplicities the resolutions obtained hold for any algebraically closed field independent of the characteristic the algorithm works by giving a formula in nice cases and a reduction to the nice cases otherwise the algorithm which does not involve grobner bases is very fast partial information is also obtained in certain cases with n 8

conditions imposed by tacnodes and cusps the study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress the horace method introduced by a hirschowitz has been successfully exploited to prove many h1 vanishing theorems even in higher dimension other specialization techniques which include degenerations of the plane are due to z ran and c ciliberto and r miranda g m greuel c lossen and e shustin use a local specialization procedure together with the horace method to give the first asymptotically proper general existence criterion for singular curves of low degree in this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by greuel lossen and shustin for curves with tacnodes and cusps

on bounding the number of generators for fat point ideals on the projective plane abstract let x be the surface obtained by blowing up general points p1 pn of the projective plane over an algebraically closed ground field k and let l be the pullback to x of a line on the plane if c is a rational curve on x with c l d then for every t there is a natural map c o c t x o x l c o c t d given by multiplication on simple tensors the ranks of such maps are determined as a function of t d and m where m is the largest multiplicity of c at any of the points pi if i is the ideal defining the fat point subscheme z m1p1 mnpn p2 and is the least degree in which i has generators then the ranks of the maps c o c t x o x l c o c t d can be used for bounding the number of generators of i in degrees t 1

on a class of special linear systems of mathbb p 3 in this paper we deal with linear systems of p 3 through fat points we consider the behavior of these systems under a cubic cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones

on the cohen macaulay type of s lines in an 1 abstract let a be the coordinate ring of s lines through the origin in a n 1 k we discuss what it means for these lines to be in sufficiently general position and then with this restriction on the lines we attempt to verify our belief that the cohen macaulay type of a depends only on s and n we characterize those s and n for which a is a gorenstein ring i e of c m type 1 and explicitly calculate the cohen macaulay type in several other cases we then extend some of our results to the case of an arbitrary reduced curve whose tangent cone consists of lines in sufficiently general position finally we calculate the hubert samuel polynomials of the curves we have been considering

linear systems of plane curves through fixed fat points of p2 abstract given any s points p1 ps in p 2 and s positive integers m1 ms let sn be the linear system of plane curves of degree n through pi with multiplicity at least mi 1 i s we give numerical bounds for the regularity of sn in the following cases a the points pi are non singular points of an integral curve of degree d b the pi s are in general position c the pi s are in uniform position d the pi s are generic points of p 2 we also study the sharpness of such bounds

reduced order models based on local pod plus galerkin projection a method is presented to accelerate numerical simulations on parabolic problems using a numerical code and a galerkin system obtained via pod plus galerkin projection on a sequence of interspersed intervals the lengths of these intervals are chosen according to several basic ideas that include an a priori estimate of the error of the galerkin approximation several improvements are introduced that reduce computational complexity and deal with a updating the pod manifold instead of calculating it at the end of each galerkin interval b using only a limited number of mesh points to calculate the right hand side of the galerkin system and c introducing a second error estimate based on a second galerkin system to account for situations in which qualitative changes in the dynamics occur during the application of the galerkin system the resulting method called local pod plus galerkin projection method turns out to be both robust and efficient for illustration we consider a time dependent fisher like equation and a complex ginzburg landau equation

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

regularity of linear systems of plane curves the subject of this paper is the study of linear systems of plane curves which are given by assigning a set of fixed multiple points this study not only has an interest per se but is connected to other subjects for example results on rational surfaces and curves on them can be found in ro a and gil a linear system of plane curves of projective dimension r is a subvector space i s

on nagata s conjecture abstract this paper gives an improved lower bound on the degrees d such that for general points p 1 p n p 2 and m 0 there is a plane curve of degree d vanishing at each p i with multiplicity at least m

on linear systems of p 3 through multiple points abstract we prove a conjecture about the dimension of linear systems of surfaces of degree d in p 3 through at most eight multiple points in general position

the geometric interpretation of froberg iarrobino conjectures on infinitesimal neighbourhoods of points in projective space abstract the study of infinitesimal deformations of a variety embedded in projective space requires at ground level that of deformation of a collection of points as specified by a zero dimensional scheme further basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes an optimal hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of r froberg and a iarrobino we discuss these conjectures from a geometric point of view they give for each such collection a function based on dimension number of points and order of each neighbourhood which should serve as an upper bound to its hilbert function weak conjecture the strong conjecture predicts when the upper bound is sharp in the case of equal order throughout in general we refer to the equality of the hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the strong hypothesis we interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points we develop techniques and insight toward the conjectures verification and refinement the main result gives an upper bound on the hilbert function of a collection of infinitesimal neighbourhoods in p n based on hilbert functions of certain such subschemes of p n 1 further equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand it follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis this observation is then applied to show that the weak conjecture does hold valid in p n for n 3 the main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in p 2 and further p 2 presents special exceptional cases consequences of the main result in higher dimension are then examined we note then that the full weight of the strong conjecture and validity of the strong hypothesis are not necessary toward using the main theorem in the next dimension we end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information namely on the structure of the minimal free resolution of an ideal generated by linear forms

acceleration techniques for reduced order models based on proper orthogonal decomposition this paper presents several acceleration techniques for reduced order models based on the proper orthogonal decomposition pod method the techniques proposed herein are i an algorithm for splitting the database of snapshots generated by the full order model ii a method for solving quasi symmetrical matrices iii a strategy for reducing the frequency of the projection the acceleration techniques were applied to a pod based reduced order model of the two phase flows in fluidized beds this reduced order model was developed using numerical results from a full order computational fluid dynamics model of a two dimensional fluidized bed using these acceleration techniques the computational time of the pod model was two orders of magnitude shorter than the full order model

linear systems of plane curves with base points of equal multiplicity in this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m a conjecture of harbourne and hirschowitz implies that when d 3m the dimension is equal to the expected dimension given by the riemann roch theorem also systems for which the dimension is larger than expected should have a fixed part containing a multiple 1 curve we reformulate this conjecture by explicitly listing those systems which have unexpected dimension then we use a degeneration technique developed to show that the conjecture holds for all m 12

linear systems of plane curves with a composite number of base points of equal multiplicity in this article we study linear systems of plane curves of degree d passing through general base points with the same multiplicity at each of them these systems are known as homogeneous linear systems we especially investigate for which of these systems the base points with their multiplicities impose independent conditions and which homogeneous systems are empty such systems are called non special we extend the range of homogeneous linear systems that are known to be non special a theorem of evain states that the systems of curves of degree d with 4 h base points with equal multiplicity are non special the analogous result for 9 h points was conjectured both of these will follow as corollaries from the main theorem proved in this paper also the case of 4 h 9 k points will follow from our result the proof uses a degeneration technique developed by c ciliberto and r miranda

dimension theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes introduction let k be an algebraically closed ground field of characteristic zero the following result is well known see severi 1 anhang f p 317 if en d is a complete i e maximal irreducible algebraic system defined over k of plane algebraic not necessarily irreducible curves of a given order n such that the general curve c of en d k has d nodes and no other singularities then the dimension of en d is equal to 3n p 1 where p n 1 n 2 2 d is the effective genus of c we note that 3n p 1 is equal to n n 3 2 d here n n n 3 2 is the dimension of the linear system of all plane curves of order n this is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes in non assigned position imposes d independent algebraic non linear conditions on the curve the above result is easily proved and will be included by us in a more general theorem the proof of which is the main object of this paper theorem 2 section 3 and which deals with irreducible algebraic systems e of curves of order n in which the general curve c of e k has arbitrary singularities but is free from multiple components equivalently c is a reduced curve this theorem asserts that if p is the effective genus of c the definition of the effective genus of a reducible but reduced curve is given in section 3 formula 2 then dim e c 3n p 1 with equality if and only if c has only nodes as singularities it is easily seen see severi loc cit that if 0 c d n n 1 2 then there always exists curves of order n having d nodes the maximum

positivity of line bundles on general blow ups of p2 abstract let x be the blow up of p 2 at r general points p 1 p r p 2 we study line bundles on x given by plane curves of degree d passing through p i with multiplicity at least m i motivated by results in 39 we establish conditions for ampleness very ampleness and global generation of such line bundles

breakdown free version of ilu factorization for nonsymmetric positive definite matrices in this paper a new ilu factorization preconditioner for solving large sparse linear systems by iterative methods is presented the factorization which is based on a biorthogonalization process is well defined for a general positive definite matrix numerical experiments illustrating the performance of the preconditioner are presented a comparison with the well known preconditioner rif p of benzi and tuma is also included

new breakdown free variant of ainv method for nonsymmetric positive definite matrices this paper proposes a new breakdown free preconditioning technique called sainv ns of the ainv method of benzi and tuma for nonsymmetric positive definite matrices the resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for qmr bicgstab and gmres m methods the preconditoner is reliable pivot breakdown can not occur and effective at reducing the number of iterations some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the ainv a algorithm

plane curves of minimal degree with prescribed singularities we prove that there exists a positive such that for any integer d 3 and any topological types s1 sn of plane curve singularities satisfying

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

constrained reduced order models based on proper orthogonal decomposition abstract a novel approach is presented to constrain reduced order models rom based on proper orthogonal decomposition pod the karush kuhn tucker kkt conditions were applied to the traditional reduced order model to constrain the solution to user defined bounds the constrained reduced order model c rom was applied and validated against the analytical solution to the first order wave equation c rom was also applied to the analysis of fluidized beds it was shown that the rom and c rom produced accurate results and that c rom was less sensitive to error propagation through time than the rom

nonsingular curves passing through points of p2 in generic position i abstract the least integer is looked for such that there exists a nonsingular curve of degree less than or equal to passing through any assigned set of n points of p 2 in generic position here generic takes different meanings and the generic n tuples of points describe nonempty open subsets u of p 2 n indeed we find i for u 1 n tuples in general position u 1 n 2 ii if n d 2 2 h 0 h d 1 for u2 n tuples in n generic position iii with the notations of ii for u3 n tuples in uniform position in any case a nonempty open subset u u3 is found in which u d 1

error estimation for reduced order models of dynamical systems the use of reduced order models to describe a dynamical system is pervasive in science and engineering often these models are used without an estimate of their error or range of validity in this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition we show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method most importantly the proposed approach allows the assessment of regions of validity for reduced models i e ranges of perturbations in the original system over which the reduced model is still appropriate numerical examples validate our approach the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude

reduced order modeling of unsteady viscous flow in a compressor cascade a simultaneously coupled viscous inviscid interaction vii analysis is used to model the unsteady viscous separated flow through a subsonic compressor the inner viscous flow around the airfoil and in the wake is modeled using a finite difference discretization of the boundary layer equations and a one equation turbulence transport model the outer inviscid flow is modeled using a variational finite element discretization of the compressible full potential equation the viscous and inviscid regions are simultaneously coupled using a injection type boundary condition along the airfoil and wake the resulting nonlinear unsteady equations are linearized about the nonlinear steady flow to obtain a set of linear equations that describe the unsteady small disturbance behavior of the viscous flow through the cascade the discretized small disturbance vii equations are used to form a generalized quadratic non hermitian eigenvalue problem that describes the eigenmodes natural modes and eigenvalues natural frequencies of fluid motion about the cascade using a lanczos algorithm the eigeninformation is computed efficiently for various steady flow inflow angles and unsteady interblade phase angles

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

robust incomplete factorization for nonsymmetric matrices in this paper a new incomplete lu factorization preconditioner for nonsymmetric matrices is being considered which is also breakdown free no zero pivots occurs for positive definite matrices to construct this preconditioner only the information of matrix a is used and just one of the factors of the ainv process is computed the l factor is extracted as a by product of the ainv process the pivots of the ainv process are used as diagonal entries of u the new preconditioner has left and right looking versions to improve the efficiency of the preconditioner we have used the inverse based dropping strategies for both l and u factors numerical experiments show that the left looking version of the preconditioner is significantly faster than its right looking version in terms of preconditioning time and both are equally effective to reduce the number of iterations comparisons of the new preconditioner with ainv and ilut preconditioners are also presented

an efficient method of solving the navier stokes equations for flow control a new method of solving the navier stokes equations efficiently by reducing their number of modes is proposed in the present paper it is based on the karhunen loeve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system employing these empirical eigenfunctions as basis functions of a galerkin procedure one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena and consequently reduce the navier stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom the present algorithm is well suited for the problems of flow control or optimization where one has to compute the flow field repeatedly using the navier stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables the low dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the navier stokes equation 1998 john wiley sons ltd

eigenanalysis of unsteady flow about airfoils cascades and wings a general technique for constructing reduced order models of unsteady aerodynamic flows about two dimensional isolated airfoils cascades of airfoils and three dimensional wings is developed the starting point is a the domain computational model of the unsteady small disturbance flow for illustration purposes we apply the technique to an unsteady incompressible vortex lattice model the eigenmodes of the system which may be thought of as aerodynamic states are computed and subsequently used to construct computationally efficient reduced order models of the unsteady flowfield only a handful of the most dominant eigenmodes are retained in the reduced order model the effect of the remaining eigenmodes is included approximately using a static correction technique an important advantage of the present method is that once the eigenmode information has been computed reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method finally we show how the reduced order model may be incorporated into an aeroelastic flutter model

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

linear systems on the blow up of p1 n abstract in this note we study linear systems on the blow up of p 1 n at r points in very general position we prove that the fibers of the projections p 1 n p 1 s 1 s n 1 can give contribution to the speciality of the linear system this allows us to give a new definition of expected dimension of a linear system in p 1 n which we call fiber dimension finally we state a conjecture about linear systems on p 1 3

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

resolutions of ideals of quasiuniform fat point subschemes of the notion of a quasiuniform fat point subscheme z p 2 is introduced and conjectures for the hilbert function and minimal free resolution of the ideal i defining z are put forward in a large range of cases it is shown that the hilbert function conjecture implies the resolution conjecture in addition the main result gives the first determination of the resolution of the mth symbolic power i m n of an ideal defining n general points of p 2 when both m and n are large in particular for infinitely many m for each of infinitely many n and for infinitely many n for every m 2 resolutions in other cases such as fat points with tails are also given except where an explicit exception is made all results hold for an arbitrary algebraically closed field k as an incidental result a bound for the regularity of i m n is given which is often a significant improvement on previously known bounds

a robust ilu with pivoting based on monitoring the growth of the inverse factors abstract an incomplete lu decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of l u the information on the growth of the inverse factors is used as feedback for dropping entries in l and u this method often yields a robust and effective preconditioner especially when the system is highly indefinite numerical examples demonstrate the effectiveness of this approach

analytic order of singular and critical points we deal with the following closely related problems i for a germ of a reduced plane analytic curve what is the minimal degree of an algebraic curve with a singular point analytically equivalent isomorphic to the given one ii for a germ of a holomorphic function in two variables with an isolated critical point what is the minimal degree of a polynomial equivalent to the given function up to a local holomorphic coordinate change classically known estimates for such a degree d in these questions are 1 d 1 where is the milnor number our result in both the problems is d a with an absolute constant a as a corollary we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces

karhunen loeve decomposition approach to analyzing complex network synchronization in this paper the approach of the karhunen lo ve decomposition known also as the proper orthogonal modes poms is taken to analyze phase synchronization of various complex networks with different topologies namely the classic kuramoto model coupled chaotic maps with gaussian delays and a chain of diffusively coupled bistable oscillators in the case of the kuramoto model the poms reveal the tendency and the level of synchronization with the increase of the coupling strength for globally coupled networks and scale free networks while periodic poms are found in nearest neighbor coupled networks furthermore for cluster networks on the kuramoto model the first leading poms based on different time intervals reveal that different sub groups of nodes synchronize gradually to different levels eventually leading to the complete phase synchronization in the case of coupled chaotic maps some properties of phase synchronization change with the coupling strength value in the case of the chain of diffusively coupled bistable oscillators several main poms not only determine the network phase synchronization but also provide good reconstruction of the network responses

betti numbers for fat point ideals in the plane a geometric approach we consider the open problem of determining the graded betti numbers for fat point subschemes z supported at general points of p 2 we relate this problem to the open geometric problem of determining the splitting type of the pullback of p2 to the normalization of certain rational plane curves we give a conjecture for the graded betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of p 2 we also prove our betti number conjecture in a broad range of cases an appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded betti numbers in certain degrees it also demonstrates how to derive explicit conjectural values for the betti numbers and how to compute splitting types

a method to generate computationally efficient reduced order models a new method is presented to generate reduced order models roms in fluid dynamics problems the method is based on the expansion of the flow variables on a proper orthogonal decomposition pod basis calculated from a limited number of snapshots which are obtained via computational fluid dynamics cfd then the pod mode amplitudes are calculated as minimizers of a properly defined overall residual of the equations and boundary conditions the residual can be calculated using only a limited number of points in the flow field which can be scattered either all over the whole computational domain or over a smaller projection window this means that the process is both computationally efficient reconstructed flow fields require less than 1 of the time needed to compute a full cfd solution and flexible the projection window can avoid regions of large localized cfd errors also various definitions of the residual are briefly discussed along with the number and distribution of snapshots the number of retained modes and the effect of cfd errors to conclude that the method is numerically robust this is because the results are largely insensitive to the definition of the residual to cfd errors and to the cfd method itself which may contain artificial stabilizing terms thus the method is amenable for practical engineering applications

crout versions of ilu for general sparse matrices this paper presents an efficient implementation of the incomplete lu ilu factorization derived from the crout version of gaussian elimination at step k of the elimination the kth row of u and the kth column of l are computed using previously computed rows of u and columns of l the data structure and implementation borrow from already known techniques used in developing both sparse direct solution codes and incomplete cholesky factorizations this version of ilu can be computed much faster than standard threshold based ilu factorizations computed rowwise or columnwise in addition the data structure allows efficient implementations of more rigorous and effective dropping strategies

seshadri constants and very ample divisors on algebraic surfaces a broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given it works for all surfaces in all characteristics for any n this construction is used to obtain substantial improvements for currently known lower bounds for n point seshadri constants remarks are included about a range of applications to classical problems involving linear systems on p 2