linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

excellent algebraic groups i abstract the notion of an excellent quadratic form has been introduced by m knebusch proc london math soc 33 1976 65 93 34 1977 1 31 it has an obvious analogue for a semisimple linear algebraic group g over an arbitrary field k the group g is excellent if for any field extension k of k the anisotropic kernel of the k group g k k can be defined by a k group the aim of this paper is to investigate excellence properties of special linear and orthogonal groups

integral equation formulations for characteristic modes of dielectric and magnetic bodies five types of surface integral equation ie formulations are presented to determine the characteristic modes of dielectric and magnetic bodies it should be noted that these five formulations are derived in a unified manner instead of directly using the ie coefficient matrices of scattering problems this paper presents the formulations in a more natural and rigorous way the ie formulations result in the generalized real symmetric eigenvalue equations for the characteristic modes it is found that the eigenvalues of the first four formulations indicate the negative or positive imaginary part of the complex power of the characteristic modes which are zero at resonance the first two ie formulations result in two new generalized eigenvalue equations while the last three ie formulations lead to the same generalized eigenvalue equations as in the references moreover the first two formulations can be immune from spurious modes from which the third and fourth formulations suffer

the group sk 2 of a biquaternion algebra m rost has proved the existence of an exact sequence relating the group sk1 kernel of the reduced norm for k1 of a biquaternion algebra d whose center is a field f with galois cohomology groups of f in this paper we relate the group sk2 kernel of the reduced norm for k2 of d with galois cohomology of f through an exact sequence

clifford algebras of hyperbolic involutions for as a central simple algebra of even degree with hyperbolic orthogonal involution we describe the canonically induced involution underline sigma on the even clifford algebra c 0 a sigma underline sigma of a sigma when deg a equiv 0 mod 8 a cong m 2 b and the interesting part of underline sigma is isomorphic to the canonical involution on an exterior power algebra of b as a corollary we get some properties of the involution on the exterior power algebra

additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives abstract i panin proved in the nineties that the algebraic k theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras later merkurjev and panin described the algebraic k theory of toric varieties as a direct summand of the algebraic k theory of separable algebras in this article making use of the recent theory of noncommutative motives we extend panin and merkurjev panin s computations from algebraic k theory to every additive invariant as a first application we fully compute the cyclic homology and all its variants of twisted projective homogeneous varieties as a second application we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the brauer classes of the associated central simple algebras are trivial along the way we construct a fully faithful functor from merkurjev panin s motivic category to kontsevich s category of noncommutative chow motives which is of independent interest

a variety associated to an algebra with involution abstract a variety iv a is canonically associated to a central simple algebra a with orthogonal involution of the first kind the properties of this variety are the main topic of this paper the paper is divided into two parts we first define this variety to be the subvariety of the grassmannian of the n dimensional subspaces of the n2 dimensional vector space a consisting of all the n dimensional right ideals i of a such that i i 0 the geometric properties of iv a are investigated in particular the generic property of this variety is identified further the kernel of the scalar extension map br f br k where k is the function field of iv a is also computed lastly a criterion on the embeddability of the field k into the function field of the brauer severi variety of a is given in the second part the quillen k theory of iv a is computed the k theory result then leads to an index reduction formula for the function field k

the group of k1 zero cycles on the second generalized severi brauer variety of an algebra of index 4 abstract in this manuscript it is shown that the group of k 1 zero cycles on the second generalized severi brauer variety of an algebra a of index 4 is given by elements of the group k 1 a together with a square root of their reduced norm utilizing results of krashen concerning exceptional isomorphisms we translate our problem to the computation of cycles on involution varieties work of chernousov and merkurjev then gives a means of describing such cycles in terms of clifford and spin groups and corresponding r equivalence classes we complete our computation by giving an explicit description of these algebraic groups

r equivalence in adjoint classical groups over fields of virtual cohomological dimension 2 let f be a field of characteristic not 2 whose virtual cohomological dimension is at most 2 let g be a semisimple group of adjoint type defined over f let rg f denote the normal subgroup of g f consisting of elements r equivalent to identity we show that if g is of classical type not containing a factor of type d n g f rg f 0 if g is a simple classical adjoint group of type d n we show that if f and its multi quadratic extensions satisfy strong approximation property then g f rg f 0 this leads to a new proof of the r triviality of f rational points of adjoint classical groups defined over number fields

witt groups of smooth projective quadrics abstract let k be a commutative ring containing 1 2 in this paper we construct exact sequences connecting witt groups of smooth projective quadrics over k and clifford algebras the exact sequences have an application to a classical problem in quadratic form theory the witt kernel of function fields of quadrics we also apply the exact sequences to compute witt groups of certain kinds of smooth projective quadrics

a classification theorem for albert algebras let k be a field whose characteristic is different om 2 and 3 and let l k be a quadratic extension in this paper we prove that for a fixed degree 3 central simple algebra b over l with an involution ff of the second kind over k the jordan algebra j b a 11 u obtained through tits second co nstruction is determined up to isomorphism by the class of 11 u in h1 k su b a thus settling a question raised by petersson and racine as a consequence we derive a skolem noether type theorem for albert algebras we also show that the cohomological invariants determine the isomorphism class of j b a u u if b cr is fixed introduction let k be a field with characteristic different from 2 and 3 exceptional simple jordan algebras over k are called albert algebras there are rational constructions due to tits of all albert algebras over k referred to as the frst and the second constructions we begin by recalling the second construction briefly let l k be a quadratic extension bar denottng its nontrivial k automorphisme let b be a degree 3 central simple associative algebra over l with an involution a of the second kind over k let u be a unit of b with v u u and n u l i u e l where n denotes the reduced norm on b let h b x e blv x x let j b a u 1 xt ba d3 b be the jordan algebra with the multiplication given by bo b bo bt bo bo b l5 b btu5 b bob bob y ff b x o b u l where x y 2 xy yx z 2 t x x and x x y x y 2 t x y 2 t v x 2 t x t v t x tt t denoting the reduced trace of b the cubic norm of an element aoaa of j b c u is given by 81 n ao a n ao hn a hn ff a t aoau5 a for any unit w e b we have an isomorphism j b c u o j b c wu5 w n w h given by ao a l ao aw the following converse problem is posecs in p r question if j b c u j b c u y then does there e is 1 unit w e b such that u wuv w n w y received by the editors june 12 1996 1991 mathematics subject classification primary 17c40

mom volume integral equation solution for an isolated metal dielectric resonator with the edge based basis functions the objective of the present study is to show that the method of volume integral used in conjunction with the zeroth order edge based basis functions de carvalho and de souza mendes 1999 gives an accurate and stable solution for a metal dielectric resonator when the number of the basis functions remains relatively small special attention is paid to the resonator in the form of a thin dielectric layer backed by two metal sheets the metal surface is described in terms of the rao wilton glisson rwg basis functions

projective modules over the real algebraic sphere of dimension 3 let a be a commutative noetherian ring of krull dimension 3 we give a necessary and sufficient condition for a projective modules of rank 2 to be free using this we show that all the finitely generated projective modules over the algebraic real 3 sphere are free c 2010 elsevier inc all rights reserved

on the determination of resonant modes of dielectric objects using surface integral equations although surface integral equations have been extensively used for solving the scattering problem of arbitrarily shaped dielectric objects when applied to the resonance problem there are still some issues not fully addressed by the literature in this paper the method of moments with rao wilton glisson basis functions is applied to the electric field integral equation efie for solving the resonance problem of dielectric objects the resonant frequency is obtained by searching for the minimum of the reciprocal of the condition number of the impedance matrix in the complex frequency plane and the modal field distribution is obtained through singular value decomposition svd the determinant of the impedance matrix is not used since it is difficult to find its roots for the exterior efie the original basis functions are used as testing functions for the interior efie the basis functions rotated by 90 spl deg are used as testing functions to obtain an accurate modal field solution the impedance matrix needs to be reduced by half before svd is applied to it numerical results are given and compared with those obtained by using the volume integral equation

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

classification of algebras arising from the tits process in the preceding paper lo we generalized the second tits construction of simple exceptional jordan algebras 6 to a construction over an arbitrary ring of scalars which we called the tits process and claimed that it plays for jordan rings a role akin to that of the cayley dickson process for alternative rings in this paper we assume that the scalars form a field f and determine the semisimple jordan algebras arising from the tits process we see that except for predictable problems in low characteristics all simple jordan algebras of degree 3 are obtainable by starting from fl and repeating the tits process review let be a unital commutative associative ring a module recall that jv n 1 is a cubic norm structure on f if 1 n f is cubic form 2 f is a quadratic map 1 is an element of 2 2 is an udjoint for n

on the formulation of hybrid finite element and boundary integral methods for 3 d scattering this paper studies in detail a variety of formulations for the hybrid finite element and boundary integral fe bi method for three dimensional 3 d electromagnetic scattering by inhomogeneous objects it is shown that the efficiency and accuracy of the fe bi method depends highly on the formulation and discretization of the boundary integral equation bie used a simple analysis of the matrix condition number identifies the efficiency of the different fe bi formulations and an analysis of weighting functions shows that the traditional fe bi formulations cannot produce accurate solutions a new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance finally the multilevel fast multipole algorithm mlfma is employed to significantly reduce the memory requirement and computational complexity of the proposed fe bi method

absolute continuity of symmetric markov processes we study girsanov s theorem in the context of symmetric markov processes extending earlier work of fukushima takeda and fitzsimmons on girsanov transformations of gradient type we investigate the most general girsanov transformation leading to another symmetric markov process this investigation requires an extension of the forward backward martingale method of lyons zheng to cover the case of processes with jumps

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

efficient simulation of rectangular dielectric resonators using volume mpie mom formulation with combined entire domain and subdomain basis functions an efficient method of moments mom formulation using combined entire domain and subdomain basis functions is proposed for simulating the resonant modes of a rectangular dielectric resonator in free space an approximate analytical solution for the resonant mode of the dr is used as the entire domain basis function while a set of tetrahedral basis functions are used as subdomain basis functions since the main profile of the resonant mode can be well represented by the entire domain basis function only small number of subdomain basis functions are needed for further refinement of the representation compared with the mom formulations using only subdomain or entire domain basis functions this method is much faster for the same accuracy numerical results are given and compared with other methods

method of moments solution for a printed patch slot antenna on a thin finite dielectric substrate using the volume integral equation in this paper a volume integral equation vie based modeling method suitable for a patch or slot antenna on a thin finite dielectric substrate is developed and tested two new key features of the method are the use of proper dielectric basis functions and proper vie conditioning close to the metal surface where the surface boundary condition of the zero tangential e component must be extended into adjacent tetrahedra the extended boundary condition is the exact result for the piecewise constant dielectric basis functions the latter operation allows one to achieve a good accuracy with one layer of tetrahedra for a thin dielectric substrate and thereby greatly reduces computational cost the use of low order basis functions also implies the use of low order integration schemes and faster filling of the impedance matrix for some common patch slot antennas the vie based modeling approach is found to give an error of about 1 or less in the resonant frequency for one layer tetrahedral meshes with a relatively small number of unknowns this error is obtained by comparison with fine finite element method fem simulations or with measurements or with the analytical mode matching approach hence it is competitive with both the method of moments surface integral equation approach and with the fem approach for the printed antennas on thin dielectric substrates

strongly inner anisotropic forms of simple algebraic groups the definition of the index of a semisimple algebraic group over a field will be recalled in 1 55 more precisely we show in proposition 2 b cf 3 1 that if over a given field k there exists a central division algebra of degree 4 i e of dimension 16 and of order 4 in the brauer group then there exists a group of type ei t over a purely transcendental extension of degree 1 k t of k this will be deduced in two different ways from theorem 1 of section 2 whose precise statement is too technical to be given in this introduction but whose spirit is that it enables one from the existence of groups of a given type with preassigned properties of a geometric and cohomological nature over a field k to infer the existence of groups of a more complicated type meeting similar requirements over k t that theorem has other interesting applications to classification problems for instance we also deduce from it again in two different ways proposition 3 b in 3 2 that the existence of a division algebra of degree 3 over k implies the existence of groups with index

classifying nonsingular quadratic forms of rank three abstract the determinant on 2 by 2 matrices of trace zero gives a quadratic form linear algebra shows that its orthogonal group is the matrix algebra automorphisms together with scalars of square 1 as this holds over all commutative rings descent theory can be applied to it to get a quick classification of all nonsingular quadratic forms of rank 3 further descent arguments show how this classification yields the witt invariant and discriminant

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

pfaffians and quadratic forms by a quadratic module over a commutative ring r we mean a pair m q with m a finitely generated projective r module and q m t r a quadratic form we say that m q is a quadratic space if the polar form b x v q x 4 q x q y is nonsingular i e induces an isomorphism m hom m r in kos and kp a classification of quadratic spaces of rank 4 with trivial discriminant or more generally trivial arf invariant is given in terms of separable quaternion algebras in this paper we develop a similar theory for spaces of rank 6 let a be an azumaya algebra with an involution of symplectic type and let a be the r module of even elements as remarked independently by jacobson frohlich and tamagawa the restriction of the reduced norm na to a has a square root n this generalizes the pfallian as a square root of the determinant for alternating matrices if rank a 16 then n is a quadratic form on a and a n is a quadratic space of rank 6 the clifford algebra of a n is isomorphic to m2 a as a graded algebra we recall that the clifford algebra of a quadratic space of even rank is an azumaya algebra that the even clifford algebra c is an azumaya algebra over its centre z and that z is a separable quadratic r algebra the isomorphism class of z is called the arf invariant of m q and m q is said to have trivial arf invariant if z iy r x r thus a n is a quadratic space of rank 6 with trivial arf invariant furthermore n represents 1 since 1 e a and n 1 1 conversely we show that if m q is a quadratic space of rank 6 with trivial arf invariant and which represents 1 then m q a n for some azumaya algebra a of rank 16 with a symplectic involution this result yields classification theorems for spaces of rank 6 as well as generalizations of results of albert and jacobson on tensor products of quaternion algebras a similar theory holds for forms of rank 5 here we do not assume that the quadratic modules are nonsingular this would imply f r but only that they are semiregular we say that a quadratic module m q of rank

determination of resonant modes of dielectric resonators using mom sie with combined entire domain and subdomain basis functions method of moments mom combined with surface integral equation sie is a favorable approach for determining resonant modes of dielectric resonators to decrease the number of unknowns the subdomain basis functions are combined with an entire domain basis function which is an approximate solution of the resonant mode since the main profile of the current distribution can be represented by the entire domain basis function only small refinements are needed for better depiction of the current distribution as an example the te sub 11 spl delta mode of a rectangular dielectric resonator is simulated and compared with other methods

three isomorphic vector spaces ii abstract three isomorphic vector spaces b n k c n k and d n k are defined the interplay of these vector spaces leads to easy proofs for multinomial identities using automorphism each multinomial identity is recast into k 1 1 more identities distributions arising out of some stopping rules in drawing balls of k colors from an urn with and without replacement are connected so that one can easily go from one to the other identities involving joint cdf s of order statistics are generalized to those coming from arbitrary multivariate distributions discrete distribution similar to multinomial is defined where all the calculations can be done on usual multinomial distribution and transferred to the general

transcendental obstructions to weak approximation on general k3 surfaces abstract we construct an explicit k3 surface over the field of rational numbers that has geometric picard rank one and for which there is a transcendental brauer manin obstruction to weak approximation to do so we exploit the relationship between polarized k3 surfaces endowed with particular kinds of brauer classes and cubic fourfolds

zero cycles on homogeneous varieties abstract in this paper we study the group a 0 x of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety x to do this we translate rational equivalence of 0 cycles on a projective variety into r equivalence on symmetric powers of the variety for certain homogeneous varieties we then relate these symmetric powers to moduli spaces of etale subalgebras of central simple algebras which we construct this allows us to show a 0 x 0 for certain classes of homogeneous varieties for groups of each of the classical types extending previous results of swan karpenko of merkurjev and of panin

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

degree three unramified cohomology of adjoint semisimple groups we compute degree 3 unramified cohomology of adjoint semisimple group new classes of non rational adjoint semisimple groups are found

switched diffusion processes and systems of elliptic equations a dirichlet space approach the switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a dirichlet space approach and is applied to prove the existence theorem of the cauchy initial problem for the system a representation theorem for the solution of the dirichlet boundary value problem and a generalised skorohod decomposition for the reflecting switched diffusion process are obtained

del pezzo surfaces of degree 6 over an arbitrary field abstract we give a characterization of all del pezzo surfaces of degree 6 over an arbitrary field f a surface is determined by a pair of separable algebras these algebras are used to compute the quillen k theory of the surface as a consequence we obtain an index reduction formula for the function field of the surface

motivic decomposition of isotropic projective homogeneous varieties we give a decomposition of the chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties as an application we prove a generalization of rost s nilpotence theorem

a note on exponential stability of the nonlinear filter for denumerable markov chains abstract asymptotic stability of the optimal filter with respect to its initial conditions is investigated in this paper under the assumption that the observation function is one to one and the observation noise is sufficiently small it is shown that exponential stability of the nonlinear filter holds for a large class of denumerable markov chains including all finite markov chains throughout this paper ergodicity of the signal process is not assumed

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

on the dual basis for solving electromagnetic surface integral equations a powerful technique for solving electromagnetic em surface integral equations sies for inhomogenous objects by the method of moments mom involves the well known rao wilton glisson rwg basis function to represent the electric current and for field orthogonality and numerical stability reasons a variation of the rwg basis known as the ntilde x rwg basis where ntilde is a unit normal vector at the object surface to represent the magnetic current though this combination provides a numerically efficient and effective solution that has been demonstrated on a variety of structures one cannot feel entirely comfortable because of the presence of fictitious magnetic current associated with the modified basis chen and wilton proposed a different smoother basis in 1990 that avoids the fictitious line charges but because of computational cost issues it has not been used beyond chen s dissertation recently this basis was rediscovered and has received considerable attention our work reexamines the dual basis exploring issues not addressed by chen and wilton and showing accurate solutions for a variety of em scattering structures

bott periodicity and calculus of euler classes on spheres a variety of computations regarding the euler class group e an an and the grothendieck group k0 an of the algebraic sphere spec an is done the euler class of the algebraic tangent bundle on spec an is computed it is also investigated whether every element in the euler class group e an an is the euler class of a projective an module of rank n

prescribed behavior of central simple algebras after scalar extension abstract 1 let a 1 a n be central simple disjoint algebras over a field f let also l i exp a i m i ind a i l i m i and for each i 1 n let l i and m i have the same sets of prime divisors then there exists a field extension e f such that exp a i e l i and ind a i e m i i 1 n 2 let a be a central simple algebra over a field k with an involution of the second kind we prove that there exists a regular field extension e k preserving indices of central simple k algebras such that a k e is cyclic and has an involution of the second kind extending

r equivalence in spinor groups the notion of r equivalence in the set x f of f points of an algebraic variety x defined over a field f was introduced by manin in 11 and studied for linear algebraic groups by colliot thelene and sansuc in 3 for an algebraic group g defined over a field f the subgroup rg f of r trivial elements in the group g f of all f points is defined as follows an element g belongs to rg f if there is a rational morphism f al g over f defined at the points 0 and 1 such that f o 1 and f 1 g in other words g can be connected with the identity of the group by the image of a rational curve the subgroup rg f is normal in g f and the factor group g f rg f g f r is called the group of r equivalence classes the group of r equivalence classes is very useful while studying the rationality problem for algebraic groups the problem to determine whether the variety of an algebraic group is rational or stably rational we say that a group g is r trivial if g e r 1 for any field extension e f if the variety of a group g is stably rational over f then g is r trivial thus if one can establish non triviality of the group of r equivalence classes g e r just for one field extension e f the group g is not stably rational over f the group of r equivalence classes for adjoint semisimple classical groups was computed in 15 this computation was used to obtain examples of non rational adjoint algebraic groups consider simply connected algebraic groups of classical types let g gl1 a be the algebraic group of invertible elements of a central simple f algebra a of dimension n2 and let h sl1 a be the subgroup in g of the reduced norm 1 elements the group h is a simply connected group of inner type a 1 v voskresenskiy 29 has shown that the natural homomorphism ax k1 a induces an isomorphism

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

cohomological invariants of simply connected groups of rank 3 let g be a linear algebraic group defined over a field f one can define an equivalence relation called r equivalence on the group g f of points over f as follows cf 4 9 14 two points g0 g1 g f are r equivalent if there is a rational morphism f x 1 f g of algebraic varieties over f defined at points 0 and 1 such that f 0 g0 and f 1 g1 the group of r equivalence classes is denoted by g f r for example if g sl1 a is the special linear group of a central simple f algebra a then the group of r equivalence classes g f r is equal to the reduced whitehead group in algebraic k theory cf 27

the monoid structure on homotopy obstructions abstract let a be a commutative noetherian ring containing a field k with 1 2 k dim a d and let p be a projective a module with r a n k p n let l o p denote the set of all pairs i where i is an ideal of a and p i i 2 is a surjective map the homotopy relations on l o p induced by l o p t leads to a set 0 l o p of equivalence classes in l o p there are two distinguished elements e 0 e 1 0 l o p respectively the images of 0 0 and a 0 define the obstruction class e p e 0 0 l o p to be called the nori homotopy class of p the following results are under suitable smoothness or regularity hypotheses we prove if 2 n d 2 then 0 l o p has a natural structure of a monoid which is a group if p q a when 2 n d 3 we prove p q a e p e 1 the additive zero further we give a definition of a euler class group e p under suitable smoothness hypotheses we prove if p q a and 2 n d 3 then there is natural isomorphism e p 0 l o p of groups

extensions of levy khintchine formula and beurling deny formula in semi dirichlet forms setting abstract the levy khintchine formula or more generally courrege s theorem characterizes the infinitesimal generator of a levy process or a feller process on r d for more general markov processes the formula that comes closest to such a characterization is the beurling deny formula for symmetric dirichlet forms in this paper we extend these celebrated structure results to include a general right process on a metrizable lusin space which is supposed to be associated with a semi dirichlet form we start with decomposing a regular semi dirichlet form into the diffusion jumping and killing parts then we develop a local compactification and an integral representation for quasi regular semi dirichlet forms finally we extend the formulae of levy khintchine and beurling deny in semi dirichlet forms setting through introducing a quasi compatible metric

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

surjectivity of the total clifford invariant and brauer dimension abstract a celebrated theorem of merkurjev that the 2 torsion of the brauer group is represented by clifford algebras of quadratic forms is in general false when the base is no longer a field the first counterexamples when the base is among certain arithmetically subtle hyperelliptic curves over local fields were constructed by parimala scharlau and sridharan we prove that considering clifford algebras of all line bundle valued quadratic forms such counterexamples disappear and we recover merkurjev s theorem in these cases for any smooth curve over a local field or any smooth surface over a finite field the 2 torsion of the brauer group is always represented by clifford algebras of line bundle valued quadratic forms

some new examples in the theory of quadratic forms we construct a 6 dimensional anisotropic quadratic form phi and a 4 dimensional quadratic form psi over some fieldf such that phi becomes isotropic over the function field f psi but every proper subform of phi is still anisotropic over f psi it is an example of non standard isotropy with respect to some standard conditions of isotropy for 6 dimensional forms over function fields of quadrics known previously besides of that we produce an 8 dimensional quadratic form phi with trivial determinant such that the index of the clifford invariant of phi is 4 but phi can not be represented as a sum of two 4 dimensional forms with trivial determinants using this we find a 14 dimensional quadratic form with trivial discriminant and clifford invariant which is not similar to a difference of two 3 fold pfister forms the proofs are based on computations of the topological filtration on the grothendieck group of certain projective homogeneous varieties to do these computations we develop several methods covering a wide class of varieties and being to our mind of independent interest

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

alternative surface integral equation formulations for characteristic modes of dielectric and magnetic bodies in this paper the physical meanings and various computing methods of characteristic modes cms of dielectric and magnetic bodies are investigated two types of surface integral equation sie formulations based on the dependent relationships between the equivalent surface electric and magnetic currents are proposed following the two types of sie formulations two generalized eigenvalue equations are established to determine the cms of dielectric or magnetic bodies this is in contrast to the approach using induced volume currents it is found that the eigenvalues possess clear physical meanings similar to those of perfectly electric conducting bodies the cms can be used to describe the resonant features including resonant frequencies and resonant field distributions numerical results are presented to validate the proposed formulations

isotropy of six dimensional quadratic forms over function fields of quadrics abstract letfbe a field of characteristic different from 2 and let be an anisotropic six dimensional quadratic form overf we study the last open cases in the problem of describing the quadratic forms such that becomes isotropic over the function fieldf

numerical equivalence defined on chow groups of noetherian local rings in the present paper we define a notion of numerical equivalence on chow groups or grothendieck groups of noetherian local rings which is an analogue of that on smooth projective varieties under a mild condition it is proved that the chow group modulo numerical equivalence is a finite dimensional vector space as in the case of smooth projective varieties numerical equivalence on local rings is deeply related to that on smooth projective varieties for example if grothendieck s standard conjectures are true then a vanishing of chow group of local rings modulo numerical equivalence can be proven using the theory of numerical equivalence the notion of numerically roberts rings is defined it is proved that a cohen macaulay local ring of positive characteristic is a numerically roberts ring if and only if the hilbert kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength numerically roberts rings satisfy the vanishing property of intersection multiplicities we shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized chern characters

r equivalence and special unitary groups abstract a norm homomorphism for the group of r equivalence classes of all simply connected semisimple classical algebraic groups is constructed the group of r equivalence classes for special unitary groups su b is computed it is proved that the variety of su b is rational if ind b 3 and the stable birational type of su b depends only on the brauer class of b and does not depend on the involution

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles