monodromy group for a strongly semistable principal bundle over a curve let x be a geometrically irreducible smooth projective curve defined over a field k assume that x has a k rational point fix a k rational point x e x from these data we construct an affine group scheme x defined over the field k as well as a principal x bundle s1755069607000151inline1 over the curve x the group scheme x is given by a xs0211a graded neutral tannakian category built out of all strongly semistable vector bundles over x the principal bundle s1755069607000151inline1 is tautological let g be a linear algebraic group defined over k that does not admit any nontrivial character which is trivial on the connected component containing the identity element of the reduced center of g let e g be a strongly semistable principal g bundle over x we associate to e g a group scheme m defined over k which we call the monodromy group scheme of e g and a principal m bundle e m over x which we call the monodromy bundle of e g the group scheme m is canonically a quotient of x and e m is the extension of structure group of s1755069607000151inline1 the group scheme m is also canonically embedded in the fiber ad e g x over x of the adjoint bundle

analysis of atkinson s variable transformation for numerical integration over smooth surfaces in ℝ 3 recently a variable transformation for integrals over smooth surfaces in 3 was introduced in a paper by atkinson this interesting transformation which includes a grading parameter that can be fixed by the user makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner some analysis of the approximations thus produced was provided by atkinson who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter in a recent report by atkinson and sommariva this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface and optimal results are given for special values of the grading parameter in the present work we give a complete analysis of atkinson s method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere we obtain optimal results that explain the actual rates of convergence and we achieve this for all values of the grading parameter

homogeneous mean values weights and asymptotics 1 the weights w1 wn associated with a mean value m x1 xn are defined as partial derivatives of m and identified for various homogeneous means 2 we prove theorem 1 if a1 an are real and x then m a1 x an x x wiai o 1x provided only that m satisfies very general hypotheses in particular this asymptotic formula originally discovered by l hoehn and i niven math mag 58 1985 151 156 in special cases is valid for power means dresher means mixed means and hypergeometric means 3 we prove theorem 2 if x1 x2 is an infinite sequence of positive numbers lying in a finite interval 0 xi and if the arithmetic mean of the first n values approaches a limit as n then the hypergeometric mean approaches the same limit this theorem was first established for the special case of the logarithmic mean of any order by a o pittenger j math anal appl 123 1987 281 291

semi sigmoidal transformations for evaluating weakly singular boundary element integrals accurate numerical integration of line integrals is of fundamental importance to reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights here a co ordinate transformation technique is introduced for evaluating weakly singular integrals which after some initial manipulation of the integral uses the same integration nodes and weights as those of the regular integrals the transformation technique is based on newly defined semi sigmoidal transformations which cluster integration nodes only near the singular point the semi sigmoidal transformations are defined in terms of existing sigmoidal transformations and have the benefit of evaluating integrals more accurately than full sigmoidal transformations as the clustering is restricted to one end point of the interval comparison of this new method with existing coordinate transformation techniques shows that more accurate evaluation of weakly singular integrals can be obtained based on observation of several integrals considered guidelines are suggested for the type of semi sigmoidal transformation to use and the degree to which nodes should be clustered at the singular points copyright 2000 john wiley sons ltd

semistable principal g bundles in positive characteristic let x be a normal projective variety defined over an algebraically closed field k of positive characteristic let g be a connected reductive group defined over k we prove that some frobenius pullback of a principal g bundle admits the canonical reduction ep such that its extension by p p rup is strongly semistable see theorem 5 1 then we show that there is only a small difference between semistability of a principal g bundle and semistability of its frobenius pullback see theorem 6 3 this and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable rational principal g bundles

moduli spaces of framed instanton bundles on cp 3 and twistor sections of moduli spaces of instantons on c 2 abstract we show that the moduli space m of framed instanton bundles on cp 3 is isomorphic as a complex manifold to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on r 4 we then use this characterization to prove that m is equipped with a torsion free affine connection with holonomy in sp 2 n c

semistable sheaves on projective varieties and their restriction to curves let x be a nonsingular projective variety of dimension n over an algebraically closed field k let h be a very ample line bundle on x if v is a torsion free coherent sheaf on x we define deg v to be cl v c h and v deg v rk v we call v sernistable resp stable if for all proper subsheaves w of v we have w v resp w v cf 7 14 in this paper we prove that if v is semistable on x then its restriction to a general complete intersection curve of sufficiently high degree is semistable theorem 6 1 to give an idea of the proof assume x is a surface and v a vector bundle of rank 2 the restriction of v to a general curve c of degree m is not semistable if and only if it is not semistable on the generic curve ym defined over the function field of iph h let s be the line bundle on y contradicting the semistability of vi y cf sects 4 1 and 4 2 first we show that l extends uniquely to a line bundle l m on x proposition 2 1 if we can get l as a subbundle of v we are through for then l would contradict the semistability of v so we would like the restriction map h hom lm v h hom l m v to be surjective now for fixed l it follows from the lemma of enriques severi proposition 3 2 6 corollary 7 8 that h horn l v h horn l v is surjective for large m therefore it is enough if the l remain the same line bundle l for infinitely many m to prove that l l we construct a degenerating family of curves d f s x x s 3 d p x such that the generic fibre is a curve ct 1 of degree 2 1 and the special fibre is a reduced curve with two nonsingular components ci of degree 2 cf sect 5 let m denote 2 extending the subbundle lt ct to a subsheaf of p v on d and restricting the extension to ci gives a lower bound for the maximal degree of a line subbundle of v cim in terms of that for v ctm i proposition 4 3 this implies that degl is bounded lemma 6 5 1 so that for an infinite subsequence of m degl is constant if degltm r deglt by refining the above argument with the degenerating family one can prove that lt ci

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

harder narasimhan reduction for principal bundles over a compact kahler manifold generalizing the harder narasimhan filtration of a vector bundle it is shown that a principal g bundle e g over a compact kahler manifold admits a canonical reduction of its structure group to a parabolic subgroup of g here g is a complex connected reductive algebraic group in the special case where g rm gl n mathbb c this reduction is the harder narasimhan filtration of the vector bundle associated to e g by the standard representation of rm gl n mathbb c the reduction of e g in question is determined by two conditions if p denotes the parabolic subgroup l its levi factor and e p subset e g the canonical reduction then the first condition says that the principal l bundle obtained by extending the structure group of the p bundle e p using the natural projection of p to l is semistable denoting by mathfrak u the lie algebra of the unipotent radical of p the second condition says that for any irreducible p module v occurring in mathfrak u mathfrak u mathfrak u the associated vector bundle e p times p v is of positive degree here mathfrak u mathfrak u mathfrak u is considered as a p module using the adjoint action the second condition has an equivalent reformulation which says that for any nontrivial character chi of p which can be expressed as a nonnegative integral combination of simple roots with respect to any borel subgroup contained in p the line bundle associated to e p for chi is of positive degree the equivalence of these two conditions is a consequence of a representation theoretic result proved here

multi variable extended mean values on considere des fonctions dont les valeurs sont des valeurs moyennes d une suite x x 0 x n de nombre reels pour une bonne paire f g de fonctions on definit une valeur moyenne e f g x de chaque suite x a valeurs dans cette intervalle

principal bundles over the projective line abstract let g be a connected reductive linear algebraic group defined over a field k and e g a principal g bundle over the projective line p k 1 satisfying the condition that e g is trivial over some k rational point of p k 1 if the field k is algebraically closed then it is known that the principal g bundle e g admits a reduction of structure group to the multiplicative group g m we prove this for arbitrary k this extends the results of harder 1968 10 and mehta and subramanian 2002 14

stability of rank 2 vector bundles along isomonodromic deformations we are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact riemann surfaces which are provided with irreducible meromophic tracefree connections in the case of a logarithmic connection on the riemann sphere such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles according to a result of a bolibruch in the general case of meromorphic connections over riemann surfaces of arbitrary genus we prove that the vector bundle will be semi stable up to a small isomonodromic deformation more precisely the vector bundle underlying the universal isomonodromic deformation is generically semi stable along the deformation and even maximally stable for curves of genus g 2 this result is non trivial even in the case of non singular connections

the euler maclaurin expansion and finite part integrals in this paper we compare g p the mellin transform together with its analytic continuation and overline overline g p the related hadamard finite part integral of a function g x which decays exponentially at infinity and has specified singular behavior at the origin except when p is a nonpositive integer these coincide when p is a nonpositive integer overline overline g p is well defined but g p has a pole we show that the terms in the laurent expansion about this pole can be simply expressed in terms of the hadamard finite part integral of a related function this circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the euler maclaurin expansion for the quadrature error functional

numerical integration of functions with boundary singularities abstract in this paper we deal with the problem of constructing efficient rules for the numerical evaluation of integrals of functions which are very smooth everywhere in the domain of integration except at the boundaries where they possess mild singularities in particular we consider integrals defined on bounded intervals or on triangles integrals of this type appear for example in the numerical solution of singular and weakly singular integral equations by boundary element methods

einstein hermitian connections on polystable principal bundles over a compact kahler manifold given a compact kahler manifold m and a connected reductive algebraic group g over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i a prinlcipal g bundle over m admits an einstein hermitian connection if and only if the principal bundle is polystable if m is a projective manifold a higgs g bundle over m admits an einstein hermitian connection if and only if the higgs bundle is polystable

means that minimize relative error and an associated integral equation abstract determinations are made of the means that minimize various relative errors in the sense that the harmonic mean of a and b minimizes the traditional relative error on a b the general problem for averaged relative error leads to a nonlinear integral equation for which we prove existence and uniqueness results as well as constructive solution procedures the inverse problem of finding the measure of relative error corresponding to a given mean is also analyzed in detail these studies shed light on both new and previously known inequalities for specific means

numerical integration over smooth surfaces in r3 via class sm variable transformations part i smooth integrands class s m variable transformations with integer m for finite range integrals were introduced by the author about a decade ago these transformations periodize the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy especially with even m in a recent work by the author these transformations were extended to arbitrary real m and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail it was shown that with m chosen appropriately exceptionally high accuracy can be achieved by the trapezoidal rule for example if the integrand function is smooth on the interval of integration including the endpoints and vanishes at the endpoints then excellent results are obtained by taking 2m to be an odd integer in the present work we consider the use of these transformations in the computation of integrals on surfaces of simply connected bounded domains in r 3 in conjunction with the product trapezoidal rule we assume these surfaces are smooth and homeomorphic to the surface of the unit sphere and we treat the cases in which the integrands are smooth we propose two approaches one in which the product trapezoidal rule is applied with the integrand as is and another in which the integrand is preprocessed before the rule is applied we give thorough analyses of the errors incurred in both approaches which show that surprisingly high accuracies can be achieved with suitable values of m we also illustrate the theoretical results with numerical examples

stability of special instanton bundles on ℙ 2n 1 we prove that the special instanton bundles of rank 2n on p2n 1 c with a symplectic structure studied by spindler and trautmann are stable in the sense of mumford takemoto this implies that the generic special instanton bundle is stable moreover all instanton bundles on p5 are stable we get also the stability of other related vector bundles

flat holomorphic connections on principal bundles over a projective manifold let g be a connected complex linear algebraic group and r u g its unipotent radical a principal g bundle e g over a projective manifold m will be called polystable if the associated principal g r u g bundle is so a g bundle e g over m is polystable with vanishing characteristic classes of degrees one and two if and only if e g admits a at holomorphic connection with the property that the image in g r u g of the monodromy of the connection is contained in a maximal compact subgroup of g r u g

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

asymptotic expansions of gauss legendre quadrature rules for integrals with endpoint singularities let jh1f x dx where c l l and let gn f sili wmf xni be the n point gauss legendre quadrature approximation to in this paper we derive an asymptotic expansion as n oo for the error en f i f gn f when f x has general algebraic logarithmic singularities at one or both endpoints we assume that f x has asymptotic expansions of the forms f x jr u8 og l x l xr ass 1 s 0

the theorem of grauert mulich spindler the boundedness of semi stable sheaves on p is a simple corollary to the above see 13 if one looks into the proof of this theorem by spindler and the techniques by van de ven 14 and barth 1 he can generalize immediately the theorem in the following way let e be a torsion free coherent sheaf on a non singular projective variety x over an algebraically closed field k and ox 1 a very ample invertible sheaf on x if g l y n dimx are sufficiently general member of a very ample linear subsystem l of iox 1 l then y yl i1 is a non singular subvariety of codimension n in x and elf is torsion free

extension of a class of periodizing variable transformations for numerical integration class s m variable transformations with integer m for numerical computation of finite range integrals were introduced and studied by the author in the paper a sidi a new variable transformation for numerical integration numerical integration iv 1993 h brass and g hammerlin eds pp 359 373 a representative of this class is the sin m transformation that has been used with lattice rules for multidimensional integration these transformations periodize the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy especially with even m in the present work we extend these transformations to arbitrary values of m and give a detailed analysis of the resulting transformed trapezoidal rule approximations we show that with suitable m they can be very useful in different situations we prove for example that if the integrand function is smooth on the interval of integration add vanishes at the endpoints then results of especially high accuracy are obtained by taking 2m to be an odd integer such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration we also illustrate some of the results with numerical examples via the extended sin m transformation

some improvements of aczel s inequality and popoviciu s inequality in this paper we show several new generalized and sharpened versions of aczel s inequality and popoviciu s inequality our results contain as special cases the improvement of certain known results on aczel s inequality and popoviciu s inequality as application an integral inequality of aczel popoviciu type is obtained

isomonodromic deformations and very stable vector bundles of rank two for the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two consider the family of holomorphic vector bundles over curves underlying this universal deformation in a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable here we prove that the vector bundle corresponding to a general parameter is in fact very stable meaning it does not admit any nonzero nilpotent higgs field

inequalities for indefinite forms abstract letn 2 be a fixed integer and p x x 1 p x 2 p x n p 1 p x r p whererpdenotes the set of all vectorsx x1 xn for whichxi 0 i 1 n x1p x2p xnpifp 0 andxi 0 i 1 n x1p x2p xnpifp three inequalities are presented for p the first is a comparison theorem the second is a holder like generalization of aczel s inequality an extension of popoviciu s inequality while the third is a generalization of bellman s inequality to all possible values ofp the proofs show that the above inequalities are consequences of some well known inequalities for power means and power sums

periodizing transformations for numerical integration the order of the trapezoidal rule can be raised by making a substitution that transforms the integrand into a new integrand which has a smooth periodic extension the exponent in the rate at which the transformed weights approach zero near the endpoints is called the damping power of the transformation currently used polynomial transformations of damping power m have order m 1 if m is odd and order m 2 if m is even we prove that the highest possible order is m 1 if m is odd but 2m 2 if m is even and give new polynomial transformations that achieve the optimal order

restriction theorems for principal bundles let e be a semistable or stable principal bundle over a smooth complex projective variety x and let d x be a complete intersection we study the semi stability of the restriction e d some of the results known for vector bundles such as grauert mulich flenner and mehta ramanathan theorems are generalized to principal bundles

inequalities for hyperbolic functions abstract several inequalities involving hyperbolic functions are derived some of them are obtained with the aid of stolarsky and gini means

euler maclaurin expansions for integrals with endpoint singularities a new perspective in this note we provide a new perspective on euler maclaurin expansions of offset trapezoidal rule approximations of the finite range integrals i f baf x dx where f c a b but can have general algebraic logarithmic singularities at one or both endpoints these integrals may exist either as ordinary integrals or as hadamard finite part integrals we assume that f x has asymptotic expansions of the general forms where ps y and qs y are some polynomials in y here the s and s are complex in general and different from 1 2 the results we obtain in this work generalize and include as special cases those pertaining to the known special cases in which f x x a log x a pga x b x log b x qgb x where p and q are nonnegative integers and ga c a b and gb c a b in addition they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f x as x a and x b with h b a n where n is a positive integer and with one of these results reads as h 0 where z is the riemann zeta function

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

variable transformations and gauss legendre quadrature for integrals with endpoint singularities gauss legendre quadrature formulas have excellent convergence properties when applied to integrals 1 0 f x dx with f c 0 1 however their performance deteriorates when the integrands f x are in c 0 1 but are singular at x 0 and or x 1 one way of improving the performance of gauss legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker singularities than those of f x thus if x t is a variable transformation that maps 0 1 onto itself we apply gauss legendre quadrature to the transformed integral 1 0 f t t dt whose singularities at t 0 and or t 1 are weaker than those of f x at x 0 and or x 1 in this work we first define a new class of variable transformations we denote p q where p and q are two positive parameters that characterize it we also give a simple and easily computable representative of this class next by invoking some recent results by the author concerning asymptotic expansions of gauss legendre quadrature approximations as the number of abscissas tends to infinity we present a thorough study of convergence of the combined approximation procedure with variable transformations from p q we show how optimal results can be obtained by adjusting the parameters p and q of the variable transformation in an appropriate fashion we also give numerical examples that confirm the theoretical results

sigmoidal transformations and the euler maclaurin expansion for evaluating certain hadamard finite part integrals starting with some results of lyness concerning the euler maclaurin expansion of cauchy principal value integrals over left 0 1 right it is shown how by the use of sigmoidal transformations good approximations can be found for the hadamard finite part integral int scriptstyle 0 1 left phi left x right left x c right 2 right dx where 0 c 1 the analysis is illustrated by some numerical examples

semistable and numerically effective principal higgs bundles we study miyaoka type semistability criteria for principal higgs g bundles e on complex projective manifolds of any dimension we prove that e has the property of being semistable after pullback to any projective curve if and only if certain line bundles obtained from some characters of the parabolic subgroups of g are numerically effective one also proves that these conditions are met for semistable principal higgs bundles whose adjoint bundle has vanishing second chern class in a second part of the paper we introduce notions of numerical effectiveness and numerical flatness for principal higgs bundles discussing their main properties for non higgs principal bundles we show that a numerically flat principal bundle admits a reduction to a levi factor which has a flat hermitian yang mills connection and as a consequence that the cohomology ring of a numerically flat principal bundle with coefficients in r is trivial to our knowledge this notion of numerical effectiveness is new even in the case of non higgs principal bundles

moduli space of principal sheaves over projective varieties let g be a connected reductive group the late ramanathan gave a notion of semi stable principal g bundle on a riemann surface and constructed a projective moduli space of such objects we generalize ramanathan fs notion and construction to higher dimension allowing also objects which we call semistable principal g sheaves in order to obtain a projective moduli space a principal g sheaf on a projective variety x is a triple p e o where e is a torsion free sheaf on x p is a principal g bundle on the open set u where e is locally free and o is an isomorphism between e u and the vector bundle associated to p by the adjoint representation we say it is semi stable if all filtrations e of e as sheaf of killing orthogonal algebras i e filtrations with e ui e i 1 and ei ej e e e i j have pei rke pe rkei 0 where pei is the hilbert polynomial of ei after fixing the chern classes of e and of the line bundles associated to the principal bundle p and characters of g we obtain a projective moduli space of semistable principal g sheaves we prove that in case dimx 1 our notion of semi stability is equivalent to ramanathan fs notion

optimal evaluations of some seiffert type means by power means let us consider the second trigonometric mean t defined by seiffert and the hyperbolic mean m defined by neuman and sandor there are some known inequalities between these means and some power means a p we prove that the evaluationsa l n 2 l n l n 3 2 2

isomonodromic deformations of logarithmic connections and stability let x 0 be a compact connected riemann surface of genus g with d 0 subset x 0 an ordered subset of cardinality n and let e g be a holomorphic principal g bundle on x 0 where g is a reductive affine algebraic group defined over mathbb c that is equipped with a logarithmic connection nabla 0 with polar divisor d 0 let mathcal e g nabla be the universal isomonodromic deformation of e g nabla 0 over the universal teichmuller curve mathcal x mathcal d longrightarrow text teich g n where text teich g n is the teichmuller space for genus g riemann surfaces with n marked points we prove the following see sect 5 1 assume that g ge 2 and n 0 then there is a closed complex analytic subset mathcal y subset text teich g n of codimension at least g such that for any t in text teich g n setminus mathcal y the principal g bundle mathcal e g vert mathcal x t is semistable where mathcal x t is the compact riemann surface over t 2 assume that g ge 1 and if g 1 then n 0 also assume that the monodromy representation for nabla 0 does not factor through some proper parabolic subgroup of g then there is a closed complex analytic subset mathcal y subset text teich g n of codimension at least g such that for any t in text teich g n setminus mathcal y the principal g bundle mathcal e g vert mathcal x t is semistable 3 assume that g ge 2 assume that the monodromy representation for nabla 0 does not factor through some proper parabolic subgroup of g then there is a closed complex analytic subset mathcal y subset text teich g n of codimension at least g 1 such that for any t in text teich g n setminus mathcal y the principal g bundle mathcal e g vert mathcal x t is stable in 12 the second named author proved the above results for g text gl 2 mathbb c

application of class i m variable transformations to numerical integration over surfaces of spheres class im variable transformations with integer m for finite range integrals were introduced by the author numerical integration iv international series of numerical mathematics basel 1993 pp 359 373 about a decade ago these transformations periodize the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy especially with even m in a recent work by the author math comp 2005 these transformations were extended to arbitrary m and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail it was shown that with m chosen appropriately exceptionally high accuracy can be achieved by the trapezoidal rule in the present work we make use of these transformations in the computation of integrals on surfaces of spheres in conjunction with the product trapezoidal rule we treat integrands that have point singularities of the single layer and double layer types we propose different approaches and provide full analyses of the errors incurred in each we show that surprisingly high accuracies can be achieved with suitable values of m we also illustrate the theoretical results with numerical examples finally we also recall analogous procedures developed in another work by the author appl math comput 2005 for regular integrands

d2lri a nonadaptive algorithm for two dimensional cubature we describe a nonadaptive automatic cubature routine for integration over a wide variety of two dimensional domains including infinite regions the underlying algorithm first maps the region onto the unit square applies a periodizing sixth order sidi transformation and then generates a sequence of approximations based on embedded lattice rules numerical experiments suggest that the routine is reliable and efficient for a wide range of integrand types and that it is often more effective than other published routines for integrands with a singularity along the boundary and for integration over infinite domains

on acoustic and electric faraday cages two dimensional problems involving many identical small circles are considered the circles are the cross sections of parallel wires modelling a cage or a grating both electrostatic and acoustic fields are considered the main emphasis is on periodic configurations of n circles distributed evenly around a large circle a ring foldy s theory is used for acoustic problems and then adapted for electrostatic problems in both cases circulant matrices are encountered the fields can be calculated explicitly then the limit n is studied a connection between the n circle problem and the limiting problem fields exterior to the ring is established using known results on the convergence of a defective form of the trapezoidal rule defective in that endpoint contributions are ignored because the integrand has logarithmic singularities at those points this shows that the solution of the limiting problem is approached very slowly as n 1 log n as n

on equivariant principal bundles over wonderful compactifications abstract let g be a simple algebraic group of adjoint type over c and let m be the wonderful compactification of a symmetric space g h take a g equivariant principal r bundle e on m where r is a complex reductive algebraic group and g is the universal cover of g if the action of the isotropy group h on the fiber of e at the identity coset is irreducible then we prove that e is polystable with respect to any polarization on m further for wonderful compactification of the quotient of psl n c n 4 respectively psl 2 n c n 1 by the normalizer of the projective orthogonal group respectively the projective symplectic group we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification

means generalized divided differences and intersections of osculating hyperplanes abstract we define means innvariables by taking the intersection point inrnofnosculating hyperplanes to a given curve inrn these planes are the natural extensions of the osculating plane inr3 more precisely letcbe a curve inrn and let 0

relative bogomolov s inequality and the cone of positive divisors on the moduli space of stable curves an interesting point of the above theorem is that even if the weak positivity of disx y e at y is a global property on y it can be derived from the local assumption the goodness of xg and the semistability of eg this gives a great advantage to our applications in order to understand the intuition underlying the theorem let us consider a simulated case namely we suppose that f x y is a smooth surface fibred over a curve and the fiber is general bogomolov s instability theorem 1 says that if e is semistable then the codimension two cycle 2rc2 e r 1 cl e 2 has nonnegative degree so if we push it down to a codimension one cycle on y then one can rephrase bogomolov s theorem as saying that the semistability of ey implies the non negativity of disx y e

on a certain quadrature formula traduction anglaise d un article en japonais publie en aout 1970 dans la serie kokyuroku vol 91 pp 82 118

restriction theorems for principal bundles in arbitrary characteristic the aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic let g be a reductive algebraic group over any field k k let x be a smooth projective variety over k let h be a very ample line bundle on x and let e be a semistable resp stable principal g bundle on x w r t h the main result of this paper is that the restriction of e to a general smooth curve which is a complete intersection of ample hypersurfaces of sufficiently high degrees is again semistable resp stable

further extension of a class of periodizing variable transformations for numerical integration class s m variable transformations with integer m for accurate numerical computation of finite range integrals via the trapezoidal rule were introduced and studied by the author a representative of this class is the sin m transformation in a recent work of the author this class was extended to arbitrary noninteger values of m and it was shown that exceptionally high accuracies are achieved by the trapezoidal rule in different circumstances with suitable values of m in another recent work by monegato and scuderi the sin m transformation was generalized by introducing two integers p and q instead of the single integer m we denote this generalization as the sin p q transformation here when p q m the sin p q transformation becomes the sin m transformation unlike the sin m transformation which is symmetric the sin p q transformation is not symmetric when p q and this offers an advantage when the behavior of the integrand at one endpoint is quite different from that at the other endpoint in view of the developments above in the present work we generalize the class s m by introducing a new class of nonsymmetric variable transformations which we denote as s p q where p and q can assume arbitrary noninteger values such that the sin p q transformation is a representative of this class and s email protected s m m we provide a detailed analysis of the trapezoidal rule approximation following a variable transformation from the class s p q and show that with suitable and not necessarily integer p and q it achieves an unusually high accuracy when the integrand has algebraic endpoint singularities we also illustrate our results with numerical examples via the sin p q transformation finally we discuss the computation of surface integrals in r 3 containing point singularities with the help of class s p q transformations