on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

on the negative k theory of schemes in finite characteristic abstract we show that if x is a d dimensional scheme of finite type over an infinite perfect field k of characteristic p 0 then k i x 0 and x is k i regular for i d 2 whenever the resolution of singularities holds over k this proves the k dimension conjecture of weibel c weibel k theory and analytic isomorphisms invent math 61 1980 177 197 2 9 except for d 1 i d 2 in all characteristics assuming the resolution of singularities

nil k theory maps to cyclic homology algebraic k theory breaks into two pieces nil k theory and karoubi villamayor k theory karoubi has constructed chern classes from the latter groups into cyclic homology we construct maps from nil k theory to cyclic homology which are compatible with karoubi s maps but with a degree shift several recent results show that in characteristic zero our map is often an isomorphism ever since a connes discovered cyclic homology in 1982 it has been widely held that cyclic homology is closely related to algebraic k theory supporting this belief karoubi defined chern classes limep kn a limhcn 2p a in k1 and goodwillie defined a related map ey kn a hcn a in g other maps between k theory and variants of cyclic homology have been constructed by loday unpublished burghelea bur 1 ogle ogle 1 and others of course all of these maps are related to the dennis trace map to hochschild homology d kn a hhn a which r k dennis discovered in 1975 d in this paper we prove three results the first is that karoubi s map factors through goodwillie s theorem a for every associative algebra there is a commutative diagram kn a

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

a smart nonstandard finite difference scheme for second order nonlinear boundary value problems a new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems an artificial parameter is introduced in the scheme symbolic computation is proposed for the construction of the scheme local truncation error of the method is discussed numerical examples are illustrated numerical results show that the method is very effective

nonstandard finite difference equations for odes and 1 d pdes based on piecewise linearization a method for the solution of initial and boundary value problems in nonlinear ordinary differential equations and for one dimensional partial differential equations which provides c 1 solutions is presented the method is based on the linearization of the differential equation in intervals which contain only two grid points and provides three point nonstandard finite difference equations for the nodal amplitudes the method is applied to steady reaction diffusion equations two point singularly perturbed boundary value problems and the steady burgers equation and compared with standard finite difference and finite element formulations for one dimensional partial differential equations the temporal derivatives are first discretized and the resulting ordinary differential equation accounts for both the temporal and spatial stiffnesses and is solved by means of piecewise linearization since the linearization includes a jacobian matrix it may be easily employed to refine the mesh where steep gradients occur

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

trigonometric polynomial or exponential fitting approach the application of a trigonometric polynomial and an exponential fitting approach is compared for a three point formula for second order derivatives for simpson s quadrature rule and for numerov s scheme for second order differential equations the expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied the errors for specific problems obtained in both the approaches as a function of the frequency are compared

polylogarithms regulators and arakelov motivic complexes we construct an explicit regulator map from bloch s higher chow group complex to the deligne complex of a complex algebraic variety x we define the arakelov motivic complex as the cone of this map shifted by one its last cohomology group is a version of the arakelov chow group defined by h gillet and c soule we relate the grassmannian n logarithms defined as in g5 to the geometry of symmetric space sln c su n for n 2 we recover lobachevsky s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm using the relationship with symmetric spaces we construct the borel regulator on k2n 1 c via the grassmannian n logarithms we study the chow dilogarithm and prove a reciprocity law which strengthens suslin s reciprocity law for milnor s group k 3 on curves our note g5 can serve as an introduction to this paper

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

one point pseudospectral collocation for the one dimensional bratu equation abstract the one dimensional planar bratu problem is u xx exp u 0 subject to u 1 0 because there is an analytical solution this problem has been widely used to test numerical and perturbative schemes we show that over the entire lower branch and most of the upper branch the solution is well approximated by a parabola u x u 0 1 x 2 where u 0 is determined by collocation at a single point x the collocation equation can be solved explicitly in terms of the lambert w function as u 0 w 1 2 2 1 2 where both real valued branches of the w function yield good approximations to the two branches of the bratu function we carefully analyze the consequences of the choice of we also analyze the rate of convergence of a series of even chebyshev polynomials which extends the one point approximation to arbitrary accuracy the bratu function is so smooth that it is actually poor for comparing methods because even a bad inefficient algorithm is successful it is however a solution so smooth that a numerical scheme the collocation or pseudospectral method yields an explicit analytical approximation we also fill some gaps in theory of the bratu equation we prove that the general solution can be written in terms of a single parameter free x without knowledge of the explicit solution the analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly we give three overlapping perturbative approximations to the eigenrelation allowing the analytical solution to be easily evaluated throughout the entire parameter space

a nonlinear shooting method for two point boundary value problems abstract we study a new nonlinear shooting method for solving two point boundary value problems and show numerical experiments with various initial velocity conditions we discuss and analyze the numerical solutions which are obtained by the shooting method

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

comparison between algebraic and topological k theory of locally convex algebras abstract this paper is concerned with the algebraic k theory of locally convex c algebras stabilized by operator ideals and its comparison with topological k theory we show that if l is locally convex and j a frechet operator ideal then all the different variants of topological k theory agree on the completed projective tensor product l j and that the obstruction for the comparison map k l j k top l j to be an isomorphism is absolute algebraic cyclic homology we prove the existence of an exact sequence theorem 6 2 1 we show that cyclic homology vanishes in the case when j is the ideal of compact operators and l is a frechet algebra whose topology is generated by a countable family of sub multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family theorem 8 3 3 this proves the generalized version of karoubi s conjecture due to mariusz wodzicki and announced in his paper m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 we also consider stabilization with respect to a wider class of operator ideals called sub harmonic every frechet ideal is sub harmonic but not conversely for example the schatten ideal l p is sub harmonic for all p 0 but is frechet only if p 1 we prove a variant of the exact sequence above which essentially says that if a is a c algebra and j is sub harmonic then the obstruction for the periodicity of k a c j is again cyclic homology theorem 7 1 1 this generalizes to all algebras a result of wodzicki for h unital algebras announced in m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 the main technical tools we use are the diffeotopy invariance theorem of cuntz and the second author which we generalize in theorem 6 1 6 and the excision theorem for infinitesimal k theory due to the first author

the nilpotence conjecture in k theory of toric varieties it is shown that all nontrivial elements in higher k groups of toric varieties are annihilated by iterations of the natural frobenius type endomorphisms this is a higher analog of the triviality of vector bundles on affine toric varieties

a constructive monotone iterative method for second order bvp in the presence of lower and upper solutions this paper concerns the monotone approximations of solutions of boundary value problems such as u f t u u 0 u 0 u 1 0 we consider linear iterative scheme in case f is lipschitz in u and satisfies a one sided lipschitz condition in u the initial approximations are lower and upper solutions which can be ordered one way a email protected we also consider the periodic and the dirichlet problems

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

iterative method for solving a nonlinear boundary value problem in this paper a boundary value problem for a nonlinear second order ordinary differential equation is studied by means of the maximum principle we established the existence and the uniqueness of a solution of the problem then for finding the solution an iterative method is proposed it is proved that this method converges much faster than the picar successive approximations and in a particular case it gives two sided monotone approximations to the exact solution of the problem finally some illustrative examples are considered to confirm the efficiency of the method

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

regulators on additive higher chow groups as an attempt to understand motives over k x x m we define the cubical additive higher chow groups with modulus for all dimensions extending the works of s bloch h esnault and k r ulling on 0 dimensional cycles we give an explicit construction of regulator maps on the groups of 1 cycles with an aid of the residue theory of a parshin and v lomadze

excision in algebraic k theory introduction 1 excision and the homology of affine groups 2 volodin s spaces 3 excision for rings with the triple factorization property 4 lie analogue of volodin s construction 5 the homology of nilpotent groups and nilpotent lie algebras 6 a proof of the main theorems 7 the tensor product of h unital algebras 8 the matrix ring of a small category 9 the acyclicity of triangular complexes 10 a proof of karoubi s conjecture

higher topological cyclic homology and the segal conjecture for tori abstract we investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory higher topological cyclic homology is constructed from the fixed points of a version of topological hochschild homology based on the n dimensional torus and we propose it as a computationally tractable cousin of n fold iterated algebraic k theory the fixed points of toral topological hochschild homology are related to one another by restriction and frobenius operators we introduce two additional families of operators on fixed points the verschiebung indexed on self isogenies of the n torus and the differentials indexed on n vectors we give a detailed analysis of the relations among the restriction frobenius verschiebung and differentials producing a higher analog of the structure hesselholt and madsen described for 1 dimensional topological cyclic homology we calculate two important pieces of higher topological cyclic homology namely topological restriction homology and topological frobenius homology for the sphere spectrum the latter computation allows us to establish the segal conjecture for the torus which is to say to completely compute the cohomotopy type of the classifying space of the torus

torsion and divisibility for reciprocity sheaves and 0 cycles with modulus the notion of modulus is a striking feature of rosenlicht serre s theory of generalized jacobian varieties of curves it was carried over to algebraic cycles on general varieties by bloch esnault park rulling krishna levine recently kerz saito introduced a notion of chow group of 0 cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields we study the non homotopy invariant part of the chow group of 0 cycles with modulus and show their torsion and divisibility properties modulus is being brought to sheaf theory by kahn saito yamazaki in their attempt to construct a generalization of voevodsky suslin friedlander s theory of homotopy invariant presheaves with transfers we prove parallel results about torsion and divisibility properties for them

the bellman kalaba lakshmikantham quasilinearization method for neumann problems in this paper we develop a generalized quasilinearization method for a nonlinear neumann problem and obtain a sequence of approximate solutions converging monotonically and quadratically to a solution of the problem

higher chow groups with modulus and relative milnor theory let x be a smooth variety over a field k and d an effective divisor whose support has simple normal crossings we construct an explicit cycle map from z r x d nis the nisnevich motivic complex of the pair x d to km r x d nis r a shift of the relative milnor k sheaf of x d we show that this map induces an isomorphism h m nis x d z r h xnis k r x d nis for all i dimx this generalizes the well known isomorphism in the case d 0 we use this to prove a certain zariski descent property for the motivic cohomology of the pair ak m 1 0

duality via cycle complexes we show that bloch s complex of relative zero cycles can be used as a dualizing complex over perfect fields and number rings this leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields finite fields local fields of mixed characteristic and rings of integers in number rings generalizing results which so far have only been known for smooth schemes or in low dimensions and unifying the p adic and l adic theory as an application we generalize rojtman s theorem to normal projective schemes

the optimal exponentially fitted numerov method for solving two point boundary value problems second order boundary value problems are solved by means of exponentially fitted numerov methods these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe l gr ixaru g vanden berghe exponential fitting kluwer academic publishers dordrecht 2004 special attention is paid to the expression of the error term of such methods an algorithm concerning the choice of the best suited method and its parameter is discussed several numerical examples are given to sustain the theory

riemann roch for equivariant k theory abstract we show that for a linear algebraic group g acting on a smooth quasi projective scheme x over a field there is a chern character map k i g x r g r g c h x g ch g x i s g s g with rational coefficients which is an isomorphism this establishes the equivariant version of the riemann roch isomorphism between the higher algebraic k theory and the higher chow groups of smooth quasi projective schemes

replacing the finite difference methods for nonlinear two point boundary value problems by successive application of the linear shooting method abstract this paper studies numerical solution of nonlinear two point boundary value problems for second order ordinary differential equations first it establishes a connection between the finite difference method and the quasi linearization method we prove that using finite differences to discretize the sequence of linear differential equations arising from quasi linearization newton method on operator level leads to the usual iteration formula of the newton finite difference method from the provided derivation it can easily be inferred that such a relation holds also for the picard and the constant slope methods based on this result we propose a way of replacing the newton picard and constant slope finite difference methods by respective successive application of the linear shooting method this approach has a number of advantages it removes the necessity of solving systems of algebraic equations hence working with matrices altogether compared to the usual finite difference method with general solver it reduces the number of computational operations from o n 3 where n is the number of mesh points to only o n

additive polylogarithms and their functional equations let k varepsilon 2 k varepsilon varepsilon 2 the single valued real analytic n polylogarithm mathcal l n mathbb c to mathbb r is fundamental in the study of weight n motivic cohomology over a field k of characteristic 0 in this paper we extend the construction in unver algebra number theory 3 1 34 2009 to define additive n polylogarithms li n k varepsilon 2 to k and prove that they satisfy functional equations analogous to those of mathcal l n under a mild hypothesis we show that these functions descend to an analog of the nth bloch group b n k varepsilon 2 defined by goncharov adv math 114 197 318 1995 we hope that these functions will be useful in the study of weight n motivic cohomology over k e 2

an exponentially fitted and trigonometrically fitted method for the numerical solution of periodic initial value problems abstract an explicit symmetric multistep method is presented in this paper the new method is exponentially fitted and trigonometrically fitted and is of algebraic order eight the effectiveness of the exponential fitting is proved by the application of the new method and the classical one with constant coefficients to well known periodic problems

excision in cyclic homology and in rational algebraic k theory for a precise definition see 1 below by replacing everywhere k by k q one obtains the corresponding notion in rational algebraic k theory the above definition has an obvious counterpart for cyclic homology and algebras over a fixed field in the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable due to the well known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring an extension of k algebras will be called pure if it is pure as an extension of k modules in the sense of p m cohn 6 cf also appendix a 3 below the class of pure extensions contains e g i extensions which admit a k module splitting ii extensions a e r s with s flat over k in fact one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k modules is an inductive limit of split extensions see theorem a 4 of appendix a below everywhere in this paper the word excision used in the context of cyclic homology will mean excision with respect to the class of pure extensions the second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology before stating the corresponding result we need the following definition let us

nonstandard finite differences for a truncated bratu picard model abstract in this paper we consider theoretical and numerical properties of a nonlinear boundary value problem which is strongly related to the well known gelfand bratu model with parameter when approximating the nonlinear term in the model via a taylor expansion we are able to find new types of solutions and multiplicities depending on the final index n in the expansion the number of solutions may vary from 0 1 2 to in the latter case of infinitely many solutions we find both periodic and semi periodic solutions numerical experiments using a non standard finite difference nsfd approximation illustrate all these aspects we also show the difference in accuracy for different denominator functions in nsfd when applied to this model a full classification is given of all possible cases depending on the parameters n and

bass nk groups and cdh fibrant hochschild homology the k theory of a polynomial ring r t contains the k theory of r as a summand for r commutative and containing q we describe k r t k r in terms of hochschild homology and the cohomology of kahler differentials for the cdh topology we use this to address bass question whether kn r kn r t implies kn r kn r t1 t 2 the answer to this question is affirmative when r is essentially of finite type over the complex numbers but negative in general

application of local linear embedding to nonlinear exploratory latent structure analysis in this paper we discuss the use of a recent dimension reduction technique called locally linear embedding introduced by roweis and saul for performing an exploratory latent structure analysis the coordinate variables from the locally linear embedding describing the manifold on which the data reside serve as the latent variable scores we propose the use of semiparametric penalized spline methods for reconstruction of the manifold equations that approximate the data space we also discuss a crossvalidation strategy that can guide in selecting an appropriate number of latent variables synthetic as well as real data sets are used to illustrate the proposed approach a nonlinear latent structure representation of a data set also serves as a data visualization tool

a simple construction of nonstandard finite difference schemes for small nonlinear systems applied to sir models we present a simple mathematical construction for nonstandard finite difference nsfd schemes for small systems of nonlinear differential equations using standard differential equation approximation techniques such as introducing artificial viscosity and a predictor corrector scheme the construction creates an explicit scheme we begin by considering the nsfd scheme of mickens and that of erdogan and ozis for first order equations the latter s method formulates a method to calculate some of the denominator functions and nonlocal approximations that characterize mickens scheme we extend the analysis of erdogan and ozis by presenting a construction for second order nonlinear autonomous functions that naturally generates the numerator and denominator functions for mickens equivalent schemes we extend this result to systems of up to three differential equations and demonstrate the simply constructed scheme s computational advantages to handle transitions between linear and nonlinear problems as well as with the inclusion or exclusion of nonhomogeneous constant terms we include applications using the sir model for whooping cough with a constant and nonconstant for example seasonally affected transmission rate

k theory of endomorphism rings and of rings of invariants abstract this paper gives the following description ofk0of the endomorphism ring of a finitely generated projective module t heorem let t be a ring and p a finitely generated projective t module let i be the trace ideal of p then k0 end pt is isomorphic to a subgroup of k0 t i if further the natural map k1 t k1 t i is surjective then k0 end pt is isomorphic to the subgroup of k0 t generated by the direct summands of pn for n n as a corollary we can determinek0of the ring of invariants for many free linear actions in particular the following result is proved t heorem let v be a fixed point free linear representation of a finite group g over a field k of characteristic zero and let s v be the symmetric algebra of v let k be any finite dimensional k vector space then k0 s v g ks k s v g ks k similar results are given for suitable noncommutative versions ofs v

chebyshev finite difference approximation for the boundary value problems this paper presents a numerical technique for solving linear and non linear boundary value problems for ordinary differential equations this technique is based on using matrix operator expressions which applies to the differential terms it can be regarded as a non uniform finite difference scheme the values of the dependent variable at the gauss lobatto points are the unknown one solves for the application of the method to boundary value problems leads to algebraic systems the method permits the application of iterative method in order to solve the algebraic systems the effective application of the method is demonstrated by four examples

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

infinitesimal cohomology and the chern character to negative cyclic homology there is a chern character from k theory to negative cyclic homology we show that it preserves the decomposition coming from adams operations at least in characteristic zero

k regularity cdh fibrant hochschild homology and a conjecture of vorst it is a well known fact that algebraic if theory is homotopy invariant as a functor on regular schemes if x is a regular scheme then the natural map kn x kn x x 1 is an isomorphism for all n ez this is false in general for nonregular schemes and rings to express this failure bass introduced the terminology that for any contravari ant functor v defined on schemes a scheme x is called v regular if the pullback maps v x v x x ar are isomorphisms for all r 0 if x spec i we also say that r is p regular thus regular schemes are ifn regular for every n in contrast it was observed as long ago as in 2 that a nonreduced affine scheme can never be k regular in particular if a is an artinian ring that is a o dimensional noetherian ring then a is regular that is reduced if and only if a is ifi regular in 17 vorst conjectured that for an affine scheme x of finite type over a field f and of dimension d regularity and k i regularity are equivalent vorst proved this conjecture for d 1 by proving that regularity implies normality in this paper we prove vorst s conjecture in all dimensions provided the char acteristic of the ground field f is zero in fact we prove a stronger statement we say that x is regular in codimension n in x note that for all n z if a ring r is ifn regular then it is kn i regular this is proved in 17 for n 1 and in 6 4 4 for n kh x where k x is the algebraic if theory spectrum of x and kh x is the homotopy if theory of x defined in 19 we write fk r for fk spec r theorem 0 1 let r be a commutative ring which is essentially of finite type over a field f of characteristic 0 then the following hold a if tk r is n connected then r is regular in codimension khi x is an isomorphism for i n and a surjection for i n 1 so that fk x

motives of reductive groups our aim is to prove that the geometric motive with rational coefficients of a split reductive group g over a perfect field decomposes as the symmetric co algebra object of certain split mixed tate motives which are computable in terms of the root data of g moreover the isomorphism respects the hopf algebra structures in a weight graded sense we will also provide explicit decompositions in the case of classical groups

constructive proof for existence of nonlinear two point boundary value problems owing to the importance of differential equations in physics the existence of solutions for differential equations has been paid much attention in this paper the existence of solution are obtained for the nonlinear second order two point boundary value problem in the reproducing kernel space under certain assumptions on right hand side we propose constructive proof for the existence result and a method is presented to obtain the exact solution expressed by the form of series this paper is a extension of previous paper wei jiang minggen cui the exact solution and stability analysis for integral equation of third or first kind with singular kernel appl math comput 202 2 2008 666 674 which extends a method of solving linear problems to present method for solving nonlinear problems

an efficient algorithm for solving troesch s problem abstract a new algorithm is presented for solving troesch s problem the numerical scheme based on the modified homotopy perturbation technique is deduced some numerical experiments are made compared with the variational iteration method and the adomian decomposition method the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions finally the algorithm is applied to other problems

local multidimensional scaling for nonlinear dimension reduction graph drawing and proximity analysis in the past decade there has been a resurgence of interest in nonlinear dimension reduction among new proposals are local linear embedding isomap and kernel principal components analysis which all construct global low dimensional embeddings from local affine or metric information we introduce a competing method called local multidimensional scaling lmds like lle isomap and kpca lmds constructs its global embedding from local information but it uses instead a combination of mds and force directed graph drawing we apply the force paradigm to create localized versions of mds stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces we solve the problem of tuning parameter selection with a meta criterion that measures how well the sets of k nearest neighbors agree between the data and the embedding tuned lmds seems to be able to outperform mds pca lle isomap and kpca as illustrated with two well known image datasets the meta criterion can also be

noether lefschetz locus for beilinson hodge cycles i we discuss the noether lefschetz locus for beilinson s hodge cycles here beilinson s hodge cycles mean the cohomology class of bloch s higher chow cycles on smooth varieties the main result is to give an estimate of the dimension of the maximal component of the noether lefschetz locus for the universal family of open complete intersections

a variational approach to nonlinear two point boundary value problems abstract a variational approach is applied to nonlinear two point boundary value problems in this method an initial solution is chosen with some unknown parameters which can be identified by making j y stationary with respect to these parameters comparison with the exact solution shows that the method is very effective and convenient