polynomial nonlinearities in differential equations abstract adomian and his collaborators have found solutions for nonlinear stochastic or deterministic in the limiting case differential equations involving polynomial non linearities using the first author s a n polynomials and recently developed methods of calculating these polynomials it becomes very easy to write solutions for non linear terms such as y m for positive integers m the cases of negative integer m or even decimal powers will appear elsewhere

quasi cohen macaulay properties of local homomorphisms abstract for a large class of local homomorphisms r s including those of finite g dimension studied by avramov and foxby proc london math soc 75 1997 241 270 we assign a new numerical invariant called the quasi cohen macaulay defect of and a local homomorphism is called quasi cohen macaulay if it is of finite g dimension and has trivial quasi cohen macaulay defect we show among other things the following g dim spec spec g dim

an efficient algorithm for the multivariable adomian polynomials abstract in this article the sum of the series of multivariable adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable taylor expansion of an analytic function of the decomposition series of solutions u 1 u 2 u m about the initial solution components u 1 0 u 2 0 u m 0 of course the multivariable adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations the index matrices and their simplified forms of the multivariable adomian polynomials are introduced we obtain the recurrence relations for the simplified index matrices which provide a convenient algorithm for rapid generation of the multivariable adomian polynomials another alternative algorithm for term recurrence is established in these algorithms recurrence processes do not require complicated operations such as parametrization expanding and regrouping derivatives etc as practiced in prior art the mathematica program generating the adomian polynomials based on the algorithm in this article is designed

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

adomian decomposition method for solving the volterra integral form of the lane emden equations with initial values and boundary conditions in this paper we use the systematic adomian decomposition method to handle the integral form of the lane emden equations with initial values and boundary conditions the volterra integral form of the lane emden equation overcomes the singular behavior at the origin x 0 we confirm our belief that the adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well finally we consider the modified decomposition method of rach adomian and meyers for the volterra integral form

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

on the dynamic behavior of a functionally graded piezoelectric strip with periodic cracks vertical to the boundary abstract the dynamic anti plane problem for a functionally graded piezoelectric strip containing a periodic array of parallel cracks which are perpendicular to the boundary is considered integral transforms techniques are employed to reduce the problem to the solution of singular integral equations numerical results are presented to show the influences of geometry electromechanical combination factor and material gradient parameter on the fracture behavior

nonlinear differential equations with negative power nonlinearities abstract differential equations involving the term y m where m is a positive integer are solved by the decomposition method 1 3 441 452

families of auslander reiten components for simply connected differential graded algebras peter jorgensen introduced the auslander reiten quiver of a simply connected poincare duality space he showed that its components are of the form mathbb z a infty and that the auslander reiten quiver of a d dimensional sphere consists of d 1 such components we show that this is essentially the only case where finitely many components appear more precisely we construct families of modules where for each family each module lies in a different component depending on the cohomology dimensions of the differential graded algebras which appear this is either a discrete family or an n parameter family for all n

totally reflexive modules and poincare series abstract we study cohen macaulay non gorenstein local rings r m k admitting certain totally reflexive modules more precisely we give a description of the poincare series of k by using the poincare series of a non zero totally reflexive module with minimal multiplicity our results generalize a result of yoshino to higher dimensional cohen macaulay local rings moreover from a quasi gorenstein ideal satisfying some conditions we construct a family of non isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators

p rings and p homomorphisms all rings considered in this paper are unitary commutative and noetherian the notations are those of n bourbaki a grothendieck and h matsumura in particular for a semi local ring a 2 means the radical adic completion of a let r be a property of a local noetherian ring and p the following property associated to r of a noetherian algebra a over a field k is 7 3 i the k algebra a satisfies p if and on1 v for any finite extension k of k the local rings of a mj k sati the property r a ring homomorphism q a f r is called a p homomorphism if it is flat and if for any prime ideal p of a the k p algebra b 0 k p satisfies p by a p ring we mean a ring a such that for any prime ideal p of a the canonical homomorphism a p a p is a p homomorphism this paper is devoted for a nice property r to the following problems asked by a grothendieck 8 7 4 7 5 i

semi dualizing complexes and their auslander categories let r be a commutative noetherian ring we study r modules and complexes of such with excellent duality properties while their common properties are strong enough to admit a rich theory we count among them such potentially diverse objects as dualizing complexes for r on one side and on the other the ring itself in several ways these two examples constitute the extremes and their well understood properties serve as guidelines for our study however also the employment in recent studies of ring homomorphisms of complexes lying between these extremes is incentive

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

homological properties of cochain differential graded algebras abstract consider a local chain differential graded algebra such as the singular chain complex of a pathwise connected topological group in two previous papers a number of homological results were proved for such an algebra an amplitude inequality an auslander buchsbaum equality and a gap theorem these were inspired by homological ring theory by the so called looking glass principle one would expect that analogous results exist for simply connected cochain differential graded algebras such as the singular cochain complex of a simply connected topological space indeed this paper establishes such analogous results

the crack problem for nonhomogeneous materials under antiplane shear loading a displacement based formulation abstract this paper presents a displacement based integral equation formulation for the mode iii crack problem in a nonhomogeneous medium with a continuously differentiable shear modulus which is assumed to be an exponential function i e g x g0exp x this formulation leads naturally to a hypersingular integral equation the problem is solved for a finite crack and results are given for crack profiles and stress intensity factors the results are affected by the parameter describing the material nonhomogeneity this study is motivated by crack problems in strain gradient elasticity theories where higher order singular integral equations naturally arise even in the slope based formulation

biem analysis of dynamically loaded anti plane cracks in graded piezoelectric finite solids anti plane cracks in finite functionally graded piezoelectric solids under time harmonic loading are studied via a non hypersingular traction based boundary integral equation method biem the formulation allows for a quadratic variation of the material properties in two directions the boundary integral equation bie system is treated by using the frequency dependent fundamental solution based on radon transforms its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor sif and the electrical displacement intensity factor edif are determined several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters

cohen macaulay properties of ring homomorphisms abstract numerical invariants which measure the cohen macaulay character of homomorphisms r s of noetherian rings are introduced and studied comprehensive results are obtained for homomorphisms which are locally of finite flat dimension they provide a point of view from which a variety of phenomena receive a unified treatment the conceptual clarification and technical versatility of this approach leads among other things to a determination of those homomorphisms which preserve the cohen macaulay character of the rings to the discovery of new classes of homomorphisms with remarkable stability properties and to solutions of some problems on flat homomorphisms raised by grothendieck

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

adomian decomposition method for solving fractional nonlinear differential equations in this article we have discussed a new application of adomian decomposition method on time fractional nonlinear fractional differential equations three models with fractional time derivative of order a 0 a 1 are considered and solved by means of adomian decomposition method the behaviour of adomian solutions and the effects of different values of a are investigated numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

bass series of local ring homomorphisms of finite flat dimension nontrivial relations between bass numbers of local commutative rings are established in case there exists a local homomorphism r s which makes s into an r module of finite flat dimension in particular it is shown that an inequality r i depth r s i depth s holds for all i z this is a consequence of an equality involving the bass series i r m t i z r i m t i of a complex m of r modules which has bounded above and finite type homology and the bass series of the complex of s modules m r s where denotes the derived tensor product it is prove that there is an equa lity of formal laurent series i s m rs t i r m t i f t where f is the fiber of considered as a homomorphism of commutative differential graded rings

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

descent via koszul extensions abstract let r be a commutative noetherian local ring with completion r we apply differential graded dg algebra techniques to study descent of modules and complexes from r to r where r is either the henselization of r or a pointed etale neighborhood of r we extend a given r complex to a dg module over a koszul complex we describe this dg module equationally and apply artin approximation to descend it to r this descent result for koszul extensions has several applications when r is excellent we use it to descend the dualizing complex from r to a pointed etale neighborhood of r this yields a new version of p roberts theorem on uniform annihilation of homology modules of perfect complexes as another application we prove that the auslander condition on uniform vanishing of cohomology ascends to r when r is excellent henselian and cohen macaulay

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

solving nonlinear differential equations with decimal power nonlinearities abstract the decomposition method stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press is applied to the solution of nonlinear differential equations of the form ly ny x t where l is a linear differential operator and ny is a nonlinear term of the form y with a decimal number

locally complete intersection homomorphisms and a conjecture of quillen on the vanishing of cotangent homology classical deflnitions of locally complete intersection l c i homomor phisms of commutative rings are limited to maps that are essentially of flnite type or at the concept introduced in this paper is meaningful for homo morphisms ri s of commutative noetherian rings it is deflned in terms of the structure of in a formal neighborhood of each point of specs w e characterize the l c i property by difierent conditions on the vanishing of the andre quillen homology of the r algebra s one of these descriptions estab lishes a very general form of a conjecture of quillen that was open even for homomorphisms of flnite type if s has a flnite resolution by at r modules and the cotangent complex l sjr is quasi isomorphic to a bounded complex of at s modules then is l c i the proof uses a mixture of methods from commutative algebra difierential graded homological algebra and homotopy theory the l c i property is shown to be stable under a variety of operations including composition decomposition at base change localization and com pletion the present framework allows for the results to be stated in proper generality many of them are new even with classical assumptions for in stance the stability of l c i homomorphisms under decomposition settles an open case in fulton s treatment of orientations of morphisms of schemes