matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

a hyperelastic fractional damage material model with memory abstract in this paper a scalar damage model for hyperelastic materials is considered the novelty of the proposed approach lies in the evolution law for the damage variable that is formulated with the application of fractional calculus in this way damage evolution includes the memory or in other words the current intensity of damage evolution which is based on information from the past whose length is included in the fractional operator based on illustrative examples the flexibility of the model to mimic experimentally observed material behaviour is presented

on conformable fractional calculus recently the authors khalil et al 2014 introduced a new simple well behaved definition of the fractional derivative called conformable fractional derivative in this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus the fractional versions of chain rule exponential functions gronwall s inequality integration by parts taylor power series expansions laplace transforms and linear differential systems are proposed and discussed

approximations of fractional integrals and caputo fractional derivatives abstract in this paper we propose two algorithms for numerical fractional integration and caputo fractional differentiation we present a modification of trapezoidal rule that is used to approximate finite integrals the new modification extends the application of the rule to approximate integrals of arbitrary order 0 we then using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order 0 where the fractional derivative based on caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points the study is conducted through illustrative examples and error analysis

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

hierarchic models for laminated plates and shells the definition essential properties and formulation of hierarchic models for laminated plates and shells are presented the hierarchic models satisfy three essential requirements approximability asymptotic consistency and optimality of convergence rate aspects of implementation are discussed and the performance characteristics are illustrated by examples

self correspondences of the generic fibre of lefschetz pencils and the leray filtration abstract in this paper we give a detailed analysis of the interaction between homological self correspondences of the general fibre y k t of the lefschetz fibration x p 1 of a lefschetz pencil on a smooth projective variety x k and the leray filtration of we derive the result that if the standard conjecture b y holds then the operator x p x n 1 is algebraic where p x n 1 is defined as the inverse of l on l p n 1 x and 0 on l k p j x for 1 n 1 k j in the course of our proof we see that under the above assumption the kunneth projectors x i for i n 1 n n 1 are algebraic

a universal physically consistent definition of material damage abstract this paper proposes a physically consistent definition of material damage the definition is based on the physical concepts of inter atomic energy and the breaking and re establishing of the atomic bonds the states of material damage are physically determined in a broad sense by the configuration of the atomic bonds the meso characteristics such as vacancies dislocations pores slips microcavities microcracks and so on could be quantified through the use of subsystems each corresponding to a special configuration the constitutive equations referring to the material damage represented by the definition could be developed by following both nonlocal and local theories

l algebre de hopf et le groupe de galois motiviques d un corps de caracteristique nulle ii this is the second article of a series of two aiming at constructing and studying motivic galois groups in the context of triangulated motives these motivic galois groups were constructed in the first article and their algebras of regular functions were described concretely in terms of differential forms or algebraic cycles in the present article we gather important complements to the first one in the first part of this article we describe the link between the motivic galois group and the usual galois group of a subfield of in the second part we develop the basis of a theory of ramification for the motivic galois groups by constructing motivic versions of the decomposition and inertia groups associated to a geometric valuation we also introduce the relative motivic galois group of an extension k k which measures the difference between the motivic galois groups of k and k the latter is more accessible than its absolute counterpart in effect we show that it is a quotient of the pro algebraic completion of the topological fundamental pro group of a the pro variety homk k at least when the extension k k is of finite type

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

beam elements in instability problems this paper deals with the formulation of beam elements for the numerical analysis of instability phenomena in frame type structures total versus co rotational approaches are discussed comparatively for both two dimensional and three dimensional problems and the similarities between the two types are outlined in the context of 3d beam elements special attention is given to the parameterisation of the orthogonal transformation used to define the rotational field of the beam the technique advocated in the paper is based on the so called rotational vector this leads to symmetric stiffness matrices and avoids the need for special updating procedures for the rotational variables a set of test problems for which the critical behaviour is governed by fold cusp and butterfly catastrophes is used to assess the performances of the considered element types it is shown that analytically verified identities in element formulation also hold in numerical application the examples also show how complex instability behaviour can be reproduced by all elements where sufficient accuracy is introduced into the kinematic expressions the analytical derivation of element expressions with symbolic manipulations from stated basic assumptions is consistently used in the paper

motivation for hodge cycles given two smooth projective varieties x and y x is defined to motivate y if the motive of y is contained in the tensor category generated by x some techniques are given for checking this condition it is shown that in a number of cases moduli spaces of sheaves over curves or surfaces are motivated by the underlying curve or surface this is used to check the hodge and related conjectures for some of these examples

motives and filtrations on chow groups in this paper sequential to sa we discuss a philosophy of bloch and beilinson concerning filtrations on chow groups of algebraic vareties which provides us with a good understanding on the structure of at least certain portions of chow groups let k be a fixed ground field and let c be the category of smooth projective schemes over k with a fixed suitable weil cohomology theory cf notations x in mathcal c rightsquigarrow h x mathop oplus limits i in z h i x

noncommutative motives numerical equivalence and semi simplicity making use of hochschild homology we introduce the correct category nnum k f of noncommutative numerical motives over a base ring k and with coefficients in a field f we prove that nnum k f is abelian semi simple and that grothendieck s category num k q of numerical motives embeds into nnum k q after being factored out by the action of the tate object as an application we obtain an alternative proof of jannsen s celebrate semi simplicity result which uses the noncommutative world instead of a classical weil cohomology

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

noncommutative motives in positive characteristic and their applications abstract let k be a base field of positive characteristic making use of topological periodic cyclic homology we start by proving that the category of noncommutative numerical motives over k is abelian semi simple as conjectured by kontsevich then we establish a far reaching noncommutative generalization of the weil conjectures originally proven by dwork and grothendieck in the same vein we establish a far reaching noncommutative generalization of the cohomological interpretations of the hasse weil zeta function originally proven by hesselholt as a third main result we prove that the numerical grothendieck group of every smooth proper dg category is a finitely generated free abelian group as claimed without proof by kuznetsov then we introduce the noncommutative motivic galois super groups and following an insight of kontsevich relate them to their classical commutative counterparts finally we explain how the motivic measure induced by berthelot s rigid cohomology can be recovered from the theory of noncommutative motives

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

large deformation analysis of flexible beams presented here are numerical verifications of a geometrically exact curved beam model which fully accounts for large rotations large displacements initial curvatures and extensionality a multiple shooting method is used to solve the two point boundary value problem of flexible beams undergoing large elastic rotations and displacements in three dimensional space numerically exact large static deformations of eight beams subjected to different loading and or boundary conditions are obtained these solutions can be used to verify the performance of general finite element codes in analyzing large structural deformations

deformation and vibration of a spatial elastica with fixed end slopes abstract in this paper we study the static deformations and vibration frequencies of a clamped clamped spatial elastica one clamp is fixed in space and the other is attached to a slider which is allowed to slide without friction on a linear track emphasis is placed on the effect of the slope difference between the two clamps on the response of the elastica when the slider is under the action of an edge thrust the equations of motion of the elastica are formulated within the framework of director theory and the static deformations and vibration frequencies are calculated by shooting method an experimental set up is designed to examine the theoretical predictions when the slope difference between the two clamps is smaller than 68 two separate load deflection curves can be found one for spatial deformation and the other for purely planar deformation in force control the elastica slider assembly may jump between planar and spatial deformations as the edge thrust increases or decreases monotonically in displacement control on the other hand no jump will occur in the case when the slope difference is greater than 68 only planar deformation exists

dynamic response and instability of frame structures abstract this paper presents a geometrically non linear dynamic instability analysis for both two and three dimensional frames which may be subjected to finite rotations the finite element displacement method based on the beam column approach is employed to derive the non linear equations governing the behaviour of plane and spatial frames a co rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted the governing dynamic equilibrium equations are obtained from the static equations by adding the inertia and damping terms the implicit newmark time integration with the newton raphson nr iteration method is employed dynamic critical loads are defined by the budiansky roth criterion several numerical examples are illustrated to demonstrate the effectiveness of the present method

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

hierarchical finite element analyses of geometrically non linear vibration of beams and plane frames geometrically non linear vibrations of beams and plane frameworks are analyzed by the hierarchical finite element method hfem two main points are of interest the first is to compare polynomials trigonometric functions and beam eigenfunctions as displacement shape functions for beam hierarchical finite elements the second is to examine the suitability of the hfem for time domain non linear analyses

geometrically non linear beam element for dynamics simulation of multibody systems summary the main objective of the present paper is to derive an explicit expression for the stiffness term in the dynamics equations of an elastic beam based on the non linear kinematics of deformation in particular we develop a closed form expression for the non linear stiffness matrix which operates on the total elastic deformations to give the vector of internal forces the derivation begins with the description of the deformation field and the exact non linear strain4isplacement relations for a conventional beam element the strain energy is formulated in accordance with the slender beam theory and various non linear terms are identified the non linear stiffness matrix is then obtained in two stages the first step involves application of castigliano s theorem to generate the vector of nodal internal forces s from the strain energy of the element in the second step we expand s in the nodal co ordinates q using the taylor formula because of the particular form of s this expansion is exact and consists of only three terms linear quadratic and cubic it allows us to directly evaluate the respective three terms of the non linear stiffness matrix by setting the closed form expressions for the elemental matrices xg and x are provided in terms of the nodal co ordinates these were determined using the maple symbolic computation program which was also employed to generate the latex format of the results and the corresponding fortran code the paper is concluded with two numerical examples demonstrating the performance of the dynamics models incorporating the proposed non linear stiffness term the first example illustrates the response of a cantilever beam to a prescribed moment at the base the second example is modelled after an experimental planar manipulator driven by two direct drive actuators the results illustrate the differences between three dynamics models formulated with the linear second order and third order approximations for the stiffness term

emergent spacetime from modular motives the program of constructing spacetime geometry from string theoretic modular forms is extended to calabi yau varieties of dimensions three and four as well as higher rank motives modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the l functions associated to omega motives of calabi yau varieties generated by their holomorphic n forms via galois representations the modular forms that emerge in this way are related to characters of the underlying rational conformal field theory the converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication the emerging picture suggests that the l function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

polarizations and grothendieck s standard conjectures we prove that grothendieck s hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the hodge conjecture holds for complex abelian varieties of cm type for abelian varieties with no exotic algebraic classes we prove the hodge standard conjecture unconditionally

integral motives and special values of zeta functions for each field k we define a category of rationally decomposed mixed motives with z coefficients when k is finite we show that the category is tannakian and we prove formulas relating the behaviour of zeta functions near integers to certain ext groups

non linear vibration of beams with internal resonance by the hierarchical finite element method the hierarchical nite element hfem and the harmonic balance methods hbm are used to investigate the geometrically non linear free and steady state forced vibrations of uniform slender beams the beam analogue of von kah rmah n s non linear strain displacement relationships are employed and the middle plane in plane displacements are included in the model the equations of motion are developed by applying the principle of virtual work and are solved by a continuation method 1 3 and 1 5 internal resonances are discovered and their consequences are discussed the convergence properties of the hfem are analyzed and it is demonstrated that the hfem model requires far fewer degrees of freedom than the h version of the fem models presented in the literature 1999 academic press

on the inclusion principle for the hierarchical finite element method the inclusion principle provides a qualitative characterization of the eigenvalues of a matrix the principle has been shown to apply to systems described by a single hermitian matrix the most important of which being the real symmetric matrix self adjoint distributed systems when discretized by either the classical rayleigh ritz method or by the finite element method lead to algebraic eigenvalue problems described in terms of two real symmetric matrices the algebraic eigenvalues problem derived by the classical rayleigh ritz method possesses the embedding feature required by the inclusion principle but that derived by the finite element method in general does not this paper demonstrates that the inclusion principle can be extended to discretized systems derived by the hierarchical finite element method

skew polynomial rings with binomial relations abstract in this paper we continue the study of a class of standard finitely presented quadratic algebrasaover a fixed fieldk called binomial skew polynomial rings we consider some combinatorial properties of the set of defining relationsfand their implications for the algebraic properties ofa we impose a condition called onfand prove that in this caseais a free module of finite rank over a strictly ordered noetherian domain we show that an analogue of the diamond lemma is true for one sided ideals of a skew polynomial ringawith condition we prove also that if the set of defining relationsfis square free then condition is necessary and sufficient for the existence of a finite groebner basis of every one sided ideal ina and for left and right noetherianness ofa as a corollary we find a class of finitely generated non commutative semigroups which are left and right noetherian

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

beam p version finite element for geometrically non linear vibrations in space abstract a beam p version hierarchical finite element for geometrically non linear vibrations in space is presented stresses are obtained by adding the effects of traction torsion and bending a simplified version of green s strain tensor and the generalised hooke s law are used the mass and stiffness matrices are obtained by the principle of the virtual work and the ensuing non linear equations of motion are solved by newmark s method some examples of linear spacial motion and non linear planar motion are tested in order to verify the element derived and it is demonstrated that this element requires a small number of degrees of freedom for accuracy non linear vibrations in space are then studied

motives and admissible representations of automorphism groups of fields some of basic properties of the groups of automorphisms of algebraically closed fields and of their smooth representations are studied in characteristic zero grothendieck motives modulo numerical equivalence are identified with a full subcategory in the category of graded smooth representations of certain automorphism groups of algebraically closed fields

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved

noncommutative geometry and motives the thermodynamics of endomotives we combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of l functions the analogue in characteristic zero of the action of the frobenius on l adic cohomology is the action of the scaling group on the cyclic homology of the cokernel in a suitable category of motives of a restriction map of noncommutative spaces the latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive a noncommutative generalization of artin motives semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives with an action of the absolute galois group the semigroup of endomorphisms of the multiplicative group yields the bost connes system from which one obtains through the above procedure the desired cohomological interpretation of the zeros of the riemann zeta function in the last section we also give a lefschetz formula for the archimedean local l factors of arithmetic varieties

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