an invariant action for noncommutative gravity in four dimensions two main problems face the construction of noncommutative actions for gravity with star products the complex metric and finding an invariant measure the only gauge groups that could be used with star products are the unitary groups i propose an invariant gravitational action in d 4 dimensions based on the constrained gauge group u 2 2 broken to u 1 1 u 1 1 no metric is used thus giving a naturally invariant measure this action is generalized to the noncommutative case by replacing ordinary products with star products the four dimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed

gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

hamiltonian gravity and noncommutative geometry a version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry the adm version of the gravitational action is expressed in terms of these variables it is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action

discrete spectral triples and their symmetries we classify zero dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure we investigate also whether such spectral triples admit a symmetry arising from the hopf algebra structure of the finite algebra we discuss examples of commutative algebras and group algebras

noncommutative geometry topology and the standard model vacuum as a ramification of a motivational discussion for previous joint work in which equations of motion for the finite spectral action of the standard model were derived we provide a new analysis of the results of the calculations therein switching from the perspective of spectral triple to that of fredholm module and thus from the analogy with riemannian geometry to the premetrical structure of the noncommutative geometry using a suggested noncommutative version of morse theory together with algebraic k theory to analyze the vacuum solutions the first two summands of the algebra for the finite triple of the standard model arise up to morita equivalence we also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix

dynamic fracture of a rectangular limited permeable crack in magneto electro elastic media under a time harmonic elastic p wave abstract the dynamic behavior of a rectangular limited permeable crack deeply embedded in transversely isotropic magneto electro elastic media was investigated under an incident harmonic stress wave using the generalized almansi s theorem and the schmidt method with the help of the fourier transform this problem was formulated into three pairs of dual integral equations with the jumps in displacement across the crack surfaces as the unknown variables by directly expanding the jumps in displacement across the crack surfaces into a series of jacobi polynomials the solution to the dual integral equations was derived finally the dynamic response of a rectangular crack under a harmonic wave was analyzed and the effects of the electric permittivity and the magnetic permeability of air inside the crack the geometric shape of the rectangular crack and the characteristics of the harmonic wave on the stress the electric displacement and the magnetic flux intensity factors in magneto electro elastic media were concluded

connes duality in pseudo riemannian geometry the connes formula giving the dual description for the distance between points of a riemannian manifold is extended to the lorentzian case it resulted that its validity essentially depends on the global structure of space time the duality principle classifying space times is introduced the algebraic account of the theory is suggested as a framework for quantization along the lines proposed by connes

remarks on mathematical structure of brst theories a number of rigorous results concerning the mathematical structure of quantum systems with the brst symmetry are presented for studying model independent properties of such systems a generalization of the strocchi wightman theory is suggested which is a wightman type theory in the krein space with an opj algebra of physical and ghost fields it is proved that the brst charge operatorq is either bounded with q 0 or unbounded with q a number of conditions equivalent to the no ghost theorem which states that the kernel ofq is non negative are derived

the algebra of grand unified theories the standard model is the best tested and most widely accepted theory of elementary particles we have today it may seem complicated and arbitrary but it has hidden patterns that are revealed by the relationship between three grand unied theories theories that unify forces and particles by extending the standard model symmetry group u 1 su 2 su 3 to a larger group these three are georgi and glashow s su 5 theory georgi s theory based on the group spin 10 and the pati salam model based on the group su 2 su 2 su 4 in this expository account for mathematicians we explain only the portion of these theories that involves nite dimensional group representations this allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve

localizing gauge theories from noncommutative geometry abstract we recall the emergence within noncommutative geometry of a generalized gauge theory from a real spectral triple for a unital c algebra a this includes a gauge group determined by the unitaries in a and gauge fields arising from a so called perturbation semigroup which is associated to a our main new result is the construction of an upper semi continuous c bundle on a hausdorff base space x whose space of continuous sections is isomorphic to a the gauge group acts by vertical automorphisms on this c bundle and can under some conditions be identified with the space of continuous sections of a group bundle on x moreover in some cases the gauge group is found to coincide with the group of inner automorphisms of a we discuss two classes of examples of our construction yang mills theory and toric noncommutative manifolds and we show that they actually give rise to continuous c bundles moreover in these examples the gauge groups coincide with the inner automorphism groups and can be realized as spaces of sections of group bundles that we explicitly determine

some prime factorization results for type ii1 factors we prove several unique prime factorization results for tensor products of type ii1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected lie groups of real rank 1 in particular we show that if mathcal r overline otimes mathcal l mathbb f r 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f r m is isomorphic to a subfactor in mathcal r overline otimes mathcal l mathbb f s 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f s n for some 2 ri sj then m n

eigenvalue inequalities and poincar e duality in noncommutative geometry in the context of connes noncommutative geometry eigenvalue inequalities of the type discovered by vafa and witten are shown to be a characteristic feature of those spectral geometric spaces of finite topological type that satisfy rational poincare duality in k theory

multiple collinear cracks in a piezoelectric material abstract in this paper an exact solution of multiple collinear cracks in piezoelectric material is obtained the permittivity of air environment is considered two cases have been studied in the first case the permittivity of air is far less than that of piezoelectric material therefore the electric induction in the air is negligible in the second case the permittivity of air is comparable with that of piezoelectric material the problem is deduced into riemann hilbert problem and solved by the way the electric boundary conditions are discussed this result demonstrates that the consideration of air in crack gap reduces the stress intensity factor

finite part integral and boundary element method to solve three dimensional crack problems in piezoelectric materials using the hypersingular integral equation method based on body force method a planar crack in a three dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed this crack problem is reduced to solve a set of hypersingular integral equations compare with the crack problems in elastic isotropic materials it is shown that for the impermeable crack the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials based on the exact analytical solution of the singular stresses and electrical displacements near the crack front the numerical method of the hypersingular integral equation is proposed by the finite part integral method and boundary element method which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied finally the numerical solutions of the stress and electric field intensity factors of some examples are given

on the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks this paper investigates the dynamic behaviour of a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady state electro mechanical loads both the permeable and impermeable boundary conditions are examined and discussed based on the use of integral transform techniques the problem is reduced to a set of singular integral equations which can be solved using chebyshev polynomial expansions numerical results are provided to show the effect of the geometry of interacting collinear cracks the applied electric fields the electric boundary conditions along the crack faces and the loading frequency on the resulting dynamic stress intensity and electric displacement intensity factors

two parallel symmetry permeable cracks in functionally graded piezoelectric piezomagnetic materials under anti plane shear loading abstract in this paper the behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric piezomagnetic materials subjected to an anti plane shear loading is investigated to make the analysis tractable it is assumed that the material properties c44 e15 e11 q15 d11 and 11 vary exponentially with coordinate parallel to the crack by using the fourier transform the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces these equations are solved using the schmidt method the normalized stress the electrical displacement and the magnetic flux intensity factors are determined for different geometric for the permeable electric boundary conditions numerical examples are provided to show the effect of the geometry of the interacting crack and the parameter describing the functionally graded materials upon the stress intensity factor of the cracks

general aspects of pt symmetric and p self adjoint quantum theory in a krein space in our previous work we proposed a mathematical framework for symmetric quantum theory and in particular constructed a krein space in which symmetric operators would naturally act in this work we explore and discuss various general consequences and aspects of the theory defined in the krein space not only spectral properties and symmetry breaking but also several issues crucial for the theory to be physically acceptable such as time evolution of state vectors probability interpretation uncertainty relation classical quantum correspondence completeness existence of a basis and so on in particular we show that for a given real classical system we can always construct the corresponding symmetric quantum system which indicates that symmetric theory in the krein space is another quantization scheme rather than a generalization of the traditional hermitian one in the hilbert space we propose a postulate for an operator to be a physical observable in this framework

integral equation methods for elliptic problems with boundary conditions of mixed type laplace s equation with mixed boundary conditions that is dirichlet conditions on parts of the boundary and neumann conditions on the remaining contiguous parts is solved on an interior planar domain using an integral equation method rapid execution and high accuracy is obtained by combining equations which are of fredholm s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results this opens up for the rapid and accurate solution of several elliptic problems of mixed type

admissibility of weights on non normed algebras the notion of weights on topological algebras is defined and studied the primary purpose is to define the notions of admissibility and approximate admissibility of weights and to investigate when a weight is admissible or approximately admissible the results obtained are applied to vector weights and tracial weight on unbounded operator algebras as well as to weights on smooth subalgebras of a c algebra

string geometry and the noncommutative torus we construct a new gauge theory on a pair of d dimensional noncommutative tori the latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra and the noncommutative torus we show that the tachyon algebra of is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus we construct the corresponding real spectral triples and determine their morita equivalence classes using string duality arguments these constructions yield simple proofs of the o d d morita equivalences between d dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory we classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group we compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality symmetric the duality invariant gauge theory is manifestly covariant but contains highly non local interactions we show that it also admits a new sort of particle antiparticle duality which enables the construction of instanton field configurations in any dimension the duality non symmetric on shell projection of the field theory is shown to coincide with the standard non abelian yang mills gauge theory minimally coupled to massive dirac fermion fields

finite geometries and clifford algebras ii the pleasant incidence properties of m dimensional projective geometry enable us upon employing the finite geometry over the field rm f rm 2 to deal nicely with certain commutativity anti commutativity aspects of clifford algebras cl o d in dimensions d 2m 1 1 m 2 3

three dimensional green s functions in anisotropic piezoelectric solids abstract explicit expressions for three dimensional extended green s displacements in general anisotropic piezoelectric solids are derived a very efficient procedure for the numerical evaluation of the derivatives of the extended green s displacements is also proposed numerical comparisons are carried out for a transversely isotropic piezoelectric solid for which exact closed form solutions are available it is found that the extended green s displacements and their derivatives obtained with the present explicit formulation are in perfect agreement with the exact closed form solutions these green s functions can be used in the boundary integral equations for piezoelectric solids of general anisotropy and for subsequent numerical solutions of these equations by means of the boundary element method

noncommutative integration calculus a noncommutative integration calculus arising in the mathematical description of schwinger terms of fermion yang mills systems is discussed the differential complexes of forms u0 e u1 e un with e a grading operator on a hilbert space h and ui bounded operators on h which naturally contains the compactly supported de rham forms on rd i e e is the sign of the free dirac operator on rd and h is a l2 space on rd are considered an elementary proof is presented in which the integral of d forms rdtrn x0dx1 dxd for xi c0 rd gln is equal up to a constant to the conditional hilbert space trace of x0 e x1 e xd where 1 for d odd and d 1 5 matrix a spin matrix anticommuting with e for d even this result provides a natural generalization of integration of de rham forms to the setting of connes noncommutative geometry which involves the ordinary hilbert space trace rather than the dixmier trace

controlling parameter of the stress intensity factors for a planar interfacial crack in three dimensional bimaterials numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three dimensional bimaterials subjected to tension the problem is formulated as a system of singular integral equations on the basis of the body force method in the numerical analysis unknown body force densities are approximated by the products of the fundamental density functions and power series where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly the calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios the present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary the stress intensity factors are given with varying the material combination and aspect ratio of the crack it is found that the stress intensity factors ki and kii are determined by the bimaterial constant e alone independent of elastic modulus ratio and poisson s ratio

stress intensity factors of a rectangular crack meeting a bimaterial interface using the hypersingular integral equation method based on body force method a planar crack meeting the interface in a three dimensional dissimilar materials is analyzed the singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main part analytical method of hypersingular integral equations then the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method which the crack opening dislocation is approximated by the product of basic density functions and polynomials numerical solutions of the stress intensity factors of some examples are given

the closed form solution of a mode i crack in the piezoelectric piezomagnetic materials abstract in this paper the behavior of a mode i crack in the piezoelectric piezomagnetic materials subjected to a uniform tension loading is investigated by the generalized almansi s theorem through the fourier transform the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces to solve the dual integral equations the displacement jumps are directly expanded in a series of jacobi polynomials then the closed form solution of this problem can be obtained

noncommutative geometry with graded differential lie algebras starting with a hilbert space endowed with a representation of a unitary lie algebra and an action of a generalized dirac operator we develop a mathematical concept towards gauge field theories this concept shares common features with the connes lott prescription of noncommutative geometry differs from that however by the implementation of unitary lie algebras instead of associative algebras the general scheme is presented in detail and is applied to functions matrices

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

the spectral action for moyal planes extending a result of vassilevich we obtain the asymptotic expansion for the trace of a spatially regularized heat operator l f e t where is a generalized laplacian defined with moyal products and l f is moyal left multiplication the moyal planes corresponding to any skewsymmetric matrix being spectral triples the spectral action introduced in noncommutative geometry by chamseddine and connes is computed this result generalizes the connes lott action previously computed by gayral for symplectic

connes distance function on fuzzy sphere and the connection between geometry and statistics an algorithm to compute connes spectral distance adaptable to the hilbert schmidt operatorial formulation of non commutative quantum mechanics was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the moyal plane revealing a deep connection between geometry and statistics in this paper using the same algorithm the connes spectral distance has been calculated in the hilbert schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su 2 algebra this has been computed for both the discrete and the perelemov s su 2 coherent state here also we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum hilbert space of a particular fuzzy sphere indexed by n 2

equivariant poincare duality for quantum group actions we extend the notion of poincare duality in kk theory to the setting of quantum group actions an important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products along the way we discuss general properties of equivariant kk theory for locally compact quantum groups including the construction of exterior products as an example we prove that the standard podle sphere is equivariantly poincare dual to itself

noncommutative structure of singularities in general relativity initial and final singularities in the closed friedman world model are typical examples of malicious singularities they form the single point of schmidt s b boundary of this model and are not hausdorff separated from the rest of space time the method of noncommutative geometry developed by a connes and his co workers is applied to this case we rephrase schmidt s construction in terms of the groupoid g of orthonormal frames over space time and carry out the desingularization process we define the line bundle 1 2 g over g and change the space of its cross sections into an involutive algebra this algebra is represented in the space of operators on a hilbert space and with the norm inherited from these operators it becomes a c algebra the initial and final singularities of the closed friedman model are given by two distinct representations of this c algebra in the space of operators acting on the hilbert space l2 o 3 1

integral identities based on symmetric and skew symmetric weight functions for a semi infinite interfacial crack in anisotropic magnetoelectroelastic bimaterials abstract in this paper we address a semi infinite interfacial crack problem in an anisotropic magnetoelectroelastic mee bimaterial system subjected to a magnetoelectromechanical asymmetric load on the crack surface first the symmetric and skew symmetric weight functions are derived for a two dimensional 2 d deformation problem using these weight functions and extending the betti formula to mee materials the integral identities are further obtained and the present crack problem is formulated in terms of singular integral equations which establish the relationship between the applied external load and the generalized displacement jump across the crack faces the illustrative examples in relation to mode iii and mode i and mode ii problems show that the method developed in this study avoids the use of green s function and is very convenient for the fracture analysis of mee solids in which a multi field coupled effect is observed

groupoid approach to noncommutative quantization of gravity we propose a new scheme for quantizing gravity based on a noncommutative geometry our geometry corresponds to a noncommutative algebra a gc g c of smooth compactly supported complex functions with convolution as multiplication on the groupoid g e being the semidirect product of a structured space e of constant dimension or a smooth manifold and a group in the classical case e is the total space of the frame bundle and is the lorentz group the differential geometry is developed in terms of a z a submodule v of derivations of a and a noncommutative counterpart of einstein s equation is defined a pair a where is a subset of derivations of a satisfying the noncommutative einstein s equation is called an einstein pair we introduce the representation of a in a suitable hilbert space by completing a with respect to the corresponding norm change it into a c algebra and perform quantization with the help of the standard c algebraic method hermitian elements of this algebra are interp

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

metrics on the real quantum plane using the frame formalism we determine some possible metrics and metric compatible connections on the noncommutative differential geometry of the real quantum plane by definition a metric maps the tensor product of two 1 forms into a function on the quantum plane it is symmetric in a modified sense namely in the definition of symmetry one has to replace the permutator map with a deformed map fulfilling some suitable conditions correspondingly also the definition of the hermitian conjugate of the tensor product of two 1 forms is modified but reduces to the standard one if coincides with the permutator the metric is real with respect to such modified structure

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

a spectral quadruple for de sitter space a set of data supposed to give possible axioms for spacetimes with a sufficient number of isometries in spectral geometry is given these data are shown to be sufficient to obtain 1 1 dimensional de sitter spacetime the data rely at the moment somewhat on the guidance given by a required symmetry in part to allow explicit calculations in a specific model the framework applies also to the noncommutative case finite spectral triples are discussed as an example

space and time dimensions of algebras with application to lorentzian noncommutative geometry and quantum electrodynamics an analogy with real clifford algebras on even dimensional vector spaces suggests assigning an ordered pair s t of space and time dimensions or equivalently an ordered pair m n of metric and ko dimensions modulo 8 to any algebraic structure that we call cpt corepresentation represented over a hilbert space by two self adjoint involutions and an anti unitary operator having specific commutation relations it is shown that this assignment is compatible with the tensor product the space and time dimensions of the tensor product of two cpt corepresentations are the sums of the space and time dimensions of its factors and the same holds for the metric and ko dimensions this could provide an interpretation of the presence of such algebras in pt symmetric hamiltonians or the description of topological matter this construction is used to build an indefinite i e pseudo riemannian version of the spectral triple of noncommutative geometry defined over a krein space and classified by the pair m n instead of the ko dimension only within this framework we can express the lagrangian both bosonic and fermionic of a lorentzian almost commutative spectral triple we exhibit a space of physical states that solves the fermion doubling problem the example of quantum electrodynamics is described an analogy with real clifford algebras on even dimensional vector spaces suggests assigning an ordered pair s t of space and time dimensions or equivalently an ordered pair m n of metric and ko dimensions modulo 8 to any algebraic structure that we call cpt corepresentation represented over a hilbert space by two self adjoint involutions and an anti unitary operator having specific commutation relations it is shown that this assignment is compatible with the tensor product the space and time dimensions of the tensor product of two cpt corepresentations are the sums of the space and time dimensions of its factors and the same holds for the metric and ko dimensions this could provide an interpretation of the presence of such algebras in pt symmetric hamiltonians or the description of topological matter this construction is used to build an indefinite i e pseudo riemannian version of the spectral triple of noncommutative geometry defined over a krein space and classified by the pair m n

pythagoras s theorem on a two dimensional lattice from a natural dirac operator and connes s distance formula one of the key ingredients of connes s noncommutative geometry is a generalized dirac operator which induces a metric connes s distance on the pure state space we generalize such a dirac operator devised by dimakis et al whose connes distance recovers the linear distance on an one dimensional lattice to the two dimensional case this dirac operator has the local eigenvalue property and induces a euclidean distance on this two dimensional lattice which is referred to as natural this kind of dirac operator can be easily generalized into any higher dimensional lattices

quantum isometries of the finite noncommutative geometry of the standard model we compute the quantum isometry group of the finite noncommutative geometry f describing the internal degrees of freedom in the standard model of particle physics we show that this provides genuine quantum symmetries of the spectral triple corresponding to m f where m is a compact spin manifold we also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries

magneto thermo electro elastic transient response in a piezoelectric hollow cylinder subjected to complex loadings abstract the article presents an analytical solution for magneto thermo electro elastic problems of a piezoelectric hollow cylinder placed in an axial magnetic field subjected to arbitrary thermal shock mechanical load and transient electric excitation using an interpolation method solves the volterra integral equation of the second kind caused by interaction among magnetic thermal electric and mechanical fields the electric displacement is determined thus the exact expressions for the transient responses of displacement stresses electric displacement electric potential and perturbation of the magnetic field vector in the piezoelectric hollow cylinder are obtained by means of hankel transforms laplace transforms and inverse laplace transforms from sample numerical calculations it is seen that the present method is suitable for a piezoelectric hollow cylinder subjected to arbitrary thermal shock mechanical load and transient electric excitation and the result carried out may be used as a reference to solve other transient coupled problems of magneto thermo electro elasticity

noncommutative geometry and fundamental physical interactions the lagrangian level historical sketch and description of the present situation these notes comprise i a descriptive account of the history of the subject showing how physics and mathematics interwove to develop a mathematical concept of quantum manifold relevant to elementary particle theory ii a detailed technical description from scratch of the spectral action formalism and computation

on multiple circular inclusions in plane magnetoelasticity abstract a general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper by the use of complex variable theory and laurent series expansion method the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived expanding the definition of the airy s stress function of pure elastic field into the magnetoelastic field and applying the superposition method the general expression then can be reduced to a set of linear algebraic equations and solved in a series form an approximate closed form solution for the case of two arbitrarily located inclusions is also provided for illustrating the effect of the pertinent parameters the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form

the magnetoelastic problem of cracks in bonded dissimilar materials abstract in this paper we consider the magnetoelastic problem of straight cracks lying along interface of two dissimilar soft ferromagnetic materials subjected to remote uniform magnetic induction based on the hilbert problem formulation and the technique of analytical continuation closed form solution for magnetic fields and both the magnetoelastic stresses and the maxwell stresses are obtained it is found that the singularity of magnetoelastic stresses has similar trig log character as those obtained for pure elastic problems which were free from the discontinuous jumps of the magnetic properties and fields across the interface for illustrating the use of present approach the detailed results for a single crack case are provided and verified by comparison with the existing ones under special cases the numerical examples of magnetoelastic stress distribution for different material properties are presented graphically expressions of the stress intensity factors in the vicinity of crack tip and crack opening condition are also derived it is shown that the crack open assumption is valid except a limiting range of distance measured from the crack tip

on a classification of irreducible almost commutative geometries a second helping we complete the classification of almost commutative geometries from a particle physics point of view given by iochum schucker and stephan j math phys to be published four missing krajewski diagrams will be presented after a short introduction into irreducible nondegenerate spectral triples

application of hypersingular integral equation method to three dimensional crack in electromagnetothermoelastic multiphase composites a three dimensional crack problem in electromagnetothermoelastic multiphase composites emte mcs under extended loads is investigated in this paper using green s functions the extended general displacement solutions are obtained by the boundary element method this crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations in which the unknown functions are the extended displacement discontinuities then the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main part analysis method of hypersingular integral equations analytical solutions for the extended singular stresses the extended stress intensity factors sifs and the extended energy release rate near the crack front in emte mcs are provided also a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials in addition distributions of extended sifs varying with the shape of the crack are presented the results show that the present method accurately yields smooth variations of extended sifs along the crack front

deformation quantization and the baum connes conjecture alternative titles of this paper would have been index theory without index or the baum connes conjecture without baum

riemannian geometry of quantum groups and finite groups with nonuniversal differentials we construct noncommutative riemannian manifold structures on dual quasitriangular hopf algebras such as q su 2 with its standard bicovariant differential calculus using the quantum frame bundle approach introduced previously the metric is provided by the braided killing form on the braided lie algebra on the tangent space and the n bein by the maurer cartan form we also apply the theory to finite sets and in particular to finite group function algebras g with differential calculi and killing forms determined by a conjugacy class the case of the permutation group s 3 is worked out in full detail and a unique torsion free and cotorsion free or levi civita connection is obtained with noncommutative ricci curvature essentially proportional to the metric an einstein space we also construct dirac operators in the metric background including on finite groups such as s 3 in the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form

gauge theories with graded differential lie algebras we present a mathematical framework of gauge theories that is based upon a skew adjoint lie algebra and a generalized dirac operator both acting on a hilbert space

quantizing yang mills theory on a two point space we perform the batalin vilkovisky quantization of yang mills theory on a two point space discussing the formulation of connes lott as well as connes real spectral triple approach despite the model s apparent simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the gribov problem

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

rieffel type discrete deformation of finite quantum groups we introduce a discrete deformation of rieffel type for finite quantum groups using this we give an example of a finite quantum group a of order 18 such that neither a nor its dual can be expressed as a crossed product of the form c g1 g2 with g1 and g2 ordinary finite groups we also give a deformation of finite groups of lie type by using their maximal abelian subgroups

analysis of 3d fluid driven crack propagation problem in co seismic slip under p and s waves by hybrid hypersingular integral method this work reports a new and accurate way of theoretical and numerical description of the extended 3d fluid electromagnetic and flow driven crack progression in co seismic slip under p and s waves first based on the viscous fluid flow reciprocal work theorem the hybrid hypersingular integral equation hie method proposed by the author was defined by combined with the coupled extended wave time domain hie and the extended diffused interface phase field method the general extended 3d fluid flow velocity wave solutions are obtained by the extended wave time domains green s function method the 3d extended dynamic fluid driven crack modeling under fully coupled electromagnetothermoelastic p and s wave and flow field was established then the problem is reduced to solving a set of extended hybrid hies coupled with nonlinear boundary domain integral equations in which the unknown functions are the general extended flow velocity discontinuity waves the behavior of the general extended singular stress indices around the crack front terminating is analyzed by hybrid time domain main part analysis the general extended singular pore stress waves spsws and the extended dynamic stress intensity factors dsifs on the fluid driven crack surface are obtained from closed form solutions in addition a numerical method for the problem is proposed in which the extended velocity discontinuity waves are approximated by the product of time domain density functions and polynomials the extended dsifs and general extended spsws are calculated and the results are presented toward demonstrating the applicability of the proposed method

heat kernel and number theory on nc torus the heat trace asymptotics on the noncommutative torus where generalized laplacians are made out of left and right regular representations is fully determined it turns out that this question is very sensitive to the number theoretical aspect of the deformation parameters the central condition we use is of a diophantine type more generally the importance of number theory is made explicit in a few examples we apply the results to the spectral action computation and revisit the uv ir mixing phenomenon for a scalar theory although we find non local counterterms in the nc phi 4 theory on mathbb t 4 we show that this theory can be made renormalizable at least at one loop and maybe even beyond

on summability of distributions and spectral geometry modulo the moment asymptotic expansion the cesaro and parametric behaviours of distributions at infinity are equivalent on the strength of this result we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators we show how cesaro developments lead to efficient calculations of the expansion coefficients of counting number functionals and green functions the bosonic action functional proposed by chamseddine and connes can more generally be validated as a cesaro asymptotic development

the noncommutative lorentzian cylinder as an isospectral deformation we present a new example of a finite dimensional noncommutative manifold namely the noncommutative cylinder it is obtained by isospectral deformation of the canonical triple associated with the euclidean cylinder we discuss connes character formula for the cylinder in the second part we discuss noncommutative lorentzian manifolds here the definition of spectral triples involves krein spaces and operators on krein spaces a central role is played by the admissible fundamental symmetries on the krein space of square integrable sections of a spin bundle over a lorentzian manifold finally we discuss isospectral deformation of the lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder

a short survey of noncommutative geometry we give a survey of selected topics in noncommutative geometry with some emphasis on those directly related to physics including our recent work with dirk kreimer on renormalization and the riemann hilbert problem we discuss at length two issues the first is the relevance of the paradigm of geometric space based on spectral considerations which is central in the theory as a simple illustration of the spectral formulation of geometry in the ordinary commutative case we give a polynomial equation for geometries on the four sphere with fixed volume the equation involves an idempotent e playing the role of the instanton and the dirac operator d it is of the form e 12 d e 4 5 and determines both the sphere and all its metrics with fixed volume form the expectation x is the projection on the commutant of the algebra of 4 by 4 matrices we also show using the noncommutative analog of the polyakov action how to obtain the noncommutative metric in spectral form on the noncommutative tori fr

lipschitz isometries of compact quantum metric spaces abstract a compact quantum metric space is a complete order unit space endowed with a lip norm we investigate the category morphism between compact quantum metric spaces we show that a unital positive linear map between compact quantum metric spaces with lower semicontinuous lip norms is lipschitz if and only if it preserves the lipschitz elements we characterize when a unital positive linear map on a compact quantum metric space with lower semicontinuous lip norm is a lipschitz isometry

smooth dense subalgebras and fourier multipliers on compact quantum groups we define and study dense frechet subalgebras of compact quantum groups realised as smooth domains associated with a dirac type operator with compact resolvent further we construct spectral triples on compact matrix quantum groups in terms of clebsch gordon coefficients and the eigenvalues of the dirac operator mathcal d grotendieck s theory of topological tensor products immediately yields a schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo differential operators on them it is also shown that regular pseudo differential operators are closed under compositions as a by product we develop elements of the distribution theory and corresponding fourier analysis we give applications of our construction to obtain sufficient conditions for lp lq boundedness of coinvariant linear operators we provide necessary and sufficient conditions for algebraic differential calculi on hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra c infty mathcal d we check explicitly that these conditions hold true on the quantum su2q for both its 3 dimensional and 4 dimensional calculi

noncommutative geometry and conformal geometry iii vafa witten inequality and poincare duality abstract this paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint in this paper we reformulate the inequality of vafa witten 42 in the setting of twisted spectral triples this involves a notion of poincare duality for twisted spectral triples our main results have various consequences in particular we obtain a version in conformal geometry of the original inequality of vafa witten in the sense of an explicit control of the vafa witten bound under conformal changes of metrics this result has several noncommutative manifestations for conformal deformations of ordinary spectral triples spectral triples associated with conformal weights on noncommutative tori and spectral triples associated with duals of torsion free discrete cocompact subgroups satisfying the baum connes conjecture

a new construction of factors of type iii1 abstract we give in this paper a new construction of factors of type iii 1 under certain assumptions we can thanks to a result by popa give a complete classification for this family of factors although these factors are never full we can nevertheless in many cases compute connes invariant we obtain a new example of an uncountable family of pairwise non isomorphic factors of type iii 1 with the same invariant

general solutions for coupled equations for piezoelectric media abstract three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium these solutions are expressed in terms of the two functions and f where satisfies a second degree partial differential equation and f a sixth degree partial differential equation respectively if the terms concerning the derivatives of time are removed the results become three general solutions for the corresponding equilibrium equations in which the function f can be represented by functions fi i 1 2 3 each of which satisfies a second degree partial differential equation by utilizing a generalized almansi theorem and the solution wang and zheng int j solids structures32 105 115 1995 obtained is proved to be consistent with one case of one of the three general solutions when the constants e15 e31 e33 0 the piezo electric coupling is absent then two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium one of which is the result hu acta scientia sin 2 2 145 151 1953 obtained the other one has not been published last the solution in the limiting explicit form for the problem for a half space with concentrated loads at the boundary is obtained by utilizing the general solutions

gravity on finite groups gravity theories are constructed on finite groups g a self consistent review of the differential calculi on finite g is given with some new developments the example of a bicovariant differential calculus on the nonabelian finite group s3 is treated in detail and used to build a gravity like field theory on s3

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

the graded product of real spectral triples forming the product of two geometric spaces is one of the most basic operations in geometry but in the spectral triple formulation of non commutative geometry the standard prescription for taking the product of two real spectral triples is problematic among other drawbacks it is non commutative non associative does not transform properly under unitaries and often fails to define a proper spectral triple in this paper we explain that these various problems result from using the ungraded tensor product by switching to the graded tensor product we obtain a new prescription where all of the earlier problems are neatly resolved in particular the new product is commutative associative transforms correctly under unitaries and always forms a well defined spectral triple

on the classification of full factors of type iii we introduce a new invariant s m for type iii factors m with no almost periodic weights we compute this invariant for certain free araki woods factors we show that connes invariant r cannot distinguish all isomorphism classes of free araki woods factors we show that there exists a continuum of mutually non isomorphic free araki woods factors each without almost periodic weights

reformulation of the standard model in a generalized differential geometry on the discrete space m4 x2 the standard model is reconstructed in a generalized differential geometry gdg on the product space m4 x2 by reformulating the work of coquereaux et al that dealt with the same theme based on the noncommutative geometry ncg a gdg on m4 x2 is constructed by adding the basis n n 1 2 of a differential form on the discrete space x2 to the ordinary basis dx on minkowski space m4 and so it is a direct generalization of the differential geometry on the continuous manifold a gdg is a version of ncg the yang mills higgs lagrangian and the dirac lagrangian are reconstructed by using the fermion representation similar to that in a so 10 grand unified model

development of bem for convective viscous flow problems abstract three kinds of boundary element approaches for an unsteady flow problem of incompressible viscous fluid are presented the first approach which is the so called boundary domaintype is based on the use of fundamental solution for the only linear differential operator and the newton raphson iterative procedure the second one is based on the time splitting technique of the governing equations in which the fundamental equations are split into the convection equation and the stokes equations the third one is based on the well known fractional step fs method which is one of the time splitting techniques in order to show the applicability and effectiveness of our approaches numerical results of the driven cavity flow example are demonstrated through a comparison with other numerical results

fracture analysis in continuously nonhomogeneous magneto electro elastic solids under a thermal load by the mlpg abstract a meshless method based on the local petrov galerkin approach is proposed to solve initial boundary value problems of magneto electro elastic solids with continuously varying material properties stationary and transient thermal problems are considered in this paper the mechanical 2 d fields are described by the equations of motion with an inertial term nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity the spatial variation of displacements electric and magnetic potentials is approximated by the moving least squares mls scheme after performing the spatial integrations one obtains a system of ordinary differential equations for certain nodal unknowns that system is solved numerically by the houbolt finite difference scheme as a time stepping method

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes