on resolving the multiplicity of arbitrary tensor products of the u n groups representations of u n are realised as right translations on holomorphic hilbert bargmann spaces of n n complex variables r fold tensor product spaces of irreducible representations of u n are shown to be isomorphic to subspaces of the holomorphic hilbert spaces maps are exhibited which carry an irreducible representation of u n into these subspaces the algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity several examples from u 3 are explicitly worked out

kaplansky classes and derived categories we put a monoidal model category structure on the category of chain complexes of quasi coherent sheaves over a quasi compact and semi separated scheme x the approach generalizes and simplifies the method used by the author in trans am math soc 356 8 3369 3390 2004 and trans am math soc 358 7 2855 2874 2006 to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space indeed much of the paper is dedicated to showing that in any grothendieck category mathcal g any nice enough class of objects mathcal f induces a model structure on the category ch mathcal g of chain complexes the main technical requirement on mathcal f is the existence of a regular cardinal such that every object f in mathcal f satisfies the following property each generated subobject of f is contained in another generated subobject s for which s f s in mathcal f such a class mathcal f is called a kaplansky class kaplansky classes first appeared in enochs and lopez ramos rend sem mat univ padova 107 67 79 2002 in the context of modules over a ring r we study in detail the connection between kaplansky classes and model categories we also find simple conditions to put on mathcal f which will guarantee that our model structure is monoidal we will see that in several categories the class of flat objects form such kaplansky classes and hence induce monoidal model structures on the associated chain complex categories we will also see that in any grothendieck category mathcal g the class of all objects is a kaplansky class which induces the usual non monoidal injective model structure on ch mathcal g

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

the additivity of traces in triangulated categories abstract we explain a fundamental additivity theorem for euler characteristics and generalized trace maps in triangulated categories the proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation the refinement consists of several new axioms relating products and distinguished triangles the axioms hold in the examples and shed light on generalized homology and cohomology theories

canonical orthonormal wigner supermultiplet basis the explicit construction of an orthonormal basis for states of good spin isospin and su 4 wigner supermultiplet symmetry is given in a bargmann representation space a complete set of quantum labels is provided by a sp 3 r contains implies u 3 complementary symmetry

some mean values related to the quasi arithmetic mean in this article we study new mean values and their properties related to the quasi arithmetic mean and their properties in the spirit of h haruki and th m rassias

a new vector coherent state construction of su 3 supset so 3 wigner coefficients the new rotor coherent state construction of the su 3 algebra is used to derive simple expressions for su 3 so 3 wigner coefficients

on resolving the multiplicity of tensor products of irreducible representations of symplectic groups the multiplicity problem of r fold tensor products of irreducible representations of sp 2k c are considered the arbitrary r fold tensor product is shown to be isomorphic to a subspace of the holomorphic hilbert bargmann space of n 2k complex variables maps are constructed which carry an irreducible representation of sp 2k c into this subspace an algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity

wigner eckart theorem and infinitesimal operators of group representations the new notion of a tensor operator which transforms under a representation of the compact lie group is introduced this tensor operator is a linear space v of operators in order to obtain the traditional tensor operator it is sufficient to choose a basis in the space v the wigner eckart theorem is valid for the tensor operators the notion of irreducibility and the equivalence relation for tensor operators are formulated necessary and sufficient conditions of irreducibility and of equivalence are proved using the wigner eckart theorem the author gives the method of evaluation of infinitesimal operators of the representations of compact and noncompact lie groups the new invariants of irreducible representations of compact groups are found their quantity is equal to that of independent casimir operators

canonical solution of the state labelling problem for su n so n and littlewood s branching rule i general formulation the internal state labelling problem for the d row irreducible representations of su n where 2d or n when reduced with respect to so n is shown to amount to the external state labelling problem for u d the canonical solution of the latter due to biedenharn et al 1967 provides a canonical solution of the former which reflects the operation of littlewood s branching rule for u n contains implies o n in a very simple way

canonical solution of the state labelling problem for su n so n and littlewood s branching rule iii su 3 so 3 case for pt ii see ibid vol 17 no 4 p 777 89 1984 parts i and ii discussed in general terms a new solution to the state labelling problem for the d row irreducible representations irreps of su n when reduced with respect to so n this solution was termed canonical because it reflects the operation of littlewood s branching rule for u n o n in a straightforward way the su 3 so 3 case is now worked out in detail explicit expressions of the canonical basis states of so 3 irreps l belonging to an su 3 irrep h1h2 are obtained the matrix of the transformation from the bargmann moshinsky basis to the canonical one is also calculated it is shown that in both cases the extra label necessary to completely specify the states can be chosen as the label js characterising an intermediate su 2 irrep in the reduction of the produce representation jl js into su 2 irreps j where j 1 2 h1 h2 and jl 1 2l or 1 2 l 1 whenever h1 h2 l is even or odd

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

some new axioms for triangulated categories and this mapping cone should by rights be a triangle it is for instance true that any homological functor applied to gives a long exact sequence unfortunately the world of triangulated categories is a bad one and need not be a triangle it is however true that givenf and g there exists an h for which is a triangle see theorem 1 8 not all morphisms of triangles are equal some are better than others it turns out that theorem 1 8 is equivalent to the octahedral axiom in the first two sections of this article we study two possible notions of good morphisms between triangles and we quickly decide that neither is satisfactory the morphisms do not compose well the composite of good morphisms need not be good worse still the non category of good morphisms is non additive the sum of two good morphisms need not be good the problem goes right back to the definition of a triangulated category

abstract representation theory of dynkin quivers of type a abstract we study the representation theory of dynkin quivers of type a in abstract stable homotopy theories including those associated to fields rings schemes differential graded algebras and ring spectra reflection functors partial coxeter functors and serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules certain explicitly constructed spectral bimodules in fact these universal tilting modules are spectral refinements of classical tilting complexes as a consequence we obtain split epimorphisms from the spectral picard groupoid to derived picard groupoids over arbitrary fields these results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators as further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field this includes yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral serre duality result finally using abstract representation theory of linearly oriented a n quivers we construct canonical higher triangulations in stable derivators and hence a posteriori in stable model categories and stable categories

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

a resolution of the su 3 outer multiplicity problem and computation of wigner coefficients for su 3 a simple algorithm is given for the resolution of the su 3 multiplicity problem and the computation of su 3 wigner coefficients using a complete set of u 2 x u 3 bargmann tensors classified by operator patterns null space properties of these tensors are easily derived their structure is such that a direct one to one correspondence is shown to exist between the bargmann tensors and the terms in the clebsch gordan series for su 3 as derived by o reilly 1982 finally the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by hecht

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

new characterizations of some mean values the purpose of this paper is to give new characterizations of some mean values of two positive real numbers the arithmetic geometric and harmonic means of two positive real numbers are the fundamentals of this paper

vector coherent state theory in a group with non commuting raising generators vector coherent state theory is used to give an explicit construction of the irreducible representations of a group with non commuting raising generators the simple example of the compact unitary symplectic group usp 4 contains implies usp 2 usp 2

canonical orthonormal basis for su 3 contains implies so 3 iii complete set of su 3 tensor operators for pt ii see ibid vol 18 p 1905 1985 a complete set of tensor operators for su 3 is given in a model space generated by two bargmann vectors use is made of modified operator patterns to classify these tensors their tensorial properties are then discussed in the context of the su 3 tensor algebra developed by biedenharn and collaborators 1957 the authors also given analytical semi analytical expressions for important classes of wigner coefficients in gel fand rotational basis

canonical orthonormal basis for su 3 so 3 ii reduced matrix elements of the su 3 generators for pt i see ibid vol 18 p 1891 1985 a simple algorithm is given for the calculation of reduced matrix elements of the generators of the su 3 algebra in the canonical su 3 contains implies so 3 basis introduced in i

on some further extensions of the characterizations of mean values by h haruki and th m rassias abstract the purpose of this paper is to give a characterization of some mean values of two positive real numbers in a real valued function m a b the arithmetic harmonic and some other means of two positive real numbers are the fundamentals of this paper

models for su 3 in terms of so n 2 and so 2n algebras mathematical and physical models for su 3 are given in terms of unirreps of the so n 2 and so 2n algebras vector coherent state representations are given for all the discrete series unirreps of so n 2 and so 2n traceless bosons are shown to arise naturally in these representations a new and elegant realisation of the so 8 approximately so 6 2 model for su 3 is given and it is shown how to perform calculations in the so 8 model using the powerful vector coherent state representations formalism

some inequalities involving means and their converses using an idea of j sandor and v e s szabo an inequality of ky fan and its generalizations are proved we also establish some converses of these inequalities as consequences some well known inequalities are obtained

an approximation formula for the kappa matrix elements of the symplectic algebra sp 6 r a simple approximation formula valid to the high degree of accuracy of the toronto approximation is given for both the off diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of sp 6 r

canonical solution of the state labelling problem for su n so n and littlewood s branching rule ii use of modification rules for pt i see ibid vol 16 p 2095 1983 in part i it was shown that when d 1 2n case where littlewood s branching rule 1950 for u n contains implies o n has to be supplemented with newell s modification rules 1951 the explicit solution of the state labelling problem for v 1 row irreducible representations of both su 2v 1 and su 2v is given and is shown to reflect the operation of littlewood s modified branching rule in a direct way the generalisation to higher d values is then outlined

picard groups grothendieck rings and burnside rings of categories abstract we discuss the picard group the grothendieck ring and the burnside ring of a symmetric monoidal category and we consider examples from algebra homological algebra topology and algebraic geometry

new perspective on the u n wigner racah calculus iii applications to u 2 and u 3 for pt ii see ibid vol 20 no 14 p 4613 35 1987 using the framework introduced in parts i and ii the author derives an interesting relationship between u 3 symmetric tensors and u 2 racah coefficients closed formulae for generic u 2 wigner and racah coefficients are given in terms of alternative parametrisations for such coefficients

tilting theory via stable homotopy theory we show that certain tilting results for quivers are formal consequences of stability and as such are part of a formal calculus available in any abstract stable homotopy theory thus these results are for example valid over arbitrary ground rings for quasi coherent modules on schemes in the differential graded context in stable homotopy theory and also in the equivariant motivic or parametrized variant thereof in further work we will continue developing this calculus and obtain additional abstract tilting results here we also deduce an additional characterization of stability based on goodwillie s strongly co cartesian n cubes as applications we construct abstract auslander reiten translations and abstract serre functors for the trivalent source and verify the relative fractionally calabi yau property this is used to offer a new perspective on may s axioms for monoidal triangulated categories