The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

White Noise Approach to the Construction of Φ4 4 Quantum Fields

In this paper by Sobolev imbedding theorem and characterization theorem of generalized operators the existence of ₄ ⁴ quantum fields as generalized operators is obtained with space cut-off and a rigorous mathematical interpretation of renormalization procedure is given under white noise theory.     

Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2

Let {B _t ,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p,q for all , provided the integrand u belongs to the space . Moreover, if u is bounded and belongs to for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p, ^H .     

-calculus for submarkovian generators

Let be the generator of a symmetric submarkovian semigroup in . In this note we show that on the operator admits a bounded functional calculus on the sector for each \psi_p^*$"> with

This improves a result due to M. Cowling. We apply our result to obtain maximal regularity for parabolic equations and evolutionary integral equations.

     

Clifford Space as the Arena for Physics

A new theory is considered according to which extended objects in n-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of center of mass coordinates. While the usual center of mass is a point, by generalizing the latter concept, we associate with every extended object a set of r-loops, r=0,1,...,n–1, enclosing oriented (r+1)-dimensional surfaces represented by Clifford numbers called (r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called C-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but C-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in C-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of C-space.     

Pinning class of the Wiener measure by a functional: related martingales and invariance properties

For a given functional Y on the path space, we define the pinning class of the Wiener measure as the class of probabilities which admit the same conditioning given Y as the Wiener measure. Using stochastic analysis and the theory of initial enlargement of filtration, we study the transformations (not necessarily adapted) which preserve this class. We prove, in this non Markov setting, a stochastic Newton equation and a stochastic Noether theorem. We conclude the paper with some non canonical representations of Brownian motion, closely related to our study.Mathematics Subject Classification (2000): 60G44, 60H07, 60H20, 60H30     

Smoothness and Locality for Nonunital Spectral Triples

To deal with technical issues in noncommutative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. These algebras and modules allow us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral triples are closed under the C functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincaré Duality and that the algebras of smooth spectral triples are H-unital.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Diverse acoustic wave propagation to confirmable time–space fractional KP equation arising in dusty plasma

In this study, the (3+1)-dimensional fractional time–space Kadomtsev–Petviashivili (FTSKP) equation is considered and analyzed analytically, which propagates the acoustic waves in an unmagnetized dusty plasma. The fractional derivatives are studied in a confirmable sense. The new modified extended direct algebraic (MEDA) approach is adopted to investigate the diverse nonlinear wave structures. A variety of new families of hyperbolic and trigonometric solutions are obtained in single and different combinations. The obtained results are also constructed graphically with the different parametric choices.