196,172 research outputs found
(Co)bordism groups in quantum PDE's
No full textA theory of noncommutative manifolds (\textit{quantum manifolds}) is formulated, and for such manifolds a geometric theory of quantum PDEs (QPDEs) is formulated. In particular, a criterion of formal integrability is given that
extends to QPDEs previous one given by H. Goldschmidt for PDEs and by A. Prastaro for super PDEs. A general theory of integral (co)bordism for QPDEs is developed, that extends previous one for PDEs formulated by A. Prastaro.
Then, non-commutative Hopf algebras, (\textit{full quantum -Hopf algebras, }), are canonically associated to any QPDE whose elements represent all the
possible invariants that can be recognized for such a structure.
Many examples of QPDEs are considered where we apply our theory. In particular, applications to quantum supergravity are considered. Existence of (quantum) tunnel
effects for quantum superstrings in supergravity is proved
Results on the J.d'Alembert equation
No full textA new -d'Alembert equation, , is introduced in the category of quantum manifolds (in the sense introduced by A.Prastaro), that extends the commutative generalized d'Alembert equation previously considered by the same authors. For such a new equation
we give theorems of existence of local and global solutions
Geometrodynamics of non-relativistic continuous media, I.
No full textIn order to formulate the non-relativistic continuum mechanics as a unified field theory on Galilei space-time , the geometrical structure of is considered and the space time resolution of bundles of geometric objects on are analysed in detail. In particular, the conceptof geometric object gives rigorous meaning to the concept of observed physical quantity. It clarifies the ambiguity of why ''frame dependent'' quantities are useful, even essential, in the kinematic of description of continuum mechanical bodies. Moreover, it clarifies the paradosical nature of ''frame indifferent statements about frame dpendent quantities''. These turn out to be simply statements about fields of geometric objects which are not tensor fields
A geometric point of view for the quantization of non-linear field theories.
No full textThe fundamental geometric structure of any field theory is a fiber bundle beside a PDE . So we shall recognize in this geometric structure suitable properties to interpretate the meaning of quantization. This communication shortly describe our new point
of view in this field, showing how it is possible to read the
meaning of the quantization in the formal properties of PDEs
Geometrodynamics of non-relativistic continuous media, II.
No full textAn intrinsic formulation of Continuum Mechanics on the affine Galielan space-time is given, emphasizing the role of the dynamic equation as a geometric structure. In particular, a continuous body is described as a geometric structure on . Thus, the study of symnmetry properties of this structure allows us to give useful classifications of continuous bodies and to state generalized forms of Noether's theorem. These considerations are applied to incompressible fluids. Existence and uniqueness theorems for regular solutions are obtained
On the geometric approach to an equation of J.D'Alembert.
No full textHere we announce some firt results on
the J. D'Alembert equation . More precisely, by using a geometric framework we
prove that the set of smooth functions of two variables ,
solutions of the J. D'Alembert equation, is larger than the set of
functions of the form
Geometrodynamics of some non-relativistic incompressible fluids.
No full textIn some papers we proposed a geometric formulation of continuum mechanics, where a continuum body is seen as a suitable differentiable fiber bundle on the Galilean space-time , beside a differential equation of order , , on and the assignement of a frame on . In the present paper we apply this general theory to some incompressible fluids. The scope is to demonstrate that also for these more simple materials our theory is a suitable tool in order to understand better the fundamental principles of continuum mechanics
A geometric approach to a noncommutative generalized d'Alembert equation
No full textIn this paper the authors provide an account of some of their results concerning the J. D'Alembert equation especially in a suitable category of noncommutative manifolds, proving that the geometric theory of PDE's introduced by A. Pr\'astaro is an handable framework where problems in the theory of partial differential equations find their natural solutions. In fact, the J. d'Alembert equation is one such applications
On the set of solutions of the generalized d'Alembert equation
No full textBy using a geometric approach we prove that the set of solutions of the generalized d'Alembert equation , considered in the domain of the -space , is larger that the set of the functions that can be represented in the form as . Here the recent general method introduced by A. Pr\'astaro to calculate integral and quantum (co)bordism groups in PDE's is used. This method is very useful in order to prove existence of tunneling effects in PDE's, i.e., existence of solutions that change their sectional topology
Dirac quantization
No full textA panorama on the modern developments of quantizations of PDEs is given. In particular it is emphasized
that on the framework of a geometric theory of PDEs, A. Pr\'astaro has given a formulation of canonical quantization of partial differential equations, without assuming that these should be of variational type and/or linear. Furthermore, the generalization of
this geometric approach to PDEs in the category of quantum manifolds, given more recently by A. Pr\'astaro, has been considered. Relations with other recent works in noncommutative geometry are given
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