Percutaneous treatment of patients with heart diseases: selection, guidance and follow-up. A review

Aortic stenosis and mitral regurgitation, patent foramen ovale, interatrial septal defect, atrial fibrillation and perivalvular leak, are now amenable to percutaneous treatment. These percutaneous procedures require the use of Transthoracic (TTE), Transesophageal (TEE) and/or Intracardiac echocardiography (ICE). This paper provides an overview of the different percutaneous interventions, trying to provide a systematic and comprehensive approach for selection, guidance and follow-up of patients undergoing these procedures, illustrating the key role of 2D echocardiography

(Co)bordism groups in PDEs.

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. In this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicitly proof that the usual Thom-Pontrjagin construction in (co)bordism theory can be generalized also to the case of singular integral (co)bordism in the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates in a natural way Hopf algebras (\textit{full $p$-Hopf algebras, $0\le p\le n-1$}), to any PDE $E_k\subset J^k_n(W)$, just introduced in \cite{PRA28, PRA52}, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier-Stokes equation $(N)$ and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of $(NS)$ with change of sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full $p$-Hopf algebras, $0\le p\le 3$, for the Navier-Stokes equation are determined

ELEMENTI DI MECCANICA RAZIONALE

This monography is addressed to Italian university students in Mathematics, Physics and Engineering. It develops with a modern geometric language the methods of classical mechanics and geometry of (partial) differential equations. The presentation, even if elementary, gives the actual mathematics situation in classical mechanics

Geometrodynamics of non-relativistic continuous media, I.

In order to formulate the non-relativistic continuum mechanics as a unified field theory on Galilei space-time $M$, the geometrical structure of $M$ is considered and the space time resolution of bundles of geometric objects on $M$ 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

On the singular solutions of PDE's.

In the geometric formal theory of PDE's we recognize also the problem of existence of singular solutions with singularities of Thom-Boardman type, i.e., singularities that can be resolved by means of prolongations. Scope of this paper is to give a short account of some fundamental results in these directions and apply them to some important classic equations of fluid mechanics: Euler equation $(E)$ and Navier-Stokes equation $(NS)$. Quantum tunneling effects can be described by means of such singular solutions. Furthermore, we show also as singular solutions enter in the description of canonical quantization of PDE's. We shall specialize, for sake of coincision, on equations $(E)$ and $(NS)$

Wholly cohomological PDE's.

PDE's are geometric objects to which one can associate conservation laws in relation to their symmetry properties . Then, the wholly-cohomological character of a PDE is its possibility to represent any $(n-1)$-dimensional cohomological class of the $n$-dimensional basis manifold by means of a conservation law. In this paper we resume some recent results in this direction obtained by the author and also announce some new further results for PDEs defined in the category of supermanifolds

Geometrodynamics of non-relativistic continuous media, II.

An intrinsic formulation of Continuum Mechanics on the affine Galielan space-time $M$ 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 $M$. 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

Gauge geometrodynamics.

In this paper a self-contained unitary geometric development of the methods and structures on which the gauge theories are based is developed. This geometrical framework is useful for a more clear understanding of continuum physics. Since the framework is sufficiently generalized, it can be applied to all the situations of physical interest. Contents: Functors and fibre bundles. Derivative spaces and differential equations. Derivative spaces and variational calculus. Connections and derivative spaces. Geometrodynamics of gauge continuum systems and symmetries properties. Classification of gauge continuum systems. Spinor superbundles of geometric objects and dynamics

Integral bordisms and Green kernels in PDEs

Integral bordisms of (nonlinear) PDEs are characterized by means of geometric Green kernels and prove that these are invariant for the classic limit of statistical sets of formally integrable PDEs. Such geometric characterization of Green kernels is related to the geometric approach of canonical quantization of (nonlinear) PDEs, previously introduced by us. Some applications are given where particle fields on curved space-times having physical or unphysical masses, (i.e., bradions, luxons and massive neutrinos), are canonically quantized respecting microscopic causality