arXiv subjects

Agostino Prástaro

Publications and source records attributed to Agostino Prástaro.

12 recordsLinked to original sources

arXiv2015

The Maslov index in PDEs geometry

It is proved that the Maslov index naturally arises in the framework of PDEs geometry. The characterization of PDE solutions by means of Maslov index is given. With this respect, Maslov index for Lagrangian submanifolds is given on the ground of PDEs geometry. New formulas to calculate bordism groups of $(n-1)$-dimensional compact sub-manifolds bording via $n$-dimensional Lagrangian submanifolds of a fixed $2n$-dimensional symplectic manifold are obtained too. As a by-product it is given a new proof of global smooth solutions existence, defined on all $\mathbb{R}^3$, for the Navier-Stokes PDE. Further, complementary results are given in Appendices concerning Navier-Stokes PDE and Legendrian submanifolds of contact manifolds.

math.GM↗

arXiv2015

The Riemann hypothesis proved

The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between quantum $1$-spheres with quantum algebra $A=\mathbb{C}$, in the sense of A. Prástaro \cite{PRAS01, PRAS02}. Algebraic topologic properties of quantum-complex manifolds and suitable bordism groups of morphisms in the category $\mathfrak{Q}_{\mathbb{C}}$ of quantum-complex manifolds are utilized.

math.GM↗

arXiv2015

The congruent number problem and the Birch-Swinnerton-Dyer conjecture

By introducing a new point of view in Algebraic Topology relating elliptic curves in $\mathbb{R}^2$ and suitable bordism groups, the congruent number problem is solved showing that the Tunnell's theorem is also sufficient. This could be considered also an indirect proof that the Birch Swinnerton-Dyer conjecture is true.

math.GM↗

arXiv2015

The Landau's problems. I-II

This is a work in two parts devoted to solutions of the so-called {\em four Landau's problems} in Number Theory, listed by Edmund Landau at the 1912 International Congress of Mathematics. In Part I the {\em Goldbach's conjecture} is proved. (This is the first Landau's problems.) In Part II the other three Landau's problems are considered and solved too.

math.GM↗

arXiv2015

Strong reactions in quantum super PDEs. III: Exotic quantum supergravity

Following the previous two parts, of a work devoted to encode strong reaction dynamics in the A. Prástaro's algebraic topology of quantum super PDE's, nonlinear quantum propagators in the observed quantum super Yang-Mills PDE, $\hat{(YM)}[i]$, are further characterized. In particular, nonlinear quantum propagators with non-zero defect quantum electric-charge, are interpreted as {\em exotic-quantum supergravity} effects. As an application, the recently discovered bound-state called $Zc(3900)$, is obtained as a neutral quasi-particle, generated in a $Q$-quantum exotic supergravity process. {\em Quantum entanglement} is justified by means of the algebraic topologic structure of nonlinear quantum propagators. Quantum Cheshire cats are considered as examples of quantum entanglements. Existence theorem for solutions of $\hat{(YM)}[i]$ admitting negative local temperatures ({\em quantum thermodynamic-exotic solutions}) is obtained too and related to quantum entanglement. Such exotic solutions are used to encode Universe at the Planck-epoch. It is proved that the Universe's expansion at the Planck epoch is justified by the fact that it is encoded by a nonlinear quantum propagator having thermodynamic quantum exotic components in its boundary. This effect produces also an increasing of energy in the Universe at the Einstein epoch: {\em Planck-epoch-legacy} on the boundary of our Universe. This is the main source of the Universe's expansion and solves the problem of the non-apparent energy-matter ({\em dark-energy-matter}) in the actual Universe. Breit-Wheeler-type processes have been proved in the framework of the Prástaro's algebraic topology of quantum super Yang-Mills PDEs. Numerical comparisons of nonlinear quantum propagators with Weinberg-Salam electroweak theory in Standard Model are given.

math.GM↗

arXiv2015

Strong reactions in quantum super PDE's. I-II

This is a work in three parts, devoted to encode strong reactions of the high energy physics, in the algebraic topologic theory of quantum super PDE's, (previously formulated by A. Prástaro). In particular strong reactions are characterized by means of boundary value problems in quantum super PDE's. In such a way one obtains representations of quantum nonlinear propagators in quantum super PDE's, by means of elementary ones (quantum handle decompositions of quantum nonlinear propagators). These are useful to encode nuclear and subnuclear reactions in quantum physics. Prástaro's geometric theory of quantum PDE's allows us to obtain constructive and dynamically justified answers to some important open problems in high energy physics.

math.GM↗

arXiv2012

Quantum Exotic PDE's

Following the previous works on the A. Prástaro's formulation of algebraic topology of quantum (super) PDE's, it is proved that a canonical Heyting algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prastaro's geometric theory of quantum PDE's is applied to the new category of {\em quantum hypercomplex manifolds}, related to the well-known Cayley-Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE's in this new category of noncommutative manifolds. Finally the extension of the concept of exotic PDE's, recently introduced by A.Prástaro, has been extended to quantum PDE's. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by A. Prástaro.

math.GM↗

arXiv2011

Quantum extended crystal PDE's

Our recent results on {\em extended crystal PDE's} are generalized to PDE's in the category $\mathfrak{Q}_S$ of quantum supermanifolds. Then obstructions to the existence of global quantum smooth solutions for such equations are obtained, by using algebraic topologic techniques. Applications are considered in details to the quantum super Yang-Mills equations. Furthermore, our geometric theory of stability of PDE's and their solutions, is also generalized to quantum extended crystal PDE's. In this way we are able to identify quantum equations where their global solutions are stable at finite times. These results, are also extended to quantum singular (super)PDE's, introducing ({\em quantum extended crystal singular (super) PDE's}).

math.GM↗

arXiv2011

Exotic PDE's

In the framework of the PDE's algebraic topology, previously introduced by A. Prástaro, are considered {\em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or such that their boundaries $\partial N$ are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth ($4$-dimensional) Poincaré conjecture is proved. This allows to complete the previous Theorem 4.59 in \cite{PRA17} also for the case $n=4$.

math.GM↗

arXiv2010

Exotic Heat PDE's.II

Exotic heat equations that allow to prove the Poincaré conjecture and its generalizations to any dimension are considered. The methodology used is the PDE's algebraic topology, introduced by A. Prástaro in the geometry of PDE's, in order to characterize global solutions. In particular it is shown that this theory allows us to identify $n$-dimensional {\em exotic spheres}, i.e., homotopy spheres that are homeomorphic, but not diffeomorphic to the standard $S^n$.

math.GM↗

arXiv2010

Exotic n-d'Alembert PDE's and Stability

In the framework of the PDE's algebraic topology, previously introduced by A. Prástaro, {\em exotic $n$-d'Alembert PDE's} are considered. These are $n$-d'Alembert PDE's, $(d'A)_n$, admitting Cauchy manifolds $N\subset (d'A)_n$ identifiable with exotic spheres, or such that $\partial N$, can be exotic spheres. For such equations local and global existence theorems and stability theorems are obtained.

math.GM↗

arXiv2010

Exotic heat PDE's

Exotic heat equations that allow to prove the Poincaré conjecture, some related problems and suitable generalizations too are considered. The methodology used is the PDE's algebraic topology, introduced by A. Prástaro in the geometry of PDE's, in order to characterize global solutions.

math.GM↗