Related papers: The Dirac--Goodman--Pollack Conjecture
As a continuation of previous investigations, the formalism used there is extended to the case when an external electric field is present and the covariant formulation is performed again. The equation system obtained allows no restriction…
We propose a symmetry of the Dirac equation under the interchange of signs of eigenvalues of the Dirac's $K$ operator. We show that the only potential which obeys this requirement is the Coulomb one for both vector and scalar cases.…
It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller…
Let $\{G_n\}_{n=1}^{\infty}$ be a sequence of transitive infinite connected graphs with $\sup\limits_{n\geq 1} p_c(G_n) < 1,$ where each $p_c(G_n)$ is bond percolation critical probability on $G_n.$ Schramm (2008) conjectured that if $G_n$…
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…
A restriction was found in the mathematics of the Dirac equation for a free neutrino type of particle. The basic assumption here is the equivalence of the four variables of spacetime. A perspective is defined as a metric tensor format. We…
A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…
Recently Braga and Mello conjectured that for a given natural number n there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper we prove a result from which the conjecture is an immediate…
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also…
If we pick two elements of a non-abelian group at random, the odds this pair commutes is at most 5/8, so there is a "gap" between abelian and non-abelian groups \cite{G}. We prove a "topological" generalization estimating the odds a word…
The Hahn-Banach theorem states that onto each line in every normed space, there is a unitary projection, and Kadec and Snobar proved (using John's ellipsoid) that onto each $n$-dimensional subspace of any real normed space, there is a…
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric…
We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines. We…
One solution to the so-called problem of time is to construct certain Dirac observables, sometimes called evolving constants of motion. There has been some discussion in the literature about the interpretation of such observables, and in…
It is shown that Connes' generalized gauge field in non-commutative geometry is derived by simply requiring that Dirac lagrangian be invariant under local transformations of the unitary elements of the algebra, which define the gauge group.…
Soon after the Yang-Mills work, the gauge invariance became one of the basic principles in the elementary particles theory. The gauge invariance idea is that Lagrangian has to be invariant not only with respect to the coordinates…
We investigate using plane fronted gravitational wave space-times as model systems to study loop quantization techniques and dispersion relations. In this classical analysis, we start with planar symmetric space-times in the real connection…
The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with…
We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of…
In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $\delta$-discretized…