arXiv subjects

Vedad Pasic

Publications and source records attributed to Vedad Pasic.

11 recordsLinked to original sources

arXiv2020

Analyzing the spectral (a)symmetry of the massless Dirac operator on the 3-torus

We analyze the spectrum of the massless Dirac operator on the 3-torus $\mathbb{T}^3$. It is known that it is possible to calculate this spectrum explicitly, that it is symmetric about zero and that each eigenvalue has even multiplicity. However, for a general oriented closed Riemannian 3-manifold $(M,g)$ there is no reason for the spectrum of the massless Dirac operator to be symmetric. Using perturbation theory, we derive the asymptotic formulae for its eigenvalues and prove that by the perturbation of the Euclidean metric on the 3-torus, it is possible to obtain spectral asymmetry of the massless Dirac operator in the axisymmetric case.

math.SP↗

arXiv2017

Diferencijalna geometrija krivih i površi u trodimenzionalnom euklidskom prostoru

This manuscript, titled Differential Geometry of Curves and Surfaces in Three-dimensional Euclidean Space, is intended for undergraduate students of mathematics and other sciences with the need of use of Euclidean differential geometry. We deal with the differential geometry of curves and surfaces in three dimensions, covering all the main definitions, ideas and concepts needed for successful understanding and further application of this fundamental branch of mathematics. Many concrete examples with solutions are given with illustrations. ----- Ova knjiga namijenjena je dodiplomskim studentima matematike i drugih nauka sa potrebom upotrebe euklidske diferencijalne geometrije. Upoznat ćemo se sa diferencijalnom geometrijom krivih i površi u tri dimenzije, pokrivajući sve glavne definicije, ideje i koncepte potrebne za razumijevanje i uspješnu dalju primjenu ove fundamentalne grane matematike. Dati su mnogi konkretni problemi sa rješenjima i mnoštvom ilustracija.

math.HO↗

arXiv2017

Axial torsion waves in metric-affine gravity

We construct new explicit vacuum solutions of quadratic metric-affine gravity. The approach of metric-affine gravity in using an independent affine connection produces a theory with 10+64 unknowns, which implies admitting torsion and possible nonmetricity. Our spacetimes are generalisations of classical pp-waves, four-dimensional Lorentzian spacetimes which admit a nonvanishing parallel spinor field. We generalize this definition to metric compatible spacetimes with pp-metric and purely axial torsion. It has been suggested that one can interpret that the axial component of torsion as the Hodge dual of the electromagnetic vector potential. We compare these solutions with our previous results and other solutions of classical models describing the interaction of gravitational and neutrino fields.

gr-qc↗

arXiv2017

Physical interpretation of pp-waves with axial torsion

We consider generalised pp-waves with purely axial torsion, which we previously showed to be new vacuum solutions of quadratic metric-affine gravity. Our analysis shows that classical pp-waves of parallel Ricci curvature should not be viewed on their own. They are a particular representation of a wider class of solutions, namely generalised pp-waves of parallel Ricci curvature. We compare our pp-waves with purely axial torsion to solutions of Einstein-Weyl theory, the classical model describing the interaction of gravitational and massless neutrino fields.

gr-qc↗

arXiv2017

A New Representation of the Field Equations of Quadratic Metric-Affine Gravity

We deal with quadratic metric-affine gravity (QMAG), which is an alternative theory of gravity and present a new explicit representation of the field equations of this theory. In our previous work we found new explicit vacuum solutions of QMAG, namely generalised pp-waves of parallel Ricci curvature with purely tensor torsion. Here we do not make any assumptions on the properties of torsion and write down our field equations accordingly. We present a review of research done thus far by several authors in finding new solutions of QMAG and different approaches in generalising pp-waves. We present two conjectures on the new types of solutions of QMAG which the ansatz presented in this paper will hopefully enable us to prove.

gr-qc↗

arXiv2017

Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an $\varepsilon$-uniform convergence of such gained the approximate solutions, in the maximum norm of the order $\mathcal{O}\left(N^{-1}\right)$ on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has $\varepsilon$--uniform convergence, but now of order $\mathcal{O}\left(\ln^2N/N^2\right)$ on $[0,1].$ In the end a numerical experiment is presented to confirm previously shown theoretical results.

math.NA↗

arXiv2016

Continuity of modulus of noncompact convexity for minimalizable measures of noncompactness

We consider the modulus of noncompact convexity $Δ_{X,ϕ}(\varepsilon)$ associated with the minimalizable measure of noncompactness $ϕ$. We present some properties of this modulus, while the main result of this paper is showing that $Δ_{X,ϕ}(\varepsilon)$ is a subhomogenous and continuous function on $[0,ϕ(\bar{B}_X))$ for an arbitrary minimalizable measure of compactness $ϕ$ in the case of a Banach space $X$ with the Radon-Nikodym property.

math.FA↗

arXiv2015

Torsion Wave Solutions in Yang-Mielke Theory of Gravity

The approach of metric-affine gravity initially distinguishes it from Einstein's general relativity. Using an independent affine connection produces a theory with 10+64 unknowns. We write down the Yang-Mills action for the affine connection and produce the Yang-Mills equation and the so called complementary Yang-Mills equation by independently varying with respect to the connection and the metric respectively. We call this theory the Yang-Mielke theory of gravity. We construct explicit spacetimes with pp-metric and purely axial torsion and show that they represent a solution of Yang-Mills theory. Finally we compare these spacetimes to existing solutions of metric-affine gravity and present future research possibilities.

gr-qc↗

arXiv2014

A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem

In this paper we are considering a semilinear singular perturbation reaction -- diffusion boundary value problem, which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is $\epsilon$-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

math.NA↗

arXiv2014

PP-waves with Torsion - a Metric-affine Model for the Massless Neutrino

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. Further, we introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime, which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein-Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein-Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.

gr-qc↗

arXiv2005

PP-waves with torsion and metric-affine gravity

A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. We generalise this definition to metric compatible spacetimes with torsion and describe basic properties of such spacetimes. We use our generalised pp-waves for constructing new explicit vacuum solutions of quadratic metric-affine gravity.

gr-qc↗