International Journal of Geometric Methods in Modern Physics

The tunneling effect of two-dimensional Dirac fermions in a constant magnetic field is studied. This can be done by using the continuity equation at some points to determine the corresponding reflexion and transmission coefficients. For this, we consider a system made of graphene as superposition of two different regions where the second is characterized by an energy gap t'. In fact, we treat concrete systems to practically give two illustrations: barrier and diode. For each case, we discuss the transmission in terms of the ratio of the energy conservation and t'. Moreover, we analyze the resonant tunneling by introducing a scalar Lorentz potential where it is shown that a total transmission is possible.

In this paper, we will show how particular formation of G-invariant metrics on irreducible symmetric spaces leads us to have a different approach to Ricci flow as a weakly parabolic system which happens to become a ODE under these spaces.

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to Eⁿ⁺¹ × Sⁿ(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

Complete spacelike surfaces with constant mean curvature (CMC) and bounded hyperbolic angle in Generalized Robertson–Walker (GRW) spacetimes, obeying certain natural curvature assumptions, are studied. This boundedness assumption arises as a natural extension of the notion of bounded hyperbolic image of a spacelike surface in the 3-dimensional Lorentz–Minkowski spacetime. The results obtained apply to complete CMC spacelike surfaces lying between two spacelike slices in an GRW spacetime, in the steady state spacetime and in a static GRW spacetime. As an application, uniqueness and non-existence theorems for certain CMC spacelike surface differential equations in a wide family of open GRW spacetimes are given.

In this paper, we discuss the local existence and the construction of helicoidal CDPC-surfaces, i.e. the helicoidal surfaces with constant difference of the principal curvatures, in Minkowski 3-space . In general, there exist helicoidal surfaces with conjugate complex principal curvatures, which have no counterparts in R³. We prove this is still true for helicoidal CDPC-surfaces but not true for rotation surfaces in .

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F₀ = CP¹ × CP¹ Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

We study general properties of dynamical maps preserving Hermiticity and quasi-Hermiticity, and prove that the generator of a dynamical semigroup is always a pseudo-Hermitian operator. Moreover, improving a previous proposal by Jakob and Stenholm, we introduce two new Lyapunov functionals for degenerate open systems, and apply such results to a physical example.

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model.

In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.

We consider the dynamics of moving curves in three-dimensional Minkowski space and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space curve in , we show that the defocusing nonlinear Schrödinger equation(NLSE) can be identified with the spin chain, thereby giving a geometrical interpretation of it. The associated linear eigenvalue problem is also obtained in a geometrical way.

Linear entropy as a measure of entanglement is applied to explain conditions for minimal and maximal entanglement of bipartite nonorthogonal pure states. We formulate this measure in terms of the amplitudes of coherent states in the case of entangled coherent states and calculate the conditions. We generalize this formalism to the case of bipartite mixed states and show that the entanglement measure is also a function of the probabilities.

The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊_ρ S¹-bundles where LG is the group of smooth loops in G. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an LG ⋊_ρ S¹-bundle.