Geometric Explanation Research Articles

Zero Field Hall Effect in (2+1)-dimensional QED

In QED of two space dimensions, a quantum Hall effect occurs in the absence of any magnetic field. We give a simple and transparent explanation. In solid state physics, the Hall conductivity for non-degenerate ground state is expected to be given by an integer, the Chern number. In our field-free situation, however, the conductivity is $\pm 1/2$ in natural units. We fit this half-integral result into the topological setting and give a geometric explanation reconciling the points of view of QFT and solid state physics. For quasi-periodic boundary conditions, we calculate the finite size correction to the Hall conductivity. Applications to graphene and similar materials are discussed.

Unstable Solutions of Nonautonomous Linear Differential Equations

The fact that the eigenvalues of the family of matrices $A(t)$ do not determine the stability of nonautonomous differential equations $x'=A(t)x$ is well known. This point is often illustrated using examples in which the matrices $A(t)$ have constant eigenvalues with negative real part, but the solutions of the corresponding differential equation grow in time. Here we provide an intuitive, geometric explanation of the idea that underlies these examples. The discussion is accompanied by a number of animations and easily modifiable Mathematica programs. We conclude with a discussion of possible extensions of the ideas that may provide suitable topics for undergraduate research.

Geometric insight into the challenges of solving high-dimensional reliability problems

In this paper we adopt a geometric perspective to highlight the challenges associated with solving high-dimensional reliability problems. Adopting a geometric point of view we highlight and explain a range of results concerning the performance of several well-known reliability methods. We start by investigating geometric properties of the N -dimensional Gaussian space and the distribution of samples in such a space or in a subspace corresponding to a failure domain. Next, we discuss Importance Sampling (IS) in high dimensions. We provide a geometric understanding as to why IS generally does not work in high dimensions [Au SK, Beck JL. Importance sampling in high dimensions. Structural Safety 2003;25(2):139–63]. We furthermore challenge the significance of “design point” when dealing with strongly nonlinear problems. We conclude by showing that for the general high-dimensional nonlinear reliability problems the selection of an appropriate fixed IS density is practically impossible. Next, we discuss the simulation of samples using Markov Chain Monte Carlo (MCMC) methods. Firstly, we provide a geometric explanation as to why the standard Metropolis–Hastings (MH) algorithm does “not work” in high-dimensions. We then explain why the modified Metropolis–Hastings (MMH) algorithm introduced by Au and Beck [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 2001;16(4):263–77] overcomes this problem. A study of the correlation of samples obtained using MMH as a function of different parameters follows. Such study leads to recommendations for fine-tuning the MMH algorithm. Finally, the MMH algorithm is compared with the MCMC algorithm proposed by Katafygiotis and Cheung [Katafygiotis LS, Cheung SH. Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study, Structural Safety 2006 (in print)] in terms of the correlation of samples they generate.

Generalized Chaplygin gas as geometrical dark energy

The generalized Chaplygin gas provides an interesting candidate for the present accelerated expansion of the universe. We explore a geometrical explanation for the generalized Chaplygin gas within the context of brane world theories where matter fields are confined to the brane by means of the action of a confining potential. We obtain the modified Friedmann equations, deceleration parameter, and age of the universe in this scenario and show that they are consistent with the present observational data.

Realistic Yukawa couplings through instantons in intersecting brane worlds

The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. In order to do this we provide the necessary quantities for instanton calculus in this class of background.

From ten to four and back again: how to generalize the geometry

We discuss the four-dimensional N = 1 effective approach in the study of warped type II flux compactifications with SU(3) × SU(3)-structure to AdS4 or flat Minkowski space-time. The non-trivial warping makes it natural to use a supergravity formulation invariant under local complexified Weyl transformations. We obtain the classical superpotential from a standard argument involving domain walls and generalized calibrations and show how the resulting F-flatness and D-flatness equations exactly reproduce the full ten-dimensional supersymmetry equations. Furthermore, we consider the effect of non-perturbative corrections to this superpotential arising from gaugino condensation or Euclidean D-brane instantons. For the latter we derive the supersymmetry conditions in N = 1 flux vacua in full generality. We find that the non-perturbative corrections induce a quantum deformation of the internal generalized geometry. Smeared instantons allow to understand KKLT-like AdS vacua from a ten-dimensional point of view. On the other hand, non-smeared instantons in IIB warped Calabi-Yau compactifications `destabilize' the Calabi-Yau complex structure into a genuine generalized complex one. This deformation gives a geometrical explanation of the non-trivial superpotential for mobile D3-branes induced by the non-perturbative corrections.

Electron-phonon interaction in graphite intercalation compounds

Motivated by the recent discovery of superconductivity in Ca- and Yb-intercalated graphite (CaC$_{6}$ and YbC$_{6}$) and from the ongoing debate on the nature and role of the interlayer state in this class of compounds, in this work we critically study the electron-phonon properties of a simple model based on primitive graphite. We show that this model captures an essential feature of the electron-phonon properties of the Graphite Intercalation Compounds (GICs), namely, the existence of a strong dormant electron-phonon interaction between interlayer and $\pi ^{\ast}$ electrons, for which we provide a simple geometrical explanation in terms of NMTO Wannier-like functions. Our findings correct the oversimplified view that nearly-free-electron states cannot interact with the surrounding lattice, and explain the empirical correlation between the filling of the interlayer band and the occurrence of superconductivity in Graphite-Intercalation Compounds.

A new geometrical approach to one-parameter spatial motion

The geometry and kinematics of one-parameter dual spherical motions is presented using Study’s dual-line coordinates. The relations between invariants of the axodes are expressed in simple form with geometrical reasoning and explanation. In terms of this, the dual version of associated space curves is demonstrated for a ruled surface to be associated with the axodes of the motion. The relative motion between the axodes is used for deriving the geometry and kinematics of the paths instantaneously traced in the fixed space by a line associated with the moving axode. Especially, the distribution parameter and the inflection-line congruence are investigated. Furthermore, a new metric is developed and used to investigate the geometrical properties and kinematics of line trajectory as well as Disteli axis. Finally, as an application an example is put forward and explained in detail.

A geometric characterisation of resonance in Hopf bifurcation from relative equilibria

We give a new characterisation of resonance in Hopf bifurcation from relative equilibria in systems with compact symmetry group. This characterisation provides a full geometric explanation of the resonance phenomenon. In addition, we develop techniques based on normal form theory to give a complete solution to the associated bifurcation problem.

On the Size of Structures in the Solar Corona

Fine-scale structure in the corona appears not to be well resolved by current imaging instruments. Assuming this to be true offers a simple geometric explanation for several current puzzles in coronal physics, including: the apparent uniform cross-section of bright threadlike structures in the corona; the low EUV contrast (long apparent scale height) between the top and bottom of active region loops; and the inconsistency between loop densities derived by spectral and photometric means. Treating coronal loops as a mixture of diffuse background and very dense, unresolved filamentary structures address these problems with a combination of high plasma density within the structures, which greatly increases the emissivity of the structures, and geometric effects that attenuate the apparent brightness of the feature at low altitudes. It also suggests a possible explanation for both the surprisingly high contrast of EUV coronal loops against the coronal background, and the uniform ``typical'' height of the bright portion of the corona (about 0.3 solar radii) in full-disk EUV images. Some ramifications of this picture are discussed, including an estimate (10-100 km) of the fundamental scale of strong heating events in the corona.

Self-Pulsed Electron Transmission through a Finite Waveguide in a Transversal Magnetic Field

The distribution of scattering delay times is analyzed for classical electrons which are transmitted through a finite waveguide. For non-zero magnetic field the distribution shows a regular pattern of maxima (logarithmic singularities). Although their location follows from a simple commensurability condition, there is no straightforward geometric explanation of this self-pulsing effect. Rather it can be understood as a time-dependent analog of transverse magnetic focusing, in terms of the stationary points of the delay time. We also discuss the possibility of singularities in the delay-time distribution for generic 2D scattering systems.

Geometry in Canonical Correlations

In this article, a useful proposition relating the canonical correlations in multi-way layout to the singular values of a specific matrix is proved and also a geometrical explanation of canonical correlations as an angle between subspaces is given.

Characterization of bamboo foam films by neutron and X-ray experiments

We report on small angle neutron and X-ray scattering experiments on single common black films in ordered bamboo foams stabilized by sodium dodecyl sulfate and dodecanol. We observed anisotropic scattering perpendicular to the film. The integrated signal exhibits clear Kiessig fringes over a wide range of diffusion vector q. We were able to fit our data by a single layer (neutron with contrast matching water/SDS) or a “three slabs” (X-ray) reflectivity. We also discuss about the wide range of reflected angle, despite the narrow distribution of incident angles. Though thermal surface fluctuations may induce off-specular reflectivity, we show here a simple geometrical explanation of the phenomenon. This angle distribution is indeed compatible with the curvature of the film.

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε ∈ [ 0 , 1 ] to ε ∈ [ − 1 , 1 ] . It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε ∈ [ − 1 , 0 ) ∪ ( 0 , 1 ] . We give a geometric explanation for ε-starlike mapping with ε ∈ [ − 1 , 1 ] and prove that the generalized Roper–Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε ∈ [ − 1 , 1 ] . We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε ∈ [ − 1 , 1 ] in Banach spaces or C n . Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.

On variational approach to differential invariants of rank two distributions

We construct differential invariants for generic rank 2 vector distributions on n-dimensional manifolds, where n ⩾ 5 . Our method for the construction of invariants is completely different from the Cartan reduction-prolongation procedure. It is based on the dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the theory of unparameterized curves in the Lagrange Grassmannian, developed in [A. Agrachev, I. Zelenko, Geometry of Jacobi curves I, J. Dynam. Control Syst. 8 (1) (2002) 93–140; II, 8 (2) (2002) 167–215]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ⩾ 5 . In the next paper [I. Zelenko, Fundamental form and Cartan's tensor of (2,5)-distributions coincide, J. Dynam. Control. Syst., in press, SISSA preprint, Ref. 13/2004/M, February 2004, math.DG/0402195] we show that in the case n = 5 our fundamental form coincides with the Cartan covariant biquadratic binary form, constructed in 1910 in [E. Cartan, Les systemes de Pfaff a cinque variables et les equations aux derivees partielles du second ordre, Ann. Sci. Ecole Normale 27 (3) (1910) 109–192; reprinted in: Oeuvres completes, Partie II, vol. 2, Gautier-Villars, Paris, 1953, pp. 927–1010]. Therefore first our approach gives a new geometric explanation for the existence of the Cartan form in terms of an invariant degree four differential on an unparameterized curve in Lagrange Grassmannians. Secondly, our fundamental form provides a natural generalization of the Cartan form to the cases n > 5 . Somewhat surprisingly, this generalization yields a rational function on the fibers of the appropriate vector bundle, as opposed to the polynomial function occurring when n = 5 . For n = 5 we give an explicit method for computing our invariants and demonstrate the method on several examples.

On the geometry of dark energy

Experimental evidence suggests that we live in a spatially flat, accelerating universe composed of roughly one-third of matter (baryonic + dark) and two-thirds of a negative-pressure dark component, generically called dark energy. The presence of such energy not only explains the observed accelerating expansion of the universe but also provides the remaining piece of information connecting the inflationary flatness prediction with astronomical observations. However, despite its good observational indications, the nature of dark energy still remains an open question. In this paper we explore a geometrical explanation for such a component within the context of braneworld theory without mirror symmetry, leading to a geometrical interpretation for dark energy as a warp in the universe given by the extrinsic curvature. In particular, we study the phenomenological implications of the extrinsic curvature of a Friedmann–Robertson–Walker universe in a five-dimensional constant curvature bulk, with signatures (4,1) or (3,2), as compared with the x-matter (XCDM) model. From the analysis of the geometrically modified Friedmann's equations, the deceleration parameter and the weak energy condition, we find a consistent agreement with the presently known observational data on inflation for the de Sitter bulk, but not for the anti-de Sitter case.

Method of convex rigid frames and applications in studies of multipartite quNit pure states

In this Letter we suggest a method of convex rigid frames in the studies of the multipartite quNit pure-states. We illustrate what are the convex rigid frames and what is the method of convex rigid frames. As the applications we use this method to solve some basic problems and give some new results (three theorems): The problem of the partial separability of the multipartite quNit pure-states and its geometric explanation; The problem of the classification of the multipartite quNit pure-states, and give a perfect explanation of the local unitary transformations; Thirdly, we discuss the invariants of classes and give a possible physical explanation.

A Novel Modified Secant Method for Computing the Fair Share Rate

In this paper, we propose a novel modified secant algorithm to compute the flow fair share rate within the framework of the core-stateless fair queueing [1]. The geometric explanation and numerical results demonstrate that the proposed algorithm possesses better performance in terms of accuracy and convergence than that proposed in [1].

Existence and Bifurcation of Canards in $\mathbbR^3$ in the Case of a Folded Node

We give a geometric analysis of canards of folded node type in singularly perturbed systems with two-dimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition $\mu\notin\mathbb{N}$ is satisfied, where $\mu=\lambda_1/\lambda_2$ denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd $\mu$. Furthermore, we show numerically the existence of a pitchfork bifurcation for even $\mu$ and a novel turning point bifurcation close to $\mu\in\mathbb{N}$. We conclude the existence of $[(\mu-1)/2]$ secondary canards away from the resonances. Finally, we apply our results to a network of Hodgkin--Huxley neurons with excitatory synaptic coupling and explain the observed slowing of the firing rate of the synchronized network due to the existence of canards of folded node type.

Hyperbolic Schrödinger equation

Clifford algebra corresponds to Minkwoski space. The coupling between real object particles and light quantums can be discussed by Minkowski space’s directional strangeness. We introduce Galilei transfomation and Schrodinger equation into Minkowski space and give a geometrical explanation for classical quantum theory.