Geometric Explanation Research Articles

Generation of maximally entangled states with subluminal Lorentz boosts

Recent work has studied entanglement between the spin and momentum components of a single spin-1/2 particle and showed that maximal entanglement is obtained only when boosts approach the speed of light. Here we extend the boost scenario to general geometries and show that, intriguingly, maximal entanglement can be achieved with boosts less than the speed of light. Boosts approaching the speed of light may even decrease entanglement. We also provide a geometric explanation for this behavior.

The geometry of finite difference discretizations of semilinear elliptic operators

Discretizations by finite differences of some semilinear elliptic equations lead to maps , given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y − tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f′i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer–McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates.Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions.

Adsorbate-induced reconstruction by C60on close-packed metal surfaces: Mechanism for different types of reconstruction

Recent studies reveal that reconstruction of close-packed metal surfaces induced by C${}_{60}$ adsorption is the rule rather than the exception. Two types of reconstruction are reported for C${}_{60}$ on different surfaces: (1) C${}_{60}$ sinks into a 7-atom hole, such as on Cu(111); and (2) C${}_{60}$ sits over a 1-atom hole, such as on Ag(111) and Pt(111). An explanation for the preferred reconstruction type for different metals has been lacking. Here, we propose a criterion that predicts which reconstruction type should be expected: Namely, the formation of a 1- or 7-atom hole is determined only by the substrate geometric structure, including the surface lattice constant and the interlayer spacing; remarkably, the reconstruction type appears not to depend on the substrate electronic structure. Our intuitive geometrical explanation is validated by comparative first-principles calculations of the energetics of C${}_{60}$ on Cu(111), Ru(0001), Pt(111), and Ag(111) surfaces, listed here with increasing surface lattice constant. This provides a uniform explanation for the different reconstruction types that are observed experimentally for C${}_{60}$ on different close-packed metal surfaces. Moreover, our results provide a better explanation for the decomposition behavior of C${}_{60}$ on Ru(0001).

Tsallis entropy induced metrics and [formula omitted] spaces

Generalizing the group structure of the Euclidean space, we construct a Riemannian metric on the deformed set Rqn induced by the Tsallis entropy composition property. We show that the Tsallis entropy is a “hyperbolic analogue” of the “Euclidean” Boltzmann/Gibbs/Shannon entropy and find a geometric interpretation for the nonextensive parameter q. We provide a geometric explanation of the uniqueness of the Tsallis entropy as reflected through its composition property, which is provided by the Abe and the Santos axioms. For two, or more, interacting systems described by the Tsallis entropy, having different values of q, we argue why a suitable extension of this construction is provided by the Cartan/Alexandrov/Toponogov metric spaces with a uniform negative curvature upper bound.

A Geometrical Explanation of Stein Shrinkage

Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key development toward this was the discovery of the inadmissibility of the usual estimator of a multivariate normal mean. This article develops a geometrical explanation for this inadmissibility. By exploiting the spherical symmetry of the problem it is possible to effectively conceptualize the multidimensional setting in a two-dimensional framework that can be easily plotted and geometrically analyzed. We begin with the heuristic explanation for inadmissibility that was given by Stein [In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954--1955, Vol. I (1956) 197--206, Univ. California Press]. Some geometric figures are included to make this reasoning more tangible. It is also explained why Stein's argument falls short of yielding a proof of inadmissibility, even when the dimension, $p$, is much larger than $p=3$. We then extend the geometric idea to yield increasingly persuasive arguments for inadmissibility when $p\geq3$, albeit at the cost of increased geometric and computational detail.

Moduli spaces of flat connections and Morita equivalence of quantum tori

We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3-dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.

Strain induced shift of Dirac points and the pseudo-magnetic field in graphene

We prove that the strain induced shift of the Dirac points in graphene is a curl field if the strain is nonuniform. This curl field provides a geometrical explanation of the strain induced pseudo-magnetic field. We also prove that the Dirac points must be confined within two triangles, each one having one-eighth the area of the Brillouin zone.

Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus

The fractional complex transform is suggested to convert a fractional differential equation with Jumarieʼs modification of Riemann–Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically.

Spectral clustering with discriminant cuts

Recently, many k-way spectral clustering algorithms have been proposed, satisfying one or both of the following requirements: between-cluster similarities are minimized and within-cluster similarities are maximized. In this paper, a novel graph-based spectral clustering algorithm called discriminant cut (Dcut) is proposed, which first builds the affinity matrix of a weighted graph and normalizes it with the corresponding regularized Laplacian matrix, then partitions the vertices into k parts. Dcut has several advantages. First, it is derived from graph partition and has a straightforward geometrical explanation. Second, it emphasizes the above requirements simultaneously. Besides, it is computationally feasible because the NP-hard intractable graph cut problem can be relaxed into a mild eigenvalue decomposition problem. Toy-data and real-data experimental results show that Dcut is pronounced comparing with other spectral clustering methods.

Convergence of normalized iterative identification of Hammerstein systems

An iterative identification algorithm of Hammerstein systems needs a proper initial condition to guarantee its convergence. In this paper, we propose a new algorithm by fixing the norm of the parameter estimates. The normalized algorithm ensures the convergence property under arbitrary nonzero initial conditions. The proofs of the property also give a geometrical explanation on why the normalization guarantees the convergence. An additional contribution is that the static function in the Hammerstein system is extended to square-integrable functions.

$\mathcal D$-bundles and integrable hierarchies

We study the geometry of \mathcal D - bundles —locally projective \mathcal D -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of \mathcal D -bundles; in particular, we prove that the local structure of \mathcal D -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by \mathcal D -bundles on cubic curves E , that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1 . We develop a Fourier-Mukai transform describing \mathcal D -modules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle {E^{\natural}} over E . We then apply this transform to classify \mathcal D -bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in {E^{\natural}} ). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.

Wavelet frame based surface reconstruction from unorganized points

Applications of wavelet frames to image restoration problems (e.g. image deblurring and inpainting) have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images (see e.g. [13,14,8]). However, wavelet frames have not yet been used for surface reconstruction problems. Recently in [6], connections between one of the wavelet frame based image restoration model [8,28,48] and variational models (e.g. the ROF model [45]) were rigorously established. Such connections not only grant new insights to wavelet frame based image restorations, it also case a geometric explanation to wavelet frame based approaches. This leads us to a wavelet frame based model, as well as a fast algorithm, to reconstruct implicit surfaces from unorganized point sets in R^3. We will demonstrate the effectiveness of the proposed model using several commonly used examples.

Cosmological dark fluid from five-dimensional vacuum

In the framework of the induced matter theory of gravity, we derive a 5D solution of the field equations that can describe a 4D cosmological scenario where the dark fluid (dark matter plus dark energy) equation of state has a geometrical origin. There is no a natural separation of the dark sector into different components, and then the model may provide a geometrical explanation for the existence of a dark degeneracy in cosmological scenarios.

The Refined Analysis on the Convergence Behavior of Harmonic Map Sequence from Cylinders

In this paper, we are concerned with the convergence behavior of a sequence of harmonic maps from long cylinders with uniformly bounded energy. If the bubbling phenomenon does not occur, we give the length formula of the limit map (i.e., geodesic in the target manifold). Furthermore, we provide a geometric explanation of the energy identity for a sequence of harmonic maps from degenerating Riemann surfaces with uniformly bounded energy, proved by M. Zhu.

The geometrical origin of the strain-twist coupling in double helices

A simple geometrical explanation for the counterintuitive phenomenon when twist leads to extension in double helices is presented. The coupling between strain and twist is investigated using a tubular description. It is shown that the relation between strain and rotation is universal and depends only on the pitch angle. For pitch angles below 39.4° strain leads to further winding, while for larger pitch angles strain leads to unwinding. The zero-twist structure, with a pitch angle of 39.4°, is at the unique point between winding and unwinding and independent of the mechanical properties of the double helix. The existence of zero-twist structures, i.e. structures that display neither winding, nor unwinding under strain is discussed. Close-packed double helices are shown to extend rather than shorten when twisted. Numerical estimates of this elongation upon winding are given for DNA, chromatin, and RNA.

Orbifolds, the A, D, E family of caustic singularities, and gravitational lensing

We provide a geometric explanation for the existence of magnification relations for the \documentclass[12pt]{minimal}\begin{document}$A_n (n \ge 2), D_n (n \ge 4), E_6, E_7, E_8$\end{document}An(n≥2),Dn(n≥4),E6,E7,E8 family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any noncaustic target point, the total signed magnification of the corresponding preimages vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multidimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.

Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

The large-time behavior of solutions to the Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this “metastable” manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.

EXPLICIT SOLITON SOLUTIONS OF THE KAUP-KUPERSHMIDT EQUATION THROUGH THE DYNAMICAL SYSTEM APPROACH

In this paper, we study the traveling wave solutions of the Kaup-Kupershmidt (KK) equation through using the dynamical system approach, which is an integrable fifth-order wave equation. Based on Cosgrove's work [3] and the phase analysis method of dynamical systems, infinitely many soliton solutions are presented in an explicit form. To guarantee the existence of soliton solutions, we discuss the parameters range as well as geometrical explanation of soliton solutions.

基于聚类的自适应图像稀疏表示算法及其应用

A dictionary training algorithm was proposed for spare representation of images and its convergence was proved.The geometrical explanation of the algorithm is to approximate the hyperspherical cap with least hyperplanes.The algorithm clustered the error vectors of each step,and signed the cluster center as new atoms which made the dictionary more suitable for spare representation of samples.Compared with the traditional algorithm,the new one has higher adaptability,lower requirement of sample number and dictionary size,higher convergence rate,and lower complexity.Finally,the experiment of compressive sensing and denoising demonstrates that dictionary training by this algorithm has good effect.

Imaging properties of bright-field and annular-dark-field scanning confocal electron microscopy

Imaging properties of scanning confocal electron microscopy (SCEM) were studied by calculating simple model systems using the multislice method. A simple geometrical explanation was given, particularly for the difference between bright field (BF) and annular dark field (ADF) SCEM. It is demonstrated that the BF-SCEM image contrast consists of two features. One gradually changes over a wide defocus range and depends on the lateral size of the object. Another appears only near the focus and is independent of sample size. On the contrary, ADF-SCEM image contrast does not depend on the lateral size of the object. Therefore, the ADF-SCEM will provide more readily interpretable image contrast.