Nondegenerate Case Research Articles

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

Bias-Dependent Intrinsic RF Thermal Noise Modeling and Characterization of Single-Layer Graphene FETs

In this article, the bias dependence of intrinsic channel thermal noise of single-layer (SL) graphene field-effect transistors (GFETs) is thoroughly investigated by experimental observations and compact modeling. The findings indicate an increase of the specific noise as drain current increases, whereas a saturation trend is observed at very high carrier density regime. Besides, short-channel effects, such as velocity saturation (VS) also result in an increment of noise at higher electric fields. The main goal of this work is to propose a physics-based compact model that accounts for and accurately predicts the above experimental observations in short-channel GFETs. In contrast to long-channel MOSFET-based models adopted previously to describe thermal noise in graphene devices without considering the degenerate nature of graphene, in this work, a model for short-channel GFETs embracing the 2-D material’s underlying physics and including a bias dependence is presented. The implemented model is validated with deembedded high-frequency data from two short-channel devices at quasi-static (QS) region of operation. The model precisely describes the experimental data for a wide range of low-to-high drain current values without the need of any fitting parameter. Moreover, the consideration of the degenerate nature of graphene reveals a significant decrease of noise in comparison with the nondegenerate case and the model accurately captures this behavior. This work can also be of utmost significance from the circuit designers’ aspect since noise excess factor, a very important figure of merit for RF circuits implementation, is defined and characterized for the first time in graphene transistors.

Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.

Analysis of Linear Stability and Bifurcations of Central Configurations in the Planar Restricted Circular Four-Body Problem

The planar restricted four-body problem is considered. That is, motion of an infinitesimal body P in the gravitational field of three attracting bodies is studied. It is supposed, that the above three body form a stable equilateral Lagrange triangle and all four bodies move in a plane. In this case there are eight central configurations formed by the bodies. An analysis of stability and bifurcation of the central configurations is performed. In particular, it is shown that the bifurcation is only possible in cases of degeneracy, when mass of an attracting body vanishes. It is also established that in non-degenerate cases five central configurations are unstable and three central configurations can be both stable and unstable. Domains of stability in linear approximation are constructed in the plane of parameters.

Moser's theorem for hyperbolic-type degenerate lower tori in Hamiltonian system

In this paper, we give a Moser-type theorem for C ℓ -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P ( x , y , u , v ) , ( x , y , u , v ) ∈ T n × R n × R 2 , which is associated with the standard symplectic structure, with d ≥ 1 . Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the ℓ -regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d . Under arbitrary small perturbation P , we prove that if ℓ ≥ ( 5 d + 2 ) ( 8 τ + 3 ) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 2 . Our result can be seen a generalization of paper [42] from analytic case to C ℓ -smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case.

Distributed statistical inference for massive data

This paper considers distributed statistical inference for general symmetric statistics in the context of massive data with efficient computation. Estimation efficiency and asymptotic distributions of the distributed statistics are provided, which reveal different results between the nondegenerate and degenerate cases, and show the number of the data subsets plays an important role. Two distributed bootstrap methods are proposed and analyzed to approximation the underlying distribution of the distributed statistics with improved computation efficiency over existing methods. The accuracy of the distributional approximation by the bootstrap are studied theoretically. One of the methods, the pseudo-distributed bootstrap, is particularly attractive if the number of datasets is large as it directly resamples the subset-based statistics, assumes less stringent conditions and its performance can be improved by studentization.

Sharp Schauder estimates for some degenerate Kolmogorov equations

We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic Holder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.

Existence of solutions to a doubly degenerate fourth-order parabolic equation

The existence and uniqueness of solutions for a doubly degenerate fourth-order parabolic equation ωt+∇·(ρα|∇Δω|p−2∇Δω)=0 with initial-boundary conditions on a bounded domain are considered. For this purpose, the time variable is discretized for the problem with non-degenerate coefficient case. By solving a series of semi-discrete elliptic problems, the existence and uniqueness of weak solutions for the corresponding parabolic problem are gained by energy method and two limit process. Furthermore, the regularization method is applied to the problem with degenerate coefficient at boundary and the existence and uniqueness can be performed by uniform estimates. Finally, the regularity for weak solutions is also shown.

Dielectric response of electron–hole systems. Nondegenerate case and quantum corrections

Analytical results for the dielectric function in RPA are derived for three-, two-, and one-dimensional semiconductors in the weakly-degenerate limit. Based on this limit, quantum corrections are derived. Further attention is devoted to systems with linear carrier dispersion and the resulting Dirac-cone physics.

Asymptotic spreading of a time periodic diffusion equation with degenerate monostable nonlinearity

This paper deals with the asymptotic spreading for a class of diffusion equations with degenerate monostable nonlinearity, where the initial value is asymptotically front-like. By the method of squeezing technique and super- and sub-solutions, we prove that the speed of asymptotic spreading may be finite or infinite, which depends on the degeneracy of nonlinearity as well as the decaying rate of the initial value. Different from the non-degenerate case, the traveling wave solution with the critical speed is globally asymptotically stable if the initial value has light tails, while an acceleration propagation may occur with the initial value having heavy tails in the degenerate case. Furthermore, if the initial value has heavy tails but lighter than algebraic, then the speed of the asymptotic spreading is finite, and the balance between finite or infinite speed is established if the initial value has algebraic tails. Our results reflect that the degeneracy may be harmful to the acceleration propagation.

Revisiting Einstein's diffusion-mobility relation for universal quantum materials: A generalized paradigm

We have revisited the Einstein diffusion-mobility ratio, on the basis of the generalized Einstein relation, which quantifies both the quantum and classical levels of understanding on electron-hole dynamics in ordered and disordered materials. While the Einstein relation (diffusion-mobility ratio, D/) works only for classical systems under equilibrium cases, our generalized paradigm is quite general and works for systems ranging from classical to quantum systems in any dimensions. The proposed D/ formalism relies on the chemical potential which provides the cooperative nature between electronic and temperature counterparts on fundamental transport. Our generalized paradigm is applicable to both degenerate and non-degenerate cases, and explains the transition from linear to nonlinear relationship between the charge density and chemical potential. The symmetrical nature of electron-hole transport in highly degenerate two-dimensional quantum materials gives linear dispersion, preserving the time-reversal invariance property, while the symmetry is found to be broken in the nonlinear regime. Our generalized diffusion-mobility relation explains both the relativistic (Dirac particles) and non-relativistic regimes (Schrödinger particles) from low temperature to high temperature, and reproduces the classical Einstein relation in an appropriate limit.

On stretch-limited elastic strings

Motivated by the increased interest in modelling non-dissipative materials by constitutive relations more general than those from Cauchy elasticity, we initiate the study of a class of stretch-limited elastic strings : the string cannot be compressed smaller than a certain length less than its natural length nor elongated larger than a certain length greater than its natural length. In particular, we consider equilibrium states for a string suspended between two points under the force of gravity (catenaries). We study the locations of the supports resulting in tensile states containing both extensible and inextensible segments in two situations: the degenerate case when the string is vertical and the non-degenerate case when the supports are at the same height. We then study the existence and multiplicity of equilibrium states in general with multiplicity differing markedly from strings satisfying classical constitutive relations.

Homogeneous non-degenerate 3-(\u03b1,\u03b4)-Sasaki manifolds and submersions over quaternionic K\xe4hler spaces

We show that every 3-(alpha ,delta )-Sasaki manifold of dimension 4n + 3 admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature 16n(n+2)alpha delta. In the non-degenerate case we describe all homogeneous 3-(alpha ,delta )-Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If alpha delta > 0, this yields a complete classification of homogeneous 3-(alpha ,delta )-Sasaki manifolds. For alpha delta < 0, we provide a general construction of homogeneous 3-(alpha , delta )-Sasaki manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.

Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups

AbstractSet-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.

Darboux dressing transformation and superregular breathers for a coupled nonlinear Schrödinger system with the negative coherent coupling in a weakly birefringent fibre

For a coupled nonlinear Schrödinger system with the negative coherent coupling, which describes two orthogonally polarized pulses in a weakly birefringent fibre, we construct the Darboux dressing transformation and the N-th-order breather solutions with N as a positive integer. When the retarded time tends to be infinite, limits of the ratios between the N-th-order breather solutions and seed solutions are obtained. For the first-order breathers, we present the condition to distinguish the degenerate and nondegenerate cases. For the nondegenerate breathers, we analyse whether the breathers could be kink-type based on the above limits. For the second-order breathers, superregular breathers (SRBs) are derived, where each SRB could consist of two (1) kink-type, (2) single-hump or (3) double-hump quasi-Akhmediev breathers (quasi-ABs). Before and after the interaction, profiles of two quasi-ABs change for Case (1) but keep unchange for Case (2) or (3).

Optomechanically induced transparency, amplification, and Fano resonance in a multimode optomechanical system with quadratic coupling

We explore the optical response of a multimode optomechanical system with quadratic coupling to a weak probe field, where the cavity is driven by a strong control field and the two movable membranes are, respectively, excited by weak coherent mechanical driving fields. We study the two cases that the two movable membranes are degenerate and nondegenerate. For the degenerate case, it is shown that only one transparency window occurs and the transition between optomechanically induced transparency and Fano resonance can be realized by tuning the cavity-control field detuning. For the nondegenerate case, two transparency windows are observed and the absorption spectrum can switch between a single Fano resonance and double Fano resonances. Furthermore, we show that the output probe field can be greatly amplified or completely suppressed due to the complex interference effect by tuning the amplitude and phase of the mechanical driving fields. Our results can be extended to the optomechanical system with multiple membranes, which enables us to control the light propagation more flexibly.

Gravitational Constraints on a Lightlike Boundary

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1-dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case.

On the Mean Convexity of a Space-and-Time Neighborhood of Generic Singularities Formed by Mean Curvature flow

We consider a nondegenerate case of generic singularity when the limit cylinder of the rescaled mean curvature flow is \({\mathbb {R}}^3 \times {\mathbb {S}}^1\). We obtain a detailed description of a possibly small, but fixed, neighborhood of the singularity, up to (and including) the blowup time, and find that it is mean convex and the singularity is isolated. In the considered nondegenerate cases, we confirm the conjectures of mean convexity and isolation of singularities.

Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor psi is a spinor that satisfies the equation nabla _Xpsi =AXcdot psi with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product {mathbb {R}}times N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to {mathbb {S}}^2times {mathbb {R}}^2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.