Lower Dimensional Versions Research Articles

Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity

Recently, it was discovered that lower-dimensional versions of Lovelock gravity exist as scalar-tensor theories that are examples of Horndeski gravity. We study the thermodynamics of the static black hole solutions in these theories up to cubic order through Euclidean methods. Considering solutions with spherical, planar and hyperbolic event horizons (k=+1,0,−1), we show that the universality of the thermodynamics for planar black holes (k=0) and the extended first law that include the variation of the couplings together with their associated potentials hold also in lower dimensions. We find that in D=4,6 where the 2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively, the Smarr relation is modified since the entropy is not a homogenous function in these dimensions. We also present a derivation of the Smarr relation and its modified version based on the global scaling properties of the reduced action that is used to obtain the solutions consistently. Unlike the other hairy black hole solutions that have been analyzed before, despite the terms in the reduced action that break the scaling symmetry, the derivation still follows from a conserved Noether charge.

Excitation of flatland surface-wave leakage on an equivalent reactance surface by a single-wire transmission line

Recently, researchers have demonstrated, both theoretically and experimentally, that low-dimensional version of conventional leaky waves, known as “flatland leaky waves” (FLWs). FLWs can be supported by planar junctions of metasurfaces and are completely confined to a plane. Here, we design a structure that places an equivalent reactance surface near the transmission line to excite the FLWs. By properly designing the geometry of the unit cell, conditions can be created for faster surface waves (SWs) to excite slower SWs on the surface, resulting in the desired form of two-dimensional (2D) leaky-wave radiation. A sample was fabricated and measured, and both the simulation results and the experimental results yielded the desired results. Finally, we design the equivalent impedance surface model on which we can also observe the 2D FLWs. Our research not only deepens the understanding of the phenomenon of FLWs but also brings the application of equivalent reactance surfaces in the field of electromagnetic phenomena research to a higher level.

Dimension Reduction of Collective Attention Networks

Recently, the dynamics of the collective attention of various cultural products are typically modeled by mathematical models. In this article, we propose a simple collective attention model for capturing the dynamics of coupled cultural products, which is represented by a complex dynamical network. In particular, the coupling mechanism of the model involves one of the cooperative, exploitative competitive, and appropriative terms. To facilitate the analysis of the higher-dimensional complex dynamical network, we employ and extend the existing dynamical dimension reduction techniques to reduce the network to a simplified lower-dimensional version. It can then be used to describe the collective dynamics of the original system, such as the emergency of the bifurcation of the collective attention received by cultural products. We test the dimension reduction techniques on several collective attention dynamical networks. Our results indicate that articulating the complex dynamical models as well as their advanced theories and tools may open up a new avenue for the dynamics study of collective attention.

Single-walled zeolitic nanotubes.

We report the synthesis and structure of single-walled aluminosilicate nanotubes with microporous zeolitic walls. This quasi-one-dimensional zeolite is assembled by a bolaform structure-directing agent (SDA) containing a central biphenyl group connected by C10 alkyl chains to quinuclidinium end groups. High-resolution electron microscopy and diffraction, along with other supporting methods, revealed a unique wall structure that is a hybrid of characteristic building layers from two zeolite structure types, beta and MFI. This hybrid structure arises from minimization of strain energy during the formation of a curved nanotube wall. Nanotube formation involves the early appearance of a mesostructure due to self-assembly of the SDA molecules. The biphenyl core groups of the SDA molecules show evidence of π stacking, whereas the peripheral quinuclidinium groups direct the microporous wall structure.

Chaos in coupled Kerr-nonlinear parametric oscillators

A Kerr-nonlinear parametric oscillator (KPO) can generate a quantum superposition of two oscillating states, known as a Schr\"{o}dinger cat state, via quantum adiabatic evolution, and can be used as a qubit for gate-based quantum computing and quantum annealing. In this work, we investigate complex dynamics, i.e., chaos, in two coupled nondissipative KPOs at a few-photon level. After showing that a classical model for this system is nonintegrable and consequently exhibits chaotic behavior, we provide quantum counterparts for the classical results, which are quantum versions of the Poincar\'{e} surface of section and its lower-dimensional version defined with time integrals of the Wigner and Husimi functions, and also the initial and long-term behavior of out-of-time-ordered correlators. We conclude that some of them can be regarded as quantum signatures of chaos, together with energy-level spacing statistics (conventional signature). Thus, the system of coupled KPOs is expected to offer not only an alternative approach to quantum computing, but also a promising platform for the study on quantum chaos.

A two-dimensional topological representation theorem for matroid polytopes of rank 4

The Folkman–Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank r can be represented as a pseudosphere arrangement on the (r−1)-dimensional sphere Sr−1, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank 4. We introduce 2-weak configurations of points and pseudocircles (2-weak PPC configurations) on S2 and prove that every uniform matroid polytope of rank 4 can be represented by a 2-weak PPC configuration. As an application, we provide a proof of Las Vergnas conjecture on simplicial topes for the case of uniform matroid polytopes of rank 4.

Spectral Dimension Reduction of Complex Dynamical Networks

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.

Precursor for wind reversal in a square Rayleigh-Bénard cell.

We investigate large-scale circulation reversals in a two-dimensional Rayleigh-Bénard cell using a proper orthogonal decomposition (POD)-based, five-mode model. The Rayleigh number considered is Ra=5×10^{7} and the Prandtl number is Pr=4.3. A precursor event, corresponding to the action of a mode L_{*} which disconnects the core region from the boundary layers before the onset of the reversal, is identified in the simulation. The five-mode model predicts correctly the behavior of the POD modes observed in the simulation, and in particular that of mode L_{*}. The presence of mode L_{*}, which was not included in an earlier, lower-dimensional version of the model [Podvin and Sergent, J. Fluid Mech. 766, 172 (2015)JFLSA70022-112010.1017/jfm.2015.15], is found to be instrumental for the reversal dynamics of the model, which suggests that it may also be important for those of the simulation. Reversals can therefore be characterized by three time scales: the transition duration, the interreversal time, and the precursor duration, which separates the precursor event from the onset of the reversal. The distribution of the time scales is found to agree well with the simulation when small-scale intermittency is taken into account through the introduction of noise in the model coefficients.

Double Primary Relaxation in a Highly Anisotropic Orientational Glass-Former with Low-Dimensional Disorder

The freezing of the cooperative reorientational motions in orientationally disordered (OD) molecular crystals marks the so-called “glassy” transition, which may be considered a lower-dimensional version of the structural glass transition. Although structural glasses display both positional and orientational disorder, in orientational glasses, however, the disorder involves exclusively the orientational degrees of freedom of the constituent molecules, while the molecular centers of mass form an ordered lattice. We report here on a glass-forming system with even fewer degrees of freedom, namely, the OD phase of a dipolar benzene derivative, pentachloronitrobenzene (C6Cl5NO2). We probe the orientational dynamics of pentachloronitrobenzene as a function of temperature and pressure by means of dielectric spectroscopy (and high-pressure density measurements), and we show that, due to its anisotropy, the system exhibits a double primary relaxation feature associated with two distinct motions of the molecular dip...

A Manifold Learning Approach for Personalizing HRTFs from Anthropometric Features

We present a new anthropometry-based method to personalize head-related transfer functions (HRTFs) using manifold learning in both azimuth and elevation angles with a single nonlinear regression model. The core element of our approach is a domain-specific nonlinear dimensionality reduction technique, denominated Isomap, over the intraconic component of HRTFs resulting from a spectral decomposition. HRTF intraconic components encode the most important cues for HRTF individualization, leaving out subject-independent cues. First, we modify the graph construction procedure of Isomap to integrate relevant prior knowledge of spatial audio into a single manifold for all subjects by exploiting the existing correlations among HRTFs across individuals, directions, and ears. Then, with the aim of preserving the multifactor nature of HRTFs (i.e. subject, direction and frequency), we train a single artificial neural network to predict low-dimensional HRTFs from anthropometric features. Finally, we reconstruct the HRTF from its estimated low-dimensional version using a neighborhood-based reconstruction approach. Our findings show that introducing prior knowledge in Isomap’s manifold is a powerful way to capture the underlying factors of spatial hearing. Our experiments show, with p-values less than 0.05, that our approach outperforms using, either a PCA linear reduction, or the full HTRF, in its intermediate stages.

Credit-driven investment, heterogeneous labor markets and macroeconomic dynamics

In this paper we set up a baseline, but nevertheless advanced and complete model representing detailed goods market dynamics, heterogeneous labor markets, dual and cross-dual wage-price adjustment processes, as well as counter-cyclical government policies. The cyclical movements of output generate, through Okun’s law, employment variations in the heterogeneous labor market. The core of the resulting Keynesian macrodynamics is however given by credit-financed investment behavior and loan-rate setting by credit suppliers. The framework is constructed in such way that simplified, lower dimensional versions of the model can be obtained by setting parameters describing specific feedback effects from one sector to another equal to zero. Starting from such low dimensional sub-dynamics, we show the local stability of the full 7D model through a “cascade of stable matrices” approach if the feedback chains are sufficiently tranquil in their transmission mechanisms. However, local stability is the point of departure for the numerical investigation of local explosiveness and the forces that can bound such a behavior.

Feedback control of a redundant wheeled snake mechanism using transverse functions SO(4)

The Transverse Function (TF) approach is applied to the control of a nonholonomic three-segments/snake-like wheeled mechanism, similar to the planar low-dimensional version of Hirose's Active Cord Mechanism (ACM) previously studied by the authors with the same control approach, but with two additional internal degrees of freedom (d.o.f.) whose actuation yields more flexible and efficient control solutions. From a theoretical point of view, these complementary d.o.f. modify the Control Lie Algebra of the system so that only first-order Lie brackets of the control vector fields are needed to satisfy the Lie Algebra Rank Condition (LARC). The fact that four independent (angular velocity) control inputs are used also implies for this system the existence of Transverse Functions (TF) defined on the six-dimensional special orthogonal group(4). Several examples of mechanisms whose control involve TF defined on(3) have been pointed out in the past. Beyond the specific control problem addressed here, a motivation for the present study is to illustrate for the first time how functions defined on the larger set(4) can be determined and used for the control of a physical system. This study is complemented with recalls concerning the parametrization of(4) by pairs of isoclinic quaternions and with the derivation of complementary differential calculus relations associated with this parametrization.

Lower-dimensional Hořava–Lifshitz gravity

We consider Horava-Lifshitz gravity in both 1+1 and 2+1 dimensions. These lower-dimensional versions of Horava-Lifshitz gravity are simple enough to be explicitly tractable, but still complex enough to be interesting. We write the most general (non-projectable) action for each case and discuss the resulting dynamics. In the 1+1 case we utilize the equivalence with 2-dimensional Einstein-aether theory to argue that, even though non-trivial, the theory does not have any local degrees of freedom. In the 2+1 case we show that the only dynamical degree of freedom is a scalar, which qualitatively has the same dynamical behaviour as the scalar mode in (non-projectable) Horava-Lifshitz gravity in 3+1 dimensions. We discuss the suitability of these lower-dimensional theories as simpler playgrounds that could help us gain insight into the 3+1 theory. As special cases we also discuss the projectable limit of these theories. Finally, we present an algorithm that extends the equivalence with (higher order) Einstein-aether theory to full Horava-Lifshitz gravity (instead of just the low energy limit), and we use this extension to comment on the apparent naturalness of the covariant formulation of the latter.

A balanced excited random walk

The following random process on Z 4 is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.

Dimension reduction for regression estimation with nearest neighbor method

In regression with a high-dimensional predictor vector, dimension reduction methods aim at replacing the predictor by a lower dimensional version without loss of information on the regression. In this context, the so-called central mean subspace is the key of dimension reduction. The last two decades have seen the emergence of many methods to estimate the central mean subspace. In this paper, we go one step further, and we study the performances of a k-nearest neighbor type estimate of the regression function, based on an estimator of the central mean subspace. In our setting, the predictor lies in ℝp with fixed p, i.e. it does not depend on the sample size. The estimate is first proved to be consistent. Improvement due to the dimension reduction step is then observed in term of its rate of convergence. All the results are distributions-free. As an application, we give an explicit rate of convergence using the SIR method. The method is illustrated by a simulation study.

3Dscalar model as a4Dperfect conductor limit: Dimensional reduction and variational boundary conditions

Under dimensional reduction, a system in $D$ spacetime dimensions will not necessarily yield its $D\ensuremath{-}1$-dimensional analog version. Among other things, this result will depend on the boundary conditions and the dimension $D$ of the system. We investigate this question for scalar and Abelian gauge fields under boundary conditions that obey the symmetries of the action. We apply our findings to the Casimir piston, an ideal system for detecting boundary effects. Our investigation is not limited to extra dimensions and we show that the original piston scenario proposed in 2004, a toy model involving a scalar field in $3D$ ($2+1$) dimensions, can be obtained via dimensional reduction from a more realistic $4D$ electromagnetic (EM) system. We show that for perfect conductor conditions, a $D$-dimensional EM field reduces to a $D\ensuremath{-}1$ scalar field and not its lower-dimensional version. For Dirichlet boundary conditions, no theory is recovered under dimensional reduction and the Casimir pressure goes to zero in any dimension. This ``zero Dirichlet'' result is useful for understanding the EM case. We then identify two special systems where the lower-dimensional version is recovered in any dimension: systems with perfect magnetic conductor (PMC) and Neumann boundary conditions. We show that these two boundary conditions can be obtained from a variational procedure in which the action vanishes outside the bounded region. The fields are free to vary on the surface and have zero modes, which survive after dimensional reduction.

Quasinormal modes of black holes localized on the Randall-Sundrum 2-brane

We investigate conformal scalar, electromagnetic, and massless Dirac quasinormal modes of a brane-localized black hole. The background solution is the four-dimensional black hole on a 2-brane that has been constructed by Emparan, Horowitz, and Myers in the context of a lower-dimensional version of the Randall-Sundrum model. The conformally transformed metric admits a Killing tensor, allowing us to obtain separable field equations. We find that the radial equations take the same form as in the four-dimensional braneless Schwarzschild black hole. The angular equations are, however, different from the standard ones, leading to a different prediction for quasinormal frequencies.

A Model Free Approach to Combining Biomarkers

For most diseases, single biomarkers do not have adequate sensitivity or specificity for practical purposes. We present an approach to combine several biomarkers into a composite marker score without assuming a model for the distribution of the predictors. Using sufficient dimension reduction techniques, we replace the original markers with a lower-dimensional version, obtained through linear transformations of markers that contain sufficient information for regression of the predictors on the outcome. We combine the linear transformations using their asymptotic properties into a scalar diagnostic score via the likelihood ratio statistic. The performance of this score is assessed by the area under the receiver-operator characteristics curve (ROC), a popular summary measure of the discriminatory ability of a single continuous diagnostic marker for binary disease outcomes. An asymptotic chi-squared test for assessing individual biomarker contribution to the diagnostic score is also derived.

Two gravitational shock waves on theAdS3brane

A gravitational shock wave is a solution to Einstein equations describing the gravitational field of a massless particle. We obtain such a geometry for a particle moving on a AdS_3 brane embedded in a AdS_4 bulk (the lower dimensional version of the locally localized gravity model of Karch and Randall). In order to do this, we use two different techniques. First, we solve directly Einstein equations sourced by a massless particle. Then we boost to the speed of light the AdS_3 brane black hole solution of Emparan et al while sending its mass parameter to zero. Surprisingly, we obtain two different results. We discuss the origin of these two different solutions both in the bulk and in the CFT picture. As a by-product, we derive the expression for the shock wave associated to a transversally accelerating particle in AdS_4.

Testing predictor contributions in sufficient dimension reduction

We develop tests of the hypothesis of no effect for selected predictors in regression, without assuming a model for the conditional distribution of the response given the predictors. Predictor effects need not be limited to the mean function and smoothing is not required. The general approach is based on sufficient dimension reduction, the idea being to replace the predictor vector with a lower-dimensional version without loss of information on the regression. Methodology using sliced inverse regression is developed in detail.