Canonical Deformation Research Articles

Revealing Urban Deformation Patterns through InSAR Time Series Analysis with TCN and Transfer Learning

Abstract. Current multi-epoch InSAR techniques heavily rely on the assumption of linear deformation. This can sometimes overlook crucial deformation signals when using velocities for evaluation. The process of interpreting InSAR time series is not only time-consuming and labor-intensive but also requires a certain level of expertise. This study refines existing InSAR deformation categories, such as stable, linear, step, piecewise linear, power, and undefined, to define 'canonical deformation time series patterns.' We propose an innovative approach for InSAR post-processing using Temporal Convolutional Networks (TCN) and transfer learning. Due to the limited availability of real data, we use simulated data to train a pre-existing model. We then assess the effectiveness of our method in identifying urban deformation patterns. This research could significantly improve our understanding of large-scale InSAR time series deformation and reveal the underlying patterns.

Fibre stretch models of transversely isotropic incompressible hyperelasticity

A new strain–energy function for incompressible transversely isotropic non-linearly hyperelastic materials is obtained by assuming that the deviatoric stress in the preferred direction is a function only of the stretch in that direction. The model is an additive combination of two arbitrary functions of the strain invariants, one of which is an arbitrary function of the fibre stretch. The behaviour of the model for the canonical deformations of simple tension and shear is analysed. It is shown that the stress–stretch response in simple tension along and perpendicular to the fibres is determined by the fibre stretch function alone. It is also shown that a linearisation of the general model containing only three arbitrary constants is an excellent fit to a comprehensive experimental data set of other authors for simple tension along and perpendicular to the fibres and simple shear along the fibres.

Cosmology with non-minimal coupled gravity: dynamical study of the inflationary universe in the deformed phase space scenario

In this work, we consider a noncommutative (NC) inflationary model with a homogeneous scalar field non-minimally coupled to gravity. We propose to use a canonical deformation between momenta in a spatially flat Friedmann–Le maitre–Robertson–Walker universe. This particular choice of noncommutativity allows interesting dynamics that other NC models seem not to allow. In this approach, the dimensional parameter $$\theta $$ , which considered as the length of Planck, presents the quantum regime. But also, it is crucial to recover the standard results by taking the appropriate limits of this parameter. To note that the $$1\mathrm{st}$$ Friedmann equation remains unaffected while both the Friedmann acceleration and the Klein-Gordon equations are affected by an additional term linear in the NC parameter.

Invariant Translators of the Heisenberg Group

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in {mathbb {R}}^3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems

This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolution equations of classical mechanical systems are described. There are studied projections of such F- and W-functionals on Lorentz spacetime manifolds and three-dimensional spacelike hypersurfaces. These functionals are used for elaborating relativistic thermodynamic models for Lagrange–Hamilton geometric evolution and respective generalized Hamilton geometric flow and nonholonomic Ricci flow equations. The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannon entropy and the quantum von Neumann entropy. There are considered geometric flow generalizations of the approaches based on classical and quantum relative entropy, conditional entropy, mutual information, and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow information theory are elaborated using the formalism of density matrices and measurements with quantum channels for the evolution of quantum mechanical systems.

Canonical deformation of N=2 AdS4 supergravity

It is known that one can define a consistent theory of extended, $N=2$ anti-de Sitter (AdS) Supergravity (SUGRA) in $D=4$. Besides the standard gravitational part, this theory involves a single $U(1)$ gauge field and a pair of Majorana vector-spinors that can be mixed into a pair of charged spin-$3/2$ gravitini. The action for $N=2$ $AdS_{4}$ SUGRA is invariant under $SO(1,3)\times U(1)$ gauge transformations, and under local SUSY. We present a geometric action that involves two "inhomogeneous" parts: an orthosymplectic $OSp(4\vert 2)$ gauge-invariant action of the Yang-Mills type, and a supplementary action invariant under purely bosonic $SO(2,3)\times U(1)\sim Sp(4)\times SO(2)$ sector of $OSp(4\vert 2)$, that needs to be added for consistency. This action reduces to $N=2$ $AdS_{4}$ SUGRA after gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical deformation is performed by using the Seiberg-Witten approach to noncommutative (NC) gauge field theory with the Moyal product. The NC-deformed action is expanded in powers of the deformation parameter $\theta^{\mu\nu}$ up to the first order. We show that $N=2$ $AdS_{4}$ SUGRA has non-vanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action. For comparison, simple $N=1$ Poinacar\'{e} SUGRA can be obtained in the same manner, directly from an $OSp(4\vert 1)$ gauge-invariant action. The first non-vanishing NC correction is quadratic in $\theta^{\mu\nu}$ and therefore exceedingly difficult to calculate. Under Wigner-In\"{o}n\"{u} (WI) contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincar\'{e} superalgebra, and it is not clear whether this relation holds after canonical deformation. We present the linear NC correction to $N=2$ $AdS_{4}$ SUGRA explicitly, discuss its low-energy limit, and what remains of it after WI contraction.

Canonical and Lie-algebraic twist deformations of Carroll, para-Galilei and Static Hopf algebras

In this paper, we provide five canonical and three Lie-algebraic twist-deformations of Carroll, para-Galilei and Static Hopf algebras. Particularly, in the case of Carroll Hopf structure, we get the same result with the use of contraction schemes of corresponding canonically and Lie-algebraically twisted Poincaré Hopf algebras as well.

Yang–Mills theory in the SO(2,3)⋆ model of noncommutative gravity

According to the standard cosmological model, thermodynamic conditions of the early Universe were such that nuclear matter existed in the state of quark–gluon plasma, rather than hadrons. On the other hand, it is generally believed that quantum gravity effects become ever more stronger as we approach the Big Bang, in particular, we expect that the phenomenon of space–time noncommutativity will be significant. Thus we are led to consider the properties of quarks and gluons in noncommutative space–time. For this, we employ the [Formula: see text] model of noncommutative gravity. As a first step towards the full theoretical treatment of the effects of noncommutativity on quark–gluon plasma, our main goal in this paper is to consistently incorporate Yang–Mills gauge fields in the [Formula: see text] framework and investigate their coupling to gravity that arises due to space–time noncommutativity. We construct an action that is invariant under deformed [Formula: see text] gauge transformations and expand it perturbatively in orders of the canonical deformation parameter [Formula: see text] via Seiberg–Witten map. In particular, we analyze the flat-space–time limit and demonstrate that residual noncommutativity induces various new couplings of quarks and gluons.

Identification of deformation pattern changes caused by enhanced oil recovery (EOR) using InSAR

ABSTRACT Continuous hydrocarbon production and steam/water injection cause compaction and expansion of the reservoir rock, leading to irregular downward and upward ground movements. Detecting such anthropogenic ground movements is of importance, as they may significantly influence the safety and sustainability of hydrocarbon production activities, in particular, enhanced oil recovery (EOR) and even lead to local hazards, e.g. earthquakes and sinkholes. As InSAR (Interferometric Synthetic Aperture Radar) can routinely deliver global ground deformation observations on a weekly basis, with millimetre-level precision, it can be a cost-effective, and less labour intensive tool to monitor surface deformation changes due to hydrocarbon production activities. Aimed at identifying the associated deformation pattern changes, this study focuses on InSAR deformation model optimization, in order to automatically detect irregularities, both spatially and temporally. We apply multiple hypothesis testing to determine the best model based on a library of physically realistic canonical deformation models. We develop a cluster-wise constrained least-squares estimation method for parameter estimation, in order to directly introduce contextual information, such as spatio-temporal correlation, into the mathematical model. Here a cluster represents a group of spatially correlated InSAR measurement points. Our approach is demonstrated over an enhanced oil recovery site using a stack of TerraSAR-X images.

Inflationary universe in deformed phase space scenario

We consider a noncommutative (NC) inflationary model with a homogeneous scalar field minimally coupled to gravity. The particular NC inflationary setting herein proposed, produces entirely new consequences as summarized in what follows. We first analyze the free field case and subsequently examine the situation where the scalar field is subjected to a polynomial and exponential potentials. We propose to use a canonical deformation between momenta, in a spatially flat Friedmann–Lemaî tre–Robertson–Walker (FLRW) universe, and while the Friedmann equation (Hamiltonian constraint) remains unaffected the Friedmann acceleration equation (and thus the Klein–Gordon equation) is modified by an extra term linear in the NC parameter. This concrete noncommutativity on the momenta allows interesting dynamics that other NC models seem not to allow. Let us be more precise. This extra term behaves as the sole explicit pressure that under the right circumstances implies a period of accelerated expansion of the universe. We find that in the absence of the scalar field potential, and in contrast with the commutative case, in which the scale factor always decelerates, we obtain an inflationary phase for small negative values of the NC parameter. Subsequently, the period of accelerated expansion is smoothly replaced by an appropriate deceleration phase providing an interesting model regarding the graceful exit problem in inflationary models. This last property is present either in the free field case or under the influence of the scalar field potentials considered here. Moreover, in the case of the free scalar field, we show that not only the horizon problem is solved but also there is some resemblance between the evolution equation of the scale factor associated to our model and that for the R2 (Starobinsky) inflationary model. Therefore, our herein NC model not only can be taken as an appropriate scenario to get a successful kinetic inflation, but also is a convenient setting to obtain inflationary universe possessing the graceful exit when scalar field potentials are present.

Affine cubic surfaces and character varieties of knots

It is known that the fundamental group homomorphism π1(T2)→π1(S3∖K) induced by the inclusion of the boundary torus into the complement of a knot K in S3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL2-character varieties of the corresponding fundamental groups. In our earlier work [3], we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C3. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings.

Anti De Sitter Deformation of Quaternionic Analysis and the Second-Order Pole

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy’s formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum polarization diagrams in four-dimensional quantum field theory. In order to add some flexibility, especially when dealing with Cauchy’s formula for the second order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the five-dimensional Euclidean space. We show that many results of quaternionic analysis – including the Cauchy-Fueter formula – admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogues of Cauchy’s formula for the second order pole.

W-algebras and chiral differential operators at the critical level

Let Acrit be the chiral algebra corresponding to the affine Kac–Moody algebra at the critical level gˆcrit. Let Zcrit be the center of Acrit. The commutative chiral algebra Zcrit admits a canonical deformation into a non-commutative chiral algebra Wℏ. In this paper we will express the resulting first order deformation via the chiral algebra Dcrit of chiral differential operators on G((t)) at the critical level.

Classical Oscillator Model on Canonical, Lie-Algebraic Deformation and Heisenberg-Weyl Algebra Deformation Nonrelativistic Space

The oscillator model on nonrelativistic Canonical (soft), Lie-Algebraic deformation noncommutative space and deformed Heisenberg-Weyl Algebra noncomutative phase space are analyzed. For canonical deformation the additional dynamical effects are absent. For two kinds of Lie-Algebraic deformation space the additional effects are generated. Remarkably angular frequency ω ρ is stepfunction, non-periodic function, and contains imaginary frequency. The corresponding particle trajectory varies greatly along with the deformation and time parameter $\hat{k}$ , t. For deformed Heisenberg-Weyl Algebra noncommutative phase space the additional corrections are generated. K ρ ,ω ρ and particle trajectory have constant second-order correction with deformation parameters Θ and Π, but the particle still keep trajectory of classical oscillator.

Twisted acceleration-enlarged Newton-Hooke space-times and conservative force terms

There are analyzed two classical systems defined on twist-deformed acceleration-enlarged Newton-Hooke space-times - nonrelativistic particle moving in constant field force $\vec{F}$ and harmonic oscillator model. It is demonstrated that only in the case of canonical twist deformation the force terms generated by space-time noncommutativity remain conservative for both models.

Vowel and consonant contributions to vocal tract shape

The purpose of this study was to develop a method by which a vowel-consonant-vowel (VCV) utterance based on x-ray microbeam articulatory data could be separated into a vowel-to-vowel transition and a consonant superposition function. The result is a model that represents a vowel sequence as a time-dependent perturbation of the neutral vocal tract shape governed by coefficients of canonical deformation patterns. Consonants were modeled as superposition functions that can force specific portions of the vocal tract shape to be constricted or expanded, over a specific time course. The three VCVs [pa], [ta], and [ka], produced by one female speaker, were analyzed and reconstructed with the developed model. They were shown to be reasonable approximations of the original VCVs, as assessed qualitatively by visual inspection and quantitatively by calculating rms error and correlation coefficients. This establishes a method for future modeling of other speech material.

Quantization of a symplectic manifold associated to a manifold with projective structure

Let X be a complex manifold equipped with a projective structure P. There is a holomorphic principal C∗-bundle LP′ over X associated with P. We show that the holomorphic cotangent bundle of the total space of LP′ equipped with the Liouville symplectic form has a canonical deformation quantization. This generalizes the construction in the work of and Ben-Zvi and Biswas [“A quantization on Riemann surfaces with projective structure,” Lett. Math. Phys. 54, 73 (2000)] done under the assumption that dimC X=1.

CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF κ-POINCARÉ AND CONTRACTIONS TO κ-GALILEI ALGEBRAS

We propose canonical and Lie-algebraic twist deformations of κ-deformed Poincaré Hopf algebra which leads to the generalized κ-Minkowski space–time relations. The corresponding deformed κ-Poincaré quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.

Radiative processes as a condensation phenomenon and the physical meaning of deformed canonical structures

We study the radiative corrections of QED 3 from the dual point of view and show that this process is the exact dual to the Julia–Toulouse mechanism introduced by Quevedo and Trugenberger [F. Quevedo, C.A. Trugenberger, Nucl. Phys. B 501 (1997) 143] some years ago. We discuss the physics behind this mechanism that involves condensation of topological defects. It is shown that the dual Stuckelberg mechanism is responsible for the “rank-jump” phenomenon that transforms the scalar field (dual to Maxwell in this dimensionality) into the vectorial self-dual field. This phenomenon is studied using the ideas of noncommutative fields theory that examines possible deformations of the canonical structure of some well-known models in ( 2 + 1 ) D . A deformation is constructed linking the massless scalar field theory with the self-dual theory. This is the exact dual of the known deformation connecting the Maxwell theory with the Maxwell–Chern–Simons theory. Duality, radiative corrections, the Julia–Toulouse mechanism and canonical deformations are then used to establish a web of relations between the mentioned theories and to propose a physical picture of the deformation procedure adopted.

Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space

The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces.