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Estimating the 6D pose and size of objects is crucial in the task of visual grasping for robotic arms. Most current algorithms still require the 3D CAD model of the target object to match with the detected points, and they are unable to predict the object's size, which significantly limits the generalizability of these methods. In this paper, we introduce category priors and extract high-dimensional abstract features from both the observed point cloud and the prior to predict the deformation matrix of the reconstructed point cloud and the dense correspondence between the reconstructed and observed point clouds. Furthermore, we propose a staged geometric correction and dense correspondence refinement mechanism to enhance the accuracy of regression. In addition, a novel lightweight attention module is introduced to further integrate the extracted features and identify potential correlations between the observed point cloud and the category prior. Ultimately, the object's translation, rotation, and size are obtained by mapping the reconstructed point cloud to a normalized canonical coordinate system. Through extensive experiments, we demonstrate that our algorithm outperforms existing methods in terms of performance and accuracy on commonly used benchmarks for this type of problem. Additionally, we implement the algorithm in robotic arm-grasping simulations, further validating its effectiveness. [ABSTRACT FROM AUTHOR]

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators. [ABSTRACT FROM AUTHOR]

Statistician C. R. Rao made many contributions to multivariate analysis over the span of his career. Some of his earliest contributions continue to be used and built upon almost 80 years later, while his more recent contributions spur new avenues of research. The present article discusses these contributions, how they helped shape multivariate analysis as we see it today, and what we may learn from reviewing his works. Topics include his extension of linear discriminant analysis, Rao's perimeter test, Rao's U statistic, his asymptotic expansion of Wilks' Λ statistic, canonical factor analysis, functional principal component analysis, redundancy analysis, canonical coordinates, and correspondence analysis. The examination of his works shows that interdisciplinary collaboration and the utilization of real datasets were crucial in almost all of Rao's impactful contributions. [ABSTRACT FROM AUTHOR]

In this paper, we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space ℝ13 using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in ℝ24 and the pairs of null curves in ℝ13. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in ℝ24 and the pairs of minimal Lorentz surfaces in ℝ13. As an application, we give examples showing how we can obtain a minimal Lorentz surface in ℝ24 using a pair of minimal Lorentz surfaces in ℝ13. [ABSTRACT FROM AUTHOR]

We propose and theoretically substantiate an algorithm for conducting a generalized Chaos game with an arbitrary jump on finite convex polygons of the extended hyperbolic plane whose components in the Cayley–Klein projective model are the Lobachevsky plane and its ideal domain. In particular, the defining identities for a point dividing an elliptic, hyperbolic, or parabolic segment in a given ratio are proved, and formulas for calculating the coordinates of such a point at a canonical frame of the first type are obtained. The results of a generalized Chaos game conducted using the advanced software package pyv are presented. [ABSTRACT FROM AUTHOR]

Curvature theory, a fundamental subject in kinematics, is typically addressed through the concepts of instantaneous invariants and canonical coordinates, which are pivotal for the generation of mechanical paths. This paper delves into this subject with a higher-order analysis of screws, employing both canonical and natural coordinates. Through this exploration, a new Euler–Savary equation is derived, one that does not rely on canonical coordinates. Additionally, the paper provides a comprehensive classification of the degenerate conditions of the cubic of stationary curves of four-bar linkages at rotational positions. A thorough examination of four-bar linkages in translational positions—the couple links translate instantaneously—is also presented, with analyses extending up to the sixth order. The findings reveal that the Burmester's points at translational positions can be extended to Burmester's points with excess one, provided that all pivot points are symmetrically distributed about the pole norm with the two cranks in corotating senses. However, the extension to Burmester's points with excess two is not possible. Similarly, the Ball's point with excess one does not progress to Ball's point with excess two. The paper also highlights that the traditional method, which is based on canonical coordinates, is ineffective when the four-bar linkage forms a parallelogram. Fortunately, this scenario can be effectively analyzed using the screw-based approach. Ultimately, the results presented can also serve as analytical solutions for 3-RR platforms with higher-order shakiness. [ABSTRACT FROM AUTHOR]

KAM theory owes most of its success to its initial motivation: the application to problems of celestial mechanics. The masterly application was offered by Arnold in the 60s who worked out a theorem, that he named the "Fundamental Theorem" (FT), especially designed for the planetary problem. However, FT could be really used at that purpose only when, about 50 years later, a set of coordinates constructively taking the invariance by rotation and close-to-integrability into account was used. Since then, some progress has been done in the symplectic assessment of the problem, and here we review such results. [ABSTRACT FROM AUTHOR]

We study the Poisson geometrical formulation of quantum mechanics for finite dimensional mixed and pure states. Equivalently, we show that quantum mechanics can be understood in the language of classical mechanics. We review the symplectic structure of the Hilbert space and identify its canonical coordinates. We extend the geometric picture to the space of density matrices D N + . We find it is not symplectic but admits a linear s u (N) Poisson structure. We identify Casimir surfaces of D N + and show that the space of pure states P N ≡ C P N − 1 is one of its symplectic submanifolds which is an intersection of primitive Casimirs. We identify generic symplectic submanifolds of D N + and calculate their dimensions. We find that D N + is singularly foliated by the symplectic leaves of varying dimensions, also known as coadjoint orbits. We also find an ascending chain of Poisson submanifolds D N M ⊂ D N M + 1 for 1 ≤ M ≤ N − 1. Each such Poisson submanifold D N M is obtained by tracing out the C M states from the bipartite system C N × C M and is an intersection of N − M primitive Casimirs of D N + . Their Poisson structure is induced from the symplectic structure of the bipartite system. We also show their foliations. Finally, we study the positive semi-definite geometry of the symplectic submanifold E N M consisting of the mixed states with maximum entropy in D N M . [ABSTRACT FROM AUTHOR]

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp. [ABSTRACT FROM AUTHOR]

The investigation of expansive homeomorphisms in metric spaces began with Utz in 1950. Thereafter, several authors have extensively studied this concept for different motivations. In this current article, we study expansive homeomorphism in the context of quasi-pseudometric spaces. This is motivated by the fact that any expansive homeomorphism on quasi-pseudometric space is again expansive homeomorphism on its induced pseudometric space but the converse is not true in general. Moreover, the study of orbit structures has been taken to consideration in this article. For instance, we investigate the denseness of orbits in the context of quasi-metric spaces.

The authors show that any semicanonical or noncanonical third-order linear delay dynamic equation on a time scale can be written in canonical form without imposing any additional conditions on the coefficient functions. Since this is true for any time scale, this means it holds for differential and difference equations. The implication of this is the significant result that any set of conditions which show that a related equation in canonical form is oscillatory will guarantee that the semicanonical or noncanonical equation is also oscillatory. Several examples of the application of the results are incorporated into the paper. [ABSTRACT FROM AUTHOR]

The investigation of expansive homeomorphisms in metric spaces began with Utz in 1950. Thereafter, several authors have extensively studied this concept for different motivations. In this current article, we study expansive homeomorphism in the context of quasi-pseudometric spaces. This is motivated by the fact that any expansive homeomorphism on quasi-pseudometric space is again expansive homeomorphism on its induced pseudometric space but the converse is not true in general. Moreover, the study of orbit structures has been taken to consideration in this article. For instance, we investigate the denseness of orbits in the context of quasi-metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of classification of quasi‐homogeneous formal power series providing solutions of the oriented associativity equations. Such a classification is performed by introducing a system of monodromy local moduli on the space of formal germs of homogeneous semisimple flat F$F$‐manifolds. This system of local moduli is well defined on the complement of the strictly doubly resonant locus, namely, a locus of formal germs of flat F$F$‐manifolds manifesting both coalescences of canonical coordinates at the origin, and resonances of their conformal dimensions. It is shown how the solutions of the oriented associativity equations can be reconstructed from the knowledge of the monodromy local moduli via a Riemann–Hilbert–Birkhoff boundary value problem. Furthermore, standing on results of B. Malgrange and C. Sabbah, it is proved that any formal homogeneous semisimple flat F$F$‐manifold, which is not strictly doubly resonant, is actually convergent. Our semisimplicity criterion for convergence is also reformulated in terms of solutions of Losev–Manin commutativity equations, growth estimates of correlators of F$F$‐cohomological field theories, and solutions of open Witten–Dijkgraaf–Verlinde–Verlinde equations. [ABSTRACT FROM AUTHOR]

The mammalian intestine is one of the most rapidly self-renewing tissues, driven by stem cells residing at the crypt bottom. Paneth cells form a major element of the niche microenvironment providing various growth factors to orchestrate intestinal stem cell homeostasis, such as Wnt3. Different Wnt ligands can selectively activate β-catenin-dependent (canonical) or -independent (noncanonical) signaling. Here, we report that the Dishevelled-associated activator of morphogenesis 1 (Daam1) and its paralogue Daam2 asymmetrically regulate canonical and noncanonical Wnt (Wnt/PCP) signaling. Daam1/2 interacts with the Wnt inhibitor RNF43, and Daam1/2 double knockout stimulates canonical Wnt signaling by preventing RNF43-dependent degradation of the Wnt receptor, Frizzled (Fzd). Single-cell RNA sequencing analysis revealed that Paneth cell differentiation is impaired by Daam1/2 depletion because of defective Wnt/PCP signaling. Together, we identified Daam1/2 as an unexpected hub molecule coordinating both canonical and noncanonical Wnt, which is fundamental for specifying an adequate number of Paneth cells. [ABSTRACT FROM AUTHOR]

Stellarator power plants require a plan for the removal of the particles and the heat that are exhausted across the plasma edge. Unless a flowing liquid metal can be used to carry the helium exhaust to places where it can be removed from the plasma chamber, the particle exhaust must be magnetically diverted into pumping chambers. Studies are required to determine how magnetic features relate to the required divertor properties, how these magnetic features can be produced, and how they can be controlled. General studies are clarified and simplified by the use of the magnetic field line Hamiltonian ψ p (ψ , θ , φ) and a vector x → (ψ , θ , φ) that gives the point in space associated with each point in the (ψ , θ , φ) canonical coordinates, a flux and two angles. The non-resonant Fourier terms in ψp can be removed by a canonical transformation, so only resonant Fourier terms can determine the field line properties in the plasma edge and divertor. This paper discusses the important divertor properties and explains how ψ p (ψ , θ , φ) and x → (ψ , θ , φ) can be obtained numerically in a special form for any stellarator magnetic field, B → (x →). This form holds between an arbitrary magnetic surface and the chamber walls with the non-resonant terms eliminated. Studies based on variations in the terms in such derived field-line Hamiltonians can determine what magnetic features are mathematically possible and how they could be produced and controlled by the external magnetic field coils. [ABSTRACT FROM AUTHOR]

Otoliths grow throughout a fish's life and are important for identifying fish stocks and fish age. This study aims to differentiate different stocks of Schizothorax grahami (Regan, 1904) in the Chishui River, an upper reach of the Yangtze River, by otolith morphology. The otolith morphology of S. grahami from three different river sections was analyzed using the Shape Index, Fourier coefficients, and wavelet coefficients. The composite discrimination success rate of the Shape Index was 59.7%, and it was difficult to distinguish in the scatter plots. In contrast, canonical principal coordinate scatter plots clearly showed three distinguished stocks. The above results indicate that otolith morphology can discriminate between stocks in plateau endemic fish, and several S. grahami stocks may be separately managed in the Chishui River. [ABSTRACT FROM AUTHOR]

Despite the rapid development of crowd-sourced photogrammetric reconstruction, a huge database of 3D models is still given in arbitrary scale, position, and orientation. To alleviate the issue, model alignment methods have been proposed to identify good viewpoints for 3D models. This work introduces a symmetry-aware alignment method for photogrammetric 3D models. Our key idea is to associate the best orientation with global symmetry and place objects in a way they commonly appear in our surroundings. Compared to the previous alignment methods that only infer upright orientation, we also determine frontal orientation. The proposed method works in four steps. First, we compute symmetry features using point global symmetry descriptors (PGSD). Second, a set of reflective symmetry planes are generated by feature aggregation. Third, we extract global symmetry and the principal axis based on a subset of symmetry planes with smaller detection errors. Finally, upright and frontal orientations are determined by the principal axis and information entropy successively. We evaluated the proposed method on synthetic and real photogrammetric datasets. Experiments show that the proposed PGSD has achieved higher efficiency and accuracy than curvature, FPFH, and deep learning-based frameworks. Our method outperforms the existing symmetry detection and upright orientation methods, where the average recall of main symmetry plane is 71.13%, and 78.50% of the models are well-aligned. [Display omitted] • We associate the human-accustomed inherent poses with global symmetry. • We present a symmetry-aware alignment method for photogrammetric 3D models. • We propose PGSD for symmetry matching and EM for symmetry plane evaluation. • Our method achieves higher accuracy in symmetry matching and detection. • Our method enables the alignment of inherent poses with the canonical coordinate. [ABSTRACT FROM AUTHOR]

Category-level object pose estimation aims to predict the 6D object pose and size of arbitrary objects from known categories. It remains a challenge due to the large intra-class shape variation. Recently, the introduction of the shape prior adaptation mechanism into the normalized canonical coordinates (i.e., NOCS) reconstruction process has been shown to be effective in mitigating the intra-class shape variation. However, existing shape prior adaptation methods simply map the observed point cloud to the normalized object space, and the extracted object descriptors are not sufficient for the perception of the object pose. As a result, they fail to predict the pose of objects with complex geometric structures (e.g., cameras). To this end, this paper proposes a novel shape prior adaption method named MSSPA-GC for category-level object pose estimation. Specifically, our main network takes the observed instance point cloud converted from the RGB-D image and the prior shape point cloud pre-trained on the object CAD models as inputs. Then, a novel 3D graph convolution network and a PointNet-like MLP network are designed to extract pose-aware object features and shape-aware object features from these two inputs, respectively. After that, the two-stream object features are aggregated through a multi-scale feature propagation mechanism to generate comprehensive 3D object descriptors that maintain both pose-sensitive geometric stability and intra-class shape consistency. Finally, by leveraging object descriptors aware of both object pose and shape when reconstructing the NOCS coordinates, our approach elegantly achieves state-of-the-art performance on the widely used REAL275 and CAMERA25 datasets using only 25% of the parameters compared with existing shape prior adaptation models. Moreover, our method also exhibits decent generalization ability on the unconstrained REDWOOD75 dataset. [Display omitted] • A novel shape prior adaptation method for category-level object pose estimation. • A 3D-GCN to learn multi-scale pose-sensitive features from observed point clouds. • A MLP network to learn multi-scale shape-aware features from prior point clouds. • A feature propagation mechanism to fuse pose-aware features and shape-aware features. • State-of-the-art performance on the NOCS datasets with only 5.9M parameters. • Good generalization ability on the REDWOOD75 dataset without orientation constraints. [ABSTRACT FROM AUTHOR]

The invariance method, known as Lie analysis, consists of finding a group of transformations that leave a difference equation invariant. This powerful tool permits one to lower the order, linearize and more importantly, obtain analytical solutions of difference and differential equations. In this study, we obtain the solutions and periodic solutions for some family of difference equations. We achieve this by performing an invariance analysis of this family. Eventually, symmetries are derived and used to construct canonical coordinates required for the derivation of the solutions. Moreover, periodic aspects of these solutions and the stability character of the equilibrium points are investigated. [ABSTRACT FROM AUTHOR]

We revisit the Kerr black hole as cast in the Boyer–Lindquist, Kerr–Schild and Weyl canonical coordinates, and calculate its total mass/energy and total angular momentum using linearized gravity along with its background Killing isometries. We argue that the integration of the relevant gravitational flux does not depend on the geometry of the closed and simply connected spatial boundary provided it is also piecewise smooth. [ABSTRACT FROM AUTHOR]

In practical applications, we are more likely to face semi-supervised data with a small amount of independent class label or constraint information and many unlabeled instances. For semi-supervised clustering, taking advantage of the small portion of preliminary label information can significantly improve the discriminability of representations. Spectral clustering has the benefits of handling any shape data distribution and converging to the optimal global solution but is susceptible to noisy data. However, it is inevitable to contain noise for real-world applications that significantly reduce clustering performance. Motivated by this, we propose a novel Robust Semi-supervised Spectral Clustering method (named RSSC) to address clustering on noise semi-supervised datasets. Specifically, in terms of data transductive warping, we map the entire semi-supervised dataset into a new data space where labeled data is close to the canonical coordinate system, and unlabeled data with similar characteristics should be close to those labeled data. The noise data is close to the origin of the coordinate and form the noise cluster because there is no guidance. Finally, samples in the same cluster are close, and different clusters are separated. Extensive experimental results on sixteen real-world datasets demonstrate that RSSC outperforms other state-of-the-art clustering methods on performance and robustness. [ABSTRACT FROM AUTHOR]

The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics. [ABSTRACT FROM AUTHOR]

In this paper, the categorization of nonlocal symmetries of the (1+1)-dimensional nonlinear Vakhnenko equation is presented. The governing equation is turned into a set of invertibly equivalent partial differential equations (PDE) via the invertible transformation of the canonical coordinates. The nonlocal symmetries of the aforementioned PDE were derived by establishing an inverse potential system and a locally related subsystem of the system of invertibly equivalent PDEs. In addition, using similarity variables and group invariant solutions connected to nonlocal symmetry, the exact solution of the nonlinear Vakhnenkov equation is achieved. Finally, the conservation laws for the nonlinear Vakhnenko equation were developed using the multiplier method. [ABSTRACT FROM AUTHOR]

A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum o s p ( m ′ / n ′ ) ⊕ s n e ( n ′ + 1) , of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra n o ( m ′ + 1) elements times n e ( n ′ + 1) elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra n ( m ′ / n ′ ) and belong to the semidirect product Osp(m′/n′) ⋉ Ne(n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation. [ABSTRACT FROM AUTHOR]

We consider a problem of separation of variables for Lax-integrable Hamiltonian systems governed by gl(n) ⨂ gl(n)-valued classical r-matrices r(u, v). We find a new class of classical non-skew-symmetric non-dynamical gl(n) ⨂ gl(n)-valued r-matrices rJ(u, v) for which the "magic recipe" of Sklyanin [Prog. Theor. Phys. Suppl. 118, 35 (1995)] in the theory of variable separation is applicable, i.e., for which standard separating functions A(u) and B(u) of Gekhtman [Commun. Math. Phys. 167, 593 (1995)] and Scott ["Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain," arXiv:hep-th 940303] produce a complete set of canonical coordinates satisfying the equations of separation. We illustrate the corresponding separation of variable theory by the example of the anisotropically deformed An Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices rJ(u, v) and by the generalized Gaudin models [T. Skrypnyk, Phys. Lett. A 334(5–6), 390 (2005)] governed by the same classical r-matrices. The n = 2 and n = 3 cases are considered in detail. [ABSTRACT FROM AUTHOR]

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations. [ABSTRACT FROM AUTHOR]

The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of small quantum cohomology of complex Grassmannians are studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, and the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function. [ABSTRACT FROM AUTHOR]

A formal series transformation to Birkhoff–Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincaré–Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a non-autonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An example is given for an axis with constant torsion near a 1:3 resonance. [ABSTRACT FROM AUTHOR]

Neural Radiance Fields (NeRF) are able to reconstruct scenes with unprecedented fidelity, and various recent works have extended NeRF to handle dynamic scenes. A common approach to reconstruct such non-rigid scenes is through the use of a learned deformation field mapping from coordinates in each input image into a canonical template coordinate space. However, these deformation-based approaches struggle to model changes in topology, as topological changes require a discontinuity in the deformation field, but these deformation fields are necessarily continuous. We address this limitation by lifting NeRFs into a higher dimensional space, and by representing the 5D radiance field corresponding to each individual input image as a slice through this "hyper-space". Our method is inspired by level set methods, which model the evolution of surfaces as slices through a higher dimensional surface. We evaluate our method on two tasks: (i) interpolating smoothly between "moments", i.e., configurations of the scene, seen in the input images while maintaining visual plausibility, and (ii) novel-view synthesis at fixed moments. We show that our method, which we dub HyperNeRF, outperforms existing methods on both tasks. Compared to Nerfies, HyperNeRF reduces average error rates by 4.1% for interpolation and 8.6% for novel-view synthesis, as measured by LPIPS. Additional videos, results, and visualizations are available at hypernerf.github.io. [ABSTRACT FROM AUTHOR]

We use Dirac’s method for the quantization of constrained systems in order to quantize a spatially flat Friedmannâ€"LemaĂ®treâ€"Robertsonâ€"Walker spacetime in the context of f (Q) cosmology. When the coincident gauge is considered, the resulting minisuperspace system possesses second class constraints. This distinguishes the quantization process from the typical Wheelerâ€"DeWitt quantization, which is applied for cosmological models where only first class constraints are present (e.g. for models in general relativity or in f (R) gravity). We introduce the Dirac brackets, find appropriate canonical coordinates and then apply the canonical quantization procedure. We perform this method both in vacuum and in the presence of matter: a minimally coupled scalar field and a perfect fluid with a linear equation of state. We demonstrate that the matter content changes significantly the quantization procedure, with the perfect fluid even requiring to put in use the theory of fractional quantum mechanics in which the power of the momentum in the Hamiltonian is associated with the fractal dimension of a LĂ©vy flight. The results of this analysis can be applied in f (T) teleparallel cosmology, since f (Q) and f (T) theories have the same degrees of freedom and same dynamical constraints in cosmological studies. [ABSTRACT FROM AUTHOR]

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard's approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms. [ABSTRACT FROM AUTHOR]

Observed clusters should be modelled by considering the distribution function (DF) to be a random variable that quantifies the degree of excitation of the system's normal modes. A system of canonical coordinates for the space of DFs are identified so DFs can be weighted in a consistent way. [ABSTRACT FROM AUTHOR]

The excessive deformation of deep-sea sediments caused by the vibration of the mining machine will adversely affect the efficiency and safety of mining. Combined with the deep-sea environment, the coupled thermo-hydro-mechanical problem for saturated porous deep-sea sediments subject to the vibration of the mining vehicle is investigated. Based on the Green-Lindsay (G-L) generalized thermoelastic theory and Darcy's law, the model of thermo-hydro-mechanical dynamic responses for saturated porous deep-sea sediments under the vibration of the mining vehicle is established. We obtain the analytical solutions of non-dimensional vertical displacement, excess pore water pressure, vertical stress, temperature, and change in the volume fraction field with the normal mode analysis method, and depict them graphically. The normal mode analysis method uses the canonical coordinate transformation to solve the equation, which can quickly decouple the equation by ignoring the modal coupling effect on the basis of the canonical mode. The results indicate that the vibration frequency has obvious influence on the vertical displacement, excess pore water pressure, vertical stress, and change in volume fraction field. The loading amplitude has a great effect on the physical quantities in the foundation, and the changes of the physical quantities increase with the increase in loading amplitude. [ABSTRACT FROM AUTHOR]

We present an automated method to track and identify neurons in C. elegans, called 'fast Deep Neural Correspondence' or fDNC, based on the transformer network architecture. The model is trained once on empirically derived semi-synthetic data and then predicts neural correspondence across held-out real animals. The same pre-trained model both tracks neurons across time and identifies corresponding neurons across individuals. Performance is evaluated against hand-annotated datasets, including NeuroPAL (Yemini et al., 2021). Using only position information, the method achieves 79.1% accuracy at tracking neurons within an individual and 64.1% accuracy at identifying neurons across individuals. Accuracy at identifying neurons across individuals is even higher (78.2%) when the model is applied to a dataset published by another group (Chaudhary et al., 2021). Accuracy reaches 74.7% on our dataset when using color information from NeuroPAL. Unlike previous methods, fDNC does not require straightening or transforming the animal into a canonical coordinate system. The method is fast and predicts correspondence in 10 ms making it suitable for future real-time applications. [ABSTRACT FROM AUTHOR]

This study proposes a novel model selection criterion for dimensionality estimation in canonical correlation analysis (CCA), which can be used to estimate the number of correlated components between two sets of multivariate vectors, particularly in the context of canonical coordinates. The proposed method has a different form compared to existing model selection criteria, which typically use the Bartlett-Lawley measure as a goodness-of-fit term. Alternatively, we propose to use instead a goodness-of-fit term based on the sum of squares of the smallest canonical correlation coefficients. The asymptotic properties of the proposed goodness-of-fit term have been laid out and an appropriate penalty term with proof of consistency is derived based on these properties. Numerical examples are presented in the context of direction-of-arrival estimation using canonical coordinates for two channel array systems. It is shown that our proposed order selection criterion outperforms conventional criteria and is robust to various noise and interference scenarios.

As one of the main components of the East Asian winter monsoon, the winter Siberian High (SH) plays an important role in the variability of East Asian climate. However, the Climate Forecast System, version 2 (CFSv2), shows limited prediction skill for the winter SH. To improve the prediction skill, a hybrid ensemble canonical correlation (ECC) prediction model is established for the winter SH and SH intensity index (SHI) basing the year‐to‐year increment method and an efficient downscaling approach. Hence, three preceding predictors from observation/reanalysis [sea‐ice concentration (SIC), snow‐cover extent (SCE), and sea surface temperature (SST)] and one integrated current predictor from CFSv2 [surface air temperature and sea level pressure (SAT&SLP)] are selected based on their fundamental physical roles. Considering these individual predictors, four separate downscaling schemes are constructed. The regional‐mean temporal anomaly correlation coefficient (ACC) of the winter SH for each scheme is 0.54 (SIC‐scheme‐SH), 0.50 (SCE‐scheme‐SH), 0.72 (SST‐scheme‐SH) and 0.42 (SAT&SLP‐scheme‐SH) [four schemes exceed significant at the 1% level]. However, the skill of each scheme differs in regional and temporal distribution. Thus, a hybrid ECC prediction model is proposed by employing multiple linear regression. The regional‐mean temporal ACC of the winter SH between the observed and predicted results increases from −0.12 (CFSv2; not significant at the 10% level) to 0.85 (significant at the 1% level). Besides, the correlation coefficient between the observation and hybrid ECC scheme for winter SHI is 0.90 (significant at the 1% level). Furthermore, the strongest winter SH in 2012 is reproduced well by ECC‐scheme‐SH. Key Points: A hybrid ensemble canonical correlation prediction model is developed for the winter Siberian High with high prediction skillAn efficient downscaling method and the year‐to‐year increment approach are applied to the prediction modelThree preceding predictors with fundamental mechanisms and a simultaneous integrated predictor with high prediction skill are considered [ABSTRACT FROM AUTHOR]

Conventional neural networks are universal function approximators, but they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here, we prepend a conventional neural network to a Hamiltonian neural network and show that the combination accurately forecasts Hamiltonian dynamics from generalised noncanonical coordinates. Examples include a predator–prey competition model where the canonical coordinates are nonlinear functions of the predator and prey populations, an elastic pendulum characterised by nontrivial coupling of radial and angular motion, a double pendulum each of whose canonical momenta are intricate nonlinear combinations of angular positions and velocities, and real-world video of a compound pendulum clock. [ABSTRACT FROM AUTHOR]

This is a first introduction to unprojection methods, and more specifically to Tom and Jerry unprojections. These two harmless tricks deserve to be better known, since they answer many practical questions about constructing codimension 4 Gorenstein subschemes. In particular, we discuss here the two smoothing components of the anticanonical cone over P(1, 2, 3). [ABSTRACT FROM AUTHOR]

We propose a local conformal autoencoder (LOCA) for standardized data coordinates. LOCA is a deep learning-based method for obtaining standardized data coordinates from scientific measurements. Data observations are modeled as samples from an unknown, nonlinear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized, latent variables. We assume a repeated measurement sampling strategy, common in scientific measurements, and present a method for learning an embedding in Rd that is isometric to the latent variables of the manifold. The coordinates recovered by our method are invariant to diffeomorphisms of the manifold, making it possible to match between different instrumental observations of the same phenomenon. Our embedding is obtained using LOCA, which is an algorithm that learns to rectify deformations by using a local z-scoring procedure, while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA in various model settings and observe that it exhibits promising interpolation and extrapolation capabilities, superior to the current state of the art. Finally, we demonstrate LOCA's efficacy in single-site Wi-Fi localization data and for the reconstruction of three-dimensional curved surfaces from two-dimensional projections. [ABSTRACT FROM AUTHOR]

The parasitic worm, Schistosoma mansoni, expresses unusual fucosylated glycans in a stage‐dependent manner that can be recognized by the human innate immune receptor DC‐SIGN, thereby shaping host immune responses. We have developed a synthetic approach for mono‐ and bis‐fucosylated LacdiNAc (LDN‐F and LDN‐DF, respectively), which are epitopes expressed on glycolipids and glycoproteins of S. mansoni. It is based on the use of monosaccharide building blocks having carefully selected amino‐protecting groups, facilitating high yielding and stereoselective glycosylations. The molecular interaction between the synthetic glycans and DC‐SIGN was studied by NMR and molecular modeling, which demonstrated that the α1,3‐fucoside of LDN‐F can coordinate with the Ca2+‐ion of the canonical binding site of DC‐SIGN allowing for additional interactions with the underlying LDN backbone. The 1,2‐fucoside of LDN‐DF can be complexed in a similar manner, however, in this binding mode GlcNAc and GalNAc of the LDN backbone are placed away from the protein surface resulting in a substantially lower binding affinity. Glycan microarray binding studies showed that the avidity and selectivity of binding is greatly enhanced when the glycans are presented multivalently, and in this format Lex and LDN‐F gave strong responsiveness, whereas no binding was detected for LDN‐DF. The data indicates that S. mansoni has developed a strategy to avoid detection by DC‐SIGN in a stage‐dependent manner by the addition of a fucoside to a number of its ligands. [ABSTRACT FROM AUTHOR]

We discuss dynamical aspects of an analysis of the two–centre problem started in [15]. The perturbative nature of our approach allows us to foresee applications to the three–body problem. [ABSTRACT FROM AUTHOR]

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Featured Application: Inventory Management. The ABC classification problem is approached as a ranking problem by the most current classification models; that is, a group of inventory items is expressed according to its overall weighted score of criteria in descending order. In this paper, we present an extended version of the Hadi-Vencheh model for multiple-criteria ABC inventory classification. The proposed model is one based on the nonlinear weighted product method (WPM), which determines a common set of weights for all items. Our proposed nonlinear WPM incorporates multiple criteria with different measured units without converting the performance of each inventory item, in terms of converting each criterion into a normalized attribute value, thereby providing an improvement over the model proposed by Hadi-Vencheh. Our study mainly includes various criteria for ABC classification and demonstrates an efficient algorithm for solving nonlinear programming problems, in which the feasible solution set does not have to be convex. The algorithm presented in this study substantially improves the solution efficiency of the canonical coordinates method (CCM) algorithm when applied to large-scale, nonlinear programming problems. The modified algorithm was tested to compare our proposed model results to the results derived using the Hadi-Vencheh model and demonstrate the algorithm's efficacy. The practical objectives of the study were to develop an efficient nonlinear optimization solver by optimizing the quality of existing solutions, thus improving time and space efficiency. [ABSTRACT FROM AUTHOR]

A wide class of coupled-cluster methods is introduced, based on Arponen's extended coupled-cluster theory. This class of methods is formulated in terms of a coordinate transformation of the cluster operators. The mathematical framework for the error analysis of coupled-cluster methods based on Arponen's bivariational principle is presented, in which the concept of local strong monotonicity of the flipped gradient of the energy is central. A general mathematical result is presented, describing sufficient conditions for coordinate transformations to preserve the local strong monotonicity. The result is applied to the presented class of methods, which include the standard and quadratic coupled-cluster methods, and also Arponen's canonical version of extended coupled-cluster theory. Some numerical experiments are presented, and the use of canonical coordinates for diagnostics is discussed. [ABSTRACT FROM AUTHOR]

The normal modes of a trapped ion crystal are derived using an approach based on the Hermitian properties of the system's dynamical matrix. This method is equivalent to the standard Bogoliubov method, but for classical systems, it is arguably simpler and more general in that canonical coordinates are not necessary. The theory is developed for stable, unstable, and neutrally stable systems. The method is then applied to ion crystals in a Penning trap. Reduced eigenvalue problems for the case of large applied magnetic fields are developed, for which the spectrum breaks into E × B drift modes, axial modes, and cyclotron modes. Thermal fluctuation levels in these modes are analyzed and shown to be consistent with the Bohr–van-Leeuwen theorem, provided that neutrally stable modes associated with crystal rotations are included in the analysis. An expression for the rotational inertia of the crystal is derived, and a magnetic contribution to this inertia, which dominates in large magnetic fields, is described. An unusual limit is discovered for the special case of spherically symmetric confinement, in which the rotational inertia does not exist and changes in angular momentum leave the rotation frequency unaffected. [ABSTRACT FROM AUTHOR]

We describe classes of local coordinates on the Carnot–Carathéodory spaces of lower smoothness which permit the homogeneous approximation of quasimetrics and basis vector fields. We establish the minimal smoothness that is required for these classes to coincide with the class of the already-described privileged coordinates in the infinite smoothness case. Moreover, we apply these results to prove the analogs of the available theorems in the case of the canonical coordinates of the second kind. Also, we prove some convergence theorems in quasimetric spaces. [ABSTRACT FROM AUTHOR]

Let X = XΣ be a toric surface and let (X, W) be its Landau-Ginzburg (LG) mirror where W is the Hori-Vafa potential as shown in their preprint. We apply asymptotic analysis to study the extended deformation theory of the LG model (X, W), and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W. For X = P2, our construction reproduces Gross' perturbed potential Wn [Adv. Math. 224 (2010), pp. 169-245] which was proven to be the universal unfolding of W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of Wn across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross in the same work (in the case of X = P2). [ABSTRACT FROM AUTHOR]

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete. [ABSTRACT FROM AUTHOR]