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18 pages, 340 KiB

Open AccessArticle

On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

by Alexander A. Balinsky, Victor A. Bovdi and Anatolij K. Prykarpatski

Symmetry 2024, 16(1), 54; https://doi.org/10.3390/sym16010054 - 30 Dec 2023

Viewed by 1373

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix [...] Read more.

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows. Full article

14 pages, 344 KiB

Open AccessArticle

Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry

by Koichi Yamawaki

Symmetry 2023, 15(12), 2209; https://doi.org/10.3390/sym15122209 - 18 Dec 2023

Cited by 3 | Viewed by 1200

Abstract

The rho meson has long been successfully identified with a dynamical gauge boson of Hidden Local Symmetry (HLS)

H_{local}

in the non-linear sigma model

G / H

gauge equivalent to the model having the symmetry

G_{global} \times H_{local}

, with [...] Read more.

The rho meson has long been successfully identified with a dynamical gauge boson of Hidden Local Symmetry (HLS)

H_{local}

in the non-linear sigma model

G / H

gauge equivalent to the model having the symmetry

G_{global} \times H_{local}

, with

G = [S U {(2)}_{L} \times S U {(2)}_{R}] ≃ O (4), H = S U {(2)}_{V} ≃ O (3)

. However, under a hitherto unproven assumption that its kinetic term is dynamically generated, together with an ad hoc choice of the auxiliary field parameter “

a = 2

”, we prove this assumption, thereby solving the long-standing mystery. The rho meson kinetic term is generated simply by the large N limit of the Grassmannian model

G / H = O (N) / [O (N - 3) \times O (3)]

gauge equivalent to

O {(N)}_{global} \times {[O (N - 3) \times O (3)]}_{local}

, extrapolated to

N = 4

,

O {(4)}_{global} \times O {(3)}_{local}

, with all the phenomenologically successful “

a = 2

results”, i.e.,

ρ

-universality, KSRF relation, and the Vector Meson Dominance, realized independently of the parameter “a”. This in turn establishes validity of the large N dynamics at the quantitative level directly by the experiments. The relevant cutoff reads

Λ ≃ 4 π F_{π}

for

N = 4

, which is regarded as a matching scale of the HLS as a “magnetic dual” to QCD. Skyrmion is stabilized by such a dynamically generated rho meson without recourse to the underlying QCD, a further signal of the duality. The unbroken phase with a massless rho meson may be realized as a novel chiral-restored hadronic phase in the hot/dense QCD. Full article

(This article belongs to the Special Issue Symmetries and Ultra Dense Matter of Compact Stars II: Emergence of Symmetries in Strong Nuclear Correlations)

14 pages, 1775 KiB

Open AccessArticle

Action Recognition via Adaptive Semi-Supervised Feature Analysis

by Zengmin Xu, Xiangli Li, Jiaofen Li, Huafeng Chen and Ruimin Hu

Appl. Sci. 2023, 13(13), 7684; https://doi.org/10.3390/app13137684 - 29 Jun 2023

Viewed by 1268

Abstract

This study presents a new semi-supervised action recognition method via adaptive feature analysis. We assume that action videos can be regarded as data points in embedding manifold subspace, and their matching problem can be quantified through a specific Grassmannian kernel function while integrating [...] Read more.

This study presents a new semi-supervised action recognition method via adaptive feature analysis. We assume that action videos can be regarded as data points in embedding manifold subspace, and their matching problem can be quantified through a specific Grassmannian kernel function while integrating feature correlation exploration and data similarity measurement into a joint framework. By maximizing the intra-class compactness based on labeled data, our algorithm can learn multiple features and leverage unlabeled data to enhance recognition. We introduce the Grassmannian kernels and the Projected Barzilai–Borwein (PBB) method to train a subspace projection matrix as a classifier. Experiment results show our method has outperformed the compared approaches when a few labeled training samples are available. Full article

(This article belongs to the Special Issue Recent Advances in Image Processing)

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12 pages, 747 KiB

Open AccessArticle

Extrinsic Bayesian Optimization on Manifolds

by Yihao Fang, Mu Niu, Pokman Cheung and Lizhen Lin

Algorithms 2023, 16(2), 117; https://doi.org/10.3390/a16020117 - 15 Feb 2023

Cited by 1 | Viewed by 2559

Abstract

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function [...] Read more.

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices. Full article

(This article belongs to the Special Issue Gradient Methods for Optimization)

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13 pages, 3764 KiB

Open AccessArticle

Aerodynamic Shape Optimization with Grassmannian Shape Parameterization Method

by Yang Zhang, Bo Pang, Xiankai Li and Gang Chen

Energies 2022, 15(20), 7722; https://doi.org/10.3390/en15207722 - 19 Oct 2022

Cited by 7 | Viewed by 2237

Abstract

The conventional method of optimizing the aerodynamic performance of an airfoil heavily depends on the confines of the design space. The design variables create a non-normalized space that is fragmented into several different clusters of airfoils. An approach that is data-driven and deforms [...] Read more.

The conventional method of optimizing the aerodynamic performance of an airfoil heavily depends on the confines of the design space. The design variables create a non-normalized space that is fragmented into several different clusters of airfoils. An approach that is data-driven and deforms airfoils over a Grassmannian submanifold is utilized in the work that is being presented here. The affine deformation, which includes camber and thickness, can be uncoupled from the method that is currently in use, and the operations that are performed on the airfoil shape can be made smooth enough to prevent unreasonable shapes from being produced. The CST method is also a part of the current study so that a comparison can be made between the two. A new method to describe the airfoil geometries over the Grassmannian space was generated using a dataset that contained 7007 different shapes of airfoils. These two methods are used to parameterize the subsonic (NACA0012) and transonic (RAE2822) airfoils, and the new method cuts the number of design variables from twelve to six, resulting in a reduction in overall complexity. The findings demonstrate that the new method maintains a high degree of consistency regardless of the flow conditions. Full article

(This article belongs to the Special Issue Advanced Numerical Simulation Methods of Aerodynamics and Heat Transfer)

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21 pages, 20107 KiB

Open AccessArticle

Knowledge Distillation of Grassmann Manifold Network for Remote Sensing Scene Classification

by Ling Tian, Zhichao Wang, Bokun He, Chu He, Dingwen Wang and Deshi Li

Remote Sens. 2021, 13(22), 4537; https://doi.org/10.3390/rs13224537 - 11 Nov 2021

Cited by 6 | Viewed by 2870

Abstract

Due to device limitations, small networks are necessary for some real-world scenarios, such as satellites and micro-robots. Therefore, the development of a network with both good performance and small size is an important area of research. Deep networks can learn well from large [...] Read more.

Due to device limitations, small networks are necessary for some real-world scenarios, such as satellites and micro-robots. Therefore, the development of a network with both good performance and small size is an important area of research. Deep networks can learn well from large amounts of data, while manifold networks have outstanding feature representation at small sizes. In this paper, we propose an approach that exploits the advantages of deep networks and shallow Grassmannian manifold networks. Inspired by knowledge distillation, we use the information learned from convolutional neural networks to guide the training of the manifold networks. Our approach leads to a reduction in model size, which addresses the problem of deploying deep learning on resource-limited embedded devices. Finally, a series of experiments were conducted on four remote sensing scene classification datasets. The method in this paper improved the classification accuracy by 2.31% and 1.73% on the UC Merced Land Use and SIRIWHU datasets, respectively, and the experimental results demonstrate the effectiveness of our approach. Full article

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13 pages, 956 KiB

Open AccessArticle

Quantum Computation and Measurements from an Exotic Space-Time R⁴

by Michel Planat, Raymond Aschheim, Marcelo M. Amaral and Klee Irwin

Symmetry 2020, 12(5), 736; https://doi.org/10.3390/sym12050736 - 5 May 2020

Cited by 7 | Viewed by 3774

Abstract

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state

|ψ⟩

in d [...] Read more.

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state

|ψ⟩

in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group

π_{1} (V)

of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time)

R^{4}

(that is homeomorphic and isometric, but not diffeomorphic to the Euclidean

R^{4}

). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture

G Q (2, 2)

(at index 15), and the combinatorial Grassmannian Gr

(2, 8)

(at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time)

R^{4}

s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’. Full article

(This article belongs to the Special Issue Symmetry in Quantum Systems)

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25 pages, 7034 KiB

Open AccessArticle

An Evolve-Then-Correct Reduced Order Model for Hidden Fluid Dynamics

by Suraj Pawar, Shady E. Ahmed, Omer San and Adil Rasheed

Mathematics 2020, 8(4), 570; https://doi.org/10.3390/math8040570 - 11 Apr 2020

Cited by 19 | Viewed by 4295

Abstract

In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and unknown components. In the known part, we first utilize [...] Read more.

In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and unknown components. In the known part, we first utilize an intrusive Galerkin method projected on a set of basis functions obtained by proper orthogonal decomposition. We then present two variants of correction formula based on the assumption that the observed data are a manifestation of all relevant processes. The first method uses a standard least-squares regression with a quadratic approximation and requires solving a rank-deficient linear system, while the second approach employs a recurrent neural network emulator to account for the correction term. We further enhance our approach by using an orthonormality conforming basis interpolation approach on a Grassmannian manifold to address off-design conditions. The proposed framework is illustrated here with the application of two-dimensional co-rotating vortex simulations under modeling uncertainty. The results demonstrate highly accurate predictions underlining the effectiveness of the evolve-then-correct approach toward real-time simulations, where the full process model is not known a priori. Full article

(This article belongs to the Special Issue Machine Learning in Fluid Dynamics: Theory and Applications)

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19 pages, 322 KiB

Open AccessArticle

New Characterizations of the Clifford Torus and the Great Sphere

by Sun Mi Jung, Young Ho Kim and Jinhua Qian

Symmetry 2019, 11(9), 1076; https://doi.org/10.3390/sym11091076 - 27 Aug 2019

Cited by 3 | Viewed by 2966

Abstract

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we [...] Read more.

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

32 pages, 461 KiB

Open AccessArticle

The Role of Spin(9) in Octonionic Geometry

by Maurizio Parton and Paolo Piccinni

Axioms 2018, 7(4), 72; https://doi.org/10.3390/axioms7040072 - 12 Oct 2018

Cited by 4 | Viewed by 4124

Abstract

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical [...] Read more.

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical 8-form and its analogies with quaternionic geometry as well as the role of

Spin (9)

both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel

Spin (9)

manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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