Equivariant Research Articles

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

Harmonic maps between pseudo-Riemannian surfaces

We study locally harmonic maps between a Riemann surface or a Lorentz surface M and a Riemann or Lorentz surface N. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.

Pseudorandomness of the Schrödinger Map Equation

Pseudorandomness of the Schrödinger Map Equation

Some formulas for Bessel functions related to diag(1, −1, −1)-matrices and an intertwining operator

In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group.

Canonical surgeries in rotationally invariant Ricci flow

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of Kleiner and Lott [Acta Math. 219 (2017), pp. 65–134]. Results of Bamler-Kleiner [Acta Math. 228 (2022), pp. 1–215] and Haslhofer [Proc. Amer. Math. Soc. 150 (2022), pp. 5433–5437] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

Commuting mode choice and work from home in the later stages of COVID-19: Consolidating a future focussed prediction tool to inform transport and land use planning

As COVID-19 slowly dissipated after 30 months since March 2020, working from home (WFH) continues to be popular, with support from notable numbers of employees and employers. There is growing evidence that we have either reached, or appear to be close to, a level of WFH − to some extent − that might be described as an equilibrium outcome going forward. Hence, there is a need to update any travel behaviour response models that have been developed during the passage of the pandemic. This paper sets out a commuter model for each day of the week for the Greater Sydney Metropolitan Area as of September 2022; where the alternative responses are to not work, WFH, or commute by one of ten modes available. Given the probability of WFH obtained from a mixed logit model, we construct a mapping equation to establish the sources of statistical variance by origin and destination locations that influence the incidence of WFH. The evidence is used as the basis of commenting on what this means for longer-term structural changes in network performance, the likely move to a greater focus on living locally, and the need (or otherwise) for employers to retain office space. A significant return to the office has recently been promoted by some employers, and we comment on this as to what it might mean for the future of a hybrid working location model.

Trends of burning mouth syndrome: a bibliometric study.

This study utilizes bibliometric analysis to map the current research landscape and forecast emerging trends within the domain of Burning Mouth Syndrome (BMS). A comprehensive review of literature related to BMS was conducted, drawing from the Web of Science Core Collection (WoSCC) from 2008 to 2023. The analysis included both publication types "Article" and "Review Article." Advanced quantitative techniques and visual analytics tools, including CiteSpace, VOSviewer, Tableau, and the Map Equation online platform were utilized to analyze the academic publications within this domain. Our analysis incorporated 497 articles on BMS. The data exhibit a progressive increase in the annual volume of publications from 2008 to 2023. In terms of geographic and institutional contributions, the United States of America (with 80 publications) and Nihon University (with 26 publications) emerged as leading entities in BMS research, while the Netherlands and England were identified as central to international collaboration efforts. Prominent researchers in this field include Adamo Daniela (18 publications) and Sun Andy (16 publications). Furthermore, the most cited works were authored by Jääskeläinen SK. An examination of the journals in which these articles were published showed a dominance of dental journals, highlighting significant interest and research efforts in BMS within the dental research community. The steady growth in BMS research signifies the formation of a robust core of researchers and demonstrates the maturation of the field. Despite this progress, the findings highlight a notable deficiency in cross-institutional and cross-regional collaborative efforts. Keyword cluster analysis has revealed "management" as a persistently relevant theme, with "pain modulation" emerging as the current focal interest. Additionally, "blood profile," "pernicious anemia," and "folate" have been identified as prospective areas of growing interest, suggesting important directions for future investigations. This bibliometric analysis reveals the research landscape of BMS, aiming to highlight potential collaborative opportunities and define future research directions. These insights are invaluable for guiding subsequent investigations and carving new paths in the exploration of BMS.

Effect of temperature and strain rate on the hot workability behaviour of Ni–25Cr–14W superalloy: An approach using processing map and constitutive equation

Effect of temperature and strain rate on the hot workability behaviour of Ni–25Cr–14W superalloy: An approach using processing map and constitutive equation

Equivariant embedding of finite-dimensional dynamical systems

AbstractWe prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

Steady solutions for the Schrödinger map equation

In this paper we use bifurcation methods to construct a new family of solutions of the binormal flow, also known as the vortex filament equation, which do not change their form. Our examples are complementary to those obtain by S. Kida in 1981, and therefore they are also related, thanks to the so-called Hasimoto transformation, to traveling wave solutions of the 1d cubic non-linear Schrödinger equation.

A classification of automorphic Lie algebras on complex tori

Abstract We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$ . For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager’s algebra.

Designing robust trajectories by lobe dynamics in low-dimensional Hamiltonian systems

Modern space missions with uncrewed spacecraft require robust trajectory design to connect multiple chaotic orbits by small controls. To address this issue, we propose a control scheme to design robust trajectories by leveraging a geometrical structure in chaotic zones, known as a . Our scheme shows that appropriately selected lobes reveal possible paths to traverse chaotic zones in a short time. The effectiveness of our method is demonstrated through trajectory design in both the standard map and Hill's equation. Published by the American Physical Society 2024

Reversible image secret sharing based on quantum logistic mapping and Chinese remainder theorem

A reversible image secret sharing algorithm is proposed based on quantum logistic mapping and Chinese remainder theorem. In this algorithm, the hash value of the original image is generated using the SHA-3 algorithm, and then encrypted using the RSA algorithm to obtain the encrypted hash value and the public key, which are the initial values in the quantum logistic mapping equations. Next, the quantum logistic mapping algorithm will be utilized to generate the chaotic sequence. After scrambling the original image matrix with the chaotic sequence, the pre-encryption part will be completed. Finally, the cover images are binarized and then the pre-encrypted secret image is embedded into them, adding the Chinese remainder theorem. According to the simulation results, the present algorithm improves the key sensitivity of Chinese remainder theorem-based image secret sharing and has high robustness to salt and pepper noise and cut-off attack.

Unifying O(3) equivariant neural networks design with tensor-network formalism

Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU(2)-symmetric quantum many-body problems, to design new spatial equivariant components for neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term ‘fusion blocks,’ serve as universal approximators of any continuous equivariant function defined on the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.

Multivortex traveling waves for the Schrödinger map equation

We construct traveling wave solutions for the Schrödinger map equation in R2. These solutions have n(n + 1)/2 pairs of degree ±1 vortices. The locations of those vortices are symmetric in the plane and determined by the roots of a special class of Adler–Moser polynomials. With a few modifications, a similar construction allows for the creation of traveling wave solutions of the Schrödinger map equation in R3. These solutions have the shape of 2n + 1 vortex rings, whose locations are given by a sequence of polynomials with rational coefficients and are far away from each other.

Improved phase-to-height mapping method combine with device attitude

Phase-to-height mapping is one of the important processes in three dimensional phase measurement profilometry. But, in traditional phase-to-height mapping method, the measurement accuracy is affected by device attitude, so it needs saving a large amount of mapping equations to achieve high-quality phase-to-height mapping. In order to improve that, this paper proposes an improved phase-to-height mapping method combine with device attitude. Firstly, we get the unwrapped phase of the target. Then, using generalized regression neural network is used to reduce the offset of phase information at the same height due to the randomness of device attitude. Last, the phase-to-height mapping is completed by substituting the unwrapped phase (the difference between having detected object and no detected object) of eliminate the offset into improved phase-to-height mapping method. Experimental results show that the proposed method could achieve high-quality phase-to-height mapping with less mapping equation and less memory space. Compared with the nonlinear phase-to-height mapping method (probabilistic neural network to eliminate phase offset), its accuracy is improved by 44.30%. Compared with the nonlinear phase-to-height mapping method (radial basis function neural network to eliminate phase offset), the accuracy is improved by 39.58%.

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

On Gibbs measures and topological solitons of exterior equivariant wave maps

We consider k -equivariant wave maps from the exterior spatial domain \mathbb{R}^{3} \setminus B(0,1) into the target \mathbb{S}^{3} . This model has infinitely many topological solitons Q_{n,k} , which are indexed by their topological degree n\in \mathbb{Z} . For each n\in \mathbb{Z} and k\geq 1 , we prove the existence and invariance of a Gibbs measure supported on the homotopy class of Q_{n,k} . As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.

On uniqueness for half-wave maps in dimension 𝑑≥3

Extending an argument by Shatah and Struwe [Int. Math. Res. Not. 11 (2002), pp. 555–571] we obtain uniqueness for solutions of the half-wave map equation in dimension d ≥ 3 d \geq 3 in the natural energy class.