Search

We show there are no extremal metrics for the eigenvalues of the Neumann Laplacian on any compact manifold. Nonetheless, we construct examples of conformally extremal metrics for the eigenvalues of this operator in any annulus and characterise these special metrics in the general case of a compact manifold of dimension n ≥ 2 . As for the Dirichlet Laplacian, we prove non existence of extremal metrics on any compact surface. [ABSTRACT FROM AUTHOR]

This research investigates the dynamic behavior of functionally graded graphene platelet honeycomb sandwich (FG-GPLHS) plates coupled with stiffeners under moving loads, a critical area lacking in current research. Utilizing sandwich equivalence theory, a model of the FG-GPLHS plate is constructed, and the coupling of stiffeners is achieved through displacement continuity conditions combined with displacement coordinate transformation. Artificial virtual spring methods are employed to simulate boundary conditions and establish the force matrix for moving loads. The spectral geometry method (SGM)-Chebyshev method is employed to solve for the model, yielding insights into its dynamic behavior. This research takes into account a number of factors such as honeycomb parameters, boundary conditions, loading velocities, stiffeners cross-section properties, graphene platelet (GPL) distribution pattern and mass fraction to evaluate their effects on the vibration damping effectiveness of the structure. In particular, the effect of honeycomb dimensions and GPL parameters on the vibration resistance and stability of the model under moving loads, which can realize a wider range of application values for this structure in engineering. [ABSTRACT FROM AUTHOR]

Objectives: Understanding human tooth structure fluorescence aids clinical and forensic dentistry, enabling tissue/material differentiation and the creation of esthetic restorative materials. Material manufacturers seek to replicate natural tooth fluorescence, necessitating the development of novel techniques to detect them. Procuring human teeth for research is challenging due to ethical and infection control standards, prompting a search for alternative models. Material and Methods: This study compares visible light‐induced fluorescence of porcine and human teeth to assess the value of porcine teeth as human analogs. Using a pulsed laser, an optimal fluorescence‐inducing wavelength was determined, followed by comparing fluorescence spectra between species. Results: Luminescence sensitivity and lifetimes were comparable between species, but spectral geometry differed. Conclusion: Porcine teeth, commonly used for dental material investigations, may not be suitable for dental fluorescence studies due to spectral differences. Accurately mimicking human tooth fluorescence remains complex. Further research is needed to develop reliable alternatives for dental fluorescence investigations that will advance clinical and forensic dentistry. [ABSTRACT FROM AUTHOR]

We introduce a novel data-driven approach aimed at designing high-quality shape deformations based on a coarse localized input signal. Unlike previous data-driven methods that require a global shape encoding, we observe that detail-preserving deformations can be estimated reliably without any global context in certain scenarios. Building on this intuition, we leverage Jacobians defined in a one-ring neighborhood as a coarse representation of the deformation. Using this as the input to our neural network, we apply a series of MLPs combined with feature smoothing to learn the Jacobian corresponding to the detail-preserving deformation, from which the embedding is recovered by the standard Poisson solve. Crucially, by removing the dependence on a global encoding, every point becomes a training example, making the supervision particularly lightweight. Moreover, when trained on a class of shapes, our approach demonstrates remarkable generalization across different object categories. Equipped with this novel network, we explore three main tasks: refining an approximate shape correspondence, unsupervised deformation and mapping, and shape editing. Our code is made available at https://github.com/sentient07/LJN. [ABSTRACT FROM AUTHOR]

The study of protein molecular surfaces enables to better understand and predict protein interactions. Different methods have been developed in computer vision to compare surfaces that can be applied to protein molecular surfaces. The present work proposes a method using the Wave Kernel Signature: Protein LOcal Surficial Similarity Screening (PLO3S). The descriptor of the PLO3S method is a local surface shape descriptor projected on a unit sphere mapped onto a 2D plane and called Surface Wave Interpolated Maps (SWIM). PLO3S allows to rapidly compare protein surface shapes through local comparisons to filter large protein surfaces datasets in protein structures virtual screening protocols.

Fermion functional integrals are calculated for the Dirac operator of a finite real spectral triple. Complex, real and chiral functional integrals are considered for each KO-dimension where they are non-trivial, and phase ambiguities in the definition are noted. [ABSTRACT FROM AUTHOR]

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry. [ABSTRACT FROM AUTHOR]

This paper presents a semi-analytical approach to analyze the dynamic response characteristics of lattice composite sandwich plates with varying lattice truss cores. Under the premise of satisfying the classical laminate theory assumptions of Allen, the lattice truss core is considered an equivalent homogenized structure. Then, the displacement equations for the sandwich plate utilize the first-order shear deformation theory (FSDT). The energy equation for the sandwich plate is constructed using the Lagrange energy method and solved through the Rayleigh–Ritz method. To mimic arbitrary boundary conditions, artificial boundary spring techniques are utilized. The spectral geometry method (SGM) is utilized to describe the displacement field of dynamic equations and boundary conditions. Legendre polynomials are chosen to construct admissible displacement functions for the structure, supplemented by auxiliary polynomials to eliminate discontinuities at the boundaries. The method’s validity is verified through comparisons with relevant literature and finite element method (FEM). Based on free vibration analysis, both steady-state and transient displacement responses of laminated composite sandwich plates are examined, considering the influences of structural parameters and boundary conditions on dynamic response characteristics. These works can provide suitable theoretical references for the engineering field. [ABSTRACT FROM AUTHOR]

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that S A C ⊂ S G . Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve P C ⊂ 0,1 2 differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension D = ln ⁡ 3 / ln ⁡ 2 , their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension d l S A C = d = 1 , whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, d l S G = D . Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted. [ABSTRACT FROM AUTHOR]

Our research explores the application of Riemannian geometry and spectral embedding in the context of electroencephalogram (EEG) signal analysis for cognitive state classification. Leveraging the PyRiemann library and the AlphaWaves dataset, our study employs covariance estimation and the minimum distance to mean (MDM) classifier within a machine learning pipeline. The classification accuracy is assessed through stratified k-fold cross-validation. Furthermore, we introduce a novel visualization approach by calculating the spectral embedding of covariance matrices, providing insights into the underlying structure of the EEG epochs. Our findings showcase the potential of Riemannian geometry and spectral embedding as powerful tools in the domain of EEG-based cognitive state classification, contributing to the broader field of brain signal analysis and paving the way for automated and advanced neurocognitive studies. [ABSTRACT FROM AUTHOR]

We present a review of recent results about a class of Schrödinger operators usually called soft quantum waveguides, with the focus on relations between their spectral properties and the geometry of the confinement. We also mention a number of open problems in this area. [ABSTRACT FROM AUTHOR]

This study conducts an in-depth analysis of clustering small molecules using spectral geometry and deep learning techniques. We applied a spectral geometric approach to convert molecular structures into triangulated meshes and used the Laplace–Beltrami operator to derive significant geometric features. By examining the eigenvectors of these operators, we captured the intrinsic geometric properties of the molecules, aiding their classification and clustering. The research utilized four deep learning methods: Deep Belief Network, Convolutional Autoencoder, Variational Autoencoder, and Adversarial Autoencoder, each paired with k-means clustering at different cluster sizes. Clustering quality was evaluated using the Calinski–Harabasz and Davies–Bouldin indices, Silhouette Score, and standard deviation. Nonparametric tests were used to assess the impact of topological descriptors on clustering outcomes. Our results show that the DBN + k-means combination is the most effective, particularly at lower cluster counts, demonstrating significant sensitivity to structural variations. This study highlights the potential of integrating spectral geometry with deep learning for precise and efficient molecular clustering. [ABSTRACT FROM AUTHOR]

This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group of unitary operators and the sphere where these operators act. [ABSTRACT FROM AUTHOR]

While biological intelligence grows organically as new knowledge is gathered throughout life, Artificial Neural Networks forget catastrophically whenever they face a changing training data distribution. Rehearsal-based Continual Learning (CL) approaches have been established as a versatile and reliable solution to overcome this limitation; however, sudden input disruptions and memory constraints are known to alter the consistency of their predictions. We study this phenomenon by investigating the geometric characteristics of the learner's latent space and find that replayed data points of different classes increasingly mix up, interfering with classification. Hence, we propose a geometric regularizer that enforces weak requirements on the Laplacian spectrum of the latent space, promoting a partitioning behavior. Our proposal, called Continual Spectral Regularizer for Incremental Learning (CaSpeR-IL), can be easily combined with any rehearsal-based CL approach and improves the performance of SOTA methods on standard benchmarks. • We study the geometry of a model's latent space in a Continual Learning setting. • We propose Continual Spectral Regularizer, a geometrically motivated regularizer. • We combine CaSpeR with SOTA rehearsal-based CL approaches in standard benchmarks. • We compare our proposal with recent contrastive-based CL approaches. • We reveal that CaSpeR achieves increased accuracy and reduced forgetting. [ABSTRACT FROM AUTHOR]

A solid object in R 3 can be represented by its smooth boundary surface which can be equipped with an intrinsic metric to form a 2-Riemannian manifold. In this paper, we analyze such surfaces using multiple metrics that give birth to multi-spectra by which a given surface can be characterized. Their relative sensitivity to different types of local structures allows each metric to provide a distinct perspective of the shape. Extensive experiments show that the proposed multi-metric approach significantly improves important tasks in geometry processing such as shape retrieval and find similarity and corresponding parts of non-rigid objects. [ABSTRACT FROM AUTHOR]

This paper aims to establish a comprehensive model for evaluating the response characteristics of functionally graded graphene platelet-reinforced composite (FG-GPLRC) rectangular plates under stationary random acceleration excitation. Utilizing the first-order shear deformation theory (FSDT) and Halpin–Tsai assumptions, a dynamic model for the rectangular plate is derived, followed by obtaining the variational solution using the Rayleigh–Ritz method. To describe displacement components associated with dynamic equations and boundary conditions, a spectral geometry method (SGM) is employed. Additionally, a pseudo-excitation method (PEM) is utilized for the analysis of stationary random vibration. The proposed calculation method is validated through convergence analysis and comparative evaluation with existing literature and finite element models. Furthermore, the study investigates the influence of various factors, such as boundary conditions, the number and size of functionally graded rectangular plate layers, GPL distribution types, and material parameters, on the stationary random response attributes of FG rectangular plates. This research contributes to a deeper understanding of the dynamic response of functionally graded materials in structural configurations and offers a valuable analytical approach for researchers in this field. [ABSTRACT FROM AUTHOR]

We investigate the fluctuation properties in the eigenfrequency spectra of flat cylindrical microwave cavities that are homogeneously filled with magnetized ferrite. These studies are motivated by experiments in which only small pieces of ferrite were embedded in the cavity and magnetized with an external static magnetic field to induce partial time-reversal ( T ) invariance violation. We use two different shapes of the cavity, one exhibiting an integrable wave dynamics, the other one a chaotic one. We demonstrate that in the frequency region where only transverse-magnetic modes exist, the magnetization of the ferrites has no effect on the wave dynamics and does not induce T -invariance violation whereas it is fully violated above the cutoff frequency of the first transverse-electric mode. Above all, independently of the shape of the resonator, it induces a chaotic wave dynamics in that frequency range in the sense that for both resonator geometries the spectral properties coincide with those of quantum systems with a chaotic classical dynamics and same invariance properties under application of the generalized T operator associated with the resonator geometry. [ABSTRACT FROM AUTHOR]

As key subjects in spectral geometry and combinatorial graph theory, respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of "Hodge Laplacians on graphs" in the literature. Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches. For example, the spectral decomposition of a vector field on a simple point cloud using combinatorial Laplacians defined on some commonly used simplicial complexes does not give rise to the same curl-free and divergence-free components that one would obtain from the spectral decomposition of a vector field using either the continuous Hodge Laplacians defined on differential forms in manifolds or the discretized Hodge Laplacians defined on a point cloud with boundary in the Eulerian representation or on a regular mesh in the Eulerian representation. To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce boundary-induced graph (BIG) Laplacians using tools from discrete exterior calculus (DEC). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences among the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as levelset functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes. [ABSTRACT FROM AUTHOR]

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry. The strong connection between the dynamics of a physical system and its Hamiltonian's spectrum has scarcely been applied in the non-Hermitian case. Here, the authors use a photonic quantum walk to confirm and expand previous theoretical analyses connecting self-acceleration dynamics with non-trivial point-gap topology. [ABSTRACT FROM AUTHOR]

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields, including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel inductive mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean. [ABSTRACT FROM AUTHOR]

A (convex) polytope P ⊂ R d and its edge-graph G P can have very distinct symmetry properties, in that the edge-graph can be much more symmetric than the polytope. In this article we ask whether this can be "rectified" by coloring the vertices and edges of G P , that is, whether we can find such a coloring so that the combinatorial symmetry group of the colored edge-graph is actually isomorphic (in a natural way) to the linear or orthogonal symmetry group of the polytope. As it turns out, such colorings exist and some of them can be constructed quite naturally. However, actually proving that they "capture polytopal symmetries" involves applying rather unexpected techniques from the intersection of convex geometry and spectral graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group H of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant η -invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant η -invariants and equivariant Maslov triple indices in the splitting of manifolds. [ABSTRACT FROM AUTHOR]

Given a closed Riemannian manifold M and b ≥ 2 closed connected submanifolds N j ⊂ M of codimension at least 2, we prove that the first nonzero eigenvalue of the domain Ω ε ⊂ M obtained by removing the tubular neighbourhood of size ε around each N j tends to infinity as ε tends to 0. More precisely, we prove a lower bound in terms of ε , b, the geometry of M and the codimensions and the volumes of the submanifolds and an upper bound in terms of ε and the codimensions of the submanifolds. For eigenvalues of index k = b , b + 1 , … , we have a stronger result: their order of divergence is ε - 1 and their rate of divergence is only depending on m and on the codimensions of the submanifolds. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the first eigenvalue of the fractional Dirichlet–Laplacian of order s$s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman, and Taylor, in the limit as s$s$ goes to 1. The limit as s$s$ goes to 1/2$1/2$ is carefully analyzed, as well. [ABSTRACT FROM AUTHOR]

In 1996 Erdös showed that among planar domains of fixed area, the smallest principal eigenvalue of the Dirichlet Laplacian with a constant magnetic field is uniquely achieved on the disk. We establish a quantitative version of this inequality, with an explicit remainder term depending on the field strength that measures how much the domain deviates from the disk. [ABSTRACT FROM AUTHOR]

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati–Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy. [ABSTRACT FROM AUTHOR]

We present a comprehensive study of new discoveries on the spectral patterns of elastic transmission eigenfunctions, including boundary localization, surface resonance, and stress concentration. In the case where the domain is radial and the underlying parameters are constant, we give rigorous justifications and derive a thorough understanding of those intriguing geometric and physical patterns. We also present numerical examples to verify that the same results hold in general geometric and parameter setups. [ABSTRACT FROM AUTHOR]

We are concerned with a coupled-physics spectral problem arising in the coupled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic transmission eigenvalue problem. There are two major contributions in this work which are new to the literature. First, under a mild condition on the medium parameters, we prove the existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric rigidity result of the transmission eigenfunctions by showing that they tend to localize on the boundary of the underlying domain. Moreover, we also consider interesting implications of the obtained results to the effective construction of metamaterials by using bubbly elastic structures and to the inverse problem associated with the fluid-structure interaction. [ABSTRACT FROM AUTHOR]

The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. Pólya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk. [ABSTRACT FROM AUTHOR]

A spectral geometry utility awareness, with specific reference to isospectralisation and art painting analytics, is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical quantitative disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such mathematized interdisciplinarities and synergies. It more specifically extends the matrix algebra based 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the general Graph Laplacian (i.e. Laplace-Beltrami) operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension. Another is a continuing demonstration that the understanding and analysis of art sculptures dovetails with mathematics-art studies. [ABSTRACT FROM AUTHOR]

By quantifying the distance between two collider events, one can triangulate a metric space and reframe collider data analysis as computational geometry. One popular geometric approach is to first represent events as an energy flow on an idealized celestial sphere and then define the metric in terms of optimal transport in two dimensions. In this paper, we advocate for representing events in terms of a spectral function that encodes pairwise particle angles and products of particle energies, which enables a metric distance defined in terms of one-dimensional optimal transport. This approach has the advantage of automatically incorporating obvious isometries of the data, like rotations about the colliding beam axis. It also facilitates first-principles calculations, since there are simple closed-form expressions for optimal transport in one dimension. Up to isometries and event sets of measure zero, the spectral representation is unique, so the metric on the space of spectral functions is a metric on the space of events. At lowest order in perturbation theory in electron-positron collisions, our metric is simply the summed squared invariant masses of the two event hemispheres. Going to higher orders, we present predictions for the distribution of metric distances between jets in fixed-order and resummed perturbation theory as well as in parton-shower generators. Finally, we speculate on whether the spectral approach could furnish a useful metric on the space of quantum field theories. [ABSTRACT FROM AUTHOR]

We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling. [ABSTRACT FROM AUTHOR]

This paper establishes several results for coalgebraic structure in 8-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of En-ring spectra: A B, the associated 8-category of descent data is equivalent to the 8-category of comodules over BA B, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the 8-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors. [ABSTRACT FROM AUTHOR]

A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior generative model of geometric disentanglement for 3D shapes, wherein the space of object geometry is factorized into rigid orientation, non-rigid pose, and intrinsic shape. The resulting model can be trained from raw 3D shapes, without correspondences, labels, or even rigid alignment, using a combination of classical spectral geometry and probabilistic disentanglement of a structured latent representation space. Our improvements include more sophisticated handling of rotational invariance and the use of a diffeomorphic flow network to bridge latent and spectral space. The geometric structuring of the latent space imparts an interpretable characterization of the deformation space of an object. Furthermore, it enables tasks like pose transfer and pose-aware retrieval without requiring supervision. We evaluate our model on its generative modelling, representation learning, and disentanglement performance, showing improved rotation invariance and intrinsic-extrinsic factorization quality over the prior model. [ABSTRACT FROM AUTHOR]

Spectral geometry is a rapidly developing field with new classes of operators, boundary value problems and geometric objects arising in different applications. At the same time, classical problems continue gaining novel flavors. The main focus of the workshop was on some of the most significant recent developments in geometric spectral theory including geometry of eigenvalues and eigenfunctions, singular spectral problems, and spectral optimization. The talks were complemented by three thematic open problem sessions on the main topics of the meeting. [ABSTRACT FROM AUTHOR]

We study the question of whether the Brauer group is isomorphic to the cohomological one in spectral algebraic geometry. For this, we prove the compact generation of the derived category of twisted sheaves in the setting of spectral algebraic stacks. In particular, we obtain the compact generation of the ∞$\infty$‐category of quasi‐coherent sheaves and the existence of compact perfect complexes with prescribed support for such stacks. We extend these results to derived algebraic geometry by studying the relationship between derived and spectral algebraic stacks. [ABSTRACT FROM AUTHOR]

We present results of magneto-optical investigations of (CoFeZr) x (Al2O 3 ) 1 0 0 − x film nanocomposites in transverse Kerr effect (TKE) geometry in the spectral range 0.5–4.0 eV and magnetic field up to 3.0 kOe. Nanocomposites were deposited onto a glass-ceramic substrate by ion-beam sputtering. The TKE response at room temperature strongly depends on the wavelength of light, applied magnetic field H and the volume metallic fraction. From the analysis of the field dependences of TKE at different wavelengths, it follows that in the as-deposited samples, the interaction between nanoparticles at x < 3 0 at.% is small and the nanocomposite is an ensemble of superparamagnetic particles; as x increases to 32 at.%, a superspinglass-type state arises, then, in the vicinity of 34 at.%, along with individual superparamagnetic particles, superferromagnetic regions appear. Long-range ferromagnetic order arises at concentrations x less than the percolation threshold for conductivity x per = 4 2. 6 at.%. In the presence of two different magnetic states in the samples, TKE is not proportional to the magnetization. Both the field dependences at near-infrared region and the spectral dependences of TKE change significantly after annealing of the samples, while the changes in the field dependences of the magnetization are almost imperceptibly. [ABSTRACT FROM AUTHOR]

[Display omitted] • Standard solar collectors' absorber geometry modification by inserting V-grooves. • Optimizing solar collectors based on the geometric modification was demonstrated. • Importance of the absorbing surface geometry combined with spectral selectivity. • Improvement of the efficiency of the collectors despite increasing the thermal losses. • Efficiency improvement up to 5.61%, considering a V-groove opening angle of 60°. Solar energy plays an essential role in the process of the energy transition to renewable sources, and so too do technological innovations for exploring these energy sources, and for improving the thermal efficiency of conventional flat solar collectors. This study set out to model, simulate, and investigate the effect of modifying the geometry of the absorber plate of standard solar collectors for heating liquids, by inserting V corrugations. This leads to making consequent modifications to geometrical and optical parameters that improve the efficiency of the equipment. This paper presents the application of a mathematical-graphical model that seeks an optimum incidence angle to corrugate an absorber plate. This model, based on available optical experimental data, was built to improve the efficiency of solar collectors. During its application, an investigation could be made of the effect of the corrugation on the radiation and on the convection coefficients, the correlations of which were introduced into the mathematical model, and some of them were neither mentioned nor used in previous publications. A case study was carried out to investigate the behavior of three different collectors and the improvement in the efficiency of these collectors. It was shown that by using V grooves the efficiency was improved for all three flat solar collectors studied. As seen, the geometric modification contributes to improving efficiency when associated with selective absorption layers of solar collectors. The combination of both strategies can enhance the thermal efficiency of solar collectors and this can be further explored by researchers and manufacturers. [ABSTRACT FROM AUTHOR]

We introduce a graph Γ which is roughly isometric to the hyperbolic plane, and we study the Steklov eigenvalues of a subgraph with boundary Ω of Γ . For (Ω l) l ≥ 1 a sequence of subgraphs of Γ such that | Ω l | ⟶ ∞ , we prove that for each k ∈ N , the k th eigenvalue tends to 0 proportionally to 1 / | B l | . The idea of the proof consists in finding a bounded domain N of the hyperbolic plane which is roughly isometric to Ω , giving an upper bound for the Steklov eigenvalues of N and transferring this bound to Ω via a process called discretization. [ABSTRACT FROM AUTHOR]

Ocean wave climate, including wind waves and swells, is essential to human marine activities and global or regional climate systems, and is highly related to harnessing wave energy resources. In this study, a global 3-hourly instantaneous wave dataset was established with the third-generation wave model MASNUM-WAM and wind forcings derived from the products of the First Institute of Oceanography-Earth System Model version 2.0, the climate model coupled with wave model, under the unified framework of the Coupled Model Intercomparison Project phase 6. This dataset contains 17 wave parameters, including the information associated with wave energy and spectral shape geometries, from one historical (1950–2014) simulation and three future (2015–2100) scenario experiments (ssp125, ssp245, and ssp585). Moreover, all the parameters can be accessed separately in the form of wind waves and swells. The historical results show that the simulated wave characteristics agree well with satellite observations and the ERA5 reanalysis products. This dataset can provide the community with a unique and informative data source for wave climate and wave energy resource research. [ABSTRACT FROM AUTHOR]

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form -Δ + V with suitable (electric) potential V, which is taken as a fixed, underlying function on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum Σ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any k ∈ ..., the infimal energy among all admissible k-partitions is bounded from above by Σ, and if it is strictly below Σ, then a spectral minimal k-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space. [ABSTRACT FROM AUTHOR]

Flavin is involved in a wide range of life processes. Flavin can exist in different forms due to its ability to transfer electron(s) in different physiological conditions. It is difficult to experimentally obtain the absorption and fluorescence spectra at the vibrational level. A systematically theoretical simulation is a necessary way to explore the fine structure of these spectra of all forms of flavin molecules. In this study, the vibrationally resolved absorption spectra and fluorescence spectra of five forms of flavins (fully oxidized form FLox, neutral radical form HFLsq∙, anionic radical form FLsq−∙, neutral reduced form H2FLred, and anionic reduced form HFLred−) are simulated by adiabatic or vertical methods combined with time‐dependent or time‐independent (TD/TI) formalisms, with displace, Duschinsky, and Herzberg−Teller (HT) effects under the framework of the Franck‐Condon approximation. The statistical vibronic transition analysis reveals the unity of the spectrum transition property, the relevant normal modes, and the primary geometrical variations. The correlation between geometry and spectral calculation is explained. [ABSTRACT FROM AUTHOR]

A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere Y we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of Y. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its Pin (2) -equivariant refinement). We apply this to study the subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying certain natural geometric constraints. [ABSTRACT FROM AUTHOR]

The paper investigates the vibration characteristics of functionally graded graphene platelet reinforced composites (FG-GPLRC) stiffened plates in the presence of coupled Dynamic vibration absorbers (DVAs) and the optimization of the parameters of the DVAs. The FG-GPLRC plate is used as a basis for coupling arbitrary numbers of stiffeners at arbitrary angles and positions by imposing a displacement continuity condition supplemented with displacement coordinate transformations. The artificial virtual spring method is used to simulate the various boundary conditions by setting the spring stiffness and coupling the simplified DVAs to a spring-damped system. The unknown displacement coefficients were expanded using spectral geometry method to obtain the dynamic response of the coupled model. The reliability of the current model is confirmed by comparison with literature, the finite element method (FEM), and experiments. Based on the presented model, the different dynamic behaviors of plates with different FG-GPLRC distribution types at different stiffening parameters are analyzed, and it is found that different GPL distribution types are not equally sensitive to changes in the location and size of stiffeners. It will provide greater structural strength and design flexibility for the engineering of significant watercraft and critical vehicles. The vibration control of FG-GPLRC stiffened plates is developed using DVA and the DVA parameters are optimized using Artificial Bee Colony algorithm to minimize the model energy. These results can extend the structural life, which will increase the potential of FG-GPLRC stiffened plates in a wider range of engineering applications. • A dynamic model of FG-GPLRC stiffened plates coupled with multiple dynamic vibration absorbers is presented. • The ABC algorithm is applied to optimize the parameters of DVAs. • The effect of various distribution types, volume fractions, stiffener locations, and cross-section sizes are illustrated. [ABSTRACT FROM AUTHOR]

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus. [ABSTRACT FROM AUTHOR]

In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]