Geometric Counterpart Research Articles

Remarks on geometric engineering, symmetry TFTs and anomalies

Geometric engineering is a collection of tools developed to establish dictionaries between local singularities in string theory and (supersymmetric) quantum fields. Extended operators and defects, as well as their higher quantum numbers captured by topological symmetries, can be encoded within geometric engineering dictionaries. In this paper we revisit and clarify aspects of these techniques, with special emphasis on ’t Hooft anomalies, interpreted from the SymTFT perspective as obstructions to the existence of Neumann boundary conditions. These obstructions to gauging higher symmetries are captured via higher link correlators for the SymTFT on spheres. In this work, we give the geometric engineering counterpart of this construction in terms of higher links of topological membranes. We provide a consistency check in the context of 5D SCFTs with anomalous 1-form symmetries, where we give two independent derivations of the anomaly in terms of higher links, one purely field theoretical and the other purely geometrical. Along the way, we also recover the construction of non-invertible duality defects in 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM from a geometric engineering perspective.

Enhancing healthcare supply chain management through artificial intelligence-driven group decision-making with Sugeno–Weber triangular norms in a dual hesitant q-rung orthopair fuzzy context

The healthcare industry faces numerous challenges in managing its supply chain efficiently, where critical decisions must be made promptly to ensure the availability of essential medical resources. This research introduces a novel artificial intelligence (AI) approach, utilizing the “Sugeno–Weber (SW) t-conorms and t-norms” (SWt-CNs&t-Ns) for decision-making in a Dual Hesitant q-Rung Orthopair Fuzzy (DHq-ROF) context. The SWt-CNs&t-Ns are chosen for their adaptability in data unification, serving as prominent operations for union and intersection processes. Developing a set of fundamental operations is imperative to effectively utilize SWt-CNs&t-Ns and hybrid aggregation operators in DHq-ROF settings. Following the introduction of these processes, several aggregating operators have been provided. These operators include DHq-ROF SW weighted averaging, ordered weighted averaging, hybrid averaging, and their geometric counterparts utilizing DHq-ROF data. The SW triangular norm-based approach aggregates group preferences, facilitating a systematic decision-making process. Triangular norms ensure a realistic representation of interrelationships among decision criteria, leading to optimal healthcare supply chain management solutions. Furthermore, the SW triangular norm-based approach aggregates group preferences, enabling a systematic and comprehensive decision-making process. Choosing the best healthcare supply chain management solutions is easier when you use triangular norms because they give a more accurate picture of how the decision criteria affect each other. The effectiveness of the proposed AI framework is demonstrated through a series of experiments and case studies, showcasing its ability to enhance decision accuracy, reduce uncertainty, and improve overall supply chain performance. The research findings underscore the potential of AI-driven solutions to revolutionize healthcare supply chain management, ultimately leading to better resource allocation, cost efficiency, and improved patient care.

Time-lapse attenuation variations using distributed acoustic sensing vertical seismic profile data during CO2 injection at CaMI Field Research Station, Alberta, Canada

For seismic monitoring of geologic [Formula: see text] storage, it is useful to measure time-lapse (TL) variations of seismic attenuation. Seismic attenuation directly connects to different petrophysical parameters within the [Formula: see text] storage complex. We have developed an approach to derive smooth time-variant amplitude spectra from seismic signals by using the sparse strongest signal peaks, and then measuring two different attributes (conventional Q factor and its “geometric” counterpart) characterizing the path effects of seismic attenuation from the smooth spectra. This approach is straightforward and does not require sophisticated algorithms or parameterization schemes. We apply this approach to TL distributed acoustic sensing (DAS) vertical seismic profile (VSP) data from the Field Research Station (FRS) [Formula: see text] injection project in southern Alberta, Canada. High-quality stacked reflection records are obtained from the baseline and monitor DAS VSP surveys at the FRS and TL attenuation-attribute differences are derived from these reflection records. TL variations of attenuation are observed within the injection zone at the FRS, which are interpreted as being related to the injected [Formula: see text]. Although there is always a significant trade-off between the accuracy and temporal resolution of the measured attenuation parameters, reliable attenuation measurements around the injection zone are still achieved with an in-use reflection signal of a sufficient length and bandwidth. Attenuation attributes measured from this approach can be an advantageous tool for monitoring the distribution and migration of [Formula: see text] plumes.

Back to heterotic strings on ALE spaces. Part II. Geometry of T-dual little strings

This work is the second of a series of papers devoted to revisiting the properties of Heterotic string compactifications on ALE spaces. In this project we study the geometric counterpart in F-theory of the T-dualities between Heterotic ALE instantonic Little String Theories (LSTs) extending and generalising previous results on the subject by Aspinwall and Morrison. Since the T-dualities arise from a circle reduction one can exploit the duality between F-theory and M-theory to explore a larger moduli space, where T-dualities are realised as inequivalent elliptic fibrations of the same geometry. As expected from the Heterotic/F-theory duality the elliptic F-theory Calabi-Yau we consider admit a nested elliptic K3 fibration structure. This is central for our construction: the K3 fibrations determine the flavor groups and their global forms, and are the key to identify various T-dualities. We remark that this method works also more generally for LSTs arising from non-geometric Heterotic backgrounds. We study a first example in detail: a particularly exotic class of LSTs which are built from extremal K3 surfaces that admit flavor groups with maximal rank 18. We find all models are related by a so-called T-hexality (i.e. a 6-fold family of T-dualities) which we predict from the inequivalent elliptic fibrations of the extremal K3.

Inversion maps and torus actions on rational homogeneous varieties

Complex projective algebraic varieties with {{mathbb {C}}}^*-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a {{mathbb {C}}}^*-action with no proper isotropy subgroups.

Counting conjectures and e-local structures in finite reductive groups

We prove new results in generalised Harish-Chandra theory providing a description of the so-called Brauer–Lusztig blocks in terms of information encoded in the ℓ-adic cohomology of Deligne–Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the ℓ-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between ℓ-structures and their geometric counterpart. Finally, using the description of Brauer–Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisations of generalised Harish-Chandra series.

Intrinsic incremental evolution of hypoelastic discrete mechanical systems

AbstractThis paper provides an explicit geometric and coordinate‐free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle . Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle is fully used in order to get the geometric counterpart κ of the so‐called tangent stiffness matrix. Subject to specific conditions in , the incremental evolution is then a well‐founded question. An hypoelastic four‐grains granular system illustrates in detail these general results.

Leading-edge bluntness effects on the hypersonic flow over the double wedge at multiple aft-wedge angles

The present numerical investigation focuses on the leading-edge bluntness effects on the double wedge with varied aft-wedge angles exposed to low enthalpy hypersonic free stream conditions. The bluntness ratio in this study varies, ranging from R/L1 = 0 (sharp leading edge) to R/L1 = 0.577 (maximum allowable bluntness), along with the aft-wedge angle varying between θ2 = 45° and 60°. Noticeably, even a small bluntness ratio can completely change the shock interaction pattern compared to its sharp geometrical counterpart due to a detached leading-edge shock, enlarged separation bubble, and location of various shock waves concerning it. Critical bluntness ratios exist for the low aft-wedge θ2 = 45° angle, but increasing the aft-wedge angle makes the flow field highly unsteady for some bluntness ratios. Nevertheless, these bluntness ratios for such double-wedge configurations are reported using the mean of separation bubble size. Moreover, this work unravels the cause of such unsteadiness for the unsteady flow fields using the spatial-temporal evolution of the wall pressure distribution and fast Fourier transform of the pressure fluctuation signal at the compression corner and supports the deduced observation with the help of energy-based proper orthogonal decomposition. The increased shock–boundary layer interaction strength moves the separation point upstream beyond the junction of cylindrical bluntness and inclined fore-wedge surface, accompanying sudden change in its direction of motion that perturbs the shear layer that set to a self-sustained, highly unsteady flow field.

BCINet: Bilateral cross-modal interaction network for indoor scene understanding in RGB-D images

Depth cue has proven to be useful information in the indoor scene understanding of RGB-D images for providing a geometric counterpart to RGB representation. However, because of the differences between RGB-D image pairs, utilizing cross-modal data effectively is a key issue. Most methods exclusively leverage depth data to unilaterally complement RGB data for better feature representation; they invariably ignore the fact that RGB and depth data can bilaterally complement each other. Herein, a novel RGB-D scene-understanding network called BCINet is presented, in which RGB and depth data bilaterally complement each other via a proposed bilateral cross-modal interaction module (BCIM). The BCIM helps to capture cross-modal complementary cues by crossly fusing enhanced features from one modality to the counterpart modality through a feature enhanced module. Meanwhile, exploiting the long-range dependencies of RGB-D features is also significant for accurate scene understanding. Specifically, we design a hybrid pyramid dilated convolution module to enlarge the receptive fields along both the vertical and horizontal spatial directions to adaptively capture diverse contexts with different shapes. Additionally, we propose a context-guided module to aggregate these diverse higher-level contexts with lower-level features in the encoder to guide the information flow for progressively refining the segmentation map. Experimental results on two indoor scene datasets demonstrate the superiority and effectiveness of the proposed BCINet over several state-of-the-art approaches.

Real-Time Passable Area Segmentation With Consumer RGB-D Cameras for the Visually Impaired

Passable area segmentation is an essential component of navigation assistance for the visually impaired people. Pixel-wise semantic segmentation of RGB images can be advanced by exploiting informative features from depth modality. Simple fusion, however, may not lead to satisfactory results as actual depth data are generally noisy, which may worsen the accuracy as the networks go deeper. In this paper, we construct a wearable system with a novel Lightweight Multi-Modality Collaborative Network (LM-MCN), which is performing real-time passable area segmentation to assist people walking alone more safely. Firstly, a normal map estimation (NME) module, which can convert dense depth to a normal map with high accuracy and efficiency, is introduced. Moreover, we propose a multi-modality feature fusion module (M-MFFM), which can recalibrate and fuse the features from both RGB images and the inferred depth information for accurate passable area segmentation. Furthermore, we publish a new passable area segmentation dataset, named SensingAI-Easy-to-Walk (SETW), collected from sidewalk views in urban areas. Extensive experiments demonstrate that depth information is more appealing since it provides a geometric counterpart to the RGB representation. In addition, the experimental results report our method achieves a 96.50% MaxF with real-time inference speed of 50 FPS for KITTI road, which demonstrates the validity and superiority of LM-MCN.

Geometric quantum thermodynamics.

Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We focus on microcanonical and canonical ensembles, looking at the geometric counterpart of Gibbs ensembles for distributions on the space of quantum states. We show that one can define quantum heat and work in an intrinsic way, including single-trajectory work. We reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the first and second laws of thermodynamics and Jarzynki's fluctuation theorem. Overall, this results in a more transparent physics than conventionally available. The mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned. The experimental relevance is brought out via a stochastic model for chiral molecules (in the two-state approximation) and Josephson junctions. Numerically, we demonstrate this invariably leads to the emergence of the geometric canonical ensemble.

DiffCo: Autodifferentiable Proxy Collision Detection With Multiclass Labels for Safety-Aware Trajectory Optimization

The objective of trajectory optimization algorithms is to achieve an optimal collision-free path between start and goal states. In real-world scenarios, where environments can be complex and nonhomogeneous, a robot needs to be able to gauge whether a state will be in collision with various objects in order to meet some safety metrics. The collision detector should be computationally efficient and, ideally, analytically differentiable to facilitate stable and rapid gradient descent during optimization. However, methods today lack an elegant approach to detect collision differentiably, relying rather on numerical gradients that can be unstable. We present DiffCo, the first, fully autodifferentiable, nonparametric model for collision detection. Its nonparametric behavior allows one to compute collision boundaries on the fly and update them, requiring no pretraining and allowing it to update continuously in dynamic environments. It provides robust gradients for trajectory optimization via backpropagation and is often 10–100 times faster to compute than its geometric counterparts. DiffCo also extends trivially to modeling different object collision classes for semantically informed trajectory optimization.

Involute evolute curve family induced by the coupled dispersionless equations

In this paper, we derive the coupled dispersionless (CD) equation from the motion of the involute evolute curve pairs, which provides a favourable interpretation for the integrability conditions. This nonlinear differential equation allows us to introduce the Lax pair providing the integrability of this equation. The gauge equivalence between the CD equation and the Zhaidary-III equation is established. Integrable generalized Heisenberg ferromagnet type equations for the tangent, principal normal, and binormal unit vectors T,N,B are presented. Each of these three integrable nonlinear equations for the orthogonal frame is, separately, geometrical equivalent counterparts of the CD equation. Finally, we study the link between the involute evolute curves and the CD equation.

A GEOMETRIC STEINBERG FORMULA

We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg–Mirković conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic.

Small bandwidth ℂ*-actions and birational geometry

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity—measured in terms of two invariants of the action, called bandwidth and bordism rank—and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.

Community detection and percolation of information in a geometric setting

AbstractWe make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

Flat telescope based on an all-dielectric metasurface doublet enabling polarization-controllable enhanced beam steering

Abstract A flat telescope (FTS), which incorporates an all-dielectric metasurface doublet (MD) based on hydrogenated amorphous silicon nanoposts, is proposed and demonstrated to achieve flexibly magnified angular beam steering that is sensitive to both light polarization and deflection direction. Specifically, for transverse-electric-polarized incident beams, the MD exhibits deflection magnification factors of +5 and +2, while for transverse magnetic polarization, the beam is steered in reverse to yield magnification factors of −5 and −2 in the horizontal and vertical directions, respectively. The proposed MD comprises cascaded metalenses, which can invoke polarization-selective transmission phases. The MD which emulates a set of convex and concave lenses renders positively increased beam deflection, whereas the case corresponding to a pair of convex lenses facilitates negatively amplified beam deflection. The essential phase profiles required for embodying the MD are efficiently extracted from its geometric lens counterpart. Furthermore, the implemented FTS, operating in the vicinity of a 1550 nm wavelength, can successfully enable enhanced beam steering by facilitating polarization-sensitive bidirectional deflection amplifications. The proposed FTS can be applied in the development of a miniaturized light detection and ranging system, where the beam scanning range can be effectively expanded in two dimensions.

Imaginary images and Stokes phenomena in the weak plasma lensing of coherent sources

ABSTRACT The study of astrophysical plasma lensing, such as in the case of extreme scattering events, has typically been conducted using the geometric limit of optics, neglecting wave effects. However, for the lensing of coherent sources such as pulsars and fast radio bursts, wave effects can play an important role. Asymptotic methods, such as the so-called Eikonal limit, also known as the stationary phase approximation, have been used to include first-order wave effects; however, these methods are discontinuous at Stokes lines. Stokes lines are generic features of a variety of lens models, and are regions in parameter space where imaginary images begin to contribute to the overall intensity modulation of lensed sources. Using the mathematical framework of Picard–Lefschetz theory to compute diffraction integrals, we argue that these imaginary images contain as much information as their geometric counterparts, and may potentially be observable in data. Thus, weak-lensing events where these imaginary images are present can be as useful for inferring lens parameters as strong-lensing events in which multiple geometric images are present.

Normal‐Driven Spherical Shape Analogies

AbstractThis paper introduces a new method to stylize 3D geometry. The key observation is that the surface normal is an effective instrument to capture different geometric styles. Centered around this observation, we cast stylization as a shape analogy problem, where the analogy relationship is defined on the surface normal. This formulation can deform a 3D shape into different styles within a single framework. One can plug‐and‐play different target styles by providing an exemplar shape or an energy‐based style description (e.g., developable surfaces). Our surface stylization methodology enables Normal Captures as a geometric counterpart to material captures (MatCaps) used in rendering, and the prototypical concept of Spherical Shape Analogies as a geometric counterpart to image analogies in image processing.

Scattered subspaces and related codes

After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered $${{\mathbb {F}}}_{q}$$ -subspaces of $$V(r,q^n)$$ and maximum rank distance (MRD) codes has been established in the extremal cases $$h=1$$ and $$h=r-1$$ . In this paper, we propose a connection for any $$h\in \{1,\ldots ,r-1\}$$ , extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.