Geometric Explanation Research Articles

Phase space properties of cosmological models in f(Q, T) gravity

The Weyl type f(Q, T) gravity is the modification of f(Q) gravity and f(Q, T) theories, where non-metricity is denoted by Q and T is the trace of the energy–momentum tensor. Together with a geometric explanation for dark energy, the theory can provide an exact interpretation of the transformation of the late-time Universe and the observable data. In this study, we present an accelerated cosmic model of the Universe in f(Q, T) gravity. We consider the model of f(Q, T) gravity as, f(Q,T)=-Q+ϕ(Q,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(Q,T) = -Q+\\phi (Q,T)$$\\end{document}. We examine the energy condition for the model of f(Q, T) gravity and find out that our model satisfies the null and strong energy conditions at the same time, it violates the weak and dominant energy conditions. After that, we perform the phase-space study of our cosmological model with and without interaction independently. In case of the absence of interaction, we get six critical points out of which three critical points are stable critical points while the rest three critical points are saddle. When we perform the stability analysis in the presence of interaction, we get three critical points out of which one critical point is stable while the rest two are saddle points. The phase-plot analysis also shows the cosmological models in f(Q, T) gravity.

Nonconvex Factorization and Manifold Formulations Are Almost Equivalent in Low-Rank Matrix Optimization

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish a sandwich relation on the spectrum of Riemannian and Euclidean Hessians at first-order stationary points (FOSPs). As a result of that, we obtain an equivalence on the set of FOSPs, second-order stationary points, and strict saddles between the manifold and factorization formulations. In addition, we show that the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another. Similarities and differences in the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and factorization formulations for handling rank constraints, and it provides a geometric explanation for the similar empirical performance of factorization and manifold approaches in low-rank matrix optimization observed in the literature. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections (in particular, the sandwich relation), we are able to solve unanswered questions in the literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing. Funding: This work was supported by the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107 and 62141407], and the Shanghai Science and Technology Program [Grant 21JC1400600]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0030 .

Containment control of systems with constraints and bounded disturbances

AbstractThis article studies the distributed containment control of a second‐order discrete‐time multi‐agent system with convex constraints on positions, nonconvex constraints on velocities, bounded disturbances on static points and switching topologies. A distributed control algorithm with disturbance information and local information is proposed. It is proved that when the static points with disturbances and the joint graph have a directed spanning forest among each time interval, all agents are gradually driven to the neighborhood of a convex hull spanned by the static points, and an upper bound of the neighborhood is obtained. The most innovative point is to utilize a rigorous mathematical theory to deal with convex constraint error rather than a geometric explanation. Meanwhile, another control algorithm with static point information and local information is proposed to converge all agents to the convex hull in the absence of external disturbances. Additionally, a numerical example is conducted to illustrate this theory.

The mass gap in five dimensional Einstein–Gauss–Bonnet black holes: a geometrical explanation

It is well known that, unlike in higher dimensional general relativity (GR), we cannot have a black hole with an arbitrarily small mass in five dimensional Einstein–Gauss–Bonnet gravity. When we study the dynamical black hole formation via the radiation collapse in the radiating Boulware–Deser spacetime in five dimensions, the central zero mass singularity is weak, conical and naked, and the horizon forms only when a finite amount of matter, that depends on the coupling constant of the Gauss–Bonnet term, falls into the central singularity. To understand this phenomenon transparently and geometrically, we study the radiating Boulware–Deser spacetime in five dimensions using a 1+1+3 spacetime decomposition, for the first time. We find that the geometric and thermodynamic quantities can be expressed in terms of the gravitational mass and the Gauss–Bonnet (GB) parameter and separate each of them into their Gauss–Bonnet and matter parts. Drawing comparisons with five dimensional GR at every step, we explicitly show how the mass gap arises for a general mass function M(v) and what functions for M(v) make certain geometrical quantities well defined at the central singularity. We show in the case of self-similar radiation collapse in the modified theory, the central singularity is not a sink for timelike geodesics and is extendable. This clearly demonstrates how the GB invariant affects the nature of the final state of a continual collapse in this modified theory.

A Geometrical Explanation for Change in Pelvic Tilt (or Lack of Change) Following Long Spinal Fusions.

Retrospective cohort. Examine the relationship between compensatory pelvic retroversion, positive sagittal imbalance (measured by C2 tilt), and the C2 pelvic angle (C2PA) in patients before long spinal fusions; and to determine the association between changes in C2PA and pelvic tilt (PT) following long spinal fusions. Adult spinal deformity surgical goals often include a PT target, yet patients frequently demonstrate persistent compensatory pelvic retroversion following surgery. Adults above 18 years old undergoing long spinal fusions (>4 levels) with standing preoperative and postoperative radiographs were included. To examine drivers of preoperative sagittal balance, regression models were fit to estimate the association between preoperative C2PA and pelvic incidence with preoperative PT and C2 tilt. To predict postoperative change in PT, multivariable regression was used to estimate change in PT, adjusting for change in C2PA and preoperative C2 tilt. Among the 80 patients identified, the median age was 61 (IQR: 45-72) and 46 (58%) were female. The median number of levels fused was 10 (IQR: 8-13) and 55 (69%) were instrumented to the sacrum/pelvis. Preoperative C2PA had a significant nonlinear association with preoperative PT ( r2 =0.81, P <0.001) and preoperative C2 tilt ( r2 =0.41, P =0.002). Postoperative change in PT was strongly associated with change in C2PA (β=0.81; P <0.001) and preoperative C2 tilt (β=0.55; P <0.001). Following long spinal fusions, change in PT (or lack thereof) can be reliably predicted based on change in C2PA and preoperative C2 tilt. In patients with normal preoperative C2 tilt, the change in C2PA is nearly equivalent to the change in PT, but in patients with more positive C2 tilt (sagittal imbalance), a greater change in C2PA will be required to achieve an equivalent change in PT. 3.

D. R. Cox: aspects of scientific inference

Abstract The year 2022 marked 100 years since R. A. Fisher’s landmark paper on the foundations of statistics, and 50 years since D. R. Cox’s seminal work on the proportional hazards model. At a 1-day meeting in recognition of the latter, held at the London School of Hygiene and Tropical Medicine, I attempted to place some of D. R. Cox’s most influential work in its appropriate historical context, reconstructing some of his ideas from unifying principles and providing a different exposition to that of the original papers. The present article is an elaboration of that talk. Additional material includes a geometric explanation of conditional inference in models with binary outcomes, relating the logistic analysis of matched comparisons presented by Cox (1958c, Two further applications of a model for binary regression. Biometrika, 45, 562–565) to the protracted and sometimes polemical discourse over the analysis of 2×2 contingency tables.

Visualizing the fibre texture of satin spar using laboratory 2D X-ray diffraction.

The suitability of point focus X-ray beam and area detector techniques for the determination of the uniaxial symmetry axis (fibre texture) of the natural mineral satin spar is demonstrated. Among the various diffraction techniques used in this report, including powder diffraction, 2D pole figures, rocking curves looped on φ and 2D X-ray diffraction, a single simple symmetric 2D scan collecting the reciprocal plane perpendicular to the apparent fibre axis provided sufficient information to determine the crystallographic orientation of the fibre axis. A geometrical explanation of the 'wing' feature formed by diffraction spots from the fibre-textured satin spar in 2D scans is provided. The technique of wide-range reciprocal space mapping restores the 'wing' featured diffraction spots on the 2D detector back to reciprocal space layers, revealing the nature of the fibre-textured samples.

Quantum ghost imaging of a vector field.

Quantum ghost image technique utilizing position or momentum correlations between entangled photons can realize nonlocal reconstruction of the image of an object. In this work, based on polarization entanglement, we experimentally demonstrate quantum ghost imaging of vector images by using a geometric phase object. We also provide a corresponding theoretical analysis. Additionally, we offer a geometrical optics path explanation of ghost imaging for vector fields. The proposed strategy offers new insights into the fundamental development of ghost imaging and also holds great promise for developing complex structured ghost imaging techniques. Our work expanding the principle of ghost imaging to spatially varying vector beams will lead to interesting developments of this field.

Understanding the contagiousness of Covid-19 strains: A geometric approach

Protein–protein interaction occurs on surface patches with some degree of complementary geometric and chemical features. Building on this understanding, this study endeavors to characterize the spike protein of the SARS-CoV-2 virus at the morphological and geometrical levels in its Alpha, Delta, and Omicron variants. In particular, the affinity between different SARS-CoV-2 spike proteins and the ACE2 receptor present on the membrane of the human respiratory system cells is investigated. To achieve an adequate degree of geometrical accuracy, the 3D depth maps of the proteins in exam are filtered by developing an ad-hoc convolutional filter with a kernel implemented as a sphere of varying radius, simulating a ball rolling on the surface (similar to the 'rolling ball' filter). This ball ideally models a hypothetical molecule that could interface with the protein and is inspired by the geometric approach to macromolecule–ligand interactions proposed by Kuntz et al. in 1982. The aim is to mitigate the imperfections and to obtain a smoother surface that could be studied from a geometrical perspective for binding purposes. A set of geometric descriptors, borrowed from the 3D face analysis context is then mapped point-by-point onto protein depth maps. Following a feature extraction phase inspired by Histogram of Oriented Gradients and Local Binary Patterns, the final histogram features are used as input for a Support Vector Machine classifier to automatically classify the proteins according to their surface affinity, where a similarity in shape is observed between ACE2 and the spike protein of the SARS-CoV-2 Omicron variant. Finally, Root Mean Square Error analysis is used to quantify the geometrical affinity between the ACE2 receptor and the respective Receptor Binding Domains of the three SARS-CoV-2 variants, culminating in a geometrical explanation for the higher contagiousness of Omicron relative to the other variants under study.

Closer look at cosmological consequences of interacting group field theory

Group field theory has shown to be a promising framework to derive cosmological predictions from full quantum gravity. In this brief note, we revisit the background dynamics when interaction terms are taken into account and conclude that, although the bounce is clearly robust, providing a geometrical explanation for inflation seems to be very difficult. We consider possible improvements and modifications of the original scenario and derive several limits on the parameters of the model, trying to exhaustively investigate the relevant features and weaknesses of the approach.

G.B. Riccioli’s geo-heliocentric use of Epicepicycles, ellipses and spirals

According to Giovanni Battista Riccioli (1598–1671), planets describe orbits in a fluid heaven in a helio-geocentric model of cosmos. In his Almagestum Novum (1651) he stresses the need for a novel and rigorous geometrical explanation for the motion of heavenly bodies, which considers a separate Primum Mobile as an unnecessary hypothesis. Riccioli rejects the standard eccentric-epicycle theory as unsatisfactory and argues for what he calls “Eccentrepicyclos” or “Epicepicycles.” He takes the Keplerian elliptical theory into account and includes a variable oscillation of both the mobile eccentric center and the variation of the epicycle’s diameter. As a result, planets move along spiral orbits that have variable sizes. The inequalities, which astronomers had always tried to explain, are now justified: the spiral trajectory, obtained by means of the oscillation of the eccentric center, warrants the first inequality, about the variation of the planets’ velocity. The variable amplitude of the spirals, obtained by the variation of the epicycle’s diameter, explains the second inequality, namely the apparent retrograde or progressive planetary motion. In the later work Astronomia Reformata (1665), Riccioli uses explicitly the term “ellipse.” Riccioli’s innovations are of great interest and help to understand the complexity of the astronomical debates about the best world-system.

Duality viewpoint of criticality

In this work, we study quantum many-body systems which are self-dual under duality transformation connecting different symmetry protected topological (SPT) phases. We provide a geometric explanation of the criticality of these self-dual models. More precisely, we show a ground state (quasi-)degeneracy under the periodic boundary conditions,i.e., the ingappability of the bulk spectrum. Equivalently, the symmetry group at criticality, including the duality symmetry, has a mixed 't Hooft anomaly. This approach can not only predict the spectrum of the self-dual model with ordinary 0-form symmetry, but also be applied to that with generalized symmetry, such as higher form and subsystem symmetry. As an application, we illustrate our results with several examples in one and two dimensions, which separate two different SPTs.

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in Pn with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree n. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

On the regularity of the distance near the boundary of an obstacle

We study the regularity of the Euclidean distance function from a given point-wise target of a n-dimensional vector space in the presence of a compact obstacle bounded by a smooth hypersurface. It is known that such a function is semiconcave with fractional modulus one half. We provide a geometrical explanation of the exponent one half. Furthermore, under a natural (weak) assumption on the position of the point-wise target relatively to the obstacle, we show that there exists a point in the boundary of the obstacle so that no better regularity result holds near such a point. As a consequence of this result, we show that the Euclidean metric cannot be extended to a tubular neighborhood of the obstacle, as a Riemannian metric, keeping the property that the associated distances coincide outside the obstacle.

The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models

By making use of an isothermal crystallization kinetics formulation in which growth and nucleation rates are functions of the degree of crystallization, we describe a process consisting of birth and growth of spherical crystalline domains. It is found that beyond the early stages of crystallization, the role played by the nucleation rate becomes quite insignificant as compared to the role of the crystalline domain growth rate. This is in contrast to what we would expect from the simplified Avrami exponential equation, where both rates play a symmetrical role. We argue that the asymmetric roles of the growth and nucleation rates greatly contribute to the wide applicability of Avrami-type expressions. A geometric explanation of the diminishing role of nucleation in the later stages of transformation is given. Future theoretical developments might benefit from the arguments and results presented.

Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

On Hyperbolic Complex Numbers

For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion.

New optical radial direction for optical modeling with Minkowski extended frame

Numerous geometric applications have been improved to functionalize optical fibers and geometric fabrics. Optical uniform motions are designed by electromagnetic directions. In this article, we manufacture optical important conditions for extended fields. Also, we design new uniformly optical approach for unchanged direction optical motion, uniformly accelerated optical motion, and uniformly circular optical motion. Also, we study magnetic curves and their optical modeling and give optical relations between Fermi–Walker parallel transportation and uniformly optical modeling in four dimensional Minkowski space. Finally, the results also indicated that applied geometric explanation for radial direction of uniformly motion in Minkowski 4-space.

Extremal transitions via quantum Serre duality

Two varieties Z and $${\widetilde{Z}}$$ are said to be related by extremal transition if there exists a degeneration from Z to a singular variety $${\overline{Z}}$$ and a crepant resolution $${\widetilde{Z}} \rightarrow {\overline{Z}}$$ . In this paper we compare the genus-zero Gromov–Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum D-module of $${\widetilde{Z}}$$ , after analytic continuation and restriction of a parameter, recovers the quantum D-module of Z. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem.

Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane

In this paper, we take advantage of envelope theory and singularity theory to study the evolutoids and pedaloids in Minkowski plane. We illustrate an internal correlation from algebraic and geometric viewpoints, and give the geometric explanation of evolutoids and pedaloids. Then, we generalize the notions of evolutoids and pedaloids to the category of frontals in Minkowski plane. Furthermore, we apply the technique of singularity theory, using the discriminants and versal unfolding tools, to consider evolving evolutoids and give the classifications of singularities of the evolutoids, and explain when cusps and inflexions occur and how evolutoids evolving. Besides, we found that there is a close relationship between the evolutoids and pedaloids and good correspondence between their singularities.