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Autore: Grafiati

Pubblicato: 22 giugno 2024

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1

Widder, Nathan. "The Mathematics of Continuous Multiplicities: The Role of Riemann in Deleuze's Reading of Bergson". Deleuze and Guattari Studies 13, n. 3 (agosto 2019): 331–54. http://dx.doi.org/10.3366/dlgs.2019.0361.

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A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut.

2

VACARU, SERGIU I. "FINSLER AND LAGRANGE GEOMETRIES IN EINSTEIN AND STRING GRAVITY". International Journal of Geometric Methods in Modern Physics 05, n. 04 (giugno 2008): 473–511. http://dx.doi.org/10.1142/s0219887808002898.

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We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

3

Itin, Yakov. "Pseudo-Riemann’s quartics in Finsler’s geometry—two-dimensional case". Journal of Physics: Conference Series 2482, n. 1 (1 maggio 2023): 012007. http://dx.doi.org/10.1088/1742-6596/2482/1/012007.

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Abstract Finsler’s geometry usually describes an extension of Riemmann’s geometry into a direction-dependent geometric structure. Historically, the well-known Riemann’s quartic length element example served as the inspiration for this construction. Surprisingly, the covariant Fresnel equation—a fundamental dispersion relation in solid-state electrodynamics—emerges as the exact same quartic expression. As a result, Riemann’s quartic length expression can be regarded of as a mathematical representation of a well-known physical phenomenon. In this study, we offer numerous Riemann’s quartic examples that show Finsler’s geometry, even in the situation of a positive definite Euclidean signature space, is too restrictive for many applications. The strong axioms of Finsler’s geometry are violated in a substantially greater number of distinctive subsets for the spaces having an indefinite (Minkowski) signature. We suggest a weaker characterization of Finsler’s structure based on explicitly calculated two-dimensional examples. In tangential vector space, this concept permits singular subsets. Only open subsets of a manifold’s tangent bundle are required to satisfy the strong axioms of Finsler’s geometry. We demonstrate the distinctive unique subsets of the Riemann’s quartic in two dimensions and briefly discuss their possible physical origins.

4

Lesfari, A. "Riemann-Roch theorem and Kodaira-Serre duality". Annals of West University of Timisoara - Mathematics and Computer Science 58, n. 1 (1 giugno 2022): 4–17. http://dx.doi.org/10.2478/awutm-2022-0002.

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Abstract The Riemann-Roch theorem is of utmost importance and a vital tool to the fields of complex analysis and algebraic geometry, specifically in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this paper is to give two proofs of this important theorem and explore some of its numerous consequences. As an application, we compute the genus of some interesting algebraic curves or Riemann surfaces.

5

Mattes, M., e M. Sorg. "Riemann - Cartan Geometry of Trivializable Gauge Fields". Zeitschrift für Naturforschung A 44, n. 3 (1 marzo 1989): 222–38. http://dx.doi.org/10.1515/zna-1989-0309.

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A Riemann-Cartan structure can be associated to any SO (4) trivializable gauge field. Under certain integrability conditions, this non-Riemannian geometry may be replaced by a strictly Riemannian one. The Yang-Mills equations guarantee the existence of such a Riemannian structure. The general SO(4) trivializable solution for the SO(3) Yang-Mills equations is discussed within the geometric approach.

6

Hoare, Graham. "Bernhard Riemann’s legacy of 1859". Mathematical Gazette 93, n. 528 (novembre 2009): 468–75. http://dx.doi.org/10.1017/s0025557200185213.

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The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.

7

Katanaev, Mikhail O., e Alexander V. Mark. "Combined Screw and Wedge Dislocations". Universe 9, n. 12 (29 novembre 2023): 500. http://dx.doi.org/10.3390/universe9120500.

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Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge and screw dislocations.

8

Kagan, V. F. "Riemann's Geometric Ideas". American Mathematical Monthly 112, n. 1 (1 gennaio 2005): 79. http://dx.doi.org/10.2307/30037389.

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9

Kagan, V. F. "Riemann's Geometric Ideas". American Mathematical Monthly 112, n. 1 (gennaio 2005): 79–86. http://dx.doi.org/10.1080/00029890.2005.11920172.

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10

Guo, Enli, e Xiaohuan Mo. "Riemann-Finsler geometry". Frontiers of Mathematics in China 1, n. 4 (dicembre 2006): 485–98. http://dx.doi.org/10.1007/s11464-006-0023-9.

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11

Solikin, Agus. "Konsep Kesejajaran Garis dalam Geometri Euclid dan Geometri Riemann serta Aplikasinya dalam Kajian Ilmu Falak". MUST: Journal of Mathematics Education, Science and Technology 2, n. 2 (28 dicembre 2017): 243. http://dx.doi.org/10.30651/must.v2i2.865.

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Kesejajaran garis dalam geometri Euclid dan Riemann dalam kajian matematika memiliki perbedaan. Perbedaan dalam konsep kesejajaran garis tersebut, tentunya akan memberikan perbedaan pada kajian-kajian berikutnya. Berdasarkan hal tersebut, maka penelitian ini dirancang untuk mengkaji berkenaan konsep kesejajaran garis dalam geometri Euclid dan Riemann, kemudian aplikasinya dalam kajian ilmu Falak Berdasarkan fokus pembahasan tersebut, maka metode penelitian direncanakan dalam bentuk deskriptif kualitatif, dengan sumber data literatur-literatur yang terkait dengan fokus penelitian dan data dikumpulkan dengan cara penelaahan dokumen-dokumen tersebut. Selanjutnya data yang terkumpul dianalisis dengan cara deskriptif analitis induktif yang menggunakan pendekatan grounded theory. Berdasarkan penelitian ini diperoleh suatu penjelasan yang utuh tentang konsep kesejajaran garis dalam geometri Euclid dan Riemann, bahwa dalam geometri Euclid dikenal kesejajaran garis, sedangkan dalam geometri Riemann tidak kenal kesejajaran garis. Berkenaan dengan aplikasi dari kesajajaran garis dalam geometri Euclid pada kajian ilmu falak dapat dilihat pada konsep dip (kerendahan ufuk), sedangkan kesejajaan garis dalam geometri Reemann pada kajian ilmu falak dapat diketemukan pada konsep lintang geografis pada suatu tempat

12

Sukestiyarno, Yohanes Leonardus, Khathibul Umam Zaid Nugroho, Sugiman Sugiman e Budi Waluya. "Learning trajectory of non-Euclidean geometry through ethnomathematics learning approaches to improve spatial ability". Eurasia Journal of Mathematics, Science and Technology Education 19, n. 6 (1 giugno 2023): em2285. http://dx.doi.org/10.29333/ejmste/13269.

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Non-Euclidean geometry is an abstract subject and difficult to learn, but mandatory for students. The ethnomathematics approach as a learning approach to improve students’ spatial abilities. The aim of this research is to discover new elements of the spatial abilities of non-Euclidean geometry; determine the relationship between spatial abilities for Euclid, Lobachevsky, and Riemann geometry. This study used the micro genetic method with a 2×2 factorial experimental research design. The sample of this research is 100 students of mathematics education. There are three valid and reliable research instruments through expert trials and field trials. Data collection was carried out in two ways, namely tests and observations. Quantitative data were analyzed through ANCOVA, and observational data were analyzed through the percentage of implementation of the learning trajectory stages. The result is that the spatial ability of students who are given the ethnomathematics learning approach is higher than students who are given the conventional learning approach for Lobachevsky geometry material after controlling for the effect of Euclidean geometry spatial ability. Also, the same thing happened for the spatial abilities of Riemann geometry students. The learning trajectory is conveying learning objectives (learning objective); providing ethnomathematics-based visual problems; students do exploration; students make conclusions and summaries of exploration results; and ends with students sharing conclusions/summaries about concepts and principles in geometric systems. It was concluded that learning non-Euclid geometry through learning paths with an ethnomathematics approach had a positive impact on increasing students’ spatial abilities.

13

Berardini, Elena, Alain Couvreur e Grégoire Lecerf. "A Proof of the Brill-Noether Method from Scratch". ACM Communications in Computer Algebra 57, n. 4 (dicembre 2023): 200–229. http://dx.doi.org/10.1145/3653002.3653004.

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In 1874 Brill and Noether designed a seminal geometric method for computing bases of Riemann-Roch spaces. From then, their method has led to several algorithms, some of them being implemented in computer algebra systems. The usual proofs often rely on abstract concepts of algebraic geometry and commutative algebra. In this paper we present a short self-contained and elementary proof that mostly needs Newton polygons, Hensel lifting, bivariate resultants, and Chinese remaindering.

14

Mo, Xiaohuan. "On Riemann-Finsler geometry". Chinese Science Bulletin 43, n. 6 (marzo 1998): 447–50. http://dx.doi.org/10.1007/bf02883805.

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15

De, K., e U. C. De. "Riemann solitons on para-Sasakian geometry". Carpathian Mathematical Publications 14, n. 2 (17 novembre 2022): 395–405. http://dx.doi.org/10.15330/cmp.14.2.395-405.

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The goal of the present article is to investigate almost Riemann soliton and gradient almost Riemann soliton on 3-dimensional para-Sasakian manifolds. At first, it is proved that if $(g, Z,\lambda)$ is an almost Riemann soliton on a para-Sasakian manifold $M^3$, then it reduces to a Riemann soliton and $M^3$ is of constant sectional curvature $-1$, provided the soliton vector $Z$ has constant divergence. Besides these, we prove that if $Z$ is pointwise collinear with the characteristic vector field $\xi$, then $Z$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, the almost Riemann soliton is expanding. Furthermore, it is established that if a para-Sasakian manifold $M^3$ admits gradient almost Riemann soliton, then $M^3$ is locally isometric to the hyperbolic space $H^{3}(-1)$. Finally, we construct an example to justify some results of our paper.

16

Shen, Zhongmin. "Riemann-Finsler Geometry with Applications to Information Geometry". Chinese Annals of Mathematics, Series B 27, n. 1 (gennaio 2006): 73–94. http://dx.doi.org/10.1007/s11401-005-0333-3.

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17

FERNÁNDEZ, V. V., W. A. RODRIGUES, A. M. MOYA e R. DA ROCHA. "CLIFFORD AND EXTENSOR CALCULUS AND THE RIEMANN AND RICCI EXTENSOR FIELDS OF DEFORMED STRUCTURES (M, ∇′, η) AND (M, ∇, g)". International Journal of Geometric Methods in Modern Physics 04, n. 07 (novembre 2007): 1159–72. http://dx.doi.org/10.1142/s021988780700248x.

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Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

18

Lyzzaik, Abdallah. "The geometry of open continuous mappings having two valences between Riemann surfaces". Mathematical Proceedings of the Cambridge Philosophical Society 120, n. 2 (agosto 1996): 309–29. http://dx.doi.org/10.1017/s0305004100074879.

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AbstractAn open continuous function from an open Riemann surface with finite genus and finite number of boundary components into a closed Riemann surface is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in the image surface either p or q times, counting multiplicity, with possibly a finite number of exceptions.The object of this paper is to prove that the geometry of any (p, q)-map resembles that of a (p, q)-map whose q-set (the set of image points of f that are taken on exactly q times, counting multiplicity), constitutes a finite set of Jordan arcs or curves (loops). This leads to interesting geometrie results regarding (p, q)-maps without exceptional points. Further, it yields that every (p, q)-map is homotopic to a simplicial (p, q)-map having the same covering properties.

19

Chern, Shiing-Shen, e Shanyu Ji. "Projective geometry and Riemann's mapping problem". Mathematische Annalen 302, n. 1 (maggio 1995): 581–600. http://dx.doi.org/10.1007/bf01444509.

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20

Cheng, Jih-Hsin. "Submanifolds in Cauchy Riemann Geometry". Journal of Mathematical Study 53, n. 4 (giugno 2020): 471–92. http://dx.doi.org/10.4208/jms.v53n4.20.04.

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21

Chen, Chuanmiao. "Geometric Proof of Riemann Conjecture". Advances in Pure Mathematics 11, n. 04 (2021): 334–45. http://dx.doi.org/10.4236/apm.2021.114021.

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22

Xue, Fangxiu, Xiaowei Zhang, Zepeng Wang, Jian Wen, Cheng Guan, Hongyan Han, Jingcheng Zhao e Na Ying. "Analysis of Imaging Internal Defects in Living Trees on Irregular Contours of Tree Trunks Using Ground-Penetrating Radar". Forests 12, n. 8 (29 luglio 2021): 1012. http://dx.doi.org/10.3390/f12081012.

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The outer contours of living trees are often considered as a standard circle during non-destructive testing (NDT) of internal defects using ground-penetrating radar (GPR). However, the detection of classical cross-sections (circular) lacks consideration of irregular contours, making it difficult to accurately locate the radar image of the target. In this paper, we propose a method based on the image affine transformation and the Riemann mapping principle to analyze the effect of irregular detection routes on the geometric characteristics of target reflection hyperbola. First, for the similar output phenomenon in the “hyperbola fitting”, geometric analysis and numerical simulation were performed. Then, the conversion of irregular trunk radar images and physical domain radar images was implemented using the method of image affine transformation and the Riemann mapping principle. Finally, the influence of irregular detection routes on the geometry of the target reflection curve was investigated in detail through numerical simulations and actual experiments. The numerical simulation and measurement results demonstrated that the method in this study could better reflect the imaging characteristics of the target reflection hyperbola under the irregular detection pattern. This method provides assistance to further study the defects of irregular living trees and prevents the misjudgment of targets as a result of hyperbolic distortion, resulting in a greater prospect of application.

23

Hitching, George H. "Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor". MATHEMATICA SCANDINAVICA 112, n. 1 (1 marzo 2013): 61. http://dx.doi.org/10.7146/math.scand.a-15233.

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Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

24

Bojarski, B., e G. Khimshiashvili. "Global Geometric Aspects of Riemann–Hilbert Problems". gmj 8, n. 4 (dicembre 2001): 713–26. http://dx.doi.org/10.1515/gmj.2001.713.

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Abstract We discuss some global properties of an abstract geometric model for Riemann–Hilbert problems introduced by the first author. In particular, we compute the homotopy groups of elliptic Riemann–Hilbert problems and describe some connections with the theory of Fredholm structures which enable one to introduce more subtle geometrical and topological invariants for families of such problems.

25

Debbasch, Fabrice. "Discrete Geometry from Quantum Walks". Condensed Matter 4, n. 2 (11 aprile 2019): 40. http://dx.doi.org/10.3390/condmat4020040.

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A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and 2-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.

26

Eceizabarrena, Daniel. "Geometric differentiability of Riemann's non-differentiable function". Advances in Mathematics 366 (giugno 2020): 107091. http://dx.doi.org/10.1016/j.aim.2020.107091.

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27

Chen, Chuanmiao. "Local Geometric Proof of Riemann Conjecture". Advances in Pure Mathematics 10, n. 10 (2020): 589–610. http://dx.doi.org/10.4236/apm.2020.1010036.

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Chen, Chuanmiao. "Geometric Proof of Riemann Conjecture (Continued)". Advances in Pure Mathematics 11, n. 09 (2021): 771–83. http://dx.doi.org/10.4236/apm.2021.119051.

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29

SHEN, YiBing. "Several problems in Riemann-Finsler geometry". SCIENTIA SINICA Mathematica 45, n. 10 (1 settembre 2015): 1611–18. http://dx.doi.org/10.1360/n012015-00214.

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Willmore, Tom. "Riemann extensions and affine differential geometry". Results in Mathematics 13, n. 3-4 (maggio 1988): 403–8. http://dx.doi.org/10.1007/bf03323255.

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31

Gu, Xianfeng, Yalin Wang e Shing-Tung Yau. "Geometric Compression Using Riemann Surface Structure". Communications in Information and Systems 3, n. 3 (2003): 171–82. http://dx.doi.org/10.4310/cis.2003.v3.n3.a2.

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32

Giddings, Steven B., e Philip Nelson. "The geometry of super Riemann surfaces". Communications in Mathematical Physics 116, n. 4 (dicembre 1988): 607–34. http://dx.doi.org/10.1007/bf01224903.

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33

Friedman, Yaakov, Tzvi Scarr e Joseph Steiner. "A geometric relativistic dynamics under any conservative force". International Journal of Geometric Methods in Modern Physics 16, n. 01 (gennaio 2019): 1950015. http://dx.doi.org/10.1142/s0219887819500154.

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Riemann’s principle “force equals geometry” provided the basis for Einstein’s General Relativity — the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton’s First Law — the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new Relativistic Newtonian Dynamics for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

34

Ma, Wen-Xiu. "Trigonal curves and algebro-geometric solutions to soliton hierarchies II". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, n. 2203 (luglio 2017): 20170233. http://dx.doi.org/10.1098/rspa.2017.0233.

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This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel–Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker–Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker–Akhiezer functions. We emphasize that we analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

35

Zhang, Yachao, Xuan Lai, Yuan Xie, Yanyun Qu e Cuihua Li. "Geometry-Aware Discriminative Dictionary Learning for PolSAR Image Classification". Remote Sensing 13, n. 6 (23 marzo 2021): 1218. http://dx.doi.org/10.3390/rs13061218.

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In this paper, we propose a new discriminative dictionary learning method based on Riemann geometric perception for polarimetric synthetic aperture radar (PolSAR) image classification. We made an optimization model for geometry-aware discrimination dictionary learning in which the dictionary learning (GADDL) is generalized from Euclidian space to Riemannian manifolds, and dictionary atoms are composed of manifold data. An efficient optimization algorithm based on an alternating direction multiplier method was developed to solve the model. Experiments were implemented on three public datasets: Flevoland-1989, San Francisco and Flevoland-1991. The experimental results show that the proposed method learned a discriminative dictionary with accuracies better those of comparative methods. The convergence of the model and the robustness of the initial dictionary were also verified through experiments.

36

Gursky, Matthew J., e Jeffrey Streets. "A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow". Geometric Flows 4, n. 1 (1 gennaio 2019): 30–50. http://dx.doi.org/10.1515/geofl-2019-0003.

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Abstract We define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

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Pokhariyal, Ganesh Prasad. "Geometric and Physical Properties of Curvature Tensors -A Review". JOURNAL OF INTERNATIONAL ACADEMY OF PHYSICAL SCIENCES 27, n. 03 (30 agosto 2023): 193–205. http://dx.doi.org/10.61294/jiaps2023.2731.

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Bernard Riemann was the first to define curvature tensor. Most of the curvature tensors are defined with the help of Riemann curvaturetensor, Ricci tensor and metric tensor.It has been observed that different combinations of Ricci tensor and metric tensor in the defined tensors lead to some of the different geometrical and physical properties.M.S.C. 2010:53C25, 53C43.

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Tao, Mengshuang, e Huanhe Dong. "Algebro-Geometric Solutions for a Discrete Integrable Equation". Discrete Dynamics in Nature and Society 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/5258375.

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With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

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Bowers, Philip L., e Kenneth Stephenson. "The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense". Mathematical Proceedings of the Cambridge Philosophical Society 111, n. 3 (maggio 1992): 487–513. http://dx.doi.org/10.1017/s0305004100075575.

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W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurston's conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurston's work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincaré – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

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Stack, George J. "Riemann’s Geometry and Eternal Recurrence as Cosmological Hypothesis". International Studies in Philosophy 21, n. 2 (1989): 37–40. http://dx.doi.org/10.5840/intstudphil198921266.

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Eceizabarrena, Daniel. "Some geometric properties of Riemann's non-differentiable function". Comptes Rendus Mathematique 357, n. 11-12 (novembre 2019): 846–50. http://dx.doi.org/10.1016/j.crma.2019.10.007.

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çevik, Dinçer. "Riemann’s Philosophy of Geometry and Kant’s Pure Intuition". Organon F 31, n. 2 (31 maggio 2024): 114–40. http://dx.doi.org/10.31577/orgf.2024.31202.

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CASTRO, CARLOS, e JORGE MAHECHA. "FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS". International Journal of Geometric Methods in Modern Physics 01, n. 06 (dicembre 2004): 751–93. http://dx.doi.org/10.1142/s0219887804000393.

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Abstract (sommario):

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert–Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet–Bandrauk–Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D=1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn=nπ and which coincide with the imaginary parts of the zeros of the function sinh (s). Finally, we discuss the relationship to the theory of 1/f noise.

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Blaga, Adara. "Geometric solitons in a D-homothetically deformed Kenmotsu manifold". Filomat 36, n. 1 (2022): 175–86. http://dx.doi.org/10.2298/fil2201175b.

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Abstract (sommario):

We consider almost Riemann and almost Ricci solitons in a D-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure, and explicitly obtain the Ricci and scalar curvatures for some cases. We also provide a lower bound for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton.

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Elbasraoui, Abdelkrim, e Abdellah Sebbar. "Equivariant Forms: Structure and Geometry". Canadian Mathematical Bulletin 56, n. 3 (1 settembre 2013): 520–33. http://dx.doi.org/10.4153/cmb-2011-195-2.

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Abstract.In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of SL2(ℤ) by means of the cross-ratio, weight 2 modular forms, quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

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Yavari, Arash, e Alain Goriely. "Riemann–Cartan geometry of nonlinear disclination mechanics". Mathematics and Mechanics of Solids 18, n. 1 (23 marzo 2012): 91–102. http://dx.doi.org/10.1177/1081286511436137.

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Kong, De-Xing, Kefeng Liu e De-Liang Xu. "The Hyperbolic Geometric Flow on Riemann Surfaces". Communications in Partial Differential Equations 34, n. 6 (14 maggio 2009): 553–80. http://dx.doi.org/10.1080/03605300902768933.

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TAKHTAJAN, LEON A. "LIOUVILLE THEORY: QUANTUM GEOMETRY OF RIEMANN SURFACES". Modern Physics Letters A 08, n. 37 (7 dicembre 1993): 3529–35. http://dx.doi.org/10.1142/s0217732393002269.

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Abstract (sommario):

Inspired by Polyakov’s original formulation1,2 of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that the total quantum correction to the central charge of Liouville theory is given by one-loop contribution, which is equal to 1. Applied to the bosonic string, this result ensures the vanishing of total conformal anomaly along the lines different from those presented by KPZ3 and Distler-Kawai.4

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Alan Kostelecký, V., N. Russell e R. Tso. "Bipartite Riemann–Finsler geometry and Lorentz violation". Physics Letters B 716, n. 3-5 (ottobre 2012): 470–74. http://dx.doi.org/10.1016/j.physletb.2012.09.002.

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Kostelecký, V. Alan. "Riemann–Finsler geometry and Lorentz-violating kinematics". Physics Letters B 701, n. 1 (giugno 2011): 137–43. http://dx.doi.org/10.1016/j.physletb.2011.05.041.

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