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Journal articles on the topic 'Geometric law'

Author: Grafiati

Published: 7 June 2025

Last updated: 2 August 2025

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1

Gilman, Robert H., Yuri Gurevich, and Alexei Miasnikov. "A geometric zero-one law." Journal of Symbolic Logic 74, no. 3 (2009): 929–38. http://dx.doi.org/10.2178/jsl/1245158092.

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AbstractEach relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. If x is an element of X, let Bn(x) be the ball of radius n around x. Suppose that X is infinite, connected and of bounded degree. A first-order sentence ϕ in the language of X is almost surely true (resp. a.s. false) for finite substructures of X if for every x ∈ X, the fraction of substructures of Bn(x) satisfying ϕ approaches 1 (resp. 0) as n approaches infinity. Suppose further that, for every finite substructure, X has a disjoint isomorphic substructure. Then every ϕ is a.s.

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2

Malthouse, Edward C., and Kalyan Raman. "The Geometric Law of Annual Halving." Journal of Interactive Marketing 27, no. 1 (2013): 28–35. http://dx.doi.org/10.1016/j.intmar.2012.08.001.

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3

Henschel, V., and W. D. Richter. "Geometric Generalization of the Exponential Law." Journal of Multivariate Analysis 81, no. 2 (2002): 189–204. http://dx.doi.org/10.1006/jmva.2001.2001.

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4

Li, Bin, Qian Shou Liu, Di Min Wu, Zi Hui Zhang, and Yang Ke Zhou. "Control Law Design Based on Geometric Algebra." Applied Mechanics and Materials 347-350 (August 2013): 496–500. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.496.

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A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

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5

Hill, Richard. "A geometric proof of a reciprocity law." Nagoya Mathematical Journal 137 (March 1995): 77–144. http://dx.doi.org/10.1017/s0027763000005080.

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In this paper we prove the reciprocity law for a Kummer extension of an algebraic number field K. The proof is similar to the proof of the same theorem by Kubota [14, 15]. Such methods were applied by Gauss [6, 7] to the cases K ═ Q, and by Habicht [8] to the case . We now discuss informally the structure of the proof. All definitions and statements made at this stage are only approximations to the truth, and shouldn’t be used as references for the later chapters.

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6

Urusov, V. S. "Geometric model for deviations from Vegard's law." Journal of Structural Chemistry 33, no. 1 (1992): 68–79. http://dx.doi.org/10.1007/bf00753064.

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7

Anitori, Laura, Rajan Srinivasan, and Muralidhar Rangaswamy. "Envelope-Law and Geometric-Mean STAP Detection." IEEE Transactions on Aerospace and Electronic Systems 46, no. 1 (2010): 184–92. http://dx.doi.org/10.1109/taes.2010.5417155.

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8

Kifer, Yuri, and Ariel Rapaport. "Geometric law for multiple returns until a hazard." Nonlinearity 32, no. 4 (2019): 1525–45. http://dx.doi.org/10.1088/1361-6544/aafbd4.

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9

Fuming, Chu, and Fang Fukang. "Geometric approach to Fokker-Planck equation, conservation law." Chinese Physics Letters 3, no. 7 (1986): 333–35. http://dx.doi.org/10.1088/0256-307x/3/7/012.

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10

Dufresne, Daniel. "The integral of geometric Brownian motion." Advances in Applied Probability 33, no. 1 (2001): 223–41. http://dx.doi.org/10.1017/s0001867800010715.

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This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

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11

Körpınar, T., R. Cem Demirkol, Z. Körpınar, and V. Asil. "Maxwellian evolution equations along the uniform optical fiber in Minkowski space." Revista Mexicana de Física 66, no. 4 Jul-Aug (2020): 431. http://dx.doi.org/10.31349/revmexfis.66.431.

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We firstly discuss the geometric phase rotation for an electromagnetic wave traveling along the optical fiber in Minkowski space. We define two types of novel geometric phases associated with the evolution of the polarization vectors in the normal and binormal directions along the optical fiber by identifying the normal-Rytov parallel transportation law and binormal-Rytov parallel transportation law and derive their relationships with the new types of Fermi-Walker transportation law in Minkowski space. Then we describe a novel approach of solving Maxwell's equations in terms of electromagnetic

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12

Wu, Di Min, Zhen Jing Li, Bin Li, Yu Xia Chen, and Li Li. "Position and Attitude Tracking Control Law Design Using Geometric Algebra." Applied Mechanics and Materials 602-605 (August 2014): 1113–16. http://dx.doi.org/10.4028/www.scientific.net/amm.602-605.1113.

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A position and attitude tracking control law is developed using geometric algebra (GA). The rigid body motion can be represented by the screw versor (or motor) in GA. Using the kinematics of the motor, the tracking control law of the rigid body motion can be formulated similar to the proportional control law. This paper provides a GA-based position and attitude tracking control law by using the negative feedback of the motor logarithm. The stability of the control law is validated by the numerical simulation.

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13

Kımuya, Alex. "Re-examining the law of energy conservation-A Euclidean geometric proof." Eurasian Journal of Science Engineering and Technology 6, no. 1 (2025): 1–35. https://doi.org/10.55696/ejset.1559047.

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The law of energy conservation is a cornerstone of physics, limiting energy use and dictating the efficiency of thermodynamic processes. The primary objective of this paper is to challenge the traditional acceptance of the law of energy conservation as an unprovable axiom by presenting a novel, provable, and purely geometric approach within the framework of Euclidean geometry, thereby re-evaluating its theoretical and empirical foundations. Driven by the ongoing pursuit of solutions to energy crises, the paper critically examines attempts to disprove the law and the search for alternative ener

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14

Miztani, Euich. "Magnetic Helix-Relativistic Ampere-Maxwell Law from Geometric Viewpoint." Communications in Applied Geometry 3, no. 1 (2016): 17. http://dx.doi.org/10.37622/cag/3.1.2016.17-19.

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15

Cho, Namhoon, Youdan Kim, and Sanghyuk Park. "Three-Dimensional Nonlinear Differential Geometric Path-Following Guidance Law." Journal of Guidance, Control, and Dynamics 38, no. 12 (2015): 2366–85. http://dx.doi.org/10.2514/1.g001060.

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16

SHIN, J. K., and G. S. SHIN. "DEVIATION OF THE POWER-LAW BY GEOMETRIC AGING EFFECT." Fractals 15, no. 02 (2007): 139–49. http://dx.doi.org/10.1142/s0218348x07003514.

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An agent-based model is employed for the study of the group size distributions. A fixed number of homogeneous agents are distributed on a two-dimensional lattice system. The dynamics of the agents is described in terms of the inverse distance potential and the friction factor. From a random initial distribution, the agents move forming groups until all the agents come to a stationary position. For a squared system with L × L cells, the group size distribution showed a well defined power-law behavior up to the cut-off size. But when the system changed to an L × H non-squared one, a "geometric a

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17

MORET, M. A., V. de SENNA, M. C. SANTANA, and G. F. ZEBENDE. "GEOMETRIC STRUCTURAL ASPECTS OF PROTEINS AND NEWCOMB–BENFORD LAW." International Journal of Modern Physics C 20, no. 12 (2009): 1981–88. http://dx.doi.org/10.1142/s0129183109014874.

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The major factor that drives a protein toward collapse and folding is the hydrophobic effect. At the folding process a hydrophobic core is shielded by the solvent-accessible surface area of the protein. We study the behavior of the numbers in 5526 protein structures present in the Brookhaven Protein Data Bank. The first digit of mass, volume, average radius and solvent-accessible surface area are measured independently and we observe that most of these geometric observables obey the Newcomb–Benford law. That is volume, mass and average radius obey the Newcomb–Benford law. Nevertheless, the dig

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18

Barreto-Souza, Wagner. "Bivariate gamma-geometric law and its induced Lévy process." Journal of Multivariate Analysis 109 (August 2012): 130–45. http://dx.doi.org/10.1016/j.jmva.2012.03.004.

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19

Marcott, Craig, and Robert Riley. "The phillips curve and Okun's law: A geometric interpretation." International Advances in Economic Research 2, no. 2 (1996): 195. http://dx.doi.org/10.1007/bf02295063.

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20

Dass, Amala. "Nano-scaling law: geometric foundation of thiolated gold nanomolecules." Nanoscale 4, no. 7 (2012): 2260. http://dx.doi.org/10.1039/c2nr11749e.

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21

Li, Chao-Yong, and Wu-Xing Jing. "Geometric approach to capture analysis of PN guidance law." Aerospace Science and Technology 12, no. 2 (2008): 177–83. http://dx.doi.org/10.1016/j.ast.2007.04.007.

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22

Mondai, S., and R. Padhi. "Formation Flying using GENEX and Differential geometric guidance law." IFAC-PapersOnLine 48, no. 9 (2015): 19–24. http://dx.doi.org/10.1016/j.ifacol.2015.08.053.

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23

WANG, LIQIU. "SECOND LAW OF THERMODYNAMICS AND ARITHMETIC-MEAN–GEOMETRIC-MEAN INEQUALITY." International Journal of Modern Physics B 13, no. 21n22 (1999): 2791–93. http://dx.doi.org/10.1142/s0217979299002678.

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The application of the second law of thermodynamics to a typical irreversible process of a thermally isolated system shows that the Arithmetic-mean–geometric-mean (AM–GM) inequality, a powerful mathematical inequality, follows logically from the second law of thermodynamics, a powerful physical law.

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24

Li, Zhengfang, Zhengyuan Gao, Zhiguo An, et al. "Effect Analysis of Process Parameters on Geometric Dimensions during Belt-Heated Incremental Sheet Forming of AA2024 Aluminum Alloy." Coatings 14, no. 7 (2024): 889. http://dx.doi.org/10.3390/coatings14070889.

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The effect of process parameters on geometric dimensions is complex during belt-heated incremental sheet forming, leading to control difficulties in the geometric dimension of parts. In this study, the geometric errors were analyzed under the action of different process parameters, and the corresponding influence law was obtained through macro- and microexperiments. On this basis, the micromorphology of the deformation region section was analyzed in detail, and the effect of process parameters on micromorphology characteristics was obtained during belt-heated incremental sheet forming. Meanwhi

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25

Michel, Jesse, Sushruth Reddy, Rikhav Shah, Sandeep Silwal, and Ramis Movassagh. "Directed random geometric graphs." Journal of Complex Networks 7, no. 5 (2019): 792–816. http://dx.doi.org/10.1093/comnet/cnz006.

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Abstract Many real-world networks are intrinsically directed. Such networks include activation of genes, hyperlinks on the internet and the network of followers on Twitter among many others. The challenge, however, is to create a network model that has many of the properties of real-world networks such as power-law degree distributions and the small-world property. To meet these challenges, we introduce the Directed Random Geometric Graph (DRGG) model, which is an extension of the random geometric graph model. We prove that it is scale-free with respect to the indegree distribution, has binomi

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26

Kang, Jeong Gi. "In Newton’s proof of the inverse square law, geometric limit analysis and Educational discussion." Korean School Mathematics Society 24, no. 2 (2021): 173–90. http://dx.doi.org/10.30807/ksms.2021.24.2.001.

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This study analyzed the proof of the inverse square law, which is said to be the core of Newton's <Principia>, in relation to the geometric limit. Newton, conscious of the debate over infinitely small, solved the dynamics problem with the traditional Euclid geometry. Newton reduced mechanics to a problem of geometry by expressing force, time, and the degree of inertia orbital deviation as a geometric line segment. Newton was able to take Euclid's geometry to a new level encompassing dynamics, especially by introducing geometric limits such as parabolic approximation, polygon approximatio

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27

Zhang, Yilei, Zhili Hu, Jin Li, and Mingliang Dai. "Exploring Filling Law of Small Fillet of Dual Clutch Hub." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 36, no. 3 (2018): 464–70. http://dx.doi.org/10.1051/jnwpu/20183630464.

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In extruding the dual clutch hub with small fillet gear, the filling of small fillet gear is not complete due to the large forming resistance. A method of preforming the blank into a concave shape is proposed. The mechanical analysis and finite element simulation are used to analyze the principle that the blank of concave shape can promote the filling of small fillet gear, stamping and extruding are both used to form the dual clutch hub and the preform is used to simulate the fillet gear’s extruding to analyze the influence of the geometric parameters of the preform on the forming quality. The

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28

ZHOU, LING, and SHENG-JUN WANG. "FRACTAL AGGREGATES ON GEOMETRIC GRAPHS." Fractals 26, no. 03 (2018): 1850038. http://dx.doi.org/10.1142/s0218348x1850038x.

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We study the aggregation process on the geometric graph. The geometric graph is composed by sites randomly distributed in space and connected locally. Similar to the regular lattice, the network possesses local connection, but the randomness in the spatial distribution of sites is considered. We show that the correlations within the aggregate patterns fall off with distance with a fractional power law. The numerical simulation results indicate that the aggregate patterns on the geometric graph are fractal. The fractals are robust against the randomness in the structure. A remarkable new featur

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29

Kim, Joo-Cheol, and Sang-Jin Lee. "Hack's Law and the Geometric Properties of Catchment Plan-form." Journal of Korea Water Resources Association 42, no. 9 (2009): 691–702. http://dx.doi.org/10.3741/jkwra.2009.42.9.691.

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30

Yatsko, V. A. "The interpretation of Bradford’s law in terms of geometric progression." Automatic Documentation and Mathematical Linguistics 46, no. 2 (2012): 112–17. http://dx.doi.org/10.3103/s0005105512020094.

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31

Cohen, Joel E., and Daniel Courgeau. "Modeling distances between humans using Taylor’s law and geometric probability." Mathematical Population Studies 24, no. 4 (2017): 197–218. http://dx.doi.org/10.1080/08898480.2017.1289049.

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32

Bovier, Anton, and Pierre Picco. "A Law of the Iterated Logarithm for Random Geometric Series." Annals of Probability 21, no. 1 (1993): 168–84. http://dx.doi.org/10.1214/aop/1176989399.

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33

Clement Fernández, R., J. M. Echarri Hernández, and E. J. Gómez Ayala. "A geometric proof of Kummer's reciprocity law for seventh powers." Acta Arithmetica 146, no. 4 (2011): 299–318. http://dx.doi.org/10.4064/aa146-4-1.

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34

Kröplin, B., and D. Dinkler. "A material law for coupled local yielding and geometric instability." Engineering Computations 5, no. 3 (1988): 210–16. http://dx.doi.org/10.1108/eb023738.

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35

Gonzalez-Gascon, F. "Geometric foundations of a new conservation law discovered by Hojman." Journal of Physics A: Mathematical and General 27, no. 2 (1994): L59—L60. http://dx.doi.org/10.1088/0305-4470/27/2/010.

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36

Paik, Kyungrock, and Praveen Kumar. "Power-Law Behavior in Geometric Characteristics of Full Binary Trees." Journal of Statistical Physics 142, no. 4 (2011): 862–78. http://dx.doi.org/10.1007/s10955-011-0125-y.

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37

Wolf, Kurt Bernardo. "The Euclidean root of Snell’s law I. Geometric polarization optics." Journal of Mathematical Physics 33, no. 7 (1992): 2390–408. http://dx.doi.org/10.1063/1.529608.

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38

Cao, Weiming, Weizhang Huang, and Robert D. Russell. "A Moving Mesh Method Based on the Geometric Conservation Law." SIAM Journal on Scientific Computing 24, no. 1 (2002): 118–42. http://dx.doi.org/10.1137/s1064827501384925.

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39

Dhananjay, N., D. Ghose, and M. S. Bhat. "Capturability of a Geometric Guidance Law in Relative Velocity Space." IEEE Transactions on Control Systems Technology 17, no. 1 (2009): 111–22. http://dx.doi.org/10.1109/tcst.2008.924561.

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40

Núñez, Manuel. "Generalized Ohm’s law and geometric optics: Applications to magnetosonic waves." International Journal of Non-Linear Mechanics 110 (April 2019): 21–25. http://dx.doi.org/10.1016/j.ijnonlinmec.2019.01.007.

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41

Yan, Xinghui, Yuzhong Tang, Yulei Xu, Heng Shi, and Jihong Zhu. "Multi-Constrained Geometric Guidance Law with a Data-Driven Method." Drones 7, no. 10 (2023): 639. http://dx.doi.org/10.3390/drones7100639.

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A data-driven geometric guidance method is proposed for the multi-constrained guidance problem of variable-velocity unmanned aerial vehicles (UAVs). Firstly, a two-phase flight trajectory based on a log-aesthetic space curve (LASC) is designed. The impact angle is satisfied by a specified straight-line segment. The impact time is controlled by adjusting the phase switching point. Secondly, a deep neural network is trained offline to establish the mapping relationship between the initial conditions and desired trajectory parameters. Based on this mapping network, the desired flight trajectory c

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42

Fowkes, Nicholas. "An elementary geometric proof of Snell’s law from Fermat’s principle." Mathematical Gazette 109, no. 574 (2025): 27–33. https://doi.org/10.1017/mag.2025.5.

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In the long history of the study of the refraction of light, Willebrord Snell is credited with being the first to show that the ratio of the sines of the angles of incidence and reflection is constant for any given pair of media.

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43

Kinugasa, T., C. Chevallereau, and Y. Aoustin. "Effect of circular arc feet on a control law for a biped." Robotica 27, no. 4 (2009): 621–32. http://dx.doi.org/10.1017/s0263574708005006.

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SUMMARYThe purpose of our research is to study the effects of circular arc feet on the biped walk with a geometric tracking control. The biped studied is planar and is composed of five links and four actuators located at each hip and each knee thus the biped is underactuated in single support phase. A geometric evolution of the biped configuration is controlled, instead of a temporal evolution. The input-output linearization with a PD control law and a feed forward compensation is used for geometric tracking. The controller virtually constrains 4 degrees of freedom (DoF) of the biped, and 1 Do

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44

Yukich, J. E. "Ultra-small scale-free geometric networks." Journal of Applied Probability 43, no. 3 (2006): 665–77. http://dx.doi.org/10.1239/jap/1158784937.

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We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.

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45

Fuchs, Michael, and Mehri Javanian. "Limit behavior of maxima in geometric words representing set partitions." Applicable Analysis and Discrete Mathematics 9, no. 2 (2015): 313–31. http://dx.doi.org/10.2298/aadm150619013f.

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We consider geometric words ?1....?n with letters satisfying the restricted growth property ?k ? d + max{?0,...,?k-1}, where ?0 := 0 and d ? 1. For d = 1 these words are in 1-to-1 correspondence with set partitions and for this case, we show that the number of left-to-right maxima (suitable centered) does not converge to a fixed limit law as n tends to infinity. This becomes wrong for d ? 2, for which we prove that convergence does occur and the limit law is normal. Moreover, we also consider related quantities such as the value of the maximal letter and the number of maximal letters and show

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46

Heydarzadeh, M., M. Mohsenzadeh, M. Abbasian-Motlaq, and E. Yusofi. "Asymptotic de Sitter inflation in different geometric backgrounds." International Journal of Geometric Methods in Modern Physics 18, no. 08 (2021): 2150118. http://dx.doi.org/10.1142/s0219887821501188.

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In this paper, we will show that a power law asymptotic de Sitter inflation will be established for all values of Hankel indices [Formula: see text] that relate to the different background spacetimes of the universe in their evolution. First, we calculate the relations for the Hubble, deceleration, slow roll and the equation of state parameters in terms of the Hankel function index [Formula: see text]. We then show that the obtained relations and figures correspond to the conventional limit conditions for pure de Sitter inflation. As an important result, power law quasi-de Sitter inflation is

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47

Shi, Peijian, David A. Ratkowsky, and Johan Gielis. "The Generalized Gielis Geometric Equation and Its Application." Symmetry 12, no. 4 (2020): 645. http://dx.doi.org/10.3390/sym12040645.

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Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the

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48

Kronberg, E. A., and P. W. Daly. "Spectral analysis for wide energy channels." Geoscientific Instrumentation, Methods and Data Systems Discussions 3, no. 2 (2013): 533–46. http://dx.doi.org/10.5194/gid-3-533-2013.

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Abstract. For energetic particle measurements whose spectra follow a power law, it is often challenging to define a characteristic ("effective") energy of an energy channel. In order to avoid time-consuming calculations, the geometric mean is often used as an approximation for the effective energy. This approximation is considered to be pretty good. It is, however, potentially inadequate in cases with wide energy channels and soft spectral slopes. In order to determine the limits of the goodness of the approximation, we derive formulas to calculate the deviation of the effective energy, phase

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49

Kronberg, E. A., and P. W. Daly. "Spectral analysis for wide energy channels." Geoscientific Instrumentation, Methods and Data Systems 2, no. 2 (2013): 257–61. http://dx.doi.org/10.5194/gi-2-257-2013.

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Abstract. For energetic particle measurements whose spectra follow a power law, it is often challenging to define a characteristic ("effective") energy of an energy channel. In order to avoid time-consuming calculations, the geometric mean is often used as an approximation for the effective energy. This approximation is considered to be pretty good. It is, however, potentially inadequate in cases with wide energy channels and soft spectral slopes. In order to determine the limits of the goodness of the approximation, we derive formulas to calculate the deviation of the effective energy, phase

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50

Yukich, J. E. "Ultra-small scale-free geometric networks." Journal of Applied Probability 43, no. 03 (2006): 665–77. http://dx.doi.org/10.1017/s0021900200002011.

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We consider a family of long-range percolation models (G p ) p&gt;0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd log pd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤ d ∩ [0, n] d ), resulting in two large, disconnected subgraphs.

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