Thiago Trafane Oliveira Santos, Daniel Oliveira Cajueiro
Zipf's law states that the probability of a variable being larger than $s$ is roughly inversely proportional to $s$. In this paper, we evaluate Zipf's law for the distribution of firm size by the number of employees in Brazil. We use publicly available binned annual data from the Central Register of Enterprises (CEMPRE), which is held by the Brazilian Institute of Geography and Statistics (IBGE) and covers all formal organizations. Remarkably, we find that Zipf's law provides a very good, although not perfect, approximation to data for each year between 1996 and 2020 at the economy-wide level and also for agriculture, industry, and services alone. However, a lognormal distribution also performs well and even outperforms Zipf's law in certain cases.
Road network representation learning aims to learn compressed and effective vectorized representations for road segments that are applicable to numerous tasks. In this paper, we identify the limitations of existing methods, particularly their overemphasis on the distance effect as outlined in the First Law of Geography. In response, we propose to endow road network representation with the principles of the recent Third Law of Geography. To this end, we propose a novel graph contrastive learning framework that employs geographic configuration-aware graph augmentation and spectral negative sampling, ensuring that road segments with similar geographic configurations yield similar representations, and vice versa, aligning with the principles stated in the Third Law. The framework further fuses the Third Law with the First Law through a dual contrastive learning objective to effectively balance the implications of both laws. We evaluate our framework on two real-world datasets across three downstream tasks. The results show that the integration of the Third Law significantly improves the performance of road segment representations in downstream tasks.
Kepler's 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained "later on" as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler's 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the "time=area" statement quantitatively, in the way of Kepler's equation, M = E - e sin E (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars' next opposition. This method is partially based on Kepler's original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.
We construct the CFT dual of the first law of spherical causal diamonds in three-dimensional AdS spacetime. A spherically symmetric causal diamond in AdS$_3$ is the domain of dependence of a spatial circular disk with vanishing extrinsic curvature. The bulk first law relates the variations of the area of the boundary of the disk, the spatial volume of the disk, the cosmological constant and the matter Hamiltonian. In this paper we specialize to first-order metric variations from pure AdS to the conical defect spacetime, and the bulk first law is derived following a coordinate based approach. The AdS/CFT dictionary connects the area of the boundary of the disk to the differential entropy in CFT$_2$, and assuming the `complexity=volume' conjecture, the volume of the disk is considered to be dual to the complexity of a cutoff CFT. On the CFT side we explicitly compute the differential entropy and holographic complexity for the vacuum state and the excited state dual to conical AdS using the kinematic space formalism. As a result, the boundary dual of the bulk first law relates the first-order variations of differential entropy and complexity to the variation of the scaling dimension of the excited state, which corresponds to the matter Hamiltonian variation in the bulk. We also include the variation of the central charge with associated chemical potential in the boundary first law. Finally, we comment on the boundary dual of the first law for the Wheeler-deWitt patch of AdS, and we propose an extension of our CFT first law to higher dimensions.
Over the last decade, the continuing decline in the cost of digital computing technology has brought about a dramatic transformation in how digital instrumentation for radio astronomy is developed and operated. In most cases, it is now possible to interface consumer computing hardware, e.g. inexpensive graphics processing units and storage devices, directly to the raw data streams produced by radio telescopes. Such systems bring with them myriad benefits: straightforward upgrade paths, cost savings through leveraging an economy of scale, and a lowered barrier to entry for scientists and engineers seeking to add new instrument capabilities. Additionally, the typical data-interconnect technology used with general-purpose computing hardware -- Ethernet -- naturally permits multiple subscribers to a single raw data stream. This allows multiple science programs to be conducted in parallel. When combined with broad bandwidths and wide primary fields of view, radio telescopes become capable of achieving many science goals simultaneously. Moreover, because many science programs are not strongly dependent on observing cadence and direction (e.g. searches for extraterrestrial intelligence and radio transient surveys), these so-called "commensal" observing programs can dramatically increase the scientific productivity and discovery potential of an observatory. In this whitepaper, we detail a project to add an Ethernet-based commensal observing mode to the Jansky Very Large Array (VLA), and discuss how this mode could be leveraged to conduct a powerful program to constrain the distribution of advanced life in the universe through a search for radio emission indicative of technology. We also discuss other potential science use-cases for the system, and how the system could be used for technology development towards next-generation processing systems for the Next Generation VLA.
Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratric strong law for the center of mass of the elephant random walk. The asymptotic normality of the center of mass, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.
Recently, Boehm and Stefan constructed duplicial (paracyclic) objects from distributive laws between (co)monads. Here we define the category of factorisations of a distributive law, show that it acts on this construction, and give some explicit examples.
Two fundamental issues surrounding research on Zipf's law regarding city sizes are whether and why this law holds. This paper does not deal with the latter issue with respect to why, and instead investigates whether Zipf's law holds in a global setting, thus involving all cities around the world. Unlike previous studies, which have mainly relied on conventional census data such as populations, and census-bureau-imposed definitions of cities, we adopt naturally (in terms of data speaks for itself) delineated cities, or natural cities, to be more precise, in order to examine Zipf's law. We find that Zipf's law holds remarkably well for all natural cities at the global level, and remains almost valid at the continental level except for Africa at certain time instants. We further examine the law at the country level, and note that Zipf's law is violated from country to country or from time to time. This violation is mainly due to our limitations; we are limited to individual countries, or to a static view on city-size distributions. The central argument of this paper is that Zipf's law is universal, and we therefore must use the correct scope in order to observe it. We further find Zipf's law applied to city numbers; the number of cities in the first largest country is twice as many as that in the second largest country, three times as many as that in the third largest country, and so on. These findings have profound implications for big data and the science of cities. Keywords: Night-time imagery, city-size distributions, head/tail division rule, head/tail breaks, big data
Using martingale convergence theorem, we prove a law of large numbers for monotone convolutions $μ_{1}\trianglerightμ_{2}\triangleright\cdots\trianglerightμ_{n}$, where $μ_{j}$'s are probability laws on $\mathbb{R}$ with finite variances but not required to be identical.
In this paper we analyze Gresham's Law, in particular, how the rate of inflow or outflow of currencies is affected by the demand elasticity of arbitrage and the difference in face value ratios inside and outside of a country under a bimetallic system. We find that these equations are very similar to those used to describe drift in systems of free charged particles. In addition, we look at how Gresham's Law would play out with multiple currencies and multiple countries under a variety of connecting topologies.
In this paper, we present a possible theoretical explanation for benford's law. We develop a recursive relation between the probabilities, using simple intuitive ideas. We first use numerical solutions of this recursion and verify that the solutions converge to the benford's law. Finally we solve the recursion analytically to yeild the benford's law for base 2.
In 1958 Jeffreys proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys-Lomnitz law of creep by allowing its power law exponent $α$, usually limited to the range [0,1] to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotonic derivative, with a related spectrum of retardation times. The complete range $α\le 1$ yields a continuous transition from a Hooke elastic solid with no creep ($α\to -\infty$) to a Maxwell fluid with linear creep ($α=1$) passing through the Lomnitz viscoelastic body with logarithmic creep ($α=0$), which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys-Lomnitz creep law extended to all $α\le 1$.
Zipf's law states that the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but require additional constraints. We show that Gibrat's rule, at all firm levels, yields Zipf's law under a balance condition between the effective growth rate of incumbent firms (which includes their possible demise) and the growth rate of investments in entrant firms. Remarkably, Zipf's law is the signature of the long-term optimal allocation of resources that ensures the maximum sustainable growth rate of an economy.
Using our recent attempt to formulate second law of thermodynamics in a general way into a language with a probability density function, we derive degenerate vacua. Under the assumption that many coupling constants are effectively ``dynamical'' in the sense that they are or can be counted as initial state conditions, we argue in our model behind the second law that these coupling constants will adjust to make several vacua all having their separate effective cosmological constants or, what is the same, energy densities, being almost the \underline{same} value, essentially zero. Such degeneracy of vacuum energy densities is what one of us works on a lot under the name "The multiple point principle" (MPP).
A simple proof of a strengthened form of the first law of black hole mechanics is presented. The proof is based directly upon the Hamiltonian formulation of general relativity, and it shows that the the first law variational formula holds for arbitrary nonsingular, asymptotically flat perturbations of a stationary, axisymmetric black hole, not merely for perturbations to other stationary, axisymmetric black holes. As an application of this strengthened form of the first law, we prove that there cannot exist Einstein-Maxwell black holes whose ergoregion is disjoint from the horizon. This closes a gap in the black hole uniqueness theorems.
Shinya Komugi, Yoshiaki Sofue, Hiroyuki Nakanishi
et al.
We have combined Halpha and recent high resolution CO(J=1-0) data to consider the quantitative relation between gas mass and star formation rate, or the so-called Schmidt law in nearby spiral galaxies at regions of high molecular density. The relation between gas quantity and star formation rate has not been previously studied for high density regions, but using high resolution CO data obtained at the NMA(Nobeyama Millimeter Array), we have found that the Schmidt law is valid at densities as high as $10^3 \mathrm{M_\odot} \mathrm{pc}^{-2}$ for the sample spiral galaxies, which is an order of magnitude denser than what has been known to be the maximum density at which the empirical law holds for non-starburst galaxies. Furthermore, we obtain a Schmidt law index of $N=1.33\pm0.09$ and roughly constant star formation efficiency over the entire disk, even within the several hundred parsecs of the nucleus. These results imply that the physics of star formation does not change in the central regions of spiral galaxies. Comparisons with starburst galaxies are also given. We find a possible discontinuity in the Schmidt law between normal and starburst galaxies.