scholarly journals Advancing Shannon Entropy for Measuring Diversity in Systems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
R. Rajaram ◽  
B. Castellani ◽  
A. N. Wilson
Keyword(s):  
Shannon Entropy ◽  
Probability Distributions ◽  
Power Laws ◽  
Cumulative Probability ◽  
Energy Distributions ◽  
Scale Free ◽  
Subatomic Particles ◽  
Measuring Diversity ◽  
Bose Einstein ◽  
Science Part

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy H has been used to rethink diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon’s approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a renormalization of probability distributions based on the notion of case-based entropy Cc as a function of the cumulative probability c. Given a probability density p(x), Cc measures the diversity of the distribution up to a cumulative probability of c, by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of p^c(x) up to cumulative probability c. We illustrate the utility of our approach by renormalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for energy of subatomic particles. The comparison shows that Cc is a vast improvement over H as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions.

Opposite Effects for Cations in Enhancing and Suppressing Nucleation of Pathology Crystallization of Gout

CrystEngComm ◽  
2021 ◽  
Author(s):  
anqi da ◽  
aijun ren ◽  
Rong Chen ◽  
Xingde Zhang ◽  
yonglan pan ◽  
...  
Keyword(s):  
Probability Distributions ◽  
Crystal Nucleation ◽  
Cumulative Probability ◽  
Monosodium Urate

In this paper, we focused on the influence of common serum cations on monosodium urate monohydrate (MSUM) crystals nucleation determined from cumulative probability distributions (CPD) of the solute crystal nucleation...

scholarly journals RECIPROCITY AND THE EMERGENCE OF POWER LAWS IN SOCIAL NETWORKS

2006 ◽  
Vol 17 (07) ◽  
pp. 1067-1076 ◽  
Author(s):  
MICHAEL SCHNEGG
Keyword(s):  
Social Networks ◽  
Power Laws ◽  
Scaling Exponent ◽  
Policy Makers ◽  
Public Health Programs ◽  
Scale Free ◽  
The Past ◽  
Naturally Occurring ◽  
Growing Networks ◽  
Law Degree

Research in network science has shown that many naturally occurring and technologically constructed networks are scale free, that means a power law degree distribution emerges from a growth model in which each new node attaches to the existing network with a probability proportional to its number of links (= degree). Little is known about whether the same principles of local attachment and global properties apply to societies as well. Empirical evidence from six ethnographic case studies shows that complex social networks have significantly lower scaling exponents γ ~ 1 than have been assumed in the past. Apparently humans do not only look for the most prominent players to play with. Moreover cooperation in humans is characterized through reciprocity, the tendency to give to those from whom one has received in the past. Both variables — reciprocity and the scaling exponent — are negatively correlated (r = -0.767, sig = 0.075). If we include this effect in simulations of growing networks, degree distributions emerge that are much closer to those empirically observed. While the proportion of nodes with small degrees decreases drastically as we introduce reciprocity, the scaling exponent is more robust and changes only when a relatively large proportion of attachment decisions follow this rule. If social networks are less scale free than previously assumed this has far reaching implications for policy makers, public health programs and marketing alike.

scholarly journals Vortex and half-vortex dynamics in a nonlinear spinor quantum fluid

2015 ◽  
Vol 1 (11) ◽  
pp. e1500807 ◽  
Author(s):  
Lorenzo Dominici ◽  
Galbadrakh Dagvadorj ◽  
Jonathan M. Fellows ◽  
Dario Ballarini ◽  
Milena De Giorgi ◽  
...  
Keyword(s):  
Vortex Dynamics ◽  
Strong Nonlinearity ◽  
Phase Detection ◽  
Ultrafast Imaging ◽  
Half Integer ◽  
Fundamental Particles ◽  
Subatomic Particles ◽  
Phase Singularities ◽  
The Moment ◽  
Bose Einstein

Vortices are archetypal objects that recur in the universe across the scale of complexity, from subatomic particles to galaxies and black holes. Their appearance is connected with spontaneous symmetry breaking and phase transitions. In Bose-Einstein condensates and superfluids, vortices are both point-like and quantized quasiparticles. We use a two-dimensional (2D) fluid of polaritons, bosonic particles constituted by hybrid photonic and electronic oscillations, to study quantum vortex dynamics. Polaritons benefit from easiness of wave function phase detection, a spinor nature sustaining half-integer vorticity, strong nonlinearity, and tuning of the background disorder. We can directly generate by resonant pulsed excitations a polariton condensate carrying either a full or half-integer vortex as initial condition and follow their coherent evolution using ultrafast imaging on the picosecond scale. The observations highlight a rich phenomenology, such as the spiraling of the half-vortex and the joint path of the twin charges of a full vortex, until the moment of their splitting. Furthermore, we observe the ordered branching into newly generated secondary couples, associated with the breaking of radial and azimuthal symmetries. This allows us to devise the interplay of nonlinearity and sample disorder in shaping the fluid and driving the vortex dynamics. In addition, our observations suggest that phase singularities may be seen as fundamental particles whose quantized events span from pair creation and recombination to 2D+t topological vortex strings.

scholarly journals Identifications of Faint IRAS Sources

1996 ◽  
Vol 171 ◽  
pp. 402-402
Author(s):  
M.W. Kümmel ◽  
S.J. Wagner
Keyword(s):  
Broad Band ◽  
Far Infrared ◽  
Power Laws ◽  
Point Sources ◽  
Spectral Energy ◽  
Energy Distributions ◽  
The North ◽  
Luminosity Functions ◽  
Error Circle ◽  
Radio Band

From overlapping scans in the IRAS all-sky survey and additional pointed observations the deepest far infrared survey before ISO exists in the region around the North Ecliptic Pole (NEP) (Hacking P. and Houck J.R., ApJS 63 p. 311). This survey contains detections up to 10 and fluxes up to 100 times fainter than the IRAS survey. In the central square degree around the NEP we combine the far IR-survey with deep radio data at 151 MHz and 1.5 GHz (Visser, A.E. et al., A&AS 110 p. 419, Kollgaard, R.I. et al., ApJS 93 p. 145) and own observation at 2.2μm (K′) and 435nm (B). The error circle around the IRAS source was chosen to include the true source with 85% probability (1.4 sigma). For 29 of the 32 IRAS sources we found at least one possible counterpart. Ten of the objects have multiple (up to four) counterparts in K′. Four of the IRAS sources have counterparts in the 1.5 GHz survey. The higher accuracy of the radio position (∼ 1″) allowed an unambiguous identification of the K′ counterpart. None of the IRAS sources could be found in the 151 MHz survey. The broad band spectra of the three galaxies with measured radio flux exhibit maximum emission between the radio band and 100μm which corresponds to emission by cool dust (< 50 K). Contrary to the infrared luminosity functions at 12μm and 60μm which show power laws, the K′ luminosity function is bimodal. The brightest K′ objects are all point sources. Due to the small number statistics the power law indices of the luminosity functions can not be distinguished. We find a linear relationship between the K′ flux and the flux at 60μm and 12μm over at least one decade. The large deviations by individual sources make an identification of the correct counterpart through this relation impossible. The spectral energy distributions of unambiguously identified sources span only one decade in energy (vSv), i.e. they have flat energy distributions. This suggests an identification of K′ objects with flat energy distribution in case of multiple counterparts.

scholarly journals Scale-free power-laws as interaction between progress and diffusion

Complexity ◽  
2013 ◽  
Vol 19 (4) ◽  
pp. 56-65 ◽  
Author(s):  
Martin Hilbert
Keyword(s):  
Power Laws ◽  
Scale Free ◽  
And Diffusion

Suppression of Bose-Einstein condensation in one-dimensional scale-free random potentials

2010 ◽  
Vol 82 (17) ◽  
Author(s):  
I. N. de Oliveira ◽  
F. A. B. F. de Moura ◽  
R. A. Caetano ◽  
M. L. Lyra
Keyword(s):  
Einstein Condensation ◽  
Random Potentials ◽  
One Dimensional ◽  
Scale Free ◽  
Bose Einstein Condensation ◽  
Bose Einstein

scholarly journals The effect of reported high-velocity small raindrops on inferred drop size distributions and derived power laws

2010 ◽  
Vol 10 (4) ◽  
pp. 9121-9151 ◽  
Author(s):  
H. Leijnse ◽  
R. Uijlenhoet
Keyword(s):  
High Velocity ◽  
Drop Size ◽  
Doppler Radar ◽  
Probability Distributions ◽  
Power Laws ◽  
Size Distributions ◽  
Optical Links ◽  
Gamma Distributions ◽  
Raindrop Size Distribution ◽  
Drop Size Distributions

Abstract. It has recently been shown that at high rainfall intensities, small raindrops may fall with much larger velocities than would be expected from their diameters. These were argued to be fragments of recently broken-up larger drops. In this paper we quantify the effect of this phenomenon on raindrop size distribution measurements from a Joss-Waldvogel disdrometer, a 2-D Video Distrometer, and a vertically-pointing Doppler radar. Probability distributions of fall velocities have been parameterized, where the parameters are functions of both rainfall intensity and drop size. These parameterizations have been used to correct Joss-Waldvogel disdrometer measurements for this phenomenon. The effect of these corrections on fitted scaled drop size distributions are apparent but not major. Fitted gamma distributions for three different types of rainfall have been used to simulate drop size measurements. The effect of the high-velocity small drops is shown to be minor. Especially for the purpose of remote sensing of rainfall using radar, microwave links, or optical links, the errors caused by using the slightly different retrieval relations will be masked completely by other error sources.

A Comparison of Probability Bounds Analysis and Sensitivity Analysis in Environmentally Benign Design and Manufacture

2006 ◽  
Author(s):  
Jason Matthew Aughenbaugh ◽  
Scott Duncan ◽  
Christiaan J. J. Paredis ◽  
Bert Bras
Keyword(s):  
Sensitivity Analysis ◽  
Epistemic Uncertainty ◽  
Probability Distributions ◽  
Environmentally Benign ◽  
Cumulative Probability ◽  
Uncertain Parameters ◽  
Probability Bounds ◽  
Irreducible Uncertainty ◽  
Design And Manufacture ◽  
Selection Of

There is growing acceptance in the design community that two types of uncertainty exist: inherent variability and uncertainty that results from a lack of knowledge, which variously is referred to as imprecision, incertitude, irreducible uncertainty, and epistemic uncertainty. There is much less agreement on the appropriate means for representing and computing with these types of uncertainty. Probability bounds analysis (PBA) is a method that represents uncertainty using upper and lower cumulative probability distributions. These structures, called probability boxes or just p-boxes, capture both variability and imprecision. PBA includes algorithms for efficiently computing with these structures under certain conditions. This paper explores the advantages and limitations of PBA in comparison to traditional decision analysis with sensitivity analysis in the context of environmentally benign design and manufacture. The example of the selection of an oil filter involves multiple objectives and multiple uncertain parameters. These parameters are known with varying levels of uncertainty, and different assumptions about the dependencies between variables are made. As such, the example problem provides a rich context for exploring the applicability of PBA and sensitivity analysis to making engineering decisions under uncertainty. The results reveal specific advantages and limitations of both methods. The appropriate choice of an analysis depends on the exact decision scenario.

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