scholarly journals The Centroid Solid Angle and Probability Models of Square Prism Dice Rolls

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Statistical Analysis ◽  
Solid Angle ◽  
Probability Model ◽  
Z Score ◽  
Probability Models ◽  
Mathematical Probability ◽  
Square Prism ◽  
Proposed Model ◽  
Solid Angles ◽  
Spherical Projection

Past studies have indicated that the centroid solid angle is related to probabilities of square prism dice rolls. We explain how it is relevant to these probabilities and how to use the spherical projection to calculate the centroid solid angles for the faces on a square prism. These values are then used in a statistical analysis in the quest of constructing a mathematical probability model. The proposed model is based on the principle that the probability of ending up on a particular resting aspect is proportional to the centroid solid angle of that aspect and inversely proportional to a power of the centroid height in that aspect. Using a power of 2.427, this proposed model fits our data of over 60,000 non-symmetrical square prism dice rolls of various sizes (unequal heights and widths) with the largest magnitude Z-score of 1.01. Different powers can potentially describe other situations; e.g. different surfaces, larger dice, heavier dice, etc.

scholarly journals Bell Could Become the Copernicus of Probability

2016 ◽  
Vol 23 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Probability Model ◽  
Quantum Probability ◽  
Bell’S Inequality ◽  
Probability Models ◽  
Bell's Inequality ◽  
Mathematical Probability ◽  
Model Interpretation ◽  
Born’S Rule ◽  
Quantum Phenomena

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as the concrete example of the non-Kolmogorovian model of probability, similarly to the Lobachevskian model — the first example of the non-Euclidean model of geometry. This is the “probability model” interpretation of the violation of Bell’s inequality. We also criticize the standard interpretation—an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.

scholarly journals A new approach for modeling covid-19 death data

2021 ◽  
pp. 1-9
Author(s):  
Muhammad Farooq ◽  
Qamar-uz-zaman ◽  
Muhammad Ijaz
Keyword(s):  
Probability Model ◽  
Developed Countries ◽  
Likelihood Method ◽  
Data Sets ◽  
Probability Models ◽  
New Approach ◽  
Proposed Model ◽  
Rate Functions ◽  
Day By Day ◽  
Death Data

The Covid-19 infections outbreak is increasing day by day and the mortality rate is increasing exponentially both in underdeveloped and developed countries. It becomes inevitable for mathematicians to develop some models that could define the rate of infections and deaths in a population. Although there exist a lot of probability models but they fail to model different structures (non-monotonic) of the hazard rate functions and also do not provide an adequate fit to lifetime data. In this paper, a new probability model (FEW) is suggested which is designed to evaluate the death rates in a Population. Various statistical properties of FEW have been screened out in addition to the parameter estimation by using the maximum likelihood method (MLE). Furthermore, to delineate the significance of the parameters, a simulation study is conducted. Using death data from Pakistan due to Covid-19 outbreak, the proposed model applications is studied and compared to that of other existing probability models such as Ex-W, W, Ex, AIFW, and GAPW. The results show that the proposed model FEW provides a much better fit while modeling these data sets rather than Ex-W, W, Ex, AIFW, and GAPW.

Dimensions of plane and solid angles and their units in the International System of Units (SI)

2020 ◽  
pp. 26-32
Author(s):  
M. I. Kalinin ◽  
L. K. Isaev ◽  
F. V. Bulygin
Keyword(s):  
Solid Angle ◽  
International System ◽  
International Committee ◽  
Plane Angle ◽  
Arc Length ◽  
Weights And Measures ◽  
International System Of Units ◽  
System Of Units ◽  
In Series ◽  
Solid Angles

The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.

Study on Hazard Analysis Method of Earthquake Secondary Fire Disaster in Urban Area Based on GIS

2012 ◽  
Vol 204-208 ◽  
pp. 3457-3461
Author(s):  
Tian Qi Li ◽  
Fei Geng
Keyword(s):  
Urban Areas ◽  
Hazard Analysis ◽  
Probability Model ◽  
Probability Models ◽  
Housing Density ◽  
Earthquake Excitation ◽  
Fire Disaster ◽  
High Hazard ◽  
The City ◽  
And Diffusion

In order to study the probability of occurrence of secondary fire after the earthquake in urban areas, the probability model of the hazard analysis that the fire occurred and the spread is established and applied. Probability models need to consider the destruction level of buildings under earthquake excitation as well as the probability of the leakage and diffusion of combustible material in the buildings in the corresponding destruction level, combination of weather, season, housing density and other factors to determine the probability of the single building earthquake secondary fire. On this basis , the natural administrative areas in the city as a unit , considering the factors of regional hazard analysis such as population density , property distribution and density within a region , to calculate the hazard indicator and determine the high hazard areas of secondary fire in the city. The Geographic Information System was used as the platform, to division of urban earthquake secondary fire high-hazard areas.

Analytical failure probability model for generic gravity dam classes

2017 ◽  
Vol 231 (5) ◽  
pp. 546-557 ◽  
Author(s):  
Mohammad Amin Hariri-Ardebili
Keyword(s):  
Failure Probability ◽  
Safety Assessment ◽  
Statistical Approach ◽  
Probability Model ◽  
Gravity Dam ◽  
Preliminary Design ◽  
Failure Model ◽  
Proposed Model ◽  
Dimensional Gravity ◽  
Demand And Capacity

Risk analysis of concrete dams and quantification of the failure probability are important tasks in dam safety assessment. The conditional probability of demand and capacity is usually estimated by numerical simulation and Monte Carlo technique. However, the estimated failure probability (or the reliability index) is dam-dependent which makes its application limited to some case studies. This article proposes an analytical failure model for generic gravity dam classes which is optimized based on large number of nonlinear finite element analyses. A hybrid parametric–probabilistic–statistical approach is used to estimate the failure probability as a function of dam size, material distributional models and external hydrological hazard. The proposed model can be used for preliminary design and evaluation of two-dimensional gravity dam models.

A Probabilistic Routing Protocol in VANET

2010 ◽  
Vol 2 (4) ◽  
pp. 21-37 ◽  
Author(s):  
Gongjun Yan ◽  
Stephan Olariu ◽  
Shaharuddin Salleh
Keyword(s):  
Ad Hoc Networks ◽  
Routing Protocol ◽  
High Speed ◽  
Vehicular Ad Hoc Networks ◽  
Ad Hoc ◽  
Probability Model ◽  
Proposed Model ◽  
Probabilistic Routing ◽  
Hoc Networks ◽  
Manet Networks

The key attribute that distinguishes Vehicular Ad hoc Networks (VANET) from Mobile Ad hoc Networks (MANET) is scale. While MANET networks involve up to one hundred nodes and are short lived, being deployed in support of special-purpose operations, VANET networks involve millions of vehicles on thousands of kilometers of highways and city streets. Being mission-driven, MANET mobility is inherently limited by the application at hand. In most MANET applications, mobility occurs at low speed. By contrast, VANET networks involve vehicles that move at high speed, often well beyond what is reasonable or legally stipulated. Given the scale of its mobility and number of actors involved, the topology of VANET is changing constantly and, as a result, both individual links and routing paths are inherently unstable. Motivated by this latter truism, the authors propose a probability model for link duration based on realistic vehicular dynamics and radio propagation assumptions. The paper illustrates how the proposed model can be incorporated in a routing protocol, which results in paths that are easier to construct and maintain. Extensive simulation results confirm that this probabilistic routing protocol results in more easily maintainable paths.

scholarly journals On the S-instability and degeneracy of discrete deep learning models

2019 ◽  
Vol 9 (3) ◽  
pp. 627-655 ◽  
Author(s):  
Andee Kaplan ◽  
Daniel J Nordman ◽  
Stephen B Vardeman
Keyword(s):  
Probability Model ◽  
Correlated Data ◽  
Data Sequence ◽  
Probability Models ◽  
Large Classes ◽  
Learning Contexts ◽  
Probability Structure ◽  
Model Probability ◽  
Parameter Sequence ◽  
Outcome Result

Abstract A probability model exhibits instability if small changes in a data outcome result in large and, often unanticipated, changes in probability. This instability is a property of the probability model, given by a distributional form and a given configuration of parameters. For correlated data structures found in several application areas, there is increasing interest in identifying such sensitivity in model probability structure. We consider the problem of quantifying instability for general probability models defined on sequences of observations, where each sequence of length $N$ has a finite number of possible values that can be taken at each point. A sequence of probability models, indexed by $N$, and an associated parameter sequence result to accommodate data of expanding dimension. Model instability is formally shown to occur when a certain log probability ratio under such models grows faster than $N$. In this case, a one component change in the data sequence can shift probability by orders of magnitude. Also, as instability becomes more extreme, the resulting probability models are shown to tend to degeneracy, placing all their probability on potentially small portions of the sample space. These results on instability apply to large classes of models commonly used in random graphs, network analysis and machine learning contexts.

scholarly journals A Combined Quantitative Evaluation Model for the Capability of Hyperspectral Imagery for Mineral Mapping

Sensors ◽  
2019 ◽  
Vol 19 (2) ◽  
pp. 328 ◽  
Author(s):  
Na Li ◽  
Xinchen Huang ◽  
Huijie Zhao ◽  
Xianfei Qiu ◽  
Kewang Deng ◽  
...  
Keyword(s):  
Remote Sensing ◽  
Statistical Analysis ◽  
Imaging System ◽  
Evaluation Model ◽  
Spectral Response ◽  
Influence Factors ◽  
Hyperspectral Data ◽  
Statistical Characteristics ◽  
Mineral Mapping ◽  
Proposed Model

To analyze the influence factors of hyperspectral remote sensing data processing, and quantitatively evaluate the application capability of hyperspectral data, a combined evaluation model based on the physical process of imaging and statistical analysis was proposed. The normalized average distance between different classes of ground cover is selected as the evaluation index. The proposed model considers the influence factors of the full radiation transmission process and processing algorithms. First- and second-order statistical characteristics (mean and covariance) were applied to calculate the changes for the imaging process based on the radiation energy transfer. The statistical analysis was combined with the remote sensing process and the application performance, which consists of the imaging system parameters and imaging conditions, by building the imaging system and processing models. The season (solar zenith angle), sensor parameters (ground sampling distance, modulation transfer function, spectral resolution, spectral response function, and signal to noise ratio), and number of features were considered in order to analyze the influence factors of the application capability level. Simulated and real data collected by Hymap in the Dongtianshan area (Xinjiang Province, China), were used to estimate the proposed model’s performance in the application of mineral mapping. The predicted application capability of the proposed model is consistent with the theoretical analysis.

Reply by author to Dr. LaFehr

Geophysics ◽  
1977 ◽  
Vol 42 (4) ◽  
pp. 877-877
Author(s):  
Shri Krishna Singh
Keyword(s):  
Closed Form ◽  
Solid Angle ◽  
Short Note ◽  
Circular Disc ◽  
Closed Form Expression ◽  
Gravitational Attraction ◽  
Form Expression ◽  
Practical Usefulness ◽  
Solid Angles ◽  
Classic Paper

It is difficult to include all references when dealing with a subject so well studied as the gravitational attraction of a circular disc. Although the practical usefulness of Nettleton’s paper can not be denied by anyone, it nevertheless gives no details (except for some references) of the computation of solid angles subtended by a disc from which his graphs (Geophysics, 1942, Figure 4) result. My short note deals with (in what I consider an easy way of) obtaining a closed form expression for the solid angle. For applications of the result the reader would do well to look up Nettleton’s classic paper.

Viewing the World from Different Angles: Plato’s Timaeus 54E-55A

Apeiron ◽  
2013 ◽  
Vol 46 (3) ◽  
pp. 244-269
Author(s):  
Ernesto Paparazzo
Keyword(s):  
Present Article ◽  
Solid Angle ◽  
Alternative Interpretation ◽  
Plane Geometry ◽  
A Value ◽  
Solid Geometry ◽  
The World ◽  
Euclid’S Elements ◽  
Solid Angles ◽  
The Universe

Abstract The present article investigates a passage of the Timaeus in which Plato describes the construction of the pyramid. Scholars traditionally interpreted it as involving that the solid angle at the vertex of the pyramid is equal, or nearly so, to 180°, a value which they took to be that of the most obtuse of plane angles. I argue that this interpretation is not warranted, because it conflicts with both the geometrical principles which Plato in all probability knew and the context of the Timaeus. As well as recalling the definitions and properties of plane angles and solid angles in Euclid’s Elements, I offer an alternative interpretation, which in my opinion improves the comprehension of the passage, and makes it consistent with both the immediate and wider context of the Timaeus. I suggest that the passage marks a transition from plane geometry to solid geometry within Plato’s account of the universe.

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