Journal articles: 'Paire of Distribution Function'

1

YONEDA, Yasuhiro. "Atomic Pair Distribution Function (PDF) Analysis of Ferroelectric Materials." Nihon Kessho Gakkaishi 54, no. 3 (2012): 155–58. http://dx.doi.org/10.5940/jcrsj.54.155.

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Tarasov, Vasily E. "Nonlocal Probability Theory: General Fractional Calculus Approach." Mathematics 10, no. 20 (October 17, 2022): 3848. http://dx.doi.org/10.3390/math10203848.

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Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density.

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Modarres, Reza. "Estimating the distribution function of symmetric pairs." Communications in Statistics - Theory and Methods 46, no. 4 (March 16, 2016): 1843–54. http://dx.doi.org/10.1080/03610926.2015.1030421.

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Hansen, Niels Richard. "Asymptotics for local maximal stack scores with general loop penalty function." Advances in Applied Probability 39, no. 3 (September 2007): 776–98. http://dx.doi.org/10.1239/aap/1189518638.

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A stack is a structural unit in an RNA structure that is formed by pairs of hydrogen bonded nucleotides. Paired nucleotides are scored according to their ability to hydrogen bond. We consider stack/hairpin-loop structures for a sequence of independent and identically distributed random variables with values in a finite alphabet, and we show how to obtain an asymptotic Poisson distribution of the number of stack/hairpin-loop structures with a score exceeding a high threshold, given that we count in a proper, declumped way. From this result we obtain an asymptotic Gumbel distribution of the maximal stack score. We also provide examples focusing on the computation of constants that enter in the asymptotic distributions. Finally, we discuss the close relation to existing results for local alignment.

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Hansen, Niels Richard. "Asymptotics for local maximal stack scores with general loop penalty function." Advances in Applied Probability 39, no. 03 (September 2007): 776–98. http://dx.doi.org/10.1017/s0001867800002044.

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A stack is a structural unit in an RNA structure that is formed by pairs of hydrogen bonded nucleotides. Paired nucleotides are scored according to their ability to hydrogen bond. We consider stack/hairpin-loop structures for a sequence of independent and identically distributed random variables with values in a finite alphabet, and we show how to obtain an asymptotic Poisson distribution of the number of stack/hairpin-loop structures with a score exceeding a high threshold, given that we count in a proper, declumped way. From this result we obtain an asymptotic Gumbel distribution of the maximal stack score. We also provide examples focusing on the computation of constants that enter in the asymptotic distributions. Finally, we discuss the close relation to existing results for local alignment.

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Saboor, Abdus, Hassan S. Bakouch, Fernando A. Moala, and Sheraz Hussain. "Properties and methods of estimation for a bivariate exponentiated Fréchet distribution." Mathematica Slovaca 70, no. 5 (October 27, 2020): 1211–30. http://dx.doi.org/10.1515/ms-2017-0426.

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AbstractIn this paper, a bivariate extension of exponentiated Fréchet distribution is introduced, namely a bivariate exponentiated Fréchet (BvEF) distribution whose marginals are univariate exponentiated Fréchet distribution. Several properties of the proposed distribution are discussed, such as the joint survival function, joint probability density function, marginal probability density function, conditional probability density function, moments, marginal and bivariate moment generating functions. Moreover, the proposed distribution is obtained by the Marshall-Olkin survival copula. Estimation of the parameters is investigated by the maximum likelihood with the observed information matrix. In addition to the maximum likelihood estimation method, we consider the Bayesian inference and least square estimation and compare these three methodologies for the BvEF. A simulation study is carried out to compare the performance of the estimators by the presented estimation methods. The proposed bivariate distribution with other related bivariate distributions are fitted to a real-life paired data set. It is shown that, the BvEF distribution has a superior performance among the compared distributions using several tests of goodness–of–fit.

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Fritzsch, B., and A. Zehe. "Distribution function of donor-acceptor pairs in nipi-structures." Superlattices and Microstructures 12, no. 1 (January 1992): 43–46. http://dx.doi.org/10.1016/0749-6036(92)90217-s.

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Rea, H. J., R. Sung, J. R. Corney, D. E. R. Clark, and N. K. Taylor. "Interpreting Three-Dimensional Shape Distributions." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 219, no. 6 (June 1, 2005): 553–66. http://dx.doi.org/10.1243/095440605x31427.

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Effective content-based shape retrieval systems would allow engineers to search databases of three-dimensional computer-aided design (CAD) models for objects with specific geometries or features. Much of the academic work in this area has focused on the development of indexing schemes based on different types of three-dimensional to two-dimensional ‘shape functions’. Ideally, the shape function used to generate a distribution should be easy to compute and permit the discrimination of both large and small features. The work reported in this paper describes the properties of three new shape distributions based on computationally simple shape functions. The first shape function calculates the arithmetic difference between distributions derived (using the original D2 distance shape function) from both a three-dimensional model and its convex hull. The second shape function is obtained by sampling the angle between random pairs of facets on the object. The third shape function uses the surface orientation to filter the results of a distance distribution. The results reported in this paper suggest that these novel shape functions improve significantly the ability of shape distributions to discriminate between complex engineering parts.

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Chen, L. F., and S. R. Liang. "A Modified Pulsar Model Green Function Period Distribution." Symposium - International Astronomical Union 125 (1987): 62. http://dx.doi.org/10.1017/s0074180900160486.

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The downward accelerated e− in the “unfavorable” zone, like that in the “favorable” one, will emit γ-photons, which in turn convert into e± pairs in some places near the surface of stars. But what happens, which is different from that in the “favorable” zone, is that some γ-photons will travel through a long distance before their conversion. This makes it possible that some γ-photons arrive at the “diode” district in the “favorable” zone. The magnetic conversion of pairs is much easier to happen than that occured in the “favorable” zone, where the γ-photons are created by the primary e− beam. The existence of dense e± plasma near the surface of stars makes the E, vanish at places where such plasma is present.

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Klein, Ingo, and Monika Doll. "(Generalized) Maximum Cumulative Direct, Residual, and Paired Φ Entropy Approach." Entropy 22, no. 1 (January 12, 2020): 91. http://dx.doi.org/10.3390/e22010091.

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A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative Φ entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents so-called contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey λ and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey λ distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily S&P500 returns and time durations between mine explosions.

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Di Crescenzo, Antonio, Barbara Martinucci, and Julio Mulero. "A QUANTILE-BASED PROBABILISTIC MEAN VALUE THEOREM." Probability in the Engineering and Informational Sciences 30, no. 2 (December 9, 2015): 261–80. http://dx.doi.org/10.1017/s0269964815000376.

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For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

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Coelho, A. A., P. A. Chater, and A. Kern. "Fast synthesis and refinement of the atomic pair distribution function." Journal of Applied Crystallography 48, no. 3 (May 22, 2015): 869–75. http://dx.doi.org/10.1107/s1600576715007487.

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A fast method for calculating the atomic pair distribution function is described in the context of performing refinements of structural models. Central to the speed of synthesis is the approximation of Gaussian functions of varying full widths at half-maximum using a narrower Gaussian with a fixed full width at half-maximum. The initial Gaussians are first laid down as delta functions which are then convoluted with the narrower Gaussian to form the final pattern. The net result is an algorithm, which has been included in the Rietveld refinement computer programTOPAS, that synthesizes and refines structural parameters a factor of 300–1000 times faster than alternative algorithms/programs, with speed advantages increasing as the number of atomic pairs increases.

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13

Wang, Lidong, Piotr Oprocha, and Hui Wang. "DC3 Pairs and the Set of Discontinuities in Distribution Functions." Discrete Dynamics in Nature and Society 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/673578.

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14

Nadaf, Tayyebeh, Taher Lotfi, and Stanford Shateyi. "Revisiting the Copula-Based Trading Method Using the Laplace Marginal Distribution Function." Mathematics 10, no. 5 (March 1, 2022): 783. http://dx.doi.org/10.3390/math10050783.

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Pairs trading under the copula approach is revisited in this paper. It is well known that financial returns arising from indices in markets may not follow the features of normal distribution and may exhibit asymmetry or fatter tails, in particular. Due to this, the Laplace distribution is employed in this work to fit the marginal distribution function, which will then be employed in a copula function. In fact, a multivariate copula function is constructed on two indices (based on the Laplace marginal distribution), enabling us to obtain the associated probabilities required for the process of pairs trade and creating an efficient tool for trading.

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Nasiru, Suleman, Abdul Ghaniyyu Abubakari, and Christophe Chesneau. "The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data." Mathematical and Computational Applications 28, no. 1 (February 14, 2023): 25. http://dx.doi.org/10.3390/mca28010025.

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The usefulness of (probability) distributions in the field of biomedical science cannot be underestimated. Hence, several distributions have been used in this field to perform statistical analyses and make inferences. In this study, we develop the arctan power (AP) distribution and illustrate its application using biomedical data. The distribution is flexible in the sense that its probability density function exhibits characteristics such as left-skewedness, right-skewedness, and J and reversed-J shapes. The characteristic of the corresponding hazard rate function also suggests that the distribution is capable of modeling data with monotonic and non-monotonic failure rates. A bivariate extension of the AP distribution is also created to model the interdependence of two random variables or pairs of data. The application reveals that the AP distribution provides a better fit to the biomedical data than other existing distributions. The parameters of the distribution can also be fairly accurately estimated using a Bayesian approach, which is also elaborated. To end the study, the quantile and modal regression models based on the AP distribution provided better fits to the biomedical data than other existing regression models.

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Kuznetsov, M. M., Y. D. Kuleshova, A. A. Perov, and L. V. Smotrova. "Hypersonic approximation for distribution function of pairs of molecules in shock wave." Journal of Physics: Conference Series 1560 (June 2020): 012056. http://dx.doi.org/10.1088/1742-6596/1560/1/012056.

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Dietrich, Jeremy, Renu Malhotra, and Dániel Apai. "Statistical Distribution Function of Orbital Spacings in Planetary Systems." Astronomical Journal 167, no. 2 (January 8, 2024): 46. http://dx.doi.org/10.3847/1538-3881/ad1244.

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Abstract The minimum orbital separation of planets in long-stable planetary systems is often modeled as a step function, parameterized with a single value Δ min (measured in mutual Hill radius of the two neighboring planets). Systems with smaller separations are considered unstable, and planet pairs with greater separations are considered stable. Here we report that a log-normal distribution function for Δ min , rather than a single threshold value, provides a more accurate model. From our suite of simulated planetary systems, the parameters of the best-fit log-normal distribution are μ = 1.97 ± 0.02 and σ = 0.40 ± 0.02, such that the mean, median, and mode of Δ min are 7.77, 7.17, and 6.11, respectively. This result is consistent with previous estimates for Δ min threshold values in the range ∼5–8. We find a modest dependence of the distribution of Δ min on multiplicity within the system, as well as on planetary mass ratios of the closest planet pair. The overall distribution of nearest-neighbor planetary orbital spacings (measured in the mutual Hill radii and denoted simply as Δ) in long-term stable systems is also well fit with a log-normal distribution, with parameters μ = 3.14 ± 0.03 and σ = 0.76 ± 0.02. In simulations of sets of many planets initially packed very close together, we find that the orbital spacings of long-term stable systems is statistically similar to that in the observed Kepler sample of exoplanetary systems, indicating a strong role of sculpting of planetary architectures by dynamical instabilities.

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Lin, Wei, Mingyue Zheng, Yunhui Chen, Qian He, Adeel Khoja, Mingyue Long, Jiaxin Fan, et al. "Dose Correlation of Panax ginseng and Atractylodes macrocephala Koidz. Drug Pairs in the Chinese Medicine Prescription Based on the Copula Function." Evidence-Based Complementary and Alternative Medicine 2021 (August 24, 2021): 1–14. http://dx.doi.org/10.1155/2021/9933254.

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Objective. Panax ginseng and Atractylodes macrocephala Koidz. (AMK) are widely used in treating various diseases; however, research is insufficient on measuring the relationship that exists by combining this drug pair using the copula function. Methods. In this study, 279 traditional Chinese medicine prescriptions containing the Panax ginseng and AMK drug pair were extracted from the prescription database for three types of screened indications, namely, diabetes mellitus, diarrhea, and insomnia. Following the principle of dose conversion, each dynasty unit was uniformly converted into a modern unit. Then, the kernel density distribution of Panax ginseng and AMK was fitted with their empirical distribution functions. Finally, the optimal copula function was selected from the copula function family as a t-copula function. Results. The empirical distribution and probability density functions of Panax ginseng and AMK were obtained. From the results, their Kendall rank correlation coefficients with indications of diabetes mellitus, insomnia, and diarrhea were 0.8689, 0.7858, and 0.7403, whereas their Spearman rank correlation coefficients were 0.9563, 0.9276, and 0.8958. Results also indicated that the use of the t-copula function can better reflect the correlation between Panax ginseng and AMK doses. Conclusion. From the three indications, the dose between Panax ginseng and AMK was positively correlated. This study, therefore, confirms the medicinal principle of Chinese medicine “combining” from the perspective of mathematical statistics. Results from this study are practical to evaluate the correlation between the drug pair doses, Panax ginseng and AMK, using the copula function model, which provides a new model for the scientific explanation of compatibility for Chinese medicines. This study also provides a methodological basis for more drug measurement studies and clinical medications.

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Kraleva, Liliya, Vincent Rijmen, and Nikolai L. Manev. "Correlation Distribution Analysis of a Two-Round Key-Alternating Block Cipher." Tatra Mountains Mathematical Publications 73, no. 1 (August 1, 2019): 109–30. http://dx.doi.org/10.2478/tmmp-2019-0009.

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Abstract In this paper we study two-round key-alternating block ciphers with round function f (x)= x(2t+1)2s, where t, s are positive integers. An algorithm to compute the distribution weight in respect to input and output masks is described. Also, in the case t = 1 the correlation distributions depending on input and output masks are completely determined for arbitrary pairs of masks.

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Chapman, Karena W., Saul H. Lapidus, and Peter J. Chupas. "Applications of principal component analysis to pair distribution function data." Journal of Applied Crystallography 48, no. 6 (October 13, 2015): 1619–26. http://dx.doi.org/10.1107/s1600576715016532.

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Developments in X-ray scattering instruments have led to unprecedented access toin situand parametric X-ray scattering data. Deriving scientific insights and understanding from these large volumes of data has become a rate-limiting step. While formerly a data-limited technique, pair distribution function (PDF) measurement capacity has expanded to the point that the method is rarely limited by access to quantitative data or material characteristics – analysis and interpretation of the data can be a more severe impediment. This paper shows that multivariate analyses offer a broadly applicable and efficient approach to help analyse series of PDF data from high-throughput andin situexperiments. Specifically, principal component analysis is used to separate features from atom–atom pairs that are correlated – changing concentration and/or distance in concert – allowing evaluation of how they vary with material composition, reaction state or environmental variable. Without requiring prior knowledge of the material structure, this can allow the PDF from constituents of a material to be isolated and its structure more readily identified and modelled; it allows one to evaluate reactions or transitions to quantify variations in species concentration and identify intermediate species; and it allows one to identify the length scale and mechanism relevant to structural transformations.

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Wei, Tongli, and Xiansheng Cao. "Microscopic derivation of the collision properties of molecules in two systems at thermal equilibrium." American Journal of Physics 92, no. 1 (January 1, 2024): 29–35. http://dx.doi.org/10.1119/5.0128644.

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The mean relative speed and relative speed distribution function of molecules play important roles in kinetics and related topics. College textbooks frequently present a simplified derivation of these quantities that, while yielding the correct result, are not well justified. This paper presents a detailed physical picture of collision-related processes based on the microscopic kinetic theory. Formulas for the mean relative speed and mean collision rate along with the relative speed distribution function and the distribution function of relative speed of the paired particles in collisions (collision possible distribution function) are derived for molecules from two different systems that are each at thermal equilibrium.

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MIKHEEV, N. V., and A. YA PARKHOMENKO. "DOUBLE RADIATIVE DECAY $Z \to f\bar f\gamma \gamma $ IN THE STANDARD MODEL." Modern Physics Letters A 09, no. 13 (April 30, 1994): 1175–81. http://dx.doi.org/10.1142/s0217732394000988.

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Z-boson decay [Formula: see text] in the Standard Model is analyzed. The distribution function on the invariant masses of the photon and fermion pairs is calculated in the leading logarithmic approximation. It is shown that this distribution function has a specific shape of a “crest.”

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Guse, B., T. Hofherr, and B. Merz. "Introducing empirical and probabilistic regional envelope curves into a mixed bounded distribution function." Hydrology and Earth System Sciences Discussions 7, no. 4 (July 6, 2010): 4253–90. http://dx.doi.org/10.5194/hessd-7-4253-2010.

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Abstract. A novel approach to consider additional spatial information in flood frequency analyses, especially for the estimation of discharges with recurrence intervals larger than 100 years, is presented. For this purpose, large flood quantiles, i.e. pairs of a discharge and its corresponding recurrence interval, as well as an upper bound discharge, are combined within a mixed bounded distribution function. Large flood quantiles are derived using probabilistic regional envelope curves (PRECs) for all sites of a pooling group. These PREC flood quantiles are introduced into an at-site flood frequency analysis by assuming that they are representative for the range of recurrence intervals which is covered by PREC flood quantiles. For recurrence intervals above a certain inflection point, a Generalised Extreme Value (GEV) distribution function with a positive shape parameter is used. This GEV asymptotically approaches an upper bound derived from an empirical envelope curve. The resulting mixed distribution function is composed of two distribution functions, which are connected at the inflection point. This method is applied to 83 streamflow gauges in Saxony/Germany. Our analysis illustrates that the presented mixed bounded distribution function adequately considers PREC flood quantiles as well as an upper bound discharge. The introduction of both into an at-site flood frequency analysis improves the quantile estimation. A sensitivity analysis reveals that, for the target recurrence interval of 1000 years, the flood quantile estimation is less sensitive to the selection of an empirical envelope curve than to the selection of PREC discharges and of the inflection point between the mixed bounded distribution function.

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Guse, B., Th Hofherr, and B. Merz. "Introducing empirical and probabilistic regional envelope curves into a mixed bounded distribution function." Hydrology and Earth System Sciences 14, no. 12 (December 9, 2010): 2465–78. http://dx.doi.org/10.5194/hess-14-2465-2010.

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Abstract. A novel approach to consider additional spatial information in flood frequency analyses, especially for the estimation of discharges with recurrence intervals larger than 100 years, is presented. For this purpose, large flood quantiles, i.e. pairs of a discharge and its corresponding recurrence interval, as well as an upper bound discharge, are combined within a mixed bounded distribution function. The large flood quantiles are derived using probabilistic regional envelope curves (PRECs) for all sites of a pooling group. These PREC flood quantiles are introduced into an at-site flood frequency analysis by assuming that they are representative for the range of recurrence intervals which is covered by PREC flood quantiles. For recurrence intervals above a certain inflection point, a Generalised Extreme Value (GEV) distribution function with a positive shape parameter is used. This GEV asymptotically approaches an upper bound derived from an empirical envelope curve. The resulting mixed distribution function is composed of two distribution functions which are connected at the inflection point. This method is applied to 83 streamflow gauges in Saxony/Germany. Our analysis illustrates that the presented mixed bounded distribution function adequately considers PREC flood quantiles as well as an upper bound discharge. The introduction of both into an at-site flood frequency analysis improves the quantile estimation. A sensitivity analysis reveals that, for the target recurrence interval of 1000 years, the flood quantile estimation is less sensitive to the selection of an empirical envelope curve than to the selection of PREC discharges and of the inflection point between the mixed bounded distribution function.

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Kensche, Philip R., Vera van Noort, Bas E. Dutilh, and Martijn A. Huynen. "Practical and theoretical advances in predicting the function of a protein by its phylogenetic distribution." Journal of The Royal Society Interface 5, no. 19 (May 30, 2007): 151–70. http://dx.doi.org/10.1098/rsif.2007.1047.

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The gap between the amount of genome information released by genome sequencing projects and our knowledge about the proteins' functions is rapidly increasing. To fill this gap, various ‘genomic-context’ methods have been proposed that exploit sequenced genomes to predict the functions of the encoded proteins. One class of methods, phylogenetic profiling, predicts protein function by correlating the phylogenetic distribution of genes with that of other genes or phenotypic characteristics. The functions of a number of proteins, including ones of medical relevance, have thus been predicted and subsequently confirmed experimentally. Additionally, various approaches to measure the similarity of phylogenetic profiles and to account for the phylogenetic bias in the data have been proposed. We review the successful applications of phylogenetic profiling and analyse the performance of various profile similarity measures with a set of one microsporidial and 25 fungal genomes. In the fungi, phylogenetic profiling yields high-confidence predictions for the highest and only the highest scoring gene pairs illustrating both the power and the limitations of the approach. Both practical examples and theoretical considerations suggest that in order to get a reliable and specific picture of a protein's function, results from phylogenetic profiling have to be combined with other sources of evidence.

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Reynolds, Robert, and Allan Stauffer. "Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function." Symmetry 13, no. 10 (October 18, 2021): 1962. http://dx.doi.org/10.3390/sym13101962.

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In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by ∫−∞∞∫−∞∞emx−my−ex−ey+y(log(a)+x−y)kdxdy in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus, a table of integral pairs is given for interested readers. All of the results in this work are new.

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Tokarchuk, M. V. "To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables." Mathematical Modeling and Computing 9, no. 2 (2022): 440–58. http://dx.doi.org/10.23939/mmc2022.02.440.

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Based on a chain of BBGKI equations with a modified boundary condition that takes into account multiparticle correlations, kinetic equations in the approximate "pairs" collisions and in the polarization approximation, taking into account the interaction through the third particle, obtained. The specifics of the model representation of the pair potential of particle interaction through short-range and long-range parts were taken into account. In the case of the short-range potential in the form of the potential of solid spheres, the contribution of Enskog's revised theory to the complete integration of the collision of the kinetic equation is obtained. The collision integrals include paired quasi-equilibrium distribution functions that depend on the nonequilibrium mean values of the particle number density and the inverse temperature. The method of collective variables Yukhnovskii is applied for the calculation of pair quasi-equilibrium distribution function with an allocation of short-range and long-range parts in the potential of the interaction of particles. In this case, the system with short-range interaction is considered as a frame of reference.

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Balistrocchi, M., and B. Bacchi. "Modelling the statistical dependence of rainfall event variables by a trivariate copula function." Hydrology and Earth System Sciences Discussions 8, no. 1 (January 18, 2011): 429–81. http://dx.doi.org/10.5194/hessd-8-429-2011.

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Abstract. In many hydrological models, such as those derived by analytical probabilistic methods, the precipitation stochastic process is represented by means of individual storm random variables which are supposed to be independent of each other. However, several proposals were advanced to develop joint probability distributions able to account for the observed statistical dependence. The traditional technique of the multivariate statistics is nevertheless affected by several drawbacks, whose most evident issue is the unavoidable subordination of the dependence structure assessment to the marginal distribution fitting. Conversely, the copula approach can overcome this limitation, by splitting the problem in two distinct items. Furthermore, goodness-of-fit tests were recently made available and a significant improvement in the function selection reliability has been achieved. Herein a trivariate probability distribution of the rainfall event volume, the wet weather duration and the interevent time is proposed and verified by test statistics with regard to three long time series recorded in different Italian climates. The function was developed by applying a mixing technique to bivariate copulas, which were formerly obtained by analyzing the random variables in pairs. A unique probabilistic model seems to be suitable for representing the dependence structure, despite the sensitivity shown by the dependence parameters towards the threshold utilized in the procedure for extracting the independent events. The joint probability function was finally developed by adopting a Weibull model for the marginal distributions.

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Zhang, Jianhua, Jinzhu Pu, and Mifeng Ren. "Molecular Weight Distribution Control for Polymerization Processes Based on the Moment-Generating Function." Entropy 24, no. 4 (April 1, 2022): 499. http://dx.doi.org/10.3390/e24040499.

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The molecular weight distribution is an important factor that affects the properties of polymers. A control algorithm based on the moment-generating function was proposed to regulate the molecular weight distribution for polymerization processes in this work. The B-spline model was used to approximate the molecular weight distribution, and the weight state space equation of the system was identified by the subspace state space system identification method based on the paired date of B-spline weights and control inputs. Then, a new performance criterion mainly consisting of the moment-generating function was constructed to obtain the optimal control input. The effectiveness of the proposed control method was tested in a styrene polymerization process. The molecular weight distribution of the styrene polymers can be approximated by the B-spline model effectively, and it can also be regulated towards the desired one under the proposed control method.

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Moughli, H., B. Azeddine, Z. Tiouti, and M. Rajczyk. "Study and modeling of a CdS /PbS betavoltaic cell by Monte Carlo simulation." Chalcogenide Letters 20, no. 3 (April 2023): 227–33. http://dx.doi.org/10.15251/cl.2023.203.227.

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In this paper, we present simulations of the concentration of electron-hole pairs generated from each point in solid targets under Ni-63 source bombardment of a CdS/PbS-based betavoltaic cell. This model is an accurate representation of the electronic interaction has been reported. We can obtain the distribution of the electron-hole pairs generated in the CdS/PbS junction as a function of the depth by Monte Carlo simulation, this distribution allowed us to find the concentrations of excess minority carriers as a function of the thickness, which can be function and injection into the continuity equations to determine the diffusion current and then the selected petavoltage properties. The model was tested for the Ni-63 CdS/PbS structure, with energy of 17 keV.

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Liu, Hao Xue, Gui Hua Cui, Min Huang, Bing Wu, and Yu Liu. "Color-Difference Threshold for Printed Images." Applied Mechanics and Materials 469 (November 2013): 236–39. http://dx.doi.org/10.4028/www.scientific.net/amm.469.236.

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Five ISO 400 images were used as test images and a method of limits psychophysical experiment was designed to test color-difference threshold in printed images. The color appearance of each original image was modified by an exponential function for CIELAB lightness and chroma, an offset function for CIELAB hue at 20 steps for each attributes respectively. The modified images and their originals were paired to form the test image pairs. The mean color differences of image pairs, ranged from 0 to 4 CIELAB units, were calculated by CIELAB color-difference formula and nearly uniformly divided into 21 grades for each attributes. The test image pairs were assessed in a CPC-8n lighting booth. 12 normal color vision observers took part in the experiment. The experimental results showed that the mean color-differences threshold for lightness, chroma and hue attributes were 1.49, 1.53 and 0.78 CIELAB units showing the threshold for hue was apparently smaller than that of lightness and chroma, and the thresholds of different images were dependent on the image content or color distribution.

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Harrison, P. G., and J. Bor. "Response Time Distribution in a Tandem Pair of Queues with Batch Processing." Journal of the ACM 68, no. 4 (June 30, 2021): 1–41. http://dx.doi.org/10.1145/3448973.

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Response time density is obtained in a tandem pair of Markovian queues with both batch arrivals and batch departures. The method uses conditional forward and reversed node sojourn times and derives the Laplace transform of the response time probability density function in the case that batch sizes are finite. The result is derived by a generating function method that takes into account that the path is not overtake-free in the sense that the tagged task being tracked is affected by later arrivals at the second queue. A novel aspect of the method is that a vector of generating functions is solved for, rather than a single scalar-valued function, which requires investigation of the singularities of a certain matrix. A recurrence formula is derived to obtain arbitrary moments of response time by differentiation of the Laplace transform at the origin, and these can be computed rapidly by iteration. Numerical results for the first four moments of response time are displayed for some sample networks that have product-form solutions for their equilibrium queue length probabilities, along with the densities themselves by numerical inversion of the Laplace transform. Corresponding approximations are also obtained for (non-product-form) pairs of “raw” batch-queues—with no special arrivals—and validated against regenerative simulation, which indicates good accuracy. The methods are appropriate for modeling bursty internet and cloud traffic and a possible role in energy-saving is considered.

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Binder, Benjamin J., and Matthew J. Simpson. "Spectral analysis of pair-correlation bandwidth: application to cell biology images." Royal Society Open Science 2, no. 2 (February 2015): 140494. http://dx.doi.org/10.1098/rsos.140494.

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Images from cell biology experiments often indicate the presence of cell clustering, which can provide insight into the mechanisms driving the collective cell behaviour. Pair-correlation functions provide quantitative information about the presence, or absence, of clustering in a spatial distribution of cells. This is because the pair-correlation function describes the ratio of the abundance of pairs of cells, separated by a particular distance, relative to a randomly distributed reference population. Pair-correlation functions are often presented as a kernel density estimate where the frequency of pairs of objects are grouped using a particular bandwidth (or bin width), Δ>0. The choice of bandwidth has a dramatic impact: choosing Δ too large produces a pair-correlation function that contains insufficient information, whereas choosing Δ too small produces a pair-correlation signal dominated by fluctuations. Presently, there is little guidance available regarding how to make an objective choice of Δ. We present a new technique to choose Δ by analysing the power spectrum of the discrete Fourier transform of the pair-correlation function. Using synthetic simulation data, we confirm that our approach allows us to objectively choose Δ such that the appropriately binned pair-correlation function captures known features in uniform and clustered synthetic images. We also apply our technique to images from two different cell biology assays. The first assay corresponds to an approximately uniform distribution of cells, while the second assay involves a time series of images of a cell population which forms aggregates over time. The appropriately binned pair-correlation function allows us to make quantitative inferences about the average aggregate size, as well as quantifying how the average aggregate size changes with time.

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BEC, J., L. BIFERALE, M. CENCINI, A. S. LANOTTE, and F. TOSCHI. "Intermittency in the velocity distribution of heavy particles in turbulence." Journal of Fluid Mechanics 646 (March 8, 2010): 527–36. http://dx.doi.org/10.1017/s0022112010000029.

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The statistics of velocity differences between pairs of heavy inertial point particles suspended in an incompressible turbulent flow is studied and found to be extremely intermittent. The problem is particularly relevant to the estimation of the efficiency of collisions among heavy particles in turbulence. We found that when particles are separated by distances within the dissipative subrange, the competition between regions with quiet regular velocity distributions and regions where very close particles have very different velocities (caustics) leads to a quasi bi-fractal behaviour of the particle velocity structure functions. Contrastingly, we show that for particles separated by inertial-range distances, the velocity-difference statistics can be characterized in terms of a local roughness exponent, which is a function of the scale-dependent particle Stokes number only. Results are obtained from high-resolution direct numerical simulations up to 20483 collocation points and with millions of particles for each Stokes number.

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Navarro, Jorge, Francisco J. Samaniego, and N. Balakrishnan. "Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components." Advances in Applied Probability 45, no. 4 (December 2013): 1011–27. http://dx.doi.org/10.1239/aap/1386857855.

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The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

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Navarro, Jorge, Francisco J. Samaniego, and N. Balakrishnan. "Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components." Advances in Applied Probability 45, no. 04 (December 2013): 1011–27. http://dx.doi.org/10.1017/s0001867800006741.

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The signature of a system is defined as the vector whoseith element is the probability that the system fails concurrently with theith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

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Miyaji, Takamitsu, M. Krumpe, A. Coil, H. Aceves, and B. Husemann. "Clustering and Halo Occupation Distribution of Active Galactic Nuclei." Proceedings of the International Astronomical Union 9, S304 (October 2013): 243. http://dx.doi.org/10.1017/s1743921314003937.

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AbstractWe present the results of our series of studies on correlation function and halo occupation distribution of AGNs utilizing data the ROSAT All-Sky Survey (RASS) and the Sloan Digital Sky Survey (SDSS) in the redshift range of 0.07<z<0.36. In order to improve the signal-to-noise ratio, we take cross-correlation approach, where cross-correlation functions (CCF) between AGNs and much more numerous AGNs are analyzed. The calculated CCFs are analyzed using the Halo Occupation Distribution (HOD) model, where the CCFs are divided into the term contributed by the AGN-galaxy pairs that reside in one dark matter halo (DMH), (the 1-halo term) and those from two different DMHs (the 2-halo term). The 2-halo term is the indicator of the bias parameter, which is a function of the typical mass of the DMHs in which AGNs reside. The combination of the 1-halo and 2-halo terms gives, not only the typical DMH mass, but also how the AGNs are distributed among the DMHs as a function of mass separately for those at the center of the DMHs and satellites. The main results are as follows: (1) the range of typical mass of the DMHs in various sub-samples of AGNs log (MDMH/h−1MΘ) ~ 12.4–13.4, (2) we found a dependence of the AGN bias parameter on the X-ray luminosity of AGNs, while the optical luminosity dependence is not significant probably due to smaller dynamic range in luminosity for the optically-selected sample, and (3) the growth of the number of AGNs per DMH (N (MDMH)) with MDMH is shallow, or even may be flat, contrary to that of the galaxy population in general, which grows with MDMH proportionally, suggesting a suppression of AGN triggering in denser environment. In order to investigate the origin of the X-ray luminosity dependence, we are also investigating the dependence of clustering on the black hole mass and the Eddington ratio, we also present the results of this investigation.

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Wang, Hong-Guang, Jian-Ling Chen, Zhi-Gang Wen, and Fei-Peng Pi. "On the mode switching timescales of pulsar PSR B0329+54." Proceedings of the International Astronomical Union 8, S291 (August 2012): 549–51. http://dx.doi.org/10.1017/s1743921312024830.

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AbstractChen et al. (2011) found that the durations (timescales) of the normal and abnormal modes of PSR B0329+54 follow a gamma distribution, and constrained the parameters of the distribution function. In this paper, we perform a further analysis on the relationship between the timescales of the two modes. The ratio between the durations of a normal mode and the succeeding abnormal mode is calculated for 54 such pairs. It is found that the cumulative distribution function (CDF) of the ratio is consistent with the CDF obtained by assuming random mode switching, suggesting that the two modes work independently.

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Wahlberg, Patrik. "The random Wigner distribution of Gaussian stochastic processes with covariance inS0(ℝ2d)." Journal of Function Spaces and Applications 3, no. 2 (2005): 163–81. http://dx.doi.org/10.1155/2005/252415.

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The paper treats time-frequency analysis of scalar-valued zero mean Gaussian stochastic processes onℝd. We prove that if the covariance function belongs to the Feichtinger algebraS0(ℝ2d)then: (i) the Wigner distribution and the ambiguity function of the process exist as finite variance stochastic Riemann integrals, each of which defines a stochastic process onℝ2d, (ii) these stochastic processes onℝ2dare Fourier transform pairs in a certain sense, and (iii) Cohen's class, ie convolution of the Wigner process by a deterministic functionΦ∈C(ℝ2d), gives a finite variance process, and ifΦ∈S0(ℝ2d)thenW∗Φcan be expressed multiplicatively in the Fourier domain.

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Zwierzycki, Wiesław, Bolesław Wojciechowicz, Arkadiusz Stachowiak, and Krzystof Bieńczak. "The Sliding Pair's Reliability Calculation. General Approach." Solid State Phenomena 113 (June 2006): 348–52. http://dx.doi.org/10.4028/www.scientific.net/ssp.113.348.

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This paper shows the possibilities of sliding pair’s calculation evaluation with the use of the generalized wear model where the wear rate v is a function of friction parameters p, v and complex constants K and x. It also estimates the inaccuracy (uncertainty) scale of methods making up the reliability prediction algorithm, especially in connection with the choice of: the method of transformation of random arguments distribution into the function distribution of the kinematic pairs stability, pseudorandom-number generators, contact stress characteristics.

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Pelessoni, Renato, Paolo Vicig, Ignacio Montes, and Enrique Miranda. "Bivariate p-boxes." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24, no. 02 (April 2016): 229–63. http://dx.doi.org/10.1142/s0218488516500124.

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A p-box is a simple generalization of a distribution function, useful to study a random number in the presence of imprecision. We propose an extension of p-boxes to cover imprecise evaluations of pairs of random numbers and term them bivariate p-boxes. We analyze their rather weak consistency properties, since they are at best (but generally not) equivalent to 2-coherence. We therefore focus on the relevant subclass of coherent p-boxes, corresponding to coherent lower probabilities on special domains. Several properties of coherent p-boxes are investigated and compared with those of (one-dimensional) p-boxes or of bivariate distribution functions.

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Bakaev, Nikolai Yu. "Resolvent estimates of elliptic differential and finite-element operators in pairs of function spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 5 (2004): 217–38. http://dx.doi.org/10.1155/s0161171204303200.

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We present some resolvent estimates of elliptic differential and finite-element operators in pairs of function spaces, for which the first space in a pair is endowed with stronger norm. In this work we deal with estimates in (Lebesgue, Lebesgue), (Hölder, Lebesgue), and (Hölder, Hölder) pairs of norms. In particular, our results are useful for the stability and error analysis of semidiscrete and fully discrete approximations to parabolic partial differential problems with rough and distribution-valued data.

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Zhang, Xu-Hong, and Fu-Hu Liu. "Statistical Behavior of Lepton Pair Spectrum in the Drell-Yan Process and Signal from Quark-Gluon Plasma in High-Energy Collisions." Advances in High Energy Physics 2021 (June 1, 2021): 1–21. http://dx.doi.org/10.1155/2021/9548737.

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We analyze the transverse momentum ( p T ) spectra of lepton pairs ( ℓ ℓ ¯ ) generated in the Drell-Yan process, as detected in proton-nucleus (pion-nucleus) and proton-(anti)proton collisions by ten collaborations over a center-of-mass energy s N N or s if in a simplified form) range from ~ 20 GeV to above 10 TeV. Three types of probability density functions (the convolution of two Lévy-Tsallis functions, the two-component Erlang distribution, and the convolution of two Hagedorn functions) are utilized to fit and analyze the p T spectra. The fit results are approximately in agreement with the collected experimental data. Consecutively, we obtained the variation law of related parameters as a function of s and invariant mass Q . In the fit procedure, a given Lévy-Tsallis (or Hagedorn) function can be regarded as the probability density function of transverse momenta contributed by a single quark ( q ) or anti-quark ( q ¯ ). The Drell-Yan process is then described by the statistical method.

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Bobrowski, Adam. "Asymptotic behavior of a Feller evolution family involved in the Fisher-Wright model." Advances in Applied Probability 40, no. 03 (September 2008): 734–58. http://dx.doi.org/10.1017/s0001867800002779.

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We study the evolution in time of the joint distribution of a pair of Feller processes, related by the fact that some random time ago they were identical, evolving as a single Feller process; from that time on, they began to evolve independently, conditional on a state at the time of split, according to the same Feller transition probabilities. Such processes are involved in the Fisher-Wright model: the distribution of the time counted backwards from the present to the time of split in the past is a function of deterministic but time-varying effective size 2Nof the population from which the two processes are sampled. In terms of a corresponding family of Feller operators, assuming asymptotic stability or ergodicity of the process of mutation, we find the limit form of the distribution of such pairs of processes sampled from decaying, asymptotically constant, and growing populations. In the case where mutation is not asymptotically stable or ergodic, limit distributions are found for the distribution of relative differences.

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Sutherland, Will. "The 3-D Distribution of Abell Clusters." Symposium - International Astronomical Union 130 (1988): 538. http://dx.doi.org/10.1017/s0074180900136678.

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The Struble & Rood catalogue (Ap. J. Supp, 63, 543) of all measured Abell cluster redshifts is analysed, with corrections for the selection biases. This contains 533 redshifts with |b| ≥ 30°, z ≤ 0.3 compared with 104 in the sample of Bahcall & Soneira (Ap. J.270, 20). Although the catalogue contains biases in angular position (redshifts are preferentially measured in apparent “supercluster” regions) the information on redshift clustering is effectively unbiased since one cannot tell a priori whether pairs of clusters close on the sky are really associated in redshift. Thus the distribution of redshift differences for pairs of given angle & distance classes, f(Δz|θ,D1,D2), is a fair sample of the true distribution. Then by normalising to the “correct” angular correlation function, we obtain the joint distribution f(Δz,θ) and hence ξ(r). In practice, in the estimation of ξ we assign each pair a weight a where where wA is the correlation or cross correlation for the appropriate subset of the whole Abell catalogue. This gives ξ(r) ≈ (r/20 h−1Mpc)−1.8, just slightly smaller than the result of Bahcall & Soneira, but here ξ(r) ≈ 0 for r ≥ 50−1Mpc.

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TANNENBAUM, M. J. "WHY THE xE DISTRIBUTION TRIGGERED BY A LEADING PARTICLE DOES NOT MEASURE THE FRAGMENTATION FUNCTION BUT DOES MEASURE THE RATIO OF THE TRANSVERSE MOMENTA OF THE AWAY-SIDE JET TO THE TRIGGER-SIDE JET." International Journal of Modern Physics E 16, no. 07n08 (August 2007): 2011–17. http://dx.doi.org/10.1142/s0218301307007386.

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Hard-scattering of point-like constituents (or partons) in p-p collisions was discovered at the CERN-ISR1 in 1972 by measurements utilizing inclusive single or pairs of hadrons with large transverse momentum (pT). Due to the steeply falling power-law pT spectrum of the hard-scattered partons, the inclusive single particle (e.g. π0) pTt spectrum from parton fragmentation to a jet is dominated by trigger fragments with large zt ~ 0.7–0.8, where zt = pTt/pTjet is the fragmentation variable. It was generally assumed, following Feynman, Field and Fox,2 as shown by data from the CERN-ISR experiments, that the pTa distribution of away side hadrons from a single particle trigger [with pTt], corrected for zt, would be the same as that from a jet-trigger and follow the same fragmentation function as observed in e+e− or DIS. PHENIX3 attempted to measure the fragmentation function from the away side xE ~ PTa/pTt distribution of charged particles triggered by a π0 in p − p collisions at RHIC and showed by explicit calculation that the xE distribution is actually quite insensitive to the fragmentation function. Illustrations of the original arguments and ISR results will be presented. Then the lack of sensitivity to the fragmentation function will be explained, and an analytic formula for the xE distribution given, in terms of incomplete Gamma functions, for the case where the fragmentation function is exponential. The away-side distribution in this formulation has the nice property that it both exhibits xE scaling and is directly sensitive to the ratio of the away jet [Formula: see text] to that of the trigger jet, [Formula: see text], and thus can be used, for example, to measure the relative energy loss of the two jets from a hard-scattering which escape from the medium in A + A collisions. Comparisons of the analytical formula to RHIC measurements will be presented, including data from STAR4,5 and PHENIX,3,6 leading to some interesting conclusions.

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Tiouti, Z., A. Talhi, B. Azeddine, and A. Helmaoui. "Modeling of betavoltaic cells GaN using a Monte Carlo calculation." Journal of Ovonic Research 18, no. 5 (November 3, 2022): 691–97. http://dx.doi.org/10.15251/jor.2022.185.691.

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In this study, we use a Monte Carlo calculation code to simulate the concentration of electron-hole pairs generated of each point in the solid targets under a bombardment of Ni63 source for betavoltaic cells; this model is reported to be an accurate representation of electron interaction.From this simulation we can obtain the distribution of electron-hole pairs generated in GaN/GaN junction as a function of the depth, this distribution allowed us to find the concentrations of minority carriers excess depending on the thickness, which can represent as function and inject into the continuity equations for determine the diffusion current and then the characteristics of betavoltaic chosen. The model has been tested for Ni-63/GaN/GaN structure, with energy 17 KeV.

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Buot, F. A. "Quantum Distribution-function Transport Equations in Non-normal Systems and in Ultra-fast Dynamics of Optically-excited Semiconductors." VLSI Design 8, no. 1-4 (January 1, 1998): 265–73. http://dx.doi.org/10.1155/1998/98486.

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The derivation of the quantum distribution-function transport equations combines the Liouvillian super-Green's function technique and the lattice Weyl-Wigner formulation of the quantum theory of solids. A generating super-functional is constructed which allows an algebraic and straightforward application of quantum field-theoretical techniques in real time to derive coupled quantum-transport, condensate, and pairwavefunction equations. In optically-excited semiconductors, quantum distributionfunction transport equations are given for phonons, plasmons, photons, and electron-hole pairs and excitons by transforming the Bethe-Salpeter equation into a multi-time evolution equation. The virtue of quantum distribution function is that it allows easy application of ‘device-inflow’ subsidiary boundary conditions for simulating femtosecond device-switching phenomena.

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Barbarani, Vito. "Combinatorial Models of the Distribution of Prime Numbers." Mathematics 9, no. 11 (May 27, 2021): 1224. http://dx.doi.org/10.3390/math9111224.

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This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

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HAYS, CHRISTOPHER, ASHUTOSH KOTWAL, and OLIVER STELZER-CHILTON. "NEW TECHNIQUES IN THE SEARCH FOR Z′ BOSONS AND OTHER NEUTRAL RESONANCES." Modern Physics Letters A 24, no. 30 (September 28, 2009): 2387–403. http://dx.doi.org/10.1142/s021773230903179x.

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The search for neutral resonances at the energy frontier has a long and illustrious history, resulting in multiple discoveries. The canonical search scans the reconstructed invariant mass distribution of identified fermion pairs. Two recent analyses from the CDF experiment at the Fermilab Tevatron have applied novel methods to resonance searches. One analysis uses simulated templates to fit the inverse mass distribution of muon pairs, a quantity with approximately constant resolution for momenta measured with a tracking detector. The other analysis measures the angular distribution of electron pairs as a function of dielectron mass, gaining sensitivity over a probe of the mass spectrum alone. After reviewing several models that predict new neutral resonances, we discuss these CDF analyses and potential future applications.

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Journal articles on the topic 'Paire of Distribution Function'

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