Following a recent proof of Shannon's entropy power inequality (EPI), a comprehensive framework for deriving various EPIs for the Rényi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation. Simple arguments are given to recover the previously known Rényi EPIs and derive new ones, by unifying a multiplicative form with constant c and a modification with exponent α of previous works. In particular, for log-concave densities, we obtain a simple transportation proof of a sharp varentropy bound.

Femtoscopy is a unique tool to investigate the space-time geometry of the matter created in ultra-relativistic collisions. If the probability density distribution of hadron emission is parametrized, then the dependence of its parameters on particle momentum, collision energy, and collision geometry can be given. In recent years, several measurements came to light that indicated the adequacy of assuming a Lévy-stable shape for the mentioned distribution. In parallel, several new phenomenological developments appeared, aiding the interpretation of the experimental results, or providing tools for the measurements. In this paper we review and discuss some of these advances, phenomenological and experimental.

In this paper we establish information theoretical bridges between 1) Thermal Heisenberg uncertainties $\Delta x \Delta p$ (at temperature $T$), and 2) LMC structural quantifiers. After having achieved such purpose, we determine to what an extent our bridges can be extended to both the semi classical and classical realms. Also, we find a strict bound relating a special LMC structural quantifier to quantum uncertainties.

Vaccination has been the most effective way to control the outbreak of the Covid pandemic. The numbers and types of vaccines have reached considerable proportions, even if the question of vaccine procedures and frequency still needs to be resolved. We learn the necessity of defining a vaccination distribution strategies with regard to Covid that could be used for any future pandemics of similar gravity. In fact, vaccine monitoring implies the existence of a strategy that should be measurable in terms of input and output, based on a mathematical model, including death rates, spread of infections, symptoms, hospitalization and so on. This paper addresses the issue of vaccine diffusion and strategies for monitoring the pandemic. It provides a description of the importance and take up of vaccines and the links between procedures and the containment of Covid variants, as well as the long-term effects. Finally, the paper focuses on the global scenario in a world undergoing profound social and political change, with particular attention on current and future health provision. This contribution would represent an example of vaccination experiences which can be useful in other pandemic or epidemiological contexts.

Multicriteria decision aid requires a database as a decision matrix, in which two or more alternatives are evaluated according to two or more variables selected as decision criteria. Several problems of this nature use measures by Likert scales. Depending on the method, parameters from these data (e.g. means, modes or medians) are required for calculations. This parameterization of data in ordinal scales has fueled controversy for decades between authors who favor mathematical/statistical rigor and argue against the procedure, stating that ordinal scales should not be parameterized, and scientists from other areas who have shown gains from the process that compensate for this relaxation. The aim of this article is to demonstrate the advantages of the Composition of Probabilistic Preferences (CPP) method in multicriteria problems with data from Likert scales. The CPP is capable of allaying the protests raised and obtaining more accurate results than descriptive statistics or parametric models can bring. The proposed algorithm in R-code involves the use of CPP with empirical distributions and fitting histograms of data measured by Likert scales. Two case studies with simulated datasets having peculiar characteristics and a real case illustrate the advantages of the CPP.

We propose two methods for obtaining estimations on the minimum distance and covering radius of orthogonal arrays. Both methods are based on knowledge about the (feasible) sets of distance distributions of orthogonal arrays with given length, cardinality, factors and strength. New bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of orthogonal arrays with parameters in doable ranges.

In this study, we present different methods of estimating fuzzy reliability of a two-parameter Rayleigh distribution via the maximum likelihood estimator, median first-order statistics estimator, quartile estimator, L-moment estimator, and mixed Thompson-type estimator. The mean-square error MSE as a measurement for comparing the considered methods using simulation through different values for the parameters and unalike sample sizes is used. The results of simulation show that the fuzziness values are better than the real values for all sample sizes, as well as the fuzzy reliability at the estimation of the Maximum

Grossversuch IV is a large and well documented experiment on hail suppression by silver iodide seeding. The original 1986 evaluation remained vague, although indicating a tendency to increase hail when seeding. The strategy to deal with distributions of hail energy far from normal was not optimal. The present re-evaluation sticks to the question asked and avoids both misleading transformations and unsatisfactory meteorological predictors. The raw data show an increase by about a factor of 3 for the hail energy when seeding. This is the opposite of what seeding is supposed to do. The probability to obtain such a result by chance is below 1%, calculated by permutation and bootstrap techniques applied on the raw data. Confidence intervals were approximated by bootstrapping as well as by a new method called "correlation imposed permutation" (CIP).

Understanding swash zone dynamics is of crucial importance for coastal management as the swash motion, consisting of the uprush of the wave on the beach face and the subsequent downrush, is responsible for driving changes the beach morphology trough sediment exchanges between the sub-aerial and sub-aqueous beach. Improved understanding of the probabilistic characteristics of these motions has the potential to allow coastal engineers to develop improved sediment transport models which, in turn, can be further developed into coastal management tools. In this paper, novel descriptors of swash motions are obtained by combining field data and statistical modelling. Our results indicate that the probability distribution function (PDF) of shoreline height (p(ζ)) and trough-to-peak swash heights (p(ρ)) measured at a high energy, sandy beach were both inherently multimodal. Based on the observed multimodality of these PDFs, Gaussian Mixtures are shown to be the best method to statistically model them. Further, our results show that both offshore and surf zone dynamics are responsible for driving swash zone dynamics, which indicates unsaturated swash. The novel methods and results developed in this paper, both data collection and analysis, could aid coastal managers to develop improved swash zone models in the future.

Developments of new probability models for data analysis are keen interest of importance for all fields. The log-Dagum distribution has a prominent role in the theory and practice of statistics. In this article, a new family of continuous distributions generated from a log Dagum random variable called the log-Dagum Weibull distribution is proposed. The key properties of the proposed distribution are derived. Its density function can be symmetrical, left-skewed, right-skewed and reversed-J shaped and can have increasing, decreasing, bathtub hazard rates shaped. The model parameters are estimated by the method of maximum likelihood and illustrate its importance by means of applications to real data sets.

Wind energy powered pumps could be an alternative to conventional fuel powered pumps for water abstraction because they rely on a free energy and they are environmentally friendly. The objective of this study was to assess the potential of wind energy to operate water abstraction systems in Teso sub-region of Uganda for livestock watering Daily mean wind speeds recorded at a height of 10 m for a period of ten years (2005–2015) were collected from Amuria and Soroti Meteorological stations in the study area. Data were analyzed using Weibull distribution to evaluate the annual wind speed frequency distributions and consequently assess their potential for water abstraction. The results indicated that warmer months (January, February and March) have higher mean wind speeds than the cold months (August, September and October). High wind speeds in the dry seasons corresponded to the periods of high water demand. The highest shape parameter (k) of 3.07 was registered in 2009 and scale parameter (c) of 3.78 in 2012. The highest wind power density of 43 W/m² was obtained the year 2012 while the lowest wind power density of 15.47 W/m² was obtained for Soroti district in the year 2009. The maximum power extractable in Amuria in 2012 was 324 W/m² which is potentially enough for water abstraction. Maximum discharges of 1.86 m³/s and 1.52 m³/s were obtained for Amuria and Soroti districts respectively at mean wind speeds of 5 m/s. Therefore, Teso sub region winds have potential for water abstraction and Amuria district better sites for livestock watering using wind energy.

In this work, the nearest neighbor distances and Voronoi cell features of the Cu-Ag deposits were analyzed and fitted with Lognormal, Weibull, and Gamma distributions. The nearest neighbor distance distributions of the samples were compared with those of complete spatially random points, showing spatial inhomogenity due to the nucleation exclusion effect. The radial distribution function was calculated, showing both influences from the grain size and the nucleation exclusion effect. Voronoi cells were generated based on the shape of the grains. The size, occupancy, and coordination of the Voronoi cells were examined and fitted. The results show that despite the Cu-Ag deposits seem to be governed by instantaneous nucleation mode, the spatial distribution of the nuclei is more impacted by the nucleation exclusion effect than the Cu-only samples. This behavior is also justified by the grain size distribution generated with Voronoi cell size and occupancy distributions.

Marine species are declining at an unprecedented rate, catalyzing many nations to adopt conservation and management targets within their jurisdictions. However, marine species and the biophysical processes that sustain them are naive to international borders. An understanding of the prevalence of cross-border species distributions is important for informing high-level conservation strategies, such as bilateral or regional agreements. Here, we examined 28,252 distribution maps to determine the number and locations of transboundary marine plants and animals. Over 90% of species have ranges spanning at least two jurisdictions, with 58% covering more than ten jurisdictions. All jurisdictions have at least one transboundary species, with the highest concentrations of transboundary species in the USA, Australia, Indonesia, and the Areas Beyond National Jurisdiction. Distributions of mapped biodiversity indicate that overcoming the challenges of multinational governance is critical for a much wider suite of species than migratory megavertebrates and commercially exploited fish stocks—the groups that have received the vast majority of multinational management attention. To effectively protect marine biodiversity, international governance mechanisms (particularly those related to the Convention on Biological Diversity, the Convention on Migratory Species, and Regional Seas Organizations) must be expanded to promote multinational conservation planning, and complimented by a holistic governance framework for biodiversity beyond national jurisdiction.

Plants grown under fluctuating light impact plant developments compared with those grown under non-fluctuating light conditions. However, our knowledge on the underlying regulatory mechanisms is still quite limited, particularly from the transcriptional perspective. In order to investigate the influence of different light intensity distributions on tomato plant development, we designed three fluctuating light intensity distributions with the non-fluctuating light intensity as control and compared the transcriptional differences after five weeks of treatment. We found plant height and aerial/root weight were significantly reduced under all fluctuating light treatments. Transcriptome analysis revealed that the number of up and down regulated genes had a distinct distribution pattern between different treatments and control. The largest difference between the numbers of down and up regulated genes was found between treatment 1 and 3, reaching to a total of 416 genes. The number and type of the top 20 enriched pathways differed between treatments and control. The largest number of genes enriched was involved in the biosynthesis of secondary metabolites. These results provide insights into the transcriptional regulations of tomato under different light intensity distributions.

Energy distance is used to eigenvalue distributions in this new Wishart distribution study, providing a new perspective on high-dimensional data. Multivariate statistics relies on Wishart distribution to explain covariance matrices. This analysis shows how degrees of freedom and covariance matrix configurations affect eigenvalue dispersion. We use energy distance based on eigenvalue distributions to assess Wishart distribution differences without the "curse of dimensionality" via extensive simulation simulations. This approach improves measurement accuracy and processing efficiency in distribution comparisons and enables new analytical pathways in scientific fields that need data covariance comprehension. Our results show that degrees of freedom and matrix structure considerably affect energy distances, expanding the theoretical framework and practical applications of statistical models.

Matrix-variate Gaussian type or Wishart type distributions in the real domain are widely used in the literature. When the exponential trace has an arbitrary power and when a factor involving a determinant enters into the model or a matrix-variate gamma type or Wishart type model with exponential trace having an arbitrary power, is extremely difficult to handle. Evaluation of the normalizing constant in such a model is the most important part because when studying the properties of such a model, the method used in the evaluation of the normalizing constant will be the relevant steps in all the computations involved. One such model with a factor involving a trace and the exponential trace having an arbitrary power, in the real domain, is known in the literature as Kotz' model. No explicit evaluation of the normalizing constant in the model involving trace with an exponent and determinant with an exponent entering into the model and at the same time the exponential trace having an arbitrary exponent seems to be available in the literature. The normalizing constant widely used in the literature and interpreted as the normalizing constant in the general model and refers to as a Kotz' model does not seem to be correct. Corresponding model in the complex domain, with the correct normalizing constant, does not seem to be available in the literature. One of the main contributions in this paper is the matrix-variate distributions in the complex domain belonging to Gaussian type, gamma type, type-1 and type-2 beta type when the exponential trace has an arbitrary power. All these models are believed to be new. A second main contribution is the explicit evaluation of the the normalizing constants, in the real and complex domains especially in the complex domain, in a matrix-variate model involving a determinant and a trace as multiplicative factors and at the same time the exponential trace having an arbitrary power. Another main contribution is the introduction of matrix-variate models with exponential trace having an arbitrary exponent, in the categories of type-1 beta, type-2 beta and gamma distributions or in the family of Mathai's pathway models [1], both in the real and complex domains. Another new contribution is the logistic-based extensions of models in the real and complex domains with exponential trace having an arbitrary exponent and connecting to extended zeta functions introduced by this author recently. Some properties of such models are indicated but not derived in detail in order to limit the size of the paper. The techniques and steps used at various stages in this paper will be highly useful for people working in multivariate statistical analysis as well as people applying such models in engineering problems, communication theory, quantum physics and related areas, apart from statistical applications.

We give two examples of the appearance of heavy-tailed distributions in applications to social sciences. Among these distributions are the laws of Pareto, Lotka, and some new ones. The examples are illustrated by constructing suitable toy models.

Wind power output is highly dependent on the wind speed at the selected site, therefore wind-speed distribution modeling is the most important step in the assessment of wind energy potential. This study aims at accurate evaluation of onshore wind energy potential in seven coastal cities in the south of Iran. Six Probability Distribution Functions (PDFs) were examined over representative stations. It has been deduced that the Weibull function, which was the most used PDF in similar studies, was only applicable to one station. Here, Gamma offered the best fit for three stations and for the other ones, Generalized Extreme Value (GEV) performed better. Considering the ranking of six examined PDFs and the simplicity of Gamma, it was identified as the effective function in the southern coasts of Iran bearing in mind the geographic distribution of stations. Besides, six turbine power curve functions were contributed to investigate the capacity factor. That was very important, as using only one function could cause under- or over-estimation. Then, stations were classified based on the National Renewable Energy Laboratory system. Last but not least, examining a range of wind turbines enabled scholars to extend this study into the practice and prioritize development of stations considering budget limits.

The probability distribution of the interevent time between two successive earthquakes has been subject of numerous studies due to its key role in seismic hazard assessment. In recent decades, many distributions have been considered and there has been a long debate about the possible universality of the shape of this distribution when the interevent times are suitably rescaled. In this work we aim to find out if there is a link between the different phases of a seismic cycle and the variations in the distribution that best fits the interevent times. To do this, we consider the seismic activity related to the Mw 6.1 L’Aquila earthquake that occurred on April 6, 2009 in central Italy by analyzing the sequence of events recorded from April 2005 to July 2009, and then the seismic activity linked to the sequence of the Amatrice-Norcia earthquakes of Mw 6 and 6.5 respectively and recorded in the period from January 2009 to June 2018. We take into account some of the most studied distributions in the literature: q-exponential, q-generalized gamma, gamma and exponential distributions and, according to the Bayesian paradigm, we compare the value of their posterior marginal likelihood in shifting time windows with a fixed number of data. The results suggest that the distribution providing the best performance changes over time and its variations may be associated with different phases of the seismic crisis.

Employing numerical methods to simulate manufacturing processes and study their influential factors is increasing. The development of computers and engineering software has increased the ability and accuracy of numerical methods in simulating various aspects of different processes. One of the manufacturing processes that has been considered by many industries in recent years to join metals in solid-state with unique properties is called friction stir welding. It is challenging and, in some cases, impossible to experimentally study the various aspects of this process, such as temperature distribution, stress distribution, and material flow, due to severe plastic deformation in the weld zone. In this case, numerical methods are used to investigate these parameters and better understand the process. This study first investigates various numerical methods researchers use to simulate the process. The ability, pros, and cons of these methods to simulate the process are also examined. Then, the applications of numerical models in simulating the temperature distribution during the process as well as the effects of input parameters on the temperature history of the process, are deeply considered. Next, the application of numerical methods in material flow modeling during the process is investigated. Finally, the modeling of the microstructure of the welding zone is investigated using numerical methods that help significantly in predicting the weld microstructure.

White noise is fundamentally linked to many processes, it has a flat power spectral density and a delta-correlated autocorrelation. Operators acting on white noise can result in coloured noise, whether they operate in the time domain, like fractional calculus, or in the frequency domain, like spectral processing. We investigate whether any of the white noise properties remain in the coloured noises produced by the action of an operator. For a coloured noise, which drives a physical system, we provide evidence to pinpoint the mother process from which it came. We demonstrate the existence of two indices, that is, kurtosis and codifference, whose values can categorise coloured noises according to their mother process. Four different mother processes are used in this study: Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The mother process determines the kurtosis value of the coloured noises that are produced. It maintains its value for Gaussian, never converges for Cauchy, and takes values for Laplace and Uniform that are within a range of its white noise value. In addition, the codifference function maintains its value for zero lag-time essentially constant around the value of the corresponding white noise.

X-band dual-polarization phased array radar (XPAR-D) possesses high temporal-spatial resolutions and plays a significant role in detecting meso- and micro-scale convective systems. However, the precipitation attenuation it endures necessitates an effective correction method. This study selected radar data from XPAR-D at the peak of Maofeng Mountain in Guangzhou during May 16-17, 2020 from three precipitation stages after quality control. Attenuation coefficients are calculated for different precipitation types through scattering simulations of raindrop size distribution (RSD) data. Drawing upon this, an attenuation correction algorithm (MZH-KDP method) is proposed for radar reflectivity factor (ZH) according to different raindrop types, and is compared to the ZH-KDP method currently in use. The results indicate that the attenuation amount of XPAR-D echoes depends on the attenuation path and echo intensity. When the attenuation path is shorter and the echo intensity is weaker, the amount of attenuation and correction is smaller. Difficulties arise when there are noticeable deviations in such a situation, which are challenging to solve via attenuation correction methods. Longer attenuation paths and stronger echoes highlight the advantages of the MZH-KDP method, while the ZH-KDP method tends to overcorrect the bias. The MZH-KDP method outperforms the ZH-KDP method for different precipitation types. The superior correction capability of the MZH-KDP method provides a significant advantage in improving the performance of XPAR-D for the detection of extreme weather.

Estimating the parameters of statistical distributions generally involves solving a system of nonlinear equations or a nonlinear equation, being a technical difficulty in their usual application in hydrology. This article presents improved approximations and, in some cases, new approximations for the estimation with the method of ordinary moments and the method of linear moments, which are useful for the direct calculation of the parameters, because the errors in the approximate estimation are similar to the use of iterative numerical methods. Thirteen statistical distributions of two and three parameters frequently used in hydrology are presented, for which parameter estimation was laborious. Thus, the approximate estimation of the parameters by the two methods is simple but also precise and easily applicable by hydrology researchers. The approximate forms characterized by very small errors are the result of the research conducted within the Faculty of Hydrotechnics to update the romanian normative standards in the hydrotechnical field.

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

By imposing the condition of non-vanishing expectation values for the amplitude and phase of field operators, pure quantum states are identified composed of two consecutive number states. These pure states also deliver noise-free radiation modes restricting the “half-photon noise” to the expectation value of the lowest level of dynamic and coherent number states. As a result, instantaneous phase-sensitive amplification of photons is easily controlled and direct evaluation of time or distance - varying wavefront distributions of photons and phases can be carried out for sub-Poissonian distributions of photons without the need for quasi-probabilities.

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

The Rényi entropy functional was optimised for the novel findings of this investigation under various conditions. Their research's findings suggest that the complete spectrum of Lévy stable distributions is covered by generalised t-distributions. Additionally, an exposition was undertaken to prove that the Lévy distribution generalizes Tsallisian distribution, rather than the reverse. The current study is a strong generalization of an existing research work in the literature.

The model internal year-to-year variability (hereafter, internal variability) is a significant source of uncertainty when estimating anthropogenic aerosol effective radiative forcing (ERF). In this study, we investigate the impact of internal variability using large ensemble simulations (600 years in total) with the same climate model under prescribed anthropogenic aerosol forcings. A comparison of the magnitudes (i.e., standard deviation, Std) of these influences confirms that internal variability has negligible impacts on the instantaneous radiative forcing (RF) diagnosed by double radiation calls but has considerable impacts on estimating ERF through rapid adjustments (ADJ). Approximately half of the model grids exhibit strong internal variability influence on ERF (Std > 5 W m−2). These strong internal variabilities lead to a 50% probability that 30-year linear change can reach 2 W m−2, and 10-year linear change can reach 4 W m−2. A 50-year simulation can provide a relatively stable annual mean map of ERF (ERF = ADJ + RF), but it fails for ADJ. The statistically significant areas in the annual mean maps of both ERF and ADJ from a 10-year simulation exhibit instability with evident chaotic features. These analyses contribute to the setup of comparative experiments and a more comprehensive understanding of the modeled anthropogenic aerosol ERF.

In this study, deep neural network (DNN) and transfer learning (TL) techniques were employed to predict the viscous resistance and wake distribution based on the positions of flow control fins (FCFs) applied to containerships of various sizes. Both methods utilized data collected through Computational Fluid Dynamics (CFD) analysis. The position of the flow control fin (FCF) and hull-form information were utilized as input data, and the output data included viscous resistance coefficients and components of propeller axial velocity. The base DNN model was trained and validated using a source dataset from a 1000 TEU containership. Grid search cross-validation technique was employed to optimize the hyperparameters of the base DNN model. Then, transfer learning was applied to predict the viscous resistance and wake distribution for containerships of varying sizes. To enhance the accuracy of feature prediction with a limited amount of dataset, learning rate optimization was conducted. Transfer learning involves retraining and reconfiguring the base DNN model, and the accuracy was verified based on the fine-tuning method of the learning model. The results of this study can provide hull designers for containerships with performance evaluation information by predicting wake distribution, without relying on CFD analysis.

Efficient textual data distributions (TDD) alignment and generation are open research problems in textual analytics and NLP. It is presently difficult to parsimoniously and methodologically confirm that two or more natural language datasets belong to similar distributions, and to identify the extent to which textual data possess alignment. This study focuses on addressing a segment of the broader problem described above by applying multiple supervised and unsupervised machine learning (ML) methods to explore the behavior of TDD by (i) topical alignment, and (ii) by sentiment alignment. Furthermore we use multiple text generation methods including fine-tuned GPT-2, to generate text by topic and by sentiment. Finally we develop a unique process driven variation of Kullback-Leibler divergence (KLD) application to TDD, named KL Textual Distributions Contrasts (KL-TDC) to identify the alignment of machine generated textual corpora with naturally occurring textual corpora. This study thus identifies a unique approach for generating and validating TDD by topic and sentiment, which can be used to help address sparse data problems and other research, practice and classroom situations in need of artificially generated topic or sentiment aligned textual data.

The average sizes \ {\bar{L}}_{i\ } and their dispersion W_i along the i-th axis of crystallites in powders are used to determine the X-ray diffraction sizes, D_{i\ XRD}, averaged over crystallite columns within the BWA method. Numerical calculations have been carried out for an orthorhombic lattice of crystallites, such as LiFePO4, NMC, having Lamé's g-type superellipsoid shape. For Lognormal distributions, the analytical expression for the normalized coefficient Kn is found and includes two constants. The first K_{g,0} is a constant at W→0, the second K_g is a constant depending on the g -type shape. The dependences of D_{i\ XRD} are also calculated for Normal distribution. A fairly simple equation can be obtained as a result of analytical transformations in the framework of experimentally validated approximations. However, a simpler way is to carry out numerical computer calculations with subsequent approximation of the calculated curves. Using the obtained analytical expressions to control technologies from nuclear fuel to cathode materials will improve the efficiency of flexible energy network especially storage in autonomous and standby power plants.

The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator.The notion of entangled probability distribution is reviewed. Evolution equations describing the time-dependence of probability distributions identified with quantum system states are discussed.The connection with the Schrödinger equation and the von Neumann equation is clarified.

It has been shown that the Micro-Rain Radar (MRR) can be used to derive rainfall rates every 10 m over a depth of 1.28 km using the mean vertical air velocity corrected Doppler raindrop fall speed spectra. Furthermore, it has been shown that by assuming a reasonable advection velocity for the rain, these data can be analyzed to produce spatial radial power spectra often readily fit using a power function. Previous work has shown, however, that each spectrum applies only to each particular set of data and usually lacks the statistical qualifications necessary to be considered generally applicable. However, this limitation does not preclude the potential existence of other generalizations that can be used to explore the rainfall formation processes. The intent of this study, then, is to perform an initial look for such possible behaviors using time-height profiles of the rainfall rate. It is found that once the rainfall rate, R, exceeds about 20 mm h-1, there is, apparently, an associated flattening of the spectra with increasing R so that the smaller scales play an ever increasingly important role in such rain near the ground perhaps reflecting the increasing importance of such scales in the formation of pockets of more intense convective rain. The true generality of this finding needs additional scrutiny using more data particularly from two spatially separated MRR as is currently under preparation.

A realistic approach for gathering high-resolution observation of the rainfall rate, R, in the vertical plane is to use data from vertical pointing Doppler radars. After accounting for the vertical air velocity and attenuation, it is possible to determine the fine, spatially resolved drop size spectra and to calculate R for further statistical analyses. The first such results in a vertical plane are reported here. Specifically, we present results using MRR-Pro Doppler radar observations at resolutions of ten meters in height over the lowest 1.28 km as well as ten seconds in time over four sets of observations using two different radars at different locations. Both correlation functions and power spectra are useful for translating observations and numerical model outputs of R from on one scale down to other scales that may be more appropriate to particular applications such as flood warnings and soil erosion, for example. However, it was found in all cases that while locally applicable radial power spectra could be calculated, because of statistical heterogeneity, most of the power spectra lost all generality and proper correlation functions could not be computed in general except for one 17 minute interval. Nevertheless, these results are still useful since they could be combined to develop catalogs of power spectra over different meteorological conditions and in different climatological settings and locations. Furthermore, even within the limitations of these data, this approach is being used to gain a deeper understanding of rainfall to be reported in a forthcoming paper.

It is important to understand the statistical-physical structure of the rain in the vertical so that observations aloft can be translated meaningfully into what will occur at the surface. In order to achieve this understanding, it is necessary to gather high temporal and spatial resolution observations of rain in the vertical. This can only be accomplished using radars. It can be achieved by translating radar Doppler spectra into drop size distributions which can then be integrated to calculate variables such as the rain fall rate. A long-standing difficulty in using such measurements, however, is the problem of vertical air motion which can shift the Doppler spectra, and, therefore, significantly alter the deduced drop size distributions and integrated variables. In this work, we illustrate the improvement in measured rain structures using a new approach for removing the effect of mean vertical air motion. It is demonstrated that the new approach proposed here not only produces what appear to be better estimates of the rainfall rates, but, also as a consequence, provides estimates of the temporal and spatial regionally coherent updraft and downdrafts occurring in the precipitation. Furthermore, the technique is readily applicable to other radars especially those operating at non-attenuating frequencies.

In previous studies we found that generalized functions can be smooth, discrete, periodic or discrete periodic and they can either be local or global and they are regular or generalized functions. We also saw that these properties were related to Poisson’s summation formula on one hand and to Heisenberg’s uncertainty principle on the other. In this paper, we interlink these studies and show that scalars (real or complex numbers) considered as trivial functions are discrete and periodic, local and global as well as regular and generalized, simultaneously. However, this is also a paradox because it means that Dirac’s δ and 1 (its Fourier transform) coincide. They both are unity. We show that δ and 1 coincide in the sense of scalars (real or complex numbers) but they differ in the sense of (generalized) functions. This result can moreover be related to Max Born’s principle of reciprocity. It also answers an open question in present-day quantum mechanics because it means that the Dirac delta squared is simply delta.

The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as analysis window. In this paper the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15 MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources.

Paratemporal methods based on tree expansion have proven to be effective in efficiently generating trajectories of stochastic systems. However, combinatorial explosion of branching arising from multiple choice points presents a major hurdle that must be overcome to implement such techniques. In this paper we tackle this scalability problem by developing a systems theory-based framework covering both conventional and proposed tree expansion algorithms for speeding up discrete event system stochastic simulations while preserving desired accuracy. An example is discussed to illustrate the tree expansion framework in which a discrete event system specification (DEVS) Markov stochastic model takes the form of a tree isomorphic to a free monoid over the branching alphabet. We derive the computation times for baseline, non-merging, and merging tree expansion algorithms to compute the distribution of output values at any given depth. The results show the remarkable reduction from exponential to polynomial dependence on depth effectuated by node merging. We relate these results to the similarly reduced computation time of binomial coefficients underlying Pascal’s triangle. Finally, we discuss application of tree expansion to estimating temporal distributions in stochastic simulations involving serial and parallel compositions with potential real world use cases.

In recent years, pedestrian re-identification has gained a lot of interest due to its importance for many purposes such as security and safety. Many types of solutions have been proposed to solve this problem, where the majority are based on a features extraction such Convolution Neural Networks (CNNs). These approaches assume that only the identities that are in the training data can be recognized. The pedestrians in the training data are called In distribution (ID). However, in real world scenarios, new pedestrians and objects can appear in the scene and the model should detect them as Out Of Distribution (OOD). In our previous study , we proposed a pedestrian re-identification based on von-Mises Fisher (vMF) distribution. Each identity is embedded in the unit sphere as a compact vMF distribution far from other identities distributions. The embedding is done through a base CNN. Recently, proposed a framework called Virtual Outlier Synthetic (VOS), that detects OOD based on synthesising virtual outlier in the embedding space in an online manner. Their approach assumes that the samples from the same object maps to a compact space. This assumption aligns with the vMF based approach. Therefore, in this paper, we revisited the vMF approach and merged with VOS in order to detect OOD data points. We present our approach to merge both frameworks. We conducted several experiments to evaluate our proposed framework. Results showed that our framework was able to detect new pedestrian that do not exist in the training data in the inference phase. It also slightly helped to improve the re-identification performance.

Magnesium nanoparticles of various mean diameters (53 – 239 nm) were synthesized herein via Pulsed Laser Ablation in Liquid (PLAL) from millimeter sized magnesium powders within iso-propyl alcohol. It was observed via a 3x3 full factorial DOE that the processing parameters can control the nanoparticle distribution to produce three size-distribution types (bimodal, skewed and normal). Ablation times of 2, 5, and 25 minutes where investigated. An ablation time of 2 minutes produced a bimodal distribution with the other types seen at higher periods of processing. Mg nanoparticle UV-Vis absorbance at 204 nm increased linearly with increasing ablation time, indicating an increase in nanoparticle count. The colloidal density (mg/ml) generally increased with increasing nanoparticle mean diameter as noted via increasing UV-vis absorbance. High la-ser scan speeds (within the studied range of 3000 - 3500 mm/s) tend to increase the nanoparticle count/yield. For the first time, the effect of scan speed on colloidal density, UV-vis absorbance and nanoparticle diameter from metallic powder ablation was investigated and is reported herein. The nanoparticles formed dendritic structures after being drop cast on aluminum foil as observed via FESEM analysis. Dynamic light scattering was used to measure the size of the nanoparticles. Magnesium nanoparticles have promising use in the fabrication of wearables, such as in conductive tracks or battery electrodes, owing to their low heat capacity, high melting point and bio-compatibility.

Continual learning, a crucial facet of machine learning, involves the perpetual acquisition of valuable insights from incoming data, sans the necessity for full dataset access. Esteemed as a fundamental goal in artificial intelligence, continual learning grapples with an enduring challenge—catastrophic forgetting. A proficient model must exhibit adaptability for new data assimilation and robustness to retain existing knowledge. Class-incremental learning (CIL) aids the gradual integration of knowledge from newly introduced classes, forming a universal classifier. However, directly training the model with fresh class instances triggers a problem—forgetting distinguishing features of prior classes, causing a performance decline. Addressing such issues in machine learning, this survey aims to delineate significant challenges, outcomes, and recent advancements, including our contributions to CIL techniques, especially in image classification.