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In this paper, we introduce the probabilistic Bernstein polynomials and derive new and interesting correlations among several special functions and special number sequences such as Euler polynomials, Bernoulli polynomials of higher order, Frobenius–Euler polynomials of higher order, Stirling numbers of the second kind and Bell polynomials subject to several special random variables. [ABSTRACT FROM AUTHOR]

Prunus mira and Prunus armeniaca are the two prominent fruit-bearing plant species in the Nepal Himalaya and possess notable ethnobotanical significance. These two species represent relatively unexplored edible resources in Nepalese flora and are characterized by the presence of glands in their leaves and fruits with stony endocarp. Subsequent to an on-site investigation and personal interviews with 90 respondents, we accumulated comprehensive data related to folk nomenclature, distribution, ethnobotanical, threats, and conservation status of P. mira and P. armeniaca in the Nepal Himalaya. We recorded a total of nine vernacular names for P. mira and six for P. armeniaca. P. mira was found to be distributed in six districts at an elevation of 2200–3600 m and P. armeniaca in nine districts within the range between 1500 and 3500 m. The use reports of P. mira were more than 454 and that of P. armeniaca were 367 with significant difference (W = 5485, p < 0.05). Site-wise, the use report of P. armeniaca was significantly different (χ2 = 44.317, df = 5, p < 0.01) but the use report of P. mira was less significant. Both species are economically valuable, growing in the farmyards and homestead areas. However, the modifications to agricultural practices posed an additional threat to the survival of these species. Establishing contemporary harvesting infrastructure, developing sustainable marketing goods, and disseminating conservation awareness throughout the Nepal Himalaya areas are necessary to ensure their preservation. [ABSTRACT FROM AUTHOR]

The presence of abandoned, lost or otherwise discarded fishing gears, including drifting fish aggregating devices (dFADs), in marine ecosystems poses significant ecological and socioeconomic concerns. The estimation of the number of dFADs in the marine environment is challenging due to the loss of tracking information when dFAD tracking buoys are remotely deactivated. For the first time, a data set of dFADs buoy positions, including those that had previously been remotely deactivated, has been made available for the period July–August 2020. Data from this period provide valuable insights into the life expectancy, spatial distribution, and status of deactivated dFAD buoys, enabling a more accurate assessment of dFAD presence and impacts. Deactivated buoys represented a 17.2% increase in the total number of tracked objects, and we estimate the in-water half-life of deactivated dFAD tracking buoys to be 101 days. Including deactivated buoys increases the number of strandings during the SP by 23.7%. Nevertheless, the representativity of these results is unknown given the limited spatio-temporal and numerical extent of our data, highlighting the importance of availability of comprehensive data on dFADs to effectively estimate their total numbers and mitigate their environmental impacts. [ABSTRACT FROM AUTHOR]

The novel fractional Stockwell transform is extended to the suitably defined Schwartz space and the generalized fractional Stockwell transform is introduced. The pseudo-differential operator associated with the fractional Stockwell transform is defined and it is proved to be continuous on the Schwartz space. Examples are illustrated for the generalized fractional Stockwell transform. The generalized convection-diffusion equation is solved using the fractional Stockwell transform. [ABSTRACT FROM AUTHOR]

Motivation for abandoning the 'environmental filtering' metaphor has been increasing in the literature, yet it remains a widely used conceptual tool to guide research and education within ecological disciplines. I consider the possible origins of the filtering paradigm through linkages to hypotheses about the potentially oversized role of localized competition in the early development of niche concepts, as well as assumptions about the scale dependence of factors influencing the species distributions at large scales. I believe that these perspectives may have discounted the relevance of non‐competitive interactions to the structure and maintenance of local communities, and the role of biotic interactions in large‐scale ecological processes more generally. I describe the types of ecological phenomena not readily captured by the filtering model, as well as the potential for biotic factors to interact with abiotic gradients (at scale) and influence broad ecological patterns. Lastly, I pose an alternative model of community assembly which emphasizes biotic–abiotic interactions and non‐filtering mechanisms to serve as a springboard for future renovations of the filtering metaphor. [ABSTRACT FROM AUTHOR]

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann's entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger's Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics. [ABSTRACT FROM AUTHOR]

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of -convolution. [ABSTRACT FROM AUTHOR]

This paper focuses on the fractional Schrödinger problem with the use of a new fractional derivative. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma. [ABSTRACT FROM AUTHOR]

Abstract This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann’s entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger’s Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics.

Quantile-quantile plots in the precise sense of scatterplots showing corresponding quantiles of two variables have long been supported by official command qqplot. That command is generalized here in several ways in a new command, qqplotg. In this article, I explain the major features of qqplotg and give several examples of its use. Themes include the use of quantile-quantile plots to explore the possibilities for working on a transformed scale and the value of plotting difference between quantiles versus mean quantile or plotting position. Various historical and methodological remarks are sprinkled throughout. [ABSTRACT FROM AUTHOR]

In this paper, we establish Abelian and Tauberian results that characterize the quasiasymptotic behavior of Lizorkin distributions via the asymptotic behavior of their shearlet transform. [ABSTRACT FROM AUTHOR]

We introduce the directional short-time fractional Fourier transform (DSTFRFT) and prove an extended Parseval’s identity and a reconstruction formula for it. We also investigate the continuity of both the directional short-time fractional Fourier transform and its synthesis operator on the appropriate space of test functions. Using the obtained continuity results, we develop a distributional framework for the DSTFRFT on the space of tempered distributions S ′ (R n) . We end the article with a desingularization formula. [ABSTRACT FROM AUTHOR]

Let be a finite extension of and let be a -adic Lie group. In this paper, we define the Iwasawa algebra and prove that it is isomorphic to the convolution algebra of compactly supported distributions on . This has important applications in the theory of admissible representations of on -adic Banach spaces. In particular, we prove the Frobenius reciprocity for continuous principal series representations. [ABSTRACT FROM AUTHOR]

The constraint of computational power and the huge number of degrees of freedom of the atmosphere means a sampling uncertainty exists in probabilistic ensemble forecasts. In our previous study, the uncertainty could be quantified, creating a convergence measure which converges proportional to n−1/2 in the limit of large ensemble size n. This power law can then be extrapolated to determine how sampling uncertainty would decrease with larger ensemble sizes and hence find the necessary ensemble size. It is unknown, however, how the sampling uncertainty depends on different weather regimes. This study extends the previous idealised ensemble developed, by including weak and strong forcing convective weather regimes, to look at how sampling uncertainty convergence differs in each. Two 5,000‐member ensembles were run, with weak and strong forcing respectively. Comparisons with a kilometre‐scale weather prediction model ensured realistic weak and strong forcing regimes by comparing the rain, convective available potential energy (CAPE), convective adjustment timescale, and distribution shapes throughout the diurnal cycle. Differences in distribution shape between the regimes led to differences in the convergence measure. Large differences in spread between weak and strong forcing runs throughout the 24 hr period led to large differences in sampling uncertainty of the mean and standard deviation, which could be quantified according to well‐known equations. The timing of these differences was case‐dependent. For extreme statistics such as the 0.95 quantile and for cases where there was precipitation, the moisture variables for the weak forcing case had the largest sampling uncertainty and required the most members for convergence proportional to n−1/2. This was due to the tails of the weak forcing moisture variables containing the least amount of density. Different ensemble sizes will hence be required depending on whether one is in the weak or strong forcing convective weather regime. [ABSTRACT FROM AUTHOR]

The classical Rolle's theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line. [ABSTRACT FROM AUTHOR]

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate some products of distributions in Colombeau algebra of generalized functions. In the classical theory of Schwartz distributions, multiplication of distributions is not defined for two arbitrary singular distributions. The properties of the Colombeau algebra allow us to calculate products of singular distributions which are not defined in the classical theory. The notion of association in Colombeau algebra of generalized functions allows us the results obtained in this way to be considered as products in the classical theory of distributions. The definition of the Colombeau product of distributions can be considered as generalization of their classical product in Schwartz theory. [ABSTRACT FROM AUTHOR]

In Present time, the utility of the theory of reliability and lifetesting analysis attracting many workers and theory is well established in engineering and medical sciences. Now, as the area of applicability of life testing is enlarging day by day, the variety of situations and their related data are also growing up. So, it seems not logical that this variety of data can be dealt with only two or three types of the above life testing models. It seems desirable to develop more and more realistic life testing models to deal with the variety of life testing data so that the proper inferences concerning with the reliability and other aspects can be drawn with more precision. It has been observed that in a number of situations the application of the models discussed so far can be made only after imposing a number of assumptions. So, there is a requirement of a relatively larger family of life time models, so that proper justification can be done with the daya analysis. In the present paper, an attempt has been made to develop a new life-testing model to enlarge the family of life testing models. [ABSTRACT FROM AUTHOR]

Leakage accidents of buried pipelines have become increasingly common due to the prolonged service of some pipelines which have been in use for more than 150 years. Therefore, there is an urgent need for accurate prediction of pollution scope to aid in the development of emergency remediation strategies. This study investigated the distribution of a light non-aqueous phase liquid in soils containing gas and water through numerical simulations and laboratory experiments. Firstly, a three-dimensional porous medium model was established using ANSYS FLUENT, and for the first time, the distribution of gas and groundwater in soil environments was simulated in the model. Subsequently, the distribution of the three phases of diesel, gas, and water in soil was studied with different leakage velocities and it was found that the leakage velocity played a significant role in the distribution. The areas of diesel in soils at 60 min were 0.112 m2, 0.194 m2, 0.217 m2, and 0.252 m2, with corresponding volumes of 0.028 m3, 0.070 m3, 0.086 m3, and 0.106 m3, respectively, for leakage velocities of 1.3 m/s, 3.4 m/s, 4.6 m/s, and 4.9 m/s. Calculation formulas for distribution areas and volumes were also developed to aid in future prevention and control strategies under different leakage velocities. The study also compared the distribution areas and volumes of diesel in soils with and without groundwater, and it was found that distribution scopes were larger in soils containing groundwater due to capillary force. In order to validate the accuracy of the numerical simulation, laboratory experiments were conducted to study the diffusion of oil, gas, and water under different leakage velocities. The results showed good agreement between the experiments and the simulations. The research findings are of great significance for preventing soil pollution and provide a theoretical basis for developing scientifically sound soil remediation strategies. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the probabilistic Bernstein polynomials and derive new and interesting correlations among several special functions and special number sequences such as Euler polynomials, Bernoulli polynomials of higher order, Frobenius–Euler polynomials of higher order, Stirling numbers of the second kind and Bell polynomials subject to several special random variables.

We assessed the richness, diversity, composition of the large mammal community in Ghamot National Park (GNP), Neelum valley Azad Jammu and Kashmir, Pakistan, and how these characteristics differed between four habitat types: forest, riparian zone, scrubland, and wetland, as well as between seasons. Across all locales with 66 line transect surveys (53km), indirect observation had the highest sampling (n=131; 76.60%) while direct observation had the lowest (n=40; 23.39). Fecal dropping was the most common type of indirect evidence (n=73; 55.72%), followed by foot prints (n=29; 9.94%) and dens/latrine (n=19; 14.50%). The Scrubland zone had the most indirect evidences (n=37) of any zone (elevation ranges between 2600-3600m).The forest zone (elevation ranges between 2700-3450m) had the second highest sampling (n=28), followed by the Alpine Zone (n=28), Riparian Zone (n=21), and Alpine Pasture Zone (n=16) (elevation 4000-4400m; habitat type high alpine pastures). By using direct and indirect field evidence, 14 species from 6 families were identified. Unfortunately, the number of species encountered directly was low in this study; four species (28.57%) were directly observed, including Canis aureus, Vulpes vulpes, Macaca mulatta, and Semnopithecus ajax, while the remaining ten (71.42%) were observed indirectly. Summer had the most direct and indirect observations (n=86; 50.29%), followed by winter (n=85; 49.70%). (4.28). Season had little effect on species richness, diversity, and composition across or within habitat types. However, the richness of species varied greatly across seasons among the four habitat types. The forest and riparian zone had the greatest similarity in species composition, both between and among seasons. Finally, our findings show that GNP has an essential influence in Pakistan mammal conservation. Our findings will provide as a baseline for park management to make effective conservation decisions, as well as a baseline for researchers conducting similar ecological studies.

Animal pollinators underpin the functioning and persistence of ecosystems globally. However, the vital role of pollination is being progressively eroded by the worldwide decline of pollinator species caused by human-induced environmental degradation, resulting in rising costs to biodiversity, agriculture, and economy. Most studies quantifying pollinator diversity and declines have focused on insects, whereas vertebrate pollinators remain comparatively neglected. Here, we present the first comprehensive study quantifying the macroecological patterns of species richness and extinction risk of bird and mammal pollinators in India, a region of extremely high biodiversity and increasing anthropogenic pressure. Our results reveal that hotspots of mammal pollinator diversity are restricted to the south of the Western Ghats, whereas bird pollinator diversity hotspots are scattered throughout the country. Analyses of hotspots of threatened species (based on the IUCN Red List) show that only mammal pollinators are currently classified as threatened in India, whereas multiple hotspots of population declines were observed for birds, and primarily in the Southwest for mammal pollinators. Our analyses failed to identify a role for species traits as drivers of these patterns, whereas most pollinators appear to be threatened by agriculture, logging and hunting for food, and medicinal purposes. Pollinator endangerment has widescale ecological and economic implications such as reduced food production, plant extinction, loss of functional and genetic diversity, and economic damage. We suggest protection of vertebrate pollinators should be emphasised in active conservation agendas in India. [ABSTRACT FROM AUTHOR]

This letter introduces a Discriminant Analysis-based unimodality Test (DAT) for automatically identifying whether a dataset is unimodal or multimodal, detecting deviations in time series datasets, estimating statistical parameters, and identifying the skewness of the data. DAT is effective in classifying datasets under both unimodal and multimodal conditions and is suitable for bi-classification applications. The performance of DAT was compared to two well-known unimodality tests, namely the dip and folding tests, and is shown to perform better. We then extended DAT to the development of a fault detection technique, which was tested against five different machine learning classifiers using data from the Case Western Reserve University (CWRU) ball bearing database. The results obtained show that the proposed approach is effective, achieving 99.999% accuracy for detecting small ball bearing fault sizes (0.007 inches). Our conclusion indicates a significant potential of the proposed test for improving anomaly detection in industrial and other related fields. • Introduces the Discriminant Analysis-Based Unimodality Test (DAT). • Compares DAT with Dip and Folding tests and proves superiority. • Extension and application of DAT for fault detection with 99.999% accuracy. • Shows immense potential for enhancing anomaly detection. [ABSTRACT FROM AUTHOR]

Baitul Mal Unisba (BMU), a zakat-based philanthropic institution, offers scholarships to its students, aiming not only to support their tuition but also to foster exceptional character development. This research addresses a significant challenge at BMU: the inefficiency in its administrative processes due to the absence of a robust information system. The study proposes developing a management information system to optimize scholarship distribution, precision, and accuracy of managing scholarship data - from collection and recording to processing and reporting. This improvement is anticipated to support more informed decision-making in scholarship distribution. The system was built using the Rapid Application Development process, which consists of three stages: requirements planning, workshop design, and implementation. The emphasis is on the optimal use of scholarship funds, time, and manpower. This research made an important contribution by developing a prototype of a web-based scholarship information system aimed at improving the effectiveness of distribution. The newly created scholarship management information system improves these processes by removing duplicate tasks and streamlining the overall workflow. It provides comprehensive data for the decisionmaking panel, including academic performance, activity reports, and scholarship quotas. Initial testing of the prototype shows that it meets all functional requirements, suggesting its potential effectiveness in resolving the current challenges in BMU's scholarship selection and distribution process. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the index Whittaker transform as a continuous linear map from one space of Boehmian into another. The index Whittaker transform is further investigated for the compactly supported distributions. [ABSTRACT FROM AUTHOR]

Jaiswal D, Banerjee S. 2024. Notes on Gerridae (Hemiptera: Heteroptera: Gerromorpha) from the Eastern Ghats of Telangana and Northern Andhra Pradesh, India. Nusantara Bioscience 16: 43-53. Gerridae is the family of semi-aquatic bugs found in both the lentic and lotic freshwater bodies. The present study focused on the Gerridae from the Eastern Ghats of Telangana and Northern Andhra Pradesh, India. This study documented a total number of 16 species belonging to 13 genera and 7 subfamilies under the family Gerridae. Tenagogonus nicobarensis Andersen, 1964, was earlier described from Andaman and Nicober islands and also known to be endemic to that particular geographical area. We are recording this species for the first time from the mainland of the Indian subcontinent. Ventidius aquarius Distant, 1910 was also recorded for the first time during this present study from the Eastern Ghats, an endemic species to peninsular India. Another species, Naboandelus signatus Distant, 1910 is also recorded from the Eastern Ghats of Andhra Pradesh as well as an addition to the state fauna of Andhra Pradesh. It is a widespread species and reported from both central and northeastern India. In southern India, it was reported from the two states and present record will be the additional third state to its distribution. In addition to taxonomic details, the article covers the geographic distribution of the 16 species. This study has also led to the addition of nine species of Gerridae to the state fauna of Andhra Pradesh. [ABSTRACT FROM AUTHOR]

In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions. [ABSTRACT FROM AUTHOR]

River water quality is closely related to the major ion sources and hydrological conditions. However, there is a limited cognition about the geochemical sources and the seasonal variations of major ions. Thus, in this study, a total of 90 water samples were collected from the Longjiang River and its three tributaries in the dry and wet seasons. The samples were analyzed, including major ion concentrations and physicochemical parameters. Statistical analysis, such as correlation analysis and principal component analysis (PCA), was employed to investigate the spatial and seasonal variations in major ion composition and their respective sources. Our study revealed that the predominant major ions in the studied samples are Ca2+, Mg2+, HCO − 3, and SO2 − 4. Most of ions exhibited notable spatial disparities attributable to variations in geological settings and human activities. Regions characterized by igneous rock outcrops tend to exhibit higher levels of K+ and Na+, while areas with higher population densities in the middle and downstream segments show elevated concentrations of Cl−, NO − 3, SO2 − 4, Na+, and K+. The observed peak SO2 − 4 levels may be attributed to active mining operations. Most parameters displayed higher values in flood season than those in dry season due to dilution effects. Stoichiometric analysis indicated that carbonate weathering inputs contribute to over 85% of the mean total cation concentrations in the water, followed by contributions from silicates, atmospheric deposition, and anthropogenic inputs. On the whole, although the water quality remains non-polluted and is suitable for drinking and irrigation purposes, the enrichment of SO2 − 4 and NO − 3 may contribute to water eutrophication. Caution is warranted during the dry season due to reduced water flow resulting from dam interceptions and limited dilution capacity, potentially leading to elevated pollutant concentrations. Taken together, our results provided a scientific basis for water quality managements of monsoon rivers. [ABSTRACT FROM AUTHOR]

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition g = h ⊕ m , we consider rollings of m over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle π ¯ : Q ¯ → Q over the configuration space equipped with a suitable principal connection, rollings of m over G/H can be expressed in terms of horizontally lifted curves on Q ¯ . The total space of π ¯ : Q ¯ → Q is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of m over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on Q ¯ , the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples. [ABSTRACT FROM AUTHOR]

In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions.

A topological neural network (TNN), which takes input data from a Tychonoff topological space instead of the usual finite dimensional space, is introduced. As a consequence, a distributional neural network (DNN) that takes Borel measures as data is also introduced. Combined these new neural networks facilitate things like recognizing long range dependence, heavy tails and other properties in stochastic process paths or like acting on belief states produced by particle filtering or hidden Markov model algorithms. The veracity of the TNN and DNN are then established herein by a strong universal approximation theorem for Tychonoff spaces and its corollary for spaces of measures. These theorems show that neural networks can arbitrarily approximate uniformly continuous functions (with respect to the sup metric) associated with a unique uniformity. We also provide some discussion showing that neural networks on positive-finite measures are a generalization of the recent deep learning notion of deep sets.

We prove the continuity of the fractional Stockwell transform and the corresponding synthesis operator on the spaces of highly localized functions over $ \mathbb {R} $ R and $ \mathbb {R}\times \mathbb {R}\setminus \{0\} $ R × R ∖ { 0 } , respectively, and their duals. We also provide Abelian and Tauberian-type results relate the asymptotic behaviour of distributions with the asymptotics of their fractional Stockwell transform. [ABSTRACT FROM AUTHOR]

Damaging aftershock sequences often exhibit considerable spatio‐temporal complexity. The stress changes associated with coseismic slip and aseismic afterslip are commonly proposed to drive aftershock sequences, but few systematic studies exist and do not always support strong, universal driving relationships. To investigate the roles that these two sources of stress changes may play in driving aftershocks, we assess the spatio‐temporal relationships between coseismic slip, afterslip, and on‐fault (within 5 km) aftershock density following seven Mw6.0–7.6 continental‐settings earthquakes, using available high‐quality slip models and regional seismic data. From previous empirical work and frictional considerations, near the mainshock we expect coseismic slip and afterslip to be anti‐correlated, and aftershocks to occur where coseismic slip is low/zero, near high slip gradients, and/or to migrate with afterslip. However, we find that spatial relationships between afterslip and coseismic slip, and between afterslip and aftershock density differ between earthquakes. Aftershock density correlates with coseismic slip following five of the earthquakes, and with total cumulative slip (coseismic slip + afterslip) following six: indicating that on‐fault aftershock distributions may be approximated by total slip (at current resolutions). Additionally, we find that the gradients of coseismic slip and afterslip (proxies for new stress concentrations) do not clearly correlate with aftershock distributions and that the choice of spatial domain over which relationships are tested can affect results significantly. A possible explanation of these results is that fault zones contain considerable fine‐scale structural and frictional heterogeneity. Nonetheless, the empirical evidence for frequently assumed relationships between coseismic slip, afterslip and on‐fault aftershocks is mixed. Plain Language Summary: The spatial distributions of aftershocks following major earthquakes can be complex and varied. We investigate how the locations of aftershocks on the main fault relate to the distributions of slip from the mainshock itself and from a gentle relaxation process that occurs on the fault in the following weeks, months, and years, called afterslip: we expect aftershocks to broadly occur where afterslip is and where mainshock slip is low/zero. We investigate seven interesting mainshocks that occur within comparable continental settings and show that spatial relationships between coseismic slip, afterslip, and aftershocks vary between these case studies. Aftershocks generally occur where slip of some kind (either mainshock slip or afterslip) has been modeled, which may be useful for estimating the broad distribution of on‐fault aftershocks in the future. As mainshock slip, afterslip, and aftershocks often appear to be occurring in exactly the same regions (contradicting common assumptions), we propose that fault zones contain finer scale detail than can currently be resolved. Key Points: We explore how coseismic slip and afterslip distributions may influence on‐fault aftershock density following seven key continental earthquakesTotal modeled slip often correlates well with on‐fault aftershock density thus could be used as an indicator of aftershock evolutionCoseismic slip, afterslip, and aftershocks commonly co‐locate, contrary to expectations, possibly due to unresolved slip heterogeneity [ABSTRACT FROM AUTHOR]

Sulfides with different distribution states may have different effects on steel properties. How to separately count each kind of sulfide remains a question. In the current study, sulfides in 1215 free‐cutting and 416 free‐cutting stainless steel are counted by Image‐Pro Plus image‐processing software. The sulfides can be divided into three categories, that is, cluster‐like sulfides, chain‐like sulfides, and individual sulfides. The features of each kind of sulfide are summarized and an automatic sulfide classification method is proposed. By using this method, sulfides in a metallographic photograph of 1215 steel are successfully classified into three types and counted separately. The quantity percentage of chain‐like, cluster‐like, and individual sulfides is 24.65%, 56.79%, and 18.56%, respectively, while the area percentage of different types of sulfides is 34.33%, 52.80%, and 12.87%. Individual sulfides have the highest‐quantity percentage of sulfides with a diameter smaller than 3 μm and with an area smaller than 10 μm2. Through this method, different types of sulfides can be deeply analyzed. It offers reliable data to evaluate whether the sulfides are well controlled or not, providing a guideline for the improvement of sulfide control methods. [ABSTRACT FROM AUTHOR]

We introduce the fractional short-time Fourier transform in $ \mathcal S'(\mathbb {R}) $ S ′ (R) and provide generalized asymptotics for the fractional Fourier and the short-time Fourier transforms within $ \mathcal {S}'(\mathbb {R}) $ S ′ (R). Abelian- and Tauberian-type results are given. [ABSTRACT FROM AUTHOR]