We present the first global analysis of parton distribution functions (PDFs) at approximate N Formula omittedLO in the strong coupling constant Formula omitted, extending beyond the current highest ...NNLO achieved in PDF fits. To achieve this, we present a general formalism for the inclusion of theoretical uncertainties associated with the perturbative expansion in the strong coupling. We demonstrate how using the currently available knowledge surrounding the next highest order (N Formula omittedLO) in Formula omitted can provide consistent, justifiable and explainable approximate N Formula omittedLO (aN Formula omittedLO) PDFs. This includes estimates for uncertainties due the currently unknown N Formula omittedLO ingredients, but also implicitly some missing higher order uncertainties (MHOUs) beyond these. Specifically, we approximate the splitting functions, transition matrix elements, coefficient functions and K-factors for multiple processes to N Formula omittedLO. Crucially, these are constrained to be consistent with the wide range of already available information about N Formula omittedLO to match the complete result at this order as accurately as possible. Using this approach we perform a fully consistent approximate N Formula omittedLO global fit within the MSHT framework. This relies on an expansion of the Hessian procedure used in previous MSHT fits to allow for sources of theoretical uncertainties. These are included as nuisance parameters in a global fit, controlled by knowledge and intuition based prior distributions. We analyse the differences between our aN Formula omittedLO PDFs and the standard NNLO PDF set, and study the impact of using aN Formula omittedLO PDFs on the LHC production of a Higgs boson at this order. Finally, we provide guidelines on how these PDFs should be used in phenomenological investigations.
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The Gaussian linear model provides a unique way to obtain the posterior probability distribution as well as the Bayesian evidence analytically. Considering the expansion rate data, the Gaussian ...linear model can be applied for Formula omittedCDM, wCDM and a non-flat Formula omittedCDM. In this paper, we simulate the expansion data with various precision and obtain the Bayesian evidence, then it has been used to discriminate the models. The data uncertainty is in range Formula omitted and two different sampling rates have been considered. Our results indicate that considering Formula omitted uncertainty, it is possible to discriminate 2 Formula omitted deviation in equation of state from Formula omitted. On the other hand, we investigate how precision of the expansion rate data affects discriminating the Formula omittedCDM from a non-flat Formula omittedCDM model. Finally, we perform a parameters inference in both the MCMC and Gaussian linear model, using current available expansion rate data and compare the results.
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In this work, we discuss: (i) The ratios of different parton distribution functions (PDFs), i.e., MMHT2014, CJ15, CTEQ6l1, HERAPDF15, MSTW2008, HERAPDF20 and MSHT20, and the corresponding ...Kimber-Martin-Ryskin (KMR) unintegrated parton distribution functions (UPDFs) sets versus the hard scale Formula omitted, to find out the sensibility of the KMR UPDFs with respect to the input PDFs sets. It is shown that there is not much difference between the different input-PDFs or corresponding UPDFs sets ratios. (ii) Then, the dependence of proton Formula omitted-factorization structure functions on the different UPDFs sets which can use the above PDFs sets as input, are presented. The results are compared with the experimental data of ZEUS, NMC and H1+ZEUS at the hard scale Formula omitted and Formula omitted, and a reasonable agreement is found, considering different input PDFs sets. (iii) Furthermore, by fitting a Gaussian function, which depends on the transverse momentum Formula omitted, to the KMR UPDFs and averaging over x (the fractional parton momentum), we obtain the average transverse momentum, Formula omitted, in the scale range Formula omitted, which is in agreement with the other groups predictions, Formula omitted at Formula omitted Formula omitted. (iv) Finally we explore the average transverse momentum for which, the results of proton structure function with the KMR UPDFs and that of the Gaussian Formula omitted-dependent, are consistent to each other. Through the above report, at each step the parton branching (PB) UPDFs, i.e., the transverse momentum dependent PDFs (PB TMDPDFs) are considered for comparisons with the corresponding KMR UPDFs output.
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Recently, the ATLAS data of isolated three-photon production showed that the next-to-leading order (NLO) collinear factorization is not enough to describe experimental data. Therefore, one needs to ...calculate the cross section beyond the NLO, and as showed later, these data can be well described by the NNLO calculation within the collinear factorization framework. However, it is shown that the Formula omitted-factorization can be quite successful in describing exclusive and high energy collision processes, henceforth we decided to calculate isolated three-photon production within this framework. In this work we use the Martin, Ryskin, and Watt unintegrated parton distribution functions (MRW UPDFs) at LO and NLO levels, in addition to parton branching (PB) UPDFs in order to calculate cross section which we utilize the KATIE parton level event generator. It will be shown that in contrast to collinear factorization, the Formula omitted-factorization can describe quiet well the three-photon production ATLAS data. Interestingly our results using the NLO-MRW and PB UPDFs can cover the data within their uncertainty bands, similar to the NNLO collinear results.
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The bidirectional reflectance distribution function (BRDF) factor ƒ′ provides a bridge between the inherent and apparent optical properties (IOPs and AOPs) of inland waters. The previous BRDF studies ...focused on ocean waters, while few studies examine inland waters. It is meaningful to improve the theory of remote sensing of water surface and the accuracy of image derivation in inland waters. In this study, radiative transfer simulation was applied to calculate the ƒ′ values using appropriate IOPs based on in situ combined with realistic boundary conditions (N = 11,232). This study shows that ƒ′ factor varied over the range of 0.33–16.64 in Lake Nansihu, a finite depth water, higher than the range observed for the ocean (0.3–0.6). Our results demonstrate that the factor ƒ′ depends on not only solar zenith angle (θsub.s) but also the average number of collisions (n−) and particulate backscattering ratio (b~sub.bp). The ƒ′ factor shows a continuous geometric increase as the solar zenith angle increases at 400–650 nm but is relatively insensitive to solar angle in the 650–750 nm range in which ƒ′ increases as b~sub.bp and n− decreases. To account for these findings, two empirical models for ƒ′ factor as a function of θsub.s, n− and b~sub.bp are proposed in various spectral wavelengths for Lake Nansihu waters. Our results are crucial for obtaining Hyperspectral normalized reflectance or normalized water-leaving radiance and improving the accuracy of satellite products.
This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the ...posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest is only considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed.
Stein's method has offered a completely novel way of evaluating the quality of normal approximations. This volume contains thorough coverage of the method's fundamentals. It includes a large number ...of recent developments in both theory and applications.
We present an empirical estimator for the squared Hellinger distance between two continuous distributions, which almost surely converges. We show that the divergence estimation problem can be solved ...directly using the empirical CDF and does not need the intermediate step of estimating the densities. We illustrate the proposed estimator on several one-dimensional probability distributions. Finally, we extend the estimator to a family of estimators for the family of α-divergences, which almost surely converge as well, and discuss the uniqueness of this result. We demonstrate applications of the proposed Hellinger affinity estimators to approximately bounding the Neyman-Pearson regions.
Non-standard thermostatistical formalisms derived from generalizations of the Boltzmann-Gibbs entropy have attracted considerable attention recently. Among the various proposals, the one that has ...been most intensively studied, and most successfully applied to concrete problems in physics and other areas, is the one associated with the Ssub.q non-additive entropies. The Ssub.q-based thermostatistics exhibits a number of peculiar features that distinguish it from other generalizations of the Boltzmann-Gibbs theory. In particular, there is a close connection between the Ssub.q-canonical distributions and the micro-canonical ensemble. The connection, first pointed out in 1994, has been subsequently explored by several researchers, who elaborated this facet of the Ssub.q-thermo-statistics in a number of interesting directions. In the present work, we provide a brief review of some highlights within this line of inquiry, focusing on micro-canonical scenarios leading to Ssub.q-canonical distributions. We consider works on the micro-canonical ensemble, including historical ones, where the Ssub.q-canonical distributions, although present, were not identified as such, and also more resent works by researchers who explicitly investigated the Ssub.q-micro-canonical connection.
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