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In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F12; those of Watson, Dixon, Whipple, and Saalschutz for the series F23; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey.

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

Nocturnal insects have evolved remarkable visual capacities, despite small eyes and tiny brains. They can see colour, control flight and land, react to faint movements in their environment, navigate using dim celestial cues and find their way home after a long and tortuous foraging trip using learned visual landmarks. These impressive visual abilities occur at light levels when only a trickle of photons are being absorbed by each photoreceptor, begging the question of how the visual system nonetheless generates the reliable signals needed to steer behaviour. In this review, I attempt to provide an answer to this question. Part of the answer lies in their compound eyes, which maximize light capture. Part lies in the slow responses and high gains of their photoreceptors, which improve the reliability of visual signals. And a very large part lies in the spatial and temporal summation of these signals in the optic lobe, a strategy that substantially enhances contrast sensitivity in dim light and allows nocturnal insects to see a brighter world, albeit a slower and coarser one. What is abundantly clear, however, is that during their evolution insects have overcome several serious potential visual limitations, endowing them with truly extraordinary night vision. This article is part of the themed issue ‘Vision in dim light’.

The impairment in musculoskeletal structures in patients with low back pain (LBP) is often disproportionate to their complaint. Therefore, the need arises for exploration of alternative mechanisms contributing to the origin and maintenance of non-specific LBP. The recent focus has been on central nervous system phenomena in LBP and the pathophysiological mechanisms underlying the various symptoms and characteristics of chronic pain. Knowledge concerning changes in pain processing in LBP remains ambiguous, partly due to the diversity in the LBP population. The purpose of this study is to compare quantitative sensory assessment in different groups of LBP patients with regard to chronicity. Recurrent low back pain (RLBP), mild chronic low back pain (CLBP), and severe CLBP are compared on the one hand with healthy controls (HC), and on the other hand with fibromyalgia (FM) patients, in which abnormal pain processing has previously been reported. Cross-sectional study. Department of Rehabilitation Sciences, Ghent University, Belgium. Twenty-three RLBP, 15 mild CLBP, 16 severe CLBP, 26 FM, and 21 HC participated in this study. Quantitative sensory testing was conducted by manual pressure algometry and computer-controlled cuff algometry. A manual algometer was used to evaluate hyperalgesia as well as temporal summation of pain and a cuff algometer was used to evaluate deep tissue hyperalgesia, the efficacy of the conditioned pain modulation and spatial summation of pain. Pressure pain thresholds by manual algometry were significantly lower in FM compared to HC, RLBP, and severe CLBP. Temporal summation of pain was significantly higher in FM compared to HC and RLBP. Pain tolerance thresholds assessed by cuff algometry were significantly lower in FM compared to HC and RLBP and also in severe CLBP compared to RLBP. No significant differences between groups were found for spatial summation or conditioned pain modulation. No psychosocial issues were taken into account for this study. The present results suggest normal pain sensitivity in RLBP, but future research is needed. In mild and severe CLBP some findings of altered pain processing are evident, although to a lesser extent compared to FM patients. In conclusion, mild and severe CLBP presents within a spectrum, somewhere between completely healthy persons and FM patients, characterized by pain augmentation.

The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories\"--the \"well generated triangulated categories\"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

[This corrects the article DOI: 10.3389/fnins.2022.931748.].

Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Padé summation, self-similar factor approximation, self-similar diff-log summation, self-similar Borel summation, and self-similar Borel–Leroy summation. Several typical problems are treated. The comparison of the results shows that all these methods provide close estimates for the large-variable exponents. The reliable estimates are obtained when different methods of summation are compatible with each other.

The purpose of this paper is to improve the accuracy of solving prediction tasks of the missing IoT data recovery. To achieve this, the authors have developed a new ensemble of neural network tools. It consists of two successive General Regression Neural Network (GRNN) networks and one neural-like structure of the Successive Geometric Transformation Model (SGTM). The principle of ensemble topology construction on two successively connected general regression neural networks, supplemented with an SGTM neural-like structure, is mathematically substantiated, which improves the accuracy of prediction results. The effectiveness of the method is based on the replacement of the summation of the results of the two GRNNs with a weighted summation, which improves the accuracy of the ensemble operation in general. A detailed algorithmic implementation of the ensemble method as well as a flowchart of its operation is presented. The parameters of the ensemble operation are determined by optimization using the brute-force method. Based on the developed ensemble method, the solution of the task of completing the partially missing values in the real monitoring dataset of the air environment collected by the IoT device is presented. By comparing the performance of the developed ensemble with the existing methods, the highest accuracy of its performance (by the parameters of Mean Absolute Percentage Error (MAPE) and Root Mean Squared Error (RMSE) accuracy) among the most similar in this class has been proved.

The foveated architecture of the human retina and the eye’s mobility enables prime spatial vision, yet the interplay between photoreceptor cell topography and the constant motion of the eye during fixation remains unexplored. With in vivo foveal cone-resolved imaging and simultaneous microscopic photo stimulation, we examined visual acuity in both eyes of 16 participants while precisely recording the stimulus path on the retina. We find that resolution thresholds were correlated with the individual retina’s sampling capacity, and exceeded what static sampling limits would predict by 18%, on average. The length and direction of fixational drift motion, previously thought to be primarily random, played a key role in achieving this sub-cone diameter resolution. The oculomotor system finely adjusts drift behavior towards retinal areas with higher cone densities within only a few hundred milliseconds to enhance retinal sampling.