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Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper.

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

This paper is about Dirichlet averages in the matrix-variate case or averages of functions over the Dirichlet measure in the complex domain. The classical power mean contains the harmonic mean, arithmetic mean and geometric mean (Hardy, Littlewood and Polya), which is generalized to the y-mean by de Finetti and hypergeometric mean by Carlson; see the references herein. Carlson’s hypergeometric mean averages a scalar function over a real scalar variable type-1 Dirichlet measure, which is known in the current literature as the Dirichlet average of that function. The idea is examined when there is a type-1 or type-2 Dirichlet density in the complex domain. Averages of several functions are computed in such Dirichlet densities in the complex domain. Dirichlet measures are defined when the matrices are Hermitian positive definite. Some applications are also discussed.

This Special Issue of the journal Axioms collates submissions in which the authors report their perceptions and results in the field of mathematical physics and/or physical mathematics without any preconditions of the specific research topic [...]

Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sq≡k1−∑ipiqq−1(q∈R;S1=SBG≡−k∑ipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function lnqx≡x1−q−11−q(ln1x=lnx) associated with the Sq entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations.

The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area.

In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations.

A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored.

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, and a member of the International Statistical Institute. He is a founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He was instrumental in the implementation of the United Nations Basic Space Science Initiative (1991–2012). This paper highlights research results of A.M. Mathai in the period of time from 1962 to 2015. He published over 300 research papers and over 25 books.

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.

After collecting data from observations or experiments, the next step is to analyze the data to build an appropriate mathematical or stochastic model to describe the data so that further studies can be done with the help of the model. In this article, the input-output type mechanism is considered first, where reaction, diffusion, reaction-diffusion, and production-destruction type physical situations can fit in. Then techniques are described to produce thicker or thinner tails (power law behavior) in stochastic models. Then the pathway idea is described where one can switch to different functional forms of the probability density function through a parameter called the pathway parameter. The paper is a continuation of related solar neutrino research published previously in this journal.

This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.

This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in physics and astronomy. Nuclear reaction rate probability integrals in nuclear physics, Krätzel integrals in applied mathematical analysis, inverse Gaussian distributions, generalized type-1, type-2, and gamma families of distributions in statistical distribution theory, Tsallis statistics and Beck−Cohen superstatistics in statistical mechanics, and Mathai’s pathway model are all shown to be connected to the integral under consideration. Representations of the integral in terms of Fox’s H-function are pointed out.

We are going back to the roots of the original solar neutrino problem: the analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and II data reveals that they represent a non-Gaussian signal. The Hurst exponent is different from the scaling exponent of the probability density function, and both the Hurst exponent and scaling exponent of the probability density function of the Super-Kamiokande data deviate considerably from the value of 0.5, which indicates that the statistics of the underlying phenomenon is anomalous. To develop a road to the possible interpretation of this finding, we utilize Mathai’s pathway model and consider fractional reaction and fractional diffusion as possible explanations of the non-Gaussian content of the Super-Kamiokande data.

An entropy for the scalar variable case, parallel to Havrda-Charvat entropy, was introduced by the first author, and the properties and its connection to Tsallis non-extensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In the current paper, we extend the entropy to cover the scalar case, multivariable case, and matrix variate case. Then, this measure is optimized under different types of restrictions, and a number of models in the multivariable case and matrix variable case are obtained. Connections of these models to problems in statistical and physical sciences are pointed out. An application of the simplest case of the pathway model to the interpretation of solar neutrino data by applying standard deviation analysis and diffusion entropy analysis is provided.

Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.

Background: Cystic fibrosis bone disease (CFBD) is a common comorbidity in adult people with cystic fibrosis (pwCF), resulting in an increased risk of bone fractures. This study evaluated the capacity of artificial intelligence (AI)-assisted low-dose chest CT (LDCT) opportunistic screening for detecting low bone mineral density (BMD) in adult pwCF. Methods: In this retrospective single-center study, 65 adult pwCF (mean age 30.1 ± 7.5 years) underwent dual-energy X-ray absorptiometry (DXA) of the lumbar vertebrae L1 to L4 to determine BMD and corresponding z-scores and completed LDCTs of the chest within three months as part of routine clinical care. A fully automated CT-based AI algorithm measured the attenuation values (Hounsfield units [HU]) of the thoracic vertebrae Th9-Th12 and first lumbar vertebra L1. The ability of the algorithm to diagnose CFBD was assessed using receiver operating characteristic (ROC) curves. Results: HU values of Th9 to L1 and DXA-derived BMD and the corresponding z-scores of L1 to L4 showed a strong correlation (all p < 0.05). The area under the curve (AUC) for diagnosing low BMD was highest for L1 (0.796; p = 0.001) and Th11 (0.835; p < 0.001), resulting in a specificity of 84.9% at a sensitivity level of 75%. The HU threshold values for distinguishing normal from low BMD were <197 (L1) and <212 (Th11), respectively. Conclusions: Routine LDCT of the chest with the fully automated AI-guided determination of thoracic and lumbar vertebral attenuation values is a valuable tool for predicting low BMD in adult pwCF, with the best results for Th11 and L1. However, further studies are required to define clear threshold values.

Non-contrast computed tomography (CT) is commonly used for the evaluation of various pathologies including pulmonary infections or urolithiasis but, especially in low-dose protocols, image quality is reduced. To improve this, deep learning-based post-processing approaches are being developed. Therefore, we aimed to compare the objective and subjective image quality of different reconstruction techniques and a deep learning-based software on non-contrast chest and low-dose abdominal CTs. In this retrospective study, non-contrast chest CTs of patients suspected of COVID-19 pneumonia and low-dose abdominal CTs suspected of urolithiasis were analysed. All images were reconstructed using filtered back-projection (FBP) and were post-processed using an artificial intelligence (AI)-based commercial software (PixelShine (PS)). Additional iterative reconstruction (IR) was performed for abdominal CTs. Objective and subjective image quality were evaluated. AI-based post-processing led to an overall significant noise reduction independent of the protocol (chest or abdomen) while maintaining image information (max. difference in SNR 2.59 ± 2.9 and CNR 15.92 ± 8.9, p < 0.001). Post-processing of FBP-reconstructed abdominal images was even superior to IR alone (max. difference in SNR 0.76 ± 0.5, p ≤ 0.001). Subjective assessments verified these results, partly suggesting benefits, especially in soft-tissue imaging ( p < 0.001). All in all, the deep learning-based denoising-which was non-inferior to IR-offers an opportunity for image quality improvement especially in institutions using older scanners without IR availability. Further studies are necessary to evaluate potential effects on dose reduction benefits.

Background : This study aimed to evaluate the impact of an AI-assisted fracture detection program on radiology residents' performance in pediatric and adult trauma patients and assess its implications for residency training. Methods : This study, conducted retrospectively, included 200 radiographs from participants aged 1 to 95 years (mean age: 40.7 ± 24.5 years), encompassing various body regions. Among these, 50% (100/200) displayed at least one fracture, totaling one hundred thirty-five fractures, assessed by four radiology residents with different experience levels. A machine learning algorithm was employed for fracture detection, and the ground truth was established by consensus among two experienced senior radiologists. Fracture detection accuracy, reporting time, and confidence were evaluated with and without AI support. Results : Radiology residents' sensitivity for fracture detection improved significantly with AI support (58% without AI vs. 77% with AI, p < 0.001), while specificity showed minor improvements (77% without AI vs. 79% with AI, p = 0.0653). AI stand-alone performance achieved a sensitivity of 93% with a specificity of 77%. AI support for fracture detection significantly reduced interpretation time for radiology residents by an average of approximately 2.6 s ( p = 0.0156) and increased resident confidence in the findings ( p = 0.0013). Conclusion : AI support significantly enhanced fracture detection sensitivity among radiology residents, particularly benefiting less experienced radiologists. It does not compromise specificity and reduces interpretation time, contributing to improved efficiency. This study underscores AI's potential in radiology, emphasizing its role in training and interpretation improvement.

The aim of this study is to examine the precision of semi-automatic, conventional and automatic volumetry tools for pulmonary nodules in chest CT with phantom N1 LUNGMAN. The phantom is a life-size anatomical chest model with pulmonary nodules representing solid and subsolid metastases. Gross tumor volumes (GTV i s) were contoured using various approaches: manually (0); as a means of semi-automated, conventional contouring with (I) adaptive-brush function; (II) flood-fill function; and (III) image-thresholding function. Furthermore, a deep-learning algorithm for automatic contouring was applied (IV). An intermodality comparison of the above-mentioned strategies for contouring GTV i s was performed. For the mean GTV ref (standard deviation (SD)), the interquartile range (IQR)) was 0.68 mL (0.33; 0.34-1.1). GTV segmentation was distributed as follows: (I) 0.61 mL (0.27; 0.36-0.92); (II) 0.41 mL (0.28; 0.23-0.63); (III) 0.65 mL (0.35; 0.32-0.90); and (IV) 0.61 mL (0.29; 0.33-0.95). GTV ref was found to be significantly correlated with GTV i s (I) p < 0.001, r = 0.989 (III) p = 0.001, r = 0.916, and (IV) p < 0.001, r = 0.986, but not with (II) p = 0.091, r = 0.595. The Sørensen-Dice indices for the semi-automatic tools were 0.74 (I), 0.57 (II) and 0.71 (III). For the semi-automatic, conventional segmentation tools evaluated, the adaptive-brush function (I) performed closest to the reference standard (0). The automatic deep learning tool (IV) showed high performance for auto-segmentation and was close to the reference standard. For high precision radiation therapy, visual control, and, where necessary, manual correction, are mandatory for all evaluated tools.

Purpose: To evaluate the use of pulmonary computed tomography (CT) angiography during initial admission at an emergency department (ED), to identify COVID-19 patients with accompanying pulmonary embolism (PE) and its impact on clinical management. Methods: We performed a retrospective analysis of COVID-19 patients that underwent pulmonary CT angiography at the ED. CT scans were evaluated for the presence and extent of PE and for imaging changes suspicious of COVID-19. Patients were subdivided into two groups: (1) Group A consisted of patients with proven COVID-19 based on real-time polymerase chain reaction (RT-PCR), and (2) Group B of patients suspected for COVID-19, comprising patients positive on RT-PCR and/or COVID-19-suspicious CT findings. To assess the differences between patients with and without pulmonary embolism, Fisher’s exact test was used. Results: A total of 308 patients were admitted to the ED for diagnostic work-up of dyspnea and suspected COVID-19, and 95 patients underwent pulmonary CT angiography. PE was detected in 13.6% (3/22) of patients in Group A and 20.7% (6/29) in Group B. No significant differences were observed between patients with and without PE concerning hospitalization (Group B: 100% (6/6) vs. 91.3% (21/23)), the necessity of oxygen therapy (Group B: 66% (4/6) vs. 43.5% (10/23)), and death (Group B: 33% (2/6) vs. 4.3% (1/23) p > 0.05, respectively). Conclusions: In 20.7% of COVID-19 patients, PE was detected upon admission to the ED. Although the incorporation of early pulmonary CT angiography in patients suspicious of COVID-19 may be beneficial to identify concomitant PE, further studies are necessary to corroborate these findings.

Short tau inversion recovery (STIR) sequences are frequently used in magnetic resonance imaging (MRI) of the spine. However, STIR sequences require a significant amount of scanning time. The purpose of the present study was to generate virtual STIR (vSTIR) images from non-contrast, non-fat-suppressed T1- and T2-weighted images using a conditional generative adversarial network (cGAN). The training dataset comprised 612 studies from 514 patients, and the validation dataset comprised 141 studies from 133 patients. For validation, 100 original STIR and respective vSTIR series were presented to six senior radiologists (blinded for the STIR type) in independent A/B-testing sessions. Additionally, for 141 real or vSTIR sequences, the testers were required to produce a structured report of 15 different findings. In the A/B-test, most testers could not reliably identify the real STIR (mean error of tester 1-6: 41%; 44%; 58%; 48%; 39%; 45%). In the evaluation of the structured reports, vSTIR was equivalent to real STIR in 13 of 15 categories. In the category of the number of STIR hyperintense vertebral bodies ( p = 0.08) and in the diagnosis of bone metastases ( p = 0.055), the vSTIR was only slightly insignificantly equivalent. By virtually generating STIR images of diagnostic quality from T1- and T2-weighted images using a cGAN, one can shorten examination times and increase throughput.

Background: To investigate the diagnostic performance of simultaneous 18 F-fluoro-deoxyglucose ([ 18 F]-FDG) PET/MR enterography in assessing and grading endoscopically active inflammation in patients with ulcerative colitis. Methods: 50 patients underwent PET/MR 24 h before ileocolonoscopy. Inflammatory activities of bowel segments were evaluated with both Mayo endoscopic subscore and Nancy histologic index. MR, DWI (Diffusion-weighted imaging) and PET were utilized as qualitative parameters for detecting endoscopically active inflammation. SUVmaxQuot in each segment (maximum of standard uptake value relative to liver) was calculated to quantify inflammation. Results: In the study arm without bowel purgation, combined reading of PET and MR resulted in significantly increased specificity against each submodality alone (0.944 vs. 0.82 for MR and 0.843 for PET, p < 0.05) and highest overall accuracy. In the study arm with bowel purgation, the significantly lower specificity of PET (0.595) could be markedly improved by a combined reading of PET and MR. Metabolic conditions in bowel segments with both endoscopic and histological remission were significantly lower than in segments with endoscopic remission but persistent microscopic inflammation (SUVmaxQuot 0.719 vs. 0.947, p < 0.001). SUVmaxQuot correlated highly with Mayo endoscopic subscore (ρ = 0.718 and 0.606) and enabled grading of inflammatory activity. Conclusions: Simultaneous [ 18 F]-FDG PET/MR may be considered as an alternative to endoscopy in clinical trials.