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Unloading knee braces represent a conservative treatment option for non-pharmalogical management of unicompartmental osteoarthritis of the knee. Though there is consensus on the clinical effectiveness of unloading, the effect mechanism of bracing remains part of a debate. Our study was designed to assess the effect of unloader bracing on damaged cartilage via MRI cartilage mappings.Fourteen patients (7 female, 7 male, mean age 43.1 ± 9.4 years) with unicompartmental cartilage wear in knees with varus or valgus malalignment were enrolled. Clinical scores, radiographs and MR-graphic properties (T2/T2* mapping, T1 Delayed Gadolinium Enhanced MRI of the cartilage (dGEMRIC) mapping, high-resolution PDw sequences) of knee cartilage were recorded before and three months after brace use.Bracing the knees for a mean of 14.4 ± 2.0 weeks (range 11 to 18 weeks) resulted in significant pain reduction (VAS changed from 5.9 ± 2.0 to 2.0 ± 1.3, p 0.001) and improvement in knee function (KOOS increased from 42.1 ± 22.7 to 64.8 ± 18.7, p 0.001). In the affected cartilage regions T2 relaxation times significantly decreased from 56.1 ± 11.4 ms to 46.5 ± 11.2 ms (p 0.05). No changes in T1-dGEMRIC and T2* relaxation times, thickness or the extent of the damaged cartilage area could be detected.Our results suggest, that unloader bracing improves the biochemical properties of the damaged cartilage by increasing collagen and proteoglycan concentration as well as decreasing the cartilage edema.

The original version of the book was inadvertently published with incorrect abstract. This has now been amended in the chapters.

A brief history of the Centre for Mathematical and Statistical Sciences, Kerala, India, is given and an overview of Mathai’s research and education programs in the following topics is outlined: Fractional Calculus; Functions of Matrix Argument—M-transforms, M-convolutions; Kratzel integrals; Pathway Models; Geometrical Probabilities; Astrophysics—reaction rate theory, solar neutrinos; Special Functions—G and H-functions; Multivariate Analysis; Algorithms for Non-linear Least Squares; Characterizations—characterizations of densities, information measure, axiomatic definitions, pseudo analytic functions of matrix argument and characterization of the normal probability law; Mathai’s Entropy—entropy optimization; Analysis of Variance; Population Problems and Social Sciences; Quadratic and Bilinear Forms; Linear Algebra; Probability and Statistics.

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.

Based on resolutions of the United Nations General Assembly, UN-affiliated Regional Centres for Space Science and Technology Education were established in China, India, Morocco, Nigeria, Brazil/Mexico, and Jordan. Simultaneously, education curricula at the university level were developed for the core disciplines of remote sensing, satellite communications, satellite meteorology, space and atmospheric science, and global navigation satellite systems. In 2017, these education curricula were supplemented by the publication of two open access books covering full university courses in linear algebra and probability & statistics.

The present paper contains an overview and summary on the achievements of the basic space science initiative in terms of donated and provided planetariums, astronomical instruments, and space weather instruments, particularly operating in developing nations. These instruments have been made available to respective host countries, particularly developing nations, through the series of twenty basic space science workshops, organized through the United Nations Programme on Space Applications since 1991. Organized by the United Nations, the European Space Agency (ESA), the National Aeronautics and Space Administration (NASA) of the United States of America, and the Japan Aerospace Exploration Agency (JAXA), the basic space science workshops were organized as a series of workshops that focused on basic space science (1991-2004), the International Heliophysical Year 2007 (2005-2009), and the International Space Weather Initiative (2010-2012) proposed by the Committee on the Peaceful Uses of Outer Space on the basis of discussions of its Scientific and Technical Subcommittee, as reflected in the reports of the Subcommittee. Workshops on the International Space Weather Initiative in the series were hosted by the Government of Egypt in 2010 (see A/AC.105/994), the Government of Nigeria in 2011, and the Government of Ecuador in 2012 (see A/AC.105/1030). Workshops on the International Heliophysical Year 2007 were hosted by the United Arab Emirates in 2005 (see A/AC.105/856), India in 2006 (see A/AC.105/882), Japan in 2007 (see A/AC.105/902), Bulgaria in 2008 (see A/AC.105/919) and the Republic of Korea in 2009 (see A/AC.105/964). Workshops on basic space science were hosted by the Governments of India (see A/AC.105/489), Costa Rica and Colombia (see A/AC.105/530), Nigeria (see A/AC.105/560/Add.1), Egypt (see A/AC.105/580), Sri Lanka (see A/AC.105/640), Germany (see A/AC.105/657), Honduras (see A/AC.105/682), Jordan (see A/AC.105/723), France (see A/AC.105/742), Mauritius (see A/AC.105/766), Argentina (see A/AC.105/784) and China (see A/AC.105/829). All workshops were co-organized by the International Astronomical Union (IAU) and the Committee on Space Research (COSPAR).

A status report for an attempt to implement a project of education, teaching, and research in space science and technology in the spirit of the United Nations is briefly summarized in this Viewpoint.

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.

Based on the results of the diffusion entropy analysis of Super-Kamiokande solar neutrino data, a generalized entropy, introduced earlier by the first author is optimized under various conditions and it is shown that Maxwell–Boltzmann distribution, Raleigh distribution and other distributions can be obtained through such optimization procedures. Some properties of the entropy measure are examined and then Maxwell–Boltzmann and Raleigh densities are extended to multivariate cases. Connections to geometrical probability problems, isotropic random points, and spherically symmetric and elliptically contoured statistical distributions are pointed out.

In Chaps. 2, 3, 4, and 5 we considered the real scalar variable case, real multivariate case, real one matrix-variate case, real several matrix-variate case. In the present chapter we will look into fractional calculus in the complex domain. Since we will be dealing with p × p Hermitian positive definite matrices, for p = 1 Hermitian positive definite means a real scalar positive variable. Hence we start with p ≥ 2. Fractional calculus of one real scalar variable case is the one most frequently appearing in various theoretical and applied areas. Fractional calculus in the complex domain was considered only recently, see Mathai [2]. The following discussion is based on this work.

The laws of nature are fundamentally random. This Springer Briefs in Mathematical Physics is an attempt to illustrate elements of a research programme in mathematics and statistics applied to selected problems in physics, particularly the relations between solar neutrinos, diffusion, entropy, and fractional calculus as they appear in neutrino astrophysics since the 1970s. The original research programme was published in three monographs [18–20]. An update of this research programme and selected results achieved since the 1970s is contained in Mathai and Haubold [21] and Mathai, Saxena, and Haubold [22]. The research programme connects mathematics and statistics to theoretical physics with the following historical background in mind.

General notations on matrices, determinants, traces etc. are given in the introduction to Chap. 2 and hence they will not be repeated here. Before starting the discussion, we will need some Jacobians of matrix transformations here. For results on Jacobians, see Mathai [3]. For the real matrix-variate case, the determinant of X will be denoted by either det(X) or by |X|. When complex matrices are involved we will use the notation det(X) for determinant because we would like to reserve the notation |(⋅)| for the absolute value of (⋅). In this case the absolute value of the determinant of \(\tilde {X}\) will be written as \(|\mathrm {det}(\tilde {X})|\), denoting a matrix X in the complex domain as \(\tilde {X}\). All matrices appearing in this chapter are p × p real positive definite unless stated otherwise. Some Jacobians of matrix transformations will be stated here as lemmas without proofs. For proofs and other details, see Mathai [3].