Rational Arithmetic Mathematica Functions to Evaluate the One-sided One-sample K-S Cumulative Sample Distribution

One of the most widely used goodness-of-fit tests is the Kolmogorov-Smirnov (KS) family of tests which have been implemented by many computer statistical software packages. To calculate a p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the one-sided one-sample K-S test, this paper identifies two direct formulae and five recursion formulae that can be used to calculate a p value and then develops two additional direct formulae and four iterative versions of the direct formulae for a total of thirteen formulae. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each formula is implemented in rational arithmetic. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval bandwidth). Extensive tables of bandwidths are presented for sample sizes up to 2, 000. The results confirm the hypothesis that as the number of digits in the numerator and denominator integers of the rational number test statistic increases, the computation time also increases. In comparing the computational times of the thirteen formulae, the direct formulae are slightly faster than their iterative versions and much faster than all the recursion formulae. Computational times for the fastest formula are given for sample sizes up to fifty thousand.

The co-production of historical knowledge: implications for the history of identities

This essay argues that understanding people’s lives, emotions and intellectual reasoning is crucial to exploring national identity and that ‘the co-production of historical knowledge’ provides an approach or methodology that allows for a deeper comprehension of people’s self-identities by encouraging a diverse range of people to participate in the research process. We argue that many academic historians have maintained an intellectual detachment between university history and public and community history, to the detriment of furthering historical knowledge. We argue for a blurring of the boundaries between university and communities in exploring modern British history, and especially the history of national identities. It includes extracts of writing from community partners and a brief photographic essay of projects related to exploring identities

Rational Arithmetic Mathematica Functions to Evaluate the Two-Sided One Sample K-S Cumulative Sampling Distribution

One of the most widely used goodness-of-fit tests is the two-sided one sample Kolmogorov-Smirnov (K-S) test which has been implemented by many computer statistical software packages. To calculate a two-sided p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the two-sided one sample K-S test, this paper identifies an exact formula for sample sizes up to 31, six recursion formulae, and one matrix formula that can be used to calculate a p value. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each of these formulae is implemented in rational arithmetic. For the six recursion formulae and the matrix formula, computational experience for sample sizes up to 500 shows that computational times are increasing functions of both the sample size and the number of digits in the numerator and denominator integers of the rational number test statistic. The computational times of the seven formulae vary immensely but the Durbin recursion formula is almost always the fastest. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval half-width) and tables of calculated half-widths are presented for sample sizes up to 500. Using calculated half-widths as input, computational times for the fastest formula, the Durbin recursion formula, are given for sample sizes up to two thousand

Arbitrary Precision Mathematica Functions to Evaluate the One-Sided One Sample K-S Cumulative Sampling Distribution

Efficient rational arithmetic methods that can exactly evaluate the cumulative sampling distribution of the one-sided one sample Kolmogorov-Smirnov (K-S) test have been developed by Brown and Harvey (2007) for sample sizes n up to fifty thousand. This paper implements in arbitrary precision the same 13 formulae to evaluate the one-sided one sample K-S cumulative sampling distribution. Computational experience identifies the fastest implementation which is then used to calculate confidence interval bandwidths and p values for sample sizes up to ten million

Transport and Elastic Properties of Fractal Media

We investigate the influence of fractal structure on material properties. We calculate the statistical correlation functions of fractal media defined by level-cut Gaussian random fields. This allows the modeling of both surface fractal and mass fractal materials. Variational bounds on the conductivity, diffusivity and elastic moduli of the materials are evaluated. We find that a fractally rough interface has a relatively strong influence on the properties of composites. In contrast a fractal volume (mass) has little effect on material properties.Comment: 10 pages, 6 figure

A bioprinted cardiac patch composed of cardiac-specific extracellular matrix and progenitor cells for heart repair

Congenital heart defects are present in 8 of 1000 newborns and palliative surgical therapy has increased survival. Despite improved outcomes, many children develop reduced cardiac function and heart failure requiring transplantation. Human cardiac progenitor cell (hCPC) therapy has potential to repair the pediatric myocardium through release of reparative factors, but therapy suffers from limited hCPC retention and functionality. Decellularized cardiac extracellular matrix hydrogel (cECM) improves heart function in animals, and human trials are ongoing. In the present study, a 3D-bioprinted patch containing cECM for delivery of pediatric hCPCs is developed. Cardiac patches are printed with bioinks composed of cECM, hCPCs, and gelatin methacrylate (GelMA). GelMA-cECM bioinks print uniformly with a homogeneous distribution of cECM and hCPCs. hCPCs maintain >75% viability and incorporation of cECM within patches results in a 30-fold increase in cardiogenic gene expression of hCPCs compared to hCPCs grown in pure GelMA patches. Conditioned media from GelMA-cECM patches show increased angiogenic potential (>2-fold) over GelMA alone, as seen by improved endothelial cell tube formation. Finally, patches are retained on rat hearts and show vascularization over 14 d in vivo. This work shows the successful bioprinting and implementation of cECM-hCPC patches for potential use in repairing damaged myocardium

Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors

We present an efficient expression for the analytic continuation to arbitrary complex frequencies of the complex optical and AC conductivity of a homogeneous superconductor with arbitrary mean free path. Knowledge of this quantity is fundamental in the calculation of thermodynamic potentials and dispersion energies involving type-I superconducting bodies. When considered for imaginary frequencies, our formula evaluates faster than previous schemes involving Kramers--Kronig transforms. A number of applications illustrates its efficiency: a simplified low-frequency expansion of the conductivity, the electromagnetic bulk self-energy due to longitudinal plasma oscillations, and the Casimir free energy of a superconducting cavity.Comment: 20 pages, 7 figures, calculation of Casimir energy adde