The question of how many shuffles randomise a pack of playing cards has fascinated mathematicians, scientists, and the general public. Different theoretical approaches have led to statistically different answers. This paper reports a comprehensive test of manual and mechanical shuffles examined in regard to runs, rising sequences, entropy & information loss, and other statistical measures.

M P Silverman, "Progressive Randomization of a Deck of Playing Cards: Experimental Test and Statistical Analysis of the Riffle Shuffle," Open Journal of Statistics 9 (2019) 268-298

Predicting the forces of reaction on a simple ladder has long remained an unresolved physical problem. The difficulty is that the system is statically indeterminate and therefore requires complementary information. Part I analyses three fundamental models of a ladder and reports a comprehensive experiment test that reveals the correct model. Part II examines the role of friction at the base and top supports of the ladder. Many thousands of accidents annually involve falls from ladders, and many of these cases are litigated in the courts. The solution to the ladder problem provides guidelines for safe use of ladders and a reliable theoretical basis for ascertaining whether collapsed support structures met statutory building codes.

Part I: M P Silverman, "Reaction Forces on a Fixed Ladder in Static Equilibrium: Analysis and Definitive Experimental Test of the Ladder Problem", World Journal Mechanics 8 (2018) 311-342 - Full paper

Part II: M P Silverman, "The Role of Friction in the Static Equilibrium of a Fixed Ladder: Theoretical Analysis and Experimental Test", World Journal of Mechanics 8 (2018) 445-463 - Full paper

Numerous objects, either natural (e.g. trees) or designed (e.g. fishing rods), take the form of a flexible rod (cantilever). This paper presents a comprehensive analysis and experimental test of the exact theory of the bending of a tapered rod. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the theory or neglect the tapering of the rod.

See: M P Silverman and J Farrah, "Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory", World Journal Mechanics 8 (2018) 272-300 - Full paper

Brownian motion can be analysed by either of two equivalent methods (Fokker-Planck equation or Langevin equation) if the motion is continuous. However, the diffusion of radioactive particles is not a continuous process, since their paths terminate when the particles decay. Nevertheless, this paper demonstrates that the two methods yield identical statistical information in the case of a constant nuclear decay rate (Rutherford-Soddy law).

See: M P Silverman and A Mudvari, "Brownian Motion of Radioactive Particles: Derivation and Monte Carlo Test of Spatial and Temporal Distributions", World Journal of Nuclear Science and Technology 8 (2018) 86-119

Methods to detect and monitor radioactive gases require understanding the combined effects of diffusion and decay. Derivation and solution of the Fokker-Planck and Langevin equations appropriate to these two processes reveal important differences in the spatial and temporal statistics of stable and unstable particles.

See: M P Silverman, "Brownian Motion of Decaying Particles: Transition Probability, Computer Simulation, and First-Passage Times", Journal of Modern Physics 8 (2017) 1809-1849

Measurement of radon concentration by passive diffusion depends on an accurate assessment of how many atoms pass through the detection region in a given time. An alternative, but equivalent, micro-statistical approach is to determine the mean time spent by one atom in the detection region. The solution of this problem, besides validating the original macro-statistical approach, addressed a number of interesting issues concerning the random walk of a particle subject to random decay.

See: M P Silverman, “Analysis of Residence Time in the Measurement of Radon Activity by Passive Diffusion in an Open Volume: A Micro-Statistical Approach”, World Journal of Nuclear Science and Technology 7 (2017) 252-273

Mn-54 decays to Cr-54 by the capture of an orbital electron. Interactions that affect the inner electrons could conceivably affect the Mn-54 decay rate. In response to published claims that Mn-54 decay on Earth is influenced by the Sun, I performed a battery of statistical tests on a long record of Mn-54 decays. The outcome was an exquisite agreement with standard nuclear physics.

See: M P Silverman, "Search for Anomalies in the Decay of Radioactive Mn-54", Europhysics Letters 114 (2016) 62001

See: Editorial highlight by Hamish Johnston in Physics World:

Radon is a radioactive gas that seeps into homes and workplaces and is the leading cause of lung cancer among non-smokers. Many methods exist to measure indoor radon concentration, but accurate ones require specialised equipment and long waiting times, since samples are sent to commercial labs for processing. I have developed a method that is accurate, uses the simplest, overall least expensive nuclear counters, and can be implemented in situ by those who wish to make their own measurements.

See: M P Silverman, "Method to Measure Indoor Radon Concentration in an Open Volume with Geiger-Mueller Counters: Analysis from First Principles", World Journal of Nuclear Science and Technology 6 (2016) 232-260

Reports of periodic variations in the decay rate of certain unstable nuclei are believed to violate known physical laws and have been disputed, but the effects are weak and the situation remains ambiguous. I propose alternative methods to search for non-standard radioactive decay that are much more sensitive to variations in the decay rate.

See: M P Silverman, "Search for Non-Standard Radioactive Decay Based on Distribution of Activities", Europhysics Letters 110 (2015) 52001

See: M P Silverman, Effects of a Periodic Decay Rate on the Statistics of Radioactive Decay: New Methods to Search for Violations of the Law of Radioactive Change, Journal of Modern Physics 6 (2015) 1533-1553

Cheating or Coincidence?

Correspondences between a student's solutions to a set of homework problems and the instructor's solutions may suggest that the student copied the work from an illicit source. A statistical analysis can help determine how likely it is that the points of similarity are coincidental. Since effects of randomness can be counter-intuitive and easily misjudged, the analysis must be done objectively and accurately to reduce the chance of a false accusation.

See: M P Silverman, "Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism", Open Journal of Statistics 5 (2015) 143-157

The Half-Life of Radioactive Nuclei

Estimating the half-life of an unstable nucleus from just two measurements does not give an accurate result. However, a histogram of a large number of two-point estimates leads to a nearly perfect Cauchy distribution centred at the true value.

See: M P Silverman, “Theory of nuclear half-life determination by statistical sampling”, Europhysics Letters 105 (2014) 22001 p1-p5. doi: 10.1209/0295-5075/105/22001

Rising Temperatures and Urban Heat Stress

The Earth’s surface temperature is increasing. However, measurements of subterranean temperature variations, which are largely insensitive to the vicissitudes of weather, indicate that the rate of temperature rise in cities can far outpace the rate of global or regional warming.

See: M P Silverman, “Statistical analysis of subsurface diffusion of solar energy with implications for urban heat stress”, Journal of Modern Physics, 5 (2014) 751-762

Recurrent Patterns in Random Events

The common perception (see Wikipedia) of randomness is that it is devoid of all order, pattern, or predictability. However, that is a misperception. It is precisely the existence of patterns of recurrence that characterise a random process and serve usefully in many ways, among which is to test the randomness of quantum events.

See: M P Silverman, “Numerical Procedures for Calculating the Probabilities of Recurrent Runs”, Open Journal of Statistics 4 (2014) 122-131