Power partitions and a generalized eta transformation property

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

The Erdős–Rényi Law and Strong Limit Theorems of Probability

Fifty years ago P. Erdős and A. Rényi published their famous paper on the new law of large numbers. In this survey, we describe numerous results and achievements which are related with this paper or motivated by it during these years.

Construction of Consistent Comparison Matrix by Macharis’ Method Revisit

A famous paper that has been cited more than four hundred times tried to combine (a) the preference ranking organization method for enrichment evaluations (PROMETHEE) and (b) the analytic hierarchy process (AHP) to construct a new method for multicriteria decision-making problems. The paper developed a consistent comparison matrix for their AHP by the defined first row and then they allowed the expert to change several entries in the comparison matrix. Hence, how to construct a new comparison matrix that is (i) consistent and (ii) satisfying the assigned values by the expert becomes a challenging problem. A recent article provided a reply to the above problem by the construction of all entries for the comparison matrix. However, they did not follow the original design proposed by the famous paper. In this paper, we present a new approach with a proposition that satisfies the original design of the famous paper and also achieves two goals (i) and (ii). The research gap of proof is fulfilled by this paper. Our findings explain that the original construction of the famous paper to develop a consistent comparison matrix only by the first row with several assigned values by an expert is indeed workable under two additional restrictions proposed by the recent article. We believe that after our proposition, researchers have the confidence to execute the original design of the paper that has been cited more than four hundred times.

Information Theory: Deep Ideas, Wide Perspectives, and Various Applications

The history of information theory, as a mathematical principle for analyzing data transmission and information communication, was formalized in 1948 with the publication of Claude Shannon’s famous paper “A Mathematical Theory of Communication” [...]

A Study of The Effect of Black Swan Events on Stock Markets – and Developing a Model for Predicting and Responding to them

Black Swan events are rare and seemingly random in nature. In the famous paper by Nassim Nicholas Taleb, it is posited that Black Swan events cannot be reliably predicted, and it is instead important to be prepared for them at all times (Musgrave, 2009). This study aims to understand various Black Swan events in recent history from the point of view of equity markets, and performs a comparative study between different events across time and geography in order to understand if there are any standard early indicators. In this study, a total of seven global events have been observed within the selected period from FY 1997 to FY 2019. All events have been considered from the standpoint of their impact on S&P 500 index. Apart from assessing the sensitivity factor, the impact of each event was observed for any statistical similarity through a One-Way ANOVA test of their normalised values.

Programming Away Human Rights and Responsibilities? “The Moral Machine Experiment” and the Need for a More “Humane” AV Future

Dilemma situations involving the choice of which human life to save in the case of unavoidable accidents are expected to arise only rarely in the context of autonomous vehicles (AVs). Nonetheless, the scientific community has devoted significant attention to finding appropriate and (socially) acceptable automated decisions in the event that AVs or drivers of AVs were indeed to face such situations. Awad and colleagues, in their now famous paper “The Moral Machine Experiment”, used a “multilingual online ‘serious game’ for collecting large-scale data on how citizens would want AVs to solve moral dilemmas in the context of unavoidable accidents.” Awad and colleagues undoubtedly collected an impressive and philosophically useful data set of armchair intuitions. However, we argue that applying their findings to the development of “global, socially acceptable principles for machine learning” would violate basic tenets of human rights law and fundamental principles of human dignity. To make its arguments, our paper cites principles of tort law, relevant case law, provisions from the Universal Declaration of Human Rights, and rules from the German Ethics Code for Autonomous and Connected Driving.

Answering Mermin’s challenge with conservation per no preferred reference frame

In 1981, Mermin published a now famous paper titled, “Bringing home the atomic world: Quantum mysteries for anybody” that Feynman called, “One of the most beautiful papers in physics that I know.” Therein, he presented the “Mermin device” that illustrates the conundrum of quantum entanglement per the Bell spin states for the “general reader.” He then challenged the “physicist reader” to explain the way the device works “in terms meaningful to a general reader struggling with the dilemma raised by the device.” Herein, we show how “conservation per no preferred reference frame (NPRF)” answers that challenge. In short, the explicit conservation that obtains for Alice and Bob’s Stern-Gerlach spin measurement outcomes in the same reference frame holds only on average in different reference frames, not on a trial-by-trial basis. This conservation is SO(3) invariant in the relevant symmetry plane in real space per the SU(2) invariance of its corresponding Bell spin state in Hilbert space. Since NPRF is also responsible for the postulates of special relativity, and therefore its counterintuitive aspects of time dilation and length contraction, we see that the symmetry group relating non-relativistic quantum mechanics and special relativity via their “mysteries” is the restricted Lorentz group.

Stokes drift in coral reefs with depth-varying permeability

In his famous paper of 1847 (Stokes GG. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8 , 441–455), Stokes introduced the drift effect of particles in a fluid that is undergoing wave motion. This effect, now known as Stokes drift, is the result of differences between the Lagrangian and Eulerian velocities of the fluid element and has been well-studied, both in the laboratory and as a mechanism of mass transport in the oceans. On a smaller scale, it is of vital importance to the hydrodynamics of coral reefs to understand drift effects arising from waves on the ocean surface, transporting nutrients and oxygen to the complex ecosystems within. A new model is proposed for a class of coral reefs in shallow seas, which have a permeable layer of depth-varying permeability. We then note that the behaviour of the waves above the reef is only affected by the permeability at the top of the porous layer, and not its properties within, which only affect flow inside the porous layer. This model is then used to describe two situations found in coral reefs; namely, algal layers overlying the reef itself and reef layers whose permeability decreases with depth. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

Forgetting an Argument

The notion of forgetting, as considered in the famous paper by Lin and Reiter in 1994 has been extensively studied in classical logic and more recently, in non-monotonic formalisms like logic programming. In this paper, we convey the idea of forgetting to another major AI formalism, namely Dung-style argumentation frameworks. Our approach is axiomatic-driven and not limited to any specific semantics: we propose semantical and syntactical desiderata encoding different criteria for what forgetting an argument might mean; analyze how these criteria relate to each other; and check whether the criteria can be satisfied in general. The analysis is done for a number of widely used argumentation semantics. Our investigation shows that almost all desiderata are individually satisfiable. However, combinations of semantical and/or syntactical conditions reveal a much more interesting landscape. For instance, we found that the ad hoc approach to forgetting an argument, i.e., by the syntactical removal of the argument and all of its associated attacks, is too restrictive and only compatible with the two weakest semantical desiderata. Amongst the several interesting combinations identified, we showed that one satisfies a notion of minimal change and presented an algorithm that given an AF F and argument x, constructs a suitable AF G satisfying the conditions in the combination.

Gibbs’ third paper

In his third and most famous paper, Gibbs created chemical potential to enable analysis of equilibrium in multi-species / multi-phase systems, introduced his eponymous phase rule, and developed the conceptual framework for composite properties of matter. By combining math and science, he demonstrated the usefulness of calculus in thermodynamics.