Time-Reversibility, Causality and Compression-Complexity

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number π and heart interbeat interval variability.

Jak pokazać to, czego pokazać nie można? O obrazowaniu liczb niewymiernych

In this article, I would like to draw attention to the cognitive potential of an image, showing how significant the role of visual imagination in mathematics is. I will focus here mainly on the possibilities of visualizing irrational numbers.Our starting point is the intuitive case of the square root of two, observed in the diagonal of a square. We will also discuss a simple, geometrical method of constructing the square roots of all integers. Next, we move over to the golden ratio, hidden in a regular pentagon. We will use a looped, endless animation to visualize the irrational number φ. Then we will have a closer look at the famous number π and discuss two different attempts to find its visual representation. In the last two sections of the article, we consider the possibility of indicating rational and irrational real numbers and also grasp the whole set of real numbers.All the issues discussed in this article have inspired visual artists to create artworks that can help to understand relatively advanced mathematical problems.

The Dynamics of Digits: Calculating Pi with Galperin’s Billiards

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number π . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of π in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be π itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of π with Galperin billiards, including curious cases with irrational number bases.

Mysterious Circle Numbers.Does πp,q Approach πp When q Is Tending to p ?

This paper aims to introduce a mathematical-philosophical type of question from the fascinating world of generalized circle numbers to the widest possible readership. We start with recalling well-known (in part from school) properties of the polygonal approximation of the common circle when approximating the famous circle number π by convergent sequences of upper and lower bounds being based upon the lengths of polygons. Next, we shortly refer to some results from the literature where suitably defined generalized circle numbers of l p - and l p , q -circles, π p and π p , q , respectively, are considered and turn afterwards over to the approximation of an l p -circle by a family of l p , q -circles with q converging to p, q → p . Then we ask whether or not there holds the continuity property π p , q → π p as q → p . The answer to this question leads us to the answer of the question stated in the paper’s title. Presenting here for illustration true paintings instead of strong technical or mathematical drawings intends both to stimulate opening heart and senses of the reader for recognizing generalized circles in his real life and to suggest the philosophical challenge of the consequences coming out from the demonstrated non-continuity property.

Atividade de inferência informal para avaliar π

Resumo: Inferência é a parte da Estatística que trata de previsões, para uma população, a partir das informações de uma amostra. O assunto é discutido de maneira formal em disciplinas de Estatística de cursos superiores, pois envolve conceitos matemáticos avançados. Entretanto, fazer previsões é algo presente na vida de todos e com os estudantes da Educação Básica não é diferente. É importante que os estudantes desse nível de ensino tenham oportunidade de fazer previsões, ainda que limitadas e informais, como uma experiência inicial do uso dessa parte fundamental da Estatística. Neste artigo, aproveitamos a mística envolvendo o número π para discutir, de modo informal, aspectos da Inferência Estatística. Descrevemos uma atividade, presente no portal AtivEstat- Atividades de Estatística, que usa conceitos simples de Geometria para avaliar o valor do número irracional π. Palavras-chave: Inferência informal; Atividade; Valor de π. Informal inference activity to estimate π Abstract: Inference is the part of Statistics that deals with forecasts for a population from the information in a sample. The subject is discussed formally in higher education statistics disciplines as it involves advanced mathematical concepts. However, making predictions is something present in everyone's life and with students at theBasic Education it is not different. It is important that students, at this educational level, have the opportunity to make predictions, even if they are limited and informal, as an initial experience of using this key part of Statistics. In this article, we take advantage of the mystique involving the number π, to informally discuss aspects of Statistical Inference. We describe an activity, included in the Portal AtivEstat - Statistics Activities, that uses simple concepts of Geometry to evaluate the value of the irrational number π. Keywords: Informal inference; Activity; π value.

A study on the pendant number of graph products

A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

Optimizing the number of gates in quantum search

In its usual form, Grover’s quantum search algorithm uses O( √ N) queries and O( √ N log N) other elementary gates to find a solution in an N-bit database. Grover in 2002 showed how to reduce the number of other gates to O( √ N log log N) for the special case where the database has a unique solution, without significantly increasing the number of queries. We show how to reduce this further to O( √ N log(r) N) gates for every constant r, and sufficiently large N. This means that, on average, the circuits between two queries barely touch more than a constant number of the log N qubits on which the algorithm acts. For a very large N that is a power of 2, we can choose r such that the algorithm uses essentially the minimal number π 4 √ N of queries, and only O( √ N log(log? N)) other gates.