Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System

Orbital systems are often self-organized and/or characterized by harmonic relations. Inspired by music theory, we rewrite the Geddes and King-Hele (QJRAS, 24, 10–13, 1983) equations for mirror symmetries among the distances of the planets of the Solar System in an elegant and compact form by using the 2/3rd power of the ratios of the semi-major axis lengths of two neighboring planets (eight pairs, including the belt of the asteroids). This metric suggests that the Solar System could be characterized by a scaling and mirror-like structure relative to the asteroid belt that relates together the terrestrial and Jovian planets. These relations are based on a 9/8 ratio multiplied by powers of 2, which correspond musically to the interval of the Pythagorean epogdoon (a Major Second) and its addition with one or more octaves. Extensions of the same model are discussed and found compatible also with the still hypothetical vulcanoid asteroids versus the transneptunian objects. The found relation also suggests that the planetary self-organization of our system could be generated by the 3:1 and 7:3 resonances of Jupiter, which are already known to have shaped the asteroid belt. The proposed model predicts the main Kirkwood asteroid gaps and the ratio among the planetary orbital parameters with a 99% accuracy, which is three times better than an alternative, recently proposed harmonic-resonance model for the Solar System. Furthermore, the ratios of neighboring planetary pairs correspond to four musical “consonances” having frequency ratios of 5/4 (Major Third), 4/3 (Perfect Fourth), 3/2 (Perfect Fifth) and 8/5 (Minor Sixth); the probability of obtaining this result randomly has a p < 0.001. Musical consonances are “pleasing” tones that harmoniously interrelate when sounded together, which suggests that the orbits of the planets of our Solar System could form some kind of gravitationally optimized and coordinated structure. Physical modeling indicates that energy non-conserving perturbations could drive a planetary system into a self-organized periodic state with characteristics vaguely similar of those found in our Solar System. However, our specific finding suggests that the planetary organization of our Solar System could be rather peculiar and based on more complex and unknown dynamical structures.

Expansion for the critical point of site percolation: the first three terms

We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

Natural geometric unit system and electron magnetic moment anomaly

Consequences of implementation of the natural geometric unit system (the SG) based on the pre-2019 SI system, in which four fundamental physical constants undergo joint numerical and dimensional normalization to unity c = G= k = h = 1, with only one base geometric unit u equal to √|h · G/c 3 | m, where the Newtonian gravitational constant G ≈ 6.673 655 205 · 10 -11 m3/(kg · s 2 ), are further explored. In addition to the earlier hypothesized simple electron mass to charge ratio formula me = e/(2 9πα), and formulas for stable quarks rest masses: quark u mu = √(⅔) / (2 7π √(πα)) u, equivalent of 2.360 MeV/c 2 and quark d md = √(⅓) -1 / (2 7π √(πα)) u, equivalent of 5.007 MeV/c 2 , a simple formula for electron magnetic moment anomaly is proposed α/2π - (α/2π) 2 - 2 8 (α/2π) 3 - 2 12 (α/2π) 4 - 2 16 (α/2π) 5 - 2 24 (α/2π) 6 ≈ 0.001 159 652 180. The finding supports the research area of purely geometric modelling of the fundamental physical forces and their unification. It seems plausible, that in the SG, with use of half integer powers of 2, 3, π and α only,all the fundamental properties of stable matter and electromagnetic radiation could be described

Modified Hilbert Curve for Rectangles and Cuboids and Its Application in Entropy Coding for Image and Video Compression

In our previous work, by combining the Hilbert scan with the symbol grouping method, efficient run-length-based entropy coding was developed, and high-efficiency image compression algorithms based on the entropy coding were obtained. However, the 2-D Hilbert curves, which are a critical part of the above-mentioned entropy coding, are defined on squares with the side length being the powers of 2, i.e., 2n, while a subband is normally a rectangle of arbitrary sizes. It is not straightforward to modify the Hilbert curve from squares of side lengths of 2n to an arbitrary rectangle. In this short article, we provide the details of constructing the modified 2-D Hilbert curve of arbitrary rectangle sizes. Furthermore, we extend the method from a 2-D rectangle to a 3-D cuboid. The 3-D modified Hilbert curves are used in a novel 3-D transform video compression algorithm that employs the run-length-based entropy coding. Additionally, the modified 2-D and 3-D Hilbert curves introduced in this short article could be useful for some unknown applications in the future.

On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

In this paper, we show that if [Formula: see text] is the [Formula: see text]th solution of the Pell equation [Formula: see text] for some non-square [Formula: see text], then given any integer [Formula: see text], the equation [Formula: see text] has at most [Formula: see text] integer solutions [Formula: see text] with [Formula: see text] and [Formula: see text], except for the only pair [Formula: see text]. Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.

Associative spectra of graph algebras I

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

Analysis of Microarray Data by Empirical Wavelet Transform for Cancer Classification Using Block by Block Method

In this study, DNA microarray data is analyzed from a signal processing perspective for cancer classification. An adaptive wavelet transform named Empirical Wavelet Transform (EWT) is analyzed using block-by-block procedure to characterize microarray data. The EWT wavelet basis depends on the input data rather predetermined like in conventional wavelets. Thus, EWT gives more sparse representations than wavelets. The characterization of microarray data is made by block-by-block procedure with predefined block sizes in powers of 2 that starts from 128 to 2048. After characterization, a statistical hypothesis test is employed to select the informative EWT coefficients. Only the selected coefficients are used for Microarray Data Classification (MDC) by the Support Vector Machine (SVM). Computational experiments are employed on five microarray datasets; colon, breast, leukemia, CNS and ovarian to test the developed cancer classification system. The obtained results demonstrate that EWT coefficients with SVM emerged as an effective approach with no misclassification for MDC system.

Soma iterada de algarismos de um número racional

The digital roots S* (x), of a n positive integer is the digit 0 ≤ b ≤ 9 obtained through an iterative digit sum process, where each iteration is obtained from the previous result so that only the b digit remains. For example, the iterated sum of 999999 is 9 because 9 + 9 + 9 + 9 + 9 + 9 = 54 and 5 + 4 = 9. The sum of the digits of a positive integer, and even the digital roots, is a recurring subject in mathematical competitions and has been addressed in several papers, for example in Ghannam (2012), Ismirli (2014) or Lin (2016). Here we extend the application Sast to a positive rational number x with finite decimal representation. We highlight the following result: given a rational number x, with finite decimal representation, and the sum of its digits is 9, so when divided x by powers of 2, the number resulting also has the sum of its digits 9. Fact that also occurs when the x number is divided by powers of 5. Similar results were found when the x digit sum is 3 or 6.

Natural geometric unit system and electron magnetic moment anomaly

Consequences of implementation of the natural geometric unit system (the SG) based on the pre-2019 SI system, in which four fundamental physical constants undergo joint numerical and dimensional normalization to unity c = G = k = h = 1, with only one base geometric unit u equal to √|h · G/c3| m, where the Newtonian gravitational constant G ≈ 6.673 655 205 · 10-11 m3/(kg · s2), are further explored. In addition to the earlier hypothesized simple electron mass to charge ratio formula me = e/(29πα), and formulas for stable quarks rest masses: quark u mu = √(⅔) / (27π √(πα)) u, equivalent of 2.360 MeV/c2 and quark d md = √(⅓)-1 / (27π √(πα)) u, equivalent of 5.007 MeV/c2, a simple formula for electron magnetic moment anomaly is proposed α/2π - (α/2π)2 - 28(α/2π)3 - 212(α/2π)4 - 216(α/2π)5 - 224(α/2π)6 ≈ 0.001 159 652 180. The finding supports the research area of purely geometric modelling of the fundamental physical forces and their unification. It seems plausible, that in the SG, with use of half integer powers of 2, 3, π and α only, all the fundamental properties of stable matter and electromagnetic radiation could be described.