arXiv subjects

Vladimir Pletser

Publications and source records attributed to Vladimir Pletser.

At least 19 recordsLinked to original sources

arXiv2026

Evolution of Mean Orbital Spacing in Planetary and Satellite Systems under Tidal Dissipation and Nebular Drag

The approximately geometric spacing of orbital distances in planetary and regular satellite systems has long been recognized, yet its dynamical evolution remains poorly constrained. In this paper, we investigate the secular evolution of the mean distance ratio of secondaries under the combined effects of primary tidal dissipation and nebular gas drag. A general analytical framework is derived linking the initial and final mean distance ratios to system parameters and to the physical characteristics of the dominant dissipative processes. Applying this formalism to the Solar System planets and to the regular satellites of Jupiter, Saturn, and Uranus, we show that primary tidal interactions produce only negligible changes in mean distance ratios over timescales comparable to the age of the Solar System. Similarly, nebular gas drag during the protoplanetary and circumplanetary disk phases leads to limited deviations over a broad range of disk models and lifetimes. These results suggest that, within the assumptions adopted, the mean distance ratio evolves only weakly and may preserve information about primordial system configurations established during early disk evolution. The approximate conservation of this quantity may therefore provide a useful diagnostic for constraining the formation and early dynamical evolution of planetary and satellite systems, with potential implications for the architecture of exoplanetary systems.

astro-ph.EP↗

arXiv2023

Enabling Astronaut Self-Scheduling using a Robust Advanced Modelling and Scheduling system: an assessment during a Mars analogue mission

Human long duration exploration missions (LDEMs) raise a number of technological challenges. This paper addresses the question of the crew autonomy: as the distances increase, the communication delays and constraints tend to prevent the astronauts from being monitored and supported by a real time ground control. Eventually, future planetary missions will necessarily require a form of astronaut self-scheduling. We study the usage of a computer decision-support tool by a crew of analog astronauts, during a Mars simulation mission conducted at the Mars Desert Research Station (MDRS, Mars Society) in Utah. The proposed tool, called Romie, belongs to the new category of Robust Advanced Modelling and Scheduling (RAMS) systems. It allows the crew members (i) to visually model their scientific objectives and constraints, (ii) to compute near-optimal operational schedules while taking uncertainty into account, (iii) to monitor the execution of past and current activities, and (iv) to modify scientific objectives/constraints w.r.t. unforeseen events and opportunistic science. In this study, we empirically measure how the astronauts, who are novice planners, perform at using such a tool when self-scheduling under the realistic assumptions of a simulated Martian planetary habitat.

cs.RO↗

arXiv2023

Annular structures in perturbed low mass disc-shaped gaseous nebulae I : general and standard models

Abstract This is the first of two papers where we study analytical solutions of a bidimensional low mass gaseous disc slowly rotating around a central mass and submitted to small radial periodic perturbations. Hydrodynamics equations are solved for the equilibrium and perturbed configurations. A wave-like equation for the gas perturbed specific mass is deduced and solved analytically for several cases of exponents of the power law distributions of the unperturbed specific mass and sound speed. It is found that, first, the gas perturbed specific mass displays exponentially spaced maxima, corresponding to zeros of the radial perturbed velocity; second, the distance ratio of successive maxima of the perturbed specific mass is a constant depending on disc characteristics and, following the model, also on the perturbation's frequency; and, third, inward and outward gas flows are induced from zones of minima toward zones of maxima of perturbed specific mass, leading eventually to the possible formation of gaseous annular structures in the disc. The results presented may be applied in various astrophysical contexts to slowly rotating thin gaseous discs of negligible relative mass, submitted to small radial periodic perturbations.

astro-ph.EP↗

arXiv2022

Euler's and the Taxi Cab relations and other numbers that can be written twice as sums of two cubed integers

We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.

math.NT↗

arXiv2022

Annular structures in perturbed low mass disc-shaped gaseous nebulae II : general and polytropic models

This is the second of two papers where we study additional analytical solutions of a bidimensional low mass gaseous disc rotating around a central mass and submitted to small radial perturbations. In a first Paper, hydrodynamics equations were solved for the equilibrium and perturbed configurations and a wave-like equation for the gas perturbed specific mass was deduced and solved analytically for several cases of exponents of the power law distributions of the unperturbed specific mass and sound speed. In this paper, two other general cases of exponents, including a polytropic case, are solved analytically for small frequencies of the perturbations. Similar conclusions to the ones of Paper I are found, namely that the maxima of the gas perturbed specific mass are exponentially spaced and that their distance ratio is a constant, function of disc characteristics and of the perturbations frequency. Gaseous annular structures would eventually be formed in the disc by inward and outward gas flows from zones of minima toward zones of maxima of perturbed specific mass.

astro-ph.EP↗

arXiv2021

Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations

For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. General algebraic expressions of fundamental solutions of the Pell equations are found for some values of the multiplier k in function of the closest natural square. Further, among the expressions of Pell equation solutions, a set of recurrent relations is identical to those found previously without the Pell equation solving method. It is found also that two constants of the problem of multiples of triangular numbers are directly related to the fundamental solutions of the simple Pell equation, which is an unexpected result as it means that simple Pell equation fundamental solutions in all generality are related to constants in recurrent relations of the problem of finding triangular numbers multiple of other triangular numbers.

math.NT↗

arXiv2021

Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers

It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $ξ$ of triangular numbers that are multiples of other triangular numbers. We show that the remainders in the congruence relations of $ξ$ modulo k come always in pairs whose sum always equal $\left(k-1\right)$, always include 0 and $\left(k-1\right)$, and only 0 and $\left(k-1\right)$ if $k$ is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier $k$ is twice the triangular number of $n$, the set of remainders includes also $n$ and $\left(n^{2}-1\right)$ and if $k$ has integer factors, the set of remainders include multiples of a factor following certain rules. Finally, algebraic expressions are found for remainders in function of $k$ and its factors. Several exceptions are noticed and superseding rules exist between various rules and expressions of remainders. This approach allows to eliminate in numerical searches those $\left(k-\upsilon\right)$ values of $ξ_{i}$ that are known not to provide solutions, where $\upsilon$ is the even number of remainders. The gain is typically in the order of $k/\upsilon$, with $\upsilon\ll k$ for large values of $k$.

math.GM↗

arXiv2021

Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers

Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value. Instead of recurrent relations, we develop in this paper closed form equations to calculate directly the values of triangular numbers and their indices without the need of knowing the previous solutions. We develop the theoretical equations for four cases of ranks from 1 to 4 and we give several examples for non-square multipliers 2, 3, 5 and 8.

math.GM↗

arXiv2020

Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and recurrent relations are deduced theoretically. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value.

math.NT↗

arXiv2018

Orbital Period Ratios and Fibonacci Numbers in Solar Planetary and Satellite Systems and in Exoplanetary Systems

It is shown that orbital period ratios of successive secondaries in the Solar planetary and giant satellite systems and in exoplanetary systems are preferentially closer to irreducible fractions formed with Fibonacci numbers between 1 and 8 than to other fractions, in a ratio of approximately 60% to 40%. Furthermore, if sets of minor planets are chosen with gradually smaller inclinations and eccentricities, the proximity to Fibonacci fractions of their period ratios with Jupiter or Mars$'$ period tends to increase. Finally, a simple model explains why the resonances with ratios of orbital periods $P_{1}$ and $P_{2}$ of successive secondaries being equal to ratio of small integers $p$ and $(p+q)$, $P_{1}/P_{2}=p/(p+q)$, are stronger and more commonly observed.

physics.pop-ph↗

arXiv2017

Fibonacci Numbers and the Golden Ratio in Biology, Physics, Astrophysics, Chemistry and Technology: A Non-Exhaustive Review

Fibonacci numbers and the golden ratio can be found in nearly all domains of Science, appearing when self-organization processes are at play and/or expressing minimum energy configurations. Several non-exhaustive examples are given in biology (natural and artificial phyllotaxis, genetic code and DNA), physics (hydrogen bonds, chaos, superconductivity), astrophysics (pulsating stars, black holes), chemistry (quasicrystals, protein AB models), and technology (tribology, resistors, quantum computing, quantum phase transitions, photonics).

physics.pop-ph↗

arXiv2017

Lecar's visual comparison method to assess the randomness of Bode's law: an answer

The usual main objection against any attempt in finding a physical cause for the planet distance distribution is based on the assumption that similar distance distribution could be obtained by sequences of random numbers. This assumption was stated by Lecar in an old paper (1973). We show here how this assumption is incorrect and how his visual comparison method is inappropriate.

physics.pop-ph↗

arXiv2017

Exponential Distance Relation and Near Resonances in the Trappist-1 Planetary System

We report in this paper a new exponential relation distance of planets in the newly discovered exoplanetary system of the Trappist-1 star, and we comment on near orbital mean motion resonances among the seven planets. We predict that possible smaller planets could be found inside the orbit of the innermost discovered Planet b.

astro-ph.IM↗

arXiv2016

Contradictions and narrowness of views in "The fables of Ishango, or the irresistible temptation of mathematical fiction", answers and updates (in English and in French)

We answer to criticisms of O. Keller about our interpretation work on the Ishango rod, the oldest mathematical tool of humankind. Our hypothesis, that is widely accepted, is that this prehistoric rod is the first mankind manifestation of a basic arithmetic intention, with simple arithmetic operations and possibly showing passages between 10 and 12 bases.

math.HO↗

arXiv2015

On some general solutions of the simple Pell equation

Two theorems are demonstrated giving analytical expressions of the fundamental solutions of the Pell equation $X^{2}-DY^{2}=1$ found by the method of continued fractions for two squarefree polynomial expressions of radicands of Richaud-Degert type $D$ of the form $D=\left(f\left(u\right)\right)^{2}\pm2^αn$, where $D$, $n>0$, $α\geq0,\in\mathbb{Z}$, and $f\left(u\right)>0,\in\mathbb{Z}$, any polynomial function of $u\in\mathbb{Z}$ such that $f\left(u\right)\equiv0\left(mod\,\left(2^{α-1}n\right)\right)$ or $f\left(u\right)\equiv\left(2^{α-2}n\right)\left(mod\,\left(2^{α-1}n\right)\right)$.

math.NT↗

arXiv2015

General solutions of sums of consecutive cubed integers equal to squared integers

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the difference and sum of the triangular numbers of $\left(a+M-1\right)$ and $\left(a-1\right)$ in the product of their greatest common divisor $g$ and remaining square factors $δ^{2}$ and $σ^{2}$, yielding $c=gδσ$. Further, the condition that $g$ must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all integer solutions $\left(M,a,c\right)$ showing that these solutions appear recurrently. In particular, it is found that there always exist at least one solution for the cases of all odd values of $M$, of all odd integer square values of $a$, and of all even values of $M$ equal to twice an integer square.

math.NT↗

arXiv2014

Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=μM+1$ with $μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,η,δ$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $η\equiv1\left(mod\,2\right)$ and $δ\equiv0,1$ or $5\left(mod\,6\right)$, and if $δ\equiv0\left(mod\,6\right)$, $M\equiv0\left(mod\,12\right)$, while if $δ\equiv1$ or $5\left(mod\,6\right)$, $M\equiv2$ or $11\left(mod\,12\right)$ with $a_{1}$ and $a_{2}$ being of different or same parities.

math.NT↗

arXiv2014

Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials

Square roots $s$ of sums of $M$ consecutive integer squares starting from $a^{2}\geq1$ are integers if $M\equiv0,9,24$ or $33(mod\,72)$; or $M\equiv1,2$ or $16(mod\,24)$; or $M\equiv11(mod\,12)$ and cannot be integers if $M\equiv3,5,6,7,8$ or $10(mod\,12)$. Finding all solutions with $s$ integer requires to solve a Diophantine quadratic equation in variables $a$ and $s$ with $M$ as a parameter. If $M$ is not a square integer, the Diophantine quadratic equation in variables $a$ and $s$ is transformed into a generalized Pell equation whose form depends on the $M(mod\,4)$ congruent value, and whose solutions, if existing, yield all the solutions in $a$ and $s$ for a given value of $M$. Depending on whether this generalized Pell equation admits one or several fundamental solution(s), there are one or several infinite branches of solutions in $a$ and $s$ that can be written simply in function of Chebyshev polynomials evaluated at the fundamental solutions of the related simple Pell equation. If $M$ is a square integer, it is known that $M\equiv1(mod\,24)$ and $M=(6n-1)^{2}$ for all integers $n$; then the Diophantine quadratic equation in variables $a$ and $s$ reduces to a simple difference of integer squares which yields a finite number of solutions in $a$ and $s$ to the initial problem.

math.NT↗