An Introduction to Diophantine Equations

Andreescu, Titu; Andrica, Dorin; Cucurezeanu, Ion

doi:10.1007/978-0-8176-4549-6

Cited by 64 publications

(61 citation statements)

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“…This case corresponds to a linear Diophantine equation which can be completely solved using Bezout's lemma, see, e.g., Theorems 2.1.1 and 2.1.2 in [1].…”

Section: Integral Equationsmentioning

confidence: 99%

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Locally homogeneous nearly Kähler manifolds

Cortés

Vásquez

2015

Ann Glob Anal Geom

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We construct locally homogeneous six-dimensional nearly Kähler manifolds as quotients of homogeneous nearly Kähler manifolds M by freely acting finite subgroups of Aut 0 (M). We show that non-trivial such groups do only exists if M = S 3 × S 3 . In that case, we classify all freely acting subgroups of Aut 0 (M) = SU(2) × SU(2) × SU(2) of the form A × B, where A ⊂ SU(2) × SU(2) and B ⊂ SU(2).

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“…This case corresponds to a linear Diophantine equation which can be completely solved using Bezout's lemma, see, e.g., Theorems 2.1.1 and 2.1.2 in [1].…”

Section: Integral Equationsmentioning

confidence: 99%

“…Moreover, to verify if either 1 2 ∈ W(C) or − 1 2 ∈ W(C) amounts to solve each of the following equations,…”

Section: Type III Groupsmentioning

confidence: 99%

Locally homogeneous nearly Kähler manifolds

Cortés

Vásquez

2015

Ann Glob Anal Geom

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“…Since Eq. (1) is Diophantine [43] with four unknowns, it is necessary to enforce some additional relationships between the different types of obstacles to ensure that the solution is unique. In particular, we consider three different situations:…”

Section: Stochastic Simulationsmentioning

confidence: 99%

Modeling transport through an environment crowded by a mixture of obstacles of different shapes and sizes

Ellery

Baker

McCue

et al. 2016

Physica A: Statistical Mechanics and its Applications

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h i g h l i g h t s• Stochastic simulations of individual and collective motion through a crowded environment. • Crowded environments populated by mixtures of obstacles of different shapes and sizes.• Transport properties depend on the obstacle volume fraction and details of the obstacle shape and size distribution. • Standard fractional diffusion equation descriptions ought to be used with care. a b s t r a c tMany biological environments are crowded by macromolecules, organelles and cells which can impede the transport of other cells and molecules. Previous studies have sought to describe these effects using either random walk models or fractional order diffusion equations. Here we examine the transport of both a single agent and a population of agents through an environment containing obstacles of varying size and shape, whose relative densities are drawn from a specified distribution. Our simulation results for a single agent indicate that smaller obstacles are more effective at retarding transport than larger obstacles; these findings are consistent with our simulations of the collective motion of populations of agents. In an attempt to explore whether these kinds of stochastic random walk simulations can be described using a fractional order diffusion equation framework, we calibrate the solution of such a differential equation to our averaged agent density information. Our approach suggests that these kinds of commonly used differential equation models ought to be used with care since we are unable to match the solution of a fractional order diffusion equation to our data in a consistent fashion over a finite time period.

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“…7 shows the pro mission error for d The solid curve sho ). The equation 4 is known as Diophantine equation [21][22][23][24]. …”

Section: Proof By Substitutimentioning

confidence: 99%

Diophantine equation based model of data transmission errors caused by interference generated by DC-DC converters with deterministic modulation

Bojarşki¹,

Smoleński²,

Leżyński³

et al. 2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The assurance of the electromagnetic compatibility of sensitive smart metering systems and power electronic converters, which introduce high-level electromagnetic interference is important factor conditioning reliable operation of up to date power systems. Presented experimental results have shown that currently binding, frequency domain tests are ineffective for the evaluation of data transmission error hazards. The proposed in this paper mathematical, time-domain model, based on Diophantine equation, enables evaluation of data transmission errors caused by interference introduced by DC-DC power electronic interfaces with deterministic modulation. In the paper there have been presented possible application areas for the proposed model. Thus, evaluation of the probability of the appearance of data transmission errors, caused by interference generated by power electronic converters, is important for both cognitive and technical reasons. Elaboration of the presented mathematical model has enabled explanation of significant differences in experimentally obtained distributions of awaiting times for data transmission errors caused by interference generated by a DC-DC converter with random [9-11] and deterministic modulation. The evaluated probability of the appearance of data transmission error may become a useful factor supporting the development of data transmission systems.The DC-DC converter was selected as an interference source for mathematical analyses of data transmission errors caused by interference generated by converters with deterministic modulation. A detailed description of the interference introduced by a DC-DC converter [12][13][14] has been presented in our previous paper [15]. Deterministic vs. random modulation of power electronic interfacesRandom modulation of power electronic interfaces is often recommended as an EMI reduction technique [16,17]. In fact the utilization of random modulation does provide ostensible reduction [18,19] of the interference levels measured, in accordance with EMC standards, in the frequency domain. Fig. 1 shows the experimental results for conducted emission generated by a DC-DC converter with deterministic and random modulation, measured using average detector, in accordance with the EN 55022 standard, with the upper limit of the frequency range reduced from 30 MHz to 5 MHz for better clarity of the results.

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An Introduction to Diophantine Equations

Cited by 64 publications

References 0 publications

Locally homogeneous nearly Kähler manifolds

Locally homogeneous nearly Kähler manifolds

Modeling transport through an environment crowded by a mixture of obstacles of different shapes and sizes

Diophantine equation based model of data transmission errors caused by interference generated by DC-DC converters with deterministic modulation

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