On radiuses of convergence of q-metallic numbers and related q-rational numbers

Ren, Xiaomin

doi:10.1007/s40993-022-00336-7

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…We are not sure whether these computer results are reliable, so this has to be studied further. We also mention that Ren [17] has proved similar claim for the convergents of metallic numbers [m, m, m, . .…”

Section: Roots Of the Pell Polynomialsmentioning

confidence: 52%

“…but we cannot prove this yet. This was claimed in the earlier preprint version of this work (and repeated in [17]), but our proof was incorrect.…”

Section: R0mentioning

confidence: 69%

“…Note that in the rational case the radius of convergence of the Taylor series can also be understood as the radius of the maximal disc around 0 that contains no zeros of the polynomial S(q) in the denominator of (1.1). We refer also to the recent work [17], where our technique was used to prove the conjecture for a very particular set of numbers (called metallic).…”

Section: Introductionmentioning

confidence: 99%

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On Radius of Convergence of q-Deformed Real Numbers

Leclere,

Morier-Genoud,

Ovsienko

et al. 2024

MMJ

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We study analytic properties of "q-deformed real numbers", a notion recently introduced by two of us. A q-deformed positive real number is a power series with integer coefficients in one formal variable q. We study the radius of convergence of these power series assuming that q is a complex variable. Our main conjecture, which can be viewed as a q-analogue of Hurwitz's Irrational Number Theorem, claims that the q-deformed golden ratio has the smallest radius of convergence among all real numbers. The conjecture is proved for certain class of rational numbers and confirmed by a number of computer experiments. We also prove the explicit lower bounds for the radius of convergence for the q-deformed convergents of golden and silver ratios.

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Section: Roots Of the Pell Polynomialsmentioning

confidence: 52%

“…but we cannot prove this yet. This was claimed in the earlier preprint version of this work (and repeated in [17]), but our proof was incorrect.…”

Section: R0mentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Radius of Convergence of q-Deformed Real Numbers

Leclere,

Morier-Genoud,

Ovsienko

et al. 2024

MMJ

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“…Nitrogen is metabolized by soil bacteria to nitrate, useless for plant nutrition, while urea (an alternative to ammonia and ammonium nitrate) is processed by urease enzymes to CO 2 and gaseous ammonia, which unable to function as a nutrient . Inhibiting the conversion of nitrogen to nitrate and of urea to ammonia gas via the addition of urease and nitrification inhibitors to fertilizer formulations thus reduces the amount of excess nitrogen needed in fertilizers . The addition of living microorganisms to fertilizers which colonize the soil or plants (biofertilizers) can increase the bioavailability and supply of nutrients to crops, also providing a means to reduce the amount of fertilizer applied and increase the efficiency of nutrient use. , …”

Section: Possible Pathways To Sustainable Fertilizersmentioning

confidence: 99%

Sustainable Fertilizers: Publication Landscape on Wastes as Nutrient Sources, Wastewater Treatment Processes for Nutrient Recovery, Biorefineries, and Green Ammonia Synthesis

Babcock-Jackson

Konovalova

Krogman

et al. 2023

J. Agric. Food Chem.

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The ability of modern agriculture to meet future food demand imposed by accelerating growth of the world’s population is a major challenge, and fertilizers play a key role by replacing nutrients in agricultural soil. Given the need for fertilizers, their cost in nonrenewable resources and energy, and the consequences of the greenhouse gas emissions required to make them, people have begun to explore ways to make fertilizer manufacturing and use more sustainable. Using data from the CAS Content Collection, this review examines and analyzes the academic and patent literature on sustainable fertilizers from 2001 to 2021. The breakdown of journal and patent literature publication over time on this topic, country or region of publications, the substances included in published research, among other things allow us to understand the general progress in the field as well as the classes of materials and concepts driving innovation. We hope that this bibliometric analysis and literary review will assist researchers in relevant industries to discover and implement ways to supplement conventional fertilizers and nutrient sources while improving the efficiency and sustainability of waste management and ammonia production.

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“…They further extended this notion to arbitrary real numbers [14] by some number-theoretic properties of irrational numbers. These works are related to many directions including Jones polynomial of rational knots [7,9,16,13], Teichmüller spaces [4], the Markov-Hurwitz approximation theory [6,8,11,20], the modular group and the Picard group [10,19], combinators of posets [12,17,18] and triangulated category [2].…”

Section: Introductionmentioning

confidence: 99%

On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers

Ren¹

2022

Preprint

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The left and right q-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right q-deformed rational numbers. In the present paper, we focus on negative continued fractions and defined left q-deformed negative continued fractions. We give a formula for computing the q-deformed Farey sum of the left q-deformed rational numbers based on it. We use this formula to give a combinatorial proof of the relationship between the left q-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. Finally, we combine their work and the q-deformed Farey sum, and give a homological interpretation of the q-deformed Farey sum. We also give an approach to finding a relationship between real quadratic irrational numbers and homological algebra.

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On radiuses of convergence of q-metallic numbers and related q-rational numbers

Cited by 5 publications

References 9 publications

On Radius of Convergence of q-Deformed Real Numbers

On Radius of Convergence of q-Deformed Real Numbers

Sustainable Fertilizers: Publication Landscape on Wastes as Nutrient Sources, Wastewater Treatment Processes for Nutrient Recovery, Biorefineries, and Green Ammonia Synthesis

On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers

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