Spiral growth inNephrolepidina: evidence of “golden selection”

Paleobiology ◽  
10.1666/12057 ◽  
2014 ◽  
Vol 40 (2) ◽  
pp. 151-161 ◽  
Author(s):  
Andrea Benedetti
Keyword(s):  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Spiral Growth ◽  
Regular Growth ◽  
Mean Values ◽  
Evolutionary Selection ◽  
Golden Angle ◽  
New Type ◽  
New Evidence ◽  
Interpretative Model

Examination of the neanic apparatuses of known populations ofNephrolepidina praemarginata,N. morgani, andN. tournouerireveals that the equatorial chamberlets are arranged in spirals, along the direction of connection of the oblique stolons, giving the optical effect of intersecting curves. InN. praemarginatacommonly 34 left- and right-oriented primary spirals occur from the first annulus to the periphery, 21 secondary spirals from the third to fifth annulus, 13 ternary spirals from the fifth to eighth annulus, following the Fibonacci sequence.The number of the spirals increases in larger specimens and in more embracing morphotypes, and especially in trybliolepidine specimens; the secondary and ternary spirals from the investigatedN. praemarginatatoN. tournoueripopulations tend to start from more distal annuli. An interpretative model of the spiral growth ofNephrolepidinais attempted.The angle formed by the basal annular stolon and distal oblique stolon in equatorial chamberlets ranges from 122° inN. praemarginatato mean values close to the golden angle (137.5°) inN. tournoueri.The increase in the Fibonacci number of spirals during the evolution of the lineage, along with the disposition of the stolons between contiguous equatorial chamberlets, provides new evidence of evolutionary selection for specimens with optimally packed chamberlets.Natural selection favors individuals with the most regular growth, which fills the equatorial space more efficiently, thus allowing these individuals to reach the adult stage faster. We refer to this new type of selection as “golden selection.”

scholarly journals Contributions of Flexible-Arch Configurations in Shimenzi Arch Dam: New Evidence from Field Measurements

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jianwen Zhong ◽  
Enzhi Wang ◽  
Yuande Zhou ◽  
Qingbin Li ◽  
Penghui Li
Keyword(s):  
Field Measurements ◽  
Arch Dam ◽  
Artificial Joints ◽  
The Past ◽  
Numerical Studies ◽  
Force Transfer ◽  
New Type ◽  
Contraction Joint ◽  
Arch Structure ◽  
New Evidence

This paper presents a retrospective investigation into the performance of a new type of flexible-arch configurations in Shimenzi arch dam based on the past ten-year-long field measurements. The flexible-arch configurations are mainly comprised of artificial short joints at the middle downstream surface and a middle contraction joint with hinged well and enlarged arch ends with bending joints. Fundamental design considerations of these components are provided, and their contributions to the performance of Shimenzi arch dam are discussed in detail using the monitoring data from joint meters, strain gauges, and thermometers. Some elementary numerical studies have been conducted on a typical arch structure with different arrangements of artificial joints. Both the field data and numerical results prove well the effectiveness of the purposely built short joints and the middle contraction joint on the relaxation of tensile stress mobilization. Field survey data also clearly demonstrate the significance of the hinged well at the upstream side of the middle joint for a continuous arch force transfer.

On the local behavior of the order of appearance in the Fibonacci sequence

2014 ◽  
Vol 10 (04) ◽  
pp. 915-933 ◽  
Author(s):  
Florian Luca ◽  
Carl Pomerance
Keyword(s):  
Positive Integer ◽  
Rational Function ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Asymptotic Density ◽  
Infinitely Many Solutions ◽  
Local Behavior ◽  
Sieve Methods ◽  
Density Zero

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the k th Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

scholarly journals Resilience and sustainability of creative industries businesses

2021 ◽  
Vol 115 ◽  
pp. 02004
Author(s):  
Helena Majdúchová ◽  
Daniela Rybárová
Keyword(s):  
Negative Impact ◽  
Slovak Republic ◽  
Creative Industries ◽  
Point Of View ◽  
Growth Potential ◽  
Added Value ◽  
Creative Economy ◽  
Mean Values ◽  
Unlimited Growth ◽  
New Type

The importance of the creative economy for the development of society not only from an economic but also from a social and cultural point of view has been the subject of numerous scientific and professional studies. The connecting link between them is the observation that creativity has been considered in recent years as a new type of competitive advantage. As a generator of added value for all businesses, the creative industries have unlimited growth potential and are thus seen as a route to sustainable development. One of the problems of the creative industry is to obtain resources for business development. This paper examines the financial structure of selected areas of the creative industries in the Slovak Republic, using mean values for 2016-2019, correlation and regression analysis of financial data in 2019 (before the negative impact of COVID19). The findings point to the absence of credit sources, which confirms the fact that these businesses are perceived as risky and unstable by banks and financial institutions in the Slovak Republic.

scholarly journals On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

2011 ◽  
Vol 2011 ◽  
pp. 1-4 ◽  
Author(s):  
Diego Marques
Keyword(s):  
Natural Number ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Numbers

LetFnbe thenth Fibonacci number. The order of appearancez(n)of a natural numbernis defined as the smallest natural numberksuch thatndividesFk. For instance, for alln=Fm≥5, we havez(n−1)>z(n)<z(n+1). In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

scholarly journals DIGITALIZACIÓN DE FORMAS DE LA NATURALEZA COMO RECURSO MORFOLÓGICO

2020 ◽  
pp. 133-149
Author(s):  
Santiago Javier Santamaría-Bedón
Keyword(s):  
Golden Ratio ◽  
Three Dimensional ◽  
Fibonacci Sequence ◽  
Vertical Axis ◽  
3D Models ◽  
Morphological Variations ◽  
Geometric Patterns ◽  
Golden Angle ◽  
Dense Point ◽  
Photo Alignment

La biomimética imita los procesos naturales para plantear posibles soluciones a problemas humanos y ha sido aplicado por varias profesiones, convirtiéndose en una práctica recurrente para adoptar nuevas morfologías o principios funcionales. Leonardo de Pisa empleó estos fenómenos y desarrolló una serie matemática llamada sucesión de Fibonacci, dando lugar posteriormente a la proporción áurea. Estos descubrimientos matemáticos se manifiestan en forma concurrente en patrones geométricos de los procesos y estructuras de la naturaleza como en la distribución de hojas en las plantas. El objetivo de este artículo es proponer recursos formales procedentes de la aplicación de la biomimética de manera superficial/reductiva, que adopta la morfología de la planta Sempervivum tectorum L. para futuras soluciones formales y funcionales del ámbito del diseño. Para ello, se aplica la fotogrametría como técnica para capturar información bidimensional mediante fotografías y convertirlas en información tridimensional, obteniendo un modelo idéntico al original. La digitalización del elemento natural cumple con la siguiente metodología: preparación del entorno, toma de datos, alineación de fotografías, creación de nube de puntos densa, creación de malla y la textura. Posteriormente se realiza la construcción de modelos 3D, siguiendo el recorrido helicoidal digitalizado. Esta trayectoria crea un patrón en forma de hélice que se repite en sentido radial, tomando como referencia el eje vertical, de esta manera se crea un vórtice en la corona del sólido. A partir de esta estructura se realiza variaciones morfológicas, respetando la dirección de la hélice matriz. Palabras clave: Morfología, fotogrametría, ángulo de oro, sucesión de Fibonacci, diseño. AbstractBiomimetics mimics natural processes to propose possible solutions to human problems and has been applied by various professions, making it a recurring practice to adopt new morphologies or functional principles. Leonardo of Pisa used these phenomena and developed a mathematical series called the Fibonacci sequence, later giving rise to the golden ratio. These mathematical discoveries are manifested concurrently in geometric patterns of the processes and structures of nature as in the distribution of leaves in plants. The objective of this article was to propose formal resources from the application of biomimetics in a superficial / reductive way, which adopts the morphology of the Sempervivum tectorum L. plant for future formal and functional solutions in the field of design. To this end, photogrammetry was applied as a technique to capture two-dimensional information through photographs and convert them into three-dimensional information, obtaining a model identical to the original. The digitization of the natural element complied with the following methodology: environment preparation, data collection, photo alignment, dense point cloud creation, mesh and texture creation. Subsequently, the construction of 3D models is carried out, following the digitized helical path. This path created a helix-shaped pattern that was repeated radially, taking the vertical axis as a reference, thus creating a vortex in the crown of the solid. Morphological variations were made from this structure, respecting the direction of the parent helix. Keywords: Morphology, digital photogrammetry, golden angle, Fibonacci sequence, design.

scholarly journals Decay of Mean Values of Multiplicative Functions

2003 ◽  
Vol 55 (6) ◽  
pp. 1191-1230 ◽  
Author(s):  
Andrew Granville ◽  
K. Soundararajan
Keyword(s):  
Upper Bound ◽  
Multiplicative Function ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Mean Values ◽  
Absolute Value ◽  
New Type ◽  
The Absolute

AbstractFor given multiplicative function f , with |f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) Σn≤xf(n) converges. Halász showed that this depends on the minimum M (over y ∈ ℝ) of Σp≤x (1 – Re(f(p)p–iy )/p, and subsequent authors gave the upper bound ⪡ (1 + M)e–M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.

scholarly journals On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1073 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Prime Numbers ◽  
All Solutions ◽  
Adic Valuation

Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

Representing Numbers Using Fibonacci Variants

2015 ◽  
Author(s):  
Stephen K. Lucas
Keyword(s):  
Natural Number ◽  
Greedy Algorithm ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Fixed Number ◽  
Natural Numbers ◽  
Form Versus ◽  
Entire Number

This chapter introduces the Zeckendorf representation of a Fibonacci sequence, a form of a natural number which can be easily found using a greedy algorithm: given a number, subtract the largest Fibonacci number less than or equal to it, and repeat until the entire number is used up. This chapter first compares the efficiency of representing numbers using Zeckendorf form versus traditional binary with a fixed number of digits and shows when Zeckendorf form is to be preferred. It also shows what happens when variants of Zeckendorf form are used. Not only can natural numbers as be presented sums of Fibonacci numbers, but arithmetic can also be done with them directly in Zeckendorf form. The chapter includes a survey of past approaches to Zeckendorf representation arithmetic, as well as some improvements.

scholarly journals The Proof of a Conjecture Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2638
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Positive Integer ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

The Ionian Agora

1942 ◽  
Vol 62 ◽  
pp. 21-32 ◽  
Author(s):  
R. E. Wycherley
Keyword(s):  
Open Space ◽  
General Account ◽  
History Of ◽  
New Ideas ◽  
New Type ◽  
Systematic Planning ◽  
In The Beginning ◽  
New Evidence

The agora, the nucleus of all Greek cities, was in the beginning simply a convenient open space, around which buildings were irregularly placed. With the growth of systematic planning in Ionia a new type was evolved, and henceforth the old-fashioned agora and the Ionian existed side by side. Several years ago F. J. Tritsch wrote an account of the old type of agora, taking Elis (Fig. 1) as the best example. Since then Athens has yielded richer and more interesting material. The new evidence clarifies and confirms the picture drawn by Tritsch, which may still be accepted as true in principle. One might, however, attempt a brief general account of the new or Ionian agora, which has not perhaps been given the place it deserves in the history of Hellenic architecture. Finally, since remarkable Hellenistic developments have been revealed in the Athenian agora, one is prompted to ask whether the influence of new ideas and methods produced modifications in the older type.

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