scholarly journals Periodic Coefficients and Random Fibonacci Sequences

10.37236/3204 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Karyn McLellan

The random Fibonacci sequence is defined by $t_1 = t_2 = 1$ and $t_n = \pm t_{n-1} + t_{n-2}$, for $n \geq 3$, where each $\pm$ sign is chosen at random with probability $P(+) = P(-) = \frac{1}{2}$. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate $1.13198824\ldots$. We will consider what happens to random Fibonacci sequences when we remove the randomness; specifically, we will choose coefficients which belong to the set $\{1, -1\}$ and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length.

1998 ◽  
Vol 18 (5) ◽  
pp. 1139-1157 ◽  
Author(s):  
DEBORAH HEICKLEN

In this paper, the structure of $r$ equivalence, which was introduced by Vershik and which classifies group actions of the group $G=\sum_{n=1}^\infty{\Bbb Z}\slash r_n{\Bbb Z}$, $r_n\in{\Bbb N}\setminus\{1\}$, is examined. This is an equivalence relation that naturally arises from looking at certain sequences of $\sigma$-algebras. Vershik proved that if a sequence $r=(r_1,r_2,\ldots)$ does not satisfy a super-rapid growth rate, then entropy is an invariant for $r$ equivalence. In this paper, a strong converse of this is proven: for any $r$ which does satisfy this super-rapid growth rate, we can find a zero entropy action in every $r$ equivalence class.


2021 ◽  
Vol 36 (Supplement_1) ◽  
Author(s):  
L Worsfold ◽  
L Marriott ◽  
S Johnson ◽  
J Harper

Abstract Study question Are period trackers giving women accurate information about their periods and ovulation? Summary answer The top 10 period trackers gave conflicting information on period dates, ovulation day and the fertile window. What is known already Period tracking applications allow women to track their menstrual cycles and receive a prediction for their periods. The majority of applications also provide predictions of day of ovulation and the fertile window. Previous research indicates applications are basing predictions on assuming women undergo a textbook 28-day cycle with ovulation occurring on day 14 and a fertile window between days 10 and 17. Study design, size, duration An audit of menstrual cycle apps was conducted on the Apple app store using menstrual cycle tracker/period tracker as the search terms. The top ten apps that followed the inclusion and exclusion criteria were analysed and used for this study. All apps had the ability to allow retrospective data entry giving future cycle predictions and fertile window, and nine of the apps predicted a day of ovulation. Participants/materials, setting, methods Five women’s profiles for 6 menstrual cycles were created and entered into each app. Cycle length (CL) and ovulation day (OD) for the 6th cycle were; Woman 1 – Constant 28 day CL, 0D 16, Woman 2 – Average 23 day CL, OD 13, Woman 3 – Average 28 day CL, OD 17, Woman 4 – Average 33 day CL, OD 20 and Woman 5 – Irregular, average 31 day CL, OD 14. Main results and the role of chance For cycle length, the apps all predicted woman 1’s cycles correctly but for women 2–5, the apps predicted 0 to 8 days shorter or longer than expected. For day of ovulation; for woman 1, no apps predicted this correctly; the apps ranged from day 13–15. For woman 2, 1 app was correct and overall the apps showed a lot of variation from day 8 to 13. For woman 3, no apps got it right, with a range of day 13–16. For woman 4, 2 apps got it right, but the apps ranged from day 13–20. For woman 5, no apps got right; the apps ranged from day 13–21. Irrespective of cycle length, 7 apps predicted a fertile window of 7 days in almost all cases; except 1 app that predicted 6 days for woman 2 and a different app which predicted 8 days for woman 4. For the remaining 3 apps, one always predicted a 10 day fertile window. One app predicted an 11 day fertile window in 4/5 women. One app predicted a 12 day fertile window in 4/5 women. Limitations, reasons for caution The five profiles created spanned a range of observed cycle characteristics, but many permutations are possible. A Monte Carlo type analysis could be conducted to examine these possibilities to provide more precise assessment of app performance, but as data had to be added manually into apps, this was not possible. Wider implications of the findings: The apps do not use the same algorithm and show variation. The information given by these apps is not 100% accurate, especially for the day of ovulation and the fertile window which can only be predicted if using a marker of ovulation, such as basal body temperature or ovulation sticks. Trial registration number Not applicable


Author(s):  
Justine Beaulieu ◽  
Johanna Del Castillo Munera ◽  
Yilmaz Balci

Five Phytophthora species comprising a total of 243 isolates (77 P. cinnamomi, 23 P. citrophthora, 18 P. multivora, 18 P. pini, and 107 P. plurivora) were screened for sensitivity to mefenoxam, fosetyl-Al, dimethomorph, dimethomorph + ametoctradin and fluoxastrobin using amended agar assays. Mefenoxam-insensitive isolates were detected within P. cinnamomi (4%), P. multivora (11%), and P. plurivora (12%) even at approximately 2.5x the recommended label rate. These isolates were also insensitive to higher (off-label) concentrations of fluoxastrobin. Concentrations of dimethomorph (400 g/mL) and dimethomorph + ametoctradin (100 g/mL) were mostly effective in mycelial growth inhibition, but two P. plurivora isolates were insensitive, suggesting that resistance management is required. All mefenoxam-insensitive isolates were sensitive to fosetyl-Al at the label rate. Surprisingly, the populations of P. cinnamomi from mid-Atlantic oak forests included insensitive isolates. With almost all species, isolates recovered from asymptomatic hosts (e.g., soil/potting media collected of randomly selected asymptomatic hosts) had a significantly greater relative growth rate when compared to isolates recovered from symptomatic hosts (e.g., isolates recovered from lesions or wilted plants). These findings suggest that mefenoxam should no longer be used to manage oomycetes in Maryland ornamental nurseries and that the use of fluoxastrobin should be limited.


1972 ◽  
Vol 78 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Marca Burns

SUMMARYBirthcoat samples from Lincoln and Welsh Mountain lambs born in an ova transfer experiment (Weiner & Slee, 1965) were examined to determine the fibre type arrays and any effect of ova transfer on manifestation of the prenatal check or the base (Dry, 1965).In the Lincoln breed arrays were Truncated Ravine or Truncated Valley thus showing central checking. Pre-curly tip fibres were mainly confined to central primary follicles. Post-natal medullation was plentiful, especially in peak curly-tips, but kemp was absent. In the Welsh lambs all identifiable arrays were either Plateau or Saddle, with the lateral primary fibres more checked than the centrals. Pre-curly tip fibres occupied almost all the primary follicles, with only a few curly-tips in primary laterals. Post-natal medullation was strongest in pre-curly tips and their successors.The effect of ova transfer was to decrease the manifestation of check in Lincoln-in-Welsh and increase it in Welsh-in-Lincoln, without affecting the base. Although this, in each case, changed the check of the lamb in the direction of that of the foster breed, it is suggested that this may not be due to any influence of maternal check on foetal check. The effect on pre-natal check may be due to foetal size, and in particular skin expansion, during a short period immediately following completion of primary follicle initiation. As compared to their respective controls, the Welsh transfer lambs had a higher S/P ratio at birth, whilst that of the transferred Lincolns was reduced.If increased prenatal fibre check is associated with increased foetal growth rate at the relevant period, it may also be correlated with increased lamb size at birth and therefore have been favoured in selection for meat production, at least in some breeds.


1968 ◽  
Vol 11 (2) ◽  
pp. 285-288
Author(s):  
U.S.R. Murty

Let A be an m×n (0, l )-matrix. Let C1, C2, …, Cn denote its columns. A sequence of distinct columns is said to form a chain if the inner product of and (for 1 ≤ t ≤ k-l) is at least one. k-1 is called the length of the chain and this chain is said to connect are said to be connected. As can be easily seen, connectedness is an equivalence relation on the set of columns. A matrix is called connected if all its columns belong to the same equivalence class. If Ci and Cj belong to the same equivalence class, then s(Ci, Cj) will denote the length of the shortest chain between Ci and Cj We define the distance between any two columns Ci and Cj to be denoted by d(Ci, Cj), in the following manner.


Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Pavel Trojovský

Many mathematicians have been interested in the study of recursive sequences. Among them, a class of “chaotic” sequences are named “meta-Fibonacci sequences.” The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q-sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a “Mertens’ version” of the Hofstadter sequence, defined (for x>0) by x↦∑n≤xμnQn, where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.


Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.


2009 ◽  
Vol 42 (8) ◽  
pp. 085005 ◽  
Author(s):  
Élise Janvresse ◽  
Benoît Rittaud ◽  
Thierry de la Rue

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1928
Author(s):  
Pavel Trojovský

Let (Fn)n≥0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z:Z≥1→Z≥1 is defined as z(n):=min{k≥1:Fk≡0(modn)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1).


Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2912
Author(s):  
Eva Trojovská ◽  
Venkatachalam Kandasamy

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.


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