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implies that the cumulant equations are satisfied to within approximately [MATH] 4.3 Optimal closure Optimal closure In addition to the relatively simple truncations discussed in §§ 4.1 4.2 , corresponding to assumption that [MATH] , the projections in figure also display the gradients that are obtained by varying the ... |
[MATH] described in § 3.4 . The resulting family of gradients produce a fan of gradient vectors lying between the limit points associated with second-order truncation [MATH] ) and the third-order truncation [MATH] ). A single member of the family corresponds to the eddy damping that is optimal, in the sense of equation... |
Figure displays orthogonal slices through the functional [MATH] to illustrate its partial dependence on the parameters [MATH] and [MATH] . The gradients that are obtained by using the optimal model approach described in § 3.4 |
are displayed in comparison with those that were obtained by linear regression analysis (Wang, 2013 . The optimal model approach yields a reasonably good agreement with the observed gradients of [MATH] at [MATH] . The optimal value of [MATH] was found to be 8.96. A summary of the results, including the dependence of th... |
4.4 Test optimisation problem Test optimisation problem In practice, local gradient information can be used in a gradient-based optimisation routine. To demonstrate, we define the functional [MATH] where [MATH] |
corresponds to the desired value of [MATH] . For convenience we define [MATH] as the value of [MATH] corresponding to the parameters [MATH] and attempt to solve the inverse problem of determining an a priori unknown [MATH] from the known value [MATH] . During each iteration of the optimisation procedure, we calculate t... |
corresponding to [MATH] . We set [MATH] and [MATH] and select [MATH] as an initial guess for [MATH] . Within four iterations the optimisation routine finds [MATH] and [MATH] to within a tolerance of less than [MATH] . This is an interesting, albeit contrived, example of a problem for which the use of sub-optimal gradie... |
5 Three-dimensional convection ( [MATH] ) Three-dimensional convection ( [MATH] A logical extension of the model for two-dimensional Boussinesq convection analysed in the previous section is the model for three-dimensional Boussinesq convection studied by |
Reiterer et al. 1998 . Like its two-dimensional counter part, the system is a truncated Galerkin representation of the full dynamics. Unlike its two-dimensional counter part, the system evolves on a [MATH] dimensional, rather than [MATH] dimensional, phase space and therefore yields statistics that exhibit a more compl... |
[EQUATION] where [EQUATION] The parameters [MATH] and [MATH] continue to represent the Prandtl number and the renormalised Rayleigh number. In addition, equation ( 41 ) defines a set of geometrical parameters, as a function of the wave number [MATH] , which correspond to [MATH] in the previous problem. To within consta... |
Reiterer et al. 1998 . To aid comparison with the results presented in Reiterer et al. 1998 , we choose [MATH] [MATH] and vary [MATH] . The statistics were obtained over a dimensionless time |
[MATH] As described in Reiterer et al. 1998 , when [MATH] for [MATH] and [MATH] , the system is chaotic. When projected onto the |
[MATH] plane the attractor consists of two lobes either side of the hyperplane [MATH] , as can be seen in figure . As [MATH] increases the deviation of the horizontally averaged temperature from the linear behaviour associated with pure conduction increases. |
The precise relationship between [MATH] and [MATH] for the parameters [MATH] and [MATH] is displayed in figure 10 . In spite of the discontinuities resulting from the use of a finite time average for each value of [MATH] , the relationship indicates that [MATH] tends to increase as [MATH] |
increases. At a glance, a linear relationship between [MATH] and [MATH] over [MATH] appears to provide a reasonable first description of the sensitivity. However, closer inspection reveals that [MATH] varies significantly on scales of approximately [MATH] , in contrast to the equivalent relationship for the Lorenz syst... |
), for which [MATH] is approximately constant over a large range of [MATH] Since the dynamical system has [MATH] degrees of freedom, the number of cumulants up to order [MATH] is given by equation ( 22 ): |
[EQUATION] The derivative of [MATH] with respect to [MATH] was computed by truncating the cumulant equations at [MATH] and invoking the optimal eddy-damping closure described in 3.4 . As is evident from figure 10 , the computed gradients appear to under estimate the underlying exact gradients in general, but neverthele... |
[MATH] . Figure 10 displays gradients corresponding to the optimal eddy damping parameter [MATH] , along with lines whose gradients are |
[MATH] to indicate the sensitivity of the results to changes in [MATH] . It is interesting that at [MATH] , we observe that [MATH] , which indicates that the computed gradient is insensitive to changes in [MATH] |
As discussed in § 3.4 , different values of [MATH] correspond to different assumptions about the involvement of third-order cumulants in the statistical equilibrium. Picking an arbitrary value of [MATH] in equation 38 ) might result in the adjoint operator being close to singular and therefore yielding gradients that d... |
11 shows evaluations of the derivative of [MATH] with respect to [MATH] using equation ( 38 ) for values of [MATH] in the vicinity of the optimal value [MATH] as determined by equation ( 35 ). When [MATH] and [MATH] it is evident that some choices of [MATH] result in a singular or near-singular adjoint operator and, th... |
[MATH] 6 Conclusions Conclusions We have described a systematic means of obtaining approximate forward and adjoint sensitivity information from a chaotic system using a truncated system of cumulant equations. Unlike linearisation of the underlying evolution equations for individual trajectories, the cumulant equations ... |
We combined data from direct simulation with tangent linear and adjoint equations for the system’s statistical state dynamics. These equations can be obtained from the original system systemically using a cumulant generating function. Whilst the method is approximate, because it relies on truncation of the cumulant equ... |
The extraction of gradient information from functionals of chaotic dynamical systems is a stringent test for modelling and closure schemes. A given model can be tuned to adequately represent a given problem. However, unless it accurately describes the underlying physics, it is unlikely to yield accurate information abo... |
Although we have focused on relatively low-dimensional dynamical systems, the idea of using cumulant expansions was motivated by the need to analyse high-dimensional dynamical systems. The challenge in the successful application of the method to large systems lies in the acquisition of a large number of accurate high o... |
Appendix A Derivation of the cumulant equations The Hopf generating functional (Hopf, 1952 is defined according to [EQUATION] where [MATH] . The moment [MATH] can therefore be generated as |
[EQUATION] where [MATH] is a multi-index, such that [MATH] and [MATH] . A moment [MATH] can be decomposed into a sum of products of cumulants [MATH] , containing all possible factorisations of the monomial [MATH] |
[EQUATION] where [MATH] is a multiset that decomposes a multi-index into addends. For example, if [MATH] then [MATH] would be one such decomposition. The multiset [MATH] consists of all such decompositions. For example, if [MATH] , then |
[MATH] , and [EQUATION] in which the exponents denote set multiplicities. In the example above, the set multiplicities arise from the different ways that a set consisting of [MATH] elements can be partitioned. According to ( 45 ) and ( 46 ), the moment [MATH] can be expressed in terms of cumulants as |
[EQUATION] The decomposition ( 45 ) is identical to that which arises when partial derivatives are applied to composite functions. Indeed, using [MATH] |
[EQUATION] which shows the logarithm of the moment generating function is the cumulant generating function. Appendix B Observed cumulant gradients |
The gradients used to compute the truncation errors displayed in figures were obtained from simulations of the Lorenz equations for [MATH] values of [MATH] uniformly distributed over a unit interval centred on [MATH] . An approximation of the partial derivative of non-zero cumulants up to order [MATH] was obtained by m... |
Acknowledgements The author gratefully acknowledges funding from an EPSRC Doctoral Prize under grant number EP/M507878/1 and an Imperial College Junior Research Fellowship. The work benefited from discussions with Davide Lasagna at a SIG meeting for Flow Modelling, Instability and Control on March 29-30 2017, as part o... |
# Source: arxiv 1806.09534 # Title: The number of valid factorizations of Fibonacci prefixes # Sections: all # Downloaded: 2026-03-03T02:41:10.568619+00:00 |
The number of valid factorizations of Fibonacci prefixes Abstract We establish several recurrence relations and an explicit formula for [MATH] , the number of factorizations of the length- [MATH] prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In particula... |
keywords: numeration systems , Fibonacci numeration system , Fibonacci word MSC: 68R15, 11B39 url] frid/ url] shallit/ Introduction |
In the classical Fibonacci, or Zeckendorf, numeration system , a positive integer is represented as a sum of Fibonacci numbers: [EQUATION] |
where [MATH] and, as usual, [MATH] [MATH] , and [MATH] for all [MATH] For example, [MATH] , where a digit in brackets is [MATH] if the respective Fibonacci number appears in the sum, and [MATH] otherwise. Here a representation ends by the digits corresponding to [MATH] [MATH] and [MATH] |
Under the condition that [MATH] and [MATH] are never consecutive, that is, [MATH] , or, equivalently, that the Fibonacci numbers [MATH] |
are chosen greedily, such a canonical representation is unique, and the language [MATH] of all canonical representations is given by the regular expression [MATH] , where the empty word [MATH] is the representation of 0. At the same time, if consecutive Fibonacci numbers are allowed, but at most once each, the number o... |
. Its values oscillate between [MATH] (on numbers of the form [MATH] ) and [MATH] (on numbers of the form [MATH] For example, since |
[EQUATION] the number of legal representations of 16 is 4. Each legal representation of [MATH] can be obtained from a canonical one by a series of replacements |
[EQUATION] corresponding to the replacement of a Fibonacci number [MATH] by [MATH] In this paper, we allow even more freedom in Fibonacci representations of [MATH] , allowing the transformations |
[EQUATION] for all [MATH] [MATH] . Note that the introduced transformation corresponds to passing from a sum of the form [MATH] to the sum [MATH] , and, in particular, does not change the represented number. |
The representations that can be obtained from the canonical one by a series of transformations as in ( ) are called valid , and were introduced in |
in a more general setting because of their link to the Fibonacci word and factorizations of its prefixes, as explained below. Clearly, each legal representation is valid, but the opposite is not true. For example, starting from the legal representation [MATH] , we can find two more valid representations |
[EQUATION] and starting from the legal representation [MATH] , we find a new representation [EQUATION] so that the total number of valid representations of 16 is 7. |
Let [MATH] denote the number of valid representations of [MATH] . The goal of this paper is to prove a precise formula for [MATH] , given below in Theorem . Our formula demonstrates that the values of [MATH] are determined by the shuffle of two straight lines of irrational slope; see Fig. |
Notation and Sturmian representations We use notation common in combinatorics on words; the reader is referred, for example, to for an introduction. Given a finite word [MATH] , we denote its length by [MATH] . The power [MATH] just means the concatenation |
[MATH] . The [MATH] ’th symbol of a finite or infinite word [MATH] is denoted by [MATH] , so that [MATH] . A factor [MATH] of a finite or infinite word [MATH] , or, more precisely, its occurrence starting from position [MATH] of [MATH] , is denoted by [MATH] In particular, for [MATH] , the word |
[MATH] is the prefix of [MATH] of length [MATH] The standard Fibonacci sequence [MATH] of words over the binary alphabet [MATH] is defined as follows: |
[EQUATION] The word [MATH] is called also the standard word of order [MATH] . In particular, [MATH] [MATH] [MATH] [MATH] , and so on. From the definition, we easily see that the length of [MATH] is the Fibonacci number [MATH] |
The infinite word [EQUATION] is called the Fibonacci infinite word. Here we index it starting with [MATH] In the Fibonacci , or Zeckendorf numeration system , a non-negative integer [MATH] is represented as a sum of Fibonacci numbers |
[EQUATION] where [MATH] for [MATH] . In the canonical version of the definition, the following condition holds: [EQUATION] Under this nonadjacency condition, the representation of [MATH] is unique up to leading zeros. However, by removing the nonadjacency condition, we can get multiple representations: for example, [MA... |
Let [MATH] denote the number of legal representations of [MATH] . The sequence [MATH] is well-studied (see, e.g., ) and listed in the OEIS as sequence A000119 . In particular, [MATH] , and both bounds are precise |
The following lemma is a particular case of , Prop. 2] Lemma 1 For all [MATH] such that [MATH] , the word [MATH] is a prefix of the Fibonacci word [MATH] |
So [MATH] is also the number of ways to factor the prefix [MATH] of the Fibonacci word as a sequence of standard words in strictly decreasing order. |
To expand this definition, in this note we consider all factorizations of Fibonacci prefixes [MATH] as a concatenation of standard words in (non-strictly) decreasing order. We write [MATH] and call this representation of [MATH] |
valid if [MATH] for all [MATH] and [MATH] . Note that according to the previous lemma, every legal representation is valid, but not the other way around. For example, [MATH] , making the representation [MATH] valid. Theorem 1 of |
says, in particular, that valid representations are exactly those that can be obtained from the canonical one by a series of transformations ( ). |
Note that a digit of a valid representation cannot exceed 3 since the Fibonacci word does not contain a factor of the form [MATH] for any non-empty word [MATH] |
The number of valid representations of [MATH] is denoted by [MATH] , and this note is devoted to the study of the sequence [MATH] , recently listed in the OEIS as sequence A300066 . Clearly, [MATH] , and moreover, we prove an explicit formula for |
[MATH] that implies its linear growth. Result As is well-known, the Fibonacci infinite word [EQUATION] is the fixed point of the Fibonacci morphism [MATH] ; moreover, for each [MATH] , we have [MATH] . Consequently, if [MATH] , then Lemma |
implies that [EQUATION] Let [MATH] denote the golden ratio: [MATH] . It is important that the Fibonacci word is a Sturmian word of slope [MATH] and zero intercept (see Example 2.1.24 of |
), that is, for all [MATH] , we have [EQUATION] Here [MATH] denotes the fractional part of [MATH] Proposition 1 If [MATH] , all valid representations of [MATH] end with an even number of 0s. If [MATH] , all of them end with an odd number of 0s. |
Proof. It suffices to consult the definition of a valid representation and notice that [MATH] ends with [MATH] if and only if [MATH] is even. |
We now state our main result. Theorem 1 If [MATH] , then [MATH] or, equivalently, [MATH] is equal to the number of occurrences of [MATH] in [MATH] , plus one. If [MATH] , then [MATH] , or, equivalently, [MATH] is equal to the number of occurrences of [MATH] in |
[MATH] , plus one. To prove the theorem, we will need several more propositions. Proposition 2 (a) [MATH] for all [MATH] (b) For all [MATH] and all [MATH] , we have [MATH] |
Proof. (a): Consider a factorization [MATH] Applying the Fibonacci morphism [MATH] to both sides, we get the factorization [MATH] So the number of factorizations of [MATH] |
(which is equal to [MATH] ) is at least as large as the number of factorizations of [MATH] (which is equal to [MATH] ). (b) If, in addition [MATH] for some [MATH] , we see that |
[MATH] ends with [MATH] and [MATH] ends with [MATH] which in turn ends with [MATH] . From Proposition no factorization of [MATH] ends with [MATH] ; that is, such a factorization must be of the form [MATH] Taking the [MATH] -preimage, we get the factorization [MATH] thus establishing a bijection and the equality [MATH] |
Proposition 3 We have [EQUATION] for all [MATH] and all [MATH] Proof. Proposition tells us that [MATH] , and moreover, since [MATH] the prefix of length [MATH] of [MATH] ends with [MATH] which is a suffix of [MATH] Consider a valid factorization [MATH] If [MATH] , then [MATH] since [MATH] ends with [MATH] so the factor... |
[MATH] Taking the [MATH] -preimage, we get a factorization [MATH] of [MATH] . Moreover, [MATH] is a bijection between all the factorizations of |
[MATH] and the factorizations of [MATH] with [MATH] On the other hand, if [MATH] , then [MATH] since the word that we factor ends with [MATH] . Removing this last occurrence of [MATH] , we get the prefix of [MATH] of length [MATH] From Proposition the number of valid factorizations of [MATH] |
is equal to that of [MATH] Combining the two possibilities, we get the statement of the proposition. Proposition 4 For all [MATH] and for all [MATH] , we have |
[EQUATION] Proof. If [MATH] is odd, then [MATH] and the prefix [MATH] was considered in the previous proposition. It ends with [MATH] , and the symbol added to get |
[MATH] is also [MATH] So [MATH] ends with [MATH] , and all valid factorizations end with [MATH] This means that the number of valid factorizations of |
[MATH] is equal to that of [MATH] that is, [MATH] If [MATH] is even, [MATH] , then [MATH] ends with [MATH] In particular, the last factor of any valid factorization of |
[MATH] is either [MATH] , or [MATH] Indeed, [MATH] and thus for all [MATH] the [MATH] do not have a common suffix with [MATH] So, letting [MATH] denote the number of factorizations of [MATH] of the form [MATH] , we get |
[EQUATION] Here the last equality follows from Proposition (for the first addend) and by taking [MATH] of each factorization (for the second one). |
Propositions to give a full list of recurrence relations sufficient to compute [MATH] for every [MATH] , starting from [MATH] . Before using them to prove the main theorem, we consider two particular cases. |
Corollary 1 For all [MATH] we have [EQUATION] and [EQUATION] Proof. For [MATH] , the equalities can be easily checked: [MATH] , and [MATH] We also observe that [MATH] |
[MATH] , and [MATH] Now we assume that the equalities hold for [MATH] , and use Propositions and to prove they hold for [MATH] [EQUATION] |
Corollary 2 For all [MATH] , we have [EQUATION] Proof. For [MATH] , the equalities can be easily checked: [MATH] . Suppose the equalities hold for [MATH] ; let us prove them for [MATH] . With Proposition , we have |
[EQUATION] and with Proposition , we have [EQUATION] Proposition 5 Let [MATH] and [MATH] be such that [MATH] Then [MATH] Proof. Let us write the canonical Fibonacci representation of |
[MATH] as [MATH] , where [MATH] Since [MATH] , from Proposition we get that [MATH] is even. Now [MATH] where [MATH] [MATH] So [EQUATION] |
and [EQUATION] implying that [EQUATION] The difference between the two values is [EQUATION] where [EQUATION] Let us estimate [MATH] . Since [MATH] [MATH] is even and [MATH] , an upper bound for [MATH] is |
[EQUATION] whereas a lower bound is [EQUATION] So [EQUATION] Dividing by [MATH] , we get [EQUATION] Together with ( ), meaning that [MATH] , the last inequality implies the statement of the Proposition. |
Proof of Theorem .. Let us start with the case of [MATH] and proceed by induction starting with [MATH] . For [MATH] , there are three subcases: |
(a) [MATH] [MATH] (b) [MATH] [MATH] odd; (c) [MATH] [MATH] even. From now on we suppose that the statement of the theorem holds for all [MATH] |
(a) Since [MATH] and [MATH] , Proposition gives [MATH] Write [MATH] and [MATH] Note that Proposition gives [MATH] At the same time, [MATH] and thus [MATH] Now ( ) implies that [MATH] |
and [MATH] Also, the Fibonacci representation of [MATH] is obtained from that of [MATH] by a one-symbol shift to the left. So, summing up [MATH] and [MATH] due to the Fibonacci recurrence relation, we get the number with the same representation but shifted to the left yet another position, meaning that [MATH] |
Let us consider the sum [MATH] From the inclusions above, we see that [MATH] belongs to the interval [MATH] . But we also know that [MATH] , since [MATH] . So |
[EQUATION] which is equivalent to [MATH] and to [MATH] (since [MATH] ). Since all the numbers under consideration are irrational, and thus every ceiling is just the floor plus 1, we get |
[EQUATION] To establish the statement of the theorem for this subcase, it is sufficient to use Proposition and the induction hypothesis: [MATH] and [MATH] |
(b): Here [MATH] and [MATH] It suffices to refer to the previous subcase and to Proposition [MATH] It remains to notice that [MATH] , since [MATH] |
(c): Here [MATH] and [MATH] We use Proposition [MATH] . As above, write [MATH] and [MATH] ; then [MATH] , whereas [MATH] . By the induction hypothesis, [MATH] and [MATH] |
We have [MATH] and [MATH] , implying from ( ) that [MATH] and thus [MATH] . At the same time, [MATH] implies [MATH] and thus [EQUATION] |
Comparing it to [MATH] , we see that [EQUATION] But since [MATH] and [MATH] for every [MATH] , this also means that [EQUATION] Finally, since [MATH] is not an integer for any integer [MATH] we have |
[MATH] , so that [EQUATION] It remains to use the induction hypothesis to establish [EQUATION] which was to be proved. To complete the part of the proof concerning [MATH] , it remains to notice that [MATH] is equal to the number of [MATH] s in [MATH] plus one, due to ( ). |
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