scholarly journals A Matrix Dynamics Approach to Golomb's Recursion

10.37236/1301 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Edward J. Barbeau ◽  
John Chew ◽  
Stephen Tanny
Keyword(s):  
Asymptotic Behaviour ◽  
Multiple Solutions ◽  
Explicit Construction ◽  
Solution Approach ◽  
Unpublished Note ◽  
All Solutions ◽  
Matrix Dynamics

In an unpublished note Golomb proposed a family of "strange" recursions of metafibonacci type, parametrized by $k$. Previously we showed that contrary to Golomb's conjecture, for each $k$ there are many increasing solutions, and an explicit construction for multiple solutions was displayed. By reformulating our solution approach using matrix dynamics, we extend these results to a characterization of the asymptotic behaviour of all solutions of the Golomb recursion. This matrix dynamics perspective is also used to construct what we believe is the first example of a "nontrivial" nonincreasing solution, that is, one that is not eventually increasing.

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

scholarly journals Characterization of Degenerate Configurations in Attitude Determination of Three-Vehicle Heterogeneous Formations

Sensors ◽  
2021 ◽  
Vol 21 (14) ◽  
pp. 4631
Author(s):  
Pedro Cruz ◽  
Pedro Batista
Keyword(s):  
Unique Solution ◽  
Multiple Solutions ◽  
Attitude Determination ◽  
Line Of Sight ◽  
Number Of Solutions ◽  
Relative Line ◽  
Heterogeneous Formations ◽  
Heterogeneous Formation

The existence of multiple solutions to an attitude determination problem impacts the design of estimation schemes, potentially increasing the errors by a significant value. It is therefore essential to identify such cases in any attitude problem. In this paper, the cases where multiple attitudes satisfy all constraints of a three-vehicle heterogeneous formation are identified. In the formation considered herein, the vehicles measure inertial references and relative line-of-sight vectors. Nonetheless, the line of sight between two elements of the formation is restricted, and these elements are denoted as deputies. The attitude determination problem is characterized relative to the number of solutions associated with each configuration of the formation. There are degenerate and ambiguous configurations that result in infinite or exactly two solutions, respectively. Otherwise, the problem has a unique solution. The degenerate configurations require some collinearity between independent measurements, whereas the ambiguous configurations result from symmetries in the formation measurements. The conditions which define all such configurations are determined in this work. Furthermore, the ambiguous subset of configurations is geometrically interpreted resorting to the planes defined by specific measurements. This subset is also shown to be a zero-measure subset of all possible configurations. Finally, a maneuver is simulated to illustrate and validate the conclusions. As a result of this analysis, it is concluded that, in general, the problem has one attitude solution. Nonetheless, there are configurations with two or infinite solutions, which are identified in this work.

Efficiency: an underlying principle of learning?

2018 ◽  
Vol 29 (2) ◽  
pp. 183-197 ◽  
Author(s):  
Sean Commins
Keyword(s):  
Multiple Solutions ◽  
Metabolic Cost ◽  
Descriptive Term ◽  
Underlying Principle ◽  
All Solutions ◽  
Metabolic Costs ◽  
Cost Reductions ◽  
The Brain

AbstractLearning is essential. It allows animals to change circumstances, deal with new situations and adapt to environments. Here, we argue that learning, at behavioral and neural levels, involves efficiency, reflected in metabolic cost reductions. Behaviourally, although multiple solutions to a novel problem may be available, all solutions are not learnt – it is too costly. Furthermore, once a strategy has been selected, it is reinforced producing an efficiency that leads to a maximisation of performance and metabolic cost reductions. Learning can be represented in the brain through many mechanisms; however, if learning is truly efficient, then, all such mechanisms should also be accompanied by a reduction in measurable metabolic costs. By thinking about learning in terms of efficiency, not simply as a descriptive term but rather in terms of metabolic costs, it allows learning to be examined more carefully and provides predictions that can be easily tested (and indeed refuted).

scholarly journals Dyes from the Ashes: Discovering and Characterizing Natural Dyes from Mineralized Textiles

Molecules ◽  
2020 ◽  
Vol 25 (6) ◽  
pp. 1417
Author(s):  
Alessandro Ciccola ◽  
Ilaria Serafini ◽  
Francesca Ripanti ◽  
Flaminia Vincenti ◽  
Francesca Coletti ◽  
...  
Keyword(s):  
Material Culture ◽  
Integrated Approach ◽  
Solution Approach ◽  
Left Behind ◽  
Surface Enhanced ◽  
Surface Enhanced Raman ◽  
Archeological Evidence ◽  
Enhanced Raman Scattering ◽  
Purple Color

Vesuvius eruption that destroyed Pompeii in AD 79 represents one of the most important events in history. The cataclysm left behind an abundance of archeological evidence representing a fundamental source of the knowledge we have about ancient Roman material culture and technology. A great number of textiles have been preserved, rarely maintaining traces of their original color, since they are mainly in the mineralized and carbonized state. However, one outstanding textile sample displays a brilliant purple color and traces of gold strips. Since the purple was one of the most exclusive dyes in antiquity, its presence in an important commercial site like Pompeii induces us to deepen the knowledge of such artifacts and provide further information on their history. For this reason, the characterization of the purple color was the main scope of this research, and to deepen the knowledge of such artifacts, the SERS (Surface Enhanced Raman Scattering) in solution approach was applied. Then, these data were enriched by HPLC-HRMS analyses, which confirmed SERS-based hypotheses and also allowed to hypothesize the species of the origin mollusk. In this context, a step-by-step integrated approach resulted fundamental to maximize the information content and to provide new data on textile manufacturing and trade in antiquity.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (04) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

UHF FLOWS AND COCYCLES

2012 ◽  
Vol 23 (01) ◽  
pp. 1250018
Author(s):  
A. KISHIMOTO
Keyword(s):  
Explicit Construction ◽  
Inductive Limits ◽  
Full Matrix ◽  
Matrix Algebras

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra Mk∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (4) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

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