scholarly journals A generalization of Zeckendorf’s theorem via circumscribed m-gons

2017 ◽  
Vol 10 (1) ◽  
pp. 125-150 ◽  
Author(s):  
Robert Dorward ◽  
Pari L. Ford ◽  
Eva Fourakis ◽  
Pamela E. Harris ◽  
Steven J. Miller ◽  
...  
Keyword(s):  
Zeckendorf’S Theorem

scholarly journals Generalizing Zeckendorf's Theorem to f -decompositions

2014 ◽  
Vol 141 ◽  
pp. 136-158 ◽  
Author(s):  
Philippe Demontigny ◽  
Thao Do ◽  
Archit Kulkarni ◽  
Steven J. Miller ◽  
David Moon ◽  
...  
Keyword(s):  
Zeckendorf’S Theorem

scholarly journals A New Pooling Approach Based on Zeckendorf’s Theorem for Texture Transfer Information

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 279
Author(s):  
Vincent Vigneron ◽  
Hichem Maaref ◽  
Tahir Q. Syed
Keyword(s):  
Neural Network ◽  
Deep Learning ◽  
Number Series ◽  
High Frequencies ◽  
Texture Transfer ◽  
Segmentation Task ◽  
Zeckendorf’S Theorem ◽  
Learning Architectures ◽  
Data Variation ◽  
Traditional Image

The pooling layer is at the heart of every convolutional neural network (CNN) contributing to the invariance of data variation. This paper proposes a pooling method based on Zeckendorf’s number series. The maximum pooling layers are replaced with Z pooling layer, which capture texels from input images, convolution layers, etc. It is shown that Z pooling properties are better adapted to segmentation tasks than other pooling functions. The method was evaluated on a traditional image segmentation task and on a dense labeling task carried out with a series of deep learning architectures in which the usual maximum pooling layers were altered to use the proposed pooling mechanism. Not only does it arbitrarily increase the receptive field in a parameterless fashion but it can better tolerate rotations since the pooling layers are independent of the geometric arrangement or sizes of the image regions. Different combinations of pooling operations produce images capable of emphasizing low/high frequencies, extract ultrametric contours, etc.

scholarly journals Individual Gap Measures from Generalized Zeckendorf Degompositions

2017 ◽  
Vol 12 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Robert Dorward ◽  
Pari L. Ford ◽  
Eva Fourakis ◽  
Pamela E. Harris ◽  
Steven. J. Miller ◽  
...  
Keyword(s):  
Positive Integer ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Linear Recurrence ◽  
The Individual ◽  
Geometric Decay ◽  
Almost All ◽  
Zeckendorf’S Theorem ◽  
General Conditions

Abstract Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations.

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