In this paper, we study a class of discrete Morse functions, coming from Discrete Theory, that are equivalent to simplicial stacks, Mathematical Morphology. We show that, as in can see the gradient vector field stack (seen function) only relevant information should consider. Last, but not least, also Minimum Spanning Forest dual graph is induced by initial function. This result allows computing...

This paper introduces a novel partition-based regression approach that incorporates topological information. Partition-based regression typically introduce a quality-of-fit-driven decomposition of the domain. The emphasis in this work is on a topologically meaningful segmentation. Thus, the proposed regression approach is based on a segmentation induced by a discrete approximation of the Morse-...

Measurements of the back scattering by bluntly-truncated tilted cylindrical shells in water reveal a dependence on the aspect angle which can be interpreted using geometrically described coupling mechanisms [S.F. Morse, P.L. Marston, and G. Kaduchak, J. Acoust. Soc. Am. 103, 785-794 (1998)]. By exciting a shell with a suitable acoustic pulse localized in time, the recorded backscattering respon...

The Morse-Smale complex is a topological structure that captures the behavior of the gradient of a scalar function on a manifold. This paper discusses scalable techniques to compute the Morse-Smale complex of scalar functions defined on large three-dimensional structured grids. Computing the Morse-Smale complex of three-dimensional domains is challenging as compared to two-dimensional domains b...

We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the l-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian compl...

The Thue-Morse sequence is a classical example of an almost periodic (or uniformly recurrent) sequence in the sense that its associated symbolic dynamical system is minimal. We prove that the subsequence along squares of the Thue-Morse sequence is normal.

In 1657, French lawyer and amateur mathematician Pierre de Fermat became interested in positive integer solutions u and v to the equation u − 61 v = 1. He posed a sadistic challenge to established mathematicians of the day, such as the Englishman William Brouncker and John Wallis, asking if they could find the solutions he found – without telling them the answer, of course. See, Fermat had foun...

The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ : R×X → X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R × X × X. Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metr...

We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmet...

In this paper, a novel shore-based S-band microwave Doppler coherent wave radar (Microwave Ocean Remote SEnsor (MORSE)) is designed to improve wave measurements. Marine radars, which operate in the X band, have been widely used for ocean monitoring because of their low cost, small size and flexibility. However, because of the non-coherent measurements and strong absorption of X-band radio waves...