Related papers: Discrepancy of generalized $LS$-sequences
Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line. Syndetic sets were defined by Gottschalk and Hendlund. For a…
The goal of constraint-based sequence mining is to find sequences of symbols that are included in a large number of input sequences and that satisfy some constraints specified by the user. Many constraints have been proposed in the…
We present an explicit construction of infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots)$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{{\rm…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
The L_2-discrepancy measures the irregularity of the distribution of a finite point set. In this note we prove lower bounds for the L_2 discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper…
This thesis is devoted to studying estimates of the least common multiple of some integer sequences. Our study focuses on effective bounding of the $\mathrm{lcm}$ of some class of quadratic sequences, as well as arithmetic progressions and…
Recently Sasane defined a notion of evaluating a distribution at a point using delta sequences. In this paper, we explore the relationship between generalizations of his definition and the standard definition of distributional point values.…
Classification of sequences of temporal intervals is a part of time series analysis which concerns series of events. We propose a new method of transforming the problem to a task of multivariate series classification. We use one of the…
Upper bounds for the $L_p$-discrepancies of point distributions in compact metric measure spaces for $0<p\le\infty$ have been established in the paper [6] by Brandolini, Chen, Colzani, Gigante and Travaglini. In the present paper we show…
In the present paper we introduce a new class of sequences called GM(b,r), which is the generalization of a class considered by Tikhonov. Moreover, we obtained in this note sufficient and necessary conditions for uniform convergence of sine…
We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are…
A novel non-linear approach to fast and effective comparison of sequences is presented, compared to the traditional cross-correlation operator, and illustrated with respect to DNA sequences.
The Super Learner (SL) is a widely used ensemble method that combines predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty…
We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue…
The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a…
We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…
We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent,…
In this short paper, we prove, by only using elementary tools, general cases when $U_n(P,Q) \neq \square$, where $U_n(P,Q)$ is the Lucas sequence of the first type.
The aims of this paper are twofold. First, it discusses the Littlewood conjecture and its variants with respect to uniformly distributed sequences. The second aim is to determine the exact order of the discrepancy of the van der…
We introduce a new four-parameters sequence that simultaneously generalizes some well-known integer sequences, including Fibonacci, Padovan, Jacobsthatl, Pell, and Lucas numbers. Combinatorial interpretations are discussed and many…