Related papers: Measurable Sequences
Linear theory provides a reasonable description of the velocity correlations of biased tracers both perpendicular and parallel to the line of separation, provided one accounts for the fact that the measurement is almost always made using…
Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number.…
The pure $O$-sequences of the form $(1,a,a,\ldots)$ are classified.
This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and…
The aim of this paper is to present an elementary computable theory of random variables, based on the approach to probability via valuations. The theory is based on a type of lower-measurable sets, which are controlled limits of open sets,…
The correlation between a random sequence and its transformed sequences is studied. In the case of a permutation operation or, in other word, the shuffling operation, it is shown that the correlation can be so small that the sequences can…
The relation between the expectation values computed in the random walk theory, and the heat kernel method for the diffusion equation is explained concretely. The random walk is also realized by simulations and their statistical…
The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note we observe…
This paper is devoted to three topics. First, proving a measurability theorem for multifunctions with values in non-metrizable spaces, which is required to show that solutions to stochastic wave equations with interval parameters are random…
Pyrosequencing is emerging as one of the important next-generation sequencing technologies. We derive the statistical distributions of this technique in terms of nucleotide probabilities of the target sequences. We give exact distributions…
We discuss the problem of determining the dimension of self-similar sets and measures on $\mathbf{R}$. We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of…
This paper deals with (finite or infinite) sequences of arbitrary independent events in some probability space. We find sharp lower bounds for the probability of a union of such events when the sum of their probabilities is given. The…
We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum…
In this paper we introduce and study the notion of I-convergence of sequences in a metric-like space, where I is an ideal of subsets of the set N of all natural numbers. Further introducing the notion of I*-convergence of sequences in a…
A method for estimating the merit factors of sequences will be provided. The result is also effective in determining the nonexistence of certain infinite collections of cyclic difference sets and cyclic matrices and associated binary…
We survey current developments in the approximation theory of sequence modelling in machine learning. Particular emphasis is placed on classifying existing results for various model architectures through the lens of classical approximation…
In this paper, we define an ordering relation for a set of complex numbers, and research the properties and theorems of the ordering, solve some simple complex inequalities with the ordering.
Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…
We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…
We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence which only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic…