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Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let…
We consider the inverse-folding problem for RNA secondary structures: for a given (pseudo-knot-free) secondary structure find a sequence that has that structure as its ground state. If such a sequence exists, the structure is called…
We introduce a complexity measure for symbolic sequences. Starting from a segmentation procedure of the sequence, we define its complexity as the entropy of the distribution of lengths of the domains of relatively uniform composition in…
In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~\(f\) to have exactly…
Suppose that M is countable, binary, primitive, homogeneous, and simple, and hence 1-based. We prove that the SU-rank of the complete theory of M is~1. It follows that M is a random structure. The conclusion that M is a random structure…
We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.
We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq…
We show that the expected value for the linear complexity of $m$-multisequences of length $n$ is E_n^{(m)} = n m/(m+1) + O(1).
We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion…
In this paper, we discuss tensegrity from the perspective of nonlinear algebra in a manner accessible to undergraduates. We compute explicit examples and include the SAGE and Julia code so that readers can continue their own experiments and…
The merit factor of a $\{-1, 1\}$ binary sequence measures the collective smallness of its non-trivial aperiodic autocorrelations. Binary sequences with large merit factor are important in digital communications because they allow the…
Making predictions in an unseen environment given data from multiple training environments is a challenging task. We approach this problem from an invariance perspective, focusing on binary classification to shed light on general nonlinear…
Many empirical time series are genuinely symbolic: examples range from link activation patterns in network science, DNA coding or firing patterns in neuroscience to cryptography or combinatorics on words. In some other contexts, the…
This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time…
An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…
We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We…
We define the probability of an equation in a finite algebra as the proportion of tuples in its domain that satisfy it. We call the probabilistic spectrum of an algebra the set of probability values obtained when the equation varies. We…
Segmental structure is a common pattern in many types of sequences such as phrases in human languages. In this paper, we present a probabilistic model for sequences via their segmentations. The probability of a segmented sequence is…