A new family of binary sequences with low 4-value autocorrelation of length 4p based on cyclotomic classes of order four is proposed, as well as the linear complexity and 2-adic complexity of these sequences are determined. The results show that the linear complexity of s is LC(s) = 4p−1, and the 2-adic complexity of s is Φ2s = log2 ((24p−1)/3), both of which are greater than half of the period. The sequences are safe enough to resist Berlekamp–Massey algorithm attacks and rational approximation algorithm attacks, which indicate that they are good sequences in communication.

In this work, we establish local limit theorems for q - multinomial distributions of the first and second kind and of their discrete limits multiple Heine and multiple Euler distributions respectively. Specifically, the pointwise convergence of the q -multinomial distribution of the first kind, as well as for its discrete limit, the multiple Heine distribution, to a multivariate Stieltjes–Wigert type distribution, are provided. Moreover, the pointwise convergence of the q -multinomial distribution of the second kind, as well as for its discrete limit, the multiple Euler distribution, to a multivariate deformed Gaussian distribution, are proved. Interesting applications of the asymptotic behaviour of q-multinomials distributions of the two kinds are presented.

The rapid growth of online information transmission has made the secure transmission and storage of sensitive visual information crucial, particularly in light of the continuous increase in cyber threats. Traditional encryption algorithms are not very effective on images due to their very high pixel correlation, larger file sizes, and intrinsic redundancy. This paper presents a new framework of secure image encryption that will make use of advanced chaotic maps and cellular automata dynamics. It creates deterministic randomness, high sensitivity to initial conditions, and unpredictability through chaotic maps. CA introduces a series of dynamic, rule-based transformations that ensure robust properties for diffusion and confusion. During encryption, chaotic maps create pseudo-random sequences that control both the CA-based pixel scrambling and value transformation. This provides a very secure encryption method with many layers. It ensures high entropy in key generation, resistance against brute-force attacks, and high sensitivity to initial parameters. The scheme works because it can withstand differential, statistical, and noise-based attacks and show performance metrics like entropy analysis, correlation coefficients, histogram uniformity, and robustness. The suggested method is characterized by large-scale use, adaptability, and safety from modern cryptographic threats.

Sombor index is a popular vertex-degree-based topological index recently. In this paper, the maximum values of Sombor index for the class of all bicyclic graphs with a given matching number are wholly determined, then the extremal graphs with these maximum values are also characterized.

We define the notions of strong and strict recurrency for actions of countable ordered groups on σ-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

We study the properties of the ternary infinite word p = 012102101021012101021012⋯, that is, the fixed point of the map h : 0 → 01, 1 → 21, 2 → 0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.

Canonical expressions represent the implicative propositions (i.e., the propositions with only implications) up-to renaming of variables. Using a Monte-Carlo approach, we explore the model of canonical expressions in order to confirm the paradox that says that asymptotically almost all classical theorems are intuitionistic. Actually we found that more than 96.6% of classical theorems are intuitionistic among propositions of size 100.

In this paper we undertake a general study of the complexity function of the generalized Fibonacci words which are generated by the morphism defined by σl,m(a) = albm and σl,m(a) = a.

Input-driven pushdown automata (IDPDA) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the input symbol. Nowadays such devices are usually defined such that popping from the empty pushdown does not block the computation but continues it with empty pushdown. Here, we consider IDPDAs that have a more balanced behavior concerning pushing and popping. Digging input-driven pushdown automata (DIDPDA) are basically IDPDAs that, when forced to pop from the empty pushdown, dig a hole of the shape of the popped symbol in the bottom of the pushdown. Popping further symbols from a pushdown having a hole at the bottom deepens the current hole furthermore. The hole can only be filled up by pushing symbols previously popped. We study the impact of the new behavior of DIDPDAs on their power and compare their capacities with the capacities of ordinary IDPDAs and tinput-driven pushdown automata which are basically IDPDAs whose input may be preprocessed by length-preserving finite state transducers. It turns out that the capabilities are incomparable. We address the determinization of DIDPDAs and their descriptional complexity, closure properties, and decidability questions.

We introduce a new variant of insertion systems, namely bonded insertion systems. In such systems, words are not only formed by usual letters but also by bonds between letters. Words which can be inserted, have “free” bonds at their ends which control at which positions in a word they can be inserted (namely only there, where the bonds “fit”). Two kinds of bonded insertion systems are defined in this paper: so-called bonded sequential insertion systems and bonded parallel insertion systems. In a sequential system, there is only one word inserted at a time. In a parallel system, there is a word inserted at every possible position in parallel in one time step. We investigate the generative capacity of those two kinds and relate the families of generated languages to some families of the Chomsky hierarchy and to families of languages generated by Lindenmayer systems. Additionally, we investigate some closure properties.