Let k >= 7 be fixed, and v = (55k + 556)/(26k + 672),1 < d < c < v, 1 < mu < nu < k(1-d/c) and mu <= N-2/N-1(d/c) <= v. Suppose that N '(1), N '(2) > 0 are real numbers, and that N-1, N-2 are real numbers satisfying N-1 > N '(1) and N-2 > N '(2). Then we prove the system of two Diophantine inequalities |p(1)(c) + . . . + p(k)(c )- N-1| < N-1(-(1/c)(v-c)) log(195) N-1, |p(1)(d) + . . . + p(k)(d) - N-2 | < N(2)(-(1/d)(v-d) )log(195) N-2 has prime solutions p(1), . . . ,p(k).
In Chapter 3 of his second notebook, Ramanujan defined numbers a (n, k) such that a (2, 0) = 1 and for n >= 2, a (n + 1, k) = (n - 1) a (n, k - 1) + (2n - 1 - k) a (n, k), where a(n, k) = 0 when k< 0 or k>n - 2. These numbers are expressed in terms of Stirling numbers of the first kind and associated Stirling numbers of the second kind, and satisfy certain divisibility properties. In this paper, we obtain further properties for a(n, k), including a new characterization of prime numbers. We also derive several congruences for the numbers of Ramanujan modulo p(2), some of which lead to new conditions for a prime number to be a Wilson prime.
This paper establishes explicit evaluations of the 2k-th power mean for generalized cubic Gauss sums. By exploiting analytic techniques and fundamental properties of classical Gauss sums, we derive closed-form expressions for these means. Furthermore, we develop a computationally efficient framework for analyzing higher-order moments of such sums.
In this paper, we consider one of graph invariants, called a general position number, of scale-free graphs, where the scale-free property is one of key properties of real-world networks. To understand the tendency of general position numbers of scale-free graphs, we primarily deal with deterministic artificial scale-free graphs, since such graphs are very manageable and artificial models of complex networks mimic many properties of real-world networks. We derive estimates for the general position number of three well-known hierarchical scale-free graphs generated by deterministic models: one model produces fractal graphs, while the other two generate non-fractal graphs.
The average of the fc-fold divisor function d(k)(n) =#{ (a(1),& mldr;a(k)) is an element of & Zopf;(k)(>0):a(1)& ctdot;a(k) = n } has been widely studied. The Piatetski-Shapiro sequences are of the form N (c) = (& LeftFloor;n (c)& RightFloor;)(n=1)(infinity) with c > 1, c is not an element of & Nopf;. In this article, we estimate the average of the k-fold divisor function over the intersection of Piatetski-Shapiro sequences.
In this study, we aim to detect the generating functions of some special infinite matrices created with the help of special number sequences, then to obtain a formula for the inputs of the infinite matrix. Moreover, we study to determine some known or unknown sequences in each row and column of the infinite matrix. Also, we specify a countable many of sequences with a single formula.
Determinism is a central concept of computations. In some computational models, e.g., (traditional one-head) finite automata, deterministic and nondeterministic variants characterize the same language class. Considering other models, e.g., the 2-head finite automata known as sensing 5 ' -> 3 ' Watson-Crick automata, the deterministic variants are weaker than the nondeterministic ones, i.e., the former variants characterize a proper subset of the language class accepted by the latter ones. Watson-Crick finite automata emerged as a model of DNA computing: they work on a Watson-Crick tape, on double stranded DNA molecules, thus they have two heads, one for each strand. The heads of sensing 5 ' -> 3 ' Watson-Crick automata start from the two extremes of the input, read the input in opposite direction and the computation finishes when the heads meet. Variants based on restrictions on the set of states (e.g., stateless) and on the (head movements in) transitions (e.g., 1-limited), as well as, nondeterministic, deterministic and the recently investigated state-deterministic variants have been studied. In this paper a new concept, quasi-determinism is investigated, that is, in each configuration of a computation (if it is not finished yet), the next state is uniquely determined although the next configuration may not be, in case various transitions are enabled at the same time. We prove that this new concept is a generalisation of both of usual determinism and state-determinism both for finite one-head and two-head automata. More precisely, the class of quasi-deterministic sensing 5 ' -> 3 ' Watson-Crick automata is a superclass of both of the mentioned other classes of sensing 5 ' -> 3 ' Watson-Crick automata. The new sublinear class of languages characterized by the new model is not closed under the regular operations, under intersection, nor under complement. Hierarchy of language classes accepted by various subclasses of quasi-deterministic sensing 5 ' -> 3 ' Watson-Crick automata and also some other well-known classes is presented.
There are many references to matrices or determinants that have Fibonacci numbers as elements. In this paper, we find several determinants expressing the Fibonacci and related polynomials.
We discuss certain matrices associated with Christoffel words, and show that they have a group structure. We compute their determinants and show a relationship between the Zolotareff symbol from number theory.
We revisit the standard bisimulation equalities in process models free of the restriction operator. As is well-known, in general the weak bisimilarity is coarser than the strong bisimilarity because it abstracts from internal actions. In absence of restriction, those internal actions become somewhat visible, so one might wonder if the weak bisimilarity is still 'weak'. We show that in CCScore (i.e., Milner's standard CCS without tau-prefix, summation and relabelling) the weak bisimilarity indeed remains weak, i.e., still strictly coarser than the strong bisimilarity, even without the restriction operator. Essentially, this is due to the existence of the replication operation, which can keep a process retaining its state (i.e., the capacity of interaction). By virtue of these observations, we examine a variant of the weak bisimilarity, called quasi-strong bisimilarity. This quasi-strong bisimilarity requires the matching of internal actions to be conducted in the strong manner, as for the strong bisimilarity, and the matching of visible actions to have no trailing internal actions. We exhibit that in CCScore without the restriction operator, the weak bisimilarity exactly collapses onto this quasi-strong bisimilarity, which is moreover shown to coincide with the branching bisimilarity. These results reveal that in absence of the restriction operation, some ingredient of the weak bisimilarity indeed turns into strong, particularly the matching of internal actions.
The existence of fractional factors characterizes the fractional flows in a network, and hence indirectly characterizes the feasibility of data transmission. The minimum degree and isolated toughness characterize network topology from the perspectives of sparsity and stability, which serves as the theoretical conditions for fractional factors. This article reveals from a theoretical perspective that if the minimum degree condition increases, the corresponding tight isolated toughness variant bound will decrease. This infinite number of parameter combinations cause a "choice dilemma" for decision-makers. To solve this problem, we regard these two parameters as the Pareto front of the bi-objective optimization problem, and a knot point calculation approach is designed to determine the optimal combination.
In this paper, we present an explicit formula for the generating function that enumerates smooth convex polyiamonds with perimeter n. In particular, we show that the number of such polyiamonds grows at a rate given by 2.54442495 & ctdot;.
We provide a bijection between the set of the q-decreasing binary words (in the case where q is an irreducible positive rational number) and a set of binary words avoiding some patterns whose lengths depends on the number q. Moreover, for such a set we give the details of the construction, the generating function according to the length of its words, and the recurrence relation of the enumerating sequence, depending on q.
In 2013, Aaron Williams introduced the notion of a greedy Gray code algorithm and reinterpreted known Gray codes in a unified manner using greedy algorithms. Recently, this notion was further generalized and investigated by Merino, M & uuml;tze, and Williams in 2022, and by Merino and M & uuml;tze in 2024, in the context of generating the bases of a matroid or the spanning trees of a graph, among other combintorial structures. In this article, we investigate the existence of homogeneous greedy Gray codes for Fibonacci words and generalized Dyck prefixes. We also establish useful properties and provide efficient generation algorithms for them.
The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper, we show that, for any fixed integer k > 2, the number of polyominoes of area n and degree of convexity at most k can be computed in polynomial time using O(n(4)) space.
Given a directed graph (digraph) G with vertex set V, a Feedback Vertex Set (FVS) is a subset of vertices whose removal eliminates all circuits in G. Finding a minimum feedback vertex set (MFVS) is NP-hard, but digraph reductions can reduce graph size while preserving at least one MFVS. This raises questions about the ordering in which reductions are applied and the existence of an optimal order that maximizes size reduction. The Church-Rosser property (confluence) ensures reductions can be applied in any order, leading to a unique reduced digraph up to isomorphism. In this work, we focus on arc reduction and its confluence within a broader set of known confluent reductions. We introduce Superfluous Arcs, which can be removed without affecting MFVS solutions, and propose a new parametrized reduction, chordk, to identify and remove specific superfluous arcs in polynomial time for bounded integer k. We establish the confluence of a set of reductions that includes chordk, creating the largest known confluent reduction system for MFVS, which improves preprocessing techniques for solving the MFVS problem efficiently.
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.
Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons (H), boats (B) and stars (S) were soon exhibited, and many physicists published on what they later called HBS tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. However, this set yields exactly the strict HBS tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings, for instance. Another new tileset, called Gemstones, is also presented, derived from the Star tileset.
The Zagreb index of a graph is the sum of the squared degrees of all nodes in the graph. In this note, we study the Zagreb index of exponential plane-oriented recursive trees. We first show the convergence in L2 for the root degree. Then we calculate the first two moments of the Zagreb index from a recurrence. Finally, the limit law for the Zagreb index of an exponential plane-oriented recursive tree is characterized by an application of the contraction method.
