Annals of Communications in Mathematics

Abstract:Let \( G = (V(G), E(G)) \) be a connected graph and let \( f : V(G) \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). If for each \( k \in \{0,1,2\} \), \( V_k = \{x \in V(G) : f(x) = k\} \), then \( f = (V_0, V_1, V_2) \). A function \( f \) is an outer-convex hop Roman dominating function (OConHRDF) on \( G \) provided that for every \( v \in V_0 \), there exists \( u \in V_2 \) such that \( v \in N_G^2(u) \) and \( V_0 \) is a convex set. The weight of OConHRDF \( f \) on \( G \) is denoted by \( \widetilde{\omega}_G^{conhR}(f) \) and is defined as \( \widetilde{\omega}_G^{conhR}(f) = \sum_{v \in V(G)} f(v) \).The smallest weight of an OConHRDF \( f \) on \( G \), denoted by \( \widetilde{\gamma}_{conhR}(G) \), is called the outer-convex hop Roman domination number, which can be written as \( \widetilde{\gamma}_{conhR}(G) = \min \{ \widetilde{\omega}_G^{conhR}(f) : f \text{ is an OConHRDF on } G \} \). Every OConHRDF \( f \) on \( G \) satisfying the condition \( \widetilde{\omega}_G^{conhR}(f) = \widetilde{\gamma}_{conhR}(G) \) is so-called a \( \widetilde{\gamma}_{conhR} \)-function on \( G \). This paper introduces a new parameter of a hop Roman dominating function in graphs, called outer-convex hop Roman dominating function and presents initial investigation.

Abstract:This paper introduces statistical gauge convergence as a refinement of statistical convergence in metric spaces, where deviations from the limit are controlled by positive continuous functions rather than fixed constants. We provide equivalent density based characterizations and examine their relationship with both classical and statistical convergence, showing that the corresponding implications are strict in general. Further more, we investigate the topology generated by this convergence and prove that it is typically finer than the underlying metric topology. Several examples are included to clarify the hierarchical structure among the considered notions of convergence.

Abstract:In this study, we investigate specific kernel structures within interval valued topological spaces. We introduce the notions of interval-valued Λ-sets and interval-valued λ-closed sets and discuss their essential properties. The connections between these concepts and existing notions in interval-valued topology are examined in detail. Various characterizations and foundational results are presented to clarify their structural behavior. This work aims to enhance and extend the theoretical framework of interval-valued topology.

Abstract:Let \( G = (V(G), E(G)) \) be a simple non-complete graph and let \( \xi : V \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). For each \( j \in \{0,1,2\} \), let \( V_j = \{x \in V(G) : \xi(x) = j\} \). Then \( \xi = (V_0, V_1, V_2) \). A function \( \xi \) is an interior hop Roman dominating function (InHRDF) on \( G \) if for each \( v \in V_0 \), there exists \( u \in V_2 \) such that \( d_G(u,v) = 2 \), and either \( V_1 = V(G) \) or for every \( v \in V_2 \), \( v \) is an interior vertex of \( G \). The weight of InHRDF \( \xi \) is denoted by \( \omega_G^{\text{InhR}}(\xi) \) and is defined as \( \omega_G^{\text{InhR}}(\xi) = \sum_{u \in V(G)} \xi(u) = |V_1| + 2|V_2| \).The minimum weight of an InHRDF \( \xi \) on \( G \), denoted as \( \gamma_{\text{InhR}}(G) = \min \{ \omega_G^{\text{InhR}}(\xi) : \xi \text{ is an InHRDF on } G \} \), is called the interior hop Roman domination number. Every InHRDF \( \xi \) on \( G \) satisfying the condition \( \omega_G^{\text{InhR}}(\xi) = \gamma_{\text{InhR}}(G) \) is called a \( \gamma_{\text{InhR}} \)-function on \( G \). In this paper, we investigate a new restricted parameter of a hop Roman dominating function in graphs called the interior hop Roman domination and present some combinatorial results.

Abstract:The distinguishing number is an important invariant used to measure the extent to which symmetries of graphs and permutation group actions can be broken by vertex labelings. In this paper, we investigate distinguishing labelings arising from permutation group actions with particular emphasis on Cartesian power constructions and wreath product actions. We establish structural bounds for distinguishing numbers in terms of orbit structure, stabilizers, and base size of permutation groups. Furthermore, we analyze the behavior of distinguishing numbers under Cartesian powers of sets and derive bounds for wreath product actions of the form \( G \wr S_m \) acting on \( X^m \). Several examples involving symmetric groups are presented to illustrate the theoretical results. These results contribute to a deeper understanding of symmetry breaking in permutation group actions and provide new insights into the interplay between distinguishing numbers and algebraic structures arising from wreath products.

Abstract:The Hilbert integral inequality is a well-known and widely studied result in analysis that has inspired many refinements and modifications. In this paper, we present a new logarithmic modification of this inequality. Our approach is based on a trigonometric method that offers a fresh perspective on existing standard techniques. As a consequence, we also derive another integral inequality. All arguments are presented in full detail.

Abstract:The Hilbert integral inequality is a well-known and widely studied result in analysis that has inspired many refinements and modifications. In this paper, we present a new logarithmic modification of this inequality. Our approach is based on a trigonometric method that offers a fresh perspective on existing standard techniques. As a consequence, we also derive another integral inequality. All arguments are presented in full detail.

Abstract:A total product cordial labeling of a graph \( G \) is a function \( f : V \rightarrow \{0,1\} \). For each \( xy \), assign the label \( f(x)f(y) \); \( f \) is called total product cordial labeling of \( G \) if it satisfies the condition that \( |v_f(0)+e_f(0)-v_f(1)-e_f(1)| \leq 1 \) where \( v_f(i) \) and \( e_f(i) \) denote the set of vertices and edges which are labeled with \( i = 0,1 \), respectively. A graph with a total product cordial labeling defined on it is called a total product cordial graph. In this paper, we determined the total product cordial labeling of the snake graphs \( T_n \), \( A(T_n) \), \( D(T_n) \), \( DA(T_n) \), \( Q_n \), \( A(Q_n) \), \( D(Q_n) \), and \( DA(Q_n) \).

Abstract:A set \( S \subseteq V(G) \) of the connected graph \( G = (V(G), E(G)) \) is said to be a connected secure dominating set if \( S \) is a dominating set, \( S \) is a secure set, and \( \langle S \rangle_G \) is a connected graph. The connected secure domination polynomial of \( G \) is \( D_s^c(G,x) = \sum_{i=\gamma_s^c(G)}^{n} d_s^c(G,i)x^i \) where \( \gamma_s^c(G) = \min \{|S| : S \text{ is a connected secure dominating set of } G\} \) and \( d_s^c(G,i) \) is the number of connected secure dominating sets with cardinality \( i \). In this paper, we will determine the connected secure domination polynomial of path graph \( P_n \), cycle graph \( C_n \), complete graph \( K_n \), star graph \( K_{1,n} \), and corona graph \( G \circ K_1 \).

Abstract:Here we investigate further the univariate fuzzy ordinary and fractional quantitative approximation of fuzzy real valued functions on a compact interval. This is done by quasi-interpolation sigmoid multicomposite activation functions based infinitely many specific and amplified multicomposite fuzzy neural network operators. These approximations are derived by establishing fuzzy multicomposite Jackson type inequalities involving the fuzzy moduli of continuity of the function, or of the right and left Caputo fuzzy fractional derivatives of the involved function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed-forward fuzzy multicomposite neural networks are with one hidden layer. We study in particular the fuzzy integer derivative and just fuzzy continuous cases. Our fuzzy fractional multicomposite approximation result using higher order fuzzy differentiation converges better than in the multicomposite fuzzy just continuous case. All these approximations are generated by 7 specific and basic activation functions.

Abstract:We study the algebra of all binary operations on a finite set, with composition defined as follows: for two binary operations A and B, their composition at an ordered pair x and y is B applied to the pair consisting of A(x,y) and A(y,x). A canonical family of automorphisms arises by conjugation with permutations of the underlying set. We characterize how these permutation-induced automorphisms act on several natural invariant subsets and show that the permutation they induce on constant operations determines every operation’s values on the diagonal. Beyond conjugation by permutations, we identify additional symmetries that are not conjugate to these permutation conjugations, and we show that conjugation by permutations need not be surjective onto the full automorphism group of the algebra under the given composition. For small underlying sets we obtain complete descriptions: when the set has two elements the automorphism group is isomorphic to the two-element symmetric group, and when the set has three elements it is isomorphic to the direct product of the three-element symmetric group with a cyclic group of order two. The paper concludes with algorithmic remarks and open problems toward a full classification of the automorphism group.

Abstract:Locating-hop domination refines hop domination by requiring vertices outside a hop dominating set to have nonempty and pairwise distinct distance-two signatures. Domination sequences, on the other hand, measure how long domination can be built through legal vertex choices. This paper introduces Grundy locating-hop domination sequences, in which a vertex choice is legal when it either footprints a previously undominated vertex through a closed hop neighborhood or strictly reduces an ambiguity potential that counts indistinguishable outside vertex pairs. The associated invariant is denoted by \( \gamma_{gr}^{\ell h}(G) \). General bounds are established, including \( \gamma_{\ell h}(G) \leq \gamma_{gr}^{\ell h}(G) \leq n(G) \). The parameter is additive over disjoint unions, and a hop-graph reduction identifies \( \gamma_{gr}^{\ell h}(G) \) with the corresponding locating-dominating sequence parameter on the hop graph \( G^{(2)} \). Exact values are obtained for complete graphs and stars. In particular, \( \gamma_{gr}^{\ell h}(K_n) = n \) and \( \gamma_{gr}^{\ell h}(K_{1,n}) = n \) for \( n \geq 2 \). Stars also give an infinite separation family: for \( n \geq 3 \), \( \gamma_{\ell h}(K_{1,n}) = 2 \) while \( \gamma_{gr}^{\ell h}(K_{1,n}) = n \).

Abstract:The purpose of this paper is to introduce and study the notion of various important forms of general type-2 fuzzy grammars, namely, the general type-2 fuzzy grammar in normal form, general type-2 fuzzy linear grammar, general type-2 fuzzy left linear grammar, and general type-2 fuzzy right linear grammar. We establish that each of these forms is equivalent to a general type-2 fuzzy regular grammar (except for an empty string) in the sense that they induce the same class of general type-2 fuzzy languages. These results not only generalizes classical equivalence theorems from fuzzy grammar theory but also provide a theoretical basis for the analysis and future development of general type-2 fuzzy grammar theory.

Abstract:In this paper, we investigate the structural properties of symmetric bi-semiderivations in the setting of prime and semiprime rings. By adapting classical derivation identities through the use of associated ω-homomorphisms, we obtain several characterization results that generalize known theorems from derivations to the bi-additive context. Particular attention is given to the interplay between Jordan-type structures and standard bi-semiderivations. In this direction, we prove that if R is a prime ring with char(R) ≠ 2, then every mapping satisfying the symmetric Jordan bi-semiderivation identity is in fact a symmetric bi-semiderivation. To illustrate our results, we present explicit examples constructed from matrix rings and polynomial rings, which also highlight the necessity of the imposed algebraic conditions.

Abstract:The adequacy of a statistical model ultimately rests on how faithfully the chosen distribution captures the structural features of the data at hand, including skewness, kurtosis, multi-modality, and the shape of the HRF. Consequently, the construction of new and more flexible distributions remains an active and consequential area of statistical research. In the same vein, this study introduces the Power Reduced Kies distribution (PRKD), a new two-parameter bounded distribution obtained through the power transformation of random variable of the Reduced Kies distribution. The inclusion of the power parameter endows the PRKD with a substantially richer variety of density shapes, including decreasing, right-skewed, left-skewed, reversed-J, and approximately symmetric forms, as well as a wider range of hazard rate behaviours, including increasing and bathtub-shaped forms, none of which can be collectively achieved by the single-parameter Reduced Kies distribution. Several important statistical properties of the PRKD are derived, including its linear representation, reliability characteristics, quantile function, moments, moment generating function, mode, and order statistics. The model parameters are estimated using the maximum likelihood and maximum product of spacings methods. A Monte Carlo simulation study is conducted to examine the finite-sample performance of the estimators under different parameter settings and sample sizes using mean estimates and mean squared errors. The results indicate that both estimators perform satisfactorily, although the maximum product of spacings method generally provides more reliable estimates for smaller sample sizes. The practical applicability of the PRKD is demonstrated using four real datasets from environmental, insurance, and health sciences. Comparative analyses with twelve competing unit distributions reveal that the PRKD provides highly competitive fits according to several model selection criteria and goodness-of-fit measures.

Abstract:This article assembles a detailed paper-level dependency chain for Gödel’s First and Second Incompleteness Theorems while identifying the effectiveness, consistency, and base-theory assumptions used at each stage. It separates computable enumerability from primitive-recursive axiomatizability; develops Gödel-style prime-power coding; fixes an explicit Hilbert calculus; gives capture-avoiding substitution with separate primitive-recursive majorants depending on all relevant variable indices; and proves primitive recursiveness of witness-annotated proof verification for a merely computably enumerable axiom set. The syntactic \( \Delta_0 \) and \( \Sigma_1 \) classes are fixed effectively, making the induction axioms of \( I\Sigma_1 \) primitive-recursively recognizable. The representability theorem is proved in \( Q \) by structural induction, using a finite-unfolding lemma, explicit \( \Sigma_1 \) closure under composition, and a displayed witness history for primitive recursion. For provability theory, the article defines a canonical trace-based proof predicate and gives detailed construction schemas, with explicit \( \Sigma_1 \)-induction invariants, for proof transformations and bounded proof synthesis in \( I\Sigma_1 \). These yield formalized \( \Sigma_1 \)-completeness, the Hilbert–Bernays–Löb derivability conditions, Löb’s theorem, and the Second Incompleteness Theorem. The article also proves Rosser’s strengthening, states Tarski’s theorem as the nondefinability of \( \mathrm{Th}(\mathbb{N}) \) in arithmetic, and marks the limits of anti-mechanist interpretations. Its contribution is expository and organizational rather than a new incompleteness theorem.