We discuss a SAT-based approach for solving Suguru puzzles---one-player puzzles similar to Sudoku that were confirmed NP-complete in 2022. We first discuss the formal rules of the puzzles and provide a rigorous technique to translate such rules into propositional formulas in conjunctive normal form (CNF). The resulting formulas form what is called a SAT encoding, and we prove that the number of clauses and variables in our encoding is polynomially proportional to the puzzle's dimension. This encoding allows one to reduce Suguru puzzles to SAT problems, and we use this encoding to construct a declarative SAT-based program without any search algorithm in C++ for solving general Suguru puzzles. We perform experiments involving 230 test cases for Suguru instances of size n \times n where 6 \leq n \leq 15. Finally, we derive some empirical results from these experiments and argue that, in terms of running time, our SAT-based approach outperforms the previously proposed backtracking technique in solving larger puzzles.
Consider an undirected, simple graph \( G = (V(G), E(G)) \). A graceful labeling of graph \( G \) is an injective function \(f: V(G) \to \{0, 1, 2, \dots, |E(G)|\} \) such that the induced edge labels, defined by $f^*(uv)=|f(u)-f(v)|$ for every edge $uv \in E(G)$, are all distinct. In this paper, we introduce a new type of graceful labeling, called \textbf{graceful vit labeling}. A graceful labeling $f$ of graph $G$ is called a graceful vit labeling if the vertex weight function $w_f: V(G) \to \mathbb{N}$, defined by $w_f(v)=f(v)+\sum_{uv \in E(G)}f^*(uv)$ for every $v \in V(G)$, assigns pairwise distinct weights to all vertices in graph $G$. In other words, no two vertices have the same sum of their own label and the labels of all edges incident to them. In this paper we present various examples of graphs that admit graceful vit labeling and explores structural properties and necessary conditions for their existence.
This paper introduces a modified version of Heun's method for solving initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed method replaces the standard arithmetic mean used for the average slope in Heun's method with a novel combination of harmonic mean and root mean square (RMS) for the slopes of the tangent lines. This alteration is designed to enhance the accuracy and performance of the method. Through theoretical analysis, we establish that the modified method retains stability and consistency properties, crucial for reliable numerical solutions. Extensive numerical experiments demonstrate that the method performs well across a range of step sizes, showing notable improvements in accuracy for both small and large step sizes when compared to the standard Heun's method. These results suggest that the new approach offers a robust alternative for solving IVPs, particularly in scenarios where adaptive step sizing or high precision is required.
The effect of amplitude modulated signal on the primary resonance on a Duffing-Helmholtz oscillator has been investigated using ($i$) a multiple time scale perturbation method involving two different time scales i.e., regular time ($T_0$) and a slow time, ($T_1$), to analyze the system response and ($ii$) its modified version which is valid for larger values of expansion parameter viz., asymptotic ordering parameter, incorporating Lindstedt-Poincare transformation method that allows the interaction of the amplitude and frequency of the response. The modification of the amplitude- frequency plots, exhibiting bi-stable behavior, caused by different values of modulation amplitude and damping has been obtained both analytically and numerically for various parameters of the system near the primary resonance. The peak amplitude of the response is shown to get lowered in strongly nonlinear Duffing-type system. Phase plane trajectories of stable and unstable manifolds are presented in different instances.
Let R and R' be rings with identity and let S be a strictly ordered monoid equipped with twisting homomorphisms into the endomorphism rings of R and R'. Using these data, we consider two skew generalized power series rings associated with R and R'. Starting from an (R,R')-module M, we construct an induced module consisting of generalized power series with coefficients in M and show that it naturally becomes a module over the two skew generalized power series rings through a suitable trilinear action. We then study homomorphisms in this framework. In particular, we prove that every (R,R')-module homomorphism between two modules induces a corresponding homomorphism between their associated generalized power series modules. Furthermore, we provide sufficient conditions describing when a generalized power series belongs to the kernel of the induced homomorphism in terms of the coefficientwise behavior of the original map. These results extend the theory of skew generalized power series modules from the classical single-ring setting to a two-ring context and provide a foundation for further developments, including generalized isomorphism results and related algebraic applications.
In this paper, we study atomic elements in the lattices of radical classes. A minimal element of the lattice of all \(N\)-radicals (respectively, supernilpotent radicals and special radicals) is called an \(N\)-atom (respectively, a supernilpotent atom and a special atom). We construct a class of \(N\)-atoms generated by prime essential rings and investigate their structural properties. We show that these \(N\)-atoms are closely related to supernilpotent atoms and special atoms, and we clarify the precise relationships among these three types of atoms. In particular, we prove that for a prime essential ring \(R\), the associated radicals \(\widetilde{\mathit{l}}_{R}\) and \(\overline{\mathit{l}}_{R}\) generate, respectively, a special atom and a supernilpotent atom. This result provides a positive answer to an open question concerning which prime essential rings give rise to atomic elements in the lattices of all $N-$ radical, special radicals and supernilpotent radicals.
This study employs the continuous hidden Markov model (HMM) to model the index data of the US Dollar index from 2018 to 2024. HMM is used to predict and analyze hidden patterns that generated the data. The data is modelled using a continuous HMM with 11 hidden states and lognormal distributions with different parameters for each hidden state. The accuracy of the continuous HMM is measured by MAPE. The MAPE value for both training and testing data is very low, less than 4%. This means that the continuous HMM can be used to model the data accurately. The plot shows accurate predictions between simulated data and real data, and furthermore, the model can capture the fluctuations of the data.
The coprime probability of a finite group has been defined, and the coprime probability of a p-group for any prime number p has also been obtained. In this paper, we refine the concept by introducing the coprime degree of a finite group. We calculate the coprime degrees of cyclic groups, nilpotent groups, dihedral groups, and general quaternion groups.
This research work introduces the geometry of the SG (Screen Generic)-lightlike submanifolds of a locally bronze semi-Riemannian manifold endowed with an (l,m)-type connection. The characterization theorems on geodesicity of such submanifolds with respect to the integrability and parallelism of the distributions are established. It is shown that there exists no coisotropic, isotropic or totally proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold. Assertions for the smooth transversal vector fields in totally umbilical proper SG-lightlike submanifold are obtained. Furthermore, the structure of a minimal SG-lightlike submanifold of a locally bronze semi-Riemannian manifold is detailed with an example.
Let \textsl{G} be a group and \textsl{H} be a non-normal cyclic subgroup of \textsl{G}. The non-normal cyclic subgroup graph, denoted as $\Gamma^{NN}_{H}(G)$, is defined as a directed graph with vertex set elements of \textsl{G} such that for two distinct elements \textsl{x} and \textsl{y} in \textsl{G}, \textsl{x} is the initial vertex and \textsl{y} is the terminal vertex of an edge if their product is in \textsl{H}. In 2020, the idea of the non-normal subgroup graph, which is the extension of the subgroup graph was developed. This research identifies the non-normal cyclic subgroup graphs connected to order 16 quasidihedral groups. Firstly, the Groups, Algorithms and Programming (GAP) software is used to identify the non-normal cyclic subgroups. The definition of $\Gamma^{NN}_{H}(G)$ is then used to calculate the adjacency and direction of the vertices. Lastly, Maple 2016 software will be used to visualize these graphs.
Power graphs of groups and semigroups constitute a significant class of graphs in algebraic graph theory, bridging algebraic structures and graph theoretic concepts. Further, labeling of graphs, particularly those endowed with an algebraic structure as their vertex set, is a vibrant area of research. This article contributes to this burgeoning field by exploring new classes of finite groups whose power graphs admit or forbid an important type of labeling called graceful labeling.
The present work investigates the existence and uniqueness of solutions for a fractional stochastic differential equations with $\psi$-Caputo fractional derivatives. We establish important theoretical conclusion and examine the energy growth bounds and asymptotic behavior of the stochastic solution using the Banach's fixed point theorem. Additionally, we consider our analysis into single-valued function derived from multivalued mappings. We provide examples to validate in our methodology.
We study the anti-trace—the sum of anti-diagonal entries—of powers of 3-by-3 matrices. Unlike the ordinary trace, the anti-trace has no spectral characterization, so closed forms for the anti-trace of matrix powers are valuable. Starting from the Cayley--Hamilton recurrence for matrix powers and solving a trivariate generating function, we derive explicit closed-form expressions whose coefficients are signed multinomial combinations of the sums of principal minors and their anti-principal counterparts. The formulas are purely algebraic and remain valid over any commutative ring. As a structural application, we analyze block anti-diagonal matrices of size 3m-by-3m.
Let $G(V,E)$ be a graph with $V(G)$ as the set of vertices and $E(G)$ as the set of edges. A labeling is a bijection $f:E(G)\to\{1,2,\cdots, |E(G)|\}$. For each vertex \(v\), let \(\phi(v)\) be the sum of the labels on the edges incident to \(v\). If all values \(\phi(v)\) are distinct, the labeling is antimagic, and the graph is antimagic if such a labeling exists. This paper explores the concept of antimagic labeling in graph theory, with a particular focus on the union of various tree structures, including stars, single brooms, and double brooms. We apply extended Skolem sequences to prove that for a wide range of parameters, unions of multiple 3-paths with appropriately many 4-cycles yield antimagic graphs. Additionally, we analyze combinations of 4-cycles paired with different tree structures.
In this article, we discuss an $n$-norm, with $n \ge 2$, which is defined through bounded linear functionals on $p$-summable sequence spaces. We also introduce a new norm induced by this \( n \)-norm, which will be examined for its equivalence to the usual norm. Next, we demonstrate the relationships between various mappings, including the \( n \)-norm, \( (\!n\!\!-\!\!1\!) \)-norm, $\cdots,$ \( 2 \)-norm, and the usual norm. Finally, our results show that the \( p \)-summable sequence spaces, equipped with these mappings are complete spaces.
The $\Psi$-divisible graph of a finite group $G$, denoted by $\Psi_G$ is a special type of simple undirected graph, in which the set of vertices contains non-trivial subgroups of $G$ and two distinct vertices $u$ and $v$ are adjacent if and only if $u$ is a proper subgroup of $v$ such that $\Psi(u)|\Psi(v)$ or $v$ is a proper subgroup of $u$ such that $\Psi(v)|\Psi(u)$. The existence of extreme vertices in the $\Psi$-divisible graph of the finite cyclic group $\mathbb{Z}_{p^n}$ is described in this article.
This study develops a mathematical model to investigate the transmission dynamics of Marburg Virus Disease (MVD), integrating two time-dependent control strategies, primarily focusing on isolating infected individuals. Optimal control strategies are designed using Pontryagin's Maximum Principle to minimize the spread of the infection. The basic reproduction number is calculated to identify the disease-free and endemic equilibrium points. The analysis highlights the effectiveness of interventions such as personal protective measures (e.g., wearing masks, maintaining hand hygiene, and avoiding risky dietary practices) combined with supportive hospital therapy in curbing disease transmission. Model simulations validate the impact of these optimal control strategies, while a cost-effectiveness analysis identifies the most efficient approach for disease management. Additionally, the study examines how individuals with robust immune systems can influence the overall dynamics of Marburg virus transmission.
For a molecular graph Lambda, the inverse degree index (IDI) is defined by taking the sum over all vertices of the reciprocal degree of each vertex. The IDI is a structurally sensitive topological invariant that emphasizes low-degree vertices, making it effective for analyzing branching, irregularity, and deviations in molecular structures. Graph transformations prove to be an effective tool to construct new chemically meaningful structures. In this work, we study the behavior of IDI under four newly defined graph transformations. We first introduce families of n-vertex networks based on complete graphs of order l >= 3, derive transformed networks, and establish inequalities involving the IDI. We then present extremal results for the transformed graphs and support our findings with computations on several representative examples.
This paper presents a novel framework for addressing complex multicriteria decision analysis (MCDA) problems by integrating Prospect Theory with the recently developed interval-valued intuitionistic quadri-partitioned neutrosophic soft set (IVIQPNSS) structure. The proposed model effectively incorporates both subjective preferences and objective criteria weights by leveraging Prospect Decision Theory, which models human behavior under risk and uncertainty based on gains and losses relative to a reference point. A newly formulated score function (SF) is introduced to transform the quadri-partitioned interval-valued neutrosophic information-comprising truth, indeterminacy, contradiction, and falsity-into precise numerical measures. This enables a refined ranking mechanism among alternatives. The proposed methodology merges expert judgment with data-driven insights, offering a robust algorithmic structure for decision-makers. The practical efficiency and reliability of the approach are demonstrated through a real-world application, making it a promising tool in the context of uncertain and ambiguous decision-making environments.
Multi-strain epidemic dynamics have been analyzed using two-strain SEIR models. We have taken into consideration a two-strain SEIR model that incorporates behavioural alterations for strain 2 among the susceptibles, with non-monotone incidence for strain 2 and bilinear incidence for strain 1. We have focused on the co-dynamics of both strains and analytically derived a sufficient condition, 1+R-k(& lowast;) < R-01 < R-02, for the existence of the two-strain endemic equilibrium, where R-01 and R-02 are basic reproduction numbers of respective strains and, R-k(& lowast;) > 0 is a value derived from endemic state of the model. Under this sufficient condition, we establish the global stability of the two-strain endemic equilibrium using the Lyapunov function method in terms of reproduction numbers and behavioural parameter k. Additionally, the condition, R-02 = R-01 > 1, result in instability of the two-strain endemic equilibrium. Lastly, to support these analytical results regarding the existence and stability of the two-strain endemic equilibrium, we have presented numerical simulations. Interestingly, the study reveals that the principle of competitive exclusion relaxing in this model, as behavioural heterogeneity among susceptibles effectively partitions the population. This partitioning reduces inter-strain competition and enables strain coexistence even when R-02 > R-01 > 1.
