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Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning 1-4 , owing to their reputed difficulty among the world's best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges 1,5 , resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004., (© 2024. The Author(s).)

In this paper, we propose an eco-epidemiological mathematical model in order to describe the effect of migration on the dynamics of a prey–predator population. The functional response of the predator is governed by the Holling type II function. First, from the perspective of mathematical results, we develop results concerning the existence, uniqueness, positivity, boundedness, and dissipativity of solutions. Besides, many thresholds have been computed and used to investigate the local and global stability results by using the Routh–Hurwitz criterion and Lyapunov principle, respectively. We have also established the appearance of limit cycles resulting from the Hopf bifurcation. Numerical simulations are performed to explore the effect of migration on the dynamic of prey and predator populations.

Recently it has been proven that the multiplication module is a Weak-multiplication module (W −M M ). In this paper, we present some applications of W − M M in neutrosophy theory. n-refined neutrosophic module M denoted by Mn(I). We proved that Mn(I) is regular and C.P.module. Also, every n-refined neutrosophic finitely generated ideal is a principal ideal of n-refined the ring R (Rn(I)). We proved that if Mn(I) is multiplication module over ring Rn(I) and Mn(I) is of type S2; then every f. generated An(I) of Mn(I) is n-refined multiplication module. Also, if Rn(I) is n-refined neutrosophic ring, so Rn(I) is principal ideal ring if and only if every n-refined neutrosophic multiplication module Mn(I) is of type S. Finally, several results and applications have been presented in this paper with some new definitions, examples, and other properties.

The present paper introduces the novel subfamilies of regular and bi-univalent functions Σ. The prominent group of Fibonacci polynomial ˜Vn(x, y) is utilized with subordination between regular functions in order to shape these subfamilies. In addition we derive coefficients inequalities for functions belonging to these subfamilies. Various results are exposed as separate cases of the current conclusions.

Sylow's theorems are fundamental theorems in classical group theory that are of paramount importance. The extension of these theorems into diverse fuzzy contexts emerges as a compelling area of exploration. This study introduces the novel concept of the conjunctive complex fuzzy conjugate element within the conjunctive complex fuzzy subgroup of a group, elucidating numerous crucial properties of this concept. Additionally, it propounds the notion of the conjunctive complex fuzzy p-subgroup within the conjunctive complex fuzzy subgroup (CCFSG) and delineates various indispensable characteristics associated with this construct. Additionally, the paper formulates the conjunctive complex fuzzy version of the Cauchy theorem for finite groups. Lastly, it defines the concept of the conjunctive complex fuzzy Sylow p-subgroup for a finite group and conducts a generalization of Sylow's theorems within a conjunctive complex fuzzy environment.

In this paper, new results were presented on the symmetry reduction of gas dynamics system of partial differential equations following the general framework of Lev Ovsyannikov's article "The "podmodeli" program, Gas dynamics." We considered the gas dynamics equations with an equation of state prescribing the pressure as the sum of density function and entropy function. This system has a 12-dimensional Lie algebra and we considered its certain three-dimensional subalgebra generated by space translations, Galilean translations and pressure translation. For this subalgebra, the symmetry reduction of the original system leads to a system of ordinary differential equations. We obtained a family of exact solutions for this system, which describes the motion of particles with a linear velocity field and non-homogeneous deformation in the 3D-space. For these solutions, the trajectories of all points are either parabolas or rays. At $ t = 0 $ an instantaneous collapse occurs when all of the particles accumulate in a plane with infinitely many particles at every point of the plane. For a fixed period of time, the particles were emitted from the same point on a plane and ended up on the same line. The gas motion was vortex. A one-dimensional subalgebra embedded into three-dimensional subalgebra was considered. The invariants were written in a consistent form. It was shown that the submodel of rank one was embedded in the submodel of rank three.

This paper aims to study bounded algebras in another perspective-dual ideals of bounded $ L $-algebras. As the dual concept of ideals in $ L $-algebras, dual ideals are designed to characterize some significant properties of bounded $ L $-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient $ L $-algebras on bounded $ L $-algebras. Naturally, in order to construct the first isomorphism theorem between bounded $ L $-algebras, the relationship between dual ideals and morphisms between bounded $ L $-algebras is investigated and that the kernels of any morphisms between bounded $ L $-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded $ L $-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded $ L $-algebras. Therefore, first, the properties of dual ideals in some special bounded $ L $-algebras are studied; then, some special bounded $ L $-algebras are characterized by dual ideals. For example, a good $ L $-algebra is a $ CL $-algebra if and only if every dual ideal is $ C $ dual ideal is proven.

In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.

In this paper, we introduced $ s $-index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum $ H $-eigenvalue of $ s $-index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of $ s $-index weakly positive tensors. The final numerical example shows the effectiveness of the algorithm.

In this paper, we initiate the study of existence and uniqueness of solutions for a coupled system involving Hilfer fractional quantum derivatives with nonlocal boundary value conditions containing $ q $-Riemann-Liouville fractional derivatives and integrals. Our results are supported by some well-known fixed-point theories, including the Banach contraction mapping principle, Leray-Schauder alternative and the Krasnosel'skiǐ fixed-point theorem. Examples of these systems are also given in the end.

Traditional theories of term structure of interest rate consist of four major classical theories, including Pure Expectation Theory, Liquidity Preference Theory, Preferred Habitat Theory and Market Segmentation Theory. However, they cannot be well interpreted by the traditional static term structure of interest rate methods such as polynomial spline and exponential spline. To address problems on low precision and weak stability of traditional methods in constructing static interest rate term structure curve, in this paper, we introduce the tension interpolation spline based on a fourth-order differential equation with local tension parameters calculated by Generalized Reduced Gradient (GRG) algorithm. Our primary focus is to illustrate its better prediction effect and stability with an empirical study conducted using datum of treasury bonds. Then, we divided the datum into intra-sample datum for estimating tension parameters and out-of-sample datum for evaluating their robustness of predicting stochastics collected from Shanghai Stock Exchange on $ {2^{{\rm{nd}}}} $ February, 2019. According to the principle of total least squares and total least absolute deviations, the result shows that the tension interpolation spline model has better precision and stronger stability in prediction of out-of-sample treasury bonds prices compared with the model established by polynomial spline and exponential spline. In addition, it can better explain the Liquidity Preference Theory, which confirms that it is suitable for constructing the static term structure of interest rates in the securities exchange market.

This paper investigated the master-slave synchronization for uncertain neural networks with time-delay by using the sliding mode control method. The uncertain parts in this neural network only needs to be bounded other than any structure condition. An integral sliding mode surface and sliding mode controller were designed such that the state trajectories of the neural networks could reach the sliding mode surface in finite time. Moreover, the computing method of the controller gain was proposed. Finally, a numerical example was provided to show the effectiveness of the obtained results.

The discrete Weibull model can be adapted to capture different levels of dispersion in the count data. This paper takes into account the direct relationship between explanatory variables and the median of discrete Weibull response variable. Additionally, it provides the Bayesian estimate of the discrete Weibull regression model using the random walk Metropolis algorithm. The prior distributions of the coefficient predictors were carried out based on the uniform non-informative, normal and Laplace distributions. The performance of the Bayes estimators was also compared with the maximum likelihood estimator in terms of the mean square error and the coverage probability through the Monte Carlo simulation study. Meanwhile, a real data set was analyzed to show how the proposed model and the methods work in practice.

A connected acyclic graph in which the degree of every vertex is at most four is called a molecular tree. A number associated with a molecular tree that can help to approximate the physical or chemical properties of the corresponding molecule is called a topological index. It is of great importance to investigate the relation between the structure and the thermodynamic properties of those molecules. In this paper, we investigated the extreme value of the first reformulated Zagreb index with a given order and degree of a graph. Further, we investigated the molecular trees that attain the maximum and minimum values. As an application, we presented the regression models to predict the acentric factor and entropy of octane isomers. Our extremal graphs give the minimum and the maximum acentric factor and entropy, which satisfied the experimental values.

In this paper, a new-type time-varying asymmetric integral barrier function is designed to handle the state constraint of nonlinear systems. The barrier Lyapunov function is developed by building an integral upper limit function with respect to transformation errors over an open set to cope with the position constraint of the robotic system. We know that the symmetric time-invariant constraint is only a particular situation of the asymmetric time-variant constraint, and thus compared to existing methods, it is capable of handling more general and broad practical engineering issues. We show that under the integral barrier Lyapunov function combining a disturbance observer-based tracking controller, the position vector tracks a desired trajectory successfully, while the constraint boundary is never violated. It can certify the exponential asymptotic stability of the robotic tracking system by using the given inequality relationship on barrier function and Lyapunov analysis. Finally, the feasibility of the presented algorithm is indicated by completing the simulations.

In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.

This paper presents a novel approach for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems by using the unification of the Adomian decomposition method and ZZ transformation. The suggested method combines the Aboodh transform and the Adomian decomposition method, both of which are trustworthy and efficient mathematical tools for solving fractional differential equations (FDEs). This method's theoretical analysis is addressed for nonlinear FDE systems. To find exact solutions to the equations, the method is applied to fractional Kersten-Krasil'shchik linked KdV-mKdV systems. The results show that the suggested method is efficient and practical for solving fractional Kersten-Krasil'shchik linked KdV-mKdV systems and that it may be applied to other nonlinear FDEs. The suggested method has the potential to provide new insights into the behavior of nonlinear waves in fluid and plasma environments, as well as the development of new mathematical tools for modeling and studying complicated wave phenomena.

With the continuous development of the fuzzy set theory, neutrosophic set theory can better solve uncertain, incomplete and inconsistent information. As a special subset of the neutrosophic set, the single-valued neutrosophic set has a significant advantage when the value expressing the degree of membership is a set of finite discrete numbers. Therefore, in this paper, we first discuss the change values of single-valued neutrosophic numbers when treating them as variables and classifying these change values with the help of basic operations. Second, the convergence of sequences of single-valued neutrosophic numbers are proposed based on subtraction and division operations. Further, we depict the concept of single-valued neutrosophic functions (SVNF) and study in detail their derivatives and differentials. Finally, we develop the two kinds of indefinite integrals of SVNF and give the relevant examples.

In this paper, we introduce the notion of single-valued neutrosophic soft uniform spaces as a view point of the entourage approach. We investigate the relationship among single-valued neutrosophic soft uniformities, single-valued neutrosophic soft topologies and single-valued neutrosophic soft interior operators. Also, we study several single-valued neutrosophic soft topologies induced by a single-valued neutrosophic soft uniform space.

In this research, the progressive first-failure censored data (PFFC) from the Kumaraswamy modified inverse-Weibull distribution (KMIWD) were used to obtain the recurrence relations and characterizations for single and product moments. The recurrence relationships allow for a rapid and efficient assessment of the means, variances and covariances for any sample size. Additionally, the paper outcomes can be boiled down to the traditional progressive type-II censoring. Also, some special cases are limited to some lifetime distributions as the exponentiated modified inverse Weibull and Kumaraswamy inverse exponential.

Latin square, also known as Latin square matrix, refers to a kind of $ n\times n $ matrix, in which there are exactly $ n $ different symbols and each symbol appears exactly once in each row and column. A Latin square graph $ \Gamma(L) $ is a simple graph associated with a Latin square $ L $. This paper studied the relationships between the (total) Roman domination number and (total) domination number of Latin square graph $ \Gamma(L) $. We showed that $ \gamma_{R}(\Gamma(L)) = 2\gamma(\Gamma(L)) $ or $ \gamma_{R}(\Gamma(L)) = 2\gamma(\Gamma(L))-1 $, and $ \gamma_{tR}(\Gamma(L))\ge \frac{8\gamma_t(\Gamma(L))}{5} $ for $ n\ge 2 $. In 2021, Pahlavsay et al. proved $ \gamma(\Gamma(L))\ge \lceil \frac{n}{2}\rceil $ and $ \gamma_t(\Gamma(L))\ge \lceil \frac{4n-2}{7}\rceil $ for $ n\ge 2 $. In this paper, we showed that $ \gamma_R(\Gamma(L))\ge 2\lceil \frac{n}{2}\rceil $ (equality holds if, and only if, $ \gamma(\Gamma(L)) = \lceil \frac{n}{2}\rceil $) and $ \gamma_t(\Gamma(L)) > \frac{4n}{7} $ for $ n\ge 2 $. Since $ \gamma_R(G)\le 2\gamma(G) $ and $ \gamma_{tR}(G)\le 2\gamma_t(G) $ for any graph $ G $, our results can deduce or improve Pahlavsay et al.'s results. Moreover, we characterized these Latin squares for $ \gamma_R(\Gamma(L)) = 2\lceil \frac{n}{2}\rceil $, which is equal to $ \gamma(\Gamma(L)) = \lceil \frac{n}{2}\rceil $.

Threshold selection is challenging when analyzing tail data with a generalized Pareto distribution. Data below the threshold was not used in the model, resulting in incomplete characterization of the whole data. This paper applied the Gamma distribution, Weibull distribution, and lognormal distribution to fit the central data separately, and a generalized Pareto distribution (GPD) was used to analyze the tail data. In such composite models, the thresholds are estimated directly as parameters. We proposed an empirical distribution function-based parameter estimation method. The absolute value of the difference between the empirical distribution function and the composite distribution function was used as a loss function to obtain an estimate of the parameter. This parameter estimation method is suitable for complex multiparameter distributions. The estimation method based on the empirical distribution function was verified to be feasible through simulation studies. The composite model and the estimation method based on the empirical distribution function were applied to study the earthquake magnitude data to provide a reference for earthquake hazard analysis.

In this paper, the finite element method is applied to solve the unsteady elastic equations, C-Bézier basis functions are used to construct the shape function spaces, the semi-discrete scheme of the unsteady elastic equations is obtained by Galerkin finite element method and then the fully discretized Galerkin method is obtained by further discretizing the time variable with $ \theta $-scheme finite difference. Furthermore, for several numerical examples, the accuracy of approximate solutions are improved by 1–3 order-of magnitudes compared with the Lagrange basis function in $ L^\infty $ norm, $ L^2 $ norm and $ H^1 $ semi-norm, and the numerical examples show that the method proposed possesses a faster convergence rate. It is fully demonstrated that the C-Bézier basis functions have a better approximation effect in simulating unsteady elastic equations.

In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.

This research paper investigated fixed point results for almost ($ \zeta-\theta _{\rho } $)-contractions in the context of quasi-metric spaces. The study focused on a specific class of ($ \zeta -\theta _{\rho } $)-contractions, which exhibit a more relaxed form of contractive property than classical contractions. The research not only established the existence of fixed points under the almost ($ \zeta -\theta _{\rho } $)-contraction framework but also provided sufficient conditions for the convergence of fixed point sequences. The proposed theorems and proofs contributed to the advancement of the theory of fixed points in quasi-metric spaces, shedding light on the intricate interplay between contraction-type mappings and the underlying space's quasi-metric structure. Furthermore, an application of these results was presented, highlighting the practical significance of the established theory. The application demonstrated how the theory of almost ($ \zeta -\theta _{\rho } $)-contractions in quasi-metric spaces can be utilized to solve real-world problems.

In this paper, we present a comprehensive analysis of the lower and upper bounds of solutions for a nonlinear second-order ordinary differential equation governing the electrohydrodynamic flow of a conducting fluid in cylindrical conduits. The equation describes the radial distribution of the flow velocity in an "ion drag" configuration, which is affected by an externally applied electric field. Our study involves the establishment of rigorous analytical bounds on the radial distribution, taking into account the Hartmann number $ H $ and a parameter $ \alpha. $ An analytic approximate solution is obtained as an improvement of the a priori estimates and it is found to exhibit strong agreement with numerical solutions, particularly when considering small Hartmann numbers. Further, estimates for the central velocity $ w(0) $ of the fluid occurring at the center of the cylindrical conduit were also established, and some interesting relationships were found between $ H $ and $ \alpha. $ These findings establish a framework that illuminates the potential range of values for the physical parameter within the conduit.

In this paper, we discussed some qualitative properties of solutions to a thermostat system in the framework of a novel mathematical model designed by the new $ (p, q) $-derivatives in fractional post-quantum calculus. We transformed the existing standard model into a new control thermostat system with the help of the Caputo-like $ (p, q) $-derivatives. By the properties of the $ (p, q) $-gamma function and applying the fractional Riemann-Liouville-like $ (p, q) $-integral, we obtained the equivalent $ (p, q) $-integral equation corresponding to the given Caputo-like post-quantum boundary value problem ($ (p, q) $-BOVP) of the thermostat system. To conduct an analysis on the existence of solutions to this $ (p, q) $-system, some theorems were proved based on the fixed point methods and the stability analysis was done from the Ulam-Hyers point of view. In the applied examples, we used numerical data to simulate solutions of the Caputo-like $ (p, q) $-BOVPs of the thermostat system with respect to different parameters. The effects of given parameters in the model will show the performance of the thermostat system.

The social division of labor has become increasingly specialized, and there are more and more group decision-making problems participated by multiple decision-makers. With respect to the multi-attribute group decision making problem, including two-tuple linguistic information, based on the theory and method of group decision making, Steiner point constraint and plant growth simulation algorithm, we establish a novel multi-attribute group decision making approach based on two-tuple linguistic information aggregation. We introduce Steiner points into group consensus decision making and use the PGSA algorithm to seek the global optimal point. The method seeks set points that are both mathematically and geometrically meaningful to reduce set bias. In this paper, to begin with, according to the constraints of multi-dimensional Steiner point, we map the evaluation vectors of the group experts over the alternatives into multi-dimensional space and then we propose a two-tuple linguistic information aggregation model. Moreover, we construct a comprehensive evaluation decision making approach and then design a plant growth simulation algorithm to select the optimal alternative. Finally, a case verifies the validity and rationality of the proposed model.

In this paper, an adaptive neural network control problem for nonstrict-feedback nonlinear systems with an unknown backlash-like hysteresis and bounded disturbance was presented. Radial basis function neural networks (RBFNN) were used to approximate the unknown functions and the problem of the explosion of complexity problem was handled by utilizing the command filter method. Furthermore, the influence of an unknown backlash-like hysteresis input was addressed by approximating an intermediate variable. Based on the backstepping method and the command filter technique, an adaptive neural network controller was designed via the approximation abilities of RBFNN. With the help of the Lyapunov stability theory, the proposed controller ensures that all of the signals in closed-loop systems are bounded and that the tracking error fluctuates close to the origin within a bounded area. Finally, a real-world example based on the single-link manipulator was shown to demonstrate the viability of the presented approach.

Establishing a reasonable and effective feature system is the basis of credit risk early warning. Whether the system design is appropriate directly determines the accuracy of the credit risk evaluation results. In this paper, we proposed a feature system through a validity index with maximum discrimination and commercial banks' loan profit maximization. First, the first objective function is the minimum validity index constructed by the intra-class, between-class, and partition coefficients. The maximum difference between the right income and wrong cost is taken as the second objective function to obtain the optimal feature combination. Second, the feature weights are obtained by calculating the change in profit after deleting each feature with replacement to the sum of all change values. An empirical analysis of 3, 425 listed companies from t-1 to t-5 time windows reveals that five groups of feature systems selected from 614 features can distinguish between defaults and non-defaults. Compared with 14 other models, it is found that the feature systems can provide at least five years' prediction and enable financial institutions to obtain the maximum profit.

Reduced order modelling relies on representing complex dynamical systems using simplified modes, which can be achieved through the Koopman operator(KO) analysis. However, computing Koopman eigenpairs for high-dimensional observable data can be inefficient. This paper proposes using deep autoencoders(AE), a type of deep learning technique, to perform nonlinear geometric transformations on raw data before computing Koopman eigenvectors. The encoded data produced by the deep AE is diffeomorphic to a manifold of the dynamical system and has a significantly lower dimension than the raw data. To handle high-dimensional time series data, Takens' time delay embedding is presented as a preprocessing technique. The paper concludes by presenting examples of these techniques in action.

The equivalence between multi-transitive mean sensitivity and multi-transitive mean n-sensitivity for linear dynamical systems was demonstrated in this study. Furthermore, this paper presented examples that highlighted the disparities among mean sensitivity, multi-transitive mean sensitivity, and syndetically multi-transitive mean sensitivity.

The concept of Lindel$ \ddot{o} $f proximate order has been used extensively to study the functions of completely regular growth. The main drawback of this approach is that it completely ignores the value of lower order. To overcome this problem, Chyzykov et al. introduced the concept of generalized proximate order for irregular growth. In this paper we studied the existence of generalized proximate order for every functions analytic on the unit disc with some new results for functions having irregular growth.

This paper marks a significant advancement in the field of chemoinformatics with the introduction of two novel topological indices: the forgotten eccentric neighborhood index (FENI) and the modified forgotten eccentric neighborhood index (MFENI). Uniquely developed for predicting the boiling points of various chemical substances, these indices offer groundbreaking tools in understanding and interpreting the thermal properties of compounds. The distinctiveness of our study lies in the in-depth exploration of the discriminative capabilities of FENI and MFENI. Unlike existing indices, they provide a nuanced capture of structural features essential for determining boiling points, a key factor in drug design and chemical analysis. Our comprehensive analyses demonstrate the superior predictive power of FENI and MFENI, highlighting their exceptional potential as innovative tools in the realms of chemoinformatics and pharmaceutical research. Furthermore, this study conducts an extensive investigation into their various properties. We present explicit results on the behavior of these indices in relation to diverse graph types and operations, including join, disjunction, composition and symmetric difference. These findings not only deepen our understanding of FENI and MFENI but also establish their practical versatility across a spectrum of chemical and pharmaceutical applications. Thus the introduction of FENI and MFENI represents a pivotal step forward in the predictive analysis of boiling points, setting a new standard in the field and opening avenues for future research advancements.

In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.

In this paper, we studied a double-phase eigenvalue problem with large variable exponents. Let $ \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))} $ be the first eigenvalues and $ u_{n} $ be the first eigenfunctions, normalized by $ \|u_{n}\|_{\mathcal{H}_{n}} = 1 $. Under some assumptions on the variable exponents $ p_{n}(\cdot) $ and $ q_{n}(\cdot) $, we showed that $ \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))} $ converges to $ \Lambda_{\infty} $, $ u_{n} $ converges to $ u_{\infty} $ uniformly in the space $ C^{\alpha}(\Omega)\, (0 < \alpha < 1) $ and $ u_{\infty} $ is a nontrivial viscosity solution to a Dirichlet $ \infty $-Laplacian problem. Even in the case where the variable exponents reduce to the constant exponents, our work is the first one dealing with a double-phase eigenvalue problem with large exponents.

In this paper, we explore the existence of fixed points, local dynamics at fixed points, bifurcations and chaos of a discrete prey-predator fishery model with harvesting. More specifically, it is proved that, for all involved parameters, the model has trivial fixed point, but it has semitrivial and interior fixed points under definite parametric condition(s). We study the local behavior at fixed points by applying the theory of linear stability. Furthermore, it is shown that flip bifurcation does not occur at semitrivial and trivial fixed points, but that the model undergoes Neimark-Sacker bifurcation at interior fixed point. It is also proved that, at interior fixed point, the model undergoes the flip bifurcation. By using a feedback control strategy, the chaos control is also examined. Finally, to illustrate the theoretical findings, detailed numerical simulations are provided.

This paper is concerned with the relationships between best linear minimum biased predictors (BLMBPs) in the context of a general linear model (GLM) and its transformed general linear models (TGLMs). We shall establish a mathematical procedure by means of some exact and analytical tools in matrix theory that were developed in recent years. The coverage includes constructing a general vector composed of all unknown parameters in the context of a GLM and its TGLMs, deriving the exact expressions of the BLMBPs through the technical use of analytical solutions of a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and discussing a variety of theoretical performances and properties of the BLMBPs. We also give a series of characterizations of relationships between BLMBPs under a given GLM and its TGLMs.

The main features of scientific effort in physics and engineering are the development of models for various physical issues and the development of solutions. In this paper, we investigate the numerical solution of time-fractional non-linear dispersive K(m, n, 1) type equations using two innovative approaches: the homotopy perturbation transform method and Yang transform decomposition method. Our suggested approaches elegantly combine Yang transform, homotopy perturbation method (HPM) and adomian decomposition method (ADM). With the help of the Yang transform, we first convert the problem into its differential partner before using HPM to get the He's polynomials and ADM to get the Adomian polynomials, both of which are extremely effective supports for non-linear issues. In this case, Caputo sense is used for defining the fractional derivative. The derived solutions are shown in series form and converge quickly. To ensure the effectiveness and applicability of the proposed approaches, the examined problems were analyzed using various fractional orders. We analyze and demonstrate the validity and applicability of the solution approaches under consideration with given initial conditions. Two and three dimensional graphs reflect the outcomes that were attained. To verify the effectiveness of the strategies, numerical simulations are presented. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. The results of this study demonstrate that the studied methods are effective and strong in solving nonlinear differential equations that appear in science and technology.

An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface.

Let $ \gamma(G) $ denote the domination number of a graph $ G $. A vertex $ v\in V(G) $ is called a critical vertex of $ G $ if $ \gamma(G-v) = \gamma(G)-1 $. A graph is called vertex-critical if its every vertex is critical. In this paper, we correspondingly introduce two such definitions: (i) A set $ S\subseteq V(G) $ is called a strong critical vertex-set of $ G $ if $ \gamma(G-S) = \gamma(G)-|S| $; (ii) A graph $ G $ is called strong $ l $-vertex-set-critical if $ V(G) $ can be partitioned into $ l $ strong critical vertex-sets of $ G $. Therefrom, we give some properties of strong $ l $-vertex-set-critical graphs by extending the previous results of vertex-critical graphs. As the core work, we study on the existence of this class of graphs and prove that there exists a strong $ l $-vertex-set-critical connected graph if and only if $ l\notin\{2, 3, 5\} $.

In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.

In this paper, we consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie group. We mainly consider the solvable indecomposable six-dimensional Lie algebras with co-dimension two abelian nilradical that have an abelian complement. In dimension six, there are nineteen such algebras, namely, $ A_{6, 1} $–$ A_{6, 19} $ in Turkowski's list. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields, and finally we identify the symmetry Lie algebra from standard lists.

In this paper, an investment risk model with bilateral jumps was considered, assuming the insurer invested the surplus in two types of assets, namely, risk-free and risky ones, in a certain proportion. First, the integral-differential equations of the Gerber-Shiu function related to ruin and penalty were obtained, then, the sinc approximation method was used to obtain a numerical solution. Furthermore, we presented a special example for finding the explicit solutions (ES). By calculating the relative errors of the approximate solution (SA) and ES, we verified the superiority of the sinc method. Finally, several examples under different kinds of jumps were provided to show the impact of parameters such as investment ratio, discount factor or intensity of Poisson process on the ruin probability.

In this paper two kinds of identities involving derangement polynomials and $ r $-Bell polynomials were established. The identities of the first kind extended the identity on derangement numbers and Bell numbers due to Clarke and Sved and its generalizations due to Du and Fonseca. The identities of the second kind extended some of the results on derangement polynomials and Bell polynomials due to Kim et al.

In this paper, we investigated an optimal investment problem of a defined contribution (DC) pension plan under a joint Value-at-Risk (VaR) and an expected shortfall (ES) constraint. By using a martingale method, we transformed a dynamic optimization problem to a static pointwise optimization problem and derived the closed-form representations of the optimal wealth and portfolio processes in terms of the state price density. Numerical results showed that in comparison to only an ES constraint or a VaR constraint, the joint VaR-ES constraint can not only improve risk management for the bad economic states but also lower the volatility of the optimal terminal wealth.

Let $ \mathbb{F}_q $ be the finite field of order $ q $ where $ q = p^{k} $, $ k $ is a positive integer and $ p $ is an odd prime. Let $ \mathbb{F}_q^* $ represent the nonzero elements of $ \mathbb{F}_{q} $. For $ f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, \cdots, x_n] $, we used $ N\big(f(x_1, \cdots, x_n) = 0\big) $ to denote the number of $ \mathbb{F}_q $-rational points of the affine hypersurface $ f(x_1, \cdots, x_n) = 0 $. In 2020, Zhao et al. obtained the explicit formulae for $ N(x_1^4+x_2^4 = c) $, $ N(x_1^4+x_2^4+x_3^4 = c) $ and $ N(x_1^4+x_2^4+x_3^4+x_4^4 = c) $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for $ N(a_1x_1^4+a_2x_2^4 = c) $ and $ N(b_1x_1^4+b_2x_2^4+b_3x_3^4 = c) $ with $ a_i, b_j\in \mathbb{F}_q^* (1\leq i \leq 2, 1 \leq j \leq 3) $ and $ c\in \mathbb{F}_q $. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface $ a_1x_1^4+a_2x_2^4+\cdots+a_nx_n^4 = c $ of $ n\geq 4 $ variables with $ a_i\in\mathbb{F}_q^* $ $ (1\leq i\leq n) $, $ c\in\mathbb{F}_q $ and $ p\equiv1({\rm{mod}} \ 4) $, can also be deduced. These extended and improved earlier results.

In this paper, we introduced a fuzzy model for calculating complexity based on universality, aiming to measure the complexity of natural languages in terms of the degree of universality exhibited in their rules. We validated the model by conducting experiments on a corpus of 143 languages obtained from Universal Dependencies 2.11. To formalize the linguistic universals proposed by Greenberg, we employed the Grew tool to convert them into a formal rule representation. This formalization enables the verification of universals within the corpus. By analyzing the corpus, we extracted the occurrences of each universal in different languages. The obtained results were used to define a fuzzy model that quantifies the degree of universality and complexity of both the Greenberg universals and the languages themselves, employing the mathematical theory of evaluative expressions from fuzzy natural logic (FNL). Our analysis revealed an inversely proportional relationship between the degree of universality and the level of complexity observed in the languages. The implications of our findings extended to various applications in the theoretical analysis and computational treatment of languages. In addition, the proposed model offered insights into the nature of language complexity, providing a valuable framework for further research and exploration.