Abstract In this paper, we define a new subclass of bi-univalent functions of complex order  which is defined by subordination in the open unit disc D  by using  operator. Furthermore, using the Faber polynomial expansions, we get upper bounds for the coefficients of function belonging to this class. It is known that the calculus without the notion of limits is called q-calculus which has influenced many scientific fields due to its important applications. The generalization of derivative in q-calculus that is q-derivative was defined and studied by Jackson. A function  is said to be bi-univalent in D  if both F  and F −1  are univalent in D . The class consisting of bi-univalent functions is denoted by σ . The Faber polynomials play an important role in various areas of mathematical sciences, especially in geometric function theory. The purpose of our study is to obtain bounds for the general coefficients  by using Faber polynomial expansion under certain conditions for analytic bi-univalent functions in subclass  and also, we obtain improvements on the bounds for the first two coefficients  and  of functions in this subclass. In certain cases, our estimates improve some of those existing coefficient bounds.

Abstract This study explores a multiple-security, high-risk pricing model where the implied volatility has been portrayed through Generalized Wishart affine processes. The presence of dual dependency matrices distinctively characterizes this multifaceted model. These matrices encapsulate the relationship between the generalized Wishart processes and the evolving dynamics of several securities. The adaptability of the proposed model makes it a perfect fit for high-frequency market data, whether dealing with either long or short-term maturities of calls. The main objective paper is on its derivation and addressing the call option pricing problem within the context of the volatility mode using generalized Wishart stochastic. A combination of Fourier transforms techniques and perturbation methods are utilized, mainly focusing on pricing European call options. The model proposed in this study is theoretical and practical, showcasing the strong potential for real-world applications within the financial derivative market.

Abstract We consider conditions of three types of stability: Lyapunov, formal and weak of a stationary solution, and of a periodic solution in a Hamiltonian system with a finite number of degrees of freedom. The conditions contain restrictions on the order of resonances and some inequalities for initial coefficients of the normal forms of the Hamiltonian functions. We show that the number-theoretical analysis of frequencies can help in proof of stability. We also estimate the orders of solutions’ divergence from the stationary or the periodic ones under lack of formal stability.

Abstract In this paper, by introducing predator-taxis into the diffusive predator-prey system with spatial memory, then we study the inhomogeneous spatial patterns of this system. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation for the reaction-diffusion system with delay can’t be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation for this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also studied by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations to verify our theoretical analysis results, and two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found.

Abstract The development of number system has gone through a long and difficult process, and many landmark concepts and theorems were put forward. By briefly reviewing the development of hypercomplex systems, the constructing rules of the unit elements are discussed. As a vector space defining multiplication, division and norm of vectors, hypercomplex numbers synthesize the advantages of mathematical tools such as algebra, geometry and analysis, faithfully describe the intrinsic properties of space-time and physical systems, and provide a unified language and a powerful tool for basic theories and engineering technology. In the application of hypercomplex numbers, the group-like properties of the basis vectors are the most important, and the zero factor has little influence on the algebraic operation. The multiplication table of the basis vectors fully describes the intrinsic properties of the hypercomplex system, and the matrix A constructed from the multiplication table satisfies the structure equation A 2 = nA, and thus obtains a set of faithful matrix representations of the basis elements. Th/is paper also uses typical examples to show the simple and clear concepts and wide application of hypercomplex numbers. Therefore, hypercomplex numbers are worth of learning in basic education and appling in scientific research and engineering technology

Abstract Gear drive dominates in mechanical transmissions and has comprehensive advantages of high precision, high efficiency, high reliability, long life and the ability to realize large drive ratios. The main purpose of this article is to review research development in meshing theory, which involves the meshing theory of the one degree-of-freedom (1DOF) line-conjugate tooth surface couple, the 1DOF point-conjugate tooth surface couple and the two degree-of freedom (2DOF) conjugate tooth surface couple. Some compendious discussions are made on the significant results and progresses in meshing theory developments of gearing.