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In this paper, Bianchi type-V cosmological model in perfect fluid with quadratic equation of state p= αρ 2 -ρ, where α≠ 0, has been studied in modified theory of gravity. The general solution of the f ( R, T ) equations for Bianchi type-V space-time has been obtained by considering a law of variation of scale factor which yields a time dependent deceleration parameter. The model represents the transition of the universe from early deceleration phase to the recent accelerated expansion phase.

The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.

Abstract We investigate the arithmetic geometry of classical quadratic curves over finite fields, extending prior work on the unit circle. Using classical methods from algebra and number theory, we derive exact counts of solutions to the quadratic equations in two variables defining conic sections and given by elliptic, hyperbolic, parabolic and mixed-term quadratic forms. Special attention is given to the quadratic equations with mixed-term, where solution counts exhibit a rich variation depending on the characteristic and the degree of the finite field. Our results, which are grounded in rigorous mathematical techniques and inspired by computational insights, contribute to a deeper understanding of Diophantine geometry in discrete settings.

Let G be the first Grigorchuk group. We show that the commutator width of G is 2: every element g∈[G,G] is a product of two commutators, and also of six conjugates of a. Furthermore, we show that every finitely generated subgroup H≤G has finite commutator width, which however can be arbitrarily large, and that G contains a subgroup of infinite commutator width. The proofs were assisted by the computer algebra system GAP.

Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is re-examined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton–Jacobi equation, Hamilton–Jacobi inequality, and Hamilton–Jacobi–Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

The modulus-based matrix splitting (MMS) algorithm is effective to solve linear complementarity problems (Bai in Numer Linear Algebra Appl 17: 917–933, 2010). This algorithm is parameter dependent, and previous studies mainly focus on giving the convergence interval of the iteration parameter. Yet the specific selection approach of the optimal parameter has not been systematically studied due to the nonlinearity of the algorithm. In this work, we first propose a novel and simple strategy for obtaining the optimal parameter of the MMS algorithm by merely solving two quadratic equations in each iteration. Further, we figure out the interval of optimal parameter which is iteration independent and give a practical choice of optimal parameter to avoid iteration-based computations. Compared with the experimental optimal parameter, the numerical results from three problems, including the Signorini problem of the Laplacian, show the feasibility, effectiveness and efficiency of the proposed strategy.

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann–Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

The Diophantine equation is a strong research domain in number theory with extensive cryptography applications. The goal of this paper is to describe certain geometric properties of positive integral solutions of the quadratic Diophantine equation x12+x22=y12+y22(x1,x2,y1,y2>0), as well as their use in communication protocols. Given one pair (x1,y1), finding another pair (x2,y2) satisfying x12+x22=y12+y22 is a challenge. A novel secure authentication mechanism based on the positive integral solutions of the quadratic Diophantine which can be employed in the generation of one-time passwords or e-tokens for cryptography applications is presented. Further, the constructive cost models are applied to predict the initial effort and cost of the proposed authentication schemes.

Estimation of the traditional transverse electric (TE) and transverse magnetic (TM) impedances of the magnetotelluric tensor for two-dimensional structures can be decoupled from the estimation of the strike direction with significant implications when dealing with galvanic distortions. Distortion-free data are obtainable by combining a quadratic equation with the phase tensor. In the terminology of Groom–Bailey, the quadratic equation provides amplitudes and phases that are immune to twist, and the phase tensor provides phases immune to both, twist and shear. On the other hand, distortion-free strike directions can be obtained using Bahr's approach or the phase tensor. In principle, this is all that is needed to proceed to a two-dimensional (2D) interpretation. However, the resulting impedances are strike ignorant because they are invariant under coordinate system rotation, and if they are to be related to a geological strike, they must be linked to a particular direction. This is an additional ambiguity to the one of 90° arising in classic strike-determination methods, which must be resolved independently. In this work, we use the distortion model of Groom–Bailey to resolve the ambiguity by bringing back the coupling between impedances and strike in the presence of galvanic distortions. Our approach is a hybrid between existing numerical and analytical methods that reduces the problem to a binary decision, which involves associating the invariant impedances with the correct TE and TM modes. To determine the appropriate association, we present three algorithms. Two of them require optimizing the fit to the data, and the third one requires a comparison of phases. All three keep track of possible crossings of the phase curves providing a clear-cut solution. Synthetic and field data illustrate the performance of the three schemes.

This paper investigates the problem of the orbital stability of dn periodic wave solutions of the Boussinesq equation with quadratic-cubic nonlinear terms. First, the dn periodic wave solution of the studied equation is solved by using the integral method and the knowledge of elliptic functions, and the existence of smooth curves of dn periodic wave solutions with fixed period L is proved. Then the Floquet theory and Wely’s essential spectrum theorem are applied to the spectral analysis of the operator, and obtain its spectral properties. Finally, according to the ideas for proving the stability of solitary wave solutions from Benjamin and Bona et al., by overcoming the complexity caused by the quadratic-cubic nonlinear terms in the studied equation, we prove the dn periodic wave solution of the studied equation is orbitally stable under small perturbations of the L 2 norm.