Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework

<abstract><p>Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.</p></abstract>

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

Representing squares by cubic forms

Let [Formula: see text] be an arbitrary cubic form. In this paper, we show that for non-homogeneous cubic equations like [Formula: see text], there exists a non-trivial solution when [Formula: see text].

Linearized Homotopy Perturbation Method for Two Nonlinear Problems of Duffing Equations

In the paper, we solve two nonlinear problems related to the Duffing equations in space and in time. The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution. The second problem is an unconventional periodic problem of Duffing equation in time to determine period and periodic solution. To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight factor, such that the Duffing equation is linearized as the Mathieu type differential equation. The constant preceding the displacement is expanded in powers of homotopy parameter and the coefficients are determined to avoid secular solutions appeared in the derived sequence of linear differential equations. Consequently, after setting the homotopy parameter equal to unity and solving the amplitude equation, the higher order bifurcated solutions can be derived explicitly. For the second problem, we can determine the period and periodic solution in closed-form, which are very accurate. For the sake of comparison the results obtained from the fourth-order Runge-Kutta numerical method are used to evaluate the presented analytic solutions.

Topologically non-trivial solution in a dissipative φ4 model with Lorentz-invariance violation

In this paper a (1+1)-dimension equation of motion for φ4-theory is considered for the case of simultaneously taking into a account of the processes of dissipation and violation the Lorentz-invariance. A topological non-trivial solution of one-kink type for this equation is constructed in an analytical form. To this end, the modified direct Hirota method for solving the nonlinear partial derivatives equations was used. A modification of the method lead to special conditions on the parameters of the model and the solution.

Alternative Cauchy equation in three unknown functions

In this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial $${\mathcal {C}}^{\infty }$$ C ∞ solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented.

Warped AdS2 and SU(1, 1|4) symmetry in Type IIB

We investigate the existence of solutions with 16 supersymmetries to Type IIB supergravity on a spacetime of the form AdS2× S5× S1 warped over a two-dimensional Riemann surface Σ. The existence of the Lie superalgebra SU(1, 1|4) ⊂ PSU(2, 2|4), whose maximal bosonic subalgebra is SO(1, 2)⊕SO(6)⊕SO(2), motivates the search for half-BPS solutions with this isometry that are asymptotic to AdS5×S5. We reduce the BPS equations to the Ansatz for the bosonic fields and supersymmetry generators compatible with these symmetries, then show that the only non-trivial solution is the maximally supersymmetric solution AdS5× S5. We argue that this implies that no solutions exist for fully back-reacted D7 probe or D7/D3 intersecting branes whose near-horizon limit is of the form AdS2× S5× S1× Σ.

Lyapunov - Кrasovskii functionals for homogeneous systems with multiple delays

In this article, explicit constructions of Lyapunov - Кrasovskii functionals are proposed for homogeneous systems with multiple constant delays and homogeneity degree of the right- hand sides strictly greater than one. The constructions are based on the Lyapunov functions suitable for the analysis of corresponding systems with all delays equal to zero. The letter systems are assumed to be asymptotically stable. It is proved that the proposed functionals satisfy the conditions of the Кrasovskii theorem, and hence it allows us to establish the asymptotic stability of the trivial solution for arbitrary values of delays. The functionals are applied to the estimation of the attraction region of the trivial solution.

The Solvability of Polynomial Pell’s Equation

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

Lightweight Public Key Encryption With Equality Test Supporting Partial Authorization in Cloud Storage

Public key encryption with equality test (PKEET) can check whether two ciphertexts are encrypted from the same message or not without decryption. This attribute enables PKEET to be increasingly utilized in cloud storage, where users store their encrypted data on the cloud. In traditional PKEET, the tester is authorized by the data receiver to perform equality test on its ciphertexts. However, the tester can only test one ciphertext or all ciphertexts of one receiver with one authorization. It means that the receiver cannot adaptively authorize the test right of any number of ciphertexts to the tester. A trivial solution is authorizing one ciphertext each time and repeating multiple times. The corresponding size of trapdoor in this method is linear with the number of authorized ciphertexts. This will incur storage burden for the tester. To solve the aforementioned problem, we propose the concept of PKEET supporting partial authentication (PKEET-PA). We then instantiate the concept to a lightweight PKEET-PA, which achieves constant-size trapdoor. Besides, we prove the security of our PKEET-PA scheme against two types of adversaries. Compared with other PKEET schemes that can be used in trivial solution, our PKEET-PA is more efficient in receivers’ computation and has lower trapdoor size.