As establishing a foundation for a new line of investigations on threshold secret sharing schemes with geometric algebra (GA), we propose a variation of a well-known threshold secret sharing scheme introduced by Adi Shamir in 1979, a cryptographic solution that allows a secret input to be divided into multiple random shares which are then sent, each one, to distinct parties. The computation of these shares is done so that there are proper subsets of these shares that allow reconstructing the secret input using polynomial interpolation. To reconstruct the secret input, Shamir’s scheme requires a minimum number of shares, smaller than the total number of shares, referred to as a threshold. Any number of shares smaller than the threshold reveals nothing about the input secret. The random shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant replaces the algebra in which the original secrets lie from integers to GA while preserving fundamental properties in Shamir’s scheme, such as perfect secrecy and idealness (both secret and random shares are members of the same space). As a direct result, any application in GA dealing with multivectors can immediately add threshold security using our scheme. Non-GA applications can also benefit from our results by using multivectors as a vessel for sharing multiple secrets at once.

Different ways of representing the group $SU(3)$ within a Geometric Algebra approach are explored. As part of this we consider characteristic multivectors for $SU(3)$, and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime --- the STA. As part of this, a representation of $SU(3)$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by G\"unaydin and G\"ursey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme, and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing $SU(3)$, such as $G_2$ and $SU(8)$, are also considered.

There are different dynamic characteristics in each stage of rail transit network development. It can provide guidance for rail transit network planning and phased construction,Evaluating the evolution degree of urban rail transit network and accurately dividing the evolution stages. Starting from the allometric growth relationship between transfer nodes and ordinary nodes in urban rail transit network, this paper defines the evolution level of rail transit network and the growth rate difference between transfer and ordinary nodes, and deduces the logistic dynamic equation of urban rail transit network evolution level with the help of mathematical derivative and logarithmic relationship, This paper reveals the dynamic law of the evolution process of urban rail transit network from the theoretical level, analyzes the evolution dynamic process on this basis, finds the Threshold for phasing the evolution stage, and divides it into four evolution stages. Finally, the theoretical derivation results are verified with the evolution data of Beijing Rail Transit Network (1984-2020) for 45 years. The verification results are highly consistent with the theoretical derivation results, which proves the correctness of the theoretical model and its practical guiding significance for the construction of rail transit network.

In this paper we present a new discrete retarded Gronwall-Bellman type inequality. As applications, the dynamics of some delay difference equations are studied. First, the asymptotic behavior of solutions for scalar difference equation $\Delta x(n)=-a(n)x(n)+B(n,x_n)$ is discussed, and some new criterion on the asymptotic stability of the zero solution are obtained under weaker assumptions. Then the dissipativity of a nonautonomous delay difference system with superlinear nonlinearities is investigated. By using the inequalities established here, it is shown that the discrete set-valued process generated by the system possesses a unique global pullback attractor.

In this work, we study the very weak solution for the stationary exterior Stokes problem with Navier slip boundary condition. We try to investigate some results of the existence and the uniqueness of solutions related to this problem in an exterior domain in $L^p-$theory where $1

In this article, we consider the blowup phenomenon of smooth solutions to the isentropic compressible quantum hydrodynamic model(QHD) with the initial density of compact support in arbitrary space dimensions. This result is an evolution of Xin’s work

We show the existence of global-in-time solutions to an initial-boundary value problem in one space dimension of a phase-field model on the martensitic phase transformations, which consists of the equations of degenerate and a nondegenerate nonlinear parabolic equation of second order.

We consider the equations of motion of a bar, with given density, infinite in both directions, subjected to longitudinal vibrations under the action of an external load, and a stress-strain relation represented by a fractional order operator.  Using three types of fractional operators, the initial-boundary value problems associated with the described phenomenon are posed and solved. Through the bivariate Mittag-Leffer function, which has been recently introduced, we find the fundamental solution of these problems and calculate their moments.

A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.

In this paper, we investigate nonhomogeneous incompressible Navier-Stokes-Landau-Lifshitz system in 2-D. This system consists of Navier-Stokes equations coupled with Landau-Lifshitz-Gilbert equation, an evolutionary equation for the magnetization vector. We establish a blowup criterion for the two-dimensional incompressible Navier-Stokes-Landau-Lifshitz system with finite positive initial density.

The aim of this article is to investigate the well-posedness and stability problems of the so-called Kawahara equation under the presence of an interior delayed damping. The system is shown to be well-posed. Furthermore, we prove that the trivial solution is exponentially stable in spite of the delay effect. Specifically, local and semi-global stability results are established according to the properties of the spatial distribution of the delay term.

A document by Xueqin Peng. Click on the document to view its contents.

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

A document by Yuk Leung. Click on the document to view its contents.

In this paper, to investigate the synthetic effect of PrEP (pre-exposure prophylaxis) and ART (antiretrovial therapy) on HIV transmission among MSM (men who have sex with men) in heterogenous environment, an realistic HIV epidemic model with spatial diffusion is established. Here, HIV infectious people are divided into three immunity based compartments, i.e., CD4+ T cell count less than 350, between 350 and 500, and more than 500, respectively. The basic reproduction number $R_0$ is established and proved as a threshold parameter: The global asymptotic stability of the disease-free steady state holds for $R_0<1$, and the disease will be present if $R_0>1$. Considering the substantial advantages of PrEP and ART in controlling HIV transmissions among MSM, the optimal control problem is presented for the case of positive constant diffusion coefficients, which minimize the total population of susceptible individual and HIV infected individual, the cost of PrEP and ART thearpy. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct an optimal control case study where model parameters are estimated from the demographic and epidemiological data from China. This work suggests: (1) Spatial factors cannot be ignored during the HIV intervention; (2)Taking the PrEP intervention measure for HIV transmissions among MSM as early as possible will help to improve the control efficiency and reduces its cost; (3) Reducing the PrEP drug costs will promote the efficiency of PrEP treatment in preventing the spread of HIV among MSM.

In the present paper, we study the existence of nontrivial solutions of the following one-dimensional fractional Schr\“{o}dinger equation $$ (-\Delta)^{1/2}u+V(x)u=f(x,u), \ \ x\in \R, $$ where $(-\Delta)^{1/2}$ stands for the $1/2$-Laplacian, $V(x)\in \mathcal{C}(\R, (0,+\infty))$, and $f(x,u):\R\times\R\to \R$ is a continuous function with an exponential critical growth. Comparing with the existing works in the field of exponential-critical-growth fractional Schr\”{o}dinger equations, we encounter some new challenges due to the weaker assumptions on the reaction term $f$. By using some sharp energy estimates, we present a detailed analysis of the energy level, which allows us to establish the existence of nontrivial solutions for a wider class of nonlinear terms. Furthermore, we use the non-Nehari manifold method to establish the existence of Nehari-type ground state solutions of the one-dimensional fractional Schr\”{o}dinger equations.

This paper investigates the boundary finite-time stabilization of fractional reaction-diffusion systems (FRDSs). First, a distributed controller is designed, and sufficient conditions are obtained to ensure the finite-time stability (FTS) of FRDSs under the designed controller. Then, a boundary controller is presented to achieve the FTS. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are presented to ensure the FTS of FRDSs via the designed boundary controller. The effect of diffusion term of FRDSs on the FTS is also investigated. Both Neumann and mixed boundary conditions are considered. Moreover, the robust finite-time stabilization of uncertain FRDSs is studied when there are uncertainties in the system’s coefficients. Under the designed boundary controller, sufficient conditions are presented to guarantee the robust FTS of uncertain FRDSs. Finally, numerical examples are presented to verify the effectiveness of our theoretical results.

This article proposes an adaptive neural output feedback control scheme in combination with state and disturbance observers for uncertain fractional-order nonlinear systems containing unknown external disturbance, input saturation and immeasurable state. The radial basis function neural network (RBFNN) approximation is used to estimate unknown nonlinear function, and a state observer as well as a fractional-order disturbance observer is developed simultaneously by using the approximation output of the RBFNN to estimate immeasurable states and unknown compounded disturbances, respectively. Then, a fractional-order auxiliary system is constructed to compensate the effects caused by the saturated input. In addition, by introducing a dynamic surface control strategy, the tedious analytic computation of time derivatives of virtual control laws in the conventional backstepping method is avoided. The proposed method guarantees that the boundness of all signals in the closed loop system and the tracking errors converge to a small neighbourhood around the origin. Finally, two examples are provided to verify the effectiveness of the proposed control method.