Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

Error analysis in stochastic solutions of population balance equations

Stochastic simulation of population balance equations (PBEs) is robust and flexible; however, it exhibits intrinsic stochastic errors which decreases at a very slow rate when increasing the computational resolution. Generally, these stochastic methods can be classified into two groups: (i) the classical Gillespie method and (ii) weighted flow algorithm. An analytical relationship is derived for the first time to connect the variances in these two groups. It also provides a detailed analysis of the resampling process, which has not been given appropriate attention previously. It is found that resampling has a profound effect on the numerical precision. Moreover, by comparing the time evolutions between systematic errors (i.e., errors in the mean value) and stochastic errors (i.e., variances), it is found that the former grows considerably faster than the latter; thus, systematic errors eventually dominate. The present findings facilitate the choice of the most suitable stochastic method for a specific PBE a priori in order to balance numerical precision and efficiency.     

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.     

Self-reciprocal and self-conjugate-reciprocal irreducible factors of xn − λ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.     

Low dimensional ordering on a lattice model

A simple d-dimensional lattice model is proposed, incorporating some degree of frustration and thus capable of describing some aspects of molecular orientation in covalently bound molecular solids. For d = 2 the model is shown to be equivalent to the standard two-dimensional Ising model, while for d > 2 it describes a peculiar transition from an isotropic high temperature phase to a low-dimensional anisotropic low temperature state. A general mean field analysis is presented and compared to some exact limit properties.     

Reflection of plane waves at the free surface of a monoclinic thermoelastic solid half-space

Lord–Shulman and Green–Lindsay theories of generalized thermoelasticity are applied to study the reflection from a thermally insulated stress-free thermoelastic solid half-space of monoclinic type. A particular model is chosen for the numerical computations of reflection coefficients. Effects of anisotropy and relaxation times are observed on reflection coefficients.     

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.     

A new class of generalized almost perfect nonlinear power function

As a generalization of almost perfect nonlinear functions, generalized almost perfect nonlinear (GAPN) functions defined over finite fields of odd characteristic have also important applications in cryptography and finite geometry. In this paper, we mainly focus on GAPN power functions, and a new class of such function (up to EA equivalence) is obtained. Furthermore, using our method, almost all the known GAPN power functions can be confirmed in a simple way.