Using Normal Form of Nilpotent Matrices over Finite Fields to Construct Cartesian Authentication Codes

Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the probabilities PI and PS, of a successful im-personation attack and of a successful substitution attack respectively, of these codes are also computed.

Interval finite difference method for steady-state temperature field prediction with interval parameters

A new numerical technique named interval finite difference method is proposed for the steady-state temperature field prediction with uncertainties in both physical parameters and boundary conditions. Interval variables are used to quantitatively describe the uncertain parameters with limited information. Based on different Taylor and Neumann series, two kinds of parameter perturbation methods are presented to approximately yield the ranges of the uncertain temperature field. By comparing the results with traditional Monte Carlo simulation, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed method for solving steady-state heat conduction problem with uncertain-but-bounded parameters.

AN ANZAHL THEOREM IN SINGULAR PSEUDO-SYMPLECTIC GEOMETRY OVER FINITE FIELDS

This paper obtains formulas for the number of subspaces of a given type contained in a given subspace in singular pseudo-symplectic geometry.

SOME PRIMITIVE POLYNOMIALS OVER FINITE FIELDS

This paper proves that if qn is large enough, for each element a and primitive element b of Fq, there etists a primitive polynomial of degree n ≥5 over the finite field Fq having a as the coefficient of xn-1 and b as the constant term. This proves that if qn is large enongh, for each element a ∈Fq, there exists a primitive polynomial of degree n ≥ 5 over Fq having a as the coefficient of x.

Simulation of Temperature and Flow Field by Three Dimension Finite ElementMethod for Castex Process of AS Wire

Castex of AS wire is a new technology of near net shape. To study the variation of temperature and velocity of liquid (or semisolid) aluminum during dynamic solidification the numerical simulation was carried out with the theory of heat-transfer and hydrodynamics by means of 3-dimensional finite element method. From simulation results, it is found that the variation of temperature and velocityis mainly influenced by the casting temperature of aluminum, rotating speed of Castex wheel and flow of cooling water. Among theseinfluencing factors, the casting temperature distributes most to the length of liquid phase metal. Moreover, the faster the metal solidifies,the higher the metal there moves with the overall trend of descending from the bottom of the wheel to the shoe wall as well as from sidewalls to the center of wheel groove. In comparison with the practical value, the simulation is reliable.

On the Number of Solutions of Some Type Equations in Finite Fields

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.

Construction of PBIB Desings by Using Subspace of Vector Space over Finite Fields

A new transitivity theorem of the general linear group GLn(Fq)is proved. A kind of new PBIB desings is constructed.

Some New Results about Trigonometry in Finite Fields

In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an extension of K, if K⊆F or there exists a monomorphism f: K→F. Recall that , F[x] is the ring of polynomial over F. If (means that F is an extension of K), an element is algebraic over K if there exists such that f(u) = 0 (see [1]-[4]). The algebraic closure of K in F is , which is the set of all algebraic elements in F over K.

An Infinite Family of Number Fields with No Inert Primes

The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZk decomposes into a product of prime ideals of Zk as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each fθ(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05

Three-Dimensional Phase Field Simulations of Hysteresis and Butterfly Loops by the Finite Volume Method

Three-dimensional simulations of ferroelectric hysteresis and butterfly loops are carried out based on solving the time dependent Ginzburg-Landau equations using a finite volume method. The influence of externally mechanical loadings with a tensile strain and a compressive strain on the hysteresis and butterfly loops is studied numerically. Different from the traditional finite element and finite difference methods, the finite volume method is applicable to simulate the ferroelectric phase transitions and properties of ferroelectric materials even for more realistic and physical problems.

Approximate Synchronization of Multi-Agent Systems over Finite Fields

In this paper,the approximate synchronization of leader-follower multiagent systems(MASs) over finite fields is studied in regard to local and global synchronization.First,the approximately synchronous state set(ASSS) is obtained.Second,combined with ASSS and transient periods,some criteria for the local and global approximate synchronization of systems are given.Moreover,the algorithms for calculating the maximum approximately synchronous basin(MASB) and the maximum control invariant set(MCIS) are presented.Third,the global approximate synchronization of the system is achieved by designing the state feedback control,and a design algorithm of the controller using the truth matrix method is proposed.Moreover,the results of approximate synchronization are degenerated to complete synchronization.Last,two examples are shown to demonstrate the results of this paper.

Dielectric Constant Calculation of Poly(vinylidene fluoride)Based on Finite Field and Density Functional Theory

In this study,we proposed a novel method that integrates density functional theory(DFT)with the finite field method to accurately estimate the polarizability and dielectric constant of polymers.Our approach effectively accounts for the influence of electronic and geometric conformation changed on the dielectric constant.We validated our method using polyethylene(PE)and polytetrafluoroethylene(PTFE)as benchmark materials,and found that it reliably predicted their dielectric constants.Furthermore,we explored the impact of conformation variations in poly(vinylidene fluoride)(PVDF)on its dielectric constant and polarizability.The resulting dielectric constants ofα-andγ-PVDF(3.0)showed excellent agreement with crystalline PVDF in experiments.Our findings illuminate the relationship between PVDF’s structural properties and its electrical behavior,offering valuable insights for material design and applications.

Zeros of Complete Symmetric Polynomials over Finite Fields

Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author improves Wan-Zhang's bound from three aspects.The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial.And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.

Thermo-hydro-mechanical-air coupling finite element method and its application to multi-phase problems

In this paper, a finite element method （FEM）-based multi-phase problem based on a newly proposed thermal elastoplastic constitutive model for saturated/unsaturated geomaterial is discussed. A program of FEM named as SOFT, adopting unified field equations for thermo-hydro-mechanical-air （THMA） behavior of geomaterial and using finite element-finite difference （FE-FD） scheme for so/l-water-air three-phase coupling problem, is used in the numerical simulation. As an application of the newly proposed numerical method, two engineering problems, one for slope failure in unsaturated model ground and another for in situ heating test related to deep geological repository of high-level radioactive waste （HLRW）, are simulated. The model tests on slope failure in unsaturated Shirasu ground, carried out by Kitamura et al. （2007）, is simulated in the framework of soil-water-air three-phase coupling under the condition of constant temperature. While the in situ heating test reported by Munoz （2006） is simulated in the same framework under the conditions of variable temperature hut constant air pressure.

ALGORITHM DESIGN AND IMPLEMENTATION OF FINITE WAVELET GROW TREE

Based on grow tree composite model, Finite Field Wavelet Grow Tree (FW-GT) was proposed in this paper. FW-GT is a novel framework to be used in data encryption enhancing data security. It is implemented by replacement operator and wavelet operator. Forward integration and inverse decomposition of FW-GT are performed by replacement, inverse wavelets and its corresponding replacement, wavelet transforms. Replacement operator joined nonlinear factor, wavelet operator completed data transformation between lower dimensional space and higher dimensional space. FW-GT security relies on the difficulty of solving nonlinear equations over finite fields. By using FW-GT, high security of data could be obtained at the cost of low computational complexity. It proved FW-GT algorithm’s correctness in this paper. The experimental result and theory analysis shows the excellent performance of the algorithm.

Finite dimensional factor algebras of F_(2)[X_(1),…,X_(n)]and their fixed point subalgebras

Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field F2 (with respect to all F2-algebra automorphisms) are fully described.

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.

On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.

A Class of Quadratic Matrix Equations over Finite Fields

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials.We also find a generating set for the vanishing ideal of these orbits.

Normal Bases and Their Dual-Bases over Finite Fields

In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.