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Different public key exchange protocols can be employed to design a cryptosystem. The most famous example is the ElGamal cryptosystem which is based on the Diffie–Hellman key establishment to do encryption and decryption. In the same fashion, we propose a public key cryptosystem using a public key exchange protocol. This variant uses the polynomials over noncommutative matrix ring as underlying work structure. The useful feature of the proposed cryptosystem is that it provides favorable security because of use of the generalized decomposition problem over the noncommutative ring of matrices. The issues regarding the choice of parameters and platform are discussed. Further, a brief note on security analysis of the proposed cryptosystem is also presented.

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space.

To investigate the model of fatigue crack growth rate da/dN based on crack tip opening displacement (CTOD) as the driving parameter, the variation ranges $\mathrm{\Delta }{\delta }_M$ by linear elastic fracture mechanics and $\mathrm{\Delta }{\delta }_G$ by geometric principle of CTOD were deduced. The fitting relationships of $\mathrm{\Delta }{\delta }_M-da/dN$ and $\mathrm{\Delta }{\delta }_G-da/dN$ were attained through da/dN experiment by taking 17Cr16Ni2 stainless steel, and their discrepancies were compared. The results showed that at the early and middle periods, the relationships of $\mathrm{\Delta }{\delta }_M-da/dN$ and $\mathrm{\Delta }{\delta }_G-da/dN$ could be expressed by Paris law, and da/dN driven by $\mathrm{\Delta }{\delta }_M$ was faster than that by $\mathrm{\Delta }{\delta }_G$ notably; at the later period, the experimental data of $\mathrm{\Delta }{\delta }_M-da/dN$ and $\mathrm{\Delta }{\delta }_G-da/dN$ deviated from the fitted Paris curves and were above them with being crossed with each other; in the total variation range of CTOD, it was better to fit the relationships of $\mathrm{\Delta }{\delta }_M-da/dN$ and $\mathrm{\Delta }{\delta }_G-da/dN$ by polynomial method rather than Paris law.

Excitable media represent an important class of spatio-temporal systems which can be studied using cellular automata models. In the present study examples of excitable media behavior generated using simple cellular automata models are introduced. A mutual information algorithm is then derived to determine the neighborhood, the excitation threshold, and the number of excitation states. Based on this information two methods of identifying the rule which describes the excitable media pattern using a multimodel and a polynomial model are introduced. The results are illustrated using simulated examples and real data from a Belousov–Zhabotinskii experiment.

In this paper we give sufficient conditions under which there exists a polytope containing all compact invariant sets for one class of natural polynomial Hamiltonian systems. As an example, we compute bounds of a polytope with this property for a class of natural polynomial Hamiltonian systems with a cubic potential and two degrees of freedom.

This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into . The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.

In this paper, cubic polynomial and nonpolynomial splines are developed to solve solutions of 10th- and 12th-order nonlinear boundary value problems (BVPs). Such types of BVPs occur when a consistent magnetized force field is applied crosswise the fluid in the substance of gravitational force. We will amend our problem into such a form that converts the system of $10$th- $\&$$12$th-order BVPs into a new system of $2$nd-order BVPs. The appropriate outcomes by using CP Spline and CNP Spline are compared with the exact root. To show the efficiency of our results, absolute errors calculated by using CP Spline and CNP Spline have been compared with other methods like differential transform method, Adomian decomposition method, variational iteration method, cubic B-spline, homotopy perturbation method, $5$th- and $6$th-order B-spline and our results are very encouraging. Graphs and tables are also presented in the numerical section of this paper.

An algorithm that provides direct, efficient ND polynomial factorization is presented to solve the numerical issues that arise during the direct inversion of helium atom scattering (HAS) diffraction spectra. For an n-variate polynomial the algorithm directly deflates the polynomial to n-single variable equations by evaluating the ratio of pairs of polynomial coefficients. Error estimation of the coefficients of the 1D polynomials is then performed automatically using standard 1D search techniques. The effectiveness of the technique is demonstrated against bi- and trivariate polynomials and an approximate range of validity for error prone polynomials is demonstrated. To demonstrate the effectiveness of the technique, HAS diffraction spectra for the low coverage (2 × 1)-H/Pd(311) system have been analyzed using direct inversion and have revealed that H binds in a three-fold hollow site.

A Fischer pair (FP) for a vector space E is a pair (u, v) of linear maps on E, not necessarily everywhere defined, such that E=ker u⊕Im v (Fischer decomposition). Thus, in particular, every densely defined closed operator u on a Hilbert space E forms a Fischer pair together with its adjoint u*, whenever Im u, or equivalently, Im u* is closed since then Im u*=ker u^⊥. The question of when a given pair of maps (u, v) is a FP is related to the well-posedness of the (abstract) Cauchy–Goursat problem for u, v in E.

We establish some Fischer pairs, for spaces that are built up by homogeneous Hilbert–Schmidt polynomials on a Hilbert space, consisting of differential and multiplication operators. In particular we study Fischer decompositions of the space of entire functions of Hilbert–Schmidt type. As a basis we generalize Fischers theorem for homogeneous polynomials in n variables to Hilbert–Schmidt polynomials.

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal$\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.

For a field $k$ of characteristic $0$, we present an algorithm for deciding if a morphism $\varphi :k[X_1,\dots ,X_m]\to k[X_1,\dots ,X_m]$ has an inverse. The algorithm also shows how to find the inverse when it exists.

Additive decompositions over finite fields were extensively studied by Brawely and Carlitz. In this paper, we study the additive decomposition of polynomials over unique factorization domains.

Research on refinable functions in wavelet theory is mostly focused to localized functions. However it is known, that polynomial functions are refinable, too. In our paper we investigate on conversions between refinement masks and polynomials and their uniqueness.

In this paper, we discuss the relationship between the generation polynomial of a linear sequence and its decimation sequence over the finite field 𝔽₂. Then we propose an upper bound of the number of terms in the generation polynomial of a decimation sequence of a linear sequence whose generation polynomial is trinomial. Finally, we suggest a method to calculate the terms of a decimation polynomial and their number directly, which can be used in the construction of irreducible polynomials with a controlled number of terms, when the source trinomial and the decimation distance satisfy a certain condition.

We use the Smith normal form of the augmented degree matrix to estimate the number of rational points on a toric hypersurface over a finite field. This is the continuation of a previous work by Cao in 2009.

Let 𝔽_q be the finite field of q elements and f be a nonzero polynomial over 𝔽_q. For each b ϵ 𝔽_q, let N_q(f = b) denote the number of 𝔽_q-rational points on the affine hypersurface f = b. We obtain the formula of N_q(f = b) for a class of hypersurfaces over 𝔽_q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2ⁿ + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xⁿ + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽_p, and in that situation, we show 1 ∈ S for all primes p.

We discuss the following question: How close to each other can two distinct roots of an integer polynomial be? We summarize what is presently known on this and related problems, and establish several new results on root separation of monic, integer polynomials.

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.