Narrow Results

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 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.

For any positive integer $k\ge 1$, let $p_{1/k}(n)$ be the number of solutions of the equation $n=[\sqrt[k]{a_1}]+\cdots +[\sqrt[k]{a_d}]$ with integers $a_1\ge \cdots \ge a_d\ge 1$, where $[t]$ is the integral part of real number $t$. Recently, Luca and Ralaivaosaona gave an asymptotic formula for $p_{1/2}(n)$. In this paper, we give an asymptotic development of $p_{1/k}(n)$ for all $k\ge 1$. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

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.

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.

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.

Let f be a polynomial in n variables over the finite field 𝔽_q and N_q(f) denote the number of 𝔽_q-rational points on the affine hypersurface f = 0 in 𝔸ⁿ(𝔽_q). A φ-reduction of f is defined to be a transformation σ : 𝔽_q[x₁, …, x_n] → 𝔽_q[x₁, …, x_n] such that N_q(f) = N_q(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on N_q(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for N_q(f).

On the ABC conjecture, we get an asymptotic estimate for the number of squarefree values of a polynomial at prime arguments. A key tool in our argument is a result by Tao and Ziegler (improving a previous result by Green and Tao) concerning arithmetic progressions of primes.

Denote by ${\tau }_k(n)$, $\omega (n)$ and ${\mu }_2(n)$ the number of representations of $n$ as a product of $k$ natural numbers, the number of distinct prime factors of $n$ and the characteristic function of the square-free integers, respectively. Let $[t]$ be the integral part of real number $t$. For $f=\omega ,2^{\omega },{\mu }_2,{\tau }_k$, we prove that

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.

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.

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(gcd(f(n),g(n)))$ over all integers $n$. It is known that $S\le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $p{s}^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.

Stochastic weather generators are widely used to produce large ensembles of climate time series for assessing risk-based environmental impacts. However, they often perform poorly at generating extreme values since the fitting of traditionally used distribution functions is limited by short historical records. In such cases, extreme values are generated by extrapolating the fitted distributions far outside of observations, and can result in values outside of the physically possible range. This work uses a curve-fitting approach constrained on the probable maximum precipitation (PMP) to allow for the generation of realistic precipitation over the entire range of daily precipitation amounts. The method differs from the traditional parametric approach which assumes that the daily precipitation follows a specific probability distribution. Instead, the curve-fitting approach uses a second-degree polynomial to fit the Weibull experimental frequency distribution of observed daily precipitation. In this process, the PMP is specifically represented with its associated probability of occurrence, thus ensuring the realistic representation of extreme precipitation events. The proposed algorithm is compared to three distribution functions (of varied complexity) for simulating daily precipitation amounts at 35 stations dispersed across central and southern Quebec, Canada. The curve-fitting approach is presented in two versions: with and without constraint on the PMP. The results show that compound distribution functions perform better than their single distribution counterparts at representing the overall distribution of daily precipitation amounts, especially when simulating the upper tail. The unconstrained curve-fitting approach consistently performs better than all of the distribution functions with respect to preserving the statistical characteristics (e.g., mean, standard deviation and overall distribution) of daily precipitation amounts. Constraining the second-degree polynomial to the PMP is an effective way to generate the entire range of daily precipitation amounts with no risk of generating physically impossible values for events with extremely small probability. However, its overall performance is slightly less than that of its unconstrained counterpart.