Examples and Counterexamples最新文献

Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation $f\left(x\right)=0.$ The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity $m>1$. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.

Main aim of this research was to apply multiple approaches for the development of time delay functions on three highways in Bahrain, namely; Dry Dock Highway, Arad Highway and Zallaq Highway. Four equations were obtained from previous studies and two equations were, additionally, tailored for each of the three highways. The results were used to obtain two parameters that aid in design, optimum flow rate and level of service.

By using the generalization of the gamma function ($p$-gamma function: ${\Gamma }_p(.)$), we introduce a generalization of the fractal–fractional calculus which is called $p$-fractal fractional calculus. We extend the proposed operators into the symmetric complex domain, specifically the open unit disk. Normalization for each operator is formulated. This allows us to explore the most important geometric properties. Examples are illustrated including the basic power functions.

In this short note, it is propounded an extension for quadrant dependence, and shown that some of the original proprieties of this popular concept remain valid, while others are necessarily generalized. A second Borel–Cantelli lemma due to Petrov (Statist. Probab. Lett. 58: 283–286, 2002) is revisited for events enjoying this new dependence notion and demonstrated by means of simpler arguments.

This paper provides a method of finding 2-periodical solutions for the first-order non-homogeneous differential equations with piecewise constant arguments. All existence conditions are described for 2-periodical solutions and obtained explicit formula for these solutions. An example for the problem that has infinitely many solutions is constructed.

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

In this paper we give an algebraic characterization of assemblies in terms of bands of groups. We also consider substructures and homomorphisms of assemblies. We give many examples and counterexamples.

There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, $P+Q$, in relation to the eigenvalues of $P$ and $Q$. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, ${\lambda }_{min}(P+Q)$, under the conditions that matrices $P$ and $Q$ are symmetric positive semi-definite and their sum $P+Q$ is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices $P$ and $Q$, which are denoted by $R\left(P\right)$ and $R\left(Q\right)$, respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on ${\lambda }_{min}(P+Q)$ is the distinction of the case where $R\left(P\right)$ and $R\left(Q\right)$ have no principal angles between 0 and $\frac{\pi }{2}$. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace $R\left(Q\right)$, which is employed to formulate the lower bounds for ${\lambda }_{min}(P+Q)$. At last, an example that showcases the potential for improving these two lower bounds is presented.

It was conjectured by Asmiati (2018) that the generalized Petersen graph $P\left(n,k\right)$ has a locating chromatic number 4 if and only if $(noddandk=1)$ or $(n=4andk=2)$. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if $(5\le n\le 12)$ and $(2\le k\le \lfloor \frac{n-1}{2}\rfloor )$ and $(n,k)\ne (12,5)$, then ${\chi }_{{}_L}\left(P(n,k)\right)=4$.

A code $C\subseteq {\{0,1,2\}}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T\left(n\right)$ we denote the maximum cardinality of trifferent codes with length $n$. The values $T\left(5\right)=10$ and $T\left(6\right)=13$ were recently determined (Fiore et al., 2022). Here we determine $T\left(7\right)=16$, $T\left(8\right)=20$, and $T\left(9\right)=27$. For the latter case $n=9$ there also exist linear codes attaining the maximum possible cardinality 27.