Examples and Counterexamples最新文献

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.

The research introduces the Modified Sumudu Decomposition Method (MSDM) as a novel approach for solving non-linear ordinary differential equations. Stemming from the Sumudu Transformation (ST), proposed by Watugala in the 1990s, MSDM demonstrates its efficacy through the solution of a specific third-order non-homogeneous nonlinear ordinary differential equation. This method is particularly highlighted for its application in fluid mechanics, specifically addressing a boundary layer problem. Furthermore, the study employs Pade´ Approximants to evaluate a crucial parameter, ρ=φ''(0), and compares the results with other established methods, including Modified Laplace Decomposition Method (MLDM), Modified Adomian Decomposition Method (MADM), Modified Variational Iteration Method (MVIM), and the Homotopy Perturbation Method (HPM). The findings not only contribute to the advancement of mathematical techniques for solving complex differential equations but also provide a comparative analysis, elucidating the strengths and limitations of different methodologies. This research is anticipated to have significant implications for researchers and practitioners in the field, offering a valuable toolkit for tackling a wide range of mathematical modeling challenges.

As proved by Marchenkov and Mazzanti, every Kalmar function can be represented by arithmetic terms. We display one of such terms to represent the factorial function, and as a consequence, we get an example of an arithmetic term which represents a function whose image is the set of primes.