Examples and Counterexamples最新文献

In this paper, several examples of novel equations and inequalities for various forms of the non-linear term in the Navier–Stokes equations (NSE) are provided. The NSE are formulated from the conservation of linear momentum and mass conservation. However, they are also known to conserve energy, angular momentum, enstrophy in 2D, helicity in 3D, among other important physical quantities (Gresho and Sani, 1998) [1]. Depending on the desired quantity of interest, there are various representations of nonlinear term $u\cdot \nabla u$ (e.g. convective, skew symmetric, rotational etc.) that can be implemented.

In this paper we consider the class of graphs which are redundantly $d$-rigid and $(d+1)$-connected but not globally $d$-rigid, where $d$ is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that $K_{5,5}$ is indeed the smallest counterexample to the conjecture.

In this paper, we extend the result of Mittal and Sharma (Bull. Korean Math. Soc. 2022) on Wedderburn decomposition (WD) of a finite semisimple group algebra. It is known that, under certain conditions, WD of a finite semisimple group algebra $F_{q}G$ can be computed from WD of its subalgebra $F_q(G/H)$, where $H$ is a normal subgroup of $G$ of prime order and $q=p^k$ for some prime $p$ and positive integer $k$. We extend this result to any normal subgroup $H$ of $G$ of order $n$.

We report a new batch of wave solutions for the coupled Drinfel’d–Sokolov–Wilson equation which represents a coupled system of nonlinear partial differential equations (NLPDEs). Firstly by making a travelling wave ansatz, we decouple the system and obtain a second-order ordinary differential equation (ODE). Thereafter we perform phase space and bifurcation analysis of that second-order ODE and proceed to construct the general solution for the envelope of the wave packet. The solutions are expressed in terms of the Jacobi elliptic sine function from which one can obtain solitary wave (particular) solutions by imposing appropriate conditions on the roots of certain quartic polynomials as discussed thereafter.

Non-existence of distributions with constant coefficient of variation(CV) is investigated within the discrete Power Series and Modified Power Series families of distributions. The development is used to revisit and comment on the problem of existence of a better but biased estimator.

Partial Group structures occur naturally in several topological and geometrical contexts. We formulate the basic definitions, and present some results and examples. The objective is to provide a step toward the development of a theory of partial groups, and to motivate the search for further applications.

In this paper, some examples of matrix generators for generalized commutative Jacobsthal quaternions were given. The generating matrices are useful tools for the number sequences satisfying a recurrence relation. They can be used for an algebraic representation and for obtaining some identities.

We explore a new view of the Rainbow Cascades Conjecture using permutations. Infinitely many new 6-satisfactory colorings are found, and evidence is provided that suggests only finitely many 7-satisfactory colorings exist.

The $A$-energy of a graph $G$, denoted by $E_A\left(G\right)$, is defined as sum of the absolute values of eigenvalues of adjacency matrix of $G$. Nikiforov in Nikiforov (2016) proved that $E_A\left(\overline{G}\right)-E_A\left(G\right)\le 2{\overline{\mu }}_1$ and $E_A\left(G\right)-E_A\left(\overline{G}\right)\le 2{\mu }_1$ for any graph $G$ and posed a problem to find best possible upper bound for $E_A\left(G\right)-E_A\left(\overline{G}\right)$, where ${\mu }_1$ and $\overline{{\mu }_1}$ are the largest adjacency eigenvalues of $G$ and its complement $\overline{G}$ respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than $-1$. The obtained results also improves some of the other existing results.