Examples and Counterexamples最新文献

We consider a convex combination of two classes of Lotka–Volterra operators defined on 2-dimensional simplex. Earlier, the dynamics of a particular case of the considered operators has been investigated. However, its bijective property was not studied. In this paper, we are able to establish that such maps are homeomorphism of the simplex.

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number $A_{12}$ of codewords of minimum weight 12, and raised the question about the existence of codes for other values of $A_{12}$. In this note, we use symmetric 2-$(47,23,11)$ designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of $A_{12}$.

In this paper, we determine some degree-based topological indices, such as the Sombor index, the first and second Zagreb indices, the forgotten topological index, the Narumi–Katayama index, the first and second multiplicative Zagreb indices, the atom-bond connectivity index, and eccentricity-based topological indices such as total eccentricity, the first and second Zagreb eccentricity indices, and the eccentricity connectivity index of the zero divisor graph with vertex set non-zero zero divisors of the reduced ring of the direct product of three finite fields.

With multiple time delays, we investigated a Caputo fractional order dynamical system involving susceptible, exposed, infected, and recovered individuals. Positivity and boundedness are also theoretically demonstrated using Laplace transform and Mittag-Leffler function. The stability of the disease-free and epidemic equilibrium points has been studied for both delayed and non-delayed model. For generating numerical solutions to the model system, we used the Adam-Bashforth-Moulton predictor-corrector technique. With the help of MATLAB (2018a), we were able to conduct graphical demonstrations and numerical simulations. The system displays Hopf bifurcation and the solutions are no longer periodic beyond a certain threshold value of the time delay parameters.

The Lie symmetry analysis of a power law in-time coefficients Korteweg–de Vries (KdV) equation is performed with the aim of specifying the model parameters (powers of $t$). That is, the symmetries of the resulting subclasses of the underlying equation are obtained. Further, symmetry reductions and some exact solutions are obtained.

A methodology is proposed for identifying brain tumors by dividing the database into four parts. The results obtained from the study of sample specimens for each type of brain tumor showed a high degree of similarity in recognition. This methodology can be applied in healthcare facilities to improve the accuracy of disease diagnosis.

We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.

By means of a recurrence, we provide a family of $c$-cyclic graphs, $c\ge 0$, whose Kirchhoff index is $\Theta ({\left|V\right|}^{2}log\left|V\right|)$.

We present an optimization problem in infinite dimensions which satisfies the usual second-order sufficient condition but for which perturbed problems fail to possess solutions.

Stochastic Gradient Descent (SGD) is a widely used, foundational algorithm in data science and machine learning. As a result, analyses of SGD abound making use of a variety of assumptions, especially on the noise behavior of the stochastic gradients. While recent works have achieved a high-degree of generality on assumptions about the noise behavior of the stochastic gradients, it is unclear that such generality is necessary. In this work, we construct a simple example that shows that less general assumptions will be violated, while the most general assumptions will hold.