Examples and Counterexamples最新文献

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.

In the framework of population models, logistic growth and fractional logistic growth has been analyzed. In some situations the so-called Allee effect gives more accurate approximation. In this work, fractional Allee differential equation in the Caputo sense is considered. The solution is obtained by considering formal power series. Numerical computations are presented to compare the truncating series with the classical Allee differential equation.

Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase shifts as a function of the phase at which the input is received. When the pulse is weak, it is customary to linearize around zero pulse strength. The result is called the infinitesimal phase response function. These ideas have been used extensively in theoretical biology, and also in some areas of engineering. I give examples showing that the infinitesimal phase response function may predict that two oscillators, as they exchange pulses back and fourth, will converge to synchrony, yet this is false when the exact phase response function is used, for all positive interaction strengths. For short, the analogue of the Hartman–Grobman theorem that one might expect to hold at first sight is invalid. I give a condition under which the prediction derived using the infinitesimal phase response function does hold for the exact phase response function when interactions are sufficiently weak but of positive strength. However, I argue that this condition may often fail to hold.

The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.

A new strongly regular graph with parameters $(81,30,9,12)$ is found as a graph invariant under certain subgroup of the full automorphism group of the previously known strongly regular graph discovered in 1981 by J. H. van Lint and A. Schrijver.