Narrow Results

In this paper, we obtain bases for the center and cocenter of the q-Schur superalgebra $S(1|1,r)$, in terms of its relative norm basis and normalized basis.

Let $G$ be a finite $p$-group. Then $G$ is said to be a $CZ$-group if $H_G\le Z(G)$ for every nonnormal subgroup $H$ of $G$. In this paper, we study the $CZ$-group $G$ and get $c(G)\le 3$. It is proved that $\mathrm{\text{exp}}(G^{\prime })=p$ if $c(G)=2$ and $\left|G\right|\le p^5$ if $c(G)=3$.

The existence of maximal subrings in certain noncommutative rings, especially in rings which are integral over their centers, are investigated. We prove that if a ring $T$ is integral over its center, then either $T$ has a maximal subring or $T/J(T)$ is a commutative Hilbert ring with $|\mathrm{\text{Max}}(T)|\le 2^{{\aleph }_0}$ and $|T/J(T)|\le 2^{2^{{\aleph }_0}}$. We observe that if $T$ is an algebraic $K$-algebra over a field $K$, then either $T$ has a maximal subring or $U(T)$ is integral over the prime subring of $T$. If $T$ is a left Artinian ring which is integral over its center, then we prove that either $T$ has a maximal subring or $T$ is countable and is integral over its prime subring. We see that if $T$ is a left Noetherian ring which is integral over its center, then either $T$ has a maximal subring or $\left|T\right|\le 2^{{\aleph }_0}$. We prove that if $T$ is a domain which is integral over its center $C$ and $J(C)=0$, then either $T$ has a maximal subring or $T$ is an integral domain. If $T$ is a reduced ring which is integral over its center and the center of $T$ is a Hilbert ring, then we show that either $T$ has a maximal subring or $T$ is commutative. We see that if a ring $T$ is integral over its center and $R$ is a subring of $T$ with $J(T)\cap R\subseteq J(R)$, then either $T$ has a maximal subring or $J(R)=J(T)\cap R$ and $U(R)=U(T)\cap R$. Finally, we prove that if $T$ is direct product of an infinite family of rings ${\{T_i\}}_{i\in I}$ and each $T_i$ is integral over its center, then $T$ has a maximal subrings.

The main idea of this framework is that it is capable of overcoming the drawbacks that are always linked with conventional cloud-based methods. Computation and storage resources in Internet of Things (IoT) networks are distributed closer to the network’s edge; therefore, the amount of data processed in real time is reduced. By decreasing the distance of the data transfers, less bandwidth is used. Structural problems, data safety, interoperability, and resource allocation-related matters denote challenges preventing the successful implementation of those ideas. The proposed work is the cloud-enabled fog computing framework (C-FCF) in data center systems based on the IoT platform. It brings cloud computing to a new level of scalability by compounding the following: Scalable architecture, uniform communication interfaces, the dynamic algorithms that allocate resources, and the data-centered approach, on the one hand, and strong security protocols, on the other. The wireless sensor network (WSN) approach to this technology represents a greater versatility of this system as it can be applied to perform different tasks in various industries like smart cities, healthcare, transportation, and industrial automation services. The application of the given services illustrates C-FCF’s capability of creating innovation, modeling effectiveness, and uncovering potential for integration within the IoT network. The virtual simulation analysis is necessary to validate C-FCF’s effectiveness in real-life scenarios. The simulations provide evidence of features, including low latency, efficient resource utilization, and overall system performance, which underlines the practical aspects of applying C-FCF in different IoT settings. Developing this advanced computing architecture, which can surpass the limitations of conventional technology and the ability to entail many different use cases, will potentially change the data processing and management paradigm in IoT-enabled settings.

We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.

The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain restrictions on system parameters, giving rise to a complete classification for the local dynamical behavior of the system. In particular, a sufficient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condition is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.

In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x,y) = (1/2)y²+V(x) where V(x) = (1/2)x²+(a/4)x⁴+(b/6)x⁶ with b ≠ 0. This family of differential systems always has a center at the origin and, eventually, three other period annuli. We prove that the corresponding period functions can have at most one critical period altogether. More precisely, we show that this critical period is simple and that it corresponds to the period function associated to the center at the origin. To prove the result we use that the period function verifies a Picard–Fuchs equation. Finally we describe the bifurcation diagram of the period function of the center at the origin.

In this paper, we consider local critical periods of planar vector field. Particular attention is given to revertible systems with polynomial functions up to third degree. It is assumed that the origin of the system is a center. Symbolic and numerical computations are employed to show that the general cubic revertible systems can have six local critical periods, which is the maximal number of local critical periods that cubic revertible systems may have. This new result corrects that in the literature: general cubic revertible systems can at most have four local critical periods.

This paper considers the critical periods of third-order planar Hamiltonian systems. It is assumed that the origin of the system is a center. With the aid of symbolic and numerical computations, we show the existence of seven local critical periods. This is the maximal number of local critical periods that a cubic Hamiltonian system can have.

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.

We describe a method for studying the center and isochronicity problems for a class of differential systems in the form of linear center perturbed by homogeneous series of degree 2d + m, where d is a non-negative real number and m is a positive integer. As an application, we classify centers and isochronous centers for a particular case when m = 5.

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

We study cubic near-Hamiltonian systems obtained by perturbing a symmetric cubic Hamiltonian system with two symmetric singular points. First, following [Han, 2012], we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. A computationally efficient algorithm based on the method is established to systematically compute the coefficients of the Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be an elementary center or a nilpotent center. Under the condition for the singular point to be a center, we obtain the standard form of the Hamiltonian system near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials we study limit cycles bifurcating from the center. Finally, perturbing the symmetric Hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is the same as that of another center.

Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form $ẋ=-y,$$ẏ=x+a{x}^{5}y+b{x}^3{y}^3+cx{y}^5,$ where $x,y\in ℝ$ and $a,b,c$ are real parameters satisfying $a^2+b^2+c^2\ne 0.$ Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree $6,$ and second inside the class of all polynomial differential systems of degree $6.$

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree $4k+1$ for $2\le k\le n$. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for $n=2$, at most $5n-7$ limit cycles can bifurcate from the periodic annulus of the unperturbed system for any $n\ge 3$, and the upper bound is sharp for $n=2$ and for $n=3$.

This paper considers the center problem for quasi-homogeneous polynomial systems with a given weight degree. We provide the necessary conditions such that these systems have a center at the origin. Especially, we present the necessary and sufficient conditions on the existence of a center for some class of such systems.

We obtain the conditions for the existence of a center for a cubic planar differential system, which can be considered as a polynomial subfamily of the generalized Riccati system. We also investigate bifurcations of small limit cycles from the components of the center variety of the system.

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate $\mathrm{\Sigma }$-center, derived from a nonsmooth harmonic oscillator model, to degenerate $\mathrm{\Sigma }$-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate $(2k,2l)$-center under general polynomial perturbations of degree $n$.