Narrow Results

We construct a family of $q$ deformations of $\mathrm{\text{E}}(2)$ group for nonzero complex parameters $\left|q\right|<1$ as locally compact braided quantum groups over the circle group $𝕋$ viewed as a quasitriangular quantum group with respect to the unitary $R$-matrix $\mathrm{\text{R}}(m,n):={(q/\bar{q})}^{mn}$ for all $m,n\in \mathbb{Z}$. For real $0<\left|q\right|<1$, the deformation coincides with Woronowicz’s ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between ${\mathrm{\text{SU}}}_{\mathrm{\text{q}}}(2)$ and ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups by contracting ${\mathrm{\text{U}}}_{\mathrm{\text{q}}}(2)$ groups.

We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type $A$ to a general reductive group, and interpret it in the context of the Moore–Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalized to the hyperkähler or complex symplectic situation.

Based on the Beltrami–de Sitter spacetime, we present the Newton–Hooke model under the Newton–Hooke contraction of the BdS spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton–Hooke space–time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton–Hooke group. In order to determine uniquely the Newton–Hooke limit, we propose the Galilei–Hooke's relativity principle as well as the postulate on Newton–Hooke universal time. All results are readily extended to the Newton–Hooke model as a contraction of Beltrami–anti-de Sitter spacetime with negative cosmological constant.

Our aim is to contribute to comprehension of the phenomena of precipitation in the Al–Zn–Mg alloys. For this, we have made a comparative study of the transformations of phases using the differential scanning calorimetric and the dilatometry. This last technique is relatively new in the case of Al–Zn–Mg alloys. It consists of two opposite effects (contraction and expansion) observed on the dilatometric curves. These effects translate two opposite metallurgical phenomena which are generally the precipitation and the dissolution.

This paper is concerned with entropy methods for linear drift-diffusion equations with explicitly time-dependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the so-called Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions.

We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some large time asymptotic solutions which may depend on time. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(|x|^α∇·), we prove that the inequality relating the entropy with the entropy production term is a Hardy–Poincaré type inequality, that we establish. Here we assume that α ∈ (0,2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of time-periodic coefficients, we prove the existence of a unique time-periodic solution which attracts all other solutions.

The case of a degenerate diffusion coefficient taking the form |x|^α with α > 2 is also studied. The Gibbs state exhibits a non-integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional time-dependence restores the smoothness of the asymptotic solution.

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

An OU sequence is a cyclically ordered sequence in symbols $O$ and $U$ such that the number of $O$’s is equal to that of $U$’s. Every knot diagram defines an O’U sequence by reading $O$ and $U$ at over- and under-crossings, respectively, appeared along the diagram. In this note, we determine the OU sequences for a $2$-bridge knot arising from its diagrams with two over-bridges.

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

To construct symmetrical patterns on the unit sphere from the planar iterative function systems (IFSs), we present a method of constructing IFSs with $D_3$ symmetry which is composed of three-fold rotational symmetries together with reflections. An algorithm is developed to generate strange attractors with $D_3$ symmetry on a triangular face and then project it onto the surface of the unit sphere to form aesthetics patterns with spherical symmetry. As an illustrative example, we consider the regular inscribed icosahedron in the unit sphere which contains 20 triangular faces. This method is valid to randomly generate aesthetic spherical patterns using planar IFSs.

We find the best constant c(m, n, r), such that the inequality:

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like $W(n)$ in the theory of conservative algebras plays a similar role to the role of ${𝔤𝔩}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n\in \mathbb{N}.$ The present paper is part of a series of papers, which is dedicated to the study of the algebra $W(2)$ and its principal subalgebras.

In this paper, we consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in $L^2$ of the planar viscous shock. If the initial $L^2$-perturbation is also in $L^1$, the large perturbation converges to zero in $L^2$ as time goes to infinity with $t^{-1/4}$ decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case [M.-J. Kang and A. F. Vasseur, $L^2$-contraction for shock waves of scalar viscous conservation laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire34 (2017) 139–156] by the first author and Vasseur to the multi-dimensional case.

Pair correlation methods are able to achieve highly accurate solutions for chemical problems. Unfortunately, their applicability is generally restricted to medium-sized molecules due to storage requirements and computational costs. These restrictions can be partly overcome by local correlation methods. These methods use physical and mathematical criteria to decide which interactions are of such a long range that they do not have to be computed and saved. In our new ansatz, we define an alternative way towards local correlation. The range of interactions is strictly bound to the decay of integrals over Gaussian type geminals in the atomic orbital basis. The number of variables is reduced by orders of magnitude applying an efficient contraction scheme, leading to a naturally local representation of correlation effects. This scheme is extended by orbital optimization to describe multi-reference problems and explicit correlation to improve the basis set convergence.

Most artificial muscles currently in existence driven by negative pressure are in the form of origami. Their contractions are directly attained by extracting the air internal, and obtaining a differential pressure with the external (0.101MPa). During contraction, the artificial muscle simultaneously encounters radial and axial pressures provided by the atmosphere. If the muscle is isotropic, the radial pressure would make the artificial muscle unreliable, which would weaken the muscle’s contraction, including contraction force, ratio, and speed. This paper first introduces the contraction theory and dysfunction of the muscle. Then, an optimization method is proposed as local reinforcement, aiming at the muscle’s dysfunction that may occur during contraction. Then, by analyzing and comparing the optimization effects on the muscle contraction force and deformation with different reinforced regions, the endplates and crests are determined to be appropriate. Subsequently, the deformations of the reinforced regions when stressed are analyzed, and the adequate reinforcement dimensions are confirmed. Finally, the muscle’s model passes the simulation for ensuring its contraction without dysfunction, demonstrating the rationality of the reinforcement method proposed in this paper. Hereby, the goal of reinforcing the muscle’s radial stiffness is achieved, making the artificial muscle generate an optimal contraction performance.

In this paper, a complete classification of completely irreducible closed linear bundles of Lie algebras is given. We also apply the so-called "Raïs formula" to compute the index of each of the Lie bundles of square matrices.

Fixed point iterations of some contraction mappings with variations in cone metric space (CMS) domains have been discussed in this paper. More specifically, an increasing sequence of subsets of an order complete CMSs has been considered such that one member of the sequence is mapped into the next member of the sequence by a map for which a contraction condition is satisfied. The usual iteration method has been considered to establish generalized fixed point results in CMSs. Furthermore, some examples are provided to illustrate our main results.

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma (G)$, is the domination number of $G$. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V-S$, there exists $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a smallest super dominating set of $G$, denoted by ${\gamma }_{\mathrm{\text{sp}}}(G)$, is the super domination number of $G$. In this paper, we study super domination number of some graph classes and present sharp bounds for some graph operations.

The choking flow phenomenon caused by local lateral contraction in open channels is studied in this paper. An equation for the determination of limit opening ratio is derived from the principles of conservation of energy and water continuity. The proposed equation takes into account for energy losses associated with local contraction and can accurately predict the limit opening ratios for choking.

Adaptive finite element methods (AFEM) are a fundamental numerical tool in science and engineering. They are known to outperform classical FEM in practice and deliver optimal convergence rates when the latter cannot. This paper surveys recent progress in the theory of AFEM which explains their success and provides a solid mathematical framework for further developments.

The paper surveys recent results on the finite element approximation of Hamilton-Jacobi-Bellman equations. Various methods are analyzed and error estimates in the maximum norm are derived. Also, a finite element monotone iterative scheme for the computation of the approximate solution is given and its geometrical convergence proved.