Construction of Corner Singularities for Agmon-Douglis-Nirenberg Elliptic Systems

Corner singularities in plane domains are characterized by certain singular exponents and angular functions. Our construction of singularities is based on two Cauchy integrals at two different levels of symbolic calculus. This construction yields new information about the angular functions and also more explicit formulas for the computation of the singular exponents.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

一个新的有两对共轭指数并在全平面积分的Hilbert型不等式

引入二对共轭指数,应用权函数的方法,给出一个新的有最佳常数因子并在全平面积分的Hilbert型不等式,同时考虑了对应的等价式.     

Approximation of the functions in weighted Lebesgue spaces with variable exponent

In the present work, we investigate the approximation problems of the functions by Fejér, and Zygmund means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponents and of the functions by Fejér and Abel–Poisson sums of Faber series in weighted Smirnov classes with variable exponents defined on simply connected domains with a Dini-smooth boundary of the complex plane.     

Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions

This paper considers multi-dimensional affine processes with continuous sample paths. By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode at a given time. In these two cases, we also compute the long-term growth rate and the explosion rate for exponential moments. These results provide a handle to study implied volatility asymptotics in models where log-returns of stock prices are described by affine processes whose exponential moments do not have an explicit formula.     

Sign reversal of the characteristic exponents of solutions of a differential system with initial data on finitely many points and lines

We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R ².     

Stable maps from surfaces to the plane with prescribed branching data

We consider the problem of constructing stable maps from surfaces to the plane with branch set a given set of curves immersed (except possibly with cusps) in the plane. Various constructions are used (1) piecing together regions immersed in the plane (2) modifying an existing stable map by a sequence of codimension one transitions (swallowtails etc) or by surgeries. In (1) the way the regions are pieced together is described by a bipartite graph (an edge C* corresponds to a branch curve C with the vertices of C* corresponding to the two regions containing C). We show that any bipartite graph may be realized by a stable map and we consider the question of realizing graphs by fold maps (i.e. maps without cusps). For example, using Arnol'd's classification of immersed curves, we list all branch sets with at most two branch curves and four double points realizable by planar fold maps of the torus.     

Existence of critical modules of GK-dimension 2 over elliptic algebras

We show that over an elliptic algebra, critical modules of Gelfand-Kirillov dimension 2 exist in all multiplicities (assuming the ground field is uncountable, algebraically closed). Geometrically, this shows that in a quantum plane there exist ``irreducible curve" modules of all possible degrees.

     

On Some Classical Methods to Improve Algebraic Curves and Surfaces

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.     

Caldero–Keller Approach to the Denominators of Cluster Variables

Buan, Marsh, and Reiten proved that if a cluster-tilting object T in a cluster category 𝒞 associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in 𝒞, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T. In this article, we give an alternative proof of this result using the Caldero–Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.     

Geometric exponents and Kleinian groups

Suppose is the limit set of an analytically finite Kleinian group and that is an enumeration of the components of . Then This had been conjectured by Maskit. We also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincaré exponent. Oblatum 30-X-1995 & 11-III-1996     

A matrix Hilbert transform in Hermitean Clifford analysis

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.     

Dynamics of coupled non-identical systems with period-doubling cascade

We discuss the structure of bifurcation diagram in the plane of parameters controlling period-doublings for the system of coupled logistic maps. The analysis is carried out by computing the charts of dynamical regimes and charts of Lyapunov exponents giving showy and effective illustrations. The critical point of codimension two at the border of chaos is found. It is a terminal point for the Feigenbaum critical line. The bifurcation analysis in the vicinity of this point is presented.      

Modulus of continuity of solutions of second-order elliptic equations on a plane

We find the exponents in the Hölder boundary condition for generalized solutions of a secondorder elliptic equation of divergent form on a plane with coefficients from L_∞. The accuracy of the formula obtained is verified with an example.     

Secondary school students’ levels of understanding in computing exponents

The aim of this study is to describe and analyze students’ levels of understanding of exponents within the context of procedural and conceptual learning via the conceptual change and prototypes’ theory. The study was conducted with 202 secondary school students with the use of a questionnaire and semi-structured interviews. The results suggest that three levels of understanding can be identified. At the first level students’ interpretation of exponents is based upon exponents that symbolize natural numbers. At Level 2, students’ knowledge acquisition process is a process of enrichment of the existing conceptual structures. Students at this level are able to compute exponents with negative numbers by extending the application of prototype examples. Finally, at Level 3 students not only extend the prototype examples but also reorganize their thinking in order to compute and compare exponents with roots, a concept which is quite different from the concept of exponents with natural numbers.     

First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

A distributed exact algorithm for the multiple resource constrained sequencing problem

Sequencing problems arise in the context of process scheduling both in isolation and as subproblems for general scenarios. Such sequencing problems can often be posed as an extension of the Traveling Salesman Problem. We present an exact parallel branch and bound algorithm for solving the Multiple Resource Constrained Traveling Salesman Problem (MRCTSP), which provides a platform for addressing a variety of process sequencing problems. The algorithm is based on a linear programming relaxation that incorporates two families of inequalities via cutting plane techniques. Computational results show that the lower bounds provided by this method are strong for the types of problem generators that we considered as well as for some industrially derived sequencing instances. The branch and bound algorithm is parallelized using the processor workshop model on a network of workstations connected via Ethernet. Results are presented for instances with up to 75 cities, 3 resource constraints, and 8 workstations.