scholarly journals New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation

2012 ◽  
Vol 468 (2141) ◽  
pp. 1371-1390 ◽  
Author(s):  
C. Kalla ◽  
C. Klein
Keyword(s):  
Numerical Evaluation ◽  
Riemann Sphere ◽  
Theta Functions ◽  
Point Of View ◽  
Holm Equation ◽  
New Construction ◽  
Geometric Solutions ◽  
Hyperelliptic Surface ◽  
Independent Derivation

An independent derivation of solutions to the Camassa–Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay’s identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.

scholarly journals Leading coefficients and cellular bases of Hecke algebras

2009 ◽  
Vol 52 (3) ◽  
pp. 653-677 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Cellular Structure ◽  
Hecke Algebras ◽  
Point Of View ◽  
Reflection Groups ◽  
Cellular Basis ◽  
Complex Reflection Groups ◽  
Finite Coxeter Group ◽  
New Construction

AbstractLet H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

CONDITION AND PROBLEMS OF KAUNAS FORTRESS / KAUNO TVIRTOVĖS KULTŪROS PAVELDO OBJEKTŲ BŪKLĖ IR PROBLEMOS

2012 ◽  
Vol 36 (1) ◽  
pp. 63-72
Author(s):  
Nijolė Steponaitytė
Keyword(s):  
European Union ◽  
Cultural Heritage ◽  
Natural Environment ◽  
Point Of View ◽  
Cultural Property ◽  
Master Plan ◽  
The European Union ◽  
New Construction ◽  
The Republic

The paper discusses objects of the research on Kaunas Fortress, listing of the Fortress in the Register of Cultural Property of the Republic of Lithuania, and process for establishing respective territory and preservation zones. Some protection objects – forts, batteries and their territories – are analysed from the point of view of new construction penetration into the territories and preservation zones of cultural heritage. Creation of terriologic reservates around objects of the Fortress and their regulation influence to buildings is discussed. Natural environment planning, the European Union supported projects, their results and realisation, practical benefit, some solutions of the master plan of Kaunas, that harm objects of cultural heritage territories of Kaunas Fortress are discussed as well. Santrauka Straipsnyje aptariami Kauno tvirtovės objektų tyrimai, įtraukimas į LR nekilnojamojo kultūros paveldo vertybių registrą, teritorijų ir apsaugos zonų nustatymas. Analizuojama kai kurių Kauno tvirtovės gynybinių statinių teritorijų būklė, naujų statybų skverbimasis į kultūros paveldo objektų apsaugos zonas ir teritorijas. Aptariamas teriologinių draustinių įkūrimas tvirtovės gynybiniuose objektuose, jų nuostatų įtaka statiniams, gamtotvarkos planų ir kitų Europos Sąjungos finansuojamų projektų rezultatai ir siūlymų įgyvendinimas, praktinė nauda, kai kurie Kauno miesto Bendrojo plano sprendiniai, kenkiantys Kauno tvirtovės kultūros paveldo objektų išlikimui.

scholarly journals Energy of wave and solutions of C-Holm equation under Lagrangian point of view

2018 ◽  
Vol 6 (11) ◽  
Author(s):  
Shkelqim Hajrulla ◽  
L Bezati ◽  
F Hoxha
Keyword(s):  
Energy Density ◽  
Initial Data ◽  
Coordinate Transformation ◽  
Radon Measure ◽  
Continuous Semigroup ◽  
Point Of View ◽  
Lagrangian Point ◽  
Lagrangian Coordinates ◽  
Positive Radon Measure ◽  
Holm Equation

     Abstract: We deal with the Camassa-Holm equation   possesses a global continuous semigroup of weak conservative solutions for initial data. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure µ with . The total energy is preserved by the solution.

scholarly journals Symmetry in Engineering Sciences

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 797 ◽  
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco Manzano-Agugliaro
Keyword(s):  
Point Of View ◽  
Human Beings ◽  
Special Issue ◽  
Engineering Sciences ◽  
The Aesthetic ◽  
Geometric Solutions

The symmetry concept is mainly used in two senses. The first from the aesthetic point of view of proportionality or harmony, since human beings seek symmetry in nature. Or the second, from an engineering point of view to attend to geometric regularities or to explain a repetition process or pattern in a given phenomenon. This special issue dedicated to geometry in engineering deals with this last concept, which aims to collect both the aspects of geometric solutions in engineering, which may even have a certain aesthetic character, and the aspect of the use of patterns that explain observed phenomena.

scholarly journals On Optimal Cancellation of Policies

1977 ◽  
Vol 9 (1-2) ◽  
pp. 125-138 ◽  
Author(s):  
Hans U. Gerber
Keyword(s):  
Continuous Time ◽  
Numerical Evaluation ◽  
Infinite Horizon ◽  
Poisson Model ◽  
Analytical Form ◽  
Point Of View ◽  
Optimal Stopping Problem ◽  
The Past ◽  
Discounted Dynamic Programming ◽  
Analytical Point

One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

ROLE OF THE CONTEXT IN THE REHABILITATION OF URBAN AREAS DISTURBED BY INDUSTRIAL ACTIVITY

2018 ◽  
Vol 8 (2) ◽  
pp. 117-121
Author(s):  
Daria S. RYBAKOVA ◽  
Alexandr S. FEDOTOV
Keyword(s):  
Urban Areas ◽  
Public Life ◽  
Point Of View ◽  
Main Principle ◽  
Functional Program ◽  
Industrial Activity ◽  
New Construction ◽  
The City ◽  
Methods Of Reconstruction

The article is devoted to the study of problems of urban areas, disturbed by industrial activity, as well as methods of their rehabilitation and inclusion in the city public life. In the course of the research domestic and foreign experience was analyzed using the example of realized objects of the last 10-15 years refl ecting two principal approaches to solving this issue: demolition and complete cleaning of the territory with subsequent new construction; full or partial modifi cation of the functional program of the facility while preserving the most valuable elements from the architectural and planning point of view, and complementing them with modern architectural objects (reconstruction). The methods of reconstruction are listed and classifi ed: restoration, modernization, restructuring, revitalization and renovation. According to the results of the research the main principle that architects should follow is att ention to the context of the place in all its manifestations (historical, social, fi gurative-emotional, natural or urban).

scholarly journals Geometric Integrability of Some Generalizations of the Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Ognyan Christov
Keyword(s):  
Point Of View ◽  
Holm Equation ◽  
Pseudospherical Surfaces ◽  
Geometric Point ◽  
Geometric Integrability

We study the Camassa-Holm (CH) equation and recently introducedμCH equation from the geometric point of view. We show that Kupershmidt deformations of these equations describe pseudospherical surfaces and hence are geometrically integrable.

PEAKONS AND THEIR BIFURCATION IN A GENERALIZED CAMASSA–HOLM EQUATION

2001 ◽  
Vol 11 (03) ◽  
pp. 781-792 ◽  
Author(s):  
ZHENGRONG LIU ◽  
TIFEI QIAN
Keyword(s):  
Phase Plane ◽  
Shallow Water Equation ◽  
Point Of View ◽  
Phase Portraits ◽  
Bifurcation Parameter ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
First Derivative ◽  
Wave Solutions ◽  
Holm Equation

Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation [Formula: see text] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

scholarly journals Algebro-Geometric Solutions for a Discrete Integrable Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Mengshuang Tao ◽  
Huanhe Dong
Keyword(s):  
Integrable Systems ◽  
Theta Functions ◽  
Discrete Systems ◽  
Integrable Equation ◽  
Integrable Hierarchy ◽  
Discrete Integrable Systems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Riemann Theta Functions ◽  
Geometric Solutions

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

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