37 resultados para Strictly hyperbolic polynomial

em Bulgarian Digital Mathematics Library at IMI-BAS


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∗ Research partially supported by INTAS grant 97-1644

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2002 Mathematics Subject Classification: 35L15, 35L80, 35S05, 35S30

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2000 Mathematics Subject Classification: 12D10.

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We present some results on the formation of singularities for C^1 - solutions of the quasi-linear N × N strictly hyperbolic system Ut + A(U )Ux = 0 in [0, +∞) × Rx . Under certain weak non-linearity conditions (weaker than genuine non-linearity), we prove that the first order derivative of the solution blows-up in finite time.

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2000 Mathematics Subject Classification: 12D10.

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A strictly hyperbolic quasi-linear 2×2 system in two independent variables with C2 coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued function of the so called Riemann invariant is discussed. It is shown, through a purely geometrical approach, that there always exists simple wave solution for the general system when the coefficients are arbitrary C^2 functions depending on both, dependent and independent variables.

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2000 Mathematics Subject Classification: 12D10.

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Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

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MSC 2010: 30C10, 32A30, 30G35

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∗ Partially supported by INTAS grant 97-1644

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An antagonistic differential game of hyperbolic type with a separable linear vector pay-off function is considered. The main result is the description of all ε-Slater saddle points consisting of program strategies, program ε-Slater maximins and minimaxes for each ε ∈ R^N > for this game. To this purpose, the considered differential game is reduced to find the optimal program strategies of two multicriterial problems of hyperbolic type. The application of approximation enables us to relate these problems to a problem of optimal program control, described by a system of ordinary differential equations, with a scalar pay-off function. It is found that the result of this problem is not changed, if the players use positional or program strategies. For the considered differential game, it is interesting that the ε-Slater saddle points are not equivalent and there exist two ε-Slater saddle points for which the values of all components of the vector pay-off function at one of them are greater than the respective components of the other ε-saddle point.

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* Partially supported by Universita` di Bari: progetto “Strutture algebriche, geometriche e descrizione degli invarianti ad esse associate”.

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It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).

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Dubrovin type equations for the N -gap solution of a completely integrable system associated with a polynomial pencil is constructed and then integrated to a system of functional equations. The approach used to derive those results is a generalization of the familiar process of finding the 1-soliton (1-gap) solution by integrating the ODE obtained from the soliton equation via the substitution u = u(x + λt).

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* Dedicated to the memory of Prof. N. Obreshkoff