Second order quasilinear PDEs and conformal structures in projective space.

We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pn with coordinates p1, …, pn. The coefficient matrix fij defines on Pn a conformal structure fij(p)dpidpj. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure fij(p) dpidpj is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms

Highly Parallel Convolution Method to Compare DNA Sequences with Enforced In/Del and Mutation Tolerance

Текст статьи не публикуется в открытом доступе в соответствии с политикой журнала.New error tolerant method for the comparison and analysis of symbol sequences is proposed. The method is based on convolution function calculation, where the function is defined over the binary numeric sequences obtained by the specific transformation of original symbol sequence. The method allows highly parallel implementation and is of great value for insertion/delition mutations search. To calculate the convolution function, fast Fourier transform is used in the method implementation. Предлагается новый устойчивый к ошибкам метод сравнения и анализа символьных последовательностей. Метод основан на вычислении функции свертки, где функция определяется над двоичными числовыми последовательностями, полученными путем конкретного преобразования исходной последовательности символов. Метод допускает эффективное распараллеливание и имеет большое значение для поиска мутаций вставки / удаления. Для вычисления функции свертки в реализации метода используется быстрое преобразование Фурье

Integrable Systems

Integrable systems which do not have an \u201cobvious\u201c group symmetry, beginning with the results of Poincar\ue9 and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960\u2019s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics