Local Isometric immersions of pseudo-spherical surfaces and evolution equations

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in ${\bf E}^{3}$ from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in ${\bf E}^{3}$. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper [11], we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for $u(x,t)$ of order $k\geq 3$ describing pseudo-spherical surfaces. We show that whenever an isometric immersion in ${\bf E}^3$ exists, depending on a jet of finite order of $u$, then the coefficients of the second fundamental forms are functions of the independent variables $x$ and $t$ only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and Applications, pp.N

Assessment of evolution of neurologic violations as a sign of effciency of treatment at patients with nonspecifc purulent diseases of a backbone

<p>Studying of dynamics and role of neurologic disorders in an assessment of surgical treatment effciency of nonspe-cifc purulent diseases of a backbone was the goal of the research. As a material of research two groups of patients, check group — 73 patients, the main group — 110 patients, were treated with nonspecifc purulent diseases of a backbone served. After surgical treatment essential regression of initial neurologic disorders was observed. Considerable changes of a pain vertebrogenic syndrome and neurologic defciency are sensitive criteria of an assessment of effcien-cy of results of the surgical treatment, as it refects accurate dynamics of initial indicators by the end of hospitalization.</p&gt

Formalization of the parallel searching defects algorithms development procedure in a mutual exchange of information on the results of diagnostics

Формалізовано порядок розробки умовних алгоритмів паралельного пошуку дефектів за графічними і матричними моделями об’єктів діагностування.Формализован порядок разработки условных алгоритмов параллельного поиска дефектов по графическим и матричным моделям объектов диагностирования.Procedure of conditional algorithms development for parallel search defects on the graphic and matrix models of diagnosis object was formalized

The methodology of the group search for defects development diagnostic support in the repair of communications equipment in the field

Запропонована формалізація процесу розробки діагностичного забезпечення ремонту техніки зв’язку з різним ступенем пошкодження і реалізацією видів групового пошуку дефектів.Предложена формализация процесса разработки диагностического обеспечения ремонта техники связи с разной степенью повреждения и реализацией видов группового поиска дефектов.Formalization of the process of development of the diagnostics support of telecommunications equipment repair with different defects level and realization of the of group defects search types is suggested