Positivstellensatz and flat functionals on path *-algebras

We consider the class of non-commutative *-algebras which are path algebras of doubles of quivers with the natural involutions. We study the problem of extending positive truncated functionals on such *-algebras. An analog of the solution of the truncated Hamburger moment problem by Curto and Fialkow for path *-algebras is presented and non-commutative positivstellensatz is proved. We aslo present an analog of the flat extension theorem of Curto and Fialkow for this class of algebras.Comment: corrected typo

Reduction Operators of Linear Second-Order Parabolic Equations

The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The ``no-go'' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.Comment: 31 pages, minor misprints are correcte

Conservation laws and normal forms of evolution equations

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.Comment: 16 page

PHASE DYNAMICS OF PHYTOCOENOSIS ON THE DAMPED WASTE HEAPS OF NOVOVOLYN MINING AREA

Nineteen non-recultivated waste heaps are found on the territory of the Novovolyn mining area (Lviv-Volyn coal basin), three of which are damping. The total area of disturbed lands is 116.7 hectares. An important role for optimization of disturbed objects is played by natural colonization by vegetation, since it indicates the state of the rocks the waste heaps are made of. Studying of phytocoenoses, which are formed during self-establishment, gives an opportunity to estimate the formed groups, considering their place and role in the vegetation cover of the region and predict their further development

THE IMPACT OF COAL WASTE HEAPS ON THE ENVIRONMENT OF SOKAL DISTRICT OF LVIV REGION

The environmental situation in Sokal district is one of the most complicated in Lviv region, which is caused by man-made environmental impact of mining, coal preparation and chemical industries. Most of the territory is occupied by coal waste heaps