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In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium. [ABSTRACT FROM AUTHOR]

We present a generalization of the Navier–Stokes equations that describe the flows of homogeneous multicomponent media in the presence of various thermodynamic processes, especially chemical reactions. [ABSTRACT FROM AUTHOR]

In this paper we study a 3-dimensional filtration of real gases described by Redlich–Kwong equations of state. Thermodynamical states are considered as Legendrian (Lagrangian) submanifolds in contact (symplectic) space. Connection between singularities of their projection on the space of intensive variables and phase transitions is shown. Explicit formulae for the Dirichlet boundary problem are given and the distribution of phases in space is shown. [ABSTRACT FROM AUTHOR]

Abstract In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits. [ABSTRACT FROM AUTHOR]

In this paper we study differential invariants and give a local classification of the second order linear differential operators, acting in sections of line bundles, and a local classification of corresponding differential equations. [ABSTRACT FROM AUTHOR]

Symmetries and the corresponding algebras of differential invariants of viscid fluids on a sphere are given. Their dependence on thermodynamical states of media is studied, and a classification of thermodynamical states is given. [ABSTRACT FROM AUTHOR]

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of order k ≥ 2 on an n -dimensional manifold, n ≥ 2 , and apply the results to the equivalence problem of such operators. [ABSTRACT FROM AUTHOR]

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms. [ABSTRACT FROM AUTHOR]

Algebras of symmetries and the corresponding algebras of differential invariants for 3D-flows of viscid newtonian fluids are given. Their dependence on thermodynamical states of media is studied. [ABSTRACT FROM AUTHOR]

We show that any 4-dimensional Lorentzian manifold with nondegenerate Weyl tensor possesses the spinor structure and this structure is natural, i.e. invariant under action of diffeomorphisms. [ABSTRACT FROM AUTHOR]

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of the second order and show their application to the equivalence problem of such operators. [ABSTRACT FROM AUTHOR]

In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits. [ABSTRACT FROM AUTHOR]

In this paper we analyze the Cauchy problem for the Einstein–Maxwell equation in the case of non-characteristic initial hypersurface. To find the correct notions of characteristic and Cauchy data we introduce a complex, which we call the Einstein–Maxwell complex. Then the Cauchy problem acquires correctness in terms of an associated spectral sequence. We define a Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we describe quantizations in the monoidal categories of unitary representations of compact connected Lie groups. For the -dimensional torus we show that the set of quantizations is isomorphic to the -dimensional torus. For connected compact Lie groups of rank , we get the result that the set of extendible quantizations of -modules is isomorphic to the set of quantizations of its maximal torus invariant under action by its Weyl group. For all these cases we give explicit formulae for quantizations and apply these to quantize Hilbert–Schmidt operators. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the Cauchy problem for the Einstein equation in the case of a non-characteristic initial hypersurface. To find the correct notions for the characteristic and Cauchy data we introduce a complex, which we call the Einstein complex. Then the Cauchy problem acquires correctness in terms of the associated spectral sequence. We define the Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution. [Copyright &y& Elsevier]

Abstract: In this paper we study orbits of - and -actions on the spaces of binary and ternary polynomials as well as rational forms and find criteria for their equivalence. Similar results are also valid for real forms. [Copyright &y& Elsevier]

We consider the Euler system describing a one-dimensional inviscid flows in space along curves of a certain class. Using differential invariants for the Euler system, we obtain its quotient equation. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system. We discuss solving the quotient using asymptotic expansions of unknown functions and virial expansion of thermodynamic state equations. Thus the quotient is reduced to a series of ODE systems. [ABSTRACT FROM AUTHOR]

We are going to a further development of relations between formal measurement theory and thermodynamics. Higher-order geometrical structures on Lagrangian manifolds are introduced and their relations with an accuracy of measurement and confidence domains as well as phase transitions in thermodynamics are discussed. The approach is illustrated for the case of real gases and in detail for methane. [ABSTRACT FROM AUTHOR]

Symmetries and differential invariants of viscid flows with viscosity depending on temperature on a space curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given. [ABSTRACT FROM AUTHOR]

The problem of equivalency for linear differential operators of the first order is discussed. [ABSTRACT FROM AUTHOR]

The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally associated with differential operators, are found. These geometrical structures are used to solve the problems of local as well as global equivalence of differential operators. [ABSTRACT FROM AUTHOR]

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting on line bundles over smooth manifolds with respect to groups of automorphisms. [ABSTRACT FROM AUTHOR]

We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of automorphisms. [ABSTRACT FROM AUTHOR]

In this article we study the initial boundary value problem for two-phase heat and mass transport in porous media described by the Buckley–Leverett model. We outline a method to construct asymptotic solutions of the initial boundary problem and show how to overcome singularities in solutions and shock waves. [ABSTRACT FROM AUTHOR]