Petr Ivankov

For any topological space there is a sheaf cohomology. A Grothendieck topology is a generalization of the classical topology such that it also possesses a sheaf cohomology. On the other hand any noncommutative $C^*$-algebra is a generalization of a locally compact Hausdorff space. Here we define a Grothendieck topology of $C^*$-algebras which is a...

A rigorous algebraic definition of noncommutative coverings is developed. In the case of commutative algebras this definition is equivalent to the classical definition of topological coverings of locally compact spaces. The theory has following nontrivial applications: • Coverings of continuous trace algebras, • Coverings of noncommutative tori, •...

Any flat connection on a principal fibre bundle comes from a linear representation of the fundamental group. The noncommutative analog of this fact is discussed here.

Any finite algebraic Galois covering corresponds to an algebraic Morita equivalence. Here the $C^*$-algebraic analog of this fact is proven, i.e. any noncommutative finite-fold covering corresponds to a strong Morita equivalence.

Any oriented Riemannian manifold with a Spin-structure defines a spectral triple, so the spectral triple can be regarded as a noncommutative Spin-manifold. Otherwise for any unoriented Riemannian manifold there is the two-fold covering by oriented Riemannian manifold. Moreover there are noncommutative generalizations of finite-fold coverings. This...

This article shows that any topological finite-fold covering of a foliated manifold naturally corresponds to an algebraic noncommutative finite-fold covering of the operator algebra of the foliation. A counterexample shows that this fact is not true in case of infinite coverings.

It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a quantum group. Also there is a noncommutative generalization of a covering. The combination of these algebraic...

It is well-known that any covering space of a Riemannian manifold has the natural structure of a Riemannian manifold. This article contains a noncommutative generalization of this fact. Since any Riemannian manifold with a Spin-structure defines a spectral triple, the spectral triple can be regarded as a noncommutative Spin-manifold. Similarly ther...

The concept of quantization consists in replacing commutative quantities by noncommutative ones. In mathematical language an algebra of continuous functions on a locally compact topological space is replaced with a noncommutative C*-algebra. Some classical topological notions have noncommutative generalizations. This article is concerned with a gen...

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Similarly a spectral triple is a generalization of a Riemannian manifold. An (infinitely listed) covering of a Rie...

The Gelfand - Na\u{i}mark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants can be defined by algebraical methods. This article contai...

Gelfand -Na ˘ imark theorem supplies a one to one correspondence between commutative C * -algebras and locally compact Hausdorff spaces. So any noncommutative C * -algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan the-orem...

This article contains a noncommutative generalization of the topological path lifting problem. Noncommutative geometry has no paths and even points. However there are paths of *-automorphisms. It is proven that paths of *-automorphisms comply with unique path lifting.

A classical Wilson line is a cooresponedce between closed paths and elemets of a gauge group. However the noncommutative geometry does not have closed paths. But noncommutative geometry have good generalizations of both: the covering projection, and the group of covering transformations. These notions are used for a construction of noncommutative W...

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theo...

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theo...

This article contains is concerned with noncommutative analogue of topological finitely listed covering projections. In my previous article I have already find a family of covering projections of the noncommutative torus. This article describes all covering projections of the noncommutative torus.

If $X$ is a topological space then there is a natural homomorphism $\pi_1(X)\rightarrow K_1(X)$ from a fundamental group to a $K_1$-homology group. Covering projections depend of fundamental group. So $K_1$-homology groups are interrelated with covering projections. This article is concerned with a noncommutative analogue of this interrelationship.

Gelfand - Na\u{i}mark theorem supplies contravariant functor from a category of commutative $C^*-$ algebras to a category of locally compact Hausdorff spaces. Therefore any commutative $C^*-$ algebra is an alternative representation of a topological space. Similarly a category of (noncommutative) $C^*-$ algebras can be regarded as a category of gen...

The top-down approach of engineering software integration is considered in this parer. A set of advantages of this approach are presented, by examples. All examples are supplied by open source code. Comment: 29 pages, 44 figures, 5 references

New information technologies provide a lot of prospects for performance improvement. One of them is "Dynamic Source Code Generation and Compilation". This article shows how this way provides high performance for engineering problems.

http://www.mathframe.com/articles/noncommutative/conjecture2.pdf

A Spectral Triple was firstly introduced by Alain Connes as a generalization of Riemann manifold, or, in other words, a quantization of a Riemannian manifold. The idea is follows: Having a Riemann manifold M, one may consider an algebra of smooth functions on M. This algebra is, obviously, commutative. Roughly speaking, Alain Connes suggested to co...

New type of obstacles for the Universe collapse have been found. These obstacles appear when we replace 3D spherical model of Universe by its noncommutative approximation. Noncommutative approximation has nontrivial fundamental group that may be a cause of the obstacle. This article is a draft only. All results should be approved

A framework for simulation of dynamics of mechanical aggregates has been developed. This framework enables us to build model of aggregate from models of its parts. Framework is a part of universal framework for science and engineering.

A framework for virtual reality of engineering objects has been developed. This framework may simulate different equipment related to virtual reality. Framework supports 6D dynamics, ordinary differential equations, finite formulas, vector and matrix operations. The framework also supports embedding of external software.

An alalogue of unramified coverings [1] of spectral triples [2] has been introduced. This analogue enables us to denote fundamental group of spectral triples and unoriented spectral triples. It was shown that in commutative case this fundamental group is a profinite completion of fundamental group of corresponding Riemann manifold. Fundamental grou...

A notion of fundamental group of spectral triples has been introduced. The notion uses a noncommutative analogue of unramified coverings. It was shown that in commutative case this fundamental group is a profinite completion of fundamental group of corresponding Riemann manifold.

A method is proposed of correcting a spacecraft inertial navigation system during descent from orbit using a navigation satellite. The navigation information includes the components of apparent spacecraft acceleration and parameters of motion of its center of mass, obtained using accelerometers and the satellite navigation system. This method can b...

A method is proposed for the correction of the inertial navigation system of a spacecraft prior to the reentry stage of the flight. The navigation information used includes measurements of the apparent acceleration and state vector components carried out by accelerometers and by the satellite navigation system. The method makes it possible to deter...

A method is suggested for a solution of the navigation problem during approach to Mars, which can be used for aerodynamical deceleration at the entry into the planet's atmosphere. The results of a numerical simulation for estimating the efficiency of the method under the action of perturbations are considered.

A method is proposed for navigation in the zone of approach to Mars, using relative measurements in the two-object system: the satellite and a mobile reference object (such as a detachable satellite probe or a descending device). The method can be used for inserting a satellite into a Martian orbit. Results are presented of numerical simulations of...

The top-down approach of engineering software integration is consid- ered in this parer. A set of advantages of this approach are presented, by examples. All examples are supplied by open source code.