Smoothly parameterized Čech cohomology of complex manifolds

A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Čech theory. If however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.     

Čech Type Approach to Computing Homology of Maps

A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations, this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of necessary input. We show that the approach is particularly advantageous in the case of homology map algorithms.     

A categorical property of the Stone-Čech compactification

We study some categorical properties of the functor O _β of weakly additive functionals acting in the category Tych of the Tychonoff spaces and their continuous mappings. We show that O _β preserves the weight of infinite-dimensional Tychonoff spaces, the singleton, and the empty set, and that O _β is monomorphic and continuous in the sense of T. Banakh, takes every perfect mapping to an epimorphism, and preserves intersections of functionally closed sets in a Tychonoff space.     

Stone-Čech remainders of nowhere locally compact spaces

The remainder of a compactification αX of a space X is the space αX - X. For a nowhere locally compact X, it is shown that if αX - X is extremally disconnected, then αX = βX. Conditions on X are obtained which characterize when βX - X is extremally disconnected. Also, conditions are provided which characterize when βX - X is zero-dimensional.     

The stability of fixed points of discrete dynamical systems in the space conv ℝn

Conditions for the asymptotic Lyapunov stability of the fixed points of discrete dynamical systems in the space conv ?ⁿ are established.     

Comparison of Certain Complexes of Modules of Generalized Fractions and Čech Complexes

We establish an explicit quasi-isomorphism of complexes, which is homogeneous in graded situation, from a given ?ech complex to a certain complex of modules of generalized fractions. This leads to characterizations of local cohomology and ideal transforms of a module. Also, in this note, we study the behavior of generalized fraction formation along exact sequences and vanishing of modules of generalized fractions in certain cases.     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

A note on log etale Čech cohomolog

In this paper, we prove the comparison theorem between log etale cohomology and log etale Čech cohomology for certain log schemes, which is a generalization of a result of Artin on the comparison theorem between etale cohomology and etale Čech cohomology. It turns out that the naive generalization is not true, and we also give a counter-example for it. Received: 2 July 2000 / Revised version: 4 September 2000     

Methods for the investigation of dynamical systems with impulse action and “mortal” dynamical systems

The notions of dynamical systems with impulse action and mortal dynamical systems are introduced. Their connection with the idealizations of ordinary dynamical systems is considered. General methods for the investigation of these systems are worked out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1605–1613, November, 1991.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

Hemiring-homomorphisms,Stone Čech Compactification and Hewitt Realcompactification

In this paper the Stone-ech Compactification and Hewitt Realcompactification of a Tychonoff space X are shown as the spaces of Hemiring Homomorphisms from the hemirings C^*₊(X) and C₊(X) to IR₊ respectively.AMS Subject Classification: 2000, 54E25     

Martingale–coboundary decomposition for families of dynamical systems

We prove statistical limit laws for sequences of Birkhoff sums of the type ${\sum }_{j=0}^{n?1}{v}_n°T_n^j$ where $T_n$ is a family of nonuniformly hyperbolic transformations.The key ingredient is a new martingale–coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family $T_n$ is replaced by a fixed transformation T, and which is particularly effective in the case when $T_n$ varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family $T_n$ consists of intermittent maps, unimodal maps (along the Collet–Eckmann parameters), Viana maps, and externally forced dispersing billiards.As an application, we prove a homogenisation result for discrete fast–slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.     

The formulation of dynamical contact problems with friction in the case of systems of rigid bodies and general discrete mechanical systems—Painlevé and Kane paradoxes revisited

The dynamics of mechanical systems with a finite number of degrees of freedom (discrete mechanical systems) is governed by the Lagrange equation which is a second-order differential equation on a Riemannian manifold (the configuration manifold). The handling of perfect (frictionless) unilateral constraints in this framework (that of Lagrange’s analytical dynamics) was undertaken by Schatzman and Moreau at the beginning of the 1980s. A mathematically sound and consistent evolution problem was obtained, paving the road for many subsequent theoretical investigations. In this general evolution problem, the only reaction force which is involved is a generalized reaction force, consistently with the virtual power philosophy of Lagrange. Surprisingly, such a general formulation was never derived in the case of frictional unilateral multibody dynamics. Instead, the paradigm of the Coulomb law applying to reaction forces in the real world is generally invoked. So far, this paradigm has only enabled to obtain a consistent evolution problem in only some very few specific examples and to suggest numerical algorithms to produce computational examples (numerical modeling). In particular, it is not clear what is the evolution problem underlying the computational examples. Moreover, some of the few specific cases in which this paradigm enables to write down a precise evolution problem are known to show paradoxes: the Painlevé paradox (indeterminacy) and the Kane paradox (increase in kinetic energy due to friction). In this paper, we follow Lagrange’s philosophy and formulate the frictional unilateral multibody dynamics in terms of the generalized reaction force and not in terms of the real-world reaction force. A general evolution problem that governs the dynamics is obtained for the first time. We prove that all the solutions are dissipative; that is, this new formulation is free of Kane paradox. We also prove that some indeterminacy of the Painlevé paradox is fixed in this formulation.     

Arithmetic–geometric discrete systems

Analyzing a consensus in opinion dynamics one meets interesting open problems concerning the iteration of various means. Whereas in two dimensions the consensus can be computed by the arithmetic-geometric mean of Gauss not very much is known in higher dimensions.