Band 1, Heft 1

The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.

The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect occurs in all efficient methods of rational approximation (e.g., best approxmations, Padé approximations, multipoint Padé approximations, linear and nonlinear Padé-Chebyshev approximations, etc.), where very significant errors in the approximant coefficients do not affect the accuracy of this approximant. The reason is that the errors in the coefficients of the rational approximant are not distributed in an arbitrary way, but form a collection of coefficients for a new rational approximant to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The effect of error autocorrection indicates that variations of an approximated function under some deformations of rather a general type may have little effect on the corresponding rational approximant viewed as a function (whereas the coefficients of the approximant can have very significant changes). Accordingly, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation is not stable under small deformations of the approximated functions. This property is “individual”, in the sense that it holds for specific functions.

Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.

If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $$Eexp( - \lambda T_\varepsilon )f(\frac{{\xi (T_\varepsilon )}}{\varepsilon }) - Eexp( - \lambda T_\varepsilon )Ef(\frac{{\xi (T_\varepsilon )}}{\varepsilon }),\varepsilon \searrow 0,\lambda > 0.$$

Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote $$\mathcal{L}_A$$ to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by $$\mathcal{R}_A$$ the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with $$\mathcal{L}_A \cup \mathcal{R}_A$$ co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which $$\mathcal{L}_A \cup \mathcal{R}_A$$ is co-finite in ind A, and derive some consequences.

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.