On the topology of moduli spaces of non-negatively curved Riemannian metrics

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamentallevel which are, via Born probability law, connected to the formal randomnessof infinite sequences of QM outcomes. Recently it has been shown thatQM is algorithmic 1-random in the sense of Martin-L¨of. We extend this resultand demonstrate that QM is algorithmic ω-random and generic, precisely asdescribed by the ’miniaturisation’ of the Solovay forcing to arithmetic. Thisis extended further to the result that QM becomes Zermelo–Fraenkel Solovayrandom on infinite-dimensional Hilbert spaces. Moreover, it is more likely thatthere exists a standard transitive ZFC model M, where QM is expressed in reality,than in the universe V of sets. Then every generic quantum measurementadds to M the infinite sequence, i.e. random real r ∈ 2ω, and the model undergoesrandom forcing extensions M[r]. The entire process of forcing becomesthe structural ingredient of QM and parallels similar constructions applied tospacetime in the quantum limit, therefore showing the structural resemblanceof both in this limit. We discuss several questions regarding measurability andpossible practical applications of the extended Solovay randomness of QM.The method applied is the formalization based on models of ZFC; however,this is particularly well-suited technique to recognising randomness questionsof QM. When one works in a constant model of ZFC or in axiomatic ZFCitself, the issues considered here remain hidden to a great extent.

Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

Infinite sequences of almost symmetric non-Weierstrass numerical semigroups

Let u be any positive integer. We construct infinite sequences of almost symmetric non-Weierstrass numerical semigroups whose conductors are the genera double minus $$2u-1$$ 2 u - 1 . Moreover, let v be any non-negative integer. We give an infinite number of non-Weierstrass numerical semigroups whose conductors are the genera double minus 2v.

On orbits and the finiteness of bounded automaton groups

We devise an algorithm which, given a bounded automaton [Formula: see text], decides whether the group generated by [Formula: see text] is finite. The solution comes from a description of the infinite sequences having an infinite [Formula: see text]-orbit using a deterministic finite-state acceptor. This acceptor can also be used to decide whether the bounded automaton acts level-transitively.

DOES CONTEMPORARY ANALYSIS OF DIFFICULTIES WITH ZENO SEQUENCES CONTAIN A SOLUTION TO THE DICHOTOMY?

We analyze contemporary thought experiments with some Zeno objects and infinity machines. We show how the method of reasoning from J. Hawthorne’s paper helps to understand the structure of one of the refutations of a rather sophisticated version of Zeno’s of Elea Dichotomy. After that, we propose an improvement of this version of the Dichotomy. Further, we show that the method of operating with infinite sequences of conditional sentences – proposed in J. Hawthorne’s paper – is insufficient to refute the last variant of the Dichotomy.

Introduction to Hyperspaces

The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is to extend mathematical tools for treating infinity by considering hyperspaces and developing their theory.