Related papers: Solovay's inaccessible over a weak set theory with…

We show that under rather general circumstances, the almost everywhere pointwise inequality $|f|(x) \le Mf (x)$ is equivalent to a weak form of the Lebesgue density theorem, for totally bounded closed sets. We derive both positive and…

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

We prove for any mu = mu^{< mu}< theta < lambda, lambda large enough (just strongly inaccessible Mahlo) the consistency of 2^mu = lambda-> [theta]^2_3 and even 2^mu = lambda-> [theta]^2_{sigma,2} for sigma < mu . The new point is that…

We exactly compute the energy density of the integrable O(n) non-linear sigma model as a convergent series. This series is specifically analysed for the very important O(6) symmetry, since it was suggested to result as a peculiar limit of…

We develop a theory of \emph{sharp measure zero} sets that parallels Borel's \emph{strong measure zero}, and prove a theorem analogous to Galvin-Myscielski-Solovay Theorem, namely that a set of reals has sharp measure zero if and only if it…

The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist as much as they generate the underlying $\sigma$-algebra. This leads to the result…

We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and…

For any continuous map f on a compact manifold M, we define the SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f has observable measures, even if SRB measures do not…

Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…

A common fixed point property for semigroups is applied to show that the group algebra $L^1(G)$ of a locally compact group $G$ is $2m$-weakly amenable for each integer $m\geq 1$.

We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three…

We consider the decidability of state-to-state reachability in linear time-invariant control systems over continuous time. We analyse this problem with respect to the allowable control sets, which are assumed to be the image under a linear…

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this…

We propose a novel realization of linear seesaw model in a non-invertible selection rule with the assistance of $\mathbb Z_3$ symmetry. In our framework, Dirac mass matrices are generated at one-loop level, dynamically breaking the…

We define a minimal alpha-observability of Ilyashenko's statistical attractors. We prove that the space is always full Lebesgue decomposable into pairwise disjoint sets that are Lebesgue-bounded away from zero and included in the basins of…

For each $n\in\mathbb{N}$, let $[n]\phi$ mean "the sentence $\phi$ is true in all $\Sigma_{n+1}$-correct transitive sets." Assuming G\"odel's axiom $V = L$, we prove the following graded variant of Solovay's completeness theorem: the set of…

We construct and study the one-parameter semigroup of $\sigma$-finite measures ${\cal L}^{\theta}$, $\theta>0$, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a…

We present in this survey some results regarding Riemann_Lebesgue integrability with respect to arbitrary non-additive set functions.

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for…

We discuss the connections between the failure of the axiom of choice in set theory, and certain model-theoretic structures with enough symmetry.