Related papers: Pseudojump inversion in special r. b. $\Pi^0_1$ cl…

Projection theorems of divergences enable us to find reverse projection of a divergence on a specific statistical model as a forward projection of the divergence on a different but rather "simpler" statistical model, which, in turn, results…

The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence…

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth…

Let $X$ be a real analytic orbifold. Then each stratum of $X$ is a subanalytic subset of $X$. We show that $X$ has a unique subanalytic triangulation compatible with the strata of $X$. We also show that every ${\rm C}^r$-orbifold, $1\leq…

In his book, John Stillwell wrote "finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics." In this article, we solve Stillwell's problem by showing that…

We look at the Equivalence Theorem as a statement about the absence of polynomial infrared divergences when $m_W \rightarrow 0$. We prove their absence in a truncated toy model and conjecture that, if they exist at all, they are due to…

The non-minimal pure spinor formalism for the superstring is used to prove two new multiloop theorems which are related to recent higher-derivative $R^4$ conjectures of Green, Russo and Vanhove. The first theorem states that when $0<n<12$,…

One of the many equivalent formulation of the K\"othe's conjecture is the assertion that there exists no ring which contains two nil right ideals whose sum is not nil. We discuss several consequences of an observation that if the Koethe…

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If $X(t)$ is a one-dimensional diffusion with jumps, starting from…

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+\delta)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $\delta,t$ are small real…

We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the…

This paper classifies different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey's theorem, in terms of their uniform computational content (Weihrauch degree). It can be seen as a continuation of…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

L\'evy's Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to…

For $B\subset\mathbb{R}^k$ denote by $\Phi_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that $$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)}…

Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse…

Here is one of the results obtained in this paper: Let $X, Y$ be two convex sets each in a real vector space, let $J:X\times Y\to {\bf R}$ be convex and without global minima in $X$ and concave in $Y$, and let $\Phi:X\to {\bf R}$ be…

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $|[T]| = 1$ or $|[T]|\geq \aleph_0$ and $\forall r\in V(T)$,…

In part I we reduced the arithmetic (characteristic zero) version of the P \not \subseteq NP conjecture to the problem of showing that a variety associated with the complexity class NP cannot be embedded in the variety associated the…

A theorem of Sierpi\'nski says that every infinite set Q of reals contains an infinite number of disjoint subsets whose outer Lebesgue measure is the same as that of Q. He also has a similar theorem involving the Baire property. We give a…