Related papers: $\mathsf{RT}_2^2$ does not imply $\mathsf{WKL}_0$

Let $R=K[t;\sigma]$ be a skew polynomial ring, where $K$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $\sigma$. We show that two irreducible similar skew polynomials $f,g\in R$ are similar if and only if…

In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall \Pi^0_4$ conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$, where a $\forall \Pi^0_4$ formula consists of a universal quantifier over…

The purpose of this article is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with Arakelov theory of noncommutative arithmetic curves. Our first main result is an arithmetic Riemann-Roch formula…

Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $\kappa$ for which…

A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\ell$ such that the multiplication map by $\ell:[A]_i\to [A]_{i+1}$ has maximum rank in every degree. The linear forms satisfying this property…

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete…

We give a non-constructive proof that fusion rings attached to a simple complex Lie algebra of rank 2 are complete intersections.

We prove that for any function $f$ satisfying certain mild conditions and any Cantor set $K$ with Newhouse thickness greater than $1$, there exists $x\in K$ and $t>0$ such that \[ \{x-t,x,x+f(t)\}\subset K. \] This is an extension of…

The main goal of this paper is to develop the MRA theory along with wavelet theory in L2(Qp). Generalized scaling sets are important in wavelet theory because it determine multiwavelet sets. Although the theory of scaling set and…

We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in…

When integrating computational tools such as automatic segmentation into clinical practice, it is of utmost importance to be able to assess the level of accuracy on new data, and in particular, to detect when an automatic method fails.…

For every irrational real $\alpha$, let $M(\alpha) = \sup_{n\geq 1} a_n(\alpha)$ denote the largest partial quotient in its continued fraction expansion (or $\infty$, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as…

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…

In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can…

In this paper, we introduce the notion of Rota-Baxter Lie $2$-algebras, which is a categorification of Rota-Baxter Lie algebras. We prove that the category of Rota-Baxter Lie $2$-algebras and the category of $2$-term Rota-Baxter…

In this paper, assuming the weak Schanuel Conjecture (WSC), we prove that for any collection of pairwise non-arithmetically equivalent totally real number fields, the residues at $s=1$ of their Dedekind zeta functions form a linearly…

We provide an alternative proof that $\sqrt{2}$ is irrational that does not begin with the assumption that $\sqrt{2}$ is in fact rational.

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of…

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…

Let G be a closed subgroup of the isometry group of a proper CAT(0)-space X. We show that if G is non-elementary and contains a rank-one element then its second bounded cohomology group with coefficients in the regular representation is…