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We show that it is possible to algorithmically verify if a given pattern sequence is noncorrelated. As an application, we compute that there are exactly $2272$ noncorrelated binary pattern sequences of length $\leq 4$. If we restrict our…

Drawing inspiration from a recent paper of Heuberger, Krenn, and Lipnik, we define the class of strongly k-recursive sequences. We show that every k-automatic sequence is strongly $k$-recursive, therefore k-recursive, and discuss that the…

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. The calculi are internal as they only employ the language of the logic, plus…

We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of…

We show that minimal symplectic 4--manifolds with $b_2^+ >1$ and with residually finite fundamental groups are irreducible. We also give examples of irreducible orientable four--manifolds with indefinite intersection forms which are not…

A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been…

We show that there exist infinitely many pairwise non-isotopic splitting spheres for two unlinked, unknotted $S^2$'s in $S^4$. This answers a question posed by Hughes, Kim, and Miller.

We exhibit several families of planar graphs that are minor-minimal intrinsically spherical $3$-linked. A graph is intrinsically spherical 3-linked if it is planar graph that has, in every spherical embedding, a non-split 3-link consisting…

A list of superconformal chiral operator product expansion algebras with quadratic nonlinearity in two dimensions is completed on the basis of the known classification of little conformal Lie superalgebras. In addition to the previously…

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are either 0, 1 or -1. The only known Barker sequences have length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker sequences…

We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r. This answers a question of Abu-Khazneh, Barat, Pokrovskiy and Szabo, and shows that a long-standing unsolved…

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present a…

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…

We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are…

We produce an infinite family of $2$-complexes that are intrinsically linked when embedded into four dimensions. In particular, we show that any embedding into $\mathbb{R}^4$ of the suspension of a graph containing $K_6$ as a minor contains…

Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra}…

This paper is mainly concerned with the relation-algebraical aspects of the well-known Region Connection Calculus (RCC). We show that the contact relation algebra (CRA) of certain RCC model is not atomic complete and hence infinite. So in…

Two subgroups H and K are 4-permutable in G if <H,K> = HKHK, and H is strong 4-quasinormal in G if H is 4-permutable with every subgroups K of G. A finite group G is called Sq4T-group if strong 4-quasinormality is transitive relation among…

In a relational language consisting of a single relation $ R, $ we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation $ R $ plays a crucial role…

Avramov, Conca, and Iyengar ask whether $\beta_i^S(R) \leq \binom{g}{i}$ for all $i$ when $R=S/I$ is a Koszul algebra minimally defined by $g$ quadrics. In recent work, we give an affirmative answer to this question when $g \leq 4$ by…