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The best developed formulation of closed system quantum theory that handles multiple-time statements, is the consistent (or decoherent) histories approach. The most important weaknesses of the approach is that it gives rise to many…

The discovery of time series motifs has emerged as one of the most useful primitives in time series data mining. Researchers have shown its utility for exploratory data mining, summarization, visualization, segmentation, classification,…

The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph $G$ from some class $K$ of "pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic to a…

The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for…

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the $w$-version of this theorem, where $w$ is a hereditary torsion theory for modules over a…

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In…

We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction…

In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after…

This paper investigates the admissibility of the substitution rule in cyclic-proof systems. The substitution rule complicates theoretical case analysis and increases computational cost in proof search since every sequent can be a conclusion…

In this paper, we introduce the countable chain condition for C*-algebras and study its fundamental properties. We show independence from ZFC of the statement that this condition is preserved under the tensor products of C*-algebras.

The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $\epsilon > 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c…

The fluctuation theorem for entropy production is a remarkable symmetry of the distribution of produced entropy that holds universally in non-equilibrium steady states with Markovian dynamics. However, in systems with slow degrees of…

Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where…

Given a finite group $G$ and a prime $p$, we establish the sieve formula, which is a congruence containing as summands numbers of chains of $p$-subgroups of $G$ of certain orders. This generalises the Theorem of Sylow-Frobenius, using…

Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is…

We present an exposition of the *Chain Bounding Lemma*, which is a common generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest…

One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…

The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the…

Let w be a factor of Fibonacci sequence F=x_1x_2..., then it appears in the sequence infinitely many times. Let w_p be the p-th appearance of w and v_{w,p} be the gap between w_p and w_{p+1}. In this paper, we discuss the structure of the…

A graph class $\mathscr{C}$ is called monadically stable if one cannot interpret, in first-order logic, arbitrary large linear orders in colored graphs from $\mathscr{C}$. We prove that the model checking problem for first-order logic is…