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The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic…

We have calculated numerically geometrical means of the denominators of the continued fraction approximations to the Brun constant B2. We get values close to the Khinchin constant. Next we calculated the n-th square roots of the…

In this paper we show how one can obtain simultaneous rational approximants for $\zeta_q(1)$ and $\zeta_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that…

In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…

We propose an interpretation of the Newton's second law that is suggested by Galilean Relativity theory.

With recent advances in natural language processing, rationalization becomes an essential self-explaining diagram to disentangle the black box by selecting a subset of input texts to account for the major variation in prediction. Yet,…

Argumentation is a non-monotonic process. This reflects the fact that argumentation involves uncertain information, and so new information can cause a change in the conclusions drawn. However, the base logic does not need to be…

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent…

Can an absorbing game with rational data have an irrational limit value? Yes: In this note we provide the simplest examples where this phenomenon arises. That is, the following $3\times 3$ absorbing game \[ A = \begin{bmatrix} 1^* & 1^* &…

Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain…

Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the…

We classify all the irrational pencils over the surfaces of general type with p=q=2. This classification adds a new evidence to a Catanese conjecture which states that if S has p=q=2 but no irrational pencils then it is the double cover of…

In this paper, we propose various sufficient conditions to determine if a given real number is an irrational number or a transcendental number and also apply these conditions to some interesting examples, particularly,one of them comes from…

We construct an injection from the set of permutations of length $n$ that contain exactly one copy of the decreasing pattern of length $k$ to the set of permutations of length $n+2$ that avoid that pattern. We then prove that the generating…

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple…

In this article, we give two different proofs of why the Collatz Conjecture is false.

We consider the problem of partitioning effectively a given symmetric (and irreflexive) rational relation R into two asymmetric rational relations. This problem is motivated by a recent method of embedding an R-independent language into one…

An elementary proof of the two-sidedness of the matrix-inverse is given using only linear independence and the reduced row-echelon form of a matrix. In addition, it is shown that a matrix is invertible if and only if it is row-equivalent to…