Related papers: A determinantal approach to irrationality
We introduce the notion of density of a rational language with respect to a sequence of probability measures. We prove that if $(\mu_n)$ is a sequence of Bernoulli measures converging to a positive Bernoulli measure $\overline{\mu}$, the…
We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…
We investigate the theory PAI (Peano Arithmetic with Indiscernibles). Models of PAI are of the form (M, I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M, I) satisfies the extended induction scheme…
We consider a class of real numbers, a subset of irrational numbers and certain mathematical constants, for which the elements in the simple continued fraction appears to be random. As an illustrative example, one can consider $\pi = \{x_0,…
In this paper, authors construct a new type of sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[0], a[1], ..., a[n]} and minimal extra-super increasing sequence…
Linear recursions with integer coefficients, such as the one generating the Fibonacci sequence, have been intensely studied over millennia and yet still hide new mathematics. Such a recursion was used by Ap\'ery in his proof of the…
Anthropic reasoning is a critical tool to understand probabilities, especially in a large universe or multiverse. According to anthropic reasoning, we should consider ourselves typical among members of a reference class that must include…
One advantage of paraconsistent logic is that it can deal with inconsistencies without making the system trivial. However, unlike classical propositional calculus, its deductive system is limited, and the meaning of paraconsistent negation…
This paper expands on existing learned models of human behavior via a measured step in structured irrationality. Specifically, by replacing the suboptimality constant $\beta$ in a Boltzmann rationality model with a function over states…
In this paper we prove the following renewal-type limit theorem. Given an irrational $\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $\alpha$ which exceeds R. The main result in the paper is that the…
Our central observation is that unbounded additive recurrence establishes a homomorphism between $\mathbb{N}$ and Modus Ponens in a constructive sense. By finding sums of nonconsecutive Fibonacci indices, each inference step corresponds to…
Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We show that number-theoretical properties of $\theta$ have a strong effect on the convergence when…
Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove…
A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides…
Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian…
We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given real…
Although conventional logical systems based on logical calculi have been successfully used in mathematics and beyond, they have definite limitations that restrict their application in many cases. For instance, the principal condition for…
We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…
Basic results in combinatorial mathematics provide the foundation for a theory and calculus for reasoning about sequential behavior. A key concept of the theory is a generalization of Boolean implicant which deals with statements of the…
A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $\phi$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $\phi$. In…