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Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging…
Consider exponential Carmichael function $\lambda^{(e)}$ such that $\lambda^{(e)}$ is multiplicative and $\lambda^{(e)}(p^a) = \lambda(a)$, where $\lambda$ is usual Carmichael function. We discuss the value of $\sum \lambda^{(e)}(n)$, where…
We axiomatically define a cardinal social inefficiency function, which, given a set of alternatives and individuals' vNM preferences over the alternatives, assigns a unique number -- the social inefficiency -- to each alternative. These…
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
A notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one…
Describing complex phenomena by means of cellular automata (CA) has shown to be a very effective approach in pure and applied sciences. In fact, the number of published papers concerning this topic has tremendously increased over the last…
Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with the axioms of a Cartesian…
The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the…
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance…
Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…
It is regrettable that the quantum length of an object is rarely if ever discussed, because it provides an ideal pedagogical paradigm for understanding how a physicist uses classical intuition to define quantum properties and how such…
The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained…
Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single…
In this paper we treat the specification problem in classical realizability (as defined in [20]) in the case of arithmetical formul{\ae}. In the continuity of [10] and [11], we characterize the universal realizers of a formula as being the…
Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to…
Within the gossamer numbers which extend the real numbers to include infinitesimals and infinities we prove the Fundamental Theorem of Calculus (FTC). Riemann sums are also considered in the gossamer number system, and their non-uniqueness…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…
We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these…
We consider fluctuations of error terms $\Delta(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $\Omega$ and $\Omega_{\pm}$ estimates. We obtain $\Omega$…