Volume 1: Issue 3

Volume 1: Issue 3

**Relevant Functions to (p,q)-Gamma Function and (p,q)-Beta Function**

**Author(s): ** Areen Al-Khateeb

** DOI: **10.5281/zenodo.3471723

** Keywords : ** (p,q)-Gamma function; (p,q)-Beta function

** Cite this article: ** Areen Al-Khateeb (2019). *Relevant Functions to (p,q)-Gamma Function and (p,q)-Beta Function*. Online Mathematics Journal, 01(03), 1–14. DOI: 10.5281/zenodo.3471723. **[BibTeX Export]**

In this article, we study some new functions (namely Γ ̃_(p,q) (z)=Γ_(q/p) (z) and B ̃_(p,q) (s,t)=Γ_(q/p) (s,t)) and we will show how these functions are relevant to (p,q)-Gamma function and (p,q)-Beta function.

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**A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties**

**Author(s): ** Hend Dawood and Yasser Dawood

** DOI:** 10.5281/zenodo.3234184

** Keywords : ** Interval mathematics; Interval dependency; Functional dependence; Skolemization; Guaranteed bounds; Interval enclosures; Interval functions; Quantifiable uncertainty; Scientific knowledge; Reliability; Fuzzy mathematics; InCLosure

** Cite this article: ** Hend Dawood and Yasser Dawood (2019). *A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties*. Online Mathematics Journal, 01(03), 15–36. DOI: 10.5281/zenodo.3234184. **[BibTeX Export]**

**Supplementary Material: **http://doi.org/10.5281/zenodo.3466032

Progress in scientific knowledge discloses an increasingly paramount use of quantifiable properties in the description of states and processes of the real-world physical systems. Through our encounters with the physical world, it reveals itself to us as systems of uncertain quantifiable properties. One approach proved to be most fundamental and reliable in coping with quantifiable uncertainties is interval mathematics. A main drawback of interval mathematics, though, is the persisting problem known as the “interval dependency problem”. This, naturally, confronts us with the question: Formally, what is interval dependency? Is it a meta-concept or an object-ingredient of interval and fuzzy computations? In other words, what is the fundamental defining properties that characterize the notion of interval dependency as a formal mathematical object? Since the early works on interval mathematics by John Charles Burkill and Rosalind Cecily Young in the dawning of the twentieth century, this question has never been touched upon and remained a question still today unanswered. Although the notion of interval dependency is widely used in the interval and fuzzy literature, it is only illustrated by example, without explicit formalization, and no attempt has been made to put on a systematic basis its meaning, that is, to indicate formally the criteria by which it is to be characterized. Here, we attempt to answer this long-standing question. This article, therefore, is devoted to presenting a complete systematic formalization of the notion of interval dependency, by means of the notions of Skolemization and quantification dependence. A novelty of this formalization is the expression of interval dependency as a logical predicate (or relation) and thereby gaining the advantage of deducing its fundamental properties in a merely logical manner. Moreover, on the strength of the generality of the logical apparatus we adopt, the results of this article are not only about classical intervals, but they are meant to apply also to any possible theory of interval arithmetic. That being so, our concern is to shed new light on some fundamental problems of interval mathematics and to take one small step towards paving the way for developing alternate dependency-aware interval theories and computational methods.

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Dawood, H. & Dawood, Y. (2019), ‘Parametric Intervals: More Reliable or Foundationally Problematic?’, Online Mathematics Journal 1(3), 37–54.

Dawood, H. & Dawood, Y. (2017), ‘Investigations into a Formalized Theory of Interval Differentiation'(CU-Math-2017-03-IFTID), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2016), ‘Interval Algerbras: A Formalized Treatment'(CU-Math-2016-06-IAFT), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2016), ‘The Form of the Uncertain: On the Mathematical Structures of Uncertainty'(CU-Math-2016-09-FUMSU), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2014), ‘On Some Order-theoretic Aspects of Interval Algebras'(CU-Math-2014-06-OTAIA), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

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**Parametric Intervals: More Reliable or Foundationally Problematic?**

**Author(s): ** Hend Dawood and Yasser Dawood

** DOI: **10.5281/zenodo.3234186

** Keywords : ** Interval mathematics, Classical interval arithmetic, Parametric interval arithmetic, Constrained interval arithmetic, Overestimation-free interval arithmetic, Interval dependency, Functional dependence, Dependency predicate, Interval enclosures, S-semiring, Uncertainty, Reliability

** Cite this article: ** Hend Dawood and Yasser Dawood (2019). *Parametric Intervals: More Reliable or Foundationally Problematic?*. Online Mathematics Journal, 01(03), 37–54. DOI: 10.5281/zenodo.3234186. **[BibTeX Export]**

Interval arithmetic has been proved to be very subtle, reliable, and most fundamental in addressing uncertainty and imprecision. However, the theory of classical interval arithmetic and all its alternates suffer from algebraic anomalies, and all have difficulties with interval dependency. A theory of interval arithmetic that seems promising is the theory of parametric intervals. The theory of parametric intervals is presented in the literature with the zealous claim that it provides a radical solution to the long-standing dependency problem in the classical interval theory, along with the claim that parametric interval arithmetic, unlike Moore’s classical interval arithmetic, has additive and multiplicative inverse elements, and satisfies the distributive law. So, does the theory of parametric intervals accomplish these very desirable objectives? Here it is argued that it does not.

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Dawood, H. (2019), ‘On Some Algebraic and Order-Theoretic Aspects of Machine Interval Arithmetic’, Online Mathematics Journal 1(2), 1–13.

Dawood, H. (2018), ‘InCLosure (Interval enCLosure)–A Language and Environment for Reliable Scientific Computing’, Technical report, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, InCLosure Support: http://scholar.cu.edu.eg/henddawood/software/InCLosure.

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Dawood, H. & Dawood, Y. (2020), Universal Intervals: Towards a Dependency-Aware Interval Algebra, in S. Chakraverty, ed., ‘Mathematical Methods in Interdisciplinary Sciences’, John Wiley & Sons, Hoboken, New Jersey.

Dawood, H. & Dawood, Y. (2019), ‘A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties’, Online Mathematics Journal 1(3), 15–36.

Dawood, H. & Dawood, Y. (2017), ‘Investigations into a Formalized Theory of Interval Differentiation'(CU-Math-2017-03-IFTID), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2016), ‘Interval Algerbras: A Formalized Treatment'(CU-Math-2016-06-IAFT), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2016), ‘The Form of the Uncertain: On the Mathematical Structures of Uncertainty'(CU-Math-2016-09-FUMSU), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2014), ‘On Some Order-theoretic Aspects of Interval Algebras'(CU-Math-2014-06-OTAIA), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2013), ‘Logical Aspects of Interval Dependency'(CU-Math-2013-03-LAID), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Dawood, Y. (2013), ‘A Dependency-Aware Interval Algebra'(CU-Math-2013-09-DAIA), Technical report, Department of Mathematics, Faculty of Science, Cairo University.

Dawood, H. & Megahed, N. (2019), ‘A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives’, Punjab University Journal of Mathematics 51(11), 77–100.

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Fu, C.; Liu, Y. & Xiao, Z. (2019), ‘Interval Differential Evolution with Dimension-Reduction Interval Analysis Method for Uncertain Optimization Problems’, Applied Mathematical Modelling 69, 441–452.

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Markov, S. M. (1995), ‘On Directed Interval Arithmetic and its Applications’, Journal of Universal Computer Science 1(7), 514–526.

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Shayer, S. (1965), ‘Interval Arithmetic with Some Applications for Digital Computers'(LMSD-5136512), Technical report, Lockheed Missiles and Space Company, Lockheed Corporation, Palo Alto, CA.

Skolem, T. A. (1920), ‘Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme über dichte Mengen’, Skrifter utgit av Videnskabsselskapet i Kristiania, I. Matematisk-Naturvidenskabelig Klasse No. 4, pp. 1–36, Translated from Norwegian as “Logico-combinatorial investigations on the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by Loewenheim” by Stefan Bauer-Mengelberg..

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Tarski, A. (1994), Introduction to Logic and to the Methodology of the Deductive Sciences, Oxford University Press, New York.

Vaananen, J. (2007), Dependence Logic: A New Approach to Independence Friendly Logic, Cambridge University Press.

Young, R. C. (1931), ‘The Algebra of Many-Valued Quantities’, Mathematische Annalen 104, 260–290.

Heath, T. L., ed. (2009), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters, Cambridge University Press, Cambridge.

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