SHORT NOTE | doi:10.20944/preprints202006.0370.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: Covid-19 outbreak; SARS-Cov-2 coronavirus; reproduction numbers; deterministic SEIR models; parameter determination; robust methods
Online: 30 June 2020 (11:40:43 CEST)
We discuss the generation of various reproduction ratios or numbers to monitor the outbreak of Covid-19 or other epidemics and examine the effects of intervention/relaxation measures. A detailed SEIR algorithm is described for their computation, with applications given to the current Covid-19 outbreak in several countries in America (Argentina, Brazil, Mexico, US) and Europe (France, Italy, Spain and UK). The corresponding matlab script, complete and ready to use, is provided for free downloading.
ARTICLE | doi:10.20944/preprints202006.0366.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Cramer's conjecture; elementary proof; Firoozbakht's conjecture; Farideh Firoozbakht; Legendre conjecture; maximal prime gaps Supremum; prime gaps
Online: 30 June 2020 (10:32:40 CEST)
The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.
ARTICLE | doi:10.20944/preprints202006.0365.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: elementary proof of the Riemann's Hypothesis; prime gaps; Prim Number Theorem; Tailored logarithmic integral; Supremum of prime counting function
Online: 30 June 2020 (10:30:45 CEST)
This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.
ARTICLE | doi:10.20944/preprints202006.0355.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Polylogarithm function; Polyexponential function; Frobenius-Genocchi polynomials; Poly-Frobenius-Genocchi polynomials
Online: 30 June 2020 (07:54:52 CEST)
Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.
Mon, 29 June 2020
ARTICLE | doi:10.20944/preprints202006.0353.v1
Subject: Mathematics & Computer Science, Analysis Keywords: COVID-19; Epidemiology; COVID-19 Analysis and Forecast in Pakistan; Forecasting; Estimation; ARIMA; Prophet; SIRD; Diffusion; Analysis
Online: 29 June 2020 (10:50:47 CEST)
The COVID-19 infections in Pakistan are spreading at an exponential rate and a point may soon be reached where rigorous prevention measures would need to be adopted. Mathematical models can help define the scale of an epidemic and the rate at which an infection can spread in a community. I used ARIMA Model, Diffusion Model, SIRD Model and Prophet Model to forecast the magnitude of the COVID-19 pandemic in Pakistan and compared the numbers with the reported cases on the national database. Results depicts that Pakistan could hit peak number of infectious cases between June 2020 and July, 2020.