A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

MULTIPLICATIVE DERIVATIVE AND ITS BASIC PROPERTIES ON TIME SCALES

We define multiplicative derivative and its properties on time scales. Then, we restate many concepts for multiplicative analysis such as derivative, Rolle’s theorem, mean value theorem and increasing decreasing property on time scales. We aim to create important fields of study by carrying this most important issue of multiplicative analysis, which has applications in economics, finance and many other fields, to time scale calculus

JYOTPATTI

<p>Bhaskaracharya ,(Bhaskar II, 1114- 1185 AD), was one of the great mathematicians in India. His text “<em>Sindhant Shiromani</em>” (SS) was treated as the base of the further research oriented results by most of all the mathematicians after him.<br> SS contains of two parts:, <em>Goladhyaya and Grahaganit</em>. Jyotpatti is the last chapter in <em>Goladhyaya. Jyotpatti</em> consists of 25 <em>Shloka</em> (stanzas), all in Sanskrit language. It is a general impression that SS contains <em>Lilavat</em>i and <em>Beejganit</em> also, but that is not so.<br> <em>Jyotpatti</em> deals with trigonometry. This was a milestone in developing geometry in India. <em>Jya</em> means sine and <em>Utapatti</em> means creation. Hence the name <em>Jyotpatti ( jya + upapatti</em>). <em>Jya and Kojya( or kotijya)</em> stand for the Rsine and Rcosine ratios respectively. The trigonometry developed by Bhaskara II is based on a circle of radius R, and not on a right-angled triangle as taught in the schools.<br> After defining <em>Jya, Kotijya and Utkrama</em> (<em>verse jya) </em>etc, Bhaskara obtains these ratios for the standards angles of 30,45, 60, 36 and 28,(all in degrees) by inscribing a regular polygon in a circle of radius R. Bhaskara called these angles as <em>Panchajyaka. </em>Not only this, Bhaskara developed these results for addition and subtraction of two angles. This result was further developed for the similar results, for the multiple angles. Bhaskara compares <em>jya and kotijya</em> with the longitude –latitude of earth and those with lateral threads of a cloth.<br> Contents in Jyotpatti (Only a few mentioned here)<br> (1) R jya 45 =R, and other similar R jya values. (All in degrees)<br> (2) R jya 36 = 0.5878 approx.<br> (3) Sn = side of a regular polygon of n sides = D sin (π/n), D is the diameter of circle in which polygon is inscribed.<br> (4) Derivation of formulae for sin (θ + ϕ) and cosine (θ + ϕ) called as <em>samas bhavana</em> and <em>antar bhavana</em>.<br> (5) Concept of derivatives, that is, δ(sin θ) = (cos θ) δθ etc. Which is Rolle’s Theorem.<br> Indian mathematicians developed trigonometry in different way than that of western mathematicians. Though <em>Jyotpatti</em> is a small text, it is a landmark in development of ancient and medieval trigonometry.</p>

Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for , where (resp.) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

A simple extension of Rolle’s theorem and its relation with multiple internal rates of return (IRR)

This paper presents a simple extension of Rolle’s Theorem. This extension allows determining the amount of numbers ξi in which f'(ξi) = 0 in a given interval, using the characteristics of the function f in that interval. The extension has been proved, and the geometric interpretation has been presented. Illustrative examples have also been developed for each case that can be obtained by applying the extension. Finally, the study examines the relation of this theorem with the problem of multiple internal rates of return (IRR).

A simple extension of Rolle’s theorem and its relation with multiple internal rates of return (IRR)

It has been stated a simple extension of the Rolle?s theorem. This extension permits to determine the amount of numbers in which in a given interval using the characteristics of the function in that interval. The extension has been proved and the geometric interpretation is presented. Illustrative examples have been developed for each of the cases that can be obtained by applying the extension. The relation of this theorem with the problem of multiple internal rates of return (IRR) is presented.