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Qualitative Properties of Solutions of Finite System of Differential Equations Involving R-L Sequential Fractional Derivative
Issue:
Volume 10, Issue 2, April 2021
Pages:
38-48
Received:
26 January 2021
Accepted:
14 February 2021
Published:
26 March 2021
Abstract: In this paper, qualitative properties such as existence-uniqueness of solutions of finite system of differential equations involving R-L sequential fractional derivative with initial conditions have been studied. Lower and upper solutions are defined for the problem under investigation. Comparison results are used to develop monotone technique for finite system of differential equations involving R-L sequential fractional derivative with initial conditions when the functions on the right hand side are mixed quasi-monotone. Two convergent monotone sequences are obtained by introducing monotone operator. Lipschitz condition is the key part of the study. Minimal and maximal solutions are obtained by using developed technique. Existence and uniqueness of solutions of finite system of differential equations involving R-L sequential fractional derivative is also proved as an application of the technique.
Abstract: In this paper, qualitative properties such as existence-uniqueness of solutions of finite system of differential equations involving R-L sequential fractional derivative with initial conditions have been studied. Lower and upper solutions are defined for the problem under investigation. Comparison results are used to develop monotone technique for ...
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Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero
Peter Jean-Paul,
Shanaz Wahid
Issue:
Volume 10, Issue 2, April 2021
Pages:
49-61
Received:
16 April 2021
Accepted:
30 April 2021
Published:
8 May 2021
Abstract: Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.
Abstract: Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number...
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A Speedy New Proof of the Riemann's Hypothesis
Issue:
Volume 10, Issue 2, April 2021
Pages:
62-67
Received:
20 March 2021
Accepted:
24 April 2021
Published:
14 May 2021
Abstract: In this paper we show that Riemann's function (xi), involving the Riemann’s (zeta) function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the xi function we deduce a relation between each zero of the function xi and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.
Abstract: In this paper we show that Riemann's function (xi), involving the Riemann’s (zeta) function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series a...
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