Mathematics

The presented analysis determines several new bounds on the roots of the equation a n x n + a n−1 x n−1 +⋯+ a 0 =0 (with a n >0 ). All proposed new bounds are lower than the Cauchy bound max {1, ∑ n−1 j=0 | a j / a n |} . Firstly, the Cauchy bound formula is derived by presenting it in a new light -- through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients a 0 , a 1 ,…, a n−1 , but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. The following bounds are also found in this paper: max {1,( ∑ q j=1 B j / A l ) 1/(l−k) } , where B 1 , B 2 ,… B q are the absolute values of all of the negative coefficients in the equation, k is the highest degree of a monomial with a negative coefficient, A l is the positive coefficient of the term A l x l for which k<l≤n .

This paper presents some new inequalities, the most important of which is the inequality given in Theorem 2.1. It can solve a class of inequalities by a unified method. An important application of the inequality given in Theorem 2.1 is to derive another new general form of inequality. The famous Nesbitt's inequality is a special case of this general form of inequality when n = 3. The new inequality in Theorem 2.1 proposed in this paper is easy to use and expand, and many new inequalities can be derived and obtained by direct calculation, so it has a wide range of applications. Many known inequalities can also be directly calculated by the inequalities proposed in this paper, and the calculation is simple and convenient.

The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT) for solving steady heat-transfer problems. The proposed integral transform is a generalization of the Sumudu, and the Laplace transforms and its visualization is more comfortable than the Sumudu transform, the natural transform, and the Elzaki transform. The suggested integral transform is used to solve the steady heat-transfer problems, and results are compared with the results of the existing techniques.

In this paper, new refinements for integral and sum forms of Hölder inequality are established. We note that many existing inequalities related to the Hölder inequality can be improved via obtained new inequalities in here, we show this in an application.

New condition is found for the set of points in the plane, for which the locus is a circle. It is proved: the locus of points, such that the sum of the (2m) -th powers S (2m) n }of the distances to the vertexes of fixed regular n -sided polygon is constant, is a circle if S (2m) n >n r 2m , where m=1,2,…,n−1 and r is the distance from the center of the regular polygon to the vertex. The radius ℓ satisfies: S (2m) n =n[( r 2 + ℓ 2 ) m + ∑ k=1 [ m 2 ] ( m 2k )( r 2 + ℓ 2 ) m−2k (rℓ ) 2k ( 2k m )].

In this paper, a modified nonlinear Schrödinger equation with spatio-temporal dispersion is formulated in the senses of Caputo fractional derivative and conformable derivative. A new generalized double Laplace transform coupled with Adomian decomposition method has been defined and applied to solve the newly formulated nonlinear Schrödinger equation with spatio-temporal dispersion. The approximate analytical solutions using the proposed generalized method in the sense of Caputo fractional derivative and conformable derivatives are obtained and compared with each other graphically.

In 1966, B. O'Neill [The fundamental equations of a submersion, Michigan Math. J., Volume 13, Issue 4 (1966), 459-469.] obtained some fundamental equations and curvature relations between the total space, the base space and the fibres of a submersion. In the present paper, we define new curvature tensors along Riemannian submersions such as Weyl projective curvature tensor, concircular curvature tensor, conharmonic curvature tensor, conformal curvature tensor and M− projective curvature tensor, respectively. Finally, we obtain some results in case of the total space of Riemannian submersions has umbilical fibres for any curvature tensors mentioned by the above.

In this paper, a new estimate is obtained for the multinomial Mittag-Leffler function. This function was introduced by Yuri Luchko and Rudolfo Gorenflo as the fundamental solution of the ordinary differential equation of fractional discrete distributed order.

We prove new theorems for the polynomial expansions of x n ± y n in terms of the binary quadratic forms α x 2 +βxy+α y 2 and a x 2 +bxy+a y 2 . The paper gives new arithmetic differential approach to compute the coefficients. Also, the paper gives generalization to well-known polynomial identity in the history of number theory. The paper highlights the emergence of a new class of polynomials that unify many well-known sequences including the Chebyshev polynomials of the first and second kind, Dickson polynomials of the first and second kind, Lucas and Fibonacci numbers, Mersenne numbers, Pell polynomials, Pell-Lucas polynomials, and Fermat numbers. Also, this paper highlights the emergence of the notions of trajectories and orbits of certain integers that passes through many well-known polynomials and sequences. The Lucas-Fibonacci trajectory, the Lucas-Pell trajectory, the Fibonacci-Pell trajectory, the Fibonacci-Lucas trajectory, the Chebyshev-Dickson trajectory of the first kind, the Chebyshev-Dickson trajectory of the second kind, and others are new trajectories included in this paper. Also, the Lucas orbit, Fibonacci orbit, Mersenne orbit, Lucas-Fibonacci orbit, Fermat orbit, and others are new orbits included in this paper.

It is well known that the Leibniz rule for the integer derivative of order one does not hold for the fractional derivative case when the fractional order lies between 0 and 1. Thus it poses a great difficulty in the calculation of fractional derivative of given functions as well as in the analysis of fractional order systems. In this work, we develop a few fractional differential inequalities which involve the Caputo fractional derivative of the product of continuously differentiable functions. We establish some of their properties and propose a few propositions. We show that these inequalities play a very essential role in the Lyapunov stability analysis of nonautonomous fractional order systems.