A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in ^{[<xref ref-type="bibr" rid="b5">5</xref>]} for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

In this paper, we introduce a matrix version of the generalized heat polynomials. Some analytic properties of the generalized heat matrix polynomials are obtained including generating matrix functions, finite sums, and Laplace integral transforms. In addition, further properties are investigated using fractional calculus operators.

Certain Matrix Riemann–Liouville Fractional Integrals Associated with Functions Involving Generalized Bessel Matrix Polynomials

The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim to introduce a slightly modified matrix of Riemann–Liouville fractional integrals and investigate this matrix of Riemann–Liouville fractional integrals associated with products of certain elementary functions and generalized Bessel matrix polynomials. We also consider this matrix of Riemann–Liouville fractional integrals with a matrix version of the Jacobi polynomials. Furthermore, we point out that a number of Riemann–Liouville fractional integrals associated with a variety of functions and polynomials can be presented, which are presented as problems for further investigations.

Program Linier Parametrik

This study aims to discuss a linear program problem where its constants can change, If a constant for a linear program problem is changed, then we don't need to count from the beginning again. Next, we will examine the properties of the objective function as a result of changes in these constants. In this discussion determine the non-negative "critical value" which provides the optimal solution to the problem of parametric linear programs. Searching for critical values in parametric linear programs is done by the matrix version method.

A proof of the matrix version of Baker's conjecture in Diophantine approximation

We prove that the matrix analogue of the Veronese curve is strongly extremal in the sense of Diophantine approximation, thereby resolving a question posed by Beresnevich, Kleinbock and Margulis (2015) in the affirmative.