ARTICLE | doi:10.20944/preprints201708.0056.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: expression; analytic property; Fuss–Catalan number; Catalan–Qi function; Catalan number; monotonicity; logarithmic convexity; complete monotonicity; minimality; inequality
Online: 15 August 2017 (08:36:23 CEST)
In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.
ARTICLE | doi:10.20944/preprints202207.0137.v1
Subject: Mathematics & Computer Science, Analysis Keywords: dissipative systems; upper and lower estimates; decay estimates; monotonicity method
Online: 8 July 2022 (10:13:41 CEST)
In this work we introduce a novel approach to generate lower and upper L2 estimates for solution derivatives of arbitrary order to a general class of dissipative systems in the case that such estimates are available for the solutions themselves. Our method also works in reverse order: from the L2 estimates of solution derivatives of some (arbitrary) order we can derive lower and upper L2 estimates for the solutions and then to their derivatives of any order. This procedure is based on very simple monotonicity properties combined with standard energy estimates in physical space, following previous ideas of Kreiss, Hagstrom, Lorenz and Zingano. For simplicity, it is applied here in the context of algebraic rates, but the method can be used in other contexts as well (exponential, logarithm, and so forth).
ARTICLE | doi:10.20944/preprints202011.0315.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: necessary and sufficient condition; complete monotonicity; monotonicity; inequality; polygamma function; lower bound; sectional curvature; manifold; beta distribution; convolution theorem for the Laplace transforms and analytic techniques
Online: 10 November 2020 (13:26:15 CET)
In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.
ARTICLE | doi:10.20944/preprints201611.0146.v1
Subject: Mathematics & Computer Science, Analysis Keywords: explicit form; inhomogeneous linear ordinary differential equation; derivative; Lerch transcendent; absolute monotonicity; complete monotonicity; Bernstein function; inequality; diagonal recurrence relation; Stirling numbers of the first kind; logarithmic function
Online: 29 November 2016 (08:00:53 CET)
In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.
ARTICLE | doi:10.20944/preprints202002.0066.v3
Subject: Keywords: information theory; causal inference; causal tensors; transfer entropy; partial information decomposition; left monotonicity; identity property; unobserved common cause
Online: 27 February 2020 (10:55:05 CET)
We propose a partial information decomposition based on the newly introduced framework of causal tensors, i.e., multilinear stochastic maps that transform source data into destination data. The innovation that causal tensors introduce is that the framework allows for an exact expression of an indirect association in terms of the constituting, direct associations. This is not possible when expressing associations only in measures like mutual information or transfer entropy. Instead of a priori expressing associations in terms of mutual information or transfer entropy, the a posteriori expression of associations in these terms results in an intuitive definition of a nonnegative and left monotonic redundancy, which also meets the identity property. Our proposed redundancy satisfies the three axioms introduced by Williams and Beer. Symmetry and self-redundancy axioms follow directly from our definition. The data processing inequality ensures that the monotonicity axiom is satisfied. Because causal tensors can describe both mutual information as transfer entropy, the partial information decomposition applies to both measures. Results show that the decomposition closely resembles the decomposition of other another approach that expresses associations in terms of mutual information a posteriori. A negative synergistic term could indicate that there is an unobserved common cause.
ARTICLE | doi:10.20944/preprints202207.0166.v1
Subject: Mathematics & Computer Science, Analysis Keywords: incompressible micropolar ﬂows; vorticity and micro-rotation; dissipative systems; monotonicity method; upper and lower estimates
Online: 12 July 2022 (04:03:22 CEST)
In this work the close relation between vorticity and micro-rotation in micropolar flows in Rn (n = 2 or 3) is identified and used to explain the faster decay by t^(-1/2) of the angular velocity of the micro-rotation of fluid particles, as well as establishing its optimality. For this purpose important upper and lower bounds for Leray solutions in homogeneous Sobolev spaces are derived, using the monotonicity approach recently introduced by the authors for dissipative systems in general. Several related results of interest are also given along the discussion.
ARTICLE | doi:10.20944/preprints201708.0079.v2
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Faa di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; logarithmically absolute monotonicity; logarithmically complete monotonicity; determinantal inequality; product inequality
Online: 25 August 2017 (08:41:30 CEST)
In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
ARTICLE | doi:10.20944/preprints202011.0343.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: complete monotonicity; necessary and sufficient condition; difference; derivative; trigamma function; convolution theorem for the Laplace transforms; Bernstein's theorem for completely monotonic functions
Online: 12 November 2020 (11:58:39 CET)
In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic.