Absolute Monotonicity Research Articles

Absolute Monotonicity of Normalized Tail of Power Series Expansion of Exponential Function

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis.

On the Absolute Monotonicity of the Logarithmic of Gaussian Hypergeometric Function

On the Absolute Monotonicity of the Logarithmic of Gaussian Hypergeometric Function

On the absolute monotonicity of generalized elliptic integral of the first kind

On the absolute monotonicity of generalized elliptic integral of the first kind

Differentially Private Condorcet Voting

Designing private voting rules is an important and pressing problem for trustworthy democracy. In this paper, under the framework of differential privacy, we propose a novel famliy of randomized voting rules based on the well-known Condorcet method, and focus on three classes of voting rules in this family: Laplacian Condorcet method (CMLAP), exponential Condorcet method (CMEXP), and randomized response Condorcet method (CMRR), where λ represents the level of noise. We prove that all of our rules satisfy absolute monotonicity, lexi-participation, probabilistic Pareto efficiency, approximate probabilistic Condorcet criterion, and approximate SD-strategyproofness. In addition, CMRR satisfies (non-approximate) probabilistic Condorcet criterion, while CMLAP and CMEXP satisfy strong lexi-participation. Finally, we regard differential privacy as a voting axiom, and discuss its relations to other axioms.

Absolute monotonicity of the accuracy of Ramanujan approximations for perimeter of an ellipse

Absolute monotonicity of the accuracy of Ramanujan approximations for perimeter of an ellipse

Absolute Monotonicity Involving the Complete Elliptic Integrals of the First Kind with Applications

Absolute Monotonicity Involving the Complete Elliptic Integrals of the First Kind with Applications

Several Absolutely Monotonic Functions Related to the Complete Elliptic Integral of the First Kind

Several Absolutely Monotonic Functions Related to the Complete Elliptic Integral of the First Kind

New Wilker-type and Huygens-type inequalities

In this paper, we first determine the relationships between the first Wilker's inequality, the second Wilker's inequality, the first Huygens inequality, and the second Huygens inequality for circular functions and for hyperbolic functions, respectively. Then, we establish new Wilker-type inequalities and Huygens-type inequalities for two function pairs, $x/\sin^{-1}x$ and $x/\tan ^{-1}x$, $x/\sinh ^{-1}x$ and $x/\tanh ^{-1}x$. Finally, we obtain some more general conclusions than the first work of this paper, which reveal the absolute monotonicity of four functions involving the four inequalities mentioned above.

On sharp bounds for ratios of 𝑘-balanced hypergeometric functions

We extend recently obtained sharp bounds for ratios of zero-balanced hypergeometric functions to the general k k -balanced case, k ∈ N k\in \mathbb {N} . We also discuss the absolute monotonicity of generalizations of previously studied functions involving generalized complete elliptic integrals.

On Young's inequality

We present some inequalities for trigonometric sums. Among others, we prove the following refinements of the classical Young inequality.(1)Let m≥3 be an odd integer, then for all n≥m−1,∑k=1ncos⁡(kθ)k≥∑k=1m(−1)kk. The sign of equality holds if and only if n=m and θ=π. The special case m=3 is due to Brown and Koumandos (1997).(2)For all even integers n≥2 and real numbers r∈(0,1] and θ∈[0,π] we have∑k=1ncos⁡(kθ)krk≥−548(5+5)=−0.75375.... The sign of equality holds if and only if n=4, r=1 and θ=4π/5. We apply this result to prove the absolute monotonicity of a function which is defined in terms of the log-function.

Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

Perturbed Runge--Kutta methods (also referred to as downwind Runge--Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge--Kutta counterparts. In this paper we study, the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.

An Optimally Stable and Accurate Second‐Order SSP Runge‐Kutta IMEX Scheme for Atmospheric Applications

AbstractThe objective of this paper is to develop an optimized implicit‐explicit (IMEX) Runge‐Kutta scheme for atmospheric applications focusing on stability and accuracy. Following the common terminology, the proposed method is called IMEX‐SSP2(2,3,2), as it has second‐order accuracy and is composed of diagonally implicit two‐stage and explicit three‐stage parts. This scheme enjoys the Strong Stability Preserving (SSP) property for both parts. This new scheme is applied to nonhydrostatic compressible Boussinesq equations in two different arrangements, including (i) semiimplicit and (ii) Horizontally Explicit‐Vertically Implicit (HEVI) forms. The new scheme preserves the SSP property for larger regions of absolute monotonicity compared to the well‐studied scheme in the same class. In addition, numerical tests confirm that the IMEX‐SSP2(2,3,2) improves the maximum stable time step as well as the level of accuracy and computational cost compared to other schemes in the same class. It is demonstrated that the A‐stability property as well as satisfying “second‐stage order” and stiffly accurate conditions lead the proposed scheme to better performance than existing schemes for the applications examined herein.

A diagonal recurrence relation for the Stirling numbers of the first kind

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.

On the Laplace Transform of Absolutely Monotonic Functions

We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.

Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications

In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function.

Unconditionally Strong Stability Preserving Extensions of the TR-BDF2 Method

We analyze monotonicity, strong stability and positivity of the TR-BDF2 method, interpreting these properties in the framework of absolute monotonicity. The radius of absolute monotonicity is computed and it is shown that the parameter value which makes the method L-stable is also the value which maximizes the radius of monotonicity. In order to achieve unconditional monotonicity, hybrid variants of TR-BDF2 are proposed, that reduce the formal order of accuracy, while keeping the native L-stability property, which is useful for the application to stiff problems. Numerical experiments compare these different hybridization strategies to other methods used in stiff and mildly stiff problems. The results show that the proposed strategies provide a good compromise between accuracy and robustness at high CFL numbers, without suffering from the limitations of alternative approaches already available in literature.

SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

Abstract. By employing a refined version of the P´olya type integralinequality and other techniques, the authors establish some inequalitiesand absolute monotonicity for modified Bessel functions of the first kindwith nonnegative integer order. 1. Main resultsIt is well known that modified Bessel functions of the first kind I ±ν (z) aresolutions of the differential equationz 2 d 2 wdz 2 +zdwdz−z 2 +ν 2 w = 0.They are holomorphic functions of z throughout the z-plane cut along thenegative real axis, and are entire functions of ν for fixed z 6= 0. When ν = ±n,I ν (z) are entire functions of z. In [1, p. 375, 9.6.7], it is listed thatI ν (z) =X ∞k=0 1k!Γ(ν +k +1)z2 2k+ν , z ∈ C, ν ∈ R\{−1,−2,...},whereΓ(z) = lim n→∞ n!n z Q nk=0 (z +k), z ∈ C\{0,−1,−2,...}is the classical gamma function, see [1, p. 255, 6.1.2].On [12, p. 63], the following three double inequalities are derived:1− z/21+ z/2 0, ν ≥ −12; Received July 18, 2015.2010 Mathematics Subject Classification. Primary 33C10; Secondary 26A48, 26D15,44A10.Key words and phrases. inequality, absolute monotonicity, complete monotonic function,modified Bessel function of the first kind, P´olya type integral inequality.

Preserving positivity for matrices with sparsity constraints

Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graphGGor a family of graphsGnG_n. Our results generalize Schoenberg and Rudin’s results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning, graphical models). The only such result known in the literature is for the complete graphK2K_2. We provide the first such characterization result for a large family of non-complete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and superadditive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.

Harmonic functions on the lattice: Absolute monotonicity and propagation of smallness

In this work we establish a connection between two classical notions, unrelated so far: Harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity type and propagation of smallness results for harmonic functions on the lattice.

Heine Representations and Monotonicity Properties of Determinants and Pfaffians

We establish integral representations of Heine type for certain integrals of determinants. We use these representations to derive monotonicity properties of such determinants. New integral representations of Heine type for biorthogonal functions obtained from the general Gram–Schmidt orthonormalization process are given. We also establish the monotonicity of certain Pfaffians using the de Bruijn formula of Ishikawa and Zeng. The complete or absolute monotonicity of a number of Pfaffian functions is established.