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This paper discusses the estimation of average treatment effects in observational causal inferences. By employing a working propensity score and two working regression models for treatment and control groups, Robins et al. (1994, 1995) introduced the augmented inverse probability weighting (AIPW) method for estimation of average treatment effects, which extends the inverse probability weighting (IPW) method of Horvitz and Thompson (1952); the AIPW estimators are locally efficient and doubly robust. In this paper, we study a hybrid of the empirical likelihood method and the method of moments by employing three estimating functions, which can generate estimators for average treatment effects that are locally efficient and doubly robust. The proposed estimators of average treatment effects are efficient for the given choice of three estimating functions when the working propensity score is correctly specified, and thus are more efficient than the AIPW estimators. In addition, we consider a regression method for estimation of the average treatment effects when working regression models for both the treatment and control groups are correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. (1994, 1995). Finally, we present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification. [ABSTRACT FROM AUTHOR]

In this paper, we consider a perturbation of a Laguerre-Hahn functional by adding a derivative of a Dirac delta. This transformation leaves invariant the family of Laguerre-Hahn linear functionals. We shall also analyse the class of the perturbed linear functional. Finally, we illustrate these perturbations by adding the derivative of a Dirac delta to the first kind associated functional of the classical Laguerre functional. The expression of the new orthogonal polynomials is obtained. [ABSTRACT FROM AUTHOR]

The paper details a comprehensive system for the treatment of the topic of limits—conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers properly fits the context of limits, allowing precise and unambiguous notation in limit computation. More importantly, the near-numbers are a set of concrete mathematical objects that provide a scaffold both for building an accurate intuitive understanding of the limit concept and for connecting that understanding to the formal definition of the limit. [ABSTRACT FROM AUTHOR]

In this paper, we prove, under natural assumptions, that the semilinear elliptic system -Δu+u=g(x,v), x ∈ RN, -Δv+v=f(x,u), x∈RN (0.1) possesses at least one positive solution pair (u, v) ∈ H¹(RN) × H¹(RN), where f(x, t) and g(x, t) are continuous functions on RN × R and asymptotically linear as t → + ∞. [ABSTRACT FROM AUTHOR]

In this article, we investigate the chaotic behaviours in the fractional-order Lorenz system. By utilizing the fractional calculus techniques, we found that chaos exists in the fractional-order Lorenz system of order less than 3. The lowest order we found to have chaos in this system is 2.97. [ABSTRACT FROM AUTHOR]

The article reports on the nature of projectile motion in a uniform, downward gravitational field and resisted by a drag force due to the air through which it moves. It states that the nature of the projectile motion lies near the heart of the scientific revolution. The drag force is assumed to act in a direction opposite to that of the projectile's velocity and to depend only on its magnitude, therefore, Coriolis forces and Earth's curvature are ignored. The author notes that the projectile motion under quadratic drag was at the center of an episode in the calculus priority dispute.

In this article, we consider a general modification of the Kohn–Nirenberg domain in ℂn, namely,. In particular, we study the existence of support surface and the regularity of peak function at the point 0∈∂Ωk. [ABSTRACT FROM AUTHOR]

We investigate the Extended Proximal Auxiliary Problem Principle (EPAPP) [A. Kaplan and R. Tichatschke, Extended Auxiliary Problem Principle using Bregman distances, Optimization 53 (2004), pp. 603-623, A. Kaplan and R. Tichatschke, Bregman functions and auxiliary problem principle, Optim. Methods Softw. 23, 1 (2008), pp. 95-107] for solving variational inequalities whose operator - here called the main operator - is the sum of a maximal monotone and a continuous operator. As usual for proximal methods using Bregman distances, Kalpan and Tichatschke required that the main operator is paramonotone. There are two main purposes of the present article. First, we establish the convergence of the EPAPP method without the use of paramonotonicity by replacing the latter assumption by other (partly rather natural) ones. An example showing that the main operator does not even have to be monotone is also given. Second, we discuss the EPAPP method with a stopping criterion of fixed-relative-error type by Solodov and Svaiter [An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25 (2000), pp. 214-230]. Kaplan and Tichatschke used a stopping criterion of summable-error type was used to admit an inexact solution of the corresponding auxiliary problems. To our knowledge, such a stopping criterion has not been applied to an extended APP yet; also in this situation, the paramonotonicity hypothesis is avoided. Independent of the stopping criterion under consideration, we admit an outer approximation of the set-valued component of the main operator. Owing to the use of Bregman-like functions to construct the symmetric components of the auxiliary operators, an interior-point effect is provided, meaning that - with a certain precaution - the auxiliary problems can be treated as unconstrained ones. [ABSTRACT FROM AUTHOR]

This study examines how students in the early stages of learning about the concept of functions, describe a curve and, in particular, evaluate the appropriateness of their argument about the representation of a function. Students are offered a message game which is related to a curve drawn on a coordinate system, representing an 'imaginary function'. In this message game, one half of the class must describe a curve to the other half who will then try to reproduce it as similarly as possible. Data analysis indicates that the idea of the curve that consists of series of points is very effective, while the idea of variation does not exist or is too limited among students at the beginning of learning functions. Thus, the concept of variation is not used as a criterion to describe a curve. More surprisingly, the students are working on the curve as if it was just a picture on a grid regardless of the representation of the function. [ABSTRACT FROM AUTHOR]

The article reports on the sharpening of Wielandt's characterization of the gamma function. It presents the two conditions that is satisfied with Euler's gamma function. According to the author, the two conditions by themselves do not characterize the gamma function and presents characterization of the gamma function through theorems that Wielandt discovered. He adds that a characterization in the real domain, found in 1922 by other scholars, is well known, while Wielandt's theorem is function-theoretic.

The article presents a discussion about the Koch Curve. It states that the mathematically popular Koch curve is usually used as an example to illustrate a continuous but not differentiable function and as an example of a non-rectifiable curve. However, it emphasizes that only the fact that it is not rectifiable is proved, and the proof that it is a curve and that no point does this curve have a tangent line is disregarded. It offers a mathematical and simple geometric proof that the Koch curve is indeed an arc and that it at no point has a tangent line.

The article presents a discussion about a summation formula that is used in certain transformations on generating algebraic and differential functions and the extraction and creation of coefficients. The goal of the article is to offer the method of coefficients. Mathematical functions are manipulated in order to arrive at a single and simple mathematical expression. It mentions that generating functions have emerged as one of the most popular approaches to combanitorial problems, which includes the analysis of algorithms.

We investigate a variational approach to the Hele–Shaw flow ∂̸ t χ = ▵ u + f χ, f ≥ 0 in ℝ N , where χ is the characteristic function of an open set Ω( t ) in ℝ N and solves - ▵ u ( t , ·) = f in Ω( t ). By choosing a time step τ k = τ/2 k and iteratively solving a variational problem in ℝ N , we construct a staircase family of opens sets and a corresponding family of functions: as k → ∞, both sets and functions converge increasingly, at fixed time, to a weak solution of the problem. When the latter is not unique, the solution thus obtained is characterized by a minimality property, with respect to set inclusion, at fixed time. We also prove several monotonicity results of the solutions thus obtained, with respect to both the initial set and the forcing term f . In particular, these monotonicity properties imply that Ω( t ) has finite perimeter for every t , provided that Ω(0) has finite perimeter. Finally, under very mild assumptions, we prove that the number of connected components is nonincreasing, that Ω( t ) is connected for large t , and that it tends to fill the whole of ℝ N . [ABSTRACT FROM AUTHOR]

Exact solutions of some important non-linear partial differential equations in one and two dimensions are obtained using the tanh function method. The efficiency of the method is demonstrated by applying it to a variety of selected equations. [ABSTRACT FROM AUTHOR]

A fourth-order parabolic partial differential equation in one space variable which arises in the study of transverse vibrations of a uniform flexible beam is solved. The numerical solution is obtained by using a new three-level method based on a sextic spline in space and finite-difference discretization in time. Stability analysis of the method has been carried out. It is shown that we obtain a scheme of O ( k 2 + h 4 ) and O ( h 4 + h 2 k 2 ). The method is tested on a problem which has appeared in physical applications. Comparison with some known methods shows the superiority of the present method. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for multipliers of a convolution of a smooth generalized integration operator to be topological automorphisms in the space of continuous functions. Moreover, an equivalent form of the conditions is found in terms of cyclic elements of the considered integration operator. [ABSTRACT FROM AUTHOR]

Mild sufficient conditions for the ergodicity of the Gibbs sampler are obtained. The geometric ergodicity of the Gibbs sampler is then discussed, where the main tool is the "drift" criterion. This criterion requires the construction of a generalized energy function so that the chain is dissipating energy, at a suitable rate, whenever it lies outside the "center" of the state-space. For the "two-variable" case, a useful recipe is provided for constructing appropriate energy functions. These results are illustrated with four examples. [ABSTRACT FROM AUTHOR]

The article discusses a closed form formula for the nth derivative of a mathematical function which satisfies Riccati's differential equation with constant coefficients. The mathematical theorem which the authors introduced for the nth derivative of a function is based on mathematical induction. The definition and the basic properties of Eulerian numbers are also discussed, along with examples where the closed formula can be applied, and in which one of the examples, the authors arrived at an integral expression for Eulerian numbers.

The article presents an elementary proof of Joris's Theorem. It utilizes Rolle's theorem, L'Hospital's rule, and Taylor's formula from elementary calculus. Joris considers the conditions under which a quotient of smooth functions is again smooth. Let g(x) = f(x)m and h(x) = f(x)n. It is assumed that, since m and n are relatively prime, one of m or n is odd. Function f is continuous as the composition of the continuous functions. Function f is continuous as the composition of the continuous functions n√x and h. Let K = f -1 ({0}). Then K is closed, and 0 = R \ K is an open subset of R, hence the disjoint union of countably many open intervals. On each of these intervals, both n√x and h are smooth on O.