Dierentiable Functions Research Articles

Inference on Directionally Differentiable Functions

This paper studies an asymptotic framework for conducting inference on parameters of the form ( 0), where is a known directionally dierentiable function and 0 is estimated by ^ n. In these settings, the asymptotic distribution of the plug-in estimator ( ^ n) can be readily derived employing existing extensions to the Delta method. We show, however, that the \standard bootstrap is only consistent under overly stringent conditions { in particular we establish that dierentiability of is a necessary and sucient condition for bootstrap consistency whenever the limiting distribution of ^ n is Gaussian. An alternative resampling scheme is proposed which remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of . We illustrate the utility of our results by developing a test of whether a Hilbert space valued parameter belongs to a convex set { a setting that includes moment inequality problems and certain tests of shape restrictions as special cases.

On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals

In this article, a new general identity for twice dierentiable functions via Riemann-Liouville fractional integrals is established. By making use of this equality, author has obtained new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for functions whose second derivatives in absolute value at certain powers are, respectively, convex and quasi-convex functions via Riemann-Liouville fractional integrals.

Some inequalities of Hermite-Hadamard-like type for the functions whose second derivatives in absolute value are convex

In this article, a new general identity for continuously twice dierentiable functions is established. By making use of this equality, author has obtained new estimates on generalization of Hermite-Hadamardlike type inequalities for functions whose second derivatives in absolute value at certain powers are convex and concave. Mathematics Subject Classication: 26A33, 26A55, 26D10, 26D15

Hermite-Hadamard-like type inequalities for s-convex functions and s-Godunova-Levin functions of two kinds

In this article, the author establish new estimates on generalization of Hermite-Hadamard-like type inequalities for three times dierentiable functions whose its third derivatives in absolute value at certain powers are convex, s-convex and s-Godunova-Levin functions of two kind, via Riemann-Liouville fractional integrals. Mathematics Subject Classication: 26D15, 26A51

Smooth Peano Functions for Perfect Subsets of the Real Line

In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P 2 and, if such a function exists, how smooth can it be. We show that there exists an innitely many times dierentiable function f R R 2 which maps an unbounded perfect set P onto P 2 . At the same time, no continuously dierentiable function f R R 2 can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous function from P onto P 2 if, and only if, P has uncountably many connected components.

Hermite-Hadamard type inequalities for harmonically quasi-convex functions (I)

In this paper, we give some new Hermite-Hadamard type inequalities, which estimate the dierence between the middle and the leftmost terms in the ordinary Hermite-Hadamard inequality, for harmonically convex functions by setting up an integral identity for dierentiable functions.

The one dimensional infinite square well with variable mass

We introduce a numerical method to obtain approximate eigenvalues for some problems of Sturm-Liouville type. As an application, we consider an innite square well in one dimension in which the mass is a function of the position. Two situations are studied, one in which the mass is a dierentiable function of the position depending on a parameter b. In the second one the mass is constant except for a discontinuity at some point. When the parameter b goes to innity, the function of the mass converges to the situation described in the second case. One shows that the energy levels vary very slowly with b and that in the limit as b goes to innity,

On the Simpson and Hermite-Hadamard type inequalities for the geometrically convex functions

In this article, we establish some generalized Simpson type and Hermite-Hadamard type integral inequalities for dierentiable functions whose q-th powers of absolute values of derivatives are monotonically decreasing and geometrically convex. Mathematics Subject Classication: 26D15, 26A51

Hermite-Hadamard type inequalities for harmonically quasi-convex functions (II)

In this paper, we give some new Hermite-Hadamard type inequalities, which estimate the dierence between the middle and the leftmost terms in the ordinary Hermite-Hadamard inequality, for harmonically convex functions by setting up an integral identity for dierentiable functions.

Simpson-like and Hermite-Hadamard-like type inequalities for harmonically quasi-convex functions

In this paper, by using a generalized integral identity for dierentiable functions, the author obtain some new upper bounds of HermiteHadamard type inequalities and new Simpson-like type inequalities, for dierentiable harmonically quasi-convex functions.

Continuity of Darboux functions

In 1965, Whyburn proved that if X is locally connected and first countable, Y is Hausdor, and f: X! Y a function, then: f is continous i f preserves compactness and connectedness. Definition: A function f: X! Y is preserving path connectedness (a Darboux function) if the image of any path-connected subset of X is path connected. As an example, the derivative f ’ of a real dierentiable function f defined on an interval is path preserving although f ’ is not always continuous. In the paper we prove the following theorem Theorem: Suppose X is Hausdor,locally path-connected and 1countable, Y is Hausdor, and f: X! Y a function. Then f is continuous i f preserves compactness and f preserves path connectedness.

Jensen type inequalities for twice dierentiable functions

In this paper, we give some Jensen-type inequalities for \(\varphi: I\rightarrow\mathbb{R}, I=[\alpha,\beta ]\subset\mathbb{R}\) where \(\varphi\) is a continuous function on \(I\); twice differentiable on \(I^°=(\alpha,\beta )\) and there exists \(m = \inf _{x\in I^°} \varphi ''(x)\) or \(M = \sup_{x\in I^°}\varphi '' (x)\). Furthermore, if \(\varphi ''\) is bounded on \(I^°\) ; then we give an estimate, from below and from above of Jensen inequalities.

Neural Network-Based Trajectory Optimization for Unmanned Aerial Vehicles

A neural network approximation to direct trajectory optimization methods is presented. The method uses neural networks to approximate the dynamics and objective equations integrated over a given time interval. The trajectory is then built recursively and treated as a nonlinear programming problem. The method is compared to a direct collocation method as well as more recent pseudospectral methods and shows competitive results while being computationally faster. In addition, a neural network provides a continuously dierentiable function approximation which may be advantageous when a discontinuous objective function is used in a nonlinear solver. A surveillance trajectory planning problem for an unmanned aerial vehicle is given as an example application and results are presented for all three methods.

Non-degeneracy of perturbed solutions of semilinear partial differential equations

The equation i¢u + F(V (x);u) = 0 is considered in R n . For small > 0 it is shown to possess, under appropriate conditions, a non-degenerate solution u in H 2 (R n ). It is shown that the linearised operator T at the solution satisfies kT i1 k = O( i2 ) as ! 0. equation. A solution is non-degenerate when the linearised problem at the solution, in appropriate function spaces, defines an invertible linear operator. Throughout this article, by an invertible operator, with specified Banach spaces as domain and codomain, we shall mean a linear surjective homeomorphism. The notion of non-degeneracy depends on the choice of spaces. In this paper we shall be concerned with problems that can be posed using the Sobolev spaces W 2;2 (R n ) and L 2 (R n ) as domain and codomain respectively. For conciseness we denote these spaces by H 2 and L 2 respectively. The same problems might be posable in spaces of classically dierentiable functions and the question of non-degeneracy in this setting can also arise. There are connections between the two notions of non- degeneracy which do not seem to have been fully explored. In this paper we shall study non-degeneracy for Sobolev spaces only. It seems safe to assume that the applications of non-degeneracy, once established, are many. We could mention the stable behaviour of non-degenerate solutions under perturbations guaranteed by the implicit function theorem. Moreover for equations of the type we shall consider, non-degenerate solutions give rise to multibump solutions, see for example (1) (using spaces of dierentiable

Surjective isometries on spaces of differentiable vector-valued functions

This paper gives a characterization of surjective isometries on spaces of continuously dierentiable functions with values in a nite-dimensional real Hilbert space.

Zero-set property of o-minimal indefinitely Peano differentiable functions

Given an o-minimal expansion M of a real closed field R which is not poly- nomially bounded. Let P 1 denote the definable indefinitely Peano dierentiable functions. If we further assume that M admits P 1 cell decomposition, each definable closed subset A of R n is the zero-set of a P 1 function f : R n ! R. This implies P 1 approximation of definable continuous functions and gluing of P 1 functions defined on closed definable sets.

The Sum-of-Digits Function for Complex Bases

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed. where F q is a continuous, 1-periodic, nowhere dierentiable function with known Fourier expansion. Several more sophisticated digital functions have been studied since then and the fractal behaviour of the summatory functions appeared in many of these cases (cf. (5, 23)). Various methods were used to derive such summation formulae: an early one was developed by Delange (2) and is based on reinterpretation of the occurring sums as real integrals. In (23) and (5) it is observed that the classical technique of Dirichlet generating functions can be used to derive Delange's formula. In this paper we shall generalize some results about 'ordinary' digital expansions to positional number systems of the Gaussian integers, which were introduced by Knuth (20) and extensively studied by Gilbert in a series of papers (7-15). Again fractal structures are involved, but it requires some additional ideas to use the techniques mentioned above in the case of complex bases. In the introductory Section 2 we shall present a number of auxiliary results about digital expansions of complex integers (some of which recall results of Gilbert using a dierent approach). We also exhibit automata, which describe addition of 1 (respectively, other fixed Gaussian integers) in these positional number systems. Furthermore formulae for the sum-of-digits function with respect to complex bases are derived and we analyze the length of the expansion asymptotically.

The Cauchy-Nicoletti problem with poles

The Cauchy-Nicoletti boundary value problem for a sys- tem of ordinary dierential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and non- uniqueness of a solution in the class of continuously dierentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.

SOME TYPICAL PROPERTIES OF SYMMETRICALLY CONTINUOUS FUNCTIONS, SYMMETRIC FUNCTIONS AND CONTINUOUS FUNCTIONS

In this paper we show that the typical symmetrically continuous function and the typical symmetric function have c-dense sets of points of discontinuity. Also we show the existence of a nowhere symmetrically dierentiable function and a nowhere quasi-smooth function by showing directly such functions are typical in the space of all real continuous functions.