Nondifferentiable Points Research Articles

Classification identification and modification of the pitch curve with nondifferentiable points for N-lobed noncircular gear

A novel method for the design of discretionary N-lobed pitch curve and intermittent N-lobed pitch curve is presented. And a general formulation for classification identification model of the pitch curve with nondifferentiable points for N-lobed noncircular gear ( N-LNG) is proposed. The formulation is based on the differentiable principle. In particular, the steepest rotation modification model and method of the pitch curve with discontinuous points for any type of N-LNG are established by resorting to fundamentals of the calculus of variations. The modification model can be calculated via the Euler–Lagrange equation, and its conjugate pitch curves which contain outer contact and inner contact can be obtained in general by using the principle of noncircular gear meshing. This identification and modification model and method are implemented in several numerical examples, and simulation results demonstrate that the pitch curve with nondifferentiable points for N-LNG can be effectively identified and modified.

Approximation of fuzzy numbers using the convolution method

In this study, we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number. We show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non-differentiable points. In previous studies, this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers, the possible non-differentiable points of which were the two endpoints of the 1-cut. The construction of smoothers is a key step in the process for producing approximations. We also show that if appropriate smoothers are selected, then we can use the convolution method to provide approximations that are differentiable, Lipschitz, and that also preserve the core.

Chemical potential and reaction electronic flux in symmetry controlled reactions

In symmetry controlled reactions, orbital degeneracies among orbitals of different symmetries can occur along a reaction coordinate. In such case Koopmans' theorem and the finite difference approximation provide a chemical potential profile with nondifferentiable points. This results in an ill-defined reaction electronic flux (REF) profile, since it is defined as the derivative of the chemical potential with respect to the reaction coordinate. To overcome this deficiency, we propose a new way for the calculation of the chemical potential based on a many orbital approach, suitable for reactions in which symmetry is preserved. This new approach gives rise to a new descriptor: symmetry adapted chemical potential (SA-CP), which is the chemical potential corresponding to a given irreducible representation of a symmetry group. A corresponding symmetry adapted reaction electronic flux (SA-REF) is also obtained. Using this approach smooth chemical potential profiles and well defined REFs are achieved. An application of SA-CP and SA-REF is presented by studying the Cs enol-keto tautomerization of thioformic acid. Two SA-REFs are obtained, JA'(ξ) and JA'' (ξ). It is found that the tautomerization proceeds via an in-plane delocalized 3-center 4-electron O-H-S hypervalent bond which is predicted to exist only in the transition state (TS) region. © 2016 Wiley Periodicals, Inc.

Regularization of derivatives on non-differentiable points

The notion of derivative is limited only to the idealization of linear growth. The paper presents a formal regularization procedure for the derivatives of functions using the fractional velocity, i.e. a local fractional derivation operation. The approach can be applied in strongly non-linear or fractal settings, such as singular functions, Brownian motion, quantum paths, etc.

Compensated convex transforms and geometric singularity\\ extraction from semiconvex functions

The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale $1$-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if $f$ is a semiconvex function with general modulus and $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and can be calculated as $ \lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Frechet) subdifferential $\partial_- f(x)$ of the locally semiconvex $f$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus the limit function $ \mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$ provides a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$. At the same time, the limit also provides an asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda$ approaches $+\infty$. For a locally semiconvex function $f$ with linear modulus we show further that the limit of the gradient of the upper compensated convex transform $ \lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x)$. We also show that for a DC-function $f=g-h$, the scale $1$-edge transform, when $\lambda\to+\infty$, satisfies $ \liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the two convex functions $g$ and $h$ at $x$, respectively.

ON THE STRUCTURE OF THE SINGULAR SET OF A PIECEWISE SMOOTH MINIMAX SOLUTION OF THE HAMILTON–JACOBI–BELLMAN EQUATION

The properties of a minimax piecewise smooth solution of the Hamilton–Jacobi–Bellman equation are studied. It is known the Rankine–Hugoniot conditions are necessary and sufficient conditions for the points of nondifferentiability (singularity) of the minimax solution. We generalize this condition and describe the dimension of smooth manifolds contained in the singular set of the piecewise smooth solution in terms of state characteristics that come to this set. New structural properties of the singular set are obtained in the case where the Hamiltonian depends only on the impulse variable.

Coevolutionary Genetic Algorithm Based on the Augmented Lagrangian Function for Solving the Economic Dispatch Problem

This paper proposes a coevolutionary augmented Lagrangian method (AGCE) for solving the classic economic dispatch problem. This problem becomes non-convex and non-differentiable if valve-point loadings effects are considered in the cost curves of thermal units. In such cases, the evolutionary approaches have proven to be efficient for solving the primal economic dispatch problem; however, the great majority of these methods are not capable of solving the associated dual problem. Furthermore, the solutions obtained by these methods cannot be evaluated concerning their optimality. The AGCE works in the primal-dual subspaces and is able to calculate both primal and dual optimal values. For such a purpose, AGCE processes, in parallel, the evolution of two distinct groups of individuals, associated with primal and dual variables, respectively. The “clouds” of primal and dual points become iteratively denser, and converge to the saddle points associated with the problem, even in the presence of non-differentiability points. Therefore, AGCE makes possible the evaluation of optimality of its solution points. In the results, the AGCE is compared with a traditional interior point method and with a genetic algorithm that works only in the primal subspace.

SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

Abstract. We give the characterization of H¨older differentiability pointsand non-differentiability points of the Riesz-N´agy-Taka´cs (RNT) singularfunction Ψ a,p satisfying Ψ a,p (a) = p. It generalizes recent multifractaland metric number theoretical results associated with the RNT function.Besides, we classify the singular functions using the singularity order de-duced from the H¨older derivative giving the information that a strictlyincreasing smooth function having a positive derivative Lebesgue almosteverywhere has the singularity order 1 and the RNT function Ψ a,p hasthe singularity order g(a,p) = alogp+(1−a)log(1−p)aloga+(1−a)log(1−a) ≥1. 1. IntroductionRecently many ([8, 9, 10, 16]) studied the Cantor function, a singular func-tion which is not strictly increasing and the Minkowski’s Question Mark func-tion which is a strictly increasing singular function. Also J. Parad´isetal. ([17])studied some conditions of the null and infinite derivatives of the RNT strictlyincreasing singular function using metric number theory.Recently we ([4]) also studied multifractal characterization of the null andinfinite derivative sets and the non-differentiability set of the RNT singularfunction, which is the typical singular function related to mutifractal theory.In this paper, we employ the Ho¨lder derivative, which is a generalized form ofthe usual derivative, of the RNT function on the unit interval. This definitionextends the concept of the singularity to the general singularity for a strictlyincreasing continuous function. For every point in the unit interval, we ([4])can give some code or dyadic expansion using digit 0 and 1, generating thedistribution set determined by the frequency of the zero in its expansion. Thedistribution sets in the unit interval are the local dimension sets by a self-similar

On characterization of extremal sets of differentiation of integrals in ℝ2

The paper considers a question of characterization of the sets of nondifferentiability points of integrals with respect to the bases of rectangles and squares. In particular, a complete characterization of uncertainty sets for the integrals of functions from Lp(ℝ2), 1 ≤ p ≤ ∞, with respect to the basis of squares is obtained.

Asymptotics of Likelihood Ratio Tests for General One-sided Hypotheses in the Two-sample Normal Model

We consider the general one-sided hypotheses testing problem expressed as H0: θ1 ⩾ h(θ2) versus H1: θ1 < h(θ2), where h( · ) is not necessary differentiable. The values of the right and the left differential coefficients, h′−( · ) and h′+( · ), at nondifferentiable points play an essential role in constructing the appropriate testing procedures with asymptotic size α on the basis of the likelihood ratio principle. The likelihood ratio testing procedure is related to an intersection–union testing procedure when h′−(θ2) ⩾ h′+(θ2) for all θ2, and to a union–intersection testing procedure when there exists a θ2 such that h′−(θ2) < h′+(θ2).

Propagation of Singularities for Weak KAM Solutions and Barrier Functions

This paper studies the structure of the singular set (points of nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations associated with general mechanical systems on the n-torus. First, using the level set method, we characterize the propagation of singularities along generalized characteristics. Then, we obtain a local propagation result for singularities of weak KAM solutions in the supercritical case. Finally, we apply such a result to study the propagation of singularities for barrier functions.

Smoothness properties of a regularized gap function for quasi-variational inequalities

This article studies continuity and differentiability properties for a reformulation of a finite-dimensional quasi-variational inequality (QVI) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points. We therefore take a closer look at these nondifferentiability points and show, in particular, that under mild assumptions all locally minimal points of the reformulation are differentiability points of the regularized gap function. The results are specialized to generalized Nash equilibrium problems. Numerical results are also included and show that the regularized gap function provides a valuable approach for the solution of QVIs.

Weighted Ostrowski type inequalities for functions with one point of nondifferentiability

We present a weighted generalization involving derivatives of arbitrary order of the recently obtained Ostrowski type inequality for functions with one point of nondifferentiability.

On differentiability of convex operators

The main known results on differentiability of continuous convex operators f from a Banach space X to an ordered Banach space Y are due to J.M. Borwein and N.K. Kirov. Our aim is to prove some “supergeneric” results, i.e., to show that, sometimes, the set of Gâteaux or Fréchet nondifferentiability points is not only a first-category set, but also smaller in a stronger sense. For example, we prove that if Y is countably Daniell and the space L(X,Y) of bounded linear operators is separable, then each continuous convex operator f:X→Y is Fréchet differentiable except for a Γ-null angle-small set. Some applications of such supergeneric results are shown.

Landscapes of non-gradient dynamics without detailed balance: Stable limit cycles and multiple attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.

Application of Particle Swarm Optimization in Figuring out Non-differentiable Point of Function

Owing to the importance of non-differentiable point in a function for economy, engineering and theoretical analysis, this paper brings forward a novel methodology to figure out the non-differentiable point of function which is based on adjusting the models for global extremum and local extremum of particle swarm optimazation (PSO). The algorithm takes the difference between left and right difference quotients as the adaptive value of the particle. By them, it defines local extremum and global extremum for PSO and makes particle close to non-differentiable point of target function, and then figures the point out. The validity of the algorithm is verified by the result of numerical calculation.

A Sequential Quadratic Programming Algorithm for Nonconvex, Nonsmooth Constrained Optimization

We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on open dense subsets of $\mathbb{R}^{n}$. Our method is based on a sequential quadratic programming (SQP) algorithm that uses an $\ell_1$ penalty to regularize the constraints. A process of gradient sampling (GS) is employed to make the search direction computation effective in nonsmooth regions. We prove that our SQP-GS method is globally convergent to stationary points with probability one and illustrate its performance with a MATLAB implementation.

Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers

This paper studies a tracking control system for the attitude dynamics of a rigid body. By selecting an attitude error function carefully, we show that the proposed control system guarantees a desirable tracking performance uniformly for rotational maneuvers involving a large initial attitude error. A strict Lyapunov analysis is presented to show exponential stability, and a sufficient condition to avoid non-differentiable points of the attitude error function is also shown. The proposed control system is directly developed on the special orthogonal group to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. These are illustrated by numerical examples.

DERIVATIVE OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

We give characterizations of the differentiability points and the non-differentiability points of the Riesz-N<TEX>$\'{a}$</TEX>gy-Tak<TEX>$\'{a}$</TEX>cs(RNT) singulr function using the distribution sets in the unit interval. Using characterizations, we show that the Hausdorff dimension of the non-differentiability points of the RNT singular function is greater than 0 and the packing dimension of the infinite derivative points of the RNT singular function is less than 1. Further the RNT singular function is nowhere differentiable in the sense of topological magnitude, which leads to that the packing dimension of the non-differentiability points of the RNT singular function is 1. Finally we show that our characterizations generalize a recent result from the (<TEX>$\tau$</TEX>, <TEX>$\tau$</TEX> - 1)-expansion associated with the RNT singular function adding a new result for a sufficient condition for the non-differentiability points.

Dimensions of non-differentiability points of Cantor functions

For a probability vector $(p_0,p_1)$ there exists a corresponding self-similar Borel probability measure $\mu $ supported on the Cantor set $C$ (with the strong separation property) in ${\mathbb R}$ generated by a contractive similitude $h_i(x)=a_ix+b_i$