Unbounded Derivations Research Articles

Wave breaking in the Whitham equation

We prove wave breaking — bounded solutions with unbounded derivatives — in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of the shallow water equations, provided that the slope of the initial datum is sufficiently negative, whereby we solve a Whitham's conjecture. We extend the result to equations of Korteweg–de Vries type for a range of fractional dispersion.

On weighted conditions for the absolute convergence of Fourier integrals

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivatives.

A Piecewise Nonpolynomial Collocation Method for Fractional Differential Equations

Since the solutions of the fractional differential equations (FDEs) have unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations reflecting the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh

In this article, we approximate the solution of the weakly singular Volterra integral equation of the second kind using the reproducing kernel Hilbert space (RKHS) method. This method does not require any background mesh and can easily be implemented. Since the solution of the second kind weakly singular Volterra integral equation has unbounded derivative at the left end point of the interval of the integral equation domain, RKHS method has poor convergence rate on the conventional uniform mesh. Consequently, the graded mesh is proposed. Using error analysis, we show the RKHS method has better convergence rate on the graded mesh than the uniform mesh. Numerical examples are given to confirm the error analysis results. Regularization of the solution is an alternative approach to improve the efficiency of the RKHS method. In this regard, an smooth transformation is used to regularization and obtained numerical results are compared with other methods.

On the regularity of the generalised golden ratio function

Given a nite set of real numbers A, the generalised golden ratio is the unique real number G(A) > 1 for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that G(A) varies continuously with the alphabet A (of xed size). What is more, we demonstrate that as we vary a single parameter m within A, the generalised golden ratio function may behave like m 1=h for any positive integer h. These results follow from a detailed study of G(A) for ternary alphabets, building upon the work of Komornik, Lai, and Pedicini (2011). We give a new proof of their main result, that is we explicitly calculate the function G(f0; 1;mg). (For a ternary alphabet, it may be assumed without loss of generality that A = f0; 1;mg with m 2 (1; 2)].) We also study the set of m 2 (1; 2] for which G(f0; 1;mg) = 1 + p m; we prove that this set is uncountable and has Hausdor dimension 0. We show that the function mapping m to G(f0; 1;mg) is of bounded variation yet has unbounded derivative. Finally, we show that it is possible to have unique expansions as well as points with precisely two expansions at the generalised golden ratio.

Динамические системы с разрывными решениями и задачи с неограниченными производными

Динамические системы с разрывными решениями и задачи с неограниченными производными

SECOND-ORDER NONCOMMUTATIVE DIFFERENTIAL AND LIPSCHITZ STRUCTURES DEFINED BY A CLOSED SYMMETRIC OPERATOR

The Banach $^{\ast }$-operator algebras, exhibiting the second-order noncommutative differential structure and the noncommutative Lipschitz structure, that are determined by the unbounded derivation and induced by a closed symmetric operator in a Hilbert space, are explored.

Unbounded derivations, free dilations, and indecomposability results for II$_1$ factors

Unbounded derivations, free dilations, and indecomposability results for II$_1$ factors

Recovering Exponential Accuracy in Fourier Spectral Methods Involving Piecewise Smooth Functions with Unbounded Derivative Singularities

Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations, if the solution is analytic. If the solution is discontinuous but piecewise analytic up to the discontinuities, Fourier spectral methods produce poor pointwise accuracy, but still maintain exponential accuracy after post-processing (Gottlieb and Shu in SIAM Rev 30:644---668, 1997) . In Chen and Shu (J Comput Appl Math 265:83---95, 2014), an extended technique is provided to recover exponential accuracy for functions which have end-point singularities, from the knowledge of point values on standard collocation points. In this paper, we develop a technique to recover exponential accuracy from the first $$N$$N Fourier coefficients of functions which are analytic in the open interval but have unbounded derivative singularities at end points. With this post-processing method, we are able to obtain exponential accuracy of spectral methods applied to linear transport equations involving such functions.

Absolutely continuous invariant measure of a map from grazing-impact oscillators

Abstract In this paper we investigate a one-dimensional map with unbounded derivative. The map is the limit of the Nordmark map which is the normal form of a discrete time representation of impact oscillators near grazing, i.e. when the dissipation of the systems is large, the Nordmark map can be viewed as a perturbation of the one-dimensional map. We prove that the map has an ergodic absolutely continuous invariant probability measure in a region of parameter space by constructing an induced Markov map.

Pesin theory and equilibrium measures on the interval

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

First hyperbolic times for intermittent maps with unbounded derivative

We establish some statistical properties of the hyperbolic times for a class of non-uniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric baker’s map-type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last 30 years. The statistics of these maps are controlled by the order of tangency (linked to a single parameter α, where 0 < α < ∞) that a certain ‘cut function’ makes with the boundary of the square. Previously, a direct Young tower construction has been used to obtain optimal correlation decay rates of O(n−1/α) for Hölder observables and all values of the parameter α. A central limit theorem (CLT) is obtained when 0 < α < 1.The asymptotics of a natural hyperbolic time for this family of maps are analysed via the same Young tower. By using a large deviations result of Melbourne and Nicol, we prove that the first hyperbolic time is integrable if and only if the parameter satisfies 0 < α < 1. Furthermore, within this restricted range of parameters, concentration inequalities recently established by Chazottes and Gouëzel imply sharp O(n−1/α) bounds on the tail distribution of first hyperbolic times. As shown by Alves, Viana, and others, knowledge of the tail distribution of the hyperbolic times leads to upper bounds on the rate of decay of correlations and derivation of a CLT. Comparing to the results obtained directly for this family of maps, the latter estimates via hyperbolic times are suboptimal, even over the restricted range of parameters 0 < α < 1.

On cusps and flat tops

We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to $C^{1+\epsilon}$. We do not require the critical points to verify a non-flatness condition, so the results are applicable to $C^{1+\epsilon}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

On graded meshes for weakly singular Volterra integral equations with oscillatory trigonometric kernels

We compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular Volterra integral equation of the second kind with oscillatory trigonometric kernels. Due to, in general, unbounded derivatives at the left endpoint of the interval of integration, we should approximate the solution of the integral equation by collocation methods on graded meshes. However, we show that this non-smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory. Numerical examples are given to confirm the proposed results.

Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations

Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations

One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II1 factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II1 factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.

Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).

Optimization of Supply Chain Systems with Price Elasticity of Demand

A centralized multiechelon, multiproduct supply chain network is presented in a multiperiod setting with products that show varying demand against price. An important consideration in such complex supply chains is to maintain system performance at high levels for varying demands that may be sensitive to product price. To examine the price-centric behavior of the customers, the concept of price elasticity of demand is addressed. The proposed approach includes many realistic features of typical supply chain systems such as production planning and scheduling, inventory management, transportation delay, transportation cost, and transportation limits. In addition, the proposed system can be extended to meet unsatisfied demand in future periods by backordering. Effects of the elasticity in price demand in production and inventory decisions are also examined. The supply chain model is formulated as a convex mixed-integer nonlinear programming problem. Reformulations are presented to make the problem tractable. The differential equations are reformulated as difference equations, and unbounded derivatives in the nonlinear objective function are handled with an approximation, with guaranteed bounds on the loss of optimality. The approach is illustrated on a multiechelon, multiproduct supply chain network.

Lipschitz functions, Schatten ideals, and unbounded derivations

It is proved that, for any Lipschitz function f(t1, ..., tn) of n variables, the corresponding map fop: (A1, ...,An) → f(A1, ..., An) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal Sp, p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in Sp. It is also proved that the map fop is Frechet differentiable in the norm of Sp if f is continuously differentiable.

Calculating Confidence Intervals for Continuous and Discontinuous Functions of Estimated Parameters

The delta method is commonly used to calculate confidence intervals of functions of estimated parameters that are differentiable with non-zero, bounded derivatives. When the delta method is inappropriate, researchers usually first use a bootstrap procedure where they i) repeatedly take a draw from the asymptotic distribution of the parameter values and ii) calculate the function value for this draw. They then trim the bottom and top of the distribution of function values to obtain their confidence interval. This note first provides several examples where this procedure and/or delta method fail to provide an appropriate confidence interval. It next presents a method that is appropriate for constructing confidence intervals for functions that are discontinuous or are continuous but have zero or unbounded derivatives. In particular the coverage probabilities for our method converge uniformly to their nominal values, which is not necessarily true for the other methods discussed above.