Rth Derivative Research Articles

Monte Carlo integration of C^r functions with adaptive variance reduction: an asymptotic analysis

The theme of the present paper is numerical integration of C^r functions using randomized methods. We consider variance reduction methods that consist in two steps. First the initial interval is partitioned into subintervals and the integrand is approximated by a piecewise polynomial interpolant that is based on the obtained partition. Then a randomized approximation is applied on the difference of the integrand and its interpolant. The final approximation of the integral is the sum of both. The optimal convergence rate is already achieved by uniform (nonadaptive) partition plus the crude Monte Carlo; however, special adaptive techniques can substantially lower the asymptotic factor depending on the integrand. The improvement can be huge in comparison to the nonadaptive method, especially for functions with rapidly varying rth derivatives, which has serious implications for practical computations. In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration.

On optimal adaptive quadratures for automatic integration

In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the rth derivative is positive within precision $$\varepsilon >0$$ are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses the minimal number of function evaluations. It is also shown how to adapt the quadratures to the case where the assumption that the rth derivative has a constant sign is not fulfilled. The theoretical results are illustrated and confirmed with numerical tests.

Bandwidth choice for density derivative

The methods for choosing appropriate bandwidth (smoothing parameter) are important to approximate the regression function to the original function so that the error is as small as possible. It is always a good starting point, and that the smoothing accuracy depends only on the bandwidth, which in turn affects the degree of smoothing of the estimated curve and its proximity to the true curve. The equilibrium between bias and variance is called the smoothing parameter or the bandwidth parameter. Its value is greater than zero, which reduces the function and its value corresponds to the smallest standard, and that large values of (h) produce smoothed results, because it increases the bias and reduces the variance to estimate the original regression function. It is one of the methods to reduce the mean squares of error. This article explains the structured methods of bandwidth choice for estimating the rth derivative of an univariate kernel employing techniques methods of cross-validation. We introduced new R kedd packages based on R programming development with different kernel functions for computing bandwidth choice for density derivative. Simulation approaches are used to construct the method of bandwidth along with real-life data sets.

On alternative quantization for doubly weighted approximation and integration over unbounded domains

It is known that for a ϱ-weighted Lq approximation of single variable functions defined on a finite or infinite interval, whose rth derivatives are in a ψ-weighted Lp space, the minimal error of approximations that use n samples of f is proportional to ‖ω1∕α‖L1α‖f(r)ψ‖Lpn−r+(1∕p−1∕q)+, where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω1∕α‖L1<+∞. Moreover, the optimal sample points are determined by quantiles of ω1∕α. In this paper, we show how the error of the best approximation changes when the sample points are determined by a quantizer κ other than ω. Our results can be applied in situations when an alternative quantizer has to be used because ω is not known exactly or is too complicated to handle computationally. The results for q=1 are also applicable to ϱ-weighted integration over finite and infinite intervals.

Extensions of Móricz Classes and Convergence of Trigonometric Sine Series in L1-Norm

In this paper, the extensions of classes S, C and BV are made by defining the classes Sr, Cr and BVr,r = 0,1,2,... . It is also shown that class Sr is a subclass of Cr ∩ BVr. Moreover, the results on L1-convergence of r times differentiated trigonometric sine series have been obtained by considering the rth (r = 0,1,2,...) derivative of modified sine sum under the new extended class Cr ∩ BVr.

Applications of the Shorgin Identity to Bernstein Type Operators

By establishing an identity between a sequence of Bernstein-type operators and a sequence of Szasz–Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szasz–Mirakyan operators. As one application of this identity, we prove that whenever the parameters are conveniently chosen, if $$f\in C[0,\infty )$$ satisfies a growth condition of the form $$|f(t)|\le C e^{\alpha t}(C,\alpha \in \mathbb {R}^+)$$ , then the classical Bernstein operators $$B_{mn}(f(nu),x/n)$$ converge to the Szasz–Mirakyan operator $$S_m(f,x)$$ . This convergence generalizes the classical result of De la Cal and Liquin to unbounded functions; moreover, the rth derivative of $$B_{mn}(f(nu),x/n)$$ converges to the rth derivative of $$S_m(f,x)$$ . As another application of this identity, we derive Voronowskaja type result for the general Lototsky–Bernstein operators.

Relative widths of Sobolev classes in the uniform and integral metrics

Let W p r be the Sobolev class consisting of 2π-periodic functions f such that ‖f (r)‖ p ≤ 1. We consider the relative widths d n (W p r , MW p r , L p ), which characterize the best approximation of the class W p r in the space L p by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions g should lie in MW p r , i.e., ‖g (r)‖ p ≤ M. We establish estimates for the relative widths in the cases of p = 1 and p = ∞; it follows from these estimates that for almost optimal (with error at most Cn −r , where C is an absolute constant) approximations of the class W p r by linear 2n-dimensional spaces, the norms of the rth derivatives of some approximating functions are not less than cln min(n, r) for large n and r.

Complexity of certain nonlinear two-point BVPs with Neumann boundary conditions

We study the solution of two-point boundary-value problems for second order ODEs with boundary conditions imposed on the first derivative of the solution. The right-hand side function g is assumed to be r times (r≥1) continuously differentiable with the rth derivative being a Hölder function with exponent ϱ∈(0,1]. The boundary conditions are defined through a continuously differentiable function f. We define an algorithm for solving the problem with error of order m−(r+ϱ) and cost of order mlogm evaluations of g and f and arithmetic operations, where m∈N. We prove that this algorithm is optimal up to the logarithmic factor in the cost. This yields that the worst-case ε-complexity of the problem (i.e., the minimal cost of solving the problem with the worst-case error at most ε>0) is essentially Θ((1/ε)1/(r+ϱ)), up to a log1/ε factor in the upper bound. The same bounds hold for r+ϱ≥2 even if we additionally assume convexity of g. For r=1, ϱ∈(0,1] and convex functions g, the information ε-complexity is shown to be Θ((1/ε)1/2).

Approximation of piecewise Hölder functions from inexact information

We study the Lp-approximation of functions f consisting of two smooth pieces separated by an (unknown) singular point sf; each piece is r times differentiable and the rth derivative is Hölder continuous with exponent ϱ. The approximations use n inexact function values yi=f(xi)+ei with |ei|≤δ. Let 1≤p<∞. We show that then the minimal worst case error is proportional to max(δ,n−(r+ϱ)) in the class of functions with uniformly bounded both the Hölder coefficients and the discontinuity jumps |f(sf+)−f(sf−)|. This error is achieved by an algorithm that uses a new adaptive mechanism to approximate sf, where the number of adaptively chosen points xi is only O(lnn). The use of adaption, p<∞, and the uniform bound on the Hölder coefficients and the discontinuity jumps are crucial. If we restrict the class even further to continuous functions, then the same worst case result can be achieved also for p=∞ using no more than (r−1)+ adaptive points. The results generalize earlier results that were obtained for exact information (where δ=0) and f(r) piecewise Lipschitz (where ϱ=1).

Mean approximation of functions on the real axis by algebraic polynomials with Chebyshev-Hermite weight and widths of function classes

We obtain sharp Jackson-Stechkin type inequalities on the sets L 2, (ℝ) in which the values of best polynomial approximations are estimated from above via both the moduli of continuity of mth order and K-functionals of rth derivatives. For function classes defined by these characteristics, the exact values of various widths are calculated in the space L 2,ρ (ℝ). Also, for the classes $$W_{2,\rho }^r (\mathbb{K}_m ,\Psi )$$ , where r = 2, 3, h3, the exact values of the best polynomial approximations of the intermediate derivatives f (ν), ν = 1,..., r − 1, are obtained in L 2,ρ (ℝ).

R‐consensus control for higher‐order multi‐agent systems with digraph

AbstractThis paper studies a consensus problem for lth (l ≥ 2) order multi‐agent systems with digraph, namely, for a fixed r (0 ≤ r ≤ l − 1), the rth derivative of the states xi of agents are convergent to a constant value and, for every k (0 ≤ k ≤ l − 1), are convergent to zeros. A new concept of r‐consensus is introduced and new consensus protocols are proposed for solving such an r‐consensus problem. A sufficient and necessary condition for r‐consensus is obtained. As special cases, criteria for third‐order systems are given, in which the exact relationship between feedback gains is established. Finally, an illustrative example is given to demonstrate the effectiveness of these protocols.

Critical Values of Higher Derivatives of Twisted Elliptic L-Functions

Let be the L-function of an elliptic curve E defined over the rational field . Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the rth derivatives, L (r)(E, 1, χ), of twists by Dirichlet characters of when L(E, 1, χ)=⋅⋅⋅=L (r−1)(E, 1, χ)=0.

Normal reference bandwidths for the general order, multivariate kernel density derivative estimator

This note derives the general form of the asymptotic approximate mean integrated squared error for the q-variate, νth-order kernel density rth derivative estimator. This formula allows for normal reference rule-of-thumb bandwidths to be derived. We give tables for some of the most common cases in the literature.

On the quantum and randomized approximation of linear functionals on function spaces

We deal with quantum and randomized algorithms for approximating a class of linear continuous functionals. The functionals are defined on a Holder space of functions f of d variables with r continuous partial derivatives, the rth derivative being a Holder function with exponent ?. For a certain class of such linear problems (which includes the integration problem), we define algorithms based on partitioning the domain of f into a large number of small subdomains, and making use of the well-known quantum or randomized algorithms for summation of real numbers. For N information evaluations (quantum queries in the quantum setting), we show upper bounds on the error of order N ?(?+1) in the quantum setting, and N ?(?+1/2) in the randomized setting, where ? = (r + ?)/d is the regularity parameter. Hence, we obtain for a wider class of linear problems the same upper bounds as those known for the integration problem. We give examples of functionals satisfying the assumptions, among which we discuss functionals defined on the solution of Fredholm integral equations of the second kind, with complete information about the kernel. We also provide lower bounds, showing in some cases sharpness of the obtained results, and compare the power of quantum, randomized and deterministic algorithms for the exemplary problems.

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L1-approximations of classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes WrHω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Exact values of best approximations for classes of periodic functions by splines of deficiency 2

We obtain exact values of best L1-approximations for the classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F as well as for the classes WrHω of periodic functions whose rth derivative has a given convex (up) majorant ω(t) of the modulus of continuity by subspaces of polynomial splines of order m ≥ r + 1 of deficiency 2 with nodes at the points 2kπ/n, n ∈ ℕ, k ∈ ℤ. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding function classes.

Best linear approximation methods for functions of Taikov classes in the Hardy spaces H q,ρ , q ≥ 1, 0 &lt; ρ ≤ 1

For the Hardy spaces H q,ρ , q ≥ 1, 0 < ρ ≤ 1, we develop best linear approximation methods for classes of analytic functions W r H q Φ, r ∈ ℕ, in the unit disk (studied by L. V. Taikov) whose averaged second-order moduli of continuity of the angular boundary values of the rth derivatives are majorized by a given function ϕ satisfying certain constraints.

The homology of the cyclic coloring complex of simple graphs

Let G be a simple graph on n vertices, and let χ G ( λ ) denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, Δ ( G ) , and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of ( n − 3 ) rd homology group of Δ ( G ) is equal to ( n − ( r + 1 ) ) plus 1 r ! | χ G r ( 0 ) | , where χ G r is the rth derivative of χ G ( λ ) . We also define a complex Δ ( G ) C , whose r-faces consist of all ordered set partitions [ B 1 , … , B r + 2 ] where none of the B i contain an edge of G and where 1 ∈ B 1 . We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C [ x 1 , … , x n ] / { x i x j | i j is an edge of G } . We show that when G is a connected graph, the homology of Δ ( G ) C has nonzero homology only in dimension n − 2 , and the dimension of this homology group is | χ G ′ ( 0 ) | . In this case, we provide a bijection between a set of homology representatives of Δ ( G ) C and the acyclic orientations of G with a unique source at v, a vertex of G.

Special values of Abelian L-functions at [formula omitted

In [H.M. Stark, L-functions at s = 1 . IV. First derivatives at s = 0 , Adv. Math. 35 (3) (1980) 197–235], Stark formulated his far-reaching refined conjecture on the first derivative of abelian (imprimitive) L-functions of order of vanishing r = 1 at s = 0 . In [Karl Rubin, A Stark conjecture “over Z” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1) (1996) 33–62], Rubin extended Stark's refined conjecture to describe the rth derivative of abelian (imprimitive) L-functions of order of vanishing r at s = 0 , for arbitrary values r. However, in both Stark's and Rubin's setups, the order of vanishing is imposed upon the imprimitive L-functions in question somewhat artificially, by requiring that the Euler factors corresponding to r distinct completely split primes have been removed from the Euler product expressions of these L-functions. In this paper, we formulate and provide evidence in support of a conjecture in the spirit of and extending the Rubin–Stark conjectures to the most general (abelian) setting: arbitrary order of vanishing abelian imprimitive L-functions, regardless of their type of imprimitivity. The second author's conversations with Harold Stark and David Dummit (especially regarding the order of vanishing 1 setting) were instrumental in formulating this generalization.

Adaptive estimation of linear functionals by model selection

We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_{p}$ loss. An application to the problem of estimating a signal or its rth derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator.