Lp Estimates Research Articles

An integrated production planning model with load-dependent lead-times and safety stocks

The divergence over the years of research paradigms addressing the production planning problem has led to the development of an extensive set of techniques, each of which can address a particular aspect of the practical problem and none of which provides a complete solution. In particular, most approaches fail to address the circular, non-linear dependency between resource utilization, lead-times and safety stocks. We present a non-linear programming formulation of the integrated problem using clearing functions that determines a work release schedule guaranteeing a specified service level in the face of stochastic demand. We introduce an iterative heuristic solution procedure that solves a relaxed LP approximation of the original NLP at each iteration to determine the lead-time profile to set safety-stock levels. Computational experiments suggest that our proposed iterative procedure performs well relative to conventional LP models that assume fixed, workload-independent lead-times.

Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrodinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Holder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The Lp and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrodinger type operators H2 are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderon-Zygmund operator if V ∈ B2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x) ≤ Cm(x,V)2.

Simultaneous analysis of Lasso and Dantzig selector

We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the ℓp estimation loss for 1≤p≤2 in the linear model when the number of variables can be much larger than the sample size.

SVM LEARNING AND Lp APPROXIMATION BY GAUSSIANS ON RIEMANNIAN MANIFOLDS

We confirm by the multi-Gaussian support vector machine (SVM) classification that the information of the intrinsic dimension of Riemannian manifolds can be used to illustrate the efficiency (learning rates) of learning algorithms. We study an approximation scheme realized by convolution operators involving the Gaussian kernels with flexible variances. The essential analysis lies in the study of its approximation order in Lp (1 ≤ p < ∞) norm as the variance of the Gaussian tends to zero. It is different from the analysis for approximation in C(X) since pointwise estimations do not work any more. The Lp approximation arises from the SVM case where the approximated function is the Bayes rule and is not continuous, in general. The approximation error is estimated by imposing a regularity condition that the approximated function lies in some interpolation spaces. Then, the learning rates for multi-Gaussian regularized classifiers with general classification loss functions are derived, and the rates depend on the intrinsic dimension of the Riemannian manifold instead of the dimension of the underlying Euclidean space. Here, the input space is assumed to be a connected compact C∞ Riemannian submanifold of ℝn. The uniform normal neighborhoods of the Riemannian manifold and the radial basis form of Gaussian kernels play an important role.

ESTIMATES FOR SINGULAR INTEGRALS ALONG SURFACES OF REVOLUTION

AbstractWe prove certain Lp estimates (1<p<∞) for nonisotropic singular integrals along surfaces of revolution. The singular integrals are defined by rough kernels. As an application we obtain Lp boundedness of the singular integrals under a sharp size condition on their kernels. We also prove a certain estimate for a trigonometric integral, which is useful in studying nonisotropic singular integrals.

Analysis and application of finite volume element methods to a class of partial differential equations

The finite volume element (FVE) methods for a class of partial differential equations are discussed and analyzed in this paper. The new initial values are introduced in the finite volume element schemes, and we obtain optimal error estimates in Lp and W1,p (2⩽p⩽∞) as well as some superconvergence estimates in W1,p (2⩽p⩽∞). The main results in this paper perfect the theory of the finite volume element methods.

A new proof of L p estimates for the parabolic polyharmonic equations

In this paper we obtain local Lp estimates for the parabolic polyharmonic equations by a straightforward approach.

Bernstein–Nikolskii and Plancherel–Polya inequalities in Lp ‐norms on non‐compact symmetric spaces

AbstractBy using Bernstein‐type inequality we define analogs of spaces of entire functions of exponential type in Lp (X), 1 ≤ p ≤ ∞, where X is a symmetric space of non‐compact. We give estimates of Lp ‐norms, 1 ≤ p ≤ ∞, of such functions (the Nikolskii‐type inequalities) and also prove the Lp ‐Plancherel–Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of “measurements” in a stable way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Lp ESTIMATES FOR WEAK SOLUTIONS TO NONLINEAR SUB-ELLIPTIC SYSTEMS RELATED TO HÖRMANDER’S VECTOR FIELDS

This paper is concerned with the regularity of weak solutions to the general nonlinear sub-elliptic systems related to Hormander’s vector fields. Based on the reverse Holder inequality established by Gianazza, the Lp estimates for weak solutions to the systems under the super-quadratic controllable growth condition and the natural growth condition are established, respectively. Then we obtain the Holder continuity for weak solutions.

Estimates for singular integrals and extrapolation

We prove a sharp Lp estimate for a singular Radon transform according to a size condition of its kernel, which is useful for extrapolation.

Parabolic Littlewood–Paley operators

AbstractIn this paper, weak (1,1) and Lp estimates for the parabolic Littlewood–Paley operators on some homogeneous spaces are established, which are extensions of known results. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space \(\mathfrak{h}^{1}(M)\) and John-Nirenberg space \(\mathop{\mathrm{bmo}}(M)\) when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include \(\mathfrak{h}^{1}\) – \(\mathop{\mathrm{bmo}}\) duality, Lp estimates on an appropriate variant of the sharp maximal function, \(\mathfrak{h}^{1}\) and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on Lp-Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.

Non-isotropic weighted L p estimates for the % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGafyOaIyRbaebaaaa!3C49! $$ \bar \partial $$ -equation on piecewise smooth strictly pseudoconvex domains

In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the \( \bar \partial \)-equation on piecewise smooth strictly pseudoconvex domains under a hypothesis of complex transversality in ℂn using the explicit formula of solutions by Berndtsson-Andersson.

L p approximation capability of RBF neural networks

Lp approximation capability of radial basis function (RBF) neural networks is investigated. If g: R+1 → R1 and \( g(\parallel x\parallel _{R^n } ) \) ∈ Llocp(Rn) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in Lp(K) with any accuracy for any compact set K in Rn, if and only if g(x) is not an even polynomial.

Error Estimates for Thin Plate Spline Approximation in the Disk

This paper is concerned with approximation properties of linear combinations of scattered translates of the thin-plate spline radial basis function |·|2log|·| where the translates are taken in the unit disk D in R2. We show that the Lp approximation order for this kind of approximation is 2 + 1/p (for sufficiently smooth functions), which matches Johnson's upper bound and, thus, gives the saturation order. We also show that when one increases the density of the centers at the boundary, approximation order 4 - the best possible order in the absence of a boundary - can be obtained.

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and Lp estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions.

Why always seek the expected value? A discussion relating to the Lp norm

The process of estimating a measurand is based on the choice of an appropriate estimator. The metrologist usually makes this choice by comparing estimator properties such as unbiasedness, efficiency and robustness.We show that when dealing with asymmetric parent distributions, the expected value might not correspond to the measurand value; hence it might not be the correct parameter to estimate. In such cases, another location parameter can fit the measurand better and the properties of candidate estimators should be assessed in relation to such parameters.The paper discusses two examples with reference to the class of the Lp parameters and estimators, which include several commonly used in metrology.

L p solutions to the Cauchy problem of the BGK equation

The BGK model of the Boltzmann equation plays an important role in the kinetic theory of rarefied gases. Some existence and uniqueness results of solutions to its Cauchy problem were established for large initial data under various circumstances [see, for example, Perthame, B., “Global existence to the BGK model of Boltzmann equation,” J. Differ. Equations 82, 191–205 (1989)]. In this paper, by establishing some weighted Lp estimates of the hydrodynamical quantities of a gas, we prove the existence theorem of the Lp solutions to the Cauchy problem and establish the propagation properties of the Lp moments for this kind of solutions.

Lp APPROXIMATION CAPABILITIES OF SUM-OF-PRODUCT AND SIGMA-PI-SIGMA NEURAL NETWORKS

This paper studies the L(p) approximation capabilities of sum-of-product (SOPNN) and sigma-pi-sigma (SPSNN) neural networks. It is proved that the set of functions that are generated by the SOPNN with its activation function in $L_{loc};p(\mathcal{R})$ is dense in $L;p(\mathcal{K})$ for any compact set $\mathcal{K}\subset \mathcal{R};N$, if and only if the activation function is not a polynomial almost everywhere. It is also shown that if the activation function of the SPSNN is in ${L_{loc};\infty(\mathcal{R})}$, then the functions generated by the SPSNN are dense in $L;p(\mathcal{K})$ if and only if the activation function is not a constant (a.e.).

On moduli of smoothness and Fourier multipliers in L p , 0 &lt; p&lt; 1

We prove a theorem on the relationship between the modulus of smoothness and the best approximation in L p , 0 < p < 1, and theorems on the extension of functions with preservation of the modulus of smoothness in L p , 0 < p < 1. We also give a complete description of multipliers of periodic functions in the spaces L p , 0 < p < 1.