Estimation of the location parameter of the ℓ1-norm symmetric matrix variate distributions

An identity of integrals for the ℓ₁-norm symmetric matrix variate distributions with unknown common location parameter and unknown and possibly unequal scale parameters of the columns is established. An unbiased estimator for the location parameter is obtained and is shown to dominate the maximum likelihood estimator under the squared error loss. Under certain conditions this unbiased estimator is the uniformly minimum variance unbiased estimator.     

Asymptotics of the “minimumL 1-norm” estimates in nonparametric regression models

Consider the nonparametric regression modelY=g ₀(T)+u, whereY is real-valued,u is a random error,T ranges over a nondegenerate compact interval, say [0,1], andg ₀(·) is an unknown regression function, which ism(m≥0) times continuously differentiable and itsmth derivative,g ₀ ^(m) , satisfies a Hölder condition of order γ(m+γ>1/2). A piecewise polynomialL ₁-norm estimator ofg ₀ is proposed. Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution, it is proved that the rates of convergence of the piecewise polynomialL ₁-norm estimator are $o(n^{ - \frac{{m + \gamma - 1/2 - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ almost surely and $o(n^{ - \frac{{m + \gamma - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ in probability, which can arbitrarily approach the optimal rates of convergence for nonparametric regression, where σ is any number in (0, min((m+γ?1/2)/3,γ)).     

Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn? 4 the L_∞-norm of weak solutions of the Navier-Stokes equations on ?ⁿ with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε L_p p >n.     

L_1-norm packings from function fields

In this paper,we study some packings in a cube,namely,how to pack n points in a cube so as to maximize the minimal distance.The distance is induced by the L_1-norm which is analogous to the Hamming distance in coding theory.Two constructions with reasonable parameters are obtained,by using some results from a function field including divisor class group,narrow ray class group,and so on.We also present some asymptotic results of the two packings.     

On the convergence in L 1-norm of Cesàro means with respect to representative product systems

The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L ¹-norm, only for certain values of α which depend on some parameter of the representative product system.     

Problems on Two-dimensional Bandwidth under Distance of L_∞-norm

The two-dimensional bandwidth problem is to determine an embedding of graph G in a grid graph in the plane such that the longest edges are as short as possible. In this paper we study the problem under the distance of L∞-norm.     

A simple technique for computing theH ∞-norm of polynomials

In this note, computation of theH -norm of polynomials is considered. It is shown that direct computation of theH -norm of polynomials, based on the definition of the norm, results in a simple and inexpensive technique for computing the norm.     

Itô’s formula for the L p -norm of stochastic {W^{1}_{p}} -valued processes

We prove Itô’s formula for the L _p -norm of a stochastic ${W^{1}_{p}}$ -valued processes appearing in the theory of SPDEs in divergence form.     

On finite element approximation in theL ∞-norm of variational inequalities

Summary We are interested in the approximation in theL -norm of variational inequalities with non-linear operators and somewhat irregular obstacles. We show that the order of convergence will be the same as that of the equation associated with the non-linear operator if the discrete maximum principle is verified.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

Sampled-data minimumH ∞-norm regulation of linear continuous-time systems using multirate-output controllers

This paper deals with the problem of designing multirate-output contrlleers for sampled-dataH -optimal control of linear continuous-time systems. Two formulations of the problem are studied. In the first, the intersample behavior of the disturbance and the controlled output signals is not considered, whereas in the second the continuous-time nature of these signals is taken into account. It is shown that, in both cases and unter appropriate conditions, it is plausible to reduce the repective initial problem to an associated discrete-timeH -optimization problem for which a fictitious static state feedback controller is to be designed. This fact has a beneficial influence on the theoretical and numerical complexity of the problem, since only one algebraic Riccati equation is to be solved here, as compared to two algebraic Riccati equations needed in known techniques concerning theH -optimization problem with dynamic measurement feedback.The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

An exact asymptotics for small deviations of a nonstationary Ornstein-Uhlenbeck process in the L p -norm, p ≥ 2

A result concerning an exact asymptotics for the probability $P\{ \int\limits_0^1 {|\zeta _\gamma (t)|^p dt \leqslant \varepsilon ^p } \} ,\varepsilon \to 0$ , where p ≥ 2, is proved for a nonstationary Gaussian Markov process ζ _γ (t) of Ornstein-Uhlenbeck with zero mean and the covariance function $E\zeta _\gamma (t)\zeta _\gamma (s) = \tfrac{1}{{2\gamma }}[e^{ - \gamma |t - s|} - e^{ - \gamma |t + s|} ]$ , s, t ≥ 0, where γ > 0 is a parameter. Investigation techniques are the Laplace method for the sojourn time of continuous-time Markov processes and reduction to the case of Wiener processes.     

Homogenization of the elliptic Dirichlet problem: Error estimates in the (L2 → H1)-norm

Let      

On the complexity of the surrogate dual of 0–1 programming

It is shown that the surrogate dual of a 0–1 programming problem can be solved by 0(m ³) knapsack calls, ifm denotes the number of constraints.

---

Zusammenfassung Es wird gezeigt, daß das surrogate duale Problem zu einer linearen Optimierungsaufgabe mit binären Variablen durch 0(m ³) Rucksackprobleme gelöst werden kann. Dabei bezeichnetm die Anzahl der Nebenbedingungen.

     

The fundamental group of the complement of the branch curve of ℂℙ1 × T

Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.     

Commutator identities of the homotopes of (−1, 1)-algebras

We study the commutator algebras of the homotopes of (?1, 1)-algebras and prove that they are Malcev algebras satisfying the Filippov identity h _a (x, y, z) = 0 in the case of strictly (?1, 1)-algebras. We also proved that every Malcev algebra with the identities xy ³ = 0, xy ² z ² = 0, and h _a (x, y, z) = 0 is nilpotent of index at most 6.