Strong Regularity Condition Research Articles

Strongly Regular Differential Variational Systems

A differential variational system is defined by an ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution of a finite-dimensional variational inequality containing the state variable of the system. This paper addresses two system-theoretic topics for such a nontraditional nonsmooth dynamical system; namely, (non-)Zenoness and local observability of a given state satisfying a blanket strong regularity condition. For the former topic, which is of contemporary interest in the study of hybrid systems, we extend the results in our previous paper, where we have studied Zeno states and switching times in a linear complementarity system (LCS). As a special case of the differential variational inequality (DVI), the LCS consists of a linear, time-invariant ODE and a linear complementarity problem. The extension to a nonlinear complementarity system (NCS) with analytic inputs turns out to be non-trivial as we need to use the Lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state. Further extension to a differential variational inequality is obtained via its equivalent Karush-Kuhn-Tucker formulation. For the second topic, which is classical in system theory, we use the non-Zenoness result and the recent results in a previous paper pertaining to the B-differentiability of the solution operator of a nonsmooth ODE to obtain a sufficient condition for the short-time local observability of a given strongly regular state of an NCS. Refined sufficient conditions and necessary conditions for local observability of the LCS satisfying the P-property are obtained

On the Bohr formula for the one-dimensional Schrödinger operator with increasing potential

This article contains the justification of the quasi-classical asymptotic formula (usually attributed to N. Bohr) for the counting function N(λ) of the 1-D Schrödinger operator with potential V increasing at infinity. Results were known under strong regularity conditions on V. Using direct methods based on the phase formalism, we give much wider sufficient conditions on V for the applicability of the Bohr formula. Several counterexamples show that these conditions cannot be significantly improved.

Averaging of functional differential inclusions in Banach spaces

Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of non-existence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved. Under stronger regularity conditions, approximation orders for Krylov–Bogoliubov–Mitropolskii type right-hand sides are derived.

Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates

Let (X(t))t∈[0,1] be a centered Gaussian process with stationary increments such that IE[(X{u+t-Xu)2] = C|t|s+r(t). Assume that there exists an extra parameter β > 0 and a polynomial P of degree smaller than s + β such that |r(t)-P(t)| is bounded with respect to |t|s+β. We consider the problem of estimating the parameter s ∈ (0,2) in the asymptotic framework given by n equispaced observations in [0, 1]. Adding possibly stronger regularity conditions to r, we define classes of such processes over which we show that s cannot be estimated at a better rate than nmin(1/2, β). Then, we study increment (or, more generally, discrete variation) estimators. We obtained precise bounds of the bias of the variance which show that the bias mainly depend on the parameter β and the variance on two terms, one depending on the parameter s and one on some regularity properties of r. A central limit theorem is given when the variance term relying on s dominates the bias and the other variance term. Eventually, we exhibit an estimator which achieves the minimax rate over a wide range of classes for which sufficient regularity conditions are assumed on r.

On a local uniform bootstrap validity

Some relations between smoothness of statistical functionals and locally uniform validity of Efron's bootstrap for the estimators generated by the functionals are studied. It is shown that the Fréchet differentiability easily implies the locally uniform bootstrap validity. In an important example of empirical median it is indicated that indeed stronger regularity conditions are necessary to make the bootstrap a very reliable tool. It is also shown that a smoothing method based on Huber's M‐estimation applied to the median may lead to an improved simple bootstrap there.

An SQP method for general nonlinear programs using only equality constrained subproblems

In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numcrical tests. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.

Spin Models on Bipartite Distance-Regular Graphs

Spin models were introduced by V. Jones (Pac. J. Math.137 (1989), 311-336) to construct invariants of knots and links. A spin model will be defined as a pair S = (X, w) of a finite set X and a function w on X × X satisfying several axioms. Some important spin models can be constructed on a distance-regular graph Γ = (X, E) with suitable complex numbers t0, t1, ..., td (d is the diameter of Γ) by putting w(a, b) = t∂(a, b). In this paper we determine bipartite distance-regular graphs which give spin models in this way with distinct t1, ..., td We show that such a bipartite distance-regular graph satisfies a strong regularity condition (it is 2-homogeneous), and we classify bipartite distance-regular graphs which satisfy this regularity condition.

The Boundary Harnack Principle for Non-Divergence form Elliptic Operators

If L is a uniformly elliptic operator in non–divergence form, the boundary Harnack principle for the ratio of positive L–harmonic functions holds in Holder domains of order α if α > 1/2. A counterexample shows that 1/2 is sharp. For Holder domains of order α with α ∈ (0, 1], the boundary Harnack principle holds provided the domain also satisfies a strong uniform regularity condition. Research partially supported by NSF grants DMS 8822053 and DMS 8901255. 1

Effective model theory vs. recursive model theory

Recursive model theory is supposed to be the study of the effectiveness of constructions and theorems in model theory. This often involves getting “effective” versions of various classical model-theoretic notions. The traditional way of doing this is to restrict attention to recursive models, and recursive isomorphisms between them, etc. Thus for example the following definition appears in the literature (in [3] and [1]).Definition. Given a recursive model A and an n Є ω, a subset R ⊆ An is called intrinsically r.e. provided that for every recursive model B ≈ A, the isomorphic image in B of R is an r.e. subset of Bn.It is clear that if R is definable by a (recursive, infinitary) Σ10 formula (with finitely many parameters from A), then R is intrinsically r.e. It seems natural for the converse to be true. Indeed, provided that (A, R) is sufficiently “regular” in a sense made precise in a theorem of Ash and Nerode (see [3]), the converse is true. However, if we drop the (rather strong) regularity conditions, there exist “pathological” examples of intrinsically r.e. relations which are not definable by a Σ10 formula (see [7]).In this paper, we suggest a rather different approach to studying the effectiveness of model theory, an approach we have dubbed “effective model theory”. The basic idea is to allow arbitrary nonrecursive models, but to require all notions to be relativized to the complexity of the models involved. (Much the same notion has been used in [2] under the name “relatively recursive model theory”.) Thus for example we have the following effective model theory version of the property of being intrinsically r.e.

Kernel estimation of the survival function and hazard rate under weak dependence

Abstract Let {X(k)} be a strictly stationary stochastic process, where X(k) takes values in Rd and has a probability density f. Let F (x)= 1 -F(x) be the survival function and r(x) be the hazard rate defined by r(x)=f(x)/ F (x) , where F is the distribution function of X(k). Let X1,…,Xn ben consecutive observations of {X(k)} and F n (x) =n − 1 ∑ n j= 1 Y j (x) , where Y j >x= 1 if X j >x , and Yj(x)=0 otherwise, j = 1,…,n. As estimators of F (x) and r(x), we consider F n (x) and ⨍ n (x)/ F n (x) respectively, where ⨍ n (x) is a kernel type estimator of ⨍(x) . Appropriate conditions for the uniform consistency of the estimators are given and sharp rates of convergence are also obtained. The stochastic process is assumed to satisfy certain strong mixing and absolute regularity conditions.

A.P.O. rules are asymptotically non deficient for estimation with squared error loss

The problem considered is sequential estimation of the mean θ of a one-parameter exponential family of distributions with squared error loss for estimation error and a cost c>0 for each of an i.i.d. sequence of potential observations X 1, X 2,...A Bayesian approach is adopted, and natural conjugate prior distributions are assumed. For this problem, the asymptotically pointwise optimal (A.P.O.) procedure continues sampling until the posterior variance of θ is less than c(r0+n), where n is the sample size and r 0 is the fictitous sample size implicit in the conjugate prior distribution. It is known that the A.P.O. procedure is Bayes risk efficient, under mild integrability conditions. In fact, the Bayes risk of both the optimal and A.P.O. procedures are asymptotic to 2V 0 √c, as c→0, where V 0 is the prior expectation of the standard deviation of X 1 given θ. Here the A.P.O. rule is shown to be asymptotically non-deficient, under stronger regularity conditions: that is, the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of a single observation, as c→0. The result is illustrated in the exponential and Bernoulli cases, and extended to the case of a normal distribution with both the mean and variance unknown.

A strong regularity condition on mappings

A strong regularity condition is given which generalizes the notion of a completely regular mapping defined by Dyer and Hamstrom. These mappings are shown to be fiber spaces of various types with some additional hypotheses. Strongly regular mappings on manifolds whose point inverses are ANR or AR are considered. For example, it is shown that with AR fibers the map is either a homeomorphism or the manifold has boundary which meets all non- degenerate point inverses.

Spectrum of the Static Potential Schrodinger Equation over E n

over x e En, Xeuclidean n space, for u E L2(E,), where V2 = 82/8x1 + * * + a2/8x2, i3 is the spectral parameter, and V(x) is a prescribed real measurable function. The basic spectral theory of (1.1) for the half line, u e L2(O, o ) with a boundary condition at x = 0, has been developed by Weyl [1] and amplified by Titchmarsh [2] and the recent work of Wintner and others, [3]-[6], in the American Journal. The results here extend easily to the whole line, u e L2(E1), but for L2(En) with n > 1 the theory, [8]-[15], is much less complete, particularly results on the analysis of the spectrum. In all these papers fairly strong local regularity conditions are imposed on V(x) in order that (1.1) may have meaning. However, by using a construction suggested by p. 429 of Carleman's paper [8], we can get a Green's function for (1.1) under only a very mild global integrability condition on V(x), specifically I V 1a, < 1 for some large a, where as in (3.2)