Boundedness Assumption Research Articles

PARABOLICITY OF SPACELIKE HYPERSURFACES IN GENERALIZED ROBERTSON–WALKER SPACETIMES: APPLICATIONS TO UNIQUENESS RESULTS

We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.

无界抽样情形下不定核的系数正则化回归

We investigate the coefficient-based regularized least squares regression with unbounded sampling in a data dependent hypothesis space. The learning scheme is essentially different from the standard one in a reproducing kernel Hilbert space: we do not need the kernel to be symmetric or positive semi-definite except for continuity and boundedness, the regularizer is the <italic>l</italic>²-norm of a function expansion involving samples and the unboudedness of the sampling output. This leads to additional difficulty in the error analysis. In this paper, the goal is to investigate some concentration estimates for the error based on <italic>l</italic>²-empirical covering numbers without the assumption of uniform boundedness for sampling. By introducing a suitable reproducing kernel Hilbert space and applying concentration techniques with <italic>l</italic>²-empirical covering numbers, we derive satisfactory learning rates in terms of regularity of the regression function and capacity of the hypothesis space.

Nonsimple material problems addressed by the Lagrange’s identity

Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. By using the Lagrange identity, we prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients.

Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

Regularity of optimal transport maps on multiple products of spheres

This article addresses regularity of optimal transport maps for cost = “squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang also implies higher regularity ( C^{1, \alpha}/C^\infty ) of optimal maps for smoother ( C^\alpha/C^\infty ) densities. These are the first global regularity results which we are aware of concerning optimal maps on Riemannian manifolds which possess some vanishing sectional curvatures, beside the totally flat case of \mathbb R^n and its quotients. Moreover, such product manifolds have potential relevance in statistics and in statistical mechanics (where the state of a system consisting of many spins is classically modeled by a point in the phase space obtained by taking many products of spheres). For the proof we apply and extend the method developed in [FKM1], where we showed injectivity and continuity of optimal maps on domains in \mathbb R^n for smooth non-negatively cross-curved cost. The major obstacle in the present paper is to deal with the non-trivial cut-locus and the presence of flat directions.

Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation

Liouville type theorems play crucial roles in proving the existence results of fully nonlinear elliptic equation and elliptic system. Boundedness assumptions of solutions are often imposed in deriving such Liouville type theorems. Although Liouville type theorems with boundedness assumption are enough in proving the existence result, it is interesting to study Liouville type theorem without boundedness assumption. This paper is devoted to establish some Liouville type theorems without boundedness assumption for fully nonlinear elliptic equation and elliptic system. At the same time, we prove some decay estimates for fully nonlinear elliptic equation and elliptic system.

Finite-Rank ADI Iteration for Operator Lyapunov Equations

We give an algorithmic approach to the approximative solution of operator Lyapunov equations for controllability. Motivated by the successfully applied alternating direction implicit (ADI) iteration for matrix Lyapunov equations, we consider this method for the determination of Gramian operators of infinite-dimensional control systems. In the case where the input space is finite-dimensional, this method provides approximative solutions of finite rank. Under the assumption of infinite-time admissibility and boundedness of the semigroup, we analyze convergence in several operator norms. We show that under a mild assumption on the shift parameters, convergence to the Gramian is obtained. Particular emphasis is placed on systems governed by a heat equation with boundary control. We present that ADI iteration for the heat equation consists of solving a sequence of Helmholtz equations. Two numerical examples are presented; the first showing the benefit of adaptive finite elements and the second illustrating con...

A characterization of the mountain pass geometry for functionals bounded from below

In this paper it is proved that, when a regular functional is bounded from below, the mountain pass geometry and the existence of at least two distinct local minima are equivalent conditions. As a consequence, the classical mountain pass theorem, under the additional assumption of boundedness from below of the functional, ensures actually three distinct critical points. Moreover, as application, the existence of three solutions to Hamiltonian systems is established.

An Extended Sequential Quadratically Constrained Quadratic Programming Algorithm for Nonlinear, Semidefinite, and Second-Order Cone Programming

This paper is concerned with nonlinear, semidefinite, and second-order cone programs. A general algorithm, which includes sequential quadratic programming and sequential quadratically constrained quadratic programming methods, is presented for solving these problems. In the particular case of standard nonlinear programs, the algorithm can be interpreted as a prox-regularization of the Solodov sequential quadratically constrained quadratic programming method presented in Mathematics of Operations Research (2004). For such type of methods, the main cost of computation amounts to solve a linear cone program for which efficient solvers are available. Usually, “global convergence results” for these methods require, as for the Solodov method, the boundedness of the primal sequence generated by the algorithm. The other purpose of this paper is to establish global convergence results without boundedness assumptions on any of the iterative sequences built by the algorithm.

Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.

Strong convergence of iterative algorithms with variable coefficients for asymptotically strict pseudocontractive mappings in the intermediate sense and monotone mappings

In this article, we propose some iterative algorithms with variable coefficients for finding a common element of the set of fixed points of a uniformly continuous asymptotically κ-strict pseudocon-tractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. Some strong convergence theorems of these iterative algorithms are obtained without some boundedness assumptions and without some convergence condition. The results of the article improve and extend the recent results of Ceng and Yao, Nadezhkina and Takahashi, and several others. Mathematics Subject Classification (2000): 47H09; 47J20.

General nonexact oracle inequalities for classes with a subexponential envelope

We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy nonexact oracle inequalities in an unbounded framework, under the assumption that the class has a subexponential envelope function. The main novelty, in addition to the boundedness assumption free setup, is that those inequalities can yield fast rates even in situations in which exact oracle inequalities only hold with slower rates. We apply these results to show that procedures based on $\ell_1$ and nuclear norms regularization functions satisfy oracle inequalities with a residual term that decreases like $1/n$ for every $L_q$-loss functions ($q\geq2$), while only assuming that the tail behavior of the input and output variables are well behaved. In particular, no RIP type of assumption or "incoherence condition" are needed to obtain fast residual terms in those setups. We also apply these results to the problems of convex aggregation and model selection.

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Poláčik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.

A consistent numerical scheme for self‐gravitating fluid dynamics

AbstractIn this article, we present a numerical scheme for the 3‐D system of self‐gravitating fluid dynamics in the collisional case as well as in the non‐collisional case. Consistency in the sense of distributions is proved in 1‐D and in absence of pressure. In the other cases consistency is proved under the numerical assumptions of boundedness of the velocity field in the CFL condition and of boundedness of the gradient of the gravitation potential. In 2‐D and 3‐D, concentrations of matter in strings and points can cause a theoretical difficulty in the pressureless case although one observes that the scheme still works. The initial data are L∞ functions in velocity and L1 functions in density. Applications are given to numerical simulations of the role of dark matter and gravitational collapse in cosmology as well as Jeans theory. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Implementation with partial provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agentsʼ preferences or beliefs and agents need little information about each othersʼ preferences. Our results are robust to evidence forgery at any strictly positive cost.

Multivariate Utility Maximization with Proportional Transaction Costs and Random Endowment

In this paper we deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to modeling a currency market with proportional transaction costs). In particular, we extend the results in [L. Campi and M. Owen, Finance Stoch., 15 (2011), pp. 461--499] to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. We start by studying some basic properties of the value function (which is now defined on a space of random variables), and then we dualize the problem following some convex analysis techniques which have proven very useful in this field of research. We finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. The last section of the paper is devoted to an application of our results to utility indifference pricing.

Bounded orbits and measures on a group

We consider a definable group G acting on the space of complete types over G, in a monster model of a theory T. We discuss the notion of a bounded orbit of this action. We prove that some boundedness assumptions imply definable amenability of G.

ERM learning with unbounded sampling

The learning approach of empirical risk minimization (ERM) is taken for the regression problem in the least square framework. A standard assumption for the error analysis in the literature is the uniform boundedness of the output sampling process. In this paper we abandon this boundedness assumption and conduct error analysis for the ERM learning algorithm with unbounded sampling processes satisfying an increment condition for the moments of the output. The key novelty of our analysis is a covering number argument for estimating the sample error.

Pointwise well-posedness in set optimization with cone proper sets

This paper deals with the well-posedness property in the setting of set optimization problems. By using a notion of well-posed set optimization problem due to Zhang et al. (2009) [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed. Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem.

A Nonlinear Control Method Based on ANFIS and Multiple Models for a Class of SISO Nonlinear Systems and Its Application

This paper presents a novel nonlinear control strategy for a class of uncertain single-input and single-output discrete-time nonlinear systems with unstable zero-dynamics. The proposed method combines adaptive-network-based fuzzy inference system (ANFIS) with multiple models, where a linear robust controller, an ANFIS-based nonlinear controller and a switching mechanism are integrated using multiple models technique. It has been shown that the linear controller can ensure the boundedness of the input and output signals and the nonlinear controller can improve the dynamic performance of the closed loop system. Moreover, it has also been shown that the use of the switching mechanism can simultaneously guarantee the closed loop stability and improve its performance. As a result, the controller has the following three outstanding features compared with existing control strategies. First, this method relaxes the assumption of commonly-used uniform boundedness on the unmodeled dynamics and thus enhances its applicability. Second, since ANFIS is used to estimate and compensate the effect caused by the unmodeled dynamics, the convergence rate of neural network learning has been increased. Third, a "one-to-one mapping" technique is adapted to guarantee the universal approximation property of ANFIS. The proposed controller is applied to a numerical example and a pulverizing process of an alumina sintering system, respectively, where its effectiveness has been justified.