A general vectorial Ekeland’s variational principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.     

A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

Failure of the Hasse principle for Enriques surfaces

We construct an Enriques surface X over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer–Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.     

Numerical optimization applying trust-region reflective least squares algorithm with constraints to optimize the non-linear creep parameters of soft soil

Determination of the creep model parameters is a challenging task particularly when a non-linear elastic visco-plastic (EVP) model is adopted, mainly due to the limited test duration as well as the assumption of the reference time. Therefore, this paper presents an innovative numerical solution to find the EVP model parameters applying the trust-region reflective least square optimization algorithm. The developed approach involves several available laboratory consolidation test results in the optimization procedure with the adopted commencing time to creep as a unit of time. In this paper, the laboratory results of Ottawa clay were employed to demonstrate the limitation of the recent method to obtain model parameters. Furthermore, the developed method is verified against Skå-Edeby clay in the laboratory conditions. The EVP model parameters are obtained by applying the developed method to the available laboratory consolidation results of clay samples. The analysis results of vertical strains and excess pore water pressures demonstrate that the developed method can be a feasible tool to estimate the settlement properties of clays.     

A class of second order differential equations

We study the differential equationf″=N(f)f′ ² +M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.     

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of multiple eigenvalues of infinite tridiagonal matrices

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of ; (B) the zeros of ; (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their ``double pairs'.

     

Z-Join Spectra of Z-Supercompactly Generated Lattices

The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces.     

Invariant fields of finite irreducible reflection groups

We prove the following result: If G is a finite irreducible reflection group defined over a base field k, then the invariant field of G is purely transcendental over k, even if |G| is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant field is purely transcendental (Noether's problem) is quite natural. Received: 14 January 1998     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

Escaping points of meromorphic functions with a finite number of poles

We establish several new properties of the escaping setI(f)={z∶f ⁿ (z)→∞ andf ⁿ(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)⊂J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.     

Functions of small growth with no unbounded Fatou components

We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.     

Transcendental Kantianism,Naturalized Kantianism,and the Bounds of Psychology

Are there sensory states (“perceptions”) that, unlike mere sensory registrations, require an explanatory framework (“psychology”) that goes beyond biology? Based on a reconstruction of Kant’s a priori, transcendental psychology, contemporary Kantians answer this question in the positive but dramatically limit the scope of psychology. In contrast, naturalistically oriented deflationists answer it in the negative, thereby not giving psychology any explanatory role whatsoever. In his recent monumental book Origins of Objectivity, Burge argues against both of these approaches and seeks to develop an intermediate approach between them. This he does by embedding Kantian transcendental psychology in contemporary science of perception, thereby naturalizing the former and considerably broadening the scope of psychology. In this paper I critically examine Burge’s naturalized Kantianism, thereby defending transcendental Kantianism. To this end, I first outline Kantian transcendental psychology of perception, highlighting the features that distinguish it from biology. I then show how Burge naturalizes this psychology by embedding its most fundamental notions in contemporary science of perception. Based on all this, I conclude the paper by arguing for two closely related claims. First, that transformed into empirical psychology, Kantian transcendental psychology loses the features that distinguish it from biology. Second, that genuine perception starts at the high cognitive level for which transcendental psychology accounts and not at the rather low or elementary level on which Burge focuses.     

Mortar-Based Contact Formulation for Large Deformations

The discontinuities due to the discretization lead to some challenges. First, the normal direction of the contact surface is not steady because the discrete surface is only C₀ continuous. One might smooth the normal vector field. Second, the question of contact enforcement has to be cleared. Contact forces can be modeled with either a Lagrange multiplier method or a penalty formulation to prevent penetration. Third, there must be developed a integration scheme which is able to handle the non-steady boundary. Last, there is a strong discontinuity in measuring the penetration, where different criteria for enabling or disabling contact can be found (active set strategy). In this work, different approaches to solve this tasks are presented and brought into context. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Auxiliary functions in transcendental number theory

We discuss the role of auxiliary functions in the development of transcendental number theory. Initially, auxiliary functions were completely explicit (Sect. 1). The earliest transcendence proof is due to Liouville (Sect. 1.1) who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in one variable. Hermite’s proof of the transcendence of e (1873) is much more involved, the auxiliary function he builds (Sect. 1.2) is the first example of the Padé approximants (Sect. 1.3), which can be viewed as a far reaching generalization of continued fraction expansion (Brezinski in Lecture Notes in Math., vol. 888. Springer, Berlin, 1981; and Springer Series in Computational Mathematics, vol. 12. Springer, Berlin, 1991). Hypergeometric functions (Sect. 1.4) are among the best candidates for using Padé approximations techniques.     

On regularization methods based on dynamic programming techniques

In this article, we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of these methods is to obtain stable approximations to the solution of linear inverse ill-posed problems. We follow two different approaches and derive a continuous and a discrete regularization method. Regularization properties for both methods are proved as well as rates of convergence. A numerical benchmark problem concerning integral operators with convolution kernels is used to illustrate the theoretical results.     

A Learning Algorithm for the Control of Continuous Action Set-Point Regulator Systems

The convergence properties for reinforcement learning approaches, such as temporal differences and Q-learning, have been established under moderate assumptions for discrete state and action spaces. In practice, however, many systems have either continuous action spaces or a large number of discrete elements. This paper presents an approximate dynamic programming approach to reinforcement learning for continuous action set-point regulator problems, which learns near-optimal control policies based on scalar performance measures. The continuous-action space (CAS) algorithm uses derivative-free line search methods to obtain the optimal action in the continuous space. The theoretical convergence properties of the algorithm are presented. Several heuristic stopping criteria are investigated and practical application is illustrated by two example problems—the inverted pendulum balancing problem and the power system stabilization problem.     

Some dynamical properties of transcendental meromorphic functions

If a transcendental meromorphic function f with finitely many poles has a completely invariant domain U which is simply connected and satisfies some conditions, we prove F(f)=U. And for a transcendental meromorphic function f with the lower order μ<∞ and δ(∞,f)>0, we prove J(f) cannot be contained in any finite set of straight lines completely.     

Existence of invariant measures for transcendental subexpanding functions

We consider the problem of the existence of absolutely continuous invariant measures for transcendental meromorphic functions. We prove sufficient conditions for a subexpanding meromorphic function f to have a C-finite absolutely continuous invariant measure 7 and we find a class of functions satisfying these assumptions.