The Cascade Algorithm for the Numerical Computation of Refinable Functions

The cascade algorithm is a method which can be used for the computation of refinable functions. We prove here that in general the cascade algorithm will not inherit the accuracy and refinability of the limit refinable function. We provide conditions for which the cascade iteration does indeed preserve accuracy.     

Full rank positive matrix symbols: interpolation and orthogonality

We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case. In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive method to obtain these scaling functions. AMS subject classification (2000) 42C40, 65T60, 65D05     

Multivariate refinable Hermite interpolant

We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to different dilation matrices and symmetry groups are constructed and analyzed.

Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g., the Powell-Sabin scheme). We make some of these connections precise. A spline connection allows us to determine critical Hölder regularity in a trivial way (as opposed to the case of general refinable functions, whose critical Hölder regularity exponents are often difficult to compute).

While it is often mentioned in published articles that ``refinable functions are important for subdivision surfaces in CAGD applications", it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic developments elsewhere, give an application of vector refinability to subdivision surfaces. We briefly discuss several potential advantages offered by such Hermite subdivision surfaces.

     

A NOTE ON FINITE ELEMENT WAVELETS

1. IotroductionThe study of mu1ti-wavelets was initiated by Goodman, Lee and Tang[2]. Later, acharacterization of sca1ing functions and wavelets was established by Goodman and Lee[3].In [4] GoodInan constructed multi-wavelets that satisfy certain interpolating conditions.Geronimo, Hardin and Massopust[5] used fractal interpolation to construct orthogonal scal-ing functions gh1 and gh2 (with multiplicity r = 2) that are both symmetric and colltinuousbut nondifferentiable. Strang and Strela…     

Wavelet Decompositions of Nonrefinable Shift Invariant Spaces

The motivation for this work is a recently constructed family of generators of shift invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of dilates of the shift invariant space that are well suited for applications.     

On construction and (non)existence of c-(almost) perfect nonlinear functions

Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low c-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.     

A Construction of C1-Wavelets on the Two-Dimensional Sphere

In this paper a construction of C¹-wavelets on the two-dimensional sphere is presented. First, we focus on the construction of a multiresolution analysis leading to C¹-functions on S². We show refinability of the constructed tensor product generators. Second, for the wavelet construction we employ a factorization of the refinement matrices which leads to refinement matrices characterizing complement spaces. With this method we achieve an initial stable completion. A desired stable completion can be gained by lifting the initial stable completion. The result is a biorthogonal wavelet basis leading to C¹-functions on the sphere.     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions

In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies have provided evidence that students have difficulty with mathematical proofs. Few of these research studies, however, have focused on undergraduates’ abilities to produce proofs and counterexamples in the domain of continuous functions. The goal of this study is to contribute to research on student productions of proofs and counterexamples and to identify their abilities and mathematical understandings. The findings suggest more attention should be paid to teaching and learning proofs and counterexamples, as participants showed difficulty in writing these statements. More importantly, the analysis provides insight into the design of curriculum and instruction that may improve undergraduates’ learning in advanced mathematics courses.     

Re-examination of Bregman functions and new properties of their divergences

ABSTRACT

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the ‘Bregman function’). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have ‘Bregmanian properties’, can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.     

Affine equivalence for rotation symmetric Boolean functions with 2<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> variables

Rotation symmetric Boolean functions have been extensively studied in the last 10 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. Even the case of quadratic functions is nontrivial, and this was only completely settled in a 2009 paper of Kim, Park and Hahn. The much more complicated case of cubic functions was solved for permutations using a new concept of patterns in a 2010 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions with 2 ^k variables (this number is especially relevant in computer applications) and generated by a single monomial is examined in detail, and in particular a formula for the number of classes is proved.     

A divisive clustering method for functional data with special consideration of outliers

This paper presents DivClusFD, a new divisive hierarchical method for the non-supervised classification of functional data. Data of this type present the peculiarity that the differences among clusters may be caused by changes as well in level as in shape. Different clusters can be separated in different subregion and there may be no subregion in which all clusters are separated. In each step of division, the DivClusFD method explores the functions and their derivatives at several fixed points, seeking the subregion in which the highest number of clusters can be separated. The number of clusters is estimated via the gap statistic. The functions are assigned to the new clusters by combining the k-means algorithm with the use of functional boxplots to identify functions that have been incorrectly classified because of their atypical local behavior. The DivClusFD method provides the number of clusters, the classification of the observed functions into the clusters and guidelines that may be for interpreting the clusters. A simulation study using synthetic data and tests of the performance of the DivClusFD method on real data sets indicate that this method is able to classify functions accurately.     

A hybrid-Trefftz element containing an elliptic hole

Much work on special elements that simplify geometrical modelling of structures containing holes, cracks and/ or inclusions has been reported extensively in the literature. This paper presents a hybrid-Trefftz element containing elliptic hole formulated using Hellinger–Reissner principle by employing trial functions based on the mapping technique and the Cauchy integral method. The element presented in this paper could be regarded as an improved formulation over Piltner [Special finite elements with holes and internal cracks, Int. J. Numer. Methods Eng. 21 (1985) 1471–1485] element because the chosen trail functions in this paper have provided relatively more stable solutions. The use of the element with other ordinary displacement-based finite elements has also yielded very accurate solutions even when very coarse meshes relative to the size of the elliptic hole have been used.     

A class of test functions for global optimization

We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.This paper was written while the second author was a visiting Professor at Aachen University of Technology, funded by the Deutsche Forschungsgemeinschaft.     

Protein dynamics: Complex by itself

Biological functions are intimately rooted in biopolymer dynamics. It is commonly accepted that proteins can be considered as complex systems, but the origin of such complexity is still not fully understood. Moreover, it is still not really clear if proteins are true complex systems or complicated ones. Here, molecular dynamics simulations on a two helix bundle protein have been performed, and protein trajectories have been analyzed by using correlation functions in the frequency domain. We show that even a simple protein exhibits the hallmarks of complex systems. Moreover, the molecular bases of this complex behavior are possessed completely by the protein itself, because such complexity emerges without considering the solvent explicitly. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Six Kinds of Roughly Convex Functions

This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.     

Orthonormal bases with nonlinear phases

For adaptive representation of nonlinear signals, the bank M{\cal M} of real square integrable functions that have nonlinear phases and nonnegative instantaneous frequencies under the analytic signal method is investigated. A particular class of functions with explicit expressions in M{\cal M} is obtained using recent results on the Bedrosian identity. We then construct orthonormal bases for the Hilbert space of real square integrable functions with the basis functions from M{\cal M}.     

Strongly convex functions of higher order

The notion of strongly n-convex functions with modulus c>0 is introduced and investigated. Relationships between such functions and n-convex functions in the sense of Popoviciu as well as generalized convex functions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convex functions are also given.     

关于多维无穷可分分布的Lebesgue分解

In this paper we have investigated and proved various Lebesgue properties of the n-dimensional quasi-distributions. 2-dimensional quasi-distributions have been classified into 5 types. It is proved among others that every 2-dimensional quasi-distribution can be expressed uniquely as a sum of 5 functions belonging to different types. Some necessary and sufficient conditions for functions belonging to each given type have been analysed in detail.     

Fourier series and elliptic functions

Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.