Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.     

A characterization of p-bases of rings of constants

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Comparison of volumes by means of the areas of central sections

The Busemann–Petty problem asks whether origin-symmetric convex bodies in Rⁿ with smaller areas of all central hyperplane sections necessarily have smaller n-dimensional volume. The solution was completed in the end of the 1990s, and the answer is affirmative if n4 and negative if n5. Since the answer is negative in most dimensions, it is natural to ask what information about the volumes of central sections of two bodies does allow to compare the n-dimensional volumes of these bodies in all dimensions. In this article we give an answer to this question in terms of certain powers of the Laplace operator applied to the section function of the body.     

Behavior at infinity of solutions of an equation of Osserman-Keller type

We study the behavior at infinity of solutions of equations of the form Δu=u^p, where p>1, in dimensions n?3. In particular we extend results proved by Loewner and Nirenberg in Contribution to Analysis, 1974, pp. 245-272 for the case p=(n+2)/(n−2), n?3, to values of p in the range p>n/(n−2), n?3.     

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ ₁ ∪…∪J _r andJ ₁ …J _r, where eachJ _i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer. Partially supported by NSF grant DMS 9303230. Partially supported by NSF grant DMS 9101488.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

On the Number of Arrangements of Pseudolines

Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σ _i of the lines crossing i in the order of the crossings on line i. is a permutation of . The vector (σ ₁ ,σ ₂ , ...,σ_n) is an encoding for the arrangement. Define if and , otherwise. Let , we show that the vector (τ ₁ , τ ₂ , ... , τ_n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13. Received December 20, 1995, and in revised form March 8, 1996.     

On the number of regular vertices of the union of jordan regions

Let \C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in \C cross exactly twice, then their intersection points are called regular vertices of the arrangement \A(\C) . Let R(\C) denote the set of regular vertices on the boundary of the union of \C . We present several bounds on |R(\C)| , depending on the type of the sets of \C . (i) If each set of \C is convex, then |R(\C)|=O(n ^1.5+\eps ) for any \eps>0 . (ii) If no further assumptions are made on the sets of \C , then we show that there is a positive integer r that depends only on s such that |R(\C)|=O(n ^2-1/r ) . (iii) If \C consists of two collections \C ₁ and \C ₂ where \C ₁ is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and \C ₂ is a collection of polygons with a total of n sides, then |R(\C)|=O(m ^2/3 n ^2/3 +m +n) , and this bound is tight in the worst case. Received December 4, 1998, and in revised form June 3, 2000. Online publication Feburary 1, 2001.     

On the shape of the fringe of various types of random trees

We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures. Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely related w?(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = Σ_aw(t, a), i.e. as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel–Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too. In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries. The final section of this report briefly summarizes and improves the previously known results about the w? parameter's behavior on tries and suffix trees, originally published in one author's thesis (see Analysis of the multiplicity matching parameter in suffix trees. Ph.D. Thesis, Purdue University, West Lafayette, IN, U.S.A., May 2005; Discrete Math. Theoret. Comput. Sci. 2005; AD :307–322; IEEE Trans. Inform. Theory 2007; 53 :1799–1813). This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

均值混合正态分布统计量的性质

这篇文章基于基因遗传背景,提出了一类均值混合正态分布,它不同于通常所讨论的方差混合正态分布. 作者研究了这类均值混合正态分布统计量的性质,给出了平移变换群下不变量的稳健性,即它与正态分布下该统计量有相同的性质, 并且讨论了其它统计量的分布.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

Combinatorics of normal sequences of braids

Many natural counting problems arise in connection with the normal form of braids—and seem to have not been much considered so far. Here we solve some of them. One of the noteworthy points is that a number of different induction schemes appear. The key technical ingredient is an analysis of the normality condition in terms of permutations and their descents, in the vein of the Solomon algebra. As was perfectly summarized by a referee, the main result asserts that the size of the automaton involved in the automatic structure of Bn associated with the normal form can be lowered from n! to p(n), the number of partitions of n.     

Reconstruction of the set of branches of a graph

It is proved that the set of branches of a graphG is reconstructible except in a very special case. More precisely the set of branches of a graphG is reconstructible unless all the following hold: (1) the pruned center ofG is a vertex or an edge, (2)G has exactly two branches, (3) one branch contains all the vertices of degree one ofG and the other branch contains exactly one end-block. This is the best possible result in the sense that in the special excluded case, the reconstruction of the set of branches is equivalent to the reconstruction of the graph itself.1991Mathematics Subject Classification. Primary 05C60.     

Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

It is known that in the Minkowski sum of r polytopes in dimension d, with r<d, the number of vertices of the sum can be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

Goodness of fit of an assigned set of scores for the analysis of association in a contingency table

The problem of association between two attributes in ap×q contingency table can be looked upon as the problem of relationship between two vector variablesx andy. If there is only one true non-zero canonical correlation betweenx andy, the association between the two attributes is of rank 1 and in this case, one set of scores is adequate to describe the association completely; these scores are nothing but the coefficients in the canonical variates corresponding to the true non-zero canonical correlation. Given a set of hypothetical scoresα ₁,α₁,⋯,α_p for the rows, one is interested in testing their goodness of fit. Tests for this are suggested in this paper. For obtaining these tests, a preliminary result about direction and collinearity factors in discriminant analysis, whenS irrelevant variables are eliminated, is needed. This is derived in part one of this paper. This research was sponsored by the Office of Naval Research, Contract No. N00014-68-A-0515, Project No. NR042-260. Reproduction in whole or in part is permitted for any purpose of the United States Government.