Some remarks on estimation of parameters of uniform distribution

The best (in the sense of quadratic risk) unbiased estimators are constructed for the function f(x)=σ(2x/(n+1)−1)+μ from a sample of size n from the uniform distribution over [μ−σ, μ+σ] with unknown μ and σ. The best unbiased estimator for σ with μ being known is also presented. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 36–39, Perm, 1991.     

On trees covering chains or stars

In this paper, in the context of the “dessins d’enfants” theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of “chains” or “stars.” We also discuss some applications of this result that are related to the arithmetical theory of torsion on curves. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 207–215, 2007.     

Two faces of mathematics lessons in Korea: conventional lessons and innovative lessons

This study identifies the characteristics of mathematics classrooms in Korea. First, conventional Korean mathematics lessons are analyzed from the perspective of “theory of variation”. Second, an innovative lesson for gifted children is reported in detail and analyzed from the perspective of “Lakatos’ proofs and refutations”. Third, the classroom characteristics identified in both the conventional lessons and the innovative lesson are interpreted in terms of the underlying cultural values that they share with other East Asian countries. The study concludes that although the two faces of Korean mathematics lessons look different, they may flow from the same “heart”—that of the common Confucian heritage culture culture, and in particular East Asian pragmatism.     

The Kantor-Koecher-Tits construction for Jordan coalgebras

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra 〈L(A), Δ〉, it is possible to construct a Lie coalgebra 〈L(A), Δ_L〉. Moreover, any dual algebra of the coalgebra 〈L(A), Δ_L〉 corresponds to a Lie algebra that can be determined from the dual algebra for (A, Δ), following the Kantor-Koecher-Tits process. The structure of subcoalgebras and coideals of the coalgebra 〈L(A), Δ_L〉 is characterized. Supported by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 173–189, March–April, 1996.     

Artistic Expression Goes Green

The paper is a critical discussion of the rich and insightful final chapter of Mitchell Green’s Self-Expression. There, Green seeks to elucidate the compelling, but inchoate intuition that when we’re fully and most expertly expressing ourselves, we can ‘push out’ from within not just our inner representations, but also the ways that we feel. I question, first, whether this type of ‘qualitative expression’ is really distinct from the other expressive forms that Green explores, and also whether it’s genuinely ‘expressive’. I then scrutinize the nature of the ‘qualitative congruences’ that lie at the heart of Green’s theory; and I wonder whether they can play the role Green claims they can in providing a novel account of artistic expression.     

L-fuzzy ⋎ and ⋏ hyperoperations and the associated l-fuzzy hyperalgebras

In this paper we study two fuzzy hyperoperations, denoted by ⋎ (which can be seen as a generalization of ∨) and ⋏ (which can be seen as a generalization of ∧). ⋎ is obtained from a family of crisp ∨; _p hyperoperations and ⋏ is obtained from a family of crisp ∧ _p hyperoperations. The hyperstructure (X, ⋎, ∧) resembles ahyperlattice and the hyperstructure (X, ∨, ⋏) resembles adual hyperlattice     

On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Elementary equivalence of Chevalley groups over fields

It is proved that (elementary) Chevalley groups G _π(Φ,K) and G _π′(Φ′,K′) (or E _π(Φ,K) and E _π′(Φ′,K′)) over infinite fields K and K′ of characteristic different from 2, with weight lattices Λ and Λ′, respectively, are elementarily equivalent if and only if the root systems Φ and Φ′ are isomorphic, the fields K and K′ are elementarily equivalent, and the lattices Λ and Λ′ coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 29–77, 2006.     

On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

We prove that any C^3+β-smooth diffeomorphism preserving the orientation of a circle with rotation number from the Diophantine class D_δ, 0 < β < δ < 1, is C^2+β−δ-smoothly conjugate to a rigid rotation of the circle by a certain angle. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 268–282, February, 2008.     

The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the L _p -mean (p ≥ 0) of a 2π-periodic function with zero mean value and the L _q -norm (q ≥ 1) of the positive cutoff of its derivative. We obtain estimates of the constant from below for 0 ≤ p ≤ ∞ and from above for 1 ≤ p ≤ ∞ for an arbitrary 1 ≤ q ≤ ∞. We write out the values of the sharp constant in the cases p = 2, 1 ≤ q ≤ ∞ and p = ∞, 1 ≤ q ≤ ∞.     

The weight-decay technique in learning from data: an optimization point of view

The technique known as “weight decay” in the literature about learning from data is investigated using tools from regularization theory. Weight-decay regularization is compared with Tikhonov’s regularization of the learning problem and with a mixed regularized learning technique. The accuracies of suboptimal solutions to weight-decay learning are estimated for connectionistic models with a-priori fixed numbers of computational units. The Authors were partially supported by a PRIN grant from the Italian Ministry for University and Research, project “Models and Algorithms for Robust Network Optimization”.     

Ranking players by “power” in weighted voting systems: The multi-relation method

A new quantitative definition of the “power index” is proposed for a voter in weighted voting systems (WV-schemes), where the voters a priori do not have equal rights. The proposed “power index” is generated in a self-consistent manner from information about the number of times that a voter enters any winning coalition in the WV-scheme. Explicit formulas are derived for computing the “power index”. The discussion is illustrated with prototype examples. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 81–98, 2007.     

On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 99–115, January, 2009.     

Limit correlation functions at zero for fixed trace random matrix ensembles

The limit as N → ∞ of the eigenvalue correlation function is studied in a neighborhood of zero for N × N Hermitian matrices chosen at random from the Hilbert-Schmidt sphere of an appropriate radius. Dyson’s famous sin π(t₁ − t₂)/π(t₁ − t₂)-kernel asymptotic is extended to this class of matrix ensembles. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 68–80.     

On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain

We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ³in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas (as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage from Ω(h)₍₎ to Ω(h), the spectrum itself has perturbation O(h⁻²). Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149.     

An Infinite Presentation of the Torelli Group

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves.     

Chang’s conjecture for ℵω

We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵ_ω+1,ℵ_ω)↠(ω₁,ω₀), as well as of some other transfer properties of the type (κ⁺,κ)↠(α⁺,α), where κ is singular.     

Non-special Aronszajn trees on ℵω+1

We continue our research on the relative strength ofL-like combinatorial principles for successors of singular cardinals. In [3] we have shown that the existence of a λ⁺-special Aronszajn tree does not follow from that of a λ⁺-Souslin tree. It follows from [5], [4] and [6] that under G.C.H. □_λ does imply the existence of a λ⁺-Souslin tree. In [2] we show that □_λ does not follow from the existence of a λ⁺-special Aronszajn tree. Here we show that the existence of such a tree implies that of an ‘almost Souslin’ λ⁺-tree. It follows that the statement “All λ⁺-Aronszajn trees are special” implies that there are no λ⁺-Aronszajn trees.     

Almost periodic sets and measures on the torus

We establish some “number theoretical” results about a continuous functionh from the circleT into itself, which generalize Kronecker’s theorem in several ways. These results are used to characterize the almost periodic sets of the flow on the torusT ² generated by (θ, φ) → (θ+α, φ+h(θ)), where α is irrational. The almost periodic measures are characterized in the caseh(θ)=θ.     

Long and short paths in uniform random recursive dags

In a uniform random recursive k-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S _n is the shortest path distance from node n to the root, then we determine the constant σ such that S _n /log n→σ in probability as n→∞. We also show that max _1≤i≤n S _i /log n→σ in probability.