On extension of some generalizations of quasiconformal mappings to a boundary

This work is devoted to the investigation of ring Q-homeomorphisms. We formulate conditions for a function Q(x) and the boundary of a domain under which every ring Q-homeomorphism admits a homeomorphic extension to the boundary. For an arbitrary ring Q-homeomorphism f: D → D’ with Q ∈ L ¹(D); we study the problem of the extension of inverse mappings to the boundary. It is proved that an isolated singularity is removable for ring Q-homeomorphisms if Q has finite mean oscillation at a point.     

Boundary behavior of ring <Emphasis Type="Italic">Q</Emphasis>-homeomorphisms in metric spaces

We investigate the problem of extension of so-called ring Q-homeomorphisms between domains in metric spaces with measures to the boundary. We establish conditions for the function Q(x) and the boundary of the domain under which any ring Q-homeomorphism admits a continuous or a homeomorphic extension to the boundary. The results are applicable, in particular, to Riemannian manifolds, L?wner spaces, and Carnot and Heisenberg groups.     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.     

Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for <Emphasis Type="Italic">Q</Emphasis>-mappings

We prove that an open discrete Q-mapping has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 116–126, January, 2009.     

On the boundary behavior of imbeddings of metric spaces into a Euclidean space

We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1068–1074, August, 2007.     

Non‐isolated quasi‐degrees

We show that non‐isolated from below 2‐c.e. Q ‐degrees are dense in the structure of c.e. Q ‐degrees. We construct a 2‐c.e. Q ‐degree, which can't be isolated from below not only by c.e. Q ‐degrees, but by any Q ‐degree. We also prove that below any c.e. Q ‐degree there is a 2‐c.e. Q ‐degree, which is non‐isolated from below and from above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

Minimal extensions of Π01 classes

A minimal extension of a Π⁰₁ class P is a Π⁰₁ class Q such that P ? Q, Q – P is infinite, and for any Π⁰₁ class R, if P ? R ? Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A, and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of Solutions of a Two-Point Boundary Value Problem

A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q≥0.This improves a recent result that the problem has at least one solution for 0 ≤Q≤13.21.     

The <Emphasis Type="Italic">Q</Emphasis> Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.     

Lattices freely generated by an order and preserving certain bounds

In 1964, R. A. Dean presented a far-reaching generalization of P. M. Whitman’s solution of the word problem for free lattices. Dean considered an order Q and solved the word problem for the lattice freely generated by Q while preserving certain (not necessarily all) existing finite joins and meets in Q. We present a simpler, more natural proof of Dean’s result.     

Square-free Lucas <Emphasis Type="Italic">d</Emphasis>-pseudoprimes and Carmichael-Lucas numbers

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P, Q) such that the rank of N in U(P, Q) is exactly (N − ε(N))/d, where ε is the signature of U(P, Q). We prove here that all but a finite number of Lucas d-pseudoprimes are square free. We also prove that all but a finite number of Lucas d-pseudoprimes are Carmichael-Lucas numbers.     

Hopf Algebras of Type <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>, Twistings and the FRT-construction

We study pointed Hopf algebras of the form U(R _Q ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where R _Q is the Yang–Baxter operator associated with the multiparameter deformation of GL _n supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(R _Q ) is of type A _n in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(R _Q ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(R _Q ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(R _Q ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(R _Q ), U(R _Q′) can be obtained from each other by twisting the comultiplication if and only if In the last part we show that U _Q is always a quotient of a double crossproduct. I wish to thank UIC, where some of the work was done, for hospitality.     

On locally <Emphasis Type="Italic">A</Emphasis>-convex <Emphasis Type="Italic">H</Emphasis>*-algebras

We endow any proper A-convex H*-algebra (E, τ) with a locally pre-C*-topology. The latter is equivalent to that introduced by the pre C*-norm given by Ptàk function when (E, τ) is a Q-algebra. We also prove that the algebra of complex numbers is the unique proper locally A-convex H*-algebra which is barrelled and Q-algebra.      

On boundary behavior of generalized quasi-isometries

We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in [`(ⁿ)] = ⁿ è{ ￥} ,n 3 2\overline {{^n}} = {^{^n}} \cup \{ \infty \} ,n \ge 2, under integral constraints of the type ∫ Φ(Q ⁿ⁻¹(x))dm(x) < ∞ with a convex non-decreasing function Φ: [0,∞]→[0,∞]. Integral conditions on Φ are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in ℝ ⁿ . In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.     

A criterion for the existence of a nonlinear mapping whose Jacobian matrix commutes with a matrix ring

Let Q be a ring of constant square matrices of orderm over the field of complex numbers. We consider the problem on the existence of a nonlinear mapping u: C ^m → C ^m , m ≥ 2, whose Jacobian matrix commutes with each matrix of Q. We prove that such a mapping exists if and only if Q possesses an (r, l)-pair.     

Decomposition of Multiple Coverings into More Parts

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any locally finite α k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth. The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.     

Distribution of Distances and Triangles in a Point Set and Algorithms for Computing the Largest Common Point Sets

This paper considers the following problem, which we call the largest common point set problem (LCP): given two point sets P and Q in the Euclidean plane, find a subset of P with the maximum cardinality that is congruent to some subset of Q . We introduce a combinatorial-geometric quantity λ(P, Q) , which we call the inner product of the distance-multiplicity vectors of P and Q , show its relevance to the complexity of various algorithms for LCP, and give a nontrivial upper bound on λ(P, Q) . We generalize this notion to higher dimensions, give some upper bounds on the quantity, and apply them to algorithms for LCP in higher dimensions. Along the way, we prove a new upper bound on the number of congruent triangles in a point set in four-dimensional space. Received July 17, 1997, and in revised form March 6, 1998.     

A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.