SK<Subscript>1</Subscript> of graded division algebras

The reduced Whitehead group SK₁ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK₁ of a tame valued division algebra over a henselian field coincides with SK₁ of its associated graded division algebra. Furthermore, it is shown that SK₁ of a graded division algebra is isomorphic to SK₁ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK₁ for generic abelian crossed products.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Groups of outer type E<Subscript>6</Subscript> with trivial Tits algebras

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E₆, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?     

Power Sums Related to Semigroups S(d<Subscript>1</Subscript>, d<Subscript>2</Subscript>, d<Subscript>3</Subscript>)

The explicit formulas for the sums of positive powers of the integers unrepresentable by the triple of integers , are derived.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

A characterization of Azumaya algebras

Let R be a commutative ring with identity 1, and A a finitely generated R-algebra. It is shown that A is an Azumaya R-algebra if and only if every stalk of the Pierce sheaf induced by A is an Azumaya algebra.     

SK1 of Azumaya algebras over Hensel Pairs

Let A be an Azumaya algebra of constant rank n ² over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK₁(A) coincides with its reduction SK₁(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Coxeter group actions on <Subscript>4</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript>(1) hypergeometric series

In this paper we investigate a certain linear combination K([(x)\vec])=K(a;b,c,d;e,f,g)K(\vec{x})=K(a;b,c,d;e,f,g) of two Saalschutzian hypergeometric series of type ₄ F ₃(1). We first show that K([(x)\vec])K(\vec{x}) is invariant under the action of a certain matrix group G _K , isomorphic to the symmetric group S ₆, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ⁷:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ ₁,μ ₂,μ ₃ of a certain matrix group M _K , isomorphic to the Coxeter group W(D ₆) (of order 23040) and containing the above group G _K , there is a relation among K(m₁[(x)\vec])K(\mu_{1}\vec{x}), K(m₂[(x)\vec])K(\mu_{2}\vec{x}), and K(m₃[(x)\vec])K(\mu_{3}\vec{x}), provided that no two of the μ _j ’s are in the same right coset of G _K in M _K . The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.     

Four-manifolds with π<Subscript>1</Subscript>–free second homotopy

 We study the homotopy type of closed connected orientable topological 4–manifolds M with Λ–free second homotopy group, where Λ is the integral group ring of π₁(M). This is related with problem N.4.53 of [23], and extends some results proved for the class of closed 4–manifolds with free fundamental group [10][11]. Other applications on special classes of closed topological manifolds complete the paper. Received: 27 November 2001 / Revised version: 28 October 2002 Published online: 19 May 2003 Work performed under the auspices of the GNSAGA of the CNR of Italy and partially supported by Ministero per la Ricerca Scientifica e Tecnologica of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse, and by a research grant of the University of Modena and Reggio Emilia. Mathematics Subject Classification (2000): Primary: 57 N 65, 57 R 67; Secondary: 57 Q 10, 57 R 80     

Rectangular superpolynomials for the figure-eight knot 4<Subscript>1</Subscript>

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 4₁ in an arbitrary rectangular representation R = [r^s] as a sum over all Young subdiagrams λ of R with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 3₁, to which our results for the knot 4₁ are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.     

Simply connected symplectic 4-manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript><Superscript>+</Superscript>=1 and <Emphasis Type="Italic">c</Emphasis><Subscript>1</Subscript><Superscript>2</Superscript>=2

In this article we construct a new simply connected symplectic 4-manifold with b₂⁺=1 and c₁²=2 which is homeomorphic, but not diffeomorphic, to a rational surface by using rational blow-down technique. As a corollary, we conclude that a rational surface admits an exotic smooth structure. Mathematics Subject Classification (2000) 53D05, 14J26, 57R55, 57R57     

On generalized <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> and <Emphasis Type="Italic">R</Emphasis><Subscript>1</Subscript> spaces

Characterizations of some properties of generalized R ₀ and R ₁ topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.     

(<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

We prove that the conditions of (q₁, 1)- and (1, q₂)-quasimertricity of a distance function ρ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to ρ. It follows that the Box-quasimetric defined with the use of basis vector fields of class C₁ whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of (q₁, q₂)-quasimetrics for which there are no Lipschitz equivalent 1-quasimetrics, which in particular implies another proof of a result by V. Schröder.     

Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We give an explicit formula for the exterior powers ∧ ^k π ₁ of the defining representation π ₁ of the simple Lie algebra ?ο(2n + 1, ?). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ?). We also give a survey of main results on family algebras.     

The spectrum of the Dirac operator on SU<Subscript>2</Subscript>/Q<Subscript>8</Subscript>

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU₂/Q₈, where Q₈ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär’s upper bound for the smallest Dirac eigenvalue in the particular case where SU₂/Q₈ is a homogeneous minimal hypersurface of S ⁴.     

Exact order of relative widths of classes <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>1</Subscript><Superscript>r</Superscript></Stack> in the space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

We determine the exact order of relative widths of classes W ₁ ^r of periodic functions in the space L ₁ as n → ∞ under restrictions on higher derivatives of approximating functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1409–1417, October, 2005.     

<Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript>-<Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript> regularization of split feasibility problems

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP); see for example Censor and Elfving (Numer. Algorithms 8, 221–239 1994). Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution; see for example Chaux et al. (SIAM J. Imag. Sci. 2, 730–762 2009) and references therein. In this paper, we are interested in split feasibility problems which can be seen as a general form of Q-Lasso introduced in Alghamdi et al. (2013) that extended the well-known Lasso of Tibshirani (J. R. Stat. Soc. Ser. B 58, 267–288 1996). Q is a closed convex subset of a Euclidean m-space, for some integer m ≥ 1, that can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the Lasso. Inspired by recent works by Lou and Yan (2016), Xu (IEEE Trans. Neural Netw. Learn. Syst. 23, 1013–1027 2012), we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one is nothing else than the celebrate forward-backward algorithm, and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning; see for example Censor et al. (Phys. Med. Biol. 51, 2353–2365, 2006).