On constant <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript>2</Subscript>)-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We consider the spectral resolution of a Uq (sl ₂)-invariant solution R of the constant Yang–Baxter equation in the braid group form. It is shown that if the two highest coefficients in this resolution are not equal, then R is either the Drinfeld R-matrix or its inverse. Bibliography: 13 titles.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

Cyclic codes over <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2 ^k . Further, several optimal or near optimal binary codes are obtained from cyclic codes over R _k via this map.     

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.     

Rate of convergence to stationarity of the system <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">N</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">R</Emphasis>

We consider the M/M/N/N+R service system, characterized by N servers, R waiting positions, Poisson arrivals and exponential service times. We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and study its behaviour as a function of R, N and the arrival rate λ, allowing λ to be a function of N.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Equilateral Sets in <Emphasis Type="Italic">l</Emphasis><Stack><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript></Stack>

We show that for every odd integer p 1 there is an absolute positive constantc_p, so that the maximum cardinality of a set of vectors in Rⁿ such that the l_p distance between any pair is precisely 1, is at most c_p n log n. We prove some upper bounds for other l_p norms as well.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

On Generalized Hardy Sums <Emphasis Type="Italic">s</Emphasis><Subscript>5</Subscript>(<Emphasis Type="Italic">h,k</Emphasis>)

The aim of this paper is to study generalized Hardy sums s ₅(h, k). By using mediants and the adjacent difference of Farey fractions, we establish a relationship between s ₅(h, k) and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums s _5,p (h,k). A relationship between s _5,p (h, k) and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between s ₅(h,k) and Lambert series.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1434–1440, October, 2004.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Boundedness of general index transforms

We establish the boundedness properties in L _p for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in L _p (R ₊), 1 p 2, for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case p=2 is known as the Plancherel-type theory for this class of transformations.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 127–147, January–March, 2005.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

Orthosymplectic Quantum Function Superalgebras OSPq（2l＋ 1｜2n）

By the R-matrix of orthosymplectic quantum superalgebra U _q (osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSP _q (2l + 1|2n). Furthermore, it is shown that OSP _q (2l + 1|2n) is coquasitriangular.     

Relative <Emphasis Type="Italic">FI</Emphasis>-injective and <Emphasis Type="Italic">FI</Emphasis>-flat modules

Let R be a ring, n a fixed nonnegative integer and FP _n (F _n ) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext ¹(N,M) = 0 (resp., Tor ₁(F,N) = 0) for any N ∈ FP _n . It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FP _n -precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F _n -preenvelope K → F of a right R-module K with F projective if and only if M ∈^⊥ F _n . These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

On <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript>-supplemented subgroups of finite groups

Suppose that F is a formation of finite groups. We introduce the concept of F _h -supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and 2-maximal subgroup are F _h -supplemented, respectively. Some available results are generalized.     

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.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.