Cartesian Closed Extensions of Subcategories of CONT

In this paper we give a uniform way of proving cartesian closedness for many new subcategories of continuous posets. We define C-P to be the category of continuous posets whose D–completions are isomorphic to objects from C, where C is a subcategory of the category CONT of domains. The main result is that if C is a cartesian closed full subcategory of ALG _⊥ or BC, then C-P is also a cartesian closed subcategory of the category CONTP of continuous posets and Scott continuous functions. In particular, we have the following cartesian closed categories : BC-P, LAT-P, aL-P, aBC-P, B-P, aLAT-P, ω -B-P, ω -aLAT-P, etc.     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R ⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ? R ⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

Integral points on quadrics with prime coordinates

Let f (x ₁, . . . , x _s ) be a regular indefinite integral quadratic form, and t an integer. Denote by V the affine quadric {x : f (x) = t}, and by \({V(\mathbb {P})}\) the set of \({{\bf x}\in V}\) whose coordinates are simultaneously prime. It is proved that, under suitable conditions, \({V(\mathbb{P})}\) is Zariski dense in V as long as s ≥ 10.     

Local nonglobal minima for solving large-scale extended trust-region subproblems

We study large-scale extended trust-region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust-region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS  and that there are efficient algorithms for solving large-scale TRS  problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS  and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS  via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS  by solving at most three generalized eigenvalue problems.     

Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set

For any bounded domain Ω in ?^m, let B₁(Ω) denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple T in the Cowen-Douglas class B₁(Ω), let N_T (w) denote the restriction of T to the subspace \(\cap^m_{i,j=1}{\rm{ker}}(T_i-w_iI)(T_j-w_jI)\). This commuting m-tuple N_T (w) of m + 1 dimensional operators induces a homomorphism \({\rho _{{N_T}\left( w \right)}}\) of the polynomial ring P[z₁, · · ·, z_m], namely, \({\rho _{{N_T}\left( w \right)}}\) (p) = p(N_T (w)), p ∈ P[z₁, · · ·, z_m]. We study the contractivity and complete contractivity of the homomorphism \({\rho _{{N_T}\left( w \right)}}\). Starting from the homomorphism \({\rho _{{N_T}\left( w \right)}}\), we construct a natural class of homomorphisms \(\rho_{N^{(\lambda)}(w)}\), λ > 0, and relate the properties of \(\rho_{N^{(\lambda)}(w)}\) to those of \({\rho _{{N_T}\left( w \right)}}\). Explicit examples arising from the multiplication operators on the Bergman space of Ω are investigated in detail. Finally, it is shown that contractive properties of \({\rho _{{N_T}\left( w \right)}}\) are equivalent to an inequality for the curvature of the Cowen-Douglas bundle E_T. However, we construct examples to show that the contractivity of the homomorphism ρ_T does not follow, even if \({\rho _{{N_T}\left( w \right)}}\) is contractive for all w in Ω.     

Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T _i on U for all i ∈ I. Since T _i (f) is not contained in f, Lusztig considered two subalgebras _i f and ⁱ f of f for any i ∈ I, where _i f={x ∈ f | T _i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}\). The restriction of T _i on _i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\). The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of _i f, ⁱ f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig.     

Rings of h-deformed differential operators

We describe the center of the ring Diff _h (n) of h-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff _h (n) and the Weyl algebra W _n , extended by n indeterminates.