Partitions of the wonderful group compactification

We define and study a family of partitions of the wonderful compactification of a semisimple algebraic group G of adjoint type. The partitions are obtained from subgroups of G × G associated to triples (A₁, A₂, a), where A₁ and A₂ are subgraphs of the Dynkin graph Γ of G and a : A₁ → A₂ is an isomorphism. The partitions of of Springer and Lusztig correspond, respectively, to the triples (∅, ∅, id) and (Γ, Γ, id).     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP _n ^α _n ^β _n are studied, assuming that

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

On the spectral representation of the elementary operator and spectral mapping theorems

LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X) ⁿ ,B∈B(Y) ⁿ , the elementary operator acting onB(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS _p (L _A ,R _B )=σ(A)×σ(B) and .     

Normal Matrices and an Extension of Malyshev's Formula

Let A be a complex matrix of order n with n ≥ 3. We associate with A the 3n × 3n matrix $Q\left( {\gamma } \right) = \left( \begin{gathered} A \gamma _1 I_n \gamma _3 I_n \\ 0 A \gamma _2 I_n \\ 0 0 A \\ \end{gathered} \right)$ where $\gamma _1 ,\gamma _2 ,\gamma _3 $ are scalar parameters and γ=(γ₁,γ₂,γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ) in the decreasing order. We prove that, for a normal matrix A, its 2-norm distance from the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to $\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in \mathbb{C}} \sigma _{3n - 2} (Q\left( \gamma \right)).$ This fact is a refinement (for normal matrices) of Malyshev's formula for the 2-norm distance from an arbitrary n × n matrix A to the set of n × n matrices with a multiple zero eigenvalue.     

Estimates for the number of rational points on convex curves and surfaces

Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice satisfies the following estimates: lim inf k_n(Γ)/n^d−2 < ∞ for d ≥ 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

The paper is devoted to the study of the behavior of the following mixed problem for large values of time:

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

ℓ-group homomorphisms between reduced archimedean f-rings

Let A and B be reduced archimedean f-rings, A with identity e; let $A\,\mathop \to \limits^\gamma\,BLet A and B be reduced archimedean f-rings, A with identity e; let A \mathop ? ^g BA\,\mathop \to \limits^\gamma\,B be an ℓ-group homomorphism, and set w = γ (e). We show (with some vagaries of phrasing here) (1) γ = w·ρ for a canonical ℓ-ring homomorphism A \mathop ? ^r B (w)A\,\mathop \to \limits^\rho\,B (w), where B (w) is an extension of B in which w is a von Neumann regular element, and (2) for X _A ,X _B canonical representation spaces for A, B, γ is realized via composition with a unique partially defined continuous function from X _B to X _A .     

Criteria for monogenicity of Clifford algebra-valued functions on fractal domains

Suppose that Ω is a bounded domain with fractal boundary Γ in ${\mathbb R^{n+1}}Suppose that Ω is a bounded domain with fractal boundary Γ in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} and let \mathbb R_0,n{\mathbb R_{0,n}} be the real Clifford algebra constructed over the quadratic space \mathbb Rⁿ{\mathbb R^{n}}. Furthermore, let U be a \mathbb R_0,n{\mathbb R_{0,n}}-valued function harmonic in Ω and H?lder-continuous up to Γ. By using a new Clifford Cauchy transform for Jordan domains in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} with fractal boundaries, we give necessary and sufficient conditions for the monogenicity of U in terms of its boundary value u = U|_Γ. As a consequence, the results of Abreu Blaya et al. (Proceedings of the 6th International ISAAC Congress Ankara, 167–174, World Scientific) are extended, which require Γ to be Ahlfors-David regular.     

A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A I)n[(A I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.     

Justification of a Malyshev-Type Formula in the Nonnormal Case

Let A be a complex matrix of order n, n ≥ 3 . We associate with A the 3n $$Q(\gamma ) = \left( {\begin{array}{*{20}c} A & {_{\gamma 1} I_n } & {_{\gamma 3} I_n } \\ 0 & A & {_{\gamma 2} I_n } \\ 0 & 0 & A \\ \end{array} } \right),$$ where γ₁, γ₂, γ₃ are scalar parameters and γ = (γ₁, γ₂, γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ), in decreasing order. Under certain assumptions on A, the authors have proved earlier that the 2-norm distance from A to the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to max $$\begin{array}{*{20}c} {\max } \\ {\gamma 1,\gamma 2,\gamma 3 \in \mathbb{C}} \\ \end{array} \;^\sigma 3n - 2(Q(\gamma )).$$ Now, the justification of this formula for the distance is given for an arbitrary matrix A.     

Modulus of continuity of the matrix absolute value

Lipschitz continuity of the matrix absolute value |A| = (A*A)^1/2 is studied. Let A and B be invertible, and let M ₁ = max(‖A‖, ‖B‖), M ₂ = max(‖A ⁻¹‖, ‖B ⁻¹‖). Then it is shown that

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

A SIMPLIFIED BRAUER＇S THEOREM ON MATRIX EIGENVALUES

Let A=(a _ij)∈C ^n×n and . Suppose that for each row of A there is at least one nonzero off-diagonal entry. It is proved that all eigenvalues of A are contained in . The result reduces the number of ovals in original Brauer’s theorem in many cases. Eigenvalues (and associated eigenvectors) that locate in the boundary of are discussed. The project is supported in part by Natural Science Foundation of Guangdong.     

Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X_∞ and X_∞^d being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (A_α) α ? I of upward filtering finite σ-algebras in A. X_∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X_∞. Further on we obtain that each measurable set of the remainder of X_∞ has measure zero with respect to the limit measure P_∞ (Theorem 1). X_∞^d is the STONE representation space X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A_α, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) and x(A) where x(A) is the Stone representation space of A, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of Alexandroff Hausdorff for compact T₂ measure spaces x with regular Borel measure (Theorem 3.).     

Finiteness properties of chevalley groups overFq[t]

LetG be a simple Chevalley group of rankn and Γ=G( ). Then the finiteness length of Γ shall be determined by studying the action of Γ on the Bruhat-Tits buildingX ofG . This is always possible provided that certain subcomplexes of the links of simplices inX are spherical. As a consequence, one obtains that Γ is of typeF _n−1 but not of typeFP _n ifG is of typeA _n, B_n, C_n orD _n andq≥2²ⁿ⁻¹.