Semigroups Generated by Similarity Orbits

\noindent We investigate the semigroups in M _n (\smallbf F) generated by the similarity orbit of single matrices. February 11, 2000     

On the mean lattice point discrepancy of a convex disc

For a convex planar domain D \cal {D} , with smooth boundary of finite nonzero curvature, we consider the number of lattice points in the linearly dilated domain t D t \cal {D} . In particular the lattice point discrepancy P_D(t) P_{\cal {D}}(t) (number of lattice points minus area), is investigated in mean-square over short intervals. We establish an asymptotic formula for¶¶ ò_{T - L}^{T + L}(P_D(t))²dt \int\limits_{T - \Lambda}^{T + \Lambda}(P_{\cal {D}}(t))^2\textrm{d}t ,¶¶ for any L = L(T) \Lambda = \Lambda(T) growing faster than logT.     

(Almost) primitivity of Hecke L -functions

Let f be a cusp form of the Hecke space \frak M₀(l,k,e){\frak M}_0(\lambda,k,\epsilon) and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions [`(S)]<sup>\sharp</sup>\bar{\cal S}^\sharp . We prove that when λ ≤ 2, L <sub>f</sub> is always almost primitive, i.e., that if L <sub>f</sub> is written as product of functions in [`(S)]^\sharp\bar{\cal S}^\sharp , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if l ? {?2,?3,2}\lambda\notin\{\sqrt{2},\sqrt{3},2\} then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for l ? {?2,?3,2}\lambda\in\{\sqrt{2},\sqrt{3},2\} the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class S^\sharp{\cal S}^\sharp is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in S^\sharp{\cal S}^\sharp .     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.     

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{t\in (0,\infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(\mathbb Rn){{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (\mathbb Rn))*=Bw,L*(\mathbb Rn){({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(\mathbb Rn){L^2({\mathbb R}^n)} and Bw,L*(\mathbb Rn){B_{\omega,L^\ast}({\mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(\mathbb Rn){H_{\omega,L^\ast}({\mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){t\in (0,\infty)} and p ? (0,1]{p\in (0,1]} is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1, L(\mathbb Rn))*=HL*1(\mathbb Rn){({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}, where HL*1(\mathbb Rn){H_{L^\ast}^1({\mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

ExponentialL 2-convergence of quantum Markov semigroups on\mathcal{B}(h)

With a quantum Markov semigroup (Τ _t ) _t≥0 on , whichhas a faithful normal invariant state ρ, we associate semigroupsT ^(s) (s∈[0],[1]) on the set of Hilbert-Schmidt operators onh defined by the rule . This allows us to use spectral theory to study the infinitesimal generatorL ^(s) of the semigroupT ^(s) and deduce information on the rate of the decay to equilibrium of Τ by means of estimates of the spectral gap ofL ^(s) . Fors=1/2, this method is applied to a class of quantum Markov semigroups on . We prove simple but reasonably general sufficient conditions, as well as necessary and sufficient conditions, for the gap(L ^(1/2)) to be positive. The exact value of the gap(L ^(1/2)) is computed or estimated for a certain class of equations motivated by classical probability or physical applications. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 523–538, October, 2000.     

(Almost) primitivity of Hecke <Emphasis Type="Italic">L</Emphasis>-functions

Let f be a cusp form of the Hecke space and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions . We prove that when λ ≤ 2, L _f is always almost primitive, i.e., that if L _f is written as product of functions in , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in .     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

Varieties generated by unstable involution semigroups with continuum many subvarieties

Over the years, several finite semigroups have been found to generate varieties with continuum many subvarieties. However, finite involution semigroups that generate varieties with continuum many subvarieties seem much rarer; in fact, only one example—an inverse semigroup of order 165—has so far been published. Nevertheless, it is shown in the present article that there are many smaller examples among involution semigroups that are unstable in the sense that the varieties they generate contain some involution semilattice with nontrivial unary operation. The most prominent examples are the unstable finite involution semigroups that are inherently non-finitely based, the smallest ones of which are of order six. It follows that the join of two finitely generated varieties of involution semigroups with finitely many subvarieties can contain continuum many subvarieties.     

Criteria for the L p -dissipativity of systems of second order differential equations

Abstract  We give complete algebraic characterizations of the L^p-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form , where are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L^p-dissipative if and only if

Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

We construct simultaneous rational approximations to q-series L₁(x₁; q) and L₁(x₂; q) and, if x = x₁ = x₂, to series L₁(x; q) and L₂(x; q), where

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where

Quasi Right Factorization Structures as Presheaves

In this article the notion of quasi right factorization structure in a category X\cal{X} is given. The main result is a one to one correspondence between certain classes of quasi right factorization structures and 2-reflective subobjects of a predefined object in Lax(PrOrd^Xop) \bf{\it{L}}\it{ax}({\it{PrOrd}}^{\bf{\cal{X}}^{op}}). Also a characterization of quasi right factorization structures in terms of images is given. As an application, the closure operators are discussed and it is shown that quasi closed members of certain collections are quasi right factorization structures. Finally several examples are furnished.     

On Complementary Spaces of the Lizorkin Spaces

Let S(Rⁿ){\cal S}(R^n) be the Schwartz space on R ⁿ . For a subspace V ì S(Rⁿ)V\subset {\cal S}(R^n), if a subspace W ì S(Rⁿ)W \subset {\cal S}(R^n) satisfies the condition that S(Rⁿ){\cal S}(R^n) is a direct sum of V and W, then W is called a complementary space of V in S(Rⁿ){\cal S}(R^n). In this article we give complementary spaces of two kinds of the Lizorkin spaces in S(Rⁿ){\cal S}(R^n).     

Representation of the space of polyanalytic functions as a direct sum of orthogonal subspaces. Application to rational approximations

Suppose thatD={z:|z|<1}, L ₂ (D) is the space of functions square-integrable over area inD,A _k (D) is the set of allk-analytic functions inD, (A ₁ (D)=A(D) is the set of all analytic functions inD),A _k L ₂ (D)=L ₂ (D)∩A _k (D),A ₁ L ₂ (D)=AL ₂ (D),

Specializations of indecomposable polynomials

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial ${P(x)\in {\mathbb{Z}}[x]}$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t ₁, . . . , t _r , x) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p?=?0 or p?>?deg(f), then the one variable specialized polynomial ${f(t_1^\ast+\alpha_1^\ast x,\ldots,t_r^\ast+\alpha_r^\ast x,x)}$ is indecomposable for all ${(t_1^\ast, \ldots, t_r^\ast, \alpha_1^\ast, \ldots,\alpha_r^\ast)\in \overline k^{2r}}$ outside a proper Zariski closed subset.     

A q-analogue of the FKG inequality and some applications

Let L be a finite distributive lattice and μ: L → ℝ⁺ a log-supermodular function. For functions k: L → ℝ⁺ let

Maximality of compression semigroups

LetG be a connected semisimple Lie group andr an involution onG. Further letL be an open subgroup of the groupG ^r ofr-fixed points andP⊂-G a parabolic subgroup. The semigroupS(L,P)∶={g∈G∶gLP⊂-LP} is called the compression semigroup of theL-orbit of the base point in the flag manifoldG/P. We show that compression semigroups for open orbits and regular symmetric pairs are maximal semigroups. Supported by a DFG Heisenberg-grant.