Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻¹B₂ which are analytic and invertible on the unit circle, we characterize in terms of the operators A₁,B₁,C₁,A₂,B₂,C₂ when there exists a single rational matrix function W(λ) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. When this is the case, we give explicit formulae for A,B,C in terms of A₁,B₁,C₁,A₂,B₂,C₂. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.     

Multiplicative models for configuration spaces of algebraic varieties

Fulton and MacPherson (Ann. Math. 139 (1994) 183) found a Sullivan dg-algebra model for the space of n-configurations of a smooth complex projective variety X. K?í? (Ann. Math. 139 (1994) 227) gave a simpler model, E_n(H), depending only on the cohomology ring, H?H^*X.We construct an even simpler and smaller model, J_n(H). We then define another new dg-algebra, E_n(H^°), and use J_n(H) to prove that E_n(H^°) is a model of the space of n-configurations of the non-compact punctured manifold X^°, when X is 1-connected. Following an idea of Drinfel’d (Leningrad Math. J. 2 (1991) 829), we put a simplicial bigraded differential algebra structure on {E_n(H^°)}_n?0.     

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C₀(G) × C₀(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a K_G²-Banach ${}^1$-algebra, where K_G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ${}^1$-representations of BM(G, H) are investigated.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

Some cubic Julia sets

We figure out geometric properties of the Julia setJ _a of cubic complex polynomialC _a(z) =z ³ +az(a ∈ ?) and the smallest ellipse which surroundsJ _a.     

Чебыщевские подпространства в пространстве ХардиH 1

A subspaceY of a Banach spaceX is called a Chebyshev one if for everyx∈X there exists a unique elementP _Y (x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH ¹=H ¹ (|z|<1) be Chebyshev ones, and also the properties of the operatorP _Y are studied. These results show that the theory of Chebyshev subspaces inH ¹ differs sharply from the corresponding theory inL ¹(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH ¹ there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL ¹(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the Chebyshev subspacesY with a linear operatorP _Y inH ¹ (in contrast toL ¹(C)) is exhausted by that minimum which is necessary for any Banach space.     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

On the restriction map of the Fourier-Stieltjes algebra B(G) and Bp(G)

G is a locally compact group that contains the semidirect product J of a closed normal subgroup H and a closed connected subgroup K. Conditions on J are given that imply that the restriction map B_p(G) → B_p(H) (1 < p < ∞; G amenable if p ≠ 2) of the Fourier-Stieltjes algebras is not surjective. It is also shown that if the restriction map B(J) → B(H) is surjective, J need not be a direct product, even if H is nilpotent.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Some results on the capitulation problem

Let K be an unramified abelian extension of a number field F with Galois group G. K corresponds to a subgroup H of the ideal class group of F. We study the subgroup J of ideal classes in H which become trivial in K. There is an epimorphism from the cohomology group H^?1(G, Cl_K) to J which is an isomorphism if G is cyclic; Cl_K is the ideal class group of K. Some results on the structure of J and Cl_K are obtained.     

Jordan-Ketten und realisierungen rationaler matrizen

If the inverse of a square polynomial matrix L(s) is proper rational, then L(s)^-1 can be written as C(sI–J)^-1B. The result of this note states that if J is an nXn Jordan matrix, with n=degreedetL(s), then C consists of Jordan chains of L(s), and B^T of Jordan chains of L(s)^T. This is a generalization of the fact that each matrix which transforms a complex matrix A into Jordan form is made up of eigenvectors and generalized eigenvectors of A. The proof of our result relies on the realization theory of rational matrices.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

Hyperelliptic jacobians with real multiplication

Let K be a field of characteristic p≠2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is isomorphic to A₅. If the jacobian J(C) of the hyperelliptic curve C:y²=f(x) admits real multiplication over the ground field from an order of a real quadratic field D, then either its endomorphism algebra is isomorphic to D, or p>0 and J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when p splits in D.     

Subideals of Operators II

A subideal (also called a J-ideal) is an ideal of a B(H)-ideal J. This paper is the sequel to Subideals of Operators where a complete characterization of principal and then finitely generated J-ideals were obtained by first generalizing the 1983 work of Fong and Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. Here we determine which countably generated J-ideals are B(H)-ideals, and in the absence of the continuum hypothesis, which J-ideals with generating sets of cardinality less than the continuum are B(H)-ideals. These and some other results herein are based on the dimension of a related quotient space. We use this to characterize these J-ideals and settle additional questions about subideals. A key property in our investigation turned out to be J-softness of a B(H)-ideal I inside J, that is, IJ =? I, a generalization of a recent notion of softness of B(H)-ideals introduced by Kaftal?CWeiss and earlier exploited for Banach spaces by Mityagin and Pietsch.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

A note on Jordan-von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)?1+J²(X)/4 for James constant J(X) and von Neumann-Jordan constant CNJ(X) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejung's proof is not valid. Using his proof, the result should be . On the other hand, we give a new proof of CNJ(X)?1+J²(X)/4. As an application, we give a relation between J(X) and J(lp(X)).