Showing papers in "Open Mathematics in 2005"
TL;DR: In this article, the authors obtained optimal bounds of order O(n −1) for convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.
Abstract: We obtain optimal bounds of order O(n
−1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.
86 citations
TL;DR: In this paper, a sequence of analytic perturbations of the completely integrable Hamiltonian was constructed for a positive integer n and R>0, where R>1 and n≥4 were used to estimate the time of drift in the action space.
Abstract: For a positive integer n and R>0, we set
$$B_R^n = \left\{ {x \in \mathbb{R}^n |\left\| x \right\|_\infty< R} \right\}$$
. Given R>1 and n≥4 we construct a sequence of analytic perturbations (H
j
) of the completely integrable Hamiltonian
$$h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n $$
on
$$\mathbb{T}^n \times B_R^n $$
, with unstable orbits for which we can estimate the time of drift in the action space. These functions H
j
are analytic on a fixed complex neighborhood V of
$$\mathbb{T}^n \times B_R^n $$
, and setting
$$\varepsilon _j : = \left\| {h - H_j } \right\|_{C^0 (V)} $$
the time of drift of these orbits is smaller than (C(1/ɛ
j
)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.
45 citations
TL;DR: In this paper, a complete local description of almost cosymplectic (−1, μ, 0)-spaces is established: "models" of such spaces are constructed, and it is noted that a given almost cosyplectic (− 1, μ 0)-space is locally isomorphic to a corresponding model.
Abstract: In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be\(\mathcal{D}\)-homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, μ 0)-space is locally isomorphic to a corresponding model. In the case when μ is constant, the models can be constructed on the whole of ℝ2n+1 and it is shown that they are left invariant with respect to Lie group actions.
33 citations
TL;DR: In this paper, it was shown that the order x Veronese embedding of P n is not weakly (k−1)-defective, i.e., it can be shown that for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces of all P n singular at each point of S has dimension ( n /n+x )−1− k(n+1) (proved by Alexander and Hirschowitz)
Abstract: Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( n n+x ). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension ( n /n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.
31 citations
TL;DR: In this article, it was shown that a population is isomorphic to the flag variety of the Langlands dual Lie algebra, based on the correspondence between critical points and differential operators called the Miura opers.
Abstract: Critical points of a master function associated to a simple Lie algebra\(\mathfrak{g}\) come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra\(^t \mathfrak{g}\). The proof is based on the correspondence between critical points and differential operators called the Miura opers.
31 citations
TL;DR: In this article, the authors show that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkahler varieties, and they give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric HHDFs, which is in fact a consequence of a long-standing conjecture of Stanley.
Abstract: Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkahler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkahler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkahler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.
24 citations
TL;DR: In this article, the authors construct a universal C fixme *-algebra C¯¯ *(X,α) which agrees with the partial crossed product theory in the case α is injective, and with the crossed product by a monomorphism in the condition α is onto.
Abstract: A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C
*-dynamical systems. In this paper we construct and investigate a universal C
*-algebra C
*(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system,
$$(\tilde X,\tilde \alpha )$$
where
$$\tilde \alpha $$
is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C
*(X, α).
20 citations
TL;DR: In this article, a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous graph of global elliptic boundary value problems is considered.
Abstract: We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.
18 citations
TL;DR: In this paper, the homogeneous nilpotent orbits in certain Lie color algebras were classified and the results were generalized to the setting of a real reductive dual pair.
Abstract: We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair.
18 citations
TL;DR: In this paper, alternating sums of squares of odd and even terms of the Lucas sequence were considered and their products were represented as products of appropriate Fibonacci and Lucas numbers, respectively.
Abstract: We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.
17 citations
TL;DR: In this paper, the authors constructed the Sobolev-Morrey-W -space and proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G ∋R n satisfies the flexible λhorn condition.
Abstract: In this paper the spaces of type Sobolev-Morrey-W
(Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R
n
satisfies the flexible λ-horn condition
TL;DR: In this article, a series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology is constructed, and the general member of such family is an indecomposable bundle from the extension class Ext 1 (E, F) for a suitable choice of the rank 2 ACM bundles.
Abstract: By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P
4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext
1 (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.
TL;DR: In this paper, the authors present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem, one requires shape derivative calculus and the other depends strongly on the compatibility condition of the Neumann problem.
Abstract: The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.
TL;DR: For morphisms having nonsingular target varieties, this article showed that there exists another bivariant theory of constructible functions and a unique Chern class γ:\(tilde {\mathbb{F}} \to {H}}\).
Abstract: W. Fulton and R. MacPherson posed the problem of unique existence of a bivariant Chern class—a Grothendieck transformation from the bivariant theory F of constructible functions to the bivariant homology theory H. J.-P. Brasselet proved the existence of a bivariant Chern class in the category of embeddable analytic varieties with cellular morphisms. In general however, the problem of uniqueness is still unresolved. In this paper we show that for morphisms having nonsingular target varieties there exists another bivariant theory\(\tilde {\mathbb{F}}\) of constructible functions and a unique bivariant Chern class γ:\(\tilde {\mathbb{F}} \to {\mathbb{H}}\).
TL;DR: In this paper, the authors provide short and elementary proofs of two theorems that characterize reductive group schemes over a discrete valuation ring, in a slightly more general context, and show that they are equivalent to theorem 1.
Abstract: We include short and elementary proofs of two theorems that characterize reductive group schemes over a discrete valuation ring, in a slightly more general context.
TL;DR: In this paper, it was shown that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related.
Abstract: Using the one-to-one correspondence between Kahler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kahler-Norden manifold to be holomorphically recurrent.
TL;DR: In this paper, the complete solution of the extremal problem posed by N.G. Tchebotarov in 20th century, and explicit parametric formulae for the extremals were established.
Abstract: We give the complete solution of the extremal problem posed by N.G. Tchebotarov in 20th of the last century, and we establish explicit parametric formulae for the extremals.
TL;DR: A survey of the first homology of diffeomorphism groups which preserve a smooth G-action or a foliated structure on a smooth manifold can be found in this article.
Abstract: Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category.
TL;DR: In this paper, sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are given. But they do not consider the case of non-Riccati boundary value problems.
Abstract: Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given.
TL;DR: In this article, it was shown that either the center of R is a finitely generated F-algebra or R satisfies a polynomial identity (is PI) or R is algebraic over F.
Abstract: Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.
TL;DR: In this article, the authors generalize reduction theorems for classical connections to operateators with values in $k$-th order natural bundles, and classify all (−0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
Abstract: We generalize reduction theorems for classical connections to
operators with values in $k$-th order natural bundles. Using
the 2nd order valued reduction theorems we classify all
(0,2)-tensor fields on the cotangent bundle of a manifold with
a linear (non-symmetric) connection.
TL;DR: For a ring R and a right R−module M, a submodule N of M is said to be �- small in M if, whenever N + X = M with M/X singular, we have X =M as mentioned in this paper.
Abstract: For a ring R and a right R−module M, a submodule N of M is said to be �- small in M if, whenever N + X = M with M/X singular, we have X = M. Let } be the class of all singular simple modules. Then �(M) = P {L ≤ M | L is a �-small submodule of M } = RejM(}) = ∩{N ⊂ M : M/N ∈ }}. We call M �−coatomic module whenever N ≤ M and M/N = �(M/N) then M/N = 0. And R is called right (left) �−coatomic ring if the right (left) R−module RR(RR) is �−coatomic. In this note, we study �−coatomic modules and ring. We prove M = ⊕ n=1 Mi is �-coatomic if and only if each Mi (i = 1,...,n) is �-coatomic.
TL;DR: In this article, a cyclic q-gonal surface X of genus g ≥ 2 admits a group of automorphisms C of prime order q such that X/C has genus 0.
Abstract: A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C
q of prime order q such that X/C
q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.
TL;DR: In this article, the authors define the localization LMn-algebra of an LMnalgebra L with respect to a topology F on L and prove that the maximal LMn algebra of fractions relative to an Λ-closed system (defined in Section 2) are LMn n-algebras of localization.
Abstract: The aim of this paper is to define the localization LMn-algebra of an LMn—algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LMn-algebra of fractions (defined in [3]) and the LMn-algebra of fractions relative to an Λ—closed system (defined in Section 2) are LMn-algebras of localization.
Abstract: There is one serious error in Table 2 of [2], which lists the children of each node of the closure diagram, and several minor errors. Let us first deal with the minor ones. The formula displayed at the bottom of p. 578 defines the set Ii by giving two expressions for it. The first of these expressions is incorrect and should be omitted. The types E8(a7) ′ in Figure 6 (p. 640) and E8(b4)′ in Figure 8 (p. 642) should be deleted. The nilpotent elements E that they represent do not belong to the orbits 67 and 107, respectively, but to some orbits of lower dimension. Consequently, the following corrections to Table 1 are required: Delete the representatives of the orbits 67 and 68 of type E8(a7) ′ (p. 593) and the representative of the orbit 107 of type E8(b4) ′ on p. 597. We point out that these particular representatives were not used in any of the proofs in the paper. The diagrams of type Dn(ak) and Dn(ak) ′ given in Figure 5 and the corresponding representatives E and sl2-triples are studied in detail in [1]. There are a few misprints or inaccuracies that we have noticed: 1) In the second column of Table 3 on p. 601, when i = 98 there should be two possible values, 94 and 92, for j. 2) In Table 4 on p. 603 in the row for i = 40 and j = 37, the index −88 for the representative E should be replaced by −82. And on p. 606 in the row for i = 86 and j = 84 the index −83 occurs twice; the second one should be replaced by 103. 3) In Table 4, the type of the representative E is incorrectly given in the following cases: On p. 604 in the row for i = 66 and j = 65 the type should be D6(a2) ′ + A1. On p. 605 in the row for i = 73 and j = 69 it should be D6(a2) ′ + 2A1 and in the row for i = 78 and j = 76 it should be E7(a4) ′. On p. 607 in the row for i = 107 and j = 105 the type should be D8(a1). In all these cases, the representative E itself is correct.
TL;DR: In this paper, the homology and the cycles in tensor products of algebras of symmetric functions over Ω2 were calculated. But the homologies were not shown for the tensor product of a symmetric function.
Abstract: We calculate the homology and the cycles in tensor products of algebras of symmetric function over ℤ2
TL;DR: In this paper, the homotopy equivalence classes of objects in differential graded free Lie algebras over R have been studied, and the exact sequence of Whitehead associated with a free dgl has been used to characterize a dgl to belong to S(L(V), ϖ).
Abstract: 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 DGLnnp, 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 DGLnnp. In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGLn3n+2 and denote by S(L(V), ϖ) the class of objects homotopy equivalent to (L(V), ϖ). How we can characterize a free dgl to belong to S(L(V), ϖ)? Fix an object (L(V), ϖ) in DGLn3n+2. How many homotopy equivalence classes of objects (L(W), δ) in DGLn3n+2 such that H*(W, d′)≊H*(V, d) are there? Note that DGLn3n+2 is a subcategory of DGLnnp when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.
TL;DR: In this article, a linear programming problem is transformed to the finding an element of polyhedron with the minimal norm and an orthogonal method for solving the problem is used.
Abstract: A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.
TL;DR: In this article, the congruence lattices of standard QBCC-algebras and their connection with Congruence Loops of congruency kernels are characterized.
Abstract: We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels.
TL;DR: In this paper, sufficient conditions which guarantee that a linear integro-differential equation cannot have a positive solution are established, and sufficient conditions are established for a linear integral differential equation to have a negative solution.
Abstract: Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.