On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

Asymptotic equalities for best approximations for classes of infinitely differentiable functions defined by the modulus of continuity

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space C(L_p) for classes of periodic functions expressible as convolutions of kernels Ψ_β with Fourier coefficients decreasing to zero faster than any power sequence, and with functions ? ∈ C (? ∈ L_p) whose moduli of continuity do not exceed the given majorant of ω(t). It is proved that, in the spaces C and L₁, for convex moduli of continuity ω(t), the obtained estimates are asymptotically sharp.     

Isometries with dense windings of the torus in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">M</Emphasis>)

Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus

$$O(M) = \left\{ {f:M \to \mathbb{C} \ntrianglelefteq \left\| f \right\| = \left\| {\frac{1}{f}} \right\| = 1} \right\}$$

. If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).

     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

On lengths of <Emphasis Type="Italic">H</Emphasis>ℤ-localization towers

In this paper, the H?-length of different groups is studied. By definition, this is the length of the H?-localization tower or the length of the transfinite lower central series of H?-localization. It is proved that, for a free noncyclic group, its H?-length is ≥ ω+2. For a large class of ?[C]-modules M, where C is an infinite cyclic group, it is proved that the H?-length of the semi-direct product M ? C is ≤ ω + 1 and its H?-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M?C is finitely presented and H₂(M ? C) is finite.     

Critical configurations of planar multiple penduli

We consider the oriented area function A on the moduli space M(P) of mechanical linkage P representing a planar multiple pendulum. For generic lengths of the sides of P, it is proved that A is a Morse function on M(P) and its critical points are given by the cyclic configurations of P satisfying an additional geometric condition. For triple penduli, the main result is complemented by a rather comprehensive analysis of the structure of critical configurations. Moreover, we discuss the critical configurations of another natural function on the moduli space of a planar multiple pendulum and the image of the cross-ratio mapping into the plane. A number of related results and open problems are also presented.     

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the class of first-order pseudo-differential operators L = D_t + C(t, x,D_x), where (t, x) ∈ T × M, T is the one-dimensional torus, M is a closed manifold, and C(t, x,D_x) is a first-order pseudo-differential operator on M, smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x,D_x) = a(t)p(x,D_x) + ib(t)q(x,D_x), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.     

Extending Lipschitz maps into <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-spaces

We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M, extending recent results of Lancien and Randrianantoanina. They showed that a finite-dimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the same result holds for spaces with Gateaux smooth norm or of dimension two; a three-dimensional counterexample is also given. We also show that X is polyhedral if and only if every subset E of X has the universal isometric extension property for C(K)-spaces. We also answer a question of Naor on the extension of Hölder continuous maps.     

Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras

We define the motivic Milnor fiber of cyclic L _∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L _∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H²(D²). A closed subspace M in H²(D²) is called a submodule if z _i M ? M (i = 1, 2). An associated integral operator (defect operator) C _M captures much information about M. Using a Kre?n space indefinite metric on the range of C _M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.     

When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.     

Generalizing the Krein–Rutman theorem,measures of noncompactness and the fixed point index

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let \({C \subset D}\) be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ _C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If \({{\tilde r}_{C}(f)}\) denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if \({\rho_{C}(f) < {\tilde r}_{C}(f)}\), then there exists \({u \in C {\backslash} \, \{0\}}\) with f(u) = ru where r ? r _C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

Global Well-posedness of the Stochastic Generalized Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic generalized Kuramoto- Sivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u₀ ∈ L₂(D × Ω), this problem has a unique global solution u in the space L₂(Ω, C([0, T], L₂(D))) for any T >0, and the solution map u₀ ? u is Lipschitz continuous.     

On the well-posedness of degenerate fractional differential equations in vector valued function spaces

We characterize completely the well-posedness on the vector-valued Hölder and Lebesgue spaces of the degenerate fractional differential equation D ^α (Mu)(t) = Au(t) + f(t), t ∈ ? by using vector-valued multiplier results in the spaces C ^γ (?;X) and L ^p (?;X), where A and M are closed linear operators defined on the Banach space X, 0 < γ < 1, 1 < p < ∞, the fractional derivative is understood in the sense of Caputo and α is positive.     

On the Number of Vedernikov–Ein Irreducible Components of the Moduli Space of Stable Rank 2 Bundles on the Projective Space

We propose a method for finding the exact number of Vedernikov–Ein irreducible components of the first and second types in the moduli space M(0, n) of stable rank 2 bundles on the projective space P³ with Chern classes c₁ = 0 and c₂ = n ≥ 1. We give formulas for the number of Vedernikov–Ein components and find a criterion for their existence for arbitrary n ≥ 1.     

Counting saddle connections in flat surfaces with poles of higher order

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface X of genus at least 3. The choice of a Poincaré bundle for such a moduli space M induces an isomorphism between X and a component of the moduli space of semistable sheaves over M. We prove that \(\dim H^0(M,\, \text {End}({\mathcal {E}})\otimes TM)\,=\, 1\) for any vector bundle \(\mathcal {E}\) on M coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on \(\mathcal {E}\).