Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

On summability of weighted Lagrange interpolation. II

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C _ρ, ‖·‖_ρ) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to w, ρ, (C _ρ, ‖·‖_ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence will also be considered.     

Characterization of d.c. functions in terms of quasidifferentials

A characterization of d.c. functions f:Ω→R in terms of the quasidifferentials of f is obtained, where Ω is an open convex set in a real Banach space. Recall that f is called d.c. (difference of convex) if it can be represented as a difference of two finite convex functions. The relation of the obtained results with known characterizations is discussed, specifically the ones from [R. Ellaia, A. Hassouni, Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991), 401-416] in the finite-dimensional case and [A. Elhilali Alaoui, Caractérisation des fonctions DC, Ann. Sci. Math. Québec 20 (1996), 1-13] in the case of a Banach space.     

Erratum to: “On the HSS iteration methods for positive definite Toeplitz linear systems” [J. Comput. Appl. Math. 224 (2009) 709-718]

In [1], Gu and Tian [Chuanqing Gu, Zhaolu Tian, On the HSS iteration methods for positive definite Toeplitz linear systems, J. Comput. Appl. Math. 224 (2009) 709-718] proposed the special HSS iteration methods for positive definite linear systems Ax=b with A∈C^n×n a complex Toeplitz matrix. But we find that the special HSS iteration methods are incorrect. Some examples are given in our paper.     

Asymptotic observers for a class of evolution equations. A nonlinear approach

We consider in a Hilbert space H the system (E_u) = x = uAx+B(x); y = 〈x. c〉_H, where the control u ε L^∞([0, + ∞[, ℝ⁺) multiplies a possibly unbounded m-dissipative linear operator A. The operator B is nonlinear dissipative, and y stands for the output of the system. We prove, in this nonlinear framework, the existence of a suitable Luenberger-like observer. For this purpose, we show that the usual notions of regularly persistent inputs proposed in [7] or [4] are the appropriate concepts that allow one to generalize the main results of [9] and [8] or [7] for bilinear systems to our nonlinear general system: For each regularly persistent input, the estimation error of the observer converges weakly to zero. If in addition A generates a compact semigroup, the estimation error converges strongly to zero. A prototype of such a system is the heat exchanger system described in [9] or [8].     

Topology of compact space forms from Platonic solids. I.

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253-263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL₂(C) with compact orbit space, Canad. J. Math. 23 (1971) 451-460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329-335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497-515], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, Preprint]. In this paper we investigate the topology of closed orientable 3-manifolds from Platonic solids. Here we completely recognize those manifolds in the spherical and Euclidean cases, and state topological properties for many of them in the hyperbolic case. The proofs of the latter will appear in a forthcoming paper.     

Convexity spaces. II. Separation

This is the second of a series of three papers dealing with convexity spaces. In the first paper [1] we defined a convexity space and investigated some of its basic properties. Here we consider the separation and support of convex sets. Throughout the paper we will be dealing with a convexity space (X, ·) and the terminology and notation used will be those of [1]. In particular A^c denotes the complement of the set A in X and β is used to denote set-theoretic difference.     

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a=(a₁,…,an) and b=(b₁,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217-222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61-71] (n∈N) for real-valued functions of n variables.     

Dual pairs of sequence spaces. III

In the previous papers [J. Boos, T. Leiger, Dual pairs of sequence spaces, Int. J. Math. Math. Sci. 28 (2001) 9-23; J. Boos, T. Leiger, Dual pairs of sequence spaces. II, Proc. Estonian Acad. Sci. Phys. Math. 51 (2002) 3-17], the authors defined and investigated dual pairs (E,ES), where E is a sequence space, S is a BK-space on which a sum s is defined in the sense of Ruckle [W.H. Ruckle, Sequence Spaces, Pitman Advanced Publishing Program, Boston, 1981], and ES is the space of all factor sequences from E into S. In generalization of the SAK-property (weak sectional convergence) in the case of the dual pair (E,Eβ), the SK-property was introduced and studied. In this note we consider factor sequence spaces E|S|, where |S| is the linear span of , the closure of the unit ball of S in the FK-space ω of all scalar sequences. An FK-space E such that E|S| includes the f-dual Ef is said to have the SB-property. Our aim is to demonstrate, that in the duality (E,ES), the SB-property plays the same role as the AB-property in the case ES=Eβ. In particular, we show for FK-spaces, in which the subspace of all finitely non-zero sequences is dense, that the SB-property implies the SK-property. Moreover, in the context of the SB-property, a generalization of the well-known factorization theorem due to Garling [D.J.H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967) 997-1019] is given.     

Controlling multiparticle system on the line. II. Periodic case

As in [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] we consider classical system of interacting particles P₁,…,Pn on the line with only neighboring particles involved in interaction. On the contrast to [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] now periodic boundary conditions are imposed onto the system, i.e. P₁ and Pn are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its nonperiodic counterpart.Main result established is global controllability by means of single controlling force of the multiparticle system with a generic potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections 2, 3.     

On nonnormal products and a set of A.H. Stone

A variant of Michael's example is given to the following effect: there is a Lindelöf space M of weight ℵ₁, with all Gδ-sets open, such that M×B(ℵ₁) is nonnormal. This answers a question from [K. Alster, On the class of ω₁-metrizable spaces whose product with every paracompact space is paracompact, Topology Appl. 153 (2006) 2508-2517].     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

The uniqueness of the adjoint operation. II

Let H be a complex, finite-dimensional Hilbert space, and let L(H) denote the set of linear transformations mapping H into itself. For certain interesting subsets A(H) of L(H) [nonsingular transformations and L(H) are examples], the functions h: A(H) → L(H) which have the properties h(ST) = h(T)h(S) and h(S)S ⩾ 0 are characterized.     

On determination of jumps in terms of the Abel-Poisson mean of Fourier series. II

In this paper, we generalize two important results of Bagota and Móricz [1], and generalize our earlier results in [6] from one-variable to two-variable case. As special applications, we prove that the generalized jump of f(x, y) at some point (x ₀, y ₀) can be determined by the higher order mixed partial derivatives of the Abel-Poisson mean of double Fourier series and the higher order mixed partial derivatives of the Abel-Poisson means of the three conjugate double Fourier series.     

Topological regular variation: II. The fundamental theorems

This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semigroup of natural numbers N (e.g. de Bruijn's Representation Theorem for regularly varying functions). Other such results include Goldie's direct proof of the Uniform Convergence Theorem and Seneta's version of Kendall's theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting N-flow and R-flow versions of regular variation, and in so doing generalize these theorems to Rd. We also prove a flow version of the classical Characterization Theorem of regular variation.     

On a Conjecture of E. Dittert

Let K_n denote the set of all n X n nonnegative matrices whose entries have sum n, and let φ be a real valued function defined on K_n by φ(X) = π_iⁿ⁼¹ n, + π_j=1^c_j⁻ⁿ per X for X E K_n with row sum vector (r₁,…, r_n) and column sum vector (c_l,hellip;, c_n). For the same X, let φ_ij(X)= π_k≠ir_k + π_1≠jc₁ - per X(i| j). A ϵK_n is called a φ-maximizing matrix if φ(A) > φ(X) for all X ϵ K_n. Dittert's conjecture asserts that J_n = [1/n]_n×n is the unique (φ-maximizing matrix on K_n. In this paper, the following are proved: (i) If A = [a_ij] is a φ-maximizing matrix on K_n then φ_ij(A) = φ (A) if a_ij > 0, and φ_ij (A) ⩽ φ (A)if aij = 0. (ii) The conjecture is true for n = 3.     

Spectral integrals of operator-valued functions. II. From the study of stationary processes

This paper is a continuation of the study made in [38]. Using Douglas' operator range theorem and Crimmins' corollary we obtain several new results on the “square-integrability of operator-valued functions with respect to a nonnegative hermitian measure”. Using these facts we are able to extend in an important way theorems on the “spectral integral of an operator-valued function” which were obtained in [38], to wit, we are able to drop assumptions that functions are closed operator-valued. We apply these results to Wiener-Masani type infinite-dimensional stationary processes, representing a purely non-deterministic process as a “moving average” and obtaining a “factorization” of its spectral density. Next, anticipating global applications of our tools, we investigate the adjoint and generalized inverse of spectral integrals. Our definition of measurability for closed-operator-valued functions plays a key role here. Finally, we partially prove a conjecture (J. Multivariate Anal. (1974), 166–209) on simpler necessary and sufficient conditions on “when is a closed densely defined operator T from $\text{H}$^q to $\text{H}$^p a spectral integral T = fΦdE?”: Let q be finite and E be of countable multiplicity for $\text{H}$. Then (i) Tx ∈ $\text{S}$_x^p each x ∈ $\text{D}$_T (T is E-subordinate), and (ii) E(B)T ? TE(B) each B ∈ $\text{B}$ (T is E-commutative) implies L_x^pT ? TL_x^q each x ∈ $\text{H}$^q (T commutes with all the cyclic projections), and thus T = fΦdE.     

Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction

A 2-d or 3-d fluid-structure interaction model in its linear form is considered, for which semigroup well-posedness (with explicit generator) was recently established in [G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., 2007, pp. 15-55; G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part II: Uniform stabilization with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., in press]. This is a system which couples at the interface the linear version of the Navier-Stokes equations with the equations of linear elasticity (wave-like). In this paper, we establish a backward uniqueness theorem for such a parabolic-hyperbolic coupled PDE system. If {eAt}_t?0 is the (contraction) s.c. semigroup describing its evolution on the finite energy space H, then eATy₀=0 for some T>0 and y₀∈H, implies y₀=0. This property has implications in establishing unique continuation and controllability properties, as in the case of thermoelastic equations [M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, in: Marcel Dekker Lect. Notes Pure Appl. Math., vol. 216, February 2001, pp. 109-230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol, J.P. Zolesio (Eds). (Preliminary version is in invited paper in: A.V. Balakrishnan (Ed.), Semigroup of Operators and Applications, Birkhäuser, 2000, pp. 335-351.); M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000) 452-478; M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls, Discrete Contin. Dyn. Syst. 7 (2) (2001) 283-301].