Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

On the boundedness of classical operators on weighted lorentz spaces

Conditions on weightsu(·),v(·) are given so that a classical operatorT sends the weighted Lorentz spaceL ^Lrs (vdx) intoL ^pq (udx). HereT is either a fractional maximal operatorM _α or a fractional integral operatorI _α or a Calderón-Zygmund operator. A characterization of this boundedness is obtained forM _α andI _α when the weights have some usual properties and max(r, s) ≤ min(p, q).     

Uniqueness of weak solutions of time-dependent 3-D Ginzburg-Landau model for superconductivity

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ ₀, A ₀) ∈ L ²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L ³(Ω)) using the Lorentz gauge.      

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      8. Characterizations and Extensions of Lipschitz-α Operators   总被引：1，自引：0，他引：1   Huai Xin CAO Jian Hua ZHANG Zong Ben XU 《数学学报(英文版)》2006,22(3):671-678 In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.      9. Problem with periodicity conditions for a degenerating equation of mixed type      K. B. Sabitov  O. G. Sidorenko 《Differential Equations》2010,46(1):108-116 For the equation K(t)u xx + u tt − b 2 K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u x (0, t) = u x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.      10. On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators      El Maati Ouhabaz 《Semigroup Forum》1998,57(3):305-314 m σ/|α|=|β|=m Dα(a αβDβ) is a higher order degenerate-elliptic operator on L2(RN) and V ∈ L1(RN) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.      11. On moduli of vector bundles on surfaces with negative Kodaira dimension      E. Ballico 《Annali dell'Universita di Ferrara》1982,38(1):33-40 Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ i, 1≤i≤τ,a’, b’, a’ i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK X 2 =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ i, (resp.a’, b’, a’ i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc 2=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)red withS a relatively minimal model ofX. Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K X Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec 2=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).       12. Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges      Mátyás Barczy  Endre Iglói 《Central European Journal of Mathematics》2011,9(1):65-84 We study Karhunen-Loève expansions of the process(X t (α)) t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).      13. A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications      Valentin Keyantuo  Carlos Lizama 《Mathematische Nachrichten》2011,284(4):494-506 We study the fractional differential equation (*) Dαu(t) + BDβu(t) + Au(t) = f(t), 0 ? t ? 2π (0 ? β < α ? 2) in periodic Lebesgue spaces Lp(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R‐boundedness of the sets {(ik)α((ik)α + (ik)βB + A)?1}k∈ Z and {(ik)βB((ik)α + (ik)βB + A)?1}k∈ Z . Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset‐Boussinesq‐Oseen equation are treated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim      14. Orthogonal almost locally minimal projections on ℓ 1 n      M. Zippin 《Israel Journal of Mathematics》2000,115(1):253-268 A projectionP on a Banach spaceX with ‖P‖≤λ0 is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα2), whereD=D(λ0) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL p(G) is almost locally minimal. Orthogonal projections on ℓ 1 n are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).      15. The generalized Boardman homomorphisms      Dominique Arlettaz 《Central European Journal of Mathematics》2004,2(1):50-56 This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b n : π n (X)→H n (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π n (X)→E n (X), F n (X)→(E∧F) n (X), F n (X)→H n (X;π 0 F) and F n (X)→H n+t (X;π t F) for other cohomology theories E *(−) and F *(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.      16. Discrete spectrum of the perturbed Dirac operator      Mikhail Sh. Birman  Ari Laptev 《Arkiv f?r Matematik》1994,32(1):13-32 In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D 0−αV1 acting inL 2(R 3;C 4) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.      17. Global existence and time decay of small solutions to the landau-ginzburg type equations      Nakao Hayashi  Elena I. Kaikina  Pavel I. Naumkin 《Journal d'Analyse Mathématique》2003,90(1):141-173 We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2 n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).      18. Self-similar extremal processes      E. I. Pancheva 《Journal of Mathematical Sciences》1998,92(3):3911-3920 Given an extremal process X: [0,∞)→[0,∞)d with lower curve C and associated point process N={(tk, Xk):k≥0}, tk distinct and Xk independent, given a sequence ζ n =(τ n , ξ n ), n≥1, of time-space changes (max-automorphisms of [0,∞)d+1), we study the limit behavior of the sequence of extremal processes Yn(t)=ξ n -1 ○ X ○ τn(t)=Cn(t) V max {ξ n -1 ○ Xk: tk ≤ τn(t){ ⇒ Y under a regularity condition on the norming sequence ζn and asymptotic negligibility of the max-increments of Yn. The limit class consists of self-similar (with respect to a group ηα=(σα, Lα), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property Lα ○ Y(t) = d Y ○ αα(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.      19. A Functional LIL for <Emphasis Type="Italic">d</Emphasis>-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes      Joshua Rushton 《Journal of Theoretical Probability》2007,20(3):397-427 We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t ‖X(s)‖ of an ℝ d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X. Supported in part by NSF Grant DMS 02-05034.      20. Parabolic bundles and representations¶of the fundamental group      Tomás L. Gómez  T. R. Ramadas 《manuscripta mathematica》2000,103(3):299-311 Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π1(X) doesn't have irreducible PU(r) representations, then π1(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges

We study Karhunen-Loève expansions of the process(X _t ^(α)) _t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X ₀^(α) = 0, where α > 0, T ∈ (0, ∞), and (B _t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X ^(α). As applications, we calculate the Laplace transform and the distribution function of the L ²[0, T]-norm square of X ^(α) studying also its asymptotic behavior (large and small deviation).     

A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) D^αu(t) + BD^βu(t) + Au(t) = f(t), 0 ? t ? 2π (0 ? β < α ? 2) in periodic Lebesgue spaces L^p(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R‐boundedness of the sets {(ik)^α((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} and {(ik)^βB((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} . Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset‐Boussinesq‐Oseen equation are treated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Orthogonal almost locally minimal projections on ℓ 1 n

A projectionP on a Banach spaceX with ‖P‖≤λ₀ is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα²), whereD=D(λ₀) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL _p(G) is almost locally minimal. Orthogonal projections on ℓ ₁ ⁿ are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

A Functional LIL for <Emphasis Type="Italic">d</Emphasis>-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

We establish a functional LIL for the maximal process M(t) :=sup _0≤s≤t ‖X(s)‖ of an ℝ ^d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X. Supported in part by NSF Grant DMS 02-05034.     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000