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},      2. How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?   总被引：1，自引：0，他引：1   Anders Yeo 《Graphs and Combinatorics》1999,15(4):481-493  Let D be a semicomplete multipartite digraph, with partite sets V 1, V 2,…, V c, such that |V 1|≤|V 2|≤…≤|V c|. Define f(D)=|V(D)|−3|V c|+1 and . We define the irregularity i(D) of D to be max|d +(x)−d −(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i l(D) of D to be max|d +(x)−d −(x)| over all vertices x of D and we define the global irregularity of D to be i g(D)=max{d +(x),d −(x) : x∈V(D)}−min{d +(y),d −(y) : y∈V(D)}. In this paper we show that if i g(D)≤g(D) or if i l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i g(D)=i(D)=i l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998      3. Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems      S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298 We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α k: C p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.      4. On semi-fredholm properties of a boundary value problem inR + n      Jouko Tervo 《Israel Journal of Mathematics》1988,63(1):41-66 The paper considers a boundary value problem with the help of the smallest closed extensionL ∼ :H k →H k 0×B h 1×...×B h N of a linear operatorL :C (0) ∞ (R + n ) →L(R + n )×L(R n−1)×...×L(R n−1). Here the spacesH k (the spaces ℬ h ) are appropriate subspaces ofD′(R + n ) (ofD′(R n−1), resp.),L(R + n ) andC (0) ∞ (R + n )) denotes the linear space of smooth functionsR n →C, which are restrictions onR + n of a function from the Schwartz classL (fromC 0 ∞ , resp.),L(R n−1) is the Schwartz class of functionsR n−1 →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ∼) and for the uniqueness of solutionsL ∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.      5. Random matrix products and applications to cellular automata      Yossi Moshe 《Journal d'Analyse Mathématique》2006,99(1):267-294 Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X i) i 0/∞ over ℂ. Assume that theX i's are chosen from a finite set {D 0,D 1...,D t-1(ℂ), withP(X i=Dj)>0, and that the monoid generated byD 0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN ≠0(F(x)n,r) of nonzero coefficients inF(x) n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN ≠0(F(x)n,r) ≈n α for “almost” everyn. Supported in part by FWF Project P16004-N05      6. Carleson measures for the area Nevanlinna spaces and applications      Boo Rim Choe  Hyungwoon Koo  Wayne Smith 《Journal d'Analyse Mathématique》2008,104(1):207-233 Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N p (μ) consists of all measurable functions h on D such that log+ |h| ∈ L p (μ), and the area Nevanlinna space N α p is the subspace consisting of all holomorphic functions, in N p ((1−|z|2)α dv(z)), where α > −1 and ν is area measure on D. We characterize Carleson measures for N α p , defined to be those measures μ for which N α p ⊂ N p (μ). As an application, we show that the spaces N α p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given. The second author was supported in part by KRF-2004-015-C00019.      7. On the Matlis duals of local cohomology modules and modules of generalized fractions      Kazem Khashyarmanesh 《Proceedings Mathematical Sciences》2010,120(1):35-43 Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x 1, …, x n is a regular sequence on M contained in α, then H (x1, …,xnR n D(H a n (M))) is a homomorphic image of D(M), where H b i (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H (x1, …,xn)R n (D(H a n (M)))) ⋟ D(D(M)).      8. Grothendieck groups of unbounded complexes of finitely generated modules      Jun-ichi Miyachi 《Archiv der Mathematik》2006,86(4):317-320 Let Λ be a left Artinian ring, D+(mod Λ) (resp., D−(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K0(D+(mod Λ)), K0(D−(mod Λ)) and K0(D(mod Λ)) are trivial. Received: 7 April 2005      9. A reverse Denjoy theorem II      P. C. Fenton  J. Rossi 《Journal d'Analyse Mathématique》2010,110(1):385-395 For α satisfying 0 < α < π, suppose that C 1 and C 2 are rays from the origin, C 1: z = re i(π−α) and C 2: z = re i(π+α), r ≥ 0, and that D = {z: | arg z − π| < α}. Let u be a nonconstant subharmonic function in the plane and define B(r, u) = sup|z|=r u(z) and A D (r, u) = $ \inf _{z \in \bar D_r } $ \inf _{z \in \bar D_r } u(z), where D r = {z: z ∈ D and |z| = r}. If u(z) = (1 + o(1))B(|z|, u) as z → ∞ on C 1 ∪ C 2 and A D (r, u) = o(B(r, u)) as r → ∞, then the lower order of u is at least π/(2α).      10. Fractional Poisson equations and ergodic theorems for fractional coboundaries      Yves Derriennic  Michael Lin 《Israel Journal of Mathematics》2001,125(1):93-130 For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT α=Σ j=1 ∞ a j T j , where {a j } are the coefficients in the power series expansion (1-t)α=1-Σ j=1 ∞ a j t j in the open unit disk, which satisfya j >0 anda j >0 and Σ j=1 ∞ a j =1. The operator calculus justifies the notation(I−T) α :=I−T α (e.g., (I−T 1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) α x=y, i.e.,y is a coboundary forT α . The fractional Poisson equation forT is the Poisson equation forT α . We show that if(I−T)X is not closed, then(I−T) α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ k=1 ∞ T k y/k 1-α converges in norm, and conclude that lim n ‖(1/n 1-α)Σ k=1 n T k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL 1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) α L 1 for some 0<α<1, then the one-sided Hilbert transform Σ k=1 ∞ T k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) α L p with α>1−1/p=1/q, then Σ k=1 ∞ T k f/k 1/p converges a.e., and thus (1/n 1/p ) Σ k=1 n T k f converges a.e. to zero. Whenf∈(I−T) 1/q L p (the case α=1/q), we prove that (1/n 1/p (logn)1/q )Σ k=1 n T k f converges a.e. to zero.      11. Singularly perturbed periodic and semiperiodic differential operators      V. A. Mikhailets  V. M. Molyboga 《Ukrainian Mathematical Journal》2007,59(6):858-873 Qualitative and spectral properties of the form sums , are studied in the Hilbert space L 2(0, 1). Here, (D +) is a periodic differential operator, (D −) is a semiperiodic differential operator, D ±: u ↦ −iu′, and V(x) is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces H per −mα , α ∈ [0, 1]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 785–797, June, 2007.      12. 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.      13. Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems      T. Shibata 《Annali di Matematica Pura ed Applicata》2007,186(3):525-537 We consider the nonlinear Sturm–Liouville problem (1) where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R + × L 2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u α with ‖ u α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/u p satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ α p−1 h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim u → ∞ uh′(u). Mathematics Subject Classification (2000) 34B15      14. Penultimate limiting forms in extreme value theory      M. Ivette Gomes 《Annals of the Institute of Statistical Mathematics》1984,36(1):71-85 Summary Let {X n}n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM n=max (X 1,…,X n), suitably normalized with attraction coefficients {αn}n≧1(αn>0) and {b n}n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M n−bn)/an thanG(x), for moderaten. We thus consider differences of the formF n(anx+bn)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф α(x)=exp (−x−α), x>0] or a type III [Ψ α(x)= exp (−(−x)α), x<0] d.f. of largest values, much closer toF n(anx+bn) than the ultimate itself.      15. A Property About Minimum Edge- and Minimum Clique-Cover of a Graph      Raffaele Mosca 《Graphs and Combinatorics》2001,17(3):517-528  Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ n :Ex≤1,x≥0}, P(C)={x∈ℜ n :Cx≤1,x≥0}, α E (G)=max{1 T x subject to x∈P(E)}, and α C (G)= max{1 T x subject to x∈P(C)}. In this paper we prove that if α E (G)=α C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999      16. The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)      Fan Dashan  Yang Dachun 《分析论及其应用》1997,13(4):19-41 Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω1,ω2 ɛA 1(R n ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk q α,p (gw1,ω2) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R n ), the class of C∞(Rn)-functions with compactly support, is dense inhK q α,p (ω1,ω2) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK q α,p (ω1,ω2), of any bounded sequence inhK q α,p (ω1,ω2). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.      17. The minimality of the map \frac{x}{\|{x}\|} for weighted energy      Jean-Christophe Bourgoin 《Calculus of Variations and Partial Differential Equations》2006,25(4):469-489 In this paper, we investigate the minimality of the map from the Euclidean unit ball Bn to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫Bn f(r)‖∇ u‖p dx, where f is a non-negative function. We prove that in each of the two following cases: i)  p = 1 and f is non-decreasing, ii)  p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map minimizes Ep,f among the maps in W1,p(Bn, 핊n−1) which coincide with on ∂ Bn. We also study the case where f(r) = rα with −n+2 < α < 0 and prove that does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n. Mathematics Subject Classification (2000) 58E20; 53C43      18. A limit theorem for random coverings of a circle      Leopold Flatto 《Israel Journal of Mathematics》1973,15(2):167-184 LetN α, m equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that limα→0 P(Nα,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(Nα,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.      19. On Lp-convergence of Grunwald interpolation      Min Guohua 《分析论及其应用》1992,8(3):28-37 In this paper, the Lp-convergence of Grünwald interpolation Gn(f,x) based on the zeros of Jacobi polynomials J n (α,β) (x)(−1<α,β<1) is considered. Lp-convergence (0<p<2) of Grünwald interpolation Gn(f,x) is proved for p·Max(α,β)<1. Moreover, Lp-convergence (p>0) of Gn(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.      20. On number system constructions      L. Germán  A. Kovács 《Acta Mathematica Hungarica》2007,115(1-2):155-167 We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M −1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M −1. We shall prove further that if the polynomial f(x) = c 0 + c 1 x + ··· + c k x k ∈ Z[x], c k = 1 satisfies the condition |c 0| > 2 Σ i=1 k |c i | then there is a suitable digit set D for which (Z k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Carleson measures for the area Nevanlinna spaces and applications

Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N ^p (μ) consists of all measurable functions h on D such that log⁺ |h| ∈ L ^p (μ), and the area Nevanlinna space N _α ^p is the subspace consisting of all holomorphic functions, in N ^p ((1−|z|²)^α dv(z)), where α > −1 and ν is area measure on D. We characterize Carleson measures for N _α ^p , defined to be those measures μ for which N _α ^p ⊂ N ^p (μ). As an application, we show that the spaces N _α ^p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given. The second author was supported in part by KRF-2004-015-C00019.     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

Grothendieck groups of unbounded complexes of finitely generated modules

Let Λ be a left Artinian ring, D⁺(mod Λ) (resp., D⁻(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K₀(D⁺(mod Λ)), K₀(D⁻(mod Λ)) and K₀(D(mod Λ)) are trivial. Received: 7 April 2005     

A reverse Denjoy theorem II

For α satisfying 0 < α < π, suppose that C ₁ and C ₂ are rays from the origin, C ₁: z = re ^i(π−α) and C ₂: z = re ^i(π+α), r ≥ 0, and that D = {z: | arg z − π| < α}. Let u be a nonconstant subharmonic function in the plane and define B(r, u) = sup_|z|=r u(z) and A _D (r, u) = $ \inf _{z \in \bar D_r } $ \inf _{z \in \bar D_r } u(z), where D _r = {z: z ∈ D and |z| = r}. If u(z) = (1 + o(1))B(|z|, u) as z → ∞ on C ₁ ∪ C ₂ and A _D (r, u) = o(B(r, u)) as r → ∞, then the lower order of u is at least π/(2α).     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Singularly perturbed periodic and semiperiodic differential operators

Qualitative and spectral properties of the form sums

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.     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

The minimality of the map \frac{x}{\|{x}\|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ u‖^p dx, where f is a non-negative function. We prove that in each of the two following cases:

A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.     

On Lp-convergence of Grunwald interpolation

In this paper, the L_p-convergence of Grünwald interpolation G_n(f,x) based on the zeros of Jacobi polynomials J _n ^(α,β) (x)(−1<α,β<1) is considered. L_p-convergence (0<p<2) of Grünwald interpolation G_n(f,x) is proved for p·Max(α,β)<1. Moreover, L_p-convergence (p>0) of G_n(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.