Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action ofSL _n (K) on the affine spaceX = V^⊕m ⊕ (V*)^⊕q whereV = K ⁿ ,K an algebraically closed field of arbitrary characteristic andm,q > n. We construct a ‘standard monomial’’ basis for the ring of invariantsK[X] ^SL _n ^(K). As a consequence, we deduce thatK[X] ^SL _n ^(K) is Cohen-Macaulay. We also present the first and second fundamental theorems forSL _n (K)- actions.     

On the abelian fundamental group scheme of a family of varieties

Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) induced by the canonical morphism from X to its Albanese scheme Alb _X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS _X/S ^τ )^⋎ → π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber X _η of X can be extended to a finite and commutative pointed torsor over X.     

Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T: E → X for which dens X/E = dens X/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c ₀ is automorphic and conjectured that c ₀ and ℓ₂ are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c ₀(Γ), for Γ uncountable. That c ₀(Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of ℓ _n ^p , 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as L _p (μ), p ≠ 2, C(K) spaces not isomorphic to c ₀ for K metric compact, subspaces of c ₀ which are not isomorphic to c ₀, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic. The work of the first author has been supported in part by project MTM2004-02635     

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).

     

Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

Let R=P₁⊕P₂⊕…⊕P_n be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms ϕ: P_i→P_j is a monomorphism; (2) if subideals Q₁, Q₂ of the ideal P_j are isomorphic to the ideal P_i, then there exists a subideal Q₃⊆Q₁⊎Q₂, which is also isomorphic to P_i. It is proved that, under these asumptions, a left quotient ring of the ring R exists. This left quotient ring inherits properties(1), (2) and satisfies condition (3): any nonzero homomorphism ϕ: P_i→P_i is an automorphism of the ideal P_i. Bibliography: 2 titles. Translated fromZapiski Nauchnyk Seminarov POMI, Vol. 227, 1995, pp. 9–14.     

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d _X : K ₀(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K ₀(A) = ℚ⊕ kerd _x and K ₁(A) = K ₁(C(X)), then A is isomorphic to an inductive limit of M _{n !}(C(X)). Received April 19, 2001, Accepted April 27, 2001.     

Positive definite functions and stable random vectors

We say that a random vector X = (X ₁, …, X _n ) in ℝ ⁿ is an n-dimensional version of a random variable Y if, for any a ∈ ℝ ⁿ , the random variables Σa _i X _i and γ(a)Y are identically distributed, where γ: ℝ ⁿ → [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L ₀. This result is almost optimal, as the norm of any finite-dimensional subspace of L _p with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖ _K ) is positive definite on ℝ ⁿ , where K is an origin symmetric star body in ℝ ⁿ and f: ℝ → ℝ is an even continuous function, then either the space (ℝ ⁿ , ‖·‖ _K ) embeds in L ₀ or f is a constant function. Combined with known facts about embedding in L ₀, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions.     

On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ⁴/ℤ _m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E _i , i = 1, 2, on Y such that π(E _i ) = o and the discrepancy a(E _i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 169–184, 2005.     

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let M_x be the moduli space of stable bundles of rank 2 with determinant We prove that the Chow group CH^Q₁(M_x) of 1-cycles on M_x with rational coefficients is isomorphic to CH^Q₀(X). By studying the rational curves on M_x, it is not difficult to see that there exits a natural homomorphism CH₀(J)→CH₁(M_x) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH₀(X)→CH₁(M_x), namely, to go from the infinite dimensional object CH₀(J) to the finite dimensional object CH₀(X). This is proved by relating the degeneration of Hecke curves on M_x to the second term I^*2 of Bloch's filtration on CH₀(J). Insong Choe was supported by KOSEF (R01-2003-000-11634-0).     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.     

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ ⁿ . For an integer t, a finite subset X of ℝ ⁿ given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S _i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X _i = X∩ S _i , and w:X→ ℝ_{> 0}. (The case of X⊂ S ⁿ⁻¹ with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

The Ξ operator and its relation to Krein's spectral shift function

We explore connections between Krein's spectral shift function ζ(λ,H ₀, H) associated with the pair of self-adjoint operators (H ₀, H),H=H ₀+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K ^*(H ₀−λ−i0)⁻¹ K) associated with the operator-valued Herglotz functionJ+K ^*(H ₀−z)⁻¹ K, Im(z)>0 inH, whereV=KJK ^* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H ₀,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E _J((−∞, 0))), ℝ, whereA(λ)=Re(K ^*(H ₀−λ−i0⁻¹ K),B(λ)=Im(K ^*(H ₀−λ-i0)⁻¹ K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H ₀, H) coincides with the trindex associated with the pair (Ξ(J+K ^*(H ₀−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.     

Further Results on Finitely Generated Projective Modules

In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K ₀(π) : K ₀(R)→K ₀(R/I) is a group epimorphism with the kernel {[P]−[Q] |P = PI, Q = QI}; there is an isomorphism of ordered groups from K ₀(R) to the gorup of all such functions ƒ _P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers. Received February 2, 1999, Accepted December 9, 1999     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.