An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

Kneading sequences for unimodal expanding maps of the interval

LetP andAC be two primary sequences with min{P, AC}≥RLR ^∞,ρ(P) and ρ(AC) be the eigenvalues ofP andAC, respectively. Letf∈C ⁰ (I, I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved thatf has the kneading sequenceK(f)≥(RC) ^*m *P if λ≥(ρ(P))^1/2m, andK(f)>(RC) ^*m*AC*E for any shift maximal sequenceE if λ>(ρ(AC))^1/2m. The value of (ρ(P))^1/2m or (ρ(AC))^1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one. Project supported by the National Natural Science Foundation of China     

A New Proof of a Theorem of Suslin

In this paper we compute the group H₂(SL₂(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H₂(SL₂, (F)) is the fiber product of λ_*:K₂(F)→ I²(F)/I³(F) and σ: I²(F) → I²(F)/I³(F) where λ_* and σ map onto I²(F)/I³(F). (Received: February 2003)     

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.     

The solution of characteristic value-vector problems by Newton's method

LetT(λ) be a bounded linear operator in a Banach spaceX for eachλ in the scalar fieldS. The characteristic value-vector problemT(λ)x = 0 with a normalization conditionφ x = 1, whereφ ε X ^*, is formulated as a nonlinear problem inX xS:P(y) ≡ (T(λ)x, φ x - 1) = 0,y= (X, A). Newton's method and the Kantorovič theorem are applied. For this purpose, representations and criteria for existence ofP′(y)⁻¹ are obtained. The continuous dependence onT of characteristic values and vectors is investigated. A numerical example withT(λ) =A +λB +λ ² C is presented. Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.     

Invariants forω-categorical,ω-stable theories

In this paper we give a complete solution to the classification problem forω-categorical,ω-stable theories. More explicitly, supposeT isω-categorical,ω-stable with fewer than the maximum number of models in some uncountable power. We associate with each modelM ofT a “simple” invariantI(M), not unlike a vector of dimensions, such thatI(M)=I(N) if and only ifM≅N. The spectrum function,I(−,T), for a first-order theoryT is such that for all infinite cardinals λ,I(λ,T) is the number of nonisomorphic models ofT of cardinality λ. As an application of our “structure theorem” we determine the possible spectrum functions forω-categorical,ω-stable theories.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Relations entre isopérimétrie et trou spectral pour les chaînes de Markov finies

Let G be a finite and connected graph, we will note by l(G) the maximum length of an injective path in G. We will show (by two dictinct proofs, one using sub-trees of G and the other based on multiflows of paths) that sup _{(P,μ)∈?(G)} I(P, μ)/λ(P, μ) = l(G), where the supremum is taken over all Markovian kernels P reversible with respect to a probability μ and whose allowed transitions are given by the edges of G, and where I(P, μ) (respectively λ(P, μ)) is the isoperimetric constant (resp. the spectral gap) associated to the couple (P, μ). Then we will study more precisely the same supremum, but where the probability μ is also fixed. These results give optimal minorations of the spectral gap which are linear with respect to the isoperimetric constant (and not quadratic, as in the Cheeger inequality), and we will give several examples on infinite graphs. Re?u: 12 ao?t 1997 / Version révisée: 9 novembre 1998     

Directional derivatives for the value-function in semi-infinite programming

For the problemP(λ): Maximizec ^T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point such thatP( ) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only convergent selections of solutions ofP(λ _n ), λ _n → and their duals.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Upper bounds for the first dirichlet eigenvalue of a tube around an algebraic complex curve of ℂ<Emphasis Type="Italic">P</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>(<Emphasis Type="Italic">λ</Emphasis>)

We give an upper bound for the first Dirichlet eigenvalue of a tube around a complex curve P of ℂP ⁿ (λ) which depends only on the radius of the tube and the degrees of the polynomials defining P. The bound is sharp on a totally geodesic ℂP ¹(λ) and gives a gap between the eigenvalue of a tube around ℂP ¹(λ) and around other complex curves.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Relations between some constants associated with finite dimensional Banach spaces

We define the asymmetry constants(E) of a Banach spaceE, and show examples of finite-dimensional spaces with “large” asymmetry constants. IfE isn-dimensional,λ(E)17its projection constant and π ₁(I _E ) the absolutely summing norm of the identity operatorI _E , thenn≦λ(E)π₁(I _E )≤n(s(E))². Similar equations linking thep-absolutely summing and the nuclear norms ofI _E are established. We also obtain estimates on these norms, for example π₂(I _E )=√n. The contribution of this author is a part of a Ph.D. Thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss whose guidance and valuable suggestions are gratefully acknowledged.     

Sharp metastability threshold for two-dimensional bootstrap percolation

 In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π²/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ^′=π²/6. The existence of the thresholds λ,λ^′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2]. Received: 12 May 2002 / Revised version: 12 August 2002 / Published online: 14 November 2002 Research funded in part by NSF Grant DMS-0072398 Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B43 Key words or phrases: Bootstrap percolation – Cellular automaton – Metastability – Finite-size scaling     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.