Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones

Let T() be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT()₀ is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA ₀ + ( ₁ ^l –TR).] In particular let _a denote the enumeration function of the infinite cardinals and leta ₀ a denote the partial collapsing operation on T() which maps ordinals of T() into the countable segment T ₀ of T(). Assume that the (fast growing) extended Grzegorczyk hierarchy and the slow growing hierarchy are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way:

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

Gauss polynomials and the rotation algebra

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=ab with commuting witha andb, then the (generalized) binomial coefficient arising in the expansion

Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B _n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e ₁ , ..., e _n denoting the standard basis vectors of ⁿ and letting x _k = e ₁ + ··· + e _k (k = 1, 2, ..., n), the set

C-wellposedness of the complete second order abstract cauchy problem and applications

In this paper, we first give a sufficient and necessary condition for to generate an exponentially bounded -semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem ((ACP₂) for short) in some sense. Then we use these results and those in [1] to discuss the C-(exponential) wellposedness of a kind of (ACP₂) with application backgrounds, and develop the results in [2]. This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province, China     

Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations,II

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely,

On averaging sets

For any set ={f ₁,f ₂,...,f _s} ofC ³-functions on the interval [–1, 1], and for any weight functionw(x) satisfyingL ₁w(x)L ₂(1–|x|)(L ₁,L ₂>0, 0) and , we give a constructive proof for the existence of quadrature formulas of the type

Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain , which is either the open unit disk or the open right half-plane . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function has a bi-inner dilation mapping into ; here U ₁ and Y ₁ are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions and (namely and ) satisfying and for almost all . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ _r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ _l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.      

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have

Zero–One Law for some Brownian Functionals

We prove that for a>0, (B _t) one-dimensional standard Brownian motion and ₀=inf{t>0 : B _t=0} the following zero–one law is valid

Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times

Let be the Dirichlet integral and the Brownian motion on R. Let be a finite positive measure in the Kato class and the additive functional associated with . We prove that for a regular domain D of R ^d

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

Evaluating the Frechet derivative of the matrix exponential

LetM _n denote the space ofn×n matrices. GivenX, ZM _n define

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Symmetry of positive solutions of an almost-critical problem in an annulus

We consider the subcritical problem & 0 & \qquad\textrm{in} \; A\\ u & = & 0 & \qquad\textrm{on} \; \delta A\\ \end{array} \right.$$" align="middle" border="0"> where A is an annulus in , , is the critical Sobolev exponent and 0$" align="middle" border="0"> is a small parameter. We prove that solutions of (I) which concentrate at one or two points are axially symmetric.Received: 7 July 2003, Accepted: 10 May 2004, Published online: 16 July 2004Filomena Pacella: Research supported by MIUR, project Variational Methods and Nonlinear Differential Equations.