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:

Some numerical results on best uniform rational approximation ofx α on [0,1]

Let be a positive number, and letE _n,n (x ;[0,1]) denote the error of best uniform rational approximation from _n,n tox on the interval [0,1]. We rigorously determined the numbers {E _n,n (x ;[0,1])} _n ^=1/30 for six values of in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of , Richardson's extrapolation was applied to the products to obtain estimates of

Embedded continued fractals and their Hausdorff dimension

A continued fractal is a curve which is associated to a real number[0, 1]. Properties of the continued fraction expansion of appear as geometrical properties ofQ . It is shown how number theoretic properties of affect topological and geometric properties ofQ such as existence, continuity, Hausdorff dimension, and embeddedness.Communicated by Michael F. Barnsley.     

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

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

Fubini theorem w.r.t. stochastic measure on product measurable space

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Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

On the preservation of periodic monotonicity

A 2-periodic continuous real functionf is said to beperiodically monotone if it has the following property: there exist numbert ₁t ₂t ₃t ₁+2 such thatf is nonincreasing fort ₁t ₂ and nondecreasing int ₂tt ₃. For any 2-periodic, integrable real functiong with ₀ ² |g(t|dt) we define

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Formality Theorem with Coefficients in a Module

In this paper, X will denote a manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, and showed that it is formal. Denote by the DGLA of E-polyvector fields. Considering M, a module over E, we define the-module of E-polyvector fields with values in M. Similarly, we define the-module of E-polydifferential operators with values in M,. We show that there is a quasi-isomorphism of L _∞-modules over from to . Our result extends Calaque’s (and Kontsevich’s) result.     

Non- (Quantum) Differentiable C 1-Functions in the Spaces with Trivial Boyd Indices

If E is a separable symmetric sequence space with trivial Boyd indices and is the corresponding ideal of compact operators, then there exists a C¹-function f_E, a self-adjoint element and a densely defined closed symmetric derivation δ on such that , but      

Evaluating the Frechet derivative of the matrix exponential

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

An extension of the Erdős-Stone theorem

LetK _p(u₁, ..., u_p) be the completep-partite graph whoseith vertex class hasu _i vertices (lip). We show that the theorem of Erds and Stone can be extended as follows. There is an absolute constant >0 such that, for allr1, 0<1 and=">1/r, every graphG=G ⁿ of sufficiently large order |G|=n with at least

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions H = { f:f = z + ?_{n = 2}^￥ nb_n zⁿ ,|b_n | \leqslant 1} on |z| < 1H = \{ f:f = z + \sum\limits_{n = 2}^\infty {nb_n z^n ,|b_n |{\mathbf{ }} \leqslant 1\} {\mathbf{ }}} on{\mathbf{ }}|z|{\mathbf{ }}< {\mathbf{ }}1 . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Weakly Periodic Sequences of Bounded Linear Transformations: a Spectral Characterization

Let X and Y be two Hilbert spaces, and the space of bounded linear transformations from X into Y. Let {A } be a weakly periodic sequence of period T. Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to characterize {A _n} by a positive measure and a number T of -valued functions a ₀, . . . ,a _T–1; in the spectral form , where and is an -valued Borel set function on [0, 2) such that      

Strong law of large numbers for functionals of products of random matrices

For the sequence A₁, A₂, ..., A_n, ... of m×m independent random matrices such that for each k there exists a joint density function P_k(X) of the elements _ij ^k, we prove the following theorem: if and for some positive constants ₁ and ₂, then with probability 1,

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as