Subgraphs and well-quasi-ordering

Let ?? be a class of graphs and let ? be the subgraph or the induced subgraph relation. We call ? an ideal (with respect to ?) if ? implies that ?. In this paper, we study the ideals that are well-quasiordered by ?. The following are our main results. If ? is the subgraph relation, we characterize the well-quasi-ordered ideals in terms of exluding subgraphs. If?is the induced subgraph relation, we present three wellquasi-ordered ideals. We also construct examples to disprove some of the possible generalizations of our results. The connections between some of our results and digraphs are considered in this paper too.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Strongly τ-Decomposable and Selfdecomposable Laws on Simply Connected Nilpotent Lie Groups

 Let ? be a simply connected nilpotent Lie group with Lie Algebra ? and let τ be a contraction on ?. A probability measure μ on ? is strongly τ-decomposable iff it is representable as the limit of for some probability ν on ?. We show that such a limit exists if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on ?. This result is then generalized to the class of selfdecomposable laws on ?. We also show that selfdecomposable laws on ? correspond in a 1–1 way to operator selfdecomposable laws on the tangent space ?. Received 1 October 1998; in revised form 29 March 1999     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Heat equation on the arithmetic von Koch snowflake

We investigate the asymptotic behaviour of the heat content as the time t→ 0 for an s-adic von Koch snowflake generated by a square. We show that the heat content satisfies a functional equation which, after appropriate transformations, takes the form of an inhomogeneous renewal equation. We obtain the structure of the solution of this equation in the arithmetic case up to an exponentially small remainder in t. <!-ID="Mathematics Subject Classification (2000): 35K05, 60J65, 28A80--> <!-ID="Key words: Heat equation – Arithmetic – Snowflake--> Received: 24 March 1999 / Revised version: 14 October 1999 / Published online : 8 August 2000     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields

We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent σ-algebras. As an application we give a simple proof that the Dobrushin and Shlosmans complete analyticity condition, for a Gibbs specification with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several different techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Received: 7 July 2000 / Revised version: 10 October 2000 / Published online: 5 June 2001     

à propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Exponential actions and maximal ?-invariant ideals

Let V be an exponential ?-module, ? being an exponential Lie algebra. Put ? = exp ?. Then every orbit of V under the action of ? admits a closed orbit in its closure. If G= exp ? is a nilpotent Lie group and ? an exponential algebra of derivations of ?, then ? = exp ? acts on G, L ¹(G), (?) and the maximal ?-invariant ideals of L ¹(G), resp. of (?) coincide with the kernels Ker Ω, resp. Ker Ω∩ (?), where Ω is a closed orbit of ?^*. Received: 6 December 1996 / Revised version: 7 December 1997     

Capture time of Brownian pursuits

Let B ₀,B ₁, ⋯ ,B _n be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001     

Application of a Bernstein type inequality to rational interpolation in the Dirichlet space

We prove a Bernstein type inequality involving the Bergman and Hardy norms for rational functions in the unit disk \mathbb D {\mathbb D} that have at most n poles all of which are outside the disk \frac1r \mathbb D \frac{1}{r} {\mathbb D} , 0 < r < 1. The asymptotic sharpness of this inequality is shown as n → ∞ and r → 1—. We apply our Bernstein type inequality to an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space constrained by the H2-nom. Bibliography: 14 titles.     

A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier–Stokes equations in 2<Emphasis Type="Italic">D</Emphasis>

We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|^−α |u(x)| < ∞ as |x| → ∞ for some α ∈ [0, 1/7).^† Bibliography: 10 titles.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg I=reg I; (c) arithdeg I=indeg I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).     

The strength of Martin-Löf type theory with a superuniverse. Part II

Universes of types were introduced into constructive type theory by Martin-L?f [3]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say ?. The universe then “reflects”?. This is the second part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf.[4–6]). It is proved that Martin-L?f type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schütte ordinal Γ₀ but well below the Bachmann-Howard ordinal. Not many theories of strength between Γ₀ and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory. In this second part of the paper the concern is with the with upper bounds. Received: 8 December 1998 / Published online: 21 March 2001     

Interpolation of rational functions by simple partial fractions

We study a generalized interpolation of a rational function at n nodes by a simple partial fraction of degree n and reduce the consideration to the solvability question for a special difference equation. We construct explicit interpolation formulas in the case where the equation order is equal to 1. We show that for functions A(x − a) ^m , m ? \mathbbN m \in \mathbb{N} , it is possible to reduce the consideration to a system of m + 1 independent first order equations and construct explicit interpolation formulas. Bibliography: 6 titles.     

Q 2-free Families in the Boolean Lattice

For a family F{{\cal F}} of subsets of [n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that F{{\cal F}} is P-free if it does not contain a subposet isomorphic to P. Let ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q ₂ be the poset with distinct elements a, b, c, d, a < b,c < d; i.e., the 2-dimensional Boolean lattice. We show that 2N − o(N) ≤ ex(n, Q ₂) ≤ 2.283261N + o(N), where N = \binomn?n/2 ?N = \binom{n}{\lfloor n/2 \rfloor}. We also prove that the largest Q ₂-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.     

Hitting Half-spaces by Bessel-Brownian Diffusions

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ ^n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ) ^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.