Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

On multi-coloured lines in a linear space

We prove that if X₁, X₂,…, X_k are pairwise disjoint sets of points in a linear space, each of cardinality n, whose union ∪_{j = 1}^kX_j, is not collinear, then there are at least (k ? 1) n lines in the space, which intersect at least two of th e X_j's. Equality occurs if and only if k = n + 1 and X₁,…, X_{n + 1} are obtained by taking n + 1 concurrent lines in a projective plane of order n, and omitting, from each of them, their common point. When n = 1, this reduces to a theorem of de Bruijn and Erdos (Indag. Math.10 (1984), 421–423).     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

A note on the kernels of higher derivations

Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } _n=0 ^∞ a higher k-derivation on k[X] and D′ = {D′ _n } _n=0 ^∞ a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] ^D = k if and only if k′[X] ^D′ = k′; (2) k[X] ^D is a finitely generated k-algebra if and only if k′[X] ^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] ^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.     

On maximal antichains consisting of sets and their complements

Let 1 ? k₁ ? k₂ ? … ? k_n be integers and let S denote the set of all vectors x = (x₁, x₂, …, x_n) with integral coordinates satisfying 0 ? x_i ? k_i, i = 1, 2, …, n. The complement of x is (k₁ ? x₁, k₂ ? x₂, …, k_n ? x_n) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities x_i ? y_i, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given.     

Elasticity of factorizations in R 0 + X R1 + … + X lRl [X]

For an atomic domain R the elasticity ρ(R) is defined by ρ(R) = sup{m/n ¦ u₁ … u _m = v ₁ … v_n where u_i, v_i ∈ R are irreducible}. Let R ₀ ? … ? R _l be an ascending chain of domains which are finitely generated over ? and assume that R _l is integral over R ₀. Let X be an indeterminate. In this paper we characterize all domains D of the form D = R ₀ + XR₁ + … + X^lR_l[X] whose elasticity ρ(D) is finite.     

Limiting distributions of functionals of Markov chains

Let \s{X_n, n ? 0\s} and \s{Y_n, n ? 0\s} be two stochastic processes such that Y_n depends on X_n in a stationary manner, i.e. P(Y_n ? A\vbX_n) does not depend on n. Sufficient conditions are derived for Y_n to have a limiting distribution. If X_n is a Markov chain with stationary transition probabilities and Y_n = f(X_n,..., X_n+k) then Y_n depends on X_n is a stationary way. Two situations are considered: (i) \s{X_n, n ? 0\s} has a limiting distribution (ii) \s{X_n, n ? 0\s} does not have a limiting distribution and exits every finite set with probability 1. Several examples are considered including that of a non-homogeneous Poisson process with periodic rate function where we obtain the limiting distribution of the interevent times.     

An infinite-dimensional law of large numbers in Cesaro's sense

Let (X _k) _k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>?1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1[ the sequenceV _n =(1/A _n ^α )∑_0?k?n A _n?k ^α?1 X _k converges almost surely toE(X) if and only if ‖X‖^1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Déniel and Derriennic (case α=1/2).     

A pairwise independent stationary stochastic process

The purpose of this paper is to study pairwise independence in the context of strictly stationary stochastic processes {X_π, n = 0, ±1, …}. Our main result is an example of such a process that maximizes E(X₁X₂X₃). We also show that subject to some additional independence assumptions any two of these processes are distributionally the same. The spectral properties of this process are then analysed.     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

On extensions of polynomial functions

Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A ^o ∈ Y, A^k(x) ? A_k(x,…,x), x ∈ X, and the maps A _k: X ^k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed.     

On additive maps of prime rings satisfying the engel condition

Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n₁, n₂,…,n_k be natural numbers. Suppose that […[[(x), xn¹], xn²],…, xn^k]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n₁+n₂+ …+n_k Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.     

Distribution exacte du score local,cas markovien

Given a sequence $\text{X}\text{=(X}{}_{\mathrm{k}}\text{)}{}_{\mathrm{k?1}}$ of random variables taking values in {?v,…,0,…,+u}, let's define the local score of the sequence by H_n=max_1?i?j?n(∑_k=i^jX_k). The local score is used to analyze biological sequences pointing out regions of the sequences with interesting biological properties. In order to separate randomly events from really interesting segments, we establish here the distribution of the local score of H_n when the sequence $\text{X}$ is a Markov chain of order 1. To cite this article: S. Mercier, C. Hassenforder, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

A symmetry relationship for a class of partitions

Let S(n, k, v) denote the number of vectors (a₀,…, a_n?1) with nonnegative integer components that satisfy a₀ + … + a_{n ? 1} = k and Σ_i=0^n?1ia_i ≡ v (mod n). Two proofs are given for the relation S(n, k, v) = S(k, n, v). The first proof is by algebraic enumeration while the second is by combinatorial construction.     

Pelikán's conjecture and cyclotomic cosets

The following conjecture was recently made by J. Pelikán. Let a₀ ,…, a_n be an (n + 1)-tuple of 0's and 1's; let A_k = ?_i=0^n?ka_ia_i+k for k = 0,…, n. Then if n ? 4 some A_k is even.This paper shows that Pelikán's conjecture is false for infinitely many values of n. On the other hand it is also shown that the conjecture is true for most values of n, and a characterization is given of those values of n for which it fails.     

Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

Minimal cogenerators need not be unique

A_n = A_n(?) denotes the unique left distributive binary system on {0, 1,…,2ⁿ?1) that satisfies a ? 1 = a + 1 mod 2ⁿfor all a ? A_n, and o_n(a) = k indicates the period 2^kof a ? A_n (if b,c ? A_n, then a ? b = a ? c iff b ‵ c mod 2^kand if 0 < b < c < 2^kthen a < a ? b < a ? c). Amongs others, we prove that o_t2(2^t?2) ≤ t holds for every integer t ≥ 2, the equality taking place iff t is of the form 2^2? for an integer s ≥ 0.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``