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1.
The article is devoted to the asymptotic properties of the vector fields $\tilde X_i^g $ , i = 1, …, N, θ g -connected with C 1-smooth basis vector fields {X i } i=1,…,N satisfying condition (+ deg). We prove a theorem of Gromov on the homogeneous nilpotent approximation for vector fields of classC 1. Nontrivial examples are constructed of quasimetrics induced by vector fields {X i } i=1, …, N .  相似文献   

2.
Suppose that S1,…,SN are collections of subsets of X1,…,XN, respectively, such that ni subsets belonging to Si, and no fewer, cover Xi for all i. the main result of this paper is that to cover X1 x…x XN requires no fewer than σNi=1 (ni–1) + 1 and no more than ΠNi=1ni subsets of the form A1 x…x AN, where AiS1foralli. Moreo ver, these bounds cannot be improved. Identical bounds for the spanning number of a normal product of graphs are also obtained.  相似文献   

3.
Let X be a graph with vertices x1 ,…, xn. Let Xi be the graph obtained by removing all edges {xi, xj} of X and inserting all nonedges {xi, xk}. If n ? 0 (mod 4), then X can be uniquely reconstructed from the unlabeled graphs X1.…, Xn. If n = 4 the result is false, while for n = 4m≥8 the result remains open. The proof uses linear algebra and does not explicitly describe the reconstructed graph X.  相似文献   

4.
Let 1 ? k1 ? k2 ? … ? kn be integers and let S denote the set of all vectors x = (x1, x2, …, xn) with integral coordinates satisfying 0 ? xi ? ki, i = 1, 2, …, n. The complement of x is (k1 ? x1, k2 ? x2, …, kn ? xn) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities xi ? yi, 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.  相似文献   

5.
Let Ui = (Xi, Yi), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μx, μy) and covariance matrix
. Let Xi, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H0 : μ = 0 vs. H1 : μ ≠ 0. We prove that Hotelling's T2 test, which uses (Xi, Yi), i = 1, 2,…, n (and discards Xi, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T2-test cannot be beaten. A similar result is also proved for the problem of testing μx ? μy = 0. A Bayes test and other competitors which are similar tests are discussed.  相似文献   

6.
For i=1,…,m let Hi be an ni×ni Hermitian matrix with inertia In(Hi)= (πi, νi, δi). We characterize in terms of the πi, νi, δi the range of In(H) where H varies over all Hermitian matrices which have a block decomposition H= (Xij)i, j=1,…, m in which Xij is ni×nj and Xii=Hi.  相似文献   

7.
We prove the existence of periodic solutions in a compact attractor of (R+)n for the Kolmogorov system x′i = xifi(t, x1, , xn), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.  相似文献   

8.
Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of Gn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.  相似文献   

9.
Let Tn denote a binary tree with n terminal nodes V={υ1,…,υn} and let li denote the path length from the root to υi. Consider a set of nonnegative numbers W={w1,…,wn} and for a permutation π of {1,…,n} to {1,…,n}, associate the weight wi to the node υπ(i). The cost of Tn is defined as C(TnW)=Minπni=1wilπ(i).A Huffman tree Hn is a binary tree which minimizes C(TnW) over all possible Tn. In this note, we give an explicit expression for C(HnW) when W assumes the form: wi=k for i=1,…,n?m; wi=x for i=n?m+1,…,n. This simplifies and generalizes earlier results in the literature.  相似文献   

10.
Let 1?k1?k2?…?kn be integers and let S denote the set of all vectors x = (x1, …, xn with integral coordinates satisfying 0?xi?ki, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of ki elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities xi?yi, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k1 + k2 + …kn. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k1-x1, k2-x2, …, kn -xn). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and|(12K)X| does not exceed the number of vectors in (12K)S with first coordinate different from k1, then
i=0Ki≠12K|(i)X||(i)S|+|(12K)X||(12K-1)S|?1
.  相似文献   

11.
《Journal of Algebra》2002,247(2):509-540
Let Fm be a free group of a finite rank m  2 and let Xi, Yj be elements in Fm. A non-empty word w(x1,…,xn) is called a C-test word in n letters for Fm if, whenever (X1,…,Xn) = w(Y1,…,Yn)  1, the two n-typles (X1,…,Xn) and (Y1,…,Yn) are conjugate in Fm. In this paper we construct, for each n  2, a C-test word vn(x1,…,xn) with the additional property that vn(X1,…,Xn) = 1 if and only if the subgroup of Fm generated by X1,…,Xn is cyclic. Making use of such words vm(x1,…,xm) and vm + 1(x1,…,xm + 1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u1, u2  Fm such that every endomorphism ψ of Fm with non-cyclic image is completely determined by ψ(u1), ψ(u2).  相似文献   

12.
The present paper studies so-called R-fuzzy recursive program schemes which are finitely specified by systems of equations xi = pi, i = 1,…, n. Thereby, X = {x1,…, xn} is a finite set of variables and the pi's are polynomials built up over X unioned wih a finite set F of function symbols and with coefficients in a given semiring R that determines the different kind of fuzziness by evaluating possible choices. Using power series on F-tree with variables of X the meaning of R-fuzzy recursive program schemes can formally be computed. The main result shows the equivalence of equational (fixed point) and operational semantics of such program schemes.  相似文献   

13.
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,…, xk) lies in the convex hull of x1,…, xk), but does not coincide with one of its vertices. A sequence (xn)n∈ ? in X is called M-recursive if xn+k = M(xn, xn+1,…, xn+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.  相似文献   

14.
Let k1 ? k2? ? ? kn be given positive integers and let S denote the set of vectors x = (x1, x2, … ,xn) with integer components satisfying 0 ? x1 ? kni = 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.  相似文献   

15.
Let X1, X2 ,…, Xp be p random variables with joint distribution function F(x1 ,…, xp). Let Z = min(X1, X2 ,…, Xp) and I = i if Z = Xi. In this paper the problem of identifying the distribution function F(x1 ,…, xp), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.  相似文献   

16.
Let A denote an n×n matrix with all its elements real and non-negative, and let ri be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r1,…,rn), where D(r1,…,rn) is the diagonal matrix with ri at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of BT contains a vector d whose entries are all non-null.  相似文献   

17.
Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X1, X2,…,Xn are mutually unrelated fuzzy variables with common membership function μ and α1, α2,…,αn are real numbers satisfying αi ? o for every i and Σi=1nαi=1, when does does Z = Σi = 1nαiXi have the same membership function μ?  相似文献   

18.
For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX **. This says that for any ε>0, for anyF in the unit ball ofX **, and for anyf 1,…,f n inX *, the system of inequalities |f i(x)?F(f i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.  相似文献   

19.
Jun Tarui 《Discrete Mathematics》2008,308(8):1350-1354
A family P={π1,…,πq} of permutations of [n]={1,…,n} is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x1,…,xk∈[n], permutations πi's in P produce all k! possible orders on πi(x1),…,πi(xk). Let N*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.
  相似文献   

20.
Recently, Sloane suggested the following problem: We are given n boxes, labeled 1,2,…,n. For i=1,…,n, box i weighs (m-1)i grams (where m?2 is a fixed integer) and box i can support a total weight of i grams. What is the number of different ways to build a single stack of boxes in which no box will be squashed by the weight of the boxes above it? Prior to this generalized problem, Sloane and Sellers solved the case m=2. More recently, Andrews and Sellers solved the case m?3. In this note we give new and simple proofs of the results of Sloane and Sellers and of Andrews and Sellers, using a known connection with m-ary partitions.  相似文献   

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