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1.
Let n be an integer and Bn \mathcal B_n be the variety defined by the law [xn,y][x,yn]-1 = 1.¶ Let Bn* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that [xn,y][x,yn]-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ Bn* = Bn èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? Bn* G \in \mathcal B_n^* if and only if G ? Bn G \in \mathcal B_n .  相似文献   

2.
Let X and Y Banach spaces. Two new properties of operator Banach spaces are introduced. We call these properties "boundedly closed" and "d-boundedly closed". Among other results, we prove the following one. Let U(X, Y){\cal U}(X, Y) an operator Banach space containing a complemented copy of c0. Then we have: 1) If U(X, Y){\cal U}(X, Y) is boundedly closed then Y contains a copy of c0. 2) If U(X, Y){\cal U}(X, Y) is d-boundedly closed, then X* or Y contains a copy of c0.  相似文献   

3.
In this note we prove that every infinite group G is 3-abelian (i.e. (ab)3 = a3b3 for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)3 = x3y3.  相似文献   

4.
Summary. In this paper we deal with the extension of the following functional equation¶¶ f (x) = M (f (m1(x, y)), ..., f (mk(x, y)))        (x, y ? K) f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) , (*)¶ where M is a k-variable operation on the image space Y, m1,..., mk are binary operations on X, K ì X K \subset X is closed under the operations m1,..., mk, and f : K ? Y f : K \rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m1,..., mk, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m1,..., mk)-affine hull K* of K, such that (*) holds over K*. (The set K* is defined as the smallest subset of X that contains K and is (m1,..., mk)-affine, i.e., if x ? X x \in X , and there exists y ? K* y \in K^* such that m1(x, y), ?, mk(x, y) ? K* m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* then x ? K* x \in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.  相似文献   

5.
We study the mod 2 homology of the double loop space of SU(n)/SO(n) using the Serre spectral sequence along with the Eilenberg-Moore spectral sequence. Then we also get the homology of the double loop space of the set of all Lagrangian subspaces of the symplectic vector space R2n.  相似文献   

6.
Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x1,x2,...,xn,y1,y2,...,yn in X, there exists an x in X such that¶¶ ?i=1n f(xi,x)=?i=1n f(yi,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).  相似文献   

7.
In this note we prove the spectral mapping theorem for certain evolution semigroups. Specifically, we study the evolution semigroup on Lp(Theta,mu;X), 1≤p相似文献   

8.
Let X be a random vector with values in Rn and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on Rn. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every aRn. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.  相似文献   

9.
Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f1, f2,...,fk: X \hookrightarrow Y X \hookrightarrow Y with # {f1(x), f2(x),...,fk(x)}=k    and    (x,f1(x)), (x, f2(x)),...,(x, fk(x)) ? R \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R for all x ? X x \in X if and only if the inequality k ·# A £ ?y ? Y min(k, #{a ? A | (a,y) ? R}) k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) holds for every finite subset A of X, provided {y ? Y | (x,y) ? R} \{y \in Y \mid (x,y) \in R\} is finite for all x ? X x \in X .¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every x ? X x \in X and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdös Theorem) will conclude the paper.  相似文献   

10.
Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)n y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .  相似文献   

11.
Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (fn)n \geqq 0f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices.  相似文献   

12.
We derive explicit equations for the maximal function fields F over 𝔽 q 2n given by F = 𝔽 q 2n (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 q 2n , and where A(Y) is q-additive and deg(f) = q n  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 q 2n (i.e., the extension H/F is Galois).  相似文献   

13.
We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f 1(X))…(Y ? f n (X)) + g(X), with f, f 1, ?, f n , g univariate polynomials over an arbitrary field.  相似文献   

14.
Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P3-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P3-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x-1y-1xy, has the order one or two.  相似文献   

15.
The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let Cm and Cn be cycles. We prove that f(Cm 2Cn)hf(Cm) f(Cn) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.  相似文献   

16.
Using the theory of cyclic codes the following problem of K. Burde' on characterizing finite fields GF (qn) is solved:¶ Consider GF (qn) as a vector space over GF (q). For which GF (qn) exists for any k = 0, . . . ,n exactly one subspace C of dimension k and which is invariant under the Frobenius automorphism?  相似文献   

17.
Sufficient conditions on bounded domains D ² R d={(t,x)} of class C4 are given under which solutions of the heat equation ut=j u+f in D have continuous second-order derivatives with respect to (t,x) in D- . The equation is supplemented with C4 boundary data and it is assumed that f] C2 .  相似文献   

18.
We show that for every closed Riemannian manifold X there exists a positive number¶ $ \varepsilon_0 > 0 $ \varepsilon_0 > 0 such that for all 0< e\leqq e0 \varepsilon \leqq \varepsilon_0 there exists some¶ $ \delta > 0 $ \delta > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ d \delta the geometric e \varepsilon -complex |Ye| |Y_\varepsilon| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann [4].  相似文献   

19.
We will show that the factorization condition for the Fourier integral operators Ir m (X,Y;L )I_\rho ^\mu (X,Y;\it\Lambda ) leads to a parametrized parabolic Monge-Ampère equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain Lp-continuity for translation invariant operators in \Bbb Rn{\Bbb R}^n with n £ 4n\leq 4 and for arbitrary \Bbb Rn{\Bbb R}^n with dpX×Y|Ln+2d\pi _{X\times Y}|_\Lambda \leq n+2.  相似文献   

20.
The Euler monoid En = {(a,b,t) epsilon Z3 : a2 + b2 = tn, n S 1, is free if and only if n is odd (Theorem 1). We extend the results of Lyndon and Ullman, and Beardon concerning the set of those rational numbers mu epsilon (-2,2) for which the matrix Möbius group Gmu generated by A= and B = is not free (Theorems 2, 3, 4).  相似文献   

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