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1.
Let n be an integer greater than 1, and let G be a group. A subset {x1, x2, ..., xn} of n elements of G is said to be rewritable if there are distinct permutations p \pi and s \sigma of {1, 2, ..., n} such that¶¶xp(1)xp(2) ?xp(n) = xs(1)xs(2) ?xs(n). x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. ¶¶A group is said to have the rewriting property Qn if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q3 if and only if its derived subgroup has order not exceeding 5.  相似文献   

2.
Let f=a0(x)+a1(x)y+a2(x)y2 ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 cm·H(f)emp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e1 = 4,e2 = 6, e3 = 6 [2/3]{2}\over{3} and em = 2 m for m S 4. We also show that the exponents em are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.  相似文献   

3.
Summary. Let F, Y \Phi, \Psi be strictly monotonic continuous functions, F,G be positive functions on an interval I and let n ? \Bbb N \{1} n \in {\Bbb N} \setminus \{1\} . The functional equation¶¶F-1 ([(?i=1nF(xi)F(xi))/(?i=1n F(xi)]) Y-1 ([(?i=1nY(xi)G(xi))/(?i=1n G(xi))])  (x1,?,xn ? I) \Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I) ¶was solved by Bajraktarevi' [3] for a fixed n 3 3 n\ge 3 . Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶Y(x) = [(aF(x) + b)/(cF(x) + d)],       G(x) = kF(x)(cF(x) + d) \Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d) ¶where a,b,c,d,k are arbitrary constants with k(c2+d2)(ad-bc) 1 0 k(c^2+d^2)(ad-bc)\ne 0 . Supposing the functional equation for all n = 2,3,... n = 2,3,\dots Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function j = F°Y-1 \varphi = \Phi\circ\Psi^{-1} . We get 32 new families of solutions.  相似文献   

4.
We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers xi with 1 \leqq xi \leqq pe, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?i=1k [1/(xi)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.  相似文献   

5.
There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx1,...,xm ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.  相似文献   

6.
Let x1,..., xn be points in the d-dimensional Euclidean space Ed with || xi-xj|| £ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?ij=1n|| xi-xj|| 2\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?ij=1n|| xi-xj|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x1,...,xn are chosen under the same constraints as above.  相似文献   

7.
Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {an} be a sequence of real numbers with 0 £ an £ 10 \leq a_n \leq 1, and let x and x0 be elements of C. In this paper, we study the convergence of the sequence {xn} defined by¶¶xn+1=an x + (1-an) [1/(n+1)] ?j=0n Tj xn   x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad for n=0,1,2,...  . n=0,1,2,\dots \,.  相似文献   

8.
Résumé. On travaille dans le cadre de l‘analyse semi-classique. Considérons p(x, hDx) p(x, hD_{x}) , une perturbation de -h2D -h^{2}\Delta qui est analytique à l‘infini. On suppose que dans la surface d‘énergie E0 > 0, les points critiques du symbole p(x, x) p(x, \xi) forment une sous-variété C \mathcal C et que p est non dégénéré dans l‘espace normal à C \mathcal C .¶En utilisant les résultats de [6] et [18], on obtient une majoration du nombre de résonances dans des disques de rayon d \delta centrés en E proche de E0, où d \delta satisfait Ch < d \delta < 1/C pour une constante C > 0. En généralisant la formule de trace de Sjöstrand qui exprime la trace d‘une différence d‘opérateurs en fonction des résonances, on trouve une minoration du nombre de résonances proches de E0.  相似文献   

9.
It is proved that all wild z-automorphisms including the well-known Nagata automorphism (all wild z-coordinates including the Nagata coordinates, respectively) of the polynomial algebra F[x, y, z] over an arbitrary field F cannot be lifted to a z-automorphism (z-coordinate, respectively) of the free associative algebra Fáx,y,z?{F\langle x,y,z\rangle}. The proof is based on the following two new results, which have their own interests: degree estimate of Q*FFáx1,?,xn?{{Q*_FF\langle x_1,\ldots,x_n\rangle}} and tameness of the automorphism group AutQ(Q*FFáx,y?){{{\rm Aut}_Q(Q*_FF\langle x,y\rangle)}}. The structure of the group of all z-automorphisms of the free associative algebra Fáx,y,z?{F\langle x,y,z\rangle} over an arbitrary field F is also determined.  相似文献   

10.
Let K be a convex body in \mathbbRn \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x1,?, xN x_1,\ldots, x_N independently and uniformly from K, and write C(x1,?, xN) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR+ ? \mathbbR+ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 £ in-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °Wi) = òKK f[Wi(C(x1, ?, xN))]dxN ?dx1 E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (Wi is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 £ in-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x1,?, xN) C(x_1,\ldots, x_N) .  相似文献   

11.
In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rn ×\Bbb Rn ? \Bbb R,     a(y,x) ? áA(y)x,x?p/2-1, A ? Mn ×n(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that At(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l2/p(x) |x|2 £ áA(x)x,x? £ L 2/p(x) |x|2, \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.  相似文献   

12.
Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] áq-1,p ?\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]áq-1, p ?\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.  相似文献   

13.
Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?K|2 £ á|T|x, x?Há|T*|y, y?K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR+ ? \mathbbR+{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.  相似文献   

14.
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   

15.
A generalized Hlawka's inequality says that for any n (\geqq 2) (\geqq 2) complex numbers¶ x1, x2, ..., xn,¶¶ ?i=1n|xi - ?j=1nxj| \leqq ?i=1n|xi| + (n - 2)|?j=1nxj|. \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. ¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶ ||A - Tr A ||1 \leqq ||A||1 + (n - 2)| Tr A|. ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. ¶¶ We consider also the related inequalities for p-norms (1 \leqq p \leqq ¥) (1 \leqq p \leqq \infty) on matrices.  相似文献   

16.
Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f1, f2, f3, f4, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f1(x + t, y + s) + f2(x - t, y - s) = f3(x + s, y - t) + f4(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f1, f2, f3, and f4 are given by f1 = w + h, f2 = w - h, f3 = w + k, f4 = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of Dy,t3g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and Dx,t3g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .  相似文献   

17.
Summary. The solution of the rectangular m ×n m \times n generalized bisymmetry equation¶¶F(G1(x11,...,x1n),..., Gm(xm1,...,xmn))     =     G(F1(x11,..., xm1),...,  Fn(x1n,...,xmn) ) F\bigl(G_1(x_{11},\dots,x_{1n}),\dots,\ G_m(x_{m1},\dots,x_{mn})\bigr) \quad = \quad G\bigl(F_1(x_{11},\dots, x_{m1}),\dots, \ F_n(x_{1n},\dots,x_{mn}) \bigr) (A)¶is presented assuming that the functions F, Gj, G and Fi (j = 1, ... , m , i = 1, ... , n , m S 2, n S 2) are real valued and defined on the Cartesian product of real intervals, and they are continuous and strictly monotonic in each real variable. Equation (A) is reduced to some special bisymmetry type equations by using induction methods. No surjectivity assumptions are made.  相似文献   

18.
We consider systems of partial differential equations with constant coefficients of the form ( R(Dx, Dy)f = 0, P(Dx)f = g), f,g ? C(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(Dx, Dy)g = 0  and  W ì \Bbb Rn+1R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ¥\infty of the principle part Pm of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.  相似文献   

19.
A polynomial P(X) with coefficients {ǃ} of odd degree N - 1 is cyclotomic if and only if¶¶P(X) = ±Fp1X)Fp2Xp1) ?FprXp1 p2 ?pr-1) P(X) = \pm \Phi_{p1} (\pm X)\Phi_{p2}(\pm X^{p1}) \cdots \Phi_{p_r}(\pm X^{p1 p2 \cdots p_r-1}) ¶where N = p1 p2 · · · pr and the pi are primes, not necessarily distinct, and where Fp(X) : = (Xp - 1) / (X - 1) \Phi_{p}(X) := (X^{p} - 1) / (X - 1) is the p-th cyclotomic polynomial. This is a conjecture of Borwein and Choi [1]. We prove this conjecture for a class of polynomials of degree N - 1 = 2r pl - 1 N - 1 = 2^{r} p^{\ell} - 1 for any odd prime p and for integers r, l\geqq 1 r, \ell \geqq 1 .  相似文献   

20.
Let x1, ?, xn \xi_1, \ldots, \xi_n be random variables and U be a subset of the Cartesian product \mathbbZ+n, \mathbbZ+ \mathbb{Z}_+^n, \mathbb{Z}_+ being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if¶¶E(x1j1 ?xnjn) = E(x1j1) ?E(xnjn) ? (j1, ... ,jn) ? U. \textbf {E}\big(\xi_1^{j_1} \cdots \xi_n^{j_n}\big) = \textbf {E}\big(\xi_1^{j_1}\big) \cdots \textbf {E}\big(\xi_n^{j_n}\big) \iff (j_1, \dots ,j_n) \in U. ¶It is proved that for an arbitrary subset U \subseteqq \mathbbZ+n U \subseteqq \mathbb{Z}_+^n containing all points with 0 or 1 non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.  相似文献   

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