Let F be a field, F1 be its multiplicative group, and = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let Dn be the dihedral group of degree n, H be a nontrivial group in , and τn(H) = {α = (α1, α2,…, αn):αi?H}. For σ?Dn and α?τn(H), let P(σ, α) be the matrix whose (i,j) entry is αiδiσ(j) (i.e., a generalized permutation matrix), and . Let Mn(F) be the vector space of all n×n matrices over F and P(Dn, H) = {T:T is a linear transformation on Mn (F) to itself and T(P(Dn, H)) = P(Dn, H)}. In this paper we classify all T in P(Dn, H) and determine the structure of the group P(Dn, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper. 相似文献
For each odd prime q an integer NHq (NH3 = ?1, NH5 = ?1, NH7 = 97, NH11 = ?243, …) is defined as the norm from L to of the Heilbronn sum Hq = TrI(ζ)(ζ), where ζ is a primitive q2th root of unity and L ?- (ζ) the subfield of degree q. Various properties are proved relating the congruence properties of Hq and NHq modulo p (p ≠ q prime) to the Fermat quotient ; in particular, it is shown that NHq is even iff 2q ? 1 ≡ 1 (mod q2). 相似文献
A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by where ? ∈ , α ∈ n, 〈x, y〉 = x1y1 + ? + xnyn, e(a) = exp (2πia) for a ∈ , and s = σ + ti is a complex number. The author proves that: (1) DF(s, ?, α) has analytic continuation into the whole s-plane, (2) DF(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at . The residue at is given explicitly. (3) ? = 0, α ? n, DF(s, 0, α) is analytic for . 相似文献
Let be a finite topology. If P and Q are open sets of (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of such that Q ? R ? P. A subcollection of the open sets of is termed an i-discrete collection of provided contains every open O ∈ with the property that ? ? O ? ? , contains exactly i minimal covers of ? , and provided ? = ?{O | O ∈ and O is a minimal cover of ? }. A single open set is a O-discrete collection. The number of distinct i-discrete collections of is denoted by p(, i). If there does not exist any i-discrete collection then p(,i) = 0, and this happens trivially for the case when i is greater than the number of points on which is defined. The object of this article is to establish the theorem: For any finite topology , the quantity E() = Σi = 0∞ (?1)ip(, i) = 1. 相似文献
The “cylinder conjecture” is to suppose that, if K is a gauge, the critical constants of C(K) = K ×] ? 1, +1 [? Rn+1 and of its basis K ? Rn are equal. The connection with packing constants is studied. The concept of Za(ssenhaus)-packing is introduced. ⊕i=1hG + (i ? 1)a (G a lattice) is a linear h-lattice, , the maximum density for translates of K by a linear h-lattice if the translates form a Za-packing for ζ, a packing for η, and if this packing is strict for ^. For K a bounded central star body, it is possible to find H with ζ1(C(K)) ≤ 2 ζH′(K). H is precised for K a gauge and for K = Bn. It is proved by Woods' methods that ; a result of Cleaver is used. 相似文献
Let {a1} and {ad1} be two maximal linear sequences of period pn ? 1. The cross-correlation function is defined by Cd(t) = for t = 0, t…pn ? 2, where . We find some new general results about Cd(t). We also determine the values and the number of occurences of each value of Cd(t) for several new values of d. 相似文献
Let Ω denote a simply connected domain in the complex plane and let be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals by , where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and gn is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L2n(f) = f(2n)(0) and L2n + 1(f) = f(2n)(1), in that their sequence of generating functions {gn} are “periodic.” Set gpn + k(ζ) = hk(ζ) ζpn, where p is a positive integer and each hk (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for . DeMar previously was able to find necessary conditions [7]. Next, we generalize {Ln} in several ways and find corresponding necessary and sufficient conditions. 相似文献
Given a sequence of integers [ai]i=1n, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that ai = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of and such that if and only if ai = [iα + β] (1 ? i ? n) for suitable β = β(α). 相似文献
Three main results are obtained: (1) If is an atomic maximal Abelian subalgebra of (), is the projection of () onto and h is a complex homomorphism on , then h ° is a pure state on (). (2) If {Pn} is a sequence of mutually orthogonal projections with rank(Pn) = n and ∑ Pn = I, is the projection of () onto {Pn}″ given by P(T)=∑tracen(T)Pn and h is a homomorphism on {Pn}″ such that h(Pn) = 0 for all n then h ° induces a type II∞ factor representation of the Calkin algebra. (3) If is a nonatomic maximal Abelian subalgebra of () then there is an atomic maximal Abelian subalgebra of () and a large family {Φα} of 1-homomorphisms from onto such that for each α, Φα ° is an extreme point in the set of projections from () onto . (Here denotes the projection of () onto .) 相似文献
The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, 0 denote the set of all partitions of X possessing finite entropy, and denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 supA ? BhA(T, α) = H(α) for all B∈I and α∈Z0. (2) T is weakly mixing /a2 supAhA(T, α) = H(α) for all α∈Z0. (3) If T is partially mixing with constant , then supA ? BhA(T, α) > cH(α) for all B∈I and nontrivial α∈Z0. (4) If supA ? BhA(T, α) > 0 for all B∈I and nontrivial α∈Z0, then T is weakly mixing. 相似文献
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number , where the infimum is over all f in which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u1, …, up} and V = {v1, …, vq} such that there exist positive coefficients ai and bi for which Σi = 1paiui = Σi = 1qbivi; (3) let ; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space. 相似文献
Here quadratic and cubic σ-polynomials are characterized, or, equivalently, chromatic polynomials of the graphs of order p, whose chromatic number is p ? 2 or p ? 3, are characterized. Also Robert Korfhage's conjecture that if σ2 + bσ + a is a σ-polynomial then is verified. In general, if σ(G) = Σ0naiσi is a σ-polynomial of a graph G, then an?2 is determined. 相似文献