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1.
The symbol C(m1 n 1m2 n 2...ms n s) denotes a 2-regular graph consisting ofn i cycles of lengthm i , i=1, 2,…,s. In this paper, we give some construction methods of cyclic(K v ,G)-designs, and prove that there exists a cyclic(K v , G)-design whenG=C((4m 1) n 1(4m 2) n 2...(4m s ) n s andv ≡ 1 (mod 2¦G¦).  相似文献   

2.
We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W 2 θ l D θ , F(σ) = {s k } 1 , where W 2 θ = W 2 θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential qW 2 θ ? 1 , and l D θ is a specially constructed finite-dimensional extension of the weighted space l 2 θ ; this extension contains the regularized spectral data s = {s k } 1 for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ 1 θ in terms of the l D θ norm of the difference of the regularized spectral data ‖s ? s1 θ . The result is new even for the classical case qL 2, which corresponds to the case θ = 1.  相似文献   

3.
Let T be a free ergodic measure-preserving action of an abelian group G on (X,μ). The crossed product algebra RT=L(X,μ)? G has two distinguished masas, the image CT of L(X,μ) and the algebra ST generated by the image of G. We conjecture that conjugacy of the singular masas ST(1) and ST(2) for weakly mixing actions T(1) and T(2) of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism. Our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism γ : RT(1)RT(2) such that γ(ST(1))=ST(2) has the property that the Cartan subalgebras γ(CT(1)) and CT(2) of RT(2) are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut(RT,ST), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of ST, and show that they have a simple interpretation in terms of the spectral theory of the action T. It follows that essentially all values of the Pukanszky invariant are realized by the masas ST, and there exist non-conjugate singular masas with the same Pukanszky invariant.  相似文献   

4.
Пустьd-натуральное ч исло,Z d — множество на боров k=(k 1, ...,k d ), состоящих из неотрицательных цел ыхk j ,Z + d =kZ d :k≧1. Предположи м, что системаf k (x):k∈Z + d ? ?L2(X,A, μ) и последовател ьностьa k :k∈Z + d . таковы, чт о для всех b∈Zd и m∈Z + d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z + d положительн а и не убывает. Например, есл иf k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {ak}. В работе получены оце нки порядка роста пря моугольных частных суммS m (x)= =∑ akfk(x) при maxmj→∞ как в случ ае {ak}∈l2, таки для {ak}l2. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.  相似文献   

5.
In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L 1 n z n Δ n + ... + α1 zΔ+α0 E and L 2= z n a n (z n + ... + za 1(z)Δ+a 0(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α n ∈ ?, a k (z) ∈ A(G), 0≤kn, and the following condition is satisfied: Σ j=s n?1 α j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z s+1Δ+β(z)E, β(z) ∈ A R , s ∈ ?, and z s+1 are equivalent in the spaces A R, 0?R?-∞, if and only if β(z) = 0.  相似文献   

6.
Let Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all cFq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑xFq χ1(D(x)) where D(x)=∑ni=1 aixpαi+pβi, αi?βi for each i, is known as a Dembowski-Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1 xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γe) is odd, and h (X)=λpeγXpeγ+λpγX is a permutation polynomial over Fq.  相似文献   

7.
In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.  相似文献   

8.
We deduce in an elementary way representations for the Mellin transform of a product of Bessel functions 0F1[−a2x2] and generalized hypergeometric functions pFp+1[−b2x2] for a,b>0. As a corollary we obtain a transformation formula for p+1Fp[1] which was discovered by Wimp in 1987 by using Bailey's method for the specialization 3F2[1].  相似文献   

9.
Givena m to be them th correlation coefficient of the Rudin-Shapiro polynomials of degrees 2 n ? 1, ¦a m¦ ≤ C(2 n )3/4 and there existsk ≠ 0 such that ¦a k¦ >D(2 n )0.73 (C andD are universal constants). Here we show that the 0.73 is optimal in the upper bound case.  相似文献   

10.
《Discrete Mathematics》1985,54(3):301-311
For each sequence q = {qi} = ± 1, i = 1, …, n−1 let Nq = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σi+1σi) = qi, i = 1,…, n−1. Clearly Σq (Nq/n!) =1 but what is the probability pn = Σq (Nq/n!)2 that two random permutations have the same up-down sequence? We show that pn = (Kn−11,1) where 1 = 1(x, y) ≡ 1 and Kn−1 is the iterated integral operator with (x, y) = ∫0101 K(x, y; x′, y′)φ(x′, y′) dxdy′ on L2[0, 1] × [0, 1] where K(x, y; x′, y′) is 1 if (xx′)(yy′) > 0 otherwise, and (f, g) = ∫0101fg. The eigenexpression of K yeilds pnn as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms.  相似文献   

11.
《Journal of Number Theory》1987,27(3):273-284
Let n ≠ 4a(8b + 7) be an integer. We deal with the problem of the solvability of the equation n = x12 + x22 + x32 in integers x1, x2, x3 prime to n. By a theorem of Vila (Arch. Math. 44 (1985), 424–437), the existence of such a solution implies that every central extension of the alternating group An, for n ≡ 3 (mod 8), can be realized as a Galois group over Q.  相似文献   

12.
We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W 2 θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s 2k (q) = λ k 1/2 (q) ? k, s 2k?1(q) = μ k 1/2 (q) ? k ? 1/2, where {λ k } 1 and {μ k } 1 are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t 2 θ such that the mapping F:W 2 θ?1 t 2 θ defined by the equality F(q) = {s n } 1 is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W 2 θ?1 and t 2 θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥τCqθ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥qθ?R, but is independent of the function q varying in this ball.  相似文献   

13.
Letp be a prime number ≡ 3 mod 4,G p the unit group of ?/p?, andg a generator ofG p. Letq be an odd divisor ofp - 1 andG p 2q = {t 2q;tG pthe subgroup of index2q inG p. The groupG p 2 / p 2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h 2q of the subfieldK 2q? ?(e2#x03C0;i/p ) of degree [K 2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Фj. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh 2q - . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Фj. We obtain an explicit lower bound for σ2 in terms ofh 2q - /h 2 - . Moreover, we show that log σ2 is asymptotically equal to 21og(h 2q - h 2 - )/(q - 1) forp→∞. Three tables illustrate the results.  相似文献   

14.
Let BD denote that Drazin inverse of the n×n complex matrix B. Define the core-rank of B as rank (Bi(B)) where i(B) is the index of B. Let j = 1,2,…, and Aj and A be square matrices such that Ai converges to A with respect to some norm. The main result of this paper is that AjD converges to AD if and only if there exist a j0 such that core-rank Aj=core-rankA for j ? j0.  相似文献   

15.
In 1979 R. Apéry introduced the numbers an = Σ0n(kn)2(kn + k) and un = Σ0n(kn)2(kn + k)2 in his irrationality proof for ζ(2) and ζ(3). We prove some congruences for these numbers which generalize congruences previously published in this journal.  相似文献   

16.
Starting with Euler's theorem that any odd perfect number n has the form n = pepi2eipk2ek, where p, p1,…,pk are distinct odd primes and pe ≡ 1 (mod 4), we show that extensive subsets of these numbers (so described) can be eliminated from consideration. A typical result says: if pe, pi2ei,…,pr2er are all of the prime-power divisors of such an n with ppi ≡ 1 (mod 4), then the ordered set {e1,…,er} contains an even number or odd number of odd numbers according as eporep (mod 8).  相似文献   

17.
Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.  相似文献   

18.
qV РАБОтЕ РАссМАтРИВАУ тсь МУльтИплИкАтИВН ыЕ пРЕОБРАжОВАНИь РьДО В ФУРьЕ-хААРА. ИжУЧЕНы слЕДУУЩИЕ кл Ассы МУльтИплИкАтОР ОВ: (H p α ,H q β ), (H p α ,L q ) И (с, с).  相似文献   

19.
Let (F n ) n??0 be the Fibonacci sequence given by F n+2 = F n+1 + F n , for n ?? 0, where F 0 = 0 and F 1 = 1. There are several interesting identities involving this sequence such as F n 2 + F n+1 2 = F 2n+1, for all n ?? 0. In a very recent paper, Marques and Togbé proved that if F n s + F n+1 s is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G m ) m is a linear recurrence sequence (under weak assumptions) and G n s + ... + G n+k s ?? (G m ) m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G m .  相似文献   

20.
We introduce the counting function π 2,8 * (x) of the primes with difference 8 between consecutive primes ( ****p n,pn+1 =p n + 8) can be approximated by logarithm integralLi 2,8 * . We calculate the values of π 2,8 * (x) and the sumC 2,8(x) of reciprocals of primes with difference 8 between consecutive primes (p n,pn+1 =p n +8)) wherex is counted up to 7 x 1010. From the results of these calculations, we obtain π 2,8 * (7 x 1010) = 133295081 andC 2,8(7 x 1010) = 0.3374 ±2.6 x 10-4.  相似文献   

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