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1.
The compactness method to weighted spaces is extended to prove the following theorem:Let H2,s1(B1) be the weighted Sobolev space on the unit ball in Rn with norm
6ν612,s=B1 (1rs)|ν|2 dx + ∫B1 (1rs)|Dν|2 dx.
Let n ? 2 ? s < n. Let u? [H2,s1(B1) ∩ L(B1)]N be a solution of the nonlinear elliptic system
B11rs, i,j=1n, h,K=1N AhKij(x,u) DiuhDK dx=0
, ψ ? ¦C01(B1N, where ¦Aijhk¦ ? L, Aijhk are uniformly continuous functions of their arguments and satisfy:
|η|2 = i=1n, j=1Nij|2 ? i,j=1n, 1rs, h,K=1N AhKijηihηik,?η?RNn
. Then there exists an R1, 0 < R1 < 1, and an α, 0 < α < 1, along with a set Ω ? B1 such that (1) Hn ? 2(Ω) = 0, (2) Ω does not contain the origin; Ω does not contain BR1, (3) B1 ? Ω is open, (4) u is Lipα(B1 ? Ω); u is LipαBR1.  相似文献   

2.
Given the iterative scheme xi+1 = BTxi + r where B, T are fixed n×:n real matrices, r a fixed real n-vector and xi a real n-vector we investigate the convergence and monotonicity of schemes of the type
vi+1wi+1 = BOOBS11?S12?S21S22viwi + rr
, where Sij are n×:n real matrices related to T. The n-vector iterates vi,wi bracket in a certain sense solutions x of x = BTx + r. We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself xi+1 = BTxi + r. This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.  相似文献   

3.
Let A(x,ε) be an n×n matrix function holomorphic for |x|?x0, 0<ε?ε0, and possessing, uniformly in x, an asymptotic expansion A(x,ε)?Σr=0Ar(x) εr, as ε→0+. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion P(x,ε)?Σr=0Pr(x)εr, as ε→0+, is constructed, such that the transformation y = P(x,ε)z takes the differential equation εhdydx = A(x,ε)y,h a positive integer, into εhdzdx = B(x,ε)z, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.  相似文献   

4.
In this paper we study the linked nonlinear multiparameter system
yrn(Xr) + MrYr + s=1k λs(ars(Xr) + Prs) Yr(Xr) = 0, r = l,…, k
, where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ A ¦ (x) = detars(xr) > 0. Conditions on the polynomial operators Mr, Prs are produced which guarantee a sequence of eigenfunctions for this problem yn(x) = Πr=1kyrn(xr), n ? 1, which form a basis in L2([a, b], ¦ A ¦). Here [a, b] = [a1, b1 × … × [ak, bk].  相似文献   

5.
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by G(x, A) := 1n · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, R, p) with the same distribution Fa. Suppose that all moments E | a | k, k = 1, 2, … are finite, Ea=0 and E | a | 2. Let
M(A)=σ=1s θσPσ(A,A1)
with complex numbers θσ and finite products Pσ of factors A and A1 (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability 1 G(x, M(Ann12)) converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples ArA1r, Ar + A1r, ArA1r ± A1rAr, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form
lim supn→∞i=1ni(n)|2?6An62? 0.8228…
of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n.  相似文献   

6.
With quasicommutative n-square complex matrices A1,…,As and s-square hermitian G=(gij), relationships are given between the image Σsi,j=1g ijAiHA1j of a linear transformation on Hn being positive definite and the action of H on generalized inertial decompositions of Cn.  相似文献   

7.
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral H = ⊕L2(vt) dm(t) and the operator (L?)(t, λ) = e?iλ?(t, λ) ? 2e?iλtT ?(s, x) e(s, t) dvs(x) dm(s) on H, where e(s, t) = exp ∫stTdvλ(θ) dm(λ). Let μt be the measure defined by T?(x) dμt(x) = ∫0tT ?(x) dvs dm(s) for all continuous ?, and let ?t(z) = exp[?∫ (e + z)(e ? z)?1t(gq)]. Call {vt} regular iff for all t, ¦?t(e)¦ = ¦?(e for 1 a.e.  相似文献   

8.
The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let Xs(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(Xl), given that min(Xl) ≥ 1. The following lower bound on the variance of Xs is proved:
Var(Xs)?mmn?2ks?kns?1nk1
, where m = |E(H)| and m = (kn) ? m. This implies the following: putting t(k, l) = limn→∞T(n, l, k)(kn)?1 then t(k, l) ≥ T(s, l, k)((ks) ? 1)?1, whenever sl > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.  相似文献   

9.
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let v1n=(an1,an2,?,an,n?1), put
Xn(t)=[n(n-1)]-12k=1[(n-1)t]|vn1uf(n-1)|2,0?t?1
(bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally
Fn(x)=n-1(number of λk(n)?xn,1?k?n)
(empirical distribution function of the eigenvalues of Ann. Suppose (i) limnannn=0, (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then
Fn?W(·,C)(n→∞)
,where W is absolutely continuous with (semicircle) density
w(x,C)=(2Cπ)-1(4C-x212for|x|?2C0for|x|?2C
  相似文献   

10.
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and E{N(r, t, ?)} = Σn=1 nr?2P{|Sn| > ?nrt}. In this paper, we prove that (1) lim?→0+?α(r?1)E{N(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, K(r, t) = {2α(r?1)2Γ((1 + α(r ? 1))2)}{(r ? 1) Γ(12)}, and α = 2t(2r ? t); (2) lim?→0+G(t, ?)H(t, ?) = 0 if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N(t, t, ?)} = Σn=1nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and H(t, ?) = E{N(t, t, ?)} = Σn=1 nt?2P{| Sn | > ?n2t} → ∞ as ? → 0+, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.  相似文献   

11.
If r, k are positive integers, then Tkr(n) denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r = 1. An explicit formula for Tkr(n) is derived and it is shown that limn→∞Tkr(n)nk = 1ζ(rk).If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2psas; piS and ai ≥ 0 for all i} and Tkr(S, n) denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for Tkr(S, n) are derived and it is shown that limn→∞Tkr(S, n)nk = (p1 … pa)rkζ(rk)(p1rk ? 1) … (psrk ? 1).  相似文献   

12.
An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λi, singular values σi, and inverse B = (bij). We are interested in the four closely related problems of finding λ(n) = minA, i|λi|, σ(n) = minA, iσi, χ(n) = maxA, i, j |bij|, and μ(n) = maxAΣijb2ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between (2n)?1(n4)?n2 and cn (2.274)?n, where c is a constant, c(2.274)n?χ(n)?2(n4)n2, and c(5.172)n?μ(n)?4n2 (n4)n. We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry.  相似文献   

13.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If Ln denotes the symmetric group on {1,2,…,n} then we define the projection PL : Tn(V) → Sn(V) by the formula (n!)?1Σσ ? Ln Pσ, where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If xi ? V1, 1 ? i ? n, then x1?x2? … ?xn denotes the member of Tn(V) such that (x1?x2· ? ? ?xn)(y1,y2,…,yn) = Пni=1xi(yi) for each y1 ,2,…,yn in V, and x1·x2xn denotes PL(x1?x2? … ?xn). If B? Sn(V) and there exists x i ? V1, 1 ? i ? n, such that B = x1·x2xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.  相似文献   

14.
Asymptotic results are obtained for pA(k)(n), the kth difference of the function pA(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that
PA(k+1)(n)PA(k)(n) = O(n?1/2) (n→ ∫)
thereby verifying for these A a conjecture of Bateman and Erdös.  相似文献   

15.
Let Jωx(t) = x + ∝0tbω(s) ds, where bω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process Jωx(t): Let K be a compact plane set and let x?K. Then if ∑ 2nM1(An(x)?K) < ∞ (where An(x) = {2?n?1 ? ¦ z ? x ¦ ? 2?n} and M1 denotes one-dimensional Hausdorff content), the process Jωx(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes.  相似文献   

16.
In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if X(G) = Σμ(0, x)λr ? r(x) = Σ (?1)r ? kWkλk is the characteristic polynomial of G, then
wk?rk+nr?1k
Thus γ(G) ? 2r ? 1 (n+2), where γ(G) = Σwk. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then
wi?ri + nri+1 for i>1; w1?r+nr2 ? 1;
|μ| ? (r? 1)n; and γ ? (2r ? 1 ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, r(G) = r(G?) = 4, and n(G) = n(G?) = 3 then (w1(G))4t-G ? (w1(G?)) = (8, 20, 18, 7, 1). Further, if β is the Crapo invariant,
β(G)=dX(G)(1),
then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.  相似文献   

17.
In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) (A+A1)2 is idempotent of rank r, and trrA (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) As=At for some positive integers st, and trA=rankA iff A is idempotent. (c) A(A1A)s= A(AA1)t for some integers st iff AA1=A1A is idempotent, while A(A1A)s= A(AA1)s for some integers s≠0 iff AA1=A1A. (d) A(A1A)s=A1 (AA1)t for some integers st and rankA=trA iff A is Hermitian idempotent, while A(A1A)s= A1(AA1)s for some integer s iff A is Hermitian. Here A1 indicates the conjugate transpose of A, and P-α is defined iff (P+)α=(Pα)+ for all positive integers α and P+ is the Moore-Penrose inverse of P.  相似文献   

18.
We consider the mixed boundary value problem Au = f in Ω, B0u = g0in Γ?, B1u = g1in Γ+, where Ω is a bounded open subset of Rn whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on Ω and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on Γ+. The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset T of the reals such that, if Ds = {u ? Hs(Ω): Au = 0} then for s ? = 12(mod 1), (B0,B1): Ds → Hs ? 12?) × Hs ? 32+) is a Fredholm operator if and only if s ∈T . Moreover, T = ?xewTx, where the sets Tx are determined algebraically by the coefficients of the operators at x. If n = 2, Tx is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, Tx is either an open interval of length 1 or is empty; and finally, if n ? 4, Tx is an open interval of length 1.  相似文献   

19.
Let Ωm be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ωm, and define Xms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n(m)(t), t?[0, 1], and centering constants bms, 0?s? m, such that
Z(m)(t)=s=0n(m)(t)(Xms?bms)s=0mbms
converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.  相似文献   

20.
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