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1.
Let Ω be a finite set with k elements and for each integer n ≧ 1 let Ωn = Ω × Ω × … × Ω (n-tuple) and Ωn = {(a1, a2,…, an) | (a1, a2,…, an) ∈ Ωn and ajaj+1 for some 1 ≦ jn ? 1}. Let {Ym} be a sequence of independent and identically distributed random variables such that P(Y1 = a) = k?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in Ωn and in Ω?n with respect to the stochastic process {Ym}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.  相似文献   

2.
Let A be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of A. Furthermore, let Mk(Ω;A) be the set of A-valued functions defined in an open subset Ω of Rm+1 (1 ? m ? n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator D = ∑i = 0m ei(??xi) and k? N. The aim of this paper is to characterize the dual and bidual of Mk(Ω;A). It is proved that, if Mk(Ω;A) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits Mk(Ω;A) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.  相似文献   

3.
Let S be a Dirichlet form in L2(Ω; m), where Ω is an open subset of Rn, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let Sk be a Dirichlet form on some k-dimensional submanifold Ωk of Ω. The paper is devoted to the study of the closability of the forms E with domain C0(Ω) and defined by: (?,g)=E(?, g)+ ip=1Eki(?ki, gki) where 1 ? kp < ? < n, and where ?ki, gki denote restrictions of ?, g in C0(Ω) to Ωki. Conditions are given for E to be closable if, for each i = 1,…, p, one has ki = n ? i. Other conditions are given for E to be nonclosable if, for some i, ki < n ? i.  相似文献   

4.
This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gi(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when φ(x) = max{f(x, y) ¦ y ? Ωy}, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form miny ? Ωxmaxy ? Ωyf(x, y) under the assumption that Ωmξ={x|gi(x)?0, j=1,…,s} ∩Rn, Ωy={y|ζi(y)?0, i-1,…,t} ∩ Rm, with f, gj, ζi continuously differentiable, f(x, ·) concave, ζi convex for i = 1,…, t, and Ωx, Ωy compact.  相似文献   

5.
Let Ω denote a connected and open subset of Rn. The existence of n commuting self-adjoint operators H1,…, Hn on L2(Ω) such that each Hj is an extension of i∂∂xj (acting on Cc(Ω)) is shown to be equivalent to the existence of a measure μ on Rn such that f → \̂tf (the Fourier transform of f) is unitary from L2(Ω) onto Ω. It is shown that the support of μ can be chosen as a subgroup of Rn iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω). This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of Rn by some subgroup, i.e., iff Ω is essentially a fundamental domain.  相似文献   

6.
Let S denote a set of n points in the Euclidean plane. A subset S′ of S is termed a k-set of S if it contains k points and there exists a straight line which has no point of S on it and separates S′ from S?S′. We let fk(n) denote the maximum number of k-sets which can be realized by a set of n points. This paper studies the asymptotic behaviour of fk(n) as this function has applications to a number of problems in computational geometry. A lower and an upper bound on fk(n) is established. Both are nontrivial and improve bounds known before. In particular, fk(n) = fn?k(n) = Ω(n log k) is shown by exhibiting special point-sets which realize that many k-sets. In addition, fk(n) = fn?k(n) = O(nk12) is proved by the study of a combinatorial problem which is of interest in its own right.  相似文献   

7.
For an open set Ω ? RN, 1 ? p ? ∞ and λ ∈ R+, let W?pλ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C0(Ω) in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators U, 1 ? p, q ? ∞ and a quasibounded domain Ω ? RN. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map W?pλ(Ω) λ Lq(Ω) exists and belongs to the given Banach ideal U: Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any x ? Ω to the boundary ?Ω tends to zero as O(¦ x ¦?l) for ¦ x ¦ → ∞, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ N, μ > λ S(U; p,q:N) and v > N/l · λD(U;p,q), one has that W?pλ(Ω) λ Lq(Ω) belongs to the Banach ideal U. Here λD(U;p,q;N)∈R+ and λS(U;p,q;N)∈R+ are the D-limit order and S-limit order of the ideal U, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpnlqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω fulfills condition C1l.For an open set Ω in RN, let W?pλ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C0(Ω) in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in RN and give sufficient conditions on λ such that the Sobolev imbedding operator W?pλ(Ω) λ Lq(Ω) exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω is a quasibounded open set in RN.  相似文献   

8.
For s = σ + it, σ > 1, and integer k ? 1ζk(s) = ∑dk(n)n3. In a previous paper, Ω results for ζ(c + it), 12 ? c ? 1, where obtained in an elementary way by choosing N = N(k, c) so that the size of the single term dk(N) gave Ω results slightly better than existing ones. Here the method will be shown to give the same results for Re ζ(c + it).  相似文献   

9.
Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?2(m ? n)A + B + λ2nI)u? = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ?Ω. The estimate ∥ u? ? u ∥L2(Ω) ? C? ¦ λ ¦?2n + 1(1 + ? ¦ λ ¦)?1 ∥ f ∥L2(Ω) is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.  相似文献   

10.
If Ω denotes an open subset of Rn (n = 1, 2,…), we define an algebra g (Ω) which contains the space D′(Ω) of all distributions on Ω and such that C(Ω) is a subalgebra of G (Ω). The elements of G (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in G(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and ?∈OM(R2q)—a classical Schwartz notation—for any G1,…,GqG(σ), we define naturally an element ?G1,…,Gq∈G(σ). These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.  相似文献   

11.
12.
Consider an elliptic sesquilinear form defined on V × V by J[u, v] = ∫Ωajk?u?xk\?t6v?xj + ak?u?xkv? + αju\?t6v?xj + auv?dx, where V is a closed subspace of H1(Ω) which contains C0(Ω), Ω is a bounded Lipschitz domain in Rn, ajk, ak, αj, a ? L(Ω), and Re ajkζkζj ? κ > 0 for all ζ?Cn with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈V. Then L + λI is a maximal accretive operator in L2(Ω) for λ a sufficiently large real number. It is proved that (L + λI)12 is a bounded operator from V to L2(Ω) provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian.  相似文献   

13.
In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in L2(Ω)). The existence of n commuting self-adjoint operators H1,…, Hnin L2(Ω) such that each Hj is a restriction of ?i ββxj (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rn such that the restrictions to Ω of the functions exp iλjxj form a total orthogonal family in L2(Ω). If it is required, in addition, that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω), then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rn. The measurable sets Ω ?Rn (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rn as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rn by some subgroup Γ (essentially the dual of Λ).  相似文献   

14.
Let Ω = {1, 0} and for each integer n ≥ 1 let Ωn = Ω × Ω × … × Ω (n-tuple) and Ωnk = {(a1, a2, …, an)|(a1, a2, … , an) ? Ωnand Σi=1nai = k} for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that P(Y1 = 0) = P(Y1 = 1) = 12. For each A in Ωn, let TA be the first occurrence time of A with respect to the stochastic process {Ym}m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in Ωn, there is an element B in Ωn such that the probability that TB is less than TA is greater than 12. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ kn ? 1, each element A in Ωnk, there is an element B also in Ωnk such that the probability that TB is less than TA is greater than 12; (II) for n ≥ 4 and 1 ≤ kn ? 1, each element A = (a1, a2,…,an) in Ωnk, there is an element C also in Ωnk such that the probability that TA is less than TC is greater than 12 if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ in?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.  相似文献   

15.
It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · fk(t) and t · gk1(t), where fk(t) is the kth Euler transformation of f(t) and gk1(t) = g(t)(1 ? kt · g(t)).  相似文献   

16.
Abstract connections between integral kernels of positivity preserving semigroups and suitable Lp contractivity properties are established. Then these questions are studied for the semigroups generated by ?Δ + V and HΩ, the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ?Ω, of the form ¦η(x)¦? c?0(x) ∥HΩkη∥2, where ?0 is the unique positive L2 eigenfunction of HΩ.  相似文献   

17.
Let f(n) denote the number of square permutations in the symmetric group Sn. This paper proves a conjecture that f(2k + 1) = (2k + 1)f(2k) and provides efficient procedures for the computation of f(n). The behavior of f(n) as n → ∞ is investigated and an asymptotic result obtained which shows that f(n) ~ 2Knne?n, where K = Π1cosh(12k).  相似文献   

18.
Uniform estimates in H01(Ω) of global solutions to nonlinear Klein-Gordon equations of the form utt ? Δu + mu = g(u) in Ω, u = 0 in, where Ω is an open subset of RN, m > 0, and g satisfies some growth conditions are established.  相似文献   

19.
Let Ω ? RN be an open set with dist(x, ?Ω) = O(¦ x ¦?l) for x ? Ω and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers αn of Sobolev imbeddings
over these quasibounded domains Ω. Here
denotes the Sobolev space obtained by completing C0staggered∞(Ω) under the usual Sobolev norm. We prove αn(Ip,qm) $?n, where
. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of Ip,qm. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω is a quasibounded domain of the above type.  相似文献   

20.
Let L be a lattice over the integers of a quaternion algebra with center K which is a B-adic field. Then the unitary group U(L) equals its own commutator subgroup Ω(L) and is generated by the unitary transvections and quasitransvections contained in it. Let g be a tableau, U(g), U+(g), Ω(g), T(g) be the corresponding congruence subgroups of order g. Then U(g)U+(g) ? Xi = 1τZ2, and Ω(g) = T(g) (the subgroup generated by the unitary transvections and quasitransvections with order ≤ g). Let G be a subgroup of U(L) with o(G) = g, then G is normal in U(L) if and only if U(g) ? G ? T(g).  相似文献   

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