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1.
Two proofs are given, one combinatorial, the other by character theory, for the identity, ∏λa1! a2! … an! = ∏λ1a12a2nan, where λ ranges over all partitions λ = (1a12a2nan) of n. The two demonstrations yield a simple proof of the known formula, det2T(n) = [∏λ1a12a2nan]2, where T(n) is the matrix formed by the character table of Sn. Finally a sufficient condition is given so that the permanent of T(n) is zero.  相似文献   

2.
We consider the second order Stochastic Differential Equation dPtβ = Vtβ dt with P0β = p0, dVtβ = βVtβdt − βω2Ptβ + βdWt with V0β = v0, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that Pβ converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of Pβ at level u during the time interval [0, t] · (NtPβ (u)) to the local time of P(LtP (u)).  相似文献   

3.
We show that two oriented, topological, closed 4-manifolds M41, M42 are topologically cobordant iff σ(M41) = σ(M42) and either both of M41, M42 are stably smoothable or neither of M41, M42 is stably smoothable.  相似文献   

4.
One presentation of the alternating groupA n hasn?2 generatorss 1,…,sn?2 and relationss 1 3 =s i 2 =(s1?1si)3=(sjsk)2=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA n c )A n is introduced: It hasn?1 generatorss 1,…,S n?2,e, theA n-relations (above), and relationse 2=e, (es 1)4, (es j)2=(es j)4,es i=s i s 1 -1 es 1, wherej>1 andi≥1.  相似文献   

5.
Minimal free resolutions for prime ideals with generic zero (tn3,tn3?n10tn11,tn3?n20tn2, tn31), n1<n2<n3 positive integers, (n1,n2,n3)=1, are determined.  相似文献   

6.
Several integral representations of Euler's type for the generalized Horn functions (k)H3(n) and (k)H4(n) are given; and quadratic transformations expressing these as FD(n) and FA(n), respectively, are derived. Moreover, a reduction formula expressing (k)H3(n) in terms of (1)H3(n) is given.  相似文献   

7.
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   

8.
In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑Δn(Ltai+1nLtain)2, where Δn=(ain,ai+1n) is a sequence of subdivisions of [a,b], ain=i(ba)/2n+a, i=0,1,…,2n.  相似文献   

9.
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   

10.
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   

11.
A recursive kernel estimate i = 1n YiK⧸(x − Xi)hi)⧸∑j = 1n K((x − Xj)⧸hj) of a regression m(x) = E{Y|X = x} calculated from independent observations (X1, Y1),…, (Xn, Yn) of a pair (X, Y) of random variables is examined. ForE|Y|1 + δ < ∞, δ > 0, the estimate is weakly pointwise consistent for almost all (μ) x ∈ Rd, μ is the probability measure of X, if and only if∑i−1n hid I{hi > ɛ } ⧸ ∑j = 1n hjd → 0 as n → ∞, all ɛ > 0, and∑i = 1 hid = ∞, d is the dimension of X. For E|Y|1 + δ < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ Rd, if and only if the same conditions hold. ForE|Y|1 + δ < ∞, δ > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.  相似文献   

12.
On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on Rm, p = (p1,…,pm), pj ≥ 0, fixed integers, and for x = (x1,…,xm) in Rm, f(p)(x) = ?p1+…+pm f(x)/(?p1x1 … ?pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.  相似文献   

13.
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   

14.
We investigate the spectrum of matrices (∣xi,−xja)ni,j=1 with α>0 and distinct x1,…,xn whichare relevant to the theory of scattered data interpolation and spline functions. The main result is the non-singularity of these matrices, which is based on the property that the number of negativeand positive eigenvalues of these matrices is independent of x1,…,xn. Oscillation properties of asubset of eigenvectors of these matrices are also obtained. For 2<α<4 and points x1,…,xnR2,a sufficient condition for the non-singularity of (xixjα2)ni,j=1 is derived.  相似文献   

15.
Let w ?? A ??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H w p,q (? n ×? m ) for 0 < p ? 1, q >q w and show that the weighted Hardy space H w p (? n × ? m ) defined via Littlewood-Paley square functions coincides with H w p,q (? n × ? m ) for 0 < p ? 1, q > q w . As applications, we get a general principle on the H w p (? n × ? m ) to L w p (? n ×? m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.  相似文献   

16.
Let L ∞,∞ Δ (? m ) be the space of functions fL (? m ) such that ΔfL (? m ). We obtain new sharp Kolmogorov-type inequalities for the L -norms of the Riesz derivatives D α f of the functions fL ∞,∞ Δ (? m ) and solve the Stechkin problem of approximating an unbounded operator D α by bounded operators on the class fL (? m ) such that ‖Δf ≤ 1, and also the problem of the best recovery of the operator D α from elements of this class given with error δ.  相似文献   

17.
Given rational matrix functions ψ1(λ) = Im + C1(λIn1A1)−1B1 and ψ2(λ) = Im + C2(λIn2A2)−1B2 which are analytic and invertible on the unit circle, we characterize in terms of the operators A1,B1,C1,A2,B2,C2 when there exists a single rational matrix function W(λ) = Im + C(λInA)−1B such that WH2m = ψ 1H2mand WH2m = ψ2H2m. When this is the case, we give explicit formulae for A,B,C in terms of A1,B1,C1,A2,B2,C2. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.  相似文献   

18.
《Advances in Mathematics》1985,57(3):209-225
We show that the general bisymmetric polynomials mμG(n)q(γ1,…, γn; δ1,…, δm) are a limiting case of the bisymmetric, invariant polynomials nμG(n)q(γ1,…, γn; δ1,…, δn) which characterize U(n) tensor operators 〈p, q,…, q, 0, …, 0〉. By taking suitable limits of a pair of difference equations for nμG(n)q(γ; δ) we then deduce “transposition symmetry” for mμG(n)q(γ; δ) from the same symmetry for nμG(n)q(γ; δ). As an application of transposition symmetry for mμG(n)q(γ; δ) we derive an elegant, new contiguous relation for classical, well-poised hypergeometric series, and also prove an identity between these series and multiple hypergeometric series well-poised in SU(n).  相似文献   

19.
We analyse the trees given by sharps for Π12 sets via inner core models to give a canonical decomposition of such sets when a core model is Σ13 absolute. This is by way of analogy with Solovay's analysis of Π11 sets into ω1 Borel sets — Borel in codes for wellorders. We find that Π12 sets are also unions of ω1 Borel sets — but in codes for mice and wellorders. We give an application of this technique in showing that if a core model, K, is Σ13 absolute thenTheorem. Every real is in K iff every Π13 set of reals contains a Π13 singleton.  相似文献   

20.
The weighted Sobolev-Lions type spaces W pl,γ(Ω; E0, E) = W pl,γ(Ω; E) ∩ Lp,γ (Ω; E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω∈ Rn in W pl,γ(; E0, E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ(Ω; E0, E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.  相似文献   

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