On a partition identity

Two proofs are given, one combinatorial, the other by character theory, for the identity, ∏_λa₁! a₂! … a_n! = ∏_λ1^a₁2^a₂ … n^a_n, where λ ranges over all partitions λ = (1^a₁2^a₂ … n^a_n) of n. The two demonstrations yield a simple proof of the known formula, det²T(n) = [∏_λ1^a₁2^a₂ … n^a_n]², where T(n) is the matrix formed by the character table of S_n. Finally a sufficient condition is given so that the permanent of T(n) is zero.     

Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dP_t^β = V_t^β dt with P₀^β = p₀, dV_t^β = βV_t^βdt − βω²P_t^β + βdW_t with V₀^β = v₀, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that P^β converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P^β at level u during the time interval [0, t] · (N_t^Pβ (u)) to the local time of P(L_t^P (u)).     

Minimal free resolutions of monomial curves in P3k

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.     

4-Dimensional topological bordism

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.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)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.     

On cyclic decompositions of the complete graph into the 2-regular graphs

The symbol C(m₁ ⁿ ₁m₂ ⁿ ₂...m_s 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¦).     

Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

Let T be a free ergodic measure-preserving action of an abelian group G on (X,μ). The crossed product algebra R_T=L^∞(X,μ)? G has two distinguished masas, the image C_T of L^∞(X,μ) and the algebra S_T generated by the image of G. We conjecture that conjugacy of the singular masas S_T⁽¹⁾ and S_T⁽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 : R_T⁽¹⁾→R_T⁽²⁾ such that γ(S_T⁽¹⁾)=S_T⁽²⁾ has the property that the Cartan subalgebras γ(C_T⁽¹⁾) and C_T⁽²⁾ of R_T⁽²⁾ are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut(R_T,S_T), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of S_T, 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 S_T, and there exist non-conjugate singular masas with the same Pukanszky invariant.     

Positive powers of certain conditionally negative definite matrices

We investigate the spectrum of matrices (∣x_i,−x_j∣^a)ⁿ_i,j=1 with α>0 and distinct x₁,…,x_n 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,…,x_n. Oscillation properties of asubset of eigenvectors of these matrices are also obtained. For 2<α<4 and points x¹,…,xⁿ∈R²,a sufficient condition for the non-singularity of (∥xⁱ−x^j∥^α₂)ⁿ_i,j=1 is derived.     

Integral representations of Euler's type for certain generalized Horn functions

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 F_D⁽ⁿ⁾ and F_A⁽ⁿ⁾, respectively, are derived. Moreover, a reduction formula expressing ^(k)H₃⁽ⁿ⁾ in terms of ⁽¹⁾H₃⁽ⁿ⁾ is given.     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻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) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. 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.     

Some descriptive set theory and core models

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.     

Quasi sure quadratic variation of local times of smooth semimartingales

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(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

Some congruences for the Apery numbers

In 1979 R. Apéry introduced the numbers a_n = Σ₀ⁿ(_kⁿ)²(_k^{n + k}) and u_n = Σ₀ⁿ(_kⁿ)²(_k^{n + 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.     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

Necessary and sufficient consistency conditions for a recursive kernel regression estimate

A recursive kernel estimate ∑_{i = 1}ⁿ Y_iK⧸(x − X_i)h_i)⧸∑_{j = 1}ⁿ K((x − X_j)⧸h_j) of a regression m(x) = E{Y|X = x} calculated from independent observations (X₁, Y₁),…, (X_n, Y_n) 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 ∈ R^d, μ is the probability measure of X, if and only if∑_i−1ⁿ h_i^d I_{hi > ɛ } ⧸ ∑_{j = 1}ⁿ h_j^d → 0 as n → ∞, all ɛ > 0, and∑_{i = 1}^∞ h_i^d = ∞, d is the dimension of X. For E|Y|^{1 + δ} < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ R^d, 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.     

Optimal tristance anticodes in certain graphs

For z₁,z₂,z₃∈Zⁿ, the tristance d₃(z₁,z₂,z₃) is a generalization of the L₁-distance on Zⁿ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeA_d of diameter d is a subset of Zⁿ with the property that d₃(z₁,z₂,z₃)?d for all z₁,z₂,z₃∈A_d. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z² for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z² where d(z₁,z₂)=1 if z₁,z₂ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z² is replaced by the hexagonal lattice A₂. We also investigate optimal tristance anticodes in Z³ and optimal quadristance anticodes in Z², and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.     

Extension of isometries on the unit sphere of L p spaces

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.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

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

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

Let (M_r)_{r∈? 0} be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function ? on ?ⁿ belonging to the non quasi-analytic (M_r)-class of Roumieu type, there is an element g of the same class which is analytic on ?ⁿ F and such that D^α ?(x) = D^αg(x) for every σ ∈ ?₀ ⁿ SBAP and x ∈ F.     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

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.