Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

We consider the classical fixed-size confidence region estimation problem for the mean vectorμin theN_p(μ, Σ) population where Σ is unknown but positive definite. We writeλ₁for the largest characteristic root of Σ and assume thatλ₁is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ₁>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ₁, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient.     

An algorithmic sign-reversing involution for special rim-hook tableaux

Eğecioğlu and Remmel [Linear Multilinear Algebra 26 (1990) 59–84] gave an interpretation for the entries of the inverse Kostka matrix K⁻¹ in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK⁻¹=I but were unable to do the same for the equation K⁻¹K=I. We define an algorithmic sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193–197] to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Ser. A 62 (1993) 261–279].     

Une combinatoire sous-jacente au théorème des fonctions implicites

Using his theory of combinatorial species, [3.], 1–82 a combinatorial form of the classical multidimensional implicit function theorem. His theorem asserts the existence and (strong) unicity of species satisgying systems of combinatorial equations of a very general type. We present an explicit construction of these species by using a suitable combinatorial version of the Lie Series in the sense of [1. and 2.]. The approach constitutes a generalization of the method of “éclosions” (bloomings) which was used by the author in (J. Combin. Theory Ser. A 39, No. 1 (1985), 52–82), to study multidimensional power series reversion. Remarks concerning the applicability of the method to solve certain combinatorial differential equations are also made at the end of the work.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

Schur functions and the invariant polynomials characterizing U(n) tensor operators

We give a direct formulation of the invariant polynomials _μG_q⁽ⁿ⁾(, Δ_i,;, x_{i,i + 1},) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions S_λ known as Schur functions. To this end, we show after the change of variables Δ_i = γ_i − δ_i and x_{i, i + 1} = δ_i − δ_{i + 1} that_μG_q⁽ⁿ⁾(,Δ_i;, x_{i, i + 1},) becomes an integral linear combination of products of Schur functions S_α(, γ_i,) · S_β(, δ_i,) in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}, respectively. That is, we give a direct proof that _μG_q⁽ⁿ⁾(,Δ_i,;, x_{i, i + 1},) is a bisymmetric polynomial with integer coefficients in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials _μ^mG_q⁽ⁿ⁾(γ₁,…, γ_n; δ₁,…, δ_m). These new symmetries enable us to give an explicit formula for both _μ^mG₁⁽ⁿ⁾(γ; δ) and ₁G₂⁽ⁿ⁾(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for _μ^mG_q⁽ⁿ⁾(γ; δ).     

On Schensted's construction and the multiplication of schur functions

In this paper, certain aspects of Schensted's construction are examined and a generalization is described. From this generalization, the Littlewood-Richardson rule for the multiplication of Schur functions is derived by purely combinatorial methods.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A bijective proof of the hook-length formula

A well-known theorem of Frame, Robinson, and, Thrall states that if λ is a partition of n, then the number of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape λ, and the set of pairs (T, S), where T is a Standard Young Tableau of shape λ and S is a “Pointer” Tableau of shape λ.     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup

We study the behavior of measures obtained as a result of the action of the Ornstein-Uhlenbeck semigroup T _t associated with the Gaussian measure μ on an arbitrary probability measure ν in a separable Hilbert space as t → 0+. We prove that the densities of the parts of T _t ν absolutely continuous with respect to μ converge in the measure μ to the density of the part of ν absolutely continuous with respect to μ. For a finite-dimensional space, we prove the convergence of these densities μ-almost everywhere. In the infinite-dimensional case, we give sufficient conditions for almost-everywhere convergence. We also consider conditions on the absolute continuity of T _t ν with respect to μ in terms of the coefficients of the expansion of T _t ν in a series in Hermite polynomials (an analog of the Ito- Wiener expansion) and the connection with finite absolute continuity.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1654 – 1664, December, 2004.     

On Sperner families in which no k sets have an empty intersection, II

Let n_c,d(r, k) denote the maximal cardinality of Sperner families (i.e., XY for all X, Y ε ) on an r-element set satisfying c X d for all X ε and in which no k sets have an empty intersection. n_c,d(r, k) was determined by Frankl (J. Combin. Theory Ser. A 20 (1976), 1–11) if c r/k, and, if c = 0 and d = r, by Frankl and the author (J. Combin. Theory Ser. A 28 (1980), 54–63; Acta Cybernet. 4 (1978), 213–220 for all r and k with exception of 60 values of r if k = 3. In this paper we solve the problem of determination of n_c,d(r, 3) in nearly all unknown cases. In particular, we obtain n_0,r(r, 3) for 33 of the exceptional cases.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Intersection diagrams of distance-biregular graphs

We shall investigate distance-biregular graphs by means of intersection diagrams. First we give an alternate proof of a theorem which was obtained by Mohar and Shawe-Taylor in (J. Combin. Theory Ser. B 37 (1984), 90–100). Next we give some results on distance-biregular graphs of girth g ≡ 0 (mod 4).     

Generalization of the Gale–Ryser Theorem

We prove an inequality for the Kostka–Foulkes polynomials K_λ,μ(q) and give a criteria for the existence of a unique configuration of the given type (λ, μ). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers which is a generalization the Gale–Ryser theorem on an existence of a (0,1)-matrix with given sums of rows and columns. A new proof of the Berenstein–Zelevinsky weight-multiplicity-one criteria is given.     

A new class of symmetric polynomials defined in terms of tableaux

A new class of symmetric polynomials in n variables z = (z₁,…, z_n), denoted t_λ(z), and labelled by partitions λ = [λ₁ … λ_n] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The t_λ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions e_λ(z), which are homogeneous polynomials in the z_i, the t_λ(z) are inhomogeneous. This property is reflected in the fact that the t_λ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the t_λ(z) are proved. Two generalizations of the t_λ(z) are also given. The first generalizes the t_λ(z) to a 1-parameter family of symmetric polynomials, T_λ(α; z), where α is an arbitrary parameter. The T_λ(α; z) are shown to possess properties similar to those of the t_λ(z). The second generalizes the t_λ(z) to a class of skew-tableau symmetric polynomials, t_λ/μ(z), for which only a few preliminary results are given.