Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.     

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].     

Solution of the Ulam Stability Problem for Euler–Lagrange Quadratic Mappings

In 1940 S. M. Ulam proposed at the University of Wisconsin theproblem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968 S. U. Ulam proposed the moregeneral problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P. M. Gruber proposed theUlam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” According to P. M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982–1996 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and established analogous stability problems. In this paper we first introduce newquadratic weighted meansandfundamental functional equationsand then solve theUlam stability problemfornon-linear Euler–Lagrange quadratic mappingsQ:X → Y, satisfying a mean equation and functional equation[formula]for all 2-dimensional vectors (x₁, x₂) X², withXa normed linear space (Y a real complete normed linear space), and any fixed pair (a₁, a₂) of realsa_iand any fixed pair (m₁, m₂) of positive realsm_i(i = 1, 2), [formula]     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

Cross-validation and the smoothing of orthogonal series density estimators

We describe a class of smoothed orthogonal series density estimates, including the classical sequential-series introduced by [6], Soviet Math. Dokl. 3 1559–1562) and [16], Ann. Math. Statist. 38 1261–1265), and [23], Ann. Statist 9 146–156) two-parameter smoothing. The Bowman-Rudemo method of least-squares cross-validation (1982, Manchester-Sheffield School of Probability and Statistics Research Report 84/AWB/1; 1984, Biometrika 71 353–360; [14], Scand. J. Statist. 9 65–78), is suggested as a practical way of choosing smoothing parameters automatically. Using techniques of [18], Ann. Statist. 12 1285–1297), that method is shown to perform asymptotically optimally in the case of cosine and Hermite series estimators. The same argument may be used for other types of series.     

On the limiting Pitman efficiency of some rank tests of independence

The limiting (as the significance level approaches 0) Pitman efficiency of a new “regression-based” rank test of independence to Kendall's tau and Spearman's rho is derived. The result is based on a version of [17], Ann. Statist.4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. [7], Trans. Amer. Math. Soc.87, 173–186) result is applied to verify the main assumption of Wieand's paper. This approach is shown to be useful in some other situations, also.     

Algebras Generated by Reciprocals of Linear Forms

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K-algebra C(Δ) of rational functions generated by {1/α  α  Δ}. Then the ring ∂(V) of differential operators with constant coefficients naturally acts on C(Δ). We study the graded ∂(V)-module structure of C(Δ). We especially find standard systems of minimal generators and a combinatorial formula for the Poincaré series of C(Δ). Our proofs are based on a theorem by Brion–Vergne [4] and results by Orlik–Terao [9].     

A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85–108) -divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the -divergence approach with that of C. R. Rao and S. K. Mitra (“Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971) and derive weak convergence to a χ² distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.     

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X₁][X₂;σ₂,δ₂][X_N;σ_N,δ_N] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X₁,…,X_N) and terminating with new variables (T₁,…,T_N). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property:

Full-size image

. We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (wW), R_q[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.     

Several sufficient conditions of solvability for a nonlinear higher-order three-point boundary value problem with all derivatives

In this paper, firstly, some errors in the proof of our paper “Several sufficient conditions of solvability for a nonlinear higher-order three-point boundary value problem on time scales, Appl. Math. Comput. 190 (2007) 566–575” are pointed, and we make the corresponding correction when T=R. Then, the more general problem with all derivatives is considered. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence and uniqueness of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative and Banach fixed point theorem.     

Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity

This paper generalizes the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem of Ky Fan (“Game Theory and Related Topics,” pp. 151–156, North-Holland, Amsterdam, 1979; and Math. Ann. 266, 1984, 519–537) and the Ky Fan minimax inequality by introducing a class of the generalized closedness and continuity conditions, which are called the transfer closedness and transfer continuities. We then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions. We also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions. Thus our results generalize many of the existence theorems in the literature.     

On a discrete Korovkin theorem

In [G. A. Anastassiou, A discrete Korovkin theorem, J. Approx. Theory 45 (1985), pp. 383–388, Theorem 3], a discrete Korovkin theorem was given. We restate the theorem here and its proof, correcting a mistake in the above reference.     

Erd s–Turán Type Theorems on Quasiconformal Curves and Arcs

The theorems of Erd s and Turán mentioned in the title are concerned with the distribution of zeros of a monic polynomial with known uniform norm along the unit interval or the unit disk. Recently, Blatt and Grothmann (Const. Approx.7(1991), 19–47), Grothmann (“Interpolation Points and Zeros of Polynomials in Approximation Theory,” Habilitationsschrift, Katholische Universität Eichstätt, 1992), and Andrievskii and Blatt (J. Approx. Theory88(1977), 109–134) established corresponding results for polynomials, considered on a system of sufficiently smooth Jordan curves and arcs or piecewise smooth curves and arcs. We extend some of these results to polynomials with known uniform norm along an arbitrary quasiconformal curve or arc. As applications, estimates for the distribution of the zeros of best uniform approximants, values of orthogonal polynomials, and zeros of Bieberbach polynomials and their derivatives are obtained. We also give a negative answer to one conjecture of Eiermann and Stahl (“Zeros of orthogonal polynomials on regularN-gons,” in Lecture Notes in Math.1574(1994), 187–189).     

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.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Addendum to the Paper “The Generalized Integer Gamma Distribution—A Basis for Distributions in Multivariate Statistics”

A brief remark on the paper “The Generalized Integer Gamma Distribution— A Basis for Distributions in Multivariate Statistics,” (1998,J. Multivariate Anal.64, 86–102) and an additional result concerning the distribution of the product of some particular independent beta random variables, which broadens the scope of the results in that paper, are presented.     

On the Divergence Phenomenon in Hermite–Fejér Interpolation

Generalizing results of L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617), we show that for any nonconstant entire function f and any interpolation scheme on [−1, 1], the associated Hermite–Fejér interpolating polynomials diverge on any infinite subset of \[−1, 1]. Moreover, it turns out that even for the locally uniform convergence on the open interval ]−1, 1[ it is necessary that the interpolation scheme converges to the arcsine distribution.     

A unifying look at the Apostolico–Giancarlo string-matching algorithm

String matching is the problem of finding all the occurrences of a pattern in a text. We present a new method to compute the combinatorial shift function (“matching shift”) of the well-known Boyer–Moore string matching algorithm. This method implies the computation of the length of the longest suffixes of the pattern ending at each position in this pattern. These values constituted an extra-preprocessing for a variant of the Boyer–Moore algorithm designed by Apostolico and Giancarlo. We give here a new presentation of this algorithm that avoids extra preprocessing together with a tight bound of 1.5n character comparisons (where n is the length of the text).     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.