Strong Asymptotics for Multiple Laguerre Polynomials

We consider multiple Laguerre polynomials l_n of degree 2n orthogonal on (0,∞) with respect to the weights and , where -1 < α, 0 < β₁ < β₂, and we study their behavior in the large n limit. The analysis differs among three different cases which correspond to the ratio β₂/β₁ being larger, smaller, or equal to some specific critical value κ. In this paper, the first two cases are investigated and strong uniform asymptotics for the scaled polynomials l_n(nz) are obtained in the entire complex plane by using the Deift-Zhou steepest descent method for a (3 × 3)-matrix Riemann-Hilbert problem.     

Free Diffusions and Matrix Models with Strictly Convex Interaction

We study solutions to the free stochastic differential equation , where V is a locally convex polynomial potential in m non-commuting variables and S an m-dimensional free Brownian motion. We prove that such free processes have a unique stationary distribution μV. When the potential V is self-adjoint, we show that the law μV is the limit law of a random matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law exp. If V = V_β depends on complex parameters , we prove that the moments of the law μV are analytic in β at least for those β for which V_β is locally convex. In particular, this gives information on the region of convergence of the generating function for the enumeration of related planar maps. We prove that the solution X_t has nice convergence properties with respect to the operator norm as t goes to infinity. This allows us to show that the C* and W* algebras generated by an m-tuple with law μV share many properties with those generated by a semi-circular system. Among them is the lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II₁ factor. We show that the microstates free entropy χ(μV ) is finite when V is self-adjoint. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in under the law μV is connected, vastly generalizing the case of a single random matrix. We also deduce from this dynamical approach that the convergence of the operator norms of independent matrices from the GUE proved by Haagerup and Thorbjornsen [HT] extends to the context of matrices interacting via a convex potential. Received: February 2007, Revision: July 2007, Accepted: July 2007     

On estimation and detection of a function of infinitely many variables

We observe an unknown function of infinitely many variables f = f(t), t = (t₁, ..., t_n, ... ) ∈, [0, 1]^∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension of the function f(t) to IR ^∞. Taking a quantity σ > 0 and a positive sequence a = {a_k}, we consider the set that consists of functions f such that . We consider the cases a_k = k^α and a_k = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈ or to test the null hypothesis H₀: f = 0 against the alternatives f ∈ , where the set consists of functions f ∈ such that ∥f∥₂ ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates f _ɛ ^* that provide distiguishability in the problems. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113.     

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

Let {S _k , k ≥ 0} be a symmetric random walk on , and an independent random field of centered i.i.d. random variables with tail decay . We consider a random walk in random scenery, that is . We present asymptotics for the probability, over both randomness, that {X _n > n ^β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process , where l _n (x) is the number of visits of site x up to time n.      

Maximal Betti numbers of Cohen–Macaulay complexes with a given <Emphasis Type="Italic">f</Emphasis>-vector

Given the f-vector f = (f₀, f₁, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δ_f with f(Δ_f) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has for all i and j, where f(Δ) is the f-vector of Δ and where β _ij (I _Δ) are graded Betti numbers of the Stanley–Reisner ideal I _Δ of Δ. The first author is supported by JSPS Research Fellowships for Young Scientists. Received: 23 January 2006     

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP _n ^α _n ^β _n are studied, assuming that

Null Riemannian cubics in tension in SO(3)

^** Email: lyle{at}maths.uwa.edu.au^*** Email: popiet01{at}maths.uwa.edu.au Riemannian cubics in tension in the rotation group SO(3) arevariational curves with applications to interpolation problemsin computer graphics and rigid-body trajectory planning. Theyare related by a linking equation to Lie quadratics in tension(LQT) in the Lie algebra so(3). This paper provides a qualitativeanalysis of the null case of LQT in so(3).     

A singular perturbation free boundary problem for elliptic equations in divergence form

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem , where A = A(x) is Holder continuous, β _ɛ converges to the Dirac delta δ₀. By studying some suitable level sets of u _ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C ^1,γ surface around a.e. point on the free boundary.     

Nonexcellence of the Function Field of the Product of Two Conics

Let k₀ be a field, k₀ ≠ 2, and α, β 2-fold Pfister forms over k₀. Denote by [α], [β] the classes of the corresponding quaternion algebras in ₂Brk₀, and by X_α, X_β the corresponding projective k₀-conics. Suppose ([α] + [β]) = 4. We construct a field F over k₀ such that the field extension F(X_α × X_β)/F is not excellent. Moreover, we find a 2-fold Pfister form γ over F such that ([α ] +[β ] + [γ]) = 4 and the homology group of the complex

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields η _a = (η _b )—with b denoting the nearest-neighbor edges in —that are distributed according to the Gibbs measure proportional to . Here H = ∑ _b V(η _b ) is the Hamiltonian, V is a symmetric potential, β > 0 is the inverse temperature, and ν is the Lebesgue measure on the linear space defined by imposing the loop condition for each plaquette (b ₁,b ₂,b ₃,b ₄) in . For convex V, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex V undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature β. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., E η _b = 0.     

A Volumetric Penrose Inequality for Conformally Flat Manifolds

We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to \mathbbRⁿ\ W, n 3 3{\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}, and so that their boundary is a minimal hypersurface. (Here, W ì \mathbbRⁿ{\Omega\subset \mathbb{R}^{n}} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by \frac12(V/b_n)^(n-2)/n{\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}, where V is the Euclidean volume of Ω and β _n is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost.     

Asymptotics for the Korteweg-de Vries-Burgers Equation

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut＋uux-uxx＋uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s （R）∩L^1 （R）, where s 〉 -1/2, then there exists a unique solution u （t, x） ∈C^∞ （（0,∞）;H^∞ （R）） to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u（t）=t^-1/2fM（（·）t^-1/2）＋0（t^-1/2） as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 （x） ∈ L^1 （R）, then the asymptotics are true u（t）=t^-1/2fM（（·）t^-1/2）＋O（t^-1/2-γ） where γ ∈ （0, 1/2）.     

The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

Asymptotics of degrees of someS n-sub regular representations

The numbers % MathType!End!2!1!, λ ⊢n appear in the enumeration of various objects, as well as coefficients inS _nrepresentations associated with products of higher commutators. We study their asymptotics asn→∞ and show that if (λ₁, λ₂, …)≈(α ₁,α ₂, …)n, if (λ′₁, λ′₂, …)≈(β ₁,β ₂, …)n and ifγ=1− Σ _k⩽1(α _k⩽1+β _k⩽1), then % MathType!End!2!1!. Work partially supported by N.S.F. Grant No. DMS 94-01197.     

Closed convex hull of the family of multivalently close-to-convex functions of order β

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ₀ ^k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .     

Polynomial Regression with Censored Data based on Preliminary Nonparametric Estimation

Consider the polynomial regression model , where σ²(X)=Var(Y|X) is unknown, and ε is independent of X and has zero mean. Suppose that Y is subject to random right censoring. A new estimation procedure for the parameters β₀,...,β _p is proposed, which extends the classical least squares procedure to censored data. The proposed method is inspired by the method of Buckley and James (1979, Biometrika, 66, 429–436), but is, unlike the latter method, a noniterative procedure due to nonparametric preliminary estimation of the conditional regression function. The asymptotic normality of the estimators is established. Simulations are carried out for both methods and they show that the proposed estimators have usually smaller variance and smaller mean squared error than the Buckley–James estimators. The two estimation procedures are also applied to a medical and an astronomical data set.     

A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models

Abstract Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ² V, where σ² is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X ₁ : X ₂) is an n×(p+q) known design matrix with rank(X) = r ≤ (p+q), and β = (β′ ₁: β′₂ )′ with β₁ and β₂ being p×1 and q×1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M ₂ X ₁β₁under the model and its best linear unbiased estimators under the reduced linear models of are given, where M ₂ = I -X ₂ X ₂ ⁺ . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M ₂ X ₁β₁ under the model and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model. *This work is supported by the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 10226024), Postdoctoral Foundation of China and Lab. of Math. for Nonlinear Sciences at Fudan University. This research is supported in part by The International Organizing Committee and The Local Organizing Committee at the University of Tampere for this Workshop **The work is supported in part by an NSF grant of China. Results in this paper were presented by the first author at The Eighth International Workshop on Matrices and Statistics: Tampere, Finland, August 1999     

Extensions of Vietoris’s inequalities I

Let β ₀=0.308443… denote the Littlewood-Salem-Izumi number, i.e., the unique solution of the equation