Uniformly robust tests in errors-in-variables models

Consider an ordinary errors-in-variables model. The true level α _n (θ*) of a test at nominal level α and sample size n is said to be pointwise robust if α _n (θ*) → α as n → ∞ for each parameter θ*. Let Ω* be a set of values of θ*. Define α _n = sup _{θ* ∈Ω*}α _n (θ*). The test is said to be uniformly robust over Ω* if α _n → α as n → ∞. Corresponding definitions apply to the coverage probabilities of confidence sets. It is known that all existing large-sample tests for the parameters of the errors-in-variables model are pointwise robust. However, they might not be uniformly robust over certain null parameter spaces. In this paper, we construct uniformly robust tests for testing the vector coefficient parameter and vector slope parameter in the functional errors-in-variables model. These tests are established through constructing the confidence sets for the same parameters in the model with similar desirable property. Power comparisons based on simulation studies between the proposed tests and some existing tests in finite samples are also presented.     

A note on the distribution of the sequence (nα) with transcendental α

Let α be an irrational number and let D _N*(α) and D_N(α) denote the star-discrepancy and the discrepancy of the sequence (nα)_{n ≥1} mod 1, resp. We study properties of the maps α→ v *(α) = limsup_{N →∞} N D _N*(α)/log N and α→v(α) = limsup_{N →∞} N D _N(α)/log N where α is transcendental but not a U-number. This revised version was published online in June 2006 with corrections to the Cover Date.     

An analysis of a Monte Carlo algorithm for estimating the permanent

Karmarkar, Karp, Lipton, Lovász, and Luby proposed a Monte Carlo algorithm for approximating the permanent of a non-negativen×n matrix, which is based on an easily computed, unbiased estimator. It is not difficult to construct 0,1-matrices for which the variance of this estimator is very large, so that an exponential number of trials is necessary to obtain a reliable approximation that is within a constant factor of the correct value. Nevertheless, the same authors conjectured that for a random 0,1-matrix the variance of the estimator is typically small. The conjecture is shown to be true; indeed, for almost every 0,1-matrixA, just O(nw(n)e ^-2) trials suffice to obtain a reliable approximation to the permanent ofA within a factor 1±ɛ of the correct value. Here ω(n) is any function tending to infinity asn→∞. This result extends to random 0,1-matrices with density at leastn ^−1/2ω(n). It is also shown that polynomially many trials suffice to approximate the permanent of any dense 0,1-matrix, i.e., one in which every row- and column-sum is at least (1/2+α)n, for some constant α>0. The degree of the polynomial bounding the number of trials is a function of α, and increases as α→0. Supported by NSF grant CCR-9225008. The work described here was partly carried out while the author was visiting Princeton University as a guest of DIMACS (Center for Discrete Mathematics and Computer Science).     

Generalized entropy of the Heisenberg spin chain

We consider the XY quantum spin chain in a transverse magnetic field. We consider the Rényi entropy of a block of neighboring spins at zero temperature on an infinite lattice. The Rényi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the entropy of the large block in terms of Klein’s elliptic λ-function. We study the limit entropy as a function of its parameter α. We show that the Rényi entropy is essentially an automorphic function with respect to a certain subgroup of the modular group. Using this, we derive the transformation properties of the Rényi entropy under the map α → α ⁻¹ .     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

On the number of hyperforests

We consider m-forests with n labeled vertices and T edges. Asymptotic expressions for the number of such hyperforests are obtained as m is fixed and n, T→∞ in such a manner that either 0 < α₀ ≤T/n ≤ α₁ < 1/(m(m-1)), where α₀ and α₁ are some constants, or T/n→1/(m(m−1)) sufficiently fast. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem

Combinatorial Results for Semigroups of Order-Preserving Full Transformations

Let O_n be the semigroup of all order-preserving full transformations of a finite chain, say X_n = {1, 2, ..., n}, and for a given full transformation α: X_n → X_n let f(α) = |{x ∈ X_n: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → O_n: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → O_n: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(O_n), the set of idempotents of O_n.     

A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.     

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M _n (nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α _{n, s} denotes the s th zero of M _n (nα;δ, η) , counted from the right, and if α˜ _n,s denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α _n,s and α˜ _n,s as n→∞ . December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999.     

Admissible functions with multiple discontinuities

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ _i=1 ⁿ f(x+iα asn → α is resolved.     

Flexible Gabor-wavelet atomic decompositions for L 2-Sobolev spaces

In this paper we present a general construction of frames, which allows one to ensure that certain families of functions (atoms) obtained by a suitable combination of translation, modulation, and dilation will form Banach frames for the family of L²-Sobolev spaces on ℝ of any order. In this construction a parameter α∈[0,1) governs the dependence of the dilation factor on the frequency parameter. The well-known Gabor and wavelet frames (also valid for the same scale of Hilbert spaces) using suitable Schwartz functions as building blocks arise as special cases (α=0) and a limiting case (α→1), respectively. In contrast to those limiting cases, it is no longer possible to use group-theoretical arguments. Nevertheless, we will show how to constructively ensure that for Schwartz analyzing atoms and any sufficiently dense but discrete and well-structured family of parameters one can guarantee the frame property. As a consequence of this novel constructive technique, one can generate quasicoherent dual frames by an iterative algorithm. As will be shown in a subsequent paper, the new frames introduced here generate Banach frames for corresponding families of α-modulation spaces. Mathematics Subject Classification (2000) 42C15, 46S30, 49M27, 65T60     

Stefan problem

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H ^2+α,1+α/2.     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.     

SOLITON SOLUTIONS OF A CLASS OF GENERALIZED NONLINEAR SCHRODINGER EQUATIONS

Soliton solutions of a class of generalized nonlinear evolution equations are discussed ana-lytically and numerically. This is done by using a travelling wave method to formulate one-soliton solution and the finite difference method to the numerical solutions and the interactions betweenthe solitons for the generalized nonlinear Sehrodinger equations. the characteristic behavior of thenonlinearity admintted in the system has been investigated and the soliton states of the system in thelimit when a→Oand a→∞ have been studled. The results presented show that the soliton phe-rtomenon is charaeteristics associated with the nonlinearities of the dynamical systems.     

Asymptotic behavior of a multivariate stable law when the exponent tends to zero

The asymptotic behavior as α→0 of the function f(α)=P(0<Λ^α(X_α)<R) is studied, where R>0, X_α is a strictly stable random vector with exponent α and values in a Banach spaceB, and Λ is a non-negative measurable homogeneous functional onB. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

A consistency result on thin-very tall Boolean algebras

It was proved by Baumgartner and Shelah that Con (ZFC)→Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω₂”). In this paper we improve Baumgartner-Shelah’s theorem, showing that Con (ZFC)→Con (ZFC+“for every α<ω₃ there is a superatomic Boolean algebra of width ω and height α”). The preparation of this paper was supported by DGICYT Grant PB98-1231.     

Semilinear elliptic equations with uniform blow-up on the boundary

We prove the existence and the uniqueness of a solutionu of−Lu+h|u| ^α-1u=f in some open domain ℝ^d, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC ² compact hypersurface, lim _x→∂G (dist(x, ∂G))^2α/(α-1) f(x)=0, and lim _x→∂G u(x)=∞.