Maximizing a congruence with respect to its partition of idempotents

For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ). If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ) or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ).     

Images of non-tangential sectors under Cauchy transforms

Let μ be a complex measure on the circle. We denote byPμ andQμ the Poisson and conjugate Poisson integrals of μ in the unit disk, respectively. In this note, we study the relative asymptotic growth ofPμ andQμ near the support of μ. In particular, we show that on μ-almost every radius,Qμ grows no faster thanPμ. We also discuss related questions on the boundary behavior of analytic functions. The author is supported in part by N.S.F. grant DMS 9970151.     

Some remarks on estimation of parameters of uniform distribution

The best (in the sense of quadratic risk) unbiased estimators are constructed for the function f(x)=σ(2x/(n+1)−1)+μ from a sample of size n from the uniform distribution over [μ−σ, μ+σ] with unknown μ and σ. The best unbiased estimator for σ with μ being known is also presented. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 36–39, Perm, 1991.     

Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

Convergence of Sobolev spaces on varying manifolds

We deal with variational problems on varying manifolds in ℝⁿ. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝⁿ. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μ_h -a.e., being {μ_h} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (P_{μ h} H(μ_h)), where P_{μ h} are the projectors onto the tangent spaces T_{μ h}. When μ_h belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary.     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

Minuscule alcoves for GLn and GSp2n

Let μ be a minuscule coweight for either GL _n or GSp _2n , and regard μ as an element t _μ in the extended affine Weyl group . We say that an element is μ-admissible if there exists μ′ in the Weyl group orbit of μ such that x≤t _μ′ in the Bruhat order on . Our main result is that is μ-admissible if and only if it is μ-permissible, where μ-permissibility is defined using inequalities arising naturally in the study of bad reduction of Shimura varieties. Received: 5 July 1999     

An F. and M. Riesz Theorem on the Complex Sphere

Let μ be a measure on the complex sphere. Denote by μ_pq the projection of μ on H(p, q), the space of complex spherical harmonics. Assume that μ_pq = 0 if (p − 1)q ≠ 0, and ∥μ_1q∥_∞ → 0 as q → ∞. Then μ is absolutely continuous with respect to Lebesgue measure on the sphere.     

Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

Ising model in half-space: A series of phase transitions in low magnetic fields

For the Ising model in half-space at low temperatures and for the “unstable boundary condition,” we prove that for each value of the external magnetic field μ, there exists a spin layer of thickness q(μ) adjacent to the substrate such that the mean spin is close to −1 inside this layer and close to +1 outside it. As μ decreases, the thickness of the (−1)-spin layer changes jumpwise by unity at the points μ_q, and q(μ) → ∞ as μ → +0. At the discontinuity points μ_q of q(μ), two surface phases coexist. The surface free energy is piecewise analytic in the domain Re μ > 0 and at low temperatures. We consider the Ising model in half-space with an arbitrary external field in the zeroth layer and investigate the corresponding phase diagram. We prove Antonov’s rule and construct the equation of state in lower orders with the precision of x⁷, x = e^−7ɛ. In particular, with this precision, we find the points of coexistence of the phases 0, 1, 2 and the phases 0, 2, 3, where the phase numbers correspond to the height of the layer of unstable spins over the substrate. Dedicated to Roland L’vovich Dobrushin __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 220–261, November, 2007.     

Upper estimate of the cardinality of the set of knots generated by one-and two-dimensional braids

We give the upper estimate for the cardinality of the set Ω(n, μ) of knots generated by closed one- and two-dimensional braids with n generators of the irreducible length μ in the limit as n≫1 and μ≫1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 208–221, August, 1999.     

A two-parameter boundary problem for second order differential system

Summary This paper is concerned with second order differential systems involving two parameters with boundary conditions specified at three points. In particular, we consider the system y' = k(x, λ, μ)z, z' = -g(x, λ, μ)y, where k and g are real-valued junctions defined on X: a ≤ x ≤ c, L: L₁ < λ < L₂, and M: M₁ < μ < M₂. This system is studied together with the boundary conditions α(λ, μ)y(a) - β(λ, μ)z(a)=0, γ(λ, μ)y(b) - δ(λ, μ)z(b)=0, ε₁(μ)y(b) - φ₁(μ)z(b)=ε₂(μ)y(c) - φ₂(μ)z(c), where α, β, δ, γ, ε_i, φ_i, i=1, 2, are continuous functions of the parameters. This work establishes the existence of eigenvalue pairs for the boundary problem and the oscillatory behavior of the associated solutions. These results complement those previously obtained by the authors and B. D. Sleeman, where boundary conditions of the ? Sturm-Liouville ? type were studied. Entrata in Redazione il 5 dicembre 1977. The research for this paper was supported by a University College Reasearch Grant, University of Alabama in Birmingham.     

Spectral problems for pencils of polynomial matrices. Methods and algorithms. V

Methods and algorithms for the solution of spectral problems of singular and regular pencils D(λ, μ)=A(μ)-λB(μ) of polynomial matrices A(μ) and B(μ) are suggested (the separation of continuous and discrete spectra, the computation of points of a discrete spectrum with the corresponding, Jordan chains, the computation of minimal indices and a minimal basis of polynomial solutions, the computation of the determinant of a regular pencil). Bibliography: 13 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 26–70     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

The depth of ultraproducts of Boolean algebras

We show that if μ is a compact cardinal then the depth of ultraproducts of less than μ many Boolean algebras is at most μ plus the ultraproduct of the depths of those Boolean algebras. Received May 18, 2004; accepted in final form December 9, 2004.     

Further results on simultaneous confidence intervals for the normal distribution

Summary Based on a random sample from the normal cumulative distribution function ϕ(x; μ, σ) with unknown parameters μ and σ, one-sided confidence contours for ϕ(x; μ, σ), −∞<x<∞, and simultaneous confidence intervals for ϕ(y; μ, σ)−ϕ(x; μ, σ), −∞<x<y<∞, are constructed using the method outlined in [3]. Small sample and asymptotic distributions of the relevant statistics are provided so that the construction could be completely carried out in any practical situation.     

A nodal basis of Cμ-rational spline functions on triangulations

Let ρ be a triangulation of a polygonal domain D⊂R² with vertices V={v_i:l≤i≤N_v} and RS^k(D, ρ)={u∈C^k(D): ≠ T∈ρ, u/_T is a rational function}. The purpose of this paper is to study the existence and construction of C^μ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem H^μ(V,U)={find u∈U: D^αu(v_i)=D^αf(v_i),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RS^μ. this solution needs only the knowledge of partial derivatives of order≤μ at v_i. The explicit repesentations of all C^μ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.     

Random walks on almost connected locally compact groups: Boundary and convergence

We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL ^∞ (G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those homogeneous spaces in which the canonical projection of the random walk converges in probability.     

A geometric feature for finite element schemes

In this note we characterize the geometric feature of a (μ;r,k)—FES. Namely, for a C^μ triangular interpolation scheme with C^r vertex data, any angle of the macrotriangle must be divided into at least (μ+1)/(r+1−μ) parts.