k-NN METHOD IN PARTIAL LINEAR MODEL UNDER RANDOM CENSORSHIP

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Regelflächenscharen

Let K be any commutative field and V:=K ⁴. A collection of ruled quadrics in V is called a flock of ruled quadrics if the following holds true. (1) ⋃_{ℱ∈ G} ℱ = V; (2) There is a line S⊂V such that ℱ₁⋂ℱ₂= S for all distinct ℱ₁, ℱ₂∈ . The group ΓL(V) decomposes the set of all those flocks into equivalence classes. Besides that, we consider any cone R in V, say R:= {x∈V|x ₁ x ₃ - x ₂ ² = 0}. Let R denote the set of all regular points of R. Plane sections of R which do not contain the singular point of ℜ are called regular sections. We consider decompositions of R ^* by regular sections and their equivalence classes with respect to the symmetry group ΓL(V)R of the cone ℜ. The main result is as follows. There is a (natural) bijection between the classes of equivalent flocks of rules quadrics and the classes of equivalent decompositions of R ^* by regular sections. A brief discussion of those flocks of ruled quadrics on which the construction of the so-called Betten-Walker planes is based ends the paper. Provided that char K≠3, these planes exist if and only if x∈K→x ³∈K is bijective.      

Gumbel statistics for the longest interval of identical spins in a one-dimensional Gibbs measure

We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}^ℤ. For N ∈ ℕ let l _N denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l _c (N) of l _N converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ, lim _{N →∞} ℙ(l _c (N) ≤ α log N + βx + γ) = e ^{–e –x} . Received: 2 September 2002     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.     

On 𝒪ℒ∞ structures of nuclear C * -algebras

 We study the local operator space structure of nuclear C ^* -algebras. It is shown that a C ^* -algebra is nuclear if and only if it is an 𝒪ℒ_∞,λ space for some (and actually for every) λ>6. The 𝒪ℒ_∞ constant λ provides an interesting invariant

Cycles in Partially Square Graphs

. In this work we consider finite undirected simple graphs. If G=(V,E) is a graph we denote by α(G) the stability number of G. For any vertex x let N[x] be the union of x and the neighborhood N(x). For each pair of vertices ab of G we associate the set J(a,b) as follows. J(a,b)={u∈N[a]∩N[b]∣N(u)⊆N[a]∪N[b]}. Given a graph G, its partially squareG ^* is the graph obtained by adding an edge uv for each pair u,v of vertices of G at distance 2 whenever J(u,v) is not empty. In the case G is a claw-free graph, G ^* is equal to G ². If G is k-connected, we cover the vertices of G by at most ⌈α(G ^*)/k⌉ cycles, where α(G ^*) is the stability number of the partially square graph of G. On the other hand we consider in G ^* conditions on the sum of the degrees. Let G be any 2-connected graph and t be any integer (t≥2). If ∑ _{x ∈ S} deg _G (x)≥|G|, for every t-stable set S⊆V(G) of G ^* then the vertex set of G can be covered with t−1 cycles. Different corollaries on covering by paths are given. Received: January 22, 1997 Final version received: February 15, 2000     

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution*

Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function Φ _N,β on ℝ ^M . In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m ^* (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 < γ _a < γ _c such that (1) If √α≦γ _a (m ^* (β )) ² , then the absolute minima of Φ are located within small balls around the points ± m ^* e ^μ , where e ^μ denotes the μ-th unit vector while (2) if √α≦γ _c (m ^* (β )) ² at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. Received: 5 August 1995 / In revised form: 22 May 1996     

Nonparametric analysis of doubly truncated data

One of the principal goals of the quasar investigations is to study luminosity evolution. A convenient one-parameter model for luminosity says that the expected log luminosity, T*, increases linearly as θ ₀· log(1  +  Z*), and T*(θ ₀) = T*  −  θ ₀· log(1  +  Z*) is independent of Z*, where Z* is the redshift of a quasar and θ ₀ is the true value of evolution parameter. Due to experimental constraints, the distribution of T* is doubly truncated to an interval (U*, V*) depending on Z*, i.e., a quadruple (T*, Z*, U*, V*) is observable only when U* ≤ T* ≤ V*. Under the one-parameter model, T*(θ ₀) is independent of (U*(θ ₀), V*(θ ₀)), where U*(θ ₀) = U*  −  θ ₀· log(1  +  Z*) and V*(θ ₀) = V*  −  θ ₀· log(1  +  Z*). Under this assumption, the nonparametric maximum likelihood estimate (NPMLE) of the hazard function of T*(θ ₀) (denoted by ĥ) was developed by Efron and Petrosian (J Am Stat Assoc 94:824–834, 1999). In this note, we present an alternative derivation of ĥ. Besides, the NPMLE of distribution function of T*(θ ₀), [^(F)]{\hat F} , will be derived through an inverse-probability-weighted (IPW) approach. Based on Theorem 3.1 of Van der Laan (1996), we prove the consistency and asymptotic normality of the NPMLE [^(F)]{\hat F} under certain condition. For testing the null hypothesis H_q0: T^*(q₀) = T^*-q₀·log(1 + Z^*){H_{\theta_0}: T^{\ast}(\theta_0) = T^{\ast}-\theta_0\cdot \log(1 + Z^{\ast})} is independent of Z*, (Efron and Petrosian in J Am Stat Assoc 94:824–834, 1999). proposed a truncated version of the Kendall’s tau statistic. However, when T* is exponential distributed, the testing procedure is futile. To circumvent this difficulty, a modified testing procedure is proposed. Simulations show that the proposed test works adequately for moderate sample size.     

On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.     

Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t → ±∞, to two positive limits l _* and l ^*, and never goes below min(l _*, l ^*). Furthermore, the convergence to max(l _*, l ^*) is supposed to be not too fast (slower than a suitable exponential term).     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

Hamiltonism and Partially Square Graphs

 Given a graph G, we define its partially square graph G ^* as the graph obtained by adding edges uv whenever the vertices u and v have a common neighbor x satisfying the condition N _G[x]⊆N _G[u]∪N _G [v], where N _G[x]=N _G(x)∪{x}. In particular, this condition is satisfied if x does not center a claw (an induced K _1,3). Obviously G⊆G ^*⊆G ², where G ² is the square of G. We prove that a k-connected graph (k≥2) G is hamiltonian if the independence number α(G ^*) of G ^* does not exceed k. If we replace G ^* by G we get a well known result of Chvátal and Erdo?s. If G is claw-free and G ^* is replaced by G ² then we obtain a result of Ainouche, Broersma and Veldman. Relationships between connectivity of G and independence number of G ^* for other hamiltonian properties are also given in this paper. Received: June 17, 1996 Revised: October 30, 1998     

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ^∞(G)^* the conjugate ofl ^∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ^∞(G)^* such that γμ=0 whenever γ is an annihilator ofC ₀(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ^∞(G)^*, and each of them generates a left ideal ofl ^∞(G)^* that contains no minimal left ideal.     

Asymptotic analysis and scaling of friction parameters

We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale collinear faults periodically disposed. If β₀^* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic problem behaves as β₀^*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result is that the macroscopic critical slip D_c scales with D_c^∈/∈ (here D_c^∈ is the small scale critical slip).     

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.