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).     

MU-Estimation and Smoothing

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449–1458), the parameter under estimation is the value of θ which minimizes the expectation of what is called a discrepancy measure (DM) δ(X, θ) which is a function of θ and the underlying random variable X. Such a setting does not cover the estimation of parameters such as the multivariate median defined by Oja (1983) and Liu (1990), as the value of θ which minimizes the expectation of a DM of the type δ(X₁, …, X_m, θ) where X₁, …, X_m are independent copies of the underlying random variable X. Arcones et al. (1994, Ann. Statist.22, 1460–1477) studied the estimation of such parameters. We call such an M-type MU-estimation (or μ-estimation for convenience). When a DM is not a differentiable function of θ, some complexities arise in studying the properties of estimators as well as in their computation. In such a case, we introduce a new method of smoothing the DM with a kernel function and using it in estimation. It is seen that smoothing allows us to develop an elegant approach to the study of asymptotic properties and possibly apply the Newton–Raphson procedure in the computation of estimators.     

Interpolation of characteristic classes of singular hypersurfaces

We show that the Chern–Schwartz–MacPherson class of a hypersurface X in a nonsingular variety M ‘interpolates’ between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern–Schwartz–MacPherson class of such ‘nice’ hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern–Schwartz–MacPherson class of X in terms of its polar classes.     

A Kernel Estimator of a Conditional Quantile

Let (X₁, Y₁), (X₂, Y₂), …, be two-dimensional random vectors which are independent and distributed as (X, Y). For 0<p<1, letξ(px) be the conditionalpth quantile ofYgivenX=x; that is,ξ(px)=inf{y : P(YyX=x)p}. We consider the problem of estimatingξ(px) from the data (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n). In this paper, a new kernel estimator ofξ(px) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained.     

Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X_1:n,X_2:n,…,X_n:n the corresponding order statistics. When X₁,X₂,…,X_n represent the lifetimes of n components in a system, the order statistic X_n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X_i:n and X_j:n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X₁,X₂,…,X_n are independent and identically distributed or when they jointly have an exchangeable distribution.     

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r_k}_k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r_k}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.     

Limit cycles of quadratic systems

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our canonical system, we prove in a different way that a quadratic system has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

In this paper we give a classification of a family of symmetric graphs with complete 2-arc-transitive quotients. Of particular interest are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line, and thus may be called the second type ‘cross ratio graphs’, which are different from the ‘cross ratio graphs’ studied in [A. Gardiner, C. E. Praeger, S. Zhou, Cross-ratio graphs, J. London Math. Soc. (2) 64 (2001), 257–272]. We also give a combinatorial characterisation of such second type cross ratio graphs.     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

Szegö theorem,Carathéodory domains,and boundedness of calculating functionals

Convex Chebyshev sets M in a linear space X with norm or nonsymmetric norm ‖ · ‖ which are contained in a subspace H of X are considered. It is proved that if | { |_{H, φ} is the nonsymmetric norm on H determined by the Minkowski functional of , where B is the unit ball of X and ‖φ‖, with respect to 0, then M is a Chebyshev set in for any φ. From this result sufficient and necessary conditions for the convexity of Chebyshev sets and bounded Chebyshev sets contained in a subspace H of X are derived.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 3–15.Original Russian Text Copyright © 2005 by A. R. Alimov.     

More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

For a sample of iid observations {(X_i, Y_i)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y_[]=(Y_[1], Y_[2], …, Y_[n]) and the stochastic order of subsets of Y_[] are studied. If (X, Y) is totally positive dependent of order 2, Y_[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, h_YX(yx), is decreasing in x for every y, Y_[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order.     

One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x ₀ ∈ X and y ₀ ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x ₀, y ₀)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x ₀ and y ₀, respectively.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.     

Spectral Density for Third Order Cumulants under Strong Mixing Conditions

If X{X_v: v ^d} is a strictly stationary random field, with X₀ bounded and expressible as a sum of indicator functions satisfying certain conditions, if the mixing coefficient α(s) is summable over ^d (that, is, ∑_m m^d−1α(m)<∞), and if a mixing condition involving three sets is satisfied, then the third order cumulant Cum(X_a, X_b, X_c) of X has a continuous spectral density. We do not begin with the assumption that the cumulants are absolutely summable.     

Interpolation of Intersections Generated by a Linear Functional

Let (X ₀, X ₁) be a Banach couple such that X ₀ ∩ X ₁ is dense in X ₀ and X ₁. By (X ₀, X ₁)_θ,q , 0 < θ < 1, 1 ⩽ q < ∞, we denote the spaces of the real interpolation method. Let ψ be a nonzero linear functional defined on some linear space M ⊂ X ₀ + X ₁ and such that ψ ∈ (X ₀ ∩ X ₁)*, and let N = Ker ψ. We examine conditions under which the natural formula

Pseudo-Index of Fano Manifolds and Smooth Blow-Ups

Suppose π: X → Y is a smooth blow-up along a submanifold Z of Y between complex Fano manifolds X and Y of pseudo-indices i_X and i_Y respectively (recall that i_X is defined by i_X :=min {−K_X·C | C is a rational curve of X}). We prove that if 2 dim (Z) < dim (Y)+i_Y −1 and show that this result is optimal by classifying the ‘boundary’ cases. As expected, these results are obtained by studying rational curves on X and Y.     

Whitney constants and approximation of -quasi-linear forms by -linear forms

We discuss some relations between Whitney constants w_m(B_X,Y) for bounded functions from, the unit ball of a real normed space X into another real normed space Y. In particular, we generalize a result of Tsar’kov that

to any n-dimensional X (here denotes linearized Whitney constant).     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

The blue and minque in Gauss-Markoff model with linear transformation of the observable variables

For a singular linear model A = (y, Xβ, σ2 V) and its transformed model AF = (Fy, FXβ, σ2FVF'), where V is nonnegative definite and X can be rank-deficient,the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala (1981) are strengthened and consequences in Puntanen and Nurhonen (1992), and Puntanen (1996) are extended.     

Uniqueness of Minimal Projections in Smooth Matrix Spaces

Let X=(M(n, m), ·), where · fulfills Condition 0.3 and W=M(n, 1)+M(1, m). A formula for a minimal projection from X onto W is given in (E. W. Cheney and W. A. Light, 1985, “Approximation Theory in Tensor Product Spaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin; E. J. Halton and W. A. Light, 1985, Math. Proc. Cambridge Philos. Soc.97, 127–136; and W. A. Light, 1986, Math. Z.191, 633–643). We will show that this projection is the unique minimal projection (see Theorem 2.1).