Homogeneous Hypersurfaces in ℂ<Superscript>3</Superscript>, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ³ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ³ with finite-dimensional infinitesimal automorphism algebra \mathfrakg\mathfrak{g} and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all \mathfrakg\mathfrak{g}-homogeneous hypersurfaces in ℂ³ and also classify the corresponding local transitive actions of the model algebra \mathfrakg\mathfrak{g} on hypersurfaces in ℂ³.     

A smoothing Newton-type method for solving the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> spectral estimation problem with lower and upper bounds

This paper discusses the L ₂ spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.     

Fractional harmonic maps into manifolds in odd dimension n > 1

In this paper we consider critical points of the following nonlocal energy $$\begin{array}{ll}{\mathcal{L}}_n(u) = \int_{{I\!\!R}^n}| ({-\Delta})^{n/4} u(x)|^2 dx, \qquad(1)\end{array}$$ where ${u \in \dot{H}^{n/2}({I\!\!R}^n,{\mathcal{N}}), {\mathcal{N}} \subset {I\!\!R}^m}$ is a compact k dimensional smooth manifold without boundary and n > 1 is an odd integer. Such critical points are called n/2-harmonic maps into ${{\mathcal{N}}}$ . We prove that ${(-\Delta) ^{n/4} u\in L^p_{loc}({I\!\!R}^n)}$ for every p ≥  1 and thus ${u \in C^{0,\alpha}_{loc}({I\!\!R}^n)}$ , for every 0 < α < 1. The local Hölder continuity of n/2-harmonic maps is based on regularity results obtained in [4] for nonlocal Schrödinger systems with an antisymmetric potential and on some new 3-terms commutators estimates.     

Schur positivity and the <Emphasis Type="Italic">q</Emphasis>-log-convexity of the Narayana polynomials

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N _q (n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.     

On the inapproximability of <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">K</Emphasis>: why two moments of job size distribution are not enough

The M/G/K queueing system is one of the oldest models for multiserver systems and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (nonnumerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio, which we refer to as “the gap.” This is the first such result in the literature to address “the gap.” The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.     

Strong Normalizability of Typed Lambda-Calculi for Substructural Logics

The strong normalization theorem is uniformly proved for typed λ-calculi for a wide range of substructural logics with or without strong negation. We would like to thank the referees for their valuable comments and suggestions. This research was supported by the Alexander von Humboldt Foundation. The second author is grateful to the Foundation for providing excellent working conditions and generous support of this research. This work was also supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 20700015, 2008.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.     

Retracts in Polydisk and Analytic Varieties with the <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-Extension Property

This article mainly concerns retracts in polydisk, analytic varieties with the H ^∞-extension property and the three-point Pick problem on . Arising in the study of Nevanlinna-Pick interpolation on the bidisk, Agler and McCarthy recently discovered a remarkable theorem which characterizes subsets in the bidisk with the polynomial extension property, and in this case, these subsets are retracts. To study H ^∞-extensions of holomorphic functions from subvarieties of polydisk, one naturally is concerned with retracts in polydisk. Under certain mild assumptions, it is shown that subvarieties with H ^∞-extension property are exactly retracts. Furthermore, we apply our argument to determine those retracts whose retractions are unique. In particular, a retract in having at least two different retractions is exactly a balanced disk. As an application, we give a sufficient condition of the uniqueness of the solution for the three-point Pick problem on .      

Role of Relative <Emphasis Type="Italic">A</Emphasis>-Maximal Monotonicity in Overrelaxed Proximal-Point Algorithms with Applications

A general framework for a class of overrelaxed proximal point algorithms based on the notion of relative A-maximal monotonicity is introduced; then, the convergence analysis for solving a general class of nonlinear variational inclusion problems is explored. The framework developed in this communication is quite suitable, unlike other existing notions of generalized maximal monotonicity, including A-maximal (m)-relaxed monotonicity in literature, to generalize first-order nonlinear evolution equations/evolution inclusions based on the generalized nonlinear Yosida approximations in Hilbert spaces as well as in Banach spaces.     

Asset pricing and portfolio selection based on the multivariate extended skew-Student-<Emphasis Type="Italic">t</Emphasis> distribution

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar mean-variance frontier, and which may be implemented using quadratic programming.     

On the power of standard information for <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> approximation in the randomized setting

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L _∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n ⁻¹ when n function values are used.     

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both static (buckling) and dynamic (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions, we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires.     

Oscillation and Asymptotic Behavior for <Emphasis Type="Italic">n</Emphasis>th-order Nonlinear Neutral Delay Dynamic Equations on Time Scales

In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization

We propose and analyze a perturbed version of the classical Josephy–Newton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods. For the linearly constrained Lagrangian methods, in particular, we obtain superlinear convergence under the second-order sufficient optimality condition and the strict Mangasarian–Fromovitz constraint qualification, while previous results in the literature assume (in addition to second-order sufficiency) the stronger linear independence constraint qualification as well as the strict complementarity condition. For the sequential quadratically constrained quadratic programming methods, we prove primal-dual superlinear/quadratic convergence under the same assumptions as above, which also gives a new result.     

Signed words and permutations, V; a sextuple distribution

We calculate the distribution of the sextuple statistic over the hyperoctahedral group B _n that involves the flag-excedance and flag-descent numbers “fexc” and “fdes,” the flag-major index “fmaj,” the positive and negative fixed point numbers “ ” and “ ” and the negative letter number “neg.” Several specializations are considered. In particular, the joint distribution for the pair is explicitly derived.