The near-best solution of a polynomial minimization problem by the Carathéodory-Fejér method

For a compact set C, 1, we consider the Chebyshev problem

The Exact Estimation of the Hermite-Fejér Interpolation

The exact pointwise estimation of the Hermite-Fejér interpolation process based on the zeros of the Jacobi polynomial $P^{(\alpha,\beta)}_n(x)(-1 ‹\alpha,\beta \leq 0)$ is given. The method employed is useful for other extended H-F interpolations also.     

The Spectral Carathéodory-Fejér Problem

The problem of the title is to construct an analytic k × k matrix-valued function in the unit disc with a number of prescribed derivatives at 0 and with spectral radius bounded by 1. We show that the problem can be reduced to an interpolation problem for the symmetrized polydisc , and thereby show that, in the case of derivatives of orders 0 and 1 being prescribed, the problem is equivalent to the infinitesimal Kobayashi extremal problem for , which is solved completely in the case k = 2.     

Pseudo-Taylor expansions and the Carathéodory–Fejér problem

We give a new solvability criterion for the boundary Carathéodory–Fejér problem: given a point $x\in \mathbb{R}$ and, a finite set of target values $a^0,a^1,\dots ,a^n\in \mathbb{C}$, to construct a function f in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z\to x$ nontangentially in the upper half-plane is $a^k$ for $k=0,1,\dots ,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.     

The Mean Convergence Order of Extended Hermite-Fejér Interpolation Operators

1.IntroductionThispaperwilldiscusstheweightedmeanconvergenceextendedHerrnite-Fej6rInterpola-tionbasedonthequasiHermite-FejerInterpolation.ThequasiHermite-FejerinterPolatingpolynomialisdefinedtouniquepolynomialofde-greeatmost2n l,satisfyingitcanbewrittenas[6jThezerosoforthogonalpolynomialp.(x)aredenotedbyxAnandtheyareindexedsothatX:l>xl.>xz.>..'>x->-l,n=1,2,....(1,7)In199oPETERVERTESIandYUANXU[6Jgivensomeverygoodresultsaboutmeancon-1ther,letu(x)beanintegrableJacobiweightfunction.If…     

The nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances

In this paper, we present research for the nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances. The analytical solutions obtained by the homotopy analysis method and the Adomian decomposition technique are in close agreement. The effects of involved parameters on the wave function are analyzed and discussed.     

On the analogue of a de la Vallée Poussin's problem for arcs in the complex plane

пУстьL — кОНЕЧНАь жОР ДАНОВА ДУгА, И тОЧкАz ₀?L (НЕ сОВпАДАУЩАь НИ с ОДНИ М Иж кОНцОВL) ДЕлИтL НА ДВЕ ЧАстИL′ ИL″'. пРИ НЕкОт ОРых ОгРАНИЧЕНИьх НА гЕОМ ЕтРИУ ДУгИ НАИДЕН пОРьДОк НАИлУ ЧшИх пРИБлИжЕНИИ МНО гОЧлЕНАМИ Дль ФУНкцИИ $$f(z) = f(z,z_0 ) = \left\{ {\begin{array}{*{20}c} {z - z_0 ,z \in L';} \\ {z_0 - z,z \in L''.} \\ \end{array} } \right.$$      

The degenerate analogue of Ariki’s categorification theorem

We explain how to deduce the degenerate analogue of Ariki’s categorification theorem over the ground field \mathbbC{\mathbb{C}} as an application of Schur–Weyl duality for higher levels and the Kazhdan–Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.     

The odd Littlewood–Richardson rule

In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via an odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood–Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson–Tao hive model.     

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.     

The optimal stopping problem concerned with ultimate maximum of a Lévy process

For a Lévy process X = (X _t )₀≤t<∞ we consider the time θ = inf{t ≥ 0: sup _s≤t X _s = sup _s≥0 X _s }. We study an optimal approximation of the time θ using the information available at the current instant. A Lévy process being a combination of a Brownian motion with a drift and a Poisson process is considered as an example.     

Diffusions réfléchies réversibles dégénérées

On a domain D in ^d, for a smooth enough probability density and a diffusion matrix which can degenerate, we construct the law Q ^s of a (x)d -symmetric reflecting process in D with matrix . Therefore, we use the associated Dirichlet form and a sequence of approximating processes already used by Pardoux and R. Williams in [23]. Under mild conditions on the boundary ofD (finite Minkowski content), we prove that Q ^s is the law of a semi-martingale and provide its decomposition. Comparing with the decomposition in additive functionals, we conclude that the process is reflected in the conormal direction ^* n where n denotes Chen's normal (cf [10]), that is, the reflection direction of the Brownian motion in Kuramochi compactification.     

An optimal reinsurance problem in the Cramér–Lundberg model

In this article we consider the surplus process of an insurance company within the Cramér–Lundberg framework with the intention of controlling its performance by means of dynamic reinsurance. Our aim is to find a general dynamic reinsurance strategy that maximizes the expected discounted surplus level integrated over time. Using analytical methods we identify the value function as a particular solution to the associated Hamilton–Jacobi–Bellman equation. This approach leads to an implementable numerical method for approximating the value function and optimal reinsurance strategy. Furthermore we give some examples illustrating the applicability of this method for proportional and XL-reinsurance treaties.     

The odd–even invariant for graphs

The odd–even invariant for graphs is the graphic version of the odd–even invariant for oriented matroids. Here, simple properties of this invariant are verified, and for certain graphs, including chordal graphs and complete bipartite graphs, its value is determined. The odd–even chromatic polynomial is introduced, its coefficients are briefly studied, and it is shown that the absolute value of this polynomial at −1 equals the odd–even invariant, in analogy with the usual chromatic polynomial and the number of acyclic orientations.     

The vanishing of a higher codimension analogue of Hochster’s theta invariant

We study H. Dao’s invariant ${\eta_c^R}$ of pairs of modules defined over a complete intersection ring R of codimension c having an isolated singularity. Our main result is that ${\eta_c^R}$ vanishes for all pairs of modules when R is a graded complete intersection ring of codimension c > 1 having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are c-Tor-rigid.     

The linear symmetries of Hill’s lunar problem

Archiv der Mathematik - A symmetry of a Hamiltonian system is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. For the planar and spatial Hill lunar problem, four...