Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A ₀ in a Hilbert space ₀ are defined formally as A ₍₎=A ₀+GG ^*, where G is an injective linear mapping from = ^d to the scale space _-k(A₀)k , kN, of generalized elements associated with the self-adjoint operator A ₀, and where is a self-adjoint operator in . The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A ₍₎ in the general setting ran G – _k (A ₀), kN, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.     

One-dimensional three-body problem via symbolic dynamics

Symbolic dynamics is applied to the one-dimensional three-body problem with equal masses. The sequence of binary collisions along an orbit is expressed as a symbol sequence of two symbols. Based on the time reversibility of the problem and numerical data, inadmissible (i.e., unrealizable) sequences of collisions are systematically found. A graph for the transitions among various regions in the Poincare section is constructed. This graph is used to find an infinite number of periodic sequences, which implies an infinity of periodic orbits other than those accompanying a simple periodic orbit called the Schubart orbit. Finally, under reasonable assumptions on inadmissible sequences, we prove that the set of admissible symbol sequences forms a Cantor set. (c) 2000 American Institute of Physics.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

A generative model and a generalized trust region Newton method for noise reduction

In practical applications related to, for instance, machine learning, data mining and pattern recognition, one is commonly dealing with noisy data lying near some low-dimensional manifold. A well-established tool for extracting the intrinsically low-dimensional structure from such data is principal component analysis (PCA). Due to the inherent limitations of this linear method, its extensions to extraction of nonlinear structures have attracted increasing research interest in recent years. Assuming a generative model for noisy data, we develop a probabilistic approach for separating the data-generating nonlinear functions from noise. We demonstrate that ridges of the marginal density induced by the model are viable estimators for the generating functions. For projecting a given point onto a ridge of its estimated marginal density, we develop a generalized trust region Newton method and prove its convergence to a ridge point. Accuracy of the model and computational efficiency of the projection method are assessed via numerical experiments where we utilize Gaussian kernels for nonparametric estimation of the underlying densities of the test datasets.     

Stability considerations of a high-temperature superconductor tape at different operating temperatures

Stability of a Bi-2223/Ag multifilamentary composite conductor against fast transport current ramps was studied by using a numerical model. The model was based on the two-dimensional magnetic diffusion and heat conduction equations. Calculations were carried out both in an adiabatic mode and pool boiling modes in liquid helium, hydrogen and nitrogen. When estimating the heat load (AC losses), real temperature dependent current density–electric field characteristics were used. The results computed by the finite element method are presented and discussed with special emphasis on differences of the stability considerations between high-temperature and low-temperature superconductors.     

Analysis of amino acids in cell culture supernatant using capillary electrophoresis with contactless conductivity detection

CE-C⁴D methods for the analysis of amino acids (AAs) are presented. Combining the results from two methods with acetic acid and cyclodextrin-based BGEs, 20 proteinogenic AAs could be analyzed using CE. CE-C⁴D was also, for the first time, applied to analyze free AAs in samples of mammalian cell culture supernatant. After dilution as only sample preparation, combining the results of the two CE methods allowed monitoring the concentration changes of 17 AAs in samples taken during the cultivation of CHO cells.     

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

Arabinoxylan structure affects the reinforcement of films by microfibrillated cellulose

The chemical structure of rye arabinoxylan (rAX) was systematically modified, exploiting selective enzymes to mimic different naturally occurring xylans, i.e., its degree of substitution (DS) was decreased using α-l-arabinofuranosidase, and a controlled decrease in the degree of polymerization (DP) was performed using endo-1,4-β-d-xylanase. The arabinose to xylose ratio was decreased from 0.45 to 0.27 and the weight-average molar mass was decreased from 184,000 to 49,000 g/mol. The resulting samples were used to prepare films, as such, and with 15% (wt. − %) softwood-derived microfibrillated cellulose (MFC) to obtain novel plant-derived biocomposite materials. The enzymatic tailoring of rAX increased the crystallinity of films, evidenced by X-ray diffraction studies, and the addition of MFC to the debranched, low DS rAX induced the formation of ordered structures visible with polarizing optical microscopy. MFC decreased the moisture uptake of films and increased the relative humidity of softening of the films, detected with moisture scanning dynamic mechanical analysis. For the first time, the chemical structure of xylan was proven to significantly affect the reinforcement potential of nano-sized cellulose, as the tensile strength of films from high DP rAXs, but not that of low DP rAXs, clearly increased with the addition of MFC. At the same time, MFC only increased the Young’s modulus of films from rAX with high arabinose content, regardless of DP.