More on Comonotone Polynomial Approximation

The main achievement of this paper is that we show, what was to us, a surprising conclusion, namely, twice continuously differentiable functions in (0,1) (with some regular behavior at the endpoints) which change monotonicity at least once in the interval, are approximable better by comonotone polynomials, than are such functions that are merely monotone. We obtain Jackson-type estimates for the comonotone polynomial approximation of such functions that are impossible to achieve for monotone approximation. July 7, 1998. Date revised: May 5, 1999. Date accepted: July 23, 1999.     

Dielectric constants of acetonitrile, γ-butyrolactone,propylene carbonate,and 1,2-dimethoxyethane as a function of pressure and temperature

The dielectric constants of 1,2-dimethoxyethane, acetonitrile, -butyrolactone, and propylene carbonate were determined from capacitance measurements extrapolated to infinite frequency; ln are reported as a function of pressure up to 80 MPa at 15, 25, 35, 45°C and as a function of temperature in the range 10 to 50°C at 0.10133 MPa. The variation of ln with temperature or pressure can be expressed by a second order polynomial expression. The isothermal compressibilities of the solvents were determined at 25°C from sound velocities, densities, and heat capacities. A simple correlation can be established between ln /P and for most aprotic solvent.     

First experimental evidence of potassium ordering in layered k4co7o14

The layered P2-K4Co7O14 oxide has been prepared and characterized by means of X-ray diffraction, electrical conductivity, thermopower, and magnetic measurements. The crystal structure of K4Co7O14 (P6(3)/m space group, Z=2, a=7.5171(1) A, and c=12.371(1) A) consists of a stacking of slabs of edge-shared CoO6 octahedra with K+ ions occupying ordered positions in the interslab space, leading to a a0 radical7xa0 radical7 supercell. Potential energy calculations at 0 K are in good agreement with the ordered distribution of potassium ions in the (ab) plane. This oxide is metallic, and the magnetic susceptibility is of Pauli-type, which contrasts with the Curie-Weiss behavior of the homologous NaxCoO2 (x approximately 0.6) oxide with close alkali content. The thermopower at room temperature is about one-third that of polycrystalline Na0.6CoO2.     

Crystallographic and magnetic structure of HAVAR under high-pressure using diamond anvil cell (DAC)

Annealed (H1) and cold-rolled (H2) HAVAR has been studied using high-pressure synchrotron X-ray diffraction. A structural phase transformation was discovered at ～13 GPa at ambient temperature, transforming from m ??3 m (S.G. 225) to P 63/m m c (S.G. 194) symmetry. The transition was not reversible on pressure release. The low-pressure cubic phase was found to be more compressible than the high-pressure hexagonal phase. Conventional Mössbauer and NFS shows that the HAVAR is not magnetic at room temperature and no splitting is observed. The SQUID indicates a huge difference in the temperature dependence of the magnetic susceptibility between the cold Rolled HAVAR compared to the annealed HAVAR.     

Simultaneous approximation by greedy algorithms

We study nonlinear m-term approximation with regard to a redundant dictionary $\mathcal {D}$ in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f∈H and any dictionary $\mathcal {D}$ an expansion into a series $$f=\sum_{j=1}^{\infty}c_{j}(f)\varphi_{j}(f),\quad\varphi_{j}(f)\in \mathcal {D},\ j=1,2,\ldots,$$ with the Parseval property: ‖f‖²=∑_j|c _j(f)|². Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f ¹,.?.?.,f ^N with a requirement that the dictionary elements φ_j of these expansions are the same for all f ⁱ, i=1,.?.?.,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.     

The degree of coconvex polynomial approximation

Let change its convexity finitely many times in the interval, say times, at . We estimate the degree of approximation of by polynomials of degree , which change convexity exactly at the points . We show that provided is sufficiently large, depending on the location of the points , the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of multiplied by a constant , which depends only on .