Lattice points in a circle and divisors in arithmetic progressions

Let A(x) denote the number of lattice points in the circle u²+v²x and define as the infimum of all reals for which . The objective of this paper is to show that 35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.     

Higher order derivative tests for exponential sums incorporating the Discrete Hardy–Littlewood method

This paper provides estimates for exponential sums, combining classic tools of Van der Corput type with a deep result from the modern “discrete Hardy–Littlewood method”. As an application, an improved bound for the lattice point discrepancy of a large ellipsoid of rotation is deduced.     

Minimal surfaces with prescribed topological type on a Schwarzian chain inMc3

Similar to the investigations of unstable polygonal minimal surfaces by Courant [1] we introduce here a variational principle for the free boundary problem with prescribed topological type which produces minimal surfaces in Riemannian manifolds with constant curvature. For special boundary configurations the surfaces have no branch points. The approach can be applied to numerical algorithms since it is constructive.     

Conditional weak compactness in vector-valued function spaces

Let be an ideal of over a -finite measure space and let be the Köthe dual of with . Let be a real Banach space, and the topological dual of . Let be a subspace of the space of equivalence classes of strongly measurable functions and consisting of all those for which the scalar function belongs to . For a subset of for which the set is -bounded the following statement is equivalent to conditional -compactness: the set is conditionally -compact and is a conditionally weakly compact subset of for each , with . Applications to Orlicz-Bochner spaces are given.

     

Preparation and dielectric properties of the lead-free BaFe<Subscript>1/2</Subscript>Nb<Subscript>1/2</Subscript>O<Subscript>3</Subscript> ceramics obtained from mechanically triggered powder

In the paper the influence of mechanical activation of the powder on the final dielectric properties lead-free Ba(Fe_1/2Nb_1/2)O₃ (BFN) ceramic was examined. The BFN ceramics were obtained by 3-steps route. Firstly, the substrates were pre-homogenized in a planetary ball mill. Than, the powder was activated in vibratory mill (the shaker type SPEX 8000 Mixer Mill) for different duration between 25 h and 100 h. The influence of the milling time on the BFN powder was monitored by X-ray diffraction. The diffraction data confirmed that the milling process of the starting components is accompanied by partial synthesis of the BFN materials. The longer of the high-energy milling duration the powders results in increasing the amount of amorphous/nanocrystalline content. The mechanically activated materials were sintered in order to obtain the ceramic samples. During this temperature treatment the final crystallisation of the powder appeared what was confirmed by XRD studies. The performed dielectric measurements have revealed the reduction of the dielectric loss of the BFN ceramics compared to materials obtained by classic methods.     

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.      

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

The lipidated membrane anchor of full length N-Ras protein shows an extensive dynamics as revealed by solid-state NMR spectroscopy

Many proteins involved in signal transduction are equipped with covalently attached lipid chains providing a hydrophobic anchor targeting these molecules to membranes. Despite the considerable biological significance of this membrane binding mechanism for 5-10% of all cellular proteins, to date very little is known about structural and dynamical features of lipidated membrane binding domains. Here we report the first comprehensive study of the molecular dynamics of the C-terminus of membrane-associated full-length lipidated Ras protein determined by solid-state NMR. Fully functional lipid-modified N-Ras protein was obtained by chemical-biological synthesis ligating the expressed water soluble N-terminus with a chemically synthesized (2)H or (13)C labeled lipidated heptapeptide. Dynamical parameters for the lipid chain modification at Cys 181 were determined from static (2)H NMR order parameter and relaxation measurements. Order parameters describing the amplitude of motion in the protein backbone and the side chain were determined from site-specific measurements of (1)H-(13)C dipolar couplings for all seven amino acids in the membrane anchor of Ras. Finally, the correlation times of motion were determined from temperature dependent relaxation time measurements and analyzed using a modified Lipari Szabo approach. Overall, the C-terminus of Ras shows a versatile dynamics with segmental fluctuations and axially symmetric overall motions on the membrane surface. In particular, the lipid chain modifications are highly flexible in the membrane.     

Trifluoromethylation of alkenyl bromides and iodides (including 5-iodouracils) with (CF3)2Hg and Cu ("trifluoromethylcopper")

Bromo- and iodoalkenes undergo trifluoromethylation efficiently in DMA with "CF3Cu" generated from (CF3)2Hg and Cu. Variable stereochemical inversion is observed with substrates having a gem-carbonyl group. Substrates having gem-hydrogen, -alkyl, or -alkenyl groups give products with stereochemical retention.     

New evidence on the structure of potassium salts of 12-tungstophosphoric acid, KxH3-xPW12O40

Physicochemical properties and compositions of KxH(3-x)PW12O40 salts, where 2 < or = x < or = 3, have been investigated. It has been found that freshly prepared K2HPW12O40 salt (drying at 313 K) contains particles of heteropolyacid and particles of the neutral potassium salt, the sample being in 78.6% amorphous. On aging at room temperature, the heteropolyacid spreads to form a surface layer covering the neutral potassium salt particles K3PW12O40. Heat treatment of KxH(3-x)PW12O40 salts, where 2 < or = x < 3, from 313 K to higher temperatures induces the transformation of the heteropolyacid-covering K(3) core into a well-dispersed, amorphous surface layer. On further heating of the acidic potassium salts, the surface layer decomposes between 855 and 915 K with the formation of a PW8O26-type bronze as a new phase, the K3PW12O40 salt remaining unchanged. The latter starts to decompose at 1093 K, and in the case of all samples, the process is completed at about 1183 K. Rietveld structure refinement, XPS, and 31P NMR measurements of acidic potassium salts indicate that the core of these salts is always formed by the K3PW12O40 salt, which is covered by a heteropolyacid. Comparison of lattice parameters of the K3 salt and HPW leads to the conclusion that the layer is composed of partially or completely dehydrated heteropolyacid molecules. The coverage of the core by HPW in the K2 sample was estimated to be equal to one monolayer.