Infinitely many bounded solutions for the p-Laplacian with nonlinear boundary conditions

By applying a method due to Saint Raymond, we prove the existence of infinitely many weak solutions for a quasilinear elliptic partial differential equation, involving the p-Laplacian operator, coupled with a nonlinear boundary condition. Our main assumption is a suitable oscillatory behaviour of the nonlinearity either at infinity or at zero.     

Preparation and examination of multilayer graphene nanosheets by exfoliation of graphite in high efficient attritor mill

Graphene multilayers have been prepared by mechanical method based on milling graphite in high efficient attritor mill. The results showed that the best dispersion media is ethanol, and 10 hours of intensive milling proved to be the most efficient way to separate the graphite layers as it was shown by scanning electron microscopy and X-ray diffraction measurements.      

Range Counting over Multidimensional Data Streams

We consider the problem of approximate range counting over a stream of d-dimensional points. In the data stream model the algorithm makes a single scan of the data, which is presented in an arbitrary order, and computes a compact summary data structure. The summary, whose size depends on the approximation parameter ε, can be used to count the number of points inside a query range within additive error εn, where n is the size of the stream seen so far. We present several results, deterministic and randomized, for both rectangle and halfspace ranges.     

On large minimal blocking sets in PG(2,q)

The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non‐square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non‐prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q^4/3 + 1 or q^4/3 + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non‐prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.     

Group factorisations,uniform automorphisms,and permutation groups of simple diagonal type

We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.     

Long Non-crossing Configurations in the Plane

We revisit some maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1995). Given a set of n points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching, (b) a Hamiltonian path, and (c) a spanning tree. We obtain some new results for (b) and (c), as well as for the Hamiltonian cycle problem.     

Vibrational Spectroscopic and Theoretical Studies of Urea Derivatives with Biochemical Interest: N,N′-Dimethylurea,N,N,N′,N′-Tetramethylurea,and N,N′-Dimethylpropyleneurea

Abstract

Mid-infrared, far-infrared, and Raman vibrational spectroscopic studies were combined with density functional theory (DFT) calculations and normal coordinate force field analyses for N,N′-dimethylurea (DMU), N,N,N′,N′-tetramethylurea (TMU), and N,N′-dimethylpropyleneurea (DMPU: IUPAC name 1,3-dimethyltetrahydropyrimidin-2(1H)-one). The equilibrium molecular geometry of DMU (all three conformers), TMU, and DMPU and the frequencies, intensities, and depolarization ratios of their fundamental infrared (IR) and Raman vibrational transitions were obtained by DFT calculations. The vibrational spectra were fully analyzed by normal coordinate methods as well. A starting force field for DMPU was obtained by adapting corresponding force constants for DMU and TMU, resulting after refinements in the stretching force constants C=O (7.69, 7.30, 7.68 N·cm^?1), C–N (5.16, 5.55, 5.05 N·cm^?1), and C-Me (5.93, 4.00, 4.22 N·cm^?1) for DMU, TMU, and DMPU, respectively. The dominating conformer of liquid DMU was identified as trans-trans, strong intermolecular hydrogen bonding was verified in solid DMU, and weak dipole–dipole association was found in liquid TMU and in DMPU. Special attention was paid to analyzing the methyl group frequencies, which revealed deviations from local C_3v symmetry. A linear correlation was found between the CH stretching force constants and the inverse of the CH bond lengths (1/r ²). The averaged NH stretching frequencies of gaseous, dissolved, and solid urea and of DMU, with variations for hydrogen bonding of different strength, are linearly correlated to the NH stretching force constants. Characteristic skeletal vibrations were assigned for a broad variety of urea derivatives and also for pyrimidine derivatives, which all contain the N₂C=O entity. The very strong IR bands of C=O stretching (1,676 ± 40 cm^?1) and asymmetric CN₂ stretching (1,478 ± 60 cm^?1), and the very intense Raman feature of symmetric CN₂ stretching or ring breathing (757 ± 80 cm^?1), can be recognized as fingerprint bands also for the pyrimidine derivatives cytosine, thymine, and uracil, which all are nucleobases in DNA and RNA nucleotides.     

Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , where ${L:X \rightarrow \mathbb R}$ is a sequentially weakly lower semicontinuous C ¹ functional, ${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional ${\mathcal{E}_{\lambda^\ast}}$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result.