Existence of solutions for p(x)-Laplacian problems on a bounded domain

In this paper we study the following p(x)-Laplacian problem: -div(a(x)&VERBAR;&DEL; u&VERBAR;(p(x)-2)&DEL; u)+b(x)&VERBAR; u&VERBAR;(p(x)-2)u = f(x, u), x ε &UOmega;, u = 0, on &PARTIAL; &UOmega;, where 1< p(1) &LE; p(x) &LE; p(2) < n, &UOmega; &SUB; R-n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W-0(1,p(x)) for the p(x)-Laplacian problems in the superlinear and sublinear cases. © 2004 Elsevier Inc. All rights reserved.

Nucleophilic attacks of 1,2-diaminoethane on MeCN ligands: synthesis, x-ray structure, and spectral and electrochemical properties of [Ru(.mu.-O)(.mu.-O2CAr)2[NH2CH2CH2NHC(Me)NH]2(PPh3)2](ClO4)2(Ar = C6H4-p-X; X = H, Me, OMe, Cl)

The diruthenium(III) complex [Ru2O(O2CAr)2(MeCN)4(PPh3)2](ClO4)2 (1), on reaction with 1,2-diaminoethane (en) in MeOH at 25-degrees-C, undergoes nucleophilic attacks at the carbon of two facial MeCN ligands to form [(Ru2O)-O-III(O2CAr)2-{NH2CH2CH2NHC(Me)NH}2(PPh3)2](ClO4)2 (2) (Ar = C6H4-p-X, X = H, Me, OMe, Cl) containing two seven-membered amino-amidine chelating ligands. The molecular structure of 2 with Ar = C6H4-p-OMe was determined by X-ray crystallography. Crystal data are as follows: triclinic, P1BAR, a = 13.942 (5) angstrom, b = 14.528 (2) angstrom, c = 21.758 (6) angstrom, alpha = 109.50 (2)-degrees, beta = 92.52 (3)-degrees, gamma = 112.61 (2)-degrees, V = 3759 (2) angstrom 3, and Z = 2. The complex has an {Ru2(mu-O)(mu-O2CAr2)2(2+)} core. The Ru-Ru and average Ru-O(oxo) distances and the Ru-O-Ru angle are 3.280 (2) angstrom, 1.887 [8] angstrom, and 120.7 (4)-degrees, respectively. The amino group of the chelating ligand is trans to the mu-oxo ligand. The nucleophilic attacks take place on the MeCN ligands cis to the mu-oxo ligand. The visible spectra of 2 in CHCl3 display an absorption band at 565 nm. The H-1 NMR spectra of 2 in CDCl3 are indicative of the formation of an amino-amidine ligand. Complex 2 exhibits metal-centered quasireversible one-electron oxidation and reduction processes in the potential ranges +0.9 to +1.0 V and -0.3 to -0.5 V (vs SCE), respectively, involving the Ru(III)2/Ru(III)Ru(IV) and Ru(III)2/Ru(II)Ru(III) redox couples in CH2Cl2 containing 0.1 M TBAP. The mechanistic aspects of the nucleophilic reaction are discussed.

Boundary of the rauzy fractal sets in R x C generated by P(x)=x(4)-x(3)-x(2)-x-1

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Boundary of the rauzy fractal sets in ℝ×ℂgenerated by p(x) = x 4- x 3 - x 2 -x- 1

We study the boundary of the 3-dimensional Rauzy fractal ε ⊂ ℝ×ℂ generated by the polynomial P(x) Dx 4-x 3-x 2-x-1. The finite automaton characterizing the boundary of ε is given explicitly. As a consequence we prove that the set ε has 18 neighboors where 6 of them intersect the central tile ε in a point. Our construction shows that the boundary is generated by an iterated function system starting with 2 compact sets.

An asymmetric homodinuclear complex containing the mu-x-oxobis(mu-x-carboxylato)diruthenium(III) core: X-ray structure of [Ru20(o2ec6H4-p-OMe)3(en) (PPh3)2] (C104)" 3CHC13

Asymmetric tri-bridged diruthenium(III) complexes, [Ru2O(O(2)CR)(3)(en) (PPh(3))(2)](ClO4) (R = C6H4-p-X: X = OMe (1a), Me (1b); en=1,2-diaminoethane), were prepared and structurally characterized. Complex 1a 3CHCl(3), crystallizes in the triclinic space group P (1) over bar with a = 14.029(5), b = 14.205(5), c = 20.610(6) Angstrom, alpha= 107.26(3), beta = 101.84(3), gamma= 97.57(3)degrees, V= 3756(2) Angstrom(3) and Z = 2. The complex has an {Ru-2(mu-O)(mu-O(2)CR)(2)(2+)} core and exhibits [O4PRu(mu-O)RuPO2N2](+) coordination environments for the metal centers. The novel structural feature is the asymmetric arrangement of ligands at the terminal sites of the core which shows an Ru... Ru separation of 3.226(3) Angstrom and an Ru-O-Ru angle of 119.2(5)degrees. An intense visible band observed near 570 nm is assigned to a charge transfer transition involving the d pi-Ru(III) and p pi-mu-O Orbitals. Cyclic voltammetry of the complexes displays a reversible Ru-2(III,III) reversible arrow Ru-2(III,IV) couple near 0.8 V (versus SCE) in MeCN-0.1 M TBAP.

Axial binding of 2-methylimidazole to the tetraarylcarboxylatodiruthenium (II, III) core: X-ray crystal structure of [Ru-2(O2CC6H4-p-OMe)(4)(2-mimH)(2)](ClO4). 1.75CH(2)Cl(2).H2O

Diruthenium (II. III) complexes of the type [Ru-2(O2CAr)(4) (2-mimH)(2)](ClO4) (Ar = C6H4-p-X : X=OMe,1, X=Me, 2, 2-mimH=2-methylimidazole) have been isolated from the reaction of Ru2Cl(O2CAr)(4) with 2-mimH in CH2Cl2 followed by the addition of NaClO4. The crystal structure of 1.1.75CH(2)Cl(2).H2O has been determined. The crystal belongs to the monoclinic space group p2(1)/c with the following unit cell dimensions for the C40H40N4O16ClRu2.1.75CH(2)Cl(2).H2O (M = 1237.0) : a = 12.347(3)Angstrom, b = 17.615(5)Angstrom, c = 26.148(2)Angstrom,beta = 92.88(1)degrees. v = 5679(2)Angstrom(3). Z=4, D-c = 1.45 g cm(-3). lambda(Mo-K-alpha) = 0.7107 Angstrom, mu(Mo-K-alpha) = 8.1 cm(-1), T = 293 K, R = 0.0815 (wR(2) = 0.2118) for 5834 reflections with 1 > 2 sigma(I). The complex has a tetracarboxylatodiruthenium (II, III) core and two axially bound 2-methylimidazole ligands. The Ru-Ru bond length is 2.290(1)Angstrom. The Ru-Ru bond order is 2.5 and the complex is three-electron paramagnetic. The complex shows an irreversible Ru-2(II,III)-->Ru-2(Il,II) reduction near -0.2 V vs SCE in CH2Cl2-0. 1 MTBAP. The complexes exemplify the first adduct of the tetracarboxylatodiruthenium (II,III) core having N-donor ligands

Arene ruthenium complexes of N,N′- and N,O-donor schiff base ligands: an X-ray structure of [(η6-p-cymene) RuCl(C5,H4N-2-CH=NC6H4-p-Me)Cl·C6H6·H2O

Arene ruthenium(II) Schiff base complexes of formulations [(η -p-cymene)RuCl(C5H4N-2-CH=NC6H4-p-X)](ClO4) (1) and [(η6-p-cymene)RuCl(O-o-C6H4CH=NC6H4-p-X)] (2) (X = H, Me, OMe, NO2, Cl) were prepared by reacting [(η6-p-cymene)RuCl2]2 with corresponding pyridine-2-carboxaldimines and sodium salts of salicylaldimines in dry THF, respectively. Complex 1 is isolated as a perchlorate salt. The molecular structure of [(η6-p-cymene)RuCl(C5H4 N-2-CH=NC6H4-p-Me)]Cl·C6H6·H2O has been determined by X-ray crystallography. The complex contains an η6-p-cymene group, a chloride and a bidentate chelating Schiff base ligand.

Observation of exclusive charmonium production and gamma+gamma to mu+mu- in p+pbar collisions at sqrt{s} = 1.96 TeV

We have observed the reactions p+pbar --> p+X+pbar, with X being a centrally produced J/psi, psi(2S) or chi_c0, and gamma+gamma --> mu+mu-, in proton- antiproton collisions at sqrt{s} = 1.96 TeV using the Run II Collider Detector at Fermilab. The event signature requires two oppositely charged muons, each with pseudorapidity |eta| mu+mu-. Events with a J/psi and an associated photon candidate are consistent with exclusive chi_c0 production through double pomeron exchange. The exclusive vector meson production is as expected for elastic photo- production, gamma+p --> J/psi(psi(2S)) + p, which is observed here for the first time in hadron-hadron collisions. The cross sections ds/dy(y=0) for p + pbar --> p + X + pbar with X = J/psi, psi(2S) orchi_c0 are 3.92+/-0.62 nb, 0.53+/-0.14 nb, and 75+/-14 nb respectively. The cross section for the continuum, with |eta(mu+/-)|

Observation of exclusive charmonium production and gamma+gamma to mu+mu- in p+pbar collisions at sqrt{s} = 1.96 TeV

We have observed the reactions p+pbar --> p+X+pbar, with X being a centrally produced J/psi, psi(2S) or chi_c0, and gamma+gamma --> mu+mu-, in proton- antiproton collisions at sqrt{s} = 1.96 TeV using the Run II Collider Detector at Fermilab. The event signature requires two oppositely charged muons, each with pseudorapidity |eta| mu+mu-. Events with a J/psi and an associated photon candidate are consistent with exclusive chi_c0 production through double pomeron exchange. The exclusive vector meson production is as expected for elastic photo- production, gamma+p --> J/psi(psi(2S)) + p, which is observed here for the first time in hadron-hadron collisions. The cross sections ds/dy(y=0) for p + pbar --> p + X + pbar with X = J/psi, psi(2S) orchi_c0 are 3.92+/-0.62 nb, 0.53+/-0.14 nb, and 75+/-14 nb respectively. The cross section for the continuum, with |eta(mu+/-)|

On the maximal rate of (n+1) x n and (n+2) x n complex orthogonal designs

For p x n complex orthogonal designs in k variables, where p is the number of channels uses and n is the number of transmit antennas, the maximal rate L of the design is asymptotically half as n increases. But, for such maximal rate codes, the decoding delay p increases exponentially. To control the delay, if we put the restriction that p = n, i.e., consider only the square designs, then, the rate decreases exponentially as n increases. This necessitates the study of the maximal rate of the designs with restrictions of the form p = n+1, p = n+2, p = n+3 etc. In this paper, we study the maximal rate of complex orthogonal designs with the restrictions p = n+1 and p = n+2. We derive upper and lower bounds for the maximal rate for p = n+1 and p = n+2. Also for the case of p = n+1, we show that if the orthogonal design admit only the variables, their negatives and multiples of these by root-1 and zeros as the entries of the matrix (other complex linear combinations are not allowed), then the maximal rate always equals the lower bound.

Edge-sharing bioctahedral diruthenium(III) complexes containing methoxy and carboxylate bridges:x-ray structures, redox behavior, and core stability

Edge-sharing bioctahedral (ESBO) complexes [Ru-2(OMe)(O2CC6H4-p-X)3(1-MeIm)(4)](ClO4)2 (X = OMe (1a), Me (1b)) and [Ru-2(O2CC6H4-P-X)(4)(1-MeIm)(4)](ClO4)(2) (X = OMe (2a), Me (2b)) are prepared by reacting Ru2Cl(O(2)CR)(4) with 1-methylimidazole (1-MeIm) in methanol followed by treatment with NaClO4. Complex 2a and the PF6- salt (1a') of 1a have been structurally characterized. Crystal data for 1a.1.5MeCN. 0.5Et(2)O: triclinic, P (1) over bar, a = 13.125(2) Angstrom, b = 15.529(3) Angstrom, c 17.314(5) Angstrom, a; 67.03(2)degrees, beta 68.05(2)degrees, gamma = 81.38(1)degrees, V 3014(1) Angstrom(3), Z = 2. Crystal data for 2a: triclinic, P (1) over bar, a 8.950(1) Angstrom, b = 12.089(3) Angstrom, c = 13.735(3) Angstrom, alpha 81.09(2)degrees, beta = 72.27(1)degrees, gamma = 83.15(2)degrees, V = 1394(1) Angstrom(3), Z = 1. The complexes consist of a diruthenium(III) unit held by two monoatomic and two three-atom bridging ligands. The 1-MeIm ligands are at the terminal sites of the [Ru-2(mu-L)(eta(1):mu-O(2)CR)(eta(1):eta(1):mu-O(2)CR)(2)](2+) core having a Ru-Ru single bond (L = OMe or eta(1)-O(2)CR). The Ru-Ru distance and the Ru-O-Ru angle in the core of 1a' and 2a are 2.49 Angstrom and similar to 76 degrees. The complexes undergo one-electron oxidation and reduction processes in MeCN-0.1 M TBAP to form mixed-valence diruthenium species with Ru-Ru bonds of orders 1.5 and 0.5, respectively.