On Some Exponential Diophantine Equations

 We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000)     

Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

We relate the existence of Frobenius morphisms into the Jacobians of Drinfeld modular curves to the existence of congruences between cusp forms.     

Generalized derivations and additive theory II

We return to the theme of generalized derivations related to symmetric functions to correct the hypothesis of one of the main theorems of our first paper, so that all cases are now properly covered.     

On integral Apollonian circle packings

The curvatures of four mutually tangent circles with disjoint interiors form what is called a Descartes quadruple. The four least curvatures in an integral Apollonian circle packing form what is called a root Descartes quadruple and, if the curvatures are relatively prime, we say that it is a primitive root quadruple. We prove a conjecture of Mallows by giving a closed formula for the number of primitive root quadruples with minimum curvature −n. An Apollonian circle packing is called strongly integral if every circle has curvature times center a Gaussian integer. The set of all such circle packings for which the curvature plus curvature times center is congruent to 1 modulo 2 is called the “standard supergasket.” Those centers in the unit square are in one-to-one correspondence with the primitive root quadruples and exhibit certain symmetries first conjectured by Mallows. We prove these symmetries; in particular, the centers are symmetric around y=x if n is odd, around x=1/2 if n is an odd multiple of 2, and around y=1/2 if n is a multiple of 4.     

Quadratic descent of hermitian forms

Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

Remarks about a generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem

${(?\mathrm{\Delta }+m^2)}^{\sigma }u+Vu=(W?F(u))f(u)\text{in }{\mathbb{R}}^N,$

where $0<\sigma <1$, V is a bounded continuous potential and F the primitive of f. We also show results about the regularity of any solution of this problem.     

Monotonicity properties of ‐functions

We present some monotonicity results for a class of Dirichlet series generalizing previously known results. The fact that is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Lastly, we use similar techniques to prove another formulation of the Riemann hypothesis for the L‐function associated to the Ramanujan‐tau function.     

Phosphomolybdic acid supported on Schiff base functionalized graphene oxide nanosheets: Preparation,characterization, and first catalytic application in the multi‐component synthesis of tetrahydrobenzo[a]xanthene‐11‐ones

New Schiff base (SB) functionalized graphene oxide (GO) nanosheets containing phosphomolybdic counter‐anion H₂PMo₁₂O₄₀^¯ (H₂PMo) were successfully prepared by grafting of 3‐aminopropyltriethoxysilane (APTS) on GO nanosheets followed by condensation with benzil and finally reaction with phosphomolybdic acid (H₃PMo₁₂O₄₀, denoted as H₃PMo) and characterized using Fourier transform infrared (FT‐IR) spectroscopy, field emission scanning electron microscopy (FESEM), transmission electron microscopy (TEM), atomic force microscopy (AFM), particle size distribution, energy‐dispersive X‐ray (EDX) analysis, EDX elemental mapping, and inductively coupled plasma optical emission spectrometry (ICP‐OES). The prepared new nanomaterial, denoted as GO‐SB‐H₂PMo, was shown to be an efficient heterogeneous catalyst in one‐pot, three‐component reaction of β‐naphthol, aldehydes, and dimedone, giving high yields of tetrahydrobenzo[a]xanthene‐11‐ones within short reaction times. The catalyst is readily recovered by simple filtration and can be recycled and reused several times with no significant loss of catalytic activity.