Lecture Hall Partitions

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of lecture hall partitions of length n of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincaré series of the affine Coxeter group , and one direct proof.     

Experimental determination of optical constants in the vacuum ultra violet wavelength region between 80 and 140 nm: A reflectance versus thickness method and its application to ZnSe

The 80-120 nm spectral range is a key domain for solar physics. Below 105 nm solids do not transmit light and the reflectance of available mirrors is particularly low which makes optical measurements specifically difficult. Optical constants of the materials may consequently be unavailable or unreliable.We present here a two media reflectance method at normal incidence suited to this VUV range, in which the variable is not the incidence angle but the thickness of the top layer made of the material to be analyzed. The real (n) and imaginary (k) parts of the complex index are directly and graphically determined in the (n, k) plane as the common intersection point of isoreflectance curves corresponding to samples different only in the thickness of the top layer.The method is tested and illustrated with ZnSe films evaporated on Al covered float glass substrates. In the literature, the reflectance magnitudes measured on ZnSe crystals differ strongly from an author to the other, leading to discrepant data for ZnSe in the VUV domain.We obtain precise and reliable values of (n, k), which fit the experimental values determined on freshly cleaved ZnSe crystals by J.L. Freeouf and the theoretical values calculated from the electronic band structure of ZnSe by John P. Walter and Marvin L. Cohen, but strongly differ from the optical constants selected by E.D. Palik in his tables.     

Shape Minimization of Dendritic Attenuation

What is the optimal shape of a dendrite? Of course, optimality refers to some particular criterion. In this paper, we look at the case of a dendrite sealed at one end and connected at the other end to a soma. The electrical potential in the fiber follows the classical cable equations as established by W. Rall. We are interested in the shape of the dendrite which minimizes either the attenuation in time of the potential or the attenuation in space. In both cases, we prove that the cylindrical shape is optimal.     

Non-localization of eigenfunctions for Sturm–Liouville operators and applications

In this article, we investigate a non-localization property of the eigenfunctions of Sturm–Liouville operators $A_a=?{?}_{xx}+a(?)\mathrm{Id}$ with Dirichlet boundary conditions, where $a(?)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(?)$ on a measurable subset ω of $(0,L)$, where $e(?)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all subsets ω having a prescribed measure and all $L^{\infty }$ potential functions $a(?)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.     

Addressing Charge-Transfer and Locally-Excited States in a Twisted Biphenyl Push-Pull Chromophore

We present the synthesis and spectroscopic characterization of a twisted push–pull biphenyl molecule undergoing photoinduced electron transfer. Steady-state and transient absorption spectra suggest, in this rigid molecular structure, a subtle interplay between locally-excited and charge-transfer states, whose equilibrium and dynamics is only driven by solvation. A theoretical model is presented for the solvation dynamics and, with the support of quantum chemical calculations, we demonstrate the existence of two sets of states, having either local or charge-transfer character, that only “communicate” thanks to solvation, which is the sole driving force for the charge-separation process.     

Synthesis of Bis(arylethynyl)pyrrolo[3,2-b]pyrroles and Effect of Intramolecular Charge Transfer on Their Photophysical Behavior

3,6-Bis(arylethynyl)pyrrolo[3,2-b]pyrroles were synthesized through a two-step procedure involving double direct alkynylation of the electron-rich core followed by Sonogashira coupling. In comparison with the parent tetraarylo-pyrrolo[3,2-b]pyrroles and benzo-fused pyrrolopyrroles, these new dyes showed moderately redshifted absorption. Almost all derivatives showed positive fluorescence solvatochromism and, for the first time, red-emitting pyrrolopyrroles were obtained. Computational studies revealed that, in most cases, there is negligible change in the geometry between ground and excited states. Interestingly, there was a fundamental difference between pyrrolopyrroles possessing electron-withdrawing substituents at positions 2 and 5 and their analogs lacking these substituents. The former dyes behaved like dipolar chromophores with the lowest excited state both one-photon and two-photon allowed, which corresponds to intramolecular charge transfer occurring along the branches perpendicular to the pyrrolopyrrole long axis. In compounds lacking electron-withdrawing substituents at positions 2 and 5, intramolecular charge transfer took place along the long axis of pyrrolopyrrole and consequently the one-photon transitions are not two-photon allowed. Despite displaying quadrupolar core-to-peripheral intramolecular charge transfer, these derivatives showed two-photon absorption cross sections in the NIR1 region comparable to tetraaryl-pyrrolo[3,2-b]pyrroles lacking π-expansion (up to about 500 GM).     

Conservation laws of an electro-active polymer

Ionic electro-active polymer is an active material consisting in a polyelectrolyte (for example Nafion). Such material is usually used as thin film sandwiched between two platinum electrodes. The polymer undergoes large bending motions when an electric field is applied across the thickness. Conversely, a voltage can be detected between both electrodes when the polymer is suddenly bent. The solvent-saturated polymer is fully dissociated, releasing cations of small size. We used a continuous medium approach. The material is modelled by the coexistence of two phases; it can be considered as a porous medium where the deformable solid phase is the polymer backbone with fixed anions; the electrolyte phase is made of a solvent (usually water) with free cations. The microscale conservation laws of mass, linear momentum and energy and the Maxwell’s equations are first written for each phase. The physical quantities linked to the interfaces are deduced. The use of an average technique applied to the two-phase medium finally leads to an Eulerian formulation of the conservation laws of the complete material. Macroscale equations relative to each phase provide exchanges through the interfaces. An analysis of the balance equations of kinetic, potential and internal energy highlights the phenomena responsible of the conversion of one kind of energy into another, especially the dissipative ones : viscous frictions and Joule effect.