Sharp inequalities related to Gosper's formula

The purpose of this Note is to construct a new type of Stirling series, which extends the Gosper's formula for big factorials. New sharp inequalities for the gamma and digamma functions are established. Finally, numerical computations which demonstrate the superiority of our new series over the classical Stirling's series are given.     

Some connections of monotonicity with Weierstrass extreme value theorem

The main scope of this article is to prove that if a continuous function attains its minimum before maximum on every compact interval, than it is nondecreasing. Finally, other similar results are established.     

Error estimates of Ramanujan-type series

The aim of this paper is to establish some inequalities related to Ramanujan-type approximation series of the constant π and other classical constants such as Apéry’s constant and Catalan’s constant.     

Metric dimensional reduction at singularities with implications to Quantum Gravity

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimensional reduction would still be desirable. The main contribution of this article is to show that dimensional reduction effects are due to General Relativity at singularities, and do not need to be postulated ad-hoc.     

Photoelectrochemical Scanning Droplet Cell Microscopy (PE‐SDCM)

Principles of localised photoelectrochemistry are summarised and an experimental approach is described that allows the performance of the most important photoelectrochemical experiments within a diameter of 100 μm. Various light sources, such as a continuum emitter with a monochromator, LEDs, and lasers are coupled into a multi‐mode fibre to illuminate a small spot that is wetted by the electrolyte from a capillary. Reference electrode, counter electrode, and optical fibre are installed in the capillary system. The performance of this system is demonstrated by photocurrent measurements on n‐doped Si and p‐doped Si as model substrates. A thickness‐graded aluminium thin film for partial shadowing on Si proves the applicability for material library investigations in combinatorial materials science. Further experiments demonstrate the possibility of electrical light chopping as well as impedance spectroscopy with subsequent Mott–Schottky analysis for the determination of charge‐carrier concentration and type, flat‐band potential, and inversion layer formation. Photoelectrochemical scanning droplet cell microscopy (PE‐SDCM) is an extremely versatile tool for the screening of water splitting photoelectrodes, the characterisation of photocatalysts, and high throughput characterisation of microgram amounts of new solar cell materials.     

The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity

Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R ². If the matrix fields are respectively of class C ² and C ¹, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W ^1,∞ _loc and L ^∞ _loc, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense.     

On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory

Let ω be an open connected subset of $\text{R}{}^2$ and let $\text{θ}$ be an immersion from ω into $\text{R}{}^3$. It is established that the set formed by all rigid displacements of the surface $\text{θ}\text{(ω)}$ is a submanifold of dimension 6 and of class $\text{C}{}^{\mathrm{\infty }}$ of the space $\text{H}{}^1\text{(ω)}$. It is shown that the infinitesimal rigid displacements of the same surface $\text{θ}\text{(ω)}$ span the tangent space at the origin to this submanifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Estimating the Somos' quadratic recurrence constant

TextThe aim of this paper is to provide some estimates about the Somos' quadratic recurrence constant, using its relation with the generalized Euler constant.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3QjmHit3mC4.