Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Wimp searches with gamma rays in the Fermi era: Challenges,methods and results

The launch of the gamma-ray telescope Fermi Large Area Telescope (Fermi-LAT) started a pivotal period in indirect detection of dark matter. By outperforming expectations, for the first time a robust and stringent test of the paradigm of weakly interacting massive particles (WIMPs) is within reach. In this paper, we discuss astrophysical targets for WIMP detection and the challenges they present, review the analysis tools which have been employed to tackle these challenges, and summarize the status of constraints on and the claimed detections in the WIMP parameter space. Methods and results will be discussed in comparison to Imaging Air Cherenkov Telescopes. We also provide an outlook on short term and longer term developments.     

A Recursive Algorithm for Order Cycle-time that Minimizes Logistics Cost

A firm's logistics cost, including shipping and inventory-carrying, is a substantial percentage of its sales. Nevertheless, typical inventory-control methods ignore or insufficiently represent the shipping cost. This paper describes a recursive algorithm that determines the reorder cycle-time that minimizes total logistics cost. It allows for a realistic accounting of shipping cost, which is modelled here as a function of shipping distance and weight. The algorithm uses a relaxation procedure to identify a suitable initial approximation to the optimal order cycle-time and then, through a series of recursive steps, moves to the optimal result. We demonstrate the algorithm with a single item, with a group of items that share a common order cycle, and with multi-items when item demands are random variables. Experience with this algorithm indicates that it converges to the optimal result in a very few steps.     

Workstage repair policies for sequential manufacturing systems

Dynamic stage repair policies where the stage repair probabilities are determined by the number of items in the buffer are examined through a model of a manufacturing system with two unreliable workstages connected by a minimal intermediate storage buffer. Definitions for workstage availability and line balance are extended for sequential manufacturing systems with dynamic repair policies. A measure for stage repair inputs and a general form of dynamic stage repair policies are presented. The steady-state probabilities of the resultant state-inhomogeneous Markov chain are computed by an efficient solution procedure. Insights obtained from extensive numerical studies are illustrated and discussed.     

Hexaphyrin(1.0.1.0.0.0). A new colorimetric actinide sensor

Hexaphyrin(1.0.1.0.0.0) (isoamethyrin) undergoes a significant color change in the presence of UO₂²⁺, PuO₂²⁺, and NpO₂²⁺. The complexation of the first of these dioxo actinide cations was studied in semi-quantitative fashion in 1:1 MeOH-CH₂Cl₂. Under these conditions, the detection limit for UO₂²⁺ was found to be ca. 5.8 ppm by naked eye monitoring and <28 ppb by UV-vis spectroscopy. Isoamethyrin does not undergo a color change in the presence of most transition metals or when exposed to Gd(III). Isoamethyrin thus constitutes an attractive alternative to 2,2′(1,8-dihydroxy-3,6-disulfonaphthylene-2,7-bisazo)-bisbenzenarsonic acid (AzIII) and 2-(5-bromo-2-pyridylazo)-5-(diethylamino)phenol (BrPADAP), systems currently used as actinide cation sensors.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

On Pairs of Diagonal Quintic Forms

We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p-adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.