Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Antiandrogen. I. 2-azapregnane and 2-oxapregnane steroids.

2-Azachlormadinone acetate (5a, 17-acetoxy-6-chloro-2-azapregna-4,6-diene- 3,20-dione), 2-oxachlormadinone acetate (6, 17-acetoxy-6-chloro-2-oxapregna-4,6-diene-3,20-dione) and the derivatives were prepared as potential antiandrogenic agents. Biological evaluation showed that 5a and 6 had a potent antiandrogenic activity when tested in the castrated male rat.     

Adsorption of fission products onto soils using a fission multitracer

The adsorption behavior of fission products to various soils was studied using a multitracer. The multitracer was prepared by neutron irradiation of ²³⁵U. Distribution coefficients of fission products were obtained for seventeen kinds of Japanese soils. It was found that zirconium, niobium, and rare earth elements show high distribution coefficients for all soil samples, however, elements like alkali metals show varied values.     

Synthesis of well‐defined syndiotactic poly(methyl methacrylate) with low‐temperature atom transfer radical polymerization in fluoroalcohol

The copper‐mediated atom transfer radical polymerization of methyl methacrylate (MMA) in 1,1,1,3,3,3‐hexafluoro‐2‐propanol (HFIP) was studied to simultaneously control the molecular weight and tacticity. The polymerization using tris[2‐(dimethylamino)ethyl]amine (Me₆TREN) as a ligand was performed even at ?78°C with a number‐average molecular weight (M_n) of 13,400 and a polydispersity (weight‐average molecular weight/number‐average molecular weight) of 1.31, although the measured M_n's were much higher than the theoretical ones. The addition of copper(II) bromide (CuBr₂) apparently affected the early stage of the polymerization; that is, the polymerization could proceed in a controlled manner under the condition of [MMA]₀/[methyl α‐bromoisobutyrate]₀/[CuBr]₀/[CuBr₂]₀/[Me₆TREN]₀ = 200/1/1/0.2/1.2 at ?20°C with an MMA/HFIP ratio of 1/4 (v/v). For the field desorption mass spectrum of Cu^IBr/Me₆TREN in HFIP, there were [Cu(Me₆TREN)Br]⁺ and [Cu(Me₆TREN)OCH(CF₃)₂]⁺, indicating that HFIP should coordinate to the Cu^I/Me₆TREN complex. The syndiotacticity of the obtained poly(methyl methacrylate)s increased with the decreasing polymerization temperature; the racemo content was 84% for ?78°C, 77% for ?30°C, 75% for ?20°C, and 63% for 30°C. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1436–1446, 2006     

Thermal properties and fracture toughness of a liquid‐crystalline epoxy resin cured with an aromatic diamine crosslinker having a mesogenic group

A mesogenic‐type curing agent was synthesized to introduce a mesogenic group not only into epoxy resin backbones but also into the crosslink units. In the mesogenic curing agent system, the domain size became larger, and the network arrangement in each domain existed to a greater extent than that in a system cured with the ordinary diamine curing system according to the evidence from polarized optical micrographs and polarized Fourier transform infrared mapping measurements. Moreover, the fracture toughness of the system was considerably improved. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2486–2494, 2006     

Investigation of the drying conditions for amidosulfuric acid.

The drying conditions for primary standards of volumetric analysis have a significant effect on the titration results due to changes in the purity, stability and homogeneity. Amidosulfuric acid, a strong acid used as a reference material for volumetric analysis in Japan, was dried in a vacuum desiccator or heated at different temperatures, and then measured by Karl-Fischer titration, thermogravimetry/mass spectroscopy (TG-MS), ion chromatography and coulometric titration. The optimum drying conditions were at 50 degrees C for 24 h with crushing.     

Microlocal study of topological Radon transforms and real projective duality

Various topological properties of projective duality between real projective varieties and their duals are obtained by making use of the microlocal theory of (subanalytically) constructible sheaves developed by Kashiwara [M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985) 193-209] and Kashiwara-Schapira [M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer, Berlin-Heidelberg-New York, 1990]. In particular, we prove in the real setting some results similar to the ones proved by Ernström in the complex case [L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994) 1-21]. For this purpose, we describe the characteristic cycles of topological Radon transforms of constructible functions in terms of curvatures of strata in real projective spaces.     

Oscillation of the solutions of parabolic equations with nonlinear neutral terms

In the present paper the oscillatory properties of the solutions of parabolic equations with nonlinear neutral terms are investigated. Our approach is to reduce the multi-dimensional problem to a one-dimensional problem for delay differential inequalities.     

On the pagenumber of trivalent Cayley graphs

Book embedding of graphs is one of the graph layout problem. It is useful for the multiprocessor network layout or the fault-tolerant processor arrays. We show that the trivalent Cayley graphs proposed by Vadapalli and Srimani can be embedded in five pages, and show some additional results on cube-connected cycles.