Symmetry and uniqueness of solutions to some Liouville-type equations and systems

We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the mass associated to these problems. The argument is in the spirit of the sphere covering inequality which for the first time is used in treating different exponential nonlinearities and systems.     

Unmixedness and arithmetic properties of matroidal ideals

Let $$R=k[x_1,\ldots ,x_n]$$ be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that $$ara(I)=n-d+1$$. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).     

Quasi-interpolation and outliers removal

In this work, we present a method that will allow us to remove outliers from a data set. Given the measurements of a function f = g + e on a set of sample points \(X \subset \mathbb {R}^{d}\), where \(g \in C^{M+1}(\mathbb {R}^{d})\) is the function of interest and e is the deviation from the function g. We will say that a sample point x ∈ X is an outlier if the difference e(x) = f(x) ? g(x) is large. We show that by analyzing the approximation errors on our sample set X, we may predict which of the sample points are outliers. Furthermore, we can identify outliers of very small deviations, as well as ones with large deviations.     

Conditioning of the matrix-matrix exponentiation

If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as A ^B := e ^log(A)B . It arises, for instance, in Von Newmann’s quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. In this paper, we revisit this function and derive new related results. Particular emphasis is devoted to its Fréchet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Fréchet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of A ^B is proposed. Some numerical experiments are included.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

One-pot synthesis of $$\upgamma $$-spiroiminolactones and $$\upgamma $$-dispiroiminolactones using $$\textit{N}{,}{} \textit{N}^{\prime }$$N,N′-disubstituted parabanic acid and thioparabanic acid derivatives

A direct entry and simple process for the synthesis of \(\upgamma \)-spiroiminolactones present in a large number of natural products has been developed. In the first step, the synthesis of parabanic acid derivatives was commenced from the reaction of \(\textit{N}{,}{} \textit{N}^{\prime }\)-disubstituted urea and thiourea with oxalyl chloride, then a three-component reaction was carried out with isocyanides, acetylenic esters, and \(\textit{N}{,}{} \textit{N}^{\prime }\)-disubstituted parabanic acid derivatives. The method allows the construction of a variety of \(\upgamma \)-spiroiminolactone structures in good to high yields starting from readily available precursors. It was found that in the case of \(\textit{N}{,}{} \textit{N}^{\prime }\)-diphenyl thioparabanic acid, additional products of \(\upgamma \)-dispiroiminolactones have been formed due to the higher electrophilicity of \(\upalpha \)-dicarbonyl groups. The structures were fully established using spectroscopic analysis NMR, IR, and Mass spectrometry. The crystal structure of \(\upgamma \)-dispiroiminolactone was confirmed from single-crystal X-ray diffraction study.     

Due date assignment for multistage assembly systems

This paper is concerned with the study of the constant due-date assignment policy in a multistage assembly system. The multistage assembly system is modeled as an open queueing network. It is assumed that the product order arrives according to a Poisson process. In each service station, there is either one or infinite machine with exponentially distributed processing time. The transport times between every pair of service stations are independent random variables with generalized Erlang distributions. It is assumed that each product has a penalty cost that is some linear function of its due-date and its actual completion time. The due date is found by adding a constant to the time that the order arrives. This constant value is the constant lead time that a product might expect between time of placing the order and time of delivery. By applying the longest path analysis in queueing networks, we obtain the distribution function of manufacturing lead time. Then, the optimal constant lead time is computed by minimizing the expected aggregate cost per product. Finally, the results are verified by Monte Carlo simulation.     

Borane–Lewis Base Complexes as Homolytic Hydrogen Atom Donors

Radical stabilization energies (RSE)s have been calculated for a variety of boryl radicals complexed to Lewis bases at the G3(MP2)‐RAD level of theory. These are referenced to the B? H bond dissociation energy (BDE) in BH₃ determined at W4.3 level. High RSE values (and thus low BDE(B? H) values) have been found for borane complexes of a variety of five‐ and six‐membered ring heterocycles. Variations of RSE values have been correlated with the strength of Lewis acid–Lewis base complex formation at the boryl radical stage. The analysis of charge‐ and spin‐density distributions shows that spin delocalization in the boryl radical complexes constitutes one of the mechanisms of radical stabilization.     

Effects of shearing flow with inertia on monolayer mesoscale structure

In an experimental flow system capable of imparting a well-controlled shear-rate distribution with inertia to a monolayer consisting of coexisting phases, we have studied the resulting phase morphology and domain fragmentation. These evolve on distinct time scales: the viscous time associated with the viscosity in the bulk and the Marangoni stress and the fragmentation/relaxation time associated with the phase morphology. A relationship between the microstructure (line tension) and macroflow (shear rate) determining the meso length scale of the coexisting phase domains has been deduced from dimensional analysis and was found to correlate well with the quantitative experimental observations.