Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal \(\mathbf {x}_0\) from its noisy observations \(\mathbf {y}=\mathbf {x}_0+\mathbf {z}\). In this paper, we focus on the “structured denoising problem,” where the signal \(\mathbf {x}_0\) possesses a certain structure and \(\mathbf {z}\) has independent normally distributed entries with mean zero and variance \(\sigma ^2\). We employ a structure-inducing convex function \(f(\cdot )\) and solve \(\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}\) to estimate \(\mathbf {x}_0\), for some \(\lambda >0\). Common choices for \(f(\cdot )\) include the \(\ell _1\) norm for sparse vectors, the \(\ell _1-\ell _2\) norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate \(\mathbf {x}^*\) is the normalized mean-squared error \(\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}\). We show that NMSE is maximized as \(\sigma \rightarrow 0\) and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the \({\lambda }\)-scaled subdifferential \({\lambda }\partial f(\mathbf {x}_0)\). When \({\lambda }\) is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}\). The paper also connects these results to the generalized LASSO problem, in which one solves \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}\) to estimate \(\mathbf {x}_0\) from noisy linear observations \(\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}\). We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a “phase transition” as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.     

Synthetic utilization of polynitroaromatic compounds. 6. Remarkable regioselectivity in nucleophilic displacement of aromatic nitro groups with amines

5,7-Dinitroquinazoline-4-ones undergo nucleophilic displacement of a nitro group with N-, S-, and O-nucleophiles. In contrast to previously studied dinitro-substituted benzoannulated five- and seven-membered heterocycles (where a high degree of selectivity was observed), these quinazolines mostly yield mixtures of regioisomeric substitution products. At the same time, primary and secondary amines react selectively to afford 5-aminoquinazolones (peri-substitution). A similar effect is observed for some other polynitroaromatic compounds with adjacent nitro and carbonyl groups. This phenomenon is attributed to a stabilization of the intermediate peri-sigma-complex by intramolecular hydrogen bond N(+)-H...O double bond C.     

Hartman–Wintner‐type inequalities for a class of nonlocal fractional boundary value problems

In this paper, we establish new Hartman–Wintner‐type inequalities for a class of nonlocal fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.     

The effect of an activation solution with siliceous species on the chemical reactivity and mechanical properties of geopolymers

Precursors are critical parameters in geopolymerization mechanisms because they govern the reaction kinetics as well as the working properties of the final materials. This study focuses on the effect of alkaline solutions on geopolymer formation. Toward this end, several geopolymer samples were synthesized from the same metakaolin and various alkaline solutions. First, the solutions were characterized by thermogravimetric analysis as well as DTA–TGA, infrared spectroscopy, and MAS-NMR spectrometry. The structural evolution of the formed geopolymers was investigated using infrared spectroscopy. The measurement of mechanical strength was tested by compression. The results provide evidence of relationships between the chemical composition, the extent of depolymerization of the alkaline solutions, the kinetics of Si–O–Si bond substitution by Si–O–Al and the compressive strength. For a given aluminosilicate source, the nature and the quantity of siliceous species in the activation solution appear to lead to variation in the reactivity and, consequently, to the formation of various networks that control the kinetics of formation of geopolymers and their mechanical properties.

     

Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces

In this paper, we establish two coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. The theorems presented extend some results due to ?iri? (2009) [3]. An example is given to illustrate the usability of our results.     

Base-induced transformations of ortho-nitrobenzylketones: intramolecular displacement of nitro group versus nitro-nitrite rearrangement

Reactions of 4-bromo-5-nitrophthalonitrile (BNPN) with enolates of alkyl 4-R-2,4-dioxobutanoates gave alkyl 3-acyl-5,6-dicyanobenzofuran-2-carboxylates, 4-(2-R-2-oxoethyl)-5-nitrophthalonitriles, or 3-acyl-1,2-benzoxazole-5,6-dicarbonitriles. With a base, 4-(2-R-2-oxoethyl)-5-nitrophthalonitriles either undergo nitro-nitrite rearrangement resulting in 3-acyl-1,2-benzoxazole-5,6-dicarbonitriles or yield 2-R-benzofuran-5,6-dicarbonitriles with a nitro group displacement.     

Some Results on Best Proximity Points

We study the existence and uniqueness of best proximity points for two classes of non-self-contractive mappings: almost (φ,θ)-contractive mappings and Meir–Keeler-type contractive mappings.     

On students’ conceptions of arithmetic average: the case of inference from a fixed total

There is more to understanding the concept of mean than simply knowing and applying the add-them-up and divide algorithm. In the following, we discuss a component of understanding the mean – inference from a fixed total – that has been largely ignored by researchers studying students understanding of mean. We add this component to the list of types of reasoning needed to understand mean and discuss student responses to tasks designed to elicit this component of reasoning. These responses reveal that inference from a fixed total reasoning is rare even in advance high school students.     

Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations

In this paper, we establish certain fixed point theorems in metric spaces with a partial ordering. Presented theorems extend and generalize several existing results in the literature. As application, we use the fixed point theorems obtained in this paper to study existence and uniqueness of solutions for fourth-order two-point boundary value problems for elastic beam equations.     

Ultrasonic characterization of a fluid layer using a broadband transducer

A measurement method is proposed for the ultrasonic characterization of a fluid layer, corresponding to the resin transfer molding (RTM) manufacturing process. The ultrasonic velocity and attenuation of the silicone oil are measured in three samples having different viscosities. The measurement method is established on the basis of the attenuation of ultrasonic waves in fluids. A correction of the beam diffraction is implemented to improve measurement precision. A single element transducer with central frequency of 15 MHz is used. The tested fluids simulate the industrial resin used to manufacture composite materials. When injecting this resin, its viscosity increases until it reaches a critical state of polymerization. In this paper we focus on ultrasonic characterization of three fluids representing three intermediate cases of fluid resin during its injection before reaching the polymerization state.