A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

This paper demonstrates that if the restricted isometry constant $\delta_{K+1}$ of the measurement matrix $A$ satisfies $$\delta_{K+1} < \frac{1}{\sqrt{K}+1},$$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from A\x. By contrast, a matrix is also constructed with the restricted isometry constant $$\delta_{K+1} = \frac{1}{\sqrt{K}}$$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009

A new proof of some polynomial inequalities related to pseudo-splines

AbstractPseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], which also appeared in [A. Cohen, J.P. Conze, Régularité des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351–365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=N−1. In this note, we will give a new insight into this proposition

N 1,N 4,3,6-Tetra­methyl-1,2,4,5-tetra­zine-1,4-dicarboxamide

The asymmetric unit of the title compound, C8H14N6O2, contains two independent mol­ecules. In one mol­ecule, the amide-substituted N atoms of the tetra­zine ring deviate from the plane [maximum deviation = 0.028 (1) Å] through the four other atoms in the ring by 0.350 (2) and 0.344 (2) Å, forming a boat conformation, and the mean planes of the two carboxamide groups form dihedral angles of 10.46 (13) and 20.41 (12)° with the four approximtely planar atoms in the tetra­zine ring. In the other mol­ecule, the amide-substituted N atoms of the tetra­zine ring deviate from the plane [maximum deviation = 0.033 (1) Å] through the four other atoms in the ring by 0.324 (2) and 0.307 (2) Å, forming a boat conformation, and the mean planes of the two carboxamide groups form dihedral angles of 14.66 (11) and 17.08 (10)° with the four approximately planar atoms of the tetra­zine ring. In the crystal, N—H⋯O hydrogen bonds connect mol­ecules to form a two-dimensional network parallel to (1-1-1). Intra­molecular N—H⋯N hydrogen bonds are observed

3,6-Dimethyl-N 1,N 4-bis­(pyridin-2-yl)-1,2,4,5-tetra­zine-1,4-dicarboxamide

In the title mol­ecule, C16H16N8O2, four atoms of the tetra­zine ring are coplanar, with the largest deviation from the plane being 0.0236 (12) Å; the other two atoms of the tetra­zine ring deviate on the same side from this plane by 0.320 (4) and 0.335 (4) Å. Therefore, the central tetra­zine ring exhibits a boat conformation. The dihedral angles between the mean plane of the four coplanar atoms of the tetrazine ring and the two pyridine rings are 26.22 (10) and 6.97 (5)°. The two pyridine rings form a dihedral angle of 31.27 (8)°. In the molecule, there are a number of short C—H⋯O interactions. In the crystal, molecules are linked via a C—H⋯O interaction to form zigzag chains propagating along the [010] direction

Extraction-condition Optimization of Baicalein and Schisandrin from Hu-gan-kang-yuan Formula Using Orthogonal Array Design

Purpose: To optimize the extraction conditions for Hu-gan-kang-yuan Formula based on extraction rates of baicalein and schisandrin using an orthogonal array design.Methods: Ethanol concentration (50 - 70 %), ratio of solvent to raw material (8 - 12 mL/g), and extraction time (1 - 3 h) were examined with a three-factor and three-level L9(3)3 orthogonal array design. In addition, analysis of variance (ANOVA) was used to evaluate the statistical significance of the effects of individual factors on extraction rates of baicalein and schisandrin determined by high performance liquid chromatography (HPLC). The number of extractions was further investigated to confirm the extraction rate of target compounds based on the optimized conditions.Results: The optimized conditions were an ethanol concentration of 70 %; ratio of solvent to raw material, 12:1 mL/g; and extraction time of 60 min. Ethanol concentration and ratio of solvent to raw material showed significant effects on the extraction of the two compounds (p &lt; 0.05). The number of extraction steps, two, was reasonable. The final optimum extraction conditions resulted in 79.48 ± 1.40 and 88.55 ± 1.85 % of extraction for baicalein and schisandrin, respectively.Conclusion: The optimized extraction conditions for baicalein and schisandrin are practicable and repeatable, and can be upgraded for pilot-scale production of Hu-gan-kang-yuan preparations.Keywords: Hu-gan-kang-yuan Formula, Extract-condition optimization, Orthogonal array design, Baicalein, Schisandri