On the stability of gradient polynomial systems at infinity

We first define the notion of the infimum at infinity of a polynomial function and the notion of stability at infinity near the fiber of the gradient descent system. Then we prove that the gradient descent system is stable at infinity near the fiber of the infimum value at infinity.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

A Frank–Wolfe type theorem for nondegenerate polynomial programs

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)      

Levitin–Polyak well-posedness for set optimization problems involving set order relations

Positivity - In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set...     

Łojasiewicz inequality at infinity for polynomials in two real variables

We propose different types of Łojasiewicz inequality at infinity for polynomials in two real variables. The formulas for the Łojasiewicz exponents are given.     

Preparation of chitosan nanoparticles by TPP ionic gelation combined with spray drying,and the antibacterial activity of chitosan nanoparticles and a chitosan nanoparticle–amoxicillin complex

Chitosan nanoparticles were prepared from chitosan with various molecular weights by tripolyphosphate (TPP) ionic gelation combined with a spray drying method. The morphologies and characteristics of chitosan nanoparticles were determined by TEM, FE-SEM and from their mean sizes and zeta potentials. The effect of chitosan molecular weight (130, 276, 760 and 1200 cPs) and size of spray dryer nozzle (4.0, 5.5 and 7.0 µm) on mean size, size distribution and zeta potential values of chitosan nanoparticles was investigated. The results showed that the mean size of chitosan nanoparticles was in the range of 166–1230 nm and the zeta potential value ranged from 34.9 to 59 mV, depending on the molecular weight of chitosan and size of the spray dryer nozzles. The lower the molecular weight of chitosan, the smaller the size of the chitosan nanoparticles and the higher the zeta potential. A test for the antibacterial activity of chitosan nanoparticles (only) and a chitosan nanoparticle–amoxicillin complex against Streptococcus pneumoniae was also conducted. The results indicated that a smaller chitosan nanoparticle and higher zeta potential showed higher antibacterial activity. The chitosan nanoparticle–amoxicillin complex resulted in improved antibacterial activity as compared to amoxicillin and chitosan nanopaticles alone. Using a chitosan nanoparticle–amoxicillin complex could reduce by three times the dosage of amoxicillin while still completely inhibiting S. pneumoniae.     

Preparation of chitosan nanoparticles by spray drying,and their antibacterial activity

Chitosan nanoparticles were prepared from chitosan of different molecular weight by spray drying. The morphology of the particles was characterized by SEM, and size distribution and zeta potential were determined. The effects of chitosan solution concentration, molecular weight of chitosan, and size of the spray dryer nozzles on average size, size distribution and zeta potential of chitosan nanoparticles were investigated. The effects of chitosan nanoparticles and chitosan nanoparticles–amoxicillin complex on Staphylococcus aureus were also tested. The results showed that the average size of chitosan nanoparticles were in the range 95.5–395 nm and zeta potentials were 39.3–45.7 mV, depending on the concentration and molecular weight of the chitosan. The lower the concentration and molecular weight of the chitosan, the smaller the chitosan nanoparticles and the higher the zeta potential. Testing for antibacterial activity against S. aureus indicated that chitosan nanoparticles strongly inhibited growth of the bacteria; the minimum inhibitory concentration, 20 μg/mL, was lower than those of chitosan solution or amoxicillin. The antibacterial capacity of chitosan nanoparticles also depended on the size, zeta potential, and molecular weight of the chitosan. Complexation of chitosan nanoparticles with amoxicillin improved the antibacterial activity of amoxicillin.