THE MINIMAL OPERATOR AND WEIGHTED INEQUALITIES FOR MARTINGALES

In this article the authors introduce the minimal operator on martingale spaces, discuss some one-weight and two-weight inequalities for the minimal operator and characterize the conditions which make the inequalities hold.

On the Domains of General Ordinary Differential Operators in the Direct Sum Spaces

Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [ap,bp). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.

The Well-Posed Operators with Their Spectra in Lpw-Spaces

In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τp,q in Lpw-spaces of order n with complex coefficients and its formal adjoint τ+q',p' in Lpw-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T0 (τp,q) generated by such expression τp,q and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.

The Clinical Evaluation of Minimally Invasive Treatment for Axillary Osmidrosis

Background: Traditional treatment of axillary odour is the surgical removal of skin and subcutaneous tissue along the edge of the axillary hair in a fusiform shape, which requires the restriction of upper limb activity, has large scars and a high recurrence rate. Minimally invasive methods have developed in recent years. The subcutaneous trimming method with small incision reduces the incision compared to traditional incision, but still requires skin incision, which does not achieve true, minimally invasive treatment. Simple negative pressure suction is difficult to destroy the sweat glands and sebaceous glands in the dermis layer. Thus, we hypothesized that combination of three minimally invasive methods might be more effective. Objective: To explore the clinical effect of combined treatment of axillary osmidrosis by three minimally invasive methods of microwave, scraping & suction and scratching. Methods: From October 2015 to October 2020, 80 patients with underarm odor were enrolled and lacal swelling anesthetic was applied. Microwave was used to treat sweat glands and sebaceous glands in the dermis following the pores. The curettage-aspiration was used to scrape and suck the apocrine sweat glands in the subcutaneous fat layer and the junction between the subcutaneous fat and dermis. To achieve the purpose of minimally invasive and thorough treatment of underarm odor, the spoon scraped the remaining sweat gland tissue. Results: Among 80 patients on 160 sides, all cases (100.00%) were cured on 160 sides at 1 month post-operative follow-up, 75 cases (93.75%) were cured on 150 sides, 5 cases (6.25%) on 5 sides were ineffective at 6-month post-operative follow-up, 74 cases (92.50%) were cured on 148 sides, 6 cases (7.50%) on 5 sides were ineffective at 1-year post-operative follow-up. Complications occurred in 5 cases (6.25%) after operation: 2 cases (2.50%) with skin necrosis on 3 sides, 1 case (1.25%) with uneven skin on 2 sides, 1 case (1.25%) with axillary abscess on 2 sides, and 1 case (1.25%) with keloids on 2 sides. Conclusion: The combination of three minimally invasive methods of microwave, scraping & suction and scratching to treat underarm odor is simple operation, has high cure rate, quick postoperative recovery, and few complications. It is an effective method for minimally invasive and thorough treatment of underarm odor in clinic.

STRONG LIMIT POINT FOR LINEAR HAMILTONIAN DIFFERENCE SYSTEM

The definition of strong limit-point for singular Hamiltonian difference expressions with complex coefficients are given, and some strong limit-point criteria are established.