For a ring endomorphism α and an α-derivation δ, we introduce α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].

Background:Adhesive capsulitis is a common disease that causes pain and reduced range of motion, but vague on the shoulder. Woman are affected fewer than men, but there is no known racial or genetic tendency. Most patients with adhesive capsulitis will improve with nonsurgical treatment. Acetaminophen and nonsteroidal anti-inflammatory drugs for pain relief in patients without contraindication are first-line options. Acupuncture considered being safe and effective in reducing pain. The aim of this study was to Investigation of the effectiveness of acupuncture in the treatment of frozen shoulder.Materials and Methods:In a controlled clinical trial, patients referred to the Baqiatallah clinic in 91 years with shoulder pain, frozen shoulder diagnosed based on history and physical exam, they have been enrolled. Indicators measured in the study was included the involved joint pain, range of motion and quality of life. Patients, first at baseline, one and a half months later (end of session) and then 3 months after the examination information about each individual entered in the from of questionnaires were pre-determined and data were analyzed by SPSS 17 software.Results:In this clinical trial study total 40 patients with frozen shoulder (20 interference with the acupuncture and 20 people control) study that patients average age 55/54. Age maximum 71 years and minimum 44 years. Acupuncture in the treatment of frozen shoulder with the results achieved in the general case acupuncture may improve shoulder motion in patients. VAS index at three months after treatment compared with the control group had a greater improvement.Conclusion:In the case of acupuncture and ultimately improve the overall look of all the movement of flexion and adduction of the shoulder, but the movement has been further improved, VAS index at three months after treatment compared with the control group had a greater improvement and finally, we perform acupuncture as a way to improve shoulder motion in patients with frozen shoulder offered.

The distance signless Laplacian matrix [Formula: see text] of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

For a skew PBW extension over a right duo compatible ring, we characterize several kinds of their elements such as units, idempotent, von Neumann regular, π-regular and the clean elements. As a consequence of our treatment, we extend several results in the literature for Ore extensions and commutative rings.

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