In this paper we define φ-Connes module amenability of a dual Banach algebra A where φ is a bounded wk∗-module homomorphism from A to A. We are mainly concerned with the study of φ-module normal virtual diagonals. We show that if S is a weakly cancellative inverse semigroup with subsemigroup E of idempotents, χ is a bounded wk∗-module homomorphism from l1(S) to l1(S) and l1(S) as a Banach module over l1(E) is χ-Connes module amenable, then it has a χ-module normal virtual diagonal. In the case χ =id, the converse holds

Let A be a Banach algebra and E be a Banach A -bimodule then S=A ÅE, the l1-direct sum of A and E becomes a module extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -module if and only if l1 (ES) is amenable as Banach algebra.

Let c0 be the real vector space of all real sequences which converge to zero. For every x, yÎc0, it is said that y is block diagonal majorized by x (written y<b x) if there exists a block diagonal row stochastic matrix R such that y=Rx. In this paper we find the possible structure of linear functions T: c0®c0 preserving <b.

In this paper we define a congruence ~ on inverse semigroup S such that amenability of S is equivalent to amenability of S/ ~. We study module amenability of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.

A transitive subgroup $G\leq S_n$ is called a genus $g$ group if there exist non identity elements $x_1,...,x_r\in G$ satisfying $G=\langle x_1,x_2,...,x_r\rangle$, $\prod_{i=1}^r {x_i}=1$ and $\sum_{i=1}^r ind\, x_i=2(n+g-1)$. The Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere $\mathbb{P}^1\mathbb{C}$ with $r$ branch points and the monodromy group $G$. Isomorphisms of such covers are in one to one correspondence with genus $g$ groups.In this article, we show that $G$ possesses genus one and two group if it is diagonal type and acts primitively on $\Omega$. Furthermore, we study the connectedness of the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ for genus 1 and 2.

Background: Good physician-patient communication improves health outcomes. However, many medical students lack effective communication skills.Objectives: This study was done to evaluate a virtual education module for teaching communication skills to medical students in Iran.Methods: The pre-test-post-test experimental design was conducted with simple random sampling using random number generation software to compare the virtual module of communication skills. Iran University of Medical Sciences students' attitudes were assessed with the Communication Skills Attitude Scale (CSAS) and clinical performance was assessed with the Calgary Cambridge checklist. A total of 50 students in each group have completed the course. The virtual module was designed and developed with the steps of the Calgary Cambridge model and was taught using PowerPoint, posters, pamphlets, and podcasts. Independent t-test samples, and analysis of covariance (ANCOVA) were utilized for data analysis using SPSS software.Results: The mean age of participants in the Conventional group was 23.92±3.27, and in the virtual module group was 24.02±3.16. The mean difference in attitude toward communication between the two groups was 25.62±5.45, and performance was 6.23±1.61, which was statistically significant. The statistical analysis revealed a statistically significant difference in the mean attitude and performance of students before and after the virtual module (P<0.05).Conclusion: The virtual module significantly improved students' communication skills, attitude, and performance compared to conventional education. The implementation of virtual education modules can effectively enhance the teaching of communication skills in medical schools. Further studies are needed to assess the long-term impacts on physician-patient communication.