In this paper, we introduce the new concept of biprojectivity of a Banach algebra modulo an ideal, as a generalization of this notion in the classical case. By using it , we obtain some necessary and sufficient conditions for contractibility of Banach algebras modulo an ideal. In particular we characterize the contractibility of quotient Banach algebras. Also we study the relationship between t...

‎generalizing the notion of character amenability for banach‎ ‎algebras‎, ‎we study the concept of $varphi$-connes amenability of‎ ‎a dual banach algebra $mathcal{a}$ with predual $mathcal{a}_*$‎, ‎where $varphi$ is a homomorphism from $mathcal{a}$ onto $bbb c$‎ ‎that lies in $mathcal{a}_*$‎. ‎several characterizations of‎ ‎$varphi$-connes amenability are given‎. ‎we also prove that the‎ ‎follo...

1.1. Topological Social Choice Theory. Social choice theory aims to understand the nature of, and provide methods for, the aggregation of individual preferences to yield a social preference which is fair and satisfactory to individual voters on an axiomatic ground. An important example of this is in popular elections. In the 1980s, Chichilnisky and colleagues developed a topological approach pi...

This paper examines the determinants of intra-firm trade in US imports using detailed country-product data. We create a new measure of product contractibility based on the degree of intermediation in international trade for the product. We find important roles for the interaction of country and product characteristics in determining intra-firm trade shares. Intra-firm trade is high for products...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

This paper revisits how vertical linkages between overseas affiliates and their parents are related to intrafirm trade by shedding light on variations in contractibility across sectors the case of Japanese multinational enterprises based affiliate-level data. We confirm that is observed only a limited fraction affiliates. To include large number with zero into our regressions, we estimate model...

The Degree Contractibility problem is to test whether a given graph G can be modified to a graph of minimum degree at least d by using at most k contractions. We prove the following three results. First, Degree Contractibility is NP-complete even when d = 14. Second, it is fixed-parameter tractable when parameterized by k and d. Third, it is W[1]-hard when parameterized by k. We also study its ...

Some open problems in metric continuum theory are discussed. They concern 1) generalized homogeneity, in particular monotone homogeneity of dendrites; 2) planability of curves; 3) means on continua; 4) contractibility of continua; and 5) selectibility of continua.