Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a+im : i ∈ N0}. For positive integers a, b, c, d,m the sum of products set Rm(a)Rm(b)+Rm(c)Rm(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+lm) for some i, j, k, l ∈ Z}. It is proved that if gcd(a, b, c, d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class R...

The paper presents the basics and concepts of RM, state of the art with RM potential of RM, possible implementation of RM, and also with a practical example demonstrate current trends in product customization, where RM is enabling technology.

In this paper, we first give a description of a surjective unit-preserving real-linear uniform isometry $ T : A longrightarrow B$,  where $ A $ and $ B $ are complex function spaces on compact Hausdorff spaces $ X $ and $ Y $, respectively, whenever ${rm ER}left (A, Xright ) = {rm Ch}left (A, Xright )$ and ${rm ER}left (B, Yright ) = {rm Ch}left (B, Yright )$. Next, we give a description of $ T...

‎Let $G$ be a finite group‎. ‎An element $gin G$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $G$‎, ‎$chi(g)neq 0$‎. ‎The bi-Cayley graph ${rm BCay}(G,T)$ of $G$ with respect to a subset $Tsubseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin G‎, ‎ tin T}$‎. ‎Let ${rm nv}(G)$ be the set‎ ‎of all non-vanishi...