Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z 2(M)-inverse split provided f^(-1)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct...

In 1990, Cook, Kannan and Schrijver proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a 3-dimensional mixed-integer polytope whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the kth 1-branch split closure of these po...

the set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $ns_n$‎. ‎a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices‎, ‎and such a graph $g$ is called a realization of $pi$‎. ‎the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$‎. ‎the complete product split graph on $l‎ ...

In para-cycling, competitors are classed based on functional impairment resulting in cyclists with neurological and locomotor impairments competing against each other. In Paralympic competition, classes are combined by using a factoring adjustment to race times to produce the overall medallists. Pacing in short-duration track cycling events is proposed to utilize an "all-out" strategy in able-b...

We introduce a new sample complexity measure, which we refer to as split-sample growth rate. For any hypothesis H and for any sample S of size m, the split-sample growth rate τ̂H(m) counts how many different hypotheses can empirical risk minimization output on any sub-sample of S of size m/2. We show that the expected generalization error is upper bounded by O ( √