An r-coloring of a subset A of a finite abelian group G is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements a, b, c ∈ A with a + b = c. We investigate κr,G, the maximum number of sum-free r-colorings admitted by subsets of G, and our results show a close relationship between κr,G and largest sum-free sets of G.Given a sufficiently large abelian group G of type I, i.e., |G| has a prime divisor q with q ≡ 2 (mod 3). For r = 2, 3 we show that a subset A ⊂ G achieves κr,G if and only if A is a largest sum-free set of G. For even order G the result extends to r = 4, 5, where the phenomenon persists only if G has a unique largest sum-free set. On the contrary, if the largest sum-free set in G is not unique then A attains κr,G if and only if it is the union of two largest sum-free sets (in case r = 4) and the union of three ("independent") largest sum-free sets (in case r = 5).Our approach relies on the so called container method and can be extended to larger r in case G is of even order and contains sufficiently many largest sum-free sets.We are interested in the questions as how large κ r,G can be, given r ≥ 2 and G, and which subsets of G achieve the maximum.A straightforward lower bound for κ r,G is obtained by considering a largest sum-free set B ⊂ G, which gives κ r,G ≥ r |B| . The size of largest sum-free sets of G, denoted by µ(G), is

In this paper, we introduce the Banzhaf power indices for simple games on convex geometries. We define the concept of swing for these structures, obtaining convex swings. The number of convex swings and the number of coalitions such that a player is an extreme point are the basic tools to define the convex Banzhaf indices, one normalized and other probabilistic. We obtain a family of axioms that give rise to the Banzhaf indices. In the last section, we present a method to calculate the convex Banzhaf indices with the computer program Mathematica, and we apply this to compute power indices in the Spanish and Catalan parliaments and in the Council of Ministers of the European Union. 

As the disk formation mechanism(s) in Be stars is(are) as yet unknown, we investigate the role of rapidly rotating radiationdriven winds in this process. We implemented the effects of high stellar rotation on m-CAK models accounting for: the shape of the star, the oblate finite disk correction factor, and gravity darkening. For a fast rotating star, we obtain a two-component wind model, i.e., a fast, thin wind in the polar latitudes and an Ω-slow, dense wind in the equatorial regions. We use the equatorial mass densities to explore Hα emission profiles for the following scenarios: 1) a spherically symmetric star, 2) an oblate shaped star with constant temperature, and 3) an oblate star with gravity darkening. One result of this work is that we have developed a novel method for solving the gravity darkened, oblated m-CAK equation of motion. Furthermore, from our modeling we find a) the oblate finite disk correction factor, for the scenario considering the gravity darkening, can vary by at least a factor of two between the equatorial and polar directions, influencing the velocity profile and mass-loss rate accordingly, b) the Hα profiles predicted by our model are in agreement with those predicted by a standard power-law model for following values of the lineforce parameters: 1.5 k 3, α ∼ 0.6 and δ 0.1, and c) the contribution of the fast wind component to the Hα emission line profile is negligible; therefore, the line profiles arise mainly from the equatorial disks of Be stars.

Tibor Gallai conjectured that the edge set of every connected graph G on n vertices can be partitioned into ⌈n/2⌉ paths. Let G k be the class of all 2k-regular graphs of girth at least 2k − 2 that admit a pair of disjoint perfect matchings. In this work, we show that Gallai's conjecture holds in G k , for every k ≥ 3. Further, we prove that for every graph G in G k on n vertices, there exists a partition of its edge set into n/2 paths of lengths in {2k − 1, 2k, 2k + 1}.

Cancer cells can block the activation of T lymphocytes by deploying inhibitory signals to cell surface receptors that downregulate the immune response. Immune checkpoint inhibitors (ICI) are monoclonal antibodies that regulate the immune response by acting on these receptors. The use of ICI has been successful for cancer types that do not respond well to conventional chemotherapy, showing clinical benefit in various advanced and metastatic cancers and supporting the promise of cancer immunotherapy. However, in some cases, these treatments are associated with immune-related adverse events, many of which affect the digestive system. The treatment of immune-related adverse events depends on the affected organ and the severity of symptoms. Here, we review the commonly used US FDA-approved ICI and briefly outline their mechanism of action. We also describe the resulting collateral effects on the gastrointestinal tract, liver, and pancreas and discuss their management and prognosis.