Let m be a natural number. It is proved that the simple groups G2(q) where q = 32m+1 and m ≥ 1,is uniquely determined by the set of orders of its maximal abelian subgroups. Mathematics Subject Classification: 20D05, 20D08

‎In this paper we prove that a simplicial complex is determined‎ ‎uniquely up to isomorphism by its barycentric subdivision as well as‎ ‎its comparability graph‎. ‎We also put together several algebraic‎, ‎combinatorial and topological invariants of simplicial complexes‎.

A semigroup is a set of elements to which is related an operation usually called multiplication and an equivalence relation, such that the set is closed and associative relative to the operation. We shall discuss, briefly, finite semigroups which are uniquely factorable in the same sense as the multiplicative semigroup of all nonzero integers. Clifford' defined an arithmetic in such a way as to...

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from G to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C4-saturated graphs; there are only ten of them.

We construct a one-dimensional uniquely ergodic cellular automaton which is not nilpotent. This automaton can perform asymptotically infinitely sparse computation, which nevertheless never disappears completely. The construction builds on the self-simulating automaton of Gács. We also prove related results of dynamical and computational nature, including the undecidability of unique ergodicity,...

It is shown that all uniquely homogeneous spaces are connected. We characterize the uniquely homogeneous spaces that are semitopological or quasitopological groups. We identify two properties of homogeneous spaces called skew-2-flexibility and 2-flexibility that are useful in studying unique homogeneity. We also construct a large family of uniquely homogeneous spaces with only trivial continuou...

The interplay between viruses and host immune responses is critical to determine either a successful infection or resolution. Mammalian cells have developed different mechanisms to recognize invading pathogens via their pathogen-associated molecular patterns (PAMPs), which are often nucleic acid products (e.g., doublestranded RNAs or dsRNA) generated during viral genome replication [1]. To evad...

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k–list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...

A mixed hypergraph is a triple H = (X, C,D) where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with the same color and each D-edge has two vertices with distinct colors. H = (X, C,D) is called a mixed hypertree if there exists a tree T = (X, E) such that ...

Let ~1,22 . . . . . ~,; n/>2 be any properties of graphs. A vertex (~L, ~2 . . . . . J~,,)-partition of a graph G is a partition (V1, l~,...,/7,,) of V(G) such that for each i = 1,2 . . . . . n the induced subgraph G[Vi] has the property ~i. A graph G is said to be uniquely (~1,~2 . . . . . ~,)-partitionable if G has unique vertex (2~1, ~2 , . . . , ~,)-partition. In the present paper we invest...