The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair (X,∆) have finite, rational polyhedral fundamental domains. Let Z be an n-dimensional Fano manifold of index n − 1 such that −KZ = (n − 1)H for an ample divisor H. Let Γ be the base locus of a...

We discuss the space of complex exponential maps Eκ: z 7→ e z +κ. We prove that every hyperbolic component W has connected boundary, and there is a conformal isomorphism ΦW :W → H − which extends to a homeomorphism of pairs ΦW : (W,W ) → (H − ,H−). This solves a conjecture of Baker and Rippon, and of Eremenko and Lyubich, in the affirmative. We also prove a second conjecture of Eremenko and Lyu...

We study list homomorphism problems L-HOM(H) for the class of reflexive digraphs H (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs H, and appear to be more difficult for digraphs. However, it is known that each problem L-HOM(H) is NP-complete or polynomial time solvable. In this paper we focus on reflexive digraphs. We in...

Abstract Kreisel’s conjecture is the statement: if, for all $n\in \mathbb {N}$ , $\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$ then \forall x.\varphi (x)$ . For a theory of arithmetic T given recursive function h $T _{\leq h} $ holds if there proof $\varphi in whose code at most $h(\#\varphi )$ This notion depends on underlying coding. ${P}^h_T(x)$ predicat...

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)| |V (H)|−1 over all subgraphs H with at least two vertices. Generalizing the NashWilliams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb(G) ≤ k + d k+d+1 , then G decomposes into k + 1 forests with one having maximum degree at most d. The conjecture was previously proved for d = k+1 an...

A dominating set D for a graph G is a subset of V (G) such that any vertex in V (G)−D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G2H Vizing’s conjecture [10] states that γ(G2H) ≥ γ(G)γ(H) for every pair of graphs G, H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prov...

Let $H$ and $G$ be graphs. An $H$-colouring of is a proper edge-colouring $f:E(G)\rightarrow E(H)$ such that for any vertex $u\in V(G)$ there exists $v\in V(H)$ with $f\left (\partial_Gu\right )=\partial_Hv$, where $\partial_Gu$ $\partial_Hv$ respectively denote the sets edges in incident to vertices $u$ $v$. If admits an we say colours $G$. The question whether graph every bridgeless cubic add...

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...

In this paper‎, ‎using the discrete Fourier transform in the finite-dimensional Hilbert space C^n‎, ‎a class of nonRieszable equal norm tight frames is introduced ‎and‎ using this class, a bound for Fiechtinger Conjecture is presented. By the Fiechtinger Conjecture that has been proved recently, for given A,C>0 there exists a universal constant delta>0 independent of $n$ such that every C-equal...

Given a graphic degree sequence D, let χ(D) (respectively ω(D), h(D), and H(D)) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is D. It is proved that χ(D) ≤ h(D). Moreover, it is shown that a subdivision of a clique of order χ(D) exists where...