where {e1, · · · , en} (resp. {ξ1, · · · , ξm}) is an orthonormal basis of the tangent (resp. normal) bundle, and R (resp. R) is the curvature tensor for the tangent (resp. normal) bundle. In the study of submanifold theory, De Smet, Dillen, Verstraelen, and Vrancken [5] made the following normal scalar curvature conjecture: Conjecture 1. Let h be the second fundamental form, and let H = 1 n tr...

h = h− · h where h is the class number of Q(ζp), h + is the class number of Q(ζp + ζ̄p), and h ∈ N. We now consider the problem of showing that if p divides h, it must divide h. We remark that in fact it is a conjecture of Vandiver that p never divides the factor h, and this has been checked for primes into the millions. Washington [2, pp. 158-129] produces heuristics however showing that withou...

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and $H \leqslant G$ index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log |G:H|}$. This inequality is best possible symmetric groups, we conjecture that it family arbitrarily large finite groups.

Let us say two (simple) graphs G,G′ are degree-equivalent if they have the same vertex set, and for every vertex, its degrees in G and in G′ are equal. In the early 1980’s, S. B. Rao made the conjecture that in any infinite set of graphs, there exist two of them, say G and H, such that H is isomorphic to an induced subgraph of some graph that is degree-equivalent to G. We prove this conjecture.

Let f(r) = minH P F∈E(H) 1 2|F | , where H ranges over all 3-chromatic hypergraphs with minimum edge cardinality r. Erdős-Lovász (1975) conjectured f(r) → ∞ as r → ∞. This conjecture was proved by Beck in 1978. Here we show a new proof for this conjecture with a better lower bound: f(r) ≥ ( 1 16 − o(1)) ln r ln ln r .

We prove that 3-dimensional Schrödinger operator with slowly decaying sparse potential has an a.c. spectrum that fills R. A new kind of WKB asymptotics for Green’s function is obtained. The absence of positive eigenvalues is established as well. Consider the Schrödinger operator H = −∆+ V, x ∈ R (1) We are interested in studying the scattering properties of H for the slowly decaying potential V...

In this paper, we extend Meek’s conjecture (Meek, 1997) from directed and acyclic graphs to chain graphs, and prove that the extended conjecture is true. Specifically, we prove that if a chain graph H is an independence map of the independence model induced by another chain graph G, then (i) G can be transformed into H by a sequence of directed and undirected edge additions and feasible splits ...

In an accumulation game, the Hider secretly distributes his given total wealth h among n locations, while the Searcher picks r locations and confiscates the material placed there. The Hider wins if what is left at the remaining n − r locations is at least 1; otherwise the Searcher wins. Ruckle’s Conjecture says that an optimal Hider strategy is to put an equal amount h/k at k randomly chosen lo...

Fix k≥2 and let H be a graph with χ(H)=k+1 containing critical edge. We show that for sufficiently large n, the unique n-vertex H-free maximum number of cycles is Tk(n). This resolves both question conjecture Arman, Gunderson Tsaturian [4].