Abstract. We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: T , where P is an n × n row-stochastic matrix, E is a nonnegative n × n rank-one row-stochastic matrix, and 0

We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: "For any matrix A = [cP + (1 c)E] , where P is an n n row-stochastic matrix, E is a nonnegative nn rank

of correct reconstruction tend to 1/|A | as n → ∞? It is known that reconstruction is possible if dλ 2 2(M)> 1, where λ2(M) is the second eigen-value of M. Moreover, in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree

. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1

Abstract. We consider the eigenvalue problem for the fractional p−Laplacian in an open bounded, possibly disconnected set Ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After dis-cussing some regularity issues for eigenfuctions, we show that the second eigenvalue λ2(Ω) is well

for |λ2|. The purpose of this note is to disseminate these known results more widely and to show how they imply, as a very special case, some recent theorems of Haveliwala and Kamvar about the second eigenvalue of the Google matrix. If A = (aij) is an n × n real matrix, A has n (counting algebraic

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1.1 Representing RNA secondary structure as planar tree-graph The primary structure of a linear RNA chain molecule is the nucleotide sequence s = s1s2 …si …sL, and runs in the direction 5 ' → 3 ' terminus. L defines the number of nucleotides and si ∈ ∑ = (A, C, G, U) is the biochemical nucleotide at the i th position. The RNA molecule s folds upon itself relatively rapid into a two-dimensional RNA secondary structure S [1]. The structure S is stabilized by the canonical Watson-Crick G≡C and A=U, and wobble G=U base pairings. (Fig. S1) A planar RNA secondary structure S is mathematically described by a set of base pairings (i, j) ∈ S connecting bases si and sj, where i < j [2]. Given (i, j) and (k, l) ∈ S, a nucleotide can base pair to at most one other nucleotide i.e., i = k ⇔ j = l. A set of ∆ ∈ Z + consecutive base pairs defines a stem for stabilizing the structure against thermal fluctuations. The number of unpaired nucleotides between

Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best-known explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinear