We study the problem of finding lowest common ancestors (LCA) in trees and directed acyclic graphs (DAGs). Specifically, we extend the LCA problem to DAGs and study the LCA variants that arise in this general setting. We begin with a clear exposition of Berkman and Vishkin’s simple optimal algorithm for LCA in trees. The ideas presented are not novel theoretical contributions, but they lay the ...

We consider the problem of learning skill templates for a parameterized reinforcement learning problem class T. That is, we assume that a task, i.e., an instance of the problem class, is defined by a task parameter vector τ ∈ T ⊆ R n and an associated interpretation. Likewise, a skill is considered as a parameterized policy with parameter vector θ ∈ R m. A parameterized skill [1] is a mapping Θ...

where both m(·) and σ(·) are unknown functions defined over R, the data {(Xt, Yt)}t=1 are weakly dependent stationary time series, and et is an error process with zero mean and unit variance. Suppose that {mθ(·)|θ ∈ Θ} is a family of parametric specification to the regression function m(x) where θ ∈ R is an unknown parameter belonging to a parameter space Θ. This paper considers testing the val...

This article provides a regularity criterion for the surface quasigeostrophic equation with supercritical dissipation. This criterion is in terms of the norm of the solution in a Orlicz-Morrey space. The result shows that, if a weak solutions θ satisfies Z T 0 ‖∇θ(·, s)‖ α α−r M L2 logP L 1 + ln(e+ ‖∇⊥θ(·, s)‖L2/r ) ds <∞, for some 0 < r < α and 0 < α < 1, then θ is regular at t = T . In view o...

and Applied Analysis 3 where bi t , rj t , aik t , cil t , djk t , ejl t , ejl t i, k 1, 2, . . . ,N; j, l 1, 2, . . . ,M are assumed to be continuous ω-periodic functions and the delays τik, δjk, ηil, ξjl are assumed to be positive constants. The system 1.3 is supplemented with the initial condition: xi θ φi θ , yj θ ψi θ , θ ∈ −τ, 0 , φi 0 > 0, ψi 0 > 0, 1.4

We prove that classical Hecke operators on Maass forms are a special case of completely positive maps on II1 factors, associated to a pair of isomorphic subfactors. This representation induces several matrix inequalities on the eigenvalues of the Hecke operators Maass forms. In particular the family of eigenvalues corresponding to an eigenvector is a completely bounded multiplier of the Hecke a...

ν = q · P M = E − E′ is the lepton’s energy loss in the nucleon rest frame (in earlier literature sometimes ν = q · P ). Here, E and E′ are the initial and final lepton energies in the nucleon rest frame. Q2 = −q2 = 2(EE′−−→k · −→k ′)−m2 −m2 ′ where m (m ′) is the initial (final) lepton mass. If EE′ sin2(θ/2) m2 , m2 ′ , then ≈ 4EE′ sin2(θ/2), where θ is the lepton’s scattering angle with respe...

Let f∗(z) = ∞ ∑ j=0 aj z j be a convergent series in which {aj}j=0 are known real numbers. In this paper, by referring to Osler’s lemma [8], we obtain explicit forms of the two bivariate series ∞ ∑ j=0 an j+m r j cos(α+ j)θ and ∞ ∑ j=0 an j+m r j sin(α+ j)θ, where r, θ are real variables, α ∈ R, n ∈ N and m ∈ {0, 1, . . . , n − 1}. With some illustrative examples, we also show how to obtain the...

ν = q · P M = E − E′ is the lepton’s energy loss in the nucleon rest frame (in earlier literature sometimes ν = q · P ). Here, E and E′ are the initial and final lepton energies in the nucleon rest frame. Q2 = −q2 = 2(EE′−−→k · −→k ′)−m2 −m2 ′ where m (m ′) is the initial (final) lepton mass. If EE′ sin2(θ/2) m2 , m2 ′ , then ≈ 4EE′ sin2(θ/2), where θ is the lepton’s scattering angle with respe...