Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cusps. These minimizers are the same ones identified by Hironaka, Aaber-Dunfiled, Kin-Takasawa independently. In the case $g \equiv 6 \pmod{12}$ we find a new family of pseudo-Anosovs defined on $\varSigma_g$ with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling two cusps. We prove that if $\delta_g^+$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $\varSigma_g$, then $$\limsup_{\substack{g \equiv 6 \pmod{12} g \to \infty}} g \log \delta^+_g \le 2 \log \delta(D_5) \approx 1.0870,$$ where $\delta(D_n)$ is the minimal dilatation of pseudo-Anosovs on an $n$-punctured disk. We also study monodromies of fibrations on N(1). We prove that if $\delta_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with $n$ punctures, then $$\limsup_{n \to \infty} n \log \delta_{1,n} \le 2 \log \delta(D_4) \approx 1.6628.$$Comment: 46 pages, 14 figures; version 3: Major change in Section 2.1, and minor correction

Minimum Equivalent Precedence Relation Systems

In this paper two related simplification problems for systems of linear inequalities describing precedence relation systems are considered. Given a precedence relation system, the first problem seeks a minimum subset of the precedence relations (i.e., inequalities) which has the same solution set as that of the original system. The second problem is the same as the first one except that the ``subset restriction'' in the first problem is removed. This paper establishes that the first problem is NP-hard. However, a sufficient condition is provided under which the first problem is solvable in polynomial-time. In addition, a decomposition of the first problem into independent tractable and intractable subproblems is derived. The second problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described. The results in this paper generalize those in [Moyles and Thompson 1969, Aho, Garey, and Ullman 1972] for the minimum equivalent graph problem and transitive reduction problem, which are applicable to unweighted directed graphs

Graviton Mode Function in Inflationary Cosmology

We consider the production of gravitons in an inflationary cosmology by approximating each epoch of change in the equation of state as sudden, from which a simple analytic graviton mode function has been derived. We use this mode function to compute the graviton spectral energy density and the tensor-induced cosmic microwave background anisotropy. The results are then compared to the numerical calculations which incorporate a smooth radiation-matter phase transition. We find that the sudden approximation is a fairly good method. Besides, in determining the frequency range and amplitude of the mode function, we introduce a pre-inflationary radiation-dominated epoch and use a physically sensible regularization method.Comment: 26 pages, revtex, 4 figures (available upon request). We have added a new Section VIII with four new figures, in which Eq. (35) is used to calculate the graviton spectral energy density and the tensor-induced CMB anisotrop

Density Matrix Renormalization Group Lagrangians

We introduce a Lagrangian formulation of the Density Matrix Renormalization Group (DMRG). We present Lagrangians which when minimised yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz, as well as within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.Comment: 6 page