Algebraic Limits Research Articles

Channel-based algebraic limits to conductive heat transfer

Recent experimental advances probing coherent phonon and electron transport in nanoscale devices at contact have motivated theoretical channel-based analyses of conduction based on the nonequilibrium Green's function formalism. The transmission through each channel has been known to be bounded above by unity, yet actual transmissions in typical systems often fall far below these limits. Building upon recently derived radiative heat transfer limits and a unified formalism characterizing heat transport for arbitrary bosonic systems in the linear regime, we propose new bounds on conductive heat transfer. In particular, we demonstrate that our limits are typically far tighter than the Landauer limits per channel and are close to actual transmission eigenvalues by examining a model of phonon conduction in a 1-dimensional chain. Our limits have ramifications for designing molecular junctions to optimize conduction.

A sufficiently complicated noded Schottky group of rank three

In 1974, Marden proved the existence of non-classical Schottky groups by a theoretical and non-constructive argument. Explicit examples are only known in rank two; the first one by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to construct non-classical Schottky groups in any rank. The method assumes the knowledge of certain algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups. The aim of this paper is to provide explicitly a sufficiently complicated noded Schottky group of rank three and explain how to use it to construct explicit non-classical Schottky groups.

Clustering in structure and reactions using configuration interaction techniques

Background: Atomic nuclei are remarkable quantum many-body systems where clustering properties develop naturally from underlying interactions between the constituent nucleons. Clustering degrees of freedom manifest themselves in multiple structure and reaction observables.Purpose: Our goal is to study nuclear clustering and its emergence in many-nucleon dynamics from nucleon-nucleon interactions. Clustering is a phenomenon that is known to emerge on the boundary between structure and reactions, therefore developing appropriate techniques that bridge the structure-reaction divide and establishing connections to observables is among our principal objectives. Showing consistency and how the new techniques can be reduced to well established other methods is an important part of this work.Methods: The configuration-interaction technique based on second quantization is used to treat the quantum many-body problem assuring that fermionic antisymmetry is fully satisfied. The use of the harmonic oscillator single-particle basis allows for the center-of-mass coordinate to be separated and prepared in a desired oscillator state for each cluster. The relative motion reaction basis channels are constructed by coupling clusters in different harmonic oscillator states with respect to their relative motion. Finally, using a resonating group method strategy we solve the generalized eigenvalue problem to obtain scattering channels. Structural clustering characteristics are discussed and the modified harmonic oscillator representation for scattering equations method is used to extract scattering observables.Results: New methods for treating clustering problems have been put forward. We demonstrate broad applicability of the developed techniques. Examples highlight connections with algebraic techniques, and the role of approximations leading to algebraic limits is assessed using realistic examples. Various types of clustering characteristics are used to study alpha clustering in light nuclei that are relevant to currently ongoing experimental efforts. We demonstrate the emergence of strongly clustered bands of states in beryllium, triple alpha channels in $^{12}\mathrm{C}$, and molecular type clustering in $^{21}\mathrm{Ne}$. Starting from nucleon-nucleon interactions without any additional assumptions scattering phase shifts for alpha-alpha scattering are determined and shown to be consistent with those observed.Conclusions: In this work we put forward a new configuration-interaction-based method that targets the physics of clustering, and further unifies nuclear structure and reactions. We provide detailed discussions and many examples highlighting features and advantages of the approach.

Extending pseudo-Anosov maps into compression bodies

We show that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if its (un)stable lamination is a projective limit of meridians. The proof is through 3-dimensional hyperbolic geometry, and involves an investigation of algebraic limits of convex cocompact compression bodies.

A NOTE ON ALGEBRAIC LIMIT OF A REAL FUNCTION

A new viewpoint on the notion of the limit of a function is proposed. We define a series of algebraic limit transitions of a function and prove that every real function has a number of non-trivial algebraic limits.

Algebraic limits of geometrically finite manifolds are tame

In his 1986 article [Th2], W. Thurston proposed that one might approachMarden’s conjecture from a dynamical point of view via a study of limits in the natural deformation space: its interior consists of geometrically finite manifolds, and promoting their tameness to algebraic limits on the boundary has proven to be a successful strategy to address Marden’s conjecture in special cases. In this paper we complete this part of his approach.

On Algebraic Limits and Topological Properties of Sets of Möbius groups in $ \\bar {R}^n $

Let G be the algebraic limit of a sequence $ \\{ G_m\\} $ of r generator subgroups of $ M(\\bar {R}^n) $ . We prove that (i) if $ G_m $ is elementary, then G is elementary; (ii) if $ G_m $ is not non-elementary and non-discrete and G satisfies the Condition A, then G is not non-elementary and non-discrete. Let $ E_l = \\{ (f_1, f_2,\\ldots ,f_l): f_1,f_2,\\ldots ,f_l\\in M(\\bar {R}^2), \\langle \\,f_1,f_2,\\ldots , \\,f_l \\rangle\\ elementary\\} $ and $ D_l = \\{ (f_1,f_2,\\ldots ,f_l): f_1,f_2,\\ldots ,f_l\\in M(\\bar {R}^2), \\langle \\,f_1,f_2,\\ldots ,f_l \\rangle\\ discrete\\} $ . we obtain that the set $ E_l , D_l\\cup E_l $ and $ D_l\\cap E_l^c $ are closed in $ \\underbrace {M(\\bar {R}^2)\ imes \\cdots \ imes M(\\bar {R}^2)}\\limits _{l} $ .

Iteration of mapping classes and limits of hyperbolic 3-manifolds

Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕ i X,Y)} i=1 ∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface Dϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-Dϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and Dϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕ si X,Y)} i=1 ∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.

Algebraic limits of Kleinian groups which rearrange the pages of a book

In this paper, we give examples of two new phenomena in Kleinian groups. We rst exhibit a sequence of homeomorphic marked hyperbolic 3-manifolds whose algebraic limit is not homeomorphic to any element in the sequence. We then use this construction to exhibit situations where the space of convex co-compact representations of a given 3-manifold group has many components but its closure is connected. Let M be a compact, irreducible, oriented 3-manifold and let D( 1(M)) denote the space of all discrete, faithful representations of 1(M) into PSL2(C). A sequence of representations { n} ⊂ D( 1(M)) converging to ∈ D( 1(M)) gives rise to a sequence {N n = H= n( 1(M))} of hyperbolic 3-manifolds, each of which is homotopy equivalent to M . The sequence {N n} is said to converge algebraically to N = H= ( 1(M)). (See [7, 13, 14] for more information about algebraic convergence of Kleinian groups.) In many situations (see [1, 6, 15, 24, 25, 27]), it has been shown that N n must be homeomorphic to N for all large enough n, and we had suspected that this would always be the case. In this paper, we give a collection of examples where N n is not homeomorphic to N for any n. Our sequences are quite well-behaved: the n( 1(M)) are convex co-compact and mutually quasiconformally conjugate, and the algebraic limit ( 1(M)) is geometrically nite. In our examples, M is obtained by gluing a collection of I -bundles to a solid torus along a family of parallel annuli. These manifolds are particularly simple examples of books of I -bundles (see [9]) where, to explain the terminology, one should think of the solid torus as the binding and the I -bundles

On discrete isometry groups of negative curvature

In this paper we extend well-known results concerning the algebraic limits and deformations of groups of hyperbolic isometries of hyperbolic 3-space, H 3 , to negatively curved groups. For us these will be groups of isometries of variable negative curvature metrics satisfying a pinching condition and in particular will include the R-rank one Lie groups. We accomplish these goals, as in the hyperbolic case, by producing a version of Jorgensen's inequality for such groups. Using an appropriate normalisation we can consider algebraic limits and deformations of such groups in the homeomorphism group of the n-ball, Hom(B n ). We ask that the generators of each group move continuously or some sequence of generators have limits in Hom(B n ), but there is no such restriction on the associated negatively curved metrics

Strong convergence of Kleinian groups and Carathéodory convergence of domains of discontinuity

In the theory of Kleinian groups, important examples of Kleinian groups are frequently constructed as algebraic limits of known sequences of Kleinian groups, for example quasi-conformal deformations of a Kleinian group. It is an important problem to determine the properties of the limit Kleinian group from information on the sequence converging to the limit. The topological properties of the domain of discontinuity and the limit set of a Kleinian group provide valuable pieces of information about the Kleinian group. It is reasonable to expect that in a good situation, the domain of discontinuity of the limit Kleinian group is the Carathéodory limit of the domains of discontinuity of the sequence, or, equivalently, the limit set of the limit Kleinian group is the Hausdorif limit of the limit sets of the sequence. Our main theorem (Theorem 5) shows that this is true when the sequence strongly converges to the limit, and the Kleinian groups of the sequence and the limit preserve the parabolicity in both directions and satisfy the condition (*) introduced by Bonahon.

A symbolic limit evaluation program in REDUCE

A method for the automatic evaluation of algebraic limits is described. It combines many of the techniques previously employed, including top-down recursive evaluation, power series expansion, and L'Hôpital's rule. It introduces the concept of a special algebraic form for limits. The method has been implemented in MODE-REDUCE.