Isoperimetric Problem and Meta-fibonacci Sequences

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$ , such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as b e (i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and b v (i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of b e (i, G) (b v (i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation b c (i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T 2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T 2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that b c (i, T 2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.

Fine-Grained Layered Multicast

Traditional approaches to receiver-driven layered multicast have advocated the benefits of cumulative layering, which can enable coarse-grained congestion control that complies with TCP-friendliness equations over large time scales. In this paper, we quantify the costs and benefits of using non-cumulative layering and present a new, scalable multicast congestion control scheme which provides a fine-grained approximation to the behavior of TCP additive increase/multiplicative decrease (AIMD). In contrast to the conventional wisdom, we demonstrate that fine-grained rate adjustment can be achieved with only modest increases in the number of layers and aggregate bandwidth consumption, while using only a small constant number of control messages to perform either additive increase or multiplicative decrease.

Propriedades estáticas e dinâmicas de multicamadas magnéticas acopladas quasiperiodicamente

In this work we study, for two different growth directions, multilayers of nanometric magnetic metallic lms grown, using Fibonacci sequences, in such a way that the thickness of the non-magnetic spacer may vary from a pair of lms to another. We applied a phenomenological theory that uses the magnetic energy to describe the behavior of the system. After we found numerically the global minimum of the total energy, we used the equilibrium angles to obtain magnetization and magnetoresistance curves. Next, we solved the equation of motion of the multilayers to nd the dispersion relation for the system. The results show that, when spacers are used with thickness so that the biquadratic coupling is strong in comparison to the bilinear one, non usual behaviors for both magnetization and magnetoresistance are observed. For example, a dependence on the parity of the Fibonacci generation utilized for constructing the system, a low magnetoresistance step in low external magnetic fields and regions that show high sensibility to small variations of the applied field. Those behaviors are not present in quasiperiodic magnetic multilayers with constant spacer thickness

Can fractal-like spectra be experimentally observed in aperiodic superlattices?

We numerically investigate the effects of inhomogeneities in the energy spectrum of aperiodic semiconductor superlattices, focusing our attention on Thue-Morse and Fibonacci sequences. In the absence of disorder, the corresponding electronic spectra are self-similar. The presence of a certain degree of randomness, due to imperfections occurring during the growth processes, gives rise to a progressive loss of quantum coherence, smearing out the finer details of the energy spectra predicted for perfect aperiodic superlattices and spurring the onset of electron localization. However, depending on the degree of disorder introduced, a critical size for the system exists, below which peculiar transport properties, related to the pre-fractal nature of the energy spectrum, may be measured.

Fibonacci bimodal maps

We introduce the Fibonacci bimodal maps on the interval and show that their two turning points are both in the same minimal invariant Cantor set. Two of these maps with the same orientation have the same kneading sequences and, among bimodal maps without central returns, they exhibit turning points with the strongest recurrence as possible.

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45

Hybrid threshold adaptable quantum secret sharing scheme with reverse Huffman-Fibonacci-tree coding

With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m-bonacci sequences to detect eavesdropping. Meanwhile, we encode m-bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.

When 'listeners can't talk' : comparing active listening in opening sequences of telephone and online counselling

While there is growing use of online counselling, little is known about its interactional organisation and how it compares to telephone counselling. This is despite past research suggesting that both counsellors and clients report the impact of the different modalities on the presentation and management of the counselling interaction. This paper compares the interactional affordances of telephone and online web counselling in opening sequences on Kids Help Line, a 24-hour Australian counselling service for children and young people up to the age of 25. We examine two ways that counsellors show active listening through response tokens and formulations. The analysis describes how counsellors’ use of minimal response tokens facilitate the clients’ problem presentation and are used in the management of turn taking and sequence organisation. For example, counsellors use the response token Mm hm to show that they understand that the client’s unit of talk to is not yet complete, and to affirm or invite the client to continue speaking. Formulations in phone and web counselling are another way that counsellors display active listening to re-present stretches of the clients’ preceding talk. In phone and web counselling, however, the respective modalities can complicate matters of turn transition and sequence organisation. By examining actual phone and online counselling sessions, this paper offers empirical demonstrations of the interactional affordances of phone and online counselling, and shows how the institutional practice of active listening is accomplished across different counselling modalities