Narrow Results

In this paper, we are concerned with the $T$-Graphs, which are graphs defined based on the Topological structure of the given set. Precisely, for a given topology $T$ on a set $X$, a $T$-Graph ‘$G=(V,E)$’ is an undirected simple graph with the vertex set $V$ as $P(X)$ and the edge set $E$ as the set of all unordered pairs of nodes $u,v$ in $V$, denoted by $(u,v)\text{ or }(v,u)$, satisfying either ‘$u\in T$ and $u^{\text{c}}\cap v\in T$’ (or) ‘$v\in T$ and $v^{\text{c}}\cap u\in T$’.

The main purpose of this paper is to study the structure of $T$-Graphs for various topologies $T$ on a set $X$. Our goals in this paper are threefold. First, to show the $L_d(2,1)$ labeling number $\lambda (G,d)$ of any $T$-Graph $G$ exists finitely, if the labeling is $d$ multiple of non-negative integral values. In addition to show this labeling number $\lambda (G,d)$ is not just bounded above but bounded below as well. Second, to measure the bound values in terms of $d$ multiple of the order of the $T$-Graphs and finding a relation between the order of the $T$-Graphs and the maximum degree $\mathrm{\Delta }$ of the $T$-Graphs. Finally, third is to show that in case of $L(2,1)$$T$-graphs on a set with atleast 2 elements, the labeling number is $2(\mathrm{\Delta }+1)$ and is smaller than that of Griggs and Yeh’s conjecture value ${\mathrm{\Delta }}^2$.

The class of hypercubes is one of the most important and popular topologies for interconnection networks of multicomputer systems. This class includes the binary hypercube and generalized hypercube. Based on the observation that these two graphs can be constructed using a graph theoretic operation known as the product of graphs, we propose a new method for generating large symmetric graphs for networks of multicomputer systems. This method is essentially algebraic in nature, and makes use of the product of a class of graphs known as quasi-group graphs. We call the graphs we obtain PQG graphs. Because these graphs are constructed by an algebraic operation, it simplifies the analysis of their performance. Many of the well-known topologies can in fact be expressed as PQG graphs; this makes the method a very general one. We also investigate the problem of routing in a PQG graph, and propose a hardware implementation of the routing algorithm to reduce the delay in the routing of messages. We then apply our results to the Petersen graph and show that the product of such graphs has vastly superior topological properties than hypercubes of the same degree.

It was noticed by Harel in [Har86] that "one can define -complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular ω-language is -complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is -complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about ω-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given ω-rational function is continuous, while we show here that it is -complete to determine whether a given ω-rational function has at least one point of continuity. Next we prove that it is -complete to determine whether the continuity set of a given ω-rational function is ω-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function $F:{\mathrm{\Sigma }}^{\omega }\to {\mathrm{\Gamma }}^{\omega }$, one can construct a deterministic Büchi automaton ${𝒜}$ accepting a dense ${\mathrm{\Pi }}_2^0$-subset of ${\mathrm{\Sigma }}^{\omega }$ such that the restriction of $F$ to $L({𝒜})$ is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous $\omega$-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular $\omega$-languages.

The bifurcated, ordered Cayley tree with disorder and frustration as a model for relaxing granular media is examined numerically. Surprisingly, only exponential time dependencies are found despite frustration. Extensions to bidirectional models are discussed extensively and qualitatively new behavior is detected.

In this paper a local trade web (LTW) in the Asia-Pacific region is examined using the data derived from the United Nations and the International Monetary Fund. The topology of the LTW has been specified, based upon which the impacts of US financial crisis on the structural and behavior pattern of the LTW are further investigated. The major findings are given as follows. Firstly, the LTW is much more integrated than the global trade web; secondly, after the financial crisis, the fundamental structure of the network remains relatively stable but the strength of the web has been changed and the structure of the web has evolved over time. Economic implications for what have been observed are also discussed.

In this paper, topology monitoring of growing networks is studied. When some new nodes are added into a network, the topology of the network is changed, which needs to be monitored in many applications. Some auxiliary systems (network monitors) are designed to achieve this goal. Both linear feedback control and adaptive strategy are applied to designing such network monitors. Based on the Lyapunov function method via constructing a potential or energy function decreasing along any solution of the system, and the LaSalle's invariance principle, which is a generalization of the Lyapunov function method, some sufficient conditions for achieving topology monitoring are obtained. Illustrative examples are provided to demonstrate the effectiveness of the new method.

We study the searching efficiency of complex networks considering node's visual range within which a node can see its neighbors and knows the topology. We firstly introduce the network generating models and searching strategies. Using the generating function method, in both random networks and scale-free networks we derive the most-effective-visual-range (MEVR) which means every step of random walkers can find most of new nodes and we also obtain the searching-cost (SC) under visual range. To validate the generating function method, we perform simulations in random networks and scale-free networks. We also explain why the deviation between numerical simulation and theoretical prediction in scale-free networks is much larger than that in random networks. By studying the visual range of nodes in the networks, the results open the possibility to learn about the searching on networks with unknown topologies.

In the present paper, a variation of the widespread model of diffusion-limited aggregation (DLA) is presented. Unlike the traditional DLA model, where particles are attached to the aggregate whenever they touch it, we here restrict attachment by reducing the number of available bonds of the particles. This subtle change in the model changes the topological properties of the resulting aggregate.

By using a binary mixture of particles, with different coordination number, the fractal dimension (d_f), the spectral dimension (d_s) and the random walk dimension (d_w) are studied as a function of particle-type ratio. The behavior of the system shows non-negligible deviation from the traditional model.

We investigate vacuum transitions in lattice higgs models at finite temperature. The 2 dimensional U(1) Higgs model is used as a toy model. In the 4 dimensional SU(2) Higgs model the region of the phase transition and temperatures above it are considered. The couplings (β, κ, λ) = (2.25, 0.27, 0.5) and (8.0, 0.12996, 0.0017235) correspond to masses in lattice units (a_σm_H, a_σm_w) of (0.02, 0.05) and (0.2,0.2), respectively. The algorithm is described and a parallelized version is proposed. Taking the influence of the finite lattice into account we discuss temperature effects. We compare our results with perturbative estimates and claim that they link low and high temperature approximations.

The fastest supercomputers today such as Blue Gene/L, Blue Gene/P, Cray XT3 and XT4 are connected by a three-dimensional torus/mesh interconnect. Applications running on these machines can benefit from topology-awareness while mapping tasks to processors at runtime. By co-locating communicating tasks on nearby processors, the distance traveled by messages and hence the communication traffic can be minimized, thereby reducing communication latency and contention on the network. This paper describes preliminary work utilizing this technique and performance improvements resulting from it in the context of a n-dimensional k-point stencil program. It shows that even for simple benchmarks, topology-aware mapping can have a significant impact on performance. Automated topology-aware mapping by the runtime using similar ideas can relieve the application writer from this burden and result in better performance. Preliminary work towards achieving this for a molecular dynamics application, NAMD, is also presented. Results on up to 32,768 processors of IBM's Blue Gene/L, 4,096 processors of IBM's Blue Gene/P and 2,048 processors of Cray's XT3 support the ideas discussed in the paper.

Cosmic microwave background data shows the observable universe to be nearly flat, but leaves open the question of whether it is simply or multiply connected. Several authors have investigated whether the topology of a multiconnected hyperbolic universe would be detectable when 0.9<Ω<1. However, the possibility of detecting a given topology varies depending on the location of the observer within the space. Recent studies have assumed the observer sits at a favorable location. The present paper extends that work to consider observers at all points in the space, and (for given values of Ω_m and Ω_Λ and a given topology) computes the probability that a randomly placed observer could detect the topology. The computations show that when Ω=0.98 a randomly placed observer has a reasonable chance (~50%) of detecting a hyperbolic topology, but when Ω=0.99 the chances are low (<10%) and decrease still further as Ω approaches one.

It is shown that background fields of a topological character usually introduced as such in compactified string theories correspond to quantum degrees of freedom which parametrise the freedom in choosing a representation of the zero-mode quantum algebra in the presence of nontrivial topology.

We study inhomogeneities in the distribution of the excursion sets in the Cosmic Microwave Background (CMB) temperature maps obtained by the three years survey of the Wilkinson Microwave Anisotropy Probe (WMAP). At temperature thresholds |T| = 90 μK, the distributions of the excursion sets with over 200 pixels are concentrated in two regions, nearly at the antipodes, with galactic coordinates l = 94.7°, b = 34.4° and l = 279.8°, b = -29.2°. The centers of these two regions drift towards the equator when the temperature threshold is increased. The centers are located close to one of the vectors of ℓ = 3 multipole. The two patterns of the substructures in the distribution of the excursion sets are mirrored, with χ² = 0.7–1.5. There is no obvious origin of this effect in the noise structure of WMAP, and there is no evidence for a dependence on the galactic cut. Would this effect be cosmological, it could be an indication of an anomalously large component of horizon-size density perturbations, independent of one of the spatial coordinates, and/or of a non-trivial slab-like spatial topology of the Universe.

We study a class of topological black hole solutions in RSII braneworld scenario in the presence of a localized Maxwell field on the brane. Such a black hole can carry two types of charge, one arising from the extra dimension, the tidal charge, and the other from a localized gauge field confined to the brane. We find that the localized charge on the brane modifies the bulk geometry and in particular the bulk Weyl tensor. The bulk geometry does not depend on different topologies of the horizons. We present the temperature and entropy expressions associated with the event horizon of the braneworld black hole and by using the first law of black hole thermodynamics we calculate the mass of the black hole.

We describe the finite volume effects of CP-odd quantities, such as the neutron electric dipole moment and the anapole moment in the θ-vacuum, under different topological sectors. We evaluate the three-point Green's functions for the electromagnetic current in a fixed nontrivial topological sector in order to extract these CP-odd observables. We discuss the role of zero modes in the CP-odd Green's function and show that, in the quenched approximation, there is a power divergence in the quark mass for CP-odd quantities at finite volume.

An exact solution of the vacuum Einstein field equations over a nonsimply-connected manifold is presented. This solution is spherically symmetric and has no curvature singularity. It can be considered as a regularization of the Schwarzschild solution over a simply-connected manifold, which has a curvature singularity at the center. Spherically symmetric collapse of matter in ℝ⁴ may result in this nonsingular black-hole solution, if quantum-gravity effects allow for topology change near the center.

Certain exact solutions of the Einstein field equations over nonsimply-connected manifolds are reviewed. These solutions are spherically symmetric and have no curvature singularity. They provide a regularization of the standard Schwarzschild solution with a curvature singularity at the center. Spherically symmetric collapse of matter in ℝ⁴ may result in these nonsingular black-hole solutions, if quantum-gravity effects allow for topology change near the center or if nontrivial topology is already present as a remnant from a quantum spacetime foam.

When hadron-quark continuity is formulated in terms of a topology change at a density higher than twice the nuclear matter density $(n_0)$, the core of massive compact stars can be described in terms of quasiparticles of fractional baryon charges, behaving neither like pure baryons nor like deconfined quarks. Hidden symmetries, both local gauge and pseudo-conformal (or broken scale), emerge and give rise both to the long-standing “effective $g_A^{\ast }\approx 1$” in nuclear Gamow–Teller (GT) transitions at $\lesssim n_0$ and to the pseudo-conformal sound velocity $v_{\mathrm{\text{pcs}}}^2/c^2\approx 1/3$ at $\gtrsim 3n_0$. It is suggested that what has been referred to, since a long time, as “quenched $g_A$” in light nuclei reflects what leads to the dilaton-limit $g_A^{\mathrm{\text{DL}}}=1$ at near the (putative) infrared fixed point of scale invariance. These properties are confronted with the recent observations in GT transitions and in astrophysical observations.

In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory.

This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.