Lecture on Channel Coding

Considers channel capacity, coding rate, repetition code, Hamming code, Hamming distance

The kinetics of the 2π+2π photodimerisation reactions of single-crystalline derivatives of trans-cinnamic acid: a study by infrared microspectroscopy

The kinetics of the photodimerisation reactions of the 2- and 4-β-halogeno-derivatives of trans-cinnamic acid (where the halogen is fluorine, chlorine or bromine) have been investigated by infrared microspectroscopy. It is found that none of the reactions proceed to 100% yield. This is in line with a reaction mechanism developed by Wernick and his co-workers that postulates the formation of isolated monomers within the solid, which cannot react. β-4-Bromo and β-4-chloro-trans-cinnamic acids show approximately first order kinetics, although in both cases the reaction accelerates somewhat as it proceeds. First order kinetics is explained in terms of a reaction between one excited- and one ground-state monomer molecule, while the acceleration of the reaction implies that it is promoted as defects are formed within the crystal. By contrast β-2-chloro-trans-cinnamic acid shows a strongly accelerating reaction which models closely to the contracting cube equation. β-2-Fluoro- and β-4-fluoro-trans-cinnamic acids show a close match to first order kinetics. The 4-fluoro-derivative, however, shows a reaction that proceeds via a structural intermediate. The difference in behaviour between the 2-fluoro- and 4-fluoro-derivative may be due to different C–HF hydrogen bonds observed within these single-crystalline starting materials.

Perspex machine VII: The universal perspex machine

The perspex machine arose from the unification of projective geometry with the Turing machine. It uses a total arithmetic, called transreal arithmetic, that contains real arithmetic and allows division by zero. Transreal arithmetic is redefined here. The new arithmetic has both a positive and a negative infinity which lie at the extremes of the number line, and a number nullity that lies off the number line. We prove that nullity, 0/0, is a number. Hence a number may have one of four signs: negative, zero, positive, or nullity. It is, therefore, impossible to encode the sign of a number in one bit, as floating-, point arithmetic attempts to do, resulting in the difficulty of having both positive and negative zeros and NaNs. Transrational arithmetic is consistent with Cantor arithmetic. In an extension to real arithmetic, the product of zero, an infinity, or nullity with its reciprocal is nullity, not unity. This avoids the usual contradictions that follow from allowing division by zero. Transreal arithmetic has a fixed algebraic structure and does not admit options as IEEE, floating-point arithmetic does. Most significantly, nullity has a simple semantics that is related to zero. Zero means "no value" and nullity means "no information." We argue that nullity is as useful to a manufactured computer as zero is to a human computer. The perspex machine is intended to offer one solution to the mind-body problem by showing how the computable aspects of mind and. perhaps, the whole of mind relates to the geometrical aspects of body and, perhaps, the whole of body. We review some of Turing's writings and show that he held the view that his machine has spatial properties. In particular, that it has the property of being a 7D lattice of compact spaces. Thus, we read Turing as believing that his machine relates computation to geometrical bodies. We simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to pro-ram than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to construct a fractal of perspex machines with the cardinality of a real-numbered line or space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. We cite an experiment that has been proposed to test the physical reality of the perspex machine's model of time, but we make no claim that the physical universe works this way or that it has the cardinality of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of physical things, including manufactured computers and biological organisms, that have a cardinality no greater than the real-number line.

Optimization of apodization functions in terahertz transient spectrometry

A quadratic programming optimization procedure for designing asymmetric apodization windows tailored to the shape of time-domain sample waveforms recorded using a terahertz transient spectrometer is proposed. By artificially degrading the waveforms, the performance of the designed window in both the time and the frequency domains is compared with that of conventional rectangular, triangular (Mertz), and Hamming windows. Examples of window optimization assuming Gaussian functions as the building elements of the apodization window are provided. The formulation is sufficiently general to accommodate other basis functions. (C) 2007 Optical Society of America

Estimation of distances in virtual environments using size constancy

It is reported in the literature that distances from the observer are underestimated more in virtual environments (VEs) than in physical world conditions. On the other hand estimation of size in VEs is quite accurate and follows a size-constancy law when rich cues are present. This study investigates how estimation of distance in a CAVETM environment is affected by poor and rich cue conditions, subject experience, and environmental learning when the position of the objects is estimated using an experimental paradigm that exploits size constancy. A group of 18 healthy participants was asked to move a virtual sphere controlled using the wand joystick to the position where they thought a previously-displayed virtual cube (stimulus) had appeared. Real-size physical models of the virtual objects were also presented to the participants as a reference of real physical distance during the trials. An accurate estimation of distance implied that the participants assessed the relative size of sphere and cube correctly. The cube appeared at depths between 0.6 m and 3 m, measured along the depth direction of the CAVE. The task was carried out in two environments: a poor cue one with limited background cues, and a rich cue one with textured background surfaces. It was found that distances were underestimated in both poor and rich cue conditions, with greater underestimation in the poor cue environment. The analysis also indicated that factors such as subject experience and environmental learning were not influential. However, least square fitting of Stevens’ power law indicated a high degree of accuracy during the estimation of object locations. This accuracy was higher than in other studies which were not based on a size-estimation paradigm. Thus as indirect result, this study appears to show that accuracy when estimating egocentric distances may be increased using an experimental method that provides information on the relative size of the objects used.

On the maximal connected component of hypercube with faulty vertices

In evaluating an interconnection network, it is indispensable to estimate the size of the maximal connected components of the underlying graph when the network begins to lose processors. Hypercube is one of the most popular interconnection networks. This article addresses the maximal connected components of an n -dimensional cube with faulty processors. We first prove that an n -cube with a set F of at most 2n - 3 failing processors has a component of size greater than or equal to2(n) - \F\ - 1. We then prove that an n -cube with a set F of at most 3n - 6 missing processors has a component of size greater than or equal to2(n) - \F\ - 2.

The locally twisted cubes

This paper introduces a new variant of the popular n-dimensional hypercube network Q(n), known as the n-dimensional locally twisted cube LTQ(n), which has the same number of nodes and the same number of connections per node as Q(n). Furthermore. LTQ(n) is similar to Q(n) in the sense that the nodes can be one-to-one labeled with 0-1 binary sequences of length n. so that the labels of any two adjacent nodes differ in at most two successive bits. One advantage of LTQ(n) is that the diameter is only about half of the diameter of Q(n) We develop a simple routing algorithm for LTQ(n), which creates a shortest path from the source to the destination in O(n) time. We find that LTQ(n) consists of two disjoint copies of Q(n) by adding a matching between their nodes. On this basis. we show that LTQ(n) has a connectivity of n.

On the maximal connected component of hypercube with faulty vertices (II)

evaluating the fault tolerance of an interconnection network, it is essential to estimate the size of a maximal connected component of the network at the presence of faulty processors. Hypercube is one of the most popular interconnection networks. In this paper, we prove that for ngreater than or equal to6, an n-dimensional cube with a set F of at most (4n-10) failing processors has a component of size greater than or equal to2"-\F-3. This result demonstrates the superiority of hypercube in terms of the fault tolerance.

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(2(n), 4) enjoys several attractive topological properties. Let max_epsilon(G) (m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that max_epsilon(RC(2n,4))(m) = Sigma(i)(r)=(0)(p(i)/2 + i)2(Pi), where p(0) > p(1) > ... > p(r) are nonnegative integers defined by m = Sigma(i)(r)=(0)2(Pi). We then apply this formula to find the bisection width of RC(2(n), 4). The conclusion shows that, as n-dimensional cube, RC(2(n), 4) enjoys a linear bisection width. (c) 2005 Elsevier B.V. All rights reserved.

On the maximal connected component of a hypercube with faulty vertices III

Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.

Fault tolerance of Mobius cubes under two forbidden fault set models

An n-dimensional Mobius cube, 0MQ(n) or 1MQ(n), is a variation of n-dimensional cube Q(n) which possesses many attractive properties such as significantly smaller communication delay and stronger graph-embedding capabilities. In some practical situations, the fault tolerance of a distributed memory multiprocessor system can be measured more precisely by the connectivity of the underlying graph under forbidden fault set models. This article addresses the connectivity of 0MQ(n)/1MQ(n), under two typical forbidden fault set models. We first prove that the connectivity of 0MQ(n)/1MQ(n) is 2n - 2 when the fault set does not contain the neighborhood of any vertex as a subset. We then prove that the connectivity of 0MQ(n)/1MQ(n) is 3n - 5 provided that the neighborhood of any vertex as well as that of any edge cannot fail simultaneously These results demonstrate that 0MQ(n)/1MQ(n) has the same connectivity as Q(n) under either of the previous assumptions.

Locally twisted cubes are 4-pancyclic

The locally twisted cube is a newly introduced interconnection network for parallel computing. Ring embedding is an important issue for evaluating the performance of an interconnection network. In this paper, we investigate the problem of embedding rings into a locally twisted cube. Our main contribution is to find that, for each integer l is an element of (4,5,...,2(n)}, a ring of length I can be embedded into an n-dimensional locally twisted cube so that both the dilation and the load factor are one. As a result, a locally twisted cube is Hamiltonian. We conclude that a locally twisted cube is superior to a hypercube in terms of ring embedding capability. (C) 2004 Elsevier Ltd. All rights reserved.

Pancyclicity of Mobius cubes with faulty nodes

An interconnection network with n nodes is four-pancyclic if it contains a cycle of length l for each integer l with 4 <= l <= n. An interconnection network is fault-tolerant four-pancyclic if the surviving network is four-pancyclic in the presence of faults. The fault-tolerant four-pancyclicity of interconnection networks is a desired property because many classical parallel algorithms can be mapped onto such networks in a communication-efficient fashion, even in the presence of failing nodes or edges. Due to some attractive properties as compared with its hypercube counterpart of the same size, the Mobius cube has been proposed as a promising candidate for interconnection topology. Hsieh and Chen [S.Y. Hsieh, C.H. Chen, Pancyclicity on Mobius cubes with maximal edge faults, Parallel Computing, 30(3) (2004) 407-421.] showed that an n-dimensional Mobius cube is four-pancyclic in the presence of up to n-2 faulty edges. In this paper, we show that an n-dimensional Mobius cube is four-pancyclic in the presence of up to n-2 faulty nodes. The obtained result is optimal in that, if n-1 nodes are removed, the surviving network may not be four-pancyclic. (C) 2005 Elsevier B.V. All rights reserved.

Symmetry-break in Voronoi tessellations

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.

Top down influence on visuo-tactile interaction modulates neural oscillatory responses

Multisensory integration involves bottom-up as well as top-down processes. We investigated the influences of top-down control on the neural responses to multisensory stimulation using EEG recording and time-frequency analyses. Participants were stimulated at the index or thumb of the left hand, using tactile vibrators mounted on a foam cube. Simultaneously they received a visual distractor from a light emitting diode adjacent to the active vibrator (spatially congruent trial) or adjacent to the inactive vibrator (spatially incongruent trial). The task was to respond to the elevation of the tactile stimulus (upper or lower), while ignoring the simultaneous visual distractor. To manipulate top-down control on this multisensory stimulation, the proportion of spatially congruent (vs. incongruent) trials was changed across blocks. Our results reveal that the behavioral cost of responding to incongruent than congruent trials (i.e., the crossmodal congruency effect) was modulated by the proportion of congruent trials. Most importantly, the EEG gamma band response and the gamma-theta coupling were also affected by this modulation of top-down control, whereas the late theta band response related to the congruency effect was not. These findings suggest that gamma band response is more than a marker of multisensory binding, being also sensitive to the correspondence between expected and actual multisensory stimulation. By contrast, theta band response was affected by congruency but appears to be largely immune to stimulation expectancy.