On avoding r-repetitions in [formula omitted

A (finite or infinite) word is said to be k-th power-free if it does not contain k consecutive equal blocks. A colouring of the integer lattice points in the n-dimensional Euclidean space is power-free if there exists a positive integer k such that the sequence of colours of consecutive points on any straight line is a k-th power-free word. The Thue threshold of mathbb Z^n is the least number of colours t(n) allowing a power-free colouring of the integer lattice points in the n-dimensional Euclidean space.Answering a question of Grytczuk (2008), we prove that t(2)=t(3)=2. Moreover, we show the existence of a 2-colouring of the integer lattice points in the Euclidean plane such that the sequence of colours of consecutive points on any straight line does not contain squares of length larger than 26.In order to obtain these results, we study repetitions in Toeplitz words. We show that the Toeplitz word generated by any sequence of primitive partial words of maximal length k is k-th power-free. Moreover, adding a suitable hypothesis on the positions of the holes in the generating sequence, we obtain that also the subwords occurring in the considered Toeplitz word according to an arithmetic progression of suitable difference, are k-th power-free words.

According to the approximate theory of transport in a conducting organic material containing exponential traps, ln J versus ln V plots are straight lines with slope l=TC/T, where TC is the characteristic temperature of the trap distribution. It is assumed in this theory that the concentration pt of trapped holes is much larger than the concentration p of free holes. Our experiments and recent literature results show that at high applied voltages the observed ln J versus ln V plots deviate from the straight lines and bend down. The numerical solution presented in this article shows that at high voltages the contribution of p to the space charge does not remain negligible. Calculated ln J versus ln V plots do bend down consistent with our experimental results. The current approaches the trap-filled limit asymptotically as the applied voltage approaches infinity.

When the human motor system adapts to novel dynamics of the arm during reaching, hand trajectories tends to converge toward a roughly straight line. This straight line is thought to be the desired trajectory of the system. Trial-to-trial changes in performance are well described by a first order state-space model: errors in a given trial affect performance on the subsequent trial as a function of the distance in state space between the two trials. This function describes the generalization patterns that govern adaptation. Whereas the desired trajectory and the generalization function have been quantified for short-term adaptation, little is known about their behavior with long-term training. We report that when subjects are trained to reach in novel force fields over multiple days, the state-space model suggests that the desired trajectory undergoes systematic changes. In a constant field, the desired trajectory becomes curved. The direction of change of the desired trajectory is affected by the training protocol, such that occasional unperturbed trials (catch trials) caused subject to increasingly under-compensate for the perturbations whereas a lack of such trials caused subjects to increasingly overcompensate for the perturbations. We suggest that the desired trajectory is not constant, but is a result of an optimization that considers the success rate. In this optimization process, subjects weigh more heavily the importance of certain trials: either those that are infrequent or those that cause large errors.

Two points in a polygon are called if the straight line between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point in P is visible from some point of G. The author provides an introduction to art gallery theorems and surveys the recent results of the field. The emphasis is on the results rather than the techniques. Several new problems that have the same geometric flavor as art gallery problems are also examined.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.

Recent theoretical results have completely solved the problem of determining the minimal length path for a vehicle moving from an initial configuration to a final configuration. Time- optimal paths for a constant-speed vehicle are a subset of the minimum length paths. The time-optimal paths consist of sequences of arcs of circles and straight lines. The Pontrya gin Maximum Principle introduces concepts (dual variables, bang-bang solutions, singular solutions, and transversality conditions) that provide important insight into the nature of the time-optimal paths. We have created a module that finds the time-optimal path from an initial canfrguration to a final configuration. We have demonstrated that the paths can be followed by a large (820-kg) mobile robot.

Recent theoretical results have completely solved the problem of determining the minimum length path for a vehicle with a minimum turning radius moving from an initial configuration to a final configuration. Time optimal paths for a constant speed vehicle are a subset of the minimum length paths. This paper uses the Pontryagin maximum principle to find time optimal paths for a constant speed vehicle. The time optimal paths consist of sequences of axes of circles and straight lines. The maximum principle introduces concepts (dual variables, bang-bang solutions, singular solutions, and transversality conditions) that provide important insight into the nature of the time optimal paths. We explore the properties of the optimal paths and present some experimental results for a mobile robot following an optimal path.

The λ Bootis stars, a group of late B- to early F-type Population I stars, have surface abundances that resemble the general metal depletion pattern found in the interstellar medium. Inspired by the recent result that the fundamental parameters of these peculiar stars differ in no respect from those of a comparison sample of normal stars, the hypothesis of an interaction between a star and a diffuse interstellar cloud is considered as a possible explanation of the peculiar abundance pattern. It is found that such a scenario is able to explain the selective accretion of interstellar gas depleted in condensable elements as well as the spectral range of the λ Bootis phenomenon.

With parallel coordinates (abbr. ||-coords) the perceptual barrier imposed by our 3-dimensional habitation is breached enabling the visualization of multidimensional problems. The representation of N-dimensional points by polygonal lines is deceptively simple and additional ideas are needed to represent multivariate relations. In this talk, a panorama of highlights from the foundations to the most recent results, and interlaced with applications, are intuitively developed. This is also an opportunity to demystify some subtleties.By learning to untangle patterns from ||-coords displays (Fig. 1, 2) a powerful knowledge discovery process has evolved. It is illustrated on a real dataset together with guidelines for exploration and good query design. Realizing that this approach is intrinsically limited (see Fig. 3 -- left) leads to a deeper geometrical insight, the recognition of M-dimensional objects recursively from their (M-- 1)-dimensional subsets (Fig. 3 -- right). Behind this striking cognitive success lies a special family of planes unique to ||-coords, the superplanes, whose points are represented by straight (rather than polygonal) lines. It emerges that any linear N-dimensionsal relation is represented by (N-- 1) indexed points. Points representing lines have two indices, points representing planes 3 indices and so on. In turn, powerful geometrical algorithms (e.g. for intersections, containment, proximities) and applications including classification Fig. 4 emerge. The classifier's power is demonstrated by obtaining a rule for the recognition of hostile vehicles from afar by their noise signature.A smooth surface in 3-D is the envelope of its tangent planes each of which is represented by 2 points Fig. 6. As a result, a surface in 3-D is represented by two planar regions and in N-dimensions by (N-- 1) regions. This is equivalent to representing a surface by its normal vectors, rather than projections as in standard surface descriptions. Developable surfaces are represented by curves Fig. 7 revealing the surfaces' characteristics. Convex surfaces in any dimension are recognized by the hyperbola-like (i.e. having two assymptotes) regions from just one orientation Fig. 5 -- right, Fig. 8, Fig. 10 -- right. Non-orientable surfaces (i.e. like the Mobius strip) yield stunning patterns Fig. 9 unlocking new geometrical insights. Non-convexities like folds, bumps, coiling, dimples and more are no longer hidden Fig. 10 -- left and are detected from just one orientation. Evidently this representation is preferable for some applications even in 3-D. By the way, many of these results were first discovered visually and then proved mathematically; in the true spirit of Geometry.These state of the art examples show what has been achieved on the representation of complex relations and how they generalize to N-dimensions. The patterns persist in the presence of errors deforming in ways revealing the type and magnitude of the errors and that's good news for the applications. New vistas for multidimensional visualization are emerging. The processing is performed directly on the data and is not display bound opening the way for the exploration of massive datasets. Only the results are displayed in patterns where information is immensely concentrated (see again Fig. 3 - just the point is needed) and without any display clutter. These are the graphs of multidimensional relations within the data. The challenge is to speed up the recursive algorithm, employing among others, intelligent agents to rapidly identify relational properties. For some applications it will be worthwhile to display the patterns of partial results, during the processing, to enable computational steering. We stand on the threshold of cracking the gridlock of multidimensional visualization.

Recent results have established that Melan's theorem can be applied to discrete elastic systems governed by the Coulomb friction law only when the normal contact reactions are uncoupled from the tangential (slip) displacements. For coupled systems, periodic loading scenarios can be devised which lead to either shakedown or cyclic slip depending on the initial condition. Here we explore this issue in the simplest coupled system involving two contact nodes. The evolution of the system ‘memory’ is conveniently represented graphically by tracking the instantaneous condition in slip-displacement space. The frictional inequalities define directional straight line constraints in this space that tend to ‘sweep’ the operating point towards the safe shakedown condition if one exists. However, if the safe shakedown region is defined by a triangle in which two adjacent sides correspond to the extremal positions of the two frictional constraints for the same node, initial conditions can be found leading to cyclic slip. The critical value of a loading parameter at which this occurs can be determined by requiring that three of the four constraint lines intersect in a point. Below this value, shakedown occurs for all initial conditions. Similar concepts can be extended to multi-node systems.

In previous work, it was suggested that the sequence regularity property of cortical neurons could be accounted for if the local geometric structure of the cortex were a recapituation of the global complex logarithmic structure of the retinotopic mapping. This model is developed in detail: the excitatory and inhibitory structure of cortical receptive fields may be approximated by a complex logarithmic local geometry, coupled with an intra-cortical lateral inhibition operator which may flow unidirectionally yet still create "rotating" receptive field structure. The direction of intra-cortical lateral inhibition follows the borders of cortical ocular dominance columns, which are the approximate images under the global complex logarithmic mapping, of exponentially spaced, horizontal straight lines in the visual field. Two different topological structures are discussed for the local cortical manifold. The binocular trigger features of cortical neurons follow from the same geometric model, and the ratio of binocular to monocular cortical cells is related to the size and shape of cortical dendritic tree's by an application of integral geometry. Recent results in optical pattern recognition are cited to suggest that the rotation and size invariant properties of the cortical map are essential to any cross-correlational basis for stereopsis. Finally, a meromorphic function is presented which is both locally and globally complex logarithmic in its structure, and therefore represents the model presented in this and previous papers in a concise mathematical form. This function is closely related to the description of a Karman vortex pattern, in fluid mechanics, and leads to the suggestion that the boundary conditions of layer IV of the cortex (i.e. periodic ocular dominance columns) are causally related to the existence of sequence regularity in the cortex. The developmental implications of this statement are that the specification of neural connections in the cortex may follow directly, both locally and globally, from the detailed nature of the cortical boundary conditions (i.e. anatomy), coupled with general physico-mathematical considerations of continuity and differentiability in the neural fiber flow.

Well-specified feature detection models of visual character recognition typically assume feature sampling independence; that is, they assume that the detection of one feature is probabilistically independent of the detection of others. Recent results have suggested this assumption may be suspect with letter-like stimuli. The present study utilized very simple stimuli consisting of up to two straight-line segments that were either physically connected or separated by a gap. A strong model that assumed that features are reported if and only if they are sampled together with independence could not be rejected even when the lines were connected.

The electronic structure and vibrational frequencies of the stable C84 isomer of D2 symmetry: theory and experiment

Stable multivariate Eulerian polynomials and generalized Stirling permutations

Automatic aesthetic drawing of plane graphs has recently created intense interest due to its broad applications, and as a consequence, a number of drawing methods, such as the straight line drawing, convex drawing, orthogonal drawing, rectangular drawing and box-rectangular drawing, have come out [8,9,3,4,5,6,7, 10,11,14,16,23,29,33]. In this talk we survey the recent results on these drawings of plane graphs.KeywordsPlane GraphOuter FaceGraph DrawingStraight Line DrawingConvex DrawingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.