Dense Set Of Points Research Articles

The Complexity Of Nowhere Differentiable Continuous Functions

It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By “differentiable” we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21]) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.)

Surface reconstruction from outdoor image sequences

This paper describes the results of a study aimed at inferring the surface structure of outdoor scenes from video data acquired by a vehicle-mounted TV camera. The method is based on a decomposition of each frame of the video sequence into a dense set of feature points. Through the application of a structure-from-motion algorithm, the 3D locations of the time-consistent features are estimated and sequentially updated as each new frame is processed. The resultant point-based 3D representation has been found to be reliable, accurate and rich enough to enable the surface structure of the viewed scene to be reconstructed. Examples of the surfaces extracted by this ‘bottom-up’ approach from a 12 second video sequence are presented here. The implications of these results for autonomous vehicle navigation are discussed.

A Generalized Scanning Technique for Display of Parametrically Defined Surfaces

Special attention is given to the adaptation of the technique for efficiency to both polynomial and rational Bezier patches. The resulting algorithms are simple and robust. Their speed depends mainly on the surface area of the object being rendered, not on the number of surface patches. The technique is particularly suitable for rapid display of geometrically complex objects. Since the technique directly renders from the definitions of the surfaces without needing polygons, it can maintain a relatively compact database. Several examples with timing are included. The technique described creates smooth, shaded images for parametrically defined surfaces. It depends on a general surface scan to generate a dense set of points that represents the surface. Hidden surfaces are eliminated by sorting the sample points into a z-buffer and retaining the points nearest the viewer. Alternative schemes are given for computing the surface normal for each point in the z-buffer. Conventional illumination routines can then be applied to determine the intensities at each pixel, although the reflection-mapping technique is preferred for this article.

Continua with a dense set of end points

Continua with a dense set of end points

ON TYPICAL PROPERTIES OF LEVEL SETS OF LYAPUNOV EXPONENTS

The level sets of Lyapunov exponents of linearized systems are considered as functions of the linearized Cauchy problem. It is proved that lower semicontinuity is a typical property for these functions. Typicality is understood in the Baire sense: a property is typical if it is possessed by a dense set of points which is a countable intersection of open sets.Bibliography: 11 titles.

Singular behavior of the density of states and the Lyapunov coefficient in binary random harmonic chains

We study the integrated density of statesH(ω2) of a chain of harmonic oscillators with a binary random distribution of the masses. We show in particular that there is a dense set of values of the squared frequency for which the differenceH(ω2+ɛ)-H(ω2) has a singularity of the type ¦ɛ¦2α, multiplied by a periodic function of ln ¦ɛ¦, where the exponent α and the period depend continuously onω2. In the region where α < 1/2,H is not differentiate on a dense set of points. The same type of singularities is also present in the Lyapunov coefficient.

The collision channel without feedback

A model is proposed for the situation where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> users share a common communication resource but, because of unknown time offsets among their clocks, cannot transmit their data packets in a time-sharing mode and, because of the lack of a feedback link, can never determine these time offsets and also can never be sure of the outcomes of their individual packet transmissions. Each user is required to make his packet transmissions at times determined by a protocol signal that is independent of the data to be sent. The capacity and zero-error capacity regions of this channel are determined for both the unsynchronized and slot-synchronized cases; these four regions are shown to coincide. It is further shown that a dense set of rate points on the outer boundary of this region can be achieved in the slot-synchronized case. Specific constructions of protocol sequences for achieving these points are given, and the technique of "decimation decoding" is introduced for identifying the sender of each successfully transmitted packet. Maximum-erasure burst-correcting codes over an alphabet of arbitrary size are constructed and shown to suffice for reconstructing the packets lost in "collisions" when these protocol sequences are used.

Functions with a dense set of proper local maximum points

Let X X be any metric space. The existence of continuous real functions on X X , with a dense set of proper local maximum points, is shown. Indeed, given any σ \sigma -discrete set S ⊂ X S \subset X , the set of all f ∈ C ( X ) f \in C(X) , which assume a proper local maximum at each point of S S , is a dense subset of C ( X ) C(X) . This implies, for a perfect metric space X X , the density in C ( X , Y ) C(X,Y) of "nowhere constant" continuous functions from X X to a normed space Y Y . In this way, two questions raised in [2] are solved.

Dynamical properties of quasihyperbolic toral automorphisms

AbstractWe study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate measurably, but never topologically, and are never weak Bernoulli. The points of period k become uniformly distributed exponentially fast, and Lipschitz functions mix exponentially fast. Every reasonably smooth compact null set has the property that there is a dense set of periodic points whose entire orbit misses the set, but this is false for general compact null sets. Katznelson's property of almost weak Bernoulli can be strengthened to a certain exponential rate of independence, but breaks down at a critical number. Finally, open sets have return times that decay exponentially fast.

Applications of the Connes spectrum to C∗-dynamical systems, II

We show that for a C ∗-dynamical system ( A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to G ̂ if and only if every nonzero closed ideal in G × α A has a nonzero intersection with A. Denote by G J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G J are discrete, then G X α A is simple if and only if A is G-simple and Γ( α) = G ̂ . This result is applicable not only when G is discrete but also when G≃ R or G≃ T provided that A is not primitive. Specializing to single automorphisms (i.e., G= Z ) we show that if (the transposed of) α acts freely on a dense set of points in A ̂ , then Λ(α)= T . The converse is only proved when A is of type I.

On the computation of theoretical seismograms for multimode surface waves

Summary It is needlessly expensive to use standard computational techniques, when applying inverse Fourier transformation to construct theoretical seismograms from frequency-domain data for multimode surface waves. Such techniques require the evaluation of dispersion and attenuation information for each mode, at each of a dense set of points which are equally spaced in frequency. These evaluations are by far the most expensive part of the computation of theoretical seismograms for surface waves. By further development of a method proposed by Aki, and departing from the standard, equal-frequency-interval computational techniques, it is possible to decrease the required number of dispersion and attenuation evaluations. With our new method a decrease of a factor of almost two for the fundamental mode, and approximately five for the higher modes is obtained, i.e. we obtain an increase in computational efficiency of 200 per cent for the fundamental mode and 500 per cent for the higher modes. The basis of the new method is the approximation of the amplitude and apparent initial phase spectra by quadratic and linear functions, respectively; this permits an analytic quadrature of the Fourier integral.

Calculation of phonon-induced gap anisotropy in superconducting Zn

Leavens and Carbotte have given a microscopic theory of gap anisotropy applicable to the case of a weak-coupling superconductor. The theory relates the anisotropy in the gap directly to Fermi surface anisotropy and anisotropy in the electron-phonon interaction. We present detailed calculations of the phonon-induced anisotropy for the case of Zn. The gap is calculated for a dense set of points on the irreducible one-twenty-fourth of the Fermi surface. From this information the average gap and the pure single-crystal critical temperature are computed and compared with the dirty-limit case. The specific heat and spin-lattice relaxation are calculated as examples of thermodynamic properties.

Gap anisotropy in a weak coupling superconductor

We derive formulas for the gap edge and transition temperature of a pure-crystal anisotropic superconductor. The formulas are based on a simplified form of the Eliashberg gap equations applicable in the weak coupling limit. The kernels of these equations depend on the electron-phonon interaction. We calculate these from pseudopotential theory for the electron-ion interaction and a Born-von Kármán force constant system for the lattice dynamics. The anisotropy in the gap is calculated at a dense set of points on the irreducible ( 1 48 )th of the Fermi surface. The average gap, critical temperature and gap anisotropy distribution are all calculated. Comparison is made with specific heat data and the spin lattice relaxation is also discussed.

Computation of a class of subsonic flow patterns

For flows bounded by segments of straight lines and segments of free boundaries, the flow in the pseudo-logarithmic plane was considered in [B.2]. The method described was based on the determination of a fundamental solution possessing a singularity at a point corresponding to a source or a sink. The computation of the fundamental solution requires the evaluation of many terms which are iterated double integrals involving a functionP(?) which appears in the differential equation. The computational aspects of the evaluation of these integrals are discussed in this paper. First a numerical procedure for approximating the functionP(?) by a polynomial is described. Then the approximate evaluation of the iterated double integrals is discussed. This may be accomplished by replacing the integrals by summations over a sufficiently dense set of points or by using special systems such as FORMAC.

Holomorphic functions with dense sets of Plessner points

PROOF. First we deal with the case where a = 0, and we begin with the construction of a locally univalent function having exactly one Plessner point. Let G be a simply connected Riemann surface, of finite area, and consisting of the unit disk together with a smooth vermiform appendage that winds its way over the plane in such a way that each point of the plane is a boundary point of G. Let the function f map the disk D conformally onto G, and let it map the origin onto the point w = 0 in the portion of G that covers the unit disk in the w-plane. Then there exists exactly one point ei0 such that the image of each curve in D from 0 to ei0 passes through the full length of the appendage. The cluster set of f relative to each curve in D that terminates at ei0 is the extended plane; therefore ei0 is a Plessner point of f. By adding further appendages to G, each appropriately branched, we obtain a simply connected Riemann surface G* such that every continuous one-to-one mapping from D to G* has a dense set of Plessner points on C. If we observe the precaution of keeping the area of G* finite, then each conformal mappingf* from D to G* has a set of Fatou points of measure 2wr, hence a set of Plessner points of measure 0. The case where a = 2r is trivial, as we see by examining the function

High frequency wave-vector dependent conductivity of a degenerate electron gas

We derive formulas for the gap edge and transition temperature of a pure-crystal anisotropic superconductor. The formulas are based on a simplified form of the Eliashberg gap equations applicable in the weak coupling limit. The kernels of these equations depend on the electron-phonon interaction. We calculate these from pseudopotential theory for the electron-ion interaction and a Born-von Kármán force constant system for the lattice dynamics. The anisotropy in the gap is calculated at a dense set of points on the irreducible (148)th of the Fermi surface. The average gap, critical temperature and gap anisotropy distribution are all calculated. Comparison is made with specific heat data and the spin lattice relaxation is also discussed.

Harmonic and Killing Vector Fields in Compact Orientable Riemannian Spaces with Boundary

Publisher Summary This chapter presents the results on harmonic and killing vector fields to the case of Riemannian spaces with boundary. The harmonic tensors on such Riemannian spaces are studied, and the relations between curvature and relative Betti numbers in Riemannian spaces with boundary are discussed. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the second fundamental form is negative (positive) semi-definite, then there does not exist a harmonic (Killing, conformal Eilling) vectoy field tangential to B other than the zero vector field. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the mean curvature of B is negative (positive) or zero, then there does not exist a harmonic (Killing, conformal Killing) vector field normal to B other than the zero vector field. If an M admits a Killing vector field normal to the boundary B and vanishing only on a nowhere dense set of points on B, then the boundary is totally geodesic.

Analytic transformations of everywhere dense point sets

There is a well known theorem in the theory of point sets, due to Cantor,t to the effect that All enumerable, everywhere dense linear point sets without first and last points have the same order type as the rational numbers. That is, any set of this type can be mapped on the rational points of a line by a one to one correspondence which preserves order, and consequently any two sets of this type can be mapped on one another by such a correspondence. A correspondence of two everywhere dense point sets clearly determines at most one continuous function which maps the segments on which the given sets are everywhere dense on one another, and also generates the correspondence. The requirement that the correspondence preserve order is equivalent to the requirement that a continuous mapping function exist, so that we may state the above theorem in the following form: For any two enumerable linear point sets, each everywhere dense on an open interval, a continuous function can be found which maps the two intervals on one another, and effects a onte to one correspondence between the point sets. Since the function of this theorem is by no means uniquely determined, the question naturally arises as to whether we can place further restrictions on it without destroying the validity of the theorem. It turns out that we may always require the function which effects the mapping to be analytic and it is the demonstrationi of this fact and some related questions which occupy our attention in this paper.