One-parameter Family Of Curves Research Articles

Tracking Critical Points on Evolving Curves and Surfaces

In recent years it became apparent that geophysical abrasion can be well characterized by the time evolution N(t) of the number N of static balance points of the abrading particle. Static balance points correspond to the critical points of the particle’s surface represented as a scalar distance function r, measured from the center of mass of the particle, so their time evolution can be expressed as . The mathematical model of the particle can be constructed on two scales: on the macro (global) scale the particle may be viewed as a smooth, convex manifold described by the smooth distance function r with equilibria, while on the micro (local) scale the particle’s natural model is a finely discretized, convex polyhedral approximation r Δ of r, with equilibria. There is strong intuitive evidence suggesting that under some particular evolution models (e.g., curvature-driven flows) N(t) and N Δ(t) primarily evolve in the opposite manner (i.e. if one is increasing then the other is decreasing and vice versa). This observation appears to be a key factor in tracking geophysical abrasion. Here we create the mathematical framework necessary to understand these phenomena more broadly, regardless of the particular evolution equation. We study micro and macro events in one-parameter families of curves and surfaces, corresponding to bifurcations triggering the jumps in N(t) and N Δ(t). Based on this analysis we show that the intuitive picture developed for curvature-driven flows is not only correct, it has universal validity, as long as the evolving surface r is smooth. In this case, bifurcations associated with r and r Δ are coupled to some extent: resonance-like phenomena in N Δ(t) can be used to forecast downward jumps in N(t) (but not upward jumps). Beyond proving rigorous results in the case of evolving planar curves for the limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable. This property serves as the basis for the second, experimental part of our research where we demonstrate via computer simulations that the phenomena on evolving surfaces appear to be closely analogous to the planar case, however, they also show additional geometric features which are still not completely understood.

Огибающая семейства линий

A one-parameter family of algebraic curves has an
 envelope line, which may be imaginary in certain cases. Jakob
 Steiner was right, considering the imaginary images as creation of
 analysis. In the analysis a real number is just a part of a complex
 number and in certain conditions the initial real values can give an
 imaginary result. But Steiner was wrong in denying the imaginary
 images in geometry. The geometry, in contrast to the single analytical
 space exists in several spaces: Euclidean geometry operates
 only on real figures valid and does not contain imaginary figures
 by definition; pseudo-Euclidean geometry operates on imaginary
 images and constructs their images, taking into account its own
 features. Geometric space is complex and each geometric object in
 it is the complex one, consisting of the real figure (core) having the
 "aura" of an imaginary extension. Thus, any analytical figure of the
 plane is present at every point of the plane or by its real part or by
 its imaginary extension. Would the figure’s imaginary extension be
 visible or not depends on the visualization method, whether the
 image has been assumed on superimposed epures – the Euclideanpseudo-Euclidean
 plane, or the image has been traditionally assumed
 only in the Euclidean plane. In this paper are discussed cases when
 a family of algebraic curves has an envelope, and is given an answer
 to a question what means cases of complete or partial absence of
 the envelope for the one-parameter family of curves. Casts some
 doubt on widely known categorical st

Bimodal entanglement entropy distribution in the many-body localization transition

We introduce the cut averaged entanglement entropy in disordered periodic spin chains and prove it to be a concave function of subsystem size for individual eigenstates. This allows us to identify the entanglement scaling as a function of subsystem size for individual states in inhomogeneous systems. Using this quantity, we probe the critical region between the many-body localized (MBL) and ergodic phases in finite systems. In the middle of the spectrum, we show evidence for bimodality of the entanglement distribution in the MBL critical region, finding both volume law and area law eigenstates over disorder realizations as well as within \emph{single disorder realizations}. The disorder averaged entanglement entropy in this region then scales as a volume law with a coefficient below its thermal value. We discover in the critical region, as we approach the thermodynamic limit, that the cut averaged entanglement entropy density falls on a one-parameter family of curves. Finally, we also show that without averaging over cuts the slope of the entanglement entropy \vs subsystem size can be negative at intermediate and strong disorder, caused by rare localized regions in the system.

THE OPTICAL–INFRARED EXTINCTION CURVE AND ITS VARIATION IN THE MILKY WAY

ABSTRACT The dust extinction curve is a critical component of many observational programs and an important diagnostic of the physics of the interstellar medium. Here we present new measurements of the dust extinction curve and its variation toward tens of thousands of stars, a hundred-fold larger sample than in existing detailed studies. We use data from the APOGEE spectroscopic survey in combination with ten-band photometry from Pan-STARRS1, the Two Micron All-Sky Survey, and Wide-field Infrared Survey Explorer. We find that the extinction curve in the optical through infrared is well characterized by a one-parameter family of curves described by R(V). The extinction curve is more uniform than suggested in past works, with σ ( R ( V ) ) = 0.18 , and with less than one percent of sight lines having R ( V ) > 4 . Our data and analysis have revealed two new aspects of Galactic extinction: first, we find significant, wide-area variations in R(V) throughout the Galactic plane. These variations are on scales much larger than individual molecular clouds, indicating that R(V) variations must trace much more than just grain growth in dense molecular environments. Indeed, we find no correlation between R(V) and dust column density up to E ( B − V ) ≈ 2 . Second, we discover a strong relationship between R(V) and the far-infrared dust emissivity.

Geometric characteristics of a class of cubic curves with rational offsets

Planar Bézier curves that have rationally parameterized offsets can be classified into two classes. The first class is composed of curves that have Pythagorean hodographs (PH) and the second class is composed of curves that do not have PHs but can have rational PHs after reparameterization by a fractional quadratic transformation. This paper reveals a geometric characterization for all properly-parameterized cubic Bézier curves in the second class. The characterization is given in terms of Bézier control polygon geometry. Based on the derived conditions, we also present a simple geometric construction of G1 Hermite interpolation using such Béziercurves. The construction results in a one-parameter family of curves if a solution exists. We further prove that there exists a unique value of the parameter which minimizes the integral of the squared norm of the second order derivative of the curves.

Generalized Linear Systems on Curves and Their Weierstrass Points

Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (ℐ, ε) consisting of a torsion-free, rank-1 sheaf ℐ on C, and a map of vector spaces ε: V → Γ(C, ℐ). If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C t degenerating to C, and each family of linear systems (ℒ t , ε t ) along C t , with ℒ t invertible, degenerating to (ℐ, ε), the corresponding Weierstrass divisors degenerate to a subscheme whose associated 0-cycle is W. We show that the limit subscheme contains always an “intrinsic” subscheme, canonically associated to (ℐ, ε), but the limit itself depends on the family ℒ t .

One generalization of the class of helical functions

This paper is devoted to the study of domains whose boundaries are attainable by one-parameter families of curves formed by the rotation of a curve specially chosen for every family. We establish characteristics of analytic functions that map the unit circle on these domains. In addition, we single out subclasses of domains with rectifiable quasiconformal boundaries. We establish certain sufficient conditions for the univalence of functions that are analytic in mentioned domains.

PROFILES OF INFLATED SURFACES

We study the shape of inflated surfaces introduced in [3] and [12]. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves which we describe explicitly as the solutions of a certain differential equation.

Analytical study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data.

A two-point [formula omitted] Hermite interpolating family of spirals

A one-parameter family of spirals that can match planar, two-point G 1 Hermite data is presented. These spirals can be used as an alternative to the biarc, which is also a one-parameter family of curves that can match two-point G 1 Hermite data. Some suggestions on choosing the free parameter of the family of spirals is given. It is shown that there is a unique G 1 Hermite interpolating spiral that passes through a given point in an allowable region. Three examples of the use of these spirals are given: curve completion with spirals, design with spirals, and approximation of the clothoid by spirals.

Turbulent thermal boundary layer on a permeable flat plate

Scaling laws are established for the profiles of temperature, turbulent heat flux, rms temperature fluctuation, and wall heat transfer in the turbulent boundary layer on a flat plate with transpiration. In the case of blowing, the temperature distribution represented in scaling variables outside the viscous sublayer has a universal form known from experimental data for flows over impermeable flat plates. In the case of suction, the temperature distribution is described by a one-parameter family of curves. A universal law of heat transfer having the form of a generalized Reynolds analogy provides a basis for representation of the heat flux distributions corresponding to different Reynolds numbers and transpiration velocities in terms of a function of one variable. The results are obtained without invoking any special closure hypotheses.

The distinctiveness of a curve in a parameterized neighborhood: extraction and applications

A new feature of curves pertaining to the acceptance/rejection decision in curve detection is proposed. The feature measures a curve's distinctiveness in its neighborhood, which is modeled by a one-parameter family of curves. A computational framework based on the Hough transform for extracting the distinctiveness feature is elaborated and examples of feature extractors for the circle and the ellipse are given. It is shown that the proposed feature can be extracted efficiently and is effective in separating signals from false positives. Experimental results with circle and ellipse testing that strongly support the efficiency and effectiveness claims are obtained. The results further demonstrate that the proposed feature exhibits good noise resiliency.

Review: An investigation of single particle breakage tests for coal handling system of the gladstone port

Aspects of the literature on single particle breakage test have been reviewed in this article. The test procedures that are commonly used by the researchers in examining and measuring the breakage characteristics of the ore and coal particles are also discussed. It appears that most of the common size distribution function fitting techniques were not suitable for accurate representation of the size distributions obtained from a pendulum breaking process. The single impact test, double impact test (drop weight test, pendulum test) and slow compression test can be used to study the behaviour of the single particle breakage events. The single impact test, slow compression test and drop weight test cannot measure the energy utilization pattern in single particle breakage events, but this can be determined from the pendulum test. The energy utilized for breakage was predominantly dependent upon the size and shape of the specimen, level of input energy and the breakage properties of the specimen. This review highlights that the size distribution curves were linear in the fine particle region and have varying curvature in the coarser region, the gradient of the linear fine particle region of the size distribution curves increases with an increase in the specific comminution energy. The comminution energy increases with input energy at lower levels of input energy but at the higher levels of input energy the comminution energy did not show the same proportional increase. At a given level of input energy, the size distribution resulting from the breakage of the particles by the pendulum apparatus can be represented by a one-parameter family of curves.

Single-particle breakage tests of Gladstone Port Authority's coal by a twin pendulum apparatus

Abstract South Blackwater and Ensham coal samples were tested in single-particle breakage tests using a laser detector, computer-monitored twin pendulum breakage apparatus to measure the energy utilization pattern of the breakage particles. The single-particle breakage tests were conducted with three particle sizes (−16:0 + 13:2 mm, -13:2 + 11:2 and -11:2 + 9:5 mm) of two types of coal at five input energy levels. The results showed a small variation of comminution energy of same-sized particles when broken at one input energy because South Blackwater and Ensham coal are anthracite coals. The specific comminution energy increases linearly with the input energy for different sizes of South Blackwater coal. The breakage products of the two sizes and two types of coal are shown, where it is clear that the fine in the product increases with the specific comminution energy. The size distribution graphs are curves and not straight lines. The twin pendulum breakage product results show that the fineness of the products increases with the decrease of particle size. The results also show that South Blackwater coal degrades more easily than Ensham coal. At a particular level of input energy, the size distribution resulting from the breakage of coal particles can be represented by a one-parameter family of curves.

Some Invariants of Admissible Homotopies of Space Curves

A regular homotopy of a generic curve in a three-dimensional projective space is called admissible if it defines a generic one-parameter family of curves in which every curve has neither self-intersections nor inflection points, is not tangent to a smooth part of its evolvent, and has no tangent planes osculating with the curve at two different points. We indicate some invariants of admissible homotopies of space curves and prove, in particular, that the curve \(\user1{x} = cos t\user1{,y} = sin t,z = cos 3t\) cannot be deformed in the class of admissible homotopies into a curve without flattening points.

Simple proof of the Lagrange multiplier rule for the fixed endpoint problem in the calculus of variations

A new simple proof of the Lagrange multiplier rule is presented in this paper. The approach used involves simple analytical techniques that are very easy to follow and does not involve theorems on imbeddability in a one-parameter family of curves or matrix-rank analysis as do most of the existing techniques. The proof is here developed for the fixed-endpoint problem in a three-dimensional space.

The closure theorem of Poncelet

A report on a joint study together with C. Kers, F. Oort and D. W. Raven on historical and mathematical aspects of Poncelet's closure theorem. Proofs of the theorem by Griffiths (1976), Jacobi (1828) and Poncelet himself (1822) are discussed and a new result is reported concerning a certain one-parameter family of curves. This family of curves arises naturally from arguments in Poncelet's original proof and it offers an interesting case of strong non-commutativity of dualizing and specializing.

Highly negative electric probes in a flowing continuum plasma

The general problem of an electric probe in a flowing continuum plasma is solved using perturbation theory, together with the method of matched asymptotic expansions. The small parameter used is the inverse of the dimensionless probe potential. The lowest order results yield a quasineutral region in which the ion concentration is constant, and an ion sheath region in which the electron density is zero. The electric potential and the electric field are zero at the edge of the sheath region. The remaining problem consists of solving the sheath equations. A solution is found for spherical and cylindrical probes in the case that the sheath is thin compared with the radius of the probe, but thick compared with the viscous boundary layer. For sheaths that are not thin compared with the probe radius, an iterative method of solution is proposed. The results are presented as a one-parameter family of curves. It is shown that there exists a limiting current, which obtains when the ion density around the probe remains essentially equal to the freestream value. The theoretical results obtained are in fair agreement with available experimental data. Attention is given to other theoretical results, as well as to various improvements neededmore » for further progress.« less

BACK-DIFFUSION IN A FINITE MEDIUM WITH A MOVING BOUNDARY

A mathematical model is formulated to describe the distribution of solute concentration in a layer of a chemical solution when a free surface recedes by continuous evaporation of the solvent. It is considered that the evaporation activates a back-diffusion of solute away from the receding free surface and into the remaining liquid, resulting in an unsteady one-dimensional concentration field. Formulation of the problem is on the basis of Fick‘s laws of diffusion and is shown to be equivalent to solving a fixed-point boundary, initial-value problem for a general, second-order, linear parabolio equation. If a(t) = l(t)/D, where l(t) is the liquid depth variation in time and D is the diffusion coefficient, a solution of the boundary-value problem is required in the form of a functional on a(t). It is found that the problem has a unique solution within an interval 0 < t < T, for some T > 0, when a(t) is analytic and non-vanishing. A method for constructing an approximate mathematical solution, when 0 < /da/dt/ ≤4, is described. It is further shown that an exact solution is available when the form of the depth variation l(t) corresponds to one member of a particular one-parameter family of curves.

Parametric Analysis of Jammed Active Satellite Links

This paper is a parametric study of active satellite repeater communications links in a jamming environment. Under heavy jamming conditions with an earth coverage satellite antenna, the communication effectiveness of the link is presented as a oneparameter family of curves. A crucial parameter which is a figure of merit of the link is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PD^{2}/T</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> is the power radiated by the active satellite repeater, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> is the diameter of the receiving dish and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> is the receiving system noise temperature. Jamming-limited and thermal noise-limited conditions are identified and their implications are discussed.