Envelope of a Family of Curves

2016 ◽  
Vol 4 (4) ◽  
pp. 14-18 ◽  
Author(s):  
Гирш ◽  
A. Girsh
Keyword(s):  
Real Number ◽  
Euclidean Plane ◽  
Euclidean Geometry ◽  
Algebraic Curves ◽  
Geometric Object ◽  
Parameter Family ◽  
Visualization Method ◽  
Family Of Curves ◽  
Geometric Space ◽  
The One

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

A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ
Keyword(s):  
Parameter Family ◽  
Sufficient Condition ◽  
Simple Application ◽  
Quadratic Number ◽  
General Sufficient Condition ◽  
Family Of Curves ◽  
Unit Circles ◽  
Triple Points ◽  
Straight Lines ◽  
Special Case

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

Non-Euclidean Geometry

Space ◽  
2020 ◽  
pp. 306-311
Author(s):  
Jeremy Gray
Keyword(s):  
Euclidean Geometry ◽  
Elementary Geometry ◽  
Non Euclidean Geometry ◽  
The One

Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably axiomatized.

Nets of Conics in the Euclidean Plane and an Associated Representational Geometry

1973 ◽  
Vol 16 (4) ◽  
pp. 479-495
Author(s):  
R. Blum ◽  
A. P. Guinand
Keyword(s):  
Nineteenth Century ◽  
Euclidean Plane ◽  
Algebraic Curves ◽  
Algebraic Varieties

The study of systems of conies and other algebraic curves was initiated in the middle of the nineteenth century by Cayley, Hesse, Cremona, and others. Most of the investigations from that time to the present have been concerned with extensions to algebraic varieties and systems of higher orders or dimensions, or with associated algebraic curves such as Jacobians and Hessians.

scholarly journals K-theory for Cuntz-Krieger algebras arising from real quadratic maps

2003 ◽  
Vol 2003 (34) ◽  
pp. 2139-2146 ◽  
Author(s):  
Nuno Martins ◽  
Ricardo Severino ◽  
J. Sousa Ramos
Keyword(s):  
Transition Matrix ◽  
Parameter Family ◽  
Quadratic Maps ◽  
Markov Transition Matrix ◽  
Kneading Sequence ◽  
Markov Transition ◽  
K Theory ◽  
The One ◽  
Real Quadratic

We compute theK-groups for the Cuntz-Krieger algebras𝒪A𝒦(fμ), whereA𝒦(fμ)is the Markov transition matrix arising from the kneading sequence𝒦(fμ)of the one-parameter family of real quadratic mapsfμ.

DYNAMICAL INTEGRATION

1993 ◽  
Vol 03 (01) ◽  
pp. 217-222 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Numerical Methods ◽  
Transformation Group ◽  
Forthcoming Paper ◽  
Parameter Family ◽  
Continuous Transformation ◽  
Highly Nonlinear ◽  
One Step ◽  
Function Approximations ◽  
The One ◽  
Parameter Values

In this letter we show how to use a new form of integration, called dynamical integration, that utilizes the dynamics of a system defined by an ODE to construct a map that is in effect a one-step integrator. This method contrasts sharply with classical numerical methods that utilize polynomial or rational function approximations to construct integrators. The advantages of this integrator is that it uses only one step while preserving important dynamical properties of the solution of the ODE: First, if the ODE is conservative, then the one-step integrator is measure preserving. This is significant for a system having a highly nonlinear component. Second, the one-step integrator is actually a one-parameter family of one-step maps and is derived from a continuous transformation group as is the set of solutions of the ODE. If each element of the continuous transformation group of the ODE is topologically conjugate to its inverse, then so is each member of the one-parameter family of one-step integrators. If the solutions of the ODE are elliptic, then for sufficiently small values of the parameter, the one-step integrator is also elliptic. In the limit as the parameter of the one-step family of maps goes to zero, the one-step integrator satisfies the ODE exactly. Further, it can be experimentally verified that if the ODE is chaotic, then so is the one-step integrator. In effect, the one-step integrator retains the dynamical characteristics of the solutions of the ODE, even with relatively large step sizes, while in the limit as the parameter goes to zero, it solves the ODE exactly. We illustrate the dynamical, in contrast to numerical, accuracy of this integrator with two distinctly different examples: First we use it to integrate the unforced Van der Pol equation for large ∊, ∊≥10 which corresponds to an almost continuous square-wave solution. Second, we use it to obtain the Poincaré map for two different versions of the periodically forced Duffing equation for parameter values where the solutions are chaotic. The dynamical accuracy of the integrator is illustrated by the reproduction of well-known strange attractors. The production of these attractors is eleven times longer when using a conventional fourth-order predictor-corrector method. The theory presented here extends to higher dimensions and will be discussed in detail in a forthcoming paper. However, we caution that the theory we present here is not intended as a line of research in numerical methods for ODEs.

ON A ONE-PARAMETER FAMILY OF EXOTIC SUPERSPACES IN TWO DIMENSIONS

1991 ◽  
Vol 06 (35) ◽  
pp. 3239-3250 ◽  
Author(s):  
MURAT GÜNAYDIN
Keyword(s):  
Two Dimensions ◽  
Parameter Family ◽  
Oscillator Algebra ◽  
Jordan Superalgebras ◽  
Special Values ◽  
Finite Dimensional ◽  
Operator Realization ◽  
The One ◽  
Conformal Superalgebras ◽  
Bosonic Oscillator

Using Jordan algebraic techniques we define and study a family of exotic superspaces in two dimensions with two bosonic and two fermionic coordinates. They are defined by the one-parameter family of Jordan superalgebras JD (2/2)α. For two special values of α the JD (2/2)α can be realized in terms of a single fermionic or a single bosonic oscillator, respectively. For other values of α it can be interpreted as defining an exotic oscillator algebra. The derivation, reduced structure and Möbius superalgebras of JD (2/2)α are identified with the rotation, Lorentz and finite-dimensional conformal superalgebras of the corresponding superspaces. The conformal superalgebras turn out to be the superalgebras D(2,1;α) with the even subgroup SO(2,2)×SU(2) . We give an explicit differential operator realization of the actions of D(2,1;α) on these superspaces.

scholarly journals ON THE ESSENCE, THE PLACE, THE PART AND CHARACTER OF THE TASKS WITH PARAMETERS IN THE GEOMETRY COURSES OF THE INSTITUTIONS OF GENERAL SECONDARY EDUCATION

2020 ◽  
Vol 1 (191) ◽  
pp. 150-154
Author(s):  
Olena Synyukova ◽  
◽  
Oleh Chepok ◽  
Keyword(s):  
Secondary Education ◽  
Euclidean Plane ◽  
Euclidean Geometry ◽  
Axiomatic Theory ◽  
Geometrical Figure ◽  
Long Time ◽  
Mutual Displacement ◽  
The Given ◽  
Independent Assessment ◽  
Parameter Relation

The so-called tasks with parameters for a long time now have become an integral part as of the every to some extent profound course of algebra or of algebra and the beginnings of cultures at the institutions of general secondary education, as of the corresponding tasks of the State Final Attestation in Mathematics and the External independent assessment in mathematics. And it isn’t accidental because in the most often cases the solution of the task with a parameter turns for the student into a small investigation by his own. The realization of such investigation favors the formation of the creative practical-oriented personality. Simultaneously we must state that, despite of the existence of a lot of the high scientific and methodical level created corresponding training books, it is difficult just now to find in the methodical literature the clear answers to the natural questions of what is meant on the whole by the task with parameter (or with parameters) and its solution. At the same time, in the courses of geometry of the institutions of general secondary education to the tasks with parameters it is given next to nothing consideration. But in fact such tasks in the courses are present, their importance for the proper construction of the courses can be exaggerated. In the paper the problems of what must be understand by the task with the parameter or with the parameters and by its solution are analyzed. The essence, the part and the place of the tasks with parameters in the geometry courses of institutions of general secondary education are elucidated. Euclidean geometry as an axiomatic theory investigates the sets that in their overwhelming majority represent by themselves the mathematical abstructions of the spatial forms of the surrounding, some relations between such set and quantities that characterize such sets and relations. In the contrast to the courses of algebra, in the geometrical courses the part of parameters may be played by all of the three mentioned components. Geometrical figures can change by the size and by the form. Changing by the size bring us to the concept of the scalar quantity. Changing by the form are considered in the tasks of paving and, for example, in the tasks of finding the amount and the types of symmetries of geometrical figure in dependence of its form. The part of the parameter-relation can be played by different variants of mutual displacement of the given figures in Euclidean plane or in Euclidean space. According to their content, different geometrical tasks with parameters are considered in the work. The task of the existence of geometrical figures, the tasks, conserning the character of some geometrical places of points, the tasks of tracing with the help of a compass and a ruler are among them.

scholarly journals A Scientific Calculator for Exact Real Number Computation Based on LRT, GMP and FC++

2012 ◽  
Vol 22 ◽  
pp. 35-41
Author(s):  
Alejandra Lucatero ◽  
J. Raymundo Marcial-Romero ◽  
J. A. Hernández
Keyword(s):  
Real Number ◽  
Programming Language ◽  
Functional Programming ◽  
Operational Semantics ◽  
Trigonometric Functions ◽  
Lazy Evaluation ◽  
Evaluation Mechanism ◽  
The One ◽  
Exact Real Number Computation ◽  
Scientific Calculator

Language for Redundant Test (LRT) is a programming language for exact real number computation. Its lazy evaluation mechanism (also called call-by-need) and its infinite list requirement, make the language appropriate to be implemented in a functional programming language such as Haskell. However, a direction translation of the operational semantics of LRT into Haskell as well as the algorithms to implement basic operations (addition subtraction, multiplication, division) and trigonometric functions (sin, cosine, tangent, etc.) makes the resulting scientific calculator time consuming and so inefficient. In this paper, we present an alternative implementation of the scientific calculator using FC++ and GMP. FC++ is a functional C++ library while GMP is a GNU multiple presicion library. We show that a direct translation of LRT in FC++ results in a faster scientific calculator than the one presented in Haskell.

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