On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

1924 ◽  
Vol 22 (2) ◽  
pp. 189-199
Author(s):  
F. Bath
Keyword(s):  
Three Dimensions ◽  
Apparent Contradiction ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Lines In Space ◽  
Quadric Threefold ◽  
Quartic Curves

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

scholarly journals The configuration determined by five generators of a quadric threefold

1946 ◽  
Vol 7 (4) ◽  
pp. 183-195
Author(s):  
L. M. Brown
Keyword(s):  
Infinite System ◽  
Special Relationship ◽  
Four Dimensions ◽  
Lines In Space ◽  
Restricted Sense ◽  
Original Figure ◽  
Quadric Threefold

Through four generally placed lines in space of four dimensions there passes a doubly infinite system of quadric primals, but through five lines there pass in general no quadrics. It therefore follows that there must exist some special relationship between five lines in order that they may be generators of a quadric. This problem has been discussed by Richmond,1 who gives a condition which is in a restricted sense an extension of Pascal's Theorem. The five lines being taken in order certain points may be obtained which lie in a space. In Section I we state Richmond's criterion and show that it is sufficient as well as necessary. Section II is concerned with the twelve spaces which arise if all the different possible orders of the lines are considered. They cut by pairs in six planes whose configuration is developed. In Section III other lines connected with the configuration are introduced. It is shown that by taking crossers of the lines of our original figure in a certain manner five further generators are obtained, and that the same entire configuration of generators arises whether we begin with the five original or the five final lines. Furthermore, though the twelve spaces analogous to Pascal lines obtained from the final five are new, yet the six planes, their intersections by pairs, and the configuration dependent from them, are the same as those constructed from the original five.

On elliptic quartic curves with assigned points and chords

1931 ◽  
Vol 27 (1) ◽  
pp. 20-23 ◽  
Author(s):  
W. G. Welchman
Keyword(s):  
Finite Number ◽  
Three Dimensions ◽  
Number Of Solutions ◽  
Quartic Curve ◽  
Pass Through ◽  
Quartic Curves

1. The problem which I propose to solve is that of finding the number of quartic curves of intersection of two quadrics which pass through p points and have q lines as chords, where p + q = 8. There are ∞16 elliptic quartics in space; to contain a line as a chord is two conditions, and to pass through a point is two conditions, so we should expect a finite number of solutions. Throughout this paper I shall refer to an elliptic quartic curve in space of three dimensions simply as a “quartic.” I shall denote the number of solutions for a particular value of p by np.

Three related quartic curves in four dimensions

1932 ◽  
Vol 28 (4) ◽  
pp. 403-415 ◽  
Author(s):  
H. G. Telling
Keyword(s):  
Linear System ◽  
Simple Proof ◽  
Differential Method ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Pass Through ◽  
Quartic Curves

1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.

Polarities for the nodes of a Segre cubic primal in space of four dimensions

1936 ◽  
Vol 32 (4) ◽  
pp. 507-520 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Three Dimensions ◽  
Four Dimensions ◽  
General Terms ◽  
Lines In Space

The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

scholarly journals LARGE QUANTUM GRAVITY EFFECTS: CYLINDRICAL WAVES IN FOUR DIMENSIONS

2000 ◽  
Vol 09 (06) ◽  
pp. 669-686 ◽  
Author(s):  
MARÍA E. ANGULO ◽  
GUILLERMO A. MENA MARUGÁN
Keyword(s):  
Quantum Gravity ◽  
Symmetry Axis ◽  
Point Of View ◽  
Three Dimensions ◽  
Asymptotic Region ◽  
Four Dimensions ◽  
Cylindrical Waves ◽  
Significant Difference ◽  
Gravity Effects ◽  
Linearly Polarized

Linearly polarized cylindrical waves in four-dimensional vacuum gravity are mathematically equivalent to rotationally symmetric gravity coupled to a Maxwell (or Klein–Gordon) field in three dimensions. The quantization of this latter system was performed by Ashtekar and Pierri in a recent work. Employing that quantization, we obtain here a complete quantum theory which describes the four-dimensional geometry of the Einstein–Rosen waves. In particular, we construct regularized operators to represent the metric. It is shown that the results achieved by Ashtekar about the existence of important quantum gravity effects in the Einstein–Maxwell system at large distances from the symmetry axis continue to be valid from a four-dimensional point of view. The only significant difference is that, in order to admit an approximate classical description in the asymptotic region, states that are coherent in the Maxwell field need not contain a large number of photons anymore. We also analyze the metric fluctuations on the symmetry axis and argue that they are generally relevant for all of the coherent states.

scholarly journals Nervous Activity of the Brain in Five Dimensions

Biophysica ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 38-47
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
Norbert Jausovec ◽  
Arjuna P. H. Don ◽  
Sheela Ramanna ◽  
...  
Keyword(s):  
Fourier Analysis ◽  
Nervous Activity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Data Sets ◽  
Spatial And Temporal Patterns ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Scale Free ◽  
The Brain

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

scholarly journals How Kindergarten Teachers Assess Their Own Professional Competencies

2021 ◽  
Vol 11 (12) ◽  
pp. 769
Author(s):  
Eva Pupíková ◽  
Dalibor Gonda ◽  
Kitti Páleníková ◽  
Janka Medová ◽  
Dana Kolárová ◽  
...  
Keyword(s):  
Classroom Management ◽  
Content Knowledge ◽  
Educational Needs ◽  
Kindergarten Teachers ◽  
Further Education ◽  
Three Dimensions ◽  
Self Assessment ◽  
Teaching Competencies ◽  
Four Dimensions

One of the requirements of Education 4.0 is that students and practitioners should be involved in the creation of the content of study plans. Therefore, in the present research we focused on identifying the further educational needs of kindergarten teachers. Teachers’ educational needs were divided into four dimensions: ‘content knowledge’, ‘diagnostic knowledge’, ‘didactical knowledge’, and ‘classroom management knowledge’. In parallel, we discovered how teachers assess the level of their own teaching competencies. Based on the obtained data, we identified that teachers have the greatest need for further education in the dimension of ‘diagnostic knowledge’ and that the need for their further education in this dimension did not depend on the length of practice. In the other three dimensions, a declining trend in teachers’ educational needs has been recorded with an increasing length of practice, declining significantly in three of the four dimensions examined. The study points to the need to create in-service courses for kindergarten teachers to deepen their ‘diagnostic knowledge’ and thus ensure the sustainability of the quality of pre-school education for children. Teachers‘ self-assessment of their own teaching competencies corresponds to their educational needs, which supports the relevance of the findings on the further educational needs of kindergarten teachers. This study aimed to obtain relevant data on which the improvement of the higher education of future kindergarten teachers might be based. At the same time, this would allow the analysis and tailoring of the content of professional development courses to the needs of in-service kindergarten teachers.

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