scholarly journals A generalized vector analysis of four dimensions

1923 ◽  
Vol 103 (723) ◽  
pp. 644-663 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Four Dimensions ◽  
Straight Line ◽  
Straight Lines

The vector analysis in use up to the present, as a rule, are concerned with quantities which are represented by straight lines. The straight line AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian co-ordinates ( x, y, z ) with respect to A, we write a = i x + j y + k z , where i, j, k are fundamental vectors. An account will be given of a vector analysis in which the vector is represented by δa = Ʃ n i n δ x n .

scholarly journals A general vector analysis, with applications to electrodynamical theory

1925 ◽  
Vol 108 (747) ◽  
pp. 418-455 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Cartesian Coordinates ◽  
Straight Line ◽  
Straight Lines ◽  
General Vector

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

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.

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 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 An Improved Path-Generating Regulator for Two-Wheeled Robots to Track the Circle/Arc Passage

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Jun Dai ◽  
Naohiko Hanajima ◽  
Toshiharu Kazama ◽  
Akihiko Takashima
Keyword(s):  
Control Method ◽  
Convergence Property ◽  
Tangential Direction ◽  
Navigation Problem ◽  
Wheeled Robots ◽  
Robot Trajectory ◽  
Look Ahead ◽  
Straight Line ◽  
The Family ◽  
Straight Lines

The improved path-generating regulator (PGR) is proposed to path track the circle/arc passage for two-wheeled robots. The PGR, which is a control method for robots so as to orient its heading toward the tangential direction of one of the curves belonging to the family of path functions, is applied to navigation problem originally. Driving environments for robots are usually roads, streets, paths, passages, and ridges. These tracks can be seen as they consist of straight lines and arcs. In the case of small interval, arc can be regarded as straight line approximately; therefore we extended the PGR to drive the robot move along circle/arc passage based on the theory that PGR to track the straight passage. In addition, the adjustable look-ahead method is proposed to improve the robot trajectory convergence property to the target circle/arc. The effectiveness is proved through MATLAB simulations on both the comparisons with the PGR and the improved PGR with adjustable look-ahead method. The results of numerical simulations show that the adjustable look-ahead method has better convergence property and stronger capacity of resisting disturbance.

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.

The quantum consistency of the 12-dimensional theory

2016 ◽  
Vol 31 (04n05) ◽  
pp. 1650010
Author(s):  
Simon Davis
Keyword(s):  
Cosmological Constant ◽  
Moduli Space ◽  
Dimensional Reduction ◽  
Text Structure ◽  
Three Dimensions ◽  
Supersymmetric Theory ◽  
Particle Spectrum ◽  
Dimensional Theory ◽  
Cosmological Constant Problem ◽  
Four Dimensions

By considering the 12-dimensional superalgebra, inferences are drawn about the finiteness of the 12-dimensional theory unifying the superstring models. The dimensional reduction of the nonsupersymmetric theory in four dimensions to a supersymmetric action in three dimensions is established for the bosonic sector. It is found to be the quotient by [Formula: see text] of the integration over the fiber coordinate of a theory with [Formula: see text] supersymmetry. Consequently, a flow on the moduli space of Spin(7) manifolds from a [Formula: see text] structure with [Formula: see text] supersymmetry yielding a phenomelogically realistic particle spectrum to a [Formula: see text] holonomy manifold compatible with supersymmetry in three dimensions and a nonsupersymmetric action in four dimensions, solving the quantum cosmological constant problem, is proven to exist. The projection of the representations of the [Formula: see text] superalgebra of the 12-dimensional theory to four dimensions include nonperturbative string solitons that are more stable because the dynamics is described by supersymmetric theory with a higher degree of finiteness.

Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

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