Some Properties of Lines in Space of Four Dimensions and their Interpretation in the Geometry of the Circle in Space of Three Dimensions

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100019253 ◽

1936 ◽

Vol 32 (4) ◽

pp. 507-520 ◽

Cited By ~ 4

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.

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LARGE QUANTUM GRAVITY EFFECTS: CYLINDRICAL WAVES IN FOUR DIMENSIONS

International Journal of Modern Physics D ◽

10.1142/s0218271800000633 ◽

2000 ◽

Vol 09 (06) ◽

pp. 669-686 ◽

Cited By ~ 17

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.

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Nervous Activity of the Brain in Five Dimensions

Biophysica ◽

10.3390/biophysica1010004 ◽

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.

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A Survey on the Computation of Quaternions From Rotation Matrices

Journal of Mechanisms and Robotics ◽

10.1115/1.4041889 ◽

2019 ◽

Vol 11 (2) ◽

Cited By ~ 3

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.

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How Kindergarten Teachers Assess Their Own Professional Competencies

Education Sciences ◽

10.3390/educsci11120769 ◽

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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The quantum consistency of the 12-dimensional theory

International Journal of Modern Physics A ◽

10.1142/s0217751x1650010x ◽

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.

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How do people judge fairness in supervisor and peer relationships? Another assessment of the dimensions of justice

Human Relations ◽

10.1177/0018726719875497 ◽

2019 ◽

Vol 73 (12) ◽

pp. 1632-1663 ◽

Cited By ~ 3

Author(s):

Marion Fortin ◽

Russell Cropanzano ◽

Natàlia Cugueró-Escofet ◽

Thierry Nadisic ◽

Hunter Van Wagoner

Keyword(s):

Factor Analysis ◽

Organizational Justice ◽

Peer Relationships ◽

Three Dimensions ◽

Informational Justice ◽

Justice Research ◽

Four Dimensions ◽

Follow Up Studies ◽

Theoretical Considerations

The ultimate goal of organizational justice research is to help create fairer workplaces. This goal may have been slowed by an inattention to the criteria that workers themselves use to ascertain what they believe is fair. Referred to as ‘justice rules’, these were originally determined by theoretical considerations and organized in four dimensions (distributive, procedural, interpersonal and informational justice). There have been few attempts to investigate how far these classical norms represent fairness experiences and concerns in modern workplaces, especially in the context of working with peers. In a person-centric study, we investigate which rules people use when judging the fairness of interactions with supervisors and peers. This allows us to identify 14 new justice rules that are not taken into account by traditional measures. When subjected to factor analysis in follow-up studies, the enlarged set of rules suggests a more parsimonious structure for organizational justice, with only three dimensions apiece for supervisor and peer justice. We term these factors relationship, task, and distributive justice. Furthermore, we find that the resulting model of justice rules is a good predictor of attitudes in relation to supervisors and peers and can provide additional insights into how to understand and manage justice.

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The Impact of Cultural Intelligence on Organisational Performance: Applied Study in Sample of Jordanian Pharmaceutical Companies

Journal of Information & Knowledge Management ◽

10.1142/s0219649220500100 ◽

2020 ◽

Vol 19 (02) ◽

pp. 2050010

Author(s):

Najm Abood Najm ◽

Aya Y. Zaghari

Keyword(s):

Market Share ◽

Cultural Intelligence ◽

Three Dimensions ◽

Performance Criteria ◽

Pharmaceutical Companies ◽

Organisational Performance ◽

Administrative Practices ◽

Four Dimensions ◽

Cognitive Behavioural ◽

The Impact

The aim of this study is to determine the impact of cultural intelligence in its four dimensions (metacognitive, cognitive, motivational and behavioural) on organisational performance based on three criteria: productivity, profitability and market share in Jordanian pharmaceutical companies. The study is sought to provide a broad review of the literature of cultural intelligence and administrative practices in the field of business administration and research in different parts. The results of the study confirmed that there is an impact of three dimensions of cultural intelligence (metacognitive, cognitive, behavioural) on organisational performance (productivity, profitability, market share), while the motivational dimension has no impact on all three performance criteria in Jordanian pharmaceutical companies.

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Four dimensions of learning in designing an e-learning course scenario

Zeszyty Naukowe Akademii Sztuki Wojennej ◽

10.5604/01.3001.0010.3523 ◽

2017 ◽

Vol 105 (4) ◽

pp. 39-52

Author(s):

Małgorzata Gawlik-Kobylińska

Keyword(s):

Three Dimensions ◽

Fourth Dimension ◽

Educational Activities ◽

Four Dimensions ◽

Analysis And Design ◽

Scenario Design ◽

Dimensions Of Learning ◽

The Face ◽

E Learning ◽

Virtual Resources

The article proposes a concept for e-course scenario design, which takes into account three dimensions of learning mentioned by K. Illeris: cognitive, emotional and social one. The author notes that due to the nature of the activity in cyberspace, the process of designing virtual resources should include an additional, fourth dimension – psychomotor. In her view, the adaptation of a modified concept of Illeris will help to optimise the process of analysis and design of educational activities, especially in the face of challenges that will arise in the nearest future.

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