A six-variable identity for a ternary vector cross product in eight-dimensional space

1988 ◽  
Vol 21 (3) ◽  
pp. 593-597 ◽  
Author(s):  
R Shaw
Keyword(s):  
Dimensional Space ◽  
Cross Product ◽  
Variable Identity

Vectors

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Unit Vector ◽  
Cross Product ◽  
Nonzero Vector ◽  
Algebraic Representation ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Three Dimensional Space

The reader, even if familiar with vectors, will find it useful to work through this chapter because it introduces notation that will be used throughout this book. We will take vectors to be entities that possess magnitude, orientation, and sense in three-dimensional space. Graphically, we will represent them as arrows with the sense from tail to head, magnitude proportional to the length, and orientation indicated by the angles they form with a given set of reference directions. Two different kinds of symbol will be used to designate vectors algebraically, boldface letters (and the boldface number zero for a vector of zero magnitude), and subscripted letters to be introduced later. The first problems deal with simple vector geometry and its algebraic representation. Multiplying a vector by a scalar affects only its magnitude (length) without changing its direction. Problem 1. State the necessary and sufficient conditions for the three vectors A, B, and C to form a triangle. (Problems 1–9, 12–14, 19–23, and 25 from Sokolnikoff & Redheffer, 1958.) Problem 2. Given the sum S = A + B and the difference D = A – B, find A and B in terms of S and D (a) graphically and (b) algebraically. Problem 3. (a) State the unit vector a with the same direction as a nonzero vector A. (b) Let two nonzero vectors A and B issue from the same point, forming an angle between them; using the result of (a), find a vector that bisects this angle. Problem 4. Using vector methods, show that a line from one of the vertices of a parallelogram to the midpoint of one of the nonadjacent sides trisects one of the diagonals. Two vectors are said to form with each other two distinct products: a scalar, the dot product, and a vector, the cross product.

scholarly journals Interactions, space presentations, blocks and cross products

2021 ◽  
Vol 6 (2) ◽  
pp. 061-073
Author(s):  
Gudrun Kalmbach HE
Keyword(s):  
Dimensional Space ◽  
Cross Product ◽  
Higgs Bosons ◽  
3 Dimensional ◽  
Complex Cross ◽  
Moebius Transformation ◽  
The Standard Model ◽  
Complex Multiplications ◽  
Subspace Projections ◽  
Amplitude Density

Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex 4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

Three Dimensional structure and organization of diploid chromosomes by optical section microscopy

1987 ◽  
Vol 45 ◽  
pp. 633-635
Author(s):  
David A. Agard ◽  
Yasushi Hiraoka ◽  
John W. Sedat
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Developmental State ◽  
Mitotic Cell ◽  
Biological Properties ◽  
Dimensional Structure ◽  
Specific Gene ◽  
Three Dimensional Structure ◽  
Complementary Techniques ◽  
Dynamic Structures

In an effort to understand the complex relationship between structure and biological function within the nucleus, we have embarked on a program to examine the three-dimensional structure and organization of Drosophila melanogaster embryonic chromosomes. Our overall goal is to determine how DNA and proteins are organized into complex and highly dynamic structures (chromosomes) and how these chromosomes are arranged in three dimensional space within the cell nucleus. Futher, we hope to be able to correlate structual data with such fundamental biological properties as stage in the mitotic cell cycle, developmental state and transcription at specific gene loci.Towards this end, we have been developing methodologies for the three-dimensional analysis of non-crystalline biological specimens using optical and electron microscopy. We feel that the combination of these two complementary techniques allows an unprecedented look at the structural organization of cellular components ranging in size from 100A to 100 microns.

Diffraction contrast of dislocations in icosahedral quasicrystals

1990 ◽  
Vol 48 (4) ◽  
pp. 454-455
Author(s):  
K. Urban ◽  
Z. Zhang ◽  
M. Wollgarten ◽  
D. Gratias
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Characterization ◽  
Extinction Condition ◽  
Burgers Vector ◽  
Diffraction Contrast ◽  
Lattice Geometry ◽  
Reference Space ◽  
Icosahedral Quasicrystals ◽  
The One

Recently dislocations have been observed by electron microscopy in the icosahedral quasicrystalline (IQ) phase of Al65Cu20Fe15. These dislocations exhibit diffraction contrast similar to that known for dislocations in conventional crystals. The contrast becomes extinct for certain diffraction vectors g. In the following the basis of electron diffraction contrast of dislocations in the IQ phase is described. Taking account of the six-dimensional nature of the Burgers vector a “strong” and a “weak” extinction condition are found.Dislocations in quasicrystals canot be described on the basis of simple shear or insertion of a lattice plane only. In order to achieve a complete characterization of these dislocations it is advantageous to make use of the one to one correspondence of the lattice geometry in our three-dimensional space (R3) and that in the six-dimensional reference space (R6) where full periodicity is recovered . Therefore the contrast extinction condition has to be written as gpbp + gobo = 0 (1). The diffraction vector g and the Burgers vector b decompose into two vectors gp, bp and go, bo in, respectively, the physical and the orthogonal three-dimensional sub-spaces of R6.

Flavin radicals, conformational sampling and robust design principles in interprotein electron transfer: the trimethylamine dehydrogenase-electron-transferring flavoprotein complex

2004 ◽  
Vol 71 ◽  
pp. 1-14
Author(s):  
David Leys ◽  
Jaswir Basran ◽  
François Talfournier ◽  
Kamaldeep K. Chohan ◽  
Andrew W. Munro ◽  
...  
Keyword(s):  
Electron Transfer ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Kinetic Studies ◽  
Molecular Assembly ◽  
Conformational Sampling ◽  
Trimethylamine Dehydrogenase ◽  
Key Residues ◽  
Electron Transferring Flavoprotein ◽  
Electron Transfer Complex

TMADH (trimethylamine dehydrogenase) is a complex iron-sulphur flavoprotein that forms a soluble electron-transfer complex with ETF (electron-transferring flavoprotein). The mechanism of electron transfer between TMADH and ETF has been studied using stopped-flow kinetic and mutagenesis methods, and more recently by X-ray crystallography. Potentiometric methods have also been used to identify key residues involved in the stabilization of the flavin radical semiquinone species in ETF. These studies have demonstrated a key role for 'conformational sampling' in the electron-transfer complex, facilitated by two-site contact of ETF with TMADH. Exploration of three-dimensional space in the complex allows the FAD of ETF to find conformations compatible with enhanced electronic coupling with the 4Fe-4S centre of TMADH. This mechanism of electron transfer provides for a more robust and accessible design principle for interprotein electron transfer compared with simpler models that invoke the collision of redox partners followed by electron transfer. The structure of the TMADH-ETF complex confirms the role of key residues in electron transfer and molecular assembly, originally suggested from detailed kinetic studies in wild-type and mutant complexes, and from molecular modelling.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

Analysis method ofrecruitment needs for the regional economy: Occupational section

2017 ◽  
pp. 142-149 ◽  
Author(s):  
E. Pitukhin ◽  
S. Shabaeva ◽  
I. Stepus ◽  
D. Moroz
Keyword(s):  
Cluster Analysis ◽  
Labor Market ◽  
Comparative Analysis ◽  
Regional Economy ◽  
Dimensional Space ◽  
Analysis Method ◽  
Scalar Form ◽  
Regional Labor Market ◽  
Regional Labor ◽  
Regional Specificity

The paper deals with comparative analysis of occupations in the regional labor market. Occupation is treated as a multi-dimensional space of characte- ristics, whereas a scalar form of a characteristic makes it possible to carry out a comparative analysis of occupations. Using cluster analysis of a pilot region indicators five meaningfully interpretable clusters of occupations were identified, reflecting their regional specificity.

Acoustical Properties of Speech as Indicators of Depression and Suicidal Risk

2008 ◽  
Vol 4 ◽  
Author(s):  
Michael Joshua Landau
Keyword(s):  
Multiple Regression ◽  
Energy Band ◽  
Mental States ◽  
Cross Product ◽  
Female Group ◽  
Suicidal Risk ◽  
Energy Bands ◽  
Band Ratio ◽  
Acoustical Properties ◽  
Male Group

Acoustical properties of speech have been shown to be related to mental states such as remission and depression. The objective of this project was to relate the energy in frequency bands with the severity of the mental state using the Beck Depression Inventory (BDI). Recorded speech was obtained from male and female subjects with mental states of remission, depression, and suicidal risk. These subjects had recorded automated and spontaneous speech samples. Multiple regression analysis was used to relate the independent energy band ratio variables with the dependent BDI scores, and thus allow the determination of equitable BDI scores for future patients. For the male group, the square of the 3rd energy band and the cross-product of the 2nd and 3rd energy band were prominent in both the reading and interviewed groups. Therefore the equation with the 2nd lowest Akaike Information Criterion (AIC) score was chosen for the reading male group, and the 1st lowest AIC score was chosen for the interviewed male group. For the female group, the square and cross-product of the 1st and 2nd energy bands were prominent in both the reading and interviewed groups. Therefore the 2nd lowest AIC score was chosen for the reading female group, and the 1st lowest AIC score was chosen for the interviewed female group. The clinician could thus determine the patient’s mood or state of mind by comparing the estimated BDI score with the ranges of total BDI scores: remitted 0 – 20, depressed 15 – 38, suicidal 38 – 46. Keywords: speech, mental states, power spectra, multiple regression, information theoretic criterion

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