Geometric and Differential Features of Scators as Induced by Fundamental Embedding

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

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AN ALTERNATIVE DIMENSIONAL REDUCTION PRESCRIPTION

Modern Physics Letters A ◽

10.1142/s0217732396001077 ◽

1996 ◽

Vol 11 (13) ◽

pp. 1037-1045 ◽

Cited By ~ 3

Author(s):

J.D. EDELSTEIN ◽

C. NÚÑEZ ◽

F.A. SCHAPOSNIK ◽

J.J. GIAMBIAGI

Keyword(s):

Dimensional Reduction ◽

Dimensional Space ◽

Dimensional Case ◽

Green Functions ◽

Two Dimensional ◽

Spatial Coordinate ◽

Higher Dimensional

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to dropping the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular two-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher-dimensional space.

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TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

Parallel Processing Letters ◽

10.1142/s0129626493000162 ◽

1993 ◽

Vol 03 (02) ◽

pp. 129-138

Author(s):

STEVEN CHEUNG ◽

FRANCIS C.M. LAU

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Three Dimensional ◽

Dimensional Case ◽

Routing Problem ◽

Higher Dimensional ◽

Low Dimensional ◽

Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

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On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures II: the low-dimensional case

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.136 ◽

2017 ◽

Vol 38 (8) ◽

pp. 3042-3061

Author(s):

MICHIHIRO HIRAYAMA ◽

NAOYA SUMI

Keyword(s):

Probability Measure ◽

Closed Manifold ◽

Dimensional Case ◽

Anosov Diffeomorphism ◽

Stable And Unstable Manifolds ◽

Higher Dimensional ◽

Unstable Manifolds ◽

Low Dimensional

In this paper, we consider diffeomorphisms on a closed manifold $M$ preserving a hyperbolic Sinaĭ–Ruelle–Bowen probability measure $\unicode[STIX]{x1D707}$ having intersections for almost every pair of stable and unstable manifolds. In this context, we show the ergodicity of $\unicode[STIX]{x1D707}$ when the dimension of $M$ is at most three. If $\unicode[STIX]{x1D707}$ is smooth, then it is ergodic when the dimension of $M$ is at most four. As a byproduct of our arguments, we obtain sufficient (topological) conditions which guarantee that there exists at most one hyperbolic ergodic Sinaĭ–Ruelle–Bowen probability measure. Even in higher dimensional cases, we show that every transitive topological Anosov diffeomorphism admits at most one hyperbolic Sinaĭ–Ruelle–Bowen probability measure.

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Celestial Tapestry

10.1093/oso/9780198851950.001.0001 ◽

2020 ◽

Author(s):

Nicholas Mee

Keyword(s):

Mathematics Curriculum ◽

Dimensional Space ◽

Golden Section ◽

Single Point ◽

Historical Context ◽

Computer Art ◽

Lightning Strike ◽

Secret Message ◽

Axiomatic Method ◽

Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

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A Generative-Discriminative Framework that Integrates Imaging, Genetic, and Diagnosis into Coupled Low Dimensional Space

NeuroImage ◽

10.1016/j.neuroimage.2021.118200 ◽

2021 ◽

pp. 118200

Author(s):

Sayan Ghosal ◽

Qiang Chen ◽

Giulio Pergola ◽

Aaron L. Goldman ◽

William Ulrich ◽

...

Keyword(s):

Dimensional Space ◽

Low Dimensional

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Artifacts in Simultaneous hdEEG/fMRI Imaging: A Nonlinear Dimensionality Reduction Approach

Sensors ◽

10.3390/s19204454 ◽

2019 ◽

Vol 19 (20) ◽

pp. 4454 ◽

Cited By ~ 2

Author(s):

Marek Piorecky ◽

Vlastimil Koudelka ◽

Jan Strobl ◽

Martin Brunovsky ◽

Vladimir Krajca

Keyword(s):

Data Structure ◽

Spatial Clustering ◽

Dimensional Space ◽

Head Movements ◽

Nonlinear Dimensionality Reduction ◽

Space Resolution ◽

Additional Information ◽

Nonlinear Dimension ◽

The Time Domain ◽

Low Dimensional

Simultaneous recordings of electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI) are at the forefront of technologies of interest to physicians and scientists because they combine the benefits of both modalities—better time resolution (hdEEG) and space resolution (fMRI). However, EEG measurements in the scanner contain an electromagnetic field that is induced in leads as a result of gradient switching slight head movements and vibrations, and it is corrupted by changes in the measured potential because of the Hall phenomenon. The aim of this study is to design and test a methodology for inspecting hidden EEG structures with respect to artifacts. We propose a top-down strategy to obtain additional information that is not visible in a single recording. The time-domain independent component analysis algorithm was employed to obtain independent components and spatial weights. A nonlinear dimension reduction technique t-distributed stochastic neighbor embedding was used to create low-dimensional space, which was then partitioned using the density-based spatial clustering of applications with noise (DBSCAN). The relationships between the found data structure and the used criteria were investigated. As a result, we were able to extract information from the data structure regarding electrooculographic, electrocardiographic, electromyographic and gradient artifacts. This new methodology could facilitate the identification of artifacts and their residues from simultaneous EEG in fMRI.

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Aperiodic Crystals

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314099987 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C1-C1 ◽

Cited By ~ 1

Author(s):

Ted Janssen ◽

Aloysio Janner

Keyword(s):

Physical Properties ◽

Dimensional Space ◽

X Rays ◽

New Techniques ◽

New Family ◽

Modulated Phases ◽

Open Questions ◽

Higher Dimensional ◽

Lattice Periodicity ◽

Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

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Optimizing the structure and movement of a robotic bat with biological kinematic synergies

The International Journal of Robotics Research ◽

10.1177/0278364918804654 ◽

2018 ◽

Vol 37 (10) ◽

pp. 1233-1252 ◽

Cited By ~ 2

Author(s):

Jonathan Hoff ◽

Alireza Ramezani ◽

Soon-Jo Chung ◽

Seth Hutchinson

Keyword(s):

Principal Component Analysis ◽

Topological Structure ◽

Degrees Of Freedom ◽

Dimensional Space ◽

Principal Component ◽

Biologically Inspired ◽

Wing Kinematics ◽

Wing Motion ◽

Kinematic Synergies ◽

Low Dimensional

In this article, we present methods to optimize the design and flight characteristics of a biologically inspired bat-like robot. In previous, work we have designed the topological structure for the wing kinematics of this robot; here we present methods to optimize the geometry of this structure, and to compute actuator trajectories such that its wingbeat pattern closely matches biological counterparts. Our approach is motivated by recent studies on biological bat flight that have shown that the salient aspects of wing motion can be accurately represented in a low-dimensional space. Although bats have over 40 degrees of freedom (DoFs), our robot possesses several biologically meaningful morphing specializations. We use principal component analysis (PCA) to characterize the two most dominant modes of biological bat flight kinematics, and we optimize our robot’s parametric kinematics to mimic these. The method yields a robot that is reduced from five degrees of actuation (DoAs) to just three, and that actively folds its wings within a wingbeat period. As a result of mimicking synergies, the robot produces an average net lift improvesment of 89% over the same robot when its wings cannot fold.

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