scholarly journals Representation of Crystallographic Subperiodic Groups in Clifford’s Geometric Algebra

2013 ◽  
Vol 23 (4) ◽  
pp. 887-906 ◽  
Author(s):  
Eckhard Hitzer ◽  
Daisuke Ichikawa
Keyword(s):  
Geometric Algebra ◽  
Clifford’S Geometric Algebra

scholarly journals The double-padlock problem: Is secure classical information transmission possible without key exchange?

2014 ◽  
Vol 33 ◽  
pp. 1460355
Author(s):  
James M. Chappell ◽  
Lachlan J. Gunn ◽  
Derek Abbott
Keyword(s):  
Open Problem ◽  
Geometric Algebra ◽  
Mathematical Description ◽  
The Other ◽  
Mathematical Representation ◽  
Information Channel ◽  
Physical Realization ◽  
Classical Information ◽  
Reduced Complexity ◽  
Clifford’S Geometric Algebra

The idealized Kish-Sethuraman (KS) cipher is theoretically known to offer perfect security through a classical information channel. However, realization of the protocol is hitherto an open problem, as the required mathematical operators have not been identified in the previous literature. A mechanical analogy of this protocol can be seen as sending a message in a box using two padlocks; one locked by the Sender and the other locked by the Receiver, so that theoretically the message remains secure at all times. We seek a mathematical representation of this process, considering that it would be very unusual if there was a physical process with no mathematical description. We select Clifford's geometric algebra for this task as it is a natural formalism to handle rotations in spaces of various dimension. The significance of finding a mathematical description that describes the protocol, is that it is a possible step toward a physical realization having benefits in increased security with reduced complexity.

scholarly journals Analysis of Two-Player Quantum Games in an EPR Setting Using Clifford's Geometric Algebra

PLoS ONE ◽  
2012 ◽  
Vol 7 (1) ◽  
pp. e29015 ◽  
Author(s):  
James M. Chappell ◽  
Azhar Iqbal ◽  
Derek Abbott
Keyword(s):  
Geometric Algebra ◽  
Quantum Games ◽  
Clifford’S Geometric Algebra

scholarly journals Applications of Clifford’s Geometric Algebra

2013 ◽  
Vol 23 (2) ◽  
pp. 377-404 ◽  
Author(s):  
Eckhard Hitzer ◽  
Tohru Nitta ◽  
Yasuaki Kuroe
Keyword(s):  
Geometric Algebra ◽  
Clifford’S Geometric Algebra

scholarly journals Spheroidal Domains and Geometric Analysis in Euclidean Space

2021 ◽  
pp. 189-209
Author(s):  
Garret Sobczyk
Keyword(s):  
Artificial Intelligence ◽  
Data Analysis ◽  
Euclidean Space ◽  
Geometric Algebra ◽  
Geometric Analysis ◽  
Natural Generalization ◽  
Cauchy Kernel ◽  
Vector Algebra ◽  
Computer Scientists ◽  
Clifford’S Geometric Algebra

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers, and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad of different and often confusing notations. The geometric algebra of Euclidean 3-space, the natural generalization of both the well-known Gibbs-Heaviside vector algebra and Hamilton's quaternions, is used here to study spheroidal domains, spheroidal-graphic projections, the Laplace equation, and its Lie algebra of symmetries. The Cauchy-Kovalevska extension and the Cauchy kernel function are treated in a unified way. The concept of a quasi-monogenic family of functions is introduced and studied. 

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