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### On the Complexity of Conversion Between Classic Real Number Representations [chapter]

Lars Kristiansen, Jakob Grue Simonsen
2020 Lecture Notes in Computer Science
This raises the question of categorizing the pairs of representations between which either subrecursive conversion is possible, or is not possible.  ...  ) conversion between the representations can be performed effectively and with good subrecursive bounds.  ...  We are grateful for the meticulous comments of one of the referees; these have helped to significantly improve the paper.  ...

### Impact of modern instrumentation on the system of basic concepts in metrology

J. M. Jaworski, J. Bek, A. J. Fiok
2014 ACTA IMEKO
One of the most important new concepts introduced here is the concept of measuring metasystem which comprises not only instrumentation but also the measured object, a "measurement interface" between the  ...  The proposed system is based on understanding of measurement as an experiment of parameter identification of the model of the measured object.  ...  Roman Morawaki whose opinions and help in preparation of the final version of this paper were of great importance.  ...

### Page 5227 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews
From this, one obtains an ex- tremely interesting graphical vector analysis, since it allows one to transform scalar and vector products between any number of vectors.  ...  of two real inequivalent 8 x 8 irreducible representations (D() and D')) of SO(4,4).  ...

### Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form

Stephen J. Sangwine, Nicolas Le Bihan
2008 Advances in Applied Clifford Algebras
In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument  ...  As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of -1), but the complex phase is multiplied by a different complex root of -1 in the exponential  ...  Before proceeding, we note the analogy between this polar form, based on two complex numbers, and the Cayley-Dickson form of a quaternion q = (w+xi)+(y +zi)j = w+xi+yj +zk which is also based on two complex  ...

### On the Time Complexity of Partial Real Functions

Armin Hemmerling
2000 Journal of Complexity
It turns out that, beside the Ko Friedman theory, also the classical theory of time complexity for discrete functions over N is naturally embedded in our setting.  ...  We consider the Ko Friedman notion of (non-uniform) time complexity for real functions approximately computable in the generalized sense defined earlier.  ...  Ko hler for the critical reading of previous versions of this note and to K. Weihrauch for discussions, critical remarks, and hints to his related results.  ...

### Quantum-theoretical treatments of three-photon processes

M. K. Olsen, L. I. Plimak, M. Fleischhauer
2002 Physical Review A. Atomic, Molecular, and Optical Physics
We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations.  ...  We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency  ...  In the case of THG, we found differences between the representations when we began to investigate the statistics of the fields.  ...

### Neuronal Synchrony in Complex-Valued Deep Networks [article]

David P. Reichert, Thomas Serre
2014 arXiv   pre-print
Focusing on the latter, we demonstrate the potential of the approach in several simple experiments.  ...  We introduce a neural network formulation based on complex-valued neuronal units that is not only biologically meaningful but also amenable to a variety of deep learning frameworks.  ...  Goodfellow for helpful feedback on earlier versions of this work. We would also like to thank Elie Bienenstock and Stuart Geman for insightful discussions which motived this project.  ...

### QFT, antimatter, and symmetry

David Wallace
2009 Studies in history and philosophy of modern physics
A systematic analysis is made of the relations between the symmetries of a classical field and the symmetries of the one-particle quantum system that results from quantizing that field in regimes where  ...  The results are applied to gain a greater insight into the phenomenon of antimatter.  ...  Conversely, suppose that ϕ is a representation of G on a complex vector space W. Then trivially, ϕ is also a representation of G on the real vector space W F .  ...

### Supergroups in Critical Dimensions and Division Algebras

Čestmir Burdik, Sultan Catto, Yasemin Gürcan, Amish Khalfan, Levent Kurt, V. Kato La
2017 Journal of Physics, Conference Series
We establish a link between classical heterotic strings and the groups of the magic square associated with Jordan algebras, allowing for a uniform treatment of the bosonic and superstring sectors of the  ...  numbers while they are real or complex numbers for an element of a Jordan algebra.  ...  They are associated with the four division algebras of the Hurwitz's theorem  , i.e. R (the real numbers), C (complex numbers), H (quaternions) and O (octonions or Cayley numbers).  ...

### Algorithmic Complexity of Multiplex Networks

Andrea Santoro, Vincenzo Nicosia
2020 Physical Review X
A fundamental and still open problem is to assess if and when the multilayer representation of a system provides a qualitatively better model than the classical single-layer aggregated network.  ...  We propose an intuitive way to encode a multilayer network into a bit string, and we define the complexity of a multilayer network as the ratio of the Kolmogorov complexity of the bit strings associated  ...  ACKNOWLEDGMENTS The authors thank Lucas Lacasa for helpful conversations. V. N. acknowledges support from the EPSRC Grant No. EP/S027920/1. A.  ...

### Products of weak values: Uncertainty relations, complementarity, and incompatibility

Michael J. W. Hall, Arun Kumar Pati, Junde Wu
2016 Physical Review A
First, a 'product representation formula' allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables.  ...  the form of an upper bound on the product of the corresponding weak values.  ...  Acknowledgement: MH is supported by the ARC Centre of Excellence CE110001027. J. Wu  ...

### Quantum Mechanics Must Be Complex

Alessio Avella
2022 Physics
Two independent studies demonstrate that a formulation of quantum mechanics involving complex rather than real numbers is necessary to reproduce experimental results.  ...  The result reinforced the conjecture that complex numbers aren't necessary, but the lack of a general proof left open some paths for refuting the equivalence between "complex" and "real" quantum theories  ...  of complex numbers is equivalent, or "isomorphic," to a two-dimensional, real plane, with the two dimensions representing the real and imaginary part of complex numbers, respectively.  ...

### The Influence on Synthesis of Modern Computer Arithmetic [chapter]

Anne Mignotte, Jean Michel Muller, Olivier Peyran
1995 IFIP Advances in Information and Communication Technology
However, even if real numbers were in a floating point representation, their definition was far from the IEEE standard: the mantissa and exponent have the size of the "single precision" standard, but there  ...  is no constraint on the accuracy of the operations using floating point numbers (see section 2 for the constraints imposed by the standard).  ...  This is the reason why it is called fixed point representation. One might need to use real numbers in a large range though, and fixed point representation would require the use of huge registers.  ...

### Digital filtering using the multidimensional logarithmic number system

Vassil S. Dimitrov, Graham A. Jullien, Konrad Walus, Franklin T. Luk
2002 Advanced Signal Processing Algorithms, Architectures, and Implementations XII
We introduce the use of multidimensional logarithmic number system (MDLNS) as a generalization of the classical 1-D logarithmic number system (LNS) and analyze its use in DSP applications.  ...  The use of more than one base has at least two extra advantages.  ...  complexity of the 2-digit 2-D LNS versus the classical LNS.  ...

### A Geometric Representation of Continued Fractions

Alan F. Beardon, Ian Short
2014 The American mathematical monthly
We explore this representation using the isometric action of the group of Möbius transformations on hyperbolic space, and prove a classical theorem on continued fractions. http://dx.doi.org/10.4169/amer.math.monthly  ...  Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space.  ...  Here we follow a similar path from an elementary representation of real continued fractions by horocycles to a deeper study of the representation of complex continued fractions by horospheres in three-dimensional  ...
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