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

2020
*
Lecture Notes in Computer Science
*

This raises

doi:10.1007/978-3-030-51466-2_7
fatcat:x6dpi4lfdzhrjpl3byzamsuuz4
*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

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

2008
*
Advances in Applied Clifford Algebras
*

In this new polar

doi:10.1007/s00006-008-0128-1
fatcat:6ys2hjgfc5aqnhaaeznrap5vf4
*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

2000
*
Journal of Complexity
*

It turns out that, beside

doi:10.1006/jcom.1999.0539
fatcat:ocxamxknbbaehkrdbfgrnmh75m
*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

2002
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

We compare

doi:10.1103/physreva.65.053806
fatcat:cfuu735bbbhj7gtwlg7svpdpsa
*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]

2014
*
arXiv
*
pre-print

Focusing

arXiv:1312.6115v5
fatcat:ak4datu4wjcxvl5pxrjrcnlmai
*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

2009
*
Studies in history and philosophy of modern physics
*

A systematic analysis is made

doi:10.1016/j.shpsb.2009.07.002
fatcat:i2pq6bjojzdbvb6jxcp5b7xvoa
*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

2017
*
Journal of Physics, Conference Series
*

We establish a link

doi:10.1088/1742-6596/804/1/012009
fatcat:zkznw73sffhsxniioxul254mzu
*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 [3] , i.e. R (*the**real**numbers*), C (*complex**numbers*), H (quaternions) and O (octonions or Cayley*numbers*). ...##
###
Algorithmic Complexity of Multiplex Networks

2020
*
Physical Review X
*

A fundamental and still open problem is to assess if and when

doi:10.1103/physrevx.10.021069
fatcat:5fdu7dciozcqffhwc24oq7xdui
*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

2016
*
Physical Review A
*

First, a 'product

doi:10.1103/physreva.93.052118
fatcat:5m5n4yni3jds3e3uojezea635u
*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

2022
*
Physics
*

Two independent studies demonstrate that a formulation

doi:10.1103/physics.15.7
fatcat:ke6onqfwive2taxazy5ptzajd4
*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]

1995
*
IFIP Advances in Information and Communication Technology
*

However, even if

doi:10.1007/978-0-387-34920-6_27
fatcat:dmecf2wmxng27k5anzj6ohrx4e
*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

2002
*
Advanced Signal Processing Algorithms, Architectures, and Implementations XII
*

We introduce

doi:10.1117/12.452047
fatcat:6av5l4lfqfae3lqlefvk2lkhhu
*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

2014
*
The American mathematical monthly
*

We explore this

doi:10.4169/amer.math.monthly.121.05.391
fatcat:txyokhbnhvfmjo2remgtkwomme
*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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