6,681 research outputs found
Generation of field mediated three qubit entangled state shared by Alice and Bob
A scheme to generate shared tripartite entangled states, with two-trapped
atoms in a cavity held by Alice (qubits A1 and A2) entangled to a single
trapped atom in a remote lab owned by Bob (B), is proposed. The entanglement is
generated through interaction of trapped atoms with two mode squeezed light
shared by the two cavities. The proposed scheme is an extension of the proposal
of ref. [W. Son, M. S. Kim, J. Lee, and D. Ahn, J. Mod. Opt. 49, 1739 (2002)],
where the possibility of entangling two remote qubits using a bipartite
continuous variable state was examined. While the global negativity detects the
free entanglement of the three atom mixed state, the bound entanglement is
detected by the negativity calculated from pure state decomposition of the
state operator. The partial negativities calculated by selective partial
transposition of the three atom mixed state detect the pairwise entanglement of
qubit pairs A1B, A2B, and A1A2. The entanglement of three atoms is found to be
W-like, no GHZ like quantum correlations being generated.Comment: 14 pages, 06 figures, section IV revised, Other minor changes to
improve readabilit
Self-similar solutions to the mean curvature flow in
In this paper we make an analysis of self-similar solutions for the mean
curvature flow (MCF) by surfaces of revolution and ruled surfaces in
. We prove that self-similar solutions of the MCF by
non-cylindrival surfaces and conical surfaces in are trivial.
Moreover, we characterize the self-similar solutions of the MCF by surfaces of
revolutions under a homothetic helicoidal motion in in terms
of the curvature of the generating curve. Finally, we characterize the
self-similar solutions for the MCF by cylindrical surfaces under a homothetic
helicoidal motion in . Explicit families of exact solutions for
the MCF by cylindrical surfaces in are also given
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Integral on transcomplex numbers
The usual complex integral is defined in terms of complex numbers in Cartesian form but transcomplex numbers are defined in polar form and almost all transcomplex numbers, with infinite magnitude, have no Cartesian form. However,
there are eight infinite, transcomplex numbers which do have a Cartesian form and these can be used to define the transcomplex integral as the limit of sums of these eight numbers. Thus we introduce the transcomplex integral
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Transreal calculus
Transreal arithmetic totalises real arithmetic by defining division by zero in terms of three definite, non-finite numbers: positive infinity, negative
infinity and nullity. We describe the transreal tangent function and extend continuity and limits from the real domain to the transreal domain. With this preparation, we extend the real derivative to the transreal derivative and extend proper integration from the real domain to the
transreal domain. Further, we extend improper integration of absolutely convergent functions from the real domain to the transreal domain. This demonstrates that transreal calculus contains real calculus and operates at singularities where real calculus fails
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Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics
The IEEE 754 standard for
oating-point arithmetic is widely used in
computing. It is based on real arithmetic and is made total by adding both
a positive and a negative infinity, a negative zero, and many Not-a-Number
(NaN) states. The IEEE infinities are said to have the behaviour of limits.
Transreal arithmetic is total. It also has a positive and a negative infinity
but no negative zero, and it has a single, unordered number, nullity.
We elucidate the transreal tangent and extend real limits to transreal
limits. Arguing from this firm foundation, we maintain that there are
three category errors in the IEEE 754 standard. Firstly the claim that
IEEE infinities are limits of real arithmetic confuses limiting processes
with arithmetic. Secondly a defence of IEEE negative zero confuses the
limit of a function with the value of a function. Thirdly the definition
of IEEE NaNs confuses undefined with unordered. Furthermore we prove
that the tangent function, with the infinities given by geometrical con-
struction, has a period of an entire rotation, not half a rotation as is
commonly understood. This illustrates a category error, confusing the
limit with the value of a function, in an important area of applied mathe-
matics { trigonometry. We brie
y consider the wider implications of this
category error.
Another paper proposes transreal arithmetic as a basis for
floating-
point arithmetic; here we take the profound step of proposing transreal
arithmetic as a replacement for real arithmetic to remove the possibility
of certain category errors in mathematics. Thus we propose both theo-
retical and practical advantages of transmathematics. In particular we
argue that implementing transreal analysis in trans-
floating-point arith-
metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in
science and engineering and many in finance, medicine and other socially
beneficial applications
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Transdifferential and transintegral calculus
The set of transreal numbers is a superset of the real numbers. It
totalises real arithmetic by defining division by zero in terms of three def-
inite, non-finite numbers: positive infinity, negative infinity and nullity.
Elsewhere, in this proceedings, we extended continuity and limits from the
real domain to the transreal domain, here we extended the real derivative
to the transreal derivative. This continues to demonstrate that transreal
analysis contains real analysis and operates at singularities where real
analysis fails. Hence computer programs that rely on computing deriva-
tives { such as those used in scientific, engineering and financial applica-
tions { are extended to operate at singularities where they currently fail.
This promises to make software, that computes derivatives, both more
competent and more reliable.
We also extended the integration of absolutely convergent functions
from the real domain to the transreal domain
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Construction of the transcomplex numbers from the complex numbers
A geometrical construction of the transcomplex numbers was given
elsewhere. Here we simplify the transcomplex plane and construct the
set of transcomplex numbers from the set of complex numbers. Thus
transcomplex numbers and their arithmetic arise as consequences of their
construction, not by an axiomatic development. This simplifes transcom-
plex arithmetic, compared to the previous treatment, but retains totality
so that every arithmetical operation can be applied to any transcomplex
number(s) such that the result is a transcomplex number. Our proof
establishes the consistency of transcomplex and transreal arithmetic and
establishes the expected containment relationships amongst transcomplex,
complex, transreal and real numbers. We discuss some of the advantages
the transarithmetics have over their partial counterparts
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