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b0e9150b9e7a04222034e1b795a5c7a3e3cadb9f | subsection | 56 | 74 | Relative equilibria in | In this section we will prove a few results about fixed points, as well as elliptic
and hyperbolic relative equilibria in {\bf H}^2. We also show that parabolic relative equilibria do not exist. Since, by the Principal Axis theorem for the Lorentz group, every Lorentzian rotation (see Appendix) can be written, in some ... | {
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"raw": "M. Henle, Modern Geometries: Non-Euclidean, Projective, and Discrete, Prentice-Hall, Upper Saddle River, New Jersey, 2001.",
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... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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679444fbc5032c1e59cc85b88348e85442e8d2d3 | subsection | 57 | 74 | Relative equilibria in | Indeed, if all the bodies
are at all times on one side of a plane containing the rotation axis, then the integrals of the angular momentum are violated because the vector representing the total angular momentum cannot be zero or parallel to the z axis.Let us now define the hyperbolic relative equilibria.Definition 4 A ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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6833c14d4c776aa3be028e17801a4f9b665142ab | subsection | 58 | 74 | Fixed Points in | The simplest solutions of the equations of motion are the fixed points. They
can be seen as trivial elliptic relative equilibria that correspond to \omega =0. In
terms of the equations of motion, we can define them as follows.Definition 6 A solution of system (REF ) is called a fixed point if\overline{\nabla }_{{\bf q}... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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78576dfc61fd9e6d9f286aa2ab159bae0dd50b4e | subsection | 59 | 74 | Elliptic Relative Equilibria in | We now consider elliptic relative equilibria, and prove an analogue of Theorem REF .Theorem 10 Consider the n-body problem with equal masses in {\bf H}^2. Then,
for any m>0 and z>1, there are a positive and a negative
\omega that produce elliptic relative equilibria in which the bodies are at the
vertices of an n-gon r... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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3516ebdf29e979d3cb0075157d576efaff3477e6 | subsection | 60 | 74 | Elliptic Relative Equilibria in | To prove the other implication, we substitute into equations (REF ) a solution
of the form (REF ) with i=1,2,3,\ \rho :=\rho _1, \rho _2, \rho _3,\ z:=z_1=z_2=z_3=
(\rho ^2+1)^{1/2}, and \alpha _1=0, \alpha _2=2\pi /3, \alpha _3=4\pi /3. The
computations then lead to the system{\left\lbrace \begin{array}{ll}
m_1+m_2=\z... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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a39fe35b3100b2afab091ed26ee2b85f4c53a4c4 | subsection | 61 | 74 | Elliptic Relative Equilibria in | The sign of \omega determines the sense of rotation.Remark 6 For every \omega ^2/m>0, there is exactly one z>1 that satisfies
equation (REF ) (see Figure REF ).
[Figure: The graph of the function f(z)={4z^2+1\over 4z^3(z^2-1)^{3/2}} forz>1.] | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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8c22ac23c1773789b8550c5c4d6364f1224b8656 | subsection | 62 | 74 | Hyperbolic Relative Equilibria in | We now prove some results concerning hyperbolic relative equilibria.
We first show that, in the n-body problem, hyperbolic relative equilibria
do not exist along any given fixed geodesic of {\bf H}^2. In other words, the
bodies cannot chase each other along a geodesic and maintain the same
initial distances for all tim... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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d5619da2c32f82378860ea14207a02d68c14710f | subsection | 63 | 74 | Hyperbolic Relative Equilibria in | But every possible choice implies the existence of at least
one i and one j with i\ne j and \alpha _i=\alpha _j. For those i and j,
\sinh (\alpha _i-\alpha _j)=0, so equation (REF ) is undefined, therefore
equations (REF ) cannot have solutions of the form (REF ). Consequently
hyperbolic relative equilibria do not exis... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
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0ccc7e9282333506feea43db938af330dad24609 | subsection | 64 | 74 | Hyperbolic Relative Equilibria in | Then the analogue of equation (REF ) is{m\over x^2|x|(x^2+1)^{1/2}}+{M\over 4x^2|x|(x^2+1)^{3/2}}=\omega ^2,and it is obvious that for any m,M>0 and x\ne 0, there are a positive
and negative \omega satisfying the above equation.Remark 9 Theorem REF also works for two bodies of equal masses, m:=m_1=m_2,
of coordinatesx_... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
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] | 2,008 | en | Mathematics | [
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edd9c4ce7f0286a8c254ae41769bb17b1bfd06a5 | subsection | 65 | 74 | Parabolic Relative Equilibria in | We now show that there are no parabolic relative equilibria. More precisely, we prove the following result.Theorem 15
The n-body problem in {\bf H}^2 has no parabolic relative equilibria.Let x_i,y_i, and z_i be as in the definition of parabolic relative equilibria (REF ).
Then \dot{x}_i=-b_i+c_i, \dot{y}_i=a_i+(c_i-b_... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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bfc8dbf1795ebda4c2c353b9bd9ff885721ef27d | subsection | 66 | 74 | Saari's conjecture | In 1970, Don Saari conjectured that solutions of the classical n-body problem with constant moment of inertia are relative equilibria, , . The moment of inertia is defined in classical Newtonian celestial mechanics as {1\over 2}\sum _{i=1}^nm_i{\bf q}_i\cdot {\bf q}_i, a
function that gives a crude measure of the bodie... | {
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"raw": "D. Saari, On bounded solutions of the n-body problem, in Periodic Orbits, Stability and resonances, G.E.O., Giacaglia (Ed.), pp. 76-81, D. Riedel, Dordrecht, 1970.",
"source_ref_id": "32cbc08f... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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e06b9488fdd0f533a6d94b76ab8dcf8ce757160b | subsection | 67 | 74 | Saari's conjecture | According to the above definition of \bf I, we can assume without loss of generality that the geodesic passes through the point (0,0,1) and rotates about the z-axis
with angular velocity \omega (t)\ne 0. The angular momentum of each body is {\bf L}_i=m_i{\bf q}_i\otimes \dot{\bf q}_i, so its derivative with respect to ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
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407a731cad61e93783a7e2f0baa46c207a375ba8 | subsection | 68 | 74 | Saari's conjecture | Consequently, L_i^z=c_i,
where c_i is a constant.Let us also remark that since the angular momentum and angular velocity vectors are
parallel to the z axis, L_i^z={\bf I}_i\omega (t), where {\bf I}_i=m_i(x_i^2+y_i^2)
is the moment of inertia of the body m_i about the z-axis. Since the total moment
of inertia, {\bf I}, ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
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aba30d3b4bf3cbf497f8bee8a5025eee44a8d09a | subsection | 69 | 74 | Saari's conjecture | Thus \dot{L}_i^x=0 in either case.From here the proof proceeds as before by replacing \bf I with \bf J and the z axis with the x axis, and noticing that L_i^x={\bf J}_i\omega (t), to
show that every m_i has a constant x_i coordinate. In other words, each body is moving along a (in general non-geodesic) hyperbola given ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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f274e2810b518c2fa06b878ed55cc6c84df9b977 | subsection | 70 | 74 | The Weierstrass model | Since the Weierstrass model of the hyperbolic (Bolyai-Lobachevsky) plane is little
known, we will present here its basic properties. This model appeals for at least
two reasons: (i) it allows an obvious comparison with the sphere, both from the geometric and analytic point of view; (ii) it emphasizes the differences be... | {
"cite_spans": [
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"arxiv_id": "",
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"raw": "F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), 307-320.",
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"start": 17... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
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3534fc3dae983334829bb672d3e9470651c1126a | subsection | 71 | 74 | The Weierstrass model | Then the Principal Axis Theorem states that every Lorentzian transformation has one of the forms:A=P\begin{bmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{bmatrix} P^{-1},A=P\begin{bmatrix}
1 & 0 & 0 \\
0 & \cosh s & \sinh s \\
0 & \sinh s & \cosh s
\end{bmatrix}P^{-1},orA=P\b... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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bdc3f0dd843fc821771d570ddb0b190cf8c1a07c | subsection | 72 | 74 | The Weierstrass model | Other properties of the Weierstrass model can be found in and . The Lorentz group is treated in some detail in , but the Principal Axis Theorems for the Lorentz group contained in and fails to include parabolic rotations, and is therefore incomplete. | {
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"Ernesto Perez-Chavela",
"Manuele Santoprete"
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6b90d37e393327b11b0be26f1ae7dc5f5242850a | subsection | 73 | 74 | History of the model | The first researcher who mentioned Karl Weierstrass in connection with the hyperboloidal model of the Bolyai-Lobachevsky plane was Wilhelm Killing. In a paper published in 1880, , he used what he called Weierstrass's coordinates to describe the “exterior hyperbolic plane” as an “ideal region” of the Bolyai-Lobachevsky ... | {
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"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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074d09a3966304fb52d8259fa7217cf46b92df6e | abstract | 0 | 28 | Abstract | We investigate the influence of temperature and dissipation on the
Landau-Zener transition probability in circuit QED. Dissipation is modelled by
coupling the transmission line to a bath of harmonic oscillators. The reduced
description for the density operator is treated within Bloch-Redfield theory. A
phase-space repr... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
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49e0a3d69b95fc8300596a10aa7271f624b5b215 | subsection | 1 | 28 | Introduction | The demonstration of coherent quantum dynamics in superconducting flux
and charge qubits , ,
represents a major step towards a solid-state implementation of a
quantum computer.
Manipulation and readout of a qubit can be achieved by a controlled
interaction with an electromagnetic circuit. For charge qubits
implemented... | {
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5ad8a7f2d46251b5665e9d2d706ccf22da70fcf7 | subsection | 2 | 28 | Landau-Zener dynamics in circuit QED | Circuit QED involves a Cooper pair box that couples capacitively to a
transmission line which is described as a harmonic oscillator
. The Cooper pair box is formed by a dc SQUID such
that the effective Josephson energy = E_\mathrm {J}^0
\cos (\pi \Phi /\Phi _0) can be tuned via an external flux \Phi , where
\Phi _0 den... | {
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fe6439a19c5ab72a649bf0b536a6409fe1a0d039 | subsection | 3 | 28 | Landau-Zener dynamics in circuit QED | For the time-evolution from t=-\infty to t=\infty , the
corresponding bit-flip probability can be evaluated exactly and reads
= 1 - ^{-2 \pi g^2/ v} .Note that this generalisation of the Landau-Zener formula is also
valid for large qubit-oscillator coupling g\gg \Omega for which
more than two levels are relevant and, t... | {
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"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
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f01ab1da546836651c238fb5025c6a323bf0310e | subsection | 4 | 28 | Dissipative dynamics | Dissipative effects in an electromagnetic circuit are characterised by
an impedance Z(\omega ) which, within a quantum mechanical description,
can be modelled by coupling the circuit bi-linearly to its
electromagnetic environment . This provides the
system-bath Hamiltonian , , , = \widetilde{}
+ (a^{\dag } + a) \sum _{... | {
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2c12fbd794d35931ddaabe47fccaa86003255be1 | subsection | 5 | 28 | Dissipative dynamics | The bath enters via the
symmetric and the antisymmetric correlation functions(t)= \frac{1}{2}\langle [(t), ]_{+} \rangle _{\rm eq}
&=&
\int _{0}^{\infty }
\omega \;
J(\omega )
\coth \Big ( \frac{\hbar \omega }{2k_\mathrm {B}T}\Big )
\cos (\omega t)
\\
(t) =\frac{1}{2}\langle [(t), ] \rangle _{\rm eq}
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\int _{0}^{\i... | {
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"David Zueco",
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bac2ea3d47ed23c1d5004c621d0d7b4a810e71d6 | subsection | 6 | 28 | Dissipative dynamics | The corresponding solution of the
Heisenberg equations for the dimensionless position operator (t)
is derived in the and reads(t) = a^{\dag } ^{\Omega t} + a ^{-\Omega t}
+ \big [ (t) \sigma _{x} - (t) \sigma _{y} \big ] ,with the time-dependent functions
(t) = \Omega [ \cos (\Omega t) - \cos (\omega _\mathrm {J}t) ]
... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
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ee1ecc05d700068bb0cd9e5569cc1d839825c9d0 | subsection | 7 | 28 | Dissipative dynamics | The prefactor of the last term contains an effective force=
\frac{}{2(\Omega ^{2} - \omega _\mathrm {J}^{2})}
\Big [
\omega _\mathrm {J} \coth \Big (\frac{\hbar \omega _\mathrm {J}}{2k_\mathrm {B}T}\Big )
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... | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
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"Peter Hänggi",
"Sigmund Kohler"
] | [
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f89832b0477d691bf754016d707f3f0d5d3b4aba | subsection | 8 | 28 | Solving the master equation in phase space | The numerical solution of the quantum master equation QME-cqed
requires an appropriate basis expansion. For the qubit, we choose the
eigenstates of \sigma _z, i.e. |{\uparrow }\rangle and
|{\downarrow }\rangle . The resulting matrix elements \rho _{ij}
with i,j=, are operators in the Hilbert space of the harmonic
oscil... | {
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8777afd67949c996966b5e681f1ff20521a80c25 | subsection | 9 | 28 | Solving the master equation in phase space | The “ground state” \phi _{00} is the Wigner
representation of the density operator of the harmonic oscillator at
thermal equilibrium. Thus, if the oscillator stays close to
equilibrium, the decomposition of the density operator can be
performed with only a few basis states — irrespective of the
temperature. The resulti... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
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2ff8407bcc2cde76f4940b5042e0b82bd8818f39 | subsection | 10 | 28 | Landau-Zener tunnelling at finite temperature | Thermal effects can modify the Landau-Zener transition
probability PLZ even in the absence of dissipation, i.e. for
\gamma =0. Then the natural initial state is no longer the (initial)
ground state |,0\rangle , but rather the canonical ensemble(t=-\infty )
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8146eba551a29418d21e29a959ca2c520b137b49 | subsection | 11 | 28 | Landau-Zener tunnelling at finite temperature | For oscillator
states with n>1, two final oscillator states are possible.
Assuming that interference terms do not play any role, we find the
transition probabilityP_{,n\rightarrow } = w(2g\sqrt{n}) w(2g\sqrt{n+1})
+ [1-w(2g\sqrt{n})] [1-w(2g\sqrt{n-1})] ,which formally also holds for n=0,1.
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57e6c684e4fdb69c664e91d880307aea41d55d67 | subsection | 12 | 28 | Landau-Zener tunnelling at finite temperature | () for \gamma =0), while the solid linesrefer to the individual-crossing result Puuana. The dashedlines are a guide to the eye.]The temperature dependence of the spin-flip probability shown
in figure REF possesses an intriguing non-monotonic
behaviour: For low temperatures, T\lesssim 0.2\hbar \Omega , the
probability i... | {
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7ba7742559b1bffc21d36dda754e2bc8d6626e83 | subsection | 13 | 28 | Dissipative Landau-Zener transitions | In the previous section we have studied the consequences of thermal
excitations of the initial state for the transition
probability PLZ in the absence of an oscillator-bath coupling,
i.e. for dissipation strength =0. We next address the question
how dissipation and decoherence modify Landau-Zener tunnelling. | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
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09095c4e504dc81b65fe08333f28c403c398467c | subsection | 14 | 28 | The zero-temperature limit | For a heat bath at zero temperature, the exact solution of the
dissipative Landau-Zener problem has been derived in recent works
, . Moreover, this limit generally is
rather challenging for a master equation description of
quantum dissipation , . Therefore the
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bf777c6eceeaffeae67e702a07338f8baed44dac | subsection | 15 | 28 | The zero-temperature limit | The sum \sum _{k^{\prime }} c_{k^{\prime }}^{2} can be expressed in terms of the spectral
density Jeff, such it becomes\sum _{k^{\prime }} ^{2}
=
\frac{1}{g^2}\int \omega \, J_{{\rm eff}} (\omega )
=
\frac{1}{\pi } \left[
\arctan \left(
\frac{2\Omega ^{2} -^{2}}{\sqrt{4\Omega ^{2}-^{2}}}
\right)
+ \frac{\pi }{2} \right... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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a6057be6de2088d9147b330f9c79f4200847835b | subsection | 16 | 28 | Thermal excitations and dissipative transitions | We next turn to the generic situation in which both thermal
excitations of the initial state and dissipative transitions play a
role, i.e. we consider the situation of finite temperatures and
finite dissipation strength.
The resulting spin-flip probabilities for three different sweep
velocities are shown in figure REF ... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
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60bb6dd2c1f4ca9a084907c4a1ecfc1cdc1feb45 | subsection | 17 | 28 | Thermal excitations and dissipative transitions | Thus for large times, t\gg \Omega /v, the spectral density at the
qubit splitting \hbar vt vanishes and, consequently, the qubit is
effectively decoupled from the bath.
[Figure: Population dynamics of the states {, 0} (a) and{,1} (b) during the Landau-Zener sweep.Dissipation strength and temperature are=10^{-2}\Omega a... | {
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"en... | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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ff0b552bcd35be142cfc4689b3942ca5fb4d624b | subsection | 18 | 28 | Conclusions | We have investigated the influence of finite temperature, decoherence,
and dissipation on Landau-Zener transitions of a two-level system
(qubit) that is coupled via a harmonic oscillator to a heat bath. In
particular, we have focussed on a recent solid-state realization of
this model, namely the so-called circuit QED f... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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ee7dc0ccae57be002c44a72278b9a98d2a58d3b8 | subsection | 19 | 28 | Derivation of the quantum master equation | In this Appendix we outline the derivation of the master
equation QME-cqed starting from the general Bloch-Redfield
expression BR. | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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7cf1a32b2cca4f24b859a29d17b83c3d22be9e5d | subsection | 20 | 28 | Heisenberg coupling operator | The essential step is the solution of the
Heisenberg equation of motion for the scaled position operator = a
+ a^{\dag } of the oscillator, which will rely on approximations.
In doing so, we even address a slightly more general qubit Hamiltonian
outside the charge degeneracy point, i.e. we also consider the
charging en... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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38ecc8c4e77907ac514145a0e592d793fd21086a | subsection | 21 | 28 | Heisenberg coupling operator | In the absence of the coupling,
the time evolution of the qubit operators reads\bar{\sigma }_{x}(t) &=& \bar{\sigma }_{x} \cos (t) + \bar{\sigma }_{y} \sin (t) ,
\\
\bar{\sigma }_{z}(t) &=& \bar{\sigma }_{z} .Inserting this into the equation of motion Qddot for the
oscillator coordinate, it becomes evident that the qub... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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a96890b24055e47115499bbd2c6541f408ccb199 | subsection | 22 | 28 | Ohmic spectral density | In circuit QED, the environment of the qubit and the transmission line
is formed by electric circuits and, thus, can be characterised by an
effective impedance. In most cases, this impedance is dominated by an
Ohmic resistor, which corresponds to the Ohmic spectral density
Jw of the bath. Then the time integration in t... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
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] | 2,008 | en | Physics | [
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467572c02880fc3cfa20c665194dcc1abe38ff5e | subsection | 23 | 28 | Basis expansion | In this appendix we outline the diagonalisation of the oscillator
Liouvillian in Wigner representation, , whose eigenvectors are
used as a basis set for the numerical treatment. Since the operator
, apart from the cross-diffusion D_{xp}, is of the same form as
the Fokker-Planck operator of the corresponding problem for... | {
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... | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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45a5b7827a310ae673b97f182ef010b423a3fc4b | subsection | 24 | 28 | Diagonalisation of the oscillator Liouvillian | By solving the characteristic functions of the partial differential
equation \dot{\phi }=\phi , one finds the operators&=& \partial _{x} + \frac{}{\Omega } \partial _{p} \, ,
\\
&=&
\frac{\Omega ^{2}}{^{2}-\Omega ^{2}}
\Big ( \partial _{x} - \frac{}{\Omega } \partial _{p} + x
- \frac{}{\Omega } p \Big ) ,which commute... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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8083ec96ad2657bceb471c3e1422a954b71d9576 | subsection | 25 | 28 | Diagonalisation of the oscillator Liouvillian | Repeating the calculation from above for
^\dag , we find the left ground state _{00}=1, so that we
obtain the eigenfunctions_{nn^{\prime }}
&=&
\frac{1}{n! n^{\prime } !}
()^{n} ()^{n^{\prime }} _{00}
\\
_{nn^{\prime }} &=& ()^{n} ()^{n^{\prime }} \; 1which fulfil the ortho-normalisation relation\int x p \, _{mm^{\prim... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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733918ebbbbb457a9015cac1e2ff14796f0882f6 | subsection | 26 | 28 | Expansion of the entire Liouvillian | Besides the already diagonalized oscillator Liouvillian, the total
Liouvillian QME-cqed-ps for the qubit plus the oscillator
contains also the operatorsx &=& - - + +^*, \qquad = \frac{\Omega ^{2}}{^{2} -\Omega ^{2}},
\\
\partial _{p}
&=& -\frac{-}{2 \sqrt{1-^{2}/4 \Omega ^{2}}} .Then, the basis decomposition of the Wi... | {
"cite_spans": []
} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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ac90fcff91441126b22639965b2ab5442c8555d6 | subsection | 27 | 28 | Computation of expectation values | The expectation value of an operator can be performed directly in the
basis of the eigenfunctions without back transformation to the operator
representation of the density operator. For an observable= \otimes (x,p) ,for which S and refer to qubit and oscillator variables,
respectively, the expectation value\langle \ran... | {
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} | 10.1088/1367-2630/10/11/115012 | 0807.1748 | Landau-Zener tunnelling in dissipative circuit QED | [
"David Zueco",
"Peter Hänggi",
"Sigmund Kohler"
] | [
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b17a41ccfe6e2e3219bd06b10a7bf70fdc4bab5d | abstract | 0 | 17 | Abstract | We derive the discrete anomaly conditions for the binary tetrahedral group T'
as well as the binary dihedral groups Q_2n. The ambiguities of embedding these
finite groups into SU(2) and SU(3) lead to various possible definitions of the
discrete indices which enter the anomaly equations. We scrutinize the different
choi... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.11.025 | 0807.1749 | Discrete Anomalies of Binary Groups | [
"Christoph Luhn"
] | [
"hep-ph",
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2a1e43532316edd6f85ed6657f9f9c4159ef1139 | subsection | 1 | 17 | Introduction | The family structure of the Yukawa couplings which give rise to the masses and
mixings of quarks and leptons remains unexplained within the Standard
Model (SM). One of the most successful ideas to overcome this annoying
shortcoming consists in extending the SM gauge group with a family-dependent
U(1) symmetry. These Fr... | {
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{
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"doi":... | 10.1016/j.physletb.2008.11.025 | 0807.1749 | Discrete Anomalies of Binary Groups | [
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c37c35bdd2403c2df3316a53a8ea095064d1a496 | subsection | 2 | 17 | Introduction | It is one of the main intentions of this
letter to shed some light on this ambiguity, proving that one particular
assignment of discrete indices is sufficient to determine whether a specific
model of flavor is discrete anomaly free or not. We illustrate the procedure
by applying our results to some existing examples. | {
"cite_spans": []
} | 10.1016/j.physletb.2008.11.025 | 0807.1749 | Discrete Anomalies of Binary Groups | [
"Christoph Luhn"
] | [
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21f84422b405f1f7a0a4a73a076bf3b39983ac58 | subsection | 3 | 17 | The Binary Tetrahedral Group | The alternating group on four letters, \mathcal {A}_4, is the symmetry group of the
tetrahedron, and as such a subgroup of SO(3). It has three one-dimensional
and one three-dimensional irreps. Similar to SU(2)
being the double cover of SO(3), the binary tetrahedral group \mathcal {T}^{\prime } is
the double cover of \m... | {
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1e4537531a86997dfe0ed5250198c3e70015aee9 | subsection | 4 | 17 | The Binary Tetrahedral Group | The
resulting Kronecker products are those of \mathcal {A}_4{\bf 1_k} \otimes {\bf 1_l} ~=~ {\bf 1_{k+l}} \ , \qquad {\bf 1_k} \otimes {\bf 3} ~=~ {\bf 3} \ , \qquad {\bf 3} \otimes {\bf 3} ~=~ {\bf 1_0} + {\bf 1_1}+{\bf 1_2} + 2 \cdot {\bf 3}
\ ,plus the ones involving the doublets{\bf 2_k} \otimes {\bf 1_l} ~=~ {\bf ... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.11.025 | 0807.1749 | Discrete Anomalies of Binary Groups | [
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] | [
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33d7723e3b0ee8de46a589c56da8bbf9c2d87b9e | subsection | 5 | 17 | The Binary Tetrahedral Group | We obtain\begin{array}{llll}
\widetilde{\ell }({\bf 1_0}) ~=~0\ , &\quad \widetilde{\ell }({\bf 1_1}) ~=~x\ , &\quad \widetilde{\ell }({\bf 1_2}) ~=~16-x\ , & \\[2mm]
\widetilde{\ell }({\bf 2_0}) ~=~1\ , &\quad \widetilde{\ell }({\bf 2_1}) ~=~y\ , &\quad \widetilde{\ell }({\bf 2_2}) ~=~10-y\ , & \quad \widetilde{\ell }... | {
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cf5317ff1b61d38398c8bd51440f697b766cacfd | subsection | 6 | 17 | The Binary Tetrahedral Group | This is owed to the
breaking pattern of SU(2) down to \mathcal {T}^{\prime }: the irreps {\bf 1_1} and {\bf 1_2} always come in pairs, as do the irreps {\bf 2_1} and {\bf 2_2}. Since mass terms need to be of the form {\bf 1_1} \otimes {\bf 1_2}
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5924f83bd53af21f16658953c7c11484fe5b0c34 | subsection | 7 | 17 | The Binary Tetrahedral Group | With the hypercharges being opposite to each
other, the particles of each pair add\begin{array}{ll}
{\bf 1_1}-{\bf 1_2}: & \quad Y_{\bf 1_1} (2x-16) \ ,\\[2mm]
{\bf 2_1}-{\bf 2_2}: & \quad Y_{\bf 2_1} (2y-10) \,=\, - Y_{\bf 2_1}
(2x+8) \ ,
\end{array}to the discrete anomaly. Choosing either x=8 or x=20 their
contributi... | {
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a1715ba2fe4f3bd3dce4ba0d11a78fb33283fac3 | subsection | 8 | 17 | The Binary Tetrahedral Group | As these SU(2) irreps are the only
ones that, under \mathcal {T}^{\prime }, decompose with an odd number of {\bf 2_0} irreps,
the constraint from the Witten anomaly breaks down to
the requirement of having an even number of {\bf 2_0} irreps in the complete
theory, including the heavy degrees of freedom. From the Kronec... | {
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} | 10.1016/j.physletb.2008.11.025 | 0807.1749 | Discrete Anomalies of Binary Groups | [
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9b88e12d6a1a62fa5395a9d834e0854570d4e27c | subsection | 9 | 17 | The Binary Dihedral Groups | The dihedral group \mathcal {D}_n is the symmetry group of the planar n-polygon.
Its 2n elements can be expressed as rotations in three-dimensional space,
indicating that \mathcal {D}_n \subset SO(3). The dicyclic or binary dihedral group
\mathcal {Q}_{2n} is the double cover of \mathcal {D}_n. It is defined by the
pre... | {
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52dca1750a4a3c6efe2125f89720c43e18aeb040 | subsection | 10 | 17 | The Binary Dihedral Groups | This shows that, for odd n, \mathcal {D}_n has
only two one-dimensional irreps.The Kronecker products for the irreps of \mathcal {Q}_{2n} are{\bf 1_1} \otimes {\bf 1_1} ~=~ {\bf 1_{0}} \ , \qquad {\bf 1_1} \otimes {\bf 1_2} ~=~ {\bf 1_3} \ , \qquad {\bf 1_1} \otimes {\bf 1_3} ~=~ {\bf 1_2} \ ,\\[3mm]{\bf 1_{2}} \otimes... | {
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2188335d0babc4bac134b6ae0333afc4653bf726 | subsection | 11 | 17 | The Binary Dihedral Groups | Using Eqs. (REF ) and (REF ), one
finds the following embedding .The constraints
arising from the Witten anomaly require that the number of spinor-like irreps,
i.e. the {\bf {2_{\kappa \alpha }}} with \kappa =\mathrm {odd}, is even if all
degrees of freedom are counted. However, the Kronecker products show that
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7980175f0eb5f4540370796077e7fa3c6f432f2d | subsection | 12 | 17 | The Binary Dihedral Groups | Notice that the discrete indices for {\bf 2_{{\kappa \alpha }}},
{\bf 2_{{-\kappa \alpha }}}, and {\bf 2_{{(\kappa +2n)\alpha }}} are
identical as required by the identities of Eq. (REF ).
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7ce3a2431c77d6d6cf06aa75ba507e5d0d9c4c1a | subsection | 13 | 17 | The Binary Dihedral Groups | (REF ) gets replaced by\widetilde{\ell }({\bf 1_2}) ~=~ \widetilde{\ell }({\bf 1_3})~=~ \frac{n^2}{2} \,+\,\zeta \cdot 2n \qquad (n=\mathrm {odd}) \ ,with \zeta =0,1, while for even n we still have Eq. (REF ) with
arbitrary y. With the discrete indices defined in
Eqs. (REF )-(REF ), a model is discrete anomaly free if\... | {
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aa9d95b1418609e27bc313cb30b15e313067849a | subsection | 14 | 17 | The Binary Dihedral Groups | For odd n the term (c^{\prime }_2+c^{\prime }_3)\cdot n^2/2 has to be integer allowing us to
define 2c_n=c^{\prime }_2+c^{\prime }_3 with c_n\in \mathbb {Z}. Thus we can simplify our
equation to\!\!\!\left.\begin{array}{l}
\sum _{\kappa =1}^{n} c_\kappa \kappa ^2
\\[3mm]
c^{\prime }_1 \, \xi \,2n
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c58cfb85dd012b914e189b5dd27b6823e8ac74a6 | subsection | 15 | 17 | The Binary Dihedral Groups | For the remaining examples, the assignments of the quarks and leptons to the
irreps of \mathcal {Q}_{2n} are listed in Table REF . Other fermions that are
introduced in these models (like for example the Higgs doublets in
supersymmetric models) give no net contribution to the \mathcal {Q}_{2n}-\mathcal {Q}_{2n}-U(1)_Y ... | {
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e42c85228fb6109d1dad24210cf6d429100c9d8a | subsection | 16 | 17 | Conclusion | In this letter we have derived the discrete indices for the irreps of the
binary tetrahedral group \mathcal {T}^{\prime } as well as the binary dihedral groups \mathcal {Q}_{2n}. Despite the ambiguities of embedding the finite symmetries into
continuous SU(2) and SU(3), it is possible to define discrete indices that
en... | {
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e6b995906dc5b05367310912ab787ea1344283be | abstract | 0 | 19 | Abstract | Inspired by biological dynamics, we consider a growth Markov process taking
values on the space of rooted binary trees, similar to the Aldous-Shields
model. Fix $n\ge 1$ and $\beta>0$. We start at time 0 with the tree composed of
a root only. At any time, each node with no descendants, independently from the
other node... | {
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} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
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1063f7a9aead5b720b87196b1dad0a9cc78e4ca4 | subsection | 1 | 19 | Introduction | Growth and replicative senescence are very often
connected. In cultured cells growth is not observed indefinitely,
the division rate slows down and ultimately ceases .
A
cell clock mechanism based on the shortening of the replica with respect to
the DNA template has been proposed in 1970s and was
called marginotomy of ... | {
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"raw": "Hayflick L. The limited in vitro lifetime of human diploid cell strains. Exp. Cell Res. 1965; 37: 614-36.",
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e143f4362038444e2f5108defc7cc6e5a42a212d | subsection | 2 | 19 | Introduction | We
also estimated the mean telomere size of the cell population, which
can be actually measured in cultured cells.Is worth of mention that the results obtained here do not depend on the
particular division-counting mechanism. So, telomere or any other cell
clock may be responsible for the replicative senescence/Gompert... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
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"B. F. Svaiter"
] | [
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7a5e446e4c5bfa78250f5d6b4aedb7891e8bf0cf | subsection | 3 | 19 | The stochastic model | We will assume that each cell in the initial cell population has its
telomere with the same length L_0.
For the sake of simplicity, we will also assume that a fixed amount
of basis is lost by each telomere at each cell division, say \delta .
This is an approximation of the dynamics of telomere length .
Future research ... | {
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d4dc6bf255433e848475ba5a9c09abe841abfb85 | subsection | 4 | 19 | The stochastic model | The expected value of X(k,t) isE(X(k,t))=\sum _{j=0} ^\infty j P(X(k,t)=j).Define\begin{array}{l}
x_k(t)=E(X(k,t)),\qquad k=0,\cdots ,n.
\end{array}Thenx_k(0)=
{\left\lbrace \begin{array}{ll}
1,& k=0,\\
0,& k=1,\cdots ,n
\end{array}\right.}and\begin{array}{rcl}
d\; x_0(t)/dt&=&-\lambda _0 x_0(t),\\
d\; x_k(t)/dt&=&2\la... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
"q-bio.CB",
"math-ph",
"math.MP",
"q-bio.QM",
"stat.OT"
] | 2,008 | en | Quantitative Biology | [
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538ba15000d7d4d793f5d4b7d29c7e8473099510 | subsection | 5 | 19 | The stochastic model | The expected number of cells at time t is given by\sum _{i=0} ^n x_i(t)=[1,1,\dots ,1] {\mathbf {x}} (t)\;. | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
"q-bio.CB",
"math-ph",
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] | 2,008 | en | Quantitative Biology | [
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0.014483818784356117,
... | |
82e73f978e0e0f4dca2ba946115cf855525b2eb6 | subsection | 6 | 19 | The dynamics of the stochastic model | In this section we present the analytical solution of the differential
equations which governs the expected size of the cell
population (REF ). We compute in particular the matrix
\exp (t\mathbf {M}).In order to solve (REF ), first note that, since M is
lower-triangular, the spectrum of M is
-\lambda _0,-\lambda _1,\cd... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
"q-bio.CB",
"math-ph",
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] | 2,008 | en | Quantitative Biology | [
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... | |
ca64fb1183e219dda370935df4dc36fbe972d219 | subsection | 7 | 19 | The dynamics of the stochastic model | Using again the fact that \mathbf {M} is
lower-triangular and (REF ) we haveb_{i,k}=0, \qquad i<k.For i=k, since m_{k,k}=-\lambda _{k-1},b_{k,k}=(m_{k,k}+\lambda _{k-1})\;w_{k,k}=0.So, if k=n+1 then b=0 and the claim holds.For the case k=1,2,\cdots ,n and i>k,b_{i,k}=m_{i,i-1}\;w_{i-1,k}+(m_{i,i}+\lambda _{k-1})w_{i,k}... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
"q-bio.CB",
"math-ph",
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] | 2,008 | en | Quantitative Biology | [
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0.0092372... | |
55bc5ce5a909818e473ff9439186aea038e954a0 | subsection | 8 | 19 | The dynamics of the stochastic model | Then, for
any \eta \in \mathbb {R},[\eta ^n,\eta ^{n-1},\dots , \eta , 1] \mathbf {A}(\gamma )=
[(\eta +\gamma )^n,(\eta +\gamma )^{n-1},\dots , (\eta +\gamma ), 1]\,.Proof.Let
(u_1,\cdots ,u_{n+1})= [\eta ^n,\eta ^{n-1},\dots , \eta , 1]
\mathbf {A}(\gamma ).
Direct calculation, together with (REF ) yields
u_k&=&\s... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
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] | [
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... | |
4432fb42ddf75bd48384de087eea0cc0b0461505 | subsection | 9 | 19 | The dynamics of the stochastic model | From this expressions and (REF ), we obtain a simple expression for
the expected number of cells at time t:
\sum _{i=0} ^n x_i(t)
=[1,1,\dots ,1]\mathbf {A}(-2)\;
\exp (t\mathbf {D})
\mathbf {A}(2)\begin{bmatrix}
1\\0\\ \vdots \\0
\end{bmatrix}
To evaluate (REF ), use Lemma REF and (REF )
to obtain
[1,1,\dots ,1]\m... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
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30348fc52e9ea9ca26c89143baf91e2fc51fc250 | subsection | 10 | 19 | The dynamics of the stochastic model | Defining the normalized Growth curve
S_{norm}(\tau )=S(t_*+n\,\tau )
or, alternatively
using the change of variable
t= t_*+n \tau ,
in (REF ) we get
2^{-n} \sum _{i=0} ^{n} x_i(\tau )=
\left(1-\frac{e^{-\beta \tau }\theta _n\ln 2}{n} \right)^n
\approx \exp \left(-e^{-\beta \tau }\theta _n\ln 2 \right).
Note that t... | {
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76181c783c8ecbfbf7610d976baa7d1d197a7301 | subsection | 11 | 19 | The dynamics of the stochastic model | To evaluate this quotient,
define for \Psi :\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}
\Psi (u,t)= \sum _{k=0} ^n u^{n-k}
X(k,t)
Let \psi (u,t) be the expected value of \Psi (u,t)
\psi (u,t)&=E(\, \Psi (u,t)\,) \\
&= \sum _{k=0} ^n u^{n-k}
E(X(k,t))
As E(X(k,t))=x_k(t), using (REF ) we obtain
\psi (u,t... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
] | [
"q-bio.CB",
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06f855111d759b0c38cd3610edfcaeb8b04e0ac4 | subsection | 12 | 19 | The dynamics of the stochastic model | Using the approximation
E\left(
\frac{ \sum _{k=0} ^n (n-k)\; X(k,t)}{\sum _{k=0} ^n X(k,t))}
\right)
\approx \frac{E\left( \sum _{k=0} ^n (n-k)\; X(k,t)\right)}{E\left( \sum _{k=0} ^n
X(k,t)\right)}
we obtain the estimation of the expected mean telomere size:
E\left(
\frac{ \sum _{k=0} ^n(L_{\min }+(n-k)\delta )\; ... | {
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cbab57375c1d0ab4509b766e4460680bf238583f | subsection | 13 | 19 | The dynamics of the stochastic model | For a general review on this perspective, see | {
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da01de975fb43ed6a8beb716015fe96f14020e07 | subsection | 14 | 19 | How close are the stochastic and Gompertzian models? | First use (REF ) to determine the time t_* at which the
(expected) population size is half of the final size:n\ln \left(1-\frac{e^{-\beta t_*/n}}{2}\right)=\ln 1/2 =-\ln 2Then-\ln \left(1-\frac{e^{-\beta t_*/n}}{2}\right)=(\ln 2)/n.As h<-\ln (1-h) for 0<h<1,
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c3505a263f1c70c9242163b7cb947054eff66502 | subsection | 15 | 19 | Time evolution of the mean telomere length | In cultured cells, telomere length is not evaluated
individually. Instead, what is actually measured is the mean
telomere length of a bunch of cells , . | {
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5bf1270fd0d0803277e495ddb2ba908549a373aa | subsection | 16 | 19 | Time evolution of the mean telomere length | Therefore, in order to verify the
fitness of the stochastic model to real data, we must obtained the
expected mean
telomere length predicted by the model.In our model, the telomere length of a cell which has undergone k
mitosis isL_{\min }+(n-k)\delta .Therefore, the mean telomere length at time t is\frac{ \sum _{k=0} ... | {
"cite_spans": []
} | 0807.1750 | A Markovian growth dynamics on rooted binary trees evolving according to
the Gompertz curve | [
"C. Landim",
"R. D. Portugal",
"B. F. Svaiter"
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19b120e496fce8290adc5df716fcabdc603c4631 | subsection | 17 | 19 | Time evolution of the mean telomere length | So, it it important to remark that this random variable can be
measured in experiments of cell culture tissue. Hence, we have an observable
quantity which may agree or disagree with the model proposed here, confirming or invalidating respectively the model.Using the approximationE\left(
\frac{ \sum _{k=0} ^n (n-k)\; X(... | {
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46312def583f7c03e824b53eaf1c38a51e2c6601 | subsection | 18 | 19 | Discussion | We studied an stochastic continuous-time growth model where the
number of past divisions determine the probability of the next
division. There is biological data supplying evidence that cell do
count division, by means of telomere shortening. Moreover,
mitosis-dependent telomere shortening is well documented, as well a... | {
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efcfc92fb327b3a466727f641343582200c56402 | abstract | 0 | 18 | Abstract | Agostic interactions are covalent intramolecular interactions between an
electron deficient metal and a sigma-bond in close geometrical proximity to the
metal atom. While the classic cases involve CH sigma-bonds close to early
transition metals like titanium, many more agostic systems have been proposed
which contain C... | {
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} | 10.1016/j.ccr.2008.07.007 | 0807.1751 | Characterization of agostic interactions in theory and computation | [
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68c4403ab99ee751ef66205e23290514b2a283ca | subsection | 1 | 18 | Introduction | Transition metal compounds which exhibit a close proximity of CH systems to the metal atom were discovered relatively early in the mid 1960's and early 1970's as the quality and availability of x-ray crystallography improved , , , , , . But the dispute as to whether or not the close approach was caused by an attractive... | {
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afe1632f7c74ae59ae99d273b17d9380259af0d4 | subsection | 2 | 18 | Agostic Interactions by type | As mentioned earlier, there are several types of agostic interactions. They are usually grouped according to the connectivity between the interacting atom (usually hydrogen) and the metal along covalent bonds (See fig. REF ).
[Figure: \alpha , \beta and \gamma agostic interactions.]In this sense agostic interactions ar... | {
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e73f93f9adf75b67ac4e1752ea40cc859daccc0a | subsection | 3 | 18 | Body | The first theoretical work on \alpha -agostic interactions was done more than 20 years ago by Goddard, Hoffmann and Jemmis . The importance of this work stems from the fact that this study radically altered the view on \alpha -agostic interactions. Until then, it was thought that the direct interaction of the occupied ... | {
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d2b0730c7341f74b00ef8d9085c9089338af74fd | subsection | 4 | 18 | Body | Some authors go as far as arguing for a replacement of the term agostic interaction with agostic geometry to emphasize the fact that the structural change in those compounds does not stem from a direct interaction of the metal atom and the carbon-hydrogen bond , .Not surprisingly many of the conclusions drawn for \alph... | {
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e3120f0e3e3b49099efbb20e8a6498fc4c807511 | subsection | 5 | 18 | Agostic Interactions by method | Since many methods have been applied to the characterization of agostic interactions it is instructive to review the main computational tools that are commonly used. This section is divided into two subcategories.Firstly, the purely computational tools are reviewed. While density based methods like Bader's AIMrecently:... | {
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a3bf14d4be55c683791e3a3bd6b893fde3702d20 | subsection | 6 | 18 | AIM - Atoms in Molecules | The atoms in molecules method by Bader is an elegant way to rationalize chemical information through physical observation. Since many chemical concepts, starting with the term chemical bond, evolved when chemistry was still exclusively empirical it is often hard to translate them into physically meaningful concepts . A... | {
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721825d3de6fb43439054d8f0c3a7f145fce3a5c | subsection | 7 | 18 | AIM - Atoms in Molecules | However, the established values for hydrogen bonds lie in the range of 0.002–0.035 a.u. indicating that the interaction in these compounds is fundamentally different. Similarly, the Laplacian \nabla ^2\rho of the electron density at the bond critical point falls outside the established region for hydrogen bonds. Finall... | {
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c410c45a18796f21180adf7110f193380f551fe2 | subsection | 8 | 18 | AIM - Atoms in Molecules | They state that while there is no bond critical point indicating the existence of agostic interactions in this system the molecular orbital picture leads to the conclusion that the \beta -agostic group acts as a two-electron ligand. This means that the first step of the polymerization is dominated by the CH\cdots Ti in... | {
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4ee90b17f036a2fd0884b823b982b9486803feab | subsection | 9 | 18 | AIM - Atoms in Molecules | Other functionals may fail to locate the agostic minimum entirely. McGrady and coworkers point out that only a functional that correctly reproduces a range of physical observables is fit for an analysis of agostic bonding situations. | {
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91613c20dfe85984a7e5d507e03add3783d42418 | subsection | 10 | 18 | NBO - Natural Bond Orbitals | One of the most widely used methods for the investigation of chemical bonding is Weinhold's natural bonding orbital approach . NBO generalizes the concept of natural orbitals, which are obtained by diagonalizing the first-order density matrix, by deriving natural atomic orbitals and natural bond orbitals from the one-e... | {
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26fde70459d089cb072e615df560efe7d0d641d9 | subsection | 11 | 18 | NBO - Natural Bond Orbitals | The \rm p_z orbital has a natural population of 0.318 e and \Delta E_{i\rightarrow j^*}^{(2)} for the CH-\sigma \rightarrow B \rm p_z donation yields 8.96 kcal/mol.
A more traditional agostic interaction with a \gamma -BH\cdots Sn structure was recently reported . Here, the B-H \sigma bond orbitals interact strongly wi... | {
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6c5cbfa099d3bb26449290e6c5d44b4ba851533a | subsection | 12 | 18 | NBO - Natural Bond Orbitals | It is concluded that the low lying 6d and 5f orbitals form the foundation of the energetic preference of agostic geometries over the anagostic geometries.Calculations on agostic interactions involving silicon, both as SiC-agostic interactions as well as SiH-agostic interactions , have been performed recently.Clot, Moun... | {
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c457af456f617d3e673a8a6f4a65a3050282ecbd | subsection | 13 | 18 | NBO - Natural Bond Orbitals | Later Desiraju and Thakur expanded on the difference between agostic interactions and hydrogen bonds using NBO as a platform for their calculations .Although there is a clear preference for AIM theory for the treatment of agostic interactions among purely theoretically working groups and some experimental groups , , NB... | {
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97d093989c81ab150fc025092ab5fb745cf76c9c | subsection | 14 | 18 | other methods | Another commonly applied procedure to elucidate chemical bonding is the electron localization function or ELF. First introduced as a measure to compare the degree of electron localization in a given system compared to a uniform electron gas by Becke and Edgecombe and soon applied to chemical systems , . Soon after, Sil... | {
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bcc376c8ceada45c69a2216b7f5c5169006ac3e8 | subsection | 15 | 18 | NMR properties | One of the tell-tale signs of an agostic interaction is a significantly lower NMR \rm ^1J_{CH} coupling constant. While the CH bond length is only marginally elongated, the coupling constant can decrease by more than 50% . Because of the computational effort that is usually involved in the calculation of NMR properties... | {
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be26e02508770c532decbefa14b4deb5a444ec8a | subsection | 16 | 18 | Vibrational Constants | Because agostic bonds are usually slightly elongated compared to their anagostic counterparts the associated CH stretching frequency is lower than that of a CH bond with no additional interactions present.
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8f91e014488156d80a540f051d8cb15f56dee0be | subsection | 17 | 18 | Conclusions | Given the wealth of publications in recent years it is obvious that agostic interactions still continue to draw attention and inspire further work. The very nature of the interaction is far from completely understood and many computational tools have been used to probe a wide variety of systems. It is the aim of this r... | {
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