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6f98b99724820bb3eb03c89ec895b1443fbf2b05 | subsection | 2 | 19 | Introduction | Of course, in our setting, we are dealing with sequence length, not computing time, but the two problems have an analogous flavour.In the next section, we describe a general framework for discussing these issues, and we exhibit an (abstract) example where the sequence length required to test a discrete parameter is far... | {
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08577386b8dda59de1e28d502e8d4c062c97feb7 | subsection | 3 | 19 | Testing versus reconstructing | In this section, we describe definitions and properties of testing and reconstructing in a general setting; we will specialize our approach to the phylogenetic setting in the following section.Suppose A=A_n and U= U_n are finite sets, and that we have a random variable X=X_{(a,\theta )} taking values in U and whose dis... | {
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48965ec33ae42a78dc86fd805024682339fe371b | subsection | 4 | 19 | Definitions: Reconstruction, testing and accuracy | Throughout we will let (X_{(a, \theta )}^1, \ldots X_{(a, \theta )}^k) denote a sequence of k i.i.d. observations generated by (a, \theta ).A reconstruction method R is a random variableReconstruction is often viewed as a deterministic function, but in reality, most methods have to break ties and so allowing R to be ra... | {
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a38c6ec844c8dd4802ca3f23ad46f0192599f529 | subsection | 5 | 19 | Definitions: Reconstruction, testing and accuracy | We say that a testing process has accuracy 1-\epsilon (for k samples) if for all a \in A, and \theta \in \Theta (a), the following
two conditions hold:{\mathbb {P}}(\psi (a, (X^1_{(a,\theta )}, \ldots , X^k_{(a,\theta )})) = {\rm true}) \ge 1- \epsilon ,and for any b \ne a, and \theta ^{\prime } \in \Theta (b),{\mathbb... | {
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36311c099fe9d6252f709e7fc5bbaebe8a973dde | subsection | 6 | 19 | Definitions: Reconstruction, testing and accuracy | For Part (ii), let U = \lbrace 1, \ldots , n\rbrace and let A be a collection of subsets of U with the property that
for any two elements a,a^{\prime } \in A:\frac{|a \cap a^{\prime } |}{\min \lbrace |a|, |a^{\prime }|\rbrace } \le \epsilon .Consider the following simply-parameterized random variable X_a (a \in A) defi... | {
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597a35852c4f08aac7104f7e4cfaf820012e7947 | subsection | 7 | 19 | Definitions: Reconstruction, testing and accuracy | The following lemma follows easily from the definitions.Lemma 2.2
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2de4781a8dc07e8725bceed317fbc8af771ff2e4 | subsection | 8 | 19 | Testing versus reconstruction in phylogenetics | A phylogenetic (X–) tree is a tree , whose leaf set X is labelled and whose interior vertices are unlabeled and of degree at least 3. If, in addition, every interior vertex of has degree exactly 3 then is said to be {\em binary}. Without loss of generality, we can usually take X={1, ..., n}.
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51104902a00f08b56ab0fc95f7a30e8e609e183e | subsection | 9 | 19 | Testing for a finite-state Markov process requires at least | Finite-state Markov processes on trees underlie many approaches in molecular phylogenetics (see, for example, ). We provide a brief formal description; for more details, the reader may wish to consult or .
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846f6987962c71c5f841f3bf4ce91ee4c6408444 | subsection | 10 | 19 | Testing for a finite-state Markov process requires at least | Then any testing procedure that has accuracy > \frac{1}{2} requires k to grow at least at the rate \log (n),
even in the simply-parameterized setting where all edge lengths are equal to a fixed strictly positive value.Remark:
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b6aa7ed3c7f0775f19e8c902ac5b1626f9c09ee4 | subsection | 11 | 19 | Testing for a finite-state Markov process requires at least | Let R: U^k \rightarrow \lbrace a, *\rbrace be defined as follows:R({\bf u}) =
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79f2730cd2ef300275c7330265bbd46f37418784 | subsection | 12 | 19 | Testing for a finite-state Markov process requires at least | Note that C has the same probability distribution for any element b in A^{\prime } as it does for a, and so, in particular, we have:{\mathbb {P}}_a(C=c) = {\mathbb {P}}_*(C=c)for all choices of c.
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276b6a45093dfe83e49fff502167ac62566d32d6 | subsection | 13 | 19 | Testing for a finite-state Markov process requires at least | Then:\sum _{\bf u}|Q_a({\bf u})-Q_*({\bf u})| = \sum _{\bf u}|Q_a({\bf u})-\frac{1}{m}\sum _{b \in A^{\prime }} Q_b({\bf u})|.We can rewrite the expression on the right-hand side of (REF ) as:|\sum _{\bf u} Q_a({\bf u})(1- \frac{1}{m}\sum _{b \in A^{\prime }}\frac{Q_b({\bf u})}{Q_a({\bf u})})| \le \sum _{\bf u} Q_a({\b... | {
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49ec4a1d428cb95298e6a6740d6cf6c91fb7e010 | subsection | 14 | 19 | Testing for a finite-state Markov process requires at least | Thus, by Jensen's inequality:{\mathbb {E}}\left(|\frac{1}{m}\sum _{b \in A^{\prime }} Z_b|\right)^2 \le {\mathbb {E}}\left((\frac{1}{m}\sum _{b \in A^{\prime }} Z_b)^2\right) = \frac{1}{m^2}\sum _{b \in A^{\prime }} {\mathbb {E}}(Z_b^2).Now, since {\mathbb {E}}(Z_b)=0, we have 1 + {\mathbb {E}}(Z_b^2) = {\mathbb {E}}((... | {
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374aadaaa0b11eae1964556cc1f215db657bc649 | subsection | 15 | 19 | An O(1) test for the random cluster model | In this section, a character (on X) denotes an arbitrary partition \lbrace \alpha _1, \ldots , \alpha _m\rbrace of X into any number of disjoint subsets.In the random cluster model, one has a phylogenetic X–tree and each edge e has an associated probability p(e) that the edge of is cut. These cuts are performed indepen... | {
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23f91189d528595a7e49ad3be4b477e99aaf16aa | subsection | 16 | 19 | An O(1) test for the random cluster model | There exist induced rooted phylogenetic subtrees A, B, on leaf sets A and B, respectively, where A, B are disjoint, nonempty subsets of X, such that:(i)
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2d694929dd540ed8ffc3826af21f960b2c2fa851 | subsection | 17 | 19 | An O(1) test for the random cluster model | Let \alpha , \beta denote, respectively, the blocks of the character at vertices u, v of 1, and consider the conjunctive event E = \bigcap _{i=1}^5E_i in which:(E_1)
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afd0a7f3e7521b57b86d38e80682faf613c27f70 | subsection | 18 | 19 | Concluding comments | The reader may be curious as to where our proof for the \log (n) lower bound on sequence length for testing under the finite-state model breaks down for the random cluster model. The crucial distinction is that the random cluster model fails to satisfy condition (REF ) required in the proof for Theorem REF . That is, i... | {
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1eb3f4deded754f0a90c1298d21547115d05e14a | abstract | 0 | 28 | Abstract | We analyze the effective potential for nanoparticles trapped at a fluid
interface within a simple model which incorporates surface and line tensions as
well as a thermal average over interface fluctuations (capillary waves). For a
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6b582b35885a54d82d51e69db6f86faa645890b2 | subsection | 1 | 28 | Introduction | The effective forces between rigid objects immersed
in a fluctuating medium have attracted a steadily growing interest
because their understanding allows one to design and tune them by choosing
suitable media and boundary conditions and by varying the thermodynamic state
of the medium. Possible applications range from ... | {
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"doi"... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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"H. Lehle",
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] | [
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] | 2,008 | en | Physics | [
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71875b963c23f9572585068f43d6e1b2f6142657 | subsection | 2 | 28 | Free energy model for spherical colloids trapped at interfaces | In this section, we consider the free energy of a single, spherical colloid
with radius R trapped
at an interface between two phases I and II which is assumed to be flat in
equilibrium.
The height of the center of the colloid above the interface
is denoted by z, and we look for the free energy F of the colloid with z
f... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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9318c148491589591b9b08089a4b63856912539d | subsection | 3 | 28 | Surface tension | Let \gamma denote the surface tension of the fluid interface and
\gamma _{\rm I[II]} denote the surface tension of the colloid surface (with area
A_{\rm I[II]}) exposed to phase I [II] (see Fig. REF ).
The free energy F(z), measured with respect to the configuration
where the colloid is completely immersed in phase I, ... | {
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1e08a68816ceb3dc135f4781fac4eaf6a17ca665 | subsection | 4 | 28 | Line tension | The line tension was introduced by Gibbs , to define the
excess free energy associated to the line where three phases meet. The accurate
experimental measurement of the line tension has been a considerable challenge.
In fact, the uncertainty in the order of magnitude of the line tension has generated
a considerable num... | {
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6d1e6e34b2ccb9e3720ae3e9e7ce4ca606a9430e | subsection | 5 | 28 | Line tension | The new equilibrium position of the colloid is determined by an equation
of fourth order in \hat{z}:\hat{z}_{\rm eq}\left( \frac{\hat{\tau }}{\sqrt{1-\hat{z}_{\rm eq}^2}} -1\right) &=&
\cos \theta _0 \;.The spring constant k= F^{\prime \prime }(z_{\rm eq}) of the potential well near the
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"doi":... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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] | [
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29209c29fa2dbd476a0d6f9e18b150a1aad00b20 | subsection | 6 | 28 | Capillary waves | In thermal equilibrium, the interface is not sharp but acquires a finite
thickness through density fluctuations.
In a coarse–grained picture, these density fluctuations correspond to
fluctuations of the mean interface position u(\mathbf {x}) around the equilibrium
position u=0. Here, \mathbf {x} = (x,y) is a vector in ... | {
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869d082f81145f8101da71e952d59a43ddc34345 | subsection | 7 | 28 | Capillary waves | Therefore\mathcal {H}[u(\mathbf {x}),z] &=&
(F_{\rm men}[\lbrace u(\mathbf {x}),z\rbrace ] - F_{\rm men}[\lbrace 0,z_{\rm eq}\rbrace ])
+\gamma _{\rm I}\Delta A_{\rm I}
+\gamma _{\rm II}\Delta A_{\rm II} + \tau \Delta LThe difference in the interface areas colloid/phase I and colloid/phase II
between the configuration ... | {
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... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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] | [
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6e888900fcf0525f65dbbfecfa15189c220fe6a6 | subsection | 8 | 28 | Capillary waves | The Fourier coefficients P_{m} are referred to as
contact line multipoles below, and
since the contact line height f(\varphi ) is real,
P_m=P^*_{-m} holds. With these definitions, the boundary term
acquires the form\mathcal {H}_{\rm b}[u(\mathbf {x}),h] &=& \mathcal {H}_{\rm b,1} + \mathcal {H}_{\rm b,2}\,, \\
\mathca... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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c555e78c96d889fdb4df788a768575ababf75045 | subsection | 9 | 28 | Capillary waves | The
mean–field part solves the Euler–Lagrange equation
(-\Delta +\lambda _c^{-2})\,u_{\rm mf}=0 with the boundary condition
u_{\rm mf}\,|_{ \partial S_{\rm men, ref}}=f .
Consequently the fluctuation part
vanishes at the contact line:
v\,|_{ \partial S_{\rm men, ref}}=0.
Then the partition function
\mathcal {Z}=\mathc... | {
"cite_spans": []
} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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19a3504ba87d579bff63fcc969e3673acb47bd79 | subsection | 10 | 28 | Capillary waves | ())
is diagonal in the multipole moments P_m (see ):-\frac{\beta \gamma }{2}
\oint _{\partial S_{\rm men, ref}} \!\!\!\!\!\!\!\!\!\!
d\ell \, f({\bf x})\,( \partial _n u_{\rm mf}({\bf x}))
\stackrel{\lambda _c \gg R}{\simeq }
-\pi \beta \gamma \left( \frac{P_0^2}{\ln \hat{\lambda }_c} + 2
\sum _{m> 0} m|P_m|^2 \right)\... | {
"cite_spans": []
} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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] | [
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4f327c8c23ff660716628ff2abffc312e4b80fb4 | subsection | 11 | 28 | Capillary waves | Thus the effective potential for the colloid moving around its equilibrium
position is given byF(z) & = & - \beta ^{-1} \; \ln \mathcal {Z} \\
& = & {\rm const.} + \pi \gamma \, (z-z_{\rm eq})^2\,
\frac{ 1 - \frac{\displaystyle \hat{\tau }}{\displaystyle \hat{r}_{0,{\rm eq}}^3}}{1+\ln \hat{\lambda }_c
\left(1-\frac{\d... | {
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d57273d40bdc8718a4946c9728451be0726e708d | subsection | 12 | 28 | Capillary waves | Clearly, for arbitrarily small but negative \tau there exists a
critical multipole order m_c above which the exponent becomes
positive and thus the partition function becomes infinite. Taken at face
value, for negative line tensions the interface would become unstable
by forming ripples with small wavelengths near the ... | {
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"H. Lehle",
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] | [
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1ae568f4cef3b2f41ef44506cc6a334e5c9fce65 | subsection | 13 | 28 | Fluctuation induced forces between two colloids | The previous considerations can be extended to the case of two colloids which
are trapped at the interface at distance d. Clearly, if both colloids are at
their equilibrium position (defined by Eq. (REF )) and capillary waves
are neglected, the interface is flat and therefore no interface–mediated
interactions are pres... | {
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... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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e9b170ad042a927fddb73d95d82dff0c89d6a99e | subsection | 14 | 28 | Fluctuation induced forces between two colloids | In the present work, we will show (i) that for case (A2), fixed colloids but
unpinned contact line, the effective potential is still long–ranged,
V(d) \propto \ln (1+\ln d) and (ii) that line tensions do not change the
leading power in the long–range behavior of V(d) but the instability for
negative line tensions in th... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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e713239df0f377ce2aa22ebf4d0b32de6d1a4d44 | subsection | 15 | 28 | Fluctuation induced forces between two colloids | Via the integration domain S_{\rm men,ref} of \mathcal {H}_{\rm cw},
the total Hamiltonian and hence the partition function \mathcal {Z}(d)
of the system depends
on the distance d of the colloid centers.As before, the fluctuations over the contact lines
f_i=\sum _m \exp ({\rm i}m\varphi _i)P_{im}
are incorporated into ... | {
"cite_spans": []
} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
"M. Oettel"
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eb637008a053c61390edb6c43d96ab965438707d | subsection | 16 | 28 | Fluctuation induced forces between two colloids | Then the partition function \mathcal {Z}=\mathcal {Z}_{\rm fluc}\mathcal {Z}_{\rm mf}
factorises into a product of a fluctuation part
independent of
the boundary conditions
and a mean field part which depends on the fluctuating boundary conditions f_i
of the meniscus on the colloid surfaces:\mathcal {Z}_{\rm fluc} &=& ... | {
"cite_spans": []
} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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"cond-mat.soft"
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4632f8232cd781b761a7a920b06a1726e3593e57 | subsection | 17 | 28 | Fluctuation induced forces between two colloids | The techniques to evaluate
the fluctuation and mean–field part to the effective potential have
been presented in detail in Ref. , and we give a summary of the main
results which are necessary to discuss the influence of the line tension terms on
V(d). | {
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0.0040... |
98a7d4e3af3fce33868ec7c1783133c6efa7f75b | subsection | 18 | 28 | Fluctuation part | The fluctuation part contributes equally
for both
cases (A1) and (A2) introduced above.
The \delta -functions
in the fluctuation part of the partition function
can be removed by using their integral representation via
auxiliary fields \psi _i ({\bf x}_i) defined on the
interface boundaries \partial S^0_i . This enable... | {
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of capillary waves and line tensions | [
"H. Lehle",
"M. Oettel"
] | [
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abcfd19b1da32ae2958f7f254fc11da57fe959ac | subsection | 19 | 28 | Mean–field part | The calculation of \mathcal {Z}_{\rm mf} (Eq. (REF )) requires to determine
the solution of the differential equation(-\Delta +\lambda _c^{-2})\,u_{\rm mf}=0with the boundary conditions at the fluctuating contact line and at infinity,
respectively:\left.u_{\rm mf}({\bf x}_i)\right|_{{\bf x}_i \in \partial S^0_i}
&=&f_i... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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f36adda373dc90dd8bc2328feac539e4cf49e98b | subsection | 20 | 28 | Mean–field part | (REF ), the
mean field part of the partition function \mathcal {Z}_{\rm mf}
can be
written in a Gaussian form:\mathcal {Z}_{\rm mf} = \int \mathcal {D} f_i\; \exp \left( - \beta \mathcal {H}[u_{\rm mf},d]
\right)\;
\exp \left\lbrace -\pi \beta \gamma \left(1-\frac{\displaystyle \hat{\tau }}{\displaystyle \hat{r}_{0,{\r... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
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] | [
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cddf00ec1c4fb6f40b8d7f7e58790099a753f5d6 | subsection | 21 | 28 | Mean–field part | (REF ) that the
d–dependent part of \mathcal {Z}_{\rm mf} is given by \det {\bf E}.
For the case of vanishing line tension, the properties of \det {\bf E}
were discussed in detail in Ref. . In particular it turns out that
the four leading terms in the expansion of V_{\rm mf} and V_{\rm fluc}
cancel each other (b_m=-a_m... | {
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of capillary waves and line tensions | [
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277fdedebdc3b8fdd84f4e5b647ee45547c0be56 | subsection | 22 | 28 | Mean–field part | The dependence on \tau enters through thereduced variable x= \hat{\tau }/r_{0,{\rm eq}}^3 where \hat{\tau }= \tau /(\gamma R)and \hat{r}_{0,{\rm eq}}= r_{0,{\rm eq}}/R.](A2)
This case implies fixing the colloids at their equilibrium
positions z_{\rm eq}. Thus in the integration measure for the contact
line fluctuations... | {
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of capillary waves and line tensions | [
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f308dffa96f4dc55a1119892ad5a61c923b864df | subsection | 23 | 28 | Conclusion | In this paper, we have studied the influence of capillary waves on the stability
and interactions of colloids (with radius R) trapped at a fluid interface
with surface tension \gamma , with particular
attention to the effects of a line tension \tau .
Quite often, the stability of colloids
at a fluid interface with resp... | {
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"... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
"M. Oettel"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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d550006b258938c197f2614d2a81b6cb5cc90700 | subsection | 24 | 28 | Derivation of the boundary Hamiltonian | In this appendix we derive the boundary term
\mathcal {H}_{\rm b} which describes free energy changes upon shifting the
contact line
(cf. the result in Eqs. (REF )–() of Sec. REF ).
According to Eqs. (REF )–(REF ) the boundary term is given by\mathcal {H}_{\rm b}=
\gamma _{\rm I}\Delta A_{\rm I}+\gamma _{\rm II}\Delta ... | {
"cite_spans": []
} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
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37af050b5fc4b155e838ea331ed9cee06671383e | subsection | 25 | 28 | Derivation of the boundary Hamiltonian | Then, the
associated
changes of the free energy
can be written as\gamma _{\rm I}\Delta A_{\rm I}
+\gamma _{\rm II}\Delta A_{\rm II}
&=&
\gamma \cos \theta _0
\int _{0}^{2\pi }d\varphi \int _{r_{0,\rm eq}}^{r_0(\varphi )}
dr\,
\frac{r}{\sqrt{1-r^2/R^2}}
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&=&
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... | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
"H. Lehle",
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] | [
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db39b4726ebf36a807c906138f554eac4a49d90a | subsection | 26 | 28 | Derivation of the boundary Hamiltonian | (REF ) and using relation (REF )
for the equilibrium position of the colloid
(R\cos \theta _0 /z_{\rm eq}=\tau /(\gamma r_{0,\rm eq})-1)
we find\gamma _{\rm I}\Delta A_{\rm I}
+\gamma _{\rm II}\Delta A_{\rm II}
+\gamma \Delta A_{\rm proj}
&=&
\frac{\gamma }{2}\int _0^{2\pi }
d\varphi \,
[f-h]^2
-
\frac{\tau z_{\rm eq}}... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
of capillary waves and line tensions | [
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d748e5908d8a00a1d3edb2b8e3d9a68ed8240d25 | subsection | 27 | 28 | Derivation of the mean–field energy term in Eq. ( | Let (r,\varphi ) be polar coordinates in the equilibrium interface plane
z=0 where r=0 is the center of the circle enclosed by the reference
contact line.
The solution to the mean–field equation
(-\Delta +\lambda _c^{-2})\,u_{\rm mf}=0
with the boundary condition u_{\rm mf}(r_{0,{\rm eq}},\varphi )=
f(\varphi ) = \sum... | {
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} | 10.1088/0953-8984/20/40/404224 | 0807.1757 | Stability and interactions of nanocolloids at fluid interfaces: effects
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a1c9dbea57e3c76a953ae75c7368e9a8a5ee47ca | abstract | 0 | 21 | Abstract | We investigate the formation of carbon-enhanced metal-poor (CEMP) stars via
the scenario of mass transfer from a carbon-rich asymptotic giant branch (AGB)
primary to a low-mass companion in a binary system. We explore the extent to
which material accreted from a companion star becomes mixed with that of the
recipient, ... | {
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} | 10.1111/j.1365-2966.2008.13700.x | 0807.1758 | Thermohaline mixing and gravitational settling in carbon-enhanced
metal-poor stars | [
"Richard J. Stancliffe",
"Evert Glebbeek"
] | [
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768ba4005ab8cd539a66ff3f246c6c000d5eb7f3 | subsection | 1 | 21 | Introduction | Carbon-enhanced, metal-poor (CEMP) stars are defined as stars with [C/Fe][A/B] = \log (N_\mathrm {A}/N_\mathrm {B}) - \log (N_\mathrm {A}/N_\mathrm {B})_\odot >+1.0 , with [Fe/H]<-2 in most cases. These objects appear with increasing frequency at low metallicity . The study of CEMP stars is being used to probe conditio... | {
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a3573e0e3d7be1f1230e978753ab10d1276a8852 | subsection | 2 | 21 | Introduction | Neither of these scenarios is consistent with the extensive mixing found by .A possible source for reduced thermohaline mixing efficiency has been suggested by . These authors suggest that the action of gravitational settling will alter the composition gradient of the accreting star near its surface. Helium will settle... | {
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cc41fcd8c8b01ef9a8e7c9d2cebd3ac33e1c0acb | subsection | 3 | 21 | The stellar evolution code | Calculations in this work have been carried out using a modified version of the stars stellar evolution code originally developed by and updated by many authors . The version used here includes the nucleosynthesis routines of and , which follow the nucleosynthesis of 40 isotopes from D to ^{32}\mathrm {S} and important... | {
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cf7e3c4cb09c1f3772086170a592b43410c93c9f | subsection | 4 | 21 | AGB models | We have evolved a set of models with masses of 1, 1.5, 2, 2.5, 3 and 3.5\,M_\mathrm {\odot } from the pre-main-sequence without the use of convective overshooting. A mixing length of \alpha =2.0 has been employed throughout. This value is chosen based on calibration to a solar model. All the models were evolved using 9... | {
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98fe46ee0cafc2479aad82906716f5a69d4855c7 | subsection | 5 | 21 | AGB models | As mass loss is very rapid in the superwind phase we believe that we have missed only a few thermal pulses (and associated episodes of third dredge-up) at most. We record the final yield, the mass removed and the age of the final model for each of the masses considered. These details are needed to model the accretion o... | {
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36df89384d87274a156a19dec3a2adae9eb4355e | subsection | 6 | 21 | AGB models | The results are displayed in Figure REF . We note that there is generally good agreement between the 2\,M_\mathrm {\odot } models, with few of the isotopes showing variations of over one order of magnitude. For many of the heavier isotopes, our average compositions tend to lie between those of KL07 and H04. The maximum... | {
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} | 10.1111/j.1365-2966.2008.13700.x | 0807.1758 | Thermohaline mixing and gravitational settling in carbon-enhanced
metal-poor stars | [
"Richard J. Stancliffe",
"Evert Glebbeek"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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c00cc81e1e85f60819254a6d0b14543ba5336aae | subsection | 7 | 21 | Cessation of third dredge-up | The 1\,M_\mathrm {\odot } model displays some behaviour not found in the other models, namely that TDUP operates only between pulses 3 and 8 and ceases until the final pulse, where it occurs once more. We believe that the reason for the temporary cessation of TDUP is a consequence of the changing metallicity of the env... | {
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2ca88d3bf4d636c95ab4299bce1f84e1c3a24c09 | subsection | 8 | 21 | CEMP models | We now turn to the modelling of the secondaries in these putative AGB binaries. We assume we are free to choose the primary mass, secondary mass and amount of material accreted to produce a secondary which has the appropriate mass (about 0.8\,M_\mathrm {\odot }, see below). The primary mass determines the composition o... | {
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1e6fe01f46918aad82cbbcf5ae8ad3119485024f | subsection | 9 | 21 | CEMP models | Figure REF shows the mean molecular weight profile at varies points in the evolution of the model which accretes 0.1\,M_\mathrm {\odot } from a 1.5\,M_\mathrm {\odot } companion with thermohaline mixing taken into account. Initially, the star burns hydrogen in its core, raising the mean molecular weight in the central ... | {
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9f4f2fe13e4473680a70833449859513cac2dcb7 | subsection | 10 | 21 | CEMP models | These plots are representative of the full data set.
[Figure: The evolution of the surface [C/H] ratio as a function of luminosity when accreting 0.1\,M_\mathrm {\odot } (solid line), 0.01\,M_\mathrm {\odot } (dotted line) and 0.001\,M_\mathrm {\odot } (dashed line) of material from a 1.5\,M_\mathrm {\odot } companion.... | {
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} | 10.1111/j.1365-2966.2008.13700.x | 0807.1758 | Thermohaline mixing and gravitational settling in carbon-enhanced
metal-poor stars | [
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8e9e72875e708fab1757bb86f210ee3530b5bee6 | subsection | 11 | 21 | CEMP models | Models without thermohaline mixing struggle to populate those turn-off stars with low [C/H] values and unless a large quantity of material is accreted, they suffer too much dilution at first dredge-up. Models with thermohaline mixing do not produce the highest [C/H] values as the accreted matter is mixed with the prist... | {
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metal-poor stars | [
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66a3c6ee042d9d4ce39393dcb29ea0cf8e4be0c4 | subsection | 12 | 21 | CEMP models | This occurs because the convective envelope reaches down to those regions of the star where carbon originally present in the star has been processed to nitrogen via the CN cycle. This rise in [N/H] is exactly what would be expected in a `normal' metal-poor star.The picture is very different for models including thermoh... | {
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b91b8280d549bb4b691fafe910a6d75f52745b58 | subsection | 13 | 21 | The effect of gravitational settling | There are certainly problems with models including thermohaline mixing from an observational perspective. They do not seem to reproduce those objects with high [C/H] ratios and they imply there should be a population of low [C/H] objects that are not observed. However from the theoretical perspective there are problems... | {
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738055a1dd714837d36664f6fd7c3e3967d85664 | subsection | 14 | 21 | The effect of gravitational settling | This leads to the mean molecular weight decreasing in the surface regions (down to a fractional mass of around \mathrm {d}M/M\approx 0.01) and a mean molecular weight gradient building up (see the long-dashed line of the figure). It is this which may potentially inhibit any thermohaline mixing. Note that the mean molec... | {
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} | 10.1111/j.1365-2966.2008.13700.x | 0807.1758 | Thermohaline mixing and gravitational settling in carbon-enhanced
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695d5f92e97009394e7ea360c8029c55261a3ea0 | subsection | 15 | 21 | The effect of gravitational settling | The accreted material suffers more dilution with the H-enriched, He-depleted surface of the star which reduces its mean molecular weight, slowing the rate of thermohaline mixing. In the case that the amount of material accreted is very small, the barrier can be completely effective and mixing does not proceed to great ... | {
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} | 10.1111/j.1365-2966.2008.13700.x | 0807.1758 | Thermohaline mixing and gravitational settling in carbon-enhanced
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afca2d2cebf958c55a5d87b63bc56d626e8d1d1a | subsection | 16 | 21 | The effect of gravitational settling | This is in contrast to the situation with thermohaline mixing alone, where the model which accretes 0.001\,M_\mathrm {\odot } (the dashed line in Figure REF ) has a [C/H] value about 0.6 dex lower than that of the model which accretes 0.01\,M_\mathrm {\odot } throughout its whole evolution.With the inclusion of gravita... | {
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837bdb2c7267570ebc5d091e27b6ffc4a283b161 | subsection | 17 | 21 | Discussion | As suggested by a means of suppressing the action of thermohaline mixing is necessary to prevent the destruction of lithium in some CEMPs. Lithium is a fragile element that is easily destroyed at temperatures of around 2\times 10^6 K which are found at around 0.01\,M_\mathrm {\odot } below the stellar surface of the lo... | {
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5cc039620ab3bd20cecfd0a3f150de2c4aa7086b | subsection | 18 | 21 | Discussion | Stars are identified as turn-off objects if \log L/\mathrm {L}_\odot <0.7 (i.e. they have not yet undergone first dredge-up) and are displayed in the upper panels. Stars are defined as post-first dredge-up giants if \log L/\mathrm {L}_\odot >1.5 and are displayed in the lower panels. The total number of stars, \mathrm ... | {
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49372f3ce64b2c529326e425d38fd525fc6b370b | subsection | 19 | 21 | Conclusions | We have modelled the accretion of AGB material on to low-mass stars in order to determine what chemical signatures may be observed in CEMP stars. We have examined three specific cases: canonical evolution including only convective mixing, the inclusion of thermohaline mixing and the inclusion of both thermohaline mixin... | {
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96d102f75d470682b404129f9bead7fb9dd235a7 | subsection | 20 | 21 | Stellar yields | Here we present the final gross yields (i.e. the total mass of each isotope ejected) from each of the AGB models for all of the isotopes in the nucleosynthesis network.
[Table: Final gross yields in stellar masses for all the isotopes in the nucleosynthesis network.] | {
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9e29f6842ef7c1eb8f972bdcf57ae68132b471af | abstract | 0 | 17 | Abstract | Prediction and prevention of musculo-skeletal injuries is an important aspect
of preventive health science. Using as an example a human knee joint, this
paper proposes a new coupled-loading-rate hypothesis, which states that a
generic cause of any musculo-skeletal injury is a Euclidean jolt, or
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} | 0807.1759 | New Mechanics of Generic Musculo-Skeletal Injury | [
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3ebee83370189ef463d3ca9f6d3bcbcba30ab76d | subsection | 1 | 17 | Introduction | In this paper, we propose a new model of musculo-skeletal injury, using as an example the human knee joint. The knee joint comprises three articulations: (i) a tibio-femoral
joint) between the medial and lateral condyles of the femur and
tibia (see Figure REF ), (ii) patelo-femoral joint between
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390c9f07c0059aee366a5ae0f8358315c240d87f | subsection | 2 | 17 | Introduction | The number of sports–related injuries is
increasing because of active participation of children in
competitive sports . Children differ from adults in many areas,
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ligaments compared with growth plates, and continued growth.
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1eab9f27b827f7fc3a8ca4a60e558122491efc38 | subsection | 3 | 17 | Introduction | These knee SE(3)–jolts do not belong to the
standard Newton–Euler dynamics. The only way to monitor them
would be to measure “in vivo" the rate of the
combined (forces + torques)– rise in the knee joint (see Figure
REF ).This paper proposes a new hypothesis for generic musculo-skeletal injury, called:Coupled–loading–ra... | {
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32f4e2259e75cea281459fb275b74dca5635dd54 | subsection | 4 | 17 | Introduction | The cause of knee disclinations is an angular 3–axial jolt, the time
rate-of-change of a 3–axial torque (angular jolt = inertia moment \times
angular jerk).This decoupled framework has been implemented in the Human Biodynamics
Engine , a high-resolution neuro–musculo–skeletal dynamics simulator
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f0861eea7ed8d895d6028bf977aecc5f9c310beb | subsection | 5 | 17 | The | In the language of modern biodynamics , , the general knee motion is
governed by the Euclidean SE(3)–group of 3D motions. Within the
knee SE(3)–group we have both SE(3)–kinematics (consisting of
the knee SE(3)–velocity and its two time derivatives:
SE(3)–acceleration and SE(3)–jerk) and the knee SE(3)–dynamics
(consist... | {
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27b4575866aa527538fa2a1c797ecb2f416f2ee2 | subsection | 6 | 17 | The | With this respect, recall
that the fundamental law of biomechanics – the so–called
covariant force law , , , states:\text{Force co-vector field}=\text{Mass distribution}\times \text{Acceleration vector--field},which is formally written (using the Einstein summation convention, with
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bce73a7072aa647df18db3df37f475f929236aef | subsection | 7 | 17 | Body | Briefly, the SE(3)-group of knee motions is defined as a
semidirect (noncommutative) product of 3D knee rotations and 3D
knee micro–translations,SE(3):=SO(3)\rhd \mathbb {R}^{3}.Its most important subgroups are the following:\begin{array}{cc}
\mathbf {Subgroup} & \mathbf {Definition} \\ \hline \begin{array}{c}
SO(3),\t... | {
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91426a9d71d3296d99f1109dc4757102c4c90452 | subsection | 8 | 17 | Local joint | To support our locally–coupled loading–rate hypothesis, we formulate the
coupled Newton–Euler dynamics of the knee motions within the SE(3)- group. The forced Newton–Euler equations read in vector (boldface)
form\text{Newton} &:&~\mathbf {\dot{p}}~\mathbf {\equiv M\dot{v}=F+p\times \omega }, \\
\text{Euler} &:&~\mathbf... | {
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9a7b589b6093447fda74edc1ca307ba874f5406e | subsection | 9 | 17 | Local joint | However, for simplicity, in this paper we shall consider only the
simple case of two separate diagonal 3\times 3 matrices
(\mathbf {M,I}). defining the total moving segment mass–inertia
distribution, with principal inertia moments given in Cartesian
coordinates (x,y,z) by volume integralsI_{1}=\iiint \rho (z^{2}+y^{2})... | {
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9d0a9803f1930066617407a73c8b0b11c6491d7d | subsection | 10 | 17 | Joint injury dynamics: the | The SE(3)-jolt, the actual cause of the knee injury (in the form
of the plastic deformations), is defined as a coupled Newton+Euler
jolt; in (co)vector form the SE(3)-jolt readsNote that the derivative of the cross–product of two vectors follows the
standard calculus product–rule: \frac{d}{dt}(\mathbf {u\times v})=\mat... | {
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9f009407ff158398c672a13e399f61bf26c5bba4 | subsection | 11 | 17 | Joint injury dynamics: the | However, for the sake of simplicity and wider readability, we
stick to the same overdot notation.~\dot{F}_{i} &=&\ddot{p}_{i}-\varepsilon _{ik}^{j}\dot{p}_{j}{\omega }^{k}-\varepsilon _{ik}^{j}p_{j}{\dot{\omega }}^{k}, \qquad (i,j,k=1,2,3) \\
~\dot{T}_{{i}} &=&\ddot{\pi }_{i}~-\varepsilon _{ik}^{j}\dot{\pi }_{j}\omega ... | {
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6bda48fdedf0e3a0280fa9b0a252c780d768bba2 | subsection | 12 | 17 | Joint injury dynamics: the | \\
&& \\
\text{Euler~jolt} &:&\left\lbrace
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\dot{T}_{{1}}=\ddot{\pi }_{1}-(m_{{2}}-m_{{3}})\left( v_{{3}}\dot{v}_{{2}}+v_{{2}}\dot{v}_{{3}}\right) -(I_{{2}}-I_{{3}})\left( \omega _{{3}}\dot{\omega }_{{2}}+{\omega }_{{2}}{\dot{\omega }}_{{3}}\right) , \\
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"Vladimir G. Ivancevic"
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e4ef0fa474701d4d0bbcc15fe5894cb307666168 | subsection | 13 | 17 | Joint disclinations and dislocations caused by the | For mild knee injury (caused by internal loss of stability), the best injury predictor is considered to
be the product of localized knee strain and strain rate, which is
the standard isotropic viscoelastic continuum concept (see, e.g. ). To improve
this standard concept, in this subsection, we consider the knee
joint a... | {
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"raw": "Halloran, J.P., Easley, S.K., Petrella, A.J., Rullkoetter, P. Comparison of deformable and elastic foundation finite element simulations for predicting knee replacement mechanics. J. Biomech. Eng. 1... | 0807.1759 | New Mechanics of Generic Musculo-Skeletal Injury | [
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d572e893337d58605600224d258a2fd5519b12d5 | subsection | 14 | 17 | Joint disclinations and dislocations caused by the | According to
Edelen , , these four SE(3)-based differential forms
satisfy the following set of continuity equations:&&\mathbf {\dot{\alpha }}=\mathbf {-dJ-S,} \\
&&\mathbf {\dot{Q}}=\mathbf {-dS,} \\
&&\mathbf {d\alpha }=\mathbf {Q,} \\
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8435a25d246a6d1ebb3f8a4de98b0144927e978c | subsection | 15 | 17 | Joint disclinations and dislocations caused by the | The 3–form equation () states that the time derivative \mathbf {\dot{Q}=}\frac{1}{3!}\dot{Q}_{ijk}\,dx^{i}\wedge dx^{j}\wedge dx^{k}
of the disclination density \mathbf {Q} is the (negative) divergence of the
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060d2f6734913f4a02c99b9dfb6ba1aa5d46af20 | subsection | 16 | 17 | Conclusion | Based on the previously developed covariant force law , , its recent application to traumatic brain injury , and using as an example a human knee joint, in this
paper we have formulated a new coupled loading–rate hypothesis
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b74b5f503c68ef81d0ed57df25fb88255ca8682b | abstract | 0 | 6 | Abstract | In this paper, we investigate the long-standing gap of quantum key rate
between the Weak Coherent Pulse (WCP) and Heralded Single Photon Sources(HSPS)
implementation of quantum cryptographical protocol. We prove that, by utilizing
the Heralded Pair Coherent State (HPCS) photon sources, such a gap can be
actually filled... | {
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} | 0807.1760 | Closing the gap of secure quantum key rate with the Heralded
Pair-Coherent States | [
"ShengLi Zhang",
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"ChenHui Jin",
"GuangCan Guo"
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289e7c7a31fa607151593b27c590fd84a641538b | subsection | 1 | 6 | Body | 10Closing the gap of secure quantum key rate with the Heralded Pair-Coherent States
ShengLi Zhang^{1,2}, XuBo Zou^{1}, ChenHui Jin^{2}
and GuangCan Guo^{1}
1 Key Laboratory of Quantum Information, University of Science and
Technology of China (CAS), Hefei 230026, China.2 Zheng Zhou Information and Technology Institute,... | {
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1274b90014f93f1814ff45f88316f05fcef0d4f6 | subsection | 2 | 6 | Body | Put simply, to optimize the final performance,
one should resort to WCP sources within the threshold distance but
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d0933b45f2f3be8655cc851b0b876f21b1f1d075 | subsection | 3 | 6 | Body | If the signal state and the decoy state have
the same wavelength, timing, and many other physical characters, no
eavesdropper will be able to distinguish a decoy state from a signal
state successfully. Thus the condition probability that Bob's
detector clicks when a n-photon pulse is emitted from Alice will
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"en... | 0807.1760 | Closing the gap of secure quantum key rate with the Heralded
Pair-Coherent States | [
"ShengLi Zhang",
"XuBo Zou",
"ChenHui Jin",
"GuangCan Guo"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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37d12b183db225a21c6ca89f66e5cebf94358841 | subsection | 4 | 6 | Body | In
fact, it will be shown that only one signal state \mu and two
decoy state \nu _1,\nu _2(1>\mu >\nu _1>\nu _2 and
\nu _1^2+\nu _2^2\le \mu ^2) will be enough to give an estimation.
First of all, it is sufficient to use only \nu _1,\nu _2 and obtain
the estimation of Y_1 and e_1. Plugging Eq. (REF )
into Eq.(REF ), on... | {
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Pair-Coherent States | [
"ShengLi Zhang",
"XuBo Zou",
"ChenHui Jin",
"GuangCan Guo"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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2abe98fa31b43e7435c1a9979c0b367a4996aa65 | subsection | 5 | 6 | Body | Similarly, for SARG
protocol, we obtain
Q_{\mu ,SARG}=\left(\frac{e_{det}}{2}+\frac{1}{4}\right)\xi -\left(\frac{e_{det}}{2}+\frac{1}{4}-\frac{p_{dark}}{2}\right)\zeta ,
EQ_{\mu ,SARG}=\frac{e_{det}}{2}\xi -\left(\frac{e_{det}}{2}-\frac{p_{dark}}{4}\right)\zeta ,
E_{\mu ,SARG}=EQ_{\mu ,SARG}/Q_{\mu ,SARG}^, where \xi =... | {
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"arxiv_id": "",... | 0807.1760 | Closing the gap of secure quantum key rate with the Heralded
Pair-Coherent States | [
"ShengLi Zhang",
"XuBo Zou",
"ChenHui Jin",
"GuangCan Guo"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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45771f569b1ab07bc9c402aca24c63356f46e8cc | abstract | 0 | 19 | Abstract | We study the conditions for collisions between planetesimals to be
accretional or disruptive in turbulent disks, through analytical arguments
based on fluid dynamical simulations and orbital integrations. In turbulent
disks, the velocity dispersion of planetesimals is pumped up by random
gravitational perturbations fro... | {
"cite_spans": []
} | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.004296417348086834,
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abee99ae6a851e1a9a9f377c3d89aa2f38b77016 | subsection | 1 | 19 | Introduction | It is often considered that the evolution of protoplanetary disks and
the consequent accretion of gas by the central protostar are driven by
turbulent viscosity due to a Magneto-Rotational-Instability (MRI)
. and carried out
fluid dynamical simulations of MRI and found that the random torques
due to the turbulent densi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 576,
"openalex_id": "",
"raw": "Balbus, S. A., & Hawley, J. F. 1991, , 376, 214",
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{
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"doi": "",
"end":... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.002... |
85e5078d4f29ce844bbac6cac4334216cdac1780 | subsection | 2 | 19 | Accretion and destruction conditions | We summarize the accretion and destruction conditions below. From
energy conservation, the collision velocity (v_{\rm coll}) between
two planetesimals (labeled 1 and 2) satisfiesE = \frac{1}{2}v_{\rm coll}^2 - \frac{G(M_1 + M_2)}{R_1 + R_2}
= \frac{1}{2}v_{\rm rel}^2,where M_j and R_j are the mass and physical radius o... | {
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{
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"doi": "",
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"raw": "Ida, S. & Nakazawa, K. 1989, å, 224, 303",
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"start": 428
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"doi": "",
"end": 702,... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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9393df3b508ca69e9fe29520caa8046bb61f4cdf | subsection | 3 | 19 | Accretion and destruction conditions | The physical radius R is given byR=7.8 \times 10^8 (M/M_{\oplus })^{1/3} (\rho _{\rm p}/3{\rm gcm}^{-3})^{-1/3}
{\rm cm}.A collision results in destruction if the collision velocity
is such that the
specific kinetic energy of a collision (v_{\rm col}^2/2) exceedsQ_{\rm D} \simeq \left[ Q_0
\left(\frac{R}{1{\rm cm}}\rig... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 529,
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"raw": "Benz, W. & Asphaug, E. 1999, Icarus, 142, 5",
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"start": 121
},
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"end": 6... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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2c60b35d2223015839de17a425c616a8f7d57dd6 | subsection | 4 | 19 | Equilibrium eccentricities | We first derive the equilibrium eccentricities of
planetesimals at which the excitation by the MRI turbulence is
balanced by damping due to drag and/or collisions.
Comparing the
estimated eccentricities with e_{\rm acc} and e_{\rm dis},
we then evaluate the outcome of
collisions between planetesimals as a function of p... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 995,
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"raw": "Hayashi, C., 1981, Prog. Theor. Phys. Suppl, 70, 35",
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"start": 895
},
{
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"doi": "",
... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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f4665f2bf1eb5a5be6d20dd3892c6c5fb865e429 | subsection | 5 | 19 | Excitation | The orbital eccentricities of planetesimals are pumped up both by the
random gravitational perturbations from density fluctuations of disk
gas, as well as by mutual gravitational scattering among
planetesimals. Assuming planetesimals have equal masses, their
orbital eccentricities should be excited to at most \sim e_{\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 354,
"openalex_id": "",
"raw": "Safronov, V. 1969, Evolution of thr Protoplanetary Cloud and Formation of the Earth and Planets (Moscow: Nauka Press)",
"source_ref_id": "29e88038abc30904ed79191ec5cb5fd4ad6d07a1",
"star... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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... |
f0cefa6cea296729a3205997b4d311264c9fd4b4 | subsection | 6 | 19 | Excitation | Interestingly, with a quite
different approach, derived a similar formula
for \Delta a/a with the same dependences on r, \Sigma _g and t.
If e \simeq \Delta a/a, their formula is consistent with ours.
They suggested that \gamma \sim \alpha
or \alpha ^{1/2} h/a where h is disk scale height
and \alpha is the parameter f... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 201,
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"raw": "Johnson, E. T., Goodman, J. & Menou, K. 2006, , 647, 1413",
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{
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"doi": "",
... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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-... |
16b6ae602cda5986b0ce0eafea0e591e02cec707 | subsection | 7 | 19 | Damping | The eccentricity damping processes are
i) tidal interaction with disk gas, ii) aerodynamical gas drag,
and iii) inelastic collisions.
The tidal damping timescale (i) is derived by as\tau _{\rm tidal} \simeq 1.3
\left(\frac{M}{M_{\odot }}\right)^{-1}
\left(\frac{\Sigma _{g} r^2}{M_{\odot }}\right)^{-1}
\left(\frac{c_s}{... | {
"cite_spans": [
{
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"doi": "",
"end": 486,
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"raw": "Tanaka, H., & Ward, W. R., 2004, , 602, 388",
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"start": 134
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{
"arxiv_id": "",
"doi": "",
"end": 8... | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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27412f5dbc8455d29349197f4edc76de5e5ed928 | subsection | 8 | 19 | Equilibrium eccentricity | We now equate eq. (REF ) with
eqs. (REF ), (REF ), and (REF ),
respectively,
to obtain an equilibrium eccentricity for each damping process.
For simplicity and to a good approximation, the actual equilibrium
eccentricity can be approximated as the minimum of the three
equilibrium eccentricities.
From eqs. (REF ) and (R... | {
"cite_spans": []
} | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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... |
5542327f21a607695020e80e803cfa3652d938a6 | subsection | 9 | 19 | Equilibrium eccentricity | (REF ),\begin{array}{ll}
e_{\rm coll} & \simeq 3.2 \times 10^2 f_g (f_d \eta _{\rm ice})^{-1/2} \gamma \left(\frac{M}{M_{\oplus }}\right)^{1/6}
\left(\frac{\rho _{\rm p}}{3{\rm gcm}^{-3}}\right)^{1/3}
\left(\frac{r}{1{\rm AU}}\right)^{5/4} \\
& \simeq 3.6 f_g (f_d \eta _{\rm ice})^{-1/2} \gamma \left(\frac{R}{1{\rm km}... | {
"cite_spans": []
} | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.012103861197829247,
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0... |
6b338dddb78e9993b83985147de99267299a6a97 | subsection | 10 | 19 | Equilibrium eccentricity | \end{array}The accretion of planetesimals is possible for
M > M_{\rm acc} (R > R_{\rm acc}).
In the top panel of fig. REF
(\gamma = 10^{-3} and f_g = 1),
planetesimal accretion proceeds in a range of
R's in which the solid line (e_{\rm eq}) is
located below the dashed line (e_{\rm acc}), that is,
only if a body is lar... | {
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{
"arxiv_id": "",
"doi": "",
"end": 484,
"openalex_id": "",
"raw": "Johansen, A., Oishi, J. S., Low, M. M., Klahr, H., Thomas, H. & Youdin, A. 2007. Nature, 448, 1022",
"source_ref_id": "fd89ca012e4718a17237d8f328b751d716644103",
"start": 336
}
]
} | 10.1086/591903 | 0807.1761 | Accretion and destruction of planetesimals in turbulent disks | [
"Shigeru Ida",
"Tristan Guillot",
"Alessandro Morbidelli"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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