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0}\end{aligned}$$
By comparing the error bars in the key power laws and with the error bars in power-law relations , , and , we observe that there is less uncertainty in the first case, indicating that the quantities $||{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$, $||{\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$, $\Lambda$ and $R_\Pi$ tend to be more sensitive to approximation errors than $\R_{\E_0}({\widetilde{\mathbf{u}}_{\E_0}})$. Non-negligible error bars may also indicate that, due to modest enstrophy values attained in our computations, the ultimate asymptotic regime corresponding to $\E_0 \rightarrow
\infty$ has not been reached in some power laws.
A useful aspect of employing the average ring radius $R_{\Pi}$ as the characteristic length scale is that its observed scaling with respect to $\E_0$ can be used as an approximate indicator of the resolution $1/N$ required to numerically solve problem \[pb:maxdEdt\_E\] for large values of enstrophy. From the scaling in relation , it is evident that a two-fold increase in the value of $\E_0$ will be accompanied by a similar reduction in $R_\Pi$, thus requiring an eight-fold increase in the resolution (a two-fold increase in each dimension). This is one of the reasons why computation of extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ for large enstrophy values is a very challenging computational task. In particular, this relation puts a limit on the largest value of $\E_0$ for which problem \[pb:maxdEdt\_E\] can be in principle solved computationally at the present moment: a value of $\E_0 =
2000$, a mere order of magnitude above the largest value of $\E_0$ reported here, would require a resolution of $8192^3$ used by some of the largest Navier-Stokes simulations to date.
To summarize, as the enstrophy increases from $\E_0 \approx 0$ to $\E_0 = \O(10^2)$, the optimal vortex states change their structure from cellular to ring-like. While with the exception of $\R({\widetilde{\mathbf{u}}_{\E_0}})$ and $\K({\widetilde{\mathbf{u}}_{\E_0
| 4,101
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| 3,722
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|
by the expression $${{\mathcal J}_{\rm c} = C_4 + \frac{i}{2} e^{-\phi} J \wedge J = i \sum_i T_i \tilde{\omega}_i} \quad .$$ In the IIA O6-orientifold, instead, the action of the involution $\sigma_A$ is $$\sigma_A (x^i ) = x^i \qquad \sigma_A ( y^i ) = - y^i \quad .$$ This implies that the $\tau_i$’s are real. The real part of the $S$ and $U_i$ moduli consists of the $\tau_i$’s and the dilaton, while their imaginary part consists of the RR 3-form $C_3$. The complexified holomorphic 3-form has the expression $$\begin{aligned}
\Omega_{\rm c} & = i S (dx^1 \wedge dx^2 \wedge dx^3 ) - i U_1 ( dx^1 \wedge dy^2 \wedge dy^3 ) \nonumber \\
& - i U_2 ( dy^1 \wedge dx^2 \wedge dy^3 ) - i U_3 ( dy^1 \wedge dy^2 \wedge dx^3 ) \quad ,\end{aligned}$$ which is therefore linear in both $S$ and $U_i$. In the IIA case, it is the $B$ field that complexifies the Kähler form, so that the $T_i$ moduli are given by the expression $${ J_{\rm c} = B + i J = i \sum_i T_i {\omega}_i} \quad .$$ The two orientifold models are mapped into each other by performing three T-dualities along the $x^i$ directions, under which operation the moduli $U_i$ and $T_i$ are interchanged. This operation corresponds to mirror symmetry for this specific orbifold [@Strominger:1996it].
If one turns on the RR fluxes, it can be easily seen from eqs. and that one generates a term in the superpotential which is a cubic polynomial in the $U$ moduli from the IIB perspective and in the $T$ moduli from the IIA perspective. The RR fluxes are related by T-duality as $$F_{a b_1 ...b_p} \overset{T_a}{\longleftrightarrow} F_{b_1 ...b_p} \quad , \label{TdualityruleRRfluxes}$$ where with $a$ and $b$ we denote any of the internal directions. In IIB, only the 3-form flux $F_3$ is turned on, and performing three T-dualities along the $x$ directions, this is mapped to the various fluxes of IIA according to how many indices there are along the $x$ directions. The result is summarised in Table \[TableRRfluxes\].
[|c||c|c|]{}
---------------------------------------
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n thus dominates the growth of galaxies.
Dependence on halo mass {#sec:mass}
=======================
In Fig. \[fig:halomassz2\] we plot the same properties as in Fig. \[fig:haloradz2\] as a function of halo mass for gas at radii $0.8R_\mathrm{vir}<R<R_\mathrm{vir}$, where differences between hot- and cold-mode gas are large. Grey, dashed lines show analytic estimates and are discussed below. The dotted, grey line in the top-left panel indicates the star formation threshold, i.e. $n_{\rm H}=0.1~{\rm cm}^{-3}$. The differences between the density and temperature of the hot- and cold-mode gas increase with the mass of the halo. The average temperature, maximum past temperature, pressure, entropy, metallicity, absolute radial peculiar velocity, absolute accretion rate, and the hot fraction all increase with halo mass.
We can compare the gas overdensity at the virial radius to the density that we would expect if baryons were to trace the dark matter, $\rho_\mathrm{vir}$. We assume an NFW profile [@Navarro1996], take the mean internal density relative to the critical density at redshift $z$, $\Delta_c\langle\rho\rangle$, from spherical collapse calculations [@Bryan1998] and the halo mass-concentration relation from @Duffy2008 and calculate the mean overdensity at $R_\mathrm{vir}$. This is plotted as the dashed, grey line in the top-left panel of Fig. \[fig:halomassz2\]. It varies very weakly with halo mass, because the concentration depends on halo mass, but this is invisible on the scale of the plot. For all halo masses the median density is indeed close to this analytic estimate. While the same is true for the hot-mode gas, for high-mass haloes ($M_\mathrm{halo}\gtrsim10^{12}$ M$_\odot$) the median density of cold-mode gas is significantly higher than the estimated density and the difference reaches two orders of magnitude for $M_\mathrm{halo}\sim10^{13}$ M$_\odot$. A significant fraction of the cold-mode gas in these most massive haloes is star for
| 4,103
| 1,503
| 3,787
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s the image of a fixed element of $F_j$ under the map $\psi_j$. Since $(z_j^{\ast})_1$ can be either $0$ or $1$ by Equation (\[e42\]), $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
So far, we showed that $\psi_j$ is surjective when $j$ is even. We now show that $\psi_j$ is surjective when $j$ is odd. Recall that when $j$ is odd, $L_j$ is *free of type I* and $L_{j-1}, L_{j+1}, L_{j+2}, L_{j+3}$ are *of type II*. Recall that $\bigoplus_{i \geq 0} M_i$ is a Jordan splitting of a rescaled hermitian lattice $(L^{j-1}, \xi^{-(j-1)/2}h)$ and that $M_1=\pi^{(j-1)/2}L_1\oplus\pi^{(j-1)/2-1}L_3\oplus \cdots \oplus \pi L_{j-2}\oplus L_{j}$. We can also let $L_j=\left(\oplus H(1)\right)\oplus A(4b, 2\delta, \pi)\textit{ with $b\in A$}$ by Theorem \[210\] so that $n(L_j)=(2)$.
We write $M_1=L_j\oplus N_1$, where $N_1$ is $\pi^1$-modular so that $n(N_1)=(2)$ or $n(N_1)=(4)$. If $n(N_1)=(4)$, then the proof of the surjectivity of $\psi_j$ is similar to and simpler than that of the case $n(N_1)=(2)$. Thus we assume that $n(N_1)=(2)$. Then by Theorem \[210\], $N_1=\left(\oplus H(1)\right)\oplus A(4b', 2\delta, \pi)$ with $b'\in A$. Let $(e_1, e_2, e_3, e_4)$ be a basis for $A(4b, 2\delta, \pi)\oplus A(4b', 2\delta, \pi)$. Then the diagonal block (associated to $(e_1, e_2, e_3, e_4)$) of the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $L^{j-1}$ is $\begin{pmatrix}1+\pi x_j&0&0&0\\\pi z_j&1&0&0\\0&0&1&0\\0&0&0&1 \end{pmatrix}$. We choose another basis $$(e_1-e_3, \pi e_3+e_4, e_2+e_4, e_1-2b\pi/\delta\cdot e_2),
\textit{ denoted by $(e_1', e_2', e_3', e_4')$},$$ for the lattice $A(4b, 2\delta, \pi)\oplus A(4b', 2\delta, \pi)$ so that the associated Gram matrix is $$A(4(b+b'), -2\delta(1+4b'), \pi(1+4b'))\oplus A(4\delta, -4b(1+4b), \pi(1+4b)).$$ Here, the former lattice $A(4(b+b'), -2\delta(1+4b'), \pi(1+4b'))$ is $\pi^1$-modular with the norm $(2)$ and the latter lattice $A(4\delta, -4b(1+4b), \pi(1+4b))$ is $\pi^1$-modular w
| 4,104
| 3,578
| 3,245
| 3,655
| 3,636
| 0.771098
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|
)$ and $c(j)$ are rationally null-homologous. Thus their linking number is well-defined, establishing the first claim.
[**B1$\Longleftrightarrow$B3**]{}: $\b^n$ is defined precisely when $[c(n)]=0$ in $H_1(M;\BQ)$ which is precisely the condition under which $c(n+1)$ can be defined.
[**B1$\Longrightarrow$B4**]{}: If $n=1$ the implication is true since B4 is vacuous. Thus assume by induction that the implication is true for $n-1$, that is our inductive assumption is that, for all $s+t<n$, ${\ensuremath{\ell k}}(c(s),c^+(t))=0$. Now consider the case that $s+t=n$. Since $\b^n$ is defined $[c(n)]=0$ in $H_1(M;\BQ)$. We claim this is true precisely when $c(n)\cd c(1)=0$ (here we refer to oriented intersection number on the surface $V_x$). For suppose $\psi_x:M\to S^1$ and $\psi_y:M\to S^1$ are maps such that $\psi^{-1}_x(*)=V_x$ and $\psi^{-1}_y(*)=V_y$. Then $(\psi_x)_*([c(n)])=0$ since $c(n)\subset V_x$; and $(\psi_y)_*([c(n)])=0$ precisely when $c(n)\cd V_y=c(n)\cd c(1)=0$. But the map $\psi_x \times \psi_y$ completely detects $H_1(M)$/Torsion. Therefore, once $c(n)$ exists, $\b^n$ is defined if and only if: $$\begin{aligned}
0 = c(n)\cd c(1) &= \pm(c(1)\cd V_{c(n-1)})\\
&= \pm k_{n-1}{\ensuremath{\ell k}}(c(1),c^+(n-1))\end{aligned}$$ which establishes B4 in the case $s=1$. But we claim that, if B4 is true for $s+t<n$, then for $s+t=n$ and $s< t$, $$k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1))=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).$$ This equality can then be applied, successively decreasing $s$, to establish B4 in generality. This claimed equality is established as follows. $$\begin{aligned}
\pm k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1)) &= \pm V_{c(t-1)}\cd c(s+1)\\
&= \pm V_{c(t-1)}\cd(V_x\cap V_{c(s)})\\
&= V_{c(s)}\cd(V_x\cap V_{c(t-1)})\\
&= V_{c(s)}\cd c(t)\\
&= k_s{\ensuremath{\ell k}}(c(t),c^+(s))\\
&=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).\end{aligned}$$ The last step is justified by verifying that $c(s)\cd c(t)=0$ if $s< t$. For $$\begin{aligned}
c(s)\cd c(t) &= c(s)\cd V_{c(t-1)}\\
&= \pm k_{t-
| 4,105
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.FacebookClient
....
Note: you can change the facebook url(because already is used 2.2)
Q:
Solving logarithms with different bases?
How would I go about getting an exact value for a question like: $\log_8 4$
I know that $8^{2/3} = 4$ but how would I get that from the question?
A:
The logarithm can be rewritten as
$$\log_8(4) = x \iff 8^x = 4$$
Now note that both $8$ and $4$ are powers of $2$ to get
$$(2^3)^x = 2^2$$
or alternatively,
$$2^{3x} = 2^2$$
So $3x = 2$.
Q:
undefined method `strftime' for nil:NilClass - shows sometimes
I'm building a simple app for displaying movies using themoviedb gem. However, when I try to do a search query, it displays the error I mentioned. Now, that doesn't happen every time - only in certain cases (e.g. Matrix works fine, but Fight Club shows the error)
Here's my code:
<% @movie.each do |movie| %>
<%= link_to movie_path(movie.id) do %>
<%= image_tag("#{@configuration.base_url}w154#{movie.poster_path}") if movie.poster_path %>
<% end %>
<div class="moviesindex">
<%= link_to movie.title, movie_path(movie.id) %>
(<%= movie.release_date.to_date.strftime("%Y") %>) <br />
</div>
<% end %>
A:
Be forgiving for realease_date if movie.release_date can be nil at times using try:
<%= movie.release_date.try(:year) %>
This will give you release year if release_date is valid and gives you nil if release_date is nil.
Since release_date, I assume is already either a Date or DateTime or ActiveSupport::TimeWithZone, the to_date is not necessary. Also I think date.year is more cleaner than date.strftime("%Y").
Q:
Can the Avatar still be reincarnated?
Near to the finale of The Legend of Korra - Book 2: Spirits when Vaatu rips Raava from Korra's body and destroys her, Korra says to Tenzen that the Avatar Cycle has too been destroyed and she is the last Avatar.
At the end Korra confirms that Raava is back inside her however her connection to the previous Avatars is still gone.
I am wondering, does this mean that the Avatar can no longer be reincarnated? or is K
| 4,106
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be an $H$-set and $Y$ be an $K$-set, then: $${<}Ind_{H}^{K}X,Y{>}_{K} = {<}X,Res^{K}_{H} Y{>}_{H},$$ and $${<}Res_{H}^{K}Y,X{>}_{H} = {<}Y,Ind^{K}_{H} X{>}_{K},$$ So we have a family of linear forms $(\phi_{H})_{H\leqslant G}$ on the Burnside algebras $(RB(H))_{H\leqslant G}$ defined by: let $X\in RB(H)$, then $\phi_{H}(X):={<}X,H/H{>}$. Let $H\leqslant K$ and $X\in RB(H)$, then $$\begin{aligned}
\phi_{K}(Ind_{H}^{K}X)&={<}Ind_{H}^{K}(X),K/K{>}_{K}\\
&={<}X,Res^{K}_{H}K/K{>}_{H}\\
&={<}X,H/H{>}_{H}\\
&=\phi_{H}(X).\end{aligned}$$ The family $\big(\phi_{H}\big)_{H\leqslant G}$ is a stable by induction family of linear forms on the Burnside algebras $\big(RB(H)\big)_{H\leqslant G}$, and the bilinear forms $b_{\phi_{H}}$ are the bilinear forms ${<}-,-{>}_{H}$ so by definition they are non-degenerate.
If the Mackey algebra is symmetric, it is always possible to choose a stable by induction family of linear maps $(\phi_{H})_{H\leqslant G}$ on $(RB(H))_{H\leqslant G}$ which generalize the trace maps on $\big(RH\big)_{H\leqslant G}$ in the sense of Remark \[gene\], i.e. such that $\phi_{H}(H/1)=1$.Indeed, since the family is stable by induction, for every $H$ subgroup of $G$, we have $\phi_{H}(H/1)=\phi_{1}(1/1)$. Let us denote by $a$ the value $\phi_{H}(H/1)$. Now in the usual basis of $RB(H)$, the matrix of the bilinear form $b_{\phi_{H}}$ as a column divisible by $a$, and since this bilinear form is non degenerate, we have $a\in R^{\times}$, so one can normalize the linear forms $\phi_{H}$.
Symmetricity in the semi-simple case.
=====================================
Let $G$ be a finite group and $k$ a field of characteristic zero, or characteristic $p>0$ which does not divide the order of $G$, then it is well known that the Mackey algebra $\mu_{k}(G)$ is semi-simple, so it is clearly a symmetric algebra. One can specify a trace map for this algebra by using the previous section. Let us consider the linear form $\phi_{G}$ on $kB(G)$ defined by $$\phi(X) = \sum_{H \in [s(G)]} \frac{1}{|N_{G}(H)|} |X^{H}|,$$ where $X
| 4,107
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| 1,865
| 3,875
| null | null |
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|
$X_1,\ldots,X_\ell\subset\tilde A^{e^{O(\tilde s)}}$ of size at most $\exp(e^{O(\tilde s)}\log^{O(1)}2\tilde K)$ such that $$\tilde A\subset N\prod\{A_1,\ldots,A_r,X_1,\ldots,X_\ell\},$$ with the product taken in some order.
Here, and throughout this paper, given an ordered set $X=\{x_1,\ldots,x_m\}$ of subsets and/or elements in a group $G$, we write that a product $\Pi$ of the members of $X$ is *equal to $\prod X$ with the product taken in some order* to mean that there is a permutation $\xi\in{\text{\textup{Sym}}}(m)$ such that $\Pi=\prod_{i=1}^m x_{\xi(i)}$. If $Y=\{y_1,\ldots,y_m\}$ is another ordered set of the same number subsets and/or elements of $G$, then we say that products $\prod X$ and $\prod Y$ are *taken in the same order* if $\prod X=\prod_{i=1}^m x_{\xi(i)}$ and $\prod Y=\prod_{i=1}^m y_{\xi(i)}$ for the same permutation $\xi$.
If $\tilde A$ is abelian then the proposition is trivially true with $r=1$, $\ell=0$, $A_1=\tilde A$ and $N=\{1\}$. We may therefore assume that $s\ge\tilde s\ge2$ and, by induction, that the proposition holds for all smaller values of $\tilde s$.
We start by rewriting the part of the statement we are trying to prove as $$N\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}.$$ This is exactly equivalent to , but writing the bound in this way makes it slightly easier to keep track of through the induction. For the same reason, at various points in the argument we use the trivial observation that any quantity bounded by $O(1)$ is also bounded by $e^{O(1)}$.
Applying Proposition \[prop:ind.tor-free.post.chang\], we obtain a normal subgroup $N_0\lhd G$ with $N_0\subset A^{K^{e^{O(s)}m}}$; an integer $r_0\le\log^{O(1)}2\tilde K$; finite $\tilde K^{e^{O(1)}}$-approximate groups $\tilde A_1,\ldots,\tilde A_{r_0}\subset\tilde A^{O(1)}\subset\tilde A^{e^{O(1)}}\subset A^{e^{O(1)}m}$ such that, writing $\rho:G\to G/N$ for the quotient homomorphism, each group $\langle\rho(\tilde A_i)\rangle$ has step less than $\tilde s$; and a set $X\subset\tilde
| 4,108
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| 2,962
| 3,723
| 2,772
| 0.77707
|
github_plus_top10pct_by_avg
|
au D(x)+\sum_{y\ne x}\tau D(y)\,{{\langle \varphi_y
\varphi_x \rangle}}_\Lambda\leq2D(x)+\sum_{y\ne x}2D(y)\,G(x-y),\end{aligned}$$ where, and from now on without stating explicitly, we use the translation invariance of $G(x)$ and the fact that $G(x-y)$ is an increasing limit of ${{\langle \varphi_y\varphi_x \rangle}}_\Lambda$ as $\Lambda\uparrow{{\mathbb Z}^d}$. By [(\[eq:J-def\])]{} and the assumption in Proposition \[prp:GimpliesPix\] that $\theta_0L^{d-q}$, with $q<d$, is bounded away from zero, we obtain $$\begin{aligned}
{\label{eq:Dbd}}
D(x)\leq O(L^{-d}){\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}\leq\frac{O(L^{-d+q})}
{{\vbx{|\!|\!|}}^q}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$ For the last term in [(\[eq:tildeG-1stbd\])]{}, we consider the cases for $|x|\leq2\sqrt{d}L$ and $|x|\ge2\sqrt{d}L$ separately.
When $|x|\leq2\sqrt{d}L$, we use [(\[eq:Dbd\])]{}, [(\[eq:IR-xbd\])]{} and [(\[eq:conv\])]{} with $\frac12d<q<d$ to obtain $$\begin{aligned}
\sum_{y\ne x}D(y)\,G(x-y)\leq\sum_y\frac{O(L^{-d+q})}{{\vby{|\!|\!|}}^q}\,
\frac{\theta_0}{{\vbx-y{|\!|\!|}}^q}\leq\frac{O(\theta_0L^{-d+q})}
{{\vbx{|\!|\!|}}^{2q-d}}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$
When $|x|\ge2\sqrt{d}L$, we use the triangle inequality $|x-y|\ge|x|-|y|$ and the fact that $D(y)$ is nonzero only when $0<\|y\|_\infty\leq L$ (so that $|y|\leq\sqrt{d}\|y\|_\infty\leq\sqrt{d}L\leq\frac12|x|$). Then, we obtain $$\begin{aligned}
\sum_{y\ne x}D(y)\,G(x-y)\leq\sum_yD(y)\,\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q}
=\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$
This completes the proof of the first inequality in [(\[eq:tildeG-bd\])]{}. The second inequality can be proved similarly.
By repeated use of [(\[eq:tildeG-bd\])]{} and Proposition \[prp:conv-star\](i) with $a=b=2q$ (or Proposition \[prp:conv-star\](ii) with $x=x'$ and $y=y'$), we obtain $$\begin{aligned}
{\label{eq:psi-bd}}
\psi_\Lambda(v',v)\leq\delta_{v',v}+\frac{O(\theta_0^2)}{{\vbv-v'{|\!|\!|}}^{2q}}.\end{aligned}$$
| 4,109
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| 2,803
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github_plus_top10pct_by_avg
|
lon))$, with high probability. First assume that $2m{\leqslant}n $ and suppose that the algorithm has allocated $m$ balls to ${\mathcal{H}}^{(t)}, t=1,\ldots,m$ and let $\ell^*{\leqslant}\log_d\log n +{\mathcal{O}}(1)$ denote its maximum load. We now consider two independent balanced allocation algorithms, say ${\mathcal{A}}$ and ${\mathcal{A}}_0$, on two dynamic hypergraphs starting from step $m$. These dynamic hypergraphs are $({\mathcal{H}}^{(m)},\ldots, {\mathcal{H}}^{(n)})$ and $({\mathcal{H}}_0^{(m)},\ldots, {\mathcal{H}}_0^{(n)})$, [where ${\mathcal{H}}_0^{(t)}$ is an identical copy of ${\mathcal{H}}^{(t)}$ for $t=m,\ldots, n$]{}. Moreover, we assume that in round $m$, all bins contained in ${\mathcal{H}}^{(m)}_0$ have [exactly]{} $\ell^*$ balls. Let us couple algorithm ${\mathcal{A}}$ on ${\mathcal{H}}^{(t)}$ and algorithm ${\mathcal{A}}_0$ on ${\mathcal{H}}_0^{(t)}$. Write $V=[n]$ for the set of $n$ bins. To do so, the coupled process allocates a pair of balls to bins as follows: for $t=m+1,\ldots, 2m$, the coupling chooses a one-to-one labeling function uniformly at random, where $V$ is the ground set of both hypergraphs (i.e, set of $n$ bins) and $\{1, 2,\ldots,n\}$ is a set of labels. Next, the coupling chooses $D_{t}$ randomly from ${\mathcal{H}}^{(t)}$. [Let $D'_t$ denote the same set of $d$ bins as $D_t$ in ${\mathcal{H}}_0^{(t)}$.]{} Algorithm ${\mathcal{A}}$ allocates ball $t+1$ to a least-loaded vertex of $D_{t}$, and algorithm ${\mathcal{A}}_0$ allocates ball $t+1$ to a least-loaded vertex of $D'_t$, with both algorithms breaking ties in favour of the vertex $v$ with the smallest load [and minimum label $\sigma_t(v)$]{}. Note that algorithm ${\mathcal{A}}$ is a faithful copy of the balanced allocation process on $({\mathcal{H}}^{(m)},\ldots, {\mathcal{H}}^{(n)})$, and algorithm ${\mathcal{A}}_0$ is a faithful copy of the balanced allocation process on $({\mathcal{H}}_0^{(m)},\ldots, {\mathcal{H}}_0^{(n)})$, respectively. [(This follows as $\sigma_t$ is chosen uniformly at random.)]{} Let $X_i^{
| 4,110
| 2,654
| 1,189
| 4,039
| 1,604
| 0.7877
|
github_plus_top10pct_by_avg
|
$ be submodules of $M$ such that $V/U\iso R/P$ for some $P\in \Spec(R)$. Then there exists an irreducible submodule $W$ of $M$ such that $U=V\sect W$.
By Noetherian induction there exists a maximal submodule $W$ of $M$ such that $U=V\sect W$. We claim that $W$ is an irreducible submodule of $M$. Indeed, suppose that $W=W_1\sect W_2$. Then $U=(V\sect W_1)\sect(V\sect W_2)$ is a decomposition of $U$ in $V$. However, $U$ is irreducible in $V$ since $V/U\iso R/P$. It follows that $V\sect W_1=U$ or $V\sect
W_2=U$. Since $W$ was chosen to be maximal with this intersection property, we see that $W=W_1$ or $W=W_2$. Thus $W$ is irreducible, as desired.
(a)(b): Let ${\mathcal F}$ be a prime filtration as given in (a). We show by decreasing induction on $i<r$ that for $j=i+1,\ldots, r$ there exist irreducible $P_j$-primary submodules $N_j$ of $M$ such that $M_{i}=\Sect_{j={i+1}}^rN_j$.
For $i=r$ we may choose $N_r=M_{r-1}$, since $M/M_{r-1}\iso R/P_r$. Now let $1<i<r$, and assume that $M_{i}=\Sect_{j=i+1}^rN_j$ where $N_j$ is an irreducible $P_j$-primary submodule of $M$ for $j=i+1,\cdots, r$. Since $M_{i}/M_{i-1}\iso R/P_{i}$, it follows by Lemma \[complement\] that there exists an irreducible submodule $N_{i}$ of $M$ such that $M_{i-1}=M_{i}\sect N_{i}$. Since $R/P_{i}\iso M_{i}/M_{i-1}=M_{i}/M_{i}\sect N_{i}\subset M/N_{i}$, it follows that $\{P_{i}\}=\Ass(M_{i}/M_{i-1})\subset \Ass(M/N_{i})$. However $\Ass(M/N_{i})$ has only one element, therefore $\Ass(M/N_{i})=\{P_{i}\}$.
Clean filtrations and shellings
===============================
In this section we recall the main result of the paper of Dress [@D] (see also [@Si]), and provide some extra information. Let $\Delta$ be a simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$. Recall that $\Delta$ is [*shellable*]{}, if the facets of $\Delta$ can be given a linear order $F_1,\ldots,
F_m$ such that for all $i,j$, $1\leq i<j\leq m$, there exists some $v\in F_i\setminus F_j$ and some $k<i$ with $F_i\setminus F_k=\{v\}$.
Note that we do [*not*]{} insist that $\
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1}{\ensuremath{\ell k}}(c(s),c^+(t-1))\end{aligned}$$ which vanishes by our inductive assumption since $s+(t-1)<n$.
[**B4$\Longrightarrow$B1**]{}: Since $\b^1$ is always defined we may assume $n>1$. It follows from B4 that ${\ensuremath{\ell k}}(c(1),c^+(n-1))=0$ if $n>1$. But we saw in the proof of B1$\Longrightarrow$B4 that once $c(n)$ was defined, this was equivalent to $\b^n$ being defined.
[**B2$\Longrightarrow$B1**]{}: This is obvious.
[**B4$\Longrightarrow$B2**]{}: Since B4$\Longrightarrow$B1, we have $\b^j$ defined for $j\le n$. Now suppose $1\le j\le\[\f n2\]$. Since $\b^j={\ensuremath{\ell k}}(c(j),c^+(j)$ and $2j\le n$, this vanishes by B4.
This completes the proof of Lemma \[equivalence\].
The proof shows slightly more, namely that there is a correspondence between the infinite cyclic cover implicit in part $A$ and the class $x$ in parts $B$ and $C$. Suppose $\{M_n\}$ is a family of $n$-fold cyclic covers of $M$ corresponding to the infinite cyclic cover $M_\infty$. Note that $H_1(M_\infty;\BQ)$ is a finitely generated $\La=\BQ[t,t^{-1}]$ module (this involves a choice of generator of the infinite cyclic group of deck translations of $M_\infty$). Throughout this proof, homology will be taken with rational coefficients unless specified otherwise.
[**Step 1**]{}: $\b_1(M_n)$ grows linearly $\Longleftrightarrow H_1(M_\infty;\BQ)$ has positive rank as a $\La$-module.
As remarked above, this fact was previously known. We present a quick proof for the convenience of the reader. We are indebted to Shelly Harvey for showing us this elementary proof. Since $\La$ is a PID, $$H_1(M_\infty)\cong\La^{r_1}\oplus_j\f\La{\<p_j(t)\>}$$ where $p_j(t)\neq0$. By examining the “Wang sequence” with $\BQ$-coefficients $$H_2(M_\infty)\lra H_2(M_n)\overset{\p_*}{\lra}H_1(M_\infty)
\overset{t^n-1}{\lra}H_1(M_\infty)\overset{\pi}{\lra}H_1(M_n)
\overset{\p_*}{\lra}H_0(M_\infty)\overset{t^n-1}{\lra}$$ it is easily seen that $$\begin{aligned}
H_1(M_n) &\cong\f{H_1(M_\infty)}{\<t^n-1\>}\op\BQ\\
&\cong\(\f\La{\<t^n-1\>}
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ed curve) in the curve complex. Hence, any superinjective simplicial map is induced by the identity homeomorphism.
If $(g, n) = (1, 2)$, there are only two elements in the curve complex (see [@Sc]). They are the isotopy classes of $a$ and $b$ as shown in Figure \[figure1\]. We see that $i([a], [b]) = 1$. So, the superinjective simplicial map cannot send both of these elements to the same element. If this simplicial map fixes each of them, then it is induced by the identity homeomorphism, if it switches them then it is induced by a homeomorphism that switches the curves $a$ and $b$.
If $(g, n) = (2, 0)$, there are only three elements in the curve complex (see [@Sc]). They are the isotopy classes of $a$, $b$ and $c$ as shown in Figure \[figure1\] (ii). We see that $i([a], [b]) = 1$, $i([a], [c]) = 1$ and $i([b], [c]) = 0$. Since superinjective simplicial maps preserve geometric intersection zero and nonzero properties, $\lambda$ fixes $[a]$. If it also fixes each of $[b]$ and $[c]$ then it is induced by the identity homeomorphism, if it switches $[b]$ and $[c]$ then it is induced by a homeomorphism that switches the one sided curves $b$ and $c$, while fixing $a$ up to isotopy.
=1.2in =3.6in
=1.4in
If $(g, n) = (2, 1)$, then the curve complex is given by Scharlemann in [@Sc] as follows: Let $a$ and $b$ be as in Figure \[figure2\]. We see that $i([a], [b]) = 1$. The vertex set of the curve complex is $\{[a], [b], t_a^m ([b]) : m \in \mathbb{Z} \}$, where $t_a$ is the Dehn twist about the 2-sided curve $a$. The complex is shown in Figure \[figure2\]. Since $i([a], t_a^m ([b])) \neq 0$ for any $m$, we have $i(\lambda([a]), \lambda(t_a^m ([b])) \neq 0$ for any $m$, i.e. $\lambda([a])$ will not be connected by an edge to any other vertex. For any $m$, the elements $t_a^m ([b])$ and $t_a^{m+1} ([b])$ cannot be send to the same element by $\lambda$ since they have different intersection information with $t_a^{m-1} ([b])$, and $\lambda$ preserves geometric intersection zero and nonzero by definition. Similarly, $t_a^m
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by $A_{j,\alpha} = \alpha^{j-1}$ is a $\GF(q)$-representation of $U_{s,q}$; see Section 6.5 of \[\[oxley\]\] for more detail.
\[localrep\] There is a function $f_{\ref{localrep}}\colon \bZ^3 \to \bZ$ so that, for all integers $s,t,n \ge 2$, if $q$ is a prime power, and $M$ is a matroid with a $\PG(r(M)-1,q)$-restriction $R$, no $U_{s,2s}$-minor, and no $\PG(n-1,q')$-minor for any $q' > q$, then there is a set $C \subseteq E(M)$ so that $|C| \le f_{\ref{localrep}}(s,n,t)$, every nonloop of $M \con C$ is parallel to an element of $R$, and every rank-$t$ restriction of $M\con C$ is $\GF(q)$-representable.
Let $Q$ be the set of all prime powers less than $2s$. Given integers $s,t,n \ge 1$, let $t' = \max(s,t)$. Let $k_1 = \max_{q \in Q}f_{\ref{exppgdensity}}(s,q,n)$ and let $k_2 = \max_{q \in Q}f_{\ref{stackwin}}(s,n,q,t')$. Set $f_{\ref{localrep}}(s,n,t) = k = k_1 + s + t'k_2$.
Let $q$ be a prime power and let $M$ be a matroid with a $\PG(r(M)-1,q)$-restriction. Suppose that $M$ has no $U_{s,2s}$-minor and no $\PG(n-1,q')$-minor for $q' > q$. If $r(M) < s $ then the theorem clearly holds with $C$ equal to a basis for $M$, so we may assume that $r(M) \ge s$. If $q \ge 2s$ then $R$ has a $U_{s,2s}$-restriction, so we may also assume that $q \in Q$.
Let $F \subseteq E(M)$ be maximal so that $F$ is skew to every rank-$(s-1)$ flat of $R$. Since every rank-$(s-1)$ set of $(M \con F)|E(R)$ is also a rank-$(s-1)$ set of $R$ and thus contains at most $\tfrac{q^{s-1}-1}{q-1}$ elements of $|R|$, we have $$\tau_{s-1}((M \con F)|E(R)) \ge |R| / \left(\tfrac{q^{s-1}-1}{q-1}\right) > q^{r(M)-s} = q^{r(M \con F) + r_M(F) -s}.$$ If $r_M(F) \ge k_1+s$ then, since $k_1 \ge f_{\ref{exppgdensity}}(s,q,n)$, we obtain a contradiction from Theorem \[exppgdensity\]. Therefore $r_M(F) < k_1 + s$. By the maximality of $F$, every $f \in E(M \con F)$ is spanned by some rank-$(s-1)$ set of $(M \con F)|E(R)$. Let $h$ be maximal so that $M \con F$ has a $(q,h,t')$-stack restriction $S$. By Lemma \[stackwin\], we have $h \le k_2$ and so $r(S
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x,\hs{2ex}b_y=x\bar{h}_y+w_y.$$ The conservation law reads, $$\partial_y b_x+\partial_x b_y=0,\hs{2ex}\partial_x\bar{h}_y+\partial_y\bar{h}_x=0$$ which allows us to write the current $\bar{h}_y$ as, $$\bar{h}_y=-\partial_x w_y-\partial_y w_x.$$ From the commutation relations, we learn that $w_x$ is invariant under the Galilean boost. From the discussion on the two-point functions, we find $$\partial_{y_1}{\left\langle}w_x(x_1,y_1) w_x(x_2,y_2){\right\rangle}=0,\hs{2ex}{\left\langle}\partial_y w_x \partial_y w_x{\right\rangle}=0.$$ From our assumptions that the spectrum of the dilation operator is discrete and non-negative, the following equation is valid as an operator equation $$\partial_y w_x=0.$$ We can shift the currents without changing the canonical commutation relations $$\bar{h}_y\rightarrow \bar{h}_y+\partial_x w_y,\hs{2ex}\bar{h}_x\rightarrow \bar{h}_x-\partial_y w_y.$$ The $\bar{h}_x$ component must be changed at the same time to keep the conservation law intact. The similar shifts also happen in the currents $b_\m$. Under the above shift, we can set $$\bar{h}_y=0, \label{hbary}$$ such that \_y|[h]{}\_x=0 which implies that $\bar{h}_x$ is a function of $x$ $$\bar{h}_x=\bar{h}_x(x).$$ This leads to the existence of an infinite set of conserved charges, $$M_{\epsilon}=\int \epsilon(x)\bar{h}_x(x) dx, \label{Mcharges}$$ where $\epsilon(x)$ is an arbitrary smooth function $x$. It is easy to see that $M_1$ with $\epsilon=1$ actually generates the translation along $y$ direction, while $M_x$ with $\epsilon(x)=x$ is the boost generator. This is consistent with the discussion in the warped CFT literatures[@Hofman:2011zj; @Detournay:2012pc]. We should emphasize here that this infinite set of conserved charges are common in the 2d local Galilean field theories. Next let us turn to the dilation current. Depending on the weight $c$, we will consider $c=0$ and $c\neq 0$ separately.
### Special case: $c=0$
In this case, we have $$d_x=d y\bar{h}_x+s_x,\hs{2ex} d_y=dy\bar{h}_y+s_y.$$ The equations above can be taken
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)}.
\label{u1charges}$$
In the light–cone gauge, the free fermionic heterotic–string models in four dimensions require $20$ and $44$, left–moving and right–moving real world–sheet fermions respectively, to cancel the conformal anomaly. In the usual notation these are denoted as: $\psi^\mu, \chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}$ and $\overline{y}^{1,\dots,6},\overline{\omega}^{1,\dots,6}$, $\overline{\psi}^{1,\dots,5}$, $\overline{\eta}^{1,2,3}$, $\overline{\phi}^{1,\dots,8}$.
The $SU(4) \times SU(2) \times U(1)$ Gauge Group
------------------------------------------------
In the following we set up the necessary ingredients for the classification of the SU421 free fermionic heterotic–string models. The analysis is along similar lines to the one performed in the classification of the $SO(10)$ [@fknr]; heterotic–string Pati–Salam models [@acfkr]; and flipped $SU(5)$ models [@frs]. The novelty compared to these cases is that the SU421 models employ two basis vectors that break the $SO(10)$ symmetry, whereas the HSPSM and FSU5 models use only one. However, we argue below that this class of heterotic–string vacua cannot in fact produce phenomenologically viable models. The basis vectors that generate our $SU(4) \times SU(2) \times U(1)$ heterotic–string models are given by the following 14 basis vectors $$\begin{aligned}
\label{421}
v_1={\bf1}&=&\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}|\overline{y}^{1,\dots,6},
\overline{\omega}^{1,\dots,6},
\overline{\eta}^{1,2,3},
\overline{\psi}^{1,\dots,5},\overline{\phi}^{1,\dots,8}\},\nonumber\\
v_2=S&=&\{{\psi^\mu},\chi^{1,\dots,6}\},\nonumber\\
v_{2+i}={e_i}&=&\{y^{i},\omega^{i}|\overline{y}^i,\overline{\omega}^i\}, \
i=1,\dots,6,\nonumber\\
v_{9}={b_1}&=&\{\chi^{34},\chi^{56},y^{34},y^{56}|\overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\label{basis}\\
v_{10}={b_2}&=&\{\chi^{12},\chi^{56},y^{12},y^{56}|\overline{y}^{12},
\overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dot
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ta_Y} d\operatorname{DH}_Y(\mu') \, g(\mu')$$ for any test function $g$. In particular, this inequality would hold for $g$ the indicator function of $\Delta_X$. But this is clearly impossible, since $\operatorname{DH}_Y$ is absolutely continuous with respect to Lebesgue measure on $\Delta_Y$, for which $\Delta_X$ is a set of measure zero.
\[dimension lemma\] Let $\dim X < \dim Y$. Then, $X \subseteq Y$ implies $d_X < d_Y$.
Clearly, $X \subseteq Y$ implies that $\Delta_X \subseteq \Delta_Y$ and $R_X \leq R_Y$. If $R_X = R_Y$ then the assertion follows directly from , since $$d_X = \dim X - R_X < \dim Y - R_Y = d_Y.$$ Otherwise, if $R_X < R_Y$, it follows from combining and .
As described in the introduction, the upshot of the above is that we cannot directly deduce from a new criterion for obstructions based on the Duistermaat–Heckman measure that goes beyond what is provided by the moment polytope.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Aravind Asok, Emmanuel Briand, Peter Bürgisser, David Gross, Christian Ikenmeyer, Stavros Kousidis, Graeme Mitchison, Mercedes Rosas, Volkher Scholz, and Michèle Vergne for helpful discussions.
This work is supported by the Swiss National Science Foundation (grant PP00P2–128455 and 200021\_138071), the German Science Foundation (grants CH 843/1–1 and CH 843/2–1), and the National Center of Competence in Research ‘Quantum Science and Technology’.
[^1]: A preliminary implementation of the algorithm is available upon request from the authors.
[^2]: In the context of this paper, a quasi-polynomial function is a polynomial function with periodic coefficients; see p. for the precise definition. It should not to be confused with the notion of quasi-polynomial time complexity.
---
abstract: 'The magnetic and transport properties have been investigated for the composite polycrystalline manganites, (1-x)$La_{2/3}Ca_{1/3}MnO_3$/(x)yttria-stabilized zirconia ( (1-x)LCMO/(x)YSZ ), at various YSZ fractions, x, ranging f
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l p}{\partial x}+\Gamma p+
[\Gamma v-\bar{\omega}_{b}^{2}x]\frac{\partial p}{\partial v}+
A\frac{\partial^{2} p}{\partial v^{2}}+B\frac{\partial^{2} p}{\partial v
\partial x}\hspace{0.2cm},$$ where we have used the following abbreviations; $$A=J_{ee}-\Gamma(0)I_{ee}+[g'(0)]^{2}J_{nn}-\Gamma(0)[g'(0)]^{2}I_{nn}$$ and $$B=I_{ee}+[g'(0)]^{2}I_{nn}\hspace{0.2cm}.$$
From the last two relations we have $$A=\left[ J_{ee}+g'(0)^{2}J_{nn}\right]-\Gamma(0)B$$
Defining $A$ and $B$ as $$A=\bar{\gamma}KT \hspace{0.2cm}{\rm and}\hspace{0.2cm}B=\bar{\beta}KT$$ one obtains $$\begin{aligned}
\frac{\partial p}{\partial t}=-v\frac{\partial p}{\partial x}-\bar{\omega}_{b}
^{2}x\frac{\partial p}{\partial v}+\Gamma\frac{\partial}{\partial v}(vp)+
\bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\
\nonumber\\
+KT\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT}-\frac{\bar{\gamma}}
{\Gamma(0)}\right]\frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$
Identifying $$\bar{\Omega}^{2}=\Omega^{2}\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT}
\right]\hspace{0.2cm},$$ Eq.(46) may be rewritten as, $$\begin{aligned}
\frac{\partial}{\partial t}p(x,v,t)=-v\frac{\partial p}{\partial x}
-\bar{\omega}_{b}^{2}x\frac{\partial p}{\partial v}
+\Gamma\frac{\partial}{\partial v}(vp)+
\bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\
\nonumber\\
+KT\left[ \frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}-\frac{\bar{\gamma}}
{\Gamma(0)} \right ] \frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$
Here $\bar{\gamma}(t)$ and $\bar{\Omega}(t)$ are functions of time (due to the relaxation of the nonequilibrium modes) as defined by Eqs.(45) and (47). Or in other words nonstationary nature of the bath makes $\bar{\Omega}(t)$ time-dependent through $J_{nn}$ term which is essentially a non-Markovian modification.
Now the fluctuation-dissipation relations for equilibrium and nonequilibrium baths stated in Sec.II may be invoked. For equilibrium baths as noted earlier we have the usual result; $$J_{ee}
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tant $L>0$, $H_{3}(\frac{t}{L})+dt^2$ on $S^{4k-1}\times
[0,L]$ has positive scalar curvature. Now we have a desired $4k$-ball to be glued to the part made previously out of $M'$.
After the gluing, what we get is just $M$ with a specially devised smooth metric which we denote by $\bar{g}$. Remember that the scalar curvature of $\bar{g}$ is bigger than $s_{g}-c$.
Now we will derive a contradiction. In case that $Y(M)=0$, $$s_{\bar{g}}>s_{g}
-c=Y(M)+c> Y(M)=0,$$ which is a contradiction. In case of $Y(M)<0$, we do the surgery so that $\nu^{\frac{2}{n}}< \frac{2c}{|Y(M)+c|}$. Then noting that $s_{g}<0$, $$\begin{aligned}
-(\int_M |s^-_{\bar{g}}|^{\frac{n}{2}} d\mu_{\bar{g}})^{\frac{2}{n}}
&>& -(\int_{M'-N(\delta)} |s_{g}|^{\frac{n}{2}} d\mu_{g}+ |s_{g}-c|^{\frac{n}{2}}\nu)^{\frac{2}{n}}\\
&>& -(\int_{M'} |s_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}} + (s_{g}-c)\nu^{\frac{2}{n}}\\
&=& Y(M',[g]) + (Y(M)+c)\nu^{\frac{2}{n}} \\
&>& (Y(M)+2c)-2c\\
&=& Y(M).\end{aligned}$$ This gives a contradiction to the formula (\[form1\]), and completes a proof for the $\Bbb HP^{k}$ case.
The case of $CaP^{2}$ can be proved in the same way using the fact that $CaP^{2}$ also admits a metric of positive scalar curvature, and is the mapping cone of the (generalized) Hopf fibration $\pi: S^{15}\rightarrow S^{8}$ with $S^{7}$ fibers as explained in the previous section.
Since the smallness of $\nu$ was used only in the case of $Y(M)<0$, we will show a way of proof without using it when $Y(M)<0$. As done in LeBrun [@lb4], instead of doing surgery on $(N(\delta),g)$, we first take a conformal change $\varphi g$ of $(M',g)$ such that $\varphi \equiv 1$ outside $N(\delta)$ and the scalar curvature of $\varphi g$ is positive on a much smaller neighborhood $N(\delta')$ of $W$. Moreover one can arrange that it satisfies $$-(\int_{M'} |s^{-}_{\varphi g}|^{\frac{n}{2}} d\mu_{\varphi g})^{\frac{2}{n}} > -(\int_{M'} |s^{-}_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}}-\epsilon$$ for any $\epsilon>0$. (This is possible because the codimension of $W
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ground for the formation of expressions (\[2\]).
Since the experiments indicate absence of definite values of $CP$-parity of the decaying neutral $K$-mesons, one should consider the case when the mass term in Lagrangian (\[1\]) does not possess the mentioned $SU(3)$-invariance. Then the first step of the appropriate procedure of Weinberg gives orthonormal fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ with definite masses $m_{S}$ and $m_{L}$ specified by the formulas $$\Phi_{S}(x) = \frac{(1-\varepsilon^{*})\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)+
(1+\varepsilon^{*})\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)}
{\sqrt{2(1+|\varepsilon|^{2})}} =
\frac{\Phi_{1}(x)-\varepsilon^{*}\Phi_{2}(x)}{\sqrt{1+|\varepsilon|^{2}}},
\label{3}$$ $$\Phi_{L}(x) = \frac{(1+\varepsilon)\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)-
(1-\varepsilon)\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)}
{\sqrt{2(1+|\varepsilon|^{2})}} =
\frac{\varepsilon \Phi_{1}(x)+\Phi_{2}(x)}{\sqrt{1+|\varepsilon|^{2}}},
\label{4}$$ with $$|m_{L}^{2}-m_{S}^{2}| = \sqrt{(m_{+}^{2}-m_{-}^{2})^{2}+4|a|^{2}},
\label{5}$$ $$\varepsilon = \frac{m_{+}^{2}-m_{-}^{2}+2i{\rm Im} a}
{m_{L}^{2}-m_{S}^{2}-2{\rm Re} a}.
\label{6}$$
The next step of Weinberg’s procedure consists in finding expressions for the fields $\Phi_{\pm 1 \frac{1}{2} \mp \frac{1}{2}}(x)$ through the fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ on the basis of the relations(\[3\]) and (\[4\]) and in substituting these expressions into all terms in the initial Lagrangian describing both strong and weak interactions.
Then, in completing the Weinberg’s procedure, we should obtain Euler equations for each of the fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ from the transformed Lagrangian and perform the second quantization of the solutions of these equations, that would consist in identifying these solutions as vectors in the spaces of suitable irreducible representations of the Poincare group with neutral mesons $K_{S}^{0}$ and $K_{L}^{0}$. As a result, the processes which were thought to be accompanied by the production or are caused by an in
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operatorname}{O}_{p}(H)}^P\leq P$. This contradicts the fact ${{\operatorname}{O}_{p}(G)}=1$ and it proves the claim.
\[new\] The subgroup $\langle y \rangle$ does not normalise $N_i$, for each $i \in\{1, \ldots, r\}$. In particular, $r > 1$.
Assume that $\langle y \rangle$ normalises some $N_i$ with $i \in\{1, \ldots, r\}$. Then $\langle y \rangle$ normalises $N_i$ for each $i \in\{1, \ldots, r\}$. We can view $\langle y \rangle$ as a subgroup of ${\operatorname}{Aut}(N_i)$, because ${{\operatorname}{C}_{\langle y\rangle}(N_i)}=1$ (recall that $y$ has order $p$). By Lemma \[aut\], there exists a prime $s\in\pi(N_i)\smallsetminus\pi({\operatorname}{Out}(N_i))$ such that $(s, |{{\operatorname}{C}_{N_i}(y)}|)=1$. Therefore $s$ cannot divide $|{{\operatorname}{C}_{N}(y)}|$ as ${{\operatorname}{C}_{N}(y)}={{\operatorname}{C}_{N_1}(y)}\times \cdots \times {{\operatorname}{C}_{N_r}(y)}$.
Since each element of prime power order in $(N \cap A) \cup (N \cap B)$ centralises some Sylow $p$-subgroup (because our hypotheses), we deduce that $\pi(N\cap A)\cup\pi(N\cap B)\subseteq \pi({{\operatorname}{C}_{N}(y)})$. Thus, this last property and Lemma \[xi\] yield $s\in\pi(G/N)$.
Note that $G/N\lessapprox {\operatorname}{Out}(N)$ and ${\operatorname}{Out}(N)\cong {\operatorname}{Out}(N_1) \text{ wr } \Sigma_{r}$, , the natural wreath product of ${\operatorname}{Out}(N_1)$ with $\Sigma_r$. As $s$ does not divide ${\ensuremath{\left| {\operatorname}{Out}(N_i) \right|}}$ for any $i$, it follows that $s\in\pi(\Sigma_r)$. Using Lemma \[xi\], we obtain that ${\ensuremath{\left| N \right|}}_s$ divides ${\ensuremath{\left| G/N \right|}}_s$, and so it divides ${\ensuremath{\left| \Sigma_r \right|}}_s$. Set ${\ensuremath{\left| N_1 \right|}}_s:=s^d$. Then ${\ensuremath{\left| N \right|}}_s=s^{dr}$ divides $s^{\frac{r-1}{s-1}}$, by Lemma \[SylowSym\], so $dr\leq \frac{r-1}{s-1}$ and necessarily $d=0$, which contradicts the fact that $s$ divides ${\ensuremath{\left| N_i \right|}}$.
Set $C:={{\operatorname}{C}_{{{\operatorname}{O}_{p
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\begin{aligned}
\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]&\le& \mathbb{E}_m[d_{\mathcal{P}}(\overline{X}_n)]\\
&\le& \mathbb{E}_m[d_{P_m}(\overline{X}_n)]+r_m-R\\
&\le& r_m-R+\sqrt{\frac{m}{n}}\sqrt{\overline{\sigma}^2+4r^2_m}.\end{aligned}$$ By setting $m=n^{\frac{1}{5}}$, we have $$\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]\le (7\pi^2R+\sqrt{\overline{\sigma}^2+16R^2}) n^{-\frac{2}{5}}$$ and $$\limsup_{n\rightarrow+\infty}\Big(n^{\frac{2}{5}}\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]\Big)\le \frac{\pi^2R}{2}+\sqrt{\overline{\sigma}^2+4R^2}.$$
Proofs in Section 3
-------------------
In this subsection, we first introduce Stein’s method, which is our main tool for proving the results presented in Section 3. Then, we prove Theorem \[t1\]. Finally, we discuss the modification of the proof of Theorem \[t1\] for obtaining Theorem \[t6\].
### Stein’s method for distributional approximations
Stein’s method was introduced by [@St72] for distributional approximations. The book by [@ChGoSh10] contains an introduction to Stein’s method and many recent advances. Here, we will explain the basic ideas in the context of normal approximation.
Let $W$ be a random variable with mean $x$ and variance $t>0$, and let $Z_{x, t}\sim N(x, t)$ be a Gaussian random variable. The Wasserstein distance between their distributions is defined as Inspired by the fact that $Y\sim N(x, t)$ if and only if for all absolutely continuous functions $f$ for which the above expectations exist, we consider the following Stein equation: A bounded solution to [1]{} is known to be Hereafter, we denote the standard Gaussian random variable $Z_{0,1}$ as $Z$. Setting $w=W$ and taking the expectation on both sides of [1]{}, we have The Wasserstein distance between the distribution of $W$ and $N(x,t)$ is then bounded by using the properties of $f_\varphi$ and by exploiting the dependence structure of $W$.
We will need to use the following properties of $f_\varphi$. The first lemma provides an upper bound for $f''_\varphi$.
\[l2\] For the solu
| 4,122
| 2,604
| 2,178
| 3,925
| null | null |
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|
6,60 (4,84)
Sex ill child, boys, *n (%)* 69 (60%)
Diagnosis, *n (%)* Acute lymphoblastic leukemia (ALL) 85 (73,9%)
Acute myeloid leukemia (AML) 8 (7%)
Chronic myeloid leukemia (CML) 2 (1,7%)
Non-Hodgkin lymphoma 20 (17,4%)
Time since diagnosis in months (*SD*; Range) 6,90 (8,05; 0--33)
Family status, *n (%)* Married/Co-habiting 100 (87%)
Divorced 8 (7%)
Single parent 3 (3%)
Stepfamily 4 (3%)
Participating Ill child N 60
Family members^1^ Sex, boys, *n (%)* 34 (56,7%)
Age, mean (*SD*) 9,90 (3,76)
Parents N 172
Sex, men, *n (%)* 73 (42%)
Age, mothers mean (*SD*)
| 4,123
| 6,482
| 2,124
| 2,138
| null | null |
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|
peliotopoulos'
date: 'November 30, 2007'
title: 'Connecting the Galactic and Cosmological Scales: Dark Energy and the Cuspy-Core Problem'
---
Introduction
============
The recent discovery of Dark Energy [@Ries1998; @Perl1999] has not only broadened our knowledge of the universe, it has brought into sharp relief the degree of our understanding of it. Only a small fraction of the mass-energy density of the universe is made up of matter that we have characterized; the rest consists of Dark Matter and Dark Energy, both of which have not been experimentally detected, and both of whose precise properties are not known. Both are needed to explain what is seen on an extremely wide range of length scales. On the galactic ($\sim 100$ kpc parsec), galactic cluster ($\sim$ 10 Mpc), and supercluster ($\sim$ 100 Mpc) scales, Dark Matter is used to explain phenomena ranging from the formation of galaxies and rotation curves, to the dynamics of galaxies and the formation of galactic clusters and superclusters. On the cosmological scale, both Dark Matter and Dark Energy are needed to explain the evolution of the universe.
While the need for Dark Matter is ubiquitous on a wide range of length scales, our understanding of how matter determines dynamics on the galactic scale is lacking. Recent measurements by WMAP [@WMAP] have validated the $\Lambda$CDM model to an unprecedented precision; such is not the case on the galactic scale, however. Current understanding of structure formation is based on [@Peebles1984], and both analytical solutions [@Gunn] and numerical simulations [@JNav; @Krav; @Moore; @PeeblesRev; @Silk] of galaxy formation have been done since then. These simulations have consistently found a density profile that has a cusp-like profile [@Moore; @JNav; @Silk], instead of the pseudoisothermal profile commonly observed. Indeed, De Blok and coworkers [@Blok-1] has explicitly shown that the density profile from [@JNav] attained through simulation does not fit the density profile observed for Low Surface Brightness g
| 4,124
| 377
| 3,860
| 4,169
| null | null |
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|
\sigma (\phi )(x)\geq \varepsilon (\phi )>0
\label{ip11}$$Then, for $\kappa, q\in{\mathbb{N}}$ and $p> 1$,
$$\left\Vert \psi _{\kappa }V_{\phi }^{\ast }(\frac{1}{\psi _{\kappa }}%
f)\right\Vert _{q,p} \leq C \psi _{\kappa }(\phi (0))\times \frac{1\vee
\left\Vert \phi \right\Vert _{1,q+2,\infty }^{2dq+1+2\kappa }}{ \varepsilon
(\phi )^{q(q+1)+1/p_\ast}} \times \left\Vert f\right\Vert _{q+1,p}.
\label{ip12}$$
**Proof**. We notice first that $$\left\vert g(\phi (x))\right\vert _{q}\leq C(1\vee \left\vert \phi
(x)\right\vert _{1,q}^{q})\sum_{\left\vert \alpha \right\vert \leq
q}\left\vert (\partial ^{\alpha }g)(\phi (x))\right\vert . \label{ip13}$$Using (\[NOT3c\]) and the above inequality we obtain$$\begin{aligned}
\left\vert \frac{1}{\psi _{\kappa }(x)}V_{\phi }(\psi _{\kappa
}f)(x)\right\vert _{q} &\leq &\frac{C}{\psi _{\kappa }(x)}\left\vert V_{\phi
}(\psi _{\kappa }f)(x)\right\vert _{q} \leq \frac{C(1\vee \left\vert \phi
(x)\right\vert _{1,q}^{q})}{\psi _{\kappa }(x)}\sum_{\left\vert \alpha
\right\vert \leq q}\left\vert (\partial ^{\alpha }(\psi _{\kappa }f))(\phi
(x))\right\vert \\
&\leq &\frac{C(1\vee \left\vert \phi (x)\right\vert _{1,q}^{q})}{\psi
_{\kappa }(x)}\times \psi _{\kappa}(\phi (x))\sum_{\left\vert \alpha
\right\vert \leq q}\left\vert (\partial ^{\alpha }f)(\phi (x))\right\vert .\end{aligned}$$And using (\[NOT3d\]) this gives (\[ip10\]).
**B**. We take now $\alpha $ with $\left\vert \alpha \right\vert \leq q$ and we write$$\begin{aligned}
\left\langle \partial ^{\alpha }(\psi _{\kappa }V_{\phi }^{\ast }(\frac{1}{%
\psi _{\kappa }}f)),g\right\rangle &=&(-1)^{|\alpha |}\left\langle \frac{f}{%
\psi _{\kappa }},V_{\phi }(\psi _{\kappa }\partial ^{\alpha }g)\right\rangle
\\
&=&(-1)^{|\alpha |}\int_{{\mathbb{R}}^{d}}\frac{f}{\psi _{\kappa }}(x)(\psi
_{\kappa }\partial ^{\alpha }g)(\phi (x))dx \\
&=&(-1)^{|\alpha |}\int_{{\mathbb{R}}^{d}}g(\phi (x))H_{\alpha }\Big(\phi ,%
\frac{f}{\psi _{\kappa }}\times \psi _{\kappa }(\phi )\Big)(x)dx.\end{aligned}$$It follows that$$\left\vert \left\langle \partial
| 4,125
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can be applied in this context. The organization of our paper is as follows: Sec. II deals with the definition of gravitational entropy and Sec. III enlists the metrics of accelerating black holes considered by us. Sec. IV provides the main analysis of our paper where we evaluate the gravitational entropy and the corresponding entropy density for these black holes. We discuss our results in Sec. V and present the conclusions in Sec. VI.
Gravitational Entropy
=====================
The entropy of a black hole can be described by the surface integral [@entropy1] $$S_{\sigma}=k_{s}\int_{\sigma}\mathbf{\Psi}.\mathbf{d\sigma},$$ where $ \sigma $ is the surface of the horizon of the black hole and the vector field $\mathbf{\Psi}$ is given by $$\mathbf{\Psi}=P \mathbf{e_{r}},$$ with $ \mathbf{e_{r}} $ as a unit radial vector. The scalar $ P $ is defined in terms of the Weyl scalar ($ W $) and the Krestchmann scalar ($ K $) in the form $$\label{P_sq}
P^2=\dfrac{W}{K}=\dfrac{C_{abcd}C^{abcd}}{R_{abcd}R^{abcd}}.$$ In order to find the gravitational entropy, we need to do our computations in a 3-space. Therefore, we consider the spatial metric which is defined as $$\label{sm}
h_{ij}=g_{ij}-\dfrac{g_{i0}g_{j0}}{g_{00}},$$ where $ g_{\mu\nu} $ is the concerned 4-dimensional space-time metric and the Latin indices denote spatial components, $i, j = 1, 2, 3$. So the infinitesimal surface element is given by $$d\sigma=\dfrac{\sqrt{h}}{\sqrt{h_{rr}}}d\theta d\phi.$$ Using Gauss’s divergence theorem, we can easily find out the entropy density [@entropy1] as $$s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|.$$
Accelerating Black holes
========================
Non-rotating black hole
-----------------------
The $C$-metric in spherical type coordinates is given by $$\label{cmetric}
ds^2=\dfrac{1}{(1-\alpha r cos\theta)^2}\left(-Qdt^2+\dfrac{dr^2}{Q}+\dfrac{r^2d\theta^2}{P}+Pr^2sin^2\theta d\phi^2\right),$$ where $ P=(1-2\alpha m cos\theta)$, and $ Q=\left(1-\dfrac{2m}{r}\right)(1-\alpha^2r^2) $. This metric represents an accelerating ma
| 4,126
| 2,379
| 4,010
| 3,905
| 2,979
| 0.775642
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|
to the electron (neutron) electric dipole moment are non-zero beginning at the three (two) loop level. Surprisingly similar to the Standard Kobayashi-Maskawa Model, our model is of milliweak character but with seemingly superweak phenomenology.'
---
6.5in
-1cm
**A Simple Charged Higgs Model of Soft CP Violation**
**without Flavor Changing Neutral Currents**
David Bowser-Chao$^{(1)}$, Darwin Chang$^{(2,3)}$, and Wai-Yee Keung$^{(1)}$
*$^{(1)}$Physics Department, University of Illinois at Chicago, IL 60607-7059, USA\
$^{(2)}$Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, R.O.C.\
$^{(3)}$Institute of Physics, Academia Sinica, Taipei, R.O.C.\
*
Submitted to [*Physical Review Letters*]{}
PACS numbers: 11.30.Er, 14.80.Er
Introduction {#introduction .unnumbered}
============
-1cm
Three decades after its surprising discovery in the kaon system[@ccft], CP violation has remained mysterious. A desire for deeper insight into its origin is the driving force behind many ongoing experiments and even the construction of new machines such as the two B Factories. While a profound understanding may yet be lacking, several mechanisms have been suggested to explain observed CP violation (i.e., $\epsilon \ne 0$) within a gauge field theory. Kobayashi and Maskawa(KM)[@km] proposed a third generation of fermions, so that CP violation would arise from the mixing of the three quark generations and is manifested by a single phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. Since then, many other mechanisms have been put forth, including new gauge interactions[@gauge], neutral Higgs exchange[@neutral], supersymmetric partners[@susy], and charged Higgs exchange[@weinberg; @branco]. However, the KM model has the distinguishing feature that its mechanism is of milliweak strength, though its phenomenology is manifestly superweak[@superweak], consistent with current CP related data. Such intricate character has also been the driving force behind the desire to find non-superweak CP violation i
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| 1,528
| 3,664
| null | null |
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|
\int_{-R}^R f({\mathbf x}_i^0+s\hat{{\mathbf u}}_i) ds + \varepsilon_i,$$ where $i$ corresponds to the data point index. The corresponding inverse problem is given the noisy measurement data $\{y_i\}_{i=1}^n$ in to reconstruct the object $f$.
Gaussian processes as functional priors {#functional priors}
---------------------------------------
A Gaussian process (GP) [@Rasmussen2006] can be viewed as a distribution over functions, where the function value in each point is treated as a Gaussian random variable. To denote that the function $f$ is modeled as a GP, we formally write $$\begin{aligned}
\label{eq:fGP}
f({\mathbf x}) \sim {\GP \left(m({\mathbf x}),\,\, k({\mathbf x},{\mathbf x}') \right)}.\end{aligned}$$ The GP is uniquely specified by the *mean function* $m({\mathbf x})=\mathbb{E}[f({\mathbf x})]$ and the *covariance function* $k({\mathbf x},{\mathbf x}')=\mathbb{E}[(f({\mathbf x})-m({\mathbf x}))(f({\mathbf x}')-m({\mathbf x}'))]$. The mean function encodes our prior belief of the value of $f$ in any point. In lack of better knowledge it is common to pick $m({\mathbf x})=0$, a choice that we will stick to also in this paper.
The covariance function on the other hand describes the covariance between two different function values $f({\mathbf x})$ and $f({\mathbf x}')$. The choice of covariance function is the most important part in the GP model, as it stipulates the properties assigned to $f$. A few different options are discussed in Section \[sec:cov\_func\].
As data is collected our belief about $f$ is updated. The aim of regression is to predict the function value $f({\mathbf x}_*)$ at an unseen test point ${\mathbf x}_*$ by conditioning on the seen data. Consider direct function measurements on the form $$y_i=f({\mathbf x}_i)+\varepsilon_i,$$ where $\varepsilon_i$ is independent and identically distributed (iid) Gaussian noise with variance $\sigma^2$, that is, $\varepsilon_i\sim{\mathcal{N}}(0,\sigma^2)$. Let the measurements be stored in the vector ${\mathbf y}$. Then the mean value and the v
| 4,128
| 4,876
| 4,019
| 3,915
| 1,078
| 0.794726
|
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|
} \in A_{\varepsilon} \} \cap \{ \hat{N} \in B_{\varepsilon} \} \subset \{ \widetilde{N}+\hat{N} \in A_{3\varepsilon} \cap B_{\varepsilon} \} ~,$$ where $\widetilde{N}+\hat{N}$ denotes the superposition of the two processes $\hat{N}$ and $\widetilde{N}$. These two processes can also be assumed independent. In this case, $\widetilde{N}+\hat{N}$ is still a PPP on $\mathbb{R}^{2}$. It follows: $$\begin{aligned}
{{\mathbb P}}( N \in A_{3\varepsilon} \cap B_{\varepsilon} ) & = & {{\mathbb P}}( \widetilde{N}+\hat{N} \in A_{3\varepsilon} \cap B_{\varepsilon} ) \\
& \geq & {{\mathbb P}}( \widetilde{N} \in A_{\varepsilon} \; , \hat{N} \in B_{\varepsilon} ) \\
& \geq & {{\mathbb P}}( \widetilde{N} \in A_{\varepsilon} ) {{\mathbb P}}( \hat{N} \in B_{\varepsilon} ) \; > \; 0 ~.\end{aligned}$$ To conclude the proof, it remains to prove that the above event implies the existence of (at least) five unbounded subtrees of $\mathcal{T}$ with different colors. Actually, there will be exactly five ones since the degree of $O$ is a.s. upperbounded by $5$. Let us denote by $Y_{k}$ the point of $N$ belonging to the ball $B(r e^{\i 2k\pi/5},\varepsilon)$. On the event $N\in A_{3\varepsilon}\cap B_{\varepsilon}$, the point $X_{k}$ is a descendant of $Y_{k}$ for any $k$. Hence, the subtrees rooted at $Y_{1},\ldots,Y_{5}$ are unbounded. Finally, it suffices to remark the $Y_{k}$’s have $O$ as common ancestor. Indeed, each $Y_{k}$ is at distance from $e^{\i 2k\pi/5}$ smaller than $\varepsilon$. So, $$\begin{aligned}
| Y_{k+1} - Y_{k} | & \geq & | re^{\i 2(k+1)\pi/5} - re^{\i 2k\pi/5} | - 2 \varepsilon \\
& \geq & 2 r \sin(\pi/5)-2\varepsilon\\ & \geq & 1.17 r - 2 \varepsilon ~,\end{aligned}$$ which is larger than the maximal distance between $Y_{k}$ and $O$, i.e. $r+\varepsilon$, for $\varepsilon$ small enough (using $r\in{{\mathbb N}}^*$).
The previous construction applied to $m\in\{3,4\}$ allows us to state that with positive probability, the origin $O$ has at least $m$ descendants from which $m$ unbounded trees arise. Now, so as to en
| 4,129
| 2,769
| 2,925
| 3,672
| null | null |
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|
an inverse length scale corresponding to the typical distance diffused by the particle in the time between two resetting events, and $D$ is the diffusion constant. The steady state radial density of the particle can also be extracted from the experimental trajectories by looking at the steady-state distribution of the distance $R=\sqrt{x^2+y^2}$ from the origin. Here too, we find excellent agreement with the theoretical result (Fig. 2b).
*Stochastic resetting with non-instantaneous returns.—* We now turn our attention to more realistic pictures of diffusion with stochastic resetting. These have just recently been considered theoretically in attempt to account for the non-instantaneous returns and waiting times that are seen in all physical systems that include resetting [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @HRS; @return1; @return2; @return3; @return4; @return5]. First, we consider a case where upon resetting HOTs are used to return the particle to the origin at a constant radial velocity $v=\sqrt{v_x^2+v_y^2}$ ([Fig. \[Fig:expt\]]{}). This case naturally arises for resetting by constant force in the over-damped limit. We find that the radial steady state density is then given by [@SM] (R)=p\_D\^[c.v.]{}\_(R)+(1-p\_D\^[c.v.]{})\_(R), \[radial-constant-velocity\] where $p_D^{c.v.}=\left(1+\frac{\pi r}{2 \alpha_0 v} \right)^{-1}$ is the steady-state probability to find the particle in the diffusive phase. $\rho_{\text{diff}}(R)=\alpha_0^2 R K_0(\alpha_0R)$ and $\rho_{\text{ret}}(R)=\frac{2\alpha_0^2}{\pi}RK_1(\alpha_0R)$ stand for the conditional probability densities of the particle’s position when in the diffusive and return phases respectively. Here $K_{n}(z)$ is once again the modified Bessel function of the second kind [@Stegun]. The result in [Eq. (\[radial-constant-velocity\])]{} is in very good agreement with experimental data as shown in [Fig. \[non-inst\]]{}a and Fig. S3.
![Steady-state distributions of diffusion with stochastic resetting and non-instantaneous return
| 4,130
| 1,602
| 3,439
| 3,940
| 1,686
| 0.786763
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|
BA ansatz is written as $$\begin{aligned}
\frac{d\sigma(\Omega_{^9\text{He}})}{dE \,d \Omega_{^8\text{He}}} \sim
\frac{v_{f}}{v_{i}}\,\sqrt{E} \,
\sum \nolimits_{MM_S} \left| \sum \nolimits_J\langle \Psi^{JMM_S}_f \left|
V \right| \Psi_i \rangle \right|^2 \label{eq:sigma}\\
%
= \frac{v_f}{v_i} \sum _{MM_S} \sum _{JJ'} \sum _{M'_lM_l}\! \rho_{JM}^{J'M}
C^{J'M'}_{l'M'_lSM_S} C^{JM}_{lM_lSM_S} \, Y^{\ast}_{lM'_l}
Y_{lM_l}\,.
\nonumber\end{aligned}$$ For the density matrix the generic symmetries are $$\rho_{JM}^{J'M'}= \left(\rho^{JM}_{J'M'}\right)^* \quad ; \qquad
\rho_{JM}^{JM'}= (-)^{M+M'}\rho^{J-M}_{J-M'} \; ,$$ and properties specific to coordinate choice (spirality representation) and setup (zero geometry) are $$\rho_{JM}^{J'M'} \sim \; \delta_{M,M'}(\delta_{M,1/2}+\delta_{M,-1/2}) \; .$$ For the density matrix parametrization we use the following model for the transition matrix. The wave function (WF) $\Psi_f$ is calculated in the $l$-dependent square well (with depth parameters $V_l$). The well radius is taken $r_0=3$ fm, which is consistent with typical R-matrix phenomenology $1.4 A^{1/3}$. The energy dependence of the velocities $v_i$, $v_f$ (in the incoming $^8$He-$d$ and outgoing $^9$He-$p$ channels) and WF $\Psi_i$ is neglected for our range of $^9$He energies. The term $ V \left| \Psi_i \right. \rangle $, describing the reaction mechanism, is approximated by radial $\theta$-function: $$V \left| \Psi_i \right. \rangle \rightarrow C_l \;r^{-1}\,\theta(r_0-r) \;
[Y_{l}(\hat{r}) \otimes \chi_{S}]_{JM} \; ,$$ where $C_l$ is (complex) coefficient defined by the reaction mechanism. For $\bigl|\rho_{J\pm 1/2}^{J'\pm 1/2}\bigr|$ denoted as $A_{l'l}$ the cross section as a function of energy $E$ and $x=\cos(\theta_{^8\text{He}})$ is $$\begin{aligned}
\frac{d\sigma(\Omega_{^9\text{He}})}{dE \;d x} \sim \frac{1}{\sqrt{E}}
\left[ \rule{0pt}{12pt} 4 A_{00} + 4 A_{11} + 3(1-2x^2+5x^4) A_{22}
\right. \nonumber \\
%
+ \left. 8 \, x \cos(\phi_{10})A_{10} + 4 \sqrt{3} \, x(5x^2 - 3)
\cos(\phi_{12}) A_{12} \right] \, .
| 4,131
| 3,377
| 3,887
| 3,722
| null | null |
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|
x{\boldmath $r$}},{{\mbox{\boldmath $b$}}_p}) =
\left\{ \begin{array}{ll}
{\mathcal N}_0\, \left(\frac{ r \, Q_{s,p}}{2}\right)^{2\left(\gamma_s +
\frac{\ln (2/r Q_{s,p})}{\kappa \,\lambda \,Y}\right)} & \mbox{$r Q_{s,p} \le 2$} \\
1 - \exp \left[-A\,\ln^2\,(B \, r \, Q_{s,p})\right] & \mbox{$r Q_{s,p} > 2$}
\end{array} \right.
\label{eq:bcgc}\end{aligned}$$ with $Y=\ln(1/x)$ and $\kappa = \chi''(\gamma_s)/\chi'(\gamma_s)$, where $\chi$ is the LO BFKL characteristic function [@bfkl]. The coefficients $A$ and $B$ are determined uniquely from the condition that $\mathcal{N}_p(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_p)$, and its derivative with respect to $rQ_s$, are continuous at $rQ_s=2$. In this model, the proton saturation scale $Q_{s,p}$ depends on the impact parameter: $$Q_{s,p}\equiv Q_{s,p}(x,{{\mbox{\boldmath $b$}}_p})=\left(\frac{x_0}{x}\right)^{\frac{\lambda}{2}}\;
\left[\exp\left(-\frac{{b_p}^2}{2B_{\rm CGC}}\right)\right]^{\frac{1}{2\gamma_s}}.
\label{newqs}$$ The parameter $B_{\rm CGC}$ was adjusted to give a good description of the $t$-dependence of exclusive $J/\psi$ photoproduction. The factors $\mathcal{N}_0$, $x_0$, $\lambda$ and $\gamma_s$ were taken to be free. Recently the parameters of this model have been updated in Ref. [@amir] (considering the recently released high precision combined HERA data), giving $\gamma_s = 0.6599$, $B_{CGC} = 5.5$ GeV$^{-2}$, $\mathcal{N}_0 = 0.3358$, $x_0 = 0.00105 \times 10^{-5}$ and $\lambda = 0.2063$. As demonstrate in Ref. [@armesto_amir], this phenomenological dipole describes quite well the HERA data for the exclusive $\rho$ and $J/\Psi$ production. Moreover, the results from Refs. [@bruno1; @bruno2] demonstrated that this model allows to describe the recent LHC data for the exclusive vector meson photoproduction in $pp$ and $pPb$ collisions. Another motivation to use the bCGC model, is that this model is based on the CGC physics, which was used in Ref. [@bruno_doublegama] to estimate the double vector meson production in $\gamma \gamma$ i
| 4,132
| 3,345
| 3,831
| 3,776
| 2,902
| 0.776217
|
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|
apse in the bilayer.[]{data-label="fig:phase_diag"}](glow_compare_phase-diagr.pdf){width="\linewidth"}
We determine the density instabilities of the bilayer system by analyzing the divergences of the static response function matrix $\chi_{ij} ({{\mathbf q}},0)$. Specifically, we search for zeros of the largest inverse eigenvalue, $$\chi_+^{-1} = \frac{1}{\Pi} - v_{11} [1-G_{11} ] + |v_{12} [1-G_{12}
] | \; .
\label{eq:eigen}$$ A zero of $\chi_+^{-1} ({{\mathbf q}},0)$ at a critical wave vector ${{\mathbf q}}_c$ signals an instability towards the formation of a density wave with period set by ${{\mathbf q_c}}$. If the instability occurs for a specific direction $\phi$, then the density-wave phase corresponds to a one-dimensional modulation (or stripe phase) of period $2\pi/q_c$ oriented along $\phi$. In this way, we obtain the phase diagram plotted in Fig. \[fig:phase\_diag\] for $k_F d=2$.
For tilt angles $\theta<\theta_c \simeq 0.75$, we find a stripe phase along $\phi=\pi/2$ that is of a similar nature to the one found in a single layer (dashed line of Fig. \[fig:phase\_diag\]). In particular, it is driven by strong intralayer correlations induced by the repulsive part of $v_{11}$, as evidenced by the relative insensitivity of $q_c$ to the bilayer geometry and $\theta$ (see Fig. \[fig:schematic\]). However, the presence of the second layer can decrease the value of the critical interaction strength $U_c$ for stripe formation, as one might expect from the form of Eq. . The attractive part of $v_{12}({{\mathbf q}})$ also ensures that the density waves along $\phi=\pi/2$ in each layer are in phase. Similar results were found using the conserving Hartree-Fock (HF) approximation [@babadi2011; @block2012], but for much smaller values of $U_c$, like in the single-layer case. The shift of $U_c$ due to the other layer is relatively small for distance $k_F d=2$ (see Fig. \[fig:phase\_diag\] at small values of $\theta$), but it can become substantial for smaller $k_F d$ since Eq. depends exponentially on the
| 4,133
| 2,412
| 4,190
| 3,998
| 3,351
| 0.772972
|
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|
0.7
Hb (gr/dL) 11.6 ± 1.3 11.9 ± 1.4 0.3
Cholesterol (mg/dL) 158.5 ± 47.3 165.2 ± 44.8 0.5
LDL (mg/dL) 85.2 ± 35.1 92.5 ± 37.8 0.4
HDL (mg/dL) 38.5 ± 10.6 39.02 ± 9.4 0.8
Triglycerides (mg/dL) 181.7 ± 111.6 168.6 ± 76.7 0.6
oxLDL (ng/mL) 55.9 45.7 0.1
OxLDL/LDL (ng/mg) 58.9 \* 44.6 0.02
Albumin (gr/dL) 43.8 50.2 0.3
Glucose (mg/dL) 95.03 ± 20.9 96.6 ± 22.9 0.7
Insulin (μU/mL) 44.7 49.9 0.4
HOMA-IR (mmol/L) 44.9 49.8 0.5
hsCRP (mg/L) 12.06 ± 5.4\* 7.4 ± 5.7
| 4,134
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| 1,011
| 3,610
| null | null |
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|
ots w_kS' {\Rightarrow}w_1w_2\cdots w_k$$ where the subwords $w_i$ are derived from $S$ as in $G$.
As regards closure under intersection with regular sets and under inverse homomorphisms, the constructions to show closure of $\mathbf{CF}$ cannot be extended, since they do not keep the capacity bound. We suspect that $\mathbf{CF}_{{\mathit{cb}}}$ is not closed under any of these operations.
Control by Petri nets with place capacities {#sec:PNC}
===========================================
We will first establish the connection between context-free Petri nets with place capacities and capacity-bounded grammars. Later we will investigate the generative power of various extended context-free Petri nets with place capacities.
The proof for the equivalence between context-free grammars and grammars controlled by cf Petri nets can be immediately transferred to context-free grammars and Petri nets with capacities:
\[thm:CapacityPetriNetGrammar\] Grammars controlled by context-free Petri nets with place capacity functions generate the family of capacity-bounded context-free languages.
Let us now turn to grammars controlled by extended cf Petri nets with capacities. We will first study the generative power of capacity-bounded matrix and vector grammars, which are closely related to these Petri net grammars.
\[thm:matrixGrammarBounds\] ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf MAT}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf sMAT}}^{[{\lambda}]}_{{\mathit{cb}}}$.
We give the proof of ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$. The other equalities can be shown in an analogous way. Since ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, it suffices to prove ${{\bf V}}_{{\mathit{fin}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$ and ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$. The first inclusion is obvious because any vector grammar of finite index $k$ is equivalent to th
| 4,135
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|
hat{\psi},V}}\left(i,j\right)\right]_{i\in{{\mathcal{M}_{r}},j\in\mathcal{I}_t}}$; $\mathcal{I}_r:=\mathcal{I}_r-\left\{J\right\};$
$J:=\arg\max_{j\in \mathcal{I}_r}{\mathbf{h}}_j\left({\mathbf{I}}_{N_t}+\frac{\rho}{N_t}{\widetilde{\mathbf{H}}_{\widehat{\psi},V}}^H{\widetilde{\mathbf{H}}_{\widehat{\psi},V}}\right)^{-1}{\mathbf{h}}_j^H$;
${\mathcal{M}_{r}}:={\mathcal{M}_{r}}+\left\{J\right\}$; ${\widetilde{\mathbf{H}}_{\widehat{\psi},V}}:=\left[{\mathbf{H}_{\widehat{\psi},V}}\left(i,j\right)\right]_{i\in{{\mathcal{M}_{r}},j\in\mathcal{I}_t}}$; $\mathcal{I}_r:=\mathcal{I}_r-\left\{J\right\};$
${\mathbf{h}}_j:=j$-th column of ${\widetilde{\mathbf{H}}_{\widehat{\psi},V}}$, $\forall j\in\mathcal{I}_t$;
$J:=\arg\max_{j\in\mathcal{I}_r}{\mathbf{h}}_j^H{\mathbf{h}}_j$;
${\mathcal{M}_{t}}:=\left\{J\right\}$; ${\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}:=\left[{\widetilde{\mathbf{H}}_{\widehat{\psi},V}}\left(i,j\right)\right]_{i\in{{\mathcal{M}_{r}},j\in{\mathcal{M}_{t}}}}$; $\mathcal{I}_t:=\mathcal{I}_t-\left\{J\right\};$
$$\begin{aligned}
\label{}
& J:=\arg\max_{j\in \mathcal{I}_t}\notag\\
&{\mathbf{h}}_j^H\left({\mathbf{I}}_{L_r}-\frac{\rho}{L_t}{\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}\left({\mathbf{I}}_{l-1}+\frac{\rho}{L_t}{\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}^H{\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}\right)^{-1}{\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}^H\right){\mathbf{h}}_j;\end{aligned}$$
${\mathcal{M}_{t}}:={\mathcal{M}_{t}}+\left\{J\right\}$; ${\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}:=\left[{\widetilde{\mathbf{H}}_{\widehat{\psi},V}}\left(i,j\right)\right]_{i\in{{\mathcal{M}_{r}},j\in{\mathcal{M}_{t}}}}$; $\mathcal{I}_t:=\mathcal{I}_t-\left\{J\right\};$
To reduce the complexity of beam selection, we utilize the technique of separate transmit and receive antenna selection [@Sanayei_04_CMAHTRAS] and the incremental successive selection algorithm (ISSA) [@Alkhansari_04_fastassims]. Again due to the sparsity of mmWave channels, th
| 4,136
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|
ly given by $$\label{eq:gammadef}
\begin{aligned}
ds^2 & = ds^2_{AdS}+ \sum_{i= 1\dots 3} ( dr_i^2 + G r_i^2 d\phi_i^2) + G r_1^2 r_2^2 r_3^2 \Big( \sum_{i= 1\dots 3} \nu_i dr_i \Big)^2 \ ,\\
B & = G ( r_1^2 r_2^2 \nu_3 d\phi_1 \wedge d\phi_2 + r_1^2 r_3^2 \nu_2 d\phi_3 \wedge d\phi_1 + r_2^2 r_3^2 \nu_1 d\phi_2 \wedge d\phi_3 ) \ , \\
\end{aligned}$$ with $$G^{-1}\equiv \lambda= 1+ r_1^2 r_2^2 \nu_3^2 + r_3^2 r_1^2 \nu_2^2 + r_2^2 r_3^2 \nu_1^2 \ ,$$ where the parameters $\nu_i$ are related to the $\gamma_i$ of the field theory by a factor of the $AdS$ radius [@Frolov:2005dj], which we suppress throughout.
We would like to interpret this in terms of the centrally-extended (non-)abelian T-duality introduced in section \[sec:centralext\]. To do so we find it expedient to make a basis transformation of the Cartan generators; let us assume $\nu_3 \neq 0$ and define $$\tilde{h}_1 = h_1 - \frac{\nu_1}{\nu_3} h_3 \ , \quad \tilde{h}_2 = h_2 - \frac{\nu_2}{\nu_3} h_3 \ , \quad \tilde{h}_3 = h_3+ \frac{\nu_1}{\nu_3} h_3 + \frac{\nu_2}{\nu_3} h_3 \ .$$ In this basis the $r$-matrix simply reads $$r= \frac{\nu_3}{4} \tilde{h}_1 \wedge \tilde{h}_2 \ .$$ We also introduce a new set of angles such that $\tilde{h}_i \tilde{\phi}_i = h_i \phi_i$ (where the sum over $i$ is implicit). Written in this way it is clear that we should consider a centrally-extended (non-)abelian T-duality along the $\tilde{h}_1$ and $\tilde{h}_2$ directions. To proceed we defined a slightly exotic set of frame fields for the $S^5$, adapted to the dualisation as described $$\begin{aligned}
e^\alpha &= d \alpha \ , \quad e^\xi = \sin \alpha d\xi \ , \quad
e^1 = \frac{1}{\varphi \sqrt{\lambda-1} } \left( r_1^2 \varphi^2 d\phi_1 - r_2^2 r_3^2 \nu_1 \nu_2 d\phi_2 - r_2^2 r_3^2 \nu_1 \nu_3 d\phi_3 \right) \ ,
\\
e^2 & = \frac{1}{\varphi } \left( r_2^2 \nu_3 d\phi_2 - r_3^2 \nu_2 d\phi_3 \right) \ , \quad
e^3 = \frac{r_1 r_2 r_3}{\sqrt{\lambda -1} }\sum_{i} \nu_i d\phi_i \ ,
\end{aligned}$$ where $\varphi=(r_2^2 \nu_3^2 + r_3^3 \nu_2^2)^{\frac{1}{2}}$. Thoug
| 4,137
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xpressions – are approximated pseudo-spectrally using standard dealiasing of the nonlinear terms and with resolutions varying from $128^3$ in the low-enstrophy cases to $512^3$ in the high-enstrophy cases, which necessitated a massively parallel implementation using the Message Passing Interface (MPI). As regards the computation of the Sobolev $H^2$ gradients, cf. , we set $\ell_1 = 0$, whereas the second parameter $\ell_2$ was adjusted during the optimization iterations and was chosen so that $\ell_2 \in [\ell_{\min},
\ell_{\max}]$, where $\ell_{\min}$ is the length scale associated with the spatial resolution $N$ used for computations and $\ell_{\max}$ is the characteristic length scale of the domain $\Omega$, that is, $\ell_{\min} \sim \O( 1/N) $ and $\ell_{\max} \sim \O(1)$. We remark that, given the equivalence of the inner products corresponding to different values of $\ell_1$ and $\ell_2$ (as long as $\ell_2 \neq 0$), these choices do not affect the maximizers found, but only how rapidly they are approached by iterations . For further details concerning the computational approach we refer the reader to the dissertation by [@a14]. As was the case in the analogous 2D problem studied by @ap13a, the largest instantaneous growth of enstrophy is produced by the states with vortex cells staggered in all planes, cf. case (\[c1\]) in Table \[tab:E0\]. Therefore, in our analysis we will focus exclusively on this branch of extreme vortex states which has been computed for $\E_0 \in [10^{-3},2\times10^2]$.
The optimal instantaneous rate of growth of enstrophy $\R_{\E_0} =
\R({\widetilde{\mathbf{u}}_{\E_0}})$ and the energy of the optimal states $\K({\widetilde{\mathbf{u}}_{\E_0}})$ are shown as functions of $\E_0$ for small $\E_0$ in figures \[fig:RvsE0\_FixE\_small\](a) and \[fig:RvsE0\_FixE\_small\](b), respectively. As indicated by the asymptotic form of $\R$ in and the Poincaré limit [$\K_0=\E_0/(2\pi)^2$]{}, both of which are marked in these figures, the behaviour of $\R_{\E_0}$ and $\K({\widetilde{\mathbf{
| 4,138
| 2,118
| 1,671
| 4,205
| 3,843
| 0.769726
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|
rox - \alpha /2R^4$ at large distances and differ in the short-range limit due to the “cut-off” parameters or functions which contain parameters estimated on some reasonable assumptions.
Effects of the NN interaction on the scattering process can be investigated by solving the Schrödinger equation for the Hamiltonian (\[el-ham\]) with inclusion of the potential (\[VZZ\]). However, the solution simplifies remarkably if one considers that (\[VZZ\]) depends on $R$ alone and so $V_{NN}$ can be removed from (\[el-ham\]) by a phase transformation. The transition matrix $\mathcal{R}_{i {\mathbf{k}}}({\boldsymbol{\eta}})$ that takes the internuclear interaction into account can then be expressed as [@mcd70] $$\begin{array}{c}
\mathcal{R}_{i {\mathbf{k}}}({\boldsymbol{\eta}}) =\frac{1}{2 \pi} \int {\mathrm{d}{\boldsymbol{\rho}}\;}e^{i {\boldsymbol{\eta}}\cdot {\boldsymbol{\rho}}}
a_{i {\mathbf{k}}}({\boldsymbol{\rho}})
\label{rifn}
\end{array}$$ with $ a_{i {\mathbf{k}}}({\boldsymbol{\rho}})= e^{i \delta(\rho)} \mathcal{A}_{i {\mathbf{k}}}({\boldsymbol{\rho}})$, where $\mathcal{A}_{i {\mathbf{k}}}({\boldsymbol{\rho}})$ is the transition amplitude calculated without the internuclear interaction, and the phase due to (\[VZZ\]) is expressed as $$ \delta(\rho) =- \int_{-\infty}^{+\infty} \mathrm{d} t V_{NN}(R(t)).
\label{nnphase}$$
Results {#sec:res.}
=======
Doubly differential cross sections for the ionization of Li (2s) and Li(2p)
---------------------------------------------------------------------------
In Figures (\[fig1\])-(\[fig6\]) we compare our CDW-EIS results of $\mathrm{d} \sigma^{2} / \mathrm{d} E_e \mathrm{d}\eta$ for proton and $O^{8+}$ impact on Li(2s) and Li(2p) with measurements of LaForge $\textit{et al}$ [@LaForge2013] and Hubele *et al* [@Hubele2013] for the electron ejection energies of 2, 10 and 20 eV. The experimental data of [@LaForge2013] and [@Hubele2013] are not on the absolute scale, only the relative normalisation was fixed for different $E_e$ and for Li(2p) relative to Li(2s). F
| 4,139
| 2,352
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| 4,026
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$\partial_\nu$ is the weak normal derivative. Thus the domain of $A(t)$ is the set $$D(A(t))=\Big\{ u\in H^1(\Omega_{ext}) {\, \vert \,}{\mathop{}\!\mathbin\bigtriangleup}u\in L^2(\Omega_{ext}), \partial_\nu u(t)+\beta(t,\cdot){{u}{_{|\Gamma}}}=0 \Big\}$$ and for $u\in D(A(t)), A(t)u:=-{\mathop{}\!\mathbin\bigtriangleup}u.$ Thus similarly as in [@Ar-Mo15 Section 5] one obtains that ${\mathfrak{a}}$ satisfies (\[eq:continuity-nonaut\])-(\[square property\]) with $\gamma:=r_0+1/2$ and $\omega(t)=t^\alpha$ where $r_0\in(0,1/2)$ such that $r_0+1/2<2\alpha.$ We note that [@Ar-Mo15 Section 5] the authors considered the Robin Laplacian on the bounded Lipschitz domain $\Omega.$ The main ingredient used there is that the trace operators are bounded from $H^{s}(\Omega)$ with value in $H^{s-1/2}(\Gamma,\sigma)$ for all $1/2<s<3/4.$ This boundary trace embedding theorem holds also for unbounded Lipschitz domain, and thus for $\Omega_{ext}$, see [@Mclean Theorem 3.38] or [@Cos Lemma 3.6].
Thus applying [@Ar-Mo15 Theorem 4.1] and Theorem \[main result\] we obtain that the non-autonomous Cauchy problem
$$\label{RobinLpalacian}
\left\{
\begin{aligned}
\dot {u}(t) - {\mathop{}\!\mathbin\bigtriangleup}u(t)& = 0, \ u(0)=x\in H^1(\Omega_{ext })
\\ \partial_\nu u(t)+\beta(t,\cdot){u}&=0 \ \text{ on } \Gamma
\end{aligned} \right.$$
has $L^2$-maximal regularity in $L^2(\Omega_{ext})$ and its solution is governed by an evolution family ${{U}}(\cdot,\cdot)$ that is norm continuous on each space $V, L^2(\Omega_{ext})$ and $V'.$
Non-autonomous Schrödinger operators
------------------------------------
Let $m_0, m_1\in L_{Loc}^1({\mathbb{R}}^d)$ and $m:[0,T]\times{\mathbb{R}}^d{\longrightarrow}{\mathbb{R}}$ be a measurable function such that there exist positive constants $\alpha_1,\alpha_2$ and $\kappa$ such that $$\alpha_1 m_0(\xi)\leq m(t,\xi)\leq \alpha_2 m_0(\xi),
\quad \text{ }\
\quad {\, \vert \,}m(t,\xi)-m(s,\xi){\, \vert \,}\leq \kappa|t-s|m_1(\xi)$$ for almost every $\xi\in {\mathbb{R}}^d$ and every $t,s\in [0,T].$
| 4,140
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| 1,125
| 4,061
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|
sc surfaces in vertical directions [see e.g., @Abramowicz:1988aa; @Watarai:2000aa; @Ohsuga:2005aa; @Jiang:2014aa; @Sc-adowski:2016aa]. The first term $2 L_{\mathrm{E}}$ corresponds to the luminosity from the outer standard disc in $r > r_{\mathrm{tr}}$ (see Fig. \[fig:acc\_whole\]a), given approximately by the energy generation rate due to the gravitational energy released by $r_{\mathrm{tr}}$, $G\dot{M}/r_{\mathrm{tr}} \sim L_{\mathrm{E}}$ [e.g., @Begelman:1978aa; @Kato:1998aa].
The spectrum of the BH radiation is simply assumed to be the power-law with $L_\nu\propto \nu^{-1.5}$ for $h\nu>13.6{\,\mathrm{eV}}$, where $L=\int_{h\nu>13.6{\mathrm{eV}}}L_\nu d\nu$, as often assumed in the literature [e.g., @Park:2011aa; @Park:2012aa; @Milosavljevic:2009ab]. [@Park:2011aa] have shown that the qualitative properties of accretion do not depend on the spectral shape.
### Directional dependence {#sec:direction_dep}
We inject ionizing photons at the inner boundary with the directional dependence described below. Specifically, we multiply the anisotropy factor $\mathcal{F}(\theta)$ normalized as $\int
\mathcal{F}(\theta)d\Omega=4\pi$ with an isotropic radiation flux $L/4\pi R_{\mathrm{in}}^2$ at the inner boundary $R_{\mathrm{in}}$. With this definition, $\mathcal{F}(\theta) = 1$ represents the isotropic radiation (Fig. \[fig:dirdep\]). We use the latitudinal angle $\theta$ defined as the angle measured from the equatorial plane for our convenience.
Motivated by the expected disc structure described in Section \[sec:acc\_disc\] (also see Fig. \[fig:acc\_whole\]a), we model $\mathcal{F}(\theta)$ as $$\begin{aligned}
\mathcal{F}(\theta) &= C\, f_{\mathrm{disc}}(\theta)\, f_{\mathrm{shadow}}(\theta)\,,
\label{eq:5}\end{aligned}$$ where $C$ is the normalization factor. In this expression, the inner anisotropy factor $f_{\mathrm{disc}}$ that represents the directional dependences of the radiation emitted from the inner part of the disc is multiplied by the outer one $f_{\mathrm{shadow}}$ to take into account the outer s
| 4,141
| 3,699
| 4,293
| 3,931
| 3,701
| 0.770596
|
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|
tion of motion for test particles can be accounted for in $T_{\mu\nu}$, and we still have $R=4\Lambda_{DE}G/c^2+8\pi
GT/c^4$ in Eq. $(\ref{extendL})$. For massless particles, $v^\nu\nabla_\nu \left(\mathfrak{R}[4+8\pi
T/\Lambda_{DE}c^2]v^\mu\right)=0$ instead. With the reparametization $dt \to \mathfrak{R} dt$, the extended GEOM for massless test particles reduces to the GEOM. Our extended GEOM does not affect the motion of photons.
Because the geodesic Lagrangian is extended covariantly, Eq. $(\ref{extendL})$ explicitly satisfies the strong equivalence principal. For $T_{\mu\nu}$, we may still take $T_{\mu\nu} = (\rho+p/c^2)v_\mu v_\nu - p g_{\mu\nu}$ for an inviscid fluid with density $\rho$ and pressure $p$ [@ADS]. While for the GEOM $T^{\hbox{\scriptsize{geo-Dust}}}_{\mu\nu}=\rho v_\mu v_\nu$ for dust, for the extended GEOM the pressure does not vanish [@ADS]; it is a functional of $\rho$ and $\mathfrak{R}$. Nevertheless, in the nonrelativistic limit $p<<\rho c^2$, and $T_{\mu\nu}^{\hbox{\scriptsize{Ext-Dust}}}\approx\rho v_\mu
v_\nu$ still [@ADS]. Moreover, because $v^\mu v_\mu =c^2$ for the extended GEOM, the first law of thermodynamics still holds for the fluid, and *the standard thermodynamical analysis of the evolution of the universe under the extended GEOM follows much in the same way as before.*
All dynamical effects of extension can be interpreted as the rest energy gained or lost by the test particle due to variations in the local curvature. For these effects not to have already been seen, $\mathfrak{D}(4+8\pi T/\Lambda_{DE}c^2)$ must change very slowly at current experimental limits. As such, we take $
\mathfrak{D}(x) =
\chi(\alpha_\Lambda)\int_x^\infty(1+s^{1+\alpha_\Lambda})^{-1}ds$, where $\alpha_\Lambda \ge 1$ and $\chi(\alpha_\Lambda)$ is set by $\mathfrak{D}(0)=1$. This $\mathfrak{D}(x)$ was chosen for three reasons. First, there is only one free parameter, $\alpha_\Lambda$, to determine. Second, it ensures that the effects of the additional terms in the extended GEOM will not alre
| 4,142
| 4,743
| 3,156
| 3,534
| 3,318
| 0.773186
|
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|
1 (N159) 34.7 3.08 $\times$ 10$^{3}$ (8.67 $\times$ 10$^{3}$)$^a$ 0.37 2.38 6.9 1.2 $\times$ 10$^{5}$ 165
1a (N159E) 27.7 8.01 $\times$ 10$^{2}$ (3.40 $\times$ 10$^{3}$) 0.27 1.94 2.2 3.5 $\times$ 10$^{3}$ 55
1b 34.7 1.71 $\times$ 10$^{2}$ (6.03 $\times$ 10$^{2}$) 0.23 2.26 8.2 1.0 $\times$ 10$^{5}$ 79
1c (N159W) 31.0 7.50 $\times$ 10$^{2}$ (2.92 $\times$ 10$^{3}$) 0.41 2.32 5.1 3.9 $\times$ 10$^{4}$ 67
2 (N160) 33.7 1.15 $\times$ 10$^{3}$ (4.32 $\times$ 10$^{3}$) 0.31 2.27 7.1 1.1 $\times$ 10$^{5}$ 102
3 (N158) 33.6 7.21 $\times$ 10$^{2}$ (2.17 $\times$ 10$^{3}$) 0.32 2.31 6.5 9.4 $\times$ 10$^{4}$ 130
4 26.8 5.29 $\times$ 10$^{2}$ (1.44 $\times$ 10$^{3}$) 0.63 2.50 2.8 2.2 $\times$ 10$^{2}$ 199
5 34.7 5.14 $\times$ 10$^{2}$ (1.36 $\times$ 10$^{3}$) 0.39 2.39 4.6 1.0 $\times$ 10$^{7}$ 195
6 (N159S) 24.8 6.44 $\times$ 10$^{2}$ (1.92 $\times$ 10$^{3}$) 0.83 2.42 3.8 4.8 212
7 28.7 3.01 $\times$ 10$^{2}$ (8.65 $\times$ 10$^{2}$) 0.60 2.50 3.4 3.1 $\times$ 10$^{3}$ 99
8 27.2 2.65 $\times$ 10$^{2}$ (9.18 $\times$ 10$^{2}$) 0.54 2.24 2.7 3.9 $\times$ 10$^{3}$ 131
9 26.7 1.66 $\times$ 10$^{2}$ (4.91 $\times$ 10$^{2}$) 0.53 2.32 2.8 2.1 $\times$ 10$^{3}$ 111
10 29.7 1.34 $\times$ 10$^{2}$ (4.04 $\times$ 10$^{2}$) 0.44 2.36 4.4 1.3 $\ti
| 4,143
| 5,786
| 1,027
| 3,604
| null | null |
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|
Eq.(74) in the form $$F(u,t)=F_{s}(u)e^{-\phi(t)}\hspace{0.2cm},$$
where $F_{s}(u)$ is the steady state solution obtained in the earlier section, i.e., it satisfies $$\alpha u \frac{\partial F_{s}}{\partial u}+KT\frac{\partial^{2}F_{s}}
{\partial u^{2}}=0\hspace{0.2cm}\hspace{0.2cm}.$$
We require further $$\left.{\cal L}t\right._{t\rightarrow\infty}\phi (t)=0\hspace{0.2cm}.$$
Substituting (77) in Eq.(74) it may be shown that the ‘space’ and the time part is separable. We obtain, $$-\frac{1}{\Gamma}\frac{\partial \phi}{\partial t} e^{\frac{\gamma}{2} t}=\frac{C}
{F_{s}}\left[\lambda u\frac{\partial F_{s}}{\partial u}+\frac{aKT}{\Gamma}
\frac{\partial^{2}F_{s}}{\partial u^{2}} \right]={\rm constant}=D({\rm say})
\hspace{0.2cm},$$
where we have made use of the Eq.(78) and also $$\begin{aligned}
\Delta=Ce^{-\frac{\gamma}{2} t}\hspace{0.2cm}
{\rm with} \hspace{0.2cm}C=\frac{r \gamma_{\rm neq}}{\gamma_{\rm eq}+\gamma_{
\rm neq}[g'(0)]^{2}}\hspace{0.2cm}.\end{aligned}$$
On integration over time we obtain from Eq.(80), the solution $$\phi (t)=2 D\frac{\Gamma}{\gamma}e^{-\frac{\gamma}{2} t}$$
where $D$ is determined by the initial condition.
The time-dependent solution of Eq.(71) therefore reads as $$F(u,t)=F_{s}(u)\exp\left[-\frac{2D\Gamma}{\gamma}
e^{-\frac{\gamma}{2} t}\right]
\hspace{0.2cm}.$$
Thus the corresponding probability distribution is given by, $$\begin{aligned}
p(x,v,t)=N\left[\left(\frac{\pi KT}{2\alpha}\right)^{\frac{1}{2}}+\int_{0}^
{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right]\nonumber\\
\nonumber\\
\exp\left[-\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]
e^{-\frac{2D\Gamma}{\gamma}\left[\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm}.\end{aligned}$$
To determine $D$ we now demand that just at the moment the system (and the nonthermal bath) is subjected to external excitation at $t=0$ and $x\rightarrow
-\infty$ the distribution (75) must coincide with the usual Boltzmann distribution where the energy term in the Boltzmann factor in addition to usual kinetic and potential ter
| 4,144
| 4,325
| 3,419
| 3,755
| null | null |
github_plus_top10pct_by_avg
|
usion\] contains concluding remarks. Extra results, proofs and a discussion of another version of the bootstrap, are relegated to the Appendices.
Notation
--------
Let $Z=(X,Y)\sim P$ where $Y\in\mathbb{R}$ and $X\in \mathbb{R}^d$. We write $X = (X(1),\ldots, X(d))$ to denote the components of the vector $X$. Define $\Sigma = \mathbb{E}[X X^\top]$ and $\alpha = (\alpha(1),\ldots,\alpha(d))$ where $\alpha(j) = \mathbb{E}[Y X(j)]$. Let $\sigma = {\rm vec}(\Sigma)$ and $\psi \equiv \psi(P) = (\sigma,\alpha)$. The regression function is $\mu(x) = \mathbb{E}[Y|X=x]$. We use $\nu$ to denote Lebesgue measure. We write $a_n \preceq b_n$ to mean that there exists a constant $C>0$ such that $a_n \leq C b_n$ for all large $n$. For a non-empty subset $S\subset \{1,\ldots, d\}$ of the covariates $X_S$ or $X(S)$ denotes the corresponding elements of $X$: $(X(j):\ j\in S)$ Similarly, $\Sigma_S = \mathbb{E}[X_S X_S^\top]$ and $\alpha_S = \mathbb{E}[Y X_S]$. We write $\Omega = \Sigma^{-1}$ and $\omega = {\rm vec}(\Omega)$ where ${\rm vec}$ is the operator that stacks a matrix into one large vector. Also, ${\rm vech}$ is the half-vectorization operator that takes a symmetric matrix and stacks the elements on and below the diagonal into a matrix. $A\otimes B$ denotes the Kronecker product of matrices. The commutation matrix $K_{m,n}$ is the $mn \times mn$ matrix defined by $K_{m,n} {\rm vec}(A) = {\rm vec}(A^\top)$. For any $k\times k$ matrix $A$. $\mathrm{vech}(A)$ denotes the column vector of dimension $k(k+1)/2$ obtained by vectorizing only the lower triangular part of $k\times k$ matrix $A$.
Main Results {#section::splitting}
============
We now describe how to construct estimators of the random parameters defined earlier. Recall that we rely on data splitting: we randomly split the $2n$ data into two halves ${\cal D}_{1,n}$ and ${\cal D}_{2,n}$. Then, for a given choice of the model selection and estimation rule $w_n$, we use ${\cal D}_{1,n}$ to select a non-empty set of variables ${\widehat{S}}\subset \{
1,\ldots,d\}$ whe
| 4,145
| 2,522
| 1,964
| 3,931
| 1,963
| 0.784114
|
github_plus_top10pct_by_avg
|
. Out of 27,678 IRs in MG1655 we were able to identify 914 IRs that appear in at least 10 species. These orthologs reside in 234 NC regions and were further analyzed. For each of these 234 regions, we computed a conservation score (see [fig. 4*A* and *B*](#evy044-F4){ref-type="fig"} and Materials and Methods for full details).
{ref-type="fig"}.](evy044f4){#evy044-F4}
Next, we aimed to compare the 234 conservation scores computed from the real NC regions to the expectation under null conditions, in which IRs evolve similarly to any other NC sequence. To this end, we generated 1,000 corresponding simulated data sets for each of the 234 real NC regions. For each such NC region, the empirical distribution of the 1,000 conservation scores served as a null distribution to which we compared the conservation score of the real region. An example of one NC region, which is more conserved than the null distribution, is shown in [figure 4*C*](#evy044-F4){ref-type="fig"}.
Out of the 234 NC regions, 145 were found to have a significantly higher conservation score than the expectation under null conditions (empirical *P* \< 0.05, [fig. 5](#evy044-F5){ref-type="fig"}). Generally, the distribution of the 234 empiric
| 4,146
| 1,846
| 3,902
| 3,678
| null | null |
github_plus_top10pct_by_avg
|
s controlled by these Petri nets (for details, see [@das:tur; @tur]).
Let $G=(V, \Sigma, S, R)$ be a context-free grammar with its corresponding cf Petri net $$N=(P, T, F, \phi, \beta, \gamma, \iota).$$ Let $T_1, T_2, \ldots, T_n$ be a partition of $T$.
1\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint chains such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ An *$h$-Petri net* is a system $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if $(x,y)\in F$ and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$ and $\mu_0(p)=0$ if $p\in Q$; $\tau$ is the final marking where $\tau(p)=0$ for all $p\in P\cup Q$.
2\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint cycles such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ A *$c$-Petri net* is a system $N_c=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where $Q=\bigcup_{\rho\in\Pi}P_\rho$ and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$, and $\mu_0(p_{i,1})=1$, $\mu_0(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$; $\tau$ is the final marking where $\tau(p)=0$ if $p\in P$, and $\tau(p_{i,1})=1$, $\tau(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$.
3\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of cycles such that $T_{\rho_i}
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ore generally, for any finite linearly ordered set $S$ with $n$ elements, we have a product functor $m_S:{{\mathcal C}}^S \to {{\mathcal C}}$, where ${{\mathcal C}}^S = {{\mathcal C}}^n$ with multiples in the product labeled by elements of $S$. Then for any $[n],[n'] \in \Lambda$, and any $f:[n'] \to [n]$, we can define a functor $f_!:{{\mathcal C}}^{V([n'])} \to {{\mathcal C}}^{V([n])}$ by the same formula as in : $$\label{trans.f}
f_! = \prod_{v \in V([n])}m_{f^{-1}(v)}:{{\mathcal C}}^{V([n'])} = \prod_{v \in
V([n])}{{\mathcal C}}^{f^{-1}(v)} \to {{\mathcal C}}^{V([n])}.$$ The natural associativity isomorphism for the product on ${{\mathcal C}}$ induces natural isomorphisms $(f \circ f')_! \cong f_! \circ f'_!$, and one checks easily that they satisfy natural compatibility conditions. All in all, setting $[n] \mapsto {{\mathcal C}}^{V([n])}$, $f
\mapsto f_!$ defines a weak functor (a.k.a. lax functor, a.k.a.$2$-functor, a.k.a. pseudofunctor in the original terminology of Grothendieck) from $\Lambda$ to the category of categories. Informally, we have a “cyclic category”.
To work with weak functors, it is convenient to follow Grothendieck’s approach in [@SGA]. Namely, instead of considering a weak functor directly, we define a [*category*]{} ${{\mathcal C}}_\#$ in the following way: its objects are pairs $\langle [n],M_n
\rangle$ of an object $[n]$ of $\Lambda$ and an object $M_n \in
{{\mathcal C}}^n$, and morphisms from $\langle [n'],M_{n'} \rangle$ to $\langle
[n],M_n\rangle$ are pairs $\langle f,\iota_f\rangle$ of a map $f:[n'] \to [n]$ and a bimodule map $\iota_f:f_!(M_{n'}) \to M_n$. A map $\langle f, \iota_f \rangle$ is called [*cocartesian*]{} if $\iota_f$ is an isomorphism. For the details of this construction, – in particular, for the definition of the composition of morphisms, – we refer the reader to [@SGA].
The category ${{\mathcal C}}_\#$ comes equipped with a natural forgetful projection $\tau:{{\mathcal C}}_\# \to \Lambda$, and this projection is a [*cofibration*]{} in the sense of
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(x_1A_1+ \cdots+x_nA_n).$$ Thus we cannot directly derive an analog of Proposition \[lowprop\] for Theorem \[MSSmain\].
Consequences for strong Rayleigh measures and weak half-plane property matroids
===============================================================================
A discrete probability measure, $\mu$, on $2^{[n]}$ is called *strong Rayleigh* if its *multivariate partition function* $$P_\mu({\mathbf{x}}) := \sum_{S \subseteq [n]} \mu(\{S\}) \prod_{j \in S}x_j,$$ is *stable*, i.e., if $P_\mu({\mathbf{x}}) \neq 0$ whenever ${{\rm Im}}(x_j)>0$ for all $1 \leq j \leq n$. Strong Rayleigh measures were investigated in [@BBL], see also [@Pem; @Wag]. We shall now reformulate Theorem \[t1\] in terms of strong Rayleigh measures. The measure $\mu$ is of *constant sum* $d$ if $|S|=d$ whenever $\mu(\{S\}) \neq 0$, i.e., if $P_\mu({\mathbf{x}})$ is homogeneous of degree $d$. It is not hard to see that a constant sum measure $\mu$ is strong Rayleigh if and only if $P_\mu({\mathbf{x}})$ is hyperbolic with respect to the all ones vector ${\mathbf{1}}$ and ${\mathbb{R}}_+^n \subseteq \Lambda_+({\mathbf{1}})$, see [@BBL]. Note that if ${\mathbf{e}}_i$ is the $i$th standard basis vector then $$\tr({\mathbf{e}}_i) = \sum_{S \ni i} \mu(\{S\})= {\mathbb{P}}[S : i \in S],$$ where the trace is defined as in the introduction for the hyperbolic polynomial $P_\mu$, with ${\mathbf{e}}={\mathbf{1}}$. If $S \subseteq [n]$ we write ${\mathbf{e}}_S:=\sum_{i \in S}{\mathbf{e}}_i$. The following theorem is now an immediate consequence of Theorem \[t1\].
\[t11\] Let $k\geq 2$ be an integer and $\epsilon$ a positive real number. Suppose $\mu$ is a constant sum strong Rayleigh measure on $2^{[n]}$ such that ${\mathbb{P}}[S : i \in S] \leq \epsilon$ for all $1\leq i \leq n$. Then there is a partition $S_1 \cup \cdots \cup S_k=[n]$ such that $$\| e_{S_i} \| = {\lambda_{\rm max}}({\mathbf{e}}_{S_i}) \leq \frac 1 k \delta(k\epsilon,n)$$ for each $1 \leq i \leq n$.
Let us also see that Theorem \[t1\] easily follows from Theorem \[t1
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by more than an order of magnitude when more effective supernova feedback or AGN feedback is included (not shown). For $M_\mathrm{halo}>10^{12.5}$ M$_\odot$ the hot-mode inflow rate is slightly stronger than the cold-mode inflow rate.
The grey, dashed curve indicates the accretion rate a halo with a baryon fraction $\Omega_{\rm b}/\Omega_{\rm m}$ would need to have to grow to its current baryonic mass in a time equal to the age of the Universe at $z=2$, $$\dot{M}=\dfrac{\Omega_\mathrm{b}M_\mathrm{halo}}{\Omega_\mathrm{m}t_\mathrm{Universe}}.$$ Comparing this analytic estimate with the actual mean accretion rate, we see that they are equal for $M_\mathrm{halo}>10^{11.5}$ M$_\odot$, indicating that these haloes are in a regime of efficient growth. For lower-mass haloes, the infall rates are much lower, indicating that the growth of these haloes has halted, or that their baryon fractions are much smaller than $\Omega_{\rm b}/\Omega_{\rm m}$.
The bottom-right panel of Fig. \[fig:halomassz2\] shows that, at the virial radius, hot-mode gas dominates the gas mass for high-mass haloes. The hot fraction at $R_\mathrm{vir}$ increases from 10 per cent in haloes of $\sim 10^{10}$ M$_\odot$ to 90 per cent for $M_\mathrm{halo}\sim 10^{13}$ M$_\odot$. Note that for haloes with $M_\mathrm{halo}<10^{11.3}$ M$_\odot$ the virial temperatures are lower than our adopted threshold for hot-mode gas. The hot fraction would have been much lower without supernova feedback ($f_\mathrm{hot}<5$ per cent for $M_\mathrm{halo}<10^{10.5}$ M$_\odot$).
Inflow and outflow {#sec:inout}
==================
Figs. \[fig:haloradflux\] and \[fig:halomassflux\] show the physical properties of the gas, weighted by the radial mass flux, for all, inflowing, and outflowing gas (black, blue, and red curves, respectively). Except for the last two panels, the curves indicate the medians, i.e. half the mass flux is due to gas above the curves. Similarly, the shaded regions indicate the 16th and 84th percentiles. Note that we do not plot this separa
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=Z_B^0 \int
\mathcal{D}(\bar{\psi}_{\sigma},\psi_{\sigma}) e^{-S_F^{eff}}
\label{Z2}\end{aligned}$$ where the effective action for fermions is $$\begin{aligned}
S_F^{eff}&=&\int_{k,\tilde{\omega}}
\sum_{\sigma}\bar{\psi}_{\sigma}(k,\tilde{\omega})
\Big[-i\tilde{\omega}+\frac{k^2}{2m}-\mu\Big]\psi_{\sigma}(k,\tilde{\omega})\nonumber
\\ &+&g^2 \int_{k,\tilde{\omega}} \int_{k',\tilde{\omega}'}
\int_{K,\tilde{\Omega}}
\bar{\psi}_{\uparrow}(k,\tilde{\omega})\bar{\psi}_{\downarrow}(K-k,\tilde{\Omega}-\tilde{\omega})
\nonumber \\ &\times&
\psi_{\downarrow}(k',\tilde{\omega}')\psi_{\uparrow}(K-k',\tilde{\Omega}-\tilde{\omega}')
\nonumber\\ &\times& \Big[i\tilde{\Omega}-\frac{K^2}{4m}+2\mu-\nu
\Big]^{-1}\end{aligned}$$ and the grand potential for an ideal Bose gas of bare dimers is: $$\begin{aligned}
F_B^0&\equiv& -T\ln Z_B^0\nonumber \\ &=& TL \int \frac{dK}{2\pi} \ln
\left[1-e^{-\beta(K^2/4m-2\mu+\nu)}\right].\end{aligned}$$
Due to the fact that only quadratic terms in $\psi_B$ appear in the original model, the previous result is exact. The resulting effective interaction between the atoms, however, is non-local both in space and time. If we restrict ourselves to the case $\nu >|\epsilon_{\star}|$, together with the broad resonance requirement $|\epsilon_{\star}|\gg \epsilon_F$, we can simplify the effective action $S_F^{eff}$ to one which is local. Indeed, before resonance, $2|\mu|\simeq
|\epsilon_b|<|\epsilon_{\star}|$ (see Appendix A) and as an order of magnitude, $|i\tilde{\Omega}|\sim|K^2/4m|\sim \epsilon_F$. Therefore, the detuning dominates the denominator of the molecular propagator $i\tilde{\Omega}-\frac{K^2}{4m}+2\mu-\nu \simeq -\nu$, and the effective interaction between fermions becomes $$\begin{aligned}
-\frac{g^2}{\nu} \int_{k_1,\tilde{\omega}_1}
\int_{k_2,\tilde{\omega}_2} \int_{k_3,\tilde{\omega}_3}
\bar{\psi}_{\uparrow}(1+2-3)
\bar{\psi}_{\downarrow}(3)\psi_{\downarrow}(2)\psi_{\uparrow}(1)\nonumber\\
\left. \right.\end{aligned}$$ where $(1)$ is a short notation for $(k_1,\tilde{\omega}_1)$ and similar
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left( \bar{e} \gamma_5 \gamma^{\mu} e \right).$$ This coupling is very similar to the nucleon coupling in Eqn and leads to similar effects. The QCD axion generally has this coupling with $g_{aee} \sim \frac{1}{f_a}$, though it can be fine-tuned to zero. Astrophysics constrains $g_{aee} \lessapprox 10^{-10} \text{ GeV}^{-1}$ from bounds on the cooling of white dwarves [@Raffelt:2006cw]. This interaction also gives rise to spin dependent dipole - dipole forces between electrons. However, bounds from such searches are significantly weaker than the astrophysical limits on this coupling [@Dobrescu:2006au; @electronspin].
![ \[Fig:Electron\] ALP parameter space in pseudoscalar coupling of axion to electrons Eqn. vs mass of ALP. The green region is excluded by White Dwarf cooling rates from [@Raffelt:2006cw]. The blue region is excluded by searches for new spin dependent forces between electrons [@Dobrescu:2006au; @electronspin]. The region below the solid purple line shows the possible parameter space for a QCD axion, with the region bounded by darker purple lines being the region where the QCD axion could be all of dark matter and have $f_a < M_\text{pl}$. The frequency range of the QCD axion covered by ADMX is identical to the range plotted in Figure \[Fig:Nucleon\]. ](electronplot.pdf){width="6"}
Similar to the axial nuclear moment, in the presence of a background dark matter ALP field, the non-relativistic limit of this operator leads to the following term in the electron Hamiltonian $$H_e \supset g_{aee} \vec{\nabla} a.\vec{\sigma_e}$$ where $\sigma_e$ is the electron spin operator. An electron spin that is not aligned with the ALP dark matter “wind" will then precess due to the coupling $$\label{eqn: electron precession rate from gaee}
H_e \supset g_\text{aee} \, m_a \, a_0 \cos \left(m_a t\right) \, \vec{v}.\sigma_{e}$$ Using the constraint that the energy density in the ALP oscillations not exceed the local dark matter density, this perturbation is of size $$\label{eqn: numbers for electron precessio
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the KM mechanism is inoperative. Tree-level flavor changing neutral currents are automatically absent, and the neutral Higgs sector is CP conserving at tree level. As in the KM Model, the quark and electron EDMs are severely suppressed. The electron EDM vanishes at the two-loop level, while the first non-zero contribution to the quark EDMs is at two loops. In contrast to the Weinberg-Branco model, our model easily satisfies other experimental CP violation constraints as well as the rate for $b\to s \gamma$. Finally, the parameter $\theta_{\rm{QCD}}$ vanishes at tree-level, since we disallow hard CP breaking; we shall see that radiative corrections are mild and consistent with the limit on a non-zero $\theta_{\rm{QCD}}$.
For most of this letter, we shall assume that CP is broken softly. One can also modify our model to break CP spontaneously by introducing at least one additional CP odd scalar boson, as discussed toward the end of this work, with the bulk of the phenomenology unchanged.
General Formalism {#general-formalism .unnumbered}
=================
-1cm
The Weinberg-Branco Model augments the Standard Model (SM) with additional Higgs $SU(2)_L$ doublets, which are responsible for kaon system CP violation; in this model, then, since the charged Higgs sector must break CP, so also must the neutral Higgs sector. To mandate charged Higgs exchange as the dominant CP violation mechanism we instead introduce only additional $SU(2)_L$ singlets of quarks and scalars to the theory. The simplest model for our purposes requires two additional charged Higgs singlets, $h_\alpha (\alpha=1,2)$ and a vectorial pair of heavy quark fields, $Q_{L,R}$, of electromagnetic charge $-{4\over3}$. This vector quark charge assignment avoids fractionally charged hadrons. Relevant new terms in the Lagrangian are: $${\cal L}_{h_i} =
\left[
(g \lambda_{i\alpha} \bar Q_L d_{iR} h_\alpha
+ M_Q \bar Q_L Q_R) + \hbox{h.c.} \right]
- (m^2)_{\alpha\beta} {h_\alpha}^{\dag} h_\beta
- \kappa_{\alpha\beta}
(\phi^{\dag} \phi-|\langle
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ir paper:
> [*When fitted models are approximations, conditioning on the regressor is no longer permitted ... Two effects occur: (1) parameters become dependent on the regressor distribution; (2) the sampling variability of the parameter estimates no longer derives from the conditional distribution of the response alone. Additional sampling variability arises when the nonlinearity conspires with the randomness of the regressors to generate a $1/\sqrt{n}$ contribution to the standard errors.*]{}
Moreover, it is not possible to estimate the distribution of the conditional projection parameter estimate in the distribution free framework. To see that, note that the least squares estimator can be written as $\hat\beta(j) = \sum_{i=1}^n w_i Y_i$ for weights $w_i$ that depend on the design matrix. Then $\sqrt{n}(\hat\beta(j) - \beta(j)) = \sum_{i=1}^n w_i \epsilon_i$ where $\epsilon_i = Y_i - \mu_n(i)$. Thus, for each $j \in \{1,\ldots,d\}$ we have that $\sqrt{n}(\hat\beta(j) - \beta(j))$ is approximately $\approx N(0,\tau^2)$, where $\tau^2 = \sum_i w_i^2 \sigma_i^2$, with $\sigma_i^2 = {\rm Var}(\epsilon_i | X_1,\ldots, X_n)$. The problem is that there is no consistent estimator of $\tau^2$ under the nonparametric models we are considering. Even if we assume that $\sigma_i^2$ is constant (an assumption we avoid in this paper), we still have that $\tau^2 =\sigma^2 \sum_i w_i^2$ which cannot be consistently estimated without assuming that the linear model is correct. Again, we refer the reader to [@buja2015models] for more discussion. In contrast, the projection parameter $\beta = \Sigma^{-1}\alpha$ is a fixed functional of the data generating distribution $P$ and is estimable. For these reasons, we focus in this paper on the projection parameter rather than the conditional projection parameter.
Goals and Assumptions {#goals-and-assumptions .unnumbered}
---------------------
Our main goal is to provide statistical guarantees for each of the four random parameters of variable significance introduced above, under our
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}{\varepsilon }\frac{dv_{x}}{d\phi }-\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\frac{A}{\varepsilon }G(\phi )sin\phi \right)
\label{34} \\
\frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}Q+\frac{1}{u}\left( \frac{1}{\varepsilon }\frac{dv_{z}}{d\phi }-\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}+\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi
)sin\phi \right) \label{36}\end{aligned}$$ where
$$Q=\left( \frac{dv_{x}}{d\phi }\right) ^{2}\left( 1-\frac{v_{x}^{2}}{\gamma
^{2}}\right) +\left( \frac{dv_{z}}{d\phi }\right) ^{2}\left( 1-\frac{v_{z}^{2}}{\gamma ^{2}}\right) -\frac{2v_{x}v_{z}}{\gamma ^{2}}\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi } \label{40}$$
and $$\begin{aligned}
\gamma &=&\sqrt{1+v_{x}^{2}+v_{z}^{2}} \label{44} \\
u &=&\gamma -u_{z}\;. \label{46}\end{aligned}$$
In Ref. [@Hart], by specifying final homogeneous conditions on the acceleration and the velocity and then integrating [*backward*]{} in time, the solution to these equations were obtained at all times. We now want to show that this problem can also be solved by specifying initial conditions on the motion and integrating [*forward*]{} in time.
For the “initial” velocity of our method, we use the final velocity of the backward integration method of [@Hart]. As for the initial acceleration we employ the first terms of the series generated by expanding the equations of motion, in terms of the small quantity $\varepsilon =\omega _{0}\tau _{0}$. To obtain this series we write the two components of the equations of motion (\[34\]) and (\[36\]) as
$$\begin{aligned}
\frac{dv_{x}}{d\phi } &=&-AG(\phi )sin\phi +\varepsilon \left[ u\left( \frac{d^{2}v_{x}}{d\phi ^{2}}-v_{x}Q\right) +\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}-\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }\right] \label{50} \\
\frac{dv_{z}}{d\phi
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s $H_d$-module homomorphisms of degree zero. A [*graded standard module*]{} \[graded-standard-defn\] $\widetilde{\Delta}_d(\mu)$, isomorphic to $\Delta_d(\mu)$ as an ungraded module, is given by setting $\widetilde{\Delta}_d(\mu)_r = {\mathbb{C}}[{\mathfrak{h}}]_r\otimes \mu$. By local nilpotence and finite generation, each weight space of a module $M\in \widetilde{{\mathcal{O}}}_d$ is finite dimensional and so $M$ has a well-defined Poincaré series. There is a degree shift functor\[module-shift-defn\] $[1]$ in $\widetilde{{\mathcal{O}}}_d$ defined by $M[1]_r = M_{r-1}$. By abuse of notation, $\widetilde{{\mathcal{O}}}_d$ will also denote the corresponding category of graded $U_d$-modules.
\[standAAA\] Fix $i\geq j\geq 0$ and $\mu\in{{\textsf}{Irrep}({{W}})}$. Give $B_{ij}$ the adjoint ${\mathbf{h}}$-grading and let $B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)$ have the grading this induces. Then $B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)\in \widetilde{{\mathcal{O}}}_{c+i}$ and, as elements of that category, $$B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)
\cong e\widetilde{\Delta}_{c+i}[(i-j)(n(\mu)-n(\mu^t))].$$
Write $\nabla = B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)$ and let $\deg_{c+u} $ denote the degree function in $\widetilde{{\mathcal{O}}}_{c+u}$. By hypothesis, the graded structure of an element $b\otimes v\in \nabla$ is given by $\deg (b\otimes v) = \operatorname{{\mathbf{h}}\text{-deg}}(b)+\deg_{c+j}(v)$. Proposition \[shiftonO\] implies that (as ungraded modules) $$\label{grot3}
\nabla
=S_{c+i}\circ \cdots \circ S_{c+j+1}(e\Delta_{c+j}(\mu)) \cong
e\Delta_{c+i}(\mu).$$ Thus, under its given grading, $\nabla \in \widetilde{{\mathcal{O}}}_{c+i}$.
Unfortunately, it is not easy to write the generator $e\otimes \mu$ of $e\Delta_{c+i}(\mu)$ as an element of $\nabla$ and for this reason the shift in the grading in is subtle. In order to understand this we will use the canonical grading from and we write the corresponding degree function as $\deg_{\mathrm{can}}
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states are
- $H^m(X, \wedge^{\rm even} {\cal E})$, charge $({\rm even}-2, m-n/2)$, giving spacetime states valued in a spinor of $so(8)$.
In the (NS,R) sector, the vacuum energy $E_{{\rm Id}} = -1$. The massless charged states are
- $H^m(X, {\cal E}^* \otimes {\cal E})$, charge $(0,m-n/2)$, spacetime gauge neutral,
- $H^m(X, \wedge^2 {\cal E})$, charge $(2,m-n/2)$, spacetime gauge neutral,
- $H^m(X, {\cal O})$, charge $(0,m-n/2)$, in the adjoint representation of $so(8)$,
- $H^m(X, \wedge^2 {\cal E}^*)$, charge $(-2,m-n/2)$, spacetime gauge neutral.
Now, consider the twisted sector. Here, all of ${\cal E}$ is an eigenbundle with eigenvalue $-1$.
In the (R,R) sector, $E = -1/2$. There are no massless charged states in this sector.
In the (NS,R) sector, $E = -1/2$. Again, there are no massless charged states in this sector.
States above are listed with charges $(q_-,q_+)$. The $q_+$ charge distinguishes chiral multiplets from vector multiplets; the $q_-$ charge is the charge of the $u(1)$ that combines with $so(8)$ to build $so(10)$.
For a compactification to four dimensions, ($n=3$,) states with $q_+ = -1/2$ would be spacetime fermions in chiral multiplets (and $q_+ = +1/2$ their antichiral partners); states with $q_+ = +3/2$ would be spacetime fermions in vector multiplets (and $q_+ = -3/2$ their partners).
For a compactification to six dimensions, ($n=2$,) which is the pertinent case, states with $q_+ = \pm 1$ are spacetime fermions in vector multiplets; states with $q_+ = 0$ are spacetime fermions in hypermultiplets.
Since we have a rank 4 bundle, in principle the $E_8$ should be broken to ${\rm Spin}(10)$, which in the worldsheet theory will be assembled from representations of $so(8) \times u(1)$ (the $so(8)$ rotating the remaining free left-moving fermions in the first $E_8$, and the $u(1)$ being an overall phase rotation on the bundle fermions, which on the (2,2) locus would become the left R symmetry). Under the $so(8)\times u(1)$ subalgebra, representations of $so(10)$ decompose as
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in the set of solutions $\{ {{\bar{\a}}^\dagger}_k \}$ to the QPs in . Fortunately, to obtain the solution set $\{ {{\bar{\a}}^\dagger}_k \}$, it is sufficient to solve merely one QP in with $k = 1$, according to the following theorem.
\[theorem:aLinear\] Denote the solution to the QP in with the constraint $\a(L) = k$ as ${{\bar{\a}}^\dagger}_k$, then ${{\bar{\a}}^\dagger}_k = k {{\bar{\a}}^\dagger}_1$.
$$\begin{aligned}
{{\bar{\a}}^\dagger}_k
&= \arg \min_{ \a \in \Rbb^L, \a(L) = k }
\a^T \bsG \a\\
&= k \left( \arg \min_{ \mathbf{t} \in \Rbb^L, \mathbf{t}(L) = 1 }
(k \mathbf{t}^T) \bsG (k \mathbf{t}) \right) \quad ({\rm Let\ } \a = k \mathbf{t}.)\\
&= k \left( \arg \min_{ \mathbf{t} \in \Rbb^L, \mathbf{t}(L) = 1 }
k^2 \mathbf{t}^T \bsG \mathbf{t} \right)\\
&= k \left( \arg \min_{ \mathbf{t} \in \Rbb^L, \mathbf{t}(L) = 1 }
\mathbf{t}^T \bsG \mathbf{t} \right)\\
&= k {{\bar{\a}}^\dagger}_1. \qedhere\end{aligned}$$
The closed-form expression of ${{\bar{\a}}^\dagger}_1$ can be readily obtained by solving a linear system as stated in the following theorem, which is the key for the low complexity of our method.
\[theorem:aBarDagger1\] Let ${{\bar{\a}}^\dagger}_1$ be the optimal solution to the QP in with the constraint $\a(L) = 1$, then $${{\bar{\a}}^\dagger}_1 =
\begin{bmatrix}
\rr\\
1
\end{bmatrix},$$ where $$\label{equation:r}
\rr = - \Bigl( \bsG(1\!:\!L-1,1\!:\!L-1) \Bigr)^{-1}
\bsG(1\!:\!L-1,L).$$
The QP in has only an equality constraint, and thus is linear and particularly simple [@Luenberger2008]. We now derive the closed-form solution with the Lagrange multiplier method. Let the Lagrange multiplier associated with the constraint $\a(L) = 1$ be $\lambda \geq 0$, then the Lagrangian is $$\mathcal{L}(\a,\lambda) = \a^T \bsG \a + \lambda \left( \a(L) - 1 \right).$$ The optimal solution can be obtained by letting the derivative of the Lagrangian be zero, i.e., $$\begin{aligned}
\frac{\partial}{\partial \a}\mathcal{L}(\a,\lambda)
= (\bsG + \bsG^T) \a +
\begin{bmatrix}
\0\\
\lambda
\end{bmatrix}
= 2 \bsG \a +
| 4,158
| 2,158
| 1,167
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|
1392180.1588220.049102HV AlgorithmHEIA\*cMLSGANSGA-II\*MOEA/DMTS Average0.1748460.1744750.1744610.1740940.168158 (S.D.)0.0000270.0000390.0002880.0002140.000464\*indicates if the results are significantly different to the next lowest rank, using the Wilcoxon's rank sum with a 0.05 confidence
In order to better understand the complexity of the problem and to check whether the best possible set of solutions has been found, a set of 5 runs with 300,000 total iterations was conducted on each virtual person, utilising HEIA. In this case hardly any difference was observed between 50,000 and 300,000 iterations. Figure [6](#Fig6){ref-type="fig"}a shows some very slightly higher uniformity and diversity of the points in the high FF1 bias region with 300,000 iterations. When comparing the performance over the number of generations in Fig. [6](#Fig6){ref-type="fig"}b, virtually no improvement in performance can be seen after 50,000 generations and the highest performance gain occurred before 25,000 iterations. The low possible performance increase beyond 50,000 iterations in this case would not justify conducting optimisation of this problem with higher values, unless the virtual person is suspected to benefit from an extreme reduction in pressure over the residuum tip and the soft tissue strain around the distal tibia (FF1 bias).Fig. 6**a** The comparison of Pareto Fronts from Virtual Person 1, achieved using HEIA over 50,000 iterations ('achieved') and 300,000 iterations ('real'). **b** The performance of HEIA over 300,000 iterations on Person 1. 0 is the starting population, and 1 is the best attainable set of solutions, based on the IGD values, and the red line indicates the number of function calls utilised in this study
Discussion {#Sec7}
==========
This study aimed to explore a range of potential concepts for transtibial socket design using FE modelling, surrogate modelling and GA-based optimisation techniques, to provide a quantitatively informed starting point for the prosthetist when designing a bespoke pr
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Initiated \-
Norwegian Coronavirus Disease 2019 Study Hydroxychloroquine Sulfate Not yet recruiting \-
Post-exposure Prophylaxis for SARS-Coronavirus-2 Hydroxychloroquine Recruiting University of Minnesota, Minneapolis, Minnesota, United States
The efficacy and safety of pirfenidone capsules in the treatment of severe new coronavirus pneumonia (COVID-19) Pirfenidone \- Third Xiangya Hospital of Central South University
Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan, China Oseltamivir Recruiting Zhongnan Hospital of Wuhan Univ
| 4,160
| 5,225
| 2,244
| 3,481
| null | null |
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|
n then find $\S^{**}$ which is a suitable correct iterate of both $\S$ and $\S^*$. Notice that since $\S^{**}$ is suitable, the iteration embeddings $i : \S|(\l_\S^+)^\S\rightarrow \S^{**}|(\l_{\S^{**}}^+)^{\S^{**}}$ and $j: \S^*|(\l_{\S^*}^+)^{\S^*}\rightarrow \S^{**}|(\l_{\S^{**}}^+)^{\S^{**}}$ exists.
Suppose now that ${\gamma}\not ={\gamma}^\prime$. Let $(\bar{\b}, \bar{{\gamma}}, \bar{{\gamma}}^\prime)\in \S^{**}$ be such that letting $\zeta=max(i(\eta_\Q), j(\eta_\R))$ and $\W=\S^{**}|(\zeta^+)^{\S^{**}}$, $\pi_{\W, \infty}(\bar{\b}, \bar{{\gamma}}, \bar{{\gamma}}^\prime)=(\b, {\gamma}, {\gamma}^\prime)$. It then follows that $(\bar{\b}, \bar{{\gamma}})\in i(\pi^\T_{\P, \S}(a))$ and $(\bar{\b}, \bar{{\gamma}}^\prime)\in j(\pi^{\T^*}_{\P, \S^*}(a))$. However, $i\circ \pi^\T_{\P, \S}=j\circ \pi^{\T^*}_{\P, \S^*}$, implying that $i(\pi^\T_{\P, \S}(a))=j(\pi^{\T^*}_{\P, \S^*}(a))$ and that $S^{**}{\vDash}(\bar{b}, \bar{{\gamma}})\in i(\pi^\T_{\P, \S}(a)) \wedge (\bar{b}, \bar{{\gamma}}^\prime)\in i(\pi^\T_{\P, \S}(a))$.
Let now $(\tau, \tau^*)\in \Q$ be such that $\pi_{\Q, \infty}(\tau, \tau^*)=(\b, {\gamma})$. By elementarity of $i$, we then get that $\S{\vDash}``$there is $\tau^{**}\not =\tau^*$ such that $(\tau, \tau^{**})\in \pi_{\P, \S}(a)"$. Fix such a $\tau^{**}$ and let $\varsigma \in (\tau^{**}, \l_\S)$ be an $\S$-cardinal. Then letting $\Q^*=\S|(\varsigma^+)^\S$ we have that $(\b, \pi
_{\Q^*, \infty}(\tau^{**}))\in \phi_\a(x)$ and $\pi_{\Q^*, \infty}(\tau^{**})\not = {\gamma}$, contradiction.
The next lemma finishes the proof.
Suppose $\b<\l$, $A\in \utilde{\Delta}^2_1$ and $A\subseteq R_\b=\{ x : \exists {\gamma}<\k R_{\b, {\gamma}}(x)\}$. Then $\exists {\gamma}_0<\k$ such that $\forall x\in A\exists {\gamma}<{\gamma}_0 R_{\b, {\gamma}}(x)$.
Let $f:A\rightarrow \k$ be defined by $f(x)=\nu$ if $\nu$ is the least such that $\nu$ ends a weak gap and $J_\nu(\mathbb{R}){\vDash}x\in R_\b$. Then $f$ is $\utilde{\Sigma}_1$ over $J_{\k}(\mathbb{R})$ and hence, as $\k$ is $\mathbb{R}$-admissible, $f$ is b
| 4,161
| 3,105
| 2,424
| 3,834
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n particular, if $\mathfrak{n}=\sum y_iR+q_jR$, then $\Sigma$ is an r-sequence for the $R_{\mathfrak{n}}$-module $N(k)_\mathfrak{n}$. By the Auslander-Buchsbaum formula [@matsumura Ex. 4, p.114], $N(k)_\mathfrak{n}$ is therefore free as a $R_{\mathfrak{n}}$-module.
Finally, consider $\overline{N(k)}=N(k)/\sum q_jN(k)$. Under the induced ${\mathbf{h}}$-grading, $\overline{N(k)}$ is a finitely generated, graded ${\mathbb{C}}[{\mathfrak{h}}^*]$-module and so corresponds to a ${\mathbb{C}}^*$-equivariant coherent sheaf on ${\mathfrak{h}}^*$. As a result the locus where $\overline{N(k)}$ is not free is a ${\mathbb{C}}^*$-stable closed subvariety of ${\mathfrak{h}}^*$. If this locus is non-empty it must contain the unique ${\mathbb{C}}^*$-fixed point $\mathfrak{p}=(y_1,\dots,y_{n-1})$ for this expanding ${\mathbb{C}}^*$-action. But then $(\overline{N(k)})_{\mathfrak{p}}$ would not be free, contradicting the conclusion of the last paragraph.
{#poincare-start}
We next need to understand the graded structure of the modules $\overline{N(k)}$ and $\underline{N(k)}$ under the ${\mathbf{h}}$-grading. To do this, we express $\underline{N(0)}$ as a weighted sum of standard modules in the Grothendieck group $G_0(U_c)$ and then to use Proposition \[shiftonO\] to write $\underline{N(k)}=B_{k0}\otimes \underline{N(0)}$ in a similar manner. This is quite delicate since there are some subtle shifts involved and we first want to understand these shifts for $B_{ij}\otimes \Delta_c(\mu)$.
We will need to work with the following graded version $\widetilde{{\mathcal{O}}}_d$\[cat-O-gr-defn\] of ${\mathcal{O}}_d$ constructed in [@GGOR Section 2.4]. The objects $M$ in $\widetilde{{\mathcal{O}}}_d$ are finitely generated $H_d$-modules on which ${\mathbb{C}}[{\mathfrak{h}}^*]$ acts locally nilpotently and which come equipped with a $\mathbb{Z}$-grading $M = \bigoplus_{r\in {\mathbb{Z}}} M_{r}$ such that $p M_{r} \subseteq M_{r+\ell}$ for each $p\in H_d$ with $\operatorname{{\mathbf{E}}\text{-deg}}(p)=\ell$. The morphisms are homogeneou
| 4,162
| 3,701
| 1,855
| 3,875
| 2,188
| 0.781928
|
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|
are as follows, $$\label{eq3}
\begin{array}{r@{}l@{\qquad}l}
&m{{\ddot{x}}_{m}}={{F}_{2}}-{{F}_{1}}, \\
%\label{eq4}
&m{{\ddot{y}}_{m}}=2{{F}_{s1y}}=2{{F}_{s2y}}, \\
%\label{eq5}
&mg=2{{F}_{s1z}}=2{{F}_{s2z}},
\end{array}$$ where $$\begin{aligned}
\begin{array}{r@{}l@{\qquad}l}
{{\ddot{x}}_{m(t)}}=&-{{L}_{1}}\left( {{{\dot{\theta }}}_{1}}\cos {{\theta }_{1}}+{{{\ddot{\theta }}}_{1}}\sin {{\theta }_{1}} \right)\\&-{{L}_{2}}\left[ {{\left( {{{\dot{\theta }}}_{1}}+{{{\dot{\theta }}}_{2}} \right)}^{2}}\cos \left( {{\theta }_{1}}+{{\theta }_{2}} \right)\right]\\
&-{{L}_{2}}\left[{{\left( {{{\ddot{\theta }}}_{1}}+{{{\ddot{\theta }}}_{2}} \right)}^{2}}\sin \left( {{\theta }_{1}}+{{\theta }_{2}} \right) \right],
\end{array}
\end{aligned}$$ $$\begin{aligned}
\begin{array}{r@{}l@{\qquad}l}
{{\ddot{y}}_{m(t)}}=&{{L}_{3}}\left( {{{\ddot{\theta }}}_{3}}\cos {{\theta }_{3}}+{{{\dot{\theta }}}_{3}}^{2}\sin {{\theta }_{3}} \right)\\&-{{L}_{4}}\left[ \left( {{{\ddot{\theta }}}_{3}}+{{{\ddot{\theta }}}_{4}} \right)\cos \left( {{\theta }_{3}}+{{\theta }_{4}} \right)\right]\\
&+{{L}_{4}}\left[{{\left( {{{\dot{\theta }}}_{3}}+{{{\dot{\theta }}}_{4}} \right)}^{2}}\sin \left( {{\theta }_{3}}+{{\theta }_{4}} \right) \right],
\end{array}
\end{aligned}$$ and $g=9.8 m/s^2$. And the relationship between the applied forces and the friction forces is presented by $$\label{eq6}
\begin{array}{r@{}l@{\qquad}l}
{{F}_{s1y}}^{2}+{{(\frac{mg}{2})}^{2}}<{{(\mu {{F}_{1}})}^{2}}, \\
%\label{eq7}
{{F}_{s2y}}^{2}+{{(\frac{mg}{2})}^{2}}<{{(\mu {{F}_{2}})}^{2}},
\end{array}$$ where $\mu$ is the friction coefficient in dry condition.
If ${{\ddot{x}}_{m(t)}}\ge 0$, both the applied forces $F_1$ and $F_2$ can be computed by $$\label{eq8}
\begin{array}{r@{}l@{\qquad}l}
{{F}_{1}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\ddot{y}}}_{m}}}{2} \right)}^{2}}+{{\left( \frac{mg}{2} \right)}^{2}}}, \\
%\label{eq9}
{{F}_{2}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\dd
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es. In Sec.\[HigherGauge\], we comment on extensions of the generalized notion of gauge symmetry to higher form gauge theories.
Gauge Symmetry Algebra {#GaugeSymm}
======================
In a naive textbook introduction to non-Abelian gauge symmetry, the gauge transformation parameter $\Lam(x) = \sum_a \Lam^a(x)T_a$ is a sum of products of space-time functions and Lie algebra generators. The Lie algebra of local gauge transformations is spanned by a set of basis elements, say, T\_a(p) e\^[ipx]{} T\_a \[factor\] in the Fourier basis. In this basis, a gauge transformation parameter can be expressed as (x) = \_[a, p]{} \^a(p) T\_a(p), where the sum over $p$ is understood to be the integral $\int d^D p$ for $D$-dimensional space-time. Similarly, the gauge potential can be written as A\_(x) = \_[a, p]{} \^a\_(p) T\_a(p).
Normally, for a given finite-dimensional Lie algebra with structure constants $f_{ab}{}^c$, the algebra of gauge transformations has the commutator = \_c f\_[ab]{}\^c T\_c(p+p’), where the structure constants $f_{ab}{}^c$ only involve color indices $a, b, c$. For these cases, the inclusion of functional dependence on the space-time in the generators $T_a(p)$ is trivial, and thus often omitted in discussions.
However, for noncommutative gauge symmetries, the structure constants depend not only on the color indices $a, b, c$, but also on the kinematic parameters $p, p'$. For a noncommutative space defined by = i\^, the Lie algebra $U(N)$ gauge symmetry is = \_[p”]{} [**f**]{}\_[ab]{}\^[c]{}(p, p’, p”) T\_c(p”), \[NCUN\] where the structure constants are [^2] \_[ab]{}\^[c]{}(p, p’, p”) = \^[(D)]{}(p+p’-p”), and they involve kinematic parameters $p, p'$ and $p''$. Here $f_{ab}{}^c$ is the structure constant of $U(N)$ and $d_{ab}{}^c$ is defined by {T\_a, T\_b} = d\_[ab]{}\^c T\_c for $T_a$’s in the fundamental representation. In this gauge symmetry algebra, the $U(N)$ Lie algebra and the algebra of functions on the base space are mixed. (This is the obstacle to define noncommutativ
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in\] may be applied to any given plane curve, producing a list of its limits. In practice, one needs to find the marker germs for the curve; these determine the components of the PNC. The two examples in §\[twoexamples\] illustrate this process, and show that components of all types may already occur on curves of degree $4$. Here is a simpler example, for a curve of degree $3$.
\[exone\] Consider the irreducible cubic ${{\mathscr C}}$ given by the equation $$xyz+y^3+z^3=0\,.$$ It has a node at $(1:0:0)$ and three inflection points. According to Theorem \[mainmain\] and the list in §\[germlist\], the PNC for ${{\mathscr C}}$ has one component of type II and several of type IV. The latter correspond to the three inflection points and the node. A list of representative marker germs for the component of type II and for the component of type IV due to the node may be obtained by following the procedure explained in §\[setth\]: $${\rm II}:
\begin{pmatrix}
-2 & -t & 0\\
1 & t & 0\\
1 & 0 & t^2
\end{pmatrix};\quad
{\rm IV}:
\begin{pmatrix}
1 & 0 & 0\\
0 & t & 0\\
0 & 0 & t^2
\end{pmatrix}
\,,\,
\begin{pmatrix}
1 & 0 & 0\\
0 & t^2 & 0\\
0 & 0 & t
\end{pmatrix}\,.$$ The latter two marker germs, corresponding to the two lines in the tangent cone at the node, have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. Equations for the limits of ${{\mathscr C}}$ determined by the germs listed above are $$x(xz+2y^2)=0,\quad
y(y^2+xz)=0,\quad \text{and} \quad
z(z^2+xy)=0\,,$$ respectively: a conic with a tangent line, and a conic with a transversal line (two limits). The inflection points also contribute components of type IV; the limits in that case are cuspidal cubics.
According to Theorem \[main\], all limits of ${{\mathscr C}}$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]).
Necessary preliminary considerations, and the full statement of the main theorem, are found in §\[pr
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|
}}\}}}\equiv1$ for any ${{\bf n}}''\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ with ${\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x$. As in the derivation of [(\[eq:0th-summand3\])]{} from [(\[eq:0th-summand2\])]{}, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b
\text{ even}\}}}$ in [(\[eq:3rd-ind-prefact\])]{} using the source constraints and the fact that ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}=0$ whenever ${\overline{b}}\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$. Therefore, $$\begin{aligned}
{\label{eq:3rd-ind-fact}}
{(\ref{eq:3rd-ind-prefact})}~=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{
{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}.\end{aligned}$$ By [(\[eq:Theta-def\])]{}–[(\[eq:3rd-ind-fact\])]{}, we arrive at $$\begin{aligned}
{\label{eq:2nd-exp}}
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}=
\Theta_{v,x;{{\cal A}}}&+\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\,\tau_b\,
{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda{\nonumber}\\
&-\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\Big[\tau_b\Big({{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}^b(v)
{^{\rm c}}}\Big)\Big],\end{aligned}$$ where ${{\cal C}}^b(v)\equiv{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$ is a variable for the operation $\Theta_{v,{\underline{b}};{{\cal A}}}$. This completes the second stage of the expansion.
### Completion of the lace expansion {#sss:complexp}
For notational convenience, we def
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|
vide a business reason on a Direct Cash Flow basis for an item buried within the Indirect Cash Flow format. Call me if you have any further questions.
-----Original Message-----
From: Hayslett, Rod
Sent: Wednesday, November 14, 2001 6:45 AM
To: Kleb, Steve
Cc: Geaccone, Tracy
Subject: FW: Fw: Undeliverable: Fw: Last schedule
Can you help explain how the data in the attached file matches the detailed budgets and current estimates you have provided to me? It may simply, I just haven't had time to look.
Rod,
Thanks for the email. Can you help me match the information in this email
to your operating plan or cash flow analysis?
Kyle
<<AnnualStats.xls>>
>
> Do you have the volume and rate information on Northern Natural that
> will tie to Rod's forecast as well?
>
> Kyle
>
(See attached file: AnnualStats.xls)
Start Date: 2/28/01; HourAhead hour: 5; No ancillary schedules awarded. No
variances detected.
LOG MESSAGES:
PARSING FILE -->> O:\Portland\WestDesk\California Scheduling\ISO Final
Schedules\2001022805.txt
Jeff,
In what time frame are these decisions looking at being made? keep me posted on the July 1st roll back since it would have the biggest implications for EWS/EES existing positions.
Thank you,
Jeff Richter
-----Original Message-----
From: Dasovich, Jeff
Sent: Friday, October 19, 2001 5:02 PM
To: Dasovich, Jeff; Kean, Steven J.; Shapiro, Richard; Steffes, James D.; Kaufman, Paul; Mara, Susan; Calger, Christopher F.; Tycholiz, Barry; Richter, Jeff; Belden, Tim; Comnes, Alan; Swain, Steve; Sharp, Vicki; Blachman, Jeremy; Dietrich, Janet; Delainey, David; Leff, Dan; Hughes, Evan; Herod, Brenda F.; Parquet, David; Frazier, Lamar; Denne, Karen; O'Neil, Murray P.; Huddleson, Diann
Cc: Alamo, Joseph
Subject: California Treasurers Calls on Lynch to Immediately Set DA Suspension Date Back to July 1
The state treasurer sent a letter this afternoon to Loretta Lynch.
The treasurer claims in the letter that, by extending DA through Sept. 20th, the PUC could hav
| 4,167
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ed by E. Getzler [@getz] is that we have an analog of the Gauss-Manin connection: if ${\operatorname{Spec}}R$ is smooth, the $R$-module $HP_i(A_R/R)$ carries a canonical flat connection for every $i$.
Consider now the case when $R$ is not smooth but, on the contrary, local Artin. Moreover, assume that ${{\mathfrak m}}^2=0$, so that $R$ is itself a (commutative) square-zero extension of $k$. Then a deformation $A_R$ of $A$ over $R$ is also a square-zero extension of $A$, by the bimodule $A \otimes {{\mathfrak m}}$ (${{\mathfrak m}}$ here is taken as a $k$-vector space). But this square-zero extension is special – for a general square-zero extension ${\widetilde}{A}$ of $A$ by some $M \in A{\operatorname{\!-\sf bimod}}$, there does not exist any analog of the relative cyclic $R$-module $A_{R\#}
\in {\operatorname{Fun}}(\Lambda,R)$.
We observe the following: the data needed to define such an analog is precisely a cyclic bimodule structure on the bimodule $M$.
Namely, assume given a square-zero extension ${\widetilde}{A}$ of the algebra $A$ by some $A$-bimodule $M$, and consider the cyclic $k$-vector space ${\widetilde}{A}_\# \in {\operatorname{Fun}}(\Lambda,k)$. Let us equip ${\widetilde}{A}$ with a descreasing two-step filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ by setting $F^1{\widetilde}{A} =
M$. Then this induces a decreasing filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ on tensor powers ${\widetilde}{A}^{\otimes n}$. Since ${\widetilde}{A}$ is square-zero, $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is compatible with the multiplication maps; therefore we also have a filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ on ${\widetilde}{A}_\#$. Consider the quotient $$\overline{A_\#} = {\widetilde}{A}_\#/F^2{\widetilde}{A}_\#.$$ One checks easily that ${\operatorname{\sf gr}}^0_F{\widetilde}{A}_\# \cong A_\#$ and ${\operatorname{\sf gr}}^1_F{\widetilde}{A}_\# \cong j_!M^\Delta_\#$ in a canonical way, so that $\overline{A_\#}$ fits into a canonical short exact sequence $$\la
| 4,168
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| 3,059
| 0.775111
|
github_plus_top10pct_by_avg
|
-)}
=F(v)=\tilde{B}(\tilde{\phi},v) \\[2mm]
={}&
{\left\langle}\phi, P'(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}CS_0\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}(\Sigma-K_C)\phi,v{\right\rangle}_{L^2(G\times S\times I)} \\
{}&+{\left\langle}q_{|\Gamma_+},\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0,S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)} \\[2mm]
={}&{\left\langle}T_C\phi,v{\right\rangle}_{L^2(G\times S\times I)}
+{\left\langle}q_{|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}
+{\left\langle}\gamma_-(\phi),\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)} \\
{}&
+{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}+{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)v(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)} \\[2mm]
={}&{\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}q_{|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)} \\
{}&+{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}$$ where on the second to last phase we wrote $T_C\phi=(P(x,\omega,E,D)+CS_0+\Sigma-K_C)\phi$ and used Green’s formula , and on the last phase we made use of the already proven facts: $T_C\phi={\bf f}$, $\gamma_-(\phi)={\bf g}$, and $\phi(\cdot,\cdot,E_m)=0$. Thus it holds $${\left\langle}q_{|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}=0,$$ for all $v\in C^1(\ol{G}\times S\times I)$, which clearly implies that $q_{|\Gamma_+}=\gamma_+(\phi)$ and $p_0=\phi(\cdot,\cdot,0)$.
\(iii) By assumption, $\phi_{|\Gamma_{\pm}}\in T^2(\Gamma_{\pm})$ and $\phi(\cdot,\cdot,0)$, $\phi(\cdot,\cdot,E_{\rm m})\in L^2(G\times S)$, and moreover $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$. These properties allow us to apply the G
| 4,169
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it is a right action of the group $H$ on the set $G$ and $(g_1 g_2) \lhd h =
(g_1 \lhd h) (g_2 \lhd h)$ for all $g_1$, $g_2 \in G$ and $h\in
H$. ${{\rm Aut}\,}(H)$ is the group of automorphisms of $H$ and $C_n$ is the cyclic group of order $n$.
Let $H$ and $G$ be two groups with the multiplications $m_{H}: H
\times H\rightarrow H$, $m_{G}:G \times G\rightarrow G$, units $1_H$ and respectively $1_G$ and $R: G \times H \rightarrow H
\times G$ a map. We shall define a new multiplication on the set $H \times G$ using $R$ instead of the usual flip $\tau: G\times H
\to H\times G$, $\tau (g, h) = (h, g)$ as follows: $$m_{H \times G, R}: H \times G \times H \times G \rightarrow H
\times G, \qquad m_{H \times G, R}:= (m_{H} \times m_{G})\circ (I
\times R \times I)$$
Let $\alpha := \pi_{1} \circ R : G \times H \rightarrow H$, $\beta
:=\pi_{2} \circ R: G \times H \rightarrow G$, where $\pi_{i}$ is the projection on the $i$-component; we shall denote $\alpha (g,
h) = g \rhd h$ and $\beta (g, h) = g \lhd h$, for all $g\in G$ and $h\in H$. Then $R (g, h) = (g \rhd h, g \lhd h)$ and the multiplication $m_{H \times G, R}$ on $H \times G$ can be explicitly written as follows: $$(h_{1}, g_{1}) \cdot_{R} (h_{2}, g_{2}) = \bigl( h_{1}(g_{1} \rhd
h_{2}), \, (g_{1} \lhd h_{2})g_{2}\bigl)$$ for all $h_{1}$, $h_{2} \in H$ and $g_{1}$, $g_{2} \in G$.
It can be easily shown that $(H \times G, m_{H \times G, R})$ is a group with $(1_{H}, 1_{G})$ as a unit if and only if $(H, G,
\alpha, \beta)$ is a *matched pair* in the sense of Takeuchi ([@Takeuchi]): i.e. $\alpha$ is a left action of the group $G$ on the set $H$, $\beta$ is a right action of the group $H$ on the set $G$ and the following two compatibility conditions hold: $${\label{eq:2}}
g \rhd (h_{1} h_{2}) = (g \rhd h_{1})\bigl((g \lhd h_{1}) \rhd
h_{2}\bigl)$$ $${\label{eq:3}}
(g_{1} g_{2}) \lhd h = \bigl(g_{1} \lhd (g_{2} \rhd h)\bigl)(g_{2}
\lhd h)$$ for all $h$, $h_{1}$, $h_{2} \in H$ and $g$, $g_{1}$, $g_{2} \in
G$. It follows from [(\[eq:2\])]{} and [(\[eq:3\])]{} that: $${\
| 4,170
| 3,351
| 3,580
| 3,780
| 3,854
| 0.769686
|
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versus no abortion probabilities during a PRRSV outbreak by marker genotype. Only markers showing different abortion rates by genotype are shown.
Marker Gene Contrast Odds Ratio *p*
-------------- --------- ---------- ------------ ----------
rs80800372 *GBP1* AA/AG 2.69 0.008
rs340943904 *GBP5* GG/TG 2.76 0.003
GG/TT 4.49 0.02
rs1107556229 *CD163* AA/AG 2.58 \<0.0001
AA/GG 1.96 0.0004
−547ins+275 *MX1* DD/II 9.35 0.03
ID/II 8.63 0.04
rs325981825 *HDAC6* AA/GG 4.08 0.002
AG/GG 2.34 0.02
1. Introduction {#S1}
===============
Congenital coronary artery anomalies are defined as a coronary pattern that is found in less than 1% of the general population, with a prevalence ranging from 0.3%−5.6% \[[@R1]\]. In few types of the anomalies, there is an association with sudden death and premature coronary disease \[[@R2]\]. Congenital absence of left main coronary artery (LMCA) and anomalous origins of left anterior descending artery (LAD) and left circumflex artery (LCX) arising from right sinus of Valsalva is rarely reported. Here we are presenting a 62-year-old male who presented with non-ST NSTEMI who found to have anomalous absence of left main coronary artery and anomalous origins of left anterior descending artery and left circumflex artery from right sinus of Valsalva.
2. Report of the Case {#S2}
=====================
62 years old man with past medical history of hypertension, dyslipidemia and type II diabetes mellitus presented with acute chest pain. The pain started suddenly, pressure like, at the left side, 9/10 in intensity, radiates to left arm and was associated abdominal discomfort, nausea and diaphoresis. The pain was relieved by sublingual nitroglycerin. Electrocardiography showed tall positive T waves at inferior lea
| 4,171
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nd a primary field $\phi$: $$\begin{aligned}
j^a_{L,z}(z) \phi(w) = & -\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{z-w} + :j^a_{L,z} \phi:(w) \cr
& + {A^a}_c \log|z-w|^2 :j^c_{L,z} \phi:(w) + {B^a}_c \frac{\bar z - \bar w}{z-w} :j^c_{L,\bar z} \phi:(w) + \mathcal{O}(f^4) \cr
j^a_{L,\bar z}(z) \phi(w) = & -\frac{c_-}{c_++c_-} \frac{t^a \phi(w)}{\bar z-\bar w} + :j^a_{L,\bar z} \phi:(w) \cr
& + {D^a}_c \log|z-w|^2 :j^c_{L,\bar z} \phi:(w) + {C^a}_c \frac{z-w}{\bar z - \bar w} :j^c_{L, z} \phi:(w) + \mathcal{O}(f^4).\end{aligned}$$ We expect the coefficients ${A^a}_c$, ${C^a}_c$, ${B^a}_c$, ${D^a}_c$ to be of order $f^2$. We will check that the coefficient of the first-order poles are not modified. As explained in section \[bootstrap\] the demand of consistency with current conservation imposes that the terms in the $j^a_{L,\bar z}(z) \phi(w)$ OPE can be deduced from the terms in the $j^a_{L,z}(z) \phi(w)$: \_c + [C\^a]{}\_c = 0 = [B\^a]{}\_c + [D\^a]{}\_c. To get further constraints on the tensors ${A^a}_c$ and ${B^a}_c$ we ask for the vanishing of the first-order poles in the OPE between the operator $\phi$ and the Maurer-Cartan operator, that we write as in : \[MC.phi=0\] \[ |j\^a\_[L,z]{}(z) + i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:(z)\](w) = 0. The first part of this OPE is: $$\begin{aligned}
\bar \p j^a_{L,z}(z) \phi(w) = & {A^a}_c \frac{:j^c_{L,z}\phi:(w)}{\bar z - \bar w} + {B^a}_c \frac{:j^c_{L,\bar z}\phi:(w)}{z -w} + \mathcal{O}(f^4).\end{aligned}$$ The simple poles in the previous expression should be canceled by the simple poles in the OPE between the composite operator $ i f^2 {f^a}_{bc}
:j^c_{L,z} j^b_{L,\bar z}:$ and the operator $\phi$. Notice that because of the factors $f^{2}$ multiplying the composite operator, we only need to compute the OPE at order $f^0$. We calculate : $$\begin{aligned}
\phi(w)&[ i f^2 {f^a}_{bc} :j^c_{L,z} j^b_{L,\bar z}:(z)] = i f^2 {f^a}_{bc} \lim_{:x \to z:} \phi(w)j^c_{L,z}(x) j^b_{L,\bar z}(z) \cr
= & i f^2 {f^a}_{bc} \lim_{:x \to z:} \left \{
\left[
| 4,172
| 2,003
| 3,622
| 3,837
| null | null |
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|
ms contains the initial fluctuation of energy density $\Delta {\cal U}$ \[$\Delta {\cal U}={\cal U}(\omega,0)-\frac{1}{2} KT$\] due to excitation of the system at $t=0$ \[see Eq.(15)\].
$$\begin{aligned}
p(x,v,t)\stackrel{t\rightarrow 0}
{\longrightarrow} N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}}
e^{-2D\frac{\Gamma}{\gamma}}
e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+\tilde{V}(x)\right)}\nonumber\\
\nonumber\\
=N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}}
e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+{\tilde{V}}(x)+\Delta {\cal U}\right)},
\hspace{0.2cm}{\rm for} (x\rightarrow -\infty)\hspace{0.2cm}.\end{aligned}$$
The last equality demands that $$D=\frac{\gamma}{2\Gamma} \; \frac{\Delta {\cal U}}{KT}$$
\[for the current to be coordinate independent the parabolic approximation of $\tilde{V}(x)$ is to be used\]. $D$ is thus determined in terms of the relaxing mode parameters and fluctuations of the energy density distribution at $t=0$.
The time-dependent probability density therefore allows us to construct nonstationary current, $$j(t)=\int_{-\infty}^{+\infty}dv\hspace{0.1cm}v\hspace{0.1cm}p(x,v,t)=
j_{s}e^{-\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)}\hspace{0.2cm},$$
where $j_{s}$ is the stationary or steady state current as derived in the last section.
By Eq.(74) we have, $$p_{w}(x,v)=p(x\rightarrow -\infty,v,t=0_{-})\hspace{0.2cm},$$
which was used to calculate the number of particles $n_{a}$ initially in the well just before the system was subjected to shock at $t=0$. Thus non-stationary Kramers rate of transition is given by $$k(t)=\frac{\omega_{0}}{2\pi\bar{\omega}_{b}}\left[\left\{\left(\frac{\Gamma}
{2}\right)^{2}+\bar{\omega}_{b}^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2}
\right]e^{-\frac{E_{b}}{KT}}
e^{-\left[\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm},$$
or in terms of the steady state Kramers rate $k$ $$k(t)=k\exp\left[ -\frac{\Delta {\cal U}}{KT}
e^{-\frac{\gamma}{2} t}\right]\hspace{0.2cm} \; \; ,$$
where $\Delta {\cal U}$ is a measure of the initial depa
| 4,173
| 3,343
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|
C \left( \frac{\sqrt{k}}{\hat{u}_t ^2} + \frac{1}{\hat{u}_t} \right),$$ where $\hat{u}_t \geq (1 - t) \lambda_{\min}(\Sigma_{S}) + t \lambda_{\min}(\hat{\Sigma}_{S})$. By in and Weyl’s theorem, and using the fact that $ u > u_n$, on an event with probability at lest $1 - \frac{1}{n}$, $$\left\| G\left( 1-t)\psi_{S} + t \hat
\psi_{S} \right) \right\| \leq C \left( \frac{\sqrt{k}}{u_n ^2} + \frac{1}{u_n} \right) \leq C \frac{\sqrt{k}}{u_n^2},$$ where in the last inequality we assume $n$ large enough so that $k \geq u_n^2$. The previous bound does not depend on $t$, $j$ or $P$. The result now follows. $\Box$
Appendix 2: Proofs of the results in
=====================================
In all the proofs of the results from , we will condition on the outcome of the sample splitting step, resulting in the random equipartition $\mathcal{I}_{1,n}$ and $\mathcal{I}_{2,n}$ of $\{1,\ldots,2n\}$, and on $\mathcal{D}_{1,n}$. Thus, we can treat the outcome of the model selection and estimation procedure $w_n$ on $\mathcal{D}_{1,n}$ as a fixed. As a result, we regard ${\widehat{S}}$ as a deterministic, non-empty subset of $\{1,\ldots,d\}$ of size by $k < d$ and the projection parameter $\beta_{{\widehat{S}}}$ as a fixed vector of length $k$. Similarly, for the LOCO parameter $\gamma_{{\widehat{S}}}$, the quantities $\widehat{\beta}_{{\widehat{S}}}$ and $\widehat{\beta}_{{\widehat{S}}(j)}$, for $j \in {\widehat{S}}$, which depend on $\mathcal{D}_{1,n}$ also become fixed. Due to the independence of $\mathcal{D}_{1,n}$ and $\mathcal{D}_{2,n}$, all the probabilistic statements made in the proofs are therefore referring to the randomness in $\mathcal{D}_{2,n}$ only. Since all our bounds will depend on $\mathcal{D}_{1,n}$ through the cardinality of ${\widehat{S}}$, which is fixed at $k$, the same bounds will hold uniformly over all possible values taken on by $\mathcal{D}_{1,n}$ and $\mathcal{I}_{1,n}$ and all possible outcomes of all model selection and estimation procedures $w_n \in \mathcal{W}_n$ run on $\mathcal{D}_{1,n}$. In
| 4,174
| 3,268
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| 3,734
| 2,784
| 0.77696
|
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|
mplete hyperbolic structure. Then $\overline{X_{6,\theta}}$ can be regarded as the resultant of the hyperbolic Dehn filling of the complete hyperbolic 3-manifold $\overline{X_6}$, which lies in ${\cal H}(\overline{X_6})$ (for more detail, see [@KojimaNishiYamashita]). Thus we have a map $\Phi_6: \Theta_6 \to {\cal H}(\overline{X_6})$ assigning to each $\theta$ the hyperbolic cone-manifold $\overline{X_{6,\theta}}$.
There are 15 geodesic surfaces in $\overline{X_{6,\theta}}$ represented by a pair of markings $(ij)$ where $i \ne j \in \{1, 2, \ldots, 6\}$. They are uniquely placed on $\overline{X_{6,\theta}}$ since the geodesic representative of a surface within their proper homotopy class is unique if any. Hence a geometric cell decomposition by such surfaces, which consists of 60 hexahedra, is uniquely determined and the shapes of these hexahedra $\Delta_{p, \theta}$ are invariants of the hyperbolic (cone) structure on $\overline{X_{6,\theta}}$. Let ${\cal H}$ be the space of all hexahedra which have the property as $\Delta_{p,\theta}$ described in the first paragraph of this section. Then we have a map from the subset $\Phi_6(\Theta_6)$ of ${\cal H}$ to the direct product of sixty ${\cal H}$’s by listing hyperbolic structures of $\Delta_{p, \theta}$’s.
The parameterization of the space ${\cal H}$ can be given as follows. First let us make the definition of ${\cal H}$ precise. We use the notations of the faces or vertices of the elements in ${\cal H}$ as $\Delta_{p,\theta}$ where $p=\langle 123456\rangle$. Then ${\cal H}$ is the set of all hyperbolic hexahedra having the following properties; (i) it is bounded by six faces $(12),
(23)\ldots, (61)$, (ii) the three faces $(k\,k+1)$ meet orthogonally each other for $k=1,3,5$ and $k=2,4,6$ to make vertices $(12)(34)(56)$ and $(23)(45)(61)$ at finite distance, respectively, (iii) the faces $(k\,k+1)$ and $(k+3\,k+4)$ always meet orthogonally for $k=1,2,3$.
We present a hyperbolic hexahedron in ${\cal H}$ in the projective model located in the unit ball in the E
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we warn the reader that the homomorphism ${\psi_{d,t}}$ is called $\psi_{d,\la_{d+1}-t}$ in \[*loc. cit.*\]. The importance of the homomorphisms ${\psi_{d,t}}$ is in the following.
[ ]{}\[kit\] Suppose $\la$ is a partition of $n$. Then $$S^\la=\bigcap_{\begin{smallmatrix}d\gs1\\1\ls t\ls\la_{d+1}\end{smallmatrix}}\ker({\psi_{d,t}}).$$
This provides a clear strategy for computing ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$: find all linear combinations $\theta$ of the homomorphisms ${\hat\Theta_{T}}$ such that ${\psi_{d,t}}\circ\theta=0$ for every $d,t$. Fortunately, it is known how to compute the composition ${\psi_{d,t}}\circ{\hat\Theta_{T}}$ when $T\in {\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$. For our next few results, we need to introduce some more notation. For any multiset $A$ of positive integers, let $A_i$ denote the number of $i$s in $A$. If $A,B$ are multisets, we write $A\sqcup B$ for the multiset with $(A\sqcup B)_i=A_i+B_i$ for all $i$. Given a row-standard tableau $T$, we write $T^j$ for the multiset of entries in row $j$ of $T$.
[ ]{}\[lemma5\] Suppose $\la,\mu$ are partitions of $n$, $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, $d\in\bbn$ and $1\ls t\ls\la_{d+1}$. Let $\cals$ be the set of all row-standard tableaux which can be obtained from $T$ by replacing $t$ of the entries equal to $d+1$ in $T$ with $d$s. Then $${\psi_{d,t}}\circ\Theta_T=\sum_{S\in\cals}\prod_{j\gs1}\binom{S^j_d}{T^j_d}\Theta_S.$$
The slight difficulty with using this lemma to compute homomorphism spaces is that the tableaux $S$ in the lemma are not always semistandard; so it can be difficult to tell whether a particular linear combination is zero when restricted to $S^\mu$. To circumvent this, we recall another lemma from [@fm] which gives certain linear relations between the homomorphisms ${\hat\Theta_{T}}$, and often enables us to write a homomorphism ${\hat\Theta_{T}}$ in terms of semistandard homomorphisms.
[ ]{}\[lemma7\] Suppose $\mu$ is a partition of $n$ and $\la$ a composit
| 4,176
| 3,065
| 2,831
| 3,760
| 1,733
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interior of an edge, consider a small closed interval $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a $C^r$ trivial bundle, appearing locally around a closed interval in the interior of an edge in the Reeb graph of a map of the class $\mathcal{C}$.
2. At each vertex $p$, consider a small regular neighborhood $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a PL map $q_{C_p}$ over a small regular neighborhood of a vertex in the Reeb graph of a map $f_p$ of $\mathcal{C}$ such that the domain includes no point in the measure zero set of the map $f_p$, or $C^s$-PL equivalent to a PL map over a small regular neighborhood of a vertex in the Reeb graph of a map of $\mathcal{C}$.
A pseudo quotient map of a class $\mathcal{C}$ is said to be [*realized*]{} as a quotient map of the class $\mathcal{C}$ if by identifying the target graphs suitably, we can regard the original map as $C^r$-PL equivalent to the quotient map onto the Reeb graph of a map of the class.
Examples will be presented in the next section.
The main theorem and its proof
==============================
We introduce the main theorem, related to Problem \[prob:1\],
A [*fold*]{} map is a $C^{\infty}$ map at each singular point which is represented as a Morse function and an identity map on a $C^{\infty}$ manifold. A [*special generic*]{} map is a fold map at each singular point which is represented as a natural height function of an unit open ball and an identity map on a $C^{\infty}$ manifold.
Precise explanations on these maps are in [@golubitskyguillemin], [@saeki] and [@saeki2], for example.
For an integer $n>1$, let $L_n$ be a $1$-dimensional polyhedron represented as $${\bigcup}_{k=0}^{n-1} \{(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n}) \mid 0 <r \leq 1 \} \bigcup \{(0,0)\} \subset {\mathbb{R}}^2.$$
![$L_n$.[]{data-label="fig:1"}](ln.eps){width="30mm"}
Let $q$ be a continuous map from a $C^s$ manifold of dimension $m>1$ ont
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| 3,799
| 1,485
| 0.788958
|
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4271.345 10 1173 4.89
10 Jun 20 4272.454 30 198 2.23
11 Jul 06 4288.389 30 184 2.42
12 Jul 07 4289.347 10 1417 5.18
13 Jul 08 4290.337 10 1449 5.60
14 Jul 09 4291.351 30 34 0.30
15 Jul 10 4292.434 30 227 2.27
16 Jul 26 4308.434 30 349 3.12
17 Jul 27 4309.337 10 1071 3.69
18 Jul 30 4312.452 30 134 1.42
19 Jul 31 4313.312 30 492 5.03
20 Aug 01 4314.460 10 885 3.05
21 Aug 10 4323.327 30 46 0.40
22 Aug 13 4326.323 30 721 6.43
23 Aug 14 4327.341 10 1646 5.74
24 Aug 15 4328.315 10 1896 6.53
85.16
----- --------- --------------- ------- -------- --------
: Journal of observations of G 207-9. ‘Exp.’ is the exposure time used.[]{data-label="table:logg207"}
----- --------- --------------- -------- -------- --------
Run UT date Start time Exp. Points Length
no. (2007) (BJD-2450000) (s) (h)
01 Jan 15 4115.614 30 299 2.81
02 Jan 17 4117.622 30 154 1.54
03 Jan 26 4126.615 30 233 2.19
04 Jan 28 4128.544 30 441 4.17
05 Jan 30 4130.528 30 493 4.58
06 Feb 17 4148.551 30 375 3.53
07 Mar 15 4175.283 30 725 9.27
08 Mar 16 4176.279 30 960 9.29
09 Mar 22 4182.356 30 562 6.04
10 Mar 24 4184.496 30 410 3.76
11 Mar 25 4185.398 30 338 3.12
12 Mar 26 4186.282
| 4,178
| 5,834
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| 3,568
| null | null |
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of the central accretion disc, given by $\dot{N}_{\mathrm{ion}}=L/3h\nu_{\mathrm{T}}$ for the assumed spectral shape of $L_\nu\propto \nu^{-1.5}$. Equating this supply rate with the recombination rate, we have $$\frac{L}{3h \nu_{\mathrm{T}}}\frac{\mathcal{F}(\theta)}{4 \pi}
= \int_{R_{\mathrm{in}}}^{r_{\mathrm{HII}}}(\theta)
\alpha_{\mathrm{B}}\,n_{\mathrm{outflow}}^2\,r^2\,{\mathrm{d}}r ,
\label{eq:uve}$$ where $n_{\mathrm{outflow}}$ is given by equation . Performing the integration in equation , we finally get $$\begin{aligned}
r_{\mathrm{HII}}(\theta)
=
\begin{cases}
{\displaystyle}R_{\mathrm{in}} \exp\left[
\frac{L\,\mathcal{F}(\theta)\,T_{\mathrm{HII}}^2}{3\pi h\nu_{\mathrm{T}}\, \alpha_{\mathrm{B}}\, n_\infty^2\, r_{\mathrm{B}}^3\, T_{\mathrm{HI}}^2}
\right]&r_{\mathrm{HII}} < r_{\mathrm{B}}\\[0.4cm]
{\displaystyle}\left[
\frac{L\,\mathcal{F}(\theta)\,T_{\mathrm{HII}}^2}
{\pi h\nu_{\mathrm{T}}\,\alpha_{\mathrm{B}}\,n_\infty^2\,T_{\mathrm{HI}}^2}
- 3r_{\mathrm{B}}^3 \ln\left(\frac{r_{\mathrm{B}}}{R_{\mathrm{in}}}\right)
+ r_{\mathrm{B}}^3
\right]^{1/3} \hspace{-2cm}\\[0.4cm]
&r_{\mathrm{HII}} > r_{\mathrm{B}}
\end{cases}\,.
\label{eq:6}\end{aligned}$$ Here, we show the relation $r=r_{\mathrm{HII}}(\theta)$, or equivalently $\theta=r_{\mathrm{HII}}^{-1}(r)$, in Fig. \[fig:th\_in\_Ds\] with $L =
8.2\,L_{\mathrm{E}}$ (Fig. \[fig:mdot\]). We see that equation qualitatively reproduces $\theta_{\mathrm{inflow}} (r)$ obtained in the simulation with a small deviation of a few degrees at each $r$. Such a deviation mainly comes from approximate modelling of $n_{\mathrm{outflow}}$ in equation . For example, in an outer part of the [H[ii]{} ]{}bubble where helium is not doubly ionized, the gas is no longer heated up to $7\times10^4{\,\mathrm{K}}$ by the ${\mathrm{He^+}}$ photoionization, resulting in the higher density than that estimated by equation with $T_{\mathrm{HII}} = 7 \times 10^4{\,\mathrm{K}}$. Nonetheless, our simple modelling with equation describes the numerical results well.
![The radial d
| 4,179
| 3,448
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| 3,973
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|
----------------------------------------------
$\bar{f}'_i$ & $P^{y^i, y^i y^j y^k x^i}$ & $P^{y^i, y^i x^j x^k y^j y^k}$\
It is straightforward to deduce the rule for the $P$ fluxes that survive the orientifold projection in the IIA theory. With respect to $\Omega_P (-1)^{F_L}$, the fluxes $P^{a,b}$, $P_a^{b_1 b_2 b_3}$ and $P^{a,b_1 ...b_5}$ are even, while the fluxes $P_a^b$, $P^{a,b_1 b_2 b_3}$ and $P_a^{b_1 ...b_5}$ are odd. For the fluxes that are even, one must have an even number of $x$ and an even number of $y$ indices, while for the fluxes that are odd both the number of $x$ and $y$ indices have to be odd. The indices of the fluxes $P_a^{b_1 ....b_p}$ are always grouped in pairs of indices belonging to a given torus, while for the fluxes $P^{a,b_1 ...b_p}$ there is always one index (the $a$ index) which is always an $x$ index and it is repeated, while again the other indices are grouped in pairs belonging to a given torus.
The $a$ index in the fluxes $P_a^{b_1 ...b_p}$ in the IIA upper half of Table \[allPfluxes\] is in all cases a $y$ index, but by analogy with Table \[TableNSfluxes\] we assume that it can also be $x$, provided that again all indices are grouped in pairs as before. This leads to the first three fluxes in the lower half of Table \[allPfluxes\]. Now, by performing again three T-dualities along the $x$ directions and using the T-duality rules of eq. we see that in the IIB theory these fluxes are mapped to $P^{a, b_1 ...b_4}$. In particular, the $a$ index is $x$ and it is repeated, while the other three indices are each on a different torus. Given that in the IIB theory the orientifold projection acts in the same way on the $x$ and $y$ indices, we include the last three fluxes in the IIB side of the lower half of Table \[allPfluxes\], which are mapped in the IIA setting to the fluxes $P^{a,b_1 ...b_p}$ where now the repeated index is a $y$ index and all the others are again grouped pairwise as before.
We can now write down the superpotential which contains all the $P$ fluxes compatibl
| 4,180
| 2,643
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| 3,676
| 3,364
| 0.772852
|
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|
_0,\dots,v_j\\ v_l\ne
v_{l'}\,{{}^\forall}l\ne l'\\ v_j=v}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_0}\,\bigg(\prod_{l\ge0}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2
l}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}v_{2l+1}\}$}}}\bigg){\nonumber}\\
&\qquad\times\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_{2l}\}$}}}\bigg)\bigg(
\prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf m}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf m}}(v_{2l'})={\varnothing}\}$}}}\;{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf n}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf n}}(v_{2l'})={\varnothing}\}$}}}\bigg).\end{aligned}$$ For the three products of indicators, we repeate the same argument as in [(\[eq:psi-delta-G2\])]{}–[(\[eq:nsum-2ndbd\])]{} to derive the factor $\psi_\Lambda(v_0,v)-\delta_{v_0,v}$. As a result, we have $$\begin{aligned}
{\label{eq:Theta''-prebd1stbd2}}
&\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}
\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptsty
| 4,181
| 3,125
| 2,576
| 3,773
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|
into $L_{i+1}\oplus L_{i+2}\oplus\dots$. This completes the inductive step, and thus produces the wanted derivation $d$ on $F_{\mathcal Lie}(C[1])$.
In a similar way, we may produce the derivation $g$ of $F_{\mathcal Lie,\,C[1]}C[1]$ over $d$, by decomposing $F_{\mathcal Lie, C[1]} C[1]=L'_1\oplus L'_2\oplus\dots$, where $L'_n$ is given by the space $\left(\bigoplus_{k+l=n-1} \mathcal Lie(n) \otimes C[1]^{\otimes k} \otimes C[1] \otimes C[1]^{\otimes l}\right)_{S_{n}}$. With this notation, $g$ is written as a sum $g=g_1+g_2+\dots$, where $g_i:C[1]\to L'_i$ is lifted to $F_{\mathcal Lie, C[1]} C[1]$ as a derivation over $d$, and $(g_1+\dots+g_i)^2$ only maps into $L'_{i+1}\oplus L'_{i+2}\oplus\dots$.
Using a slight variation of the above method, we may also construct the wanted homotopy $\mathcal Comm$-inner product, i.e. the module map $f$ stated above. More precisely, we build a map $\chi:C[1]\to Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$, so that $\chi$ is a chain map under the differential $d_1$ on $C[1]$, and the differential $\delta(f)=f\circ h - (-1)^{|f|} g\circ f$ on $Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$. Since a module map is given by the components $M_n=\bigoplus_{k+l=n-2} \mathcal Lie(n)\otimes C[1]^{\otimes k}\otimes C[1]\otimes C[1]^{\otimes l}\otimes C[1]$, it is enough to construct $\chi$ as a sum $\chi=\chi_2+\chi_3+\dots$, where $\chi_i:C[1]\to M_i$. Now, the lowest component $\chi_2:C[1]\to C[1]\otimes C[1]$ is defined to be the symmetrized Alexander-Whitney comultiplication. For the induction, we assume that $\Upsilon_i:=\chi_2+\dots+\chi_{i-1}$ are local maps such that $D(\Upsilon_i):=\Upsilon_i\circ d_1-\delta\circ \Upsilon_i$ maps only into higher components $M_i\oplus M_{i+1}\oplus\dots$. Let $\epsilon_i:C[1]\to M_i$ be the lowest term of $D(\Upsilon_i)$. Since $D^2=0$ and $\delta$ has $d_1$ as its lowest component, we see that $\epsilon_i$ is $[d_1,.]$-closed, and by the hypothesis of the proposition also locally $[d_1,.]$-exact. These exact te
| 4,182
| 3,806
| 3,135
| 3,658
| 2,054
| 0.783176
|
github_plus_top10pct_by_avg
|
(\Gamma(G))}$. Moreover, if $r \in \pi(G) \setminus \pi(G/N)$, then $r$ is adjacent to $p$ in $\Gamma(N)$.
By our hypotheses, for any prime $r \in \pi(G) \setminus \{p\}$ there exists an element $x \in A \cup B$ of order $r$ such that $x \in C_G(P)$, for some $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Hence the first equality follows. Now, observe that $\pi(G)=\pi(G/N) \cup \pi(N)$ and, since $G=AN=BN=AB$, after some computations we also obtain $${\ensuremath{\left| N \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}.$$ But again our hypotheses lead to $\pi((N \cap A) \cup (N \cap B))\setminus \{p\} \subseteq \pi(C_{N}(P))$. Also, since $p \in \pi(N)$, $1 \neq Z(P) \cap N \leq C_{N}(P)$. Hence the second equality also holds.
It is clear then that $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$. Assume now that $r \in \pi(G) \setminus \pi(G/N)$, and so $r \in \pi(N)$. By the second equality, we deduce that $r \in \pi(C_N(P))$ and since $p \in \pi(N)$ the last assertion follows.
\[notan\] $N$ is not an alternating group $A_n$.
Let $N=A_n$ and assume first $n\neq 6$, so $G=N=A_{n}$ or $G=\Sigma_n$. Because our hypotheses, we may assume that $\Gamma(G)$ is connected. As in Lemma \[angraph\], let $k\geq 2$ be the largest positive integer such that $\{n, n-1, \ldots, n-k+1\}$ are consecutive composite numbers, $r:=n-k$ the largest prime divisor of $n!$, and $t$ the largest prime with $t \leq k$. Since $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$, then $p\leq t$ by Lemma \[angraph\], and so $r>\frac{n}{2}>k\geq t\geq p$.
We claim that $r\notin\pi({{\operatorname}{C}_{G}(P)})$, for $P\in{{\operatorname}{Syl}_{p}\left(G\right)}$. Let suppose first that $p\neq 2$. Assume that there exists an element $x\in G$ of order $r$ such that $P\leq {{\operatorname}{C}_{G}(x)}$. Since ${{\operatorname}{C}_{G}(x)}$ is isomorphic to a subgroup of $C_r\times \Sigma_{n-r}$ and $p\neq r$, then ${\ensuremath{\
| 4,183
| 3,302
| 1,687
| 4,032
| null | null |
github_plus_top10pct_by_avg
|
first term if the source constraints for ${{\bf m}}$ and ${{\bf n}}$ are exchanged.
Next, we consider the second term of [(\[eq:WZ-num\])]{}, whose exact expression is $$\begin{gathered}
{\label{eq:2ndterm-expl}}
\sum_{\substack{{\partial}{{\bf m}}=\{v,x\},\,{\partial}{{\bf n}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus
{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\bigg(\prod_{b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{(pJ_b)^{n_b}}{n_b!}\bigg)\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{
(pJ_b)^{m_b+n_b}}{m_b!\,n_b!}=\sum_{{\partial}{{\bf N}}=\{v,x\}}w_\Lambda({{\bf N}})\sum_{
\substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}
\equiv0}}\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{gathered}$$ The following is a variant of the source-switching lemma [@a82; @ghs70] and allows us to change the source constraints in [(\[eq:2ndterm-expl\])]{}.
\[lmm:switching\] $$\begin{aligned}
{\label{eq:switching}}
\sum_{\substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {{\bf m}}|_{
{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{aligned}$$
The idea of the proof of [(\[eq:switching\])]{} can easily be extended to more general cases, in which the source constraint in the left-hand side of [(\[eq:switching\])]{} is replaced by ${\partial}{{\bf m}}={{\cal V}}$ for some ${{\cal V}}\subset\Lambda$ and that in the right-hand side is replaced by ${\partial}{{\bf m}}={{\cal V}}{\,\triangle\,}\{v,x\}$ (e.g., [@a
| 4,184
| 2,194
| 3,344
| 4,007
| null | null |
github_plus_top10pct_by_avg
|
ome about $~10^{-10}$. In this way, the number of terms kept in the sum may vary in different plots as it is clearly depends on the specific set of parameters used to produce the plot. We indicate the truncation number in the caption of each figure.
---
abstract: 'We review the stabilization of the radion in the Randall–Sundrum model through the Casimir energy due to a bulk conformally coupled field. We also show some exact self–consistent solutions taking into account the backreaction that this energy induces on the geometry.'
address: |
IFAE, Departament de F[í]{}sica, Universitat Aut[ò]{}noma de Barcelona,\
08193 Bellaterra $($Barcelona$)$, Spain
author:
- 'Oriol Pujol[à]{}s\'
title: 'Effective potential in Brane-World scenarios'
---
epsf
**Abstract**
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ UAB-FT-504
Introduction
============
Recently, it has been suggested that theories with extra dimensions may provide a solution to the hierarchy problem [@gia; @RS1]. The idea is to introduce a $d$-dimensional internal space of large physical volume ${\cal V}$, so that the the effective lower dimensional Planck mass $m_{pl}\sim {\cal V}^{1/2} M^{(d+2)/2}$ is much larger than $M \sim TeV$- the true fundamental scale of the theory. In the original scenarios, only gravity was allowed to propagate in the higher dimensional bulk, whereas all other matter fields were confined to live on a lower dimensional brane. Randall and Sundrum [@RS1] (RS) introduced a particularly attractive model where the gravitational field created by the branes is taken into account. Their background solution consists of two parallel flat branes, one with positive tension and another one with negative tension embedded in a a five-dimensional Anti-de Sitter (AdS) bulk. In this model, the hierarchy problem is solved if the distance between branes is about $37$ times the AdS radius and we live on the negative tension brane. More recently, scenarios where additional fields propagate in the bulk
| 4,185
| 2,890
| 1,348
| 3,949
| null | null |
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|
t_1} \mu_1 \xrightarrow{t_2} \ldots \xrightarrow{t_k} \mu_k$. For each $1\leq i\leq k$, marking $\mu_i$ is called *reachable* from marking $\mu$. $\mathcal{R}(N, \mu)$ denotes the set of all reachable markings from a marking $\mu$.
A *marked* Petri net is a system $N=(P, T, F, \phi, \iota)$ where $(P, T, F, \phi)$ is a Petri net, $\iota$ is the *initial marking*. Let $M$ be a set of markings, which will be called *final* markings. An occurrence sequence $\nu$ of transitions is called *successful* for $M$ if it is enabled at the initial marking $\iota$ and finished at a final marking $\tau$ of $M$.
A Petri net $N$ is said to be $k$-*bounded* if the number of tokens in each place does not exceed a finite number $k$ for any marking reachable from the initial marking $\iota$, i.e., $\mu(p)\leq k$ for all $p\in P$ and for all $\mu\in \mathcal{R}(N, \iota)$. A Petri net is called *bounded* if it is $k$-bounded for some $k\geq 1$.
A Petri net with *place capacity* is a system $N=(P, T, F, \phi, \iota,\kappa)$ where $(P, T, F, \phi,\iota)$ is a marked Petri net and $\kappa:P \to {\mathbb{N}}$ is a function assigning to each place a number of maximal admissible tokens. A marking $\mu$ of $N$ is valid if $\mu(p)\leq \kappa(p)$, for each place $p\in P$. A transition $t \in T$ is *enabled* by a marking $\mu$ if additionally the successor marking is valid.
A *cf Petri net* with respect to a context-free grammar $G=(V,\Sigma, S, R)$ is a system $$N=(P, T, F, \phi, \beta, \gamma, \iota)$$ where
- labeling functions $\beta:P\rightarrow V$ and $\gamma:T\rightarrow R$ are bijections;
- $(p,t)\in F$ iff $\gamma(t)=A\rightarrow \alpha$ and $\beta(p)=A$ and the weight of the arc $(p,t)$ is 1;
- $(t,p)\in F$ iff $\gamma(t)=A\rightarrow \alpha$, $\beta(p)=x$ where $|\alpha|_x>0$ and the weight of the arc $(t,p)$ is $|\alpha|_x$;
- the initial marking $\iota$ is defined by $\iota(\beta^{-1}(S))= 1$ and $\iota(p) = 0$ for all $p\in P-\beta^{-1}(S)$.
Further we recall the definitions of extended cf Petri nets, and grammar
| 4,186
| 3,153
| 4,348
| 3,971
| 1,829
| 0.785299
|
github_plus_top10pct_by_avg
|
){width="85.00000%"}
### $P(\nu_{\mu} \rightarrow \nu_{e})$
In figure \[fig:Pmue\_energy\_dist\]-(a) (upper panel) and (b) (lower panel), presented are the iso-contours of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ and $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in $E - L$ space. Here, the superscript $(0)$ implies that it is calculated in zeroth-order in $W$ using (\[Schroedinger-eq-0th\]) with appropriate initial condition and final projection to flavor eigenstate. In most of the $E - L$ space $P(\nu_{\mu} \rightarrow \nu_{e})$ is small. However, we identify the two regions where $P(\nu_{\mu} \rightarrow \nu_{e})$ is relatively large, $\gsim 0.3$. One of them is at low energy, $E \lsim \text{a few hundred}$ MeV, and baseline $L \gsim$1000 km. The other one is a region $E \sim 10$ GeV and $L \sim$ 10000 km. The former may be understood as due to the solar MSW enhancement, and the latter as the atmospheric MSW enhancement [@Mikheev:1986gs; @Wolfenstein:1977ue]. Roughly speaking, the regions with relatively large $| \Delta P (\nu_{\mu} \rightarrow \nu_{e}) |$ overlap with these regions.
### $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$
In figures \[fig:Pmutau\_energy\_dist\] and \[fig:Pmumu\_energy\_dist\], the same quantities (in each upper (a) and lower (b) panel) are presented but in $\nu_{\mu} \rightarrow \nu_{\tau}$ and $\nu_{\mu} \rightarrow \nu_{\mu}$ channels, respectively. In contrast to $\nu_{\mu} \rightarrow \nu_{e}$ channel, $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ contours are globally “vacuum effect dominated”, apart from the solar MSW region, both in the standard (not shown) and the non-unitary cases. The first oscillation peak of $P(\nu_{\mu} \rightarrow \nu_{\tau})$ scales roughly as the vacuum oscillation peak does, $L / 10^3 \,\text{km} = 0.33 E / 1 \, \text{GeV}$. This f
| 4,187
| 3,434
| 4,309
| 4,013
| 2,485
| 0.779378
|
github_plus_top10pct_by_avg
|
/*!
* Bootstrap Grunt task for generating raw-files.min.js for the Customizer
* http://getbootstrap.com
* Copyright 2014-2015 Twitter, Inc.
* Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE)
*/
'use strict';
var fs = require('fs');
var btoa = require('btoa');
var glob = require('glob');
function getFiles(type) {
var files = {};
var recursive = type === 'less';
var globExpr = recursive ? '/**/*' : '/*';
glob.sync(type + globExpr)
.filter(function (path) {
return type === 'fonts' ? true : new RegExp('\\.' + type + '$').test(path);
})
.forEach(function (fullPath) {
var relativePath = fullPath.replace(/^[^/]+\//, '');
files[relativePath] = type === 'fonts' ? btoa(fs.readFileSync(fullPath)) : fs.readFileSync(fullPath, 'utf8');
});
return 'var __' + type + ' = ' + JSON.stringify(files) + '\n';
}
module.exports = function generateRawFilesJs(grunt, banner) {
if (!banner) {
banner = '';
}
var dirs = ['js', 'less', 'fonts'];
var files = banner + dirs.map(getFiles).reduce(function (combined, file) {
return combined + file;
}, '');
var rawFilesJs = 'docs/assets/js/raw-files.min.js';
try {
fs.writeFileSync(rawFilesJs, files);
} catch (err) {
grunt.fail.warn(err);
}
grunt.log.writeln('File ' + rawFilesJs.cyan + ' created.');
};
/*
* MaxentStressGTest.cpp
*
* Created on: Apr 19, 2016
* Author: Michael
*/
#include <gtest/gtest.h>
#include <vector>
#include <string>
#include <networkit/graph/Graph.hpp>
#include <networkit/viz/Point.hpp>
#include <networkit/io/METISGraphReader.hpp>
#include <networkit/io/METISGraphWriter.hpp>
#include <networkit/graph/Graph.hpp>
#include <networkit/components/ConnectedComponents.hpp>
#include <networkit/viz/MaxentStress.hpp>
#include <networkit/numerics/LAMG/Lamg.hpp>
#include <networkit/numerics/ConjugateGradient.hpp>
#include <networkit/numerics/Preconditioner/IdentityPreconditioner.hpp>
#include <networkit/numerics/Preconditioner/DiagonalPreconditioner.hpp>
#inclu
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
de <networkit/community/PLM.hpp>
#include <networkit/sparsification/LocalDegreeScore.hpp>
#include <networkit/sparsification/RandomEdgeScore.hpp>
#include <networkit/sparsification/GlobalThresholdFilter.hpp>
#include <networkit/auxiliary/Timer.hpp>
#include <networkit/auxiliary/Random.hpp>
#include <iostream>
#include <unordered_map>
#include <random>
#include <networkit/viz/PivotMDS.hpp>
#include <cstdio>
namespace NetworKit {
class MaxentStressGTest : public testing::Test {};
TEST_F(MaxentStressGTest, benchMaxentStressCoordinatesLAMG) {
std::vector<std::string> graphFiles = {"input/airfoil1.graph"};
METISGraphReader reader;
for (const std::string& graphFile : graphFiles) {
Graph graph = reader.read(graphFile);
double runtime = 0;
double fullStress = 0;
double maxentStress = 0;
Aux::Random::setSeed(Aux::Random::integer(), false);
Aux::Timer t;
t.start();
PivotMDS pivotMds(graph, 2, 30);
pivotMds.run();
MaxentStress maxentStressAlgo(graph, 2, pivotMds.getCoordinates(), 1, 0.001, MaxentStress::LinearSolverType::LAMG);
maxentStressAlgo.run();
t.stop();
runtime = t.elapsedMicroseconds();
if (graph.numberOfNodes() < 1e5) {
maxentStressAlgo.scaleLayout();
fullStress = maxentStressAlgo.fullStressMeasure();
maxentStress = maxentStressAlgo.maxentMeasure();
}
runtime /= 1000;
INFO(graphFile, "\t", maxentStress, "\t", fullStress, "\t", runtime);
}
}
TEST_F(MaxentStressGTest, benchMaxentStressConjGradIdPrecAlgebraicDistance) {
std::vector<std::string> graphFiles = {"input/airfoil1.graph"};
METISGraphReader reader;
for (const std::string& graphFile : graphFiles) {
Graph graph = reader.read(graphFile);
double runtime = 0;
double fullStress = 0;
double maxentStress = 0;
Aux::Random::setSeed(Aux::Random::integer(), false);
Aux::Timer t;
t.start();
M
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
axentStress maxentStressAlgo(graph, 2, 1, 0.001, MaxentStress::LinearSolverType::CONJUGATE_GRADIENT_IDENTITY_PRECONDITIONER, true, MaxentStress::GraphDistance::ALGEBRAIC_DISTANCE);
maxentStressAlgo.run();
t.stop();
runtime = t.elapsedMicroseconds();
if (graph.numberOfNodes() < 1e5) {
maxentStressAlgo.scaleLayout();
fullStress = maxentStressAlgo.fullStressMeasure();
maxentStress = maxentStressAlgo.maxentMeasure();
}
runtime /= 1000;
INFO(graphFile, "\t", maxentStress, "\t", fullStress, "\t", runtime);
}
}
TEST_F(MaxentStressGTest, benchMaxentStressConjGradDiagPrecond) {
std::vector<std::string> graphFiles = {"input/airfoil1.graph"};
METISGraphReader reader;
for (const std::string& graphFile : graphFiles) {
Graph graph = reader.read(graphFile);
double runtime = 0;
double fullStress = 0;
double maxentStress = 0;
Aux::Random::setSeed(Aux::Random::integer(), false);
Aux::Timer t;
t.start();
MaxentStress maxentStressAlgo(graph, 2, 1, 0.001, MaxentStress::LinearSolverType::CONJUGATE_GRADIENT_DIAGONAL_PRECONDITIONER);
maxentStressAlgo.run();
t.stop();
runtime = t.elapsedMicroseconds();
if (graph.numberOfNodes() < 1e5) {
maxentStressAlgo.scaleLayout();
fullStress = maxentStressAlgo.fullStressMeasure();
maxentStress = maxentStressAlgo.maxentMeasure();
}
runtime /= 1000;
INFO(graphFile, "\t", maxentStress, "\t", fullStress, "\t", runtime);
}
}
TEST_F(MaxentStressGTest, benchMaxentStressCoordConjGradIdPrecond) {
std::vector<std::string> graphFiles = {"input/airfoil1.graph"};
METISGraphReader reader;
for (const std::string& graphFile : graphFiles) {
Graph graph = reader.read(graphFile);
double runtime = 0;
double fullStress = 0;
double maxentStress = 0;
Aux::Random::setSeed(Aux::Random::integer(), fal
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
se);
Aux::Timer t;
t.start();
PivotMDS pivotMds(graph, 2, 30);
pivotMds.run();
MaxentStress maxentStressAlgo(graph, 2, pivotMds.getCoordinates(), 1, 0.001, MaxentStress::LinearSolverType::CONJUGATE_GRADIENT_IDENTITY_PRECONDITIONER, false, MaxentStress::GraphDistance::ALGEBRAIC_DISTANCE);
maxentStressAlgo.run();
t.stop();
runtime = t.elapsedMicroseconds();
if (graph.numberOfNodes() < 1e5) {
maxentStressAlgo.scaleLayout();
fullStress = maxentStressAlgo.fullStressMeasure();
maxentStress = maxentStressAlgo.maxentMeasure();
}
runtime /= 1000;
INFO(graphFile, "\t", maxentStress, "\t", fullStress, "\t", runtime);
}
}
} /* namespace NetworKit */
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import init.data.sigma.basic init.data.sigma.lex
#include <Python.h>
#include "script/syntax.h"
#include "app/colors.h"
ScriptHighlighter::ScriptHighlighter(QTextDocument* doc)
: QSyntaxHighlighter(doc)
{
PyObject* kwmod = PyImport_ImportModule("keyword");
PyObject* kwlist = PyObject_GetAttrString(kwmod, "kwlist");
QList<QString> keywords = {"input", "output", "title", "meta"};
// Get all of Python's keywords and add them to a list.
for (int i=0; i < PyList_Size(kwlist); ++i)
{
PyObject* kw = PyList_GetItem(kwlist, i);
wchar_t* w = PyUnicode_AsWideCharString(kw, NULL);
keywords << QString::fromWCharArray(w);
PyMem_Free(w);
}
Py_DECREF(kwlist);
Py_DECREF(kwmod);
// Make rules for all the Python keywords.
QTextCharFormat kw_format;
kw_format.setForeground(Colors::green);
for (auto k : keywords)
rules << Rule("\\b" + k + "\\b", kw_format);
QTextCharFormat quote_format;
quote_format.setForeground(Colors::brown);
// Triple-quoted (multiline) strings
// Si
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
ngle-line triple-quoted string
rules << Rule("'''.*?'''", quote_format);
rules << Rule("\"\"\".*?\"\"\"", quote_format);
// Beginning of multiline string
rules << Rule("'''.*$", quote_format, BASE, MULTILINE_SINGLE);
rules << Rule("\"\"\".*$", quote_format, BASE, MULTILINE_DOUBLE);
// End of multiline string
rules << Rule("^.*'''", quote_format, MULTILINE_SINGLE, BASE);
rules << Rule("^.*\"\"\"", quote_format, MULTILINE_DOUBLE, BASE);
// Inside of multiline string
rules << Rule("^.+$", quote_format, MULTILINE_SINGLE, MULTILINE_SINGLE);
rules << Rule("^.+$", quote_format, MULTILINE_DOUBLE, MULTILINE_DOUBLE);
// Regular strings
rules << Rule("\".*?\"", quote_format);
rules << Rule("'.*?'", quote_format);
// String that can be prepended to a regex to make it detect negative
// numbers (but not subtraction). Note that a closing parenthesis is
// needed and the desired number is the last match group.
QString neg = "(^|\\*\\*|[(+\\-=*\\/,\\[])([+\\-\\s]*";
QTextCharFormat float_format;
float_format.setForeground(Colors::yellow);
rules << Rule(neg + "\\b\\d+\\.\\d*)", float_format);
rules << Rule(neg + "\\b\\d+\\.\\d*e\\d+)", float_format);
rules << Rule(neg + "\\b\\d+e\\d+)", float_format);
QTextCharFormat int_format;
int_format.setForeground(Colors::orange);
rules << Rule(neg + "\\b\\d+\\b)", int_format);
QTextCharFormat comment_format;
comment_format.setForeground(Colors::base03);
rules << Rule("#.*", comment_format);
}
////////////////////////////////////////////////////////////////////////////////
void ScriptHighlighter::highlightBlock(const QString& text)
{
int offset = 0;
int state = previousBlockState();
while (offset <= text.length())
{
int match_start = -1;
int match_length;
Rule rule;
for (auto r : rules)
{
if (r.state_in != state)
continue;
auto match = r.regex.match(text, offset);
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
if (!match.hasMatch())
continue;
auto index = match.lastCapturedIndex();
if (match_start == -1 || match.capturedStart(index) < match_start)
{
match_start = match.capturedStart(index);
match_length = match.capturedLength(index);
rule = r;
}
}
if (match_start == -1)
break;
setFormat(match_start, match_length, rule.format);
offset = match_start + match_length;
state = rule.state_out;
}
setCurrentBlockState(state);
}
> Task :compileJava FROM-CACHE
// Jest Snapshot v1, https://goo.gl/fbAQLP
exports[`should affect error message after formatting 1`] = `
"
<<<<<< before
no error
======
ERROR: (ter-padded-blocks) /src/tslint-eslint-rules/ter-padded-blocks/test.ts[1, 11]: Block must be padded by blank lines.
ERROR: (ter-padded-blocks) /src/tslint-eslint-rules/ter-padded-blocks/test.ts[3, 1]: Block must be padded by blank lines.
>>>>>> after
"
`;
exports[`should be pretty after formatting 1`] = `
"
<<<<<< before
if (true) {
somebody.doSomething();
}
======
if (true) {
somebody.doSomething();
}
>>>>>> after
"
`;
/*
* Copyright 2009-2012 Amazon Technologies, Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at:
*
* http://aws.amazon.com/apache2.0
*
* This file is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES
* OR CONDITIONS OF ANY KIND, either express or implied. See the
* License for the specific language governing permissions and
* limitations under the License.
*/
package com.amazonaws.eclipse.ec2;
/**
* Represents the results of an attempt to copy a local file to a remote host.
*/
public class RemoteFileCopyResults {
/** The local file being copied */
private final String localFile;
/** The remote destination file */
private final String remote
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
File;
/** True if the file was successfully copied to the remote location */
private boolean succeeded;
/** An optional error message if the remote file copy was not successful */
private String errorMessage;
/** An optional exception describing why this attempt failed */
private Exception error;
/**
* An optional string containing the external output from the copy command
* such as any error messages from the command used for the copy
*/
private String externalOutput;
/**
* Creates a new RemoteFileCopyResults object describing the results of
* copying the specified local file to the specified remote file location.
*
* @param localFile
* The local file being copied.
* @param remoteFile
* The remote file location.
*/
public RemoteFileCopyResults(String localFile, String remoteFile) {
this.localFile = localFile;
this.remoteFile = remoteFile;
}
/**
* @return The local file location.
*/
public String getLocalFile() {
return localFile;
}
/**
* @return The remote file location.
*/
public String getRemoteFile() {
return remoteFile;
}
/**
* @param errorMessage
* the error message describing how this remote file copy failed.
*/
public void setErrorMessage(String errorMessage) {
this.errorMessage = errorMessage;
}
/**
* @return the error message describing how this remote file copy attempt
* failed.
*/
public String getErrorMessage() {
if (externalOutput == null || externalOutput.trim().length() == 0) {
return errorMessage;
}
return errorMessage + ":\n\t" + externalOutput;
}
/**
* @param wasSuccessful
* True if the file was successfully copied to the remote file
* location.
*/
public void setSucceeded(boolean wasSuccessful) {
| null | null | null | null | null | null |
github_plus_top10pct_by_avg
|
this.succeeded = wasSuccessful;
}
/**
* @return True if the file was successfully copied to the remote file
* location.
*/
public boolean isSucceeded() {
return succeeded;
}
/**
* @param error the error to set
*/
public void setError(Exception error) {
this.error = error;
}
/**
* @return the error
*/
public Exception getError() {
return error;
}
/**
* @param externalOutput
* The command output from the attempt to copy the file to the
* remote location.
*/
public void setExternalOutput(String externalOutput) {
this.externalOutput = externalOutput;
}
}
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