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_1(S)g_2(S)\}=g_2(S)\Dc_S g_1(S)+g_1(S) \Dc_S g_2(S).$$
Denote by $S=HLH^\top$ the eigenvalue decomposition of $S$, where $H=(h_{ij})$ is an orthogonal matrix of order $r$ and $L=\diag(\ell_1,\ldots,\ell_r)$ is a diagonal matrix of order $r$ with $\ell_1\geq \cdots\geq \ell_r$. The following lemma is provided by Stein (1973).
\[lem:diff2\] Define $\Psi(L)=\diag(\psi_1,\ldots,\psi_r)$, whose diagonal elements are differentiable functions of $L$. Then we obtain
1. $\{\Dc_S\}_{ij} \ell_k=h_{ik}h_{jk}$ $(k=1,\ldots,r)$,
2. $\Dc_S H\Psi(L)H^t=H\Psi^*(L)H^t$, where $\Psi^*(L)=\diag(\psi_1^*,\ldots,\psi_r^*)$ with $$\psi_i^*=\frac{\partial \psi_i}{\partial\ell_i}+\frac{1}{2}\sum_{j\ne i}^r\frac{\psi_i-\psi_j}{\ell_i-\ell_j}.$$
\[lem:diff3\] Let $a$ and $b$ be constants and let $C$ be a symmetric constant matrix $C$. Then it holds that
1. $\Dc_S \tr(S C)=C$,
2. $\displaystyle \Dc_S S=\frac{r+1}{2}I_r$,
3. $\displaystyle \Dc_S S^2=\frac{r+2}{2}S+\frac{1}{2}(\tr S)I_r$.
4. $\Dc_S |aI_r+bS|=b|aI_r+bS|(aI_r+bS)^{-1}$ if $aI_r+bS$ is nonsingular.
[**Proof.**]{} For proofs of Parts (i), (ii) and (iii), see Haff (1982) and Magnus and Neudecker (1999). Using (i) of Lemma \[lem:diff2\] gives that $$\begin{aligned}
\{\Dc_S |aI_r+bS|\}_{ij} &= \{\Dc_S\}_{ij} \prod_{k=1}^r(a+b\ell_k) \\
&=b\sum_{c=1}^r h_{ic}h_{jc}\prod_{k\ne c}^r(a+b\ell_k) \\
&=b|aI_r+bS|\sum_{c=1}^r h_{ic}h_{jc}(a+b\ell_c)^{-1} \\
&=b|aI_r+bS|\{(aI_r+bS)^{-1}\}_{ij},\end{aligned}$$ which implies Part (iv).
Let $\nabla_W$ be the same $r\times q$ differentiation operator matrix as in the preceding subsection. If $S=WW^\top$, then we have the following lemma, where the proof is referred to in Konno (1992).
\[lem:diff4\] Let $G$ be an $r\times r$ symmetric matrix, where all the elements of $G$ are differentiable function of $S=WW^\top$. Then it holds that
1. $\nabla_W^\top G=2W^\top \Dc_S G$,
2. $\tr(\nabla_W W^\top G)=(q-r-1)\tr G+2\tr(\Dc_S SG)$.
Admissible and minimax predictive densities {#sec:properminimax}
=========================
| 1,501
| 1,458
| 1,005
| 1,395
| 1,913
| 0.784437
|
github_plus_top10pct_by_avg
|
of an STM, the TAMR can be obtained by measuring the differential conductance (d$I$/d$V$) above an adatom or a dimer for two different magnetization directions. The TAMR is obtained from $$\begin{aligned}
\mathrm{TAMR} &= \frac{[(\mathrm{d}I/\mathrm{d}V)_\perp-(\mathrm{d}I/\mathrm{d}V)_\parallel]}{(\mathrm{d}I/\mathrm{d}V)_\perp}\; ,\end{aligned}$$ where $\perp$ and $\parallel$ denote a perpendicular magnetization and a parallel magnetization with respect to the surface, respectively. Within the Tersoff-Hamann model [@Tersoff1983; @Tersoff1985], the d$I$/d$V$ signal is directly proportional to the local density of states (LDOS), $n(z, \epsilon)$, at the tip position in the vacuum, $z$, a few [Å]{}ngströms above the surface. Hence, the TAMR can be calculated theoretically from the anisotropy of the LDOS arising due to SOC [@Bode2002; @Neel2013]. Then the TAMR can be calculated as: $$\begin{aligned}
\mathrm{TAMR} &= \frac{n_\perp(z, \epsilon)-n_\parallel(z, \epsilon)}{n_\perp(z, \epsilon)}\; .
\label{eq:TAMR}\end{aligned}$$
Results and Discussion {#sec:results}
======================
Pb and Bi adatoms on Mn/W(110) {#sbsec:adatoms}
------------------------------
### Structural and magnetic properties {#sub2sec:str_mag}
---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ -------------------------
$d_{\text{nn}}$ $d_{\text{nnn}}$ $\Delta z_{\text{nn}}$ $\Delta z_{\text{nnn}}$ $\Delta x_{\text{nn}}$ $\Delta y_{\text{nnn}}$
Pb 2.76 3.17 $-0.13$ +0.02 $-0.01$ +0.02
Bi 2.70 2.97 $-0.10$ +0.06 $-0.05$ +0.03
---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ -------------------------
: Relaxed distances (in [Å]{}) of Pb and Bi adatoms from the Mn atoms of the Mn/W(110) sur
| 1,502
| 473
| 1,633
| 1,765
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|
(1)}_{m,n}}{P^{(2)}_{m,n}} - 1,\end{gathered}$$ where $P^{(i)}_{m,n}$ are given in (\[eq:N5-P-G\]), maps system (\[eq:sys-5-red\]) to $$\begin{gathered}
\frac{\psi^{(0)}_{m,n+1}\psi^{(0)}_{m+1,n+1}+\psi^{(0)}_{m+1,n} \big(\psi^{(0)}_{m,n}+\psi^{(1)}_{m,n+1}\big)}{\psi^{(1)}_{m,n}+1} = \frac{\psi^{(0)}_{m,n+1}\psi^{(1)}_{m+1,n}-\psi^{(0)}_{m+1,n}\psi^{(1)}_{m,n+1}}{\psi^{(0)}_{m,n+1} + \psi^{(1)}_{m,n+1}}, \nonumber\\
\frac{\psi^{(1)}_{m+1,n+1}+1}{\psi^{(1)}_{m,n}+\psi^{(1)}_{m,n+1}+\psi^{(0)}_{m,n+1}+1} + \frac{\psi^{(0)}_{m+1,n}+\psi^{(1)}_{m+1,n}}{\psi^{(0)}_{m,n+1}+\psi^{(1)}_{m,n+1}} = 0,\label{eq:M-red-sys-5}\end{gathered}$$ and its symmetry (\[eq:sym-self-dual-5a\]) to the following system of polynomial equations in which we have suppressed the dependence on the second index $n$: $$\begin{gathered}
\frac{\partial_{t_2} \psi^{(0)}_m}{\psi^{(0)}_m} = \big(\psi^{(0)}_m+\psi^{(1)}_m\big) \big(\psi^{(0)}_{m+2} \psi^{(0)}_{m+1} - \psi^{(0)}_{m-1} \psi^{(0)}_{m-2} + \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} + \psi^{(1)}_{m+1}-\psi^{(1)}_{m-1}\big) \nonumber \\
\hphantom{\frac{\partial_{t_2} \psi^{(0)}_m}{\psi^{(0)}_m} =}{} - \big(\psi^{(1)}_m+1\big) \big(\psi^{(1)}_{m+2} \psi^{(1)}_{m+1} - \psi^{(1)}_{m-1} \psi^{(1)}_{m-2}\big) + \big(\psi^{(0)}_m-1\big) \big(\psi^{(1)}_{m+2} \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} \psi^{(1)}_{m-2}\big), \nonumber\\
\frac{\partial_{t_2} \psi^{(1)}_m}{\psi^{(1)}_m+1} = \big(\psi^{(0)}_m+\psi^{(1)}_m\big) \big(\psi^{(0)}_{m+2} \psi^{(0)}_{m+1} - \psi^{(0)}_{m-1} \psi^{(0)}_{m-2}\big) - \psi^{(1)}_m \big(\psi^{(1)}_{m+2} \psi^{(1)}_{m+1} - \psi^{(1)}_{m-1} \psi^{(1)}_{m-2}\big) \nonumber \\
\hphantom{\frac{\partial_{t_2} \psi^{(1)}_m}{\psi^{(1)}_m+1} =}{} + \psi^{(0)}_m \big(\psi^{(1)}_{m+2} \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} \psi^{(1)}_{m-2}\big).\label{eq:M-sys-5}
\end{gathered}$$ The above system and its symmetry can be considered as a two-component generalisation of the equation (\[eq:MX2a\]) and its symmetry (\[eq:Bog\]) in the following sense. If we set, $\psi^{(0)}_{m,n} = 0$ a
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an age was 34 years and participants had a mean body mass index (BMI) of 24.1 ± 2.78 kg/m^2^. A summary of the baseline characteristics of the participants is presented in [Table [1](#tbl1){ref-type="other"}](#tbl1){ref-type="other"}.
###### Demographic Data of Participants Included in the Study[a](#t1fn1){ref-type="table-fn"}
characteristics male (*n* = 4) female (*n* = 5)
-------------------- ---------------- ------------------
age (years) 34 ± 9 35 ± 14
weight (kg) 77.55 ± 7.91 66.00 ± 12.24
height (m) 1.76 ± 0.08 1.68 ± 0.08
BMI (kg/m^2^) 25.1 ± 2.6 23.2 ± 2.9
waist to hip ratio 0.87 ± 0.05 0.82 ± 0.09
All values shown are mean ± SD.
Examination of the principal component analysis (PCA) revealed that the interindividual variation was the dominant source of variation on the dataset. The samples of the individuals were grouped together in the PCA scores plot ([Figure [1](#fig1){ref-type="fig"}](#fig1){ref-type="fig"}A). To explore the impact of storage, we employed a row-wise centering of the data and this resulted in separation of the samples according to the storage type ([Figure [1](#fig1){ref-type="fig"}](#fig1){ref-type="fig"} B).
{#fig1}
To examine the impact of the storage further, a univariate analysis approach was employed. A total of 14 compounds from the fecal water analysis were significantly different across the three storage conditions. Significant metabolites from repeated measures ANOVA corrected by the false discovery
| 1,504
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|
bottom panel of Android Studio). Based on the build variant you choose, you will have the BASE_WEB_URL correctly updating to the right value.
In your code you simply need to replace all your "127.0.0.1" and "10.0.2.2" with
BuildConfig.BASE_WEB_URL
Q:
Where Should Exception Messages be Stored
Since I can't use Microsoft as an example for best practice since their exception messages are stored in resource files out of necessity, I am forced to ask where should exception messages be stored.
I figure it's probably one of common locations I thought of
Default resource file
Local constant
Class constant
Global exception message class
Inline as string literals
A:
I may get shot (well, downvoted) for this, but why not "where you create the exception"?
throw new InvalidDataException("A wurble can't follow a flurble");
Unless you're going to internationalize the exception messages (which I suggest you don't) do you particularly need them to be constants etc? Where's the benefit?
Q:
Reading Strings into an array, column wise
I am trying to read String objects into a 2d array, from a char array, using column-major ordering:
This is what I've tried:
int x = 0;
for (int column = 0; column < matrix[0].length; column++) {//cols
for (int row = 0; row < matrix.length; row++, x++) {//rows
if(matrix[row][column] == null) {
if (x < ciphertextCharacters.length) {
matrix[row][column] = Character.toString(inputChars[x]);
}
}
}
}
given an input array (inputChars = ['t', 't', 'g', 'e', 'i', 's', 'n']) the resulting 2D array should be:
+---+----+----+
| t | e | s |
+---+----+----+
| t | i | n |
+---+----+----+
| g | * | * |
+---+----+----+
Note that before this code runs, the "*" strings are already in the array - and that's why I'm only adding new values when the index is null.
Currently, the resulting 2D array I'm getting is:
+---+----+----+
| t | e | n |
+---+----+----+
| t | i |null|
+---+----+----+
| g | * | * |
+---+----+----+
Which is n
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|
ix instead of the Laplacian [@GW03]; the dynamics are then given by the unitary operator $U_t = e^{iHt}$ and the state of the walk at time $t$ is $\Psi_t = U_t \Psi_0$.
The following analysis makes use of the hypercube’s product graph structure; this structure will be useful again later when we consider the effects of decoherence. The analysis below diverges from that of Moore and Russell [@MR01] only in that we allow each qubit to have energy $k/n$ instead of $1/n$. The energy of the entire system is then $k$. Let $$\sigma_x = \left(\begin{matrix} 0 & k/n \\ k/n & 0 \end{matrix}
\right),$$ and let $$H = \sum_{j=1}^n \identity \otimes \cdots \otimes \sigma_x \otimes
\cdots \otimes \identity\enspace,$$ where the $j$th term in the sum has $\sigma_x$ as the $j$th factor in the tensor product. Then we have $$\begin{aligned}
U_t &= e^{iHt} = \prod_{j=1}^n \identity \otimes \cdots \otimes e^{it\sigma_x} \otimes \cdots
\otimes \identity = \left[e^{it\sigma_x} \right]^{\otimes n} \\
& = \left[\begin{matrix}\cos(kt/n) & i~\sin(kt/n) \\ i~\sin(kt/n) & \cos(kt/n) \end{matrix}\right]^{\otimes n}\enspace.\end{aligned}$$ Applying $U_t$ to the initial state $\Psi_0 = \vert 0 \rangle ^{\otimes n}$, we have $$U_t \Psi_0 = \left[ \cos\left(\frac{kt}{n}\right) \vert 0 \rangle
+ i~\sin\left(\frac{kt}{n}\right) \vert 1 \rangle
\right]^{\otimes n}$$ which corresponds to a uniform state exactly when $\frac{kt}{n}$ is an odd multiple of $\frac{\pi}{4}$.
A derivation of the superoperator
=================================
We begin by recalling a model of decoherence commonly used in the discrete model, with the intention of deriving a superoperator $U_t$, acting on density matrices, which mimics these dynamics in our continuous setting. The discrete model, described in [@KT03], couples unitary evolution according to the discrete-time quantum random walk model of Aharonov et al. [@AAKV01] with partial measurement at each step occurring with some fixed probability $p$. Specifically, the evolution of the density matrix can
| 1,506
| 5,129
| 1,513
| 1,163
| 2,005
| 0.783597
|
github_plus_top10pct_by_avg
|
hen $$\begin{aligned}
d A_t^i A_t^j
=& d A_t^T \lrp{e_i e_j^T} A_t\\
=& A_t \lrp{e_i e_j^T} \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s}^T + \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s} \lrp{e_j e_i^T} a_t^T\\
&\qquad + \frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{c_m^2 M(x_s)^{-2} + M(x_s)^{-1} N(x_s)^2 M(x_s)^{-1}}} dt
\end{aligned}$$ where the second inequality is by Ito’s Lemma applied to $f(A_t) = A_t^T e_j e_j^T A_t$. Taking expectations, $$\begin{aligned}
d \E{A^i_t A^j_t}
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{c_m^2 M(x_s)^{-2} + M(x_s)^{-1} N(x_s) N(x_s)^T \lrp{M(x_s)^{-1}}^T}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{M(x_s)^{-1} \lrp{c_m^2 I + N(x_s)^2} M(x_s)^{-1}}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{M(x_s)^{-1} \lrp{M(x_s)^2} M(x_s)^{-1}}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}}}dt\\
=& \ind{i=j} dt
\end{aligned}$$
This verifies that $A_t^i A_t^j - \ind{i=j} t$ is a martingale, and hence by Levy’s characterization, $A_t$ is a standard Brownian motion. In turn, we verify that by definition of $A_t$, $$\begin{aligned}
x_t
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s \\
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t M(x_s) \lrp{M(x_s)^{-1}\lrp{\cm dV_s + N(x_s) dW_s}}\\
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t M(x_s) dA_s
\end{aligned}$$ Since we showed that $A_t$ is a standard Brownian motion, we verify that $x_t$ as defined in has the same distribution as .
On the other hand, we can verify that $A'_t := \int_0^T (I - 2\gamma_s \gamma_s^T) V_s$ is a standard Brownian motion by the reflection principle. Thus $$\begin{aligned}
\int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s \sim \N(0, \lrp{c_m^2 I + N(y_0)^2}) = \N(0, M(y_0)^2)
\end{aligned}$$ where the equality is by definition of $N$ in .
It follows immediately that $y_t$ in has the same distribution as $y_t$ in .
| 1,507
| 4,214
| 1,039
| 1,116
| null | null |
github_plus_top10pct_by_avg
|
the learning efficiency curves (Fig.\[fig3\]) up to cost $C_{min}$ and denote them as $S_n$ (area under the curve of number of characters v.s. cost like the ones in Fig. \[fig3\]a) and similarly $S_f$ (area under the curve of accumulated usage frequency v.s. cost like those in Fig. \[fig3\]b), respectively. The ratio between the area underneath the curves $S_{n}$ ($S_{f}$) and the area of a rectangular region defined by $C_{min}N_{min}$ ($C_{min}F_{min}$, where $F_{min}$ is the maximum accumulated frequency of the curves at $C=C_{min}$) is defined as the learning efficiency index, $$\begin{aligned}
v_n=\frac{S_n}{C_{min}N_{min}},\\
v_f=\frac{S_f}{C_{min}F_{min}}.
{\label{eq:speed}}\end{aligned}$$ The sooner a curve reaches $N_{min}$ ($F_{min}$) the larger is the area and so is the ratio, the more efficient is the learning order. In this sense, the above ratios serve as indexes of efficiency of learning orders.
In Fig. \[fig4\], we plot $v_n$ and $v_f$ of the hybrid strategy (DNW) as functions of $b$. We also plot two lines, for comparison, showing the learning efficiency of the NOO (blue line) and UFO (green line). As $b$ increases, $v_n$ of the hybrid strategy approaches that of the NOO. On the other hand, when $b=0.35$, $v_f$ of hybrid strategy reaches its maximum. Thus, with respect to frequency usage the DNW with $b=0.35$ is the most efficient. However, if we consider also the number of characters the range of $b\in\left[0.35, 0.7\right]$ can be regarded as very good choices. As an example, in this work we use $b=0.35$, which shows a significant improvement over commonly used methods (Fig. \[fig3\]).
![\[fig4\] Efficient index of hybrid strategies as a function of b (dots). The two horizontal lines are the efficiency of the node-offspring order (blue line) and usage frequency order (green line). (a) Efficiency when using number of characters as the learning goal. (b) Efficiency when using accumulated usage frequency as the learning goal.](Wu_fig5.pdf){width="8.4cm"}
In order to compare the DNW strategy aga
| 1,508
| 484
| 1,305
| 1,248
| 2,498
| 0.779332
|
github_plus_top10pct_by_avg
|
res more computation labor. Although the LLL LR algorithm has known average complexity for some cases [@Daude1994; @Ling2007], its average complexity for our case is unknown, and the worst-case complexity could be unbounded [@Jalden2008].
- The quantized search (QS) method developed by Sakzad [*et al.*]{} in [@Sakzad2012]. The search consists of two phases: 1) an integer $\alpha_0$ between 1 and $\floor{P^{1/2}}$ that provides the maximum rate is selected as the initial value of the amplifying factor $\alpha$; 2) the amplifying factor is then refined by searching in $[\alpha_0-1,\alpha_0+1]$ with a step size 0.1. After the amplifying factor $\alpha$ is determined, the coefficient vector $\a$ is set as $\round{\alpha\h}$. An improved version of the QS method is the quantized exhaustive search (QES) method proposed in [@Sakzad2014], which was developed for complex-valued channels.
- The rounding method that simply sets the coefficient vector by rounding the channel vector to an integer-valued vector.
As shown in Figure \[figure:Rate\], the optimal methods, i.e., the BnB method and the SG method, always provide the highest average computation rates for all dimensions and over the whole SNR regime, as expected. The LLL method provides close-to-optimal average computation rates. Our proposed QPR method also offers close-to-optimal average computation rates for almost all the dimensions and SNR values considered, except that its performance degrades a little bit for high dimensions at high SNR as shown in Figure \[figure:Rate\_L16\]. The performance of our QPR method improves slightly compared with the version we presented in [@Zhou2014; @Wen2015]. The reason is that here we initialize the output coefficient vector as $[0,\cdots,0,1]^T$, which definitely results non-zero computation rate, while in the previous version the output coefficient vector could result zero computation rate.
\
\
Finally, we demonstrate the efficiency of the proposed QPR method by comparing the running time of finding the coefficien
| 1,509
| 574
| 907
| 1,302
| null | null |
github_plus_top10pct_by_avg
|
lphahat|_{Z_{(n)}}$ is trivial (the center $Z_{(n)}\simeq \F_q^{\times}$ consists of scalar matrices $a.I_n$).
We can show as for conjugacy classes [@hausel-letellier-villegas Lemma 2.1.2] that if the characteristic $p$ of $\F_q$ and $q$ are sufficiently large, generic tuples of irreducible characters of a given type $\lambdahat$ always exist.
Put $\mathfrak{g}:=\gl_n(\F_q)$. For $X\in\mathfrak{g}$, put $$\Lambda^1(X):=\#\{Y\in\mathfrak{g}\,|\, [X,Y]=0\}.$$
The restriction $\Lambda^1:G\rightarrow\C$ of $\Lambda^1$ to $G\subset\mathfrak{g}$ is the character of the representation $G\rightarrow\GL\left(\C[\mathfrak{g}]\right)$ induced by the conjugation action of $G$ on $\mathfrak{g}$. Fix a non-negative integer $g$ and put $\Lambda:=(\Lambda^1)^{\otimes g}$.
For a multi-partition $\muhat=(\mu^1,\dots,\mu^k)\in(\calP_n)^k$ and a generic tuple $\left(R_{L_{\mu^1}}^G(\tilde{\alphahat}_1),\dots,R_{L_{\mu^k}}^G(\tilde{\alphahat}_k)\right)$ of irreducible characters we put $$R_\muhat:=R_{L_{\mu^1}}^G(\tilde{\alphahat}_1)\otimes\cdots\otimes R_{L_{\mu^k}}^G(\tilde{\alphahat}_k).$$
For two class functions $f,g\in\C(G)$, we define $$\langle f,g\rangle:=|G|^{-1}\sum_{h\in G}f(h)\overline{g(h)}.$$
We have the following theorem [@hausel-letellier-villegas Theorem 1.4.1].
We have $$\left\langle \Lambda\otimes R_\muhat,1\right\rangle=\H_\muhat\left(0,\sqrt{q}\right)$$where $\H_\muhat(z,w)$ is the function defined in §\[Cauchy\].
\[multi\]
The multiplicity $\left\langle \Lambda\otimes
R_\muhat,1\right\rangle$ depends only on $\muhat$ and not on the choice of linear characters $(\tilde{\alphahat}_1,\dots,\tilde{\alphahat}_k)$.
### Fourier transforms {#Fourier}
Let ${\rm Fun}(\mathfrak{g})$ be the $\C$-vector space of all functions $\mathfrak{g}\rightarrow\C$ and by $\C(\mathfrak{g})$ the subspace of functions $\mathfrak{g}\rightarrow\C$ which are contant on $G$-orbits of $\mathfrak{g}$ for the conjugation action of $G$ on $\mathfrak{g}$.
Let $\Psi:\F_q\rightarrow\C^{\times}$ be a non-trivial additive character and
| 1,510
| 1,064
| 1,649
| 1,392
| 4,080
| 0.768224
|
github_plus_top10pct_by_avg
|
n reboot instead of immediately, which would disconnect the provisioner.
With the above script in place, add something like this just below the trigger definitions that we added earlier:
config.vm.provision "shell", path: "./scripts/configure-static-ip.sh"
config.vm.provision :reload
4. Provision the VM
vagrant up and choose Default Switch when prompted; this causes the VM to obtain a dynamic IPv4 address that is sufficient for Vagrant to connect to the VM via SSH, mount Shared Folders, and begin provisioning.
Now, the static IP address will be set and the VM reloaded before provisioning continues.
Sample output:
homestead: Setting static IP address for Hyper-V...
==> homestead: Running provisioner: reload...
==> homestead: Running action triggers before reload ...
==> homestead: Running trigger...
==> homestead: Setting Hyper-V switch to 'NATSwitch' to allow for static IP...
homestead: Running local script: ./scripts/set-hyperv-switch.ps1
==> homestead: Attempting graceful shutdown of VM...
==> homestead: Stopping the machine...
homestead: Configuring the VM...
==> homestead: Starting the machine...
==> homestead: Waiting for the machine to report its IP address...
homestead: Timeout: 120 seconds
homestead: IP: 192.168.0.2
==> homestead: Waiting for machine to boot. This may take a few minutes...
==> homestead: Machine booted and ready!
5. Other Thoughts
Manual input is required to choose the Default Switch after the initial vagrant up; it would be ideal to find a way around this.
The vagrant-reload provisioner cannot shutdown the machine gracefully and must halt it forcibly; not a significant concern, given that it happens only once (during initial provisioning). This happens due to the fact that changing the VM's Hyper-V switch from Default Switch to NATSwitch must be done during the before-reload event, which is, in effect, akin to pulling the Ethernet cord out of a physical jack and connecting it to a different switch.
Q:
Using JSON Serde : java.net.URISyntaxException
I am new t
| 1,511
| 3,112
| 255
| 1,570
| null | null |
github_plus_top10pct_by_avg
|
ein could be cannulated (red arrow) at a point close to the center of the left renal vein (white arrow) using CT guidance. Because the catheter was wedged, requiring a change in catheter to obtain a blood sample (B).](CCR3-5-482-g003){#ccr3875-fig-0003}
######
Adrenal venous sampling results after adrenocorticotropic hormone stimulation
Aldosterone (pg/mL) Cortisol (*μ*g/dL) Aldosterone cortisol ratio
-------------------- --------------------- -------------------- ----------------------------
Right adrenal vein 34,189 1130 30
Left adrenal vein 24,379 1100 22
Inferior vena cava 212 20 11
John Wiley & Sons, Ltd
Discussion {#ccr3875-sec-0003}
==========
The IVC can present with a multitude of anatomical variations, such as double and left IVC, which are caused by complex embryonic developments. Based on the involvement of iliac and gonadal veins, several classifications have been proposed for IVC variations [6](#ccr3875-bib-0006){ref-type="ref"}, [7](#ccr3875-bib-0007){ref-type="ref"}, [8](#ccr3875-bib-0008){ref-type="ref"}. To perform a successful AVS, knowledge of possible anatomical variations related to adrenal vein drainage is crucial. For example, in patients with double IVC, the left adrenal vein may drain either directly into the IVC or into the left renal vein. In patients with left IVC, the left adrenal vein drains directly into the IVC [6](#ccr3875-bib-0006){ref-type="ref"}. In almost all individuals, the right adrenal vein drains directly into the IVC. Alper et al. reviewed the anatomy of adrenal veins [9](#ccr3875-bib-0009){ref-type="ref"}.
Contrast‐enhanced CT is useful in planning for AVS because it reveals the positions of the adrenal veins [5](#ccr3875-bib-0005){ref-type="ref"}. In this case, a coronal section of CT was helpful in detecting the left adrenal vein. Stack et al. reported a case where the left adrenal vein drained directly into the IVC [1
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|
{\rm sth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=1}, & \sigma_{i,k}({\rm nth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=N_y},\end{aligned}$$ where $\sigma_{i,j}({\rm bot})$ denotes the screening charge on the bottom boundary, etc. Note that the screening charges have units of mass density rather than surface density, because the charge is assumed to fill a volume $\delta x \delta y \delta z$.
### Cylindrical Grid
In cylindrical coordinates, one needs to deal with only four boundary surfaces: bottom (bot; $k=0$), top (top; $k=N_z+1$), inner (inn; $i=0$), and outer (out; $i=N_R+1$): the azimuthal direction is assumed periodic. Using the discrete Laplace operators given in Section \[s:equation\], one can show that the screening charges are calculated as $$\begin{aligned}
\sigma_{i,j}({\rm bot}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=1}, & \sigma_{i,j}({\rm top}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=N_z},\nonumber\\
\sigma_{j,k}({\rm inn}) &= \frac{1+\delta R/(2R_{0})}{4\pi G (\delta R)^2} \Psi|_{i=1}, & \sigma_{j,k}({\rm out}) &= \frac{1-\delta R/(2R_{N_R+1})}{4\pi G (\delta R)^2} \Psi|_{i=N_R}\end{aligned}$$ in a uniform cylindrical grid, and $$\begin{aligned}
\sigma_{i,j}({\rm bot}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=1}, & \sigma_{i,j}({\rm top}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=N_z},\nonumber\\
\sigma_{j,k}({\rm inn}) &= \frac{1}{4\pi G (R_{0}\ln f)^2} \Psi|_{i=1}, & \sigma_{j,k}({\rm out}) &= \frac{1}{4\pi G (R_{N_R+1}\ln f)^2} \Psi|_{i=N_R}\end{aligned}$$ in a logarithmic cylindrical grid.
Discrete Green’s Function and the Potential Generated by Screening Charges {#s:dgf}
--------------------------------------------------------------------------
The gravitational potential $\Theta$ that results from the screening charges can be obtained by convolving $\sigma$ with the Green’s function of the operator that determines $\sigma$. Since $\sigma$ is obtained through the application of the *discrete* Laplace operator, the corresponding Green’s function should be the DGF r
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|
ht\Vert}_{W^2(G\times S\times I)} =0$. In addition the trace mapping $\gamma_-:W^2(G\times S\times I)\to \ L^2_{\rm loc}(\Gamma_-,|\omega\cdot\nu|\ d\sigma d\omega dE)$ such that $\gamma_-(\psi)=\psi_{|\Gamma_-}$ is continuous. Similarly one has a continuous trace mapping $\gamma_+:W^2(G\times S\times I)\to \ L^2_{\rm loc}(\Gamma_+,|\omega\cdot\nu|\ d\sigma d\omega dE)$ and so we can define (a.e. unique) the trace $\gamma(\psi)$ on $\Gamma$ for $\psi\in W^2(G\times S\times I)$.
By the Sobolev Embedding Theorem for $\psi\in C^1(\ol G\times S\times I)$ (cf. [@friedman p. 22], or [@treves p. 220]; or see in the proof of Theorem \[tth\] below) \[sobim\] [(,,E)]{}\_[L\^2(GS)]{}C(\_[L\^2(GSI)]{}+\_[L\^2(GSI)]{}),EI, and then the traces $\psi(\cdot,\cdot,0),\ \psi(\cdot,\cdot,E_{\rm m})\ \in L^2(G\times S)$ are well-defined for any $\psi\in W_1^2(G\times S\times I)$.
The trace $\gamma(\psi),\ \psi\in W^2(G\times S\times I)$ is not necessarily in the space $T^2(\Gamma)$. Hence we define the spaces \[fs12\] W\^2(GSI)={W\^2(GSI) | ()T\^2() }. which is equipped with the inner product \[fs14\] ,v\_[W\^2(GSI)]{}=,v\_[W\^2(GSI)]{}+ [()]{},(v)\_[T\^2()]{}. The space $\tilde W^2(G\times S\times I)$ is a Hilbert space (cf. [@tervo14]). Moreover, we define subspaces $\tilde{W}_{\pm,0}^2(G\times S\times I)$ of it by \_[,0]{}\^2(GSI)={W\^2(GSI) | \_()=0}.
For $v\in \tilde W^2( G\times S\times I)$ and $\psi\in\tilde W^2(G\times S\times I)$ it holds the Green’s formula $$\begin{aligned}
\label{green}
\int_{G\times S\times I}(\omega\cdot \nabla_x \psi)v\ dxd\omega dE
+\int_{G\times S\times I}(\omega\cdot \nabla_x v)\psi\ dxd\omega dE=
\int_{\partial G\times S\times I}(\omega\cdot \nu) v\ \psi\ d\sigma d\omega dE,\end{aligned}$$ which is obtained by Stokes Theorem for $v,\psi\in C^1(\ol G\times S\times I)$ and then by the limiting considerations for general $v\in \tilde W^2(G\times S\times I)$ and $\psi\in\tilde W^2(G\times S\times I)$.
\[re:W2\_0\_trace\] By Green’s formula $${\left\Vert \gamma_{\pm}(\psi)\right\Vert}_{T^2(
| 1,514
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|
the complex conjugation of $H^1(\tilde{X},\cO_{\tilde{X}})$. The following result computes the Hodge structure of $H^q(\tilde{X},\cO_{\tilde{X}})$ providing the decomposition in terms of its invariant subspaces.
\[thm:Esnault\] Under the previous notation, assume the ramification set is given by a simple $\Q$-normal crossing integral Weil divisor $D = \sum_{i=1}^r n_i D_i$ which is linearly equivalent to $nH$ for some integral Weil divisor $H$. Then $$H^q(\tilde{X},\cO_{\tilde{X}}) = \bigoplus_{k=0}^{n-1} H^q (X, \cO_X(L^{(k)})), \quad
L^{(k)} = -k H + \sum_{i=1}^r {\left \lfloor \frac{kn_i}{n} \right \rfloor} D_i,$$ where the monodromy of the cyclic covering acts on $H^q(X,\cO_X(L^{(k)}))$ by multiplication by $e^{\frac{2\pi ik}{n}}$.
Since the result is true for the smooth case [@Steenbrink77; @Esnault-Viehweg82], it is enough to show that the cohomology $H^q (X, \cO_X(L^{(k)}))$ remains the same after performing the first weighted blow-up in Hirzebruch’s resolution.
Let $\pi: Y \to X$ be the $(1,p)$-blow-up at a point of type $\frac{1}{d}(1,p)$ (see §\[sec:Qres\] for the notation) and denote by $E$ its exceptional divisor. Consider $D' = \pi^{*} D - n \pi^{*} H + n {\left \lceil \pi^{*} H \right \rceil}$ – it is a Weil divisor linearly equivalent to $n {\left \lceil \pi^{*} H \right \rceil}$ and, in addition, it is effective if $D$ is. Denote by $L'^{(k)}$ its associated Weil divisor. There exist $\rho_Y: \tilde{Y} \to Y$ another cyclic branched covering equivalent to $\rho_X:=\rho$ with ramification set given by $D'$ and $\tilde{\pi}: \tilde{Y} \to \tilde{X}$ a proper map completing the following diagram: $$\begin{tikzcd}
&\tilde{X}\ar[d,"\rho_X" left] &\ar[l,"\tilde{\pi}" above]\tilde{Y}\ar[d,"\rho_Y"] &\\
L^{(k)},D\ar[r,hook] & X &\ar[l,"\pi"] Y \ar[r,hookleftarrow]& D',L'^{(k)}
\end{tikzcd}$$ Locally, one can assume $D_1 = \{ x=0 \}$ and $D_2 = \{ y=0 \}$ with $m_1$ and $m_2$ possibly zero. Denote by $c \in \frac{1}{d}\ZZ $ the rational number such that ${\left \lceil \pi^{*}H \right \rceil} -
| 1,515
| 1,452
| 1,415
| 1,444
| null | null |
github_plus_top10pct_by_avg
|
R^{\mu\nu\alpha\beta} R_{\mu\nu\alpha\beta} =R T
\\ \mathcal{L}_5=R^{\mu\nu\alpha\beta}R_{\mu\alpha}R_{\nu\beta} \;\;\quad\;\; , \;\;\quad \mathcal{L}_6=R^{\mu\nu}R_{\nu\alpha}R^{\alpha}_{\;\,\mu}
\\ \mathcal{L}_7=R R^{\mu\nu} R_{\mu\nu} = R S \;\,\, , \,\;\quad \mathcal{L}_8=R^3
\end{array}
\right.
$\
\
\
$\left\{
\begin{array}{l}
\mathscr{L}_1=R\Box R
\quad \quad \;\;\; \quad\;\; , \;\quad
\mathscr{L}_2=R_{\mu\nu} \Box R^{\mu\nu}
\\ \mathscr{L}_3=R^{\mu\nu\alpha\beta} \nabla_\nu \nabla_\beta R_{\mu\alpha}
\; \; , \;\;\,\,\,\;
\mathscr{L}_4=R^{\mu\nu} \nabla_\mu \nabla_\nu R
\\ \mathscr{L}_5=\nabla_\sigma R_{\mu\nu} \nabla^\sigma R^{\mu\nu}
\quad \quad\,\; , \;\;\;\;
\mathscr{L}_6=\nabla_\sigma R_{\mu\nu} \nabla^\nu R^{\mu\sigma}
\\ \mathscr{L}_7=\nabla_\sigma R_{\mu\nu\alpha\beta} \nabla^\sigma R^{\mu\nu\alpha\beta}
\;\;\;\;\; , \;\;\;\;
\mathscr{L}_8=\nabla_\sigma R \nabla^\sigma R
\end{array}
\right.
$ \
Recall that our aim is to see that if we consider all these scalars, there are quite natural modified gravity Lagrangian densities that we can expect to lead to second order equations of motion and that actually do. There are linear combinations of all the scalars of the basis, but also for example square-root of the $\mathcal{R}_{\left\{ 1,1 \right\}}^0$-class, or cubic-roots of the $\mathcal{R}_{6,3}^0$ one, even if we are not going to study this last because we search here for high energy geometrical correction to the Einstein-Hilbert action. We write down this fact as : $$\begin{aligned}
\mathscr{L}= \sum \Big( R^3 + R \nabla \nabla R + \nabla R\nabla R \Big) + \sqrt{ \sum \big( \nabla R\nabla R \big) } + \sqrt[3]{\sum \big( R^3 \big) }.\end{aligned}$$ Because inside a same class, the scalars have approximatively the same terms in their expansions, we can indeed expect to cancel higher order derivatives for some specific combinations of them, and then to have second order equations of motion.
Now, let us write down some definitions that allow to find relations betw
| 1,516
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| 1,403
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github_plus_top10pct_by_avg
|
gradeName1 = "Craftmanship Lvl 1";
var UpgradePrice1 = 100;
var UpgradeContent1 = "+100% Gods Hands! <br> +20% Blessed Farmer!";
var Upgrade1 = 0;
var UpgradeName2 = "Craftmanship Lvl 2";
var UpgradePrice2 = 200;
var UpgradeContent2 = "+100% Gods Hands! <br> +20% Blessed Farmer!";
var Upgrade2 = 0;
And so on...
If it were a html code i tried to generate i would use a php while function and make it echo the same code with the changed number a specefic amount of times.
But since this is inside a javascript file i really dont think thats an option?
The javascript code from before is in a .js file.
I think a potential fix could be:
Inside a .php file:
<?php
$Upgrades = 10;
$CurrentUpgrade = 1;
while ($Upgrades >= $CurrentUpgrade){
echo "
<script>
function Upgrade1(){
//Upgrade ".$CurrentUpgrade."
if (CREDITS >= UpgradePrice".$CurrentUpgrade." && Upgrade".$CurrentUpgrade." === 0){
document.getElementById('Upgrade".$CurrentUpgrade."').style.display = 'block';}
else{document.getElementById('Upgrade".$CurrentUpgrade."').style.display = 'none';}
}
</script>
";
$CurrentUpgrade ++;
}
?>
*Sry for any typo in this part, made it quite quick.
But i would quite like to keep the javascript code inside my .js file instead of having it in my .php file.
So to sum it up i would like to (If possible):
Keep all code inside the .js file
Generate javascript code with javascript code
Repeat the same javascript element with only a number changing with each element
Try to shorten the amount of code by doing this.
Still be able to use any of the assigned variables alone somewhere like in a buy function.
I hope i gave all relevant information :)
Thanks in advance.
A:
You can create a method (in index you are passing the number in the end of your variables):
function myFunction(newUpgradePrice, newUpgrade, index) {
if (CREDITS >= newUpgradePrice && newUpgrade === 0){
document.getElementById("Upgrade" + index).style.display = "block";}
else{document.getElementById("Upgrade" + index).style.display = "n
| 1,517
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| 0.82785
|
github_plus_top10pct_by_avg
|
(w_i,w_i) \in (4)$, where $w_i \in W_i\otimes_{A}R$, and $i=2m$.
5. $f(B_i, B_i^{\perp})\in B\otimes_{A}R$ and $f(a_i, b_i^{\prime})-h(a_i, b_i^{\prime})\in B\otimes_{A}R$, where $a_i\in A_i\otimes_{A}R$ and $b_i^{\prime}\in B_i^{\perp}\otimes_{A}R$, and $i$ is odd.
**
We interpret the above conditions in terms of matrices. The matrix forms are taken with respect to the basis of $L$ fixed in Theorem \[210\] and Remark \[r211\].(a). A matrix form of the given hermitian form $h$ is described in Remark \[r33\].(1) below. We use $\sigma$ to mean the automorphism of $B\otimes_AR$ given by $b\otimes r \mapsto \sigma(b)\otimes r$. For a flat $A$-algebra $R$, $\underline{H}(R)$ is the set of hermitian matrices $$\begin{pmatrix}\pi^{max\{i,j\}}f_{i,j}\end{pmatrix}$$ of size $n\times n$ satisfying the following:
1. $f_{i,j}$ is an $(n_i\times n_j)$-matrix with entries in $B\otimes_AR$.
2. If $i$ is even and $L_i$ is *of type* $\textit{I}^o$, then $\pi^if_{i,i}$ is of the form $$\xi^{i/2}\begin{pmatrix} a_i&\pi b_i\\ \sigma(\pi \cdot {}^tb_i) &1+2\gamma_i +4c_i \end{pmatrix}.$$ Here, the diagonal entries of $a_i$ are divisible by $2$, where $a_i$ is an $(n_i-1) \times (n_i-1)$-matrix with entries in $B\otimes_AR$, etc. and $\gamma_i (\in A)$ is as chosen in Remark \[r33\].(1) below.
3. If $i$ is even and $L_i$ is *of type* $\textit{I}^e$, then $\pi^i f_{i,i}$ is of the form $$\xi^{i/2}\begin{pmatrix} a_i&b_i&\pi e_i\\ \sigma({}^tb_i) &1+2f_i&1+\pi d_i \\ \sigma(\pi \cdot {}^te_i) &\sigma(1+\pi d_i) &2\gamma_i+4c_i \end{pmatrix}.$$ Here, the diagonal entries of $a_i$ are divisible by $2$, where $a_i$ is an $(n_i-2) \times (n_i-2)$-matrix with entries in $B\otimes_AR$, etc. and $\gamma_i (\in A)$ is as chosen in Remark \[r33\].(1) below.
4. Assume that $i$ is even and that $L_i$ is *of type* $\textit{II}$. The diagonal entries of $f_{i,i}$ are divisible by $2$.
5. Assume that $i$ is odd. The diagonal entries of $\pi f_{i,i}-\pi h_i$ are are divisible by $4$.
6. Assume that $i$ is odd and that $L_i$ is *of type
| 1,518
| 1,067
| 1,270
| 1,483
| 3,647
| 0.770946
|
github_plus_top10pct_by_avg
|
e MI-phase as $v$ is varied ($F=0$, $L=N=7$).[]{data-label="fig1"}](fig1.eps){width="8cm"}
We proceed with the multi-particle case. A natural extension of the tight-binding model (\[1\]), which accounts for the repulsive interaction of the atoms, is given by the Bose-Hubbard model [@Fish89], $$H=-\frac{J}{2}\left(\sum_{l=1}^L \hat{a}^\dag_{l+1}\hat{a}_l
+h.c.\right)
+\frac{W}{2}\sum_{l=1}^L \hat{n}_l(\hat{n_l}-1)$$ $$\label{2}
+2\pi F\sum_{l=1}^L l\hat{n}_l \;.$$ In Eq. (\[2\]), $\hat{a}_l^\dag$ and $\hat{a}_l$ are the bosonic creation and annihilation operators, $\hat{n}_l= \hat{a}_l^\dag\hat{a}_l$ is the occupation number operator of the $l$th lattice site, and the parameter $W$ is proportional to the integral over the Wannier function raised to the fourth power. Since the Bose-Hubbard Hamiltonian conserves the total number of atoms $N$, the wave function of the system can be represented in the form $|\Psi\rangle=\sum_{\bf n} c_{\bf n}|{\bf n}\rangle$, where the vector ${\bf n}$, consisting of $L$ integer numbers $n_l$ ($\sum_l n_l=N$), labels the $N$-particle bosonic wave function constructed from $N$ Wannier functions. (In what follows, if not stated otherwise, $|\Psi\rangle$ refers to the ground state of the system.) As known, in the thermodynamic limit, and for $F=0$, the system (\[2\]) shows a quantum phase transition from a superfluid (SF) to a Mott insulator (MI) phase as the ratio $J/W$ is varied (see [@Sach01] and references therein). It is interesting to note that an indication of this transition can already be observed in a system of few atoms [@Jaks98]. As an example, Fig. \[fig1\] shows the diagonal elements of the one-particle density matrix, $$\label{4}
\rho(k,k')=\langle\Psi|\hat{\Phi}^\dag(k)\hat{\Phi}(k')
|\Psi\rangle \;,\quad
\hat{\Phi}(k)=\sum_{l=1}^L \hat{a}_l\phi_l(k) \;,$$ for $N=L=7$, $5\le v\le 35$, and $W=0.1\int dk \phi_l^4(k)$ (here, $\phi_l(k)$ are the Wannier states in the momentum representation, and $k=p/(2\pi\hbar/d)$ is the dimensionless momentum). Physically, this qua
| 1,519
| 1,340
| 2,032
| 1,704
| 2,276
| 0.781207
|
github_plus_top10pct_by_avg
|
24 months 57.36 67.03 63.69 73.83 67.37 62.79 66.63 61.97 64.99 68.24
36 months 68.88 70.10 70.11 72.62 69.05 71.94 70.16 69.38 69.28 72.72
*p* 0.12 0.03 0.71 0.12 0.89
Adjusted for infant sex and gestational age.
######
Association between maternal antioxidant vitamin and oxidative stress levels and infant growth percentile adjusted for covariances
Vitamin A Vitamin C Vitamin E MDA 8-OHdG[a](#TF0007){ref-type="table-fn"}
-------------------- ----------- ----------- ----------- -------- ----------------------------------------- ------- -------- ------- ------- -------
Weight
At birth 39.81 45.87 45.51 44.61 45.52 41.73 47.66 44.06 48.77 40.21
6 months 55.86 66.02 61.93 71.67 63.16 65.66 63.01 56.55 61.55 63.18
12 months 53.87 62.27 57.96 68.89 58.41 66.56 55.70 60.49 58.02 48.46
18 months 53.62 55.34 52.71 61.69 56.30 51.94 62.06 37.11 53.62 --
24 months 45.22 58.89 55.49 62.05 56.24 57.56 56.27 49.32 55.14 48.95
36 months 65.01 71.26 62.75 85.27 76.55 40.12 75.16 71.40 79.94 45.91
*p* 0.06 0.01 0.18 0.33 0.20
Height
| 1,520
| 5,848
| 479
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| null | null |
github_plus_top10pct_by_avg
|
al{L}_6 -7 \mathcal{L}_7\Big) }=\frac{3}{\sqrt{2}} \sqrt{- \curv{L}_8} + T \; \, \text{,}\end{aligned}$$ such that we can in fact consider a unique scalar (let us choose $\sqrt{- \curv{L}_8}$) made of order 6 scalars that leads to non vanishing second order differential equations. We can note that $T$ cannot be equal to $\sqrt{-J_{3,2}}$ because $\sqrt{-J_{2}}$ contains an $H^3$ terms. Therefore $T$ could be found considering higher order derivatives scalars, and other perfect powers. In this case, powers of four for order 12 scalars for example. There is then another non-linear relation between order 6 scalars and (possibly) order 12 ones.
Finally, to recapitulate this part, we have found that it is natural to consider only one perfect square made of order 6 scalars : $\curv{L}_8 = \nabla^\sigma R \nabla_\sigma R $. As a result, the action : $$\begin{aligned}
S_2= \int d^4x \sqrt{-g} \; \Bigg( \frac{1}{16 \pi}\bigg[ R +\nu \sqrt{ - \nabla^\sigma R \nabla_\sigma R } \;\bigg] + \curv{L}_m \Bigg),\end{aligned}$$ leads to a unique second order $H^3$-correction to the Friedmann equation, as it is easy to see using the same reasoning as in the previous section.
Order 8
-------
Now let us study the linear combination and squares made of order 8 scalars that lead to second order equations of motion. We do not copy all the FKWC basis for general metric, but we name the scalars according to their position in Ref [@17] where this basis is fully written. The reduced FKWC basis for order 8 scalars in FLRW space-time is the following : \
$\left\{
\begin{array}{l}
\mathcal{K}_1=R^4
\quad\; , \quad
\mathcal{K}_{10}=R^{\mu\nu}R^{\alpha\beta}R^{\sigma\rho}_{\;\,\;\,\,\mu\alpha}R_{\sigma\rho\nu\beta}
\quad\; , \quad
\mathcal{K}_{11}=R\,R^{\mu\nu\alpha\beta}R_{\mu\;\,\alpha}^{\;\,\sigma\;\,\rho}R_{\nu\sigma\beta\rho} = R \big( \frac{1}{4} \mathcal{L}_1 -\mathcal{L}_2 \big)
\\~
\\
\mathcal{K}_{12}=T^2
\quad\; , \quad
\mathcal{M}_1 = R\,\Box^2R
\quad\; , \quad
\mathcal{M}_2 =R_{\mu\nu} \nabla^{\mu}\nabla^{\
| 1,521
| 2,148
| 2,052
| 1,643
| null | null |
github_plus_top10pct_by_avg
|
The patients were classified into two groups on the basis of their average *CLU* mRNA levels. Chi-square test indicated that *CLU* level was closely related to tumor stage (*P* = 0.006) and lymph node metastasis (*P* = 0.002), but not to gender, age, tumor size, serum AFP, vascular invasion, cirrhosis or recurrence (*P* \> 0.05 for all). All data were listed in [Table 2](#T2){ref-type="table"}.
Response rate and CLU expression in HCC patients treated with OXA {#sec3-3}
-----------------------------------------------------------------
Patients with low *CLU* mRNA expression emerged more frequently in CR+PR group than in SD+PD group. It suggested that *CLU* mRNA levels were obviously related to response to OXA treatment in HCC patients ([Table 3](#T3){ref-type="table"}, *P* = 0.001). Subjects possessing high expression exhibited high OXA resistance, while low ones showed well response rate.
###### Response rate of HCC patients to OXA treatment according to *CLU* expression
Response rate CR+PR SD+PD χ^2^ *P*
----------------------------- ------------ ------------ -------- -------
*CLU* high expression (*n*) 14 (13.5%) 36 (34.6%)
*CLU* low expression (*n*) 32 (30.8%) 22 (21.2%) 10.284 0.001
Total number (*n*) 46 (44.2%) 58 (55.8%)
Abbreviations: CR, complete response; PD, progressive disease; PR, partial response; SD, stable disease.
Overall survival analysis for HCC patients treated with OXA {#sec3-4}
-----------------------------------------------------------
The average overall survival time was 20.8 months in HCC patients with high *CLU* expression, and 36.6 months in patients with low expression ([Figure 3](#F3){ref-type="fig"}), showing remarkable difference (Log rank test, *P* \< 0.001). Cox regression analysis results suggested that *CLU* was related to the outcomes of HCC patients treated with OXA (*P* \< 0.001). *CLU* could be an independent prognostic biomarker ([Table 4](#T4){ref-type="table"}, HR = 2.587
| 1,522
| 462
| 1,412
| 1,556
| null | null |
github_plus_top10pct_by_avg
|
ts,a_8,$ $b_1,\ldots,b_8$ and computing the coefficients $c(\alpha)$ defined in Sec. II. More precisely, for each example we write all probabilities entering the sums $S_1$, $S_2$, $S_3$ as the marginals of joint probability distribution. As a result we obtain the expressions of the form $$S=\sum_\alpha c(\alpha)p(\alpha)$$ where $\alpha$ runs over all $3^{16}$ configurations of the variables $a_1,\ldots,a_8,b_1,\ldots,b_8$. The results of numerical computations are summarized in the Table below.
------------- ----------------------- ------------- ----------------------- ------------- -----------------------
$c(\alpha)$ No. of configurations $c(\alpha)$ No. of configurations $c(\alpha)$ No. of configurations
1 12 960 1 9 720 1 18 360
2 159 408 2 126 576 2 115 596
3 645 408 3 510 480 3 474 696
4 1 729 188 4 1 514 862 4 1 445 778
5 3 479 760 5 3 182 904 5 3 286 224
6 5 424 408 6 5 374584 6 5 510 160
7 6 896 016 7 7 139 664 7 7 178 976
8 7 261 569 8 7 822 791 8 7 670 547
9 6 410 016 9 6 903 648 9 6 795 936
10 4 866 480 10 5 058 216 10 5 012 208
11 3 176 496 11 3 006 000 11 3 087 504
12 1 758 3
| 1,523
| 5,248
| 1,216
| 975
| null | null |
github_plus_top10pct_by_avg
|
te any of the spaces $\ell_{\infty},$ $\ell_{1}$ or $\ell_{p}$. Then, we have $\left \Vert
a\right \Vert _{X}^{\ast}=\left \Vert a\right \Vert _{X^{\beta}}$ for all $a\in
X^{\beta}$, where $\left \Vert .\right \Vert _{X^{\beta}}$ is the natural norm on the dual space $X^{\beta}$.
\[[\[3, Theorem 1.23 (a)\]]{}\]Let $X$ and $Y$ be $BK$-spaces. Then we have $(X,Y)\subset B(X,Y)$, that is, every matrix $A\in(X,Y)$ defines a linear operator $L_{A}\in B(X,Y)$ by $L_{A}(x)=Ax$ for all $x\in X$, where $B(X,Y)$ denotes the set all bounded (continuous) linear operators $L:X\rightarrow Y.$
\[[\[3, Lemma 2.2\]]{}\]Let $X\supset \phi$ be $BK$-space and $Y$ be any of the spaces $c_{0},$ $c$ or $\ell_{\infty}$. If $A\in(X,Y)$, then $$\left \Vert L_{A}\right \Vert =\left \Vert A\right \Vert _{(X,\ell_{\infty})}=\underset{n}{\sup}\left \Vert A_{n}\right \Vert _{X}^{\ast}<\infty.$$
By $M_{X},$ we denote the collection of all bounded subsets of a metric space $\left( X,d\right) .$ If $Q\in M_{X},$ then the *Hausdorff measure of noncompactness* of the set $Q,$ denoted by $\chi \left( Q\right) ,$ is defined by $$\chi \left( Q\right) :=\inf \left \{ \varepsilon>0:Q\subset \underset
{i=1}{\overset{n}{\cup}}B\left( x_{i},r_{i}\right) ,\text{ }x_{i}\in
X,\text{ }r_{i}<\varepsilon \text{ }\left( i=1,2,...,n\right) ,\text{ }n\in\mathbb{N}
-\{0\} \right \} .$$ The function $\chi:M_{X}\rightarrow \left[ 0,\infty \right) $ is called the $\mathit{Hausdorff}$ ** $\mathit{measure}$ ** $\mathit{of}$ ** $\mathit{noncompactness}$.
The basic properties of the Hausdorff measure of noncompactness can be found in \[3\]
The following result gives an estimate for the Hausdorff measure of noncompactness in the $BK$ space $\ell_{p}$ for $1\leq p<\infty.$
\[[\[24, Theorem 2.8\]]{}\]Let $1\leq p<\infty$ and $Q\in M_{\ell_{p}}.$ If $P_{m}:\ell_{p}\rightarrow \ell_{p}$ $(m\in\mathbb{N}
)$ is the operator defined by $P_{m}(x)=(x_{0},x_{1},...,x_{m},0,0,...)$ for all $x=(x_{k})\in \ell_{p}$, then we have$$\chi(Q)=\lim_{m\rightarrow \infty}\le
| 1,524
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github_plus_top10pct_by_avg
|
ance matrix estimation, it is all that is needed to apply the Gaussian comparison theorem \[thm:comparisons\], which will allow us to extend the Berry-Esseen bound established in to the case when $\Gamma$ is estimated.
\[lemma::upsilon\] Let $$\label{eq:aleph}
\aleph_n = \max \Big\{ \overline{H} B \overline{v} \sqrt{ b\frac{
\log n}{n}}, B^2
\sqrt{ b \overline{v} \frac{\log b + \log n }{n} }\Big\}.$$ There exists a $C > 0$ dependent on $A$ only such that $$\label{eq:upsilon}
\sup_{P\in {\cal P}_n}
\mathbb{P}\left(\max_{j,l} \left| \hat\Gamma(j,l)-\Gamma(j,l)\right| \geq C
\,\aleph_n\right) \leq \frac{2}{n}.$$
Now we construct the confidence set. Let $Q=(Q(1),\ldots, Q(s))$ be i.i.d. standard Normal variables, independent of the data. Let $\hat Z = \hat\Gamma^{1/2} Q$ and define $\hat{t}_\alpha$ by $$\label{eq:hat.t.alpha.berry}
\mathbb{P}( ||\hat Z||_\infty > \hat{t}_\alpha \,|\, \hat\Gamma)=\alpha.$$ Finally, let $$\label{eq::conf-rectangle}
\hat{C}_n = \Bigl\{ \theta \in \mathbb{R}^s:\ ||\theta-\hat\theta||_\infty \leq \frac{\hat{t}_\alpha}{\sqrt{n}}\Bigr\}.$$
\[thm::coverage\] There exists a $C>0$, dependent only on $A$, such that $$\inf_{P\in {\cal P}}\mathbb{P}(\theta\in R_n)= 1-\alpha -
C \left( \Delta_{n,1} + \Delta_{n,2} + \Delta_{n,3} + \frac{1}{n} \right),$$ where $$\label{eq::this-is-upsilon}
\Delta_{n,3}= \frac{\aleph_n^{1/3} (2 \log 2s)^{2/3}}{\underline{\sigma}^{2/3}}.$$
[**Remark**]{}. The additional term $\Delta_{n,3}$ in the previous theorem is due to the uncertainty in estimating $\Gamma$, and can be established by using the comparison inequality for Gaussian vectors of [@chernozhukov2015comparison], keeping track of the dependence on $\underline{\sigma}^2$; see below.
In addition to $L_\infty$ balls, we can also construct our confidence set to be a hyper-rectangle, with side lengths proportional to the standard errors of the projection parameters. That is, we define $$\label{eq:hyper:CI}
\tilde C_n = \bigotimes_{j\in S} C(j),$$ where $$C(j) = \left[ \hat\beta_S(j) - z_{
| 1,525
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| 3,798
| 0.770046
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|
ion $(1)$ of Lemma $\ref{lem:1.1}$, we have the following corollary:
\[cor:1.2\] We have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\g\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \\
\frac{\operatorname{{E}}[\Pe(Z + C)]}{\operatorname{{E}}[\Pe(Z)]}
&= \frac{1}{\G(2a)} \G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$
This corollary asserts that the expected value of the loss is reduced by correcting a predicted value $y$ to the optimized predicted value $y^{*}$. Moreover, the following holds:
\[thm:1.3\] We have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + C)] \leq \operatorname{{V}}[\Pe(Z)], \end{aligned}$$ where equality sign holds only when $C = 0$; that is, when $k_{1} = k_{2}$.
This theorem asserts that the variance of the loss is reduced by correcting the predicted value $y$ to the optimized predicted value $y^{*}$. To prove this theorem, we use the following lemma:
\[lem:1.4\] For $a > 0$ and $x > 0$, we have $$\begin{aligned}
x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x) > 0. \end{aligned}$$
To prove Lemma $\ref{lem:1.4}$, we use the following lemmas:
\[lem:1.5\] For $a > 0$, we have $$\begin{aligned}
2 \G(2a) - a \G(a)^{2} > 0. \end{aligned}$$
\[lem:1.6\] For $a > 0$, we have $$\begin{aligned}
4^{a} \G\left(a + \frac{1}{2} \right) > \sqrt{\pi} \G(a + 1). \end{aligned}$$
The remainder of this paper is organized as follows. In Section $2$, we set up the problem. In Section $3$, we introduce the expected value and the variance of $\Pe(Z + c)$, and we determine the value of $c = C$ that gives the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. In addition, we give a geometrical interpretation of the parameter $C$, and give the minimized expected value $\operatorname{{E}}[\Pe(Z + C)]$. In Section $4$, we prove Theorem $\ref{thm:1.3}$. In Section $5$, we give some inequalities for the gamma and the incomplete gamma functions, which used to d
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|
2.4,1)(1.9,0.5)(1.5,0.9)
\end{pspicture}$$ Here $\tau_{k+l+2}^{l+1}$ denotes the $(l+1)$st iteration of $\tau_{k+l+2}$, and $\epsilon=(|m^*_1|+|a^*_1|+\dots+|a^*_k|)\cdot
(|m_2|+|a^*_{k+1}|+\dots+|a^*_{k+l}|)$.
A straightforward check shows that if $d^2=0$ and $g^2=0$, then it is also $h^2=0$.
The following proposition identifies algebras over $\textbf{D} (\widehat{\mathcal O^!})$. Its proof is similar to that in [@GK Proposition (4.2.14)].
\[O\_hat\_algebras\] Let $\mathcal O$ be a cyclic quadratic operad, and let $A$ and $M$ be graded vector spaces over $k$, which are finite dimensional in every degree. Then giving $(A,M)$ the structure of an algebra over $\textbf{D} (\widehat{\mathcal O^!})$ is equivalent to the following data:
- a derivation $d\in \mathrm{Der}(F _{\mathcal O^!}\,A^*[1])$ of degree $1$, with $d^2=0$,
- a derivation $g\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M^*[1])$ over $d$ of degree $1$, with $g^2=0$, which by definition \[dual-module\] also implies a derivation $h\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M[1])$ over $d$ with $h^2=0$,
- a module map $f\in \mathrm{Mod}(F_{\mathcal O^!, A^*[1]}M[1], F_{\mathcal
O^!,A^*[1]}M^*[1])$ of degree $0$ such that $f\circ h = g \circ f$, and satisfying the following symmetry condition: let $f$ be given by maps $f_n:\,M[1]\to \bigoplus_{k+l=n-2} \mathcal O^!(k+l+1)\otimes A^*[1]^{\otimes k}\otimes \,M^*[1] \otimes A^*[1]^{\otimes l}$, then $$\begin{gathered}
\label{eq:symm}
f_n(m_2)(\alpha^*;a_1,\dots,a_i,m_1,a'_1,\dots,a'_j)=\\
= (-1)^\epsilon f_n(m_1)(\alpha^*\circ \tau_{i+j+2}^{j+1}
;a'_1,\dots,a'_j,m_2,a_1,\dots,a_i).\end{gathered}$$
where $\epsilon=(|m_2|+|a_1|+\dots+|a_i|+i+1)\cdot (|m_1|+ |a'_1|+\dots+
|a'_j|+j+1)$.
If $\mathcal O$ is cyclic quadratic and Koszul, then, by theorem \[O\_hat\_Koszul\], $\textbf{D}( \widehat{\mathcal O^!} )\cong
\widehat{\mathcal O}$ and we call $(A,M)$ a homotopy $\mathcal O$-algebra with homotopy $\mathcal O$-module and homotopy $\mathcal O$-inner product if there are deriva
| 1,527
| 1,761
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|
cal{P}}= \rho_{\ast}{\mathcal{O}}_{X_n}$\[PP-defn\] is the [*Procesi bundle*]{} on $\operatorname{Hilb(n)}$ of rank $n!$ arising from the map $\rho : X_n \to \operatorname{Hilb(n)}$ while ${\mathcal{L}}$\[LL-A-defn\] is the canonical ample line bundle ${\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$ associated to the presentation $\operatorname{Hilb(n)}\cong \operatorname{Proj}A$.
{#section}
Since ${\bf z}$ and ${\bf z}^*$ are bihomogeneous, the bigradings of to pass $\operatorname{Hilb(n)}$. Therefore, Lemma \[hi-basic-lem\](1) implies that $p(J^m, s, t)=(1-s)(1-t) p({\mathbb{J}}^m, s, t)$. Substituting this formula into Proposition \[bigr-hain\] gives:
\[bigr\] The bigraded Poincaré series of $J^d$ is $$\qquad\qquad \qquad\qquad\qquad\qquad
p(J^d, s, t) = \sum_{\mu} P_{\mu}(s,t)(1-s)(1-t)\,
\Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)}.
\hfill \qquad\qquad\qquad \qquad\hfill\qed$$
{#babyverma}
In Corollary \[gr\] we will give a singly graded analogue of Corollary \[bigr\] that will be needed in the proof of the Theorem \[mainthm-intro\]. In the proof we will need the following combinatorial formulæfor the fake degrees $f_\mu(v)$, as defined in .
Let $\mu\in{{\textsf}{Irrep}({{W}})}$. Then
[(1)]{} $ f_{\mu}(v) = v^N f_{\mu^t}(v^{-1})$, where $N=n(n-1)/2$,
[(2)]{} $f_{\mu}(v)\prod_{x\in d(\mu)} (1 - v^{h(x)})
= v^{n(\mu)}\prod_{i=1}^n (1-v^i),$ where $h(x)=1+a(x)+l(x)$ as in ,
[(3)]{} $\sum_{\lambda} v^{n(\mu)}K_{\lambda \mu}(v^{-1},v^{-1})f_{\mu}(v^{-1})
f_{\lambda}(1)
= \sum_{\lambda} f_{\lambda}(v^{-1}) f_{\mu}(1)f_{\lambda}(1).$
\(1) This is a well-known formula (see, for example, [@op p.453]).
(2,3) Up to a change of notation, these are both proved within the proof of [@babyv Theorem 6.4]—see the displayed equations immediately after, respectively immediately before [@babyv (18)].
{#gr}
The ${\mathbf{E}}$-grading from descends naturally to $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}})\cong {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$ and we will use the same notation there; th
| 1,528
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| 1,453
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| 0.782535
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|
nt cone to ${{\mathscr C}}$ at $p$, and by the choice of a side of a corresponding Newton polygon, with slope strictly between $-1$ and $0$. This procedure is explained in more detail in §\[details\].
Let $b<c$ be relatively prime positive integers such that $-b/c$ is the slope of the chosen side. Let $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is generated by a polynomial of the form $$x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b)\quad,$$ with $\rho_j \ne 0$. The number $S$ of ‘cuspidal’ factors in the limit curve is the number of segments cut out by the integer lattice on the selected side of the Newton polygon.
The germ listed above contributes a component of the PNC unless $b/c=1/2$ and the limit curve is supported on a conic union (possibly) the kernel line. The limit curves arising in this way are items (7) through (11) listed in §\[appendix\]. (In particular, the picture drawn above does not capture the possible complexity of the situation: several cuspidal curves may appear in the limit, as well as all lines of the basic triangle.) These limit curves are studied enumeratively in [@MR2002d:14083]. The limit curves contributing components to the PNC in this fashion are precisely the curves that contain nonlinear components and for which the maximal connected subgroup of the stabilizer of the union of the curve and the kernel line is the multiplicative group ${{\mathbb{G}}}_m$.
[**Type V.**]{} Assume $p$ is a singular point of the support of ${{\mathscr C}}$. Germs of type V are determined by the choice of the line $\ell$ in the tangent cone to ${{\mathscr C}}$ at $p$, the choice of a formal branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ for ${{\mathscr C}}$ at $p$ tangent to $\ell$, and the choice of a certain ‘characteristic’ rational number $C>\lambda_0$ (assuming these choices can be made). This procedure is also explained in more d
| 1,529
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4.479 6.062 2.437 2.667
Standard Deviation 8.69 10.50 3.38 3.18
Min/Max 0/40 0/40 0/10 0/10
Proportion of zero sent 0.547 0.437 0.458 0.354
Observations 48 48 48 48
**B. Allocator's behavior**
Proportion returned (*y*) 0.220 0.238 0.286 0.220
Standard Deviation 0.20 0.24 0.29 0.23
95% confidence interval (0.13, 0.31) (0.14, 0.33) (0.17, 0.40) (0.13, 0.30)
Min/Max 0/0.54 0/1 0/1 0/0.66
Proportion of zero returned 0.278 0.333 0.346 0.419
Observations 22 27 26 31
**C. Correlation coefficient between *X* and *y***
Spearman (*ρ*) -0.030 -0.230 -0.070 -0.048
Kendall (*τ*~*b*~) -0.039 -0.195 -0.056 -0.037
We observe in Panel A that the investor's amount sent (*X*) increases with her endowment (*e*~*i*~), while the proportion that is sent decreases with *e*~*i*~ as already suggested in \[[@pone.0204392.ref038]\], \[[@pone.0204392.ref039]\]. We also see that investors send less when the endowment of the allocator is lower. This complements \[[@pone.0204392.ref036]\], where it is shown that the amount sent by investors decreases when allocators are endowed. Our fi
| 1,530
| 5,350
| 2,222
| 1,291
| 4,048
| 0.768456
|
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|
at happens in a specific geographical location anchored to a specific time interval. Mathematically, given a multidimensional query : $$Q = \langle q, q_{time}, q_{geo}, g_{entity} \rangle,$$ and a subset of highly relevant documents $R \subseteq D$, the algorithm for this purpose $\textsc{EventDetect}$ should produce a set of ordered events : $$\mathcal{C} = \{ c_1, c_2, \ldots, c_k \},$$
where, $c = \langle \mathcal{E}, g, t, \mathcal{W} \rangle$. The event $c$ is hence described by the participating named entities $\mathcal{E}$, its location $g$, its time of occurrence $t$, and frequently occurring contextual terms around these semantic annotations $\mathcal{W}$. This requires proposing a probability mass function, $P(\mathcal{C}, R)$, using which we can impose a total order on $\mathcal{C}$.
As an example consider the keyword-only query `summer olympics` to the processed corpora of news articles. The designed algorithm shall then identify the important events as in Figure \[fig:event\].
**Diversifying and Summarizing Search Results** are retrieval tasks that try to address the information need underlying an ambiguous query at different levels of textual granularity. Each task tries to maximize the coverage of different information needs underlying the given ambiguous query. As information intents, we propose to use the mined set of *events*. Accomplishing these tasks would allow for automatic creation of *event timelines* or *entity biographies*. We briefly discuss an intuition of achieving the same.
When diversifying search results we would like to present users with *documents* such that the user finds *at least* one document that satisfies her information intent. For this we need to devise an algorithm $\textsc{EventDiverse}$ which considers as an input $Q$ and $R \subseteq D$. As an output it returns a set of documents $S \subset R$ which cover all events in $\mathcal{C}$.
Summarizing search results would require us to construct an algorithm $\textsc{EventSummary}$ to piece together, *sentences* $\h
| 1,531
| 1,966
| 2,464
| 1,717
| 3,088
| 0.774957
|
github_plus_top10pct_by_avg
|
ave kept running our business as usual, and here's few examples to show you that we are, indeed, doing just that:
EnronOnline did 5,866 transactions with 302 counterparties on Friday. Transaction counts remain higher than average.
EES signed a three-year fixed price agreement with Home Depot to supply power to 115 of its stores in California and 68 stores across Texas. EES also has exclusive rights to develop energy management proposals for all Home Depot facilities west of the Mississippi.
EES also signed a three-year energy management agreement with Memorial Sloan-Kettering Cancer Center's four New York City facilities.
Serious issues have been raised in the media and investment community that have put our credibility and reputation as a company in question. We're taking an introspective look at our business dealings, our core values and our organization as a whole. We're doing everything we can to deal with the issues that are affecting our company.
And that's where you come in. Look around you. Look at the excellence that you and your team represents. That's the reason we hired you - you're the best. Now, more than ever, work for Enron and especially for each other.
Thank you for your continuing support of our company and each other.
please handle for me. thanks. mhc
---------------------- Forwarded by Michelle Cash/HOU/ECT on 12/12/2000 04:26
PM ---------------------------
ARSystem@mailman.enron.com on 12/12/2000 02:59:26 PM
To: michelle.cash@enron.com
cc:
Subject: Request Submitted: Access Request for diane.goode@enron.com
You have received this email because the requester specified you as their
Manager. Please click
http://itcapps.corp.enron.com/srrs/auth/emailLink.asp?ID=000000000010092&Page=
Approval to review and act upon this request.
Request ID : 000000000010092
Request Create Date : 12/12/00 3:00:54 PM
Requested For : diane.goode@enron.com
Resource Name : Unlisted Application/Software
Resource Type : Applications
Darren,
do you know wha
| 1,532
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| 715
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| null | null |
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|
\hat{\theta}^*(j) - \hat{\theta}
(j) \leq \tilde{t}^*_j, \forall j \Big| (W_1,\ldots,W_n) \right) \geq 1 - \alpha.$$ By the union bound, each $\tilde{t}^*_j$ can be chosen to be the largest positive number such that $$\mathbb{P}\left( \sqrt{n} | \hat{\theta}^*(j) - \hat{\beta}
(j) > \tilde{t}^*_j, \Big| (W_1,\ldots,W_n) \right) \leq
\frac{\alpha}{s}.$$ Consider the following two bootstrap confidence sets: $$\label{eq:ci.boot.theta}
\hat{C}^*_{n} = \left\{ \theta \in \mathbb{R}^{s} \colon \|
\theta - \hat{\theta}
\|_\infty \leq \frac{ \hat{t}^*_{\alpha}}{\sqrt{n}} \right\} \quad
\text{and} \quad
\tilde{C}^*_{n} = \left\{ \theta \in \mathbb{R}^{s} \colon |
\theta(j) - \hat{\theta}(j)
| \leq \frac{ \tilde{t}^*_{j}}{\sqrt{n}}, \forall j \in {\widehat{S}}\right\}$$
\[theorem::boot\] Assume the same conditions of Theorem \[theorem::deltamethod\] and that and $\hat{\psi}$ and $\hat{\psi}^*$ belong to $\mathcal{S}_n$ almost surely. Suppose that $n$ is large enough so that the quantities $\underline{\sigma}^2_n = \underline{\sigma}^2 - C \aleph_n >0$ and $v_n = v - C \daleth_n$ are positive, where $C$ is the larger of the two constants in and in and $$\daleth_n =
\sqrt{ b \overline{v} \frac{ \log b + \log n }{n} }.$$ Also set $\overline{v}_n = \overline{v} + C \daleth_n$. Then, for a constant $C$ depending only on $A$, $$\label{eq::boot-cov}
\inf_{P\in {\cal P}_n}\mathbb{P}(\theta\in \hat{C}^*_n) \geq 1-\alpha -
C\left(\Delta^*_{n,1} + \Delta^*_{n,2} + \Delta_{n,3} +
\frac{1}{n}\right),$$ where $$\Delta^*_{n,1} =
\frac{1}{\sqrt{v_n}} \left( \frac{ \overline{v}_n b (\log 2bn)^7}{n} \right)^{1/6}
,\quad
\Delta^*_{n,2} =
\frac{1}{\underline{\sigma}_n}\sqrt{\frac{ b \overline{v}_n \overline{H}^2 (\log n)^2
\log b}{n}},$$ and $\Delta_{n,3}$ is given in (\[eq::this-is-upsilon\]). Similarly, $$\label{eq::boot-cov.bonf}
\inf_{P\in {\cal P}_n}\mathbb{P}(\theta\in \tilde{C}^*_n) \geq 1-\alpha -
C\left(\Delta^*_{n,1} + \Delta^*_{n,2} + \Delta_
| 1,533
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$\bar G'$ into a clique of the same size in $G$, by picking an arbitrary segment from each of the pairwise disjoint lines.) $\Box$
Ramsey-type bounds in $\mathbb{R}^2$ vs. $\mathbb{R}^3$–Proof of Theorem 4
==========================================================================
As we have pointed out in the Introduction, it is sufficient to establish Theorem 4 in $\mathbb{R}^3$. We rephrase Theorem 4 for this case in the following form.
[**Theorem 4.1.**]{}
*Let $f(m)$ be a function with the property that for any disjointness graph $G$ of a system of segments in $\mathbb{R}^2$ with $\max(\alpha(G),\omega(G))\le m$ we have $|V(G)|\le f(m).$*
Then for any disjointness graph $G$ of a system of segments in $\mathbb{R}^3$ with $\max(\alpha(G),\omega(G))$ $\le m$ we have $|V(G)|\le f(m)+m^4.$
Applying Theorem 4.1 with $f(k)=ck^{1/\beta}$, Theorem 4 immediately follows. We prove Theorem 4.1 by adapting the proof of Theorem 3.1.
[**Proof of Theorem 4.1.**]{} Let $G$ be the disjointness graph of a set of segments in $\mathbb{R}^3$ with $\omega(G)\le m$ and $\alpha(G)\le m$.
First, assume that all segments lie in the union of $k$ planes, for some $k\ge 1$. Define the sets of vertices $V_i$, $W_i$, and $Z_i$ for every $1\le i\le k$, as in the proof of Theorem 3.1, and let $V_0=V(G)\setminus\bigcup_{i=1}^kW_i$. Since all elements of $W_i$ lie in the same plane, the subgraph induced by them is a planar segment disjointness graph for every $i\ge1$. We can clearly represent these graphs by segments in a common plane $\pi$ such that two segments intersect if and only they come from the same set $W_i$ and there they intersect. In this way, we obtain a system of segments in the plane whose disjointness graph $G^*$ is the [*join*]{} of the graphs $G[W_i]$, i.e., $G^*$ is obtained by taking the disjoint union of $G[W_i]$ (for all $i\ge1$) and adding all edges between $W_i$ and $W_j$ for every pair $i\ne
j$. Clearly, we have $$\omega(G^*)=\sum_{i=1}^k\omega(G[W_i])=\sum_{i=1}^k|Z_i|\le\omega(G)\le m,$$ and $$\alpha(G^
| 1,534
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|
Women’s University\
Kita-Uoya Nishimachi\
Nara 630-8506, Japan
- |
Department of Mathematics\
Kyushu University\
33, Fukuoka 812-8581 Japan
- |
Department of Mathematical and Computing Sciences\
Tokyo Institute of Technology\
Ohokayama, Meguro\
Tokyo 152-8552 Japan
author:
- Yasushi Yamashita
- Haruko Nishi
- Sadayoshi Kojima
date: ' Version 1.0 (November 26, 1998)'
title: ' Configuration spaces of points on the circle and hyperbolic Dehn fillings, II'
---
= 22cm = 16.2cm = 0.2cm
Introduction
============
In [@KojimaNishiYamashita], we have shown that the configuration space of $n \, (\geq 5)$ marked points with weights on the real projective line up to projective transformations admits a natural hyperbolization so that the result becomes a hyperbolic cone-manifold of dimension $n-3$. In brief, we identify each component of the configuration space with the space of similarity classes of convex $n$-gons with fixed external angles in the complex plane via Schwarz-Christoffel map. Then there is a beautiful way by Thurston to hyperbolize such a space as an interior of some hyperbolic polyhedron (see [@BavardGhys; @KojimaYamashita]). Each point on the boundary can be encoded by an appropriately degenerate configuration. Pasting them together along the same degenerate configurations, we obtained the resultant cone-manifold. Kapovich and Millson discussed the same hyperbolization via their duality in [@KapovichMillson].
The external angles can be given in fact at your choice and we regarded them as the weight. A variation of the weights induces a deformation. We restricted the set of possible external angles so that the $n$-gons are convex and can be represented as the inner polygon of the star shaped $n$-gons for any marking. More concretely, we set $$\Theta_n=\{(\theta_1, \ldots, \theta_n)\,| \,
\sum_{i=1}^n \theta_i= 2\pi, \,
\theta_i>0, \, \theta_i + \theta_j < \pi \text { for any }i \ne j\}.$$ This is equivalent to say that the number of faces appeared in
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o sum up contributions that come from different order in $H_{1}$. For example, $\hat{S}_{i J}^{(3)} = \hat{S}_{i J}^{(3)} [3] + \hat{S}_{i J}^{(3)} [2]$.
- We present only the terms which are required to compute $S$ matrix elements to order $W^4$. In view of the relations between $\hat{S}$ and $S$ matrix elements given in eq. (\[Sab-hatSab\]), $\hat{S}_{I J}^{(3)}$, $\hat{S}_{I J}^{(4)}$, and $\hat{S}_{i J}^{(4)}$ (and $\hat{S}_{J i}^{(4)}$) are all unnecessary.
- We only give the results of manifestly generalized T invariant form of $\hat{S}$ matrix elements with which it must be straightforward to prove generalized T invariance.
Order $W^2$ $\hat{S}$ matrix elements from Second-order in $H_{1}$ {#sec:hatSIJ-2nd}
-------------------------------------------------------------------
The second order in $H_{1}$ contribution $\hat{S}_{I J} [2]$ (and $\hat{S}_{I I} [2]$) has both order $W^2$ and $W^4$ terms. To compute $S$ matrix elements to order $W^4$ we need only the former term because the latter produces order $W^6$ terms, as one can see in (\[Sab-hatSab\]).
Then, it is necessary to remind the readers that the expressions of order $W^2$ terms in $\hat{S}_{I J} [2]$ and $\hat{S}_{I I} [2]$ already exist as the first terms in (\[hatS-IJ(2+4)\]) and (\[hatS-2nd-order\]), respectively.
Order $W^3$ $\hat{S}$ matrix elements $\hat{S}_{i J}^{(3)}$ and $\hat{S}_{J i}^{(3)}$ {#sec:hatSiJ-2nd-]}
--------------------------------------------------------------------------------------
By adding the contribution of the third and second order in $H_{1}$ we obtain for $\hat{S}$ matrix elements $\hat{S}_{i J}^{(3)}$ at order $W^3$ $$\begin{aligned}
&& \hat{S}_{i J}^{(3)} [2+3]
\nonumber \\
&=&
- \frac{ 1 }{ \Delta_{J} - h_{i} }
\left[ (ix) e^{- i \Delta_{J} x} + \frac{ e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\right]
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&+&
\sum_{K \neq J}
\frac{ 1 }{ ( \Delta_{J} - \Delta_{K} ) ( \Delta_{J} - h_
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the intersection of the domains. Since the presumption that the subtrahend and remainder are not disjoint arrives at a contradiction, then subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are indeed disjoint.
The ensembles are complementary with respect to $\Psi$ by definition \[D:COMPLEMENTARY\_ENSEMBLES\] because they are disjoint and $\Phi \cup (\Psi \setminus \Phi) = \Psi$.
Choice space {#S:CHOICE_SPACE}
------------
Informally, a choice space is the totality of all possible combinations of variables’ values within a given ensemble. The general Cartesian product formalizes this notion.
### General Cartesian product {#S:GCP}
\[D:PROTOSET\] Let $\Psi$ be an ensemble. By definition \[D:ENSEMBLE\], each member of ${{\operatorname{ran}{\Psi}}}$ is itself a non-empty set. The *proto-set* $\Psi_\heartsuit$ is the union of all such sets: $$\Psi_\heartsuit = \bigcup_{R \in\: {{\operatorname{ran}{\Psi}}}} R.$$
Definition \[D:PROTOSET\] expresses a relation between the ensemble $\Psi$’s indexed sets and the proto-set, namely that for each $i \in {{\operatorname{dom}{\Psi}}}$, $\Psi(i) \subseteq \Psi_\heartsuit$. An alternative portrayal uses the power set, claiming $\Psi(i) \in {\mathscr{P}({\Psi_\heartsuit})}$.
\[D:PROTOSPACE\] Let $\Psi$ be an ensemble with proto-set $\Psi_\heartsuit$. The *proto-space* of $\Psi$ is the set ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$, where ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$ is the set of all mappings ${{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$.
\[D:CHOICE\] Let $\Psi$ be an ensemble with proto-set $\Psi_\heartsuit$. Let $\chi \colon {{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$ be a mapping. If $\chi$ satisfies $\chi(i) \in \Psi(i)$ for each $i \in {{\operatorname{dom}{\Psi}}}$, then $\chi$ is a *choice* mapping of $\Psi$.
\[D:CHOICE\_SPACE\] The set of all choice mappings of ensemble $\Psi$ is the *choice space* (or general Cartesian product) $\prod\Psi$.
Through the general Cartesian product, an ensemble *generates* a choice spa
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are functionally independent in a neighborhood of $x=0$ and the matrix functions $\mathcal{A}^\mu$, ${\partial}f/{\partial}r$, ${\partial}f/{\partial}\bar{r}$, ${\partial}\bar{f}/{\partial}r$, ${\partial}\bar{f}/{\partial}\bar{r}$, $Q_a$ and $K_a$ depend on $r$ and $\bar{r}$ only. So, equations (\[eq:3.26\]) (or (\[eq:3.27\])) have to be satisfied for any value of coordinates $x^{i_a}$. As a consequence, we have some constraints on these matrix functions. From the Cayley-Hamilton theorem, we know that for any $n\times n$ invertible matrix $M$, the expression $(M^{-1}\det M)$ is a polynomial in $M$ of order $(n-1)$. Thus, using the tracelessness of the expression $\mathcal{A}^\mu{\left( I_q-Q_ax^{i_a} \right)}^{-1}({\partial}f/{\partial}R)\Lambda$, we can replace equations (\[eq:3.26\]) by the following condition
\[eq:3.29\] [( \^Q[( + )]{})]{}=0,Q=(I\_q-Q\_ax\^[i\_a]{})\^[qq]{},
where $\operatorname{adj}M$ denotes the adjoint of the matrix $M$. As a consequence the matrix $Q$ is a polynomial of order $(q-1)$ in $x^{i_a}$. Taking (\[eq:3.29\]) and all its partial derivatives with respect to $x^{i_a}$ (with $r$, $\bar{r}$ fixed at $x=0$), we obtain the following conditions for the matrix functions $f(r,\bar{r})$ and $\lambda(f(r,\bar{r}))$
\[eq:3.30\] [( \^)]{}=0,=1,…,m,
\[eq:3.31\] [( \^Q\_[(a\_1.]{}…Q\_[a\_s)]{}[( + )]{})]{}=0,
where $s=1,\ldots,q-1$ and $(a_1,\ldots, a_s)$ denotes the symmetrization over all indices in the bracket. A similar procedure can be applied to system (\[eq:3.27\]) to yield (\[eq:3.30\]) and
\[eq:3.32\] [( \^K\_[(a\_1]{}…K\_[a\_s)]{})]{}=0,
where now $s=1,\ldots,2k-1$. Equations (\[eq:3.30\]) represent the initial value conditions on a surface in the space of independent variables $X$, given at $x^{i_a}=0$. Note that equations (\[eq:3.31\]) (or (\[eq:3.32\])) form the conditions required for the preservation of the property (\[eq:3.30\]) along the flows represented by the vector fields (\[eq:3.15i\]). Equation (\[eq:3.24\]) allows us to express $X_a$ in the form
\[eq:3.33\]
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e same vector grammar with capacity function constantly $k$.
To show ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, consider a capacity-bounded vector grammar $$G=(\{A_0,A_1,\ldots,A_m\},\Sigma,A_0,M,\mathbf{1}).$$ (The proof that it suffices to consider the capacity function $\mathbf{1}$ is like for usual grammars.) To construct an equivalent vector grammar of finite index, we introduce the new nonterminal symbols $B_i,B'_i$, $0\leq i\leq m$, $C$, $C'$. For any rule $r: A\to \alpha$, we define the matrix $\mu(r)=(C\to C',s_0,s_1,\ldots,s_m,r,C'\to C)$ such that $s_i=B_i \to B'_i$ if $A=A_i$ and $|\alpha|_A=0$, $s_i=B'_i \to B_i$ if $A\neq A_i$ and $|\alpha|_{A_i}=1$, and $s_i$ is empty, otherwise.
Now we can construct $G'=(V',\Sigma,S',M')$ where $M'$ contains
- for any matrix $m=(r_1,r_2, \ldots, r_k)$, the matrix $m'=(\mu(r_1), \ldots, \mu(r_k))$,
- the start matrix $(S'\to A_0 B_0 B'_1 \cdots B'_m C)$,
- the terminating matrix $(C\to {\lambda}, B'_0\to {\lambda},B'_1\to{\lambda}, \ldots, B'_m\to {\lambda})$,
and $V'=V\cup \{B_i,B'_i{:}0\leq i\leq m\} \cup \{S',C,C'\}$. The construction of $G'$ allows only derivation sequences where complete submatrices $\mu(r)$ are applied: when the sequence $\mu(r)$ has been started, there is no symbol $C$ before $\mu(r)$ is finished, and no other submatrix can be started. It is easy to see that $G'$ can generate after applying complete submatrices exactly those words $\beta \gamma C$ such that $\beta \in (V\cup \Sigma)^*$, such that $\beta$ can be derived in $G$ and $|\gamma|_{B_i}=1$ iff $|\beta|_{A_i}=1$. Moreover, $G'$ is of index $2 |V|+1$.
By constructions similar to those in [@tur] and Theorem \[thm:matrixGrammarBounds\] we can show with respect to weak capacities:
\[lem:VfinInwPNch\] For $z\in \{h,c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{wPN}^{[{\lambda}]}_{cz}$.
We give only the proof for $z=h$. The other equations can be shown using analogous arguments. By Theorem \[thm:matrixGrammarBounds\] it is suff
| 1,539
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erior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the second and fourth true classes (*good*, and *very good*) the original classifier uses a reduced domain of probabilities (indicative of underconfidence), while Dirichlet calibration is able to spread these probabilities with more meaningful values (as indicated by a reduction of the calibration losses; See Figure \[fig:nb:reldiag:adas:car\]). []{data-label="fig:nb:pos:scores:class"}](figures/results/adas_car_dirichlet_full_l2_positive_scores_per_class.pdf "fig:"){width="\linewidth"}
Experimental setup {#sec:exp}
==================
In this section we provide the detailed description of the experimental setup on a variety of non-neural classifiers and datasets. While our implementation of Dirichlet calibration is based on standard Newton-Raphson with multinomial logistic loss and L2 regularisation, as mentioned at the end of Section 3, existing implementations of logistic regression (e.g. scikit-learn) with the log transformed predicted probabilities can also be easily applied.
Datasets and performance estimation
-----------------------------------
The full list of datasets, and a brief description of each one including the number of samples, features and classes is presented in Table \[tab:data\].
Figure \[fig:ds:partition\] shows how every dataset was divided in order to get an estimated performance for every combination of dataset, classifier and calibrator. Each dataset was divided using 5 times 5-fold-cross-validation to create 25 test partitions. For each of the 25 partitions the corresponding training set was divided further with a 3-fold-cross-validation for wich the bigger portions were used to train the classifiers (and validate the calibratiors if they had h
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Indeed, the dominant diagrams for the CKM-induced quark EDMs have a rainbow topology. For instance, for the d-quark EDM:
![CKM-induced d-quark EDM[]{data-label="fig:CKMquarkEDM"}](figures/FigCKMQuarkEDM)
This EDM is tuned by the imaginary part of the 1-1 entry of a non-invariant commutator $Im(\textbf{X}^{dd}_{q})$, where: $$\textbf{X}_{q}=\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}],
\label{eq:Xq}$$ which is also proportional to $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$ as for the lepton EDMs (because we are in the SM), but not with the same proportionality factor. It turns out to be much larger by 10 orders of magnitude: $$Im\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}]^{dd}\gg\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}].$$ In the SM, the rainbow-like flavor structures are typically much larger than the invariant determinants and they are correlated (strictly proportional). Now, let us turn on neutrino masses (beyond the SM) and check whether this behavior is confirmed or not. As we do not know yet the nature of the neutrino (Dirac or Majorana particle), we will consider both scenarios for generating neutrino masses.
EDMs in the presence of neutrino masses
=======================================
Dirac neutrino masses
---------------------
The simplest way of including neutrino masses to the SM is to extend its particles content by adding three right-handed (RH) fully neutral neutrinos (one for each generation). They belong to the trivial representation of the SM gauge group: $N=\nu_{R}^{\dagger}\sim(1,1)_{0}$. We add to the SM Yukawa Lagrangian an extra Yukawa interaction for the neutrinos:
$$\mathcal{L}_{Yukawa}=\mathcal{L}_{Yukawa}^{SM}-N^{I}Y_{\nu}^{IJ}L^{J}H^{\dagger C}+h.c.$$
We have a new neutrinos-related flavor structure $Y_{\nu}$ ($3\times3$ matrix in flavor space). In the
| 1,541
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angle \;
e^{i\omega \tau} \nonumber \\
& = & \frac{1}{2} KT \; + \; e^{-\gamma t/2} \;
\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] \; \; ,\end{aligned}$$ $\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] $ is a measure of departure of energy density from thermal average at $t=0$. The exponential term implies that this deviation due to the initial excitation decays asymptotically to zero as $t\rightarrow \infty$, so that one recovers the usual fluctuation-dissipation relation for the thermal bath. With the above specification of correlation function of $\xi_{\rm neq}$ Eq.(15) thus attributes the nonstationary character of the {$q_k$}-subsystem.
In passing, we stress that the above derivation$^{10}$ is based on the assumption that $\xi_{\rm neq}$ is effectively stationary on the fast correlation of the thermal modes. This is a necessary requirement for the systematic separation of time scales involved in the dynamics. We point out that the effective dynamics sets no choice on any special form of coupling $g(x)$ between the system mode and the relaxing mode and as such this may be of arbitrary nonsingular type for our problem we have considered here.
**[III.The generalized Fokker-Planck description]{}**
Eq.(12) is the required Langevin equation for the particle moving in a modified potential ${\tilde{V}}(x)$ \[Eq.(14)\] and damped by a coordinate-dependent friction $ \Gamma (x) $ \[Eq.(13)\] due to its linear coupling to a thermal bath and nonlinear coupling to the $\{q_k\}$-subsystem characterized by fluctuations $\xi_{\rm neq} (t)$. Before proceeding further a few pertinent points are to be noted to stress some distinct and important aspects of the model.
First, depending on the system-{$q_k$}-subsystem coupling $g(x)$ both the modified potential ${\tilde{V}}(x)$ as well as $\Gamma(x)$ are, in general, nonlinear. So the stochastic differential equation (12) is nonlinear. Again, the stochasticity in Eq.(12) is composed of two parts : $\xi_{\rm eq} (t)$ is an additive noise due to thermal bath whil
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riond; @del_moriond; @l3_moriond; @op_moriond].
$n_{\rm obs}$ $n_{\rm back}$ $n_{\rm sig}$
--------------------------------------- --------------- ---------------- ---------------
ALEPH 53 44.8 13.8
DELPHI 26 31.3 10.1
L3 30 30.3 9.9
OPAL 50 43.9 12.6
Total 159 150 46.4
$\Delta{M_{\mathrm{H }}}=92-96$ [ ]{} 47 37.5 24.6
: Standard Model Higgs search. Number of observed events in the data $n_{\rm obs}$, expected number of background events $n_{\rm back}$ and expected numbers of signal events $n_{\rm sig}$ assuming ${M_{\mathrm{H }}}=95$ [ ]{}for the four LEP experiments and for their combination. Also shown are the number of events observed and expected by the four experiments combined in the mass window $\Delta{M_{\mathrm{H }}}=92-96$ [ ]{}.[]{data-label="tab:res"}
As can be observed from Table \[tab:res\], an excess of events is observed by ALEPH [@al_moriond] and OPAL [@op_moriond] which, in the case of OPAL, is concentrated in the mass region around ${M_{\mathrm{H }}}\simeq{M_{\mathrm{Z}}}$, while for ALEPH it is distributed over higher masses, typically $\geq95$ [ ]{}. These results translate into the lower limits shown in Table \[tab:lim\], together with the sensitivity (expected limit) of each experiment.
-------- --------------- --------------
Observed Expected
limit ([ ]{}) limit([ ]{})
ALEPH 90.2 95.7
DELPHI 95.2 94.8
L3 95.2 94.4
OPAL 91.0 94.9
-------- --------------- --------------
: Observed 95% C.L. lower limits on ${M_{\mathrm{H }}}$. Also shown are the limits predicted by the si
| 1,543
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omega''_i\in\Omega^{{{\bf n}}}_{v'\to a'}$, followed by the summation over $i=3,\dots,2j+1$ (cf., Figure \[fig:eventI\]). Finally, we apply Lemma \[lmm:GHS-BK\] to obtain the desired bound on the last line of [(\[eq:Theta”-2ndindbd5\])]{}.
Summarizing the above (d-1) and (d-2), we obtain $$\begin{aligned}
{(\ref{eq:Theta''-2ndindrewr})}\leq\sum_{j\ge1}\sum_{u\in{{\cal A}}}P_{
\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x).\end{aligned}$$
This together with [(\[eq:Theta”-0bdfin\])]{} in the above paragraph (b) complete the proof of the bound on $\Theta''_{y,x,v;{{\cal A}}}$ in [(\[eq:Theta’Theta”bd\])]{}.
Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ assuming the decay of $G(x)$
=================================================================================
Using the diagrammatic bounds proved in the previous section, we prove Proposition \[prp:GimpliesPix\] in Section \[ss:proof-so\], and Propositions \[prp:GimpliesPik\] and \[prp:exp-bootstrap\](iii) in Section \[ss:proof-nn\].
Bounds for the spread-out model {#ss:proof-so}
-------------------------------
We prove Proposition \[prp:GimpliesPix\] for the spread-out model using the following convolution bounds:
\[prp:conv-star\]
(i) Let $a\ge b>0$ and $a+b>d$. There is a $C=C(a,b,d)$ such that $$\begin{aligned}
{\label{eq:conv}}
\sum_y\frac1{{\vby-v{|\!|\!|}}^a}\,\frac1{{\vbx-y{|\!|\!|}}^b}\leq\frac{C}
{{\vbx-v{|\!|\!|}}^{(a\wedge d+b)-d}}.\end{aligned}$$
(ii) Let $q\in(\frac{d}2,d)$. There is a $C'=C'(d,q)$ such that $$\begin{aligned}
\sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\,\frac1{{\vbx'-z{|\!|\!|}}^q}\,\frac1{{\vbz-y{|\!|\!|}}^q}\,
\frac1{{\vbz-y'{|\!|\!|}}^q}\leq\frac{C'}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}.{\label{eq:star}}\end{aligned}$$
The inequality [(\[eq:conv\])]{} is identical to [@hhs03 Proposition 1.7(i)]. We use this to prove [(\[eq:star\])]{}. By the triangle inequality, we have $\frac12{\vbx-y{|\!|\!|}}\leq{\vbx-z{|\!|\!|}}\vee{\vbz-y{|\!|\!|}}$ and $\frac12{\vbx'-y'{|\!|\!|}}\l
| 1,544
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the self-dual case for $N=3$, $N=5$ and $N=7$. In each case, we give the lowest order symmetry $X^1$. However, this symmetry does [*not*]{} reduce to the case of (\[self-dual-equn\]), but the second symmetry, $X^2$, of the hierarchy generated by the master symmetries (\[eq:phi-sys-msym\]), is a symmetry of the reduced system.
The case $\boldsymbol{N=3}$, with level structure $\boldsymbol{(1,2;1,2)}$
--------------------------------------------------------------------------
After the transformation $\phi^{(0)}_{m,n} \rightarrow 1/\phi^{(0)}_{m,n}$, this system becomes
\[eq:3D-1212\] $$\begin{gathered}
\phi^{(0)}_{m+1,n+1} = \frac{\alpha \phi_{m+1,n}^{(1)} - \beta \phi^{(1)}_{m,n+1}}{\alpha \phi_{m+1,n}^{(0)}\phi^{(1)}_{m,n+1} - \beta \phi_{m,n+1}^{(0)}\phi^{(1)}_{m+1,n}} \frac{1}{\phi^{(0)}_{m,n}} ,\\
\phi^{(1)}_{m+1,n+1} = \frac{\alpha \phi_{m,n+1}^{(0)} - \beta \phi^{(0)}_{m+1,n}}{\alpha \phi_{m+1,n}^{(0)}\phi^{(1)}_{m,n+1} - \beta \phi_{m,n+1}^{(0)}\phi^{(1)}_{m+1,n}} \frac{1}{\phi^{(1)}_{m,n}} .
\end{gathered}$$
System (\[eq:3D-1212\]) admits two point symmetries generated by $$\begin{gathered}
\begin{cases} \partial_\epsilon \phi^{(0)}_{m,n} = \omega^{n+m} \phi^{(0)}_{m,n}, \\ \partial_\epsilon \phi^{(1)}_{m,n} = 0,\end{cases} \qquad \begin{cases} \partial_\eta \phi^{(0)}_{m,n} =0,\\ \partial_\eta \phi^{(1)}_{m,n} = \omega^{n+m} \phi^{(1)}_{m,n},\end{cases}\qquad \omega^2+\omega+1=0,\end{gathered}$$ and two local generalized symmetries. Here we present the symmetry for the $m$-direction whereas the ones in the $n$-direction follow by changing $\phi^{(i)}_{m+j,n} \rightarrow \phi^{(i)}_{m,n+j}$ $$\begin{gathered}
\partial_{t_1} \phi^{(0)}_{m,n} = \phi^{(0)}_{m,n} \frac{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} - 2 \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}} ,\\
\partial_{t_1} \phi^{(1)}_{m,n} = -\phi^{(1)}_{m,n} \frac{1-2 \phi^{(0)}_{m+1,n}\phi^{(0)}_
| 1,545
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eads the torus. Fig. 3 is analogous to Fig. 1 as described above with the notable exception that the $\nu$ splitting is non-trivial (hence fewer distinct states are shown), and level crossings occur at larger values. There is no plot analogous to Fig. 2 for larger values of $\tau_1$ with $\tau_0 = 0$ because $\phi$ dependence increases rapidly in the eigenstates even for small $\tau_1$; a contour plot is preferable. Figs. 4 and 5 show contour plot results for two states at three field strengths. Note that there is slightly more dependence in $\theta$ when $\tau_1 = 0$ for the state displayed in Fig. 6 than in Fig. 5; the integration measure acts to cancel the angular variation of the state displayed in Fig. 5.
Extension to Coulomb integrals
==============================
The two-electron problem on $T^2$ is complicated by the inability (at least by the author) to find a transformation that decouples the relative electron motion from their center of mass motion as is easily done on $R^2$ [@qdhe]. The obvious transformations do not lead to any advantage over the method adopted by workers long used in atomic and molecular physics, which is to evaluate the two-body matrix elements (supressing spin indices and physical constants) $$\int \int \Phi^*_i({\bf r}_1)\Phi^*_j({\bf
r}_2)V({\bf r}_1,{\bf r}_2)(1-P_{12})\Phi_k({\bf r}_1)\Phi_l({\bf
r}_2)d^3{\bf r}_1d^3{\bf r}_2$$ with $$V({\bf r}_1,{\bf r}_2) = 4\pi\sum_{L,M}{1\over 2L+1} {r^l_{<}
\over r^{l+1}_{>}}
Y^*_{LM}(\theta_1,\phi_1)Y^*_{LM}(\theta_2,\phi_2).$$ Eq. (20) can be adopted on $T^2$ subject to some peculiarities which are due to the restriction of ${\bf r}_1,{\bf
r}_2$ to a surface. Eq. (20) on $T^2$ with the notation employed in section 3 becomes $$\notag
\int_0^{2\pi}... \int_0^{2\pi}
\Psi^*_{P{\nu_1}}(\theta_1,\phi_1)\Psi^*_{Q{\nu_2}}(\theta_2,\phi_2)V({\bf
r}_1,{\bf r}_2)(1-P_{12})$$ $$\Psi_{R{\nu_3}}(\theta_1,\phi_1)\Psi_{S{\nu_4}}(\theta_2,\phi_2)F(\theta_1)F(\theta_2)d\theta_1
d\theta_2 d\phi_1 d\phi_2.$$ Consider the direct term; in terms of a sph
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space to minimize numerical error on the marginal likelihood estimate. Although adaptive sparse grids [@Bun03] have the potential to be more beneficial in terms of resource management and precision, care would be needed in automating the grid refinements, and it is likely that a unique grid would be associated with each distinct firing pattern.
The sequential aspect of the proposed methodology provides the opportunity for real-time inference that has the potential to provide in-lab assistance during experimentation. In this framework, an interim SMC-MUNE analysis could help in identifying the choice of stimulus to apply in order to collect the best evidence to distinguish between competing hypotheses, as in Section \[sec:SimStudy\_under\]. The limitations of the present SMC-MUNE procedure to become a wholly online algorithm are the computational aspects discussed earlier and the post-processing stage to correct for potentially negligible estimates of the expected MU twitch forces. Solutions to these outstanding problems would increase the efficiency and accuracy of SMC-MUNE and, hence, the range of application.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Dr. Christine Thomas from the Miami Project to Cure Paralysis who provided the data analysed in this paper and for specialist discussions.
Additional detail {#sec:APX}
=================
The prior sufficient statistics for the simulation and case studies are: ${\bar{m}}_0 = 0$, ${\bar{c}}_0 = 10^3$, ${\bar{a}}_0 = 0.5$, ${\bar{b}}_0 = 0.1$, ${\mathbf{m}}_0 = 40 {\mathbf{1}}_u$, $C_0 = 10^4I_u$ and $a_0 = 0.5$ where ${\mathbf{1}}_u$ is a unit $u$-vector and $I_u$ is the $u\times{u}$ identity matrix. The statistic $b_0$ is defined according to with $\delta=0.05$ and $\epsilon=0.2$. The upper bounds for the excitability parameter space are ${\eta_{\max}}= 1.1s_{\tau}$ and ${\lambda_{\max}}= 14$. In the case study, the upper bound for the scale parameter was reduced to ${\lambda_{\max}}=7$.
Resampling in Alg
| 1,547
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\rho_q(d-j(q-1)-\ell,m-j-1) + \sum_{i=1}^b\rho_q(i,m-a-1).$$
This follows using Lemma \[lem:help\] repeatedly. First applying the lemma to each sum within the double summation on the right-hand side, we see that $$\begin{gathered}
\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m-j-1) = \\
\sum_{j=0}^{a-1} \left( \rho_q(d-j(q-1),m-j)-\rho_q(d-(j+1)(q-1),m-j-1) \right) = \\
\rho_q(d,m)-\rho_q(d-a(q-1),m-a) = \rho_q(d,m)-\rho_q(b,m-a).\end{gathered}$$ Using the same lemma to rewrite the single summation on the right-hand side in Equation we see that if $m>a$ $$\sum_{i=1}^b\rho_q(i,m-a-1)=\rho_q(b,m-a)-\rho_q(0,m-a-1)=\rho_q(b,m-a)-1,$$ while if $m=a$, the single summation equals $0$ and the double summation simplifies to $\rho_q(d,m)-1$. In either case, we obtain the desired result
We can now show the following.
\[thm:genrepMac\] Let $N \ge 0$ and $d \ge 1$ be integers and $q$ a prime power. Then there exist uniquely determined integers $m_1,\dots,m_d$ satisfying
1. $N=\sum_{i=1}^d \rho_q(i,m_i),$
2. $-1 \le m_1 \le \cdots \le m_d,$
3. for all $i$ satisfying $1 \le i \le d-q+1$, either $m_{i+q-1} > m_i$ or $m_{i+q-1}=m_i=-1$.
We start by showing uniqueness. Suppose that $$\label{eq:uniqueness}
N=\sum_{i=1}^d \rho_q(i,m_i)=\sum_{i=1}^d \rho_q(i,n_i)$$ and the integers $n_1,\dots,n_d$ and $m_1,\dots m_d$ satisfy the conditions from the theorem. First of all, if $m_d=-1$ or $n_d=-1$ then $N=0$. Either assumption implies that $(m_d,\dots,m_1)=(-1,\dots,-1)=(n_d,\dots,n_1)$. Indeed $n_i\ge 0$ or $m_i \ge 0$ for some $i$ directly implies that $N>0$. Therefore we from now on assume that $m_d\ge 0$ and $n_d \ge 0$. To arrive at a contradiction, we may assume without loss of generality that $n_d \le m_d-1$.
Define $e$ to be the smallest integer such that $n_e \ge 0$. Equation can then be rewritten as $$\label{eq:uniqueness1}
N=\sum_{i=1}^d \rho_q(i,m_i)=\sum_{i=e}^d \rho_q(i,n_i)$$ Condition 3 from the theorem implies that $n_{i-q+1}<n_i$ for all $i$ satisfying $e \le i \le d $. Now write $d-e+1=a(q-1)
| 1,548
| 3,423
| 1,426
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he geometric quotient $X/R$ exists.
Proof. Consider first the case when $\dim X=1$. Let $\pi:X^n\to X$ be the normalization. We construct $X^n/R^{nd}$ as in (\[induct.plan\]). Note that since $Z$ is zero dimensional, it is finite over $S$. Let $V\subset S$ be its image. Next we make a different choice for $Z_1$. Instead, we take a subscheme $Z_2\subset X^n/R^{nd}$ whose support is $Z_1$ such that the pull back of its ideal sheaf $I(Z_2)$ to $X^n$ is a subsheaf of the inverse image sheaf $\pi^{-1}{{\mathcal O}}_X\subset {{\mathcal O}}_{X^n}$.
Then we consider the push-out diagram $$V\leftarrow Z_2\into X^n/R^{nd}$$ with universal push-out $Y$. Then $X\to Y$ is a finite morphism and $X/R$ exists by (\[quot.X/S.finite.lem\]).
The case when $\dim X=2$ and $X$ is seminormal is a direct consequence of (\[induct.plan\].2.4) since the inductive assumption (\[induct.plan\].2.1) is guaranteed by (\[quot.by.R.lowdim\].1).
If $\dim X=3$, then $X$ is already normal and $Z$ is seminormal by assumption. Thus $Z/\bigl(R|_Z\bigr)$ exists by (\[quot.by.R.lowdim\].2). Therefore $X/R$ is given by the push-out of $Z/\bigl(R|_Z\bigr)\leftarrow Z\into X$.
Quotients in positive characteristic {#pos.char.sec}
====================================
The main result of this section is the proof of (\[quot.by.R.charp\]).
[@sga5 XIV] \[frob.say\] Let $S$ be an ${{\mathbb F}}_p$-scheme. Fix $q=p^r$ for some natural number $r$. Then $a\mapsto a^q$ defines an ${{\mathbb F}}_p$-morphism $F^q:S\to S$. This can be extended to polynomials by the formula $$f=\sum a_Ix^I\mapsto f^{(q)}:= \sum a^q_Ix^I.$$
Let $U={\operatorname{Spec}}R$ be an affine scheme over $S$. Write $R={{\mathcal O}}_S[x_1,\dots,x_m]/(f_1,\dots,f_n)$ and set $$R^{(q)}:= {{\mathcal O}}_S[x^{(q)}_1,\dots,x^{(q)}_m]/(f^{(q)}_1,\dots,f^{(q)}_n)
{\quad\mbox{and}\quad} U^{(q)}:=
{\operatorname{Spec}}R^{(q)},$$ where the $x^{(q)}_i$ are new variables. There are natural morphisms $$F^q:U\to U^{(q)}{\quad\mbox{and}\quad} (F^q)^*:R^{(q)}\to R{\quad\mbox{given by}\quad}
(F^q)^*(x^{(q)
| 1,549
| 732
| 957
| 1,509
| 3,008
| 0.775481
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>0$ let $\delta_1=\varepsilon/4$ and $\delta_2=\varepsilon/(2\|K\|_V\|f\|^{1/2})$. Then, if the collections of functions $k_1^{(1)},\dots, k_{N_1}^{(1)}$ and $k_1^{(2)},\dots, k_{N_2}^{(2)}$ are $L_2(Q)$ $\delta_1$-dense respectively in the classes ${\cal K}_1$, ${\cal K}_2$, and $g_1,\dots,g_{N_3}$ are $L_1(Q)$ $\delta_2$-dense in the class $\cal J$, with optimal cardinalities $N_i=N({\cal K}_i, L_2(Q),\delta_1)$, $i=1,2$, and $N_3=N({\cal J},L_2(Q),\delta_2)$, then, by the previous inequality, the functions $(k_i^{(1)}-k_j^{(2)})g_l$ are $L_2(Q)$ $\varepsilon$-dense in $\cal F$ . Since there are at most $N_1N_2N_3$ such functions (this estimate may not be optimal), the inequality (\[ivc1\]) follows.
A similar result holds for the classes ${\cal Q}_n$ defined by (\[qu\]) in the proof of Lemma \[lemma1\], the classes of functions $\bar{\cal Q}_n$ defined by (\[qubar\]) and the classes $\{H_t(x,y):t\in D_r\}$ in the proof of Lemma \[eps1-t\], as all these classes have the same structure as $\cal F$ in Lemma \[vc1\].
The class of functions ${\cal G}:=\{g(t,\cdot):t\in D_r\}$ where $g$ is defined in (\[ge\]) in the proof of Lemma \[lemma3\], requires some extra considerations. Let $Q$ be any probability measure on the line and let $s,t\in D_r$. Then, using Hölder, we have $$E_Q(g(t,x)-g(s,x))^2\le \int E_X\left(f(X)^{-1}K^2\left(\frac{X-x}{h_{1,n}}\right)\right)\times$$ $$\times E_X\left(L\Big(\frac{t-X}{h_{2,n}}f^{1/2}(X)\Big)
I(|t-X|<h_{2,n}B)-L\Big(\frac{s-X}{h_{2,n}}f^{1/2}(X)\Big)
I(|s-X|<h_{2,n}B)\right)^2dQ(x)$$ $$= h_{1,n}\|K\|_2^2\int \left(L\Big(\frac{t-y}{h_{2,n}}f^{1/2}(y)\Big)
I(|t-y|<h_{2,n}B)-L\Big(\frac{s-y}{h_{2,n}}f^{1/2}(y)\Big)
I(|s-y|<h_{2,n}B)\right)^2f(y)dy$$ $$=h_{1,n}\|K\|_2^2E_f(\ell_t-\ell_s)^2$$ where $\ell_{s}$ and $\ell_{t}$ are functions from the class $${\cal L}:=\left\{L\left(\frac{t- \cdot}{h}f^{1/2}(\cdot)\right)I(
|t-\cdot|<hB):t\in\mathbb{R}, h>0 \right\}$$ which is VC with a constant envelope by Lemma \[vc1\]. This lemma then proves that for all $Q$, $$\label{entg}
N({\cal G}
| 1,550
| 675
| 1,049
| 1,522
| 4,154
| 0.767698
|
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from naive considerations; it is confirmed by the study of the blow-ups mentioned above. Slightly more refined phenomena (for example, involving multiplicities of the components) are not represented in this figure; in general, they can be easily established by applying the results of this paper or by analyzing the blow-ups of [@MR2002d:14083].
![[]{data-label="figure1"}](pictures/smallpichor3)
We close by pointing out one such phenomenon. In general, the union of a set of quadritangent conics and a tangent line can specialize to the union of a conic and a tangent line in two ways: (i) by type II germs aimed at a general point of one of the conic components, and (ii) by a suitable type IV germ aimed at the tangency point. The multiplicity of the conic in the limit is then the multiplicity of the selected component in case (i), and the sum of the multiplicities of all conic components in case (ii). If the curve consists solely of quadritangent conics, it degenerates to a multiple conic in case (ii). This possibility occurs in the boundary of the orbit of the quartic curve from Example \[extwo\], represented in Figure \[figure2\]. We have omitted the set of stars of four distinct lines also in this figure; in this case, it is a $6$-dimensional union of $5$-dimensional orbits.
![[]{data-label="figure2"}](pictures/exapic)
Appendix: curves with small linear orbits {#appendix}
=========================================
For the convenience of the reader, we reproduce here the description of plane curves with small linear orbits given in [@MR2002d:14084]. That reference contains a proof that this list is exhaustive, and details on the stabilizer of each type of curve (as well as enumerative results for orbits of curves consisting of unions of lines, items (1)–(5) in the following list).
Let ${{\mathscr C}}$ be a curve with small linear orbit. We list all possibilities for ${{\mathscr C}}$, together with the dimension of the orbit ${{\mathscr O_{{\mathscr C}}}}$ of ${{\mathscr C}}$. The irreducible components o
| 1,551
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ecisely, the crucial new step is:
\[thm41\] Suppose there is a model in which there is a Souslin tree $S$ and in which ****, ****, and $2^{\aleph_1}=\aleph_2$ hold. Then $S$ forces that locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff.
It will be convenient to consider the following intermediate proposition, which implies the three things that we want:
**SRP**
Strong Reflection Principle [@To2]. Suppose $\lambda\geq\aleph_2$ and $\mathfrak{Z}\subseteq\mathcal{P}_{\omega_1}(\lambda)$ and that for each stationary $T\subseteq\omega_1$, $$\{\sigma\in\mathfrak{Z}:\sigma\cap\omega_1\in T\}$$ is stationary in $\mathcal{P}_{\omega_1}(\lambda)$. Then for all $X\subseteq\lambda$ of cardinality $\aleph_1$, there exists $Y\subseteq\lambda$ such that:
- $X\subseteq Y$ and $|Y|=\aleph_1$;
- $\mathfrak{Z}\cap\mathcal{P}_{\omega_1}(Y)$ contains a set which is closed unbounded in $\mathcal{P}_{\omega_1}(Y)$.
With regard to **SCC**, Shelah [@S XII.2.2, XII.2.5] proves that:
If there is a semi-proper forcing P changing the cofinality of $\aleph_2$ to $\aleph_0$, then **** holds.
There are various versions of *Namba forcing*, e.g. two in [@S] and one in [@Lar]. All of these change the cofinality of $\aleph_2$ to $\aleph_0$. Larson states in [@Lar p.142] that his version of Namba forcing preserves stationary subsets of $\omega_1$. In [@FMS], it is shown that a principle, **SR**, implies *any forcing that preserves stationary subsets of $\omega_1$ is semi-proper*. **SR** is a consequence of MM [@FMS]. is stronger than **SR** and so:
implies .
$(S)$ implies .
**** implies **** and $2^{\aleph_1}\leq\aleph_2$.
For the proof of \[thm:paracompactcopyallmodels\] we should also remark that:
**** implies ****.
We use an equivalent formulation of **SRP** due to Feng and Jech [@FJ].
**SRP**
For every cardinal $\kappa$ and every $S\subseteq[\kappa]^\omega$, for every regular $\theta>\kappa$, there is a continuous elementary chain $\{N_\alpha:\alpha\in\omega_1\}$ (with $N_0$ containing some given element
| 1,552
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{\Lambda_H}}.$$ The following is well-known and easily follows from the definitions:
\[weight restriction\] Let $V$ be a representation of $G$ and $v \in V$ a weight vector of weight $\beta \in \Lambda^*_G$. If we restrict the action to $H$ via then $v$ is a weight vector of weight $F^*(\beta) \in \Lambda^*_H$.
Let us also fix systems of positive roots $R_{H,+}$ for $H$. This in turn determines the set of dominant weights $\Lambda^*_{H,+}$ as well as a basis of fundamental weights $(\omega^H_j)$ as described in . Let us also set $r_H = \dim T_H$.
Our strategy for solving the subgroup restriction problem for $f$ then is the following: Given an irreducible representation $V_{G,\lambda}$ of $G$, we can determine its weight multiplicities with respect to the maximal torus $T_G$ by using any of the formulas presented in . We then obtain weight multiplicities for $T_H$ by restricting according to . Finally, we reconstruct the multiplicity of an irreducible representation $V_{H,\mu}$ by using the finite-difference formula (/). If this procedure was translated directly into an algorithm, the runtime would be polynomial in the coefficients of $\lambda$ (with respect to the basis of fundamental weights), i.e., exponential in their bitlength, since the number of weights is of the order of the dimension of the irreducible representation $V_{G,\lambda}$, which according to the *Weyl dimension formula* is given by the polynomial $\prod_{\alpha \in R_{G,+}} {\langle \alpha, \lambda + \rho \rangle} / {\langle \alpha, \rho \rangle}$ (cf. the formula by Straumann [@straumann65]). We will now show that it is possible to combine the weight multiplicity formula with the restriction map $F^*$ in a way that will later give rise to an algorithm that runs in polynomial time in the bitlength of the input:
\[main theorem\] Let $f \colon H \rightarrow G$ be a homomorphism of compact connected Lie groups. Then we can find $s, s', u \in {\mathbb Z}_{\geq 0}$ and group homomorphisms $\mathcal A \colon {\mathbb Z}^{s+s'} \rightarrow {\math
| 1,553
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\cdot (z_j^{\ast})_2$ (resp. $x_j=(x_j)_1+\pi \cdot (x_j)_2$) as explained in the paragraph before Equation (\[e42\]). The Dickson invariant of $T_1$ is the same as that of $\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (z_j^{\ast})_1\\0&0& 0 & 1 \end{pmatrix}$. Here, we consider $\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (z_j)_1\\0&0& 0 & 1 \end{pmatrix}$ as an element of the orthogonal group associated to $\left( (\pi) e_1'\oplus (\pi) e_2' \right)\oplus \left((2)e_3'\oplus Be_4'\right)$. On the other hand, by Equation (\[e42\]), the equations defining $F_j$ are $$(x_j)_2=(z_j^{\ast})_1, ~~~ (x_j)_1=0, ~~~ (z_j^{\ast})_1+(z_j^{\ast})_1^2=0.$$ Since $(x_j)_1=0$, the Dickson invariant of $\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (z_j^{\ast})_1\\0&0& 0 & 1 \end{pmatrix}$ is the same as that of $\begin{pmatrix} 1 & (z_j^{\ast})_1\\ 0 & 1 \end{pmatrix}$. In order to compute the Dickson invariant, we use the scheme-theoretic description of the Dickson invariant explained in Remark 4.4 of [@C1]. The Dickson invariant of an orthogonal group of the quadratic space with dimension 2 is explicitly given at the end of the proof of Lemma 4.5 in [@C1]. Based on this, the Dickson invariant of $\begin{pmatrix} 1 & (z_j^{\ast})_1\\ 0 & 1 \end{pmatrix}$ is $(z_j^{\ast})_1$. Note that $(z_j^{\ast})_1$ is indeed an element of $\mathbb{Z}/2\mathbb{Z}$ by Equation (\[e42\]).
In conclusion, $(z_j^{\ast})_1$ is the image of a fixed element of $F_j$ under the map $\psi_j$. Since $(z_j^{\ast})_1$ can be either $0$ or $1$, $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
2. Assume that $M_1=\oplus H(1)$ and that $b+b'$ is a unit. Then, $$M_0''= \left( (\pi)e_1'\oplus (\pi)e_2' \right)\oplus$$ $$\left((2)e_3'\oplus B\left(\pi e_3'+1/\sqrt{b+b'}e_4'\right)\right) \oplus (\oplus_{\lambda}\pi H_{\lambda})\oplus M_2,$$ as explained in the argument (ii) of Step (1) in the construction of $\psi_j$. For this basis, the image of a fixed element of $F_j
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|
} | \leq t_j, \forall j) - \mathbb{P}( |\hat Z_{j,n}| \leq
t, \forall j |\hat\Gamma)\right|; \mathcal{F}_n \right] +
\mathbb{P}(\mathcal{F}_n^c) \leq C \Delta_{3,n} + \frac{1}{n}.$$ In the above expression the constant $C$ is the same as in and $\mathcal{F}_n$ is the event that $\{ \max_{j,k}
|\widehat{\Gamma} - \Gamma| \leq C \aleph_n\}$, which is of probability at least $ 1-
\frac{1}{n}$, again by . This gives the first bound in .
To prove the second bound in we let $\Xi_n = C \frac{\aleph_n}{\min_j \gamma_j}$, where $C$ is the constant in , and then notice that, on the event $\mathcal{F}_n$, $$|\hat\gamma_j - \gamma_j| =
\frac{|\hat\gamma_j^2 - \gamma_j^2|} {|\hat\gamma_j + \gamma_j|}
\leq
\frac{|\hat\gamma_j^2 - \gamma_j^2|} {\gamma_j} \leq
\frac{\max_j |\hat\gamma_j^2 - \gamma_j^2|}{\min_j \gamma_j} \leq \Xi_n.$$ Thus, $$\begin{aligned}
\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall j\right)
& = \mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall
j\right) -\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right) + \mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j, \forall j\right) \\
& \geq
\mathbb{P} \left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall
j;\mathcal{F}_n\right) -\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right) + \mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right)\\
& \geq \mathbb{P} \left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall
j;\mathcal{F}_n\right) -\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right) + (1-\alpha),
\end{aligned}$$ where in the last step we have used the union bound. Next, $$\mathbb{P} \left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall
j;\mathcal{F}_n\right) \geq \mathbb{P} \left( |Z_{n,j}| \leq z_{\alpha/(2s)}
(\gamma_j - \Xi_n), \forall
j; \mathcal{F}_n \right) \geq \mathbb{P} \left( |Z_{n,j}| \leq z_{\alpha/(2s)}
(\gamma_j - \Xi_n), \forall
j \right) - \mathbb{P}\left(
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|
ght)^4}{8 \left(u^2-1\right) \left(u^2+1\right)^3} & \frac{-3 u^4+30 u^2+h \left(u^2+1\right)^2-7}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{Ru} & -\frac{(h+1) u}{\left(u^2+1\right)^2} & \frac{2 u \left(u^2+h \left(u^2-1\right)-2\right)}{\left(u^2-1\right) \left(u^2+1\right)^2} \\
\mathcal{D}_{uu} & \frac{m^2 \left(u^2+1\right)^4+4 h^2 \left(u^2-1\right) \left(u^2+1\right)^2+8 h \left(u^2-1\right) \left(u^2+1\right)^2+8 \left(u^6-u^4+u^2-1\right)}{8 \left(u^2-1\right)^2 \left(u^2+1\right)^3} & -\frac{3 u^4-2 u^2+2 h^2 \left(u^2+1\right)^2+3 h \left(u^2+1\right)^2+3}{2 \left(u^2-1\right) \left(u^2+1\right)^3} \\
\mathcal{D}_{TR} & \frac{i m \left(u^4-2 u^2+2 h \left(u^2+1\right)^2+5\right)}{4 \left(u^2+1\right)^3} & -\frac{i m \left(u^4+6 u^2-3\right) \left(-u^4-6 u^2+h \left(u^2+1\right)^2+3\right)}{8 \left(u^2-1\right) \left(u^2+1\right)^3} \\
\mathcal{D}_{Tu} & \frac{i m u}{2-2 u^4} & \frac{i m u \left(u^4+6 u^2-3\right)}{4 \left(u^2-1\right) \left(u^2+1\right)^2} \\
\mathcal{D}_{\Phi R} & \frac{i m \left(u^4+h \left(u^2+1\right)^2+3\right)}{2 \left(u^2+1\right)^3} & -\frac{i m \left(-u^4-6 u^2+h \left(u^2+1\right)^2+3\right)}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi u} & \frac{i m u}{2-2 u^4} & \frac{i m u}{\left(u^2+1\right)^2} \\
\noalign{\bigskip}
\text{} & C_{\Phi R}(u) & C_{\Phi u}(u) \\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & -\frac{i h m \left(u^4+6 u^2-3\right)^2}{4 \left(u^2-1\right) \left(u^2+1\right)^3} & \frac{i m u \left(u^4+6 u^2-3\right)^2}{4 \left(u^2-1\right) \left(u^2+1\right)^3} \\
\mathcal{D}_{T\Phi } & -\frac{i h m \left(u^4+6 u^2-3\right)}{\left(u^2+1\right)^3} & \frac{i m u \left(u^4+6 u^2-3\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi\Phi } & -\frac{4 i h m \left(u^2-1\right)}{\left(u^2+1\right)^3} & \frac{4 i m u \left(u^2-1\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{RR} & \frac{i m \left(u^4+6 u^2-3\right)}{4 \left(u^4-1\right)} & -\frac{i m u \left(u^6+3 u^4+19 u^2-15\right)}{4 \left(u^2-1\right) \left(u^2+1\right)^2}
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|
perp$ of $R$ as a subspace of $\left(\bigoplus_{w,x,y,z\in C} \mathcal F(E)(w,x,y;z) \right) ^\vee$. Notice that $\mathcal F(E)(\vec X;x)^\vee = \mathcal F(E^\vee) (\vec X;x)$, so that $\mathcal P^!$ is generated by $E^\vee$ with relations $R^\perp$, see [@GK (2.1.9)].
Now if $\mathcal P=\mathcal F(E)/(R)$ is a quadratic colored operad, then we can follow the definition from [@GK (3.2.12)], and define the cobar dual colored operad $\textbf{D} (\mathcal P)$ of $\mathcal P$, to be given by the complexes $\textbf{D} (\mathcal P)(\vec X;x)$ concentrated in non-positive degree, $\textbf{D} (\mathcal P)(\vec X;x):=$ $$\begin{gathered}
\bigoplus_{
\substack{
\text{trees $T$ of}\\
\text{type $(\vec X;x)$},\\
\text{no internal edge}}}
\mathcal P(T)^*\otimes {\mathrm{Det}}(T)\stackrel{{\partial}}{{{\longrightarrow}}}
\bigoplus_{
\substack{
\text{trees } T \text{ of}\\
\text{type }(\vec X;x),\\
\text{1 internal edge}}}
\mathcal P(T)^*\otimes {\mathrm{Det}}(T)\stackrel{{\partial}}{{{\longrightarrow}}}\\
{{\longrightarrow}}\dots\stackrel{{\partial}}{{{\longrightarrow}}} \bigoplus_{
\substack{
\text{trees } T \text{ of}\\
\text{type }(\vec X;x),\\
\text{binary tree}}}
\mathcal P(T)^*\otimes {\mathrm{Det}}(T).\end{gathered}$$ Here, $\mathcal P(T)^*$ denotes the dual of $\mathcal P(T)$, and ${\mathrm{Det}}(T)$ denotes the top exterior power on the space $k^{Ed(T)}$, where $Ed(T)$ is the space of edges of the tree $T$. By definition, we let the furthest right space whose sum is over binary trees, be of degree zero, and all other spaces be in negative degree.
In general, the zero-th homology of this complex is always canonically isomorphic to the quadratic dual $\mathcal P^!$, i.e. $H^0(\textbf{D}(\mathcal P)(\vec X;x))\cong \mathcal P^!(\vec X;x)$, see [@GK (4.1.2)]. The quadratic operad $\mathcal P$ is then said to be Koszul if the cobar dual on the quadratic dual $\textbf{D}(\mathcal P^!)(\vec X;x)$ is quasi-isomorphic to $\mathcal P(\vec X;x)$, i.e., by the above, $\textbf{D}(\mathcal P^!)(\vec X;x)$ has h
| 1,557
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| 1,602
| 1,250
| 3,788
| 0.770087
|
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|
\}
\,.\end{aligned}$$ This entire equation is proportional to the basis function $F^{(m\,h\,k)}$, which can thus be divided out, leaving an ODE for one function, $C_{mhk}(u)$.
Specializing to the homogeneous (source-free) case, we find the ODE $$\begin{aligned}
\frac{d}{du}
\left[
(1-u^{2}) \frac{d}{du}
C_{mhk}
\right]
+ \left[h(h+1) - \Xi(u) m^2 \right]C_{mhk}=0
\,.\end{aligned}$$ This equation has two regular singularities $u=\pm 1$ and an irregular singularity of rank 1 at $u=\infty$, which falls into the class of confluent forms of Heun’s equation [@NIST:DLMF]. Explicitly, it is a spheroidal differential equation, whose standard form is $$\frac{\text{d}}{\text{d}u}\left( (1-u^2)\frac{\text{d}\varphi}{\text{d}u}\right)
+ \left(\lambda + \gamma^2(1-u^2)-\frac{\mu^2}{1-u^2}\right)\varphi = 0,
\label{eq:spheroidal-equation}$$ where we have made the substitution $\lambda = h(h+1)+2m^2$, $\gamma^2 = -m^2/4$ and $\mu^2 = m^2$. When $\gamma = 0$, Eq. reduces to the Legendre differential equation and the solutions are Legendre polynomials. Being second order, the space of solutions is two dimensional, $$\varphi(u) = a_1S^{(1)}_{n\mu}(\gamma,u) + b_1S^{(2)}_{n\mu}(\gamma,u) .$$ A solution that is regular at $u=\pm 1$ only exists for eigenvalues $\lambda = \lambda_n^m(\gamma^2)$, where $\mu = m = 0, 1, 2, \ldots,$ and $n = m, m+1, m+2,\ldots$. Thus, there are only discrete values of the irrep label $h$ which satisfy regularity at the poles $u=\pm 1$.
Maxwell system {#sec:sep-vector}
--------------
Let’s look at another system of physical importance, the Maxwell system, and verify that we can separate variables in Maxwell’s equations (the Proca equation—i.e. adding a mass term—works as well). The inhomogeneous Maxwell equations in the presence of a source vector field $\mathcal{J}$ are $$\nabla^{a}\mathcal{F}_{ab} = \mathcal{J}_{b},$$ where the electromagnetic tensor $\mathcal{F}$ is built from the vector potential $A$ according to $$\mathcal{F}_{ab} = \nabla_a A_b - \nabla_b A_a.$$ We
| 1,558
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|
mathbf{u}}}) \approx - 8\pi^2\nu|{\mathbf{k}}|^2\E_0.$$
Since the fields ${\mathbf{u}}_1$ are real-valued, their Fourier modes must satisfy $\widehat{{\mathbf{u}}}_1(-{\mathbf{k}}) =
\overline{\widehat{{\mathbf{u}}}_1({\mathbf{k}})}$, where $\overline{z}$ denotes the complex conjugate (C.C.) of $z\in\mathbb{C}$. Depending on the choice of ${\mathcal{W}}_k$, a number of different solutions of can be constructed and below we focus on the following three most relevant cases characterized by the largest values of $\R({\widetilde{\mathbf{u}}})$:
1. ${\mathcal{W}}_1 = \{ {\mathbf{k}}_1, {\mathbf{k}}_2, {\mathbf{k}}_3, -{\mathbf{k}}_1, -{\mathbf{k}}_2,
-{\mathbf{k}}_3 \}$, where ${\mathbf{k}}_i = \mathbf{e}_i$, $i=1,2,3$, is the $i^{\textrm{th}}$ unit vector of the canonical basis of $\mathbb{R}^3$; the most general solution can then be constructed as $$\label{eq:uvec_3D_k1}
{\mathbf{u}}_1({\mathbf{x}}) = \mathbf{A}{\textrm{e}^{2\pi i{\mathbf{k}}_1\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i{\mathbf{k}}_2\cdot{\mathbf{x}}}} +
\mathbf{C}{\textrm{e}^{2\pi i{\mathbf{k}}_3\cdot{\mathbf{x}}}} + \textrm{C.C.}$$ with the complex-valued constant vectors $\mathbf{A} = [0,A_2,A_3]$, $\mathbf{B} = [B_1,0,B_3]$ and $\mathbf{C} = [C_1,C_2,0]$ suitably chosen so that $\E({\mathbf{u}}_1) = 1$; hereafter we will use the values $A_2 =
A_3 = \ldots = C_2 = 1/(48\pi^2)$; it follows that $|{\mathbf{k}}|^2 =
1$ $\forall\,\,{\mathbf{k}}\in{\mathcal{W}}_1$, and the optimal asymptotic value of $\R$ obtained from equation is given by $$\label{eq:R0_kvec_3D_k1}
\R({\widetilde{\mathbf{u}}}) \approx - 8\pi^2\nu\E_0,$$ \[c1\]
2. ${\mathcal{W}}_2 = {\mathcal{W}}\cup (-{\mathcal{W}})$, where $-{\mathcal{W}}$ denotes the set whose elements are the negatives of the elements of set ${\mathcal{W}}$, for ${\mathcal{W}}= \{
{\mathbf{k}}_1 + {\mathbf{k}}_2, {\mathbf{k}}_1 - {\mathbf{k}}_2, {\mathbf{k}}_1 + {\mathbf{k}}_3, {\mathbf{k}}_1 -
{\mathbf{k}}_3, {\mathbf{k}}_
| 1,559
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delta-AliceBob}$$
We have the same simple coincidence the two measures as in the case of the Leggett-Garg systems, $$\Delta_{\min}=\Gamma_{\min}.$$
Final remarks
==============
We have discussed two ways to measure contextuality. The direct approach, named Contextuality-by-Default (CbD), assigns to each random variable an index related to their context. If a system is noncontextual, a jpd can be imposed on the random variables so that any two of them representing the same property in different contexts always have the same values. If the system is contextual, the minimum value of $\Delta$ in (\[eq:Delta-LG\])-(\[eq:delta-AliceBob\]) across all possible jpds has the interpretation of how close a variable can be in two different contexts: the larger the value the greater contextuality, zero representing a necessary and sufficient condition for no contextuality.
The other approach maintains the original set of random variables, but requires negative (quasi-)probabilities. This leads to nonmonotonicity (i.e., a set of outcomes can have a smaller probability than some of its proper subsets), which is a characteristic of quantum interference. The departure from a proper probability distribution is measured by $\Gamma_{\min}$ in the minimum L1 norm $1+\Gamma_{\min}$. Similar to the CbD approach, we use here a minimization principle that gives the closest probability distribution to an ideal (but impossible) jpd. The value of $\Gamma_{\min}$ has the interpretation of how contextual the system is: a necessary and sufficient condition for no contextuality is $\Gamma_{\min}=0$, and the larger the value of $\Gamma_{\min}$, the more contextual the system is.
As we have seen, in the case of EPR-Bell and Leggett-Garg systems the two approaches lead to simple coincidence, $\Delta_{\min}=\Gamma_{\min}$. The two measures, $\Gamma_{\min}$ and $\Delta_{\min}$, can be computed, in principle, for any given system. For detailed examples of such computations, see Appendix.
Of the two measures of contextuality, $\Gamma_{\min}$ is
| 1,560
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l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=0$ for any $j\in \mathcal{B}_1$, and the equation $\sum_{l=0}^{k_j}z_{j-l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=1$ for any $j\in \mathcal{B}_2$ yield that for each $j\in \mathcal{B}$, only one of equations of type Equations (\[24’\]) or (\[ea32\]), or $\mathcal{X}_{i,2,2}$ in Equation (\[ea27\]) is reduntant, and the associated one variable, say $z_{j}^{\ast}$ for simplicity, can be eliminated by other $z_{j-l}^{\ast}, u_{j-l}^{\ast}$.\
We now combine all cases (1)-(6) observed above.
1. By (1) and (2), we eliminate $\sum_{i<j}n_in_j$ variables.
2. By (3), we eliminate $\sum_{\textit{i:odd and $L_i$:free of type $I$}}(2n_i-3)$ variables.
3. By (4), we eliminate $\sum_{\textit{i:even and $L_i$:of type $I^o$}}(n_i-1)$ variables.
4. By (5), we eliminate $\sum_{\textit{i:even and $L_i$:of type $I^e$}}(2n_i-3)$ variables.
5. By (6), we eliminate $$\#\{\textit{i:odd such that $L_i$ is free of type I}\}+\#\{\textit{i:even such that $L_i$ is of type $I^e$}\}+$$ $$\textit{$\#$\{i:even such that $L_i$ is of type I and $L_{i+2}$ is of type II\}$+\beta-\beta$ variables.}$$ Here $\beta=\# \mathcal{B}$.\
Recall from the beginning of the proof that $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ is isomorphic to an affine space of dimension $$2\sum_{i<j}n_in_j-\sum_{\textit{i:even and $L_i$:bound of type II}}n_i+ \sum_{\textit{i:even and $L_i$:of type $I^o$}}n_i+$$ $$\sum_{\textit{i:even and $L_i$:of type $I^e$}}(3n_i-2)
+\sum_{\textit{i:odd and $L_i$:free of type $I$}}(4n_i-4).$$ Therefore, the dimension of $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ is $$\begin{gathered}
\label{ea40}
l^{\prime}=\sum_{i<j}n_in_j +\sum_{i:\mathrm{even~and~} L_i:\textit{of type }I^e}(n_i-1) +
\sum_{i:\mathrm{odd~and~}L_i:\textit{free of type I}}(2n_i-2) \\
- \sum_{i:\mathrm{even~and~} L_i:\textit{bound of type II}}n_i +\#\{i:\textit{$i$ is even and $L_i$ is of type I}\} \\
-\#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}
| 1,561
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$I^o$*) or the $(3,3)$-block (when $L_i$ is *of type $I^e$*) of the $(i, i)$-block of $h\circ \tilde{m}=\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$. Note that $c_i'$ and $z_i'$ are indeed contained in $R$ and $R$ is a $\kappa$-algebra. Thus $c_i'$ and $z_i'$ mod $(\pi\otimes 1)(B\otimes_AR)$ are naturally identified with themselves, respectively.
On the other hand, we choose an even integer $j$ such that $L_j$ is *of type I* and $L_{j+2}$ is *of type II*. For such $j$, there is a nonnegative integer $m_j$ such that $L_{j-2l}$ is *of type I* for every $l$ with $0\leq l \leq m_j$ and $L_{j-2(m_j+1)}$ is *of type II* (cf. Step (vi) of the proof of Theorem \[ta4\]). Then we have $$\label{ea28}
\sum_{0\leq l \leq m_j}c_{j-2l}'= \sum_{0\leq l \leq m_j}\left(c_{j-2l}+z_{j-2l}'\right)
= \sum_{0\leq l \leq m_j}c_{j-2l} =0$$ since the sum of equations $\sum_{0\leq l \leq m_j}\mathcal{Z}_{j-2l}'$ equals $\sum_{0\leq l \leq m_j}z_{j-2l}'=0$ as mentioned in Step (vi) of the proof of Theorem \[ta4\] and $c_i=0$ by Remark \[r33\].(2).
We now apply the argument used in the previous steps to the above equation (\[ea28\]). Let $\tilde{m}\in \mathrm{Ker~}\tilde{\varphi}(R)$ be a lift of $m$. By using an argument similar to the paragraph just before Equation (\[ea20\]) of Step (1), if we write the $(2, 2)$-block (when $L_i$ is *of type $I^o$*) or the $(3,3)$-block (when $L_i$ is *of type $I^e$*) of the $(i, i)$-block of $h\circ \tilde{m}=\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$ as $\xi^{i/2}\cdot (1+2\bar{\gamma}_i+4\mathcal{F}_i(\tilde{m}))$ or $\xi^{i/2}\cdot (2\bar{\gamma}_i+4\mathcal{F}_{i}(\tilde{m}))$ respectively, then the image of $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(\tilde{m})$ in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$ is independent of the choice of the lift $\tilde{m}$ of $m$. Therefore, we may denote this image by $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(m)$. Note that $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(\tilde{m})$ is indeed contained in $R$. Thus $\sum_{0\leq l \leq
| 1,562
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M}'$ is a quotient of group schemes with respect to the addition, the operation $\star$ is well-defined on $(\underline{M}'\otimes\kappa)/ \underline{\pi M}'$. For the proof, see the first two paragraphs from below in page 511 and the first two paragraphs in page 512 in [@C2].
To summarize, the morphism $1+ : \tilde{M}'/\underline{\pi M'}\longrightarrow \tilde{M}/\tilde{M}^{1}$ is an isomorphism of group schemes and the morphism $\bar{\iota} : \tilde{M}'/ \underline{\pi M}' \rightarrow (\underline{M}'\otimes\kappa)/ \underline{\pi M}'$ is a monomorphism preserving the operation $\star$. Therefore, each element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ is uniquely written as $1+\bar{x}$, where $\bar{x}\in (\underline{M}'\otimes\kappa)(R)/ \underline{\pi M}'(R)$. Here, by $1+\bar{x}$, we mean $(1+)\circ \bar{\iota}^{-1}(\bar{x})$. From now on to the end of this paper, we keep the notation $1+\bar{x}$ to express an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ such that $\bar{x}$ is an element of $(\underline{M}'\otimes\kappa)(R)/ \underline{\pi M}'(R)$ which is a quotient of $R$-valued points of group schemes with respect to addition. Then the product of two elements $1+\bar{x}$ and $1+\bar{y}$ is the same as $1+\bar{x}\star \bar{y}$ $(=1+(\bar{x} + \bar{y}+\bar{x} \bar{y}))$.
\[ra5\] By the above argument, we write an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ formally as $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix}\mathrm{~together~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast},$$ with $s_i,\cdots, w_i$ as in Section \[m\] such that each entry of each of the matrices $(m_{i,j})_{i\neq j}, s_i, \cdots, w_i$ is in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)\cong R$. In particular, based on the description of $\mathrm{Ker~}\tilde{\varphi}(R)$ given at the paragraph following Lemma \[la2\], we have the following conditions on $m$:
1. Assume that $L_i$ is *of type I* with $i$ even or that $L_i$ is *free of type I* with $i$ odd. Then $s_i=\mathrm{id}$.
2
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t accuracy, and there is a linear trend between the test accuracy and the logarithm. We also highlight the fact that conditioned on the noise covariance, the test accuracy is not significantly correlated with either the step size or the minibatch size. In other words, similar to the observations in prior work [@jastrzkebski2017three; @he2019control], there is a strong correlation between relative variance of an SGD sequence and its test accuracy, regardless of the combination of minibatch size and step size.
[Large-Noise SGD]{} \[ss:inject\]
In this section, we implement and examine the performance of the large-noise SGD algorithm proposed in . We select a subset of SGD runs with relatively small noise covariance in the experiment in the previous section (we call them *baseline SGD runs*), and implement large-noise SGD by injecting noise. Our goal is to see, for a particular noise covariance, whether large-noise SGD has test accuracy that is similar to SGD, *in spite of significant differences in third-and-higher moments of the noise in large-noise SGD compared to standard SGD*. Our first experiment is to add noise with the same minibatch size to the $(\delta, b)$ baseline SGD run such that the new noise covariance matches that of an $(8\delta, b)$-SGD (an SGD run with larger step size). In other words, we implement $(\delta, \sqrt{7\delta / 2}, b, b)$-large-noise SGD, whose noise covariance is $8$ times of that of the baseline. Our results are shown in Figure \[fig:lr\_matching\]. Our second experiment is similar: we add noise with minibatch size $128$ to the $(\delta, b)$ baseline SGD run with $b \in \{256, 512\}$ such that the new noise covariance matches that of a $(\delta, 128)$-SGD (an SGD run with smaller batch size). More specifically, we implement $(\delta, \sqrt{\frac{1}{2} (1-\frac{128}{b})\delta }, b, 128)$-large-noise SGD runs. The results are shown in Figure \[fig:batch\_matching\]. In these figures, each $\times$ denotes a baseline SGD run, with step size specified in the legend and minibatch
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ts Motion to Dismiss this untimely
appeal.
Respectfully submitted,
/s/ Michael M. Phillips
______________________________
Michael M. Phillips
State Bar No. 15939000
THE MICHAEL M. PHILLIPS LAW
FIRM, P.C.
P.O. Box 1030
Angleton, Texas 77516-1030
(979) 849-4382 Telephone
(979) 849-1409 Fax
michael@mphillipslaw.com
ATTORNEY FOR APPELLEE,
JAS FAMILY LIMITED
PARTNERSHIP #4, LTD
CERTIFICATE OF SERVICE
I hereby certify that a true and correct copy of this Motion to Dismiss has
been sent to the following in accordance with the Texas Rules of Appellate
Procedure on this 8th day of June, 2015:
Paula M. Miller
P.O. Box 981013
Houston, Texas 77098
Fax: (801) 618-1656
/s/ Michael M. Phillips
______________________________
No. 01-15-00286-CV
IN THE COURT OF APPEALS
FOR THE FIRST DISTRICT OF TEXAS
HOUSTON, TEXAS
Inre
PAULA M. MILLER
and JAS FAMILY LIMITED PARTNERSHIP #4, LTD
AFFIDAVIT OF MICHAEL M. PHILLIPS
STATE OF TEXAS §
§
COUNTY OF BRAZORIA
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|
- \frac{c_+}{c_++c_-} \frac{t^c \phi(w)}{x-w} + :j^c_{L,z} \phi:(w) %\right. \right. \cr
%& \quad \quad \left. + {A^c}_d \log|x-w|^2 :j^d_{L,z}\phi:(w) + {B^c}_d \frac{\bar x - \bar w}{x-w} :j^d_{L,z}\phi:(w)
+%AA \mathcal{O}(f^2)
... \right] j^b_{L,\bar z}(z) \right. \cr
& \quad \left. + j^c_{L,z}(x) \left[ -\frac{c_-}{c_++c_-}\frac{t^b \phi(w)}{\bar z - \bar w} + :j^b_{L,\bar z} \phi:(w) + %AA\mathcal{O}(f^2)
...\right] \right\} \end{aligned}$$ To proceed according to the prescription of appendix \[compositeOPEs\] we have to expand the fields in the first line (respectively the second line) in the neighborhood of the point $x$ (respectively $z$). Then we have to perform the remaining OPEs between the currents and the (derivatives of) the primary field $\phi$. Notice however that all the terms proportional to ${f^a}_{cb} t^b t^c =
\frac{i}{2} {f^a}_{cb} {f^{bc}}_d t^d$ do vanish. Only the regular term in the current-primary OPE will contribute to the result at order $f^0$. Moreover it is straightforward to check that the terms proportional to ${A^a}_c$ and ${B^a}_c$ in the previous OPE do not contribute at order $f^0$. We obtain: (w)\[ i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:(z)\] = -i f\^2 [f\^a]{}\_[bc]{} ( + + ... ). where the ellipses contains terms of order $f^4$ as well as terms of order zero in the distance between $z$ and $w$. Gathering terms, we conclude that we have the equalities: $$\begin{aligned}
{A^a}_c &=& \frac{c_-}{(c_++c_-)^2} i {f^a}_{cb} t^b + \mathcal{O}(f^4)
\nonumber \\
{B^a}_c &=& \frac{c_+}{(c_++c_-)^2} i {f^a}_{cb} t^b + \mathcal{O}(f^4).\end{aligned}$$ We note that one can reach the same conclusion by computing the OPE between a current and both sides of the equation , i.e. by demanding compatibility with the proportionality relation between the operators $\p \phi$ and $ t_a :j^a_{L,z} \phi:$.
Stress-tensor-primary OPE at order $f^2$ {#AppTphi}
----------------------------------------
Here we present the computation of the OPE between the stress-energy tensor and
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ill see shortly that $\calA_{\lambda\muhat}$ is nonzero for all $\lambda,
\muhat$; let $v(\lambda) :=v_q\left(\calA_{\lambda\muhat}(q)\right)$. The first main step toward the proof of the connectedness is the following theorem.
\[minim\] Let $\muhat=(\mu^1,\mu^2,\ldots,\mu^k)\in {\P_n}^k$ with $\delta(\muhat) \geq 0$. Then
i\) The minimum value of $v(\lambda)$ as $\lambda$ runs over the set of partitions of size $n$, is $$v((1^n))=-\Delta(\muhat).$$
ii\) There are two cases as to where this minimum occurs.
Case I: The quiver $\Gamma$ is affine and the dimension vector associated to $\muhat$ is a positive imaginary root $t\v^*$ for some $t\mid n$. In this case, the minimum is reached at all partitions $\lambda$ which are the union of $n/t$ copies of any $\lambda_0\in\calP_t$.
Case II: Otherwise, the minimum occurs only at $\lambda=(1^n)$. \[step1\]
Before proving the theorem we need some preliminary results.
$\langle h_{\mu}(\x),s_\lambda(\x\y)\rangle$ is non-zero for all $\lambda$ and $\mu$.
We have $s_\lambda(\x\y)=\sum_\nu K_{\lambda\nu}m_\nu(\x\y)$ [@macdonald I 6 p.101] and $m_\nu(\x\y)=\sum_\mu
C_{\nu\mu}(\y)\,m_\mu(\x)$ for some $C_{\nu\mu}(\y)$. Hence $$\label{m-coeff}
\langle
h_\mu(\x),s_\lambda(\x\y)\rangle=\sum_\nu
K_{\lambda\nu}C_{\nu\mu}(\y).$$ For any set of variables $\x\y=\{x_iy_j\}_{1\leq i,1\leq
j}$ we have $$\label{C-fmla}
C_{\nu\mu}(\y)=\sum m_{\rho^1}(\y)\cdots m_{\rho^r}(\y),$$ where the sum is over all partitions $\rho^1,\ldots, \rho^r$ such that $|\rho^p|=\mu_p$ and $\rho^1\cup \cdots \cup \rho^r=\nu$. In particular the coefficients of $C_{\nu\mu}(\y)$ as power series in $q$ are non-negative. We can take, for example, $\rho^p=(1^{\mu_p})$ and then $\nu=(1^n)$. Since $K_{\lambda\nu}\geq 0$ [@macdonald I (6.4)] for any $\lambda,\nu$ and $K_{\lambda,(1^n)}=n!/h_\lambda$ [@macdonald I 6 ex. 2], with $h_\lambda=\prod_{s\in \lambda}h(s)$ the product of the hook lengths, we see that $\langle h_\mu(\x),s_\lambda(\x\y)\rangle$ is non-zero and our claim follows.
In particular $\calA
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a I delta J\] $$\begin{aligned}
[\delta_I, \delta_J] A_\eta\ =&\ D_\eta A_{[\delta_I, \delta_J]}\,,
\label{delta I delta J 1}\\
A_{[\delta_I, \delta_J]}\ =&\
f\xi_0[\delta_I, \delta_J] A_\eta + D_\eta\Omega_{IJ}\,,
\label{delta I delta J 2}\end{aligned}$$
with $$\begin{aligned}
\Omega_{IJ}\ =&\
-f\xi_0[f\xi_0\delta_I A_\eta,\, f\xi_0\delta_J A_\eta]
\nonumber\\
&\
+\delta_I \Omega_{J}
-[f\xi_0\delta_I A_\eta,\, \Omega_{J}]
-\delta_J \Omega_{I}
+[f\xi_0\delta_J A_\eta,\, \Omega_{I}]
-[\Omega_{I},\, D_\eta\Omega_{J}]\,,
\label{Omega IJ}\end{aligned}$$ which can be shown by explicit calculation using (\[gen MC\]) and (\[f ns\]) if we assume (\[delta I\]) with some field-dependent $\Omega_I$. Therefore if the algebra of the transformation is closed on $A_\eta$, $$[\delta_I,\,\delta_J]A_\eta\ =\ \sum_{K\ne\Omega}\delta_K A_\eta\,,
\label{alg A_eta}$$ we have $$A_{[\delta_I,\delta_J]}\ =\ \sum_{K\ne\Omega} A_{\delta_K} + D_\eta\Omega_{IJ}\
=\ \sum_K A_{\delta_K}\,,
\label{alg AIJ}$$ or equivalently, the algebra is also closed on $e^\Phi$: $$[\delta_I,\,\delta_J]e^\Phi\ =\ \sum_K\delta_K e^\Phi\,.$$ with some field-dependent $\Omega_{IJ}$. Here in (\[alg A\_eta\]) we used that $A_\eta$ is invariant under the $\Omega$-gauge transformation, $A_{\delta_\Omega}=D_\eta\Omega$, as seen from (\[delta I 1\]).
$[\delta_{{\mathcal{S}}_1},\delta_{{\mathcal{S}}_2}]$
-----------------------------------------------------
Now let us explicitly calculate the supersymmetry algebra on $A_\eta$ and $\Psi$, which is easier to calculate than the algebra on the fundamental string fields $\Phi$ (or $e^\Phi$) and $\Psi$ due to their $\Omega$-gauge invariance and enough to know that on the fundamental string fields as was shown in the previous subsection. From (\[complete transformation\]) we find
$$\begin{aligned}
A_{\delta_{\mathcal{S}}}\ =&\ f\xi_0\delta_{\mathcal{S}}A_\eta + D_\eta\Omega_{\mathcal{S}}\,,\\
\delta_{\mathcal{S}}\Psi\ =&\ X\eta F\Xi{\mathcal{S}}A_\eta\,,
\label{susy A_eta}
\end{aligned}$$
with
$$\begin{aligned}
\
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|
+\delta_5\{\dot xxyx^2\}+\delta_5\{x\dot xyx^2\}-\delta_5\{\dot
yy^2x^2\}-\delta_5\{y\dot yyx^2\} \\
-\delta_5\{\dot xxy^3\}-\delta_5\{x\dot xy^3\}+\delta_5\{\dot
yy^4\}+\delta_5\{y\dot yy^3\}.
\end{gathered}$$
This expression must coincide with $f = \{x^3\dot x y \} -
\{y^2x\dot x y\} $. In particular, a sum of all dimonomials with the central letter $y$ must be equal to zero: $$\begin{gathered}
0= \gamma_1\{y\dot yxyx\} +(\gamma_2+\delta_3)\{y\dot yx^2y\}
+(\gamma_3+\gamma_5+\delta_4+\delta_5)\{y\dot yyx^2\} \\
+\gamma_4\{xy\dot yyx\} +\gamma_6\{xy\dot yxy\} +\gamma_1\{\dot
yyxyx\} +(\gamma_2+\delta_3)\{\dot y yx^2y\}
+(\gamma_3+\delta_5)\{\dot y y^2x^2\} \\
+\gamma_4\{x\dot yy^2x\} +(\gamma_5+\delta_4)\{x^2\dot yy^2\}
+\gamma_6 \{x\dot yyxy\} +(\delta_1-\delta_3-\delta_5)\{\dot yy^4\} \\
+(\delta_1+\delta_2-\delta_3-\delta_4-\delta_5)\{y\dot yy^3\}
+(\delta_2-\delta_4)\{y^2\dot yy^2\}.
\end{gathered}$$ All coefficients in this sum have to be zero. Solving the obtained system we have $\gamma_2= -\delta_3$, $\gamma_3=
-\delta_5$, $\gamma_5= -\delta_4$, $\delta_1= \delta_3+\delta_5$, $\delta_2=\delta_4$, $\gamma_1= \gamma_4= \gamma _6= 0$.
Substitute the obtained relations to $\phi(x,y,k)$ we get that all summands with coefficients $\gamma$ and $\delta$ are eliminated.
Further, consider the remaining summands (we divide them into two groups by $\deg_y$): $$\begin{gathered}
(\alpha_1 + \alpha_4) \{ \dot xyx^3 \} +(\alpha_2 + \alpha_6) \{
y\dot x x^3 \} +\alpha_3 \{ \dot xxyx^2 \}
+\alpha_5 \{ yx^2\dot xx \} \\
+\beta_1 \{ xy\dot x x^2 \} +\beta_2 \{ yx \dot x x^2 \} +\beta_3 \{
x\dot xyx^2 \} +\beta_4 \{ xyx^2 \dot x \}
+(\beta_5 +\beta_6)\{ yx^3\dot x \} \\
= \{x^3\dot x y\},
\end{gathered}$$ $$\begin{gathered}
\alpha_1 \{ \dot xyxy^2 \}+\alpha_2 \{ y\dot xxy^2 \}
+\alpha_3\{\dot x xy^3 \} + \alpha_5 \{ y^3\dot xx \}
+ \alpha_6 \{ y\dot xy^2x \} \\
+\beta_1 \{ xy\dot xy^2 \}+\beta_2 \{ yx \dot xy^2 \}+
\beta_3\{x\dot xy^3 \}+ (\alpha_4 +\beta_4) \{ \dot xy^3x \}
+\beta_5 \{ y^3x\dot x \}+\beta_6 \{ yx
| 1,569
| 3,214
| 1,820
| 1,463
| null | null |
github_plus_top10pct_by_avg
|
th, the results of the FNAL-E731 measurement of $(7.4 \pm 5.9) \times
10^{-4}$[@fnalE731], but further from agreement with the CERN-NA31 result of $(23\pm 7)\times 10^{-4}$[@cernNA31]. If we relax the constraint on the contribution to $\Delta m_K$ to allow ${\cal F}=-0.3$ (reflecting the uncertainty due to the large long-distance contributions), then $\epsilon^{'}/\epsilon$ rises to $1.3 \times
10^{-4}$. If the omitted electromagnetic penguin contribution $\Omega_{\rm {EMP}}$ turns out to be negative and important, it could increase the predicted value of $\epsilon^{'}$ by perhaps as much as a factor of two, still well below the experimental limit.
Another (much weaker) constraint to be considered is that from the $B^0_{s,d}$ mass splitting[@renormgroup]. Proceeding in close analogy to the calculation of the contribution to $\Delta m_K$, we obtain: $$\Delta M_{B^0} =
2 \frac{G_F^2 m_W^2 }{16\pi^2} \frac{1}{2 m_B}\eta_B
\left(\frac{2}{3}\;\hbox{Re}{\cal A}_{bd}^2\;
\frac{m_W^2}{M_Q^2}\right)
\left( \frac{8}{3} B_B F_B^2 m_B^2 \right) \, ,$$ where again $m_2 \gg m_1$, $m_1=M_Q = 300$ GeV, with the renormalization group scaling factor $\eta_B=0.55$ evaluated as for $\Delta m_K$, and $B_B=1$, $F_B=180$ MeV, $m_B= 5.28$ GeV. Given the experimental value $\Delta M_{B^0} = 3.3\times 10^{-13}$ GeV, we have $$\delta(\Delta M_{B^0}) /\Delta M_{B^0} = 1.1\times 10^{-3}\,
R_Q^2 \,
\hbox{Re}\left({\cal A}_{bd} /0.049^2\right)^2 \ .$$ Even taking ${\cal A}_{bd}= (0.13)^2$, the fractional contribution is only about $5\%$.
Other Constraints {#other-constraints .unnumbered}
=================
[*$b \to s \gamma$*]{}: Because the operator due to charged Higgs diagrams has helicity opposite to that generated in the Standard Model contribution, the two do not interfere at amplitude level. Taking $m_2 \gg m_1=M_Q\simeq 300$ GeV: $${\delta B(b\rightarrow s\gamma)\over B(b\rightarrow s\gamma)_{\rm SM}}
=3.2\times 10^{-6}
\left|0.0389 \over {V_{tb}V^*_{ts}}\right|^2
R_Q^4
\left| \frac{ {\cal A}_{bs} }{ (0.049)^2} \
| 1,570
| 2,475
| 2,717
| 1,654
| 2,114
| 0.782635
|
github_plus_top10pct_by_avg
|
ned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= (k_{1}^{2} + k_{2}^{2}) b^{3} \beta
\int_{0}^{+\infty} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) b^{3} \beta
\int_{0}^{\lvert c / b \rvert} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + 2 (k_{1}^{2} - k_{2}^{2}) b^{2} c \beta
\int_{\lvert c / b \rvert}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1}^{2} + k_{2}^{2}) b c^{2} \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) b c^{2} \beta
\int_{0}^{\lvert c / b \rvert} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Now set $t := z^{\frac{1}{a}}$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= (k_{1}^{2} + k_{2}^{2}) a b^{3} \beta \int_{0}^{+\infty} t^{3a - 1} e^{-t} dt
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b^{3} \beta \int_{0}^{c'} t^{3a - 1} e^{-t} dt \\
&\quad + 2(k_{1}^{2} - k_{2}^{2}) a b^{2} c \beta \int_{c'}^{+\infty} t^{2a - 1} e^{-t} dt \\
&\quad + (k_{1}^{2} + k_{2}^{2}) a b c^{2} \beta \int_{0}^{+\infty} t^{a - 1} e^{-t} dt
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b c^{2} \beta \int_{0}^{c'} t^{a - 1} e^{-t} dt
\allowdisplaybreaks \\
&= (k_{1}^{2} + k_{2}^{2}) a b^{3} \beta \G(3a)
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b^{3} \beta \g(3a, c') \\
&\quad + 2(k_{1}^{2} - k_{2}^{2}) a b^{2} c \beta \G(2a, c') \\
&\quad + (k_{1}^{2} + k_{2}^{2}) a b c^{2} \beta \G(a)
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b c^{2} \beta \g(a, c'), \end{aligned}$$ where $c' := \lvert c / b \rvert^{\frac{1}{a}}$. Therefore, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= \frac{(k_{1}^{2} + k_{2}^{2}) c^{2}}{2}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) c^{2}}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{\G(a)}
\G\left(2a, \left\lvert \frac
| 1,571
| 2,552
| 1,240
| 1,573
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|
(Z_\Lambda/Z_\Lambda)^{2j+2}$, following Step (ii) of the strategy described in Section \[ss:pi0bd\]. Overlapping the $2j+3$ current configurations and using Lemma \[lmm:GHS-BK\] with ${{\cal V}}=\{y,x\}$ and $k=2j+2$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd6}}
\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}&\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde
I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}\leq{{\langle \varphi_{z_1}\varphi_y
\rangle}}_\Lambda^2{{\langle \varphi_x\varphi_{z'_j} \rangle}}_\Lambda^2\\
&\times\begin{cases}
{\displaystyle}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda&(j=1),\\
{\displaystyle}{{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda{{\langle \varphi_{z_2}\varphi_{
z'_1} \rangle}}_\Lambda\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{z'_{i-1}}\varphi_{
z_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{z_{i+1}}\varphi_{z'_i} \rangle}}_\Lambda\bigg)
{{\langle \varphi_{z'_{j-1}}\varphi_x \rangle}}_\Lambda&(j\ge2).
\end{cases}{\nonumber}\end{aligned}$$ Note that, by [(\[eq:G-delta-bd\])]{}, we have $$\begin{aligned}
\left.\begin{array}{r}
\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(y,x)\\[5pt]
\sum_z{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2\sum_{l\ge1}(\tilde
G_\Lambda^2)^{*(2l-1)}(z,x)\\[5pt]
\sum_{z'}{{\langle \varphi_x\varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1}
(\tilde G_\Lambda^2)^{*(2l-1)}(y,z')
\end{array}\right\}&\leq\psi_\Lambda(y,x)-\delta_{y,x},\\[5pt]
\sum_{z,z'}{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x
\varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z,z')&\leq2\big(\psi_\Lambda(y,x)-\delta_{y,x}\big).\end{aligned}$$ Therefore, [(\[eq:Theta’-2ndindbd5\])]{} without ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ is bounded by $$\begin{aligned}
{\label{eq:Theta'-2ndindbd7}}
&{{\langle \varphi_y\varphi_x \rangle}}_\Lambda\sum_{z_1,z'_1}{{\
| 1,572
| 2,944
| 1,054
| 1,539
| null | null |
github_plus_top10pct_by_avg
|
r bound ${\|\nabla\Lrb(\ltheta^*)\|}_2$. It might be possible to tighten the upper bound, given that $\ld \leq d$. However, for $\ell \ll \kappa$, for the smallest preference score item, $i_{\min} \equiv \arg \min_{i \in [d]} \ltheta^*_i$, the upper bound $\P[\sigma^{-1}(i_{\min}) > \kappa-\ell] \leq 1$ is tight upto constant factor (Lemma \[lem:posl\_upperbound\]). Substituting $\lambda_{j,a} = 1$ and $p_{j,a} = \kappa - \ell +a$ for each $j \in [n]$, $a \in [\ell]$, in Lemma \[lem:gradient\_topl\], we have that with probability at least $1 - 2e^3 d^{-3}$, $$\begin{aligned}
\label{eq:gradient_bound_bottoml}
{\|\nabla\Lrb(\ltheta^*)\|}_2 \;\; \leq \;\; (\ell-1)\sqrt{8n\ell\log d}.\end{aligned}$$ Theorem \[thm:bottoml\_upperbound\_general\] follows from Equations , and .
### Proof of Lemma \[lem:hessian\_bottoml\]
Define $\lM^{(j)} \in \cS^{\ld}$, $$\begin{aligned}
\label{eq:limited_M_j_def}
\lM^{(j)} = \sum_{\substack{i<\i \in S_j : i,\i \in [\ld]}} \sum_{a = 1}^\ell \I_{\{(i,\i) \in G_{j,a}\}} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top,\end{aligned}$$ and let $\lM = \sum_{j = 1}^n \lM^{(j)}$. Similar to the analysis in Lemma \[lem:hessian\_positionl\], we have $\lambda_2(-H(\ltheta)) \geq \frac{e^{4b}}{(1+ e^{4b})^2} \lambda_2(\lM)$. Note that we have $e^{4b}$ instead of $e^{2b}$ as $\ltheta \in \lOmega_{2b}$. We will show a lower bound on $\lambda_2(\E[\lM])$ in and an upper bound on ${\|\lM - \E[\lM]\|}$ in . Therefore using $\lambda_2(\lM) \geq \lambda_2(\E[\lM]) - {\|\lM - \E[\lM]\|}$, $$\begin{aligned}
\label{eq:bottoml_lambda2_M}
\lambda_2(\lM) \; \geq\; \frac{e^{-4b}}{4}\underbrace{(1-\beta_1)^2\Bigg(1 - \exp\bigg(-\frac{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)^2(1-\gamma_{\beta_1})^2}{2(\kappa-2)}\bigg)\Bigg)}_{ \equiv \chi_{\beta_1}}\frac{n\ld\ell^2}{d^2} - 8\ell\sqrt{\frac{n\kappa\log d}{d}} \;.\end{aligned}$$ The desired claim follows from the assumption that $n\ell \geq \big( \frac{2^{12}e^{8b}}{\chi_{\beta_1}^2}\frac{d^2}{{\ld}^2}\frac{\kappa}{\ell} \big) d\log d$, where $\chi_{\beta_1}
| 1,573
| 727
| 572
| 1,663
| null | null |
github_plus_top10pct_by_avg
|
ith exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_1d.pdf "fig:"){width="0.8\linewidth"}\
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) one- and (b) two-dimensional substrates. Both simulations with the kinetic barrier (using $N_s$ values indicated in the legend) and the original version are shown. In (a), dashed and solid lines are power-laws with exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_2d.pdf "fig:"){width="0.8\linewidth"}
Figure \[width\] shows the time evolution of the interface width for both models in one and two dimensions. The main panels and insets present the results for the WV and DT models, respectively, including or not the kinetic barrier. The interface width is expected to scale as $w\sim t^\beta$ where $\beta$ is the growth exponent [@barabasi]. The short time dynamics of both WV and DT models is well described by the linear version of the MBE equation [@Villain; @LSarma] $$\frac{\partial h}{\partial t} = -\nu \nabla^{4}h + \lambda \nabla^{2} (\nabla h)^{2} + \eta,
\label{Eq1}$$ with $\lambda=0$ where $\eta$ is a non-conservative Gaussian noise [@Villain; @LSarma; @DasSarma1992]. This result is confirmed in Fig. \[width\] where the short time behavior is consistent with the growth exponents $\beta=3/8$ in $d=1$ and $\beta=1/4$ in $d=2$ expected for the linear MBE universality class [@barabasi]. It is worth to mention that these models may undergo crossovers to different universality classes in the asymptotic, depending on the dimension and model [@wvbogo; @Vvedensky; @Xun; @Chen2017; @AaraoReis2004b; @Punyindu]. The curves in Fig. \[width\] are consistent with crossovers to different universality classes at long times. One expects that DT is asymptotically consistent with the non-linear
| 1,574
| 171
| 2,651
| 1,590
| 838
| 0.799112
|
github_plus_top10pct_by_avg
|
contributes fractional fermion number $$\frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right]
\: - \: \frac{1}{2}$$ and a complex right-moving fermion $\psi$ with the same boundary conditions contributes fractional fermion number $$- \left(
\frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right]
\: - \: \frac{1}{2}
\right)$$ In the present case, in the sector defined by automorphism $\alpha$, we have complex left-moving fermions $\lambda_{-,n}$ coupling to bundle ${\cal E}^{\alpha}_n$, with boundary conditions $$\lambda_{-,n}(\sigma + 2 \pi) \: = \: \exp\left( 2 \pi i n / t_{\alpha} \right)
\lambda_{-,n}(\sigma)$$ and complex right-moving fermions $\psi_{+,n}$ coupling to bundle $T^{\alpha}_n$, with boundary conditions $$\psi_{+,n}(\sigma + 2 \pi) \: = \: \exp\left( 2 \pi i n/t_{\alpha}\right)
\psi_{+,n}(\sigma)$$ Putting this together, we see that from each set of $\lambda_{-,n}$, the Fock vacuum couples to $$\label{eq:left-Fock-nonzero}
\left( \det {\cal E}^{\alpha}_n \right)^{- \frac{n}{t_{\alpha}} \: - \: \left[
- \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ and from each set of $\psi_{+,n}$, the Fock vacuum couples to $$\label{eq:right-Fock-nonzero}
\left( \det T^{\alpha}_n \right)^{\frac{n}{t_{\alpha}} \: + \: \left[
- \frac{n}{t_{\alpha}} \right] \: + \: \frac{1}{2} }$$
Since the $\alpha$-sector has components which are $t_{\alpha}$ gerbes, $t_{\alpha}$-th roots of bundles might exist, though not necessarily. (See appendix \[app:canonical-roots\] for examples of bundles on ${\mathbb Z}_n$-gerbes which do and do not admit $n$th roots.) Existence of these roots is a necessary condition for the existence of the physical theories. When multiple roots exist, as will happen if the components are not simply-connected, the roots must be specified as part of the data defining the sigma model.
When there are periodic fermions, there are multiple Fock vacua, each with different (fractional) charges. The different Fock vacua are defined by which subset of the fermi zero modes annihilate.
| 1,575
| 1,496
| 1,894
| 1,443
| null | null |
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|
nsists of the applied stimulus $s_t$ and resulting WMTF $y_t$. The data set is re-ordered such that the observation $y_1, \ldots, y_{\tau-1}$ define baseline measurements with $s_t=0$ for $t=1, \ldots, \tau-1$, followed by an overall WMTF $y_\tau$ corresponding to the supramaximal stimulus $s_\tau=\max_t(s_t)$ where all $u$ MUs are known (by the clinician) to have fired. The remaining measurements appear in order of increasing stimulus. The advantages of this ordering will become evident in Section \[sec:DetailObsProc\].
The reaction of MU $j$ to stimulus $s_t$ is denoted by the indicator variable $x_{j,t}$, which is $1$ if MU $j$ fires, and hence contributes to the $y_t$ measurement, and $0$ otherwise. The $u$-vector of indicators ${\mathbf{x}}_t=(x_{1,t},\ldots,x_{u,t})^\top$ defines the firing vector of the MUs in response to stimulus $s_t$. Given the experimental set-up, it is assumed that no MUs fire for any baseline measurement, $x_{j,t}=0$ for each $j=1,\ldots,u$ and $t=1,\dots \tau-1$, and all MUs fire in response to the supramaximal stimulus, $x_{j,\tau}=1$.
A sequentially indexed set of elements, vectors or scalars shall be represented as $a_{1:t} := \{a_1, \ldots, a_t\}$. The vectors where all elements are zero or all are unity are denoted by ${\mathbf{0}}$ an ${\mathbf{1}}$ respectively. The indicator function $\mathbb{I}_{A}(x)$ is $1$ if $x \in A$ and $0$ otherwise.
The neuromuscular model {#sec:Neuro-model}
-----------------------
Following the assumptions A1–A3 of @Rid06, the state-space neuromuscular model for the WMTF observations based on a fixed $u$ number of MUs is as follows. $$\begin{aligned}
X_{j,t} | s_t, \eta_j, \lambda_j & \sim \mathrm{Bern}\left[ F \left(s_t; \eta_j, \lambda_j\right) \right], \label{eq:StateProc}\\
Y_{j,t} | \mathbf{X}_t = {\mathbf{x}}_t, {\bar{\mu}}, {\bar{\nu}}, {\boldsymbol{\mu}}, \nu & \sim \mathrm{N}\left({\bar{\mu}}+ {\mathbf{x}}_t^\top{\boldsymbol{\mu}}, {\bar{\nu}}^{-1} + \nu^{-1}{\mathbf{x}}_t^\top{\mathbf{1}}\right). \label{eq:ObsProc}\end{aligne
| 1,576
| 1,264
| 2,387
| 1,612
| 1,484
| 0.78896
|
github_plus_top10pct_by_avg
|
roblem, which is equivalent to the $\mathbf{VP}$ vs. $\mathbf{VNP}$ problem [@valiant79], as well as the complexity of matrix multiplication [@strassen69] can be formulated in this framework [@mulmuleysohoni01; @mulmuleysohoni08; @burgisserikenmeyer11; @burgisserlandsbergmaniveletal11]. More concretely, let us denote by $m_{H,X,k}(\mu)$ the multiplicity of the dual of an irreducible $H$-representation $V_{H,\mu}$ in the $k$-th graded part of the coordinate ring of $X$, and similarly for $Y$ (cf. for precise definitions). Then, $$\label{mult crit}
X \subseteq Y
\;\Rightarrow\;
m_{H,X,k}(\mu) \leq m_{H,Y,k}(\mu)$$ for all $\mu$ and $k \geq 0$. Therefore, the existence of $\mu$ and $k$ such that $m_{H,X,k}(\mu) > m_{H,Y,k}(\mu)$ proves that $X \not\subseteq Y$; such a pair $(\mu,k)$ is called an *obstruction* [@mulmuleysohoni08]. One can relax this implication further and instead compare the support of the multiplicity functions, $$X \subseteq Y
\;\Rightarrow\;
\left( m_{H,X,k}(\mu) \neq 0 \Rightarrow m_{H,Y,k}(\mu) \neq 0 \right).$$ Since computing multiplicities in general coordinate rings is a difficult problem, it is natural to instead study their asymptotic behavior. Following an idea of Strassen [@strassen], it has been proposed in [@burgisserikenmeyer11] to consider the *moment polytope*, $$\Delta_X := \overline {\bigcup_{k=1}^\infty \left\{ \frac \mu k : m_{H,X,k}(\mu) \neq 0 \right\}},$$ which is a compact convex polytope that represents the asymptotic support of the stretching function. Moment polytopes do have a geometric interpretation, which should facilitate their computation [@brion87]. Clearly, $$\label{mo po crit}
X \subseteq Y
\;\Rightarrow\;
\Delta_X \subseteq \Delta_Y.$$ However, preliminary results suggest that the right-hand side moment polytope $\Delta_Y$ might be trivially large in the cases of interest [@burgisserchristandlikenmeyer11; @burgisserikenmeyer11; @kumar11; @burgisserlandsbergmaniveletal11], and therefore insufficient for finding complexity-theoretic obstruction
| 1,577
| 1,805
| 2,700
| 1,521
| 2,326
| 0.780836
|
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|
-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}}
(K_{i_\kappa }^{-1}F_{i_\kappa })
\in U^-(\chi _\nu ){{\mathcal{U}}^0}&
\end{aligned}$$ by Eq. . Hence $$\begin{aligned}
&\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}
{T}^-(E_{\beta _1}^{m_1}\cdots E_{\beta _\nu }^{m_\nu })
{T}^-(E_{\beta _{\nu +1}}^{m_{\nu +1}}\cdots E_{\beta _n}^{m_n})\\
&\quad ={T}^-(E_{i_1}E_{\beta _\nu }
-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1})\in U^-(\chi _\nu ){{\mathcal{U}}^0}.
\end{aligned}$$ By triangular decomposition of $U(\chi )$ it follows that $a_{m_1,\dots ,m_n}=0$ for all $(m_1,\dots ,m_n)$ with $m_\kappa >0$ for some $\kappa \in \{\nu +1,\nu +2,\dots ,n\}$.
By Lemma \[le:kerderK\], $E_{\beta _\nu }\in \ker {\partial ^K}_{i_1}$. Hence $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}
\in \ker {\partial ^K}_{i_1}$ by Lemma \[le:commEFi\]. Thus Lemma \[le:kerderK\] implies that $a_{m_1,\dots ,m_n}=0$ whenever $m_1>0$.
Suppose that there exists $(m_1,\dots ,m_n)$ with $m_\nu >0$ and $a_{m_1,\dots ,m_n}\not=0$. Since $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}$ is ${\mathbb{Z}}^I$-homogeneous of degree ${\alpha }_{i_1}+\beta _\nu $, the only possibility is that $m_1=1$, $m_\nu =1$, and $m_\kappa =0$ for all $\kappa \notin \{1,\nu \}$. Since $a_{m_1,\dots ,m_n}\not=0$, this is a contradiction to the previous paragraph. Thus the theorem is proven.
Next we prove a generalization of Thm. \[th:PBW\].
\[th:PBWtau\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R^\chi _+|$ and let $\tau $ be a permutation of the set $\{1,2,\dots ,n\}$. Then the sets $$\begin{aligned}
\big\{ E_{\beta _{\tau (1)}}^{m_{\tau (1)}} E_{\beta _{\tau (2)}}^{m_{\tau
(2)}}\cdots E_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
\big\{ {\bar{E}}_{\beta _{\tau (1)}}^{m_{\tau (1)}}
{\bar{E}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots
{\bar{E}}_{\bet
| 1,578
| 3,238
| 1,920
| 1,357
| null | null |
github_plus_top10pct_by_avg
|
for non-deterministic first-order grammars, and show that its conclusion is semantically false. We then locate and analyze the flawed argument in the soundness (meta)-proof of [@Jan10].\
The grammar
===========
We consider the alphabet of actions ${\cal A}$, an intermediate alphabet of labels ${\cal T}$ and a map ${{\rm LAB}_{\cal A}}: {\cal T} \rightarrow {\cal A}$ defined by: $${\cal T} := \{ x,y,z,\ell_1\},\;\;{\cal A} := \{a,b,\ell_1\},\;\;\mbox{ and }$$ $${{\rm LAB}_{\cal A}}: x \mapsto a,\;\;y \mapsto a,\;\; z \mapsto b,\;\; \ell_1 \mapsto \ell_1.$$ (these intermediate objects ${\cal T}$, ${{\rm LAB}_{\cal A}}$ will ease the definition of ${{\rm ACT}}$ below). We define a first-order grammar ${\cal G} = ({\cal N},{\cal A},{\cal R})$ by: $${\cal N} := \{A, A', A'', B, B', B'', C, D, E, L_1\}$$ and the set of rules ${\cal R}$ consists of the following: $$\begin{aligned}
A(v) &{\stackrel{y}{\longrightarrow_{}}} & C(v)\\ A(v) &{\stackrel{x}{\longrightarrow_{}}} & A'(v)\\B(v) &{\stackrel{x}{\longrightarrow_{}}} & C(v)\\B(v) &{\stackrel{y}{\longrightarrow_{}}} & B'(v)\\C(v) &{\stackrel{x}{\longrightarrow_{}}} & D(v)\\C(v) &{\stackrel{y}{\longrightarrow_{}}} & E(v)\\A'(v) &{\stackrel{x}{\longrightarrow_{}}} & A''(v)\\B'(v) &{\stackrel{x}{\longrightarrow_{}}} & B''(v)\\A''(v) &{\stackrel{x}{\longrightarrow_{}}} & D(v)\\B''(v) &{\stackrel{x}{\longrightarrow_{}}} & E(v)\\D(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{ruleD}\\E(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{ruleE1}\\E(v) &{\stackrel{z}{\longrightarrow_{}}} & v \label{ruleE2}\\L_1 &{\stackrel{\ell_1}{\longrightarrow_{}}} & \bot \label{ruleL1}$$ Let us name rule $r_i$ (for $1 \leq i \leq 14$), the rule appearing in order $i$ in the above list. We define a map ${{\rm LAB}_{\cal T}}: {\cal R} \rightarrow {\cal T}$ by: ${{\rm LAB}_{\cal T}}(r_i)$ is the terminal letter used by the given rule $r_i$. Subsequently we define ${{\rm ACT}}(r_i):= {{\rm LAB}_{\cal A}}({{\rm LAB}_{\cal T}}(r_i))$. Namely, ${{\rm ACT}}$ maps all the rules $r_1, \ldots ,
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rgument more efficient.
The one bound that is still polynomial in \[thm:new.gen\] is the bound $H\subset A^{K^{e^{O(s)}}}$; it appears that a new idea would be required to improve this any further (see \[rem:poly.bound\], below, for further details). Note, though, that in the case where the ambient group has no torsion the subgroup $H$ is automatically trivial, leaving only the polylogarithmic bounds, as follows.
\[thm:new.tf\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be a torsion-free $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist an ordered progression $P_{\text{\textup{ord}}}(x;L)\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K}$$ and $$|P_{\text{\textup{ord}}}(x;L)|\ge\exp\left(-e^{O(s^2)}\log^{O(s)}2K\right)|A|.$$
As in the abelian case, Ruzsa’s covering argument combines with \[thm:new.gen\] to give the following variant.
\[cor:ruzsa\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$, a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K},$$ and a subset $X\subset G$ of size at most $\exp(e^{O(s^2)}\log^{O(s)}2K)$ such that $A\subset XHP_{\text{\textup{nil}}}(x;L)$.
In particular, $|H\overline P(x;L)|\le\exp(K^{e^{O(s)}})|A|$. In the torsion-free setting the subgroup $H$ is again trivial, and in that case we may conclude instead that $|\overline P(x;L)|\le\exp(e^{O(s^3)}\log^{O(s^2)}2K)|A|$.
Chang’s covering argument also allows us to recover \[thm:old\] with much more precise bounds, as follows.
\[cor:chang.ag\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nil
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)$, with $ \hat{\Sigma}_{{\widehat{S}}}$ given in \[eq:Sigma.loco\]. Notice that $\hat{\Sigma}_{{\widehat{S}}}$ is almost surely positive definite, a consequence of adding extra noise in the definition of $\gamma_{{\widehat{S}}}$ and $\hat{\gamma}_{{\widehat{S}}}$. Then, using Theorem 2.1 in [@cherno2], there exists a universal constant $C > 0$ such that $$\label{eq::secondx}
\sup_{ t = (t_j, j \in {\widehat{S}}) \in \mathbb{R}^{{\widehat{S}}}_{+}} \Bigl| \mathbb{P}(
\sqrt{n}|\hat\gamma_{{\widehat{S}}}(j) - \gamma_{{\widehat{S}}}(j) | \leq t_j, \forall j \in {\widehat{S}}) -
\mathbb{P}(|Z_n(j)| \leq t_j, \forall j \in {\widehat{S}})\Bigr| \leq C
\mathrm{E}_{1,n},$$ where $\mathrm{E}_{1,n}$ is given in . By restricting the supremum in the above display to all $t \in \mathbb{R}^{{\widehat{S}}}_+$ with identical coordinates, we also obtain that $$\label{eq::firstx}
\sup_{t > 0} \Bigl| \mathbb{P}(\sqrt{n}||\hat\gamma_{{\widehat{S}}} - \gamma_{{\widehat{S}}}||_\infty \leq t) -
\mathbb{P}\left(||Z_n||_\infty \leq t \right)\Bigr| \leq C \mathrm{E}_{1,n}.$$ In order to show and , we will use the same arguments used in the proofs of and . We first define $\mathcal{E}_n$ to be the event that $$\label{eq:loco.aleph}
\max_{i,j} \left| \widehat{\Sigma}_{{\widehat{S}}}(i,j) - \Sigma_{{\widehat{S}}}(i,j) \right| \leq
N_n,$$ where $N_n$ is as in . Each entry of $\widehat{\Sigma}_{{\widehat{S}}} - \Sigma_{{\widehat{S}}}$ is bounded in absolute value by $\left( 2(A+\tau) + \epsilon \right)^2$, and therefore is a sub-Gaussian with parameter $\left( 2(A+\tau) + \epsilon \right)^4$. Using a standard derivation for bounding the maximum of sub-Gaussian random variables we obtain that $\mathbb{P}(\mathcal{E}_n^c) \leq \frac{1}{n} $. The bound follows from the same arguments as in the proof : combine the Gaussian comparison Theorem \[thm:comparisons\] with and notice that $\epsilon/\sqrt{3}$ is a lower bound on the standard deviation of the individual coordinates of the $\delta_i$’s. In particular, the Gauss
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of Example. $$\begin{picture}(160,200) \put(20,70){\line(2,-3){40}}
\put(20,70){\line(1,0){80}} \put(20,70){\line(2,3){80}}
\put(60,10){\line(2,3){80}} \put(60,130){\line(2,-3){40}}
\put(60,130){\line(1,0){80}} \put(100,190){\line(2,-3){40}}
\linethickness{0.6mm} \put(60,50){\line(0,1){40}}
\qbezier(60,90)(80,100)(100,110) \put(100,110){\line(0,1){40}}
\thinlines \put(60,50){\line(-2,-1){40}}
\put(60,50){\vector(2,-1){40}} \put(60,90){\line(-2,1){40}}
\put(100,110){\line(2,-1){40}} \put(100,150){\line(2,1){40}}
\put(100,150){\line(-2,1){40}} \put(65,5){\small A}
\put(105,65){\small B} \put(145,125){\small C} \put(10,68){\small
F} \put(50,128){\small E} \put(90,188){\small D}
\end{picture}$$ Here $EF$ and $FA$, $AB$ and $BC$, $CD$ and $DE$ are pairs and $AB$ is the marked edge. If we identify sides that are in pairs (i.e. $EF$ with $FA$, $AB$ with $BC$ and $CD$ with $DE$), then we will obtain a triangulated genus 0 curve.
The main construction: from curve to map
========================================
Let $P$ be a $2n$-gon with marked side $M$ and triangulated by non-intersecting diagonals into $2n-2$ triangles. Sides of $P$ are divided into pairs in such way, that the identification of sides in each pair gives us a genus 0 curve. We will construct a plane tree-rooted cubic map with root edge (not in the spanning tree) in the following way.
- We put a vertex $v_i$ inside each triangle $\triangle_i$ and connect vertices in adjacent triangles — the spanning tree is constructed.
- Let sides $L$ and $L'$ be in pair. $L$ and $L'$ are sides of triangles $\triangle_i$ and $\triangle_j$, respectively (these triangles may coincide). We draw an arc that connect $v_i$ and $v_j$ in the following way: going from $v_i$ the arc intersects $L$. Its next part lies in the exterior of $P$ and connects $L$ and $L'$. After intersecting $L'$ the arc goes to $v_j$.
- An arc, that intersects $M$ will be the root edge. At intersection point it is directed from inside $P$ to outside.
$$\begin{picture}(320,120) \put(10,40){\l
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or the base clause, the case $i = 1$ is true by hypothesis, since the initial step ${\mathfrak{A}}^{\mathbb{N}}(1) = {\mathit{s}}$ is presumed consistent. For the recursive clause, definition \[D:ITERATIVE\_OPERATOR\_WALK\] provides that ${\mathit{s}}_{i+1} = {\mathfrak{A}}(\xi_{i}, {\mathit{s}}_i)$ for each $i \ge 1$. By theorem \[T:AUTOMATON\_ITERATE\_CONSISTENT\], step ${\mathfrak{A}}(\xi_{i}, {\mathit{s}}_i) = {\mathfrak{A}}^{\mathbb{N}}(i+1)$ is consistent. By the axiom of induction, step ${\mathfrak{A}}^{\mathbb{N}}(i)$ is consistent for each $i \ge 1$.
By definition \[D:CONSISTENT\_STEP\] and the conclusion that step ${\mathfrak{A}}^{\mathbb{N}}(i)$ is consistent for each $i \ge 1$, it follows that $\mho_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}(i)) \in \mho_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$, also for each $i \ge 1$.
Substituting ${\mathfrak{A}}^{\mathbb{N}}= \lbrace {\mathit{s}}_n \rbrace$ into lemma \[L:RELATED\_PROJECTION\] yields frame $(\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) = \mho_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$ and functionality $(\overline{\mho}_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) = \mho_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$. From equality it then follows that $(\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) \in (\overline{\mho}_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}))(i)$ for each $i \ge 1$. In simple language, the $i^\text{th}$ term of the process projection is a member of the $i^\text{th}$ term of the procedure projection. This satisfies the requirement of definition \[D:COVERING\_PROCEDURE\] that the procedure covers the sequence of frames: ${\mathbf{f}}_i \in {\mathit{f}}_i$ for each $i \geq 1$.
Reverse inference
-----------------
The construction of automata provides that steps unfold in sequential fashion – that is, the next step becomes known after completing the current step. Consequently automata inherit an intrinsic [forward]{} orientation. It is also reasonable to inquire what may have occurred
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adjoint field. Under the assumptions on the quantum theory stated above, the coefficients of the algebra are exact[^3]. We now move from the determination of the left-right symmetry algebra of the model to the study of the vertex operators.
The primaries {#primaries}
=============
In this section we define the concept of current algebra primaries. These fields can be understood as the elementary vertex operators of the conformal field theory. We compute the operator product expansion between a primary field and a current perturbatively, and deduce the OPE between a primary field and the stress-tensor. In particular we derive the OPEs used in [@Ashok:2009jw].
Left current algebra primaries {#left-current-algebra-primaries .unnumbered}
------------------------------
Given a representation $\mathcal{R}$ of the group $G_L$ we define a left primary field $\phi_\mathcal{R}$ with respect to the left current algebra as a field satisfying the operator product expansions: $$\begin{aligned}
\label{defPrimaries}
j_{L,z}^a(z,\bar z) \phi_\mathcal{R}(w,\bar w) &= - \frac{c_+}{c_+ + c_-} t^a \frac{\phi_\mathcal{R}(w,\bar w)}{z-w} + \text{order zero} \cr
j_{L,\bar z}^a(z,\bar z) \phi_\mathcal{R}(w,\bar w) &=- \frac{c_-}{c_+ + c_-} t^a \frac{\phi_\mathcal{R}(w,\bar w)}{\bar z-\bar w} +
\text{order zero} \end{aligned}$$ where the matrices $t^a$ are the generators of the Lie super-algebra taken in the representation $\mathcal{R}$ associated to the primary field $\phi_\mathcal{R}$. If one assumes the above form for the operator product expansions, then the coefficients of the poles are fixed by the Ward identity for the symmetry $G_L$ and the demand that the contact term vanishes in the operator product expansion between the field $\phi$ and the Maurer-Cartan operator . The Ward identity implies compatibility of the OPEs with current conservation . An example of a left current primary is the adjoint primary we discussed in the previous section.
In appendix \[WZWaffine\] it is shown that a current primary field at a given p
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as one can see from Eq. (\[439\]), the variance of this principal portfolio in the leading order is given by $(N{b}^{2}/{{W}_{N}}^{2}){\bar{{\rho}^{2}}}_{mkt}$, which is of the same order of magnitude as ${\bar{{\rho}^{2}}}_{mkt}$ (recall that $b$ is of the same approximate magnitude as a typical $\beta$ and that ${W}_{N}$ is of the order of ${N}^{1 \over 2}$). The market-aligned portfolio is therefore seen to be that principal portfolio which approximately reflects the volatility profile of the market as a whole. Moreover, since it entirely composed of purchased assets, it is neither hedged nor leveraged.
By contrast, the remaining $N-1$ market-orthogonal principal portfolios are in general hedged and leveraged, and they are quite immune to overall market fluctuations[^5]. In fact, since ${\sum}_{\mu=1}^{N} {{v}}_{\mu}^{2}={\rm tr}({{\sf
\sigma}})={\bm{\beta}} \cdot {\bm{\beta}}
{\bar{{\rho}^{2}}}_{mkt}(1+{\sum}_{i=1}^{N} {\gamma}_{i}^{2})$, and ${{v}}_{N}^{2} \simeq {\bm{\beta}} \cdot {\bm{\beta}}
{\bar{{\rho}^{2}}}_{mkt} + {\sum}_{i=1}^{N} {\hat{\beta}}_{i}^{2}
{\bar{{\alpha}_{i}^{2}}}$, we find for the the average value of the $N-1$ minor eigenvalues $${(N-1)}^{-1}{\sum}_{\mu=1}^{N-1} {{v}}_{\mu}^{2}={(N-1)}^{-
1}{\sum}_{\mu=1}^{N-1} {{W}_{\mu}}^{2}{{V}_{\mu}}^{2}
\simeq {(N-1)}^{-1}{\sum}_{i=1}^{N}(1-
{\hat{\beta}}_{i}^{2}) {\bar{{\alpha}_{i}^{2}}}. \label{441}$$ Thus the weighted average of principal variances for market-orthogonal portfolios is approximately equal to (and in fact less than) the average of the residual variances of the original asset set. Therefore, these $N-1$ market-orthogonal principal portfolios are free not only of mutual correlations with other portfolios but in general also of the volatility induced by overall market fluctuations. This feat is possible in part because of the very special structure of the single-index model which makes it possible to isolate essentially all of the systematic market volatility in one portfolio, leaving the remaining $N-1$ portfolios almost totally
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ts.
**Open Access** This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix {#Sec21}
========
######
Original values of transformed hormone data
Groups Hormones (pg mL^−1^)
--------------------- ---------------------- ---------------- ---------------------
C females (*n* = 6) 217.27 ± 67.95 2,768 ± 639 2,515.66 ± 934.92
C males (*n* = 6) 280.41 ± 42.96 3,707 ± 467 429.71 ± 290.79
E females (*n* = 6) 165.93 ± 37.63 2,171 ± 310 3,025.90 ± 1,053.65
E males (*n* = 5) 546.20 ± 332.72 5,465 ± 2,487 5,744.61 ± 2,318.67
T females (*n* = 7) 3,076.33 ± 518.14 20,749 ± 3,256 412.75 ± 175.62
T males (*n* = 5) 3,339.12 ± 882.85 22,407 ± 5,875 153.61 ± 131.35
D females (*n* = 6) 910.97 ± 313.70 2,324 ± 563 151.23 ± 99.55
D males (*n* = 6) 1,924.50 ± 732.4 11,691 ± 8,168 494.39 ± 275.14
{.9}
{.10}
Introduction {#Sec1}
============
H~2~ energy has drawn attention as an alternative energy source^[@CR1],\ [@CR2]^. Currently, the annual production of H~2~ is approximately 0.1 Gtons, of which 98% comes from the reforming of fossil fuels^[@CR3]^: 40% of H~2~ is produced from natural gas, 30% is produced from heavy oil and naphtha, 18% is produced from coal, 4% is produced from electrolysis and approximately 1% is produced from biomass^[@CR4]^. Due to the advantage of environmentally friendliness and cost-effectiveness compared with conventional chemical methods, biological H~2~ production has been extensively studied over several decades^[@CR5],\ [@CR6]^.
Formate can be produced efficiently from vario
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*sacchariflorus* 4 Japan Davey *et al*., ([2016](#gcbb12419-bib-0503){ref-type="ref"}), Purdy *et al*. ([2013](#gcbb12419-bib-0037){ref-type="ref"}, [2014](#gcbb12419-bib-0038){ref-type="ref"}, [2015](#gcbb12419-bib-0039){ref-type="ref"})
Gig‐311 *sinensis x sacchariflorus* 3 Japan
EMI‐11 *sinensis* 2 Japan
Goliath *sinensis* 3 Japan
**Name or number in each species** **Experiment: modelling crop yield as % of final harvest mass**
3 *sacchariflorus* 4
1 *sacchariflorus robustus* 2 China
5 *sinensis x sacchariflorus* 2
Goliath *sinensis* 3 Japan
3 *sinensis* Unknown
John Wiley & Sons, Ltd
Mixed population {#gcbb12419-sec-0004}
----------------
In 2004 at Aberystwyth, west Wales, United Kingdom, a total of 244 *Miscanthus* genotypes were collected and planted as spaced plants as previously described (Allison *et al*., [2011](#gcbb12419-bib-0002){ref-type="ref"}; Jensen *et al*., [2011](#gcbb12419-bib-0019){ref-typ
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G^{-1}\equiv G_0^{-1}+\Delta\end{aligned}$$ where $\Delta$ is given by: $$\begin{aligned}
&&\Delta(k,\tilde{\omega};k',\tilde{\omega}) \equiv \nonumber\\
&-&g\left(\begin{array}{cc}
0&\psi_B(k+k',\tilde{\omega}+\tilde{\omega}')\\
\bar{\psi}_B(k+k',\tilde{\omega}+\tilde{\omega}')&0\\
\end{array}\right).\end{aligned}$$ The last term in equation (\[sbeff\]) can be expanded into a sum: $$\begin{aligned}
-\ln \det(G_0G^{-1})
=\sum_{l=1}^{\infty}\frac{\text{Tr}\left[(G_0\Delta)^{2l}\right]}{2l}.
\label{suml}\end{aligned}$$ Below, we will evaluate explicitely the $l=1$ and $l=2$ terms in the sum and, following [@GT; @PS], we will give an order of magnitude estimate for the higher order terms showing that they are negligible in the broad resonance limit.
**$l=1$ term**: $$\begin{aligned}
\frac{1}{2}\text{Tr}\left[(G_0\Delta)^{2}\right]=
\int_{k,\tilde{\omega}}\bar{\psi}_B(k,\tilde{\omega})\psi_B(k,\tilde{\omega})
\Pi(k,\tilde{\omega})\end{aligned}$$ After resonance ($\nu<0$), we have $2|\mu|\simeq |\epsilon_b|>
|\epsilon_{\star}|\gg \epsilon_F$, as discussed in Appendix A. As an order of magnitude $|i\tilde{\omega}|$ gives a contribution of the order of the kinetic energy $\sim|k^2/4m|\sim \epsilon_F$. As a result, we obtain $$\begin{aligned}
\Pi(k,\tilde{\omega})\simeq \Pi(0,0)\simeq \epsilon_b-\nu\end{aligned}$$ where we used $\mu\simeq \epsilon_b/2$ and the bound state equation (\[bse\]). Thus the $l=1$ term just gives rise to an effective chemical potential for the dimers equal to $2\mu -\epsilon_b$.
**$l=2$ term**: $$\begin{aligned}
\frac{1}{4}\text{Tr}\left[(G_0\Delta)^{4}\right]&=& \frac{1}{2}
\int_{k_1,\tilde{\omega}_1} \int_{k_2,\tilde{\omega}_2}
\int_{k_3,\tilde{\omega}_3} \bar{\psi}_{B}(1+2-3)\nonumber \\ &\times&
\bar{\psi}_{B}(3)\psi_{B}(2)\psi_{B}(1)g_B(1,2,3)\end{aligned}$$ in an obvious short hand notation for the associated wave vectors and frequencies, and $$\begin{aligned}
g_B(1,2,3)&\equiv& g^4\int_{k,\tilde{\omega}}\Big[
(i\tilde{\omega}-\xi_k) \nonumber \\ &\times&
(i\tilde{\omega}_2-i\tilde{\ome
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m_\nu \ge t,\, 0\le m_\mu <{b^{\chi}} (\beta _\mu )
\,\text{for all $\mu $}}}
F_{\beta _1}^{m_1}F_{\beta _2}^{m_2}\cdots F_{\beta _n}^{m_n}.$$
\[le:P2\] For all ${\alpha }\in {\mathbb{N}}_0^I$, $$\begin{aligned}
{\alpha }\dim U^-(\chi )_{-{\alpha }}=
\sum _{\nu =1}^n \sum _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
{P}^\chi ({\alpha },\beta _\nu ;t) \beta _\nu .
\end{aligned}$$
By Thm. \[th:PBWtau\], for each ${\alpha }\in {\mathbb{N}}_0^I$ there is a basis of $U^-(\chi )_{-{\alpha }}$ parametrized by the set $$\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
\sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,\,
m_\mu <{b^{\chi}} (\beta _\mu )\,\, \text{for all $\mu $}\Big\}.
\label{eq:dimU-}$$ Each $(m_1,\dots ,m_n)$ in this set contributes to ${P}^\chi ({\alpha },\beta _\nu ;t)$ with a summand $1$, for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,m_\nu \}$. Thus the claim of the lemma follows from the decomposition of the ${P}^\chi ({\alpha },\beta _\nu ;t)$ into $1+1+\cdots +1$ by reordering the summands.
\[le:subfch\] Let $\nu \in \{1,2,\dots ,n\}$, $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, and $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that ${\rho ^{\chi}} (\beta _\nu )\Lambda (K_{\beta _\nu }
L_{\beta _\nu }^{-1})
=\chi (\beta _\nu ,\beta _\nu )^t$ and $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi}} (\beta _\mu )-1}
({\rho ^{\chi}} (\beta _\mu )
\Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1})-\chi (\beta _\mu ,
\beta _\mu )^m)\not=0.$$ Then $M^\chi (\Lambda )$ contains a $U(\chi ){\otimes }{\mathbb{K}}$-submodule $V$ with $$\begin{aligned}
{\mathrm{ch}\,V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}.
\label{eq:fchV}
\end{aligned}$$ In particular, $0\not=V\subset I^\chi (\Lambda )$.
We proceed by induction on $\nu $. Let first $\nu =1$. By Lemma \[le:chsub\], $V=U^-(\chi )F_{i_1}^t{\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\
| 1,589
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begin{aligned}
T_n^{\alpha^{-1}} & \cong & T_{-n}^{\alpha}, \: \: \:
T_0^{\alpha^{-1}} \: \cong \: T_0^{\alpha}, \\
{\cal E}_n^{\alpha^{-1}} & \cong & {\cal E}_{-n}^{\alpha}, \: \: \:
{\cal E}^{\alpha^{-1}}_0 \: \cong \: {\cal E}^{\alpha}_0,\end{aligned}$$ (in conventions where $-n$ denotes the component associated to the character of the inverse). Let us now consider the following factor in the Fock vacuum bundle, $${\cal F}^{\alpha}_+ \: = \: \otimes_{n>0} \left( \left( \det {\cal E}^{\alpha}_n \right) \left(
\det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[
- \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ (where the tensor product runs over all nontrivial representations of ${\mathbb Z}_{t^{\alpha}}$). Using relations such as ${\cal E}^{\alpha^{-1}}_n \cong {\cal E}^{\alpha}_{-n}$, we see that each factor in ${\cal F}_+^{\alpha^{-1}}$ is equivalent to a factor in ${\cal F}_+^{\alpha}$, but with an exponent of the opposite sign, hence $$\label{eq:pos-Fock-duality}
{\cal F}_+^{\alpha^{-1}} \: \cong \: \left( {\cal F}_+^{\alpha} \right)^* .$$ As the combinatorics in these exponents is slightly complicated, let us consider some special cases to explicitly confirm this prediction. When $t_{\alpha} = 2$, $$\begin{aligned}
{\cal F}_+^{\alpha} & = & \left( \left( \det {\cal E}_1^{\alpha} \right)
\left( \det T_1^{\alpha} \right)^{-1} \right)^{- \frac{1}{2} - \left[
- \frac{1}{2}\right] - \frac{1}{2} }, \\
& = & \left( \left( \det {\cal E}_1^{\alpha} \right)
\left( \det T_1^{\alpha} \right)^{-1} \right)^{0} \: \cong \: {\cal O}
\: \cong \: \left( {\cal F}_+^{\alpha^{-1}} \right)^* .\end{aligned}$$ When $t_{\alpha}=3$, $$\begin{aligned}
{\cal F}_+^{\alpha} & = & \left( \left( \det {\cal E}_1^{\alpha} \right)
\left( \det T_1^{\alpha} \right)^{-1} \right)^{-\frac{1}{3} - \left[
- \frac{1}{3} \right] - \frac{1}{2}} \otimes
\left( \left( \det {\cal E}_2^{\alpha} \right) \left( \det T_2^{\alpha}
\right)^{-1} \right)^{- \frac{2}{3} - \left[ - \frac{2}{3} \right] -
\frac{1}{2} }, \\
& = & \l
| 1,590
| 2,516
| 1,734
| 1,486
| null | null |
github_plus_top10pct_by_avg
|
above are given by \[shift-defn\] $$S_c: {U}_c {\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}: \qquad
N\mapsto Q_c^{c+1} \otimes_{{U}_c} N$$ and $$\widetilde{S}_c: H_{c} {\text{-}{\textsf}{mod}}\to H_{c+1}{\text{-}{\textsf}{mod}}: \qquad
M\mapsto H_{c+1}e_-\delta \otimes_{{U}_{c}} eM.$$
{#subsec-4.1}
When $c$ is a positive real number, the Morita equivalence between $U_c$ and $ U_{c+1}$ is given by $S_c$ and we begin with that case. The general case, proved in Corollary \[morrat-cor\], will be an easy consequence.
\[morrat\] Assume that $c\in {\mathbb{R}}_{\geq 0}$ with $c\notin
\frac{1}{2} + \mathbb{Z}$. Then both shift functors $\widetilde{S}_c: H_{c}{\text{-}{\textsf}{mod}}\to H_{c+1}{\text{-}{\textsf}{mod}}$ and $S_c: {U}_c{\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}$ are Morita equivalences.
Moreover, the idempotent functor $E_c: H_{c} {\text{-}{\textsf}{mod}}\to {U}_c{\text{-}{\textsf}{mod}}$ given by $M\mapsto eM$ is a Morita equivalence.
In order to prove that $S_c$ is an equivalence, we need to show that $Q=Q_c^{c+1}$ is a projective generator for ${U}_{c+1}{\text{-}{\textsf}{mod}}$, with endomorphism ring $\mathrm{End}_{U_{c+1}}(Q) =U_{c}$. Arguing as in [@EG Theorem 1.5(iv)] the dual $Q^\ast=\operatorname{Hom}_{{U}_{c+1}}(Q,{U}_{c+1})$ is $P = \delta^{-1} e_-H_{c+1}e.$ By the dual basis lemma, $Q$ is a projective ${U}_{c+1}$-module with $\mathrm{End}_{U_{c+1}}(Q) =
U_{c}$ if and only if $PQ={U}_c$ while $Q$ is a generator if and only if $QP={U}_{c+1}$. Substituting in the given formulæ for $Q$ and $P$ shows that we need to prove that $$\label{morrat1}
H_{c+1}e_-H_{c+1} =H_{c+1}\qquad\text{and} \qquad
H_{c+1}eH_{c+1} = H_{c+1}\quad\text{for}\ c\geq 0.$$ Similarly, as $H_{c}e$ is a projective left $H_{c}$-module, $E_c$ will be a Morita equivalence if we prove that $$\label{morrat11}
H_ceH_c = H_c \quad \text{for}\ c\geq 0.$$ Since $\widetilde{S}_{c}=E^{-1}_{c+1}\circ S_{c}\circ E_{c}$, Equations \[morrat1\] and \[morrat11\] will suffice to prove the theorem.
| 1,591
| 1,060
| 1,628
| 1,400
| 3,570
| 0.771486
|
github_plus_top10pct_by_avg
|
array}{cll}
-1-\lambda x, & & x<-1\\
(1-\lambda)x, & & -1\leq x\leq1\\
1-\lambda x, & & 1<x,\end{array}\right.\\
\\f_{\lambda\textrm{b}}(x) & = & \left\{ \begin{array}{cll}
-\lambda x, & & x<1\\
(1-\lambda x)-1, & & 1\leq x\leq2\\
1-\lambda x, & & 2<x\end{array}\right.\\
\\f_{\lambda\textrm{c}}(x) & = & \left\{ \begin{array}{cll}
-\lambda x & & x<1\\
\sqrt{x-1}-\lambda x & & 1\leq x,\end{array}\right.\\
\\f_{\lambda\textrm{d}}(x) & = & \left\{ \begin{array}{cll}
(x-1)^{2}+1-\lambda x & & 1\leq x\leq1\\
(1-\lambda)x & & \textrm{otherwise}\end{array}\right.\\
\\f_{\lambda\textrm{e}}(x) & = & \tan^{-1}(x)-\lambda x,\\
\\f_{\lambda\textrm{f}}(x) & = & \left\{ \begin{array}{cll}
1-2\sqrt{-x}-\lambda x, & & x<-1\\
(1-\lambda)x, & & -1\leq x\leq1\\
2\sqrt{x}-1-\lambda x, & & 1<x\end{array}\right.\end{array}$$ taken from @Appel2000 are shown in Fig. \[Fig: Appel\]. It is easy to verify that the Lipschitz and linear upper and lower bounds of these maps are as in Table \[Table: Appel\_bnds\].
The point spectrum defined by $$P\sigma(f)=\{\lambda\in\mathbb{C}\!:(f-\lambda)x=0\textrm{ for some }x\neq0\}$$
is the simplest to calculate. Because of the special role played by the zero element $0$ in generating the point spectrum in the linear case, the bounds $m\Vert x\Vert\leq\Vert\mathscr{L}x\Vert\leq M\Vert x\Vert$ together with $\mathscr{L}x=\lambda x$ imply $\textrm{Cl}(P\sigma(\mathscr{L}))=[\Vert\mathscr{L}\Vert_{\textrm{b}},\Vert\mathscr{L}\Vert_{\textrm{B}}]$ — where the subscripts denote the lower and upper bounds in Eq. (\[Eqn: LipNorm\]) and which is sometimes taken to be a descriptor of the point spectrum of a nonlinear operator — as can be seen in Table \[Table: Appel\_spectra\] and verified from Fig. \[Fig: Appel\]. The remainder of the spectrum, as the complement of the resolvent set, is more difficult to find. Here the convenient characterization of the resolvent of a continuous linear operator as the set of all sufficiently large $\lambda$ that satisfy $|\lambda|>M$ is of little significance a
| 1,592
| 2,100
| 2,497
| 1,589
| null | null |
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|
24.3 59.0 1006.5
6 653.4 22.8 22.1 88.3 1007.9
7 726.9 19.5 23.4 73.8 1001.8
8 909.0 36.2 25.7 71.1 1015.1
sensors-20-02119-t002_Table 2
######
*RMSE* and *MAE* of I--V curves.
Working Conditions Irradiance *G* (W/m^2^) Algorithms *MAE* (%) *RMSE* (%)
-------------------- ------------------------- ------------ ----------- ------------
1 153.7 MLP 2.0 3.0
CNN 1.9 2.1
2 237.5 MLP 6.0 6.6
CNN 1.8 3.9
3 328.7 MLP 6.7 7.0
CNN 3.3 4.8
4 445.5 MLP 2.2 4.5
CNN 1.5 3.3
5 537.9 MLP 5.0 5.3
CNN 2.1 3.4
6 653.4 MLP 11.0 11.7
CNN 2.6 4.7
7 726.9 MLP 6.0 8.0
CNN 1.5 2.1
8 909.0 MLP 11.5 19.2
CNN
| 1,593
| 6,170
| 736
| 634
| null | null |
github_plus_top10pct_by_avg
|
nsform is given by $$Tg({\bf f}) = \left( \det(\mathbf{Id+K}) \right)^{-\frac{1}{2}} \exp\left(-\frac{1}{2} ({\bf f}, \mathbf{(Id+K)^{-1}} {\bf f})\right), \quad {\bf f} \in S_{d}({\mathbb{R}}).$$ Therefore $\left( \det(\mathbf{Id+K}) \right)^{\frac{1}{2}}g$ is a generalized Gauss kernel.
- \[traceL2\] Since a trace class operator is compact, see e.g. [@RS75a], we have that $\mathbf{K}$ in the above example is diagonalizable, i.e. $$\mathbf{K}\mathbf{f} = \sum_{k=1}^{\infty} k_n (\mathbf{f},\mathbf{e}_n)\mathbf{e}_n, \quad \mathbf{f} \in L_d^2({\mathbb{R}},dx),$$ where $(\mathbf{e}_n)_{n\in {\mathbb{N}}}$ denotes an eigenbasis of the corresponding eigenvalues $(k_n)_{n\in {\mathbb{N}}}$ with $k_n \in (-\frac{1}{2}, 0 ]$, for all $n \in {\mathbb{N}}$. Since $K$ is compact, we have that $\lim\limits_{n\to \infty} k_n =0$ and since $\mathbf{K}$ is trace class we also have $\sum_{n=1}^{\infty} (\mathbf{e}_n, -\mathbf{K} \mathbf{e}_n)< \infty$. We define for ${\boldsymbol \omega }\in S_d'({\mathbb{R}})$ $$\begin{aligned}
- \langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega } \rangle := \lim_{N \to \infty} \sum_{n=1}^N \langle \mathbf{e}_n, {\boldsymbol \omega }\rangle (-k_n)\langle \mathbf{e}_n,{\boldsymbol \omega } \rangle. \end{aligned}$$ Then as a limit of measurable functions ${\boldsymbol \omega } \mapsto -\langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega } \rangle$ is measurable and hence $$\begin{aligned}
\int\limits_{S_d'({\mathbb{R}})} \exp(- \langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega }\rangle ) \, d\mu({\boldsymbol \omega }) \in [0, \infty].\end{aligned}$$ The explicit formula for the $T$-transform and expectation then follow by a straightforward calculation with help of the above limit procedure.
- In the following, if we apply operators or bilinear forms defined on $L^2_d({\mathbb{R}})$ to generalized functions from $S'_d({\mathbb{R}})$, we are always having in mind the interpretation as in \[traceL2\].
\[D:Nexp\][@BG10]$\;$ Let $\mathbf{K}: L^2_
| 1,594
| 1,820
| 1,533
| 1,451
| null | null |
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|
, as in [@GGOR Section 3.5], and write $K$ for the kernel of the associated homomorphism $\phi: P_c(\lambda) \to \Delta_c(\mu)$. By [@guay Proposition 13] there is a $\Delta$-filtration of $P_c(\lambda)$ $$P_c(\nu) = M_0 \supset M_1
\supset \cdots \supset M_t = 0$$ with each factor $M_j/M_{j+1}$ of the form $\Delta_c(\lambda_j)$ for some $\lambda_j\in {{\textsf}{Irrep}({{W}})}$. Thus there exists $i$ such that $M_i+K/K \neq 0$ but $M_{i+1}+K/K = 0$. This gives a non-zero composition $$\psi: \Delta_c(\lambda_i) \cong
M_i / M_{i+1} \longrightarrow (M_i + K)/K \longrightarrow
P_c(\lambda)/K \longrightarrow \Delta_c(\mu).$$ By Lemma \[basiccom\], $\lambda_i \leq\mu$. If $\lambda_i=\mu$ then the first remark after the statement of the lemma would imply that $\psi$ and hence $\phi$ are surjective, contradicting the fact that $\lambda\not=\mu$. Thus $\lambda_i<\mu$. By BGG reciprocity [@guay Theorem 19], $[P_c(\lambda): \Delta_c(\lambda_i)] =
[\Delta_c(\lambda_i): L_c(\lambda)]\neq 0$ and so, by induction, $\lambda \leq \lambda_i$. Thus $\lambda < \mu$.
{#order-remark}
A result analogous to Corollary \[poono\] is proved as part of the proof of [@guay Proposition 13]. However the ${\mathbb{Z}}$-strings ordering used in [@guay] is different from the dominance ordering. An explicit example where the orderings differ can be found when $n=8$, by taking $\lambda = (6,1,1)$ and $\mu = (4,4)$. In this case $\lambda$ and $\mu$ are incomparable in the dominance ordering, but comparable in the ${\mathbb{Z}}$-strings ordering.
The canonical grading on $\mathcal{O}_c$. {#cangrad}
-----------------------------------------
The final ingredient we need for the proof of Theorem \[morrat\] is a canonical grading on ${\mathcal{O}}_c$. Let ${\mathbf{h}}_c\in H_c$ be defined as in . Then, for $M\in {\mathcal{O}}_c$ and $\alpha\in{\mathbb{C}}$, set $$W_{\alpha}(M) = \{ m\in M : ({\mathbf{h}}_c - \alpha)^km = 0
\text{ for $k\gg
0$}\}.$$\[cangrad-defn\] By [@GGOR (2.4.1)] this gives the *canonical grading* $M = \sum_{\alpha\in {\m
| 1,595
| 650
| 404
| 1,749
| 2,141
| 0.782345
|
github_plus_top10pct_by_avg
|
ier, even more in order to define a consistent heterotic string compactification.
For completeness, let us now consider some possible examples.
One example is described in the paper [@dopr]. (See also [@Donagi:2000zf; @Donagi:2000zs; @Donagi:2000fw; @Ovrut:2002jk; @Ovrut:2003zj; @Braun:2004xv].) In that paper, the authors first construct an elliptically-fibered Calabi-Yau threefold $Z$ with fundamental group ${\mathbb Z}_2 \times
{\mathbb Z}_2$, built as a freely-acting[^15] ${\mathbb Z}_2 \times {\mathbb Z}_2$ quotient of a simply-connected Calabi-Yau threefold $X$: $$Z \: = \: X / ( {\mathbb Z}_2 \times {\mathbb Z}_2 )$$ together with a bundle $V$ on $X$ that is not quite equivariant with respect to the ${\mathbb Z}_2 \times {\mathbb Z}_2$ action, and so descends to a twisted bundle on $Z$.
Consider the gerbe presented as $[X/G]$, where $$1 \: \longrightarrow \: {\mathbb Z}_2 \: \longrightarrow \:
G \: \longrightarrow \:
{\mathbb Z}_2 \times {\mathbb Z}_2 \: \longrightarrow \:
1,$$ with the ${\mathbb Z}_2$ kernel acting trivially. (Explicitly, the extension above is the Heisenberg extension, and $G = D_4$ [@tonypriv].) The bundle $V$ above descends to a bundle on a gerbe. Furthermore, the entire bundle is an eigenbundle under the nontrivial element of the center of $D_4$, with eigenvalue $-1$ (since it must square to the identity and can not itself be the identity) [@tonypriv].
For completeness, let us now work through the example of [@dopr] in more detail. Their Calabi-Yau manifold $X$ is an elliptic fibration over a rational elliptic surface, and in fact can be described as the fiber product over ${\mathbb P}^1$ of two rational elliptic surfaces $B$, $B'$: $$X \: = \: B \times_{ {\mathbb P}^1 } B'$$ where $\pi: X \rightarrow B'$, $\pi': X \rightarrow B$, $\beta': B' \rightarrow {\mathbb P}^1$, $\beta: B \rightarrow B$: $$\xymatrix{
& X \ar[rd]^{\pi} \ar[ld]_{\pi'} & \\
B \ar[rd]_{\beta} & & B' \ar[ld]^{\beta'} \\
& {\mathbb P}^1 &
}$$ $B$ and $B'$ are both chosen to admit an automorphism group containing
| 1,596
| 911
| 1,500
| 1,492
| 2,886
| 0.776309
|
github_plus_top10pct_by_avg
|
1 \leq i \leq n$, ${\|X_i-X_{i-1}\|}_2 \leq c_i$, for some non-negative constant $c_i$. Then for every $\delta > 0$, $$\begin{aligned}
\P[{\|X_n\|}_2 \geq \delta] & \leq & 2e^{3}e^{-\frac{\delta^2}{2\sum_{i=1}^n c_i^2}}\,.\end{aligned}$$
It follows from the upper bound on ${\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 \leq c_i^2$ with $c_i^2=\lambda^2\big( (k_j-p_{j,a})^2 + (k_j-p_{j,a}) \big)$. In the expression , $\nabla\L_{G_{j,a}}(\theta^*)$ has one entry at $p_{j,a}$-th position that is compared to $(k_j-p_{j,a})$ other items and $(k_j-p_{j,a})$ entries that is compared only once, giving the bound $$\begin{aligned}
{\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 &\leq&
\lambda_{j,a}^2(k_j-p_{j,a})^2 + \lambda_{j,a}^2 (k_j-p_{j,a}) \;.\end{aligned}$$
### Proof of Lemma \[lem:consistency\]
Define event $E \equiv \{(i,\i) \in G_p \}$. Observe that $$\begin{aligned}
E = \Big\{\Big(\I_ {\{(\sigma^{-1}(i) = p\}} + \I_{\{\sigma^{-1}(\i)) = p\}} = 1\Big) \wedge \Big(\sigma^{-1}(i),\sigma^{-1}(\i) \geq p \Big)\Big\} \;.\end{aligned}$$ Consider any set $\Omega \subset S\setminus\{i,\i\}$ such that $|\Omega| = p-1$. Let $M$ denote an event that items of the set $\Omega$ are ranked in top-$(p-1)$ positions in a particular order. It is easy to verify the following: $$\begin{aligned}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \Big| E, M\Big] &=& \frac{\P\Big[\big(\sigma^{-1}(i) < \sigma^{-1}(\i)\big), E, M\Big]}{\P\Big[E, M\Big]}\\
&=& \frac{\P\Big[\big(\sigma^{-1}(i)= p\big), M\Big]}{\P\Big[\big(\sigma^{-1}(i)= p\big), M\Big] + \P\Big[\big(\sigma^{-1}(\i) =p \big), M\Big]} \\
&=& \frac{\exp(\theta^*_i)}{\exp(\theta^*_i) + \exp(\theta^*_{\i})} = \P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \Big]\;.\end{aligned}$$ Since $M$ is any particular ordering of the set $\Omega$ and $\Omega$ is any subset of $S\setminus\{i,\i\}$ such that $|\Omega| = p -1$, conditioned on event $E$ probabilities of all the possible events $M$ over all the possible choices of set $\Omega$ sum to $1$.
### Proof of Lemma \[lem:az\_gen\]
It follows exactly alo
| 1,597
| 2,517
| 2,228
| 1,496
| null | null |
github_plus_top10pct_by_avg
|
bilayer distance. However, for smaller distances, we then encounter phases involving strong interlayer pairing [@pikovski2010; @zinner_10; @baranov2011] and the system would instead be better described in terms of interlayer bosonic dimers, as we discuss later.
![(Color online) Critical wave vector $q_c/k_F$ for the $\phi=\pi/2$ stripe phase ($\theta < \theta_c$) and the $\phi = 0$ one ($\theta > \theta_c$) — same parameters and symbol scheme as in Fig. \[fig:phase\_diag\]. The insets depict the alignment of the dipoles with the electric field ${{\mathbf E}}$ and the features of the two different stripe phases. For the $\phi = 0$ stripe phase, the density modulations in the two layers have a phase shift $\eta
\simeq 2\theta$, while the wave vector $q_c$ decreases with increasing tilt angle $\theta$ down to $q_c=0$ for $\theta=\pi/2$ (filled \[red\] diamond), where the gas collapses. For density modulations along $\phi = \pi/2$, $q_c$ appears to be fixed by the density.[]{data-label="fig:schematic"}](schematic_criticalq_d2.pdf){width="\linewidth"}
In the isotropic case ($\theta=0$), we find that the system spontaneously breaks rotational symmetry to form a stripe phase at $U
\simeq 5.74$, similarly to the single-layer case [@parish2012]. One can only observe this symmetry breaking at $\theta=0$ by starting the STLS iteration with a solution for small but finite $\theta$. This effectively corresponds to taking the limit $\theta \to 0$, which is somewhat akin to classical ferromagnetism, where one must consider the limit where magnetic field goes to zero. This stripe phase precedes Wigner crystallization which, according to quantum Monte Carlo (QMC) calculations, occurs at $U\simeq 25$ for perpendicular fermionic dipoles in a single layer [@matveeva2012].
For $\theta>\arcsin(1/\sqrt{3})$, the intralayer interaction develops an attractive sliver in the plane that can eventually lead to collapse in the single layer [@bruun2008; @yamaguchi2010; @sieberer2011; @parish2012]. Here, for large enough $U$ and $\t
| 1,598
| 689
| 2,039
| 1,738
| null | null |
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|
define the direction dependence by writing the expression for $\phi(\tau)$ in Eq.(\[smearedoperator\]) as $$\frac{d \phi(\tau)}{d \Omega} = \int d\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi})) = \phi_\Omega(\tau)
\ ,
\label{smearedoperatorang}$$ such that the ${\bm \xi}$ points in the direction of $\Theta_0$ and $\phi_0$. Integrating $\phi_\Omega(\tau)$ all over the solid angle then reproduces the smeared field operator $\phi(\tau)$ in the Schlicht case. Assuming the two level quantum system to couple linearly to $\phi_\Omega$, one can then proceed to calculate the transition rate as per formula in Eq.(\[transprobability\]) with the corresponding $W_\Omega(\tau,\tau^\prime)$ equal to $ \langle \Psi | \phi_\Omega(\tau) \phi_\Omega(\tau^\prime) | \Psi \rangle$. Equivalently, one can consider a detector whose spatial profile has the radial dependence of and depends on the angles $\Theta_0$ and $\phi_0$ through $$\begin{aligned}
f_{\epsilon}({\bm \xi},\Theta_{0})= \frac{1}{2\pi^3} \frac{\epsilon}{{(\xi^{2}+\epsilon^{2})}^2} \frac{\delta (\theta - \Theta_{0})}{\sin\Theta_{0}} \delta(\phi - \phi_0)
\ ,
\label{angprof}\end{aligned}$$ where $\theta$ and $\phi$ are measured in the ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ space. One can once again note that integrating over the solid angle $d\Omega_0 = \sin\Theta_{0} d\Theta_{0} d\phi_{0}$ yields the isotropic profile .
Following the steps in section \[schlichtsection\], the transition rate formula becomes $$\begin{aligned}
{\dot {\cal F}}_{\Theta_0}(\omega) = 2 \operatorname{Re}\int_0^\infty ds \,e^{-i \omega s} \, W_{\Theta_0}(\tau,\tau - s)
\ ,
\label{angtransitionrate}\end{aligned}$$ where $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) = \langle \Psi | \phi(\tau, \Theta_0) \phi(\tau^\prime,\Theta_0) | \Psi \rangle
\ ,
\label{angwhitmannfunction}\end{aligned}$$ and the smeared field operator reads $$\begin{aligned}
\phi(\tau, \Theta_0) = \int d^3\xi \; f_{\epsilon} ({\bm \xi}, \Theta_0) \, \phi(x(\tau, {\bm \xi}))
\ ,
\label{smearedRARF}\end{aligned
| 1,599
| 3,819
| 2,611
| 1,630
| null | null |
github_plus_top10pct_by_avg
|
=\ - \langle A_\delta, QA_\eta+(F\Psi)^2\rangle
- {\langle\!\langle}\delta\Psi, Y(Q\Psi+X\eta F\Psi){\rangle\!\rangle}\,,
\label{general variation}$$ from which we find the equations of motion, $$QA_\eta + (F\Psi)^2\ =\ 0\,,\qquad
Q\Psi + X\eta F\Psi\ =\ 0\,.
\label{equations of motion}$$
Before closing this section, we generalize several ingredients for later use. We can define $A_{\mathcal{O}}(t)$ not only for $\mathcal{O}=\partial_t, \eta,$ or $ \delta$, but also for any other derivations of the string product. Although such general $\mathcal{O}$’s are not in general commutative, we assume that they satisfy a closed algebra with respect to the graded commutator of derivations, $\{\mathcal{O}_1,\mathcal{O}_2]\ =\ \mathcal{O}_1\mathcal{O}_2
-(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2\mathcal{O}_1$. The generalized $A_{\mathcal{O}}(t)$’s satisfy the equation $$\begin{aligned}
\mathcal{O}_1A_{\mathcal{O}_2}(t)
-&(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2A_{\mathcal{O}_1}(t)
- [\![A_{\mathcal{O}_1}(t)\,, A_{\mathcal{O}_2}(t)]\!]
=\ A_{\{\mathcal{O}_1,\mathcal{O}_2]}(t)\,,\label{gen MC}\end{aligned}$$ which reduces to the Maurer-Cartan-like equation (\[MC\]) when $\{\mathcal{O}_1,\mathcal{O}_2]=0$. Using $A_{\mathcal{O}}(t)$, we can define the covariant derivative $D_{\mathcal{O}}(t)$ on a string field $A$ by $$D_{\mathcal{O}}(t) A\ =\ \mathcal{O} A
- [\![A_{\mathcal{O}}(t)\,, A]\!]\,.
$$ From (\[gen MC\]), we can show that $$[\![D_{\mathcal{O}_1}(t)\,, D_{\mathcal{O}_2}(t)]\!]\
=\
D_{\{\mathcal{O}_1, \mathcal{O}_2]}(t)\,.
\label{generalized D}$$ As an analog of the linear map $F(t)$ in the Ramond sector, we can also define the linear map $f(t)$ on a general string field $\Phi$ in the NS sector by $$\begin{aligned}
f(t)\Phi\ =&\
\frac{1}{1+\xi_0(D_\eta(t)-\eta)}\,\Phi
\nonumber\\
=&\ \Phi + \xi_0 [\![A_\eta(t), \Phi]\!]
+ \xi_0 [\![A_\eta(t),\,\xi_0[\![A_\eta(t), \Phi]\!] ]\!]\cdots\,.
\label{f ns}\end{aligned}$$ A homotopy operator for $D_\eta(t)$ in the NS sector is given by the BPZ even opera
| 1,600
| 2,756
| 1,315
| 1,534
| null | null |
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|
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