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mbda_z \\ \delta\end{smallmatrix}\right)} \}.
\end{aligned}$$ After choosing a basis of the lattice $\Lambda^*_{G_{\operatorname{ss}}}$ we arrive at the asserted formula (with $s' = \dim T_{G_{\operatorname{ss}}}$ and $u = t + r_H$).
We stress that the proof of is constructive: The maps $\mathcal A$ and $\mathcal B$, whose existence is asserted by the theorem, are defined in in terms of $A$ and $B$, whose construction is described explicitly in [@billeyguilleminrassart04 Proof of Theorem 2.1] (for the case of $\mathfrak g = {\mathfrak{su}}(d)$) and in [@bliem08 §4] (for the general case). See for an illustration in the context of the Kronecker coefficients.
If one uses the Kostant multiplicity formula instead of in the proof of then one arrives at a similar formula for the multiplicities $m^\lambda_\mu$ involving an additional alternating sum over the Weyl group of $G$. After completion of this work, we have learned of [@heckman82 Lemma 3.1] which is derived in this spirit.
Piecewise Quasi-Polynomiality {#piecewise-quasi-polynomiality .unnumbered}
-----------------------------
Let us use the fundamental weight bases fixed above to identify $\Lambda^*_G \cong {\mathbb Z}^{r_G}$ and $\Lambda^*_H \cong {\mathbb Z}^{r_H}$. The group homomorphisms $\mathcal A$ and $\mathcal B$ correspond to matrices with integer entries, which we shall denote by the same symbols. Observe that the formula in in essence amounts to counting the number $n(y) := \# \left( \Delta_{\mathcal A, \mathcal B}(y) \cap {\mathbb Z}^{s+s'} \right)$ of integral points in certain rational convex polytopes of the form $$\label{weight restriction polytope}
\Delta_{\mathcal A,\mathcal B}(y) := \{ x \in {\mathbb R}^{s+s'} : x_1, \ldots, x_s \geq 0, \mathcal A x = \mathcal B y \},$$ parametrized by $y \in {\mathbb Z}^{r_G+r_H}$. Explicitly, $$\label{concrete alternating sum}
m^\lambda_\mu = \sum_{\gamma \in \Gamma_H} c_\gamma n(\lambda, \mu + \gamma).$$ It is well-known that $n(y)$ is a *piecewise quasi-polynomial* function in $y$ [@claussloechn
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4.90 1118.00 37.21 80.63 50.92 283.50
**MGoF** 13.97 17.50 15.54 294.08 14.66 8.13 10.45 **8.01** **12.50** 3.13 5.00 250.10 18.42 8.75 11.86 **7.55**
-------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- -----------
![1st level histogram of day 5, 113 and 287, each day in a column. Distributions after centralized and equalized click farming are in 1st and 3rd rows correspondingly. And the original distributions are shown in the 2nd row.[]{data-label="fig:raw-hist-1st"}](./Raw1stLevelHist.pdf){width="\linewidth"}
![This figure shows distribution of JSD values(on 2nd level histograms) of normal and two types of click farming data. Divergences were calculated according to a reference averaged among all correct distributions.[]{data-label="fig:raw-overview"}](./RawOverview2nd.pdf){width="\linewidth"}
When classifying toward 1st level histograms, centralized click farming behaviours can be easily discovered. As displayed in the first two rows in Fig. \[fig:raw-hist-1st\], normal collections share a similar distribution while centralized click-farmed ones abruptly violated the original shape. However, as a clever click farmer, equalized click farming did not in the least distort the distribution. Most of them escaped the check under perfect disguises. But when it came to 2nd level histograms, the “clever disguise” did not work any longer. It can be clearly seen in Fig. \[fig:raw-overview\] that distribution of divergence of both click farming types shows an obvious deviation from the normal one.
The result showed that our technique outperformed MGoF in every real world cases. SDD-E provided best performance, yet it consumed the most computing power. Comparison among SDD-R revealed improvement of reference as we
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y $\mathcal{U}$ imposed on it, then Eq. **(\[Eqn: TB\]) must also be satisfied in order that the topology generated by $_{\textrm{T}}\mathcal{B}$ is indeed $\mathcal{U}$. The next theorem connects the two types of bases of Defs. A1.1 and A1.2 by asserting that although a local base of a space need not consist of open sets and a topological base need not have any reference to a point of $X$, any subcollection of the base containing a point is a local base at that point.
**Theorem A1.2.** *A collection of open sets* $_{\textrm{T}}\mathcal{B}$ *is a base for a topological space $(X,\mathcal{U})$ iff for each $x\in X$, the subcollection* $$\mathcal{B}_{x}=\{ B\in\mathcal{U}\!:x\in B\in\!\,_{\textrm{T}}\mathcal{B}\}\label{Eqn: base_local base}$$ *of basic sets containing $x$ is a local base at* $x$.$\qquad\square$
**Proof.** *Necessity.* Let $_{\textrm{T}}\mathcal{B}$ be a base of *$(X,\mathcal{U})$* and $N$ be a neighbourhood of $x$, so that $x\in U\subseteq N$ for some open set $U=\bigcup_{B\in\!\,_{\textrm{T}}\mathcal{B}}B$ and basic open sets $B$. Hence $x\in B\subseteq N$ shows, from Eq. (\[Eqn: TBx\]), that $B\in\mathcal{B}_{x}$ is a local basic set at $x$.
*Sufficiency.* If $U$ is an open set of $X$ containing $x$, then the definition of local base Eq. (\[Eqn: TBx\]) requires $x\in B_{x}\subseteq U$ for some subcollection of basic sets $B_{x}$ in $\mathcal{B}_{x}$; hence $U=\bigcup_{x\in U}B_{x}$. By Eq. (\[Eqn: TB\_topo\]) therefore, $_{\textrm{T}}\mathcal{B}$ is a topological base for $X$.$\qquad\blacksquare$
Because the basic sets are open, (TB2) of Theorem A1.1 leads to the following physically appealing paraphrase of Thm. A1.2.
**Corollary.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of open sets of* $(X,\mathcal{U})$ *is a topological base that generates* $\mathcal{U}$ *iff for each open set $U$ of $X$ and each $x\in U$ there is an open set* $B\in\!\,_{\textrm{T}}\mathcal{B}$ *such that $x\in B\subseteq U$*; *that is iff* $$x\in U\in\mathcal{U}\Longrightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!
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east one propagating part which contains $1$ and $1'$ simultaneously. In the diagram of the standard expression of a seat-plan of $\Sigma^1_{n-\frac{1}{2}}$, the vertices $1$ and $1'$ are joined by a vertical line. Shrinking this vertical line to one vertex, we have one to one correspondences between $\Sigma^1_{n-\frac{1}{2}}$ and the set of the set-partitions of order $2n-1$. (Hence we find $|\Sigma^1_{n-\frac{1}{2}}| = B_{2n-1}$, the Bell number.)
Under this preparation, we prove the theorem. Since the relations in the theorem allow us to use all the required local moves, we can show just in the course of the arguments of Section 4 that any word in the alphabet ${\cal L}^1_{n-\frac{1}{2}}$ is equal to (possibly a scalar multiple of) a standard expression in the abstract algebra $\widetilde{A_{n-\frac{1}{2}}(Q)}$.
Hence we have $$\mbox{rank}\ \widetilde{A_{n-\frac{1}{2}}}(Q)
\leq |\Sigma_{n-\frac{1}{2}}^1|.$$
As Murtin and Rollet showed in [@MR], $\Sigma_{n-\frac{1}{2}}^1$ makes a basis of ${\mbox{${\mathbb C}$}}\otimes A_{n-\frac{1}{2}}(k)
= {\mbox{${\mathbb C}$}}\otimes \psi(\widetilde{A_{n-\frac{1}{2}}(k)})$ if $k>n$. Hence $\mbox{rank}\ {\mbox{${\mathbb C}$}}\otimes A_{n}(z) = |\Sigma_{n-\frac{1}{2}}^1|$ holds as far as $z$ takes any integer value $k> n$. This implies that $\psi$ is an isomorphism and we find that the generators and the relations in the theorem characterize the subalgebra $A_{n-\frac{1}{2}}(Q)$.
Bratteli diagram of the partition algebras {#sec:bra}
==========================================
In this section, we get back to the original definition of $A_{n-\frac{1}{2}}(Q)$. ([*i. e.*]{} $A_{n-\frac{1}{2}}(Q)$ is generated by $s_1, \ldots, s_{n-2}$, $f_1, \ldots, f_{n-1}$ and $e_1,\ldots, e_{n-1}$.) Since, $A_{n-\frac{1}{2}}(Q)$ contains all the generators of ${A}_{n-1}(Q)$, it becomes a subalgebra of ${A}_{n-\frac{1}{2}}(Q)$. Hence we obtain the sequence of inclusions $A_0(Q)\subset A_\frac{1}{2}(Q) \subset \cdots \subset A_{i-\frac{1}{2}}(Q)
\subset A_{i}(Q)\subset A_{i+\frac{1}{2}}
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|
also targets the virus through increasing the endosomal pH and hinders the glycosylation process of the cellular receptors of SARS-CoV-2, which eventually blocks the viral attachment to the ACE2 receptors and inhibits the viral infection. (4) Moreover, the hydroxychloroquine obstructs the MAP-kinase pathway which results to SARS-CoV-2 virus molecular crosstalk resulting into alteration of viral assembly and also intrude the proteolytic process of the M protein of the virus.](BMB-53-191-f2){#F2}
{#F3}
######
Details about the specimens/samples used for the COVID-19 detection
Specimen Form of sample collection Volume Required Collection Material Storage Temperature Disease Condition Test can be done
------------------------ --------------------------- ----------------- ------------------------------------------------ --------------------- --------------------------------------------------- ------------------
Nasopharyngeal Swab Frozen 0.8-1.4 ml Synthetic fibre swabs with plastic shafts 2-8°C Recommended for both symptomatic and a
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ght)}+\\
& \quad +P{\left(a_2=1,b_4=1\right)}+P{\left(a_2=2,b_8=0\right)}+P{\left(a_3=0,b_8=2\right)}+P{\left(a_3=1,b_7=1\right)}+\\
& \quad +P{\left(a_3=2,b_4=2\right)}+P{\left(a_4=0,b_1=1\right)}+P{\left(a_4=1,b_2=1\right)}+P{\left(a_4=2,b_3=2\right)}+\\
& \quad +P{\left(a_5=0,b_6=2\right)}+P{\left(a_5=1,b_2=0\right)}+P{\left(a_5=2,b_1=2\right)}+P{\left(a_6=0,b_7=2\right)}+\\
& \quad +P{\left(a_6=1,b_8=1\right)}+P{\left(a_6=2,b_5=0\right)}+P{\left(a_7=0,b_1=0\right)}+P{\left(a_7=1,b_3=1\right)}+\\
& \quad +P{\left(a_7=2,b_6=0\right)}+P{\left(a_8=0,b_2=2\right)}+P{\left(a_8=1,b_6=1\right)}+P{\left(a_8=2,b_3=0\right)}+\\
& \quad +P{\left(a_1=0,b_5=1\right)}+P{\left(a_1=1,b_7=2\right)}+P{\left(a_1=2,b_4=2\right)}
+P{\left(a_2=0,b_8=2\right)}+\\
& \quad +P{\left(a_2=1,b_5=0\right)}+P{\left(a_2=2,b_4=0\right)}+P{\left(a_3=0,b_7=0\right)}+P{\left(a_3=1,b_4=1\right)}+\\
& \quad +P{\left(a_3=2,b_8=1\right)}+P{\left(a_4=0,b_2=2\right)}+P{\left(a_4=1,b_3=1\right)}+P{\left(a_4=2,b_1=2\right)}+\\
& \quad +P{\left(a_5=0,b_2=1\right)}+P{\left(a_5=1,b_1=0\right)}+P{\left(a_5=2,b_6=1\right)}+P{\left(a_6=0,b_8=0\right)}+\\
& \quad +P{\left(a_6=1,b_5=2\right)}+P{\left(a_6=2,b_7=1\right)}+P{\left(a_7=0,b_3=0\right)}+P{\left(a_7=1,b_6=2\right)}+\\
& \quad +P{\left(a_7=2,b_1=1\right)}+P{\left(a_8=0,b_6=0\right)}+P{\left(a_8=1,b_3=2\right)}+P{\left(a_8=2,b_2=0\right)}
\end{split}\label{d}$$
Example II:
$$\begin{split}
& S_2\equiv P{\left(a_1=0,b_3=2\right)}+P{\left(a_1=1,b_2=0\right)}+P{\left(a_1=2,b_6=0\right)}
+P{\left(a_2=0,b_1=1\right)}+\\
& \quad +P{\left(a_2=1,b_3=0\right)}+P{\left(a_2=2,b_6=2\right)}+P{\left(a_3=0,b_2=1\right)}+P{\left(a_3=1,b_6=1\right)}+\\
& \quad +P{\left(a_3=2,b_1=0\right)}+P{\left(a_4=0,b_7=1\right)}+P{\left(a_4=1,b_5=2\right)}+P{\left(a_4=2,b_8=0\right)}+\\
& \quad +P{\left(a_5=0,b_7=0\right)}+P{\left(a_5=1,b_8=1\right)}+P{\left(a_5=2,b_4=1\right)}+P{\left(a_6=0,b_1=2\right)}+\\
& \quad +P{\left(a_6=1,b_3=1\right)}+P{\left(a_6=2,b_2=2\right)}+P{\left(a_7=0,b_5=0\right)}+P{\left(a_7=1,b_4=0\right)}+\\
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for all $\beta \in R^\chi _{+\infty }$. The set $$V'=\{\chi '\in V^\chi _{\underline{n}}\,|\,d(\chi ')\not=0 \}$$ is open in $V^\chi _{\underline{n}}$ and contains $\chi $. Thus by Prop. \[pr:X5dense\] the set $$V''=\{\chi '\in {\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}\,|
\,d(\chi ')\not=0\}$$ is Zariski dense in all irreducible components of $V^\chi
_{\underline{n}}$ containing $\chi $. The definition of $V^\chi _{\underline{n}}$ and the choice of $\underline{n}$ yield that $R^{\chi '}_+=R^\chi
_+$ and ${b^{\chi '}}(\beta )\le {b^{\chi}} (\beta )$ for all $\chi '\in
V^\chi _{\underline{n}}$ and $\beta \in R^\chi _+$. Thus $\dim U(\chi ')_{-{\alpha }}\le \dim U(\chi )_{-{\alpha }}$ for all $\chi '
\in V^\chi _{\underline{n}}$ by Eqs. , . Hence $d(\chi ')$ is a multiple of $\det ^{\chi '}_{\alpha }$ for all $\chi '\in V^\chi _{\underline{n}}$. By Thm. \[th:Shapdet\], $$\begin{aligned}
\label{eq:d}
d(\chi ')=a(\chi ')
\prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
({\rho ^{\chi}} (\beta )K_{\beta }
-\chi (\beta ,\beta )^t L_{\beta })
^{{P}^\chi ({\alpha },\beta ;t)}
\end{aligned}$$ for all $\chi '\in V''$, where $a(\cdot )$ is some regular function on ${\overline{{\mathcal{X}}}}$ which does not vanish on $V''$. By the density of $V''$, Eq. holds for all $\chi '\in V'$ in the irreducible components of $V^\chi _{\underline{n}}$ containing $\chi $, and $a(\chi ')\not=0$ for all $\chi '\in V'$ by definition of $V'$. In particular, Eq. holds for $\chi '=\chi $. Thus the theorem is proven.
Quantized enveloping algebras {#sec:Uqg}
=============================
We adapt our main result to quantized enveloping algebras.
Let $I$ be a finite set and let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix of finite type. Let ${\mathfrak{g}}$ be the associated semisimple Lie algebra and $R_+$ the set of positive roots. For all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Assume that the numbers $d
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j_{\bar{z}}^g (z)
\nonumber \\
& &
+ \frac{g}{4} \log |z-w|^2 (\partial_z j_{\bar{z}}^g(z) - \partial_{\bar z} j^g_z(z)))
\nonumber \\
& &
+ (-1)^{ac} : j_z^c j_{\bar z}^a :(z)
\nonumber \\
& &
+ ( {(A)^{ac}}_{gh} \frac{\bar z - \bar w}{z-w} :j^{g}_{\bar z}
j^{h}_{\bar z}: (z) - ( {(B)^{ac}}_{gh} \log |z-w|^2 :j^{g}_{z}
j^{h}_{\bar z}: (z) \nonumber \\
& & + ( {(C)^{ac}}_{gh} \frac{z-w}{\bar{z}-\bar{w}}
: j_z^{g} j_z^{h}:(z)))+... %\nonumber\end{aligned}$$ The second term we take into account comes from contracting the current with the second term in the Maurer-Cartan operator: $$\begin{aligned}
\mbox{Term 2} &=& j_{\bar z}^a (z) \cdot (-)c_+ \partial_{w} j_{\bar z}^c (w)
\nonumber \\
& \sim &- c_+ \partial_{w} (c_3 \kappa^{ac} \frac{1}{(\bar z - \bar w)^2}
\nonumber \\
& &
+ {f^{ac}}_g
(\frac{c_4}{\bar{z}-\bar{w}} j^g_{\bar z} (w) + \frac{(c_4-g)(z-w)}{(\bar z- \bar w)^2}
j^g_{z} (w)
\nonumber \\
& &
+ \frac{g}{4} \frac{z-w}{\bar z - \bar w}
(\partial_z j_{\bar{z}}^g(w) - \partial_{\bar z} j^g_z(w))
+\frac{c_4}{2} \partial_{\bar z} j^g_{\bar z}(w)
+ \frac{c_4-g}{2} \frac{(z-w)^2}{(\bar z - \bar w)^2} \partial_z j^g_z (w))
\nonumber \\
& &
+ : j_{\bar z}^a j_{\bar z}^c :(w)
\nonumber \\
& &
+ (- {(A)^{ac}}_{gh} \log |z-w|^2 :j^g_{\bar z}
j^h_{\bar z}:+ {(B)^{ac}}_{gh} \frac{z-w}{\bar{z}-\bar{w}} :j^{g}_{z}
j^{h}_{\bar z}:
\nonumber \\
& &
+ {(C)^{ac}}_{gh} \frac{(z-w)^2}{(\bar{z}-\bar{w})^2} : j_z^g j_z^h:(w)))+...
%\nonumber \end{aligned}$$ Furthermore we have the contractions with the composite piece of the Maurer-Cartan operator. Following appendix \[compositeOPEs\] we use a point-splitting procedure and write ${f^c}_{de} :j_z^e j_{\bar z}^d:(w) = \lim_{:x \to w:} {f^c}_{de} j_z^e(x) j_{\bar z}^d(w)$. Then we distinguish two terms. The simplest is the term where we contract the current component $j_{\bar z}^a$ with the part at $w$ of the split operator. We then still need to contract further while eliminating singularities as $x$ goes to $w$, but this is easily done: only regular terms survive.
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|
the pragmatics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$.
A$_{5}$. *Let a mapping* $\xi $* be given which interpretes the variable* $x$* in the rfs of* $\mathcal{L}_{Q}^{P}$* on a physical object in the state* $S$*. A proof that the rf* $E(x)$* is true (false) consists in performing one of the empirical procedures mentioned in Sec. 2.6 and showing that* $E\in \mathcal{E}_{S}$* (*$E\in \mathcal{E}_{S}^{\bot }$*).*
Assumption A$_{5}$ is obviously suggested by the intended interpretation discussed above. Taking into account A$_{1}$ and JR$_{1}$ in Sec. 3.1, it implies the following statement.
P.* Let* $E(x)$* be a rf of* $\mathcal{L}_{Q}^{P}$*, let* $\xi $* be an interpretation of the variable* $x$* on a physical object in the state* $S$*, and let* $S_{E}$* be defined as in Sec. 2.2. Then,*
$\pi _{\sigma (\xi )}(\vdash E(x))=J$* iff* $S\in $* *$S_{E}$*,*
$\pi _{\sigma (\xi )}(\vdash E(x))=U$* iff* $S\notin $* *$S_{E}$*.*
The above result specifies $\pi _{\sigma (\xi )}$ on the set of all elementary afs of $\mathcal{L}_{Q}^{P}$ and shows that it depends only on the state $S$. Hence, we write $\pi _{S}$ in place of $\pi _{\sigma (\xi )}$ in the following (for the sake of brevity, we also agree to use the intuitive statement “the physical object $x$ is in the state $S$” introduced in Sec. 2.1 in place of the more rigorous statement “the variable $x$ is interpreted on a physical object in the state $S$”).
Statement P provides the starting point for introducing a *set-theoretical pragmatics* for $\mathcal{L}_{Q}^{P}$, as follows.
Firstly, we introduce a mapping
$f:\delta \in \psi _{A}^{Q}\longrightarrow \mathcal{S}_{\delta }\in \mathcal{P(S)}$
which associates a *pragmatic extension* $\mathcal{S}_{\delta }$ with every assertive formula $\delta \in \psi _{A}^{Q}$, defined by the following recursive rules.
\(i) *For every* $E(x)\in \psi _{R}^{Q}$*,* $f(\vdash
E(x))=S_{\vdash E(x)}=S_{E}$*.*
\(ii) *For every* $\delta $* *$\in \psi _{A}^{Q}$*,
| 709
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|
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|
\mathbf{y}_*}$ is strictly diagonal.
Reference methods
-----------------
The GPR point estimates are compared with a state-of-the-art kNN algorithm. We select ten predictors from the (transformed) data using a simulated annealing -based optimization approach of [@Packalen2012] and use the most similar neighbor (MSN) method for selecting the neighbors. The number of neighbors is chosen to be $k=5$, as in [@packalen2009; @Packalen2012]. The predictor selection is done using the whole data set and leave-one-out cross-validation.
The prediction credible intervals provided by GPR are compared with the Bayesian inference approach [@varvia]. In the Bayesian approach the posterior predictive density: $$\label{alsextposterior}
\pi(\mathbf{y}_*|\mathbf{x}) \propto \begin{cases} \mathcal{N}(\mathbf{x}|\hat{\mathbf{A}}\boldsymbol{\phi}(\mathbf{y}_*)+\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}},\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}) &\\
\qquad\qquad\quad\;\cdot\:\mathcal{N}(\mathbf{y}_*|\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}},\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}), & \mathbf{y}_*\geq 0 \\
0, & \mathbf{y}_*< 0, \end{cases}$$ is constructed based on the training data and the new measurement. The model matrix $\hat{\mathbf{A}}$, conditional (residual) error statistics $\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}$, and the prior statistics $\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}$ are learned from the training data. The density is then sampled using a Markov chain Monte Carlo method. The point estimate and 95% credible intervals are then calculated from the samples.
{width="\textwidth"}
Performance assessment
----------------------
The proposed GPR method is first evaluated using leave-one-out cross-validation (i.e. $n_t=492$). From the results, relative root mean square error (RMSE%), relative bias (bias%), and credible interval coverage (CI%) are calculated. Credible interv
| 710
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|
^+(x,y),L_x\}$ is very similar to that of Figure \[satolevine\]). This means that $\pi_1(M)$ does **not** admit an epimorphism to $\mathbb{Z}\ast \mathbb{Z}$ since that would imply that $\{ L_x,L_y\}$ were a homology boundary link. But $\b^2(x,y)=-1$ precludes this by [@C2]. Nonetheless, further $c(yy...y,x)$ may be taken to be empty since $c^+(x,y)$ and $L_y$ form a boundary link in the complement of $L_x$. Thus $\b^n(y,x)= 0$ for all $n$, indicating, by Theorem \[linear\], that the first Betti numbers will grow linearly in the family of finite cyclic covers corresponding to the map $\pi_1(M)\to \mathbb{Z}$ that sends a meridian of $L_x$ to zero and a meridian of $L_y$ to one.
(123,73) (10,10)[![Example with linear growth in cyclic covers but no map to $\mathbb{Z}\ast\mathbb{Z}$[]{data-label="example3"}](example3.eps "fig:")]{} (72,39)[$c^+(x,y)$]{} (-5,39)[$L_y$]{} (117,17)[$L_x$]{} (65,0)[$M$]{} (36,46)[$0$]{} (119,68)[$0$]{}
\[Example2\] Consider the family of manifolds $M_k$, shown in Figure \[example2\] and Figure \[example2b\], obtained from $0$-framed surgery on a two component link.
(135,98) (10,10)[![Example with $\b^1(x,y)=0$,$\b^2(x,y)=-k$, $\b^2(y,x)= -1$[]{data-label="example2"}](example2.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (65,0)[$M$]{}
(135,98) (10,10)[![The circle $c(x,y)$[]{data-label="example2b"}](example2b.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (69,61)[$c(x,y)$]{} (65,0)[$M$]{}
If $V_x$ denotes the capped-off Seifert surface (obtained using Seifert’s algorithm) for the link component, $L_x$, on the right-hand side and $V_y$ denotes the capped-off Seifert surface for the link component, $L_y$, on the left-hand side, then the dashed circle in Figure \[example2b\] is $c(x,y)=V_x\cap V_y$. The circle $c(y,x)$ is merely this circle with opposite orientation. Since it lies on an untwisted band of $V_x$, $\b^1(x,y)=0=\b^1(y,x)$. Therefore the Lescop invariant of $M$ vanishes. But the link $\{c(x,y),L_x\}$ is the link of Figure \[satolevine\]
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| 0.774389
|
github_plus_top10pct_by_avg
|
by Fourier-transform of the slow-time phase fluctuations of individual comb lines $\tilde{A}_\mu(t)$, as $\delta\!f_\mu(t) = \frac{d}{dt} \arg(\tilde{A}_\mu(t))$.
As according to [@hendry_spontaneous_2018], the critical amplitude $F_C$ to which a soliton locks for this detuning ($\delta\omega>3\kappa$), based on pure intensity-based trapping, is close to the minimum amplitude required for DKS existence $|F_c|^2 \geq \frac{2\kappa^2\delta\omega}{\pi^2g\kappa_\mathrm{ex}}$.
---
abstract: 'We analize a possible explanation of the pulsar motions in terms of resonant neutrino transitions induced by a violation of the equivalence principle (VEP). Our approach, based on a parametrized post-Newtonian (PPN) expansion, shows that VEP effects give rise to highly directional contributions to the neutrino oscillation length. These terms induce anisotropies in the linear and angular momentum of the emitted neutrinos, which can account for both the observed translational and rotational pulsar motions. The violation needed to produce the actual motions is completely compatible with the existing bounds.'
address: |
$^{\dagger }$Instituto de Ciencias Nucleares, UNAM, Ap. Postal 70-543, 04510\
México DF, Mexico\
$^{*}$ Theoretical Physics, University of Oxford, 1 Keble Road, Oxford\
OX13NP, United Kingdom\
$^{\ddagger}$ Instituto Balseiro and CAB, Universidad Nacional de Cuyo and CNEA,\
8400 Bariloche, Argentina
author:
- 'M. Barkovich$^{\dagger }$, H. Casini$^{*}$, J.C. D’Olivo$^{\dagger }$, R.Montemayor$^{\ddagger }$'
title: Pulsar motions from neutrino oscillations induced by a violation of the equivalence principle
---
It is very difficult to obtain precise evidence on the characteristics of the gravitational interaction beyond the range where the Newtonian approximation holds. Only systems with very large densities of mass in rapid motion can provide suitable laboratories for such a phenomenology. One well known example is the orbital behavior of binary pulsars, which gives support to the produc
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In this section we will prove the following theorem.
### Main Theorem {#main-theorem .unnumbered}
There exists an integer $l_0$ such that for an arbitrary integer $l \ge l_0$, $n=2^l-2$ the Kervaire invariant given by the formula (1) is trivial. $$$$
### Proof of Main Theorem {#proof-of-main-theorem .unnumbered}
Take the integer $k$ from the equation $n-4k=62$. Consider the diagram (5). By the Retraction Theorem \[A2\], Section 8 there exists an integer $l_0$ such that for an arbitrary integer $l \ge l_0$ an arbitrary element $[(f,\Xi,\kappa)]$ in the 2-component of the cobordism group $Imm^{sf}(\frac{3n+q}{4},\frac{n-q}{4})$ admits a retraction of order $62$. By Theorem 2 in the cobordism class $\delta[(f,\Xi,\kappa)]$ there exists a $\D_4$-framed immersion $(g,\Psi,\eta)$ with an $\I_4$-structure.
Take the self-intersection manifold $L^{62}$ of $g$ and let $L_0^{10} \subset L^{62}$ be the submanifold dual to the cohomology class $\kappa_a^{28}\mu_a^{\ast}(\tau)^{12} \in H^{52}(L^{62};\Z/2)$. By a straightforward calculation the restriction of the normal bundle of $L^{62}$ to the submanifold $L_0^{10} \subset L^{62}$ is trivial and the normal bundle of $L_0^{10}$ is the Whitney sum $12 \kappa_a \oplus 12 \mu_a$, where $\kappa_a$ is the line $\Z/2$-bundle, $\mu_a$ is the plane $\Z/4$-bundle with the characteristic classes $\kappa_a$, $\mu_a^{ast}(\tau)$ described in the formula (8). By Lemma 6.1 (in the proof of this lemma we have to assume that the normal bundle of the manifold $L^{10}_0$ is as above) and by Lemma 7.1 \[A2\] the characteristic class (8) is trivial. The Main Theorem is proved.
$$$$ $$$$
Moscow Region, Troitsk, 142190, IZMIRAN.
pmakhmet@mi.ras.ru $$$$
[99]{}
P.M.Akhmet’ev, [*Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariants*]{}, a talk at the M.M.Postnikov Memorial Conference (2007)– “Algebraic Topology: Old and New” and at the Yu.P.Soloviev Memorial Conference (2005) “Topology, analysis and applications to mathematical physics” arXiv
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\end{aligned}$$
Where we use properties of $h$ from Lemma \[l:hproperties\].
The last claim follows immediately from Lemma \[l:hproperties\].4.
[Defining q]{} \[ss:defining-q\]
In this section, we define the function $q$ that is used in Lemma \[l:fproperties\]. Our construction is a slight modification to the original construction in [@eberle2011reflection].
Let $\aq$ and $\Rq$ be as defined in . We begin by defining auxiliary functions $\psi(r)$, $\Psi(r)$ and $\nu(r)$, all from $ \Re^+$ to $\Re$: $$\begin{aligned}
\label{d:psietal}
&\psi(r) := e^{- \aq \tau(r)}\,, \qquad
&\Psi(r) := \int_0^r \psi(s) ds\,, \qquad \nu(r) := 1- \frac{1}{2}
\frac{\int_0^{r}\frac{ \mu(s) \Psi(s)}{\psi(s)} ds}{\int_0^{4\Rq}\frac{ \mu(s) \Psi(s)}{\psi(s)}ds}\,,
\end{aligned}$$
Where $\tau(r)$ and $\mu(r)$ are as defined in Lemma \[l:tau\] and Lemma \[l:mu\] with $\R = \Rq$.
Finally we define $q$ as $$\label{d:f}
q(r) := \int_0^r \psi(s) \nu(s) ds.$$ We now state some useful properties of the distance function $q$.
\[l:qproperties\] The function $q$ defined in has the following properties.
1. For all $r\leq \Rq $, $q''(r) + \aq q'(r) \cdot r \leq - \frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(r)$ \[f:contraction\]
2. For all $r$, $\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\cdot r \leq q(r) \leq r$ \[f:q(r)\_bounds\]
3. For all $r$, $\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\leq q'(r) \leq 1$ \[f:q’(r)\_bounds\]
4. For all $r$, $q''(r) \leq 0$ and $\lrabs{q''(r)} \leq \lrp{\frac{5\aq\Rq}{4} + \frac{4}{\Rq}}$ \[f:q”(r)\_bounds\]
5. For all $r$, $\lrabs{q'''(r)} \leq 5\aq + 2\aq\lrp{\aq\Rq^2 + 1} + \frac{2(\aq\Rq^2 + 1)}{\Rq^2}$ \[f:q”’(r)\_bounds\]
**Proof of \[f:contraction\]** It can be verified that $$\begin{aligned}
\psi'(r)
=& \psi(r) (-\aq \tau'(r))\\
\psi''(r)
=& \psi(r) \lrp{\lrp{\aq \tau'(r)}^2 + \aq \tau''(r)}\\
\nu'(r)=& -\frac{1}{2} \frac{\frac{\mu(r)\Psi(r)}{\psi(r)}}{\int_0^{4\Rq} \fr
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\prod_{k=1}^{i-1}
\frac{(1-zT^k)}{(1-T^k)}.
\label{F}$$
By the Cauchy $q$-binomial theorem the sum equals $$\frac{1}{(1-w)}\prod_{n\geq
1}\frac{(1-wzT^n)}{(1-wT^n)}.$$Also$$\sum_\lambda
T^{|\lambda|}=\prod_{n\geq 1}(1-T^n)^{-1}.$$If we divide Formula (\[F\]) by this we finally get
$$1-(1-z)(1-w)\prod_{n\geq
1}(1-T^n)\sum_\lambda\phi_\lambda(z,w)T^{|\lambda|}=\prod_{n\geq
1}\frac{(1-wzT^n)(1-T^n)}{(1-zT^n)(1-wT^n)}.$$
Putting now $(z,w)=(q,1/q)$ we find that
$$1-(1-q)(1-1/q)\prod_{n\geq
1}(1-T^n)\sum_\lambda\phi_\lambda(q,1/q)T^{|\lambda|}=\prod_{n\geq
1}\frac{(1-T^n)^2}{(1-qT^n)(1-q^{-1}T^n)}.\label{phi}$$
From Formula (\[H-specializ\]) we have $\calH_\lambda(\sqrt{q},1/\sqrt{q})=1$ and so $$\begin{aligned}
&A_1\left(\sqrt{q},\frac{1}{\sqrt{q}};T\right)
=\sum_\lambda\phi_\lambda\left(q,\frac{1}{q}\right)T^{|\lambda|}\\
&A_0\left(\sqrt{q},\frac{1}{\sqrt{q}};T\right)=\sum_\lambda
T^{|\lambda|}=\prod_{n\geq 1}(1-T^n)^{-1}\end{aligned}$$ Hence, under the specialization $(z,w)\mapsto(\sqrt{q},1/\sqrt{q})$ , the left hand side of Formula (\[comb\]) agrees with the left hand side of Formula (\[phi\]).
Finally, it is straightforward to see that if we put $(z,w)=(\sqrt{q},1/\sqrt{q})$, then the right hand side of Formula (\[comb\]) agrees with the right hand side of Formula (\[phi\]), hence the theorem.
Connection with modular forms
-----------------------------
For a positive, even integer $k$ let $G_k$ be the standard Eisenstein series for $SL_2(\Z)$ $$\label{Eisenstein-defn}
G_k(T)=\frac{-B_k}{2k}+\sum_{n\geq 1}\sum_{d\,|\, n} d^{k-1}T^n,$$ where $B_k$ is the $k$-th Bernoulli number.
For $k>2$ the $G_k$’s are modular forms of weight $k$; i.e., they are holomorphic (including at infinity) and satisfy $$\label{quasi-mod-2}
\begin{split}
&G_k \left( \frac{a\tau +b}{c\tau +d}\right)
= (c\tau +d)^k G_k (\tau)\\
\noalign{\vskip6pt}
& \text{for }\ \begin{pmatrix}a&b\\ c&d\end{pmatrix} \in SL_2(\Z)\ ,
\quad T = e^{2\pi i\tau}\ ,\quad \Im \tau >0\ .
\end{split}$$ For $k=2$ we have a similar transfo
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`gi 87162179 ref` `415` `AKSEVWRQMMSD`
`AKSEVWRQM+SD`
`AKSEVWRQMISD`
`SRR022865_51952` `-36` `gaaggtccaatg`
`cagatggattca`
2638027 NONSYN A:5 C:112 C:36 `gataagtagtat` gluconate kinase
`gi 87161981 ref` `47` `DIAVVDIMMDGM`
`DIAVVDIMMD M`
`DIAVVDIMMDVM`
`SRR022865_55616` `-36` `gaggggaaagga`
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and $\Phi$. Together, cases A and B show that $\prod\Theta \not= \prod\Phi$ implies $\Theta \not= \Phi$. Applying the contrapositive principle[^12] gives $\prod\Theta = \prod\Phi$ if $\Theta = \Phi$.
\[T:SPACE\_UNIQ\_ENSEMBLE\] Any choice space has one unique generating ensemble: let $\Theta$ and $\Phi$ be two ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$ respectively. If $\prod\Theta = \prod\Phi$, then $\Theta = \Phi$.
To show the contrapositive, suppose $\Theta \not= \Phi$. From this premise there must exist either A: $(i,P) \in \Theta$ such that $(i,P) \notin \Phi$, or B: $(j,Q) \in \Phi$ such that $(j,Q) \notin \Theta$.
The first case A decomposes into two sub-cases: either A1: $i \in S$ and $i \in S'$, or A2: $i \in S$ and $i \notin S'$.
In sub-case A1, we must have $\Theta(i) \not= \Phi(i)$ to support $\Theta \not= \Phi$. For this there must be either A1a: there is $u \in \Theta(i)$ such that $u \notin \Phi(i)$, or A1b: there is $v \in \Phi(i)$ such that $v \notin \Theta(i)$.
In sub-sub-case A1a, there must exist by definitions \[D:CHOICE\_SPACE\] and \[D:CHOICE\], a choice $\alpha \in \prod\Theta$ such that $\alpha(i) = u$. However, there can be no choice $\beta \in \prod\Phi$ such that $\beta(i) = u$, since $u \notin \Phi(i)$. Therefore $\alpha \notin \prod\Phi$, with the consequence that $\prod\Theta \not= \prod\Phi$.
In sub-sub-case A1b there is $v \in \Phi(i)$ such that $v \notin \Theta(i)$. Similarly to A1a, there exists $\beta \in \prod\Phi$ such that $\beta(i) = v$, but no $\alpha \in \prod\Theta$ such that $\alpha(i) = v$, again concluding that $\prod\Theta \not= \prod\Phi$.
In sub-case A2, we have $i \in {{\operatorname{dom}{\Theta}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$. By definitions \[D:CHOICE\_SPACE\] and \[D:CHOICE\], for any $x \in \Theta(i)$, there exists choice $\alpha_x \in \prod\Theta$ such that $\alpha_x(i) = x$. However, there is no choice $\beta \in \prod\Phi$ such that $\beta(i) = x$ because $i \notin {{\operatorname{dom}{\Phi}}} = {{\operatorname{
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\pm)}(k,r) =
\bar{N}_{2}
\biggl( 1 \mp \displaystyle\frac{iW(r)}{k\alpha} \biggr)
e^{\pm ikr}
\label{eq.3.2.1.3}$$ and $$\bar{N}_{2} = i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}}
\label{eq.3.2.1.4}$$
In accordance with main statements of quantum mechanics, for applying such form of the radial wave function to the description of scattering of the particle in the field of the potential $V_{2}(r)$, it needs to achieve a boundary requirement $\chi_{l=0}^{(2)}(k,r) \to 0$ at $r \to 0$, which gives a finiteness of the wave function (\[eq.2.1.1\]) at $r=0$ (and $S_{l=0}^{(2)}$ must have finite values and be not zero). One can see from (\[eq.3.2.1.2\]), that it is fulfilled only in case ($W(r)$ is real): $$\begin{array}{lcl}
Re (S_{l=0}^{(2)}) =
\displaystyle\frac{k^{2}\alpha^{2}-W^{2}(0)}
{k^{2}\alpha^{2}+W^{2}(0)}, &
Im (S_{l=0}^{(2)}) =
\displaystyle\frac{2W(0)k\alpha}{k^{2}\alpha^{2}+W^{2}(0)},
\end{array}
\label{eq.3.2.1.5}$$ where $$W(0) =
-\displaystyle\frac{\alpha}{r_{0} + \displaystyle\frac{1}{C\alpha}}.
\label{eq.3.2.1.6}$$ For the partial components of the S-matrix the following property $|S_{l=0}|^{2} = 1$ is fulfilled also. In limit $r \to 0$ we obtain the following expression for the radial wave function: $$\chi_{l=0}^{(2)}(k,r) =
\bar{N}_{2} \biggl(1 + \displaystyle\frac{iW(0)}{k\alpha}\biggr)
\biggl(e^{-ikr} - S_{l=0}^{(2)}
\displaystyle\frac{k\alpha - iW(0)}{k\alpha + iW(0)}
e^{ikr}\biggl),
\label{eq.3.2.1.7}$$ which in its form coincides with Exp. (\[eq.3.2.1.1\]) for the wave function for the potential $V_{1}(r)$ from (\[eq.3.1.4\]) with zero value.
In Fig. \[fig.3\] real and imaginary parts of the wave function near to the point $r=0$ are shown (here the starting formulas (\[eq.3.2.1.2\])–(\[eq.3.2.1.3\]) are taken). From Fig. \[fig.3\] (a, b) one can see a deformation of the imaginary part of this wave function with change of the wave vector $k$ and the parameter $C$. The real part of the wave function in its behavior looks lik
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|
${abs}(|\mathcal{C}_{i1}| - |\mathcal{C}_{i2}|) \leq 1$ [@dong2005ensembles]. This has been shown empirically to have little effect on the accuracy in most cases, while reducing the time taken to train nested dichotomies. Balanced selection has greater benefits for problems with many classes.
It is clear that the sample space of random nested dichotomies is larger than that of class balanced nested dichotomies, but it is still large enough to ensure sufficient ensemble diversity. The growth function for class balanced nested dichotomies is given by $$\begin{aligned}
T_{CB}(n) =
\begin{cases}
\frac{1}{2} \binom{n}{n/2} T_{CB}(\frac{n}{2}) T_{CB}(\frac{n}{2}), & \text{if } n \text{ is even} \\
\binom {n}{(n+1)/2} T_{CB}(\frac{n+1}{2}) T_{CB}(\frac{n-1}{2}), & \text{if } n \text{ is odd} \\
\end{cases}\end{aligned}$$
where $T_{CB}(2) = T_{CB}(1) = 1$ [@dong2005ensembles].
Dong *et. al.* also explored a form of balancing where the amount of data in each subset is roughly equal, which gave similar results for datasets with unbalanced classes [@dong2005ensembles].
Random-Pair Selection
---------------------
Random-pair selection provides a non-deterministic method of creating $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ that groups similar classes together [@leathart2016building]. In random-pair selection, the base classifier is used directly to identify similar classes in $\mathcal{C}_i$. First, a random pair of classes $c_1, c_2 \in \mathcal{C}_i$ is selected, and a binary classifier is trained on just these two classes. Then, the remaining classes are classified with this classifier, and its predictions are stored as a confusion matrix $M$. $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ are constructed by $$\begin{aligned}
\mathcal{C}_{i1} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} \leq M_{c, c_2} \} \cup \{c_1\} \\
\mathcal{C}_{i2} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} > M_{c, c_2} \} \cup \{c_2\}\end{aligned}$$ where $M_{i,j}$ is de
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| 0.810185
|
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|
s a unique strong solution $\Phi$ satisfying the initial and boundary values. Then $\phi(x,\omega,E):=\Phi(x,\omega,E_{\rm m}-E)$ is the solution of $(I+P_0)\phi=f$. This completes the proof of (\[inf4\]). The inequality (\[inf5\]) can be shown similarly as above and so the conclusion follows.
\[md-gene\] The previous theorem has the following generalization which can be applied for more general transport problems. Let $$P(x,\omega,E,D)\phi= -S_0{{\frac{\partial \phi}{\partial E}}}+F_1\cdot\nabla_x\phi
+F_2\cdot\nabla_{\tilde\omega}\phi+a\phi$$ be the first order partial differential operator with coefficients $S_0,\ F_1,\ F_2,\ a\in C^1(\ol G\times S\times I)$. Assume that $\partial G$ is in the class $C^2$ and that $\Gamma=\Gamma_{+}\cup\Gamma_{-}\cup\Gamma_{0}$ where $\Gamma_{0}$ has the zero surface measure. In addition, we assume that $\Gamma_{\pm}$ are open in $\partial G\times S\times I^\circ$. Finally, suppose that $$\begin{aligned}
& {{\frac{\partial S_0}{\partial E}}}-{\rm div}_x(F_1)-{\rm div}_{\omega}(F_2)+2a\geq 0, \label{flp1-aa} \\[2mm]
& \inf_{(x,\omega,E)\in G\times S\times I}S_0(x,\omega,E)>0, \label{flp2-aa}\end{aligned}$$ and that \[flp3-aa\] F\_10 [on]{} \_[+]{}. Then \[inf4-aa\] R( I+P\_0)=L\^2(GSI) and \[inf5-aa\] P\_0,\_[L\_2(GSI)]{}0,D(P\_0).
\[dis-re\] In certain cases, the $m$-dissipativity of $-\tilde P_0$ -like operator can even be proved by using explicit formulas for the solution. We only sketch the idea here.
Suppose that $S_0=S_0(E)$ is independent of $x$ and that $a=a(x,\omega)\in C^1(\ol G\times S)$ is independent of $E$. Let $$P(x,\omega,E,D)\phi=-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x \phi+a\phi.$$ Assume that ${ f}\in C_0^\infty(G\times S\times I^\circ)$. Then by the formula (\[comp17:1\]) below, the solution of the problem \[S0const\] P(x,,E,D)=[f]{},\_[|\_-]{}=0,(,,E\_[m]{})=0 is given by \[comp17:1-a\] (x,,E) = [1]{}( \_0\^[r(x,,E)]{} e\^[-\_0\^s a(x-,)d]{}(x-s,,R(E)+s) ds), where $$r(x,\omega,E):={}&\min\{R(E_m)-R(E),t(x,\omega)\}, \\[2mm]
R(E)
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|
pha)=\alpha l$ induces a bijection between the set $$\{ \alpha \in [K^{x}\backslash L /Q]\ ; x\alpha y \in H\},$$ and the set $$\{w\in [K^{x}\backslash L\cap H^{x}/ Q^{l}]\}.$$
- Let $\alpha\in L$ such that $x\alpha y \in H$. Since $y=lx^{-1}h$ we have: $$\begin{aligned}
x\alpha y \in H &\Leftrightarrow x\alpha l x^{-1} h \in H\\
&\Leftrightarrow \alpha l \in H^{x},\end{aligned}$$ so $\alpha l \in L\cap H^{x}$.
- The map $f$ is well defined: if $\alpha$ and $\alpha'$ are in the same double coset, there are $k\in J$ and $q\in Q$ such that $\alpha'=x^{-1}kx\alpha q$, and $$f(\alpha')=x^{-1}kx\alpha q l = x^{-1}kx\alpha l l^{-1}q l,$$ so $f(\alpha)$ and $f(\alpha')$ are in the same double coset.
- The map $f$ is injective: if $f(\alpha)=f(\alpha')$ then there are $k\in K$ and $q\in Q$ such that $\alpha l = x^{-1}kx \alpha' l l^{-1}q l = x^{-1}kx \alpha' q l$, so $\alpha$ and $\alpha'$ are in the same double coset.
- The map $f$ is surjective: let $w\in L\cap H^{x}$, then $wl^{-1}\in L$ and $f(wl^{-1}) = w$.
So, we have: $$\begin{aligned}
tr_{\phi_{G}}(t^{H}_{K}xR^{L}_{K^{x}}t^{L}_{Q}yR^{H}_{Q^{y}})&=\sum_{\alpha\in[K^{x}\backslash L /Q]} \delta_{x\alpha y, H} tr_{\phi_{G}}\big(G/(K\cap \ ^{x\alpha}Q)\big)\\
&=\sum_{w\in [K^{x}\backslash L\cap H^{x}/Q^{l}]} \phi_{G}(G/K\cap\ ^{xw}(Q^{l}))\\
&=\sum_{w\in [K^{x}\backslash L\cap H^{x}/Q^{l}]} \phi_{G}(G/K^{x}\cap\ ^{w}(Q^{l}))\\
&=\sum_{w\in [K^{x}\backslash L\cap H^{x}/Q^{l}]} \phi_{G}(Ind_{L\cap H^{x}}^{G}(L\cap H^{x}/K^{x}\cap\ ^{w}(Q^{l})))\\
&=\sum_{w\in [K^{x}\backslash L\cap H^{x}/Q^{l}]}\phi_{L\cap H^{x}}(L\cap H^{x}/K^{x}\cap\ ^{w}(Q^{l})). \end{aligned}$$ Let $\Theta= L\cap\ ^{x}H$. The basis elements which appear for the block $Bl_{\Theta,\Theta,1,1}$ of the matrix of $\phi_{\Theta}$ are the $t^{\Theta}_{A}r^{\Theta}_{A}$ for $A\leqslant \Theta$ up to conjugacy. Let $A$ and $B$ be subgroups of $\Theta$, the entry corresponding to $t^{\Theta}_{A}r^{\Theta}_{A}$ and $t^{\Theta}_{B}r^{\Theta}_{B}$ is: $$\sum_{w\in [A\backslash \Theta / B]} \
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orse function is a Morse function such that the following hold ([@kitazawa]).
1. At distinct singular points the values are distinct.
2. Inverse images of regular values are disjoint unions of standard spheres.
3. A vertex of the Reeb graph such that the inverse image includes a singular point not giving a local extremum is a vertex of degree $3$.
We consider pseudo quotient maps of the class of such functions. We regard these functions as $C^{\infty}$ functions whose measure zero sets are empty.
We present local forms of these pseudo quotient maps with several inverse images in FIGURE \[fig:5\].
![Local forms of the pseudo quotient maps with several inverse images.[]{data-label="fig:5"}](sphelocal.eps){width="30mm"}
We investigate the local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image. Consider a small regular neighborhood of the just one non-manifold point in the inverse image of the vertex. See FIGURE \[fig:6\].
![The local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image (a regular neighborhood of the just one non-manifold point in the inverse image of the vertex).[]{data-label="fig:6"}](localspheind.eps){width="15mm"}
If we remove the inverse image of the vertex, then the resulting space has $4$ connected components: two of them are in the upper part and the others are in the lower part. Moreover, the former two components are mapped onto an interval and the latter two ones are mapped onto the disjoint union of two intervals in the graph.
This leads us to the fact that a pseudo quotient map whose target is as FIGURE \[fig:7\] cannot be realized as a quotient map of the class. Arrows indicate natural local orientations induced from canonically obtained local functions. Note that around a vertex of degree $3$, the local map cannot be regared as a $D_3$-symmetric map.
![The graph represents the target of a pseudo quotient map of the class of standard-spherical functio
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{} (232,30)[(b)]{} (395,30)[(c)]{}
Target FBP Matérn
--------------- ----- -------- -- -- --
Carved cheese 0.1 12604
: Computation times of the carved cheese (in seconds)[]{data-label="Computation time cheese"}
Discussion
----------
We have presented x-ray tomography reconstructions from both simulated and real data for limited projections (i.e. sparse sampling) using an approach based on the Gaussian process. However, other limited-data problems such as limited angle tomography could be explored as well. The quality of GP reconstructions using different covariance functions looks rather the same qualitatively. However, quantitatively, the reconstruction using Matérn covariance is the best one: it has the lowest RE $23.26\%$ and the highest PSNR $22.76$. PSNR describes the similarity of the original target with the reconstructioned image (the higher value, the better of the reconstruction). Figures of merit estimates are not available for the real cheese data since there is no comparable ground truth. Nevertheless, the quality of the reconstruction can be observed qualitatively by comparing with the FBP reconstruction obtained with dense $360$ projections from $360$ degrees shown in Figure \[CheeseRec\](a). The corresponding parameter estimates for the chest phantom and the cheese are reported in Table \[GP parameters\] and \[GP parameter cheese\]. For the chest phantom case, the estimate of parameter $\sigma$ using Matérn, Laplacian and Tikhonov kernels tend to be close to the true value $\sigma_{true}$. As for the SE covariance, the standard deviation of noise is overestimated.
The reconstructions produced by the FBP benchmark algorithm using sparse projections are overwhelmed by streak artefacts due to the nature of backprojection reconstruction, as shown in Figure \[fig:ChestPhantomRec\](b) for the chest phantom and Figure \[CheeseRec\](b) the for cheese target. The edges of the target are badly reconstructed. Due to the artefacts, especially for the chest
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-\lambda(t-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds - \int_0^t e^{-\lambda(t-s)} G_s dA_s.
\end{aligned}$$ By taking derivatives, we see that $$\begin{aligned}
d\mathcal{L}_t
\leq& -\lambda f(z_t) dt + \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2}dt + G_t dA_t\\
&\qquad + \lambda \lrp{\int_0^t e^{-\lambda(t-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds} dt - \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2} dt\\
&\qquad + \lambda \lrp{\int_0^t e^{-\lambda(t-s)} G_s dA_s} dt - G_t dA_t\\
=& -\lambda \mathcal{L}_t dt
\end{aligned}$$
We can then apply Gronwall’s Lemma to $\mathcal{L}_t$, so that $$\begin{aligned}
\mathcal{L}_T \leq e^{-\lambda T} \mathcal{L}_0,
\end{aligned}$$ which is equivalent to $$\begin{aligned}
f(z_T) - \int_0^T e^{-\lambda(T-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds - \int_0^T e^{-\lambda(t-s)} G_s dA_s \leq e^{-\lambda T} f(z_0).
\end{aligned}$$ Observe that $G_s$ is measurable wrt the natural filtration generated by $A_s$, so that $\int_0^T e^{-\lambda (T-s)} G_s dA_s$ is a martingale. Thus taking expectations, $$\begin{aligned}
\E{f(z_T)} \leq e^{-\lambda T} \E{f(z_0)} + \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} + L \E{\lrn{y_s - y_0}_2} ds
\end{aligned}$$ By Lemma \[l:divergence\_yt\], $\E{\lrn{y_t - y_0}_2^2} \leq t^2 L^2 \E{\lrn{y_0}_2^2} + t\beta^2$, so that $$\begin{aligned}
& \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} ds \leq \frac{T^3 L_N^2 L^2}{\epsilon}\E{\lrn{y_0}_2^2} + \frac{T^2L_N^2}{\epsilon}\beta^2\\
& L \E{\lrn{y_s - y_0}_2} \leq T^2 L^2 \sqrt{\E{\lrn{y_0}_2^2}} + T^{3/2} L \beta
\end{aligned}$$ Furthermore, using our assumption in the Lemma statement that $T \leq \min\lrbb{\frac{\epsilon^2}{\beta^2}, \frac{\epsilon}{{6L \sqrt{R^2 + \beta^2/m}}}
| 724
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|
TNM (stage) Duration of follow-up after diagnosis (months) Site of metastasis Site of tomotherapy
--------- ----- ----- -------------------- -------------------- ----------------------------------------- ------------- ------------------------------------------------ -------------------- ---------------------
1 F 53 Head No No T4N1M0(IVA) 2.9 Pancreas
2 M 61 Body Yes Gemcitabine \#6, Cisplatin/Capecitabine T3N0M0 (II) 4.9 Pancreas
3 M 67 Tail No No T4N1M0(IVA) 1.5 Pancreas
4 F 76 Body No Gemcitabine \#5 T4N1M0(IVA) 7.6 Pancreas
5 M 57 Body No Gemcitabine/Capecitabine T4N1M1(IVB) 1.2 Liver Pancreas
6 F 64 Body, tail No No T4N1M0(IVA) 0.2 Pancreas
7 M 67 Body No No T4N1M0(IVA) 1 Pancreas
8 F 71 Body No Gemcitabine/Cisplatin \#3 T3N1M1(IVB) 8 Liver Pancreas
9 M 46 Body, tail No No T4N1M0(IVA
| 725
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es labeled by $A,B$ where $A,B\in\{T,\Phi,R\}$. They are defined by $$\begin{aligned}
\mathbf{W}_{TT}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 1}=0}\,, \\ {\nonumber}\mathbf{W}_{\Phi\Phi}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 2}=0}\,, \\ {\nonumber}\mathbf{W}_{RR}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 3}=0}\,, \\ {\nonumber}\mathbf{W}_{T\Phi}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 4}=0}\,, \\ {\nonumber}\mathbf{W}_{\Phi R}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 5}=0}\,, \\ {\nonumber}\mathbf{W}_{RT}^{(m\,h\,k)}&=\mathbf{W}^{(m\,h\,k)}\big|_{c_{\beta\neq 6}=0}\,.
\label{eq:specific-choice-of-tensor}\end{aligned}$$ This specific choice of tensor bases will be utilized to write the metric perturbation as in Eq. .
Basis functions in global coordinates {#app:global-basis}
-------------------------------------
### Scalar bases (highest-weight module) {#app:scalar-basis-highest-reps}
The scalar bases from the highest-weight module in global coordinates are given by $$F^{(m\,h\,k)} \propto (\sin\psi)^{-h} e^{i [(h-k) \tau + m \varphi] +m \psi} \times f^{(m\,h\,k)}\,,$$ where $$\begin{aligned}
f^{(m\,h\,0)} &=1 \,, \\ {\nonumber}f^{(m\,h\,1)} &=-2 (m \sin \psi -h \cos \psi ) \,, \\ {\nonumber}f^{(m\,h\,2)} &= 2 \left[h^2+m^2 +\left(h^2-h-m^2\right) \cos 2 \psi +(m-2 h m) \sin 2 \psi \right]\,.\end{aligned}$$
### Scalar bases (lowest-weight module) {#app:scalar-basis-lowest-reps}
The scalar bases from the lowest-weight module in global coordinates are given by $$F_L^{(m\,h\,k)} \propto (\sin\psi)^{+h} e^{i [(h+k) \tau + m \varphi] - m \psi} \times f_L^{(m\,h\,k)}\,,$$ where $$\begin{aligned}
f_L^{(m\,h\,0)} &=1 \,, \\ \nonumber
f_L^{(m\,h\,1)} &= -2 (m \sin \psi -h \cos \psi ), \\ \nonumber
f_L^{(m\,h\,2)} &= 2 \left[h^2+m^2 +\left(h^2+h-m^2\right) \cos 2 \psi -(m+2 h m) \sin 2 \psi \right].\end{aligned}$$
### Vector bases {#app:vector-basis-global}
The covector bases in global coordinates can be decomposed using the dual basis o
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loss of generality, we assume that $I_{n-k+1}$ corresponds to the first index (up to a permutation). We can rewrite ($\mathcal{P}_1$) (Proposition \[prop4\]) as:
$\underset{(1,v) \in \mathcal{V}_{1,k}}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $
Constraints in ($\mathcal{P}_1$) can be moved to the following equality contraints:
\[prop5\] Let $w$ be the first column of $V_{2,n}^T$. We define $W$ as the matrix $V_{2,n}^T$ whose first column $w$ has been deleted.
The minimization problem ($\tilde{\mathcal{P}}_1$)
$\underset{W v=-w}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $
with $w=V_{2,n}^{(1)}$ and $W=V_{2,n}^{(n-1)}$ has a unique solution $\tilde{v}$ equals to $\textbf{1}_{C_{n-k+1}}$ such that $\tilde{v}=(1,v)$.
We recall that $V_{2,n}$ is the restriction of the eigenvectors matrix to the $n-k$ first columns. Because the columns of this matrix form an orthogonal basis, $v\in \mathcal{V}_{1,k}$ is equivalent to $V_{2,n}^T v=0$. Thus, $\tilde{v}=(1,v)$ satisfies the equation: $V_{2,n}^{(1)} + V_{2,n}^{(n-k-1)} v$, where $V_{2,n}^{(1)}$ is the first row of $V_{2,n}$ and $V_{2,n}^{(n-k-1)}$ the matrix $V_{2,n}$ whose first row has been deleted.
For all $ \tilde{v}=(1,v), $ $$\begin{aligned}
V_{2,n}^T\tilde{v} & =V_{2,n}^{(1)} + V_{2,n}^{(n-k-1)} v\\
&=0 \\
\Leftrightarrow \qquad V_{2,n}^{(n-k-1)} v &= -V_{2,n}^{(1)}\end{aligned}$$
Note that in Proposition \[prop5\], $V_{2,n}^{(1)}$ and $V_{2,n}^{(n-k-1)}$ are denoted $w$ and $W$.
**Remark:** Constraint problem ($\tilde{\mathcal{P}}_1$) (Proposition \[prop5\]) can be equivalently written as the following penalized problem:
$\underset{v \in \mathbb{R}^{n-1}}{\arg\min} \ {\|W v+w\|}_2^2 + \lambda {\| v \|}_1$.
where $\lambda >0$ is the regularization parameter that controls the balance between the constraint and the sparsity norm, $W \in \mathbb{M}_{n-k,n-1}$ is the matrix $V_{2,n}^T$ whose first column $w$ has been deleted.
In the following, we will provide an algorithm based on the contraint problem ($\tilde{\mathcal{P}}
| 727
| 2,955
| 1,002
| 709
| 3,464
| 0.772173
|
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|
1$ and $$\label{eq:s13}
H^a_{\rm eff}=H-i\lambda_W VV^\dag\,.$$ In this case the coupling is purely imaginary. For the two cases, where the antenna is terminated by a hard wall or an open reflecting end, we may assume $\alpha=0$, resulting in $\lambda_T=\tanh(i\varphi/2)=i\tan\varphi/2$, and $$\label{eq:s14}
H^a_{\rm eff}=H+\tan\left(\frac{\varphi}{2}\right)\lambda_W VV^\dag\,.$$ In this case the coupling is purely real, and the antenna does not correspond any longer to an open channel but to a scattering center only. This is true, as long as the absorption in the antenna can really be neglected. This becomes questionable, as soon as $\varphi$ approaches $\pi$, corresponding to the excitation of a resonance within the antenna. For this singular situation the perturbative treatment of the antenna coupling applied in the derivation looses its justification.
The value of $\varphi$ depends on the length of the antenna in units of the wave length and thus on frequency. But independently of frequency the difference of the phase shift $\varphi$ for the reflection at the open end (oe) and the hard wall (hw), respectively, is always $\pi$. A phase difference of $\pi$ means a replacement of the tangent by the cotangent in Eq. (\[eq:s14\]), i.e. the coupling constants $\lambda_T$ for the two situations are related via $$\lambda_{T,\rm hw}=1/\lambda_{T,\rm oe}$$ With the above introduced total coupling constant $\lambda=\lambda_T\lambda_W$, this may be alternatively be written as $$\label{eq:s15}
\lambda_{\rm hw}\lambda_{\rm oe}=\lambda_W^2=\lambda_{50\Omega}^2$$ since $\lambda_W$ is the coupling constant for the 50$\Omega$ load, see Eq. (\[eq:s13\]). $\lambda_{\rm hw}$ and $\lambda_{\rm oe}$ denote the total coupling constants for the hard-wall and the open-end reflections. These relations allow for explicit tests of the theory.
Results and Discussion {#sec:results}
======================
In this section we want to discuss the experimental and theoretical results for the coupling fidelity decay under the p
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| 1,998
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| 0.780411
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|
-------------
All 58 animals were continuously monitored over 6 months to exclude postoperative patellar luxations, deep wound infection, or empyema.
### Patellar luxation
The animals were examined daily over the first 5 weeks and weekly for the remaining observation period by adspection for clinical signs of patellar luxation such as abnormalities in hindleg carriage, stance, or lameness. Additionally, with the sheep in a sitting position, thorough palpation of the joints was conducted and they were put through a range of motion. Digital pressure was applied to the medial border of the patella to test for its stability and exclude luxation. The different grades of patellar luxation in sheep are given in Table [1](#T1){ref-type="table"}. X-ray was performed only in the case of uncertainty upon clinical examination.
######
Grades of patellar luxation in sheep
**Grade** **Patella at examination** **Luxation** **Reposition** **Reluxation** **Lameness** **Bone deformities**
----------- --------------------------------------------- ------------------------------------------------------ -------------------------------------------------------- ------------------- ------------------------------ ----------------------
I reduced manual by digital pressure spontaneous rare mild seldom
II reduced in extension and luxated in flexion manual by digital pressure or spontaneous in flexion manual by digital pressure or spontaneous in extension upon manipulation resolvable skipping lameness sometimes
III luxated spontaneous manual by digital pressure frequent severe
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|
hedged and leveraged, and nearly free of systematic market fluctuations. Equations (\[439\])-(\[442\]) provide approximate expressions valid to first order in $1/N$ for the properties of these portfolios.*]{}
3. Single-Index Model with Constant Residual Variance {#single-index-model-with-constant-residual-variance .unnumbered}
=====================================================
To provide an explicit illustration of the principal portfolio structure within the single-index model described in the preceding section, we now turn to an exactly solvable, albeit oversimplified, version of that model. This model is defined by the assumption that the residual variance of the $i$th asset in the original set, ${\bar{{\alpha}_{i}^{2}}}$, is equal to ${\bar{{\alpha}^{2}}}$ for all assets. Observe that this assumption does not affect the expected rate of return for the $i$th asset, which is given by ${r}_{i}={\bar{{\alpha}_{i}}}+{\beta}_{i} {\bar{\rho}}_{mkt}$ as before. This simplification will allow us to derive an exact solution for the model and illustrate the concepts and methods of the previous section in more explicit terms. The price for this simplification is of course the unrealistic assumption of constant residual variance which defines the model.
The covariance matrix with the above simplification appears as $${\sf \sigma}_{ij}^{crv}={\bar{{\alpha}^{2}}} {\delta}_{ij}+
{\beta}_{i}{\beta}_{j} {\bar{{\rho}^{2}}}_{mkt}, \label{443}$$ whose rescaled version is $${\tilde{\sf \sigma}}_{ij}^{crv}={\gamma}^{2} {\delta}_{ij}+
{\hat{\beta}}_{i}{\hat{\beta}}_{j} \label{444}$$ These equations are of course specialized versions of Eqs. (\[432\]) and (\[433\]).
Referring to the results of the previous section, one can readily see that the spectrum of ${\tilde{\sf \sigma}}^{crv}$ consists of a major eigenvalue (exactly) equal to $1+{\gamma}^{2}$ \[cf. Eq. (\[437\])\], and $N-1$ minor eigenvalues, all equal to ${\gamma}^{2}$. Recall that these eigenvalues respectively correspond to the market-aligned and market-orthogo
| 730
| 1,125
| 1,459
| 803
| 3,063
| 0.775093
|
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|
---C50 1.389 (7)
P2---C63 1.835 (4) C49---H49 0.9500
P3---C57 1.820 (5) C50---H50 0.9500
P3---C51 1.831 (5) C51---C56 1.380 (7)
P3---C63 1.836 (4) C51---C52 1.387 (7)
P4---C39 1.823 (5) C52---C53 1.390 (8)
P4---C45 1.830 (5) C52---H52 0.9500
P4---C38 1.848 (4) C53---C54 1.369 (9)
P5---C32 1.826 (4) C53---H53 0.9500
P5---C26 1.832 (4) C54---C55 1.381 (8)
P5---C38 1.847 (4) C54---H54 0.9500
P6---C14 1.820 (5) C55---C56 1.394 (7)
P6---C20 1.828 (5) C55---H55 0.9500
P6---C13 1.844 (4) C56---H56 0.9500
C1---C6 1.387 (7) C57---C62 1.395 (6)
C1---C2 1.399 (7) C57---C58 1.403 (6)
C2---C3 1.392 (8) C58---C59 1.381 (6)
C2---H2 0.9500 C58---H58 0.9500
C3---C4 1.386 (10) C59---C60 1.387 (7)
C3---H3 0.9500 C59---H59 0.9500
C4---C5 1.369 (10) C60---C61 1.371 (7)
C4---H4 0.9500 C60---H60 0.9500
C5---C6 1.394 (8) C61---C62 1.404 (7)
C5---H5 0.9500 C61---H61 0.9500
C6---H6 0.9500 C62---H62 0.9500
C7---C12 1.395 (6) C63---H63A 0.9900
C7---C8 1.400 (6) C63---H63B 0.9900
C8---C9 1.380 (7) C64---C69 1.389 (6)
C8---H8 0.9500 C64---C65 1.385 (7)
C9---C10 1.376 (8) C65---C66 1.392 (7)
C9---H9 0.9500 C65---H65 0.9500
C10---C11 1.388 (8) C66---C67
| 731
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| null | null |
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|
(h, 1_G) j(1_H, g) =
\theta \bigl( \psi(h, 1_G)\bigl) \theta \bigl(\varphi(1_H, g)
\bigl) = \theta \bigl(\psi(h, 1_G) \varphi(1_H, g) \bigl)$$ that is $\theta$ is surjective. Thus $H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ is a quotient group of $X$.
We end the section with a problem that can be of interest for a further study:
*Let “P” be a property in the category of groups. Give a necessary and sufficient condition such that $H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ has the property “P”.*
In the following we give an example in the case that “P” is the property of being abelian or cyclic.
Let $(H, G, \alpha, \beta)$ be a matched pair of groups. Then:
1. The center of the bicrossed product $H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ is given by: $$Z\bigl(H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \bigl) = \{(h, g)
\in {\rm Fix}(H)\times {\rm Fix}(G) \mid g \rhd x = h^{-1}xh, \, y
\lhd h = gyg^{-1}, \forall x \in H, y \in G \}$$
2. $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is an abelian group if and only if $H$ and $G$ are abelian groups and $\alpha$ and $\beta$ are the trivial actions;
3. $H\,{}_{\alpha}\!\!\bowtie_{\beta} \, G$ is a cyclic group if and only if $\alpha$ and $\beta$ are the trivial actions and $H$, $G$ are finite cyclic groups of coprime orders.
An element $(h, g) \in H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ belongs to the center of the group if and only if $(h, g) (x, 1) = (x, 1) (h, g)$ and $(h, g) (1, y) = (1,
y) (h, g)$, for all $x \in H$ and $y \in G$. This is equivalent to $h(g \rhd x) = xh$, $g \lhd x = g$, $y \rhd h = h$ and $(y \lhd
h)g = gy$, for all $x \in H$, $y \in G$. Hence $h \in {\rm
Fix}(H)$, $g \in {\rm Fix}(G)$, $g \rhd x = h^{-1}xh$, $y \lhd h =
gyg^{-1}$ for all $x \in H$, $y \in G$. (2) follows from (1) and (3) follows from (2) and the Chinese lemma: a direct product of two groups is a cyclic group if and only if they are finite, cyclic of coprime order.
Deformation of a matched pair
=============================
[\[se:2\]]{}
Let $H$ be
| 732
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| 617
| null | null |
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|
lta R},\label{eq:radial_operator_uniform} \\
\Delta_\phi^2\Phi_{i,j,k} =& \frac{\Phi_{i,j-1,k}-2\Phi_{i,j,k}+\Phi_{i,j+1,k}}{R_i^2(\delta\phi)^2},\end{aligned}$$ while $\Delta_z^2$ is defined through Equation .
Logarithmic Cylindrical Grid
----------------------------
In logarithmic cylindrical coordinates, we discretize a cylindrical computational domain in the same way as in the uniform cylindrical grid, but with logarithmic radial spacing. We define the face-centered radial coordinates as $R_{i+1/2} = f^i R_{\rm min}$, with a common multiplication factor $f\equiv(R_{\rm max} / R_{\rm min})^{1/N_R} >1$. Since the radial zone width, given by $R_{i+1/2}-R_{i-1/2} = (f-1)f^{i-1}R_{\rm min}$ shrinks toward small radii, a logarithmic cylindrical grid is advantageous in resolving the central regions of a disk with high accuracy. We also define the cell-centered radial coordinates using the volumetric centers as $$R_i \equiv \frac{\int_{R_{i-1/2}}^{R_{i+1/2}} R^2 dR}
{\int_{R_{i-1/2}}^{R_{i+1/2}} R dR} =
%\frac{2}{3} \frac{R_{i+1/2}^3 - R_{i-1/2}^3}{R_{i+1/2}^2 - R_{i-1/2}^2} =
\frac{2(f^2+f+1)}{3(f+1)} f^{i-1}R_{\rm min},$$ for $i=1,2,\cdots,N_R$. Note that the radial cell spacing $\delta R_i \equiv R_{i+1} - R_i = (f-1)R_i$ increases with $R_i$.
The second-order finite-difference approximation to Equation takes the same form as Equation , but with the radial difference operator defined as $$\label{eq:radial_operator}
\Delta_R^2\Phi_{i,j,k} = \frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(R_i\ln f)^2}.$$ Appendix \[s:fd\] shows that Equation makes the finite difference approximation second-order accurate.
Calculation of Interior Potential for Dirichlet Boundary Conditions {#s:interior_solver}
===================================================================
In this section, we provide the general method that we use to obtain the interior potential within the original domain, given Dirichlet boundaries for the potential on the surface. We describe our methods for both Cartesian
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| 2,986
| 1,432
| 825
| 2,383
| 0.78034
|
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|
\pmod{3k}$, $Y \cap (S \times \{n\}) = \{x_2, x_3\} \times \{n\}$.\
We will now prove Theorem \[kodd\]. We know that if $X \subset \mathbb{Z}_{k+1}^2$ has one point in each row or column then $X$ is a hole of size $k+1$. Since $k+1$ is even, we can try to choose $X_n$ in each slice $\mathbb{Z}_{k+1}^2 \times \{n\}$ so that $\bigcup_{n\in\mathbb{Z}}X_n$ is the disjoint union of $\frac{k+1}{2}$ sets $Y_i$ of the form in Lemma \[biglemma\].
We can do this as follows:\
For $n \equiv 0, k+1, \ldots, 2k-1 \pmod{3k}$, let $X_n = \{(0,0),(1,1),\ldots,(k-1,k-1),(k,k)\}$.\
For $n \equiv k, 2k+1, \ldots, 3k-1 \pmod{3k}$, let $X_n = \{(0,0),(0,1),(2,2),(2,3),\ldots,(k-1,k-1),\newline(k-1,k)\}$.\
For $n \equiv 2k, 1, \ldots, k-1 \pmod{3k}$, let $X_n = \{(0,1),(1,1),(2,3),(3,3),\ldots,(k-1,k),(k,k)\}$.\
Then let $X = \bigcup\limits_{n\in\mathbb{Z}} (X_n \times \{n\}) \subset \mathbb{Z}_{k+1}^2 \times \mathbb{Z}$.
Each $X_n$ is a hole, so we can tile $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z})\setminus X$ with strings. Also, $X$ is the disjoint union of sets of the form $Y$ from Lemma \[biglemma\]: for $0 \leq i \leq \frac{k-1}{2}$, let $S_i = \{(2i,2i),(2i,2i+1),(2i+1,2i+1)\}$. Then $X \cap (S_i \times \mathbb{Z})$ is precisely the set $Y$ generated from $S_i$ in the proof of Lemma \[biglemma\]. Hence $T$ tiles $X$.
Since $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z})\setminus X$ can be tiled with strings, we can partially tile $\mathbb{Z}^3$ with strings, leaving a copy of $X$ empty in each copy of $\mathbb{Z}_{k+1}^2 \times \mathbb{Z}$. We can tile all of these copies of $X$ with $T$, so $T$ tiles $\mathbb{Z}^3$, completing the proof of Theorem \[kodd\].
The general case
================
We now move on to general $k$:
\[generalk\] Let $T$ be the tile $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$. Then $T$ tiles $\mathbb{Z}^4$.
We will assume throughout that $T$ is fixed and $k \geq 3$.
For even $k$, the construction used to prove Theorem \[kodd\] does not work, as all holes in $\
| 734
| 499
| 571
| 746
| 2,182
| 0.781955
|
github_plus_top10pct_by_avg
|
2 0.8665 0.3064 0.031\*
C12 0.6221 (2) 0.7741 (3) 0.35032 (18) 0.0191 (10)
H12 0.6426 0.7594 0.3262 0.023\*
C13 0.6964 (2) 0.5780 (3) 0.35824 (16) 0.0130 (9)
H13A 0.7378 0.5560 0.3592 0.016\*
H13B 0.6820 0.6220 0.3293 0.016\*
C14 0.5696 (2) 0.5181 (3) 0.34616 (17) 0.0141 (9)
C15 0.5491 (2) 0.5052 (3) 0.38825 (17) 0.0155 (9)
H15 0.5748 0.4752 0.4174 0.019\*
C16 0.4916 (2) 0.5358 (3) 0.38777 (19) 0.0201 (10)
H16 0.4786 0.5278 0.4168 0.024\*
C17 0.4532 (2) 0.5778 (3) 0.3455 (2) 0.0224 (11)
H17 0.4136 0.5978 0.3453 0.027\*
C18 0.4728 (2) 0.5907 (4) 0.30313 (19) 0.0236 (11)
H18 0.4464 0.6193 0.2738 0.028\*
C19 0.5311 (2) 0.5619 (3) 0.30380 (19) 0.0196 (10)
H19 0.5446 0.5722 0.2751 0.024\*
C20 0.6363 (2) 0.4418 (3) 0.28259 (18) 0.0152 (9)
C21 0.6730 (2) 0.4792 (3) 0.25509 (17) 0.0169 (9)
H21 0.7017 0.5278 0.2694 0.020\*
C22 0.6680 (2) 0.4458 (4) 0.20679 (19) 0.0234 (11)
H22 0.6928 0.4719 0.1882 0.028\*
C23 0.6265 (2) 0.3743 (4) 0.18612 (19) 0.0256 (11)
H23 0.6230 0.3515 0.1533 0.031\*
C24 0.5905 (3) 0.3362 (4) 0.21297 (19) 0.0269 (12)
H24 0.5630 0.2860 0.1989 0.032\*
C25
| 735
| 4,329
| 438
| 438
| null | null |
github_plus_top10pct_by_avg
|
,J,P).
eq(A,A). plus(0,B,B). plus(s(A),B,s(C)):- plus(A,B,C).
![Moded SLD-tree $eq\_plus$[]{data-label="fig:eq_plus_symbolic"}](figs/eq_plus_symbolic.pdf){width="70ex"}
Substitutions on input variables express conditions for the clause to be applicable. The edge from node $N_2$ to $N_3$ shows that clause two is applicable if the concrete term denoted by $\underline{P}$ can be unified with $0$. The substitution, $I1 \setminus \underline{I}$, shows that applying this clause unifies $I1$ with the term corresponding to $\underline{I}$.
Every derivation in a moded SLD-tree for a query $\leftarrow Q$ corresponds to a concrete derivation for a subclass of $Den(\leftarrow Q)$. The subclass of queries for which a derivation to node $N_i$ is applicable is obtained by applying all substitutions on input variables from $N_0$ to $N_i$. Our condition of [@DBLP:conf/iclp/VoetsS09] proves non-termination for every query for which the derivation to $N_3$ is applicable. The substitutions on input variables in the derivation to $N_3$ are $\underline{J}\setminus \underline{I}$ and $\underline{P} \setminus 0$. Applying these to the query proves non-termination for the queries in $Den(\leftarrow eq\_plus(\underline{I},\underline{I},0))$. $\hfill \square$
As in the example, moded SLD-trees are usually infinite. To obtain a finite analysis, a complete loop check is applied during the construction of the tree. As in our previous works, [@DBLP:conf/iclp/VoetsS09] [@VDS10], we use the complete loop check *LP-check*, [@shen_dynamic_approach]. Without proof, we state that this loop check can also be used for moded SLD-trees and refer to [@shen_dynamic_approach] for more information.
In Figure \[fig:eq\_plus\_symbolic\], LP-check cuts clause 4 at node $N_6$ and clause 3 at node $N_7$. $\hfill \square$
Combined with the loop check, a moded SLD-tree can be considered a light-weight alternative to an abstract interpretation for mode analysis.
Non-termination analysis for programs with integer arithmetics
======================
| 736
| 2,221
| 982
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| 198
| 0.818866
|
github_plus_top10pct_by_avg
|
ho_{n-1}|< r_n\}$ and $\sigma_{B_n} = \inf\{t>0: X_t\not\in B_n\}$. The algorithm comes to an end at the random index $N = \min\{n\geq 0\colon \rho_n\not\in D\}$, again using the standard understanding that $\min\emptyset \coloneqq \infty$. See for example the depiction in Figure \[4stepsonly\].
![Steps of the walks-on-sphere algorithm until exiting the convex domain $D$ in the stable setting. In this realisation, $N = 3$.[]{data-label="4stepsonly"}](4stepsonly_new){width="0.7\linewidth"}
Even though the domain $D$ may be unbounded, our main result predicts that, irrespective of the point of issue of the algorithm, there will always be at most a geometrically distributed number of steps (whose parameter also does not depend on the point of issue) before the algorithm ends.
\[main\] Suppose that $D$ is a convex domain. For all $x\in D$, there exists a constant $p = p(\alpha,d)>0$ (independent of $x$ and $D$) and a real-valued random variable $\Gamma$ such that $N\leq \Gamma$ almost surely, where $$\mathbb{P}(\Gamma = k ) = (1-p)^{k-1}p, \qquad k\in\mathbb{N}.$$
There are a number of remarks that we can make from the conclusion above.
- Although $\Gamma$ has the same distribution for each $x\in D$, it is not the same random variable for each $x\in D$. As we shall see in the proof of the above theorem, the inequality $N\leq \Gamma$ is derived by comparing each step of the walk-on-spheres algorithm with a sequence of Bernoulli random variables. This sequence of Bernoulli random variables are defined up to null sets which may be different under each $\mathbb{P}_x$. Therefore, whilst the distribution of $\Gamma$ does not depend on $x$, its null sets do.
- The stochastic domination in Theorem \[main\] is much stronger than the usual comparison of the mean number of steps. Indeed, whilst it immediately implies that $\mathbb{E}_x[N] = 1/p$, we can also deduce that there is an exponentially decaying tail in the distribution of the number of steps. Specifically, for any $x\in D$, $$\mathbb{P}(N >n ) \l
| 737
| 2,287
| 906
| 783
| null | null |
github_plus_top10pct_by_avg
|
0.000 0.000
BLB($n^{0.8}$) 0.002 0.000 0.000 0.004 0.002 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.062 0.054 0.064 0.076 0.074 0.050 0.070
3 K=50 0.060 0.054 0.058 0.040 0.040 0.040 0.058
K=100 0.060 0.048 0.048 0.036 0.032 0.046 0.058
K=150 0.074 0.044 0.050 0.054 0.040 0.046 0.056
mVC 0.054 0.064 0.084 0.070 0.058 0.066 0.058
mMSE 0.066 0.078 0.048 0.066 0.084 0.072 0.084
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.002 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.080 0.054 0.076 0.060 0.054 0.050 0.066
: Empirical sizes comparison for Cases 1-3 in Example \[example2\].
\[table4\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50
| 738
| 5,097
| 192
| 426
| null | null |
github_plus_top10pct_by_avg
|
0.40177 (12) 0.49661 (5) 0.0332 (3)
P1 0.70165 (5) 0.64377 (8) 0.41615 (4) 0.0122 (2)
P2 0.90214 (5) 0.37198 (8) 0.63471 (4) 0.0119 (2)
P3 0.86722 (5) 0.16721 (8) 0.60311 (4) 0.0123 (2)
P4 0.68282 (5) 0.11687 (8) 0.49608 (4) 0.0110 (2)
P5 0.65584 (5) 0.17597 (8) 0.38414 (4) 0.0100 (2)
P6 0.64487 (5) 0.47428 (8) 0.34787 (4) 0.0112 (2)
C1 0.7730 (2) 0.7088 (3) 0.42479 (17) 0.0187 (10)
C2 0.8271 (2) 0.6582 (4) 0.4461 (2) 0.0272 (12)
H2 0.8255 0.5942 0.4567 0.033\*
C3 0.8832 (3) 0.7019 (5) 0.4518 (2) 0.0370 (15)
H3 0.9200 0.6673 0.4655 0.044\*
C4 0.8853 (3) 0.7961 (5) 0.4374 (2) 0.0437 (18)
H4 0.9236 0.8258 0.4412 0.052\*
C5 0.8326 (3) 0.8468 (5) 0.4178 (2) 0.0433 (17)
H5 0.8345 0.9118 0.4090 0.052\*
C6 0.7761 (3) 0.8030 (4) 0.4107 (2) 0.0288 (12)
H6 0.7396 0.8377 0.3962 0.035\*
C7 0.6402 (2) 0.7312 (3) 0.39761 (17) 0.0153 (9)
C8 0.6100 (2) 0.7549 (3) 0.43287 (19) 0.0203 (10)
H8 0.6220 0.7256 0.4651 0.024\*
C9 0.5629 (3) 0.8202 (4) 0.4214 (2) 0.0282 (12)
H9 0.5431 0.8370 0.4457 0.034\*
C10 0.5449 (2) 0.8609 (4) 0.3743 (2) 0.0281 (12)
H10 0.5121 0.9051 0.3661 0.034\*
C11 0.5742 (3) 0.8380 (4) 0.3387 (2) 0.0257 (11)
H11 0.561
| 739
| 4,559
| 232
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github_plus_top10pct_by_avg
|
covariant derivative can be expressed in terms of the spin connection $\omega_{an m}$ as $\nabla_a=\partial_a+{1\over 2} \omega_{anm} \Sigma^{nm}$, where $\Sigma^{nm}={1\over 4}[\gamma^n,\gamma^m]$ are the generators of the Lorentz transformations in spin $1/2$ representation.
[^3]: One can see that for the conformally coupled case and for the spacetime we are considering the anomaly vanishes even on the branes. Since the conformal anomally is given by the Seeley-de Witt coefficient $a_{5/2}$, and this is a conformal invariant quantity, in the conformally coupled case $a_{5/2}$ is the same as the flat spacetime related problem, which is zero in this case because the branes are flat. We can also use the expressions found for $a_{5/2}$ presented in Ref. [@klaus; @vass] to show how in this case it vanishes.
---
abstract: |
The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra $A_1$ is a $\Z$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A$ (there is no restriction on the total degrees of $P$ and $Q$).\
[*Key Words: the Weyl algebra, the Dixmier Conjecture, automorphism, endomorphism, a $\Z$-graded algebra.*]{}
[*Mathematics subject classification 2010: 16S50, 16W20, 16S32, 16W50.*]{}
address:
- |
Department of Pure Mathematics\
University of Sheffield\
Hicks Building\
Sheffield S3 7RH, UK
- |
Lehrstuhl D für Mathematik\
RWTH Aachen University\
52062 Aachen, Germany
author:
- 'V. V. Bavula'
- 'V. Levandovskyy'
title: A remark on the Dixmier Conjecture
---
Introduction
============
In the paper, $K$ is a field of characteristic zero and $K^*:=K \setminus\{0\}$. The algebra $A_1 := K \langle X, Y \mid [Y, X]=1 \rangle$ is called the [*first Weyl algebra*]{} where $[Y, X]= YX - XY$. The $n$’th t
| 740
| 276
| 383
| 647
| null | null |
github_plus_top10pct_by_avg
|
0.994 0.996 0.994
K=100 0.994 0.998 0.998 0.990 0.988 0.994 0.996
K=150 0.994 1.000 0.998 0.986 0.992 0.998 0.990
mVC 0.918 0.922 0.944 0.930 0.958 0.922 0.958
mMSE 0.942 0.962 0.954 0.944 0.950 0.954 0.958
BLB($n^{0.6}$) 0.000 0.000 0.006 0.000 0.002 0.000 0.000
BLB($n^{0.8}$) 0.312 0.310 0.358 0.370 0.346 0.324 0.354
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.020 0.030 0.040 0.022 0.042 0.040 0.030
TB 0.994 0.998 0.996 0.992 0.994 0.996 0.992
: Empirical powers comparison for Cases 4-6 in Example \[example2\].
\[table7\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.037 0.037 0.037 0.037 0.037 0.037 0.037
K=100 0.037 0.037 0.037 0.037 0.037 0.037 0.037
K=150 0.038 0.038 0.037 0.037 0.037 0.037 0.037
mVC 0.048 0.048 0.048 0.048 0.048 0.048 0.048
mMSE 0.047 0.047 0.047 0.047 0.047 0.047 0.047
BLB(
| 741
| 4,952
| 260
| 434
| null | null |
github_plus_top10pct_by_avg
|
pi(C^\lambda,k)$ does not depend on the given $\Q$-resolution. Also, the following upper semi-continuity property will be useful.
\[lemma:propsM\] Under the above conditions:
1. \[lemma:propsM:epsilon\] $\cM_\pi(C^\lambda,k)=\cM_\pi(C^{\lambda+\varepsilon},k)$ for a sufficiently small $\varepsilon>0$.
2. \[lemma:propsM:plus1\] $\cM_\pi(C^{\lambda+1},k)=\cM_\pi(C^{\lambda},k-[C])$, where $[C]$ is the class of $C$.
More generally, $\cM_\pi\left(C^{\lambda+\frac{1}{\gcd(n_i)}},k\right)=\cM_\pi\left(C^{\lambda},k-\left[\frac{1}{\gcd(n_i)}C\right]\right)$.
First note that both are submodules of the same invariant module $\cO_{X,\zeta^{\s_{\lambda,k}}}$ since the class $\s_{\lambda,k}$ does not change for a small enough $\varepsilon>0$. The ring $\cO_X$ is Noetherian, thus $\cM_\pi(C^\lambda,k)$ admits a finite system of generators. Since the conditions in $$\operatorname{mult}_{E_\v} (\pi^{*}\operatorname{div}_X(g)) > \sum_{i=1}^r {\left \{ \lambda n_i \right \}} m_{\v i} - \nu_\v$$ are given by a strict inequality there is a small positive $\varepsilon'>0$ for which the inequalities $$\operatorname{mult}_{E_\v} (\pi^{*}\operatorname{div}_X(g)) > \sum_{i=1}^r {\left \{ \lambda n_i \right \}} m_{\v i} - \nu_\v + \varepsilon'$$ still hold. Taking $\varepsilon<\frac{\varepsilon'}{nmr}$ for $n:=\max\{n_j\}, m:=\max\{m_{\v j}\}$ such that $${\left \{ (\lambda+\varepsilon) n_i \right \}} m_{\v i}< {\left \{ \lambda n_i \right \}} m_{\v i} + \frac{\varepsilon'}{r}$$ the result follows.
As for part \[lemma:propsM:plus1\] note that $\s_{\lambda+1,k}=\s_{\lambda,k-[C]}$ and the set of conditions in does not change since $\left\{(\lambda+1)n_i\right\}=\left\{\lambda n_i\right\}$. In general, the result follows from $$\s_{\lambda+\frac{1}{\gcd(n_i)},k}=\s_{\lambda,k-\left[\frac{1}{\gcd(n_i)}C\right]}\text{ and }
\left\{(\lambda+\frac{1}{\gcd(n_i)})n_i\right\}=\left\{\lambda n_i\right\}.
\qedhere$$
Note that in the smooth case $X=(\CC^2,0)$ one can drop $k$ in the notation $\cM(C^\lambda,k)$ and simply write $\c
| 742
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| 740
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| null | null |
github_plus_top10pct_by_avg
|
above, the original idea is to change only the complex coupling strength $\lambda_c$ to one channel $c$, while the measuring is done on one or two different channels $a,b\neq c$. We denote the resulting scattering fidelity by *coupling fidelity*. We present below an exact RMT prediction for this quantity.
The starting point is to apply the convolution theorem for Fourier transforms to Eq. (\[eq:f\_ab\]) and relate it to the parametric cross-correlation function $\hat{C}[S_{ab},S_{ab}^{\prime *}](t)$ of the $S$-matrix elements in the time domain [@note1], $$\label{eq:ss_ft}
\langle \hat{S}_{ab}(t)\hat{S}_{ab}^{\prime *}(t)\rangle = \hat{C}[S_{ab},S_{ab}^{\prime *}](t)\,.$$ We denote the coupling constant for the variable antenna $c$ in forward and backward time evolution by $\lambda$ and $\lambda^\prime$, respectively. (We omit the lower index “c” henceforth.) In the case of unchanged coupling, $\lambda=\lambda^\prime$, the autocorrelation function $\hat{C}[S_{ab},S_{ab}^{*}](t)$ is real and its exact expression is obtained from Verbaarschot-Weidenmüller-Zirnbauer (VWZ) integral [@ver85a] and is given by $$\label{eq:VWZ}
\hat C[S_{ab},S_{ab}^*](t) = \delta_{ab} T_a^2(1-T_a) J_a(t) + (1+\delta_{ab}) T_a T_b P_{ab}(t).$$ It is convenient to use the parametrization of Ref. [@gor02a] to write down the explicit expressions for the functions $J_a(t)$ and $P_{ab}(t)$, as
$$\label{eq:J}
J_a(t) = 4\mathcal{I} \left[\left( \frac{r+T_a x}{1+2T_a r +T_a^2\, x} +
\frac{t-r}{1- T_a(t-r)}\right)^2\right]$$
and $$\begin{aligned}
\label{eq:P}
P_{ab}(t) &=& 2\mathcal{I}\left[
\frac{T_a T_bx^2 + d_{ab}(r)x+ (2r+1)r}{(1+2T_a r + T_a^2 x)(1+2T_b r +T_b^2 x)} \right.\nonumber \\
&& \left. + \frac{(t-r)(r+1-t)}{[1-T_a(t-r)][1-T_b(t-r)]} \right]\,,\end{aligned}$$ where $$x \equiv \frac{2r+1}{2u+1}u^2, \quad d_{ab}(r)\equiv T_aT_b+(T_a+T_b)(r+1)-1$$ and the shorthand $\mathcal{I}$ stands for the integral, $$\begin{aligned}
\label{eq:I}
\mathcal{I}[\cdots] &=& \int_{\max(0,t-1)}^t \!\!{\textrm d} r\int_0^r{\textrm d}
| 743
| 2,608
| 1,235
| 768
| 2,693
| 0.777818
|
github_plus_top10pct_by_avg
|
and the Rubin et. al. data set [@Rubin1980]. Except the last set, the observed velocity curves is fitted to either $v^{\hbox{\scriptsize{p-iso}}}(r)$, or to a functionally similar velocity curve [@Cour]. The last set gives only the galactic rotation curves, and they have been fitted to $v^{\hbox{\scriptsize{p-iso}}}(r)$ in [@ADS]. While the URC of [@Pers-1996] has a constant asymptotic velocity, it has a $r^{0.66}$ behavior for $r$ small. This behavior is different from $v^{\hbox{\scriptsize{ideal}}}$, and was not considered here [@ADS].
While $v_H$ is easily identified for all four data sets, determining $r_H$ is more complicated; this is determination is done in [@ADS]. The resultant values are used to obtain $v^{*}_H$ and $r^{*}_H$ for each set, which are then used to calculate the $\sigma_8$ and $\Delta\sigma_8$ for it. Results of these calculations are in Table \[summary\]. Four of the five data sets give a $\sigma_8$ that agrees with the WMAP value at the 95% CL. The Rubin et. al. set does not, but it is known that these galaxies were not randomly selected [@Rubin1980].
*Data Set* 0.1in $v^{*}_H$ 0.1in $\Delta v^{*}_H$ 0.1in $r^{*}_H$ 0.1in $\Delta r^{*}_H$ $\qquad\sigma_8$ $\qquad\Delta \sigma_8$ 0.1in *t-test*
------------------------- ----------------- ------------------------ ----------------- ------------------------ ------------------ ------------------------- ---------------- --
deBlok et. al. (53) 119.0 6.8 3.62 0.33 0.613 0.097 1.36
CF (348) 179.1 2.9 7.43 0.35 0.84 0.18 0.43
Mathewson et. al. (935) 169.5 1.9 15.19 0.42 0.625 0.089 1.34
Rubin et. al. (57) 223.3
| 744
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| 494
| 524
| null | null |
github_plus_top10pct_by_avg
|
oplus H(1)$.
For this choice of a basis, the block associated to $\tilde{M}_0\oplus M_2$ of the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $L^j$ is $$\begin{pmatrix} id&0 &0\\ 0 &\begin{pmatrix} 1&\frac{1}{1+4b''}(-2z_j^{\ast})\\
0 & 1+\frac{1}{1+4b''}(2 z_j^{\ast})\end{pmatrix} &0 \\ 0& 0 &id \end{pmatrix}.$$ Here, $id$ in the $(1,1)$-block corresponds to the direct summand $(\oplus_{\lambda}H_{\lambda})$ of $\tilde{M}_0$ and $id$ in the $(3,3)$-block corresponds to $M_2$.
We now apply the argument of Steps (i) and (ii) to the above Jordan splitting $\tilde{M}_0\oplus\tilde{M}_1\oplus(\oplus_{i\geq 2}M_i)$ of $L^j$ since $\tilde{M}_1$ is isometric to $\oplus H(1)$. As explained in Steps (i) and (ii), in order to describe the image of a fixed element of $F_j$ in the orthogonal group associated to $M_0''$, we only need the above block associated to $\tilde{M}_0\oplus M_2$. Note that $\frac{1}{1+4b''}$ is a unit. Then $(z_j^{\ast})_1$ 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 $1$ by Equation (\[e42\]), $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
3. Assume that both $M_0$ and $L_j$ are *of type $I^o$*. In this case, we will describe $\psi_j|_{F_j} : F_j \rightarrow \mathbb{Z}/2\mathbb{Z}$ explicitly in terms of a formal matrix. To do that, we will first describe a morphism from $F_j$ to the special fiber of the smooth integral model associated to $L^j$ and then to $G_j$. Recall that $G_j$ is the special fiber of the smooth integral model associated to $C(L^j)=\bigoplus_{i \geq 0} M_i^{\prime}$. Then we will describe a morphism from $F_j$ to the special fiber of the smooth integral model associated to $M_0'\oplus C(L^j)$ and to the special fiber of the smooth integral model associated to $Y(C(M_0'\oplus C(L^j)))$. Finally, we will describe a morphism from $F_j$ to a certain even orthogonal group associated to $Y(C(M_0'\
| 745
| 421
| 741
| 762
| 2,668
| 0.777997
|
github_plus_top10pct_by_avg
|
serve that for the convex set $G$ actually $\tau_{e,-}(y,\omega)=\infty$ for all $(y,\omega,E)\in \Gamma_{e,-}$.
\[tthcon\] The trace mappings $$\gamma_{e,\pm}:W^2(G_e\times S\times I)\to T^2(\Gamma_{e,\pm})$$ are (well-defined) bounded surjective operators with bounded right inverses (lifts) $L_{e,\pm}:
T^2(\Gamma_{e,\pm})\to W^2(G_e\times S\times I)$.
Again it needs only to consider the trace operator $\gamma_-$. The proof runs similarly to the proof of Theorem \[tth\] with following changes. Instead of estimate (\[trpr2\]) we utilize the inequality $$\begin{gathered}
|f(0)|^2=\Big|\int_0^\infty {d\over{dt}}(f(t)^2)dt\Big|
=\Big|\int_0^\infty 2f'(t)f(t)dt\Big|\\
\leq
2\Big(\int_0^\infty|f'(t)|^2dt\Big)^{1/2}
\Big(\int_0^\infty|f(t)|^2dt\Big)^{1/2}
\leq
\int_0^\infty|f'(t)|^2dt
+\int_0^\infty|f(t)|^2dt, \label{trpr13}\end{gathered}$$ which is valid for all $f\in C_0^1([0,\infty[)$ (note that $f(t)=0$ for sufficiently large $t$).
In addition, one applies the change of variables $H$ we have $H(W)=G_e$ where $W:=\{(v,t)\ |\ v\in V_-,\ 0<t<\infty\}$ and $V_-:=\{v\in V\ |\ \omega\cdot\nu_e(h(v))<0\}$.
Let $\lambda>0$. For any $g_{e}\in C(\Gamma_{e,-})$ such that ${{\frac{\partial g_{e}}{\partial \tilde y_i}}}\in C(\Gamma_{e,-})$ the (classical) solution $\Psi$ of the problem \[exle1\] \_x+=&0 D\_e,\
\_[|\_[e,-]{}]{}=&g\_[e]{}, where $D_e$ is the set (\[D\]) corresponding to $G_e\times S\times I$, is given explicitly by (cf. (\[trath3\])) \[exle2\] (x,,E)=
e\^[-t\_e(x,)]{}g\_[e]{}(x-t\_e(x,),,E), &[when]{} t\_e(x,) [is finite]{}\
0, &[otherwise]{}
. Similarly to the proof of Lemma \[trathle1\] we find that (\[exle1\]) holds weakly in $G_e\times S\times I$. For $g_{e}\in T^2(\Gamma_{e,-})$ we define the lift explicitly by \[trpr14\] L\_[e,-]{}g\_[e]{}:=. Then as in the proof of Lemma 5.8 given in [@tervo14] we get that $L_{e,-}g_{e}\in \tilde W^2(G_e\times S\times I)$ and (note that ${\left\Vert \omega\cdot\nabla_x\Psi\right\Vert}_{L^2(G_e\times S\times I)}=
\lambda{\left\Vert \Psi\right\Vert}_{L^2(G_e\ti
| 746
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| 749
| 876
| null | null |
github_plus_top10pct_by_avg
|
shida]{}, T. 2016, , 68, 16
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, N. I., & [Sunyaev]{}, R. A. 1973, , 24, 337
, P. R., & [Kang]{}, H. 1987, , 318, 32
, I., [Choi]{}, J.-H., [Begelman]{}, M. C., & [Nagamine]{}, K. 2016, , 456, 500
, I., [Frank]{}, J., & [Begelman]{}, M. C. 1989, , 338, 45
, J. M. 1979, , 234, 761
, J. M., & [van Steenberg]{}, M. E. 1985, , 298, 268
, K., [Coppola]{}, C. M., [Omukai]{}, K., [Galli]{}, D., & [Palla]{}, F. 2016, , 456, 270
, K., [Omukai]{}, K., & [Inoue]{}, A. K. 2014, , 445, 544
, H., [Hasegawa]{}, K., & [Tominaga]{}, N. 2014, , 792, 32
, H. R., & [Ohsuga]{}, K. 2015, , 67, 60
, K. E. I., [Nakamoto]{}, T., & [Omukai]{}, K. 2013, , 773, 155
, T., & [Haiman]{}, Z. 2009, , 696, 1798
, A., & [Umemura]{}, M. 2011, , 728, L3
| 747
| 1,023
| 2,985
| 973
| null | null |
github_plus_top10pct_by_avg
|
\
\mathcal{Z}_i&1 \end{pmatrix} & \quad \textit{if $L_i$ is \textit{bound of type II}};\\
(m_{i,i})_1 & \quad \textit{if $L_i$ is \textit{free of type II}}.
\end{array} \right.$$
Here, $$\left\{
\begin{array}{l}
\mathcal{X}_i=(v_i)_1+(\delta_{i-2}e_{i-2}\cdot (m_{i-2, i})_1+\delta_{i+2}e_{i+2}\cdot (m_{i+2, i})_1)\tilde{e_i};\\
\mathcal{Y}_i=((y_i)_1+\sqrt{\bar{\gamma}_i}(v_i)_1)+(\delta_{i-2}e_{i-2}\cdot (m_{i-2, i})_1
+\delta_{i+2}e_{i+2}\cdot (m_{i+2, i})_1)\tilde{e_i};\\
\mathcal{Z}_i=\delta_{i-1}^{\prime}e_{i-1}\cdot (m_{i-1, i})_1+\delta_{i+1}^{\prime}e_{i+1}\cdot (m_{i+1, i})_1+
\delta_{i-2}e_{i-2}\cdot (m_{i-2, i})_1+\delta_{i+2}e_{i+2}\cdot (m_{i+2, i})_1,
\end{array} \right.$$
where
1. When $j$ is even, $e_{j}=(0,\cdots, 0, 1)$ (resp. $e_j=(0,\cdots, 0, 1, 0)$) of size $1\times n_{j}$ if $L_{j}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
2. $\tilde{e_i}=\begin{pmatrix} \mathrm{id}\\0 \end{pmatrix}$ of size $n_i\times (n_{i}-1)$ (resp. $n_i\times (n_{i}-2)$), where $\mathrm{id}$ is the identity matrix of size $(n_i-1)\times (n_{i}-1)$ (resp. $(n_i-2)\times (n_{i}-2)$) if $L_{i}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
3. $\bar{\gamma}_i$ is as explained in Remark \[r33\].(2).
4. $
\delta_{j}^{\prime} = \left\{
\begin{array}{l l}
1 & \quad \textit{if $j$ is odd and $L_j$ is \textit{free of type I}};\\
0 & \quad \textit{otherwise}.
\end{array} \right.
$
5. When $j$ is odd, $e_{j}=(0,\cdots, 0, 1)$ of size $1\times n_{j}$.
Note that if the dimension of $B_i/Z_i$ is even and positive, then $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} (= \mathrm{O}(B_i/Z_i, \bar{q}_i))$ is disconnected. If the dimension of $B_i/Z_i$ is odd, then $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} (= \mathrm{SO}(B_i/Z_i, \bar{q}_i))$ is connected. The dimension of $B_i/Z_i$, as a $\kappa$-vector space, is as follows: $$\left\{
\begin{array}{l l}
n_i-1 & \quad \textit{if $L_i$ is \textit{of type} $\textit{I}^e$};\\
| 748
| 3,556
| 674
| 509
| 2,616
| 0.778415
|
github_plus_top10pct_by_avg
|
ults efficiently from practical sample sizes.
Keywords: software, safety, hazard, demonstration, operational profile, automata, confidence, statistics
author:
- '\'
title: Software Safety Demonstration and Indemnification
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Fact]{}
\[theorem\][Definition]{}
Prologue
========
Copyright {#S:COPYRIGHT}
---------
This document may be freely copied or modified in accordance with the Creative Commons Attribution license[^1].
Executive summary {#S:EXECUTIVE_SUMMARY}
-----------------
In systems of integrated hardware and software, the intangible nature of software raises the question of fitness in roles bearing safety risk. Such a safety risk in software, known as a hazard, is a region of code involving safety constraints (requirements) necessitating some degree of verification. Hazards are identified and monitored by safety engineers, and possess hypothetical (threatened) frequency and severity ratings. During its development, potentially hazardous software merits not only rigorously controlled general engineering process, but also quantitative assurance of hazards within particular products.
### Approach {#S:APPROACH}
The topic of this essay is assuring the interplay between safety constraints (requirements) and software control. Software is appreciated as a branching process whose permutations are intractably numerous to test exhaustively. Barring exhaustive testing, statistical verification remains an option.
The degree of statistical verification will be expressed as residual risk, a contravariant quantity. A software item’s total risk has many constituents. For instance, any software communicating with an operator runs human factors risk. Statistical safety risk, one constituent of total risk, focuses on hazardous code. Code is potentially hazardous if its statistical risk (numerical product of frequency of execution, probability of error, and expected safety loss per error) is sufficiently high.
The
| 749
| 160
| 1,405
| 539
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github_plus_top10pct_by_avg
|
ng are equivalent statements.*
\(1) *$D$ is dense in $X$*.
\(2) *If $F$ is any closed set of $X$ with $D\subseteq F$, then $F=X$*; *thus the only closed superset of $D$ is $X$.*
\(3) *Every nonempty (basic) open set of $X$ cuts $D;$ thus the only open set disjoint from $D$ is the empty set $\emptyset$.*
\(4) *The exterior of $D$ is empty.$\qquad\square$*
**Proof.** (3) If $U$ indeed is a nonempty open set of $X$ with $U\bigcap D=\emptyset$, then $D\subseteq X-U\neq X$ leads to the contradiction $X=\textrm{Cl}(D)\subseteq\textrm{Cl}(X-U)=X-U\neq X$, which also incidentally proves (2). From (3) it follows that for any open set $U$ of $X$, $\textrm{Cl}(U)=\textrm{Cl}(U\bigcap D)$ because if $V$ is any open neighbourhood of $x\in\textrm{Cl}(U)$ then $V\bigcap U$ is a nonempty open set of $X$ that must cut $D$ so that $V\bigcap(U\bigcap D)\neq\emptyset$ implies $x\in\textrm{Cl}(U\bigcap D)$. Finally, $\textrm{Cl}(U\bigcap D)\subseteq\textrm{Cl}(U)$ completes the proof.$\qquad\blacksquare$
**Definition A3.6.** (a) *A set $A\subseteq X$ is said to be* *nowhere dense* *in* ***$X$ if* $\textrm{Int}(\textrm{Cl}(A))=\emptyset$ *and* *residual* *in* ***$X$ if* $\textrm{Int}(A)=\emptyset$*.$\qquad\square$*
$A$ is nowhere dense in $X$ iff $$\textrm{Bdy}(X-\textrm{Cl}(A))=\textrm{Bdy}(\textrm{Cl}(A))=\textrm{Cl}(A)$$
so that $${\textstyle \textrm{Cl}(X-\textrm{Cl}(A))={(X-\textrm{Cl}(A))\bigcup\textrm{Cl}(A)=X}}$$
from which it follows that $$A\textrm{ is nwd in }X\Longleftrightarrow X-\textrm{Cl}(A)\textrm{ is dense in }X$$
and $$A\textrm{ is residual in }X\Longleftrightarrow X-A\textrm{ is dense in }X.$$
Thus $A$ is nowhere dense iff $\textrm{Ext}(A):=X-\textrm{Cl}(A)$ **is dense in *$X$,* and in particular a closed set is nowhere dense in $X$ iff its complement is open dense in $X$ with open-denseness being complimentarily dual to closed-nowhere denseness. The rationals in reals is an example of a set that is residual but not nowhere dense. The following are readily verifiable properties of subsets of $X$.
\(1) A
| 750
| 1,630
| 1,778
| 770
| 1,231
| 0.792355
|
github_plus_top10pct_by_avg
|
1
Note that only 48 children could be included in the analyses, due to the age restrictions of some of the questionnaires (FES and PSS).
2
Obtained by fitting a second model, including the subscales of the FES, instead of the FRI and FSI.
∗
p
\< 0.05,
∗∗
p
\> 0.001.
### Loneliness
The interaction effects between *family functioning (FRI and FSI)* and *family member* \[*FRI:* χ^2^(3) = 5.54, *p* = 0.14; *FSI:*χ^2^(3) = 2.79, *p* = 0.43\], between *cancer appraisal* and *family member* \[χ^2^(3) = 5.34, *p* = 0.15\] and between *family functioning* and *cancer appraisal* \[*FRI:*χ^2^(1) = 1.13, *p* = 0.29; *FSI:*χ^2^(1) = 2.30, *p* = 0.13\] were not significant and were subsequently left out of the final model. In the final model, 32% of the variance in *loneliness* was attributable to differences between family members (regardless of which family one belonged to) and 36% was attributable to differences between families. Within the same family, there was a correlation of 0.53 between the different family members in their reports of loneliness.
A significant effect of *FRI* upon loneliness was found \[χ^2^(1) = 9.03, *p* = 0.003\]: higher emotional closeness within the family (more cohesion and expressiveness, less conflict) was related to lower levels of loneliness in all family members. In addition, when refitting the model with the FES subscales instead of the two composite scores, there was a significant effect of *expressiveness* \[χ^2^(1) = 7.26, *p* = 0.007\]. In other words, when a participant perceived his/her family as more expressive, s/he reported to feel less lonely. None of the other FES subscales were significantly related to loneliness (all χ^2^ \< 3.7, all *p* \> 0.05). Furthermore, there was a significant effect of *cancer appraisal* \[χ^2^(1) = 81.83, *p* \< 0.001\]: the more one perceived the illness as uncontrollable and the less as manageable, the more s/he reported t
| 751
| 53
| 1,295
| 944
| null | null |
github_plus_top10pct_by_avg
|
e can prove Lemma $\ref{lem:4-1-3}$.
We use the following formula [@andrews_askey_roy_1999 p.22, Theorem 6.5.1]: $$\begin{aligned}
\G(2a) = \frac{2^{2a - 1}}{\sqrt{\pi}} \G(a) \G\left(a + \frac{1}{2} \right). \end{aligned}$$ From this and Lemma $\ref{lem:4-1-1}$, we have $$\begin{aligned}
2 \G(2a) - a \G(a)^{2}
&= \frac{2^{2a}}{\sqrt{\pi}} \G(a) \G\left(a + \frac{1}{2} \right)
- \G(a) \G(a + 1) \allowdisplaybreaks \\
&= \frac{1}{\sqrt{\pi}} \G(a)
\left\{4^{a} \G\left(a + \frac{1}{2} \right) - \sqrt{\pi} \G(a + 1) \right\} \\
&> 0. \end{aligned}$$ The lemma is thus proved.
Inequalities for the incomplete gamma functions
-----------------------------------------------
We will prove the following lemma:
\[lem:4-2-1\] 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 $\ref{lem:4-2-1}$, we need to prove two other lemmas:
\[lem:4-2-2\] For $a > 0$ and $x \geq 0$, we have $$\begin{aligned}
a \g(a, x) \geq x^{a} e^{-x}. \end{aligned}$$
For $a > 0$ and $x \geq 0$, we define $$\begin{aligned}
u(a, x) := a \g(a, x) - x^{a} e^{-x}. \end{aligned}$$ Then, we have $$\begin{aligned}
\frac{d}{dx} u(a, x) = x^{a} e^{-x} \geq 0. \end{aligned}$$ The lemma follows from this and $u(a, 0) = 0$.
\[lem:4-2-3\] For $a > 0$ and $b \in \mathbb{R}$, we have $$\begin{aligned}
\lim_{x \to +\infty} x^{b} \G(a, x) = 0. \end{aligned}$$
When $b \leq 0$, it is easily obtained from the definition of $\G(a, x)$. When $b > 0$, using the L’Hôpital’s rule, we obtain $$\begin{aligned}
\lim_{x \to +\infty} \frac{\G(a, x)}{x^{-b}}
&= \lim_{x \to +\infty} \frac{x^{a - 1} e^{-x}}{b x^{- b - 1}} \allowdisplaybreaks \\
&= \lim_{x \to +\infty} \frac{x^{a + b} e^{-x}}{b} \\
&= 0. \end{aligned}$$
Now, we can prove Lemma $\ref{lem:4-2-1}$.
For $a > 0$ and $x \geq 0$, we define $$\begin{aligned}
y_{1} (a, x) := x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x). \end{aligned}$$ Let us prove $y_{1} (a, x) > 0$ ($a > 0$, $x > 0$). For $a
| 752
| 2,601
| 909
| 811
| 3,284
| 0.77348
|
github_plus_top10pct_by_avg
|
!= j, i != "y" && j != "y", and both i and j are adjacent to each other.
I'm having difficulty concocting an algorithm to create an adjacency matrix provided a char[][] array. I've defined the rules, but finding constraints for iteration is problematic.
A:
Try this:
static void set(boolean[][] aM, int cols, int row0, int col0, int row1, int col1) {
int index0 = row0 * cols + col0;
int index1 = row1 * cols + col1;
aM[index0][index1] = aM[index1][index0] = true;
}
static boolean[][] adjacencyMatrix(char[][] cA) {
int rows = cA.length;
int cols = cA[0].length;
boolean[][] aM = new boolean[rows * cols][rows * cols];
for (int i = 0; i < rows; ++i) {
for (int j = 0; j < cols; ++j) {
if (cA[i][j] == 'y')
continue;
if (i + 1 < rows && cA[i + 1][j] != 'y')
set(aM, cols, i, j, i + 1, j);
if (j + 1 < cols && cA[i][j + 1] != 'y')
set(aM, cols, i, j, i, j + 1);
}
}
return aM;
}
public static void main(String[] args) {
char[][] cA = {
{'z', 'y', 'z'},
{'z', 'z', 'z'},
{'z', 'y', 'y'},
};
boolean[][] aM = adjacencyMatrix(cA);
for (boolean[] row : aM) {
for (boolean cell : row)
System.out.print(cell ? "1" : "0");
System.out.println();
}
}
The result is:
000100000
000000000
000001000
100010100
000101000
001010000
000100000
000000000
000000000
Q:
WPF Datagrid copy last row of grid to new row in grid (Devexpress)
there is thing that I want to do but I stack as a dummy. Here is the steps;
When user hit the ctrl + D button on Datagrid
The Last row of datagrid values will copied to clippopard or somewhere (eg.CopyToClipboard func.)
From clipboard or something else to again it could be a function e.g. pastToclipboard or we can use InitNewRowEventArgs which it gave us access the RowHandle funciton
Here is code I done so far.
private void dtg_tabletrial_KeyDown(object sender, KeyEventArgs e)
{
if (e.Key == Key.D && (
| 753
| 997
| 15
| 708
| 50
| 0.831505
|
github_plus_top10pct_by_avg
|
=1$.
(The order of the rules in $m_r$ is arbitrary). Additionally, $M$ contains the starting and the terminating matrices $$(S'\to [S] S \overline{A_1} \cdots \overline{A_m}) \mbox{ and } (\overline{S} \to {\lambda},\overline{A_1} \to {\lambda}, \ldots,\overline{A_m} \to {\lambda}),$$ where $V=\{S,A_1,\ldots,A_m\}$. Intuitively, $H$ generates sentential forms of the shape $[\beta] \gamma$ where$[\beta] \in ([V] \cup \Sigma)^*$ encodes a sentential form $\beta$ derivable in $G'$ and $\gamma \in (V\cup \overline{V})$ gives a count of the nonterminal symbols in $\beta$ as follows: $|\gamma|_A+|\gamma|_{\overline{A}}=1$ and $|\gamma|_A=|\beta|_A$. Formally, it can be shown by induction that a sentential form over $V_H \cup \Sigma$ can be generated after applying $k\geq 1$ matrices (except for the terminating) iff it has the form $[\beta] \gamma$ where
- $\beta \in (V\cup\Sigma)^*$ can be derived in $G'$ in $k-1$ steps,
- $\gamma \in \{S,\overline{S}\} \{A_1,\overline{A}_1\} \cdots\{A_m,\overline{A}_m\}$ and $|\gamma|_A=1$ iff $|\beta|_A=1$.
We can also show that the inverse inclusion also holds.
\[MATfinInCScb\] $\mathbf{MAT}_{{\mathit{fin}}}\subseteq \mathbf{GS}_{{\mathit{cb}}}$.
Capacity-bounded context-free grammars {#sec:nb-cfg}
======================================
In this section, we investigate capacity-bounded context-free grammars. It turns out that they are strictly between context-free languages of finite index and matrix languages of finite index. Closure properties of capacity bounded languages with respect to AFL operations are shortly discussed at the end of the section.
As a first result we show that the family of context-free languages with finite index is properly included in ${{\bf CF}}_{{\mathit{cb}}}$.
\[thm:hierarchyCapacityBounded1\] ${{\bf CF}}_{{\mathit{fin}}} \subset {{\bf CF}}_{{\mathit{cb}}}$.
Any context-free language generated by a grammar $G$ of index $k$ is also generated by the capacity-bounded grammar $(G,\kappa)$ where $\kappa$ is the capacity function constantly $k$.
| 754
| 1,862
| 1,735
| 774
| 1,300
| 0.791397
|
github_plus_top10pct_by_avg
|
nt of $F_j$ in the even orthogonal group associated to $M_0''$, where $M_0''$ is a Jordan component of $Y(C(L^j))=\bigoplus_{i \geq 0} M_i''$. As in the above case (1), there are 3 cases depending on whether $(a+a')/2$ is a unit or not, and whether $M_1=\oplus H(1)$ (possibly empty) or $M_1=A(4b'', 2\delta, \pi) \oplus (\oplus H(1))$ with $b''\in A$. We will see in Step (iii) below that the case $M_1=A(4b'', 2\delta, \pi) \oplus (\oplus H(1))$ with $b''\in A$ is reduced to the case $M_1=\oplus H(1)$ (possibly empty).\
1. Assume that $M_1=\oplus H(1)$ and $(a+a')/2\in (2)$. Then $$M_0''= \left((2)e_1\oplus B(e_1+e_2)\right) \oplus (\oplus_{\lambda}\pi H_{\lambda})\oplus M_2,$$ as explained in the argument (i) of Step (1) in the construction of $\psi_j$. For this basis, the image of a fixed element of $F_j$ in the orthogonal group associated to $M_0''/\pi M_0''$ is $$T_1=\begin{pmatrix} \begin{pmatrix} 1 & (z_j^{\ast})_1\\ 0 & 1 \end{pmatrix} &0 \\ 0 & id \end{pmatrix}.$$ Here, $(z_j^{\ast})_1$ is in $R$ such that $z_j^{\ast}=(z_j^{\ast})_1+\pi\cdot (z_j^{\ast})_2$ as explained in the paragraph before Equation (\[e42\]). Then by using a similar argument used in the above case (1), the Dickson invariant of $T_1$ is $(z_j^{\ast})_1$.
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 $1$ by Equation (\[e42\]), $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
2. Assume that $M_1=\oplus H(1)$ and that $(a+a')/2$ is a unit. Then $$M_0''= (2)e_1\oplus B\left(\pi e_1+1/\sqrt{(a+a')/2}\cdot (e_1+e_2)\right) \oplus (\oplus_{\lambda}\pi H_{\lambda})\oplus M_2,$$ 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 $F_j$ in the orthogonal group associated to $M_0''/\pi M_0''$ is $$T_1=\begin{pmatrix} \begin{pmatrix} 1 & 1/\sqrt{(a+a')/2}(z_j^{\ast})_1\\ 0 & 1 \end{pmatrix} &0 \\ 0 & id \end{p
| 755
| 804
| 945
| 792
| 3,162
| 0.77442
|
github_plus_top10pct_by_avg
|
i*. Note that a small value is added to *Y~i~* for the logarithm.
Adding epigenetic effects
-------------------------
Discrete values representing epigenetic states of a gene *i* are added to the regression model
where *H* is the type of histone mark (neither mark=1.0, H3K27me3=2.0, bivalent mark=3.0, H3K4me3=4.0), *M* is the DNA methylation (no annotation=1.0, methylation=2.0, unmethylation=3.0), *C* is the CpG island (absence=1.0, presence=2.0), and *α*, *β*, *γ* are the regression coefficients for *H*, *M*, *C*, respectively.
Fitting and reducing regression models
--------------------------------------
Explanatory variables in a regression model are log-transformed and quantile-normalized. 10 runs of 10-fold cross validation (CV) measure the average correlation coefficient (CV-*R*) and the average proportion of variation explained by the model (CV-*R*^2^). The stepwise model selection is done by stepAIC in R language with the backward and forward procedure. The regression model with higher-order interactions are reduced by a pipeline developed in house; ANOVA in R language first diagnoses the significance of each explanatory variable in the given saturated model. Next, significant variables (*p* \< 0.05 in F-test) are gathered. Finally, the best model is constructed by adding and removing the collected variables one by one in increasing order of *p*-value until CV-*R*^2^ is not improved anymore.
Competing interests
===================
The authors declare that they have no competing interests.
Supplementary Material
======================
###### Additional file 1
**Extended analysis of ChIP-seq data** This file provides tables including the summary of tag mapping (Table S1), the fold change of remapped tags over the original data (Table S2), the number of peaks in five datasets (Table S3), and the thresholds used to detect significant peaks (Table S4).
######
Click here for file
###### Additional file 2
**Comprehensive analysis of gene regulation in mouse ESC** This file provides figures inclu
| 756
| 112
| 2,109
| 874
| 1,458
| 0.789311
|
github_plus_top10pct_by_avg
|
19 (17.6) 38 (24.2) 0.75 (0.12, 4.88) C/T 27 (17.1) 5 (13.5) 2.70 (0.20, 35.75)
T/T 2 (1.9) 3 (1.9) 0.4332 1 T/T 2 (1.3) 1 (2.7) 0.7216 1
C/C + C/T 106 (98.1) 154 (98.1) 0.9723 1.03 (0.17, 6.29) C/C + C/T 156 (98.7) 36 (97.3) 0.5227 2.17 (0.19, 24.55)
**rs4648727**^**a**^ **rs4648727**^**a**^
A/A 10 (9.3) 14 (9.0) 0.94 (0.38, 2.29) A/A 16 (10.1) 4 (11.1) 1.46 (0.44, 4.83)
A/C 50 (46.3) 79 (50.6) 0.83 (0.50, 1.39) A/C 79 (50.0) 9 (25.0) 3.20 (1.39, 7.41)
C/C 48 (44.4) 63 (40.4) 0.7779 1 C/C 63 (39.9) 23 (63.9) 0.0194\* 1
A/A + A/C 60 (55.6) 93 (59.6) 0.5112 0.86 (0.52, 1.41) A/A + A/C 95 (60.1) 13 (36.1) 0.0089\* 2.67 (1.26, 5.65)
**rs12126768**^**a**^ **rs12126768**
G/G 6 (5.5) 9 (5.8) 0.76 (0.26, 2.25) G/G 6 (3.8) 4 (10.8) 0.42 (0.11, 1.61)
G/T 33 (30.6) 68 (43.6) 0.56 (0.33, 0.94) G/T 63 (39.9) 8 (21.6) 2.21 (0.94, 5.22)
T/T 69 (63.9) 79 (50.6) 0.0891 1 T/T 89 (56.3) 25 (67.6) 0.0436\* 1
G/G + G/T 39 (36.1) 77 (49.4) 0.0330\* 0.58 (0.35, 0.96) G/G + G/T 69 (43.7) 12 (32.4) 0.2118 1.62 (0.76, 3.44)
^a^: Contains 1 missing data point in the RVR
| 757
| 4,328
| 784
| 352
| null | null |
github_plus_top10pct_by_avg
|
that the $S2R_p$ policy is defined to reference the $P2P$ values regardless of whether the $P2P_p$ policy itself is active.
The separation kernels of VxWorks MILS, LynxSecure, INTEGRITY-178B and PikeOS meet the security functionalities and security assurance requirements in SKPP.
### Data Separation Properties
Data separation requires that resources of a partition must be completely independent of other partitions.
- MASK Separation Properties
The DoD of USA set out in 1997 to formally construct a separation kernel, a Mathematically Analyzed Separation Kernel (MASK) [@Martin00; @Martin02], which has been used by Motorola on its smart cards. MASK regulates communication between processes based on separation policies. The separation policies of MASK include two separation axioms: the *communication policy* and an anonymous policy. In the abstraction of the MASK separation kernel, *Multiple Cell Abstraction (MCA)* describes the system. The *Init* and *Next* operations evolve the system. *Cells* and *Single Cell Abstraction (SCA)* are domains of execution or a context, which consist of a collection of strands. Each strand is a stream of instructions to be executed when a message is input to a strand of a cell. The communication policy is as follows.
$$\label{eq:mask_comm}
\begin{aligned}
Fiber_y(MCA) \neq Fiber_y(Next_x(MCA)) \\
\Rightarrow Communicates(x,y)
\end{aligned}$$
where $Fiber_y$ determines the SCA corresponding to the CellID $y$ in the subscript, $Next_x$ advances the system state by advancing the cell indicated by the subscript $x$. The policy states that if the fiber of cell $y$ changes as the result of advancing the state of cell $x$, it must be the case that $x$ is permitted to communicate with $y$. The second separation constraint upon cells is as follows.
$$\label{eq:mask_comm2}
\begin{aligned}
Fiber_x(MCA_1) = Fiber_x(MCA_2) \Rightarrow \\
Fiber_y(MCA_1) = Fiber_y(MCA_2) \Rightarrow \\
Fiber_y(Next_x(MCA_1)) = Fiber_y(Next_x(MCA_2))
\end{aligned}$$
The policy represents that if an action
| 758
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| 0.807656
|
github_plus_top10pct_by_avg
|
scending aorta (50 mm) and diffuse atherosclerotic disease, without any obstructive lesions. Elective transfemoral TAVI was scheduled.
{#ytz069-F1}
######
Blood tests results
Parameters (unit) Pré-TAVI First day Post-TAVI Second day Post-TAVI Fourth day Post-TAVI At discharge Normal range
---------------------------------- ---------- --------------------- ---------------------- ---------------------- -------------- --------------
Haemoglobin (g/dL) 12.7 11.4 9.6 9.9 10.4 12--15.3
Leucocytes (×10^9^/L) 4.0 6.5 3.04 3.0 3.0 4.0--11.0
Platelets (×10^9^/L) 181 115 119 146 170 150--450
Partial thromboplastin time (s) 27.3 31.0
Prothrombin time (s) 11.3 11.6
Urea (mg/dL) 52 48 36 22 21 16--49
Creatinine (mg/dL) 0.99 0.9 0.89 0.75 0.83 0.51--0.95
Sodium (mmol/L) 139 140 143 139 139 135--145
Potassium (mmol/L) 4.8 4.1 3.9 3.8 4.2 3.5--5.1
C-reactive protein (mg/dL) 0.09 7.49 7.55 2.4 1.51 \<0.5
Aspartate aminotrans
| 759
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| 358
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| null | null |
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|
nd{aligned}$$ where $g$ takes values in the supergroup $G$ and $Tr'$ indicates the non-degenerate bi-invariant metric. We will use the normalization and results of [@Ashok:2009xx]. The Wess-Zumino-Witten points are given by the equation $1/f^2 = |k|$. Note that the action is invariant under group inversion $g \leftrightarrow g^{-1}$ and simultaneous orientation reversal $z \leftrightarrow \bar{z}$.
The conformal current algebra {#the-conformal-current-algebra .unnumbered}
-----------------------------
[From]{} the action we can calculate the classical currents associated to the invariance of the theory under left multiplication of the field $g$ by a group element in $G_L$ and right multiplication by a group element in $G_R$. The classical $G_L$ currents are given by $$\begin{aligned}
\label{normeqn}
j_{L,z} &= c_+ \partial g g^{-1}\cr
j_{L,\bar{z}} &= c_- \bar{\partial} g g^{-1} \,,\end{aligned}$$ where the constant $c_+$ and $c_-$ are given in terms of the couplings by: \[c+-\] c\_ = - . Similarly, we also have the left-invariant currents that generate right multiplication: $$\begin{aligned}
j_{R,z} &= -c_- g^{-1} \p g \cr
j_{R,\bar z} &= -c_+ g^{-1} \bar \p g\, .\end{aligned}$$ The operator product expansions (OPEs) satisfied by the left currents have been derived in [@Ashok:2009xx]. They read: $$\begin{aligned}
\label{euclidOPEs}
j_{L,z}^a (z) &j_{L,z}^b (0) \sim \ \kappa^{ab} \frac{c_1}{z^2}
+ {f^{ab}}_c \left[ \frac{c_2}{z} j_{L,z}^c(0)+ (c_2-g) \frac{\bar{z}}{z^2} j_{L,\bar{z}}^c(0) \right] \cr
& + {f^{ab}}_c \left[-\frac{g}{4}\frac{\bar z}{z}(\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{c_2}{2} \partial_z j_{L,z}^c(0)+ \frac{c_2-g}{2} \frac{\bar{z}^2}{z^2} \partial_{\bar z}j_{L,\bar{z}}^c(0) \right] \cr
& + :j_z^a j_z^b:(0)
+ {{A}^{ab}}_{cd} \frac{1}{2} \frac{\bar z^2}{z^2}:j_{\bar z}^{ c} j_{\bar z}^{d }:(0) + {{B}^{ab}}_{cd} \frac{\bar z}{z} :j_z^{ c} j_{\bar z}^{d }:(0) -
{{C}^{ab}}_{cd} \log |z|^2 :j^{ c}_z j_z^{e } :(0) \cr
& + ... \cr
%
j_{L,\bar{z}}^a (z)& j_{L,\bar{z
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|
{-\beta }}\Big).
\end{aligned}$$ Recall that $\beta '_\nu \not={\alpha }_{i_1}$, since $\nu >1$. Moreover, $$\begin{aligned}
&e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big(
\frac {1-e^{-\bfun{\chi _2}({\alpha }_{i_1}){\alpha }_{i_1}}}
{1-e^{-{\alpha }_{i_1}}}\Big)\\
&\quad =e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}
\frac {1-e^{\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}}
{1-e^{{\alpha }_{i_1}}}
=\frac {1-e^{-\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}}
{1-e^{-{\alpha }_{i_1}}}.
\end{aligned}$$ Therefore $$\begin{aligned}
{\mathrm{ch}\,V}=&\frac {e^{-t\beta _\nu}-e^{-\bfun{\chi }(\beta _\nu )\beta _\nu }}
{1-e^{-\beta _\nu }}
\prod _{\beta \in R_+^\chi \setminus \{\beta _\nu \}}
\frac {1-e^{-\bfun{\chi }(\beta )\beta }}
{1-e^{-\beta }}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} {P}^\chi ({\alpha },\beta _\nu ;t)
e^{-{\alpha }}
\end{aligned}$$ by Lemma \[le:P1\]. This proves Eq. .
Since $t>0$, ${\mathrm{ch}\,V}\not={\mathrm{ch}\,M}^\chi (\Lambda )$. By assumption on $t$, $V_{t\beta _\nu }\not=0$, and hence $V\not=0$. Since $V$ is a ${\mathbb{Z}}^I$-graded $U(\chi )$-submodule of $M^\chi (\Lambda )$, the lemma is proven.
\[th:Shapdet\] Let $\chi \in {\mathcal{X}}_5$. For all ${\alpha }\in {\mathbb{N}}_0^I$, the Shapovalov determinant of $U(\chi )$ is the family $(\det ^\chi _\al
)_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
\label{eq:det}
\det \nolimits ^\chi _{\alpha }=
\prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
({\rho ^{\chi}} (\beta )K_{\beta }
-\chi (\beta ,\beta )^t L_{\beta })
^{{P}^\chi ({\alpha },\beta ;t)}.
\end{aligned}$$
Let ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and let $\{F'_1,F'_2,\dots ,F'_k\}$ be a basis of $U^-(\chi )_{-{\alpha }}$. Then ${\mathrm{Sh}}(F'_i,F'_j)\in \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma
={\alpha }}{\Bbbk }K_\beta L_\gamma $ by Lemma \[le:Shfcoeffs\], and hence
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|
the ket, which will terminate upon raising the highest-weight state. Therefore the vector bases with different weights $k, k^\prime$ are orthogonal.
Since we have also proved that vector bases with different $m$ and $h-k$ are orthogonal, the proof of orthogonality for vector bases is done. It may not be obvious that the proof holds unaltered for scalars/vectors/tensors. In all the relevant steps above, we have noted where each argument works for each of the three types of fields.
Therefore we arrive at the conclusion that the scalar, vector, and symmetric tensor bases in global coordinates form orthogonal basis sets.$\qed$
Separation of variables {#sec:separation-variables}
=======================
In this section we show that with the scalar, vector, and tensor bases we have obtained, it is possible to separate variables for many physical systems in NHEK spacetime. One can show that all conclusions in this section hold for both Poincaré coordinates and global coordinates. In global coordinates the results are in general more complicated. Therefore for concreteness all results in this section are given in Poincaré coordinates.
The main result of this section can be summarized with the schematic equation: $$\begin{aligned}
\mathcal{D}_{x}
\left[
\left(
\parbox{2.cm}{\centering\tiny${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$\\ structure ($T, \Phi, R$)}
\right)^{(m,h,k)}
\times
\left(
\parbox{1.5cm}{\centering\tiny$u$ (or $\cos\theta$)\\ dependence}
\right)
\right]
= {\nonumber}\\
\left(
\parbox{2.cm}{\centering\tiny${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$\\ structure ($T, \Phi, R$)}
\right)^{(m,h,k)}
\times
\mathcal{D}^{(m,h)}_{u}
\left[
\parbox{1.5cm}{\centering\tiny$u$ (or $\cos\theta$)\\ dependence}
\right]
\,. {\nonumber}\end{aligned}$$ Here, $\mathcal{D}_{x}$ is an ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$-equivariant differential operator, which takes derivatives in the $T,\Phi,R,u$ directions. We completely specify the $T,\Phi,R$ dependence by being in a
| 762
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|
-t0A2_Table A2
######
LCMS analysis of FIX-PCC pre-filtrate following filtration with 6 μm filter.
Accession Description Score Coverage MW \[kDa\] calc. pI
----------- ----------------------------------------------- --------- ---------- ------------ ----------
P00734 Prothrombin 2584.88 71.86 70.0 5.90
P0C0L5 Complement C4-B 2441.20 77.69 192.6 7.27
P0C0L4 Complement C4-A 2393.34 74.89 192.7 7.08
P19827 Inter-alpha-trypsin inhibitor heavy chain H1 1154.64 54.34 101.3 6.79
P19823 Inter-alpha-trypsin inhibitor heavy chain H2 1136.27 55.50 106.4 6.86
P02760 Protein AMBP 337.33 25.57 39.0 6.25
Q06033 Inter-alpha-trypsin inhibitor heavy chain H3 332.27 43.15 99.8 5.74
P00740 Coagulation factor IX 151.88 54.66 51.7 5.47
P00742 Coagulation factor X 81.34 41.19 54.7 5.94
P02768 Serum albumin 60.16 37.44 69.3 6.28
P01834 Immunoglobulin kappa constant OS=Homo sapiens 48.66 81.31 11.8 6.52
P51884 Lumican 42.56 35.50 38.4 6.61
P49747 Cartilage oligomeric matrix protein 41.14 16.12 82.8 4.60
P01857 Immunoglobulin heavy constant gamma 1 40.75 41.21 36.1 8.19
P67936 Tropomyosin alpha-4 chain 31.98 36.29 28.5 4.69
P01861 Immunoglobulin heavy constant gamma 4 30.25 28.13 35.9 7.36
P01860 Immunoglobulin heavy constant gamma 3 28.46 27.06 41.3 7.90
P04004 V
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| 270
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| null | null |
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|
again by abuse of language) the partial order induced on $\phi _{AD}^{Q}/\approx $ by the preorder defined on $\phi _{AD}^{Q}$. Then, let us show that $(\phi
_{AD}^{Q}/\approx ,\prec )$ is order isomorphic to $(\mathcal{L(S)},\subset
) $.
Let us consider the mapping
$f_{\approx }:[\delta ]_{\approx }\in \psi _{AD}^{Q}/\approx
\;\longrightarrow $ $\mathcal{S}_{\delta }\in \mathcal{L(S)}$.
This mapping is obviously well defined because of the characterization of $\approx $ in Sec. 3.4. Furthermore, the following statements hold.
\(i) *For every* $\delta \in \phi _{AD}^{Q}$*, one and only one elementary af* $\vdash E(x)$* exists such that* $\vdash E(x)\in
\lbrack \delta ]_{\approx }$.
\(ii) *The mapping* $f_{\approx }$* is bijective.*
\(iii) *For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*,* $[\delta _{1}]_{\approx }\prec \lbrack \delta
_{2}]_{\approx }$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.*
Let us prove (i). Consider $[\delta ]_{\approx }$. Since $\mathcal{S}_{\delta }\in \mathcal{L(S)}$ and $\rho $ is bijective (Sec. 2.2), a property $E\in \mathcal{E}$ exists such that $E=\rho ^{-1}(\mathcal{S}_{\delta })$, hence $\mathcal{S}_{\delta }=\mathcal{S}_{E}$. It follows that $[\delta ]_{\approx }$ contains the af $\vdash E(x)$, for $\mathcal{S}_{\vdash E(x)}=\mathcal{S}_{E}$ (Sec. 3.2). Moreover, $[\delta ]_{\approx }$ does not contain any further elementary af. Indeed, let $\vdash F(x)$ be an elementary af of $\phi _{AD}^{Q}$ with $E\neq F$: then, $\mathcal{S}_{E}\neq
\mathcal{S}_{F}$, hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash
F(x)}$, which implies $\vdash F(x)\notin \lbrack \delta ]_{\approx }$. Thus, statement (i) is proved.
The proofs of statements (ii) and (iii) are then immediate. Indeed, statement (ii) follows from (i) and from the definition of $f_{\approx }$, while statement (iii) follows from (ii) and from the definition of $\prec $ on $\phi _{AD}^{Q}/\approx $.
Because of (ii) and (iii), the poset $(\phi _{AD}^{Q}/\approx ,\prec )$ is order-isomorphi
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| null | null |
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|
n as Eq. . We substitute the highest-weight metric perturbation into the left hand side of the linearized Einstein equation and the result is given by Eq. .
$$\begin{aligned}
{}
h^{(m\,h\,0)}_{ab} &=
R^h e^{im\Phi}
\left[
\begin{array}{cccc}
R^{+2}C_{TT}(u)
& R^{+1} C_{T\Phi}(u)
& R^{+0}C _{TR}(u)
& R^{+1} C_{uT}(u) \\
* & R^{+0} C_{\Phi\Phi}(u)
& R^{-1}C_{R\Phi}(u)
& R^{+0} C_{u\Phi}(u) \\
* &
* & R^{-2}C_{RR}(u)
& R^{-1}C_{uR}(u) \\
* &
* &
* & R^{+0} C_{uu}(u)
\end{array}
\right]
\label{eq:hab-highest}
\\
\intertext{}
G^{(1)}_{ab} [ h^{(m\,h\,0)} ] &=
R^h e^{im\Phi}
\left[
\begin{array}{cccc}
R^{+2}\mathcal{D}^{(m,h)}_{TT}[\mathbf{C}(u)]
& R^{+1}\mathcal{D}^{(m,h)}_{T\Phi}[\mathbf{C}(u)]
& R^{+0}\mathcal{D}^{(m,h)}_{TR}[\mathbf{C}(u)]
& R^{+1}\mathcal{D}^{(m,h)}_{uT}[\mathbf{C}(u)] \\
* & R^{+0}\mathcal{D}^{(m,h)}_{\Phi\Phi}[\mathbf{C}(u)]
& R^{-1}\mathcal{D}^{(m,h)}_{R\Phi}[\mathbf{C}(u)]
& R^{+0}\mathcal{D}^{(m,h)}_{u\Phi}[\mathbf{C}(u)] \\
* &
* & R^{-2}\mathcal{D}^{(m,h)}_{RR}[\mathbf{C}(u)]
& R^{-1}\mathcal{D}^{(m,h)}_{uR}[\mathbf{C}(u)] \\
* &
* &
* & R^{+0}\mathcal{D}^{(m,h)}_{uu}[\mathbf{C}(u)]
\end{array}
\right]
\label{eq:G1-hab}\end{aligned}$$
Again notice that the $(T,\Phi,R)$ dependence has factored straight through the differential operator, resulting in ten coupled ODEs for the ten $C$-functions, which we have collected together as $\mathbf{C}(u)$. The expressions for all these differential operators are given in App. \[app:A-B-C-matrice\].
We can easily verify that $G^{(1)}$ commutes with $\mathcal{L}_{H_-}$, therefore the linearized Einstein operator acting on a basis function with arbitrary weight can be obtained easily by repeatedly applying the lowering o
| 765
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|
{T}}$ involves a semistandard homomorphism which does not occur in any other ${\psi_{2,2}}\circ{\hat\Theta_{T'}}$ (except possibly for a tableau $T'$ already ruled out in the paragraph above). So we may restrict attention to those $T$ having at most one $2$ and one $3$ in the first row.
Now return to ${\psi_{2,1}}\circ{\hat\Theta_{T}}$, for $T$ of the form $$\begin{aligned}
&\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;{x_1}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{x_s},;2;{z_1};{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;3;k),
\\
\intertext{where $x_1,\dots,x_s,z_1,\dots,z_t,k$ are the integers $4,\dots,a$ in some order. When we express ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms, we find that the homomorphism labelled by}
&\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;{x_1}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{x_s},;2;2;{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{z_1};k)\end{aligned}$$ occurs with non-zero coefficient; but this homomorphism does not occur in any other ${\psi_{2,1}}\circ{\hat\Theta_{T'}}$ (except for $T'$ having two $3$s in its first row). So for any $T$ of the above form, the coefficient of ${\hat\Theta_{T}}$ in $\theta$ must be zero.
Now any semistandard tableau which contributes to $\theta$ must be of one of the following eight forms. $$\begin{aligned}
2
1.\quad&
\gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;3;\star_3{{\begin{tikzpic
| 766
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| 165
| 0.821369
|
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|
redict future video frames in a self-supervised manner. Remarkably, the model is able to capture a wide variety of seemingly disparate phenomena observed in visual cortex, ranging from single unit response dynamics to complex perceptual motion illusions. These results suggest potentially deep connections between recurrent predictive neural network models and the brain, providing new leads that can enrich both fields.'
author:
- |
William Lotter$^1$, Gabriel Kreiman$^1$, David Cox$^{1,2,3}$\
$^1$Harvard University, $^2$MIT-IBM Watson AI Lab, $^3$IBM Research\
[lotter.bill1@gmail.com, gabriel.kreiman@tch.harvard.edu, david.d.cox@ibm.com]{}
bibliography:
- 'main.bib'
title: A neural network trained to predict future video frames mimics critical properties of biological neuronal responses and perception
---
Introduction
============
The fields of neuroscience and machine learning have long enjoyed productive dialogue, with neuroscience offering inspiration for how artificial systems can be constructed, and machine learning providing tools for modeling and understanding biological neural systems. Recently, as deep convolutional neural networks (CNNs) have emerged as leading systems for visual recognition tasks, these same models have emerged—without any modification or tailoring to purpose—as leading models for explaining the population responses of neurons in primate visual cortex [@Yamins_2014; @Yamins_2016; @Kriegeskorte_2014]. These results suggest that the connections between artificial deep networks and brains may be more than skin deep.
However, while deep CNNs capture some important details of the responses of visual cortical neurons, they fail to explain other key properties of the brain. Notably, the level of strong supervision used to train state-of-the-art CNNs is much greater than that available to our brain. To the extent that representations in the brain are similar to those in CNNs trained on e.g. ImageNet, the brain must be arriving at these representations by different, largely un
| 767
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|
\pi t_i\\ \pi y_i&1+\pi x_i&\pi z_i\\ v_i&u_i&1+\pi w_i \end{pmatrix}=
\begin{pmatrix} \mathrm{id}+\pi s_i^{\prime}&\pi r_i^{\prime}&\pi^2t_i^{\prime}\\ \pi^2y_i^{\prime}&1+\pi^2x_i^{\prime}&\pi^2z_i^{\prime}\\ \pi v_i^{\prime}&\pi u_i^{\prime}&1+\pi^2w_i^{\prime} \end{pmatrix}.$$
4. If $i$ is even and $L_i$ is *of type II*, then $m_{i,i}=\mathrm{id}+\pi m_{i,i}^{\prime}$.
5. If $i$ is even and $L_i$ is *of type I*, then $z_i^{\ast}=\pi (z_i^{\ast})^{\prime}$. This equation yields the following (formal) equation by condition (d) of the description of an element of $\tilde{M}(R)$ given at the paragraph following Lemma \[la1\]:
$$\label{52}
z_i^{\prime}+\delta_{i-2}k_{i-2, i}^{\prime}+\delta_{i+2}k_{i+2, i}^{\prime}=0 \left( =\pi (z_i^{\ast})^{\prime} \right).$$
Here, $k_{i-2, i}=\pi k_{i-2, i}^{\prime}$ and $k_{i+2, i}=\pi k_{i+2, i}^{\prime}$, where $k_{i-2, i}$ and $k_{i+2, i}$ are as explained in (d) of the description of an element of $\tilde{M}(R)$.
6. If $i$ is odd and $L_i$ is *free of type I*, then $$m_{i,i}=\begin{pmatrix} s_i&\pi r_i&t_i\\ y_i&1+\pi x_i& u_i\\\pi v_i&\pi z_i&1+\pi w_i \end{pmatrix}
=\begin{pmatrix} \mathrm{id}+\pi s_i^{\prime}&\pi^2 r_i^{\prime}&\pi t_i^{\prime}\\ \pi y_i^{\prime}&1+\pi^2 x_i^{\prime}& \pi u_i^{\prime}\\\pi^2 v_i^{\prime}&\pi^2 z_i^{\prime}&1+\pi^2 w_i^{\prime} \end{pmatrix}.$$ If $i$ is odd and $L_i$ is *of type II* or *bound of type I*, then $m_{i,i}=\mathrm{id}+\pi m_{i,i}^{\prime}$.
7. If $i$ is odd and $L_i$ is *bound of type I*, then $$m_{i,i}^{\ast}=\pi (m_{i,i}^{\ast})^{\prime}, ~~~~~~~ m_{i,i}^{\ast\ast}=\pi (m_{i,i}^{\ast\ast})^{\prime}.$$ These two equations yield the following (formal) equations by conditions (e) and (f) of the description of an element of $\tilde{M}(R)$ given at the paragraph following Lemma \[la1\]: $$\label{ea72}
\left\{
\begin{array}{l}
\delta_{i-1}v_{i-1}\cdot m_{i-1, i}^{\prime}+\delta_{i+1}v_{i+1}\cdot m_{i+1, i}^{\prime}=0
~~~ \left(=\pi (m_{i,i}^{\ast})^{\prime}\right);\\
| 768
| 4,164
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| null | null |
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|
computed, the app sets the system in UM and manages the model transfer to the MCU. This operation allows to correctly store the personalized model in the Flash memory of the MCU and use it for real-time classification. The communication protocol sends 3 types of packets containing the following: the general configuration values: SVM type, gamma parameter, the number of classes and the number of features.the model parameters: the $\rho$ parameter, the total number of SVs, and the SVs for each class.the SVs and their coefficients, sending one packet for each SV.
When a packet has been sent, the interface waits the Acknowledgment (ACK) message sent from the board before sending a new packet. Each packet has its checksum to check its integrity. [Figure 4](#sensors-17-00869-f004){ref-type="fig"} shows the transmission scheme between the gateway app and the wearable node.
During the prosthesis control, the classification algorithm and the prosthetic controller run in real time on the embedded board, hence the interaction with the gateway is not required during continuous use.
3.5. Control Strategy {#sec3dot5-sensors-17-00869}
---------------------
For mechanical reasons, prosthetic hands execute the various grasps and movements starting always from a reset state, normally the open hand position. After the execution of a gesture, it is, therefore, necessary to return to the reset configuration before performing another movement. This strategy allows to control the prosthesis movement using only the motor current feedback provided by the integrated driver, since the system always starts from a known configuration.
The diagram of the FSM that controls the the system in CM is represented in [Figure 5](#sensors-17-00869-f005){ref-type="fig"}. In the *ADC Acquisition*, ADC peripheral extracts the mean value of 16 consecutive samples of the 4 EMG channels. The total conversion time of the peripheral is 32 $\mathsf{\mu}$s for the 16 consecutive samples and this does not affect the real time requirements of the system. Th
| 769
| 56
| 874
| 755
| 2,351
| 0.780647
|
github_plus_top10pct_by_avg
|
dth"}
### Topics navigation {#subsubsec:topicalnavi .unnumbered}
#### Wikigame topic dataset
Performing our analyses by representing Wikipedia pages by their topical categories shows a much clearer and more interesting picture as one can see in Figure \[fig:paths\_cat\]. Similar to above we can see (A) that the log likelihoods are rising with higher orders. However, in contrast to the Wikigame page dataset, we can now see (B) that several higher order Markov chain models are significantly better than lower orders. In detail, we can see that the appropriate Markov chain order is at least of order one and we can also observe a trend towards an order of two or three. Nevertheless, as pointed out in the section entitled “”, it is hard to concretely suggest one specific Markov chain order from these pairwise comparisons which is why we resort to this extended repertoire of model selection techniques described next.
The AIC (C) and BIC (D) statistics show further indicators – even though they are disagreeing – that the appropriate model is of higher order. Concretely, the suggest an order of three or two respectively by exhibiting the lowest values at these points. Not surprisingly, AIC suggests a higher order compared to BIC as the latter model selection method additionally penalized higher orders by the number of observations as stated in the section called “”.
The Bayesian inference investigations (E, F) exhibit a clear trend towards a Markov chain of order two. The results in (F) nicely illustrate the inherent Occam’s razor of the Bayesian model selection method as both priors – (a) no penalty and (b) exponential penalty for higher orders – suggest the same order[^14]. Finally, the cross validation results (G) confirm that a second order Markov chain produces the best results, while a third order model is nearly as good.
**Summary:** Overall, we can see that representing Wikigame paths as navigational sequences of corresponding topics leads to more interesting results: Higher order Markov c
| 770
| 3,253
| 1,835
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| null | null |
github_plus_top10pct_by_avg
|
ions can then be solved automatically in the constraint-based approach, based on Proposition 3 of [@DBLP:journals/corr/abs-0912-4360].
\[prop:rem\_imp\] Let $prem$ be a polynomial over $n$ variables and $conc$ a polynomial over 1 variable, both with natural coefficients, where $conc$ is not a constant. Moreover, let $p_1,\ldots,p_{n+1},q_1,\ldots,q_{n+1}$ be arbitrary polynomials with integer coefficients[^3] over the variables $\overline{X}$. If $$\forall \overline{X} \in {{\mathbb{N}}}: conc(p_{n+1})-conc(q_{n+1})-prem(p_1,\ldots,p_n)+
prem(q_1,\ldots,q_n) \geq 0$$ is valid, then $\forall \overline{X} \in {{\mathbb{N}}}: p_1 \geq q_1, \ldots,p_n\geq q_n \Longrightarrow p_{n+1}\geq q_{n+1}$ is also valid. $\hfill \square$
### Introducing the symbolic coefficients.
To represent half-open domains in the implication by symbolic coefficients, the domains are described by two symbolic coefficients, one upper or lower limit and one for the direction. Constraints of the form $Exp \in Dom_I$ in the implication, are replaced by constraints of the form $d_I * Exp \geq d_I* c_I$ with $d_I$ either $1$ or $-1$, describing the domain $\lbrace c_I, c_I-1, \ldots\rbrace$ for $d_I=-1$ and $\lbrace c_I, c_I+1, \ldots\rbrace$ for $d=1$. The values to be inferred for the integers of the query should satisfy the precondition of the implication. Off course, the symbolic coefficients $c_I$ should also be consistent with the values of the integers in the query.
In Example \[example:count\_to\_int\_cons\], we introduced constraints on the integer variable $\underline{N}$, $0 > \underline{N}$ and $0 + 1 > \underline{N}$, proving non-termination for queries in $Den(\leftarrow count\_to(\underline{N},L))$. By convention, we denote the symbolic coefficients as constants. For the integer variable $\underline{N}$, we introduce the symbolic coefficient $n$.
The implication introduced in Example \[example:count\_to\_int\_cons\], for the path from $N_5$ to $N_9$ in Figure \[fig:count\_to\], does not contain constraints on the domains. When
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# General characteristics of the study population (N = 277).
{#pone.0231480.t001g}
Characteristics Genotypes
---------------------------------- ---------------- ------------- ------------- ------------
Gender N (%) M, 174 (62.81) 28 (60.86) 114 (67.85) 33 (52.38)
F, 103 (37.18) 18 (39.13) 54 (32.14) 30 (47.61)
Age (years) Mean±SD 51.15±9.75 45.91±10.42 50.64±11.01
ALT, Mean±SD 58 ±5.32 56 ±3.05 56 ±3.43
AST Mean±SD 48±2.51 42±3.11 45±4.02
HCV Viral quantification Mean±SD 5.90±2.98 6.32±3.56 6.17±2.82
M = Male, F = Female, ALT = alanine aminotransferase, AST = aspartate aminotransferase
Prevalence of RAPs in the NS3 region {#sec010}
------------------------------------
Sequence analysis of HCV NS3 region showed overall 29.24% (81/277) of infected patients harbored RAPS at positions 36, 54, 55, 80, 122, 155, 158, 168 and 170 ([Table 2](#pone.0231480.t002){ref-type="table"}). Overall RAPs were found in 26.08% (12/46) of HCV-1a, 30.95% (52/168) of HCV-3a and 26.98% (17/63) of HCV-3b infected patients ([Fig 1](#pone.0231480.g001){ref-type="fig"}). The prevalence of RAPs was significantly higher (*P* = 0.026) in HCV-3a as compared to HCV-1a and HCV-3b. Among these, 64.19% (52/81) of individuals were found with one and 35.80% (29/81) with two RAPs.
{#pone.0231480.g001}
10.1371/journal.pone.0231480.t002
###### Pre-existing RAPs to NS3 protease inhibitors in different HCV genotypes.
{#pone.0231480.t002g}
Amino acid Position HCV-1a (N = 46) HCV-3a (N = 168) HCV-3b (N = 63)
--------------------- ------------------- ------------------ ------------------
36 V36A/G (10.87%) V36L (8.33%) V3
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s^{-q}(\alpha'_{n,j}) (-q,q) - \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q).$$
Using the equalities $(-p,p)=\s^{-p}((p,-p))$ and $(-q,q)=\s^{-q}((q,-q))$, the last equality above can be rewritten as follows $$\label{1=ab}
1 = (1-\s^{-p})(a) + (1-\s^{-q})(b)$$ where $a = \alpha_{n,i} \s^{p}(\beta'_{m,t}) (p,-p) \in K[H]$ and $b = \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q) \in K[H]$.
Recall that $P = P_{-q} + P_p$, $Q = Q_{-p} + Q_q$, $$\label{2=ab}
p = mt=ni \geq 2 \text{ and } q=ms=nj \geq 1.$$
Suppose that $p=q$, and so $P = P_{-p} + P_p$, $Q = Q_{-p} + Q_p$. Then $Q=\lambda P_p$ for some $\lambda\in K^*$. Since $1=[P,Q]=[P, Q-\lambda P]$, $m(P)=2$ and $m(Q-\lambda P)=1$. By the case (iv), the pair $(P, Q-\lambda P)$ is obtained from the pair $(Y, X)$ by applying an automorphism of the Weyl algebra $A_1$. So, either $p<q$ or $p>q$. In view of $(P,Q)$-symmetry ($1=[P,Q]=[-Q,P]$), it suffices to consider, say, the first case only. Since $p<q$, the equalities (\[2=ab\]) imply that $i<j$ and $t<s$. Then, using Equation (\[degsf\]) and the fact that $\deg (p,-p) = p$ for all $p \geq 1$, we see that $$\deg a = \deg \alpha_{n,i} + \deg \beta'_{m,t} + p -1,$$ $$\deg b = \deg \alpha'_{n,j} + \deg \beta_{m,s} + q -1.$$ Since $i<j$ and $t<s$, $\deg \alpha_{n,i} < \deg \alpha'_{n,j}$ and $\deg \beta'_{m,t} < \deg \beta_{m,s}$. In particular, $\deg a < \deg b$. This equality contradicts Equation (\[1=ab\]) since, by Equation (\[degsf\]), $$0=\deg 1 = \deg a - 1 - \deg b +1 = \deg a - \deg b >0.$$ This means that the cases $p<q$ and $p>q$ are impossible. The proof of the theorem is complete. $\Box$
\[24Sep16\] Let $P, Q$ be elements of the first Weyl algebra $A_1$ with $m(P)=1$ or $m(Q)=1$. If $[P,Q]=1$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some automorphism $\tau\in\Aut_K (A_1)$.
[**Proof:**]{} Without loss of generality we may assume $m(Q)=1$ and $m(P)\geq 3$. That is $Q = Q_q$ and $P = \sum_{i\in I} P_i$, where $I \subset \Z$ is a finite set, $q\in \Z\setminus\{0\}$ and the elements $Q_q$ and $P_i$ are homogeneous in $A_1$
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)\
&+&4 d\_F \_f $\frac{\Lambda}{4\pi T}$\^[2]{}+O\[\^6\]. The renormalized quark free-energy is given as F\_q\^r &=&- d\_F \_f(1+12\^2 ) +4 d\_F \_f where $\hat \Lambda=\Lambda/2\pi T$ and $\hat \mu=\mu/2\pi T$.
Gauge boson free-energy in a strongly magnetized medium {#gauge_boson}
-------------------------------------------------------
The general structure of gauge boson self-energy can be written from Ref. [@Karmakar:2018aig] as \^=B\^+ R\^+Q\^+N\^, where the form factors can be calculated for non-zero quark chemical potential as
$$\begin{aligned}
\alpha &=\frac{m_D^2}{\bar u^2}\left[1-\mathcal{T}_P(p_0,p)\right]-\sum_f \frac{(\delta m_{D,f}^2)_s}{\bar u^2}e^{{-p_\perp^2}/{2q_fB}}~\frac{p_3^2}{p_0^2-p_3^2}, \label{b_sf} \\
\beta&=\frac{m_D^2}{2}\left[\frac{p_0^2}{p^2}-\frac{P^2}{p^2}\mathcal{T}_P(p_0,p)\right] , \label{c_sf} \\
\gamma&= \frac{m_D^2}{2}\left[\frac{p_0^2}{p^2}-\frac{P^2}{p^2}\mathcal{T}_P(p_0,p)\right]+\sum_f \frac{(\delta m_{D,f}^2)_s}{\bar u^2} e^{{-p_\perp^2}/{2q_fB}}~ \frac{p_3^2}{p_0^2-p_3^2}, \label{d_sf}\\
\delta&=\sum_f (\delta m_{D,f}^2)_s\frac{\sqrt{\bar n^2}}{\sqrt{\bar u^2}}~ e^{{-p_\perp^2}/{2 eB}}\frac{p_0p_3}{p_0^2-p_3^2}, \label{a_sf}\end{aligned}$$
where ${\bar u}^2 = - p^2/P^2$, ${\bar n}^2 = -p_{\perp}^2/p^2$ and \_P(p\_0,p)=. The thermal and magnetic correction of the Debye screening mass is given as m\_D\^2&=&,\
(m\_[D,f]{}\^2)\_s&=&\_[-]{}\^ \
&=& ,\
(m\^s\_D)\^2&=&m\_D\^2 +\_f (m\_[D,f]{}\^2)\_s=m\_D\^2+(m\_D\^2)\_s. The total gluon free-energy expanded upto ${\mathcal O}[g^4]$ is given by F\_g &&d\_A\[Fsg\_expan\] where $d_A=N_c^2-1$.
The renormalized total gluon free-energy containing both hard and soft contributions is given as F\_g\^r&=& -.\[fg\]
Longitudinal and Transverse Pressure and corresponding Susceptibilities {#pressure}
-----------------------------------------------------------------------
Free-energy density of the quark-gluon plasma is given by F=u-Ts-n -eBM\[F\_sfa\], where $u$ is total the energy density and magnetization per unit volume is g
| 774
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_j)$. But then, $\psi_i$ and $\psi'_j$ are necessarily of degree $-s$. The exactness of $\textbf{D} (\mathcal O^!)$ at the degree $-s$ implies that $\psi$s of the degree $-s$ are exact, i.e. there are elements $\Psi_i, \Psi'_j\in \textbf{D}(\mathcal O^!)$ such that $\psi_i={\partial}(\Psi_i)$, $\psi'_j ={\partial}(\Psi'_j)$. Then, take $S:={\partial}\left( \sum_i\sum_\sigma \varphi_i *_\sigma \Psi_i+\sum_j
\sum_\sigma \varphi'_j \#_\sigma\Psi'_j\right)$, where the sum is over those $i$’s and $j$’s which have the constructed $\Psi_i$’s and $\Psi'_j$’s. As the total degree of the $\Psi$’s is $-s-1$, we see that the only terms of $S$ with degree greater or equal to $-s$ are the terms $$\sum_i \sum_\sigma \varphi_i *_\sigma {\partial}(\Psi_i) +
\sum_j\sum_\sigma \varphi'_j \#_\sigma{\partial}(\Psi'_j)
=\sum_i \sum_\sigma \varphi_i *_\sigma \psi_i+
\sum_j \sum_\sigma \varphi'_j \#_\sigma\psi'_j.$$ It follows that $\chi-S$ only contains terms of degree less or equal to $-s-1$, and $\chi$ is homologous to $\chi-S$. This concludes the inductive step.
We complete the proof of the theorem by noticing that $\textbf{D}(\mathcal O^!)
(l)$ is concentrated in finite degrees, so that the $\psi_i$s and $\psi'_j$s eventually have to be identically $0$. As a consequence every closed element $\chi$ is eventually homologous to $0$. It follows that the complex $ \textbf{D}(\widehat{\mathcal O^!})
(\vec X;\varnothing)$ has no homology in degrees $r\neq 0$.
Algebras over $\widehat{\mathcal O}_\infty$ {#homotop-ip}
===========================================
In this section, we investigate algebras over $\widehat{\mathcal O}_\infty:=\textbf{D} (\widehat{\mathcal O^!})$. The particular cases of the associative operad $\mathcal Assoc$ and the commutative operad $\mathcal Comm$ will be considered.
Before looking at $\widehat{\mathcal O}_\infty$, let us first consider algebras over $\widehat{\mathcal O}$. These are given by “algebra maps” $\mathcal O(n) \otimes A^{\otimes n}\to A$, “module maps” $\bigoplus_{r+s=n-1} \ma
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| 0.771314
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model or in an ensemble setting.'
author:
- |
Tim Leathart, Eibe Frank, Bernhard Pfahringer and Geoffrey Holmes\
Department of Computer Science, University of Waikato, New Zealand
bibliography:
- 'multi\_subset\_nd.bib'
title: Ensembles of Nested Dichotomies with Multiple Subset Evaluation
---
Introduction
============
Multi-class classification problems are commonplace in real world applications. Some models, like neural networks and random forests, are inherently able to operate on multi-class data directly, while other models, such as classic support vector machines, can only be used for binary (two-class) problems. The standard way to bypass this limitation is to convert the multi-class classification problem into a series of binary problems. There exist several methods of performing this decomposition, the most well-known including one-vs-rest [@rifkin2004defense], pairwise classification [@hastie1998classification] and error-correcting output codes [@dietterich1995solving]. Models that are directly capable of working with multi-class problems may also see improved accuracy from such a decomposition [@mayoraz1997decomposition; @furnkranz2002round; @pimenta2005study].
The use of ensembles of nested dichotomies is one such method for decomposing a multi-class problem into several binary problems. It has been shown to outperform one-vs-rest and perform competitively compared to the aforementioned methods [@frank2004ensembles]. In a nested dichotomy [@fox1997applied], the set of classes is recursively split into two subsets in a tree structure. Two examples of nested dichotomies for a four class problem are shown in Figure \[fig:nd\_example\]. At each split node of the tree, a binary classifier is trained to discriminate between the two subsets of classes. Each leaf node of the tree corresponds to a particular class. To obtain probability estimates for a particular class from a nested dichotomy, assuming the base learner can produce probability estimates, one can simply compute the product of
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|
among the AMI inpatients and there was no significant difference in mortality rates among males and females (28% *vs* 19.4%; *P*=0.10).
In our study, binary logistic regression analysis identified only three predictor variables for the prognosis of AMI inpatients \[[Table 1](#T0001){ref-type="table"}\]. Patients reporting early to the hospital, having longer stay and those with higher systolic blood pressure at admission, were likely to have better prognosis during their hospital stay. Length of hospital stay operated differently than other predictors, that is, if a patient survives the first 48 h, then his risk of dying from AMI decreases significantly with further increase in the length of stay.
######
Significant determinants of prognosis in AMI in-patients
Variable No. *N* = 321 Survived number (%) *N* = 253 Died number (%) *N* = 68 Unadjusted OR (95% CI) Adjusted OR (95% CI)
----------------------- --------------- ------------------------------- -------------------------- ---------------------------------------- ----------------------------------------
Time gap in treatment
0-6 h 146 124 (84.9) 22 (15.1) 1 (Ref) 1 (Ref)
6-12 h 44 34 (77.3) 10 (22.7) 1.66 (0.66-4.12) 2.37 (0.69-8.09)
12-24 h 18 18 (100) 0 (0) 0.00[\*](#TF0001){ref-type="table-fn"} 0.00[\*](#TF0001){ref-type="table-fn"}
24+ h 87 59 (67.8) 28 (32.2) 2.67 (1.35-5.32) 4.27 (1.63-11.17)
NA 26 18 (69.2) 8 (30.8) \-
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Load()
A:
generic_visit is called when a custom visitor (ie visit_Name) can't be found. Here's a piece of code I wrote recently with ast.NodeVisitor: https://foss.heptapod.net/pypy/pypy/-/blob/80ead76ab428100ffeb01109c7fc0d94f1048af2/py/_code/_assertionnew.py It interprets the AST nodes to gain debugging information about some of them and falls back in with generic_visit when a special implementation isn't provided.
Q:
Create adjacency matrix from char[][] array provided rules
I want to change a char[][] array (lets call it cA) into an adjacency matrix. An adjacency matrix has columns and rows equal to the number of elements in an array, and each vertex in the adjacency matrix is either true or false depending if the elements in the initial array are adjacent. I want to bend the rules a little and also constrain an adjacency matrix vertex to true iff the elements are adjacent and one of the elements is not a specific value.
Here's what cA array looks like:
z y z
z z z
z y y
The adjacency matrix (lets call it aM) for the cA array will be an int array of size [3*3][3*3]. The conditions for aM(i,j) to be true are that the elements i and j in cA array must be adjacent, but neither i or j can be "y".
Lets number the cA array elements 1 through 9.
1 2 3
4 5 6
7 8 9
aM can be described by doing the below operations:
aM(1,1) //false, no self-adjacency
aM(1,2) //false, 2 is a "y"
aM(1,3) //false, 1 is not adjacent to 3
aM(1,4) //true, 1 is adjacent to 4, neither are "y"
aM(1,5) //false, 1 is not adjacent to 5
aM(1,6) //false, 1 is not adjacent to 6
aM(1,7) through aM(1,9) //false, there is no adjacency between 1 and 7, 8, or 9
aM(2,1) through aM(2,9) //false, 2 is a "y"
...
Hopefully you get the idea. From the above, the adjacency matrix for cA is as below:
1 2 3 4 5 6 7 8 9 (i)
1 0 0 0 1 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 1 0 0 0
4 1 0 0 0 1 0 1 0 0
5 0 0 0 1 0 1 0 0 0
6 0 0 1 0 1 0 0 0 0
7 0 0 0 1 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0 0
(j)
The rule being aM(i,j) == 1 iff i
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|
^{x}L}}$ up to $(H\cap {\ ^{x}L})$-conjugacy, is an $R$-basis of $\mu_{R}(G)$.
Section $3$ of [@tw].
\[prop\_b\] The Mackey algebra $\mu_{R}(G)$ is isomorphic to $RB(\Omega_{G}^{2})$, where $\Omega_{G}$ is the $G$-set: $\sqcup_{L\leqslant G} G/L$.
The proof can be found in Proposition $4.5.1$ of [@bouc_green]. Let us recall that an explicit isomorphism $\beta$ can be defined on the generators of $\mu_{R}(G)$ by $\beta(t^{K}_{H}):=$ $$\xymatrix{
& G/H\ar[dl]_{\pi_{H}^{K}}\ar@{=}[dr]& \\
G/K & & G/H
}$$ where $\pi_{H}^{K} : G/H\to G/K$ is the canonical map. Similarly, we define $\beta(r^{K}_{H}):=$ $$\xymatrix{
& G/H\ar[dr]^{\pi_{H}^{K}}\ar@{=}[dl]& \\
G/H & & G/K
}$$ and $\beta(c_{g,H}):=$ $$\xymatrix{
& G/{^{g}H}\ar[dr]^{\gamma_{g,H}}\ar@{=}[dl]& \\
G/{^{g}H} & & G/H
}$$ where $\gamma_{g,H}(x{\ ^{g}H})=xgH$. One can check that this gives an isomorphism of algebras.
There is an equivalence of categories
$Mack_{R}(G)\cong \mu_{R}(G)$-Mod.
Burnside Trace.
---------------
There is a tensor product in the category of Mackey functors (see [@bouc_green], e.g.). With this tensor product, the category is a closed symmetric monoidal category with the Burnside functor as unit. So, using the formalism of May ([@may_trace]) where the dualizable Mackey functors are exactly the finitely generated projective Mackey functors, Bouc has defined the notion of Burnside dimension and Burnside trace for these Mackey functors ([@bouc_burnside_dim]). Let $M$ be a finitely generated projective Mackey functor. The Burnside trace, denoted by $Btr$ is a map from $End_{Mack_{R}(G)}(M)$ to $RB(G)$. Let $RB_{X}$ be the Dress construction of the Burnside functor at the finite $G$-set $X$ (see [@dress] or [@bouc_green]). It is well known that $RB_{X}$ is a finitely generated projective Mackey functor. By an adjunction property, we have an isomorphism of $R$-algebras $End_{Mack_{R}(G)}(RB_{X})\cong RB(X^2)$ where the product on this ring is defined as in Example \[burnside\]. Using these identifications, the Burnside trace on this Mackey fu
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|
om a node $N_b$ to a node $N_e$, is identified based on three properties. The path should be applicable, independent from the concrete terms represented by the input variables. Therefore, the first property states that no substitutions on input variables may occur between $N_b$ and $N_e$. The second property states that the selected atom of $N_b$ – i.e. $A_b^1$ – has to be an ancestor of $A_e^1$. These two properties prove that the sequence of clauses in the path from $N_b$ to $N_e$ is applicable to any goal with a selected atom from $Den(A_b^1)$. Therefore, non-termination is proven by requiring that $Den(A_e^1)$ is a subset of $Den(A_b^1)$. This property can be relaxed by requiring that each atom in $Den(A_e^1)$ is more general than some atom in $Den(A_b^1)$. If this is the case, $A_e^1$ is called *moded more general* than $A_b^1$. For definite logic programs, these three properties imply non-termination.
Let $A$ and $B$ be moded atoms. $A$ is *moded more general* than $B$ if
- $ \forall I \in Den(A),~ \exists J \in Den(B): I \textit{ is more general than } J$.$\hfill \square$
In Figure \[fig:eq\_plus\_symbolic\], the path from $N_3$ to $N_6$ satisfies these properties. The ancestor relation holds. There are no substitutions on input variables in the path. Finally, the selected atoms are identical and therefore denote the same concrete atoms. $\hfill \square$
The following proposition provides a practical sufficient condition to verify whether the moded more general relation holds.
\[prop:mmg\] Let $A$ and $B$ be moded atoms. Let $A_1$ and $B_1$ be renamings of these atoms such that they do not share variables. $A$ is moded more general than $B$ if $A_1$ and $B_1$ are unifiable with most general unifier $\lbrace V_1\setminus t_1,\ldots,
V_n \setminus t_n \rbrace$, $t_i \in Term_P$, $1 \leq i, \leq n$, such that for each binding $V_i \setminus t_i$, either:
- $V_i \in Var(B_1)$ and $V_i$ is labeled as input, or
- $V_i \in Var(A_1)$, $V_i$ is not labeled as input and no variable of $Var(t_i)$ is
| 780
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| 1,814
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| 0.789951
|
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|
(a_5=0,b_2=0\right)}+P{\left(a_5=1,b_1=2\right)}+P{\left(a_5=2,b_6=2\right)}+P{\left(a_6=0,b_8=1\right)}+\\
& \quad +P{\left(a_6=1,b_5=0\right)}+P{\left(a_6=2,b_7=2\right)}+P{\left(a_7=0,b_3=1\right)}+P{\left(a_7=1,b_6=0\right)}+\\
& \quad +P{\left(a_7=2,b_1=0\right)}+P{\left(a_8=0,b_6=1\right)}+P{\left(a_8=1,b_3=0\right)}+P{\left(a_8=2,b_2=2\right)}+\\
& \quad +P{\left(a_1=0,b_4=1\right)}+P{\left(a_1=1,b_5=0\right)}+P{\left(a_1=2,b_7=1\right)}
+P{\left(a_2=0,b_4=2\right)}+\\
& \quad +P{\left(a_2=1,b_8=1\right)}+P{\left(a_2=2,b_5=2\right)}+P{\left(a_3=0,b_4=0\right)}+P{\left(a_3=1,b_8=0\right)}+\\
& \quad +P{\left(a_3=2,b_7=2\right)}+P{\left(a_4=0,b_3=0\right)}+P{\left(a_4=1,b_1=0\right)}+P{\left(a_4=2,b_2=0\right)}+\\
& \quad +P{\left(a_5=0,b_1=1\right)}+P{\left(a_5=1,b_6=0\right)}+P{\left(a_5=2,b_2=2\right)}+P{\left(a_6=0,b_5=1\right)}+\\
& \quad +P{\left(a_6=1,b_7=0\right)}+P{\left(a_6=2,b_8=2\right)}+P{\left(a_7=0,b_6=1\right)}+P{\left(a_7=1,b_1=2\right)}+\\
& \quad +P{\left(a_7=2,b_3=2\right)}+P{\left(a_8=0,b_3=1\right)}+P{\left(a_8=1,b_2=1\right)}+P{\left(a_8=2,b_6=2\right)}+\\
& \quad +P{\left(a_1=0,b_8=1\right)}+P{\left(a_1=1,b_8=2\right)}+P{\left(a_1=2,b_8=0\right)}
+P{\left(a_2=0,b_7=2\right)}+\\
& \quad +P{\left(a_2=1,b_7=0\right)}+P{\left(a_2=2,b_7=1\right)}+P{\left(a_3=0,b_5=0\right)}+P{\left(a_3=1,b_5=2\right)}+\\
& \quad +P{\left(a_3=2,b_5=1\right)}+P{\left(a_4=0,b_6=2\right)}+P{\left(a_4=1,b_6=1\right)}+P{\left(a_4=2,b_6=0\right)}+\\
& \quad +P{\left(a_5=0,b_3=0\right)}+P{\left(a_5=1,b_3=2\right)}+P{\left(a_5=2,b_3=1\right)}+P{\left(a_6=0,b_4=2\right)}+\\
& \quad +P{\left(a_6=1,b_4=1\right)}+P{\left(a_6=2,b_4=0\right)}+P{\left(a_7=0,b_2=1\right)}+P{\left(a_7=1,b_2=2\right)}+\\
& \quad +P{\left(a_7=2,b_2=0\right)}+P{\left(a_8=0,b_1=2\right)}+P{\left(a_8=1,b_1=0\right)}+P{\left(a_8=2,b_1=1\right)}
\end{split}$$
Therefore, the corresponding Bell inequalities take the form $$S_1\leq 16$$ $$S_2\leq 18$$ $$S_3\leq 16.$$ They were obtained by assuming the existence of joint of random variables $a_1,\ldo
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h{b_{2}-p_{1,2}} & \ensuremath{1-a_{1}-b_{2}+p_{1,2}} & \multicolumn{1}{c|}{\ensuremath{1-a_{1}}}\tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{b_{1}} & \ensuremath{1-b_{1}} & & & & \ensuremath{b_{2}} & \ensuremath{1-b_{2}} & \tabularnewline\cline{2-3} \cline{7-8} \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & \tabularnewline\cline{2-3} \cline{7-8} & \ensuremath{\mathbf{B}_{2,1}=+1} & \ensuremath{\mathbf{B}_{2,1}=-1} & & & & \ensuremath{\mathbf{B}_{2,2}=+1} & \ensuremath{\mathbf{B}_{2,2}=-1} & \tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=+1}} & \ensuremath{p_{2,1}} & \ensuremath{a_{2}-p_{2,1}} & \multicolumn{1}{c|}{\ensuremath{a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{2,2}=+1} & \ensuremath{p_{2,2}} & \ensuremath{a_{2}-p_{2,2}} & \multicolumn{1}{c|}{\ensuremath{a_{2}}}\tabularnewline\multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=-1}} & \ensuremath{b_{1}-p_{2,1}} & \ensuremath{1-a_{2}-b_{1}+p_{2,1}} & \multicolumn{1}{c|}{\ensuremath{1-a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{2,2}=-1} & \ensuremath{b_{2}-p_{2,2}} & \ensuremath{1-a_{2}-b_{2}+p_{2,2}} & \multicolumn{1}{c|}{\ensuremath{1-a_{2}}}\tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{b_{1}} & \ensuremath{1-b_{1}} & & & & \ensuremath{b_{2}} & \ensuremath{1-b_{2}} & \tabularnewline\cline{2-3} \cline{7-8} \end{tabular}\label{eq:obs}$$
are compatible with the connections $$\begin{tabular}{c|cc|ccc|cc|cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}
\cline{2-3} \cline{7-8} & \ensuremath{\mathbf{A}_{1,2}=+1} & \ensuremath{\mathbf{A}{}_{1,2}=-1} & & \quad{} & & \ensuremath{\mathbf{B}_{2,1}=+1} & \ensuremath{\mathbf{B}_{2,1}=-1} & & \quad{}\quad{}\tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{1,1}=+1}} & \ensuremath{a_{1}-\alpha_{1}} & \ensuremath{\alpha_{1}} & \multico
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er branes in the construction. This calculation was made in detail in [@Dfoam], where we refer the interested reader for further details. Here we mention only the results relevant for the present discussion.
![*Schematic representation of a D-particle space-time foam model with bulk density $n^\star (r) $ of D-particles that may be inhomogeneous. The 10-dimensional space-time is bounded by two stacks of D-branes, each accompanied by an orientifold.*[]{data-label="fig:inhom"}](dbraneworld.eps){width="7.5cm"}
We concentrate first on D0-particle/D8-brane interactions in the type-IIA model of [@Dfoam]. During the late era of the Universe when the approximation of adiabatic motion is valid, we use a weak-string-coupling approximation $g_s \ll 1$. In such a case, the D-particle masses $\sim M_s/g_s$ are large, i.e., these masses could be of the Planck size: $M_s/g_s \sim M_P = 1.22 \cdot 10^{19}$ GeV or higher. In the adiabatic approximation for the relative motion, these interactions may be represented by a string stretched between the D0-particle and the D8-brane, as shown in Fig. \[fig:inhom\]. The world-sheet amplitude of such a string yields the appropriate potential energy between the D-particle and the D-brane, which in turn determines the relevant contribution to the vacuum energy of the brane. As is well known, parallel relative motion does not generate any potential, and the only non-trivial contributions to the brane vacuum energy come from motion transverse to the D-brane. Neglecting a velocity-independent term in the D0-particle/D8-brane potential that is cancelled for a D8-brane in the presence of orientifold $O_8$ planes [@polchinski] [^3], we find: $$\begin{aligned}
\mathcal{V}^{short}_{D0-D8} & = & - \frac{\pi \alpha '}{12}\frac{v^2}{r^3}~~{\rm for} ~~
r \ll \sqrt{\alpha '}~,
\label{short}\\
\mathcal{V}^{long}_{D0-D8} & = & + \frac{r\, v^2}{8\,\pi \,\alpha '}~~~{\rm for} ~~
r \gg \sqrt{\alpha '}~.
\label{long}\end{aligned}$$ where $v \ll 1$ is the relative velocity between the D-particle and th
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finish we give the details for $k=10$. In order to study the cokernel of $\pi^{(10)}$, let us fix a basis $\mathcal{B}_1 := \{ x^5, x^4y, x^3y^2, x^2y^3, xy^4, y^5, x^2 z, xyz, y^2z \}$ of $H^0(\PP^2_w,\mathcal{O}_{\PP^2_w}(k-5))$ and a basis $\mathcal{B}_2 = \langle 1,y_1,y_1^2,y_1^3,y_1^4,z_1 \rangle_\CC \oplus \langle 1,x_2 \rangle_\CC
\oplus \langle x_3^2 \rangle_\CC =: \mathcal{B}_{21} \oplus \mathcal{B}_{22} \oplus \mathcal{B}_{23}$ of the vector space on the right-hand side in computed above, see , , . The matrix associated with $\pi^{(10)}$ in the bases $\mathcal{B}_1$ and $\mathcal{B}_2$ becomes $$A = \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & -\lambda^2 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & -\lambda^2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \lambda^2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -\lambda^2 & \lambda
\end{pmatrix}.$$ To clarify how this matrix was built let us recall the definition of $\pi^{(10)}$. Given $F(x,y,z)$ its image under $\pi^{(10)}$ is $$F(1, y_1, \lambda^2(z_1-y_1^3))_{\mathcal{B}_{21}}
\oplus F(x_2, 1, z_2-x_2)_{\mathcal{B}_{22}} \oplus F(x_3,\lambda^2(y_3-x_3),1)_{\mathcal{B}_{23}},$$ where the subindex indicates coordinates in the corresponding basis. This way for example $\pi^{(10)}(xyz) = (\lambda^2 y_1 z_1 - \lambda^2 y_1^4)_{\mathcal{B}_{21}}
\oplus (x_2 z_2 - x_2^2)_{\mathcal{B}_{22}}
\oplus (\lambda^2 x_3 y_3 - \lambda^2 x_3^2)_{\mathcal{B}_{23}}$ which produces the vector $(0,0,0,0,-\lambda^2,0,0,0,-\lambda^2)$, namely the $8$th column of $A$.
For $\lambda = \zeta$ the rank of $A$ is $9$, thus $\operatorname{coker}\pi^{(10)} = 0$. However, for $\lambda = 1$ the rank is $8$ and then the kernel and the cokernel has dimension $1$. The kernel is generated by the quasi-homogeneous polynomial $F=xy^4 + xyz + y^2z=y(x y^3 +x z+ y z)$ and the cokernel is generated for instance by $x_3^2 \in \mathcal{B}_{
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ribution. We then have $$\begin{aligned}
\label{eq:Epsilarge}
E_\Psi &\approx \sqrt{2}\left(\left(1-\beta\right){\mathrm{erf}}^{-1}\left(1-\frac{2}{\Psi}\right)+\beta\mathrm{erf}^{-1}\left(1-\frac{2}{e\Psi}\right)\right),\end{aligned}$$ where $\beta$ denotes the Euler’s constant. Substituting into completes the proof.
Proof of Corollary \[Cor:aveGlsngrO\] {#App:proofCor:aveGlsngrO}
=====================================
From the tail region approximation for the inverse error function, we note that $$\label{eq:appinverflargex}
\mathrm{erf}^{-1}(x)= \sqrt{-\ln\left(1-x^2\right)} \quad \text{as} \quad x\rightarrow 1.$$ Based on and , as $\Psi\rightarrow\infty$, $G_{\bar{R}}(\Psi)$ is asymptotically equivalent to $$\begin{aligned}
G_{\bar{R}}(\Psi)&\sim 1+\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\left( \left(1-\beta\right)\sqrt{-\ln(4)+\ln\left(\frac{\Psi^2}{\Psi-1}\right)}\right.\notag\\
&\left.
+\beta\sqrt{1-\ln(4)+\ln\left(\frac{\Psi^2}{\Psi-1/e}\right)}\right)\notag\\
&\sim \frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\sqrt{\ln(\Psi)}.\end{aligned}$$ This completes the proof.
Proof of Proposition \[Prop:2\] {#App:proofoutGa}
===============================
With the Gaussian approximated PDF of $R_{\psi}$, we have the approximated outage probabilities for the systems without and with reconfigurable antennas as $$\label{eq:outageGsin}
{\mathbb{P}}(R_{\psi}<R)=F_{R_\psi}(R)$$ and $$\label{eq:outageGrec}
{\mathbb{P}}(R_{{\widehat{\psi}}}<R)=\left(F_{R_\psi}(R)\right)^{\Psi},$$ respectively, where $F_{R_\psi}(x)$ is given in . Substituting and into and , respectively, we can obtain the approximated $R_{{\widehat{\psi}}}^{{\mathrm{out}}}$ and $R_{\psi}^{{\mathrm{out}}}$ as $$\label{eq:derRcout}
R_{{\widehat{\psi}}}^{{\mathrm{out}}}\approx F_{R_\psi}^{-1}(\epsilon^{\frac{1}{\Psi}})={\bar{R}_{\psi}}-\sqrt{2{\sigma^2_{R_\psi}}}\mathrm{erf}^{-1}\left(1-2\epsilon^{\frac{1}{\Psi}}\right)$$ and $$\label{eq:derRout}
R_{\psi}^{{\mathrm{out}}}\approx F_{R_\psi}^{-1}(\epsilon)={\bar{R}_{\psi}
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\circ{\hat\Theta_{T}}$ over all $T\in{\calu}$, we get zero. A similar argument applies in the case $d=v$.
If $1\ls d<v$ and $t=2$, then we have ${\psi_{d,t}}\circ{\hat\Theta_{T}}=0$ for each $T\in{\calu}$ just using Lemma \[lemma5\]. Now take $2\ls d<v$ and $t=1$, and consider a tableau $T\in{\calu}$. There are a single $d$ and a single $d+1$ below row $1$. If these lie in the same row or column, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$. Otherwise, let $T'$ be the tableau obtained by interchanging the $d$ and the $d+1$ below row $1$. Then ${\psi_{d,1}}\circ({\hat\Theta_{T}}+{\hat\Theta_{T'}})$, and we are done.
We are left with the case $d=t=1$. Applying Lemma \[lemma5\], we find that ${\psi_{1,1}}\circ\theta$ is the sum of homomorphisms labelled by tableaux $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;\star;\star,;1,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star)$$ in which the $\star$s now represent the numbers from $3$ to $u$, and where the entries are strictly increasing along rows and weakly increasing down columns. Now we apply Lemma \[lemma7\] to each of these homomorphisms to move the $1$ from row $3$ to row $2$, and then to reorder rows $3,\dots,b+2$. We obtain a sum of tableaux of the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;\star,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star),$$ but each tableau occurs $b$ times in this way. Since $b$ is even, we have zero.
Now we need to check that $\sigma\neq0$, which is not obvious be
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lpha^2 F^2 \over 4}
\bigg ({\rm sin^2}\phi + {\alpha^2 \ {\rm sin^2}\theta \over
F^2}\bigg) +{\tau_0 \tau_1 \alpha^3 F \over 2}\rm sin\theta
cos\phi
\bigg] \Psi = \varepsilon\Psi\end{aligned}$$ $$\Rightarrow H_\tau\ \Psi = \varepsilon \Psi.$$
Calculational scheme
====================
To proceed with a basis set expansion, Gram-Schmidt (GS) functions orthogonal over the integration measure $ F = 1 + \alpha \ \rm
cos \theta$ must be generated. Fortunately, it is possible to construct such functions almost trivially. The method for doing so has been described elsewhere [@gst2], so only the salient results will be presented below.
The $\tau_1 = 0, \theta \rightarrow -\theta$ invariance of $H_\tau$ suggests that the solutions of the Schrodinger equation be split into even and odd functions, and the primitive basis set can be taken to possess this property; $$u_n(\theta) = {1 \over \sqrt
\pi} {\rm cos}[n\theta], \qquad v_n(\theta) = {1 \over \sqrt \pi}
{\rm sin}[n\theta].$$ The GS functions will take the form $$\psi^{\pm}_{K}(\theta) = \sum_{m}c^{\pm}_{Km} \left ( \begin{array}{c} u_m(\theta) \\
v_m(\theta) \end{array} \right)$$ with the $c_{Km}$ given by (momentarily supressing the parity superscripts [@parnote]) $$c_{Km}=(-)^{K+m}N_K(N_{K-1}\beta N_{K-1})(N_{K-2}\beta
N_{K-2})...(N_{m}\beta N_{m})$$ and the normalization factors $N_K$ determined from $$N^2_{k+1}={1 \over {1-\beta^2 N^2_{k}}}$$ starting from $N_0 = \sqrt{1/2}$ for positive parity states and $N_1 = 1$ for negative parity states. The $K\nu^{th}$ basis state is attained by appending azimuthal eigenfunctions onto the GS functions described above, $$\Psi^{\pm}_{K\nu}(\theta,\phi) = {1 \over \sqrt {2
\pi}}\sum_{m}c^{\pm}_{Km} \left ( \begin{array}{c} u_m(\theta) \\
v_m(\theta) \end{array} \right)
e^{i\nu\phi}.$$ The matrix $$H_{\tau \bar{K} K \bar{\nu} \nu}^{pq}= \big < \bar{K}^{\pm}
\bar{\nu} |H_\tau| K^{\pm} \nu \big >$$ is then easily constructed since the matrix elements can all be written in closed form (see the Appendix), and the eigenvalues a
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a.eps){width="\colwidth"} {width="\colwidth"}\
{width="\colwidth"} {width="\colwidth"}
Results {#sec:results}
=======
Detected host galaxies {#sec:resolved}
----------------------
Using the above criteria for detecting a residual host galaxy, we find nine of the 23 host galaxies to be resolved in both bands, although some lie close to the sensitivity limit. One object formally fell above the 5% level in one band but not the other; [[combo-17]{}]{} 33630 at $z=2.719$ might be marginally resolved in [F850LP]{} and shows a structure at 1distance that might be a tidal arm or a foreground object. With this object lying at the highest redshift of the sample we do not consider this a clear detection. As mentioned, for three further cases in the [F850LP]{} band the host galaxies are very faint (marked with a ‘?’ in the $Z_\mathrm{hg}$ column in Tab. \[tab:results\_vz\]). While their flux is above 5% of the total, their raw magnitudes fell 0.8–1.0 mag outside the reliability region where corrections and associated errors are still small. This low S/N is also reflected in the radial profiles (see Appendix \[sec:appendix\]).
As described above, tests with field stars show that 12% of all objects ($\sim$3 objects) might show spurious ‘host galaxies’ at the 5% flux level, and 3% (0 or 1 objects) at the 10% flux level. In [F606W]{} five of our objects fall with their host fluxes between these two values. In the [F850LP]{}-band these are four which include the three uncertain ones from above. According to statistics 1–3 of these might be spurious detections. However, including or excluding these more uncertain data points in the following analysis does not have an influence on the conclusions drawn.
For each object the host galaxy flux was determined by simple aperture photometry after subtraction of the scaled PSF, excluding resolved companion objects. The radius of the aperture was matched the used image size.
All extracted magnitudes are collected in
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me from single or double powers in $W$ in $\hat{H}_{1}$ are always accompanied by the matter potential in the form of $WA$ or $W^{\dagger} A$, as in eq. (\[H1-matrix\]). That is, perturbative effect of $W$ is always accompanied by matter potential, and hence can always be dealt with matter perturbation theory. It is the reason why the matter perturbation theory is able to yield the same condition on sterile masses as obtained in a fuller treatment of matter effect done in this paper.[^14]
The oscillation probability in fourth order in $W$ {#sec:probability-4th}
--------------------------------------------------
The oscillation probability in fourth order in $W$ contains the two terms $$\begin{aligned}
&&P(\nu_\beta \rightarrow \nu_\alpha)^{(4)} =
\left| S^{(2)}_{\alpha \beta} \right|^2
+ 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(4)}_{\alpha \beta} \right].
\label{P-beta-alpha-4th-def}\end{aligned}$$ We will show in appendix \[sec:second-order-square\] that the first term in (\[P-beta-alpha-4th-def\]), after averaging over the fast oscillations and using the suppression by energy denominator as discussed in the previous section, leaves the unique term, the probability leaking term $\mathcal{C}_{\alpha \beta}$ in (\[P-beta-alpha-ave-vac\]). We will also show in appendix \[sec:interference\] that the second term in (\[P-beta-alpha-4th-def\]), under the same treatment for the first term, gives vanishing contribution. Therefore, no matter-dependent term remains after using energy denominator suppression and averaging over the fast oscillations.
In conclusion, the oscillation probability in matter between active flavor neutrinos in the $(3+N)$ space unitary model to fourth order in $W$ in our small unitarity-violation perturbation theory can be written as in eq. (\[P-beta-alpha-final\]) in section \[sec:essence\]. We hope that it serves as a useful tool to test leptonic unitarity in various ongoing and future neutrino oscillation experiments.
Analytical and numerical methods for solv
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ight\Vert}_{L^\infty(G\times S)}^{1/2}
{\left\Vert \int_S \tilde{\sigma}(\cdot,\cdot,\omega',E) d\omega'\right\Vert}_{L^\infty(G\times S)}^{1/2},\end{aligned}$$ where ${\left\Vert K(E)\right\Vert}$ is the norm of $K(E)$ as an operator in $L^2(G\times S)$. Hence under the assumption (\[ass2a\]) the operator $K(E):L^2(G\times S)\to L^2(G\times S)$ is bounded and $${\left\Vert K(E)\right\Vert}\leq M_1^{1/2} M_2^{1/2},$$ uniformly for $E\in I$.
Consider the problem , , , where $K$ is of the particular form given in . We shall assume that the restricted cross-sections $\Sigma$, $\tilde\sigma$ and the stopping power $S_0$ satisfy somewhat different assumptions than in the previous section. We make the following change of variables and of the unknown function \[ecsd2\] (x,,E):=&(x,,E\_m-E),\
:=&e\^[-CE]{}, and denote S(x,E)=&S\_0(x,E\_[m]{}-E), \
(x,,E)=&(x,,E\_[m]{}-E)\
(x,,’,E)=&(x,,’,E\_[m]{}-E)\
f(x,,E)=&f(x,,E\_[m]{}-E)\
g(y,,E)=&g(y,,E\_[m]{}-E)\
(K)(x,,E)=&\_S(x,’,,E)(x,’,E)d’, with $\phi\in L^2(G\times S\times I)$ in the definition of $\tilde{K}$. Making the changes in , we find that the problem , , is equivalent to $\phi$ satisfying the equation \[se1a\] &[E]{}+[1]{}\_x+C+[1]{}[E]{}+[1]{}-[1]{}K=[1]{}e\^[-CE]{}f, on $G\times S\times I$, along with satisfying the following inflow boundary and initial value conditions, $$\begin{aligned}
\phi_{|\Gamma_-}={}&e^{-CE}\tilde g, \label{se2a} \\
\phi(\cdot,\cdot,0)={}&0. \hspace{1.5cm} \textrm{(on $G\times S$)}. \label{se3a}\end{aligned}$$
Furthermore, define *for any fixed* $E\in I$ and $C\geq 0$ the linear operator $A_C(E):L^2(G\times S)\to L^2(G\times S)$ with domain $D(A_C(E))$ by (here $\tilde S(E)=\tilde S(\cdot,E)$ and $\tilde \Sigma(E)=\tilde \Sigma(\cdot,\cdot,E)$), \[ecsd4\] &D(A\_C(E))=W\^2\_[-,0]{}(GS):={W\^2(GS) | \_-’()=0},\
&A\_C(E)=-([1]{}\_x+C +[1]{}(E)+[1]{}[E]{}(E)-[1]{} K(E)), and a function ${\bf f}(E):G\times S\to{\mathbb{R}}$ such that $${\bf f}(E)(x,\omega)={1\over{\tilde S(x,E)}}e^{-CE}\tilde f(x,\omega,E),$$ where $$(\tilde K(E
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\otimes_{[B]}X.$$ This defines a natural isomorphism $u^\ast\cong \big((u\times\id)^\ast\lI_B\big)\otimes_{[B]}-\colon{\sD}^B\to{\sD}^A$, thereby exhibiting $u^\ast$ as a weighted colimit.
Similarly, if we fix $X\in{\sD}(A)$, then by the partial morphism $$\label{eq:par-mor-wcolim}
(-\otimes_{[A]}X)\colon{\sV}^{A\op}\to{\sD}$$ is cocontinuous. Given a functor $u\colon A\to B$ we obtain natural isomorphisms $$u_!(X)\cong u_!(\lI_A\otimes_{[A]}X)\cong \big((u\times\id)_!\lI_A\big)\otimes_{[A]}X,$$ hence a natural isomorphism $u_!\cong\big(\big((u\times\id)_!\lI_A\big)\otimes_{[A]}-\big)\colon{\sD}^A\to{\sD}^B$, identifying $u_!$ as a weighted colimit.
If and are pointed, then is a cocontinuous morphism of pointed derivators and hence automatically pointed, hence preserves right Kan extensions along sieves [@groth:can-can Cor. 8.2]. Thus, a similar calculation as above yields for every such $u\colon A\to B$ a natural isomorphism $$u_\ast\cong \big(\big((u\times\id)_\ast\lI_A\big)\otimes_{[A]}-\big) \colon{\sD}^A\to{\sD}^B,$$ exhibiting $u_\ast$ as a weighted colimit. Similarly, if and are stable derivators, we note that is an exact morphism of stable derivators (by and [@groth:can-can Cor. 9.9]) and it hence preserves right homotopy finite right Kan extensions [@groth:can-can Thm. 9.14].
Applying \[thm:wcolim\] to the -module ${\sV}^{C\op}$, we find that for any $X\in {\sV}(B\times C\op)$ and $Y\in {\sV}(A\times C\op)$ we have $$\begin{aligned}
\big((u\times\id)^\ast\lI_B\big) \otimes_{[B]} X &\cong (u\times\id)^\ast X\\
\big((u\times\id)_!\lI_A\big) \otimes_{[A]} Y &\cong (u\times\id)_! Y\end{aligned}$$ Note that in this case, $\otimes_{[B]}$ and $\otimes_{[A]}$ are the composition in $\cProf({\sV})$; thus restriction and left Kan extension in can both be described using composition in $\cProf({\sV})$. The special objects $(u\times\id)^\ast\lI_B$ and $(u\times\id)_!\lI_A$ are sometimes called **base change objects**. Dually, for any $X\in {\sV}(E\times B\op)$ and $Y\in {\sV}(E\times A\op)$ we have $$\begi
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Delta\equiv -w_2^{-1} \mod (w_0w_1)$ and define $$\label{eq:Mj}
M_j:=(\Delta w_2+1)\deg_w \cC_j,\quad \text{ for all } j=1,..,r,$$ then $M_j$ satisfies \[prop:5\]. By construction $M=(\Delta w_2+1)\deg_w \cC $ which proves \[prop:4\] and $\frac{M}{\deg_w \cC}=\Delta w_2+1$ satisfies \[prop:6\]. Since $\Delta$ can be chosen big enough, then the result follows.
Examples and applications {#sec:examples}
=========================
In this section some applications of the global theory (Theorem \[thm:conucleo\_singular\]). Some of them are theoretical, such as the independence of $\cM_\pi(C^\lambda,k)$ on the $\Q$-resolution, others are concrete examples of calculations. In particular, we provide examples that exhibit new features in our approach. Namely, only a $\Q$-resolution is required to obtain the birational information on the ramified covers; also the special relevance of the singular points of the surface is shown in a particular case; finally, a Zariski pair in a weighted projective plane is described.
Independence of the Q-resolution: global to local
-------------------------------------------------
As announced in section \[sec:local\] we can show that $\cM_\pi(C^\lambda,k)$ is independent of the chosen $\Q$-resolution. This is an interesting application where the global theory is used to prove a local result.
\[prop:indres\] Let $\pi_i:Y_i\to X$, $i=1,2$ be two good $\Q$-resolutions of $C\subset (X,0)$, then $$\cM_{\pi_1}(C^\lambda,k)= \cM_{\pi_2}(C^\lambda,k).$$
By Lemma \[lemma:propsM\]\[lemma:propsM:plus1\] it is enough to show the result for all $0\leq \lambda <1$. Let us decompose $C=\sum n_iC_i$ where $C_i$ are the irreducible components of $C$. The surface can be written as $X=\frac{1}{w_2}(w_0,w_1)$ a cyclic singularity in a normal form, that is, $\gcd(w_i,w_j)=1$ for $i\neq j$. Consider $0\leq \lambda<1$ a rational number and $0\leq k<w_2$. By Lemma \[lemma:propsM\]\[lemma:propsM:epsilon\], for any given good resolution $\pi:Y\to X$ of $(X,C)$ there is a $\varepsilon > 0$ such that $\c
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ated as ${\mathsf{Stab}_L}(\Phi)$ for very small classes $\Phi$ of functors such as $\{\emptyset\}$ and $\{\emptyset,\ulcorner\}$. Can they also be generated as ${\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))$ for “manageable” collections $\Upsilon$ of derivators? For instance, are there “universal” pointed or stable derivators that suffice to detect whether a given functor is absolute for all pointed or stable derivators?
To attack these questions, we use the technology of enriched derivators and weighted limits. We will see that it suffices to answer the first two questions positively, but it is not quite adequate for the third in general, although in particular cases the answer is yes. In [@gs:enriched] we will use a better technology to answer the third question positively in general as well.
Enriched derivators {#sec:enriched-derivators}
===================
We begin by defining the basic notions of enriched derivators. We freely make use of the language and techniques established in [@gps:additivity], in particular the language of *monoidal derivators* as it is developed in detail in [@gps:additivity §3]. In that paper there is also a detailed discussion of *two-variable adjunctions of derivators* [@gps:additivity §§8-9].
A monoidal derivator is a pseudo-monoid object in $\cDER$ (the 2-category of derivators and pseudonatural transformations) such that the monoidal structure $\otimes\colon{\sV}\times{\sV}\to{\sV}$ preserves colimits separately in both variables. The pseudo-monoid structure precisely amounts to a lift of ${\sV}\colon\cCat\op\to\cCAT$ against the forgetful functor from the $2$-category of monoidal categories, strong monoidal functors, and monoidal transformations. The resulting monoidal structures are denoted by $({\sV}(A),\otimes_A,\lS_A)$.
We will also have occasion to consider the following weaker notions.
A **left derivator** is a prederivator satisfying all the axioms of a derivator except the existence of right Kan extensions.
A morphism of left derivators is again a ps
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-1/2$. There is a multiplicity of left vacua, arising from $\lambda^{7-14}$. Let $|m\rangle$ denote a vacuum with $m$ +’s and $8-m$ -’s, [*i.e.*]{} annihilated by $m$ $\lambda$’s and $8-m$ $\overline{\lambda}$’s, then under the action of the generator of ${\mathbb Z}_4$, it is straightforward to check that $|m=0,4,8\rangle$ are invariant, $|m=2,6\rangle$ get a sign flip, and the others are multiplied by various fourth roots of unity.
The ${\mathbb Z}_4$-invariant states in this sector are of the form
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
State Count
--------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------
$|m=0,4,8 \rangle \otimes spacetime vector, valued in ${\bf 1}$, ${\bf 1}$, $\wedge^4 {\bf 8}
\left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} = {\bf 70}$ of $su(8)$
\right)$
$|m=6,2\rangle \otimes 4 sets of scalars, in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = {\bf \overline{28}}$ of $su(8)$
\left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2}
\right)$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In the $k=2$ (R,NS) sector, fields have the following boundary conditions: $$\begin{aligned}
X^{1-2}(\sigma + 2 \pi) & = & + X^{1-2}(\sigma), \\
X^{3-4}(\sigma + 2 \pi) & = & + X^{3-4}(\sigma), \\
\psi^{1-2}(\sigma + 2 \pi) & = &
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\pi^{*}H = c E$. Note that $\pi^{*} D_1 = \hat D_1 + \frac{1}{d} E$ and $\pi^{*} D_2 = \hat D_2 + \frac{p}{d} E$. Then, $$\begin{aligned}
\pi^{*} L^{(k)} &= -k \pi^{*} H + \sum_{i=1}^r {\left \lfloor \frac{kn_i}{n} \right \rfloor} \hat D_i
+ \left( \frac{1}{d} {\left \lfloor \frac{kn_1}{n} \right \rfloor} + \frac{p}{d} {\left \lfloor \frac{kn_2}{n} \right \rfloor} \right) E, \\
D' &= \sum_{i=1}^r n_i \hat D_i + \left( \frac{1}{d} n_1 + \frac{p}{d} n_2 + cn \right) E, \\
L'^{(k)} &= - k {\left \lceil \pi^{*}H \right \rceil} + \sum_{i=1}^r {\left \lfloor \frac{kn_i}{n} \right \rfloor} \hat D_i
+ {\left \lfloor \frac{k \left(\frac{1}{d}n_1+\frac{p}{d}n_2+cn \right)}{n} \right \rfloor} E.\end{aligned}$$
The $\QQ$-Weil divisor $L'^{(k)} - \pi^{*} L^{(k)}$ has support on $E$ and its coefficient is given by $$\gamma := -kc + {\left \lfloor \frac{1}{d} \frac{kn_1}{n} + \frac{p}{d} \frac{kn_2}{n} + kc \right \rfloor}
- \frac{1}{d} {\left \lfloor \frac{kn_1}{n} \right \rfloor} - \frac{p}{d} {\left \lfloor \frac{kn_2}{n} \right \rfloor} \in \frac{1}{d}\ZZ.$$ Using the fact that $a-1 < {\left \lfloor a \right \rfloor} \leq a, \forall a \in \QQ$, one shows that $\gamma$ belongs to the open real interval $\left( -1, \frac{1+p}{d} \right)$.
Two cases arise. If $\gamma \leq 0$, then ${\left \lfloor \pi^{*}L^{(k)} \right \rfloor} = L'^{(k)} + {\left \lceil \gamma \right \rceil} E = L'^{(k)}$. Otherwise, since $K_\pi = (-1 + \frac{1+p}{d})E$, one has ${\left \lceil \pi^{*} L^{(k)} + K_\pi \right \rceil} = L'^{(k)} + {\left \lceil -\gamma-1+\frac{1+p}{d} \right \rceil}E = L'^{(k)}$. In both cases, by Proposition \[prop:comparison\_cohomology\], $H^q(Y,\cO_Y(L'^{(k)})) \simeq H^q(X,\cO_X(L^{(k)}))$ and the claim follows.
Local study: quasi-adjunction modules {#sec:local}
=====================================
The adjuntion ideals $\mathcal J_{\cC,P\frac{k}{n}}$ in Theorem \[thm:conucleo\_liso\] for cyclic covers of $\PP^2$ ramified along reduced curves were generalized by Libgober [@Libgober-characteristic] for abelian cover
| 795
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|
gle=2n(\lambda)+|\lambda|=\sum_i(\lambda'_i)^2$, where $\lambda'=(\lambda'_1,\lambda'_2,\dots)$ is the dual partition. Note also that $(\lambda\cup\mu)'=\lambda'+\mu'$. Define $$\|\lambda\|:=\sqrt{\langle\lambda',\lambda'\rangle}=\sqrt{\sum_i\lambda_i^2}.$$ The following inequality is a particular case of the theorem of §\[appendix\].
\[ineq-1\] Fix $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$. Then for every $(\nu^1,...,\nu^r)\in\calP_{\mu_1}\times\cdots\times\calP_{\mu_r}$ we have $$\mu_1\left\|\sum_p\nu^p\right\|^2-n\sum_p\left\|\nu^p\right\|^2\leq
\mu_1n^2-n\left\|\mu\right\|^2.
\label{cauchy-ineq}$$ Moreover, equality holds in if and only if either:
\(i) The partition $\mu$ is rectangular and all partitions $\nu^p$ are equal.
or
\(ii) For each $p=1,2,\ldots,r$ we have $\nu^p=(\mu_p)$.
Our claim is a consequence of the theorem of §\[appendix\]. Taking $x_{ps}=\nu^p_s$ we have $c_p:=\sum_sx_{ps}=\sum_s\nu^p_s=\mu_p$ and $c:=\max_pc_p=\mu_1$.
The following fact will be crucial for the proof of connectedness.
\[maxima\] For a fixed $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$ we have $$\mu_1n(\lambda)-nv(\lambda,\mu) \leq \mu_1n^2-n\left\|\mu\right\|^2,
\qquad \lambda \in \calP_n.$$ Equality holds only at $\lambda=(1^n)$ unless $\mu$ is rectangular $\mu=(t^{n/t})$, in which case it also holds when $\lambda$ is the union of $n/t$ copies of any $\lambda_0\in\calP_t$. \[opti\]
Given $\nu \unlhd \lambda$ write $\mu_1n(\lambda)-nv(\lambda,\mu)$ as $$\label{first-step}
\mu_1n(\lambda)-nv(\lambda,\mu)=
\mu_1 (n(\lambda)-n(\nu)) +
\mu_1n(\nu)-nv(\lambda,\mu)$$ By Lemma \[n-ineq\] the first term is non-negative. Hence $$\mu_1n(\lambda)-nv(\lambda,\mu)\leq
\mu_1n(\nu)-nv(\lambda,\mu), \qquad \qquad \nu\unlhd \lambda.$$ Combinining this with yields $$\label{ineq}
\max_{|\lambda|=n}\left[\mu_1n(\lambda)-nv(\lambda,\mu)\right]\leq
\max_{|\rho^p|=\mu_p}\left[\mu_1n(\rho^1\cup\rho^2\cup\cdots\cup\rho^r)-
(n(\rho^1)+\cdots+n(\rho^r))n \right].
$$
Take $\nu^p$ to be the dual of $\rho^p$ for $p=1,2,\ldots,r$. Then t
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in previous steps. This question is the motivation for reverse inference, which considers automata operating backwards.
### Iterative converse {#S:ITERATIVE_CONVERSE}
Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times ({\prod{\Psi}} \times {\prod{\Phi}})$. Suppose step ${\mathit{s}} = (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}}$, whence the volatile excitation of ${\mathit{s}}$ is $\xi = {{\psi}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Xi}}}}}$.
\[D:VOLATILE\_CONVERSE\] Let $\langle \Psi, \Phi \rangle$ be a basis with step space ${\mathbb{S}}$ and iterative operator $V \colon {\mathbb{S}} \to {\mathbb{S}}$. The *iterative* converse of $V$ is the mapping $\tilde{V} \colon {\mathbb{S}} \to {\mathscr{P}({{\mathbb{S}}})}$ defined by $$\tilde{V}({\mathit{s}}) = \lbrace \tilde{{\mathit{s}}} \in {\mathbb{S}} \colon V(\tilde{{\mathit{s}}}) = {\mathit{s}} \rbrace,$$ where ${\mathscr{P}({{\mathbb{S}}})}$ denotes the power set of ${\mathbb{S}}$ (in other words, the iterative converse is a mapping from a step to a set of steps).
### Converse actuated automaton {#S:CONVERSE_AUTOMATON}
An actuated automaton ${\mathfrak{A}} = \langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$ induces an iterative operator on step space. Reverse inference is identifying all immediate predecessor steps ${{\mathit{s}}_{n-1}}$ such that ${\mathit{s}}_n = {\mathfrak{A}}({\mathit{s}}_{n-1})$. We have obviously $$\tilde{{\mathfrak{A}}}({\mathit{s}}) = \lbrace \tilde{{\mathit{s}}} \in {\mathbb{S}} \;\colon\; {\mathfrak{A}}(\tilde{{\mathit{s}}}) = {\mathit{s}} \rbrace.$$
Although referring informally to the converse $\tilde{{\mathfrak{A}}}$ of an automaton ${\mathfrak{A}}$, speaking precisely we have defined the converse $\tilde{{\mathit{s}}}$ of a *step* ${\mathit{s}} \in {\mathbb{S}}$ within the step space associated with that automaton.
A system of equations ensues with subscripted
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the model is shown in Fig. \[fig:skt\].
![Sketch of the model of colloidal dumbbells (bright yellow and dark red spheres) and droplets (white spheres). Shown are the diameters of colloidal species 1, $\sigma_{1}$, colloidal species 2, $\sigma_{2}$, and droplet $\sigma_{d}$. (a) In the initial stages the droplet captures the colloidal dumbbells. (b) The droplet has shrunk and has pulled the dumbbells into a cluster. The competition between Yukawa repulsion and surface adsorption energies can lead to open cluster structures. []{data-label="fig:skt"}](fig1){width="9cm"}
The colloids in each dumbbell are separated from each other by a distance $l$ that fluctuates in the range of $\lambda\leq l\leq \lambda+\Delta $, where $\lambda=(\sigma_{1}+\sigma_{2})/2$.
The total interaction energy is given by $$\begin{aligned}
\dfrac{U}{k_\textrm{B}T} &=&\sum_{i<j}^{N_{c}} \phi_{11}\left ( \left | \mathbf{r}_{1i}-\mathbf{r}_{1j} \right | \right )+\sum_{i<j}^{N_{c}} \phi_{22}\left ( \left | \mathbf{r}_{2i}-\mathbf{r}_{2j} \right | \right )\nonumber \\
&&+\sum_{i,j}^{N_{c}} \phi_{12}\left ( \left | \mathbf{r}_{1i}-\mathbf{r}_{2j} \right | \right )+\sum_{i}^{N_c} \sum_{j}^{N_{d}} \Phi _{1\textrm{d}}\left ( \left | \mathbf{r}_{1i}-\mathbf{R}_{j} \right | \right )\nonumber \\
&&+\sum_{i}^{N_c} \sum_{j}^{N_{d}} \Phi_{2\textrm{d}}\left ( \left | \mathbf{r}_{2i}-\mathbf{R}_{j} \right | \right )\nonumber \\
&&+\sum_{i<j}^{N_{d}} \Phi_{\textrm{dd}}\left ( \left |\mathbf{R}_{i}-\mathbf{R}_{j} \right | \right ),
\label{eqn:total-energy}\end{aligned}$$ where $k_\textrm{B}$ is the Boltzmann constant; $T$ is the temperature; $\mathbf{r}_{1i}$ and $\mathbf{r}_{2i}$ are the center-of-mass coordinates of colloid 1 and colloid 2 in dumbbell $i$, respectively; $\mathbf{R}_{j}$ is the center-of-mass coordinate of droplet $j$; $\phi_{11}, \phi_{12}$ and $\phi_{22}$ are the colloid 1-colloid 1, colloid 1-colloid 2, and colloid 2-colloid 2 pair interactions, respectively; $\Phi_{1\textrm{d}}$ and $\Phi_{2\textrm{d}}$ are the colloid 1-drople
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l response value and its displacement *w.r.t.* its parent latent pattern. Please see the appendix for details.
Based on the above node definitions, we can use the AOG to parse each given image $I$ by dynamic programming in a bottom-up manner.
### Learning And-Or graphs {#sec:learnAOG}
The core of learning AOGs is to distinguish reliable latent patterns from noisy neural responses in conv-layers and select reliable latent patterns to construct the AOG.
**Training data:**[` `]{} Let ${\bf I}^{\textrm{obj}}\subset{\bf I}$ denote the set of object images of a target category. During the active question-answering, we obtain bounding boxes of the target object part in a small number of images, ${\bf I}^{\textrm{ant}}\!=\!\{I_1,I_2,\ldots,I_{M}\}\subset{\bf I}^{\textrm{obj}}$ among all objects. The other images without part annotations are denoted by ${\bf I}^{\textrm{unant}}={\bf I}^{\textrm{obj}}\setminus{\bf I}^{\textrm{ant}}$. In addition, the question-answering process collects a number of part templates. Thus, for each image $I\in{\bf I}^{\textrm{ant}}$, we annotate $(\Lambda_{top}^{*},v^{*})$, where $\Lambda_{top}^{*}$ denotes the ground-truth bounding box of the part in $I$, and $v^{*}\in Child(top)$ specifies the ground-truth template for the part.
**Which AOG parameters to learn:**[` `]{} We can use human annotations to define the first two layers of the AOG. If human annotators specify a total of $m$ different part templates during the annotation process, correspondingly, we can directly connect the top node with $m$ part templates as children. For each part template $v\in Child(top)$, we fix a constant scale for its region $\Lambda_{v}$. *I.e.* if there are $n$ ground-truth part boxes that are labeled for $v$, we compute the average scale among the $n$ part boxes as the constant scale $scale_{v}$.
Thus, the key to AOG construction is to mine children latent patterns for each part template $v$. We need to mine latent patterns from a total of $K$ conv-layers. We select $n_{k}$ latent patterns from the $k
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e-dependent measurements or measurements of correlated $D^0\overline{D}^0$ pairs.
In principle, using the above observables the system Eqs. (\[eq:decomp-1\])–(\[eq:decomp-4\]) is exactly solvable as long as the data is very precise. In the CP limit the branching ratio measurements $(i)$ and the strong phase $(ii)$ are sufficient to determine $\tilde{t}_1$, $\tilde{t}_2$ and $\tilde{s}_1$, which are the complete set of independent parameters in this limit.
For our parameter extraction with current data, we expand the observables to first nonvanishing order in the U-spin expansion. We measure the power counting of that expansion with a generic parameter $\varepsilon$, which, for nominal U-spin breaking effects is expected to be $\varepsilon \sim 25\%$. All of the explicit results that we give below have the nice feature that the parameters can be extracted from them up to relative corrections of order $\mathcal{O}(\varepsilon^2)$. Below it is understood that we neglect all effects of that order.
In terms of our parameters the ratios of branching ratios are given as $$\begin{aligned}
R_{K\pi} &= - \mathrm{Re}(\tilde{t}_1) \,, \\
R_{KK,\pi\pi} &= - 2 \tilde{s}_1\,, \\
R_{KK,\pi\pi,K\pi} &= \frac{1}{2}\left( \tilde{s}_1^2 - \frac{1}{4}\vert \tilde{t}_1\vert^2 +
\tilde{t}_2 \right)\,. \end{aligned}$$ By inserting the expressions for $R_{K\pi}$ and $R_{KK,\pi\pi}$ into Eq. (\[eq:br-ratio-3\]) we can solve the above equations for the independent parameter combinations. The result up to $\mathcal{O}(\varepsilon^2)$ is $$\begin{aligned}
\mathrm{Re}( \tilde{t}_1 ) &= - R_{K\pi}\,, \label{eq:Ret1tilde}\\
\tilde{s}_1 &= -\frac{1}{2} R_{KK,\pi\pi}\,, \label{eq:Res1tilde} \\
-\frac{1}{4} \left(\mathrm{Im}\, \tilde{t}_1\right)^2
+ \tilde{t}_2 &= 2 R_{KK,\pi\pi,K\pi} - \frac{1}{4}R_{KK,\pi\pi}^2 + \frac{1}{4} R_{K\pi}^2 \,.\label{eq:combi}\end{aligned}$$
We are then able to determine $\tilde{t}_1$ with Eq. (\[eq:Ret1tilde\]) and the strong phase between the CF and DCS mode, see also Ref. [@Bergmann:2000i
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