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------------------------------ -------- --------- ------- ---------- ------------ --------------- ------------
: Attack success rates and detector accuracy for adversarial examples on LeNet-VAE using a triplet network detector.[]{data-label="table:lenet_detector"}
Discussion
==========
Our proposed defense and the experiments we conducted on it have revealed intriguing, unexpected properties of neural networks. Firstly, we showed how to obtain an exponentially large ensemble of models by training a linear number of VAEs. Additionally, various VAE arrangements had substantially different effects on the network’s gradients w.r.t. a given input image. This is a counterintuitive result because qualitative examination of reconstructions shows that reconstructed images or activation maps look nearly identical to the original images. Also, from a theoretical perspective, VAE reconstructions in general should have a gradient of $\approx \mathbf{1}$ w.r.t. input images since VAEs are trained to approximate the identity function. Secondly, we demonstrated that reducing the transferability of adversarial examples between models is a matter of making the gradients orthogonal between models w.r.t. the inputs. This result makes more sense, and such a goal is something that can be enforced via a regularization term when training VAEs or generating new filtering operations. Using this result can also help guide the creation of more effective concordance-based detectors. For example, a detector could be made by training an additional LeNet model with the goal of making the second model have the same predictions as the first one while having orthogonal gradients. Conducting an adversarial attack that fools both models in the same way would be difficult since making progress on the first model might have the opposite effect or no effect at all on the second model.
A limitation of training models with such unconvent
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alysis on finite abelian groups which are used here and points out their immediate consequences for $G$-circulant matrices; some notation and conventions used in the remainder of the paper are established there. Section \[S:Gaussian\] investigates the spectra of some random $G$-circulant matrices whose entries are Gaussian random variables. The invariance properties of Gaussian random variables allow an easy detailed study to be undertaken which illuminates the general situation, in particular the role of the number of elements of order $2$. Finally, Section \[S:general\] determines the asymptotic behavior of the spectrum for general entries with finite variances.
The cases of $G$-circulant matrices with heavy-tailed entries, and of random $G$-circulant matrices when $G$ is a nonabelian finite group, will be investigated in future work.
Acknowledgements {#acknowledgements .unnumbered}
================
The author thanks Persi Diaconis for encouragement and pointers to the literature, John Duncan for helpful discussions about character theory, and the referee for careful reading and useful comments. This research was partly supported by National Science Foundation grant DMS-0902203.
Some Fourier analysis and notation {#S:Fourier}
==================================
For a finite abelian group $G$, we denote by $\widehat{G}$ the family of group homomorphisms $\chi : G \to {\mathbb{T}}$, where ${\mathbb{T}}$ is the multiplicative group $\{z \in {\mathbb{C}}\mid {\left\vert z \right\vert} = 1\}$. The elements of $\widehat{G}$ are called characters of $G$; $\widehat{G}$ is a group under the operation of pointwise multiplication. The multiplicative inverse of a character $\chi$ is its pointwise complex conjugate $\overline{\chi}$. From the homomorphism property it follows that for $a \in G$ and $\chi \in \widehat{G}$, $\chi(a^{-1}) =
\overline{\chi}(a)$.
We denote by $\ell^2(G)$ the space of functions $f: G \to {\mathbb{C}}$ equipped with the inner product $${\left\langle f, g \right\rangle} = \sum_{a \in G} f(a) \o
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)$ of a directed set $\mathbb{D}$ (Def. A1.7), with $x$ and $y$ in the equations below being taken to belong to the required domains, define subsets $\mathcal{D}_{-}$ of $X$ and $\mathcal{R}_{-}$ of $Y$ as $$\mathcal{D}_{-}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\mathbb{D}}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D-}$$ $$\mathcal{R}_{-}=\{ y\in Y\!:\textrm{ }(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\mathbb{D}}\textrm{ converges in }(X,\mathcal{U}))\}\label{Eqn: R-}$$
Thus:
$\mathcal{D}_{-}$ is the set of points of $X$ on which the values of a given net of functions $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $X$ on which subnets[^19] in $\textrm{Map}(X,Y)$ combine to form a net of functions that converge pointwise to a limit function $F:\mathcal{D}_{-}\rightarrow Y$.
$\mathcal{R}_{-}$ is the set of points of $Y$ on which the values of the nets in $X$ generated by the injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $Y$ on which subnets of injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ in $\textrm{Map}(Y,X)$ combine to form a net of functions that converge pointwise to a family of limit functions $G:\mathcal{R}_{-}\rightarrow X$. Depending on the nature of $(f_{\alpha})_{\alpha\in\mathbb{D}}$, there may be more than one $\mathcal{R}_{-}$ with a corresponding family of limit functions on each of them. To simplify the notation, we will usually let $G:\mathcal{R}_{-}\rightarrow X$ denote all the limit functions on all the sets $\mathcal{R}_{-}$.
If we consider cofinal rather than residual subsets of $\mathbb{D}$ then corresponding $\mathbb{D}_{+}$ and $\mathbb{R}_{+}$ can be expressed as $$\mathcal{D}_{+}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D+}$$ $$\mathcal{R}_{+}=\{ y\in Y\!:(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(X,\mathcal{U}))\}.\label{Eq
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4/3*s**2.
-(s - 2)*(4*s + 1)/3
Let m be -1 - (-4 + 0) - 2. Let g be m + 1 + -1 + -1. Factor -4/3*v**2 + 2/3*v**5 + 0*v**3 + 4/3*v**4 - 2/3*v + g.
2*v*(v - 1)*(v + 1)**3/3
Find y, given that 42*y**2 - 35*y**3 + 28*y - 60 + 83*y**2 - 108*y = 0.
-3/7, 2
Let z(g) = 9*g**2 + 24*g - 3. Let q(s) = -10*s**2 - 23*s + 4. Let j(l) = -3*q(l) - 2*z(l). Let j(k) = 0. What is k?
-2, 1/4
Suppose -10*l + 12 = -6*l. Factor o + l*o**2 - 2*o + 0*o**2 + 7*o.
3*o*(o + 2)
Let s(h) be the third derivative of -h**8/672 + h**6/120 - h**4/48 - 17*h**2. Suppose s(y) = 0. What is y?
-1, 0, 1
Let h(v) be the second derivative of -v**9/15120 + v**8/3360 - v**7/2520 - v**4/6 - 4*v. Let l(o) be the third derivative of h(o). Let l(u) = 0. What is u?
0, 1
Let z(m) be the first derivative of -m**6/3 + 3*m**5/5 - m**4/4 - 8. Factor z(a).
-a**3*(a - 1)*(2*a - 1)
Factor -40/7*h**3 + 128/7*h - 4/7*h**4 - 96/7*h**2 + 512/7.
-4*(h - 2)*(h + 4)**3/7
Let d(k) = -k**3 - 13*k**2 + 13*k - 2. Let h be d(-14). Find c, given that 7*c**2 - 3*c**3 + 8*c**2 - c - h*c**2 + c**4 = 0.
0, 1
Let r(b) be the first derivative of -2*b**5/25 + b**4/5 - 2*b**3/15 + 5. Factor r(f).
-2*f**2*(f - 1)**2/5
Let u be 4/9*10/64. Let g(r) be the third derivative of u*r**4 + 1/360*r**6 + 1/45*r**5 + 0 + 0*r + 2*r**2 + 1/9*r**3. Factor g(b).
(b + 1)**2*(b + 2)/3
Let w(n) be the second derivative of 1/105*n**6 + 5*n + 0 + 1/70*n**5 + 0*n**2 - 1/42*n**4 - 1/21*n**3. Factor w(v).
2*v*(v - 1)*(v + 1)**2/7
Suppose 5*p - 5*o = 20, -2*p - 3*o - 8 = -3*p. Factor -5 + s + 2*s + 3 - s**2 + 0*s**p.
-(s - 2)*(s - 1)
Determine t, given that 4*t + 5*t**2 - 3*t**2 - 2*t - t**2 = 0.
-2, 0
Suppose -35 = 4*m + 3*w - 46, -w = -1. Factor -4/9*r - m*r**3 + 14/9*r**2 + 10/9*r**4 - 2/9*r**5 + 0.
-2*r*(r - 2)*(r - 1)**3/9
Suppose -4*g + 12 = -4*c + 4, -3*g + 4*c = -4. Determine p, given that g - p**2 + 4 - 3 - p - 3 = 0.
-2, 1
Factor -p + 0 - 1/2*p**3 + 7/4*p**2 - 1/4*p**4.
-p*(p - 1)**2*(p + 4)/4
Let -1176/5*v**2 + 5488/5*v - 9604/5 + 112/5*v**3 - 4/5*v**4 = 0. Calculate v.
7
Let v = -7/15 - -17/15. Determin
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les.
By Thm. \[th:Coxgr\], $${T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)=
{T}_{i_1}{T}_{i_2}\cdots {T}_{i_m}(u_0)\quad
\text{for all $u_0\in {{\mathcal{U}}^0}$.}$$ Hence Lemma \[le:MLmap\] yields that $$\begin{aligned}
\Lambda '(u_0){\hat{T}}'(v_{\Lambda '})={\hat{T}}'(u_0v_{\Lambda '})
=&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)
{\hat{T}}'(v_{\Lambda '}),\\
\Lambda ''(u_0){\hat{T}}''(v_{\Lambda ''})={\hat{T}}''(u_0v_{\Lambda ''})
=&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)
{\hat{T}}''(v_{\Lambda ''}).
\end{aligned}$$ Thus $\Lambda '=\Lambda ''$ by ($*$). This proves the lemma.
In view of Thm. \[th:Coxgr\] and Lemmata \[le:VTMrel1\], \[le:VTMrel2\] we can say that Eq. defines an action of the groupoid ${\mathcal{W}}(\chi )$ on ${{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then Lemma \[le:VTinv\] says that the numbers ${\rho ^{\chi}} ({\alpha })\Lambda (K_{\alpha }L_{\alpha }^{-1})$, where ${\alpha }\in {\mathbb{Z}}^I$, are invariants of this action.
In general, the maps ${\hat{T}}_p$ and ${\hat{T}}^-_p$ are not isomorphisms.
Assume that $\Lambda (K_pL_p^{-1})\not=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$ for all $t\in \{1,2,\dots ,{b}-1\}$. Then ${\hat{T}}_p,{\hat{T}}^-_p:
\,M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^\chi (\Lambda )$ are isomorphisms of vector spaces over ${\mathbb{K}}$. \[pr:VTMiso\]
Let $q=\chi ({\alpha }_p,{\alpha }_p)$, $\chi '=r_p(\chi )$, $\Lambda '={t}_p^\chi (\Lambda )$, and ${\hat{T}}'={\hat{T}}^\chi _{p,\Lambda } {\hat{T}}^{\chi ',-}_{p,\Lambda '}$. By Lemma \[le:VTMrel1\] and since $r_p^2(\chi )=\chi $, ${\hat{T}}'$ is a $U(\chi )$-module endomorphism of $M^\chi (\Lambda )$. We calculate ${\hat{T}}'(v_\Lambda )$. $$\begin{aligned}
{\hat{T}}'(v_\Lambda )
=&{\hat{T}}_p(F_p^{{b}-1}v_{\Lambda '})
={T}_p(F_p^{{b}-1})F_p^{{b}-1}v_\Lambda \\
=&(K_p^{-1}E_p)^{{b}-1}F_p^{{b}-1}v_\Lambda
=q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})E_p^{{b}-1}F_p^{{b}-1}v_\Lambda \\
=&q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})\qfact{{
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z'_1}^{{\bf n}}\;
{{}^\exists}\omega_5\in\Omega_{z'_1\to z_3}^{{\bf n}}\cdots\\
\cdots{{}^\exists}\omega_{2j}\in\Omega_{z_j\to z'_{j-1}}^{{\bf n}}\,{{}^\exists}\omega_{
2j+1}\in\Omega_{z'_{j-1}\to x}^{{\bf n}}\;{{}^\exists}\omega_{2j+2},\omega_{2j+3}
\in\Omega_{x\to z'_j}^{{\bf n}}\\
\text{such that }~\omega_i\cap\omega_l={\varnothing}~(i\ne l)
\end{array}\!\right\},\end{aligned}$$
![\[fig:eventI\]A schematic representation of $\tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$ for $j\ge2$ consisting of $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$.](eventI)
where $\vec z_j^{(\prime)}=(z_1^{(\prime)},\dots,z_j^{(\prime)})$. Therefore, $$\begin{aligned}
{\label{eq:Theta'-2ndindbd5}}
{(\ref{eq:Theta'-2ndindbd4})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,z_j\\
z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z_i,z'_i)\bigg)\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,
\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}.\end{aligned}$$
Now we apply Lemma \[lmm:GHS-BK\] to bound [(\[eq:Theta’-2ndindbd5\])]{}. To clearly understand how it is applied, for now we ignore ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-2ndindbd5\])]{} and only consider the contribution from ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j, \vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}$. Without losing generality, we assume that $y,x,z_i,z'_i$ for $i=1,\dots,j$ are all different. Since there are $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$ as in [(\[eq:tildeI-def:=1\])]{}–[(\[eq:tildeI-def:geq2\])]{} (see also Figure \[fig:eventI\]), we multiply [(\[eq:Theta’-2ndindbd5\])]{} by $
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|
}{h}\right)\left(\frac{h}{U_b} \right)^2 \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$
Using the definition of $Ri_b$ (see Sect. 2), we re-write $Ri_g$ as follows: $$Ri_g = Ri_b \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$ Similarly, $N^2$ can be written as: $$N^2 = Ri_b \left(\frac{U_b^2}{h^2}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right).
\label{EqN2}$$ The velocity variances and TKE can be normalized as:
$$\sigma_{u_n}^2 = \frac{\sigma_u^2}{U_b^2},$$
$$\sigma_{v_n}^2 = \frac{\sigma_v^2}{U_b^2},$$
$$\sigma_{w_n}^2 = \frac{\sigma_w^2}{U_b^2},
\label{EqSigw}$$
$$\overline{e}_n = \frac{\overline{e}}{U_b^2}.
\label{Eqe}$$
Following the above normalization approach, we can also derive the following relationship for the energy dissipation rate: $$\overline{\varepsilon} = \nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n.
\label{EqEDR}$$ In order to expand $\overline{\varepsilon} = \overline{e} N$, we use Eq. \[EqN2\], Eq. \[Eqe\], and Eq. \[EqEDR\] as follows: $$\nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n = U_b^2 \overline{e}_n
Ri_b^{1/2} \left(\frac{U_b}{h}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ This equation can be simplified to: $$\overline{\varepsilon}_n = Re_b Ri_b^{1/2} \overline{e}_n \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ In a similar manner, $\overline{\varepsilon} = \overline{e} S$ can be re-written as: $$\overline{\varepsilon}_n = Re_b \overline{e}_n S_n.$$
---
bibliography:
- 'bibliography.bib'
---
---
abstract: 'Using 0.37 megaton$\cdot$years of exposure from the Super-Kamiokande detector, we search for 10 dinucleon and nucleon decay modes that have a two-body final state with no hadrons. These baryon and lepton number violating modes have the potential to probe theories of unification and baryogenesis. For five of these modes the searches are novel, and for the other five modes we improve th
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Faddeev-Popov method. Specifying to $SU(2)$, the Polyakov gauge is implemented by the gauge fixing conditions $$\label{eq:Pol1}
\partial_0 {\mathrm{tr}}\,\sigma_3 A_0=0\,, \qquad
{\mathrm{tr}}\,(\sigma_1\pm i\sigma_2) A_0 = 0\,,$$ where the $\sigma_i$ are the Pauli matrices. However, the gauge fixing is not complete. It is unchanged under time-independent gauge transformations in the Cartan sub-group. These remaining gauge degrees of freedom are completely fixed by the following conditions, $$\begin{aligned}
\nonumber
& \partial_1 \int dx_0 \, {\mathrm{tr}}\, \sigma_3 A_1 =0\,,\qquad
\partial_2 \int dx_0 dx_1 \, {\mathrm{tr}}\, \sigma_3 A_2 = 0\,,&\\
& \partial_3 \int dx_0 dx_1 dx_2\, {\mathrm{tr}}\,\sigma_3 A_3 = 0\,.&
\label{eq:Pol2}\end{aligned}$$ The gauge fixings are integral conditions and are the weaker the more integrals are involved. Basically they eliminate the corresponding zero modes. This can be seen directly upon putting the theory into a box with periodic boundary conditions, $T^4$, see e.g.[@Ford:1998bt].
The gauge fixing conditions , lead to the Faddeev-Popov determinant $$\begin{aligned}
\Delta_{FP}[A] = (2 T)^2 \left[ \prod_{x} \sin^2 \left( \frac{g A_0^3
(\vec x)}{ 2 T } \right) \right]\,,
\label{eq:FPdet}\end{aligned}$$ the computation of which is detailed in appendix \[app:FPdet\]. The integration over the longitudinal gauge fields precisely cancels the Faddeev-Popov determinant in the static approximation $\partial_i
A_0^c=0$, see Appendix \[app:FPdet\]. Finally we are left with the action $$\begin{aligned}
S_{\rm eff}[A] &\simeq & -\frac{1}{2} \beta \int d^3x\, A_0
\vec \partial{\,}^2 A_0\\\nonumber
&&\hspace{-1cm}-\frac{1}{2} \int_T d^4x\, A^a_{\bot,i} \left[(D_0^2)^{ab} +
\vec \partial^2\delta^{ab} \right] A_{\bot,i}^a +O(A_{\bot,i}^3)\end{aligned}$$ with $D_0^{ab} = \partial_0 \delta^{ab} + A^3_0 g f^{a3b}$ and transversal spatial gauge fields, $\partial_i A_{\bot,i}=0$. The generating functional of Yang-Mills theory in Polyakov gauge then reads $$\begin{alig
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at 10:00:00 PM CT thru Sat 11/17/2001 at 11:00:00 PM CT
Sat 11/17/2001 at 8:00:00 PM PT thru Sat 11/17/2001 at 9:00:00 PM PT
Sun 11/18/2001 at 4:00:00 AM London thru Sun 11/18/2001 at 5:00:00 AM London
Outage: Swap names of NAHOU-SQEFM01P and NAHOU-SQLAC01P
Environments Impacted: EFM Accounting users
Purpose: Migration of production EFM Accounting MSSQL Server from 6.5 to 2000
Backout:
Contact(s): William Mallary
Bob McCrory (713) 853-5749
Michael Kogotkov (713) 345-1677
Impact: CORP
Time: Sat 11/17/2001 at 6:00:00 PM CT thru Sun 11/18/2001 at 9:00:00 AM CT
Sat 11/17/2001 at 4:00:00 PM PT thru Sun 11/18/2001 at 7:00:00 AM PT
Sun 11/18/2001 at 12:00:00 AM London thru Sun 11/18/2001 at 3:00:00 PM London
Outage: CPR
Environments Impacted: All
Purpose: Hardware maintenance for Skywalker
Backout:
Contact(s): CPR Support 713-284-4175
Impact: CORP
Time: Sat 11/17/2001 at 6:00:00 PM CT thru Sun 11/18/2001 at 6:00:00 AM CT
Sat 11/17/2001 at 4:00:00 PM PT thru Sun 11/18/2001 at 4:00:00 AM PT
Sun 11/18/2001 at 12:00:00 AM London thru Sun 11/18/2001 at 12:00:00 PM London
Outage: Test/Dev general server maintenance for the following:
ferrari, modena, trout, cypress, bravo
Environments Impacted: ENW test and development
Purpose: Established maintenance window for Test and Development
Backout: None
Contact(s): Malcolm Wells 713-345-3716
Impact: CORP
Time: Sat 11/17/2001 at 6:00:00 PM CT thru Sun 11/18/2001 at 9:00:00 AM CT
Sat 11/17/2001 at 4:00:00 PM PT thru Sun 11/18/2001 at 7:00:00 AM PT
Sun 11/18/2001 at 12:00:00 AM London thru Sun 11/18/2001 at 3:00:00 PM London
Outage: New disk layout for server ERMS CPR server skywalker.
Environments Impacted: ERMS
Purpose: Move toward new standard disk layout.
Backout: If database doesn't work after restore to new layout we will roll back to the old mirrored copy and resync to that copy.
Contact(s): Malcolm Wells 713-345-3716
Impact: CORP
Time: Sat 11/17
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r_0}^2)<+\infty$ and this concludes the proof. $\Box$
We follow here the proof of Baccelli and Bordenave [@baccellibordenave Section 3.6], where the case of a fixed radius $R$ is considered. Recall that $\mathcal{T}$ and $\mathcal{T}_{-e_x}$ are the RST and DSF with direction $-e_x$, constructed on the same PPP $N$. We denote by $\mathcal{A}(X)$ and $\mathcal{A}_{-e_x}(X)$ the ancestors of $X$ in $\mathcal{T}$ and $\mathcal{T}_{-e_x}$. Let $r>0$, $\alpha>0$ and $\beta>0$. $$\begin{gathered}
{{\mathbb P}}\big(\mathcal{T}\cap B((r,0),r^\alpha) \neq \mathcal{T}_{-e_x}\cap B((r,0),r^\alpha)\big)= {{\mathbb P}}\Big(\bigcup_{X\in N\cap B((r,0),r^\alpha)}\big\{\mathcal{A}(X)\not=\mathcal{A}_{-e_x}(X)\big\}\Big)\\
\begin{aligned}
\leq & {{\mathbb P}}\big(N(B((r,0),r^\alpha))> r^\beta\big)+r^\beta \ C \sup_{X\in N\cap B((r,0),r^\alpha)} {{\mathbb P}}\big(\mathcal{A}(X)\not=\mathcal{A}_{-e_x}(X)\big) \nonumber\\
\leq & \exp\Big(-r^\beta \log \big(\frac{r^{\beta-2\alpha}}{e \pi}\big)\Big)+r^\beta \ C \ \frac{r^\alpha+1}{r-r^\alpha},
\end{aligned}\end{gathered}$$by using [@talagrand Lemma 11.1.1] for the first term in the r.h.s. and [@baccellibordenave Lemma 3.4] for the second term. The first term converges to 0 iff $\beta>2\alpha$ and the second term converges to 0 iff $\alpha<1$ and $\beta+\alpha<1$. As a consequence, we see that for any $\alpha<1/3$ we can choose $\beta>2/3$, so that both terms converges to 0 when $r\rightarrow +\infty$. $\Box$
Semi-infinite paths in a given direction {#section:directiondeterministe}
========================================
In this section, we fix a direction $\theta\in [0,2\pi)$ and are interested in the semi-infinite paths with asymptotic direction $\theta$. Our first result (Section \[section:unique\]) refines Theorem \[HN1\] and states that there exists a.s. a unique semi-infinite path with direction $\theta$. We deduce from this a precise description of the semi-infinite path with direction $\theta$ (Section \[section:description0\]).\
For the proof, let us introduce further not
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ntation of NHEK’s isometry group then requires $k-1-2h \neq 0$, otherwise there would be a lowest-weight state that would lead to a finite-dimensional (and hence non-unitary) representation. The same conclusion holds for either the vector or the tensor bases. The values of $h$ also depend on the regularity conditions we impose. For instance, in global coordinates, the highest-weight scalar basis is proportional to $$F^{(m\,h\,0)} \propto (\sin\psi)^{-h} \exp[i (h \tau + m \varphi) +m \psi].$$ Regularity at the boundaries $\psi=0$ and $\psi=\pi$ requires $h \le 0$. Another example is given in Sec. \[sec:sep-scalar\] when we solve for the free massless scalar wave equation in the NHEK spacetime, where $h$ must take on some fixed values due to the regularity conditions for spheroidal harmonics.
Orthogonality in global coordinates {#sec:orthogonality-basis}
===================================
In this section we present a proof that all the scalar, vector, and symmetric tensor basis functions of NHEK’s isometry group, when given in global coordinates, form orthogonal basis sets. In this proof we will use the vector basis functions defined on $\Sigma_u$ as an example. That is, they are functions of $\tau,\varphi,\psi$. As we shall see, lifting to the whole manifold $\mathcal{M}$ and extending the proof to the scalar and symmetric tensor cases will be straightforward.
Let us introduce the metric induced on the hypersurface $\Sigma_u$ as $\gamma_{ij}$, and $D$ is the unique torsion-free Levi-Civita connection that is compatible with $\gamma$. Here Latin letters in the middle of the alphabet ($i,j,k$) denote 3-dimensional tangent indices on $\Sigma_{u}$. Consider the vector basis function ${\bf u}^{(m\,h\,k)}(\tau,\varphi,\psi)$ and ${\bf v}^{(m^\prime\,h^\prime\,k^\prime)}(\tau,\varphi,\psi)$. We would like to show bases with different $m,h,k$ are orthogonal, $$\langle {\bf u}, {\bf v} \rangle \equiv \int_{\Sigma_u} {\mathrm{dVol}\,}\overline{u_i^{(m\,h\,k)}} v^i_{(m^\prime\,h^\prime\,k^\prime)}
\propto \delta
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- w_0}_2^2}\\
\leq& \E{\lrp{\lrn{w_{k\delta} - w_0 - \delta \lrp{\nabla U(w_{k\delta}) - \nabla U(w_0)}}_2 + \delta \lrn{\nabla U(w_0)}_2}^2}\\
\leq& \lrp{1 + \frac{1}{n}}\E{\lrn{w_{k\delta} - w_0 - \delta \lrp{\nabla U(w_{k\delta}) - \nabla U(w_0)}}_2^2 }\\
&\quad + (1+n)\delta^2 \E{\lrn{\nabla U(w_0)}_2^2}\\
\leq& \lrp{1 + \frac{1}{n}} \lrp{1 + \delta L}^2 \E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2}\\
\leq& e^{1/n + 2\delta L }\E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2}
\end{aligned}$$ where the first inequality is by triangle inequality, the second inequality is by Young’s inequality, the third inequality is by item 1 of Assumption \[ass:U\_properties\].
Inserting the above into gives $$\begin{aligned}
\E{\lrn{w_{(k+1)\delta} - w_0}_2^2} \leq e^{1/n + 2\delta L }\E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2} + \delta \beta^2
\end{aligned}$$
Applying the above recursively for $k=1...n$, we see that $$\begin{aligned}
& \E{\lrn{w_{n\delta} - w_0}_2^2}\\
\leq& \sum_{k=0}^{n-1} e^{(n-k) \cdot (1/n + 2\delta L)} \cdot \lrp{2n\delta^2 L^2 \E{\lrn{w_0}_2^2} + \delta \beta^2}\\
\leq& 16 \lrp{n^2 \delta^2 L^2 \E{\lrn{w_0}_2^2} + n\delta \beta^2}\\
=& 16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}
\end{aligned}$$
[Discretization Bounds]{} \[ss:discretization\_bonuds\]
\[l:vt-wt\] Let $v_{k\delta}$ and $w_{k\delta}$ be as defined in and . Then for any $\delta,n$, such that $T:= n\delta \leq \frac{1}{16L}$, $$\begin{aligned}
\E{\lrn{v_{T} - w_{T}}_2^2} \leq 8 \lrp{2T^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + T L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}
\end{aligned}$$ If we additionally assume that $\E{\lrn{v_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$, $\E{\lrn{w_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{v_T -
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l examine properties of those classes.
The first class we consider involves the special case that the gauge bundle is a pullback from the base. This is equivalent to the statement that the subgroup $G$ of the gauge group that acts trivially on the base, also acts trivially on the fibers of the gauge bundle.
In this case, we will argue that, at least for banded gerbes, the heterotic (0,2) SCFT factorizes – it is equivalent to a heterotic string on a disjoint union of spaces with bundles, following essentially the same mechanism as in (2,2) strings.
Review of (2,2) decomposition conjecture {#sect:decomp-22review}
----------------------------------------
As was reviewed earlier in section \[sect:rev-gerbes\], gauge theories in which a subgroup of the gauge group acts trivially on massless matter break cluster decomposition. However, it was argued in [@summ] that such theories are equivalent to tensor products / disjoint unions of cluster-decomposition-obeying theories. For example, a gauged nonlinear sigma model of this form is equivalent to a nonlinear sigma model on a disjoint union of ordinary spaces. The latter violates cluster decomposition, but does so in an obviously trivial fashion, and so there is no essential difficulty with the quantum field theory.
For (2,2) supersymmetric gauged nonlinear sigma models in two dimensions, this was encapsulated in [@summ] in the “decomposition conjecture,” which we shall generalize to heterotic strings. To make this paper self-contained, we take a moment here to review the statement of the decomposition conjecture.
Suppose we have a $K$-gerbe over $[X/H]$, defined by the quotient $[X/G]$ where $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \:
H \: \longrightarrow \: 1.$$ Let $\hat{K}$ denote the set of irreducible representations of $K$. There is a natural action of $H$ on $\hat{K}$, defined as follows: given $h \in H$ and $\rho \in \hat{K}$, pick a lift $\tilde{h} \in G$ of $h$, and define $h \cdot \rho$ by, $$(h \cdot \rho)(g) \: \equ
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per bounded by $\max_{i \in [d]}\big\{\ell_j\P[i \in I_j| i \in S_j]\big\} \leq {\ell_j}^2 e^{2b}/\kappa_j$. Therefore using triangle inequality, we have, $$\begin{aligned}
&&\Bigg\|\sum_{j =1}^n\E\big[(M^{(j)})^2\big]\Bigg\| \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{1}{(\kappa_j-1)^2} \Bigg(\frac{(\kappa_j^2 - {\ell_j}^2)\ell_j e^{2b}}{\kappa_j} + {\ell_j}^2 + e^{2b}(\kappa_j+\ell_j)(2\ell_j + {\ell_j}^2/\kappa_j) + \ell_j\kappa_j \Bigg)\Bigg\} \nonumber\\
&\leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(\frac{(\kappa_j^2 -{\ell_j}^2)}{(\kappa_j-1)^2} + \frac{\ell_j\kappa_j}{{(\kappa_j-1)^2}} + \frac{2(\kappa_j+\ell_j)\kappa_j}{(\kappa_j-1)^2} + \frac{(\kappa_j+\ell_j)\ell_j}{(\kappa_j-1)^2} + \frac{\kappa_j^2}{(\kappa_j-1)^2} \Bigg)\Bigg\} \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(\frac{(\kappa_j^2 -1)}{(\kappa_j-1)^2} + \frac{\kappa_j(\kappa_j-1)}{{(\kappa_j-1)^2}} + \frac{4\kappa_j^2}{(\kappa_j-1)^2} + \frac{2\kappa_j(\kappa_j-1)}{(\kappa_j-1)^2} + \frac{\kappa_j^2}{(\kappa_j-1)^2} \Bigg)\Bigg\} \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(3 + 2 + 16 + 4 + 4 \Bigg)\Bigg\} \label{eq:topl_grad1}\\
& \leq & 29 e^{2b} \max_{i \in [d]}\bigg \{\sum_{j:i \in S_j}\frac{\ell_j}{\kappa_j}\bigg\} \nonumber\\
& = & 29 e^{2b}D_{\max} \label{eq:topl_grad2}\\
& = & \frac{29 e^{2b}}{\beta d} \sum_{j = 1}^n \ell_j\;, \label{eq:topl_grad3}\end{aligned}$$ where uses the fact that $\kappa_j \geq 2$ and $1 \leq \ell_j \leq \kappa_j -1$ for all $j \in [n]$. follows from the definition of $D_{\max}$, Definition \[def:comparison\_graph1\] and follows from the Equation . Also, note that ${\|M_j\|} \leq 2$ for all $j \in [n]$. Applying matrix Bernstien inequality, we have, $$\begin{aligned}
\mathbb{P}\Big[{\|M - \E[M]\|} \geq t\Big] \leq d \,\exp\Bigg(\frac{-t^2/2}{\frac{29e^{2b}}{\beta d}\sum_{j=1}^n\ell_j + 4t/3}\Bigg). \end{aligned}$$ Therefore, w
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e maximization of KSE and the minimization of the mixing time**. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics and gives a physical meaning to the KSE. We first show that on average, the greater the KSE, the smaller the mixing time, and we correlated this result to its link with the transition matrix eigenvalues. Then, we show that the dynamics that maximises KSE is close to the one minimizing the mixing time, both in the sense of the optimal diffusion coefficient and the transition matrix.
Consider a network with $m$ nodes, on which a particle jumps randomly. This process can be described by a finite Markov chain defined by its adjacency matrix $A$ and its transition matrix $P$. $A(i,j)=1$ if and only if there is a link between the nodes $i$ and $j$ and 0 otherwise. $P=(p_{ij})$ where $p_{ij}$ is the probability for a particle in $i$ to hop on the $j$ node. Let us introduce the probability density at time $n$ $\mu_n=(\mu_n^i)_{i=1...m}$ where $\mu_n^i$ is the probability that a particle is at node $i$ at time $n$. Starting with a probability density $ \mu_0$, the evolution of the probability density writes: $\mu_{n+1}=P^t\mu_{n}$ where $P^t$ is the transpose matrix of $P$.\
Within this paper, we assume that the Markov chain is irreducible and thus has a unique stationary state.
Let us define: $$\label{eqdn}
d(n)= max{ || (P^t)^n\mu - \mu_{stat}|| \text{ } \forall \text{ } \mu },$$ where $||.||$ is a norm on $\mathbb{R}^n$. For $ \epsilon > 0$, the mixing time, which corresponds to the time such that the system is within a distance $\epsilon$ from its stationary state is defined as follows:
$$\label{eq:mix1}
t(\epsilon)= \min{ n, \, d(n) \leq \epsilon}.$$
For a Markov chain the KSE takes the analytical form [@billingsley1965ergodic]:
$$\label{eqhks}
h_{KS}=-\sum_{ij} \mu_{stat_{i}}p_{ij}\log(p_{ij}).$$
Random $m$ size Markov matrices are ge
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andidate critical solution. We discuss its global causal structure and its symmetries in relation with those of the corresponding continously self-similar solution derived in the $\Lambda=0$ case. Linear perturbations on this background lead to approximate black hole solutions. The critical exponent is found to be $\gamma = 2/5$.'
author:
- |
Gérard Clément$^{a}$ [^1] and Alessandro Fabbri$^{b}$ [^2]\
\
[$^{(a)}$Laboratoire de Physique Théorique LAPTH (CNRS),]{}\
[B.P.110, F-74941 Annecy-le-Vieux cedex, France]{}\
[$^{(b)}$Dipartimento di Fisica dell’Università di Bologna]{}\
[and INFN sezione di Bologna,]{}\
[Via Irnerio 46, 40126 Bologna, Italy]{}
date: 'March 23, 2001'
title: |
**Analytical treatment of critical collapse in 2+1 dimensional AdS spacetime:\
a toy model**
---
Introduction
============
Since its discovery, the BTZ black hole solution \[1\] of 2+1 dimensional AdS gravity has attracted much interest because it represents a simplified context in which to study the classical and quantum properties of black holes. A line of approach which has been opened only recently [@CP; @HO; @gar; @burko] concerns black hole formation through collapse of matter configurations coupled to 2+1 gravity with a negative cosmological constant. As first discovered in four dimensions by Choptuik [@chop], collapsing configurations which lie at the threshold of black hole formation exhibit properties, such as universality, power-law scaling of the black hole mass, and continuous or discrete self-similarity, which are characteristic of critical phenomena [@gund]. In the case of a spherically symmetric massless, minimally coupled scalar field, a class of analytical continously self-similar (CSS) solutions was first given by Roberts [@rob; @brady; @oshiro]. These include critical solutions, lying at the threshold between black holes and naked singularities, and characterized by the presence of null central singularities. Linear perturbations of these solutions [@fro; @hay] lead to appr
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N\] A *cone* ${\mathcal{C}}$ is a complete independent set of localized predecessor walks starting at ${\mathit{s}}_{\text{crux}}$.
[\[Stopping rule\]]{} We avoid the specificity of various stopping criteria (§\[S:CONE\_DESCRIPTION\]) by introducing the equivalent but arbitrary notion of localization.
\[S:CONE\_EDGE\_STEP\] Let ${\mathcal{C}}$ be a cone with ${\mathit{w}} \in {\mathcal{C}}$ a member localized predecessor walk. Suppose $n = {\lvert{{\mathit{w}}}\rvert}$ is the number of steps in ${\mathit{w}}$. The terminus ${\mathit{w}}(-(n-1)) = {\mathit{w}}_{-(n-1)}$ is the *edge step* of walk ${\mathit{w}}$.
\[S:CONE\_EDGE\] Let ${\mathcal{C}}$ be a cone. Its *edge*, written ${{\operatorname{edge}{{\mathcal{C}}}}}$, is the collection of edge steps of all member localized predecessor walks.
\[S:ACYLIC\_CONE\] An *acyclic* cone has no cycle (loop) in its path projection $\overline{\mho}_\Lambda$ (see §\[D:EXTENDED\_PROJECTION\]).
\[T:ACYLIC\_CONE\_CORRESPONDENCE\] The acyclic cone ${\mathcal{C}}$ and ${{\operatorname{edge}{{\mathcal{C}}}}}$ are in one-to-one correspondence via the edge step relation of a localized predecessor walk.
Assume the opposite: there are different localized predecessor walks with the same edge step. Let ${\mathit{u}}$ and ${\mathit{v}}$ be two different walks with common edge step ${\mathit{s}}_\text{common}$.
Suppose ${\lvert{{\mathit{u}}}\rvert} = m$ and ${\lvert{{\mathit{v}}}\rvert} = n$, so the indexes of ${\mathit{s}}_\text{common}$ are $-(m-1)$ and $-(n-1)$ respectively.
We assert that if ${\mathit{u}}_{-(m-1)+i} = {\mathit{v}}_{-(n-1)+i}$ for some $0 \leq i$, then ${\mathit{u}}_{-(m-1)+(i+1)} = {\mathit{v}}_{-(n-1)+(i+1)}$.
Suppose sequencing is governed by an actuated automaton ${\mathfrak{A}}$. So sequenced, the next step in predecessor walk ${\mathit{u}}$ is ${\mathit{u}}_{-(m-1)+(i+1)} = {\mathfrak{A}}({\mathit{u}}_{-(m-1)+i})$. Similarly, the next step in ${\mathit{v}}$ is ${\mathit{v}}_{-(n-1)+(i+1)} = {\mathfrak{A}}({\mathit{v}}_{-(n-1)+i})$. But if ${\mathit{u}}
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b{M}])
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& x_{\overline{\sigma_i\mathbb{M}}}
C[\sigma_i\mathbb{M}]
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=&{\cal C}(\sigma_i\mathbb{M}, \sigma, \mathbb{F}).\end{aligned}$$
\[rem:defect2\] Lemma \[lem:defect\] also holds if $M_{i+1}$ \[resp. $F_{i+1}$\] is defective.
By Lemma \[lem:defect\] and Remark \[rem:defect2\] we may assume that any crank form expression is given in normal form.
\[lem:crex\] Let ${\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ be a crank form expression of $w$ in normal form. If $M_i$ and $M_{i+1}$ are propagating then ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to another crank form expression ${\cal C}(\sigma_i\mathbb{M}, \sigma_i\sigma,\mathbb{F})$ in normal form. Similarly if $F_i$ and $F_{i+1}$ are propagating, then ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to ${\cal C}(\mathbb{M},\sigma\sigma_i,\sigma_i\mathbb{F})$.
Let $C_{\mathbb{M}}[i]$ and $C_{\mathbb{M}}[i+1]$ be $i$-th and $(i+1)$-st cranks of $C[\mathbb{M}]$. By Figure \[fig:crex\], we have $$P_{\mathbb{M},i}C_{\mathbb{M}}[i]C_{\mathbb{M}}[i+1]
=C_{\sigma_i\mathbb{M}}[i]C_{\sigma_i\mathbb{M}}[i+1]\sigma_i.$$
![Crank form exchange[]{data-label="fig:crex"}](15.eps)
Thus we obtain $$\begin{aligned}
{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
&=&
x_{\overline{\mathbb{M}}}
C[\mathbb{M}]
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=&
(x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i})
(P_{\mathbb{M},i}C[\mathbb{M}])
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
y^*_{\overline{\mathbb{F}}}\\
&=&
x_{\overline{\sigma_i\mathbb{M}}}
C[\sigma_i\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma_i\sigma]
C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& {\cal C}(\sigma_i\mathbb{M}, \sigma_i\sigma, \mathbb{F}).\end{aligned}$$
By Lemma \[lem:defect\], Remark \[rem:defect2\] and Lemma \[lem:crex\] we obtain the followin
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")) {
sendEmail(url.substring(7));
return true;
}
return false;
}
Q:
Move constructor for a map
I think I do understand "the basic IDEA" of move semantics, but now when I'm on the stage of implementing my own map I stopped and started to think about it when I was going to write a use case and walk through for move ctor of map. Correct me if I'm wrong but what I do understand how the whole business of move semantics works is that they suppose to help in avoiding unnecessary copying? Right? Now, take a map for example and just for the purpouse of this example assume that my map is modeled as:
class Map
{
Link* impl_;//THIS IS A POINTER TO A LINK WHICH HAS A parent, left and right (of Link type)
Map(Map&& tmp);//move ctor
//unnecessary code ommited
};
And here is the snag:
When I'm trying to think of move ctor for my map I cannot see a way of avoiding allocating a new space for all those links which needs to be created and then their pointers swapped with those from a tmp Map object (passed as a arg to my move ctor).
So I do have to allocate space anyway, or don't I?
A:
All you have to do is reassign the Link pointer, since all the other Link pointers are attached to it, they will now be part of the new Map.
Map(Map&& tmp) :impl_(tmp.impl_) { tmp.impl_ = nullptr; }
This is assuming no other data members.
A:
You need five operations total: the classic "big three" (copy constructor, copy assignment operator, destructor) and the two new move operations (move constructor, move assignment operator):
// destructor
~Map();
// copy constructor
Map(const Map& that);
// move constructor
Map(Map&& that)
{
impl_ = that.impl_;
that.impl_ = 0;
}
// copy assignment operator
Map& operator=(const Map& that);
// move assignment operator
Map& operator=(Map&& that)
{
using std::swap;
swap(impl_, that.impl_);
return *this;
}
The basic idea behind the move assignment operator is that swap(map1, map2) has the same observable side effect as map1 = map2 if you don't inspect map2 a
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ified intercept model was 88.8%, using REML+standard, but 95.8% using REML+KR.
Using REML+Satterthwaite gave very similar results to REML+KR. Occasionally, there was some over‐coverage using REML+KR or REML+Satterthwaite, particularly when using a low number of trials (*K* = 5). For example, coverage was close to 99% (regardless of which model was used), in a setting of *K* = 5 trials with an equal number of participants per trial (scenario A1; *n* ~*i*~ = 100), and in a setting of *K* = 5 trials with some small‐sized and some large‐sized trials (scenario B2‐A1; 2 small trials where *n* ~*i*~ *∼*U(30, 100), and 3 large trials where *n* ~*i*~ *∼*U(900, 1000)).
*(ii) Under a beta distribution intercept generating mechanism*
For the beta distribution intercept generating mechanism (Figure [2](#sim7930-fig-0002){ref-type="fig"}B and Web Table C.III), using REML+standard again gave better coverage than using ML, and using REML+KR or REML+Satterthwaite generally further improved upon this coverage (ie, moved it closer to 95%), especially with scenarios concerning at least 10 trials that had a large variation in sample sizes.
As before, under ML estimation, the random intercept model showed better estimates of between‐trial variance and improved coverage (closer to 95%) than the stratified intercept model. However, differences between the two models were generally small for estimation under REML (with or without a 95% CI correction).
### 3.2.6. Common treatment effect data generating mechanism {#sim7930-sec-0016}
Results based on a common (fixed) treatment effect data generating mechanism are shown in Web Appendix B. All fitted models assumed a common treatment effect and converged every time (ie, 100% convergence), and there was negligible difference in mean percentage bias of $\hat{\theta}$ between ML and REML estimation options for either model (stratified or random intercept), or between either model (Web Table B.1). The percentage coverage results were stable across all comparisons, ran
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E1'$)-($E5'$) omitting the ones which involve $s_{n-1}$ and adding the following relations: $$\begin{gathered}
f_{*}s_{n-2}s_{n-3}\cdots s_3s_2s_1
s_2s_3\cdots s_{n-3}s_{n-2}f_{*} \nonumber\\
\quad =\, f_{*}s_{n-2}s_{n-3}\cdots s_3s_2f
s_2s_3\cdots s_{n-3}s_{n-2}\phantom{,} \tag{$R2^*$} \\
\quad =\, s_{n-2}s_{n-3}\cdots s_3s_2f
s_2s_3\cdots s_{n-3}s_{n-2}f_{*},\nonumber\end{gathered}$$ $$\tag{$R4^*$}
ff_{*} = f_{*}f,\quad
ef_{*} = f_{*}e,\quad
f_{*}s_i = s_if_{*}\ \mbox{($1\leq i \leq n-3$)},$$ $$\tag{$E4^*$}
\begin{array}{rcl}
f_{*}s_{n-2}s_{n-3}\cdots s_1es_1\cdots s_{n-3}s_{n-2}f_{*}
&=& f_{*},\\
es_{1}s_{2}\cdots s_{n-2}f_{*}s_{n-2}\cdots s_{2}s_{1}e
&=& e.
\end{array}$$ We understand $A_{1+\frac{1}{2}}(Q) = A_{2-\frac{1}{2}}(Q)$ is defined by the generators $1$, $e$ and $f$ with the relations $e^2 = Qe$, $f^2 = f$, $efe =e$, $fef = f$. (Hence, $A_{2-\frac{1}{2}}(Q)$ is a rank 5 module with a basis $\{1, e, f, ef, fe\}$.)
The relations ($R2^*$) correspond to the relations $f_{n-1}s_{n-2}f_{n-1} = f_{n-1}f_{n-2} = f_{n-2}f_{n-1}$. We deduce $f_{*}s_{n-2}f_{*} = f_{*}s_{n-2}f_{*}s_{n-2}
= f_{*}s_{n-2}f_{*}s_{n-2}$ from ($R2^*$).
First we note that all the generators of $A_{n-\frac{1}{2}}(Q)$ have the part which contains $n$ and $n'$ simultaneously.
We consider the transpositions of indices $i\leftrightarrow n-i+1$. These transpositions make $A_{n-\frac{1}{2}}(Q)$ a subalgebra of $A_{n}(Q)$ generated by $${\cal L}_{n-\frac{1}{2}}^1 = \{f_1, \ldots, f_{n-1},
s_2, \ldots, s_{n-1}, e_2, \ldots, e_n\}.$$ By the relation ($R0$), $A_{n-\frac{1}{2}}(Q)$ is actually generated by letters $\{f_1$, $f_2$, $e_2$ and $s_2,\ldots, s_{n-1}\}$. Each of these generators has a part which includes $\{1, 1'\}$. In the following in this section, we suppose that $A_{n-\frac{1}{2}}(Q)$ is generated by the letters in ${\cal L}^1_{n-\frac{1}{2}}$. The $\mathbb{Z}[Q]$ bases of $A_{n-\frac{1}{2}}(Q)$ consist of $\Sigma^1_{n-\frac{1}{2}}$ a subset of seat-plans in $\Sigma^1_n$ which have at l
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PAGE2._oF::z._ 6)'! ~~~L•~ • ; ' j I
I
r=rz§i Z-:
! I
-=--··=--=·-=-=-·-=-=---=. =·--=--=-·=····=·--=--·=·-·=---=--=·-=·-=·=- ·=--=·=-=··=···-=-·=·--=--=--=-==··- -.=·
UNITED STATES DISTRICT COURT
FOR THE DISTRICT OF COLUMBIA
HAROLD STANLEY JACKSON, :
:
Plaintiff, : Civil Action No.: 19-1487
:
v. : Re Document No.: 24
:
STARBUCKS CORPORATION, :
:
and :
:
DAN WHITE-HUNT :
:
Defendants. :
MEMORANDUM OPINION
DEN
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{i,i;k}(\M_\muhat)\right)q^i,$$
On the other hand the mixed Hodge numbers $h^{i,j;k}(X)$ of any complex non-singular variety $X$ are zero if $(i,j,k)\notin
\{(i,j,k)|\hspace{.05cm} i\leq k,j\leq k,k\leq i+j\}$, see [@Del1]. Hence $h^{0,0;k}(\M_\muhat)=0$ if $k>0$.
We thus deduce that the constant term of $E(\M_\muhat;q)$ is $h^{0,0;0}(\M_\muhat)$.
From the above lemma and Formula (\[mainresult\]) we are reduced to prove that the coefficient of the lowest power $q^{-\frac{d_\muhat}{2}}$ of $q$ in $\mathbb{H}_\muhat(\sqrt{q},1/\sqrt{q})$ is equal to $1$.
The strategy to prove this goes in two steps. First, \[step-1\] we analyze the lowest power of $q$ in $\calA_{\lambda\muhat}(q)$, where $$\Omega\left(\sqrt{q},1/\sqrt{q}\right) = \sum_{\lambda, \muhat}
\calA_{\lambda\muhat}(q)\, m_\muhat.$$ Then \[step-2\] we see how these combine in $\Log
\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$. In both case, Lemma \[nrm-ineq\] and Lemma \[ineq-1\], we will use in an essential way the inequality of §\[appendix\]. Though very similar, the relation between the partitions $\nu^p$ in these lemmas and the matrix of numbers $x_{i,j}$ in §\[appendix\] is dual to each other (the $\nu^p$ appear as rows in one and columns in the other).
Preliminaries {#delta-non-neg}
-------------
For a multi-partition $\muhat\in \left(\calP_n\right)^k$ we define $$\label{Delta-defn}
\Delta(\muhat):=\tfrac12 d_\muhat-1= \tfrac12(2g-2+k)n^2- \tfrac12
\sum_{i,j} \left(\mu_j^i\right)^2.$$
Note that when $g=0$ the quantity $-2\Delta(\muhat)$ is Katz’s [ *index of rigidity*]{} of a solution to $X_1\cdots X_k=I$ with $X_i\in
\calC_i$ (see for example [@kostov2]\[p. 91\]).
From $\muhat$ we define as above Theorem \[MA\] a comet-shaped quiver $\Gamma=\Gamma_\muhat$ as well as a dimension vector $\v=\v_\muhat$ of $\Gamma$. We denote by $I$ the set of vertices of $\Gamma$ and by $\Omega$ the set of arrows. Recall that $\muhat$ and $\v$ are linearly related ($v_0=n$ and $v_{[i,j]}=n-\sum_{r=1}^j \mu^i_r$ for $j>1$ and conversely, $\mu_1^i=
| 2,123
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| 0.768981
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us harder than AES alone which does not have the decoding in noise problem. In particular, it is easily seen that if the Y-00 in the configuration of Fig. 2 can be broken, then each $AES_i$ itself can be broken.
\[htbp\]
[ {width="4.5in" height="2in"}]{}
The question arises as to what constitutes a fair comparison between a given stream cipher ENC versus Y-00 on top of ENC. A different configuration was given for ENC in [@nair06], where a single classical stream cipher (say AES) is used without parallelization but is adjusted to give the same clock rate for encrypting each data bit. The present scheme appears simpler in principle and more secure in practice when AES is used in ENC, because the functionality of multiple AES in parallel cannot be replaced by a single AES. However, with such parallelization for maintaining the same clock rate as AES (or ENC alone), the question arises as to whether the added security from Y-00 can be obtained from, say, nonlinearly combining the parallel AES’s. This question cannot be answered until security is precisely defined and quantified. However, it may be observed in this connection that there is no known attack developed for AES observed *in noise*, and the intrinsic nonlinearity of AES renders all known decoding attacks inapplicable.
The major qualitative advantage of Y-00 [@yuen05qph; @nair06] compared to a standard nonrandom cipher is that the quantum noise automatically provides high speed true randomization not available otherwise, thus giving it a different kind of protection from nonrandom ciphers. Furthermore, one has to attack such physics-based cryptosystem at the communication line with physical (measurement) equipment, which is not available to everyone at every place, whereas one only needs to sit at a computer terminal to attack conventional ciphers. In this connection, it may be mentioned that the high rate heterodyne attack needed on Y-00 is currently not quite technologically feasible, though it may be in the not-too-far future.
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|
e was a significant main effect of *family member* \[χ^2^(3) = 33.99, *p* \< 0.001\], as will be explained below (see section "Similarities and Differences Across Members Within One Family"). There was also a significant effect of the *ill child's age at diagnosis* \[χ^2^(1) = 5.07, *p* = 0.02\]: the older the ill child was at diagnosis, the less all family members reported to experience positive emotions. None of the other variables were significantly related to positive emotions (all χ^2^ \< 3.30, all *p* \> 0.07). Of note, when excluding the non-significant interactions (interaction with FSI, interaction with cancer appraisal, interaction between family functioning and cancer appraisal), the interaction effect between *FRI* and *family member* did no longer reach significance \[χ^2^(3) = 6.60, *p* = 0.09\].
Family Functioning, Cancer Appraisal and Quality of Life
--------------------------------------------------------
The final models for the associations between family functioning, cancer appraisal and quality of life for mothers and fathers on the one hand and patients and siblings on the other hand are shown in [Table 4](#T4){ref-type="table"}.
######
Final models for the associations between family functioning, cancer appraisal and reported quality of life.
QoL mothers and fathers (*N* = 157; 90 mothers, 67 fathers) QoL patients and siblings (*N* = 48; 20 patients, 28 siblings)^1^
------------------------------------------------------------ ------------------------------------------------------------- ------------------------------------------------------------------- --------------- -------- ------------------- ---------------
**Variables of interest**
FES -- FRI
| 2,125
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|
ma_1}{\longrightarrow}
& U_i\\
\downarrow && \downarrow\\
S& =& S.
\end{array}$$ Applying this to $X\to S$ and $R\to S$, we obtain a commutative diagram $$\begin{array}{lcl}
{{\mathcal O}}_S^{m(X)}&
\stackrel{p_1^*(s)-p_2^*(s)}{\longrightarrow}
&{{\mathcal O}}_S^{m(R)}\\
\ \downarrow && \ \downarrow\\
{{\mathcal O}}_X&\stackrel{p_1^*-p_2^*}{\longrightarrow}& {{\mathcal O}}_R.
\end{array}$$ Thus, for $\tau:={p_1^*(s)-p_2^*(s)}$, we get a morphism $$\ker\Bigl[{{\mathcal O}}_S^{m(X)}
\stackrel{\tau}{\longrightarrow}
{{\mathcal O}}_S^{m(R)}\Bigr]\to {{\mathcal O}}_{(X/R)^{cat}}.$$ The kernel on the left is $m:=|X_s/R_s|$ copies of ${{\mathcal O}}_S$, hence we obtain a factorization $$(X/R)^{cat}\to \amalg_mS\to S
{\quad\mbox{such that}\quad}
{\operatorname{red}}(X/R)^{cat}_s\to \amalg_m{\operatorname{Spec}}k(s)$$ is an isomorphism.
For later reference, we record the following straightforward consequence.
\[quot.of.sub\] Let $R\rightrightarrows X$ be a finite, set theoretic equivalence relation such that $X/R$ exists. Let $Z\subset X$ be a closed $R$-invariant subscheme. Then $Z/R|_Z$ exists and $Z/R|_Z\to X/R$ is a finite and universal homeomorphism (\[univ.homeo.defn\]) onto its image.
\[subquot.neq.sub\] Even in nice situations, $Z/R|_Z\to X/R$ need not be a closed embedding, as shown by the following examples.
(\[subquot.neq.sub\].1) Set $X:={{\mathbb A}}^2_{xy}\amalg {{\mathbb A}}^2_{uv}$ and let $R$ be the equivalence relation that identifies the $x$-axis with the $u$-axis.
Let $Z=(y=x^2)\amalg (v=u^2)$. In $Z/R|_Z$ the two components intersect at a node, but the image of $Z$ in $X/R$ has a tacnode.
In this example the problem is clearly caused by ignoring the scheme structure of $R|_Z$. As the next example shows, similar phenomena happen even if $R|_Z$ is reduced. (\[subquot.neq.sub\].2) Set $Y:=(xyz=0)\subset {{\mathbb A}}^3$. Let $X$ be the normalization of $Y$ and $R:=X\times_YX$. Set $W:=(x+y+z=0)\subset Y$ and let $Z\subset X$ be the preimage of $W$. As computed in (\[sch.quot.exmp\]), $R$ and $R|
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| 0.773349
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ar
D^{(+)}$ is well-defined by the rule $x+[J,J]\mapsto x+\langle D,D\rangle$.
$$\begin{CD}
J @>\subseteq>> D \\
@VVV @VVV \\
\bar J @>\phi>> \bar D
\end{CD}$$
It is evident that $\phi$ is injective if and only if $\langle
D,D\rangle\cap J=[J,J]$.
Let $x\in\langle D,D\rangle\cap J$. Then $x{\mathbin\vdash}y=y{\mathbin\dashv}x=0$ for every $y\in D$, hence $x{\mathbin{{}_{(\vdash)}}}y=\frac{1}{2}(x{\mathbin\vdash}y+y{\mathbin\dashv}x)=0$ in $J$ and $\bar x\bar y=\bar 0$ in $\bar J$. Take $\bar
y=1\in\bar J$ and obtain $\bar x=\bar 0$, i. e., $x\in [J,J]$. So, $\phi$ is injective and $\bar J$ is special.
Let $J$ be a Jordan algebra, $A$ be an associative algebra with a unit, then a homomorphism from $J$ to $A^{(+)}$ is called an *associative specialization* $\sigma\colon J\to A$. This is a linear mapping such that $$\sigma(ab)=\frac{1}{2}(\sigma(a)\sigma(b)+\sigma(b)\sigma(a))$$ for all $a,b\in J$.
Two associative specializations are called *commuting* if $[\sigma_1(a),\sigma_2(b)]=0$ for all $a,b\in J$.
A bimodule $M$ over $J$ is *special* if there exists an embedding of $M$ into a bimodule $N$ such that if $v\in N$, $a\in J$ then $$\label{eq:SpecBiModCond}
a\cdot v=\frac{1}{2}(\sigma_1(a)v+\sigma_2(a)v),$$ where $\sigma_1$, $\sigma_2$ are commuting associative specializations of $J$ into $\mathrm{Hom}(N,N)$.
\[thm:SpecSplitNullExtCrit\] Let $J$ be a special Jordan algebra, $M$ be a bimodule over $J$. Then the bimodule $M$ is special if and only if the split null extension $J\oplus M$ is a special Jordan algebra.
\[lemma:SpecSplitNullExt\] Let $J$ be a special Jordan dialgebra and $\bar{J}$ be a special Jordan algebra. Then $\widehat{J}=\bar{J}\oplus J$ is special too.
Since $J=(J,{{}_{(\vdash)}},{{}_{(\dashv)}})$ is special, we have $J\hookrightarrow D^{(+)}$ where $D={(D,\vdash,\dashv)}$ is an associative dialgebra. The dialgebra $J$ is a $\bar
J$-bimodule: $\bar a\cdot v=a{\mathbin{{}_{(\vdash)}}}v =v{\mathbin{{}_{(\dashv)}}}a=v\cdot\bar
a$, where $\bar a\in\bar J$, $v\in J$.
We prove that the bimodu
| 2,127
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|
{\dag}\,.$$ Here, the internal Hamiltonian $H$ of the closed system is represented by a Hermitian $N\times N$ matrix, whereas $V_k$ are $M$ vectors of length $N$ containing the information on the coupling of the levels to the continuum. The $V_k$ are assumed to be normalized to one, $V_k^{\dag}V_k=1$, and $\lambda_k$ is the coupling constant of channel $k$.
Such an approach was initially developed in nuclear physics [@mah69; @ver85a; @sok89] and since then has been successfully applied to study various aspects of open systems, including wave billiards [@stoe99; @fyo97b; @dit00; @fyo05a]. Usually, the phenomenological coupling constants $\lambda_k$ are considered as real numbers which enter the final expressions via the so-called transmission coefficients. However, in the present case of the antenna variation one has to consider the coupling to the variable antenna, $\lambda_{c}$, as a complex number, see discussion in Sec. \[subsec:heff\] below. For the sake of generality, we will treat all $\lambda_k$ as complex numbers with the only constraint on their real parts $\mathrm{Re}(\lambda_k)\geq0$, due to the causality condition on the $S$-matrix. We note that quite a similar problem of nonzero $\mathrm{Im}(\lambda_k)$ arises in shell-model calculations due to the principle value term of the self-energy operator, cf. [@mah69] and [@ver85a]. This requires proper modification of the theory which we briefly outline below.
According to the general scattering formalism [@mah69; @ver85a], the resonance part of the $S$-matrix at the scattering energy $E$ can be expressed in terms of $H_{\mathrm{eff}}$ as follows: $$\label{eq:s_cc1}
S_{ab}(E) = \delta_{ab} - 2i\sqrt{\mathrm{Re}(\lambda_a)\mathrm{Re}(\lambda_{b})}\, V_a^{\dag}\frac{1}{E-H_{\mathrm{eff}}}V_{b}\,.$$ Being interested in a reflection amplitude in channel $a$, it is possible, following [@fyo05a], to obtain another representation for an arbitrary diagonal element $S_{aa}$. To this end, it is convenient first to single out the contribution to $H_{\mathrm{eff}}
| 2,128
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| 2,038
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| 0.779203
|
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|
, L_{j+1},$ $L_{j+2}, L_{j+3}$) are *of type II* if $j$ is even (resp. odd). Choose $j, j' \in \mathcal{B}$ with $j<j'$. By using the fact that $j'-j\geq 5$ if $j$ is even and $j'-j\geq 4$ if $j$ is odd, the proof of the surjectivity of the morphism $\psi=\prod_{j\in \mathcal{B}}\psi_j$ is similar to that of Theorem 4.11 in [@C2] (cf. from 7th line of page 497 to the first paragraph of page 498 in [@C2]). The proof of the surjectivity of $\varphi \times \psi$ is similar to that of Theorem 4.11 in [@C2] explained in the second paragraph of page 498. Thus we skip them.
The maximal reductive quotient {#mred}
------------------------------
We finally have the structure theorem for the algebraic group $\tilde{G}$.
\[t412\] The morphism $$\varphi \times \psi : \tilde{G} \longrightarrow \prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} \times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is surjective and the kernel is unipotent and connected. Consequently, $$\prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} \times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is the maximal reductive quotient. Here, $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$ and $\mathrm{Sp}(B_i/Y_i, h_i)$ are explained in Section \[red\] (especially Remark \[r47\]) and $\beta$ is defined in Lemma \[l46\].
The proof is similar to that of Theorem 4.12 in [@C2] and so we skip it.
Comparison of volume forms and final formulas {#cv}
=============================================
This section is based on Section 5 of [@C2]. We refer to loc. cit. and Section 3.2 of [@GY] for a detailed explanation. Let $H$ be the $F$-vector space of hermitian forms on $V=L\otimes_AF$. Let $M'=\mathrm{End}_{B}(L)$ and let $H'=\{\textit{f : f is a hermitian form on $L$}\}$. Regarding $\mathrm{End}_EV$ and $H$ as varieties over $F$, let $\omega_M$ and $\omega_H$ be nonzero, translation-invariant forms on $\mathrm{End}_EV$ and $H$, respectively, with normalization $$\int_{M'}|\o
| 2,129
| 1,638
| 705
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| 1,708
| 0.786578
|
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|
hey are first order formulations of GR. One has to first solve the connection in terms of the vielbein before calculating any scattering amplitudes of gravitons. The calculation in those theories is not simpler than a direct computation from the Hilbert-Einstein action.
Perturbative Expansion in $A$
-----------------------------
In this section, we focus on the 3-point vertices relevant for the 3-graviton scattering. We consider 3-point vertices of the action (\[general-action\]) for the traceless part of $h_{\m\n}$.
First, using (\[FFF\]), one can easily see that the second term in the action (\[general-action\]) is (F\^[abc]{}+F\^[bca]{}+F\^[cab]{})(F\_[abc]{}+F\_[bca]{}+F\_[cab]{}) &=& H\^[(0)abc]{} H\^[(0)]{}\_[abc]{} + [O]{}(H\^[(0)]{} A\^2) + [O]{}(A\^4), where $H^{(0)}_{abc}$ is the field strength of the anti-symmetric tensor field $B_{ab}$ defined at the lowest order: H\^[(0)]{}\_[abc]{} \_a B\_[bc]{} + \_b B\_[ca]{} + \_c B\_[ab]{}. A vertex operator involving only external legs of $h$ appears at ${\cal O}(A^4)$ or higher. The 3-point vertices of $h_{\m\n}$ is thus independent of the parameter $\a$.
Secondly, the third term in (\[general-action\]) is the square of F\_[ab]{}\^b &=& \_a A\_b\^b - \_b A\_a\^b + [O]{}(A\^2), where the first term involves the trace of $h_{\m\n}$, and the second term vanishes if we impose the gauge-fixing condition \^b h\_[ab]{} = 0 \[dh=0\] for the graviton field. As a result, in this gauge (\[dh=0\]), the third term of the action (\[general-action\]) is also irrelevant to the 3-point vertex for the traceless part of $h_{\m\n}$.
Furthermore, the overall measure of integration is e = 1 + h\_a\^a + [O]{}(A\^2), which is also irrelevant for our consideration.
The 3-point vertices for the graviton can thus only come from the YM Lagrangian, and there are only two terms \^[(3)]{} F\^[abc]{}\_[(0)]{}(\[A\_a, A\_b\]\_c - F\^[(0)]{}\_[ab]{}\^[d]{}A\_[dc]{}), \[3-point-1\] where \_c A\_a\^d\_d A\_[bc]{} - A\_b\^d\_d A\_[ac]{} and $F^{(0)}_{abc}$ is the field strength at the zero-
| 2,130
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| 2,018
| null | null |
github_plus_top10pct_by_avg
|
-1)/2}{N_t},$$ respectively, $$\label{}
{\mathbf{A}}_R=\frac{1}{\sqrt{N_r}}\left[ \mathbf{a}_R\left(\ddot{\theta}_{R,1}\right),\cdots,\mathbf{a}_R\left(\ddot{\theta}_{R,N_r}\right)\right]^T$$ and $$\label{}
{\mathbf{A}}_T=\frac{1}{\sqrt{N_t}}\left[ \mathbf{a}_T\left(\ddot{\theta}_{T,1}\right),\cdots,\mathbf{a}_T\left(\ddot{\theta}_{T,N_t}\right)\right]^T$$ are unitary DFT matrices, and ${\mathbf{H}}_{\psi,V}\in{\mathbb{C}}^{N_r\times N_t}$ is the virtual channel matrix. Since $\mathbf{A}_R\mathbf{A}_R^H=\mathbf{A}_R^H\mathbf{A}_R={\mathbf{I}}_{N_r}$ and $\mathbf{A}_T \mathbf{A}_T^H= \mathbf{A}_T^H \mathbf{A}_T={\mathbf{I}}_{N_t}$, the virtual channel matrix and the physical channel matrix are unitarily equivalent, such that $$\label{}
{\mathbf{H}_{\psi,V}}=\mathbf{A}_R^H\mathbf{H}_{\psi}\mathbf{A}_T.$$
### Low-Dimensional VCR
For MIMO systems, the link capacity of the reconfiguration state $\psi$ is directly related to the amount of scattering and reflection in the multipath environment. As discussed in [@Tse_05_Fundamentals Chapter 7.3], the number of non-vanishing rows and columns of ${\mathbf{H}_{\psi,V}}$ depends on the amount of scattering and reflection. In the clustered scattering environment of mmWave MIMO, the dominant channel power is expected to be captured by a few rows and columns of the virtual channel matrix, i.e., a low-dimensional submatrix of ${\mathbf{H}_{\psi,V}}$.
The discussion above motivates the development of low-dimensional virtual representation of mmWave MIMO channels and the corresponding low-complexity beamforming designs for mmWave MIMO transceivers [@Brady_13_BeamspaceSAMAM; @Amadori_15_LowRDBStion; @Sayeed_07_maxMcsparseRAA; @Raghavan_11_SublinearSparse]. Specifically, a low-dimensional virtual channel matrix, denoted by ${\widetilde{\mathbf{H}}_{\psi,V}}\in{\mathbb{C}}^{L_r\times L_t}$, is obtained by beam selection from ${\mathbf{H}_{\psi,V}}$, such that ${\widetilde{\mathbf{H}}_{\psi,V}}$ captures $L_t$ dominant transmit beams and $L_r$ dominant receive beams of the
| 2,131
| 1,673
| 1,988
| 2,002
| 3,370
| 0.772823
|
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|
mathcal{U}}^0,{{\mathbb{K}}^\times })}$ define ${t}_p^\chi (\Lambda )\in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ by $$\begin{aligned}
{t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )=\Lambda (K_{{\sigma }_p^{r_p(\chi )}({\alpha })}
L_{{\sigma }_p^{r_p(\chi )}(\beta )})
\frac{r_p(\chi ) ({\alpha },{\alpha }_p)^{{b}-1}}{r_p(\chi )({\alpha }_p,\beta )^{{b}-1}}
\label{eq:tpLambda}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. By Eq. this is equivalent to $$\begin{aligned}
{t}_p^\chi (\Lambda )(K_{{\sigma }^\chi _p({\alpha })}L_{{\sigma }_p^\chi (\beta )})
=\Lambda (K_{\alpha }L_\beta )
\frac
{\chi ({\alpha }_p,\beta )^{{b}-1}}
{\chi ({\alpha },{\alpha }_p)^{{b}-1}}
\label{eq:tpLambda1}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$.
Recall the definition of ${\rho ^{\chi}} $ from Def. \[de:rhomap\].
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then $$\begin{aligned}
{\rho ^{r_p(\chi )}}({\sigma }_p^\chi ({\alpha }))\, {t}_p^\chi (\Lambda )
(K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1})=
{\rho ^{\chi }}({\alpha }) \, \Lambda (K_{\alpha }L_{{\alpha }}^{-1})
\end{aligned}$$ for all ${\alpha }\in {\mathbb{Z}}^I$ and $p\in I$. \[le:VTinv\]
Insert Eq. and use Lemma \[le:rho\].
Let $C$ be a symmetrizable Cartan matrix and $q\in {{\Bbbk }^\times }$, $\chi
\in {\mathcal{X}}$ as in the second part of Ex. \[ex:Cartan\]. In particular, $\chi ({\alpha }_i,{\alpha }_j)=q^{d_i c_{i j}}$. Let $p\in I$. Assume that ${b}={b^{\chi}} ({\alpha }_p)<\infty $. Then $q^{2d_p \bnd
}=1$, and hence $q^{2{b}({\alpha },{\alpha }_p)}=1$ for all ${\alpha }\in \ndZ
^I$. Further, $r_p(\chi )=\chi $.
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that $\Lambda (K_{\alpha }L_\al
^{-1})=q^{2({\alpha },\lambda )}$ for some $\lambda $ in the weight lattice. Then $$\begin{aligned}
{t}_p^\chi (\Lambda )(K_{\alpha }L_{\alpha }^{-1})
=&\Lambda (K_{{\sigma }_p^\chi ({
| 2,132
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| 2,146
| 2,009
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|
your doc object which, I assume, was created on the main thread. This might be the reason for the fault.
Generally, you should not access objects (especially UI elements) across differed threads, unless they are specifically designed to be thread-safe.
EDITED: You probably do not need a task here. Just do await savePicker.PickSaveFileAsync() and mark your outer method as async (the method which currently creates the task).
To understand better what thread you're on, it may help to add a debug trace like this:
Debug.Print("<Method name>, Thread: {0}", Thread.CurrentThread.ManagedThreadId);
Q:
translating canvas for rotating sprites that shoot
I am trying to align bullets to my different sprites guns (top down 2D for android). Right now Im using this method to draw my bullets:
public void draw(Canvas canvas){
update();
canvas.save();
canvas.rotate(angle, x + width / 2, y + height / 2);
canvas.translate(translaterX, translaterY);
canvas.drawBitmap(bitmap, x, y, null);
canvas.restore();
}
The problem here is the translater, Im always having to add or substract from it to make the bullets spawn from my gun when my sprites rotate and I cant figure it out, is there any standard value for this, like translaterX = bitmap.getwidth()/2? I dont really understand how the translater works and I feel like Ive tried everything, what am I missing? For the examples sake lets say that my gun is always in the middle of the bitmap.
Or is there even another way to do this that is simpler?
A:
you'll need to calculate the rotation of the gun from the center of the player sprite
x = (float)Math.cos(playerAngle) * radius;
y = (float)Math.sin(playerAngle) * radius;
radius should be the distance that is from the player center to the position of the gun
Q:
Aggregating daily data using quantmod 'to.weekly' function creates weekly data ending on Monday not Friday
I am trying to aggregate daily share price data (close only) to weekly share price data using the "to.weekly" function in quantmod. The xts
| 2,133
| 2,039
| 702
| 1,507
| 2,301
| 0.781048
|
github_plus_top10pct_by_avg
|
nodes of $[\mu]$ mapped to $i$. Such a tableau is usually represented by drawing $[\la]$ with a box for each node $\fkn$, filled with the integer $T(\fkn)$. $T$ is *row-standard* if the entries in this diagram are weakly increasing along the rows, and is *semistandard* if the entries are weakly increasing along the rows and strictly increasing down the columns.
We write ${\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ for the set of row-standard $\mu$-tableaux of type $\la$, and ${\calt_{\hspace{-2pt}0}}(\mu,\la)$ for the set of semistandard $\mu$-tableaux of type $\la$. For each $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, James defines a homomorphism $\Theta_T:M^\mu\to M^\la$ (over any field), whose precise definition we do not need here. The restriction of $\Theta_T$ to $S^\mu$ is denoted ${\hat\Theta_{T}}$. Now we have the following.
[ ]{}\[semi\] The set $$\lset{{\hat\Theta_{T}}}{T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)}$$ is linearly independent, and spans ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,M^\la)$ if $\mu$ is $2$-regular.
The Kernel Intersection Theorem
-------------------------------
Now return to the assumption that $\la$ is a partition. As a consequence of Theorem \[semi\], in order to compute ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$ when $\mu$ is $2$-regular, we just need to find all linear combinations $\theta$ of the homomorphisms ${\hat\Theta_{T}}$ for $T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)$ for which the image of $\theta$ lies in $S^\la$. Even when $\mu$ is not $2$-regular, homomorphisms from $S^\mu$ to $S^\la$ can very often be expressed in this way.
In order to determine whether the image of such a homomorphism $\theta$ lies in $S^\la$, we use another theorem of James which provides an alternative definition of $S^\la$. For any pair $(d,t)$ with $d\gs 1$ and $1\ls t\ls \la_{d+1}$, there is a homomorphism ${\psi_{d,t}}:M^\la\to M^\nu$, where $\nu$ is a composition depending on $\la,d,t$. Again, we refer the reader to [@j2 §17] for the definition;
| 2,134
| 1,438
| 800
| 2,085
| 2,483
| 0.779393
|
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|
H^{\ast}$ using various magnetic measurements of $H_{irr}^{ab} (T)$: (a) Hg-1223, (b) Hg-1234, (c) Hg-1245, and (d) La-112. In (b) and (d) dashed lines indicate $H_{c2}^{ab} (T)$ from TDO measurements. In (d) we also illustrate $H_{c2}^{c}$ for comparison. We note reasonable consistency among different experimental techniques, indicating strong suppression of $H^{\ast} \equiv H_{irr} ^{ab} (0)$ relative to $H_{c2} ^{ab} (0)$ (or $H_p$) in all cuprates.[]{data-label="fig2"}](Fig2){width="3.45in"}
--------- --------- -------- -------- ------------------------- ------- -------- -------- -------- -------- ------------ ------- -------- --------
Hg-1245 $0.15$ $1.30$ $0.80$ $55$ [@Hg1245comment] $25$ $0.06$ $0.3$ $23.0$ $5.0$ $-[278]$ $40$ $0.08$ $0.02$
Hg-1223 $0.15$ $1.04$ $0.92$ $52$ [@Zech96] $18$ $0.26$ $0.9$ $48.5$ $6.5$ $-[347]$ $50$ $0.14$ $0.02$
Hg-1234 $0.15$ $1.20$ $0.80$ $52$ [@Zech96] $10$ $0.13$ $0.2 $ $75.0$ $10.0$ $-[320]$ $46$ $0.23$ $0.02$
La-112 $0.10$ $1.00$ $1.00$ $13$ [@Zapf05] $4.0$ $0.77$ $2.4 $ $46.0$ $4.0$ $160[110]$ $10$ $0.42$ $0.04$
Bi-2212 $0.225$ $1.00$ $1.00$ $11$ [@Krusin-Elbaum04] $8.0$ $2.05$ $15 $ $65.0$ $10$ $100[155]$ $22$ $0.42$ $0.06$
NCCO $0.15$ $1.00$ $1.00$ $13$ [@Yeh92] $5.0$ $1.15$ $4.4 $ $40.0$ $5.0$ $77[59]$ $8.0$ $0.68$ $0.12$
Y-123 $0.13$ $1.00$ $1.00$ $7.0$ [@Zech96] $2.0$ $1.86$ $5.3 $ $210$ $50$ $600[239]$ $25$ $0.88$ $0.23$
--------- --------- -------- -------- ------------------------- ------- -------- -------- -------- -------- ------------ ------- -------- --------
\[table1\]
The physical significance of $h^{\ast}$ may be better understood by considering how the magnetic irreversibility for $H \parallel ab$ occurs. For sufficiently low $T$ and small $H$, a supercur
| 2,135
| 1,174
| 2,095
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|
mathbf{0}}$) the posterior has the same form as the prior with tractable updates; the same would be true for a non-baseline measurement (${\mathbf{x}}_t\neq{\mathbf{0}}$) if it were possible to set ${\bar{\nu}}^{-1}=0$ and to ignore the further information on ${\bar{\mathcal{A}}}$; such an approximation is described and justified in Section \[sec:DetailObsProc\].
Subject to this approximation, the posterior distribution for the observational parameters after assimilating $y_{1:t}$ and conditional on ${\mathbf{x}}_{1:t}$ is defined by the sufficient statistics and (multivariate) Gaussian-gamma distributions analogous to the prior specification: $$\begin{aligned}
{\bar{\mathcal{A}}}_t := \left\{{\bar{a}}_t,~ {\bar{b}}_t,~ {\bar{m}}_t,~ {\bar{c}}_t\right\} \quad \mathrm{and} \quad
{\mathcal{A}}_t := \left\{a_t,~ b_t,~ {\mathbf{m}}_t,~ C_t\right\}, \label{eq:SSdef}\end{aligned}$$ $$\begin{aligned}
{\bar{\nu}}|y_{1:t},{\mathbf{x}}_{1:t} & \sim \mathrm{Gam}\left({\bar{a}}_t,~ {\bar{b}}_t\right), \quad\quad {\bar{\mu}}|{\bar{\nu}},y_{1:t},{\mathbf{x}}_{1:t} \sim \mathrm{N}\left({\bar{m}}_t,~ {\bar{\nu}}^{-1}{\bar{c}}_t\right),\nonumber\\
\nu|y_{1:t},{\mathbf{x}}_{1:t} & \sim \mathrm{Gam}\left(a_t,~ b_t\right), \quad\quad {\boldsymbol{\mu}}|\nu,y_{1:t},{\mathbf{x}}_{1:t} \sim \mathrm{MVN}_u\left({\mathbf{m}}_t,~ \nu^{-1}C_t\right).\label{eq:ObsPriorApx}\end{aligned}$$
Given the assumptions leading to the marginal likelihood for the observation $y_t$ conditional on the firing vector ${\mathbf{x}}_t$ and sets ${\bar{\mathcal{A}}}_{t-1}$ and ${\mathcal{A}}_{t-1}$ has tractable form: $$\begin{aligned}
f\left(y_t|{\mathbf{x}}_t,{\bar{\mathcal{A}}}_{t-1},{\mathcal{A}}_{t-1}\right) & = \left\{
\begin{array}{ll}
\mathsf{t}\left[y_t;~ {\bar{m}}_{t-1},~ \frac{{\bar{b}}_{t-1}}{{\bar{a}}_{t-1}}\left({\bar{c}}_{t-1}+1\right),~
2{\bar{a}}_{t-1}\right] \quad & \mathrm{if} ~ {\mathbf{x}}_t={\mathbf{0}},\\
\mathsf{t}\left[y_t;~ {\bar{m}}_{t-1}+{\mathbf{x}}_t^\top{\mathbf{m}}_{t-1},~
\frac{b_{t-
| 2,136
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| 1,798
| 1,628
| null | null |
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|
/2001 at 6:00:00 PM CT thru Sun 11/18/2001 at 9:00:00 AM CT
Sat 11/17/2001 at 4:00:00 PM PT thru Sun 11/18/2001 at 7:00:00 AM PT
Sun 11/18/2001 at 12:00:00 AM London thru Sun 11/18/2001 at 3:00:00 PM London
Outage: General maintenance for ERMS CPR app server chewbacca.
Environments Impacted: ERMS CPR
Purpose: General maintenance and patching.
Backout: Backout patches and config changes and reboot to old configuration.
Contact(s): Malcolm Wells 713-345-3716
Impact: CORP
Time: Fri 11/16/2001 at 5:00:00 PM CT thru Sat 11/17/2001 at 2:00:00 PM CT
Fri 11/16/2001 at 3:00:00 PM PT thru Sat 11/17/2001 at 12:00:00 PM PT
Fri 11/16/2001 at 11:00:00 PM London thru Sat 11/17/2001 at 8:00:00 PM London
Outage: Update to new disk layout for server foxtrot.
Environments Impacted: ACTA production
Purpose: Move toward new standard in disk layout.
Backout: Restore old disk layout
restore data from disk.
Contact(s): Malcolm Wells 713-345-3716
Impact: ENPOWER
Time: Sat 11/17/2001 at 7:00:00 AM CT thru Sat 11/17/2001 at 12:00:00 PM CT
Sat 11/17/2001 at 5:00:00 AM PT thru Sat 11/17/2001 at 10:00:00 AM PT
Sat 11/17/2001 at 1:00:00 PM London thru Sat 11/17/2001 at 6:00:00 PM London
Outage: Build standby database on PWRPROD1
Environments Impacted: Enpower application User
Purpose: Improve system availability for Enpower Database
Backout: Drop the standby database.
Contact(s): Tantra Invedy 713 853 4304
Impact: EI
Time: Fri 11/16/2001 at 3:00:00 PM CT thru Fri 11/16/2001 at 6:00:00 PM CT
Fri 11/16/2001 at 1:00:00 PM PT thru Fri 11/16/2001 at 4:00:00 PM PT
Fri 11/16/2001 at 9:00:00 PM London thru Sat 11/17/2001 at 12:00:00 AM London
Outage: Decommission 3AC-17 server Room
Environments Impacted: El and Azurix
Purpose: Decommission 3AC-17 server Room
Backout: None, must be out and installed on the 35th floor of 3AC
Contact(s): Matthew James 713-345-8111
Jon Goebel 713-345-7570
Impact: CORP
Time: Sa
| 2,137
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|
c(\mu)$ has weight $m+c(n(\mu) -
n(\mu^{t}))$, where $m=(n-1)/2$.
[(2)]{} The Poincaré series of $\Delta_c(\mu)$ as a graded ${{W}}$-module is $$\label{polystand} p(\Delta_c(\mu), v, {{W}}) =
v^{m+c(n(\mu)-n(\mu^t))} \frac{\sum_{\lambda}f_{\lambda}(v)
[\lambda\otimes \mu]}{\prod_{i=2}^n (1-v^i)}.$$
\(1) We need to compute the action of ${\mathbf{h}}=\frac{1}{2}\sum_{i=1}^{n-1}(x_iy_i+y_ix_i)$ on the space $1\otimes \mu$. By the defining relations of $H_c$ from , and the fact that the $\{x_i\}$ and $\{y_i\}$ are dual bases, we obtain $$\begin{array}{rl}
{\mathbf{h}}= \sum_i x_iy_i + (n-1)/2 -\frac{1}{2}\sum_{s\in \mathcal{S}}
\sum_{i}c\alpha_s(y_i)x_i(\alpha_s^\vee)s
=& \sum x_iy_i + (n-1)/2 -\frac{c}{2}\sum_{s\in \mathcal{S}}
\alpha_s(\alpha_s^\vee)s\\
\noalign{\vskip 5pt}
=&\sum x_iy_i + m -c\sum_{s\in \mathcal{S}}s.
\end{array}$$ The action of $\sum(1-s)$ on $\lambda\in {{\textsf}{Irrep}({{W}})}$ can be derived from [@BM; @Lu]. More precisely, $\lambda$ is special by [@Lu (4.2.2)] and so $n(\lambda)=
b_\lambda=a_\lambda$ in the notation of [@Lu]. Therefore, by [@BM Section 4.21] and [@Lu Section 4.1 and (5.11.5)], $\sum_s(1-s)$ acts on $\lambda\in {{\textsf}{Irrep}({{W}})}$ with weight $N+n(\lambda)-n(\lambda^t)$, where $N=n(n-1)/2$ is the cardinality of $\mathcal{S}$. Thus $\sum_ss$ acts on $1\otimes \mu$ with weight $-(n(\mu) - n(\mu^t))$ and hence ${\mathbf{h}}$ acts with weight $m+c(n(\mu) - n(\mu^t))$.
\(2) As graded ${{W}}$-modules, $\Delta(\mu)\cong ({\mathbb{C}}[{\mathfrak{h}}]\otimes \mu)[k]$ for $k=m+c(n(\mu) - n(\mu^t))$. The shift arises from the fact that, by (1), the generator $1\otimes \mu$ of $\Delta_c(\mu)$ lives in degree $k$. The Chevalley-Shephard-Todd Theorem implies that, as graded ${{W}}$-modules, ${\mathbb{C}}[{\mathfrak{h}}]\cong
{\mathbb{C}}[{\mathfrak{h}}]^{{W}}\otimes {\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}}$. Now ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ is a polynomial ring with generators in degrees $2\leq i\leq n$ and so its Poincaré polynomial is $ \prod_{i=2}^n (1
| 2,138
| 1,426
| 2,212
| 1,971
| 3,204
| 0.774053
|
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|
label{eq:velocity_linear}\end{aligned}$$
Combining Eq. (\[eq:fp2\]) with the expressions for the density perturbations, we have that the total force can be split into the contribution from the density variations in the BEC by causes external to the particle (initial preparation, stirring forces in $V_{ext}$, ...), and the density perturbations due to the presence of the particle ${\boldsymbol{F}}_p=
{\boldsymbol{F}}^{(0)}+{\boldsymbol{F}}^{(1)}$: $$\begin{aligned}
{\boldsymbol{F}}^{(0)}(t) = - \frac{g_p}{2\pi a^2} \int d^2{\boldsymbol{r}} e^{- \frac{({\boldsymbol{r}}-{\boldsymbol{r}}_p(t))^2}{2a^2}}
\nabla \delta\rho_0({\boldsymbol{r}},t),
\label{eq:fp_unperturb}
\\
{\boldsymbol{F}}^{(1)}(t) = - \frac{g_p^2}{2\pi a^2} \int d^2{\boldsymbol{r}} e^{- \frac{({\boldsymbol{r}}-{\boldsymbol{r}}_p(t))^2}{2a^2}}
\nabla \delta\rho_1({\boldsymbol{r}},t).
\label{eq:fp_perturb}\end{aligned}$$ The perturbative splitting of the force in these two contributions is completely analogous to the corresponding classical-fluid case in the incompressible [@maxey1983equation] and in the compressible [@parmar2012equation] situations. The ${\boldsymbol{F}}^{(0)}$ contribution is the equivalent to the classical inertial or pressure-gradient force on a test particle, which does not disturb the fluid, in a inhomogeneous and unsteady flow. In the following it will be called the *inertial* force. The ${\boldsymbol{F}}^{(1)}$ contribution takes into account perturbatively the modifications on the flow induced by the presence of the particle, and it will be called the *self-induced drag* on the particle. To complete the comparison with the classical expressions [@maxey1983equation; @parmar2012equation], we need to express Eqs. (\[eq:fp\_unperturb\]) and (\[eq:fp\_perturb\]) in terms of the unperturbed velocity field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)=\delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$ and of the particle speed ${\boldsymbol{V}}_p(t)={\boldsymbol{\dot r}}_p(t)$. We are able to do so in a general situation for the inertial
| 2,139
| 619
| 795
| 2,371
| 2,594
| 0.778606
|
github_plus_top10pct_by_avg
|
ace flattens and the worldsheet theory becomes free. More precisely we obtain a theory of $d$ free bosons, where $d$ is the dimension of the adjoint representation of the super Lie algebra. Among these bosons, some are commuting and some are anti-commuting, depending on whether they can be associated to bosonic or fermionic coordinates of target space. At fixed $kf^2$ the $f^2 \to 0$ limit is the semi-classical limit of the model.
Our goal in this appendix is to evaluate the behavior at large radius (small $f^2$) of the terms appearing in the current-current and current-primary OPEs. Let us start with the action of the model: $$\begin{aligned}
S &= S_{kin} + S_{WZ}\cr
S_{kin} &= \frac{1}{ 16 \pi f^2}\int d^2 x Tr'[- \partial^\mu g^{-1}
\partial_\mu g]
\cr
S_{WZ} &= - \frac{ik}{24 \pi} \int_B d^3 y \epsilon^{\alpha \beta \gamma}
Tr' (g^{-1} \partial_\alpha g g^{-1} \partial_\beta g g^{-1} \partial_\gamma g ).\end{aligned}$$ We write the group element as: g=e\^[f X]{}=e\^[i f X\_a t\^a]{} where the $X_a$ are coordinates on the supergroup and the matrices $t^a$ are the generators of the Lie superalgebra. The kinetic term and the Wess-Zumino term become: $$\begin{aligned}
\label{action(X)}
S_{kin} &=& \frac{1}{4 \pi}
\int d^2 z \left( \partial X_a \bar{\partial} X^a -
\frac{f^2}{12} {f^a}_{fe} {f}_{acb} X^b \partial X^c X^e \bar{\partial} X^f
+ ...\right)
\nonumber \\
S_{WZ} &=& -\frac{kf^2 }{12 \pi}
\int d^2 z \left(
f f_{abc} X^c \partial X^b
\bar{\partial} X^a + ... \right).\end{aligned}$$ Written in this way the theory describes a set of interacting bosons (some of which are anti-commuting). The quadratic terms in the action give rise to the free propagator: $$\begin{aligned}
\label{freeProp}
X^a(z,\bar{z}) X^b(w,\bar{w}) &=& - \kappa^{ab} \log \mu^2 |z-w|^2,\end{aligned}$$ where $\mu$ is an infrared regulator. The propagator behaves like $\mathcal{O}(f^0)$, whereas a vertex with $p+2$ legs (i.e. Lie algebra indices) behaves as $\mathcal{O}(f^p)$. It follows that the theory reduces to a theory of fr
| 2,140
| 2,496
| 2,565
| 1,989
| 4,025
| 0.76858
|
github_plus_top10pct_by_avg
|
n explain a wide variety of seemingly unrelated phenomena observed in visual cortex and visual perception. These phenomena range from the details of responses of individual neurons, to complex visual illusions. Importantly, throughout, we used a base model trained on natural videos. Our work adds to a growing body of literature showing that deep neural networks trained to perform relevant tasks can serve as surprisingly good models of biological neural networks, often even outperforming models designed to explain neuroscience phenomena.
While we have shown that the PredNet architecture demonstrates a wide range of phenomena reminiscent of biology, we do not claim that the PredNet architecture *per se* is required to explain these phenomena. Rather, we argue that the network is *sufficient* to produce these phenomena, and we note that explicit representation of prediction errors in units within the feedforward path of the PredNet provides a straightforward explanation for the transient nature of responses in visual cortex in response to static images. That a single, simple objective—prediction—can produce such a wide variety of observed neural phenomena underscores the idea that prediction may be a central organizing principle in the brain [@Rao_1999], and points toward fruitful directions for future study in both neuroscience and machine learning.
### Acknowledgments {#acknowledgments .unnumbered}
This work was supported by IARPA (contract D16PC00002), the National Science Foundation (NSF IIS 1409097), and the Center for Brains, Minds and Machines (CBMM, NSF STC award CCF-1231216).
Supplementary Material
======================
On/Off Temporal Dynamics
------------------------
Temporal dynamics were tested with a set of $25$ objects. Examples of the objects can be seen in Fig. \[image\_pairing\_examples\]. Testing sequences consisted of a gray background for $7$ time steps, followed by an object on the background for $6$ time steps. As a general theme, we see some diversity in the response profiles of a
| 2,141
| 2,777
| 3,071
| 2,080
| null | null |
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|
}}\\
\leq& \int_0^1\int_0^s \E{\lrn{\nabla^2 f(x_T - y_T + s(y_T-v_T))}_2 \lrn{y_T - v_T}_2^2} ds dt\\
\leq& \frac{2}{\epsilon} \E{\lrn{y_T - v_T}_2^2}\\
\leq& \frac{72 d \delta \beta^2 \log^2 n}{\epsilon}
\end{aligned}$$ Where the second inequality is by $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\], the third inequality is by .
Summing these 3 terms, $$\begin{aligned}
&\E{f(x_T - v_T) - f(x_T - y_T)} \\
\leq& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta} \sqrt{\log n}} + \frac{36 d \delta \beta^2 \log n}{\epsilon}\\
=& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta}\sqrt{\log \frac{T}{\delta}}} + \frac{36 d \delta \beta^2 \log \frac{T}{\delta}}{\epsilon }
\end{aligned}$$
Let us bound the first term. We apply Lemma \[l:xlogxbound\] (with $x = \frac{T}{\delta}$ and $c = \frac{\epsilon^4}{2^{14} d\beta^4}$), which shows that $$\begin{aligned}
\frac{T}{\delta} \geq \frac{2^{14} d\beta^4}{\epsilon^4} \log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}
\quad \Rightarrow \quad
\frac{T}{\delta} \frac{1}{\log \frac{T}{\delta}} \geq \frac{2^{14} d\beta^4}{\epsilon^4L^2}
\quad \Leftrightarrow \quad
\frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta}{\log \frac{T}{\delta}}} \leq TL\epsilon
\end{aligned}$$
For the second term, we can again apply Lemma \[l:xlogxbound\] ($x = \frac{T}{\delta}$ and $c = \frac{\epsilon^2L}{36 d\beta^2}$), which shows that $$\begin{aligned}
\frac{T}{\delta} \geq \frac{36 d\beta^2}{\epsilon^2L} \log \lrp{ \frac{36 d\beta^2}{\epsilon^2L} }
\quad \Rightarrow \quad
\frac{T}{\delta} \frac{1}{\log \frac{T}{\delta}} \geq \frac{36 d\beta^2}{\epsilon^2L}
\quad \Rightarrow \quad
\frac{36 d \delta \beta^2 \log \frac{T}{\delta}}{\epsilon } \leq TL\epsilon
\end{aligned}$$ The above imp
| 2,142
| 4,059
| 1,967
| 1,713
| null | null |
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|
] the cone $C\colon{\sD}^{[1]}\to{\sD}$ and the fiber $F\colon{\sD}^{[1]}\to{\sD}$ is defined in pointed derivators only, but the same formulas make perfectly well sense in arbitrary derivators. It turns out that a derivator is pointed if and only if $C$ is a left adjoint if and only if $F$ is a right adjoint. In that case, there are adjunctions $C\dashv 1_!$ and $0_\ast\dashv F$, exhibiting $C$ and $F$ as (co)exceptional inverse image functors; see [@groth:ptstab Prop. 3.22].
In \[thm:stable-fun\] we will characterize stable derivators with a simliar list of conditions, essentially by combining \[thm:stable-lim-III,thm:stab-op\]. We could similarly have proven \[prop:char-ptd\] by combining \[prop:ptd-comm,thm:stab-op\], but we chose instead to give a proof with a closer connection to previous literature.
Let be a derivator and let $1\colon\bbone\to[1]$ classify the terminal object $1\in[1]$. In every derivator there are Kan extension adjunctions $(1_!,1^\ast)\colon{\sD}\rightleftarrows{\sD}^{[1]}$ and $(1^\ast,1_\ast)\colon{\sD}\rightleftarrows{\sD}^{[1]}$, and we hence have an adjoint triple $$1_!\dashv 1^\ast\dashv 1_\ast.$$ Similarly, associated to the functor $0\colon\bbone\to[1]$ there is the adjoint triple $$0_!\dashv 0^\ast\dashv 0_\ast.$$
\[prop:univ-sieve\] Let be a derivator and let $0,1\colon\bbone\to[1]$ classify the objects $0,1\in[1]$.
1. The morphisms $0_!,1_\ast\colon{\sD}\to{\sD}^{[1]}$ are fully faithful and induce an equivalence onto the full subderivator spanned by the isomorphisms. This equivalence is pseudo-natural with respect to arbitrary morphisms of derivators.
2. There are canonical isomorphism $0_!\cong \pi_{[1]}^\ast\cong 1_\ast\colon{\sD}\to{\sD}^{[1]}$.
Both morphisms $0_!$ and $1_\ast$ are fully faithful and the essential image consists precisely of the isomorphisms by [@groth:ptstab Prop. 3.12]. Since derivators are invariant under equivalences of prederivators, the subprederivator of isomorphisms is a derivator. The equivalence is pseudo-natural with respect to arbitr
| 2,143
| 1,446
| 1,585
| 2,069
| 1,241
| 0.792162
|
github_plus_top10pct_by_avg
|
and $a \to \infty$ (the domain loses regularity), the two agree with an $\mathcal{O}(\varepsilon^{-2})$ bound.
Fractional Poisson problem {#inhomogenous}
==========================
We are now interested in using the walk-on-spheres process to find the solution to the inhomogeneous version of (\[aDirichlet\]), namely $$\begin{gathered}
\begin{aligned}
-(-\Delta )^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in D, \\
u(x) & = {g}(x), & x & \in D^{\rm c},
\end{aligned}
\label{aDirichlet_g}
\end{gathered}$$ for suitably regular functions ${f}\colon D\to \mathbb{R}$ and ${g}\colon D^{\rm c} \to \mathbb R$. We want to identify a Feynman–Kac representation for solutions to for suitable assumptions on ${g}, {f}$ and $D$. Throughout this section, we adopt the setting of the following theorem.
\[hasacorr\] Let $d\geq 2$ and assume that $D$ is a bounded domain in $\mathbb{R}^d$. Suppose that ${g}$ is a continuous function which belongs to $L^1_\alpha(D^\mathrm{c})$. Moreover, suppose that ${f}$ is a function in $ C^{\alpha +\varepsilon}(\overline{D})$ for some $\varepsilon>0$. Then there exists a unique continuous solution to in $L^1_\alpha(\mathbb{R}^d)$ which is given by $$u(x) = \mathbb{E}_x[{g}(X_{\sigma_D})] + \mathbb{E}_x\left[\int_0^{\sigma_D} {f}(X_s)\,{\rm d}s\right], \qquad x\in D,
\label{non_homg_FK}$$ where $\sigma_D=\inf\{t>0\colon X_t\not\in D\}$.
The combinations of Theorem 2.10 and 3.2 in [@bucur] treat the case that $D$ is a ball. In the more general setting, amongst others, [@B99], [@R-O1] and [@R-O2] (see also citations therein) offer results in this direction, albeit from a more analytical perspective. We give a new probabilistic proof of Theorem \[hasacorr\] in the Appendix using a method that combines the idea of walks-on-spheres with the version of Theorem \[hasacorr\] when $D$ is a ball. It is for this reason that the (otherwise unclear) need for the assumption that ${f}\in C^{\alpha +\varepsil
| 2,144
| 999
| 1,729
| 2,062
| 2,941
| 0.775962
|
github_plus_top10pct_by_avg
|
**2 + 3502998*c**3*l + 478818*c**2*l**2 - 4*c*l**2 wrt c.
-12*l**2 + 21017988*l
Differentiate 2*g**2 + 2*g*j*t**2 + 7*g*t**2 + 40*g + 2*j*t**2 - j*t - 3*j - 104*t**2 + t with respect to g.
4*g + 2*j*t**2 + 7*t**2 + 40
Find the third derivative of -16*l**3*v - 727*l**3 - 11034*l**2*v + 4*l**2 wrt l.
-96*v - 4362
What is the second derivative of 3*q**4 - 566*q**2 + 13314*q wrt q?
36*q**2 - 1132
Differentiate -13*a**3 - 143*a + 14216.
-39*a**2 - 143
Find the first derivative of 29*d*v**4 + 3391*d*v**2 + 305915*d wrt v.
116*d*v**3 + 6782*d*v
Find the third derivative of 114816*h**5 - 2*h**3 + 3*h**2 + 12549*h.
6888960*h**2 - 12
What is the first derivative of 2*p**2*z**2 - 14*p**2*z + 170773*p**2 - 122*p*z wrt z?
4*p**2*z - 14*p**2 - 122*p
What is the derivative of 3343518*d*f**3*i**3 - 13*f**3 + 2*f**2*i**3 + 31*f**2*i**2 - 570*f*i**3 + i**3 wrt d?
3343518*f**3*i**3
Differentiate -551*c*j + 2*c + 57*j + 61799 with respect to j.
-551*c + 57
What is the third derivative of -4*j**4 - 41155*j**3 + 22464*j**2 + 8*j?
-96*j - 246930
What is the third derivative of -a**5 - 61466*a**3 + 2*a**2 + 13734?
-60*a**2 - 368796
Differentiate -8422*n**3*x**2 + 161*n*x**3 + 2*x**3 + 474*x**2 + 59 with respect to n.
-25266*n**2*x**2 + 161*x**3
What is the first derivative of 6*i**3*j**3 + 3*i**3 + 18*i*j*q + 9*j**3*q - 6*j**2 + 4*j*q - 4*j wrt i?
18*i**2*j**3 + 9*i**2 + 18*j*q
What is the second derivative of -1682*b**3 + 298*b**2 + b - 63347?
-10092*b + 596
Find the third derivative of 830*c**4 + 1250*c**3 - 23*c**2 - 7393 wrt c.
19920*c + 7500
What is the second derivative of -51411*s**3 - 2*s - 5551 wrt s?
-308466*s
Differentiate 20*b**3*k + 85770*b**2*k + 2008713*k - 1 wrt b.
60*b**2*k + 171540*b*k
What is the derivative of 2630*c**3*s**3*t - 430*c**3*s - 38*c**2*s**3*t + 254*c*s**3 wrt t?
2630*c**3*s**3 - 38*c**2*s**3
Find the third derivative of f**5 - 65362*f**4*t**3 + 5*f**2*t**2 + 674*f*t**3 - 11*f*t**2 + 2*t - 1 wrt f.
60*f**2 - 1568688*f*t**3
Find the third derivative of -3*m**3*s**2 + 295*m**3*s - 7*m**3 + 40*m**2*s**2 + 27
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boratory of Software Development Environment (NLSDE)\
School of Computer Science and Engineering, Beihang Univerisity, Beijing, China
author:
- Yongwang Zhao
title: A survey on formal specification and verification of separation kernels
---
real-time operating systems ,separation kernel ,survey ,formal specification ,formal verification
Introduction {#sec:intro}
============
The concept of “Separation Kernel” was introduced by John Rushby in 1981 [@Rushby81] to create a secure environment by providing temporal and spatial separation of applications as well as to ensure that there are no unintended channels for information flows between partitions other than those explicitly provided. Separation kernels decouple the verification of the trusted functions in the separated components from the verification of the kernels themselves. They are often sufficiently small and straightforward to allow formal verification of their correctness.
The concept of separation kernel originates the concept of Multiple Independent Levels of Security/Safety (MILS) [@Alves06]. MILS is a high-assurance security architecture based on the concepts of separation [@Rushby81] and controlled information flow [@Denning76]. MILS provides means to have several strongly separated partitions on the same physical computer/device and enables existing of different security/safety level components in the same system. The MILS architecture is particularly well suited to embedded systems which must provide guaranteed safety or security properties. An MILS system employs the separation mechanism to maintain the assured data and process separation, and supports enforced security/safety policies by authorizing information flows between system components.
The MILS architecture is layered and consists of separation kernels, middleware and applications. The MILS separation kernels are small pieces of software that divide the system into separate partitions where the middleware and applications are located, as shown in [[[Fig.]{}]{}]{} \[fig:mils\
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.
Considering the boost behaviour of $s_x,s_y$, after the shift we may have $$c h_y+d\bar{h}_x=0. \label{hy}$$ We can define a set of charges, $$L_\epsilon=\int \{c\epsilon(x) h_x(x,y)+d\epsilon'(x)y\bar{h}_x(x)\}dx+\int\{c\epsilon(x)h_y(x) \}dy, \label{Qcharges}$$ where $\epsilon(x)$ is arbitrary smooth function on $x$ and $\epsilon'(x)=\p_x \epsilon$. $h_y(x)$ depends only on $x$, since its boost charge vanishes. We denote $$q_x=c\epsilon(x) h_x(x,y)+d\epsilon'(x)y\bar{h}_x(x), \hs{2ex}q_y=c\epsilon(x)h_y(x).$$ One can check that the charges $L_\epsilon$ are indeed conserved $$\partial_y q_x+\partial_x q_y=0,$$ provided that $$\partial_yh_x+\partial_xh_y=0.$$ Note that when $\epsilon=1$, $$L_1=\int h_xdx+\int h_ydy$$ generates the translation in $x$ direction, while when $\epsilon=x$ $$L_x=\int \{cxh_x(x,y)+dy\bar{h}_x(x)\}dx+\int\{cxh_y(x) \}dy$$ generates the anisotropic scaling symmetry.
In the case that $d=0$, from we have $$h_y=0.$$ And considering the conservation law, we find that $h_x$ depends only on $x$. This is exactly the case for the warped CFTs discussed in [@Hofman:2011zj; @Hofman:2014loa].
Algebra of enhanced symmetries
------------------------------
After some calculations, we arrive at the algebra, $$[L_\epsilon,L_{\tilde{\epsilon}}]=L_{c \epsilon'\tilde{\epsilon}-c \tilde{\epsilon}'\epsilon}+\cdots,$$ $$[L_\epsilon,M_{\tilde{\epsilon}}]=M_{d\epsilon'\tilde{\epsilon}-c \tilde{\epsilon}'\epsilon}+\cdots,$$ $$[M_\epsilon,M_{\tilde{\epsilon}}]=\cdots,$$ where $\epsilon$ and $\tilde{\epsilon}$ are arbitrarily smooth functions of $x$ and the ellipsis denotes potential central extension terms allowed by the Jacobi identity.\
The algebra of the plane modes without central extension is $$\begin{aligned}
\label{algebra}
\nonumber \left[l_n,l_m\right]&=& c(n-m)l_{n+m},\\
\nonumber\left[l_n,m_m\right]&=& (dn-c m)m_{n+m} ,\\
\left[m_n,m_m\right] &=&0.\end{aligned}$$ This is the infinite dimensional spin-$\ell$ Galilean algebra, with $\ell=\frac{d}{c}$[@Henkel:1997zz].
The central extension is constra
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a^+(C_0)$,
2. for all $\beta\neq \xi$ in $\omega_1$, $\beta\notin \dot{U}(\xi,0)$,
3. for all $\xi,\alpha\in \omega_1$ $\dot{U}(\xi,\alpha)\subseteq \dot{U}(\xi,0)$ and has compact closure,
4. for each limit $\delta\in \omega_1$, the sequence $\{\dot{U}(\xi,\alpha):\alpha<\delta\}$ is a *regular filter*, i.e. each finite intersection of these includes the closure of another.
For an $S$-name $\dot{h}$ of a function from $\omega_1$ to $\omega_1$, let $\dot{U}(\xi,\dot{h})$ stand for $\dot{U}(\xi,\dot{h}(\xi))$. For limit $\delta$, let $\dot{Z}(\xi,\delta)$ denote the $S$-name of the compact $G_\delta$ equal to $\bigcap\{\dot{U}(\xi,\alpha):\alpha<\delta\}$. For a cub $C$ and ordinal $\xi$, we also use $\dot{Z}(\xi,C)$ as an abbreviation for $\dot{Z}(\xi,\xi^+(C))$.
Fix an enumeration $\{C_\gamma:\gamma\in\omega_2\}$ for a base for the cubs on $\omega_1$ (each containing only limit ordinals), chosen so that $C_0$ is as above and for $0<\lambda \in \omega_2$, $C_\lambda \subseteq {\mathrm}{Fix}(C_0)$ and $C_\lambda\setminus{\mathrm}{Fix}(C_\gamma)$ is countable for all $0\leq\gamma<\lambda$. We can do this by taking diagonal intersections, since **SRP** implies $2^{\aleph_1}=\aleph_2$.
For each $\delta\in C_0$, let $\beta(\delta) = \delta^+(C_0)$. Since $\dot{Z}(\xi,C_\gamma)\subseteq
\dot{U}(\xi,C_\gamma)$ for all $\xi\in \omega_1$ for all $\delta\in C_\gamma$, $\beta(\delta)<\delta^+(C_\gamma)$, and so it is forced that: $$\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}}
\cap\omega_1\subseteq\beta(\delta).$$
We can also assume that for all cubs $C\subseteq C_0$, there is an $S$-name $\dot A$, that is forced to be a stationary subset of ${\mathrm}{Fix}(C)$ satisfying: $$(\forall s\in S)(\forall \delta)
~~ s\Vdash \left(\delta\in \dot A \ \Rightarrow
(\exists\alpha\in[\delta,\beta(\delta)])\;
\alpha\in\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}}~\right).$$
The reason we can make this assumption is that we are assuming there is no $\sigma$-discrete expansion of $\omega_1$ by compact $G_\delta$’s. If,
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ot Rz)d\sigma(z) \right|\\
& = \frac{1}{\omega_d}\left| \int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \int_{{\mathbb{S}}^{d-1}} f(Rz)g(x\cdot z)d\sigma(z) \right|\\
& \leq \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |f(z)-f(Rz)||g(x\cdot z)|d\sigma(z) \\
& \leq \|f\|_{\mathrm{Lip}}\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} D(z,Rz)|g(x\cdot z)|d\sigma(z)\\
& = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |g(x\cdot z)|d\sigma(z)\\
& = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_n}\int_{-1}^1 |g(t)|w_\Lambda (t) dt \leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}D(x,y).\end{aligned}$$ It follows that $f\ast g\in\mathrm{Lip}({\mathbb{S}}^{d-1})$, and $\|f\ast g\|_{\mathrm{Lip}}\leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}.$
The finite dimensional space of real homogeneous harmonic polynomials of degree $n$ on ${\mathbb{R}}^d$ restricted to ${\mathbb{S}}^{d-1}$ is denoted by $\mathcal{H}^d_n$. These spaces are mutually orthogonal with respect to the inner product $$\langle f,g\rangle_{{\mathbb{S}}^{d-1}} = \frac{1}{\omega_n}\int_{{\mathbb{S}}^{d-1}} f(x)g(x)d\sigma(x)$$ and they densely span $L^2({\mathbb{S}}^{d-1})$ (see e.g. [@dai2013approximation], Theorem 2.2.2). Denoting by proj$_n$ the corresponding projection operators, we can associate the partial sum operators $$S_n f=\sum_{k=0}^n \mbox{proj}_nf.$$ These are finite-rank and satisfy $S_nf = f\ast K_n$, where $$K_n(t) = \sum_{k=0}^n \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t)$$ and $C_k^\Lambda$ are the Gegenbauer polynomials ([@dai2013approximation], Proposition 2.2.1). Fix $\delta\geq d-1$ and consider the averages $$K^\delta_n(t) = \frac{1}{A_n^\delta}\sum_{k=0}^n A_{n-k}^\delta \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t),$$ where $A_k^\delta = \binom{k+\delta}{k}=\frac{(\delta+k)(\delta+k-1)\dots(\delta+1)}{k!}$. These give rise to a sequence of finite-rank operators on $L^2({\mathbb{S}}^{d-1})$ defined by $S_n^\delta: f\mapsto f\ast K_n^\delta$. Write $\Lambda_n^\delta:=\|S_n^\delta\|_1= \sup \{\|S_n^\delta h\|_1: h\in B_{L^1({\mathbb{S}}
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uhat(-z,w)$ should actually be a polynomial with non-negative integer coefficients of degree $d_\muhat$ in each variable.
In [@hausel-letellier-villegas] we prove that (\[mainconj\]) is true under the specialization $(q,t)\mapsto (q,-1)$, namely, $$E(\M_\muhat;q):=H_c(\M_\muhat;q,-1)=
q^{\frac{1}{2}d_\muhat}\H_\muhat\left(\sqrt{q},\frac{1}{\sqrt{q}}\right).
\label{mainresult}$$
This formula is obtained by counting points of $\M_\muhat$ over finite fields (after choosing a spreading out of $\M_\muhat$ over a finitely generated subalgebra of $\C$). We compute $\M_\muhat(\F_q)$ using a formula involving the values of the irreducible characters of $\GL_n(\F_q)$ (a formula that goes back to Frobenius [@frobenius]). The calculation shows that $\M_\muhat$ is *polynomial count*; i.e., there exists a polynomial $P\in\C[T]$ such that for any finite field $\F_q$ of sufficiently large characteristic, $\#\M_\muhat(\F_q)=P(q)$. Then by a theorem of Katz [@hausel-letellier-villegas Appendix] $E(\M_\muhat;q)=P(q)$.
Recall also that the $E(\M_\muhat;q)$ satisfies the following identity
$$E(\M_\muhat;q)=q^{d_\muhat}E(\M_\muhat;q^{-1}).
\label{curious}$$
In this paper we use Formula (\[mainresult\]) to prove the following theorem.
If non-empty, the character variety $\M_\muhat$ is connected.
The proof of the theorem reduces to proving that the coefficient of the lowest power of $q$ in $\H_\muhat(\sqrt{q},1/\sqrt{q})$, namely $q^{-d_\muhat/2}$, equals $1$. This turns out to require a rather delicate argument, by far the most technical of the paper, that uses the inequality of § \[appendix\] in a crucial way.
Relations to Hilbert schemes on $\C^\times\times\C^\times$ and modular forms
----------------------------------------------------------------------------
Here we assume that $g=k=1$. Put $X=\C^\times\times\C^\times$ and denote by $X^{[n]}$ the Hilbert scheme of $n$ points in $X$. Define $\H^{[n]}(z,w)\in \Q(z,w)$ by $$\sum_{n\geq 0}\H^{[n]}(z,w)T^n:=\prod_{n\geq
1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)},$$ with the
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} + \hat{V}_{\rm l} \right)
\left|\psi_{1}\right>
&=& E_{1}\left|\psi_{1}\right> \; , \\
\left(\hat{H}_0 + \hat{V}_{\rm r} + \hat{V}_{\rm l} \right)
\left|\psi_{2}\right>
&=& E_{2}\left|\psi_{2}\right> \; . \end{aligned}$$ By symmetry, good approximations to these two defect states are given by the even and odd linear combinations $$\left|\psi_{1,2}\right>
=
\sum_{\ell=-\infty}^{\infty}
\frac 1{\sqrt{2}}\left(a_{\ell}\pm a_{-\ell}\right)
\left|\ell\right> \; .$$ From these, auxiliary wave functions $$\begin{aligned}
\label{eq:tilde}
\widetilde{\left|{\psi}_0\right>}
&\equiv&
\sum_{\ell=1}^{\infty}a_{\ell}\left|\ell\right>
\\ \mbox{and} \quad
\widetilde{\left|{\psi}_1\right>}
&\equiv&
\sum_{\ell=1}^{\infty}
\frac 1{\sqrt{2}}\left(a_{\ell}+ a_{-\ell}\right)
\left|\ell\right> \end{aligned}$$ are defined, which have nonvanishing amplitudes in the right half of the lattice only. Forming the scalar product of Eq. (\[eq:E0\]) with $\widetilde{\left<\psi_1\right|}$ then gives $$\widetilde{\left<\psi_1\right|}
\hat{H}_0 + \hat{V}_{\rm r}
\left|\psi_0\right>
= E_0\widetilde{\left<\psi_1\right|} \left.\! \psi_0\right> \; ;$$ forming that of Eq. (\[eq:E1\]) with $\widetilde{\left<\psi_0\right|}$ leads to $$\widetilde{\left<\psi_0\right|}
\hat{H}_0 + \hat{V}_{\rm r}
\left|\psi_1\right>
= E_1\widetilde{\left<\psi_0\right|} \left.\! \psi_1\right> \; ,$$ since $\widetilde{\left<\psi_0\right|} \hat{V}_{\rm l}= 0$ (see Eqs. (\[eq:V\_l\]) and (\[eq:tilde\])). By definition, $\widetilde{\left<\psi_1\right|}\left.{\psi}_0\right> =
\widetilde{\left<\psi_0\right|}\left.{\psi}_1\right>$. Hence, subtracting the above two equations yields $$\left( E_0 - E_1 \right)
\widetilde{\left<\psi_1\right|} \left.\! \psi_0\right> =
\widetilde{\left<\psi_1\right|} \hat{H}_0 \left|\psi_0\right> -
\widetilde{\left<\psi_0\right|} \hat{H}_0 \left|\psi_1\right> \; .$$ It is now stipulated that the localization of the state $\left|{\psi}_0\right>$ around the
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nergies taken from Ref. [@aud95]. We define the deformation parameter $\beta_{2}$ and average pairing gap $\langle\Delta\rangle$ [@sau81; @ben00; @dug01; @yam01] as $$\begin{aligned}
\beta_{2}^{\tau}&=\dfrac{4\pi}{5}\dfrac{\int d\bold{r} \varrho^{\tau}(\bold{r})r^{2}Y_{20}(\hat{r})}
{\int d\bold{r} \varrho^{\tau}(\bold{r})r^{2}},\\
\langle\Delta_{\tau}\rangle&=-\dfrac{\int d\bold{r} \tilde{\varrho}^{\tau}(\bold{r})
\tilde{h}^{\tau}(\bold{r})}{\int d\bold{r} \tilde{\varrho}^{\tau}(\bold{r})},\end{aligned}$$ where $\tilde{\varrho}(\boldsymbol{r})$ is the pairing density.
Fig. \[response\] shows the response functions for the isovector dipole mode in neutron-rich Ne isotopes. The isovector dipole operator used in the present calculation is $$\hat{F}_{1K}=e\dfrac{N}{A}\sum_{i}^{Z}r_{i}Y_{1K}(\hat{r}_{i})-
e\dfrac{Z}{A}\sum_{i}^{N}r_{i}Y_{1K}(\hat{r}_{i}),$$ and the response functions are calculated as $$S(E)=\sum_{i}\sum_{K} \dfrac{\Gamma/2}{\pi}\dfrac{|\langle i|\hat{F}_{1K}|0\rangle|^{2}}
{(E-\hbar \omega_{i})^{2}+\Gamma^{2}/4}.$$
$^{26}$Ne
---------
We can clearly see a resonance structure at around the excitation energy of 8-9 MeV, together with the giant resonance at $15-20$ MeV. Because of the small deformation the $K$ splitting is small and smeared out.
![Isovector dipole transition strengths in $^{26}$Ne for the $K^{\pi}=0^{-}$ (the upper) and $K^{\pi}=1^{-}$ (the lower) states. Underlying discrete states are shown together with the smeared response functions. The arrow indicates the neutron emission threshold $E_{\mathrm{th}}=6.58$ MeV. []{data-label="strength"}](fig4.eps)
----- --------------- --------------- ------------------------ -------------------------------------------- ----------------------
$E_{\alpha}+E_{\beta}$ $Q_{10,\alpha\beta}$
$\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm)
(a) $\nu[310]1/2$ $\nu[211]1/2$
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_V\|K\|_2h_{1,n}^{1/2}$ and the characteristics $A=R$ and $v=22$, and the lemma will follow by application of Talagrand’s inequality. We just need to estimate $Eg^2(t,X)$. We have, making several natural changes of variables, $$\begin{aligned}
Eg^2(t,X_1)&=&
E\Big\{\int f^{1/2}(x)K\Big(\frac{x-X_1}{h_{1,n}}\Big)L\Big(\frac{t-x}{h_{2,n}}f^{1/2}(x)\Big)
I(|t-x|<h_{2,n}B)dx\nonumber\\
&&~~~~~~~~~~\times \int f^{1/2}(y)K\Big(\frac{y-X_1}{h_{1,n}}\Big)L\Big(\frac{t-y}{h_{2,n}}f^{1/2}(y)
\Big)I(|t-y|<h_{2,n}B)
dy\Big\}\nonumber\\
&=&h_{1,n}^2\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
f^{1/2}(t-h_{1,n}v_2)\nonumber\\
&&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)
K\Big(\frac{t-u}{h_{1,n}}-v_1\Big)K
\Big(\frac{t-u}{h_{1,n}}-v_2\Big)\nonumber\\
&&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(u)dudv_1dv_2\nonumber\\
&=&h_{1,n}^3\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
f^{1/2}(t-h_{1,n}v_2)\nonumber\\
&&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)K(v)K(v+v_1-v_2)\nonumber\\
&&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(t-h_{1,n}v_1-h_{1,n}v)dvdv_1dv_2\nonumber\\
&\le& h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\notag\\
&&~~~~~~~~~~\times K(v)K(v+v_1-v_2)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdv_1dv_2\nonumber\\
&= &h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}(w+v_2)f^{1/2}(t-h_{1,n}w-h_{1,n}v_2)\Big)
L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\nonumber\\
&&~~~~~~~~~~\times K(v)K(v+w)I\Big(\frac{h_{1,n}}{h_{2,n}}|w+v_2|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdwdv_2\nonumber\\
&=&h_{1,n}^2h_{2,n}||f||_\infty^2\in
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.
@wever2018ensembles ([-@wever2018ensembles]) utilise genetic algorithms to build nested dichotomies. In their method, a population of random nested dichotomies is sampled and runs through a genetic algorithm for several generations. The final nested dichotomy is chosen as the best performing model on a held-out validation set. An ensemble of $k$ nested dichotomies is produced by initialising $k$ individual populations, independently evolving each population, and taking the best-performing model from each population.
Experimental Results\[sec:experiments\]
=======================================
All experiments were conducted in WEKA 3.9 [@hall2009weka], and performed with 10 times 10-fold cross validation. We use class-balanced nested dichotomies and nested dichotomies built with random-pair selection and logistic regression as the base learner.
For both splitting methods, we compare values of $\lambda \in \{1,3,5,7\}$ in a single nested dichotomy structure, as well as in ensemble settings with bagging [@breiman1996bagging] and AdaBoost [@freund1996game]. The default settings in WEKA were used for the `Logistic` classifier as well as for the `Bagging` and `AdaBoostM1` meta-classifiers. We evaluate performance on a collection of datasets taken from the UCI repository [@lichman2013uci], as well as the MNIST digit recognition dataset [@lecun1998gradient]. Note that for MNIST, we report results of 10-fold cross-validation over the entire dataset rather than the usual train/test split. Datasets used in our experiments, and their number of classes, instances and features, are listed in Table \[tab:datasets\].
We provide critical difference plots [@demvsar2006statistical] to summarise the results of the experiments. These plots present average ranks of models trained with differing values of $\lambda$. Models producing results that are not significantly different from each other at the 0.05 significance level are connected with a horizontal black bar. Full results tables showing RMSE for each experimental
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gned}$$ then the [*rank-breaking*]{} estimator in achieves $$\begin{aligned}
\label{eq:bottoml_3}
\frac{1}{\sqrt{\ld}}\big\|\widehat{\ltheta} - \ltheta^*\big\|_2 \; \leq \; \frac{32\sqrt{2}(1+ e^{4b})^2}{\chi_{\beta_1}}\frac{d^{3/2}}{{\ld}^{3/2}}\sqrt{\frac{d\log d}{n\ell} }\;,
\end{aligned}$$ with probability at least $1 - 3e^3d^{-3}$.
Proof is very similar to the proof of Theorem \[thm:main\]. It mainly differs in the lower bound that is achieved for the second smallest eigenvalue of the Hessian matrix $H(\ltheta)$ of $\Lrb(\ltheta)$, Equation . Equation can be rewritten as $$\begin{aligned}
\label{eq:likelihhod_bl}
\Lrb(\ltheta) = \sum_{j=1}^n \sum_{a = 1}^{\ell} \sum_{\substack{i <\i \in S_j \\ : i,\i \in [\ld]}} \I_ {\big\{(i,\i) \in G_{j,a}\big\}} \lambda_{j,a} \Big(\ltheta_i\I_{\big\{\sigma_j^{-1}(i) < \sigma_j^{-1}(\i)\big\}} + \ltheta_{\i}\I_{\big\{\sigma_j^{-1}(i) > \sigma_j^{-1}(\i)\big\}} - \log \Big(e^{\ltheta_i} + e^{\ltheta_{\i}}\Big) \Big)\;,\end{aligned}$$ where $(i,\i) \in G_{j,a}$ implies either $(i,\i)$ or $(\i,i)$ belong to $E_{j,a}$. The Hessian matrix $H(\ltheta) \in \cS^{\ld}$ with $H_{i\i}(\ltheta) = \frac{\partial^2 \Lrb(\ltheta)}{\partial\ltheta_i \partial\ltheta_{\i}}$ is given by $$\begin{aligned}
\label{eq:limited_hessian}
H(\ltheta) = -\sum_{j=1}^n \sum_{a = 1}^{\ell} \sum_{\substack{i<\i \in S_j :\\ i,\i \in [\ld]}} \I_{\big\{(i,\i) \in G_{j,a}\big\}}\Bigg( (\le_i - \le_{\i})(\le_i - \le_{\i})^\top \frac{\exp(\ltheta_i + \ltheta_{\i})}{[\exp(\ltheta_i) + \exp(\ltheta_{\i})]^2}\Bigg).\end{aligned}$$
The following lemma gives a lower bound for $\lambda_2(-H(\ltheta))$.
\[lem:hessian\_bottoml\] Under the hypothesis of Theorem \[thm:bottoml\_upperbound\_general\], with probability at least $1 - d^{-3}$, $$\begin{aligned}
\label{eq:lambda2_bound_bottoml}
\lambda_2(-H(\ltheta)) \geq \frac{\chi_{\beta_1}}{8(1+ e^{4b})^2} \frac{n\ld\ell^2}{d^2}\;.\end{aligned}$$
Observe that although $\ltheta^* \in \reals^{\ld}$, Lemma \[lem:gradient\_topl\] can be directly applied to uppe
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|
B_{ij}eH_{c+j} =eJ^{i-j}\delta^{i-j}$. Multiplying this identity on the right by $e$ and applying Lemma \[grade-elements\] and Corollary \[morrat-cor\](1) gives $$eJ^{i-j}\delta^{i-j} e = \operatorname{{\textsf}{ogr}}(B _{ij}eH_{c+j})e
=\operatorname{{\textsf}{ogr}}(B_{ij} eH_{c+j}e) =\operatorname{{\textsf}{ogr}}B_{ij} .$$ Since $\delta$ transforms under ${{W}}$ by the sign representation, Lemma \[corpar\](1) shows that $eJ^{i-j}\delta^{i-j} e= eA^{i-j}\delta^{i-j}e$. Combining these observations gives $\operatorname{{\textsf}{ogr}}B_{ij} = eA^{i-j}\delta^{i-j} e$. Therefore, $\operatorname{{\textsf}{ogr}}B= \bigoplus \operatorname{{\textsf}{ogr}}B_{ij} = e\widehat{A}e\cong \widehat{A}$, as graded vector spaces. In order to ensure that this is an isomorphism of graded ${\mathbb{Z}}$-algebras we need to check that the multiplication in $\operatorname{{\textsf}{ogr}}B$ coming from the tensor product multiplication in $B$ is the same as the natural multiplication in $\widehat{A}$. This follows from Lemma \[abstract-products\](1).
\(3) The equivalences $\operatorname{{\textsf}{ogr}}(B){\text{-}{\textsf}{qgr}}\simeq A{\text{-}{\textsf}{qgr}}\simeq
\operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ follow from (2) combined with , respectively Corollary \[hi-basic-lem2\](1).
Corollary {#order-free}
---------
[*Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and pick $i\geq j\geq 0$. Then, for $m\geq 0$, each of the modules $\operatorname{{\textsf}{ord}}^mN(i)$, $\operatorname{{\textsf}{ogr}}^mN(i)$, $\operatorname{{\textsf}{ord}}^m B_{ij}$ and $\operatorname{{\textsf}{ogr}}^m B_{ij}$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module.*]{}
By construction and Proposition \[pre-cohh\], the map $\Theta: \operatorname{{\textsf}{ogr}}N(i)\to eJ^{i}\delta^i$ is an isomorphism of $\operatorname{{\textsf}{ord}}$-graded modules. Thus $\operatorname{{\textsf}{ogr}}^mN(i)\cong \operatorname{{\textsf}{ogr}}^m eJ^{i}\delta^i$ is a free ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-m
| 2,156
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| 1,101
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nfinite chain of adjoint morphisms.\[item:sf5\]
Combining \[thm:stable-lim-III,thm:stab-op\], we see that \[item:sf1\] implies \[item:sf4\], which clearly implies \[item:sf3\], while \[item:sf3\] implies \[item:sf2\] since the cone is a composite of a right extension by zero with a pushout. And \[item:sf2\] implies \[item:sf1\] by \[thm:stable-lim-III\]\[item:sl6\], since right adjoints preserve all limits, so the first four statements are equivalent. The equivalence of \[item:sf1\] with \[item:sf2a\], \[item:sf3a\], and \[item:sf4a\] is dual. Evidently \[item:sf5\] implies \[item:sf2\]. And conversely, if is a stable derivator, then by there are natural isomorphisms $$\Sigma F\toiso C\qquad\text{and}\qquad F\toiso\Omega C.$$ Since $\Sigma$ and $\Omega$ are equivalences in stable derivators (), this shows that the outer morphisms in the adjoint $7$-tuple match up to an equivalence. This implies that the adjoint $7$-tuple can be extended to a doubly-infinite chain of adjoint morphisms and that this chain has period six (in the obvious sense).
We conclude by offering a first interpretation and visualization of this chain of morphisms.
Let be a stable derivator. Then a few additional adjoint morphisms in the doubly-infinite sequence extending are given by: $$\ldots\dashv\pi^\ast\Omega\dashv \Sigma 0^\ast\dashv 0_\ast\Omega\dashv C\dashv 1_!\dashv 1^\ast\dashv \pi^\ast\dashv 0^\ast\dashv 0_\ast\dashv F\dashv 1_!\Sigma\dashv \Omega 1^\ast\dashv \pi^\ast\Sigma\dashv\ldots$$ In fact, this is immediate from the proof of .
In order to not get lost in all these morphisms, let us recall that Barratt–Puppe sequences in a stable derivator can be thought of as refinements of the more classical distinguished triangles. More precisely, associated to $(f\colon x\to y)\in{\sD}^{[1]}$ there is the Barratt–Puppe sequence $BP(f)$ generated by $f$. This is a coherent diagram as in which vanishes on the boundary stripes and which makes all squares bicartesian.
$$\vcenter{
\xymatrix@-1pc{
\ar@{}[dr]|{\ddots}&\ar@{}[
| 2,157
| 1,984
| 1,634
| 1,886
| 3,334
| 0.773045
|
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|
{\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}
\bigg)\bigg(\prod_{\substack{l,l'\ge1\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf k}}'=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}\cap\,
{{\cal U}}_{{{\bf k}};1}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};1}{^{\rm c}}}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}u_0\}$}}}}_{\stackrel{\because{(\ref{eq:psi-delta})}\;}\leq
{{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z
\rangle}}_\Lambda}.{\label{eq:nsum-1stbd}}\end{gathered}$$ Then, by conditioning on ${{\cal U}}_{{{\bf k}};2}\equiv{\mathop{\Dot{\bigcup}}}_{l\ge2}{{\cal C}}_{{\bf k}}(u_{2l})$, following the same computation as above and using [(\[eq:G-delta-bd\])]{}, we further obtain that $$\begin{gathered}
{(\ref{eq:nsum-0thbd})}\leq{{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle
\varphi_{u_0}\varphi_z \rangle}}_\Lambda\sum_{{\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}}
({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\bigg(\prod_{l\ge2}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2
l}\}$}}}\bigg)\bigg(\prod_{\substack{l,l'\ge2\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}u_2\}$}}}}_{\leq\;\tilde G_\Lam
| 2,158
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| 1,978
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| null | null |
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|
thrm{th}}$ coordinate is $$\label{eq:new.gamma}
\gamma_{{\widehat{S}}}(j) = \mathbb{E}_{X,Y, \xi_j}\Biggl[ \left|Y- t_{\tau}\left(
\hat\beta_{{\widehat{S}}(j)}^\top X_{{\widehat{S}}(j)} \right) \right|-
\left| Y-t_{\tau}\left( \hat\beta_{{\widehat{S}}}^\top X_{{\widehat{S}}} \right) \right| +
\epsilon \xi(j) \Biggr],$$ where $\epsilon > 0$ is a pre-specified small number, $\xi = (\xi(j), j \in {\widehat{S}})$ is a random vector comprised of independent $\mathrm{Uniform}(-1,1)$, independent of the data, and $t_{\tau}$ is the hard-threshold function: for any $x \in \mathbb{R}$, $t_{\tau}(x)$ is $x$ if $|x| \leq \tau$ and $\mathrm{sign}(x) \tau$ otherwise. Accordingly, we re-define the estimator $\hat{\gamma}_{{\widehat{S}}}$ of this modified LOCO parameters as $$\label{eq:new.delta}
\hat{\gamma}_{{\widehat{S}}} = \frac{1}{n} \sum_{i \in \mathcal{I}_{2,n}} \delta_i,$$ where the $\delta_i$’s are random vector in $\mathbb{R}^{{\widehat{S}}}$ such that the $j^{\mathrm{th}}$ coordinate of $\delta_i$ is $$\Big| Y_i- t_{\tau}\left( \hat\beta_{{\widehat{S}}(j)}^\top
X_i({\widehat{S}}(j)) \right) \Big|-
\Big| Y_i - t_{\tau} \left( Y_i-\hat\beta_{{\widehat{S}}}^\top X_{i}({\widehat{S}}) \right)\Big|
+
\epsilon \xi_i(j), \quad j \in {\widehat{S}}.$$
[**Remark.**]{} Introducing additional noise has the effect of making the inference conservative: the confidence intervals will be slightly wider. For small $\epsilon$ and any non-trivial value of $\gamma_{{\widehat{S}}}(j)$ this will presumably have a negligible effect. For our proofs, adding some additional noise and thresholding the regression function are advantageous because the first choice will guarantee that the empirical covariance matrix of the $\delta_i$’s is non-singular, and the second choice will imply that the coordinates of $\hat{\gamma}_{{\widehat{S}}}$ are bounded. It is possible to let $\epsilon\to 0$ and $\tau \rightarrow \infty$ as $n\to\infty$ at the expense of slower concentration and Berry-Esseen rates. For sim
| 2,159
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that the required data for a homotopy $\mathcal Comm$-inner product consists of,
- a derivation $d\in \mathrm{Der}(F _{\mathcal Lie}\,C[1])$ of degree $1$, with $d^2=0$,
- a derivation $g\in \mathrm{Der}_d (F_{\mathcal Lie,\,C[1]}C[1])$ over $d$ of degree $1$, with $g^2=0$, which imduces a derivation $h\in \mathrm{Der}_d (F_{\mathcal Lie,\,C[1]}C^*[1])$ over $d$ with $h^2=0$,
- a module map $f\in \mathrm{Mod}(F_{\mathcal Lie, C[1]}C^*, F_{\mathcal
Lie,C[1]}C[1])$ of degree $0$ such that $f\circ h = g \circ f$.
In order to construct the derivation $d\in \mathrm{Der}(F _{\mathcal Lie}\,C[1])$ with $d^2=0$, let $F_{\mathcal Lie}C[1]=L_1\oplus L_2\oplus\dots$, where $L_n=(\mathcal Lie(n)\otimes C[1]^{\otimes n})_{S_n}$, be the decomposition of $F_{\mathcal Lie}C[1]$ by the monomial degree in $C[1]$. Then, $d:F_{\mathcal Lie}C[1]\to F_{\mathcal Lie}C[1]$ is determined by maps $d=d_1+d_2+\dots$, where $d_i:C[1]\to L_i$ is lifted to $F_{\mathcal Lie}(C[1])$ as a derivation. Let $d_1$ be the differential on $C[1]$, and $d_2$ be the symmetrized Alexander-Whitney comultiplication. For the general $d_i$, we use the inductive hypothesis that $d_1$, …, $d_{i-1}$ are local maps so that $\nabla_i:=d_1+ \dots+d_{i-1}$ has a square $\nabla_i^2$ mapping only into higher components $L_{i}\oplus L_{i+1}\oplus \dots$. Here, “local” means that every simplex maps into the sub-Lie algebra of its closure. Now, by the Jacobi-identity, it is true that $0=[\nabla_i, [\nabla_i,\nabla_i]]=
[d_1,e_i]\text{+ higher terms}$, where $e_i:C[1]\to L_i$ is the lowest term of $[\nabla_i,\nabla_i]$. Thus $e_i$ is $[d_1,.]$-closed and thus, using the contractibility hypothesis of the proposition, also locally $[d_1,.]$-exact. These exact terms can be put together to give a map $d_i$, so that $[d_1,d_i]$ vanishes on $L_1\oplus \dots \oplus
L_{i-1}$ and equals $- 1/2\cdot e_i$ on $L_i$. In other words, $(d_1+
\dots+d_i)^2=1/2\cdot [d_1+ \dots+d_i,d_1+ \dots+d_i]=1/2\cdot [\nabla_i
,\nabla_i ]+ [d_1,d_i]+\text{higher terms}$, maps only
| 2,160
| 2,420
| 2,228
| 2,006
| 2,044
| 0.783237
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|
4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}};{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})$}
\rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4\,\,\, 5$}
\end{pspicture}$$ The canonical example of a 0/1-operad is the endomorphism 0/1-operad given for $k$-vector spaces $A$ and $M$ by $$\begin{aligned}
{\mathcal E\!nd}^{A,M}(\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})=& Hom(\text{tensor products of $A$ and $M$},A)\\
{\mathcal E\!nd}^{A,M}(\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})=& Hom(\text{tensor products of $A$ and $M$},M)\\
{\mathcal E\!nd}^{A,M}(\vec X;\varnothing)=& Hom(\text{tensor products of $A$ and
$M$},k).\end{aligned}$$ With this notation $(A,M,k)$ is an algebra over the 0/1-operad $\mathcal P$ if there exists a 0/1-operad map $\mathcal P \to {\mathcal E\!nd}^{A,M}$. By slight abuse of language we will also call the tuple $(A,M)$ an algebra over $\mathcal P$.
It is our aim to define for each cyclic operad $\mathcal O$, the associated 0/1-operad $\widehat{ \mathcal O}$, which incorporates the cyclic structure as a colored operad. Before doing so, let us briefly recall the definition of a cyclic operad from [@GeK Theorem (2.2)].
Let $\mathcal O$ be a operad, i.e we have vector spaces $\mathcal O(n)$ for $n\geq 1$, composition maps $\circ_i:\mathcal O(n)\otimes \mathcal O(m)\to \mathcal O(n+m-1)$, and an $S_n$-action on $\mathcal O(n)$ for each $n$, satisfying the usual axioms, see [@GK (1.2.1)]. $\mathcal O$ is called [*cyclic*]{} if there is an action of the symmetric gr
| 2,161
| 1,403
| 2,028
| 2,041
| 1,645
| 0.787273
|
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|
Here, $\chi = \alpha^{-m}$.
More generally, over all components, we write $$c_1^{\rm rep}({\cal O}_{\mathfrak{X}}(m)) \: = \: \left( \frac{m}{k} J, \cdots,
\frac{m}{k} \alpha^{-m} J, \cdots \right).$$ Multiplication of components of ch$^{\rm rep}$ multiplies not only the cohomology classes, but also the coefficients. For example, $$\left( c_1^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times \{\alpha\} }
\right)^2 \: = \:
\left( \frac{m}{k} J \right)^2 \alpha^{-2m}.$$ Now, for a line bundle $L$ on an ordinary space, $${\rm ch}_2(L) \: = \: (1/2) c_1^2(L),$$ but here, by contrast, $$\begin{aligned}
{\rm ch}_2^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times\{\alpha\}} & = &
\frac{1}{2}
\left( \frac{m}{k} J \right)^2 \alpha^{-m}, \\
& = & \alpha^{+m} \frac{1}{2}
\left( c_1^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times \{\alpha\} }
\right)^2,\end{aligned}$$ so that the usual relation between Chern classes and Chern characters is modified on a stack. (In fact, if we were computing Chern classes of a bundle that split as several different eigenbundles, the relation would be much more complicated than just an additional complex phase.)
As a consistency check, let us compute the index of this line bundle, using Hirzebruch-Riemann-Roch. For any bundle ${\cal E} \rightarrow \mathfrak{X}$, the Hirzebruch-Riemann-Roch index theorem says $$\chi({\cal E}) \: = \: \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal E})
{\rm Td}(\mathfrak{X})$$ where $$\chi({\cal E}) \: = \: \sum_i (-)^i h^i(\mathfrak{X}, {\cal E}),$$ and $${\rm Td}(\mathfrak{X}) \: = \: \alpha_{\mathfrak{X}}^{-1} {\rm Td}( T
I_{\mathfrak{X}} ),$$ where $$\alpha_{\mathfrak{X}} \: = \: {\rm ch}( d( \lambda_q) ), \: \: \:
\lambda_q \: = \: \sum_k (-)^k \wedge^k N_q^*,$$ for $N_q$ the normal bundle. (As $\lambda_q$ is not a pullback from $\mathfrak{X}$, but rather is defined intrinsically on $I_{\mathfrak{X}}$, ${\rm ch}^{\rm rep}(\lambda_q)$ is not well-defined, so instead the pertinent Chern character is defined via the diagonalization map $d$.)
In the present case, s
| 2,162
| 2,060
| 1,983
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with two $d$s in different rows. If $T$ is $d$-bad, then we can express ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms using Lemma \[lemma5\] together with Lemma \[newsemilem\]. For example, if $$\begin{aligned}
T&=\young(11111122248{10},369{11}{12},57)
\\
\intertext{then ${\psi_{6,1}}\circ{\hat\Theta_{T}}={\hat\Theta_{T'}}$, where}
T'&=\young(11111122248{10},369{11}{12},56)\end{aligned}$$ and we can semistandardise this using Lemma \[newsemilem\], taking $$A=\{3\},\qquad B=\{5,6,7,9,11,12\},\qquad C=\emptyset$$ to express ${\hat\Theta_{T'}}$ as a sum of fourteen semistandard homomorphisms.
Doing this for each $d$-bad tableau $T$, we find that ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is a sum of homomorphisms labelled by semistandard tableaux with the same first row as $T$; furthermore, at least one of these tableaux will have two $d$s in the second row. Moreover, each $d$-bad tableau will yield a tableau of this kind which does not occur for any other $d$-bad tableau $T'$. To see this, suppose first of all that $d,d+1$ occur in the second column of $T$. Then there is no other $d$-bad tableau with the same first row as $T$, so any tableau occurring in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ with two $d$s in the second row can only possibly occur in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$. Alternatively, if $d,d+1$ occur in the first column of $T$, then $T$ has the form $${\text{\footnotesize$\gyoungx(1.2,;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;2;2;{x_1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{x_s},;d;{z_1};{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{d\!\!+\!\!1};k)$}}.$$ There are $v-2$ other $d$-bad tableaux with the same first row as $T$, and they all also have the same $(2,2)$-entry as $T$. Hence when we apply Lemma \[lemma5\] and Lemma \[newsemilem\] (or equivalently
| 2,163
| 1,809
| 1,977
| 1,964
| 1,150
| 0.793463
|
github_plus_top10pct_by_avg
|
_2}\cdot{\vec{p}})$,
$({\vec{\sigma}_1}\cdot{\vec{p}})({\vec{\sigma}_2}\cdot{\vec{q}})$, $({\vec{\sigma}_1}-{\vec{\sigma}_2})\cdot({\vec{q}}\times{\vec{p}})$
: All possible PC and PV NLO operational structures connecting the initial and final spin and angular momentum states. There are a total of 18. \[tab:contacts\]
The NLO contribution to the weak decay process, $\Lambda N\to NN$, includes contact interactions with one and two derivative operators, caramel diagrams and two-pion-exchange diagrams.
NLO contact potential {#sec:nlocontact}
---------------------
In principle the NLO contact potential should include, in the center of mass, structures involving both the initial (${\vec{p}}\,$) and final (${\vec{p}\,'}$) momenta, or independent linear combinations, e.g. ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$ and ${\vec{p}}$. Table \[tab:contacts\] lists all these possible structures. At NLO there are 18 LECs —6 PV ones at order ${\cal O}\left(q/M\right)$, 7 PC ones at order ${\cal
O}\left(q^2/M^2\right)$ and 5 PV ones at order ${\cal
O}\left(q^2/M^2\right)$—, which must be fitted to experiment. This is not feasible with current experimental data on hypernuclear decay. A reasonable way to reduce the number of LECs and render the fitting procedure more tractable is to note that the pionless weak decay mechanism we are interested in takes place inside a bound hypernucleus. Thus, one can consider that in the $\Lambda N\rightarrow NN$ transition potential the initial baryons have a fairly small momentum. Moreover, the final nucleons gain an extra momentum from the surplus mass of the $\Lambda$ ($M_\Lambda-M_N=116$ MeV), which in most cases allow to consider, ${\vec{p}\,'}\gg{\vec{p}}$. In this case, one may approximate ${\vec{q}}\simeq{\vec{p}\,'}$ and ${\vec{p}}=0$. Within this approximation, the NLO part of the contact potential reads (in units of $G_F$): $$\begin{aligned}
V_{4P} ({\vec q} \, ) &=
C_1^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q}}{2
| 2,164
| 2,473
| 2,839
| 2,028
| 3,551
| 0.771588
|
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|
$ and $\mathcal{T}_{-e_x}$ respectively denote the RST and DSF of direction $-e_x$ constructed on the same PPP $N$. Then, for $0<\alpha<1/3$: $$\lim_{r\rightarrow +\infty} {{\mathbb P}}\big(\mathcal{T}\cap B((r,0),r^\alpha)=\mathcal{T}_{-e_x}\cap B((r,0),r^\alpha)\big)=1.$$ The approximation also holds if we replace $r^\alpha$ by a constant radius $R$.
**Step 1:** Let us prove . First of all, notice that all the paths which intersect the arc $a(A_r, B_r)$ necessarily intersect the segment $[A_r,B_r]$ (the converse is not necessarily true). The segment $[A_r,B_r]$ is perpendicular to the horizontal axis, and all its points have abscissa $\tilde{r}=r \cos(1/ r)$. Its length is $2r \sin(1/r)\leq 2$.\
Heuristically, the event $\{\widetilde{\chi}_r\geq 1\}$ (consisting in the existence of at least one semi-infinite path crossing $a(A_r, B_r)$) is hence close to the existence of a path of the RST crossing the vertical segment $[A_r ,B_r]$ and then surviving until a large radius.\
For $R>0$, let us hence consider the event where there exists a path of the RST crossing $ \{\widetilde{r}\}\times [-1,1] \supset [A_r, B_r]$ before intersecting the sphere $S((\tilde{r},0),R)$.\
Our purpose is to show that the probability of this event is close to the probability that in the DSF $\mathcal{T}_{-e_x}$, there exists a path intersecting $\{0\}\times [-1,1]$ and then $S(O,R)\cap \{x>0\}$. To show that such an approximation holds, let us prove that our event is local in the sense of Lemma \[lemme:approximationDSF\].\
![\[fig:Zn\] [*Here is the sub-path $Z_{0},\ldots,Z_{n}$ of the RST crossing the vertical segment $\{\widetilde{r}\}\times [-1,1]$ (in bold) and the sphere $S((\tilde{r},0),R)$. On this picture, $Z_{0}$ and $Z_{n}$ belong to $B((\widetilde{r},0),2R)$ which occurs with high probability.*]{}](Zn.eps){width="7cm" height="6.5cm"}
Let us consider a path of the RST crossing $\{\widetilde{r}\}\times [-1,1]$ and afterwards $S((\tilde{r},0),R)$, i.e. towards descendants as is described below. From this path, we can extract a
| 2,165
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| 2,140
| 2,026
| 0.783379
|
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|
^\*\_0(\^\*,v)=[**F**]{}\_0\^\*(v),v.
Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), and (\[sda2b\]) are valid. Using integration by parts and Green’s formula we have the following.
For all $\psi,\psi^*\in \tilde {{{\mathcal{}}}H}$ one has \[adjoint7\] \_0\^\*(\^\*,)=\_0(,\^\*), where $\tilde{\bf B}_0(\cdot,\cdot)$ is the bilinear form (\[cosyst5a\]).
The relation is the justification for the term *adjoint problem*. For the existence of solutions of the adjoint problem, we formulate the following result.
\[adjointco1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) are valid, and that (\[csda3aa\]), (\[csda4aa\]) hold for $c>0$ and for $C$ given in (\[cosyst6a\]). Let $f^*\in L^2(G\times S\times I)^3$ and that $g^*\in T^2(\Gamma_+)^3$. Then the following assertions hold.
\(i) The variational equation \[vareqcoad\] \_0\^\*(\^\*,v)=[**F**]{}\_0\^\*(v)v, has a solution $\tilde{\psi}^*=(\tilde{\psi}^*_1, \tilde{\psi}^*_2, \tilde{\psi}^*_3)\in {{{\mathcal{}}}H}$. Writing $\tilde{\psi}_1^*:=(\psi_1^*,q_1^*)$, $\tilde{\psi}_j^*:=(\psi_j^*,q_j^*,p_{0j}^*,p_{{\rm m}j}^*)$, $j=2,3$, and $\psi^*=(\psi_1^*, \psi_2^*, \psi_3^*)\in L^2(G\times S\times I)^3$, then $\psi^*\in{{{\mathcal{}}}H}_{{\bf P}^*}(G\times S\times I^\circ)$ (see ) is a weak (distributional) solution of the system of equations , , and $\psi^*_1\in W^2(G\times S\times I)$.
\(ii) Suppose that additionally the assumption ${\bf TC}$ holds (p. ). Then a solution $\psi^*$ of the equations , obtained in part (i) is a solution of the problem -.
\(iii) Under the assumptions imposed in part (ii), any solution $\psi^*$ of the problem - that further satisfies \[asscl-aa-co-ad\] \^\*\_[|\_-]{}T\^2(\_-)\^3 (,,E\_[m]{})L\^2(GS)\^3. is unique and obeys the estimate \[adjoint11b\] [\^\*]{}\_[[H]{}]{}( \_[L\^2(GSI)\^3]{}+\_[T\^2(\_+)\^3]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.)
Note that if $\psi^*$ is a solution of the problem -, then it is a solution of the varia
| 2,166
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| 2,241
| 2,329
| null | null |
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|
erical coordinate system centered at the middle of the torus $( \rm r \ sin
\theta_s
cos\phi,r \ \rm sin \theta_s
sin\phi,r \ \rm cos \theta_s)$ $${\rm r \ sin\theta_s} = R + a \ \rm cos\theta$$ $${\rm r \ cos \theta_s} = a \rm \ sin\theta$$ $${\rm r } = \sqrt{R^2 + a^2 + 2 a R \ \rm cos\theta}$$ and defining $$\rho_{IJ} \equiv \psi^*_I(\theta)\psi_J(\theta)$$ gives for Eq. (22) after some manipulation $$\notag
\sum_{LM}\big(\delta_{M-\nu_1+\nu_3}\delta_{M+\nu_2-\nu_4}\big){(L-M)!
\over (L+M)!}\int_0^{2\pi}\int_0^{2\pi}
\rho_{PQ}(\theta_1)\rho_{RS}(\theta_2)$$ $$P_{LM}(\theta_{s_1}(\theta_1))P_{LM}(\theta_{s_2}(\theta_2))F(\theta_1)F(\theta_2)
{(R^2+a^2+2a R {\rm cos\theta_<})^{L/2} \over (R^2+a^2+2a R {\rm
cos\theta_>})^{(L+1)/2}} d\theta_1d\theta_2.$$ The arguments of the $P_{LM}$ are evaluated with $$\theta_{s_i}= \arctan \bigg ( {R + a \ {\rm cos}\theta_i \over a \
{\rm sin} \theta_i } \bigg ).$$ To evaluate the integral care must be taken with the $>, <$ character of the radial factor in the integrand. One way to proceed is as follows:
1\. Fix $\theta_1 = 0$. At this point $r_1$ is at its maximum; integrate the integrand of Eq.(27) numerically over $d\theta_2$ from $[0,2\pi]$ by some suitable method to attain a value labelled by, say, $G_0(\theta_1)$.
2\. Set $\theta_1 = \delta$. Integrate $d\theta_2$ with $r_> =
r_1$, $r_< = r_2$ over the interval $[\delta, 2\pi-\delta]$, then set $r_> = r_2$ and $r_< = r_1$ from $[2\pi-\delta, \delta]$. This is $G_\delta(\theta_1)$.
3\. Repeat the second step until the entire interval around the toroidal cross section is covered. A table \[$G_0(\theta_1),
G_\delta(\theta_1),G_{2\delta}(\theta_1)...]$ results that can then be integrated numerically. The exchange term proceeds similarly; only the densities need modification.
Conclusions
===========
This work presents a method to calculate the spectrum and wave functions for an electron on $T^2$ in an arbitrary static magnetic field. Aside from the character of the solutions and numerical data, perhaps the main res
| 2,167
| 3,206
| 2,044
| 2,087
| null | null |
github_plus_top10pct_by_avg
|
serious concern.
The original sample consisted of 306 Angolan respondents. For this analysis, parents who have children both in the country of origin and in the Netherlands are omitted because this would not allow exploring the different mediation paths of interest and test opposite hypotheses. Missing data and the above criterion reduced the sample to 255 Angolan respondents, including 86 transnational parents.^[2](#fn2-0192513X17710773){ref-type="fn"}^ Finally, because job absenteeism only applies to people in employment, results for the model regarding job absenteeism are based on respondents in employment at the time of interview, which resulted in 181 respondents.
Results {#section9-0192513X17710773}
=======
[Table 1](#table1-0192513X17710773){ref-type="table"} reports the descriptive statistics of the included variables. Looking at the mediating factors, it is found that the majority of Angolans indicate to be happy (77%) and experience limited family-to-work conflict (33%). On a scale from 0 to 5, the average score is 0.87 (*SD* = 0.89). Finally, on average Angolans report having changed jobs in the Netherlands twice and have missed on average 2.4 days of work in the 3 months prior to the interview.
######
Descriptive Statistics.
*M* *SD* Min Max
-------------------------------------------------------------------------------- ------- ------- ----- -----
Transnational parenting^[a](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.34 0.47 0 1
Happiness^[b](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.77 0.42 0 1
Family-to-work conflict^[c](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.33 0.50 0 1
Age 33.60 6.28 23 59
Sex^[d](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.50 0.50
| 2,168
| 368
| 2,753
| 2,337
| null | null |
github_plus_top10pct_by_avg
|
if you use fork/wait, you could get the exit code from the status code in the wait* functions:
int status;
if (wait(&status) != -1) { // similar for waitpid, wait4, etc.
if (WIFEXITED(status)) {
exit_code = WEXITSTATUS(status);
} else {
// handle other conditions, e.g. signals.
}
} else {
// wait failed.
}
You could check the example of in the man page of wait(2).
Q:
Refactoring a String Calculator
I've refactored the string calculator kata as much as I could and tried to maintain single responsibility. Is there anything I could factor differently? The specs can be reviewed here.
module StringCalculator
def self.add(string)
string.end_with?("\n") ? fail('ends in newline') : solve(string)
end
def self.solve(string)
verify(string)
custom = delimiter(string)
numerics = string.gsub(/\n/, custom).split(custom).map(&:to_i)
numerics.reject { |n| n > 1000 }.reduce(0, :+)
end
def self.delimiter(string)
string[0, 2] == '//' ? string[2, 1] : ','
end
def self.verify(string)
find = string.scan(/-\d+/)
fail("negatives not allowed: #{find.join(', ')}") if find.any?
end
private_class_method :verify, :delimiter, :solve
end
More importantly, I've thought of a case that involves the delimiter of a string that begins with // to be a number (which is considered invalid). I've made a quick way of solving that issue like so. Any suggestions on that?
Invalid case:
def self.delimiter(string)
string[0, 2] == '//' ? check(string) || string[2, 1] : ','
end
def self.check(string)
fail("invalid delimiter: #{string[2, 1]}") if string[2, 1] =~ /[0-9]/
end
A:
I think your code is pretty good. The main opportunity for improved clarity I saw was to consolidate your validation code into one place. In your code, validations are sprinkled across various methods, so that you are mixing the responsibility of "validation" and "work".
Here's a refactoring (not tested) which puts all the validation in one place, and gets it out of the way up front. T
| 2,169
| 6,176
| 51
| 1,762
| 920
| 0.797696
|
github_plus_top10pct_by_avg
|
-\al_i/2},$$ where $\al_1,\ldots,\al_r$ and $\be$ are nonnegative constants. The class $\pi_{SH}(\Th)$ includes both harmonic priors $\pi_{EM}(\Th)$ and $\pi_{JS}(\Th)$, which are given in (\[eqn:pr\_em\]) and (\[eqn:pr\_js\]), respectively. Indeed, $\pi_{SH}(\Th)$ is the same as $\pi_{EM}(\Th)$ if $\al_1=\cdots=\al_r=\al^{EM}$ and $\be=0$ and as $\pi_{JS}(\Th)$ if $\al_1=\cdots=\al_r=0$ and $\be=\be^{JS}$.
It is noted that $$\begin{aligned}
\frac{\partial}{\partial \la_k}\pi_{SH}(\Th) &= -\frac{1}{2}\Big(\frac{\al_k}{\la_k}+\frac{\be}{\sum_{i=1}^r \la_i}\Big)\pi_{SH}(\Th), \label{eqn:d_pi_g} \\
\frac{\partial^2}{\partial \la_k^2}\pi_{SH}(\Th) &=\frac{1}{2}\bigg\{\Big(\frac{\al_k}{\la_k^2}+\frac{\be}{(\sum_{i=1}^r \la_i)^2}\Big)+\frac{1}{2}\Big(\frac{\al_k}{\la_k}+\frac{\be}{\sum_{i=1}^r \la_i}\Big)^2\bigg\}\pi_{SH}(\Th) . \label{eqn:dd_pi_g}\end{aligned}$$ Combining (\[eqn:d\_pi\_g\]), (\[eqn:dd\_pi\_g\]) and Proposition \[prp:condition2\], we obtain $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top \pi_{SH}(\Th)] \non\\
&=\pi_{SH}(\Th)\sum_{i=1}^r\bigg[\{\al_i^2-(q-r-1)\al_i\}\frac{1}{\la_i}-2\sum_{j>i}^r\frac{\al_i-\al_j}{\la_i-\la_j}+\frac{2\al_i\be}{\tr(\Th\Th^\top)}\bigg] \non\\
&\qquad +\pi_{SH}(\Th)\frac{\be^2-(qr-2)\be}{\tr(\Th\Th^\top)}.
\label{eqn:dd-pi_g}\end{aligned}$$
\[exm:1\] Let $$\pi_{ST}(\Th)=\prod_{i=1}^r \la_i^{-\al_i/2},$$ where $\al_1,\ldots,\al_r$ are nonnegative constants. Assume that $\al_1\geq\cdots\geq\al_r$. Note that $$\begin{aligned}
\sum_{i=1}^r\sum_{j>i}^r\frac{\al_i-\al_j}{\la_i-\la_j}
&=\sum_{i=1}^r\sum_{j>i}^r\frac{1}{\la_i}\frac{\la_i-\la_j+\la_j}{\la_i-\la_j}(\al_i-\al_j)\\
&=\sum_{i=1}^r(r-i)\frac{\al_i}{\la_i}-\sum_{i=1}^r\frac{1}{\la_i}\sum_{j>i}^r\al_j+\sum_{i=1}^r\sum_{j>i}^r\frac{\la_j}{\la_i}\frac{\al_i-\al_j}{\la_i-\la_j}\\
&\geq \sum_{i=1}^r(r-i)\frac{\al_i}{\la_i}-\sum_{i=1}^r\frac{1}{\la_i}\sum_{j>i}^r\al_j.\end{aligned}$$ From (\[eqn:dd-pi\_g\]), it is seen that $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top \pi_{ST}(\Th)] \\
&\leq\pi_{ST}(\Th)\sum_{i=1}^r\big
| 2,170
| 1,156
| 2,064
| 2,159
| null | null |
github_plus_top10pct_by_avg
|
2}$ by $f^\alpha$ (and $f^{3/2}$ by $f^{\alpha +1}$) in the definition of $\hat f_n(t;h)$ the only $\alpha$ for which $\int_{-B}^Bw^2g''(0)dw=0$ for all $f$ twice differentiable with $f(t)\ne0$ is $\alpha=1/2$.\]. Thus, we have $$\int_{-B}^Bw^2g^{(i)}_{t,w}(0)dw=0\ \ {\rm for}\ \ i=1,2,3,$$ and we conclude, from this, (\[ftilde\]), (\[taylor\]) and (\[firstt\]), that $$\label{ftilde2}
\tilde f(t;h)=\int_{-B}^Bg_{t,w}(hw)dw=f(t)+\frac{h^4}{4!}\int_{-B}^Bw^4E_\tau g^{(4)}(\tau h w)dw.$$ Using the formula for $g^{(4)}$ in (\[deriv\]), integrating by parts and collecting terms, it is tedious but straightforward to check that $$\begin{aligned}
\label{rem}
&&\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\\
&&~~~~~~~~=\left[\frac{24(f')^4(t)}{f^5(t)}-\frac{36(f')^2(t)f''(t)}{ f^4(t)}+\frac{8f'(t)f'''(t)+6(f'')^2(t)}{f^3(t)}-\frac{f^{(4)}(t)}{ f^2(t)}\right]\int v^4K(v)dv,\notag\end{aligned}$$ and to note that $$\label{rem2}
\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left|\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\right|<\infty.$$
Now, the boundedness and uniform continuity of $K$ and its four derivatives and the facts that, for $f\in{\cal D}_{C,z}$, $f$ and its first three derivatives are Lipschitz with common constant $C$ and the fourth derivatives $f^{(4)}$ have all the same modulus of continuity $z$ at all $t$, and that $f$ is bounded away from zero in a neighborhood of $D_r$, imply that $$\label{comp}
\lim_{n\to\infty}\sup_{f\in {\cal D}_{C,z}}\sup_{0\le \tau\le 1}\sup_{w\in[-B,B]}\sup_{t\in D_r}|g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0)|=0.$$ Therefore, $$\lim_{n\to\infty}\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left|\int_{-B}^Bw^4E_\tau (g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0))dw\right|=0$$ and we have from this and (\[ftilde2\]) that $$\begin{aligned}
&&\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r} \left|h_n^{-4}(\tilde f(t;h_n)-f(t)) -{1\over 4!}\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\right|\\
&&~~~~~~~~~~~=\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left| {1\over 4!}E_\tau\int_{-B}^Bw^4 (g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0))dw\right|\to
0
| 2,171
| 1,013
| 1,647
| 2,134
| null | null |
github_plus_top10pct_by_avg
|
Pancreas
In-field tumor responses
------------------------
Twenty six lesions were targeted in nineteen patients (Table [3](#T3){ref-type="table"}). They included 15 pancreatic masses, 4 regional metastatic lymph nodes and 7 distant metastatic lesions. Of the 15 pancreatic masses, 8 showed partial responses (PR, 53.3%) and 7 stable disease (SD, 46.6%). Of the 4 regional metastatic lymph nodes, one showed PR (25.0%) and three, SD (75.0%). Of the seven distant metastatic lesions (six hepatic metastases and one pulmonary metastasis), 2 (a pulmonary lesion and a hepatic lesion) showed PR (28.6%) and 5, SD (71.4%). Although there were no complete responses (CR), the overall response rate was 42.3%. It is of interest that no target lesions showed in-field progression during the observation period. Figure [3](#F3){ref-type="fig"} illustrates a typical case of a pancreatic lesion treated with CCRT.
######
In-field tumor response rates of the target lesions after tomotherapy and concurrent capecitabine treatment
Target lesions CR PR SD PD
------------------------------ ------- ----------- ----------- -------
Pancreatic mass (n = 15) 0 (0) 8 (53.3) 7 (46.7) 0 (0)
Regional lymph nodes (n = 4) 0 (0) 1 (25.0) 3 (75) 0 (0)
Distant metastasis (n = 7) 0 (0) 2 (28.6) 5 (71.4) 0 (0)
Liver (n = 6) 0 (0) 1 (16.7) 5 (83.3) 0 (0)
Lung (n = 1) 0 (0) 1 (100) 0 (0) 0 (0)
Overall (n = 26) 0 (0) 11 (42.3) 15 (57.7) 0 (0)
CR, complete response; PR, partial response; SD, stable disease; PD, progressive disease
Numbers in parentheses are percentages
{#F3}
Prognosis and survival
----------------------
The med
| 2,172
| 332
| 2,345
| 2,583
| null | null |
github_plus_top10pct_by_avg
|
l) {
collateral.setPrice(new BigDecimal(map.get("Price")));
}
if (map.get("Par") != null) {
collateral.setPar(new BigDecimal(map.get("Par")));
}
if (map.get("mkt_val") != null) {
collateral.setMarketValue(new BigDecimal(map.get("mkt_val")));
}
if (map.get("Accrued Intr") != null) {
collateral.setAccurInterest(new BigDecimal(map.get("Accrued Intr")));
}
if (map.get("Total Market Value") != null) {
collateral.setTotMktValue(new BigDecimal(map.get("Total Market Value")));
}
A:
The simple answer to the overt question of, "can I make this more concise/terse" is "no". You're not really going to get what you're looking for in making this more terse or concise, nor would computeIfPresent really give you what you're looking for and keep your code readable.
The issue is that, when you go to retrieve a key from your map, you're putting it in a different field in your collateral instance. This means that trivial solutions such as looping over the map won't satisfy since you're not going to be able to get the exact field you need to map to without getting deep into reflection.
The code you have here, albeit verbose, is perfectly readable and reasonable to any other maintainer to understand what's going on. I see no incentive to change it.
Q:
creating a multidimensional array with random integers
I initially want to say thank you for taking the time to look at my post. Basically I have attempted to create a multidimensional array with random integers using Math.random. The code compiles and keeps returning the null pointer exception error message. I don't know what I did wrong in creating my object. Can anyone tell me what's wrong with the code?
public Table(int r, int c)
{
rows = r;
columns = c;
for (int i = 0; i < r; i++)
for (int j = 0; j < c; j++)
{
/*
| 2,173
| 1,040
| 175
| 2,221
| 752
| 0.80079
|
github_plus_top10pct_by_avg
|
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 0.070 0.074 0.070 0.060 0.074 0.064
6 K=50 0.064 0.070 0.048 0.050 0.070 0.050 0.066
K=100 0.060 0.078 0.048 0.068 0.058 0.042 0.052
K=150 0.062 0.074 0.056 0.062 0.056 0.044 0.060
mVC 0.126 0.160 0.132 0.154 0.134 0.152 0.124
mMSE 0.140 0.152 0.146 0.146 0.154 0.142 0.152
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.002 0.000 0.000 0.000 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 0.078 0.082 0.068 0.076 0.076 0.054 0.070
: Empirical sizes comparison for Cases 4-6 in Example \[example2\].
\[table5\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.
| 2,174
| 5,321
| 590
| 1,729
| null | null |
github_plus_top10pct_by_avg
|
{j}$ if $L_{j}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
- $v_i$ and $y_i$ are blocks of $m_{i, i}$ as explained in the description of an element of $\tilde{M}(R)$ above, and $\bar{\gamma}_i$ is as explained in Remark \[r33\].(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* $\textit{I}^e$).
4. If $i$ is odd and $L_i$ is *of type II* or *bound of type I*, $m_{i,i}=\mathrm{id}$ mod $\pi \otimes 1$.
5. If $i$ is odd and $L_i$ is *free of type I*, $s_i=\mathrm{id}$ mod $\pi \otimes 1$.\
It is obvious that $\mathrm{Ker~}\tilde{\varphi}$ is a closed subgroup scheme of the unipotent radical $\tilde{M}^+$ of $\tilde{M}$ and is smooth and unipotent since it is isomorphic to an affine space as an algebraic variety over $\kappa$.
For completeness of the content in this section, we repeat the argument written in from the last paragraph of page 502 to page 503 in [@C2].
Recall from Remark \[r31\] that we defined the functor $\underline{M}^{\prime}$ such that $(1+\underline{M}^{\prime})(R)=\underline{M}(R)$ inside $\mathrm{End}_{B\otimes_AR}(L \otimes_A R)$ for a flat $A$-algebra $R$. Thus there is an isomorphism of set valued functors $$1+ : \underline{M}^{\prime} \longrightarrow \underline{M}, ~~~ m\mapsto 1+m,$$ where $m\in \underline{M}^{\prime}(R)$ for a flat $A$-algebra $R$. We define a new operation $\star$ on $\underline{M}^{\prime}(R)$ such that $x\star y=x+y+xy$ for a flat $A$-algebra $R$. Since $\underline{M}^{\prime}(R)$ is closed under addition and multiplication, it is also closed under the new operation $\star$. Moreover, it has $0$ as an identity element with respect to $\star$. Thus $\underline{M}^{\prime}$ may and shall be considered as a scheme of monoids with $\star$. We claim that the above morphism $1+$ is an isomorphism o
| 2,175
| 814
| 1,892
| 2,113
| 2,811
| 0.776802
|
github_plus_top10pct_by_avg
|
le {\displaystyle (\psi,\phi_{j-})_{\mu}^{(+)}=-\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j-})_{\mu}^{(-)}}}\\
b_{j}={\displaystyle \left((\psi,\phi_{j+})_{\mu}^{(+)}+\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j+})_{\mu}^{(-)}\right)}\end{array}\label{Eqn: FRBC1}$$
where $(\phi_{j\pm})_{j=1}^{N}$ represents $(\phi_{\varepsilon}(\mu,\pm\nu_{j}))_{j=1}^{N}$, $\phi_{0\pm}=\phi(\mu,\pm\nu_{0})$, the $(+)$ $(-)$ superscripts are used to denote the integrations with respect to $\mu\in[0,1]$ and $\mu\in[-1,0]$ respectively, and $(f,g)_{\mu}$ denotes the usual inner product in $[-1,1]$ with respect to the full range weight $\mu$. While the first set of $N+1$ equations give $b_{i}^{-}$, the second set produces the required $b_{j}$ from these ‘"negative‘" coefficients. By equating these calculated $b_{i}$ with the exact half-range expressions for $a(\nu)$ with respect to $W(\mu)$ as outlined in Appendix A4, it is possible to find numerical values of $z_{0}$ and $X(-\nu)$. Thus from the second of Eq. (\[Eqn: Constant\_Coeff\]), $\{ X(-\nu_{i})\}_{i=1}^{N}$ is obtained with $b_{i\textrm{B}}\textrm{ }=a_{i\textrm{B}}$, $i=1,\cdots,N$, which is then substituted in the second of Eq. (\[Eqn: Milne\_Coeff\]) with $X(-\nu_{0})$ obtained from $a_{\textrm{A}}(\nu_{0})$ according to Appendix A4, to compare the respective $a_{i\textrm{A}}$ with the calculated $b_{i\textrm{A}}$ from (\[Eqn: FRBC1\]). Finally the full-range coefficients of Problem A can be used to obtain the $X(-\nu)$ values from the second of Eqs. (\[Eqn: Milne\_Coeff\]) and compared with the exact tabulated values as in Table \[Table: X-function\]. The tabulated values of $cz_{0}$ from Eq. (\[Eqn: extrapolated\]) show a consistent deviation from our calculations of Problem A according to $a_{\textrm{A}}(\nu_{0})=-\exp(-2z_{0}/\nu_{0})$. Since the $X(-\nu)$ values of Problem A in Table \[Table: X-function\] also need the same $b_{0\textrm{A}}$ as input that was used in obtaining $z_{0}$, it is reasonable to conclude that the ‘"exact‘" numerical integration of $z_{0
| 2,176
| 986
| 2,694
| 2,070
| 2,682
| 0.777902
|
github_plus_top10pct_by_avg
|
, \\
&=& \Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi}
-\phi\diamond a,\operatorname{ad}_{\MM{u}}\MM{w}
\Bigg\rangle
+ \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle , \\
& = & \Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\operatorname{ad}_{\MM{u}}\MM{w}\Bigg\rangle
+ \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle
= \Bigg\langle -\operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}}
+ {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle.\end{aligned}$$
These Euler-Poincaré equations with advected quantities cover all conservative fluid equations which describe the advection of material. For a large collection of examples, see Holm *et al.* (1998).
Example: Incompressible Euler equations
---------------------------------------
As an example, consider the reduced Lagrangian for the incompressible Euler equations $$\ell[\MM{u},\rho,p] = \int_\Omega \frac{\rho|\MM{u}|^2}{2} +
p(1-\rho)\,{\mathrm{d}}V(\MM{x}) \,.$$ Here $\rho(\MM{x},t)$ is the ratio of the local fluid density to the average density over $\Omega$; this is governed by the continuity equation (\[advecD\]). The pressure $p$ is a Lagrange multiplier that fixes the incompressibility constraint $\rho=1$. The variational derivatives in this case are $${\frac{\delta \ell}{\delta \MM{u}}} = \rho\MM{u}, \qquad {\frac{\delta \ell}{\delta \rho}} =
\frac{|\MM{u}|^2}{2}-p,\qquad {\frac{\delta \ell}{\delta p}} =1-\rho\,.$$ Consequently, the Euler-Poincaré equations become $$\begin{aligned}
(\rho\MM{u})_t + (\MM{u}\cdot\nabla)(\rho\MM{u}) +
\rho\MM{u}(\nabla\cdot\MM{u}) + \rho(\nabla\MM{u})^T\cdot \MM{u} &=&
\rho\nabla\left(\frac{|\MM{u}|^2}{2} -p\right), \\
\rho_t+\nabla\cdot(\rho\MM{u}) &=& 0, \\ \qquad \rho&=&1,\end{aligned}$$ and rearrangement gives the Euler fluid equations, $$\MM{u}_t + (\MM{u}\cdot\nabla)\MM{u} = -\nabla p, \qquad \nabla\cdot\MM{u}=0.$$
Inverse map multisymplectic formulation for EPDiff($H^1$) {#inverse map EPDiff}
=====================================
| 2,177
| 1,915
| 2,036
| 2,097
| null | null |
github_plus_top10pct_by_avg
|
------------------------------ --
Finally, we discuss the dipole state in $^{30}$Ne. Compared to the response functions in $^{26}$Ne and $^{28}$Ne, that for $^{30}$Ne is quite different, because this nucleus is well deformed as shown in Table \[GS\]. The giant resonance is split into $K^{\pi}=0^{-}$ and $1^{-}$ mode, and the split giant resonance has an overlap with the low-lying resonance below 10 MeV. In the right panel of Fig. \[28Ne\_strength\], we show the strength distribution below 10 MeV in $^{30}$Ne. For the $K^{\pi}=0^{-}$ mode, we can see a prominent peak at 8.1 MeV possessing a large isovector $E1$ strength of 0.48 $e^{2}$fm$^{2}$. This state is mainly generated by $\nu[211]1/2 \to [310]1/2$ (38.4%) and the neutron excitation from $[330]1/2$ to the non-resonance continuum state (37.9%), together with the proton excitation of $\pi[330]1/2 \to [220]1/2$ (6.1%).
In Fig. \[30Ne\_trans\_density\], the transition density for the $K^{\pi}=0^{-}$ state is shown. The transition density of protons are quite similar to that in Fig. \[trans\_density\]. For the neutrons, we can easily see the effect of mixing of the excitation into the continuum state; the transition density has large spatial extension. Furthermore, comparing to Fig. \[26Ne\_wf\], this $K^{\pi}=0^{-}$ state still possesses a structure similar to the low-lying dipole state in $^{26}$Ne.
----- --------------- --------------- ------------------------ -------------------------------------------- ----------------------
$E_{\alpha}+E_{\beta}$ $Q_{11,\alpha\beta}$
$\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm)
(a) $\nu[312]3/2$ $\nu[200]1/2$ 6.87 0.676 0.207
(b) $\nu[310]1/2$ $\nu[200]1/2$ 7.09 0.089 0.141
(c) $\nu[321]3/2$
| 2,178
| 1,792
| 2,672
| 2,128
| null | null |
github_plus_top10pct_by_avg
|
p. 306] the left-hand side of is also zero and hence $\beta_{\cD,P}^{(k')}=0$ for all $0\leq k'< \deg \cD$.
Finally, combining properties \[prop:4\], \[prop:5\], and \[prop:6\] in Lemma \[lemma:global-realization\] together with Step \[step1\] above, it immediately follows that $\beta_{\cC,P}^{(k)}=\beta_{\cD,P}^{(k')}$, for $k'=k\frac{\deg \cD}{\deg \cC}\in \ZZ$, since $k'\equiv k \mod (w_2)$, $\deg \cD_j\equiv \deg \cC_j=d_j \mod (w_2)$ for all $j=1,...,r$, and $\frac{k}{\deg \cC}=\frac{k'}{\deg \cD}$.
As a first example we can study the $6$-fold cyclic cover $\tilde X$ of $\PP^2$ ramified along the non-reduced divisor $\cC=\cC_1+2\cC_2+3\cC_3$, where $\cC_1,\cC_2,\cC_3$ are any three concurrent lines. Note that $$\cM(\cC^{\frac{k}{6}})=
\begin{cases}
\mathfrak{m}=(x,y) & \text{ for } k=5\\
\CC\{x,y\} & \text{ otherwise.}\\
\end{cases}$$ By Theorem \[thm:conucleo\_singular\] one needs to consider the morphism: $$\pi^{(k)}=
H^0(\PP^2,\cO_{\PP^2}(s_k-3)) \to \frac{\cO_{\CC^2,0}}{\cM(C^{\frac{k}{6}})}=
\begin{cases}
0 & \text{ if } k=0,...,4\\
\CC & \text{ if } k=5.\\
\end{cases}$$ It is also straightforward to check that $s_5=2$. Hence $\operatorname{coker}\pi^{(5)}=\CC$ is the only non-trivial cokernel and thus $\dim H^1(\tilde X,\CC)=2$.
\[lemma:global-realization\] Let $(\cC,P)$ be the topological type of a curve singularity in a cyclic quotient surface of normalized type $\frac{1}{w_2}(\bar w_0,\bar w_1)$. Then there exists a global curve $\cD\subset \PP^2_w$, $w=(w_0,w_1,w_2)$, in a weighted projective plane such that:
1. \[prop:1\] $(\PP^2_w,O=[0:0:1])$ is a surface singularity of type $\frac{1}{w_2}(\bar w_0,\bar w_1)$,
2. \[prop:2\] $(\cC,P)$ and $(\cD,O)$ have the same topological type,
3. \[prop:3\] One obtains $\cD\cap \operatorname{Sing}\PP^2_w \subseteq\operatorname{Sing}\cD=\{O\}$.
Moreover, if $\cC=\cC_1\cup...\cup\cC_r$ is already a global curve in some weighted projective plane $\PP^2_w$ and $P\in \cC$ is as above with $(\PP,P)=\frac{1}{w_2}(\bar w_0,\bar w_1)$, then for any inte
| 2,179
| 1,401
| 786
| 1,992
| null | null |
github_plus_top10pct_by_avg
|
ght)^{1/6}
\right],\end{aligned}$$ where in the second inequality we have used the fact that $\mathbb{P}(\mathcal{E}^c_n) \leq \frac{1}{n}$ by and have absorbed this lower order term into higher order terms by increasing the value of $C$. By a symmetric argument, we have $$\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t) \geq
\mathbb{P}( ||Z_n||_\infty \leq t) -C \left [\frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b} +2) +
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6}
\right].$$ The result now follows by bounding $\epsilon_n$ as in . $\Box$
The following lemma shows that the linear term $\sqrt{n}(\hat\nu - \nu)$ in has a Gaussian-like behavior and is key ingredient of our results. It is an application of the Berry-Esseen , due to [@cherno2]. The proof is in .
\[lem:hyper\] There exists a constant $C>0$, depending on $A$ only, such that $$\sup_{P\in {\cal P}}
\sup_t
\Bigl|\mathbb{P}(\sqrt{n}||\hat\nu - \nu||_\infty \leq t) -
\mathbb{P}(||Z_n||_\infty \leq t)\Bigr| \leq C
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6},$$ where $Z_n \sim N_s(0,\Gamma)$.
[**Proof of Lemma \[lemma::upsilon\].**]{} Throughout the proof, we set $G = G(\psi)$, where $\psi = \psi(P)$ for some $P \in \mathcal{P}_n$, and $\hat{G} = G(\hat{\psi})$ where $\hat{\psi} = \hat{\psi}(P)$ is the sample average from an i.i.d. sample from $P$. Recall that the matrices $\Gamma$ and $\hat{\Gamma}$ are given in Equations and , respectively. For convenience we will suppress the dependence of $\hat{\Gamma}$ and $\hat{G}$, and of $\Gamma$ and $G$ on $\hat{\psi}$ and $\psi$, respectively.
Express $\hat\Gamma - \Gamma$ as $$\begin{aligned}
(\hat G - G) V G^\top + G V (\hat G - G)^\top + & (\hat G - G) V (\hat G - G)^\top
+\\
(\hat G - G)(\hat{V} - V) G^\top + & G (\hat{V} - V) ( \hat{G}- G)^\top + G (\hat
V - V)G^\top + (\hat{G} - G) (\hat{V} - V)^\top (\hat{G} - G )^\top.\end{aligned}$$ The first, second and sixth terms are dominant, so it will be enough
| 2,180
| 1,369
| 1,572
| 1,943
| null | null |
github_plus_top10pct_by_avg
|
mod8$.
\[ab0\] Suppose $n\equiv3\ppmod8$, and $a$ is even, with $6\ls a\ls n-7$. Let $b=n-a-3$. Then $S^{(a,3,1^b)}$ has an irreducible summand of the form $S^{(u,v,2)}$.
Using Theorem \[main\], we need to show that there is a pair $u,v$ with $u+v+2=n$ such that $S^{(u,v,2)}$ is irreducible and $\mbinom{u-v}{u-a}$ is odd. By Theorem \[irrspecht\], $(u,v,2)$ is irreducible if and only if $$v\equiv1\pmod4,\qquad u-v\equiv-1\pmod{2^{l(v-2)}},$$ where $l(k)=\lceil\log_2(k+1)\rceil$ for an integer $m$.
We use induction on $n$, with our main tool being the following well-known relations modulo $2$ on binomial coefficients: $$\binom{2x}{2y}\equiv\binom{2x+1}{2y}\equiv\binom{2x+1}{2y+1}\equiv\binom xy,\qquad \binom{2x}{2y+1}\equiv0\pmod 2.$$ We consider three cases.
: In this case, take $v=5$ (so $u=n-7$). Since $n\equiv3\ppmod4$, we get $u\equiv0\ppmod4$, which means that $u-v\equiv3\ppmod4$, so $S^{(u,v,2)}$ is irreducible. Furthermore, the binomial coefficients $$\binom{u-5}0,\binom{u-5}1,\binom{u-5}2,\binom{u-5}3$$ are all odd, which means that $\mbinom{u-5}{u-a}$ must be odd.
: In this case, let $$n'=\frac{n+11}2,\quad a'=
\begin{cases}
\mfrac{a+6}2&(a\equiv2\ppmod4)\\[5pt]
\mfrac{a+4}2&(a\equiv0\ppmod4).
\end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$ (note that the conditions on $a$ mean that $n>11$). So by induction there is a pair $u',v'$ such that $$v'\equiv1\pmod4,\qquad u-v\equiv-1\pmod{2^{l(v'-2)}},\qquad\mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Note that because $u'-v'$ is odd and $u'-a'$ is even, this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd.
We let $u=2u'-4$ and $v=2v'-5$. Then $u+v+2=n$, and we have $v\equiv1\ppmod4$ and $$u-v=2(u'-v')+1\equiv-1\ppmod{2^{l(v'-2)+1}},$$ with $l(v-2)\ls l(v'-2)+1$. So $S^{(u,v,2)}$ is irreducible. Furthermore $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2,$$ and we are done.
: In this case, let $$n'=\frac{n+3}2,\quad a'=
\begin{cases}
\mfrac{a+2}2&(a\equiv2\p
| 2,181
| 3,147
| 1,775
| 1,986
| 1,818
| 0.785452
|
github_plus_top10pct_by_avg
|
}\,\langle X\rangle$ be a free associative dialgebra [@Loday:01]. Products in ${\mathrm{Alg}}\,\langle X\rangle$ and ${\mathrm{As}}\,\langle X\rangle$, also in ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ and ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ are denoted identically. There is no confusion because by the origin of elements it is clear which the product we mean. Fix $z\in X$ and introduce the following mappings.
A mapping $\mathcal{J}\colon{\mathrm{Alg}}\,\langle X\rangle\to {\mathrm{As}}\,\langle
X\rangle$ is defined by linearity, on non-associative words it is defined by induction on a length: if $x\in X$ then $\mathcal{J}(x)=x$; if $uv\in {\mathrm{Alg}}\,\langle X\rangle$ then $\mathcal{J}(uv)=\frac{1}{2}(\mathcal{J}(u)\mathcal{J}(v)+\mathcal{J}(v)\mathcal{J}(u))$. So, the value of $\mathcal{J}$ on a non-associative polynomial $f$ is equal to an associative polynomial obtained from $f$ by means of rewriting all products in $f$ as Jordan ones by the formula (\[eq:JordanProduct\]). By analogy, in the case of dialgebras a mapping $\mathcal{J}_{{\mathrm{Di}}}\colon{\mathrm{Di}}{\mathrm{Alg}}\,\langle
X\rangle\to {\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ is defined. It is linear, it acts identically on $x\in X$ and
$$\begin{gathered}
\mathcal{J}_{{\mathrm{Di}}}(u{\mathbin\vdash}v)=\frac{1}{2}(\mathcal{J}_{{\mathrm{Di}}}(u){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(v)+\mathcal{J}_{{\mathrm{Di}}}(v){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(u)),\\
\mathcal{J}_{{\mathrm{Di}}}(u{\mathbin\dashv}v)=\frac{1}{2}(\mathcal{J}_{{\mathrm{Di}}}(u){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v)+\mathcal{J}_{{\mathrm{Di}}}(v){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(u)).
\end{gathered}$$
Introduce the following notation $${\mathrm{Alg}}_z\,\langle X\rangle=\{\Phi\in
{\mathrm{Alg}}\,\langle X\rangle \mid \Phi=\sum f_i,\ f_i\text{~--- monomials,
}\deg_z f_i = 1\},$$ $${\mathrm{Di}}{\mathrm{Alg}}_z\,\langle X\rangle=\{\Phi\in
{\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle \mid \Phi=\sum f_i,\ f_i\text{~---
dim
| 2,182
| 803
| 2,086
| 2,029
| null | null |
github_plus_top10pct_by_avg
|
urthermore its peak high is increasingly reduced due to the destruction of the Kondo effect in a strong effective magnetic field. Therefore, the spectral properties shown in Fig. \[NRG\_fig-ph-asymmetry\] are consistent with those of an effective Anderson model in the attractive $U$ regime.
### Inelastic contributions
After reviewing the present understanding of the electronic spectral function in the Anderson Holstein model [@HewsonMeyer02] in its anti-adiabatic, particle-hole symmetric as well as particle-hole anti-symmetric regime, we proceed by discussing the implications for a potential STS including elastic and inelastic contributions. We assume for simplicity that the STM tip only couples to the molecular orbital excluding Fano physics. In order to eliminate the coupling parameters that need to be adjusted for a specific experimental setup, we define the following two spectral functions $$\begin{aligned}
\bar\rho^{(2)}(\w) &=& \frac{1}{(t_d \lambda^{\rm tip})^2} \tau^{(2)}(\w)\\
\bar\rho^{(1)}(\w)& = &\frac{1}{t^2_d \lambda^{\rm tip}} \tau^{(1)}(\w)\end{aligned}$$ that contain both inelastic terms. This eliminates the STM tip dependent prefactor and focuses only on the spectral features.
![All contributions to the STS spectra in the anti-adiabatic regime. (a) spectral function taken from Fig. \[NRG\_fig15\], (b) $\bar\rho^{(2)}(\w)$ for $\lambda_d$ stated in panel (a) and (c) $\bar\rho^{(1)}(\w)$ for $\lambda_d$ stated in panel (a). Parameters as in Fig. \[NRG\_fig15\]. []{data-label="NRG_fig16"}](fig25-Spectra_Ld){width="50.00000%"}
The individual spectra contributing to the total STS are shown in Fig. \[NRG\_fig16\] for the different coupling constants $\lambda_d$. Panel (a) includes some of the data contained already in Fig. \[NRG\_fig15\] for comparison. Panel (b) of Fig. \[NRG\_fig16\] depicts the contribution to $\bar\rho^{(2)}(\w)$. We observe the same narrowing of the distance between the two peaks when increasing $\lambda_d$ as plotted in Fig. \[NRG\_fig3\]. Note however, that the data
| 2,183
| 219
| 2,380
| 2,341
| 1,796
| 0.785653
|
github_plus_top10pct_by_avg
|
24% 228 23% 167 14%
CDG Definition Landriel Definition
Grade 1+2 67% 139 61% 53 32% Grade 1 Any non-life threatening deviation from normal postoperative course, not requiring invasive treatment
Grade 1 Any deviation from the normal postoperative course without the need for pharmacological treatment or surgical, endoscopic, and radiological interventions. Allowed therapeutic regimens are drugs as antiemetics, antipyretics, analgetics, diuretics, electrolytes, and physiotherapy. This grade also includes wound infections opened at the bedside. 72 22%
Grade 2 Requiring pharmacological treatment with drugs other than such allowed for grade I complications. Blood transfusions and total parenteral nutrition are also included. 145 45%
Grad
| 2,184
| 6,863
| 1,283
| 447
| null | null |
github_plus_top10pct_by_avg
|
$u$]{} (4891,-113)[$1$]{} (3226,-1823)[$1$]{} (2498,-1809)[$P$]{} (3481,-1823)[$1/P$]{}
The shape of the faces $(23)$ is also a hyperbolic pentagon, the faces $(12)$ and $(45)$ are hyperbolic quadrilaterals with three right angles, and the faces $(61)$ and $(56)$ are hyperbolic right triangles. Note that when $P=1, Q<1, R<1$, the hexahedra has an ideal vertex at $(1,0,0)$. This means that the faces $(61)$ and $(56)$ do not meet, but they are tangent at $(1,0,0)$, and also $(23)$ and $(34)$ are tangent at $(1,0,0)$. So the face $(61)$ and the face $(56)$ are hyperbolic right triangles with one ideal vertex, the face $(23)$ and the face $(34)$ quadrilaterals with one ideal vertex and three right angles. The faces $(12)$ and $(45)$ are as in the case $(a)$ or $(b)$.
Now conversely when three numbers $P$, $Q$, $R >0$ are given, by taking the inverse procedure above, we can construct a unique hyperbolic hexahedron in ${\cal H}$. That is, its combinatorial type is determined by the signs of $P-1$, $Q-1$, $R-1$, and the lengths of the edges on the $u$, $v$, $w$-axes are $p$, $q$, $r$, respectively. Since each face is either a hyperbolic right pentagon, a quadrilateral with three right angles (possibly with one ideal vertices), or a hyperbolic right triangle (possibly with ideal vertices), all the other lengths of the hexahedron are uniquely determined and the vertex where $(61)$, $(23)$ and $(45)$ meet is at finite distance. Therefore ${\cal H}$ can be identified with the set $$\{P, Q, R \, | \, P, Q, R >0\}.$$ Following [@AharaYamada], we shall show a geometric way of computing the parameters $P$, $Q$, $R$ for the hyperbolic hexahedron $\Delta_{p,\theta}$ as in the case $n=5$.
Recall that for $p=\langle i_1i_2i_3i_4i_5i_6\rangle$ the space ${\cal E}_{p,\theta}$ of hexagons with external angles $\theta_{i_1}$, $\ldots$, $\theta_{i_6}$ cyclically is a Lorentz space with coordinates $(x, u, v, w)$ given by $$x = \sqrt{{\operatorname{Area}}T_0}, \quad
u = \sqrt{{\operatorname{Area}}T_1}, \quad
v = \sqrt{{\operatornam
| 2,185
| 3,794
| 2,213
| 2,042
| null | null |
github_plus_top10pct_by_avg
|
$\pi$ as a factor be zero so that this equation is considered in $R$. Note that $\bar{h}_i$ and $\bar{h}_j$ are invertible as matrices with entries in $R$ by Remark \[r33\]. Thus $m_{i,j}'=\bar{h}_i^{-1}\cdot {}^tm_{j,i}'\cdot \bar{h}_j$. This induces that each entry of $m_{i,j}'$ is expressed as a linear combination of the entries of $m_{j,i}'$.\
When $i$ is odd and $L_i$ is *bound of type I*, we consider the above equation (\[ea3-\]) again because of the appearance of $m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}, f_{i,i}^{\ast}$. We rewrite $f_{i-1,i}'$ and $f_{i,i+1}'$ formally as follows: $$f_{i-1,i}'=\left(\sigma(1+\pi\cdot {}^tm_{i-1,i-1}')h_{i-1}\pi m_{i-1,i}'+\sigma(\pi\cdot {}^tm_{i,i-1}')h_i(1+\pi m_{i,i}')+\pi^3(\ast))\right),$$ $$f_{i, i+1}'=\left(\sigma(1+\pi\cdot {}^tm_{i,i}')h_i\pi m_{i,i+1}'+\sigma(\pi\cdot {}^tm_{i+1,i}')h_{i+1}(1+\pi m_{i+1,i+1}')+\pi^3(\ast\ast))\right).$$ Here, $(\ast)$ and $(\ast\ast)$ are certain formal polynomials with coefficients $\pi^3$, not $\pi^2$, since $m_{j, j'}=\pi m_{j,j'}^{\prime}$ when $j\neq j'$. We consider the following equations formally (cf. the second paragraph of Remark \[r35\]): $$\left\{
\begin{array}{l}
\pi (f_{i, i}^{\ast})'=\delta_{i-1}(0,\cdots, 0, 1)\cdot f_{i-1,i}'+\delta_{i+1}(0,\cdots, 0, 1)\cdot f_{i+1,i}';\\
\sigma({}^tf_{i, i+1}')=f_{i+1, i}',\\
\end{array}
\right.$$ where $(f_{i, i}^{\ast})'$ is a matrix of size $(1 \times n_i)$ with entries in $B\otimes_AR$. The above formal equation involving $(f_{i, i}^{\ast})'$ should be interpreted as follows (cf. Remark \[r35\]). We formally compute the right hand side by using the above formal expansions of $f_{i-1,i}'$ and $f_{i, i+1}'$, the equation $\sigma({}^tf_{i, i+1}')=f_{i+1, i}'$, and two formal equations (\[ea72\]) involving $(m_{i,i}^{\ast})'$ and $(m_{i,i}^{\ast\ast})'$. We then get the following: $$\pi (f_{i, i}^{\ast})'=\pi^2\left((m_{i,i}^{\ast})'-(m_{i,i}^{\ast\ast})'h_i+\dag\right)$$ after letting each term having $\pi^3$ as a factor be zero since $(f_{i, i}^{\ast})'$ is a matrix w
| 2,186
| 399
| 1,156
| 2,373
| 3,550
| 0.771596
|
github_plus_top10pct_by_avg
|
ed by $$\begin{gathered}
\left\{
\begin{aligned}
{\varepsilon }(K_i)=&\,1,\quad {\varepsilon }(E_i)=0, &
{\varepsilon }(L_i)=&\,1,\quad {\varepsilon }(F_i)=0, \\
{\varDelta }(K_i)=&\,K_i\otimes K_i,&
{\varDelta }(L_i)=&\,L_i\otimes L_i,\\
{\varDelta }(K_i^{-1})=&\,K_i^{-1}\otimes K_i^{-1},&
{\varDelta }(L_i^{-1})=&\,L_i^{-1}\otimes L_i^{-1},\\
{\varDelta }(E_i)=&\,E_i\otimes 1+K_i\otimes E_i,&
{\varDelta }(F_i)=&\,1\otimes F_i+F_i\otimes L_i
\end{aligned}
\right.
\label{eq:coprcU}\end{gathered}$$ for all $i\in I$.
Let ${\mathcal{U}}^{+0},{\mathcal{U}}^{-0}$, and ${{\mathcal{U}}^0}$ denote the commutative cocommutative Hopf subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{K_i,K_i^{-1}\,|\,i\in I\}$, $\{L_i,L_i^{-1}\,|\,i\in I\}$, and $\{K_i,K_i^{-1},L_i,L_i^{-1}\,|\,i\in I\}$, respectively. They are isomorphic to the ring of Laurent polynomials in $|I|$, $|I|$, and $2|I|$ variables, respectively, in the natural way. For any ${\alpha }=\sum _{i\in I}m_i{\alpha }_i\in {\mathbb{Z}}^I$ let $K_{\alpha }=\prod _{i\in I}
K_i^{m_i}$ and $L_{\alpha }=\prod _{i\in I}L_i^{m_i}$.
Let ${\mathcal{U}}^+(\chi )$, ${\mathcal{V}}^+(\chi )$, ${\mathcal{U}}^-(\chi )$, and ${\mathcal{V}}^-(\chi )$ denote the subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{E_i\,|\,i\in I\}$, $\{E_i,K_i,K_i^{-1}\,|\,i\in I\}$, $\{F_i\,|\,i\in I\}$, and $\{F_i,L_i,L_i^{-1}\,|\,i\in I\}$, respectively. Then ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ are Hopf subalgebras of ${\mathcal{U}}(\chi )$.
The algebra ${\mathcal{U}}(\chi )$ admits a unique ${\mathbb{Z}}^I$-grading $$\begin{gathered}
{\mathcal{U}}(\chi )=\oplus _{\beta \in {\mathbb{Z}}^I}{\mathcal{U}}(\chi )_\beta ,\\
1\in {\mathcal{U}}(\chi )_0,\quad
{\mathcal{U}}(\chi )_\beta {\mathcal{U}}(\chi )_\gamma \subset
{\mathcal{U}}(\chi )_{\beta +\gamma } \quad
\text{for all $\beta ,\gamma \in {\mathbb{Z}}^I$,}
\end{gathered}
\label{eq:Zngrading}$$ such that $K_i,K_i^{-1},L_i,L_i^{-1}\in {\mathcal{U}}(\chi )_0$, $E_i\in {\math
| 2,187
| 1,531
| 1,977
| 1,889
| null | null |
github_plus_top10pct_by_avg
|
ng to associated graded objects that maps $U_c{\text{-}{\textsf}{mod}}$ precisely to $\operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$.
{#intro-1.3}
We take our cue from the theory of semisimple Lie algebras. When $n=2$, $U_c$ is isomorphic to a factor of $U(\mathfrak{sl}_2)$ [@EG Section 8] and, for all $n$, the properties of $U_c$ are similar to those of $U(\mathfrak{g})/P$, where $P$ is a minimal primitive ideal in the enveloping algebra of a complex semisimple Lie algebra $\mathfrak{g}$ (see, for example [@ginz; @GGOR; @guay]). The intuition from the last paragraph not only holds for enveloping algebras but can also be formalised through the Beilinson-Bernstein equivalences of categories. This gives a diagram $$\begin{CD}
D_{\mathcal{B}} @< \sim << U(\mathfrak{g})/P \\
@V \operatorname{gr}VV @VV \operatorname{gr}V \\
{\mathcal{O}}_{T^*\mathcal{B}} @<\tau << {\mathcal{O}}(\mathcal{N})
\end{CD}$$ where $\mathcal{B}=G/B$ is the flag variety, the primitive ideal $P$ has trivial central character and $\tau: T^*\mathcal{B}\to \mathcal{N}$ is the Springer resolution of the nullcone $\mathcal{N}$. The Morita equivalence from the sheaf of differential operators $D_{\mathcal{B}}$ to $ U(\mathfrak{g})/P$ is obtained by taking global sections under the identification $U(\mathfrak{g})/P\cong D(\mathcal{B})$.
Ginzburg has raised the question of whether a similar phenomenon holds for Cherednik algebras (see [@GK Conjecture 1.6] for a variant on this conjecture). In other words, can one complete the following diagram? $$\begin{CD} ? @< \sim << U_c \\
@V \operatorname{gr}VV @VV \operatorname{gr}V \\
{\mathcal{O}}_{\operatorname{Hilb(n)}} @< \tau<<{\mathcal{O}}({\mathfrak{h}}\oplus {\mathfrak{h}}^*/{{W}})
\end{CD}$$
The main result of the paper gives a positive answer to this question. Given a graded ring $R$, we write $R{\text{-}{\textsf}{qgr}}$ for the quotient category of noetherian graded $R$-modules modulo those of finite length.
Main Theorem {#mainthm-intro}
------------
*There exists a graded ring $B$
| 2,188
| 1,843
| 819
| 2,224
| 2,664
| 0.778026
|
github_plus_top10pct_by_avg
|
ge with Conditional Format code
I'm very new to VBA (and any sort of programming in general), so I'm not sure how to proceed here. I'm guessing my error has something to do with overlapping ranges for my conditional formats as I also got errors when the code was set up a different way that were resolved once the ranges no longer overlapped. That might not be the case here, but I figured it'd be helpful to know.
I get a 'Subscript out of range' error with the following code:
Sub test2()
Dim rngToFormat As Range
Set rngToFormat = ActiveSheet.Range("$a$1:$z$1000")
Dim rngToFormat2 As Range
Set rngToFormat2 = ActiveSheet.Range("$k$20:$k$1000")
Dim rngToFormat3 As Range
Set rngToFormat3 = ActiveSheet.Range("$j$22:$j$1000")
Dim rngToFormat4 As Range
Set rngToFormat4 = ActiveSheet.Range("$i$22:$i$1000")
Dim rngToFormat5 As Range
Set rngToFormat5 = ActiveSheet.Range("$g$20:$g$1000")
Dim rngToFormat6 As Range
Set rngToFormat6 = ActiveSheet.Range("$d$9, $f$9")
Dim rngToFormat7 As Range
Set rngToFormat7 = ActiveSheet.Range("$G$3:$G$7,$G$11:$G$15,$E$3:$E$7,$E$11:$E$15,$N$3:$N$7,$N$11:$N$15,$L$3:$L$7,$L$11:$L$15")
rngToFormat.FormatConditions.Delete
rngToFormat.FormatConditions.Add Type:=xlExpression, _
Formula1:="=if(R[]C20=1, true(), false())"
rngToFormat.FormatConditions(1).Font.Color = RGB(228, 109, 10)
rngToFormat2.FormatConditions.Add Type:=xlExpression, _
Formula1:="=and(R[]C7=""6. Negotiate"", R[]C11<25)"
rngToFormat2.FormatConditions(2).Font.ColorIndex = 3
rngToFormat2.FormatConditions.Add Type:=xlExpression, _
Formula1:="=and(R[]C7=""4. Develop"", R[]C11<15)"
rngToFormat2.FormatConditions(3).Font.ColorIndex = 3
rngToFormat2.FormatConditions.Add Type:=xlExpression, _
Formula1:="=and(R[]C7=""5. Prove"", R[]C11<20)"
rngToFormat2.FormatConditions(4).Font.ColorIndex = 3
rngToFormat2.FormatConditions.Add Type:=xlExpression, _
Formula1:="=and(R[]C7=""7. Committed"", R[]C
| 2,189
| 6,186
| 99
| 1,505
| 27
| 0.836377
|
github_plus_top10pct_by_avg
|
2m+1}$$ $$\mathrm{and}~ M_k=L_{2m+k} \mathrm{~if~} k\geq 2.$$ Here, $M_i$ is $\pi^i$-modular. We caution that the hermitian form we use on $L^{2m}$ is not $h$, but its rescaled version $\xi^{-m}h$. Thus $M_i$ is $\pi^i$-modular, not $\pi^{2m+i}$-modular.
\[d49\] We define $C(L)$ to be the sublattice of $L$ such that $$C(L)=\{x\in L \mid h(x,y) \in (\pi) \ \ \mathrm{for}\ \ \mathrm{all}\ \ y \in B(L)\}.$$
We choose any even integer $j=2m$ such that $L_{j}$ is *of type I* and $L_{j+2}, L_{j+3}, L_{j+4}$ are *of type II* (possibly zero by our convention), and consider the Jordan splitting $$L^{j}=\bigoplus_{i \geq 0} M_i$$ defined above. The reason that we require $L_{j+2}, L_{j+3}, L_{j+4}$ to be *of type II* is explained in Step (1) which will be stated below. We stress that $$\left\{
\begin{array}{l }
\textit{$M_0$ is nonzero and \textit{of type I}, since it contains $L_{j}$ as a direct summand};\\
\textit{$M_1$ is \textit{bound of type I}, and all of $M_2 (=L_{j+2}), M_3 (=L_{j+3}), M_4 (=L_{j+4})$ are \textit{of type II}.}
\end{array} \right.$$
That $M_1$ is *bound of type I* does not guarantee that the norm of $M_1$ (=$n(M_1)$) is the ideal $(4)$ since $M_1=\pi^{j/2}L_1\oplus\pi^{j/2-1}L_3\oplus \cdots \oplus \pi L_{j-1}\oplus L_{j+1}$. If $n(M_1)=(2)$, then we choose a suitable basis of both $M_0$ and $M_1$ such that the associated Jordan splitting for $M_0\oplus M_1$ is $M_0'\oplus M_1'$ with $n(M_1')=(4)$ (cf. Lemma 2.9 and the following paragraph in [@C2]). Thus we may and do assume that $n(M_1)=(4)$.
Choose a basis $(\langle e_i\rangle, e)$ (resp. $(\langle e_i\rangle, a, e)$) for $M_0$ so that $M_0=\bigoplus_{\lambda}H_{\lambda}\oplus K$ when the rank of $M_0$ is odd (resp. even). Here, we follow the notation from Theorem \[210\]. Note that the Gram matrix associated to $(a, e)$, when the rank of $M_0$ is even, is $\begin{pmatrix} 1&1\\1&2b\end{pmatrix}$ with $b\in A$. Then $B(L^{j})$ is spanned by $$(\langle e_i\rangle, \pi e) ~(resp.~ (\langle e_i\rangle, \pi a, e)) \textit{~and~}
| 2,190
| 1,870
| 2,231
| 1,891
| 1,742
| 0.78619
|
github_plus_top10pct_by_avg
|
frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi.$$
A. In the first instance, we assume that $g=0$. Let $\tilde P_{C,0}$ be the smallest closed extension defined above. Using the these notations, the problem (\[p-s1-b\])-(\[p-s3-b\]) is equivalent to $$\begin{aligned}
\label{eq:problem_x}
(\tilde P_{C,0}+\Sigma-K_C)\phi={\bf f} \end{aligned}$$ where $\phi\in D(\tilde P_{C,0})$. Let $$T_{C,0}:=\tilde P_{C,0}+\Sigma-K_C.$$ Since the operator $-T_{C,0}+cI$ is $m$-dissipative (see the proof of Theorem \[coth2-d\], and recall that $c>0$ is a part of the assumption ), it generates a contraction $C^0$-semigroup $G_c(t)$, (cf. [@engelnagel], [@dautraylionsv5]), and hence $-T_{C,0}$ generates a (contraction) $C^0$-semigroup $G(t)=G_c(t)e^{-ct}$, $t\geq 0$, such that ${\left\Vert G(t)\right\Vert}\leq e^{-ct}$ for all $t\geq 0$. In addition, the solution $\phi$ of is given by (see [@engelnagel Chapter II, Theorem 1.10]) \[p-s8\] =T\_[C,0]{}\^[-1]{}[**f**]{} =(0-(-T\_[C,0]{}))\^[-1]{}[**f**]{} =\_0\^G(t)[**f**]{} dt.
We decompose the operator $-T_{C,0}$ as follows \[p-s9\] -T\_[C,0]{}= B\_0+A\_0-(+CS\_0 I)+K\_C, where linear operators $B_0$ and $A_0$ are defined by $$\begin{gathered}
D(A_0):=\tilde W^2_{-,0}(G\times S\times I),
\quad A_0\phi:=-\omega\cdot\nabla_x\phi\nonumber\\
D(B_0):=\{\phi\in W_1^2(G\times S\times I)\ |\ \phi(\cdot,\cdot,E_{\rm m})=0\},
\quad B_0\phi:={{\frac{\partial (S_0\phi)}{\partial E}}}.\end{gathered}$$ The semi-groups generated by the last three components are given by (below $H$ is the Heaviside function) \[p-s10\] & T\_[K\_C]{}(t)=e\^[tK\_C]{},\
& T\_[-(+CS\_0 I)]{}(t)=e\^[-t(+CS\_0I)]{},\
& (T\_[A\_0]{}(t)f)(x,,E)=H(t(x,)-t)f(x-t,,E), and they are clearly all of positive type.
Also, the semi-group $T_{B_0}(t)$ generated by $B_0$ is of positive type. Indeed, letting $(U(t))(x,\omega,E):=(T_{B_0}(t)f)(x,\omega,E)$ for a given $f\in D(B_0)$, then $U$ satisfies the Cauchy problem $${{\frac{\partial U}{\partial t}}}-B_0U=0,\quad U(0)=f,$$ or equivalently, \[p-s11\] [U[t]{}]{
| 2,191
| 625
| 1,510
| 2,201
| 3,815
| 0.769939
|
github_plus_top10pct_by_avg
|
{l-1})
T_{l-2}(\lambda_{l-1}-\lambda_{l-2})...T_{m}(\lambda_{m+1}-\lambda_{m})T_{m-1}(\lambda_{m}-s),$$ and for $\lambda_{l-1}\leq a\leq b<\lambda_{l}$ by $$\label{promenade2}U_\Lambda (t,s):= T_{l-1}(t-s).$$ By [@LASA14 Theorem 3.2] we know that $({{U}}_\Lambda)_{\Lambda}$ converges weakly in $MR(V,V')$ as $|\Lambda|\to 0$ and $$\lim\limits_{|\Lambda|\to 0}\|{{U}}_\Lambda-{{U}}\|_{MR(V,V')}=0$$ The continuous embedding of $MR(V,V')$ into $C([0,T];H)$ implies that $\lim\limits_{|\Lambda|\to 0}{{U}}_\Lambda={{U}}$ in the weak operator topology of ${\mathcal{L}}(H).$
Now, let $(t,s)\in \overline{\Delta}$ with $\lambda_{m-1}\leq s<\lambda_m<...<\lambda_{l-1}\leq t<\lambda_{l}$ be fixed. Applying the above approximation argument to ${\mathfrak{a}}_r^*$ one obtains that $$\begin{aligned}
\label{eq1 proof returned adjoint}
{{U}}^*_{\Lambda,r}(t,s)&=T_{l-1,r}^*(t-\lambda_{l-1})
T_{l-2,r}^*(\lambda_{l-1}-\lambda_{l-2})...T_{m,r}^*(\lambda_{m+1}-\lambda_{m})T_{m-1,r}^*(\lambda_{m}-s)
\\\label{eq2 proof returned adjoint}&=T_{l-1,r}^\prime(t-\lambda_{l-1})
T_{l-2,r}^\prime(\lambda_{l-1}-\lambda_{l-2})...T_{m,r}^\prime(\lambda_{m+1}-\lambda_{m})T_{m-1,r}^\prime(\lambda_{m}-s)
$$ where $T_{k,r}$ and $T_{k,r}^*$ are the $C_0$-semigroups associated with $$\label{eq1 proof Thm equalities: adjoint EVF and EVF} {\mathfrak{a}}_{k,r}(u,v):=\frac{1}{\lambda_{k+1}-\lambda_k}\int_{\lambda_k}^{\lambda_{k+1}}{\mathfrak{a}}(T-r;u,v){\rm d}r=\frac{1}{\lambda_{k+1}-\lambda_k}\int_{T-\lambda_{k+1}}^{T-\lambda_k}{\mathfrak{a}}(r;u,v){\rm d}r$$ and its adjoint ${\mathfrak{a}}_{k,r}^*$, respectively. Recall that $T_{k,r}^*=T_{k,r}^\prime.$
On the other hand, the last equality in (\[eq1 proof Thm equalities: adjoint EVF and EVF\]) implies that $T_{k,r}$ coincides with the semigroup associated with ${\mathfrak{a}}_{k,\Lambda_T}$ where $\Lambda_T$ is the subdivision $\Lambda_T:=(0=T-\lambda_{n+1}<T-\lambda_n<...<T-\lambda_1<T-\lambda_0=T).$ It follows from - and - that $$\begin{aligned}
\Big[{{U}}^*_{\Lambda,r}(t,s)\Big]^\prime&={\mathcal{
| 2,192
| 1,422
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| 2,334
| 3,785
| 0.770097
|
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|
expression for the transition probability is given as $${\cal F}(\omega) = \int_{-\infty}^{\infty} du \, \chi(u) \int_{-\infty}^{\infty} ds \, \chi(u -s) e^{- i \omega s} \, W_{\Theta_0, \epsilon}(u,u-s)
\label{transitionprobability}$$ where $\chi(\tau)$ is the smooth switching function which vanishes for $\tau < \tau_0$ and $\tau > \tau_f$ while it is unity for $\tau_0 < \tau < \tau_f$ and is smooth in its transition. For the stationary Rindler trajectory, we have $W_{\Theta_0, \epsilon}(u,u-s) = W_{\Theta_0, \epsilon}(s)$ as shown in Eqs.(\[TXRindler\]) and (\[wfinalcompact\]). Hence in the above equation Eq.(\[transitionprobability\]) for transition probability, we can interchange the sequence of integration and perform the $u$ integral first to get $${\cal F}(\omega) = \int_{-\infty}^{\infty} ds \, e^{- i \omega s} \, W_{\Theta_0, \epsilon}(s) \, Q(s)
\label{transitionprobability2}$$ where $Q(s) = \int_{-\infty}^{\infty} du \, \chi(u) \chi(u -s)$ is also a smooth analytic function in complex $s$ plane. One can then perform the contour integral in Eq.(\[transitionprobability2\]), by choosing an appropriate contour and evaluating the residues of the expression at the poles of the Wightman function $W_{\Theta_0, \epsilon}(s)$ to obtain a finite result. However, from the expression in Eq.(\[denominator\]), one can see that the combination of $\epsilon$ and $|\cos{\Theta_0}|$ always appears as product, hence evaluating the contour integral in Eq.(\[transitionprobability2\]) would preserve the product structure which would imply that taking the point-like limit $\epsilon \rightarrow 0^+$ would make the $\Theta_0$ dependence to go away. In-fact, one can also verify that even in the non-stationary case, the product feature of the $\epsilon$ and $|\cos{\Theta_0}|$ would still hold and the direction dependence would vanish again in the the point-like limit $\epsilon \rightarrow 0^+$.
A spatially extended detector in the Rindler frame {#rindlerframesection}
==================================================
Our aim
| 2,193
| 3,085
| 3,436
| 2,188
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|
above do not necessarily agree. For example, in the starburst galaxy He 2–10, \[\]/and \[\]/ indicate $>39,000$ K [@he2-10opt], whereas mid–infrared line ratios indicate $<37,000$ K [@roche]. The [ $2.06$ ]{}/ observed by @vr would indicate $=39,000$ K using the conversion of @dpj. At poor signal–to–noise, @vr measure [ $1.7$ ]{}/ and find $=36,000$ K. This few thousand Kelvin disagreement translates into a serious disagreement in stellar mass: a of $36,000$ K corresponds to approximately an O8V spectral type, which from eclipsing binaries should have a mass of $\sim 22$ to $25$ [@andersen; @ostrov; @niemela; @gies]; whereas a of $40,000$ K corresponds to an O6.5V to O7V spectral type, which should have a mass of $\sim 35$ [@gies; @niemela].
We now test diagnostics against each other in light of the stellar synthesis models detailed above. Because the line physics of [ $1.7$ ]{}/ is simple and well–understood (see § \[sec:diags\]), we assume this diagnostic is unbiased, and thus accurately reflects the ionizing continuum, within the limitations of measurement error.
Testing [ $2.06$ ]{}/ {#sec:heh_hek}
---------------------
In this section we consider the galaxies for which we obtained [ $1.7$ ]{}/ measurements, as well as three galaxies with [ $1.7$ ]{}/ measurements available in the literature: NGC 253 [@chad], for which the stellar continuum was subtracted as in this work; M82 [@fs-m82], for which representative stellar spectra were subtracted; and NGC 5253 [@vr], for which the stellar continuum is weak enough to ignore. These three galaxies, together with the six galaxies for which we observed [ $1.7$ ]{}/, we term our expanded sample. We also take measurements of [ $2.06$ ]{}/ from the literature for the galaxies in the expanded sample.
Figure \[fig:heh\_hek\] plots [ $2.06$ ]{}/versus [ $1.7$ ]{}/. NGC 3077, NGC 4861, NGC 4214, and He 2–10 all have [ $1.7$ ]{}/ ratios consistent with the saturated value of $\approx 0.3$, within the
| 2,194
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|
ubset \D_4$. The manifold $L^{n-4k}_{int}$ is divided into the disjoint union of the two manifolds (with boundaries) denoted by $(L^{n-4k}_{int,x
\downarrow}, \Lambda_{x \downarrow})$, $(L^{n-4k}_{int,y},
\Lambda_{y})$.
1\. The structure group of the framing $(\Psi_{int,x \downarrow},
\Psi_{\Lambda_{x \downarrow}})$ for the submanifold (with boundary) $(L^{n-4k}_{int,x \downarrow},\Lambda_{x \downarrow})$ is reduced to the subgroups $(\I_{2,x\downarrow},\I_{2,z})$. (In particular, the 2-sheeted cover over $L^{n-4k}_{int,x
\downarrow}$, classified by $\omega$ (denoted by $\tilde
L^{n-4k}_{int,x} \to L^{n-4k}_{int,x \downarrow}$) is, generally speaking, a non-trivial cover.)
2\. The structure group of the framing $(\Psi_{int,y},
\Psi_{\Lambda})$ for the submanifold (with boundary) $(L^{n-4k}_{int,y},\Lambda_y)$ is reduced to the subgroup $(\I_{2,y},\I_3)$. (In particular, the 2-sheeted cover $\tilde
L^{n-4k}_{int,y} \to L^{n-4k}_{int,y}$ classified by $\omega$, is the trivial cover.) Moreover, the double covering $\tilde L^{n-4k}_{x}$ over the component $L^{n-4k}_{x \downarrow}$ is naturally diffeomorphic to $\tilde
L^{n-4k}_y$ and this diffeomorphism agrees with the restriction of the automorphism $OP: \Z/2 \int \D_4 \to \Z/2 \int \D_4$ on the subgroup $\I_{2,x}$, $OP(\I_{2,x})=\I_{2,y}$.
3\. The structure group of the framing $(\Psi_{ext}, \zeta_{ext})$ for the submanifold (with boundary) $h(L^{n-4k}_{ext},
\Lambda^{n-4k}) \subset (\R^n \setminus U^{reg}_{\Delta},
\partial(U^{reg}_{\Delta}))$ is reduced to the subgroup $\I_{2,z}$. (In particular, the 2-sheeted cover $\tilde L^{n-4k}_{ext} \to L^{n-4k}_{ext}$ classified by $\omega$, is, generally speaking, a nontrivial cover.)
$$$$
### Proof of Lemma 1 {#proof-of-lemma-1 .unnumbered}
Components of the self-intersection manifold $g_1(N^{n-2k})
\setminus (g_1(N^{n-2k}) \cap U_{\Sigma})$ (this manifold is formed by double points $x \in g_1(N^{n-2k}), x \notin U_{\Sigma}$ with inverse images $\bar x_1, \bar x_2 \in M^{n-k}$) are classified by the following two t
| 2,195
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| 2,229
| null | null |
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|
R node
4 neural unit Terminal node
In the AOG, each OR node encodes a list of alternative appearance (or deformation) candidates as children. Each AND node uses its children to represent its constituent regions.
More specifically, the top node is an OR node, which represents a certain semantic part, *e.g.* the head or the tail. The semantic part node encodes some part templates as children. Each part template corresponds to a specific part appearance from a certain perspective. During the inference process, the semantic part (an OR node) selects the best part template among all template candidates to represent the object.
The part template in the second layer is an AND node, which uses its children latent patterns to represent a constituent region or a contextual region *w.r.t.* the part template. The part template encodes spatial relationships between its children.
The latent pattern in the third layer is an OR node, whose receptive field is a square block within the feature map of a specific convolutional filter. The latent pattern takes neural units inside its receptive field as children. Because the latent pattern may appear at different locations in the feature map, the latent pattern uses these neural units to represent its deformation candidates. During the inference process, the latent pattern selects the strongest activated child unit as its deformation configuration.
Given an image $I$[^1], we use the CNN to compute feature maps of all conv-layers on image $I$. Then, we can use the AOG for hierarchical part parsing. *I.e.* we use the AOG to semanticize the feature maps and localize the target part and its constituent regions in different layers.
The parsing result is illustrated as red lines in Fig. \[fig:rawMapToModel\]. From a top-down perspective, the parsing procedure 1) identifies a part template for the semantic part; 2) parses an image region for the selected part template; 3) for each latent pattern under the part template, it selects a neural unit within a specific deformat
| 2,196
| 3,449
| 1,282
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| null | null |
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|
nent, nor on the presence of further derivative operators.
To proceed we use the expression of the currents in terms of the bosonic fields $X^a$. First let us focus on the leading term in the expansion . We consider the OPE: \[dXdXOneTerm\] X\^a(z) X\^b(w) = ... + [\^[ab]{}]{} \_[a\_p a\_[p-1]{}...a\_[2]{} a\_1]{}(z-w, |z - |w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(w) +...The behavior of the tensor ${\tilde{A}^{ab}} {}_{a_p a_{p-1}...a_{2}
a_1}$ as a function of the parameter $f$ will give the behavior of the tensor ${A^{ab}}_{a_p...a_1}(z-w, \bar z - \bar w)$ defined in equation . As a first step let us consider the following three-point function: \[FeynmanDiag\] X\^a(z) X\^b(w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(x) \_[connected]{} We consider only the contribution of connected Feynman diagrams to this correlation function. Indeed, if the external operators $\p
X^a(z)$ and $\p X^b(w)$ are contracted on different pieces of a disconnected Feynman diagram, then the result contributes to the regular term $:\p X^a(x) \p X^b(w):$ on the right-hand side of the OPE . Thus to compute the non-trivial terms in this OPE one needs to consider only the Feynman diagrams for which the external operators $\p X^a(z)$ and $\p X^b(w)$ are connected. But this in turn implies that the Feynman diagram is fully connected. Indeed, if this were not the case then one connected piece of the Feynman diagram has for external lines operators coming from the composite operator $ :\p X^{a_1}:\p X^{a_2}...:\p X^{a_{p-1}}\p
X^{a_p}:...::(x)$ only. Such a piece would depend on the coordinate $x$ only, and would necessarily be zero by translation invariance. This shows that we need to consider only fully connected Feynman diagrams.
Now let us evaluate the $f$-dependence of a connected Feynman diagram contributing to . We will show by induction the following statement: a connected Feynman diagram in the theory with $p+2$ external legs behaves like $\mathcal{O}(f^p)$. This is the case for $p=
| 2,197
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|
ft(u^2+1\right)^3} & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{\Phi \Phi} & -\frac{2 \left(u^2-1\right)^2}{\left(u^2+1\right)^3} & \frac{4 \left(u^2-1\right)^2}{\left(u^2+1\right)^3} & -\frac{2 \left(u^2-1\right)^2}{\left(u^2+1\right)^3} & \frac{2 \left(u^2-1\right)^2}{\left(u^2+1\right)^3} & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{RR} & \frac{u^2-1}{2 \left(u^2+1\right)} & \frac{1-u^2}{u^2+1} & \frac{u^4+6 u^2-3}{8 \left(u^2+1\right)} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{Ru} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{uu} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{TR} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{u^2-1}{2 \left(u^2+1\right)} & 0 & 0 & 0 \\
\mathcal{D}_{Tu} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{\Phi R} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{u^2-1}{2 \left(u^2+1\right)} & 0 \\
\mathcal{D}_{\Phi u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}$$
$$\begin{array}{c|ccccc}
\mathcal{D}_{AB} & C_{TT}'(u) & C_{T\Phi}'(u) & C_{\Phi \Phi }'(u) & C_{RR}'(u) & C_{Ru}'(u)
\\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & \frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{4 u \left(u^2-3\right) \left(u^2-1\right)}{\left(u^2+1\right)^4} & -\frac{u \left(u^{10}+u^8-22 u^6+66 u^4-123 u^2+45\right)}{8 \left(u^2-1\right) \left(u^2+1\right)^4} & -\frac{u \left(u^6+u^4-13 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{h \left(u^2-1\right) \left(u^4+6 u^2-3\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{T\Phi} & \frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{4 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & \frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{2 (2 h+1) \left(u^2-1\right)^2}{\left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi \Phi} & \frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{4 u \left(u^2-3\right) \left(u^2-1\right)}{\left(u^2+1\right)^4} & \frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^
| 2,198
| 2,782
| 1,658
| 2,036
| null | null |
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|
/n)T_{K_C}(t/n)\Big]^n\right\Vert}
\leq e^{t(CM+{\left\Vert \Sigma-CS_0I\right\Vert}+{\left\Vert K_C\right\Vert})}.$$
Therefore, by Trotter’s product formula ([@engelnagel Corollary 5.8, p. 227]) we have for ${\bf f}\geq 0$, and for all $t\geq 0$ that G(t)[**f**]{}=\_[n]{}\^n[**f**]{}0, and thus $\phi\geq 0$ by . This implies that $\psi=e^{-CE}\phi\geq 0$. (See [@dautraylionsv6 Section XXI-§2, Proposition 2, pp. 226-227], and [@tervo14 Theorem 5.16]).
B. Suppose that $g\geq 0$ is more general. Moreover, in accordance with Theorem \[coth3-dd\] (or Corollary \[m-d-co1\]), we assume that $g\in H^1(I, T^2(\Gamma'_-))$ for which $g(E_{\rm m})=0$. We decompose the solution as follows. Let $u$ be the solution of the problem $$\begin{gathered}
-{{\frac{\partial (S_0u)}{\partial E}}}+\omega\cdot\nabla_x u+CS_0 u+\Sigma u=0,\nonumber\\
u_{|\Gamma_-}={\bf g},
\quad
u(\cdot,\cdot,E_{\rm m})=0, \label{eq:problem_simple_u}\end{gathered}$$ and let $w$ be the solution of the problem $$\begin{gathered}
-{{\frac{\partial (S_0w)}{\partial E}}}+\omega\cdot\nabla_x w+CS_0 w+\Sigma w-K_Cw={\bf f}+K_Cu, \nonumber \\
w_{|\Gamma_-}=0,
\quad
w(\cdot,\cdot,E_{\rm m})=0. \label{eq:problem_w_u}\end{gathered}$$ Then $\phi:=w+u$ is the solution of -.
Since ${\bf g}\geq 0$, one can show that the solution $u$ of the problem is positive, i.e. $u\geq 0$ (see Remark \[re:general\_positivity\_of\_u\] below). For example, in the special case of Example \[desolex1\] given below, where $S(x,E)=S(E)$ does not depend on $x$, and $\Sigma(x,\omega,E)=\Sigma(x,E)$ does not depend on $\omega$, the solution $u$ can be expressed explicitly in the form u=[1]{}H(R(E\_[m]{})-Q(x,,E))e\^[-\_0\^[t(x,)]{}(x-s,)ds]{}(x-t(x,),,Q(x,,E)), \[eq:explicit\_u\_in\_simplified\_case\] where $$R(E):={}& \int_0^E{1\over{S_0(\tau)}}d\tau, \\[2mm]
Q(x,\omega,E):={}&R(E)+t(x,\omega), \\[2mm]
\tilde{\bf g}(y,\omega,\eta):={}& S_0(R^{-1}(\eta)){\bf g}(y,\omega,R^{-1}(\eta)),$$ From it is clear that $u\geq 0$ whenever $g\geq 0$.
Finally, because ${\bf f}\geq 0$, and $K_Cu\geq
| 2,199
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| 1,570
| 2,097
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|
es. One set consists of the seven periods of $f_1-f_7$ observed in 2007, while we complemented this list with the period of $\mathrm{F}_2$ detected in 1975 to create another set. We selected $\mathrm{F}_2$ because it was the second largest amplitude peak in 1975, which makes it a good candidate for an additional normal mode.
In LP 133-144, we found all the previously observed frequencies in our 2007 dataset, as we show in Sect. \[sect:lp133freq\]. Thus, we cannot add more frequencies to our findings, and performed the model fits with six periods. We summarized the periods utilized for modelling in Table \[table:periods\] for both stars.
----------------- -------- ------- --------
Period Period
(s) (s)
$f_1$ 291.9 $f_1$ 209.2
$f_2$ 595.7 $f_2$ 305.9
$f_3$ 196.1 $f_3$ 270.6
$f_4$ 623.8 $f_4$ 327.3
$f_5$ 129.4 $f_5$ 140.5
$f_6$ 317.8 $f_6$ 179.5
$f_7$ 305.2
+$\mathrm{F}_2$ 557.4
----------------- -------- ------- --------
: G 207-9 and LP 133-144: periods utilized for the model fits.[]{data-label="table:periods"}
Best-fitting models for G 207-9
-------------------------------
We determined the best-matching models considering several cases: at first, we let all modes to be either $l=1$ or $l=2$. Then we assumed that the dominant peak is an $l=1$, considering the better visibility of $l=1$ modes over $l=2$ ones. At last, we searched for the best-fitting models assuming that at least four of the modes is $l=1$, including the dominant frequency.
We obtained the same model as the best-fitting asteroseismic solution both for the seven- and eight-period fits. It has $T_{\rmn{eff}}=12\,000$K, $M_*=0.870\,M_{\sun}$ and $M_\rmn{H}=10^{-4}\,M_*$. This model has the lowest $\sigma_\mathrm{{rms}}$ ($1.04-1.06$s) both if we do not apply any restricti
| 2,200
| 3,751
| 2,121
| 2,306
| null | null |
github_plus_top10pct_by_avg
|
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