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& = \begin{cases}
\displaystyle
\frac{1}{\sigma_a^2 - \sigma_n^2}\left[(\sigma_a^2\mu_n - \sigma_n^2\mu_a) \pm \sigma_a\sigma_n\sqrt{(\mu_a - \mu_n)^2 + 2(\sigma_a^2 - \sigma_n^2)ln\frac{\sigma_a}{\sigma_n}}\right], & \sigma_a \ne \sigma_n\\
\displaystyle \frac{\mu_n + \mu_a}{2}, & \sigma_a = \sigma_n
\end{cases}
\end{aligned}$$
Note: when $\sigma_a \ne \sigma_n$, keep the root s.t. $\displaystyle \frac{T - \mu_a}{\sigma_a^3}e^{-\frac{(T - \mu_a)^2}{2\sigma_a^2}} < \frac{T - \mu_n}{\sigma_n^3}e^{-\frac{(T - \mu_n)^2}{2\sigma_n^2}}$
$$\begin{aligned}
\label{equ:linear-weight}
T & = \mathop{\arg\min}_{T} \alpha\int_{0}^{T}PDF_{a}(x)dx +
(1-\alpha)\int_{T}^{\sup(D)}PDF_{n}(x)dx\nonumber\\
& \approx \mathop{\arg\min}_{T}
\alpha\int_{-\infty}^{T}
\frac{e^{-\frac{(x - \mu_a)^2}{2\sigma_a^2}}}{\sqrt{2\pi} \sigma_a}dx
+
(1-\alpha)\int_{T}^{+\infty}
\frac{e^{-\frac{(x - \mu_n)^2}{2\sigma_n^2}}}{\sqrt{2\pi} \sigma_n}dx\nonumber\\
& = \begin{cases}
\displaystyle
\frac{1}{\sigma_a^2 - \sigma_n^2}\left[(\sigma_a^2\mu_n - \sigma_n^2\mu_a) \pm \sigma_a\sigma_n\sqrt{(\mu_a - \mu_n)^2 + 2(\sigma_a^2 - \sigma_n^2)ln\frac{(1 - \alpha)\sigma_a}{\alpha\sigma_n}}\right], & \sigma_a \ne \sigma_n\\
\displaystyle
\frac{\mu_n + \mu_a}{2} + \frac{k^2ln\frac{1 - \alpha}{\alpha}}{\mu_a - \mu_n}, & \sigma_a = \sigma_n = k
\end{cases}
\end{aligned}$$
Note: when $\sigma_a \ne \sigma_n$, keep the root s.t. $\displaystyle \frac{\alpha (T - \mu_a)}{\sigma_a^3}e^{-\frac{(T - \mu_a)^2}{2\sigma_a^2}} < \frac{(1 - \alpha) (T - \mu_n)}{\sigma_n^3}e^{-\frac{(T - \mu_n)^2}{2\sigma_n^2}}$
Moreover, with an estimated ano
| 1,801
| 4,908
| 682
| 1,433
| null | null |
github_plus_top10pct_by_avg
|
more, \[n2k-3\] & (\_2(x,E’,E)((x,\_+(E’,E,),E’))\
= & (x,E’,E)(x,\_+(E’,E,),E’)\
&+ \_2(x,E’,E)(\_)(x,\_+(E’,E,),E’) (E’,E,)\
&+ \_2(x,E’,E)(x,\_+(E’,E,),E’) and similarly for the last term in (\[n2k-2\]). Since $\mu_{22}(E,E)=\mu_{22,p}(E,E)=1$ we have \[n2k-4\] & ((\_[22,2]{})(x,,E’,E))\_[|E’=E]{}\
=& (x,E,E)(x,,E)\
&+ \_2(x,E,E)(\_)(x,,E)(E,E,) + \_2(x,E,E)(x,,E)\
&+ (x,E,E)(x,,E)\
&+ \_2(x,E,E)(\_)(x,,E)(E,E,) + \_2(x,E,E)(x,,E)\
=& 2\_2(x,E,E)(x,,E)+ \_2(x,E,E)((E,E,)+(E,E,)) (\_)(x,,E)\
&+ 2(x,E,E)(x,,E). Finally, \[n2k-5\] & [E]{}(x,,E’,E)\
=& (\_2(x,E’,E)(x,\_+(E’,E,),E’) +\_2(x,E’,E)(x,\_-(E’,E,),E’) )\
=& (x,E’,E)(x,\_+(E’,E,),E’)\
& +\_2(x,E’,E)(\_)(x,\_+(E’,E,),E’)(E’,E)\
& +(x,E’,E)(x,\_-(E’,E,),E’)\
& +\_2(x,E’,E)(\_)(x,\_-(E’,E,),E’) (E’,E).
As a conclusion we see that for $n=2$ $$\begin{gathered}
(K_{22,2}\psi)(x,\omega,E)
=
{\partial\over{\partial E}}\Big(
{{{\mathcal{}}}H}_1((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E)\Big) \\
-
{{{\mathcal{}}}H}_1\big(({{\frac{\partial (\ol{{{\mathcal{}}}K}_{22,2}\psi)}{\partial E}}}(x,\omega,\cdot,E)\big)(E)
\\
+2\hat\sigma_2(x,E,E){{\frac{\partial \psi}{\partial E}}}(x,\omega,E)
+2{{\frac{\partial \hat\sigma_2}{\partial E'}}}(x,E,E)\psi(x,\omega,E) \\
+\hat\sigma_2(x,E,E)
\big({{\frac{\partial \xi_+}{\partial E'}}}(E,E,\omega)+{{\frac{\partial \xi_-}{\partial E'}}}(E,E,\omega)\big)
\cdot
(\nabla_\omega\psi)(x,\omega,E), \label{n2k-6}\end{gathered}$$ where ${{\frac{\partial (\ol{{{\mathcal{}}}K}_{22,2}\psi)}{\partial E}}}$ is computed by (\[n2k-5\]).
The corresponding collision operator can be analogously computed in the general dimension $n$ (which we omit here). The [Møller]{} collision term produces first order partial differential terms, along with a Hadamard finite part operator. The exact form of [Møller]{} collision operator allows for accessing relevant approximation schemes for which the error analysis can be carried out. We find that the CSDA-approximation does no
| 1,802
| 306
| 1,375
| 1,841
| null | null |
github_plus_top10pct_by_avg
|
erally speaking, would you say that most people can be trusted, or that you can't be too careful in dealing with people?'. Institutional trust, the mediation variable, is obtained by combining respondents' reported trust in the parliament, legal system and parties (see [Table 2](#pone.0220160.t002){ref-type="table"} and Supporting Information for more details). The homicide rate is estimated by dividing the number of homicides reported to the police by the total number of residents in the region. The result is then multiplied by 100,000. The same process is followed for property crimes, which include robberies and vehicle thefts. Our operationalization of state's fairness relies on two dimensions of the European Quality of Government Index, namely the impartiality and corruption pillars. These are based on two batteries of questions measuring the perceived degree of impartiality and corruption in the respondents' area regarding law enforcement, public health, public education and so on \[[@pone.0220160.ref016]\] (see <https://qog.pol.gu.se/data/datadownloads/qog-eqi-data> for more details). High values of corruption indicate a lack of corruption, whereas high values of impartiality indicate a strong impartiality of institutions.
10.1371/journal.pone.0220160.t002
###### Demographics and operationalization of main concepts.
{#pone.0220160.t002g}
Variables' description Mean S.D. Range n individuals (NUTS II)
---------------------------------------------------------------------------------------------------------------------------------- -------- -------- ----------------- -------------------------
Generalized Trust: 0 = Can't be too careful; 10 = Most people can be trusted 4.69 2.48 0--10 23,042 (122)
Trust in Parties: 0 = No trust at all; 10 = Complete trust
| 1,803
| 1,191
| 2,526
| 2,082
| null | null |
github_plus_top10pct_by_avg
|
\Gamma_{\pm})}\leq {\left\Vert \psi\right\Vert}_{\tilde{W}^2_{\mp}(G\times S\times I)}$$ and thus $\gamma_{\pm} :\tilde{W}^2_{\mp,0}(G\times S\times I)\to T^2(\Gamma_{\pm})$ is bounded.
Occasionally we work in the (energy independent) spaces $L^2(G\times S)$. The corresponding Hilbert spaces $W^2(G\times S)$, $T^2(\Gamma'_{\pm})$, $T^2(\Gamma')$ and $\tilde W^2(G\times S)$ are similarly defined, where $$\Gamma':={}&(\partial G)\times S, \\[2mm]
\Gamma'_{0}:={}&\{(y,\omega)\in (\partial G)\times S\ |\ \omega\cdot\nu(y)=0\}, \\[2mm]
\Gamma'_{-}:={}&\{(y,\omega)\in (\partial G)\times S\ |\ \omega\cdot\nu(y)<0\}, \\[2mm]
\Gamma'_{+}:={}&\{(y,\omega)\in (\partial G)\times S\ |\ \omega\cdot\nu(y)>0\}.$$ In addition, the trace $\gamma'(\psi):=\psi_{|\Gamma'}$ for $\psi\in \tilde{W}^2(G\times S)$ is defined as $\gamma(\psi)$ above.
In the context of CSDA-equations we need the following additional Hilbert spaces. Let $H$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product ,v\_H:=,v\_[L\^2(GSI)]{}+ (),(v)\_[T\^2()]{} The elements of $H$ are of the form $\tilde\psi=(\psi,q)\in
L^2(G\times S\times I)\times T^2(\Gamma)$. Actually, they are exactly elements of the closure of the graph of trace operator $\gamma:C^1(\ol G\times S\times I)\to
C^1(\partial G\times S\times I)$ in $L^2(G\times S\times I)\times T^2(\Gamma)$. The inner product in $H$ is $$\begin{aligned}
\label{spaceH}
{\left\langle}\tilde\psi,\tilde \psi'{\right\rangle}_{H}={\left\langle}\psi, \psi'{\right\rangle}_{L^2(G\times S\times I)}
+{\left\langle}q,q'{\right\rangle}_{T^2(\Gamma)},\end{aligned}$$ for $\tilde\psi=(\psi,q)$, $\tilde\psi'=(\psi',q')\in H$.
Furthermore, let $H_1$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product $$\begin{aligned}
\label{spaceH1}
{\left\langle}\psi,v{\right\rangle}_{H_1}:={}&{\left\langle}\psi,v{\right\rangle}_{L^2(G\times S\times I)}+
{\left\langle}\gamma(\psi),\gamma(v){\right\rangle}_{T^2(\Gamma)}\nonumber \\
&
+
{\left\langle}\psi(\cdot,\cdot,0),v(\cdot,\cdo
| 1,804
| 798
| 1,844
| 1,708
| null | null |
github_plus_top10pct_by_avg
|
. ]{}
[^17]: [\[Foot: 0=3Dphi\]If $y\notin\mathcal{R}(f)$ then $f^{-}(\{ y\}):=\emptyset$ which is true for any subset of $Y-\mathcal{R}(f)$. However from the set-theoretic definition of natural numbers that requires $0:=\emptyset$, $1=\{0\}$, $2=\{0,1\}$ to be defined recursively, it follows that $f^{-}(y)$ can be identified with $0$ whenever $y$ is not in the domain of $f^{-}$. Formally, the successor set $A^{+}=A\bigcup\{ A\}$ of $A$ can be used to write $0:=\emptyset$, $1=0^{+}=0\bigcup\{0\}$, $2=1^{+}=1\bigcup\{1\}=\{0\}\bigcup\{1\}$ $3=2^{+}=2\bigcup\{2\}=\{0\}\bigcup\{1\}\bigcup\{2\}$ etc. Then the set of natural numbers $\mathbb{N}$ is defined to be the intersection of all the successor sets, where a successor set $\mathcal{S}$ is any set that contains $\emptyset$ and $A^{+}$ whenever $A$ belongs to $\mathcal{S}$. Observe how in the successor notation, countable union of singleton integers recursively define the corresponding sum of integers. ]{}
[^18]: [See footnote \[Foot: 0=3Dphi\] for a justification of the definition when $b$ is not in $\mathcal{R}(a)$.]{}
[^19]: [\[Foot: subnet\]A subnet is the generalized uncountable equivalent of a subsequence; for the technical definition, see Appendix A1. ]{}
[^20]: [\[Foot: point\_inter\]Equation (\[Eqn: func\_bi\]) is essentially the intersection of the pointwise topologies (\[Eqn: point\]) due to $f$ and $f^{-}$. ]{}
[^21]: [\[Foot: strict reln\]If $\preceq$ is an order relation in $X$ then the]{} *strict relation $\prec$ in $X$* [corresponding to $\preceq$, given by $x\prec y\Leftrightarrow(x\preceq y)\wedge(x\neq y)$,]{} *is not an order relation* [because unlike $\preceq$, $\prec$ is not reflexive even though it is both transitive and asymmetric.]{} **
[^22]: [\[Foot: infinite\]This makes $T$, and hence $X$, inductively defined infinite sets. It should be realized that (ST3)]{} *does not mean* [that every member of $T$ is obtained from $g$, but only ensures that the immediate successor of any element of $T$ is also in $T.$ The infimum $_{\rightarrow}
| 1,805
| 3,289
| 3,005
| 1,875
| 2,043
| 0.783245
|
github_plus_top10pct_by_avg
|
$$V_{2}(r) =
\displaystyle\frac{2\alpha^{2}}
{\biggl(r + r_{0} + \displaystyle\frac{1}{C\alpha}\biggr)^{2}}.
\label{eq.3.1.5}$$ We see, that this potential is finite in the whole region of its definition at any values of the parameters $C>0$ and $r_{0} \ge 0$. Thus, we have obtained the reflectionless potential of the inverse power type with a shift to the left, which is defined on the whole positive semiaxis of $r$ (including $r=0$ and $r_{0}=0$).
In accordance with [@Maydanyuk.2005.APNYA] (see p. 452–455, sec. 5.1.2), one can construct a hierarchy of the inverse power potentials, and a general solution of the potential with arbitrary number $n$ can be written down so: $$\begin{array}{ll}
V_{n}(r) =
\displaystyle\frac{\gamma_{n} \alpha^{2}}{\bar{r}^{2}}, &
\gamma_{n \pm 1} = 1 + \gamma_{n} \pm \sqrt{4\gamma_{n}+1}.
\end{array}
\label{eq.3.1.6}$$ If to require, that the first potential $V_{1}(r)$ in this hierarchy must be constant (i. e. at $\gamma_{1}=0$ and $n=1$), then all hierarchy of the inverse power potentials (\[eq.3.1.6\]) becomes the *hierarchy of the reflectionless inverse power potentials*, and the solution (\[eq.3.1.5\]) becomes the general solution for the reflectionless inverse power potential. Note, that *when the hierarchy of the inverse power potentials becomes reflectionless, then the coefficients $\gamma_{n}$ become integer numbers*. We write its first values: $$\gamma_{n} = 0, 2, 6, 12, 20, 30, 42...
\label{eq.3.1.7}$$
Now, if to calculate $\beta_{n}$ for given $\gamma_{n}$ with number $n$ from (\[eq.3.1.7\]) from the following condition: $$\beta_{n} (\beta_{n}-\alpha) = \gamma_{n} \alpha^{2},
\label{eq.3.1.8}$$ then the first potential $V_{1}(r)$ from (\[eq.3.1.4\]) becomes reflectionless inverse power potential (at $\beta =\beta_{n}$). The second potential $V_{2}(r)$ from (\[eq.3.1.4\]) is finite in the whole region of its definition (including $r=0$) and should be reflectionless also, however it is not inverse power potential. So, substituting the coefficients $\gamm
| 1,806
| 2,723
| 2,579
| 1,820
| 1,440
| 0.789549
|
github_plus_top10pct_by_avg
|
i_\mu })^t.
\label{eq:maid1}
\end{aligned}$$ Then $$\begin{aligned}
E_i F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
\qquad &
\\
F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
\end{aligned}$$ for all $i\in I$.
By Eq. it suffices to prove that $$\begin{aligned}
F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
\qquad &
\\
F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
\end{aligned}
\label{eq:hwv}$$ for all $\nu \in \{1,2,\dots ,n\}$. Let first $\nu \in \{1,2,\dots ,\mu -1\}$. Then $$F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} =0$$ by Thms. \[th:PBW\] and \[th:EErel\] by considering the ${\mathbb{Z}}^I$-degree. Thus Eq. holds. Let now $\nu \in \{\mu ,\mu +1,\dots ,n\}$, $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$, and $\beta '_\kappa =1_{\chi _\mu }{\sigma }_{i_\mu }{\sigma }_{i_{\mu +1}}\cdots
{\sigma }_{i_{\mu +\kappa -2}}({\alpha }_{i_{\mu +\kappa -1}})$ for all $\kappa \in \{1,2,\dots ,2n\}$. Then by Lemma \[le:VTinv\], Eq. is equivalent to $${\rho ^{\chi _\mu }}({\alpha }_{i_\mu })\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1})
={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t,$$ and ${\alpha }_{i_\mu }=\beta '_1$. Thus we can apply Lemma \[le:FFv\] (with $\Lambda _\mu $ instead of $\Lambda $, $\chi _\mu $ instead of $\chi $, $\beta '_\kappa $ instead of $\beta _\kappa $, and $\mu
| 1,807
| 882
| 2,266
| 1,753
| null | null |
github_plus_top10pct_by_avg
|
consider the trace pairing ${\rm Tr}:\mathfrak{g}\times\mathfrak{g}\rightarrow\C^{\times}$. Define the Fourier transform $\calF^\mathfrak{g}:{\rm Fun}(\mathfrak{g})\rightarrow{\rm Fun}(\mathfrak{g})$ by the formula
$$\calF^\mathfrak{g}(f)(x)=\sum_{y\in\mathfrak{g}}\Psi\left({\rm Tr}\,(xy)\right)f(y)$$for all $f\in{\rm Fun}(\mathfrak{g})$ and $x\in\mathfrak{g}$.
The Fourier transform satisfies the following easy property.
For any $f\in{\rm Fun}(\mathfrak{g})$ we have:
$$|\mathfrak{g}|\cdot f(0)=\sum_{x\in\mathfrak{g}}\calF^\mathfrak{g}(f)(x).$$ \[fourprop1\]
Let $*$ be the convolution product on ${\rm Fun}(\mathfrak{g})$ defined by $$(f*g)(a)=\sum_{x+y=a}f(x)g(y)$$for any two functions $f,g\in{\rm Fun}(\mathfrak{g})$.
Recall that
$$\calF^\mathfrak{g}(f*g)=\calF^\mathfrak{g}(f)\cdot\calF^\mathfrak{g}(g).\label{conv}$$
For a partition $\lambda$ of $n$, let $\mathfrak{p}_\lambda$, $\mathfrak{l}_\lambda$, $\mathfrak{u}_\lambda$ be the Lie sub-algebras of $\mathfrak{g}$ corresponding respectively to the subgroups $P_\lambda$, $L_\lambda$, $U_\lambda$ defined in §\[finite-groups\], namely $\mathfrak{l}_\lambda=\bigoplus_i\gl_{\lambda_i}(\F_q)$, $\mathfrak{p}_\lambda$ is the parabolic sub-algebra of $\mathfrak{g}$ having $\mathfrak{l}_\lambda$ as a Levi sub-algebra and containing the upper triangular matrices. We have $\mathfrak{p}_\lambda=\mathfrak{l}_\lambda\oplus\mathfrak{u}_\lambda$.
Define the two functions $R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1), Q_{\mathfrak{l}_\lambda}^\mathfrak{g} \in\C(\mathfrak{g})$ by
$$\begin{aligned}
&R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1)(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in \mathfrak{p}_\lambda\},\\&Q_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in \mathfrak{u}_\lambda\}.\end{aligned}$$
We define the type of a $G$-orbit of $\mathfrak{g}$ similarly as in the group setting (see above Corollary \[R\]). The types of the $G$-orbits of $\mathfrak{g}$ are then also parameterized by $\mathbf{T}_n$.
From Lemma \[R=F\], we see tha
| 1,808
| 1,481
| 1,442
| 1,711
| null | null |
github_plus_top10pct_by_avg
|
Wisotzki, L., Barden, M., Beckwith, S. V. W., Bell, E. F., Borch, A., Caldwell, J. A. R., Häu[ß]{}ler, B., Jogee, S., McIntosh, D. H., Meisenheimer, K., Peng, C. Y., Rix, H.-W., Somerville, R. S., & Wolf, C. 2004, submitted to ApJ, astro-ph/0403645
Schade, D., Boyle, B. J., & Letawsky, M. 2000, MNRAS, 315, 498
Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
, E. J., [Koekemoer]{}, A. M., [Grogin]{}, N. A., [Giacconi]{}, R., [Gilli]{}, R., [Kewley]{}, L., [Norman]{}, C., [Hasinger]{}, G., [Rosati]{}, P., [Marconi]{}, A., [Salvati]{}, M., & [Tozzi]{}, P. 2001, ApJ, 560, 127
Sérsic, J. 1968, Atlas de Galaxes Australes, Observatorio Astronomico de Cordoba
Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, ApJ, 588, 65
, J., [Fosbury]{}, R. A. E., [Villar-Mart[í]{}n]{}, M., [Cohen]{}, M. H., [Cimatti]{}, A., [di Serego Alighieri]{}, S., & [Goodrich]{}, R. W. 2001, A&A, 366, 7
Wolf, C., Meisenheimer, K., Kleinheinrich, M., Borch, A., Dye, S., Gray, M., Wisotzki, L., Bell, E. F., Rix, H.-W., Cimatti, A., Hasinger, G., & Szokoly, G. 2004, submitted to A&A, astro-ph/0403666
Wolf, C., Meisenheimer, K., Rix, H.-W., Borch, A., Dye, S., & Kleinheinrich, M. 2003, A&A, 401, 73
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AGN images and surface brightness plots of AGN and stars {#sec:appendix}
========================================================
Figure \[fig:allimages\] shows plots for each of the nine resolved object plus the composite ‘stacked’ object. Two objects appear twice as they appear in overlapping areas of [[gems]{}]{}tiles. Figures \[fig:stars\_v\] and \[fig:stars\_z\] show a random selection of isolated stars used to show the zero case of a point sources without any host galaxy contribution, for comparison purposes.
{width="\fullwidth"}
{width="\fullwidth"}
[^1]: The [[galfit]{}]{} version 1.7a was recentering the PSF to
| 1,809
| 1,429
| 3,309
| 2,147
| null | null |
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|
) = 0$ and $\lambda_2(\lL)=\lambda_3(\lL)=\cdots=\lambda_{\ld}(\lL)$. Therefore, using $\lambda_2(\lL) = \Tr(\lL)/(\ld-1) = n\ld$. Using the fact that $\E[\lM]$ and $\lL$ are symmetric matrices, we have, $$\begin{aligned}
\label{eq:lambda2_bottoml_expec}
\lambda_2(\E\big[\lM\big]) \geq \frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{4} \frac{n\ld\ell^2}{d^2}. \end{aligned}$$
To get an upper bound on ${\|\lM - \E[\lM]\|}$, notice that $\lM^{(j)}$ is also given by, $$\begin{aligned}
\label{eq:bottoml_hess4}
\lM^{(j)} \;\; =\;\; \ell\, \diag(\le_{\{I_j\}}) - \le_{\{I_j\}}\le_{\{I_j\}}^\top\;,\end{aligned}$$ where $\le_{\{I_j\}} \in \reals^{\ld}$ is a zero-one vector, with support corresponding to the bottom-$\ell$ subset of items in the ranking $\sigma_j$. $I_j = \{\sigma_j(\kappa-\ell+1),\cdots, \sigma_j(\kappa)\}$ for $j \in [n]$. $(\lM^{(j)})^2$ is given by $$\begin{aligned}
\label{eq:bottoml_hess5}
(\lM^{(j)})^2 \;\; =\;\; \ell^2 \,\diag(\le_{\{I_j\}}) - \ell\, \le_{\{I_j\}}\le_{\{I_j\}}^\top\;.\end{aligned}$$ Using the fact that sets $\{S_j\}_{j \in [n]}$ are chosen uniformly at random and $\P[i \in \I_j | i \in S_j] \leq 1$, we have $\E[\diag(\le_{\{I_j\}})] \preceq (\kappa/d) \diag(\le_{\{{\boldsymbol{1}}\}})$. Maximum of row sums of $\E\big[\le_{\{I_j\}}\le_{\{I_j\}}^\top\big]$ is upper bounded by $\ell\kappa/d$. Therefore, from triangle inequality we have ${\|\sum_{j=1}^n \E[(\lM^{(j)})^2]\|} \leq 2n\ell^2\kappa/d$. Also, note that ${\|\lM^{(j)}\|} \leq 2\ell$ for all $j \in [n]$. Applying matrix Bernstien inequality, we have that $$\begin{aligned}
\label{eq:bottoml_hess6}
\mathbb{P}\Big[{\|\lM - \E[\lM]\|} \geq t\Big] \leq d\,\exp\Big(\frac{-t^2/2}{2n\ell^2\kappa/d + 4\ell t/3}\Big). \end{aligned}$$ Therefore, with probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:bottoml_error}
{\|\lM - \E[\lM]\|} \leq 4\ell\sqrt{\frac{2n\kappa\log d}{d}} + \frac{64\ell\log d}{3} \leq 8\ell\sqrt{\frac{n\kappa\log d}{d}}\;,\end{aligned}$$ where the second ineq
| 1,810
| 1,307
| 1,560
| 1,733
| null | null |
github_plus_top10pct_by_avg
|
up to $N=8000$, see Fig. \[Results Figure\]. We see that the [*fairness condition*]{} $\mbox{sup}_m \bar{P}_{ch}(m) << 1$ is satisfied (for $N=600$ we got $\mbox{sup}_m \bar{P}_{ch}(m) = \bar{P}_{ch}(92)= 0.0811$, while for $N_A=N_R$ we have $\mbox{sup}_m \bar{P}_{ch}(m) = \bar{P}_{ch}(1455)= 0.0247$ for $N=8000$), which is explicitly shown on Fig. \[Results Figure\]. Moreover, the numerical fit gives $\mbox{sup}_m \bar{P}_{ch}(m) \propto N^{-1/2}$ behavior, giving us the scaling of the complexity of the protocol, with respect to the number $N$ of exchanged messages between Alice and Bob.
![(color online) The expected probability to cheat $\bar{P}_{ch}(m)$ (upper row) and the maximal expected probability to cheat $\mbox{sup}_m \bar{P}_{ch}(m)$ (lower row) for the uniform $p(\alpha)$ on $I_{\alpha}=[0.9,0.99]$. The plots from the left column represents results for our protocol, while the right ones are for the restricted “typical” case of $N_A=N_R$. Note the scaling behavior $\mbox{sup}_m \bar{P}_{ch}(m) \propto N^{-1/2}$. \
[]{data-label="Results Figure"}](Fig34.pdf){width="8.5cm" height="6.0cm"}
V. Fairness of the protocol: General measurements and noise {#sec:fairness-general_measurements}
===========================================================
First, we consider only one-qubit orthogonal measurements. Since Alice is an honest client, $X^{\cal{A}} = \hat{A}^{\otimes m}$, we have that $P^{\cal{A}}_A (m, \alpha, X^{\cal{A}}) = 1$ and $P^{\cal{A}}_R (m, \alpha, X^{\cal{A}}) = P_R(m, \alpha)$.
Bob is a dishonest client and his strategy $X^{\cal{B}}$ consists of measuring $(m-k)$ times the Accept observable $\hat{A}$ and $k$ times observable $\hat{K}= 0\cdot |m\rangle\langle m| + 1\cdot |m^\bot\rangle\langle m^\bot|$, where $$\!\! |m\rangle \! = \! \cos\frac{\theta}{2}|0\rangle + e^{\varphi}\!\sin\frac{\theta}{2}|1\rangle \! = \! \cos\frac{\theta^\prime}{2}|-\rangle + e^{\varphi^\prime}\!\!\sin\frac{\theta^\prime}{2}|+\rangle.$$ Let $m = m_a + m_r$, where $m_a$ is the number of the Accept and $m_r$ th
| 1,811
| 1,294
| 2,066
| 1,867
| 2,926
| 0.776036
|
github_plus_top10pct_by_avg
|
e ?
Here is my c_cpp_properties.json :
{
"configurations": [
{
"name": "Win32",
"includePath": [
"${workspaceFolder}",
"C:\\Program Files (x86)\\mingw-w64\\i686-8.1.0-posix-dwarf-rt_v6-rev0\\mingw32\\lib\\gcc\\i686-w64-mingw32\\8.1.0\\include"
],
"defines": [
"_DEBUG",
"UNICODE",
"_UNICODE"
],
"intelliSenseMode": "msvc-x64"
}
],
"version": 4
}
And this is launch.json :
{
"version": "0.2.0",
"configurations": [
{
"name": "(gdb) Launch",
"type": "cppdbg",
"request": "launch",
"program": "${workspaceFolder}/a.exe",
"args": [],
"stopAtEntry": false,
"cwd": "${workspaceFolder}",
"environment": [],
"externalConsole": true,
"MIMode": "gdb",
"miDebuggerPath": "C:\\Program Files (x86)\\mingw-w64\\i686-8.1.0-posix-dwarf-rt_v6-rev0\\mingw32\\bin\\gdb.exe",
"setupCommands": [
{
"description": "Enable pretty-printing for gdb",
"text": "-enable-pretty-printing",
"ignoreFailures": true
}
]
}
]
}
A:
I think you should add prelaunched task with label of your build task to launch.json like this:
"preLaunchTask": "build" // label of your build task
This means you should have in your tasks.json following task with label build e.g.
"tasks": [
{
"label": "build",
"type": "shell",
"command": "gcc -g source.c"
"group": {
"kind": "build",
"isDefault": true
}
}
]
Also "-g" flag is important for enabling debugging
Q:
Trouble running unit test - got undefined method error
I followed the instructions on this page but couldn't get my unit test working.
http://framework.zend.com/manual/2.2/en/tutorials/unittest
| 1,812
| 2,082
| 25
| 1,949
| 123
| 0.824058
|
github_plus_top10pct_by_avg
|
f $\Gamma$ to be $$\Lambda_{\Kul}(\Gamma)=\Lambda(\Gamma)\cup L_2(\Gamma),$$
4. *Kulkarni’s discontinuity region* of $\Gamma$ to be $$\Omega_{\Kul}(\Gamma)=\mathbb{P}^n_{\mathbb{C}}\setminus\Lambda_{\Kul}(\Gamma).$$
Kulkarni’s limit set has the following properties. For a more detailed discussion of this in the two-dimensional setting, see [@CNS].
\[p:pkg\] Let $\Gamma\subset \PSL(3,\Bbb{C})$ be a complex Kleinian group. Then:
1. The sets\[i:pk2\] $\Lambda_{\Kul}(\Gamma),\,\Lambda(\Gamma),\,L_2(\Gamma)$ are $\Gamma$-invariant closed sets.
2. \[i:pk3\] The group $\Gamma$ acts properly discontinuously on $\Omega_{\Kul}(\Gamma)$.
3. \[i:pk4\] If $\Gamma$ does not have any projective invariant subspaces, then $$\Omega_{\Kul}(\Gamma)=Eq(\Gamma).$$ Moreover, $\Omega_{\Kul}(\Gamma)$ is complete Kobayashi hyperbolic and is the largest open set on which the group acts properly discontinuously.
The Geometry of the Veronese Curve {#s:gever}
===================================
Now let us define the Veronese embedding. Set $$\begin{array}{l}
\psi:\Bbb{P}^1_\Bbb{C}\rightarrow \Bbb{P}^2_\Bbb{C}\\
\psi([z,w])=[z^2,2zw, w^2].
\end{array}$$
Let us consider $\iota: \PSL(2,\Bbb{C})\rightarrow \PSL(3,\Bbb{C})$ given by $$\iota\left(\frac{az+b}{cz+d}\right )=\left [\left [
\begin{array}{lll}
a^2&ab&b^2\\
2ac&ad+bc&2bd\\
c^2&dc&d^2\\
\end{array}
\right ]\right ].$$ Trivially, $\iota$ is well defined. Note that this map is induced by the canonical action of $\SL(2,\Bbb{C})$ on the space of homogeneous polynomials of degree two in two complex variables.
\[l:mor\] The map $\iota$ is an injective group morphism.
Let $$A=\left [\left [
\begin{array}{ll}
a&b\\
c&d\\
\end{array}
\right ]\right ],\,B=\left [\left [
\begin{array}{ll}
e&f\\
g&h\\
\end{array}
\right ]\right ]\in\PSL(2,\Bbb{C}).$$ Then
$$\begin{array}{ll}
\iota (AB)
&=\iota\left [\left [
\begin{array}{ll}
ae+bg&af+bh\\
ce+dg&cf+dh\\
\end{array}
\right ]\right ]\\
&=\left [\left [
\begin{array}{lll}
(ae+bg)^2&(ae+bg)(af+bh)&( af+bh)^2\\
2(ae+bg)(ce+dg)&(cf+d
| 1,813
| 571
| 1,191
| 1,855
| null | null |
github_plus_top10pct_by_avg
|
tain a teleported state identical to the original input state. We call the channel $f_q$ reversible, if it is injective, that is, for different input state $|\Phi\rangle_A$ ($\||\Phi\rangle_A\|=1$) the corresponding output state $f_q(|\Phi\rangle_A)$ is different. We remark, that the reversibility of teleportation channels has also been investigated in Ref. [@pra55_2547]. We adopt a more general definition here. Reversibility means that every input state can be recovered (theoretically) from the output state. One can easily verify that this condition is equivalent to that the linear operator $LL_q^\dag \colon {{\mathcal H}}_A \to {{\mathcal H}}_C$ is injective.
It may be the case, however, that the channel $f_q$ is not linear. This way, the input state can be recovered from the output only using some sophisticated nonlinear transformations, which may not be realistic. Therefore, it is a natural requirement for the channel to be linear.
We show that if the teleportation channel is reversible, then its linearity is equivalent to that the probability (\[eq:p\_q\]) of the outcome $q$ is independent of the input state $|\Phi\rangle_A$. Suppose that $|\Phi\rangle_1$ and $|\Phi\rangle_2$ are linearly independent, and let $(\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2)$ be such that $\|\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2\|=1$. From the linearity condition $f_q(\alpha_1 |\Phi\rangle_1 + \alpha_2 |\Phi\rangle_2) = \alpha_1
f_q(|\Phi\rangle_1) + \alpha_2 f_q(|\Phi\rangle_2)$, one can obtain: $$\begin{gathered}
\alpha_1 \left( \frac1{\big\| LL_q^\dag (\alpha_1|\Phi\rangle_1 +
\alpha_2|\Phi\rangle_2) \big\|} - \frac1{\big\| LL_q^\dag |\Phi\rangle_1
\big\|} \right) LL_q^\dag |\Phi\rangle_1
\\
+ \alpha_2 \left( \frac1{\big\|
LL_q^\dag (\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2) \big\|} -
\frac1{\big\| LL_q^\dag |\Phi\rangle_2 \big\|} \right) LL_q^\dag
|\Phi\rangle_2 =0.
\label{eq:linfq2}\end{gathered}$$ Since $f_q$ is injective, $LL_q^\dag |\Phi\rangle_1$ and $LL_q^\dag
|\Phi\rangle_2$ a
| 1,814
| 4,930
| 381
| 1,216
| null | null |
github_plus_top10pct_by_avg
|
Y order due to the proximity to even layers. Here, however, we presume that SP scenario holds and $u_r$ vanishes at large $L$. Practically, simulations are performed at finite $u$ and we always control that the XY stiffness (helicity modulus) along the layers remains finite and independent of $L$.
Dual formulation
----------------
The dual formulation in terms of the closed loops of integer J-currents (along bonds in and between the layers) can be achieved similarly to the case $N_z=2$ by reinstating the compactness of $\phi_z$ in Eqs.(\[NNz\],\[HNNz\]) through the Villain approach: $\nabla_{ij} \phi_z \to (\nabla_{ij} \phi_z + 2\pi m_{z,ij})$ along the planes and $-u\cos(\phi_{z+1} - \phi_z) \to (u_V/2)(\phi_{z+1} - \phi_z + 2\pi m_{i,z})$ for Josephson coupling, where $m_{z,ij}$ refers to an arbitrary (oriented) integer defined on the bond $ij$ belonging to the plane $z$ and $m_{z,i}$ stands for an integer on a bond connecting site $i$ in the plane $z$ to the same site in the plane $z+1$. The partition function is obtained as a result of integration over all $\phi_z(i)$ and summations over all bond integers.
The J-currents enter through the Poisson identity $\sum_{m=0,\pm 1, \pm 2,..} f(m) \equiv \sum_{J=0,\pm 1, \pm 2,..} \int dx \exp(2\pi \i J x)f(x)$ applied to each bond integer. This allows explicit integration over all phases $\phi_z$ as well as over the bond integers $m_{z,ij}, m_{i,z}$. There are two types of J-currents: inplane $J^{(a)}_{z,ij},\, a=1,2$ within each “elementary cell” (along $z$) and between the planes $J_{i,z}$. The label $a=1$ refers to J-current defined on the bond $ij$ belonging to a plane with odd $z$. Accordingly, $J^{(2)}_{z,ij}$ stands for the inplane current on the layer with even $z$. Then, $J_{i,z}$ denotes the current from the site $i$ from the plane $z$ to the plane $z+1$. The integration over phases $\phi$ generates the Kirchhoff constraint — similarly to the bilayer case.
Finally the ensemble can be represented as Z&=& \_[{}]{} (- H\_J ),\
H\_J&=& \_[ij; z,z’]{
| 1,815
| 1,115
| 2,601
| 1,896
| 2,743
| 0.777352
|
github_plus_top10pct_by_avg
|
of the CW2 attack on the average of defense arrangements is investigated in Supplementary Material. While it may seem that our proposed defense scheme is easily fooled by a strong attack such as CW2, there are still ways of recovering from such attacks by using detection. In fact, there will always be perturbations that are extreme enough to fool any classifier, but perturbations with larger magnitude become increasingly easy for humans to notice. In this section, “classifier” is the model we are trying to defend, and “auxiliary model” is another model we train (a triplet network) that is combined with the classifier to create a detector.
Transferability of Attacks Between LeNet and Triplet Network
------------------------------------------------------------
The best case scenario for an auxiliary model would be if it were fooled only by a negligible percentage of the images that fool the classifier. It is also acceptable for the auxiliary model to be fooled by a large percentage of those images, provided it does not make the same misclassifications as the classifier. Fortunately, we observe that the majority of the perturbed images that fooled LeNet did not fool the triplet network in a way that would affect the detector’s viability. This is shown in Table \[table:transferability\_classifier\_detector\]. For example, 1060 perturbed images created for LeNet using FGS fooled the triplet network. However, only 70 images were missed by the detector due to the requirement for agreement between the auxiliary model and classifier. The columns with “Relevant” in the name show the success rate of transfer attacks that would have fooled the detector.
------------ ------- ------- ------- ------- ------- -------
(r)[2-7]{}
FGS 0.089 0.106 0.164 0.015 0.007 0.004
IGS 0.130 0.094 0.244 0.004 0.007 0.002
CW2 0.990 0.121 0.819 0.013 0.049 0.008
| 1,816
| 1,648
| 2,667
| 1,831
| null | null |
github_plus_top10pct_by_avg
|
-1\] in the proof of Proposition \[pre-cohh\] and so we will just indicate how to modify the earlier proofs to work here.
{#B-freeC}
Since $M(k)$ is a $(H_{c+k},{U}_c)$-bimodule, the embeddings ${\mathbb{C}}[{\mathfrak{h}}]\hookrightarrow
H_{c+k}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}\hookrightarrow {U}_c$ make $M(k)$ into a $({\mathbb{C}}[{\mathfrak{h}}],\, {\mathbb{C}}[{\mathfrak{h}}^*]^{{W}})$-bimodule. Let ${\mathbb{C}}$ be the trivial module over either ${\mathbb{C}}[{\mathfrak{h}}]$ or ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ and set $
\overline{M(k)} = {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]} M(k)$ and $
\underline{M(k)} = M(k) \otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}} {\mathbb{C}}.$
\[Bbar-freeC\] [(1) ]{} $M(k)$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module and a right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module.
1. $\underline{M(k)}$ is a finitely generated, free left ${\mathbb{C}}[{\mathfrak{h}}]$-module.
2. Analogously, $\overline{M(k)}$ is a finitely generated, free right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module.
\(1) By Corollary \[morrat-cor\], $M(k)$ is projective as a left $H_{c+k}$-module and as a right ${U}_c$-module. By , $H_{c+k}$ and hence $M(k)$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module. Similarly, the argument of Lemma \[Bbar-freeA\](2) shows that $U_c$ and hence $M(k)$ are free right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-modules.
\(2) This is contained in the proof of Lemma \[Bbar-freeA\](3).
\(3) Mimic the proof of Lemma \[Bbar-freeA\](4).
{#filter-injC}
Using the conventions from , each $M(k)$ and $J^{k-1}\delta^{k}e$ is ${\mathbf{E}}$-graded. Since ${\mathbb{C}}[{\mathfrak{h}}]_+$ is ${\mathbf{E}}$-graded, the ${\mathbf{E}}$-grading on $M(k)$ descends to one on $\overline{M(k)}$. Similarly, $J^{k-1}\delta^{k}e$ has the order grading $\operatorname{{\textsf}{ogr}}$ from . Write $\Theta: J^{k-1}\delta^ke\hookrightarrow \operatorname{{\textsf}{ogr}}M(k)$ for the inclusion from Lemma \[thetainjC\].
There exists an injec
| 1,817
| 1,110
| 780
| 1,744
| 3,565
| 0.771522
|
github_plus_top10pct_by_avg
|
ng a vbo and vao from some raw data, in this case a pointer to an array of floats.
RawModel* Loader::loadToVao(float* positions, int sizeOfPositions) {
unsigned int vaoID = this->createVao();
this->storeDataInAttributeList(vaoID, positions, sizeOfPositions);
this->unbindVao();
return new RawModel(vaoID, sizeOfPositions / 3);
}
unsigned int Loader::createVao() {
unsigned int vaoID;
glGenVertexArrays(1, &vaoID);
glBindVertexArray(vaoID);
unsigned int copyOfVaoID = vaoID;
vaos.push_back(copyOfVaoID);
return vaoID;
}
void Loader::storeDataInAttributeList(unsigned int attributeNumber, float* data, int dataSize) {
unsigned int vboID;
glGenBuffers(1, &vboID);
glBindBuffer(GL_ARRAY_BUFFER, vboID);
glBufferData(GL_ARRAY_BUFFER, dataSize * sizeof(float), data, GL_STATIC_DRAW);
glVertexAttribPointer(attributeNumber, 3, GL_FLOAT, false, 0, 0);
glBindBuffer(GL_ARRAY_BUFFER, 0);
unsigned int copyOfVboID = vboID;
vbos.push_back(copyOfVboID);
}
void Loader::unbindVao() {
glBindVertexArray(0);
}
The RawModel is just a class that should take in the array of floats and create a vbo and a vao. The vectors vbos and vaos that I am using are just there to keep track of all the ids so that I can delete them once I am done using all the data.
I am 90% confident that this should all work properly. However, when I go to try and just run some code that would draw it, OpenGL is exiting because it is trying to read from address 0x0000000 and it doesn't like that. I pass the raw model that I created from the code before this into a function in my renderer that looks like this:
void Renderer::render(RawModel* model) {
glBindVertexArray(model->getVaoID());
glEnableVertexAttribArray(0);
glDrawArrays(GL_TRIANGLES, 0, model->getVertexCount());
glDisableVertexAttribArray(0);
glBindVertexArray(0);
}
I have checked to make sure that the VaoID is the same when I am creating the vao, and when I am trying to retrieve the vao. It is in fact the same.
| 1,818
| 1,907
| 206
| 772
| 95
| 0.826729
|
github_plus_top10pct_by_avg
|
[ea3\]), (\[ea3’\]), (\[ea7\]), (\[ea8\]), (\[ea9\]), (\[ea10\]), (\[13\]), (\[ea14\]), (\[ea15\]), (\[ea16\]), (\[ea17\]), (\[ea19\])) stated in the proof of Theorem \[ta4\] in order to compute $h\circ (1+\pi y)$. Based on this, we enumerate equations which $m$ satisfies as follows:
1. Assume that $i<j$. By Equation (\[ea3\]) which involves an element of $\tilde{M}^1(R)$, each entry of $f_{i,j}'$ has $\pi$ as a factor so that $f_{i,j}'\equiv f_{i,j} (=0)$ mod $(\pi\otimes 1)(B\otimes_AR)$. In other words, the $(i,j)$-block of $h\circ (1+x)(1+\pi y)$ divided by $\pi^{max\{i, j\}}$ is $f_{i,j} (=0)$ modulo $(\pi\otimes 1)(B\otimes_AR)$, which is independent of the choice of $1+\pi y$. Let $\tilde{m}\in \mathrm{Ker~}\tilde{\varphi}(R)$ be a lift of $m$. Therefore, if we write the $(i, j)$-block of $\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$ as $\pi^{max\{i, j\}}\mathcal{X}_{i,j}(\tilde{m})$, where $\mathcal{X}_{i,j}(\tilde{m}) \in M_{n_i\times n_j}(B\otimes_AR)$, then the image of $\mathcal{X}_{i,j}(\tilde{m})$ in $M_{n_i\times n_j}(B\otimes_AR)/(\pi\otimes 1)M_{n_i\times n_j}(B\otimes_AR)\cong M_{n_i\times n_j}(R)$ is independent of the choice of the lift $\tilde{m}$ of $m$. Therefore, we may denote this image by $\mathcal{X}_{i,j}(m)$. On the other hand, by Equation (\[ea2\]), we have the following identity: $$\label{ea20}
\mathcal{X}_{i,j}(m)=\sum_{i\leq k \leq j} \sigma({}^tm_{k,i})\bar{h}_km_{k,j} \mathrm{~if~}i<j.$$ We explain how to interpret the above equation. We know that $\mathcal{X}_{i,j}(m)$ and $m_{k,k'}$ (with $k\neq k'$) are matrices with entries in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$, whereas $m_{i,i}$ and $m_{j,j}$ are formal matrices as explained in Remark \[ra5\]. Thus we consider $\bar{h}_k$, $m_{i,i}$, and $m_{j,j}$ as matrices with entries in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$ by letting $\pi$ be zero in each entry of formal matrices $h_k$, $m_{i,i}$, and $m_{j,j}$. Then the right hand side is computed as a sum of products of matrices (involving the usual matrix add
| 1,819
| 348
| 1,562
| 1,965
| 3,149
| 0.774483
|
github_plus_top10pct_by_avg
|
/{(\pi x)}} \exp{\left[ -ix +i{|\nu|\pi}/{2} +i{\pi}/{4} \right]}$. Classically the limit $x \gg 1 $ corresponds to advanced solution called the Bunch-Davies vacuum $e^{-ik\tau}/\sqrt{2k}$. Generally the limit obtained here differs from the classical one but can be restored taking $l=2$. Then to obtain the proper high energy limit we must choose $D_2=0$ and $D_1=\exp{\left[ i{|\nu|\pi}/{2} +i{\pi}/{4} \right]}$. Applying evaluated values of $D_1$ and $D_2$ to the solution (\[solmodes2\]) we finally obtain mode functions for the initial state $$\begin{aligned}
f_i(k,\tau) &=& \mathcal{N} \frac{1}{\sqrt{k}}\sqrt{-k\tau+k\beta} H^{(1)}_{|\nu|}(x) \\
g_i(k,\tau) &=& \mathcal{N} \sqrt{k}\sqrt{-k\tau+k\beta}
\left[ -\frac{1}{2} \frac{ H^{(1)}_{|\nu|}(x) }{-k\tau+k\beta } -\frac{2+np}{|2+np|}\sqrt{D_*\xi^n}(-\tau+\beta)^{\frac{np}{2}}
\left( \frac{|\nu|}{x}H^{(1)}_{|\nu|}(x)-H^{(1)}_{|\nu|+1}(x) \right) \right]\end{aligned}$$ where $$\mathcal{N}=e^{i\left( \frac{|\nu|\pi}{2} +\frac{\pi}{4} \right)} \sqrt{\frac{\pi}{2|2+np|}}.$$ The functions $g_i(k,\tau)$ were calculated from relation $f(k,\tau)'=g(k,\tau)$ showed earlier. We had used also the expression for derivative of the Hankel functions in the form $$\frac{dH^{(1)}_{|\nu|}(x)}{dx} =\frac{|\nu|}{x}H^{(1)}_{|\nu|}(x)-H^{(1)}_{|\nu|+1}(x).$$ Similar investigations lead to the expression for the mode functions for the final state. We must remember that it is however not only a simple set $l=2$ but also the change of the solution for the scale factor to this expressed by (\[solution2\]). In this case we obtain $$\begin{aligned}
f_f(k,\tau) &=& e^{i\frac{\pi}{4}} \sqrt{\frac{\pi}{4k}} \ \sqrt{k\tau+k\zeta}H^{(1)}_0(k\tau+k\zeta), \\
g_f(k,\tau) &=& e^{i\frac{\pi}{4}} \sqrt{\frac{\pi k}{4}}\sqrt{k\tau+k\zeta}
\left[ \frac{1}{2} \frac{H^{(1)}_0(k\tau+k\zeta)}{k\tau+k\zeta} - H^{(1)}_1(k\tau+k\zeta) \right].\end{aligned}$$
Now we are ready to consider the creation of the gravitons during the transition from some initial to final states. The initial vacu
| 1,820
| 3,530
| 1,688
| 1,620
| null | null |
github_plus_top10pct_by_avg
|
sumptions , and the operator $A_1(E)$ is bounded for any fixed $E\in I$, and collectively they obey a uniform bound, $$\begin{aligned}
\label{eq:A1E_unif_bound}
\sup_{E\in I} {\left\Vert A_1(E)\right\Vert}\leq \kappa^{-1}\Big({\left\Vert \tilde \Sigma\right\Vert}_{L^\infty(G\times S\times I)}+
{\left\Vert {{\frac{\partial \tilde S}{\partial E}}}\right\Vert}_{L^\infty(G\times I)}+{M_1}^{1/2}{M_2}^{1/2}\Big)=:C_0'<\infty,\end{aligned}$$ where $M_j\geq 0$, $j=1,2$ are as in .
The (uniform) estimate follows immediately from the assumptions and the estimate .
Using the above notations we have for $C:=C_0+C_0'$ the following decomposition \[eq:ACE\_decomp\] A\_C(E)=A\_0(E)-C\_0I-C\_0’I+A\_1(E). Recall that $C_0$ and $C_0'$ were defined in and . Since by Lemma \[csdale2\], $$&{\left\langle}(-C_0'I+A_1(E))\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}=-C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}+{\left\langle}A_1(E))\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}\nonumber\\
\leq &
-C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}+C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}=0,$$ we see that $-C_0'I+A_1(E)$ is bounded and dissipative. On the other hand, according to Lemma \[csdale1\], $A_0(E)-C_0I$ is $m$-dissipative. Hence $A_C(E)$ is $m$-dissipative for any $E\in I$ (cf. [@engelnagel Chapter III, Theorem 2.7], or [@tervo14 Theorem 4.2]). We record this observation into the next lemma.
\[le:ACE\_m\_diss\] For $C=C_0+C_0'$ and for every fixed $E\in I$, the operator $A_C(E)$ is $m$-dissipative.
We shall assume that $$\tilde{\sigma}\in C(I,L^\infty(G\times S,L^1(S')))\cap
C(I,L^\infty(G\times S',L^1(S))),$$ where $\tilde{\sigma}$ interpreted as an element of $C(I,L^\infty(G\times S,L^1(S')))$ is $$\tilde\sigma(E)(x,\omega)(\omega')=\tilde\sigma(x,\omega',\omega,E),$$ and when interpreted as an element of $C(I,L^\infty(G\times S',L^1(S)))$ it is $$\tilde{\sigma}(E)(x,\omega')(\omega) =
\tilde\sigma(x,\omega',\omega,E).$$ Furthermore, in order to avoid ambiguity, we have
| 1,821
| 1,017
| 1,149
| 1,644
| null | null |
github_plus_top10pct_by_avg
|
d by $$\|\Sigma^{-1}\|^2_{\mathrm{op}} \frac{\|
\hat{\Sigma} - \Sigma
\|_{\mathrm{op}} } { 1 - \|\hat{\Sigma} - \Sigma \|_{\mathrm{op}} \|
\Sigma^{-1}\|_{\mathrm{op}} }.$$ The matrix Bernstein inequality along with the assumption that $U \geq \eta > 0$ yield that, for a positive $C$ (which depends on $\eta$), $$\|\hat{\Sigma} - \Sigma \|_{\mathrm{op}} \leq C A \sqrt{ k U \frac{ \log k +
\log n}{n}},$$ with probability at least $1 - \frac{1}{n}$. Using the fact that $\| \Sigma^{-1} \|_{\mathrm{op}} \leq
\frac{1}{u}$ and the assumed asymptotic scaling on $B_n$ we see that $\|
\Sigma^{-1} E \|_{\mathrm{op}} \leq 1/2$ for all $n$ large enough. Thus, for all such $n$, we obtain that, with probability at least $ 1-
\frac{1}{n}$, $$T_2 \leq 2 C A
\frac{ k}{u^2}
\sqrt{ U \frac{ \log k +
\log n}{n}},$$ since $\| \hat{\alpha} \| \leq A \sqrt{k}$ almost surely. Thus we have shown that holds, with probability at least $1 - \frac{2}{n}$ and for all $n$ large enough. This bound holds uniformly over all $P \in \mathcal{P}_n^{\mathrm{OLS}}$. $\Box$
[**Proof of .**]{} In what follows, any term of the order $\frac{1}{n}$ are absorbed into terms of asymptotic bigger order.
As remarked at the beginning of this section, we first condition on $\mathcal{D}_{1,n}$ and the outcome of the sample splitting, so that ${\widehat{S}}$ is regarded as a fixed non-empty subset $S$ of $\{1,\ldots,d\}$ of size at most $k$. The bounds and are established using and from , where we may take the function $g$ as in , $s = k$, $b =
\frac{k^2 + 3k}{2} $ and $\psi = \psi_{{\widehat{S}}}$ and $\hat{\psi}_{{\widehat{S}}}$ as in and , respectively. As already noted, $\psi$ is always in the domain of $g$ and, as long as $n \geq d$, so is $\hat{\psi}$, almost surely. A main technical difficulty in applying the results of is to obtain good approximations for the quantities $\underline{\sigma}, \overline{H}$ and $B$. This can be accomplished using the bounds provided in below, which rely on matrix-calculus. Even so, the claim
| 1,822
| 1,732
| 1,712
| 1,538
| null | null |
github_plus_top10pct_by_avg
|
$ as a $\kappa$-variety so that the number of rational points is $2f$, where $f$ is the cardinality of $\kappa$.
Construction following our technique {#cfot}
------------------------------------
Let $q$ be the function defined over $L$ such that $$q : L\longrightarrow A, l\mapsto h(l,l).$$ If we write $l=x+\pi y$ such that $x, y \in A$, then $q(l)=h(x+\pi y, x+\pi y)=x^2-2\delta y^2$. Thus $q$ mod 2 is an additive polynomial over $\kappa$. Let $B(L)$ be the sublattice of $L$ such that $B(L)/\pi L$ is the kernel of the additive polynomial $q$ mod 2 on $L/\pi L$. In this case, $B(L)=\pi L$.
We define another sublattice $Z(L)$ of $L$ such that $Z(L)/\pi B(L)$ is the radical of the quadratic form $\frac{1}{2}q$ mod 2 on $B(L)/\pi B(L)$. In this case, $Z(L)=\pi B(L)=2L$.
For an étale $A$-algebra $R$ with $g\in \mathrm{Aut}_{B\otimes_AR}(L\otimes_AR, h\otimes_AR)$, it is easy to see that $g$ induces the identity on $L/B(L)=L/\pi L$. An element $g$ also induces the identity on $L/Z(L)=L/2L$ since $$q(gw-w, gw-w)=2q(w)-\left(h(gw, w)+h(w, gw)\right)=$$ $$2q(w)-\left(h(w+x, w)+h(w, w+x)\right) =-\left(h(x, w)+h(w, x)\right)$$ where $w\in L$ and $x\in B(L)=\pi L$ such that $gw=w+x$, and thus $h(x, w)\in \pi B$ and $\left(h(x, w)+h(w, x)\right)\in 4 B$.
Based on this, we construct the following functor from the category of commutative flat $A$-algebras to the category of monoids as follows. For any commutative flat $A$-algebra $R$, set $$\underline{M}(R) = \{m \in \mathrm{End}_{B\otimes_AR}(L \otimes_A R)\} ~|~ \textit{$m$ induces the identity on $L\otimes_A R/ Z(L)\otimes_A R$}\}.$$ This functor $\underline{M}$ is then representable by a polynomial ring and has the structure of a scheme of monoids. Let $\underline{M}^{\ast}(R)$ be the set of invertible elements in $\underline{M}(R)$ for any commutative $A$-algebra $R$. Then $\underline{M}^{\ast}$ is representable by a group scheme which is an open subscheme of $\underline{M}$ (Section \[m\]). Thus $\underline{M}^{\ast}$ is smooth. Indeed, in our case, $\underline{M}^
| 1,823
| 1,227
| 1,141
| 1,815
| null | null |
github_plus_top10pct_by_avg
|
by Eq. (\[Eqn: chaos1\]), serves to complete the significance of the tower by capping it with a “boundary” element that can be taken to bridge the classes of functional and non-functional relations on $X$.
We are now ready to define a *maximally ill-posed problem $f(x)=y$* for *$x,y\in X$* in terms of a *maximally non-injective map $f$* as follows.
**Definition 4.1.** ***Chaotic map.*** *Let $A$ be a non-empty closed set of a compact Hausdorff space $X.$ A function* $f\in\textrm{Multi}(X)$ **(*equivalently the sequence of functions $(f_{i})$*)** *is* *maximally non-injective* *or* *chaotic on* **$A$** *with respect to the order relation* **(\[Eqn: chaos1\])** *if*
*(a) for any $f_{i}$ on $A$ there exists an $f_{j}$ on $A$ satisfying $f_{i}\preceq f_{j}$ for every $j>i\in\mathbb{N}$.*
*(b) the set $\mathcal{D}_{+}$ consists of a countable collection of isolated singletons.$\qquad\square$*
**Definition 4.2.** ***Maximally ill-posed problem.*** *[<span style="font-variant:small-caps;">L</span>]{}et $A$ be a non-empty closed set of a compact Hausdorff space $X$ and let $f$ be a functional relation in* $\textrm{Multi}(X)$*. The problem $f(x)=y$ is* *maximally ill-posed at* **$y$** *if $f$ is chaotic on $A$*.$\qquad\square$
As an example of the application of these definitions, on the dense set $\mathcal{D}_{+}$, the tent map satisfies both the conditions of sensitive dependence on initial conditions and topological transitivity [@Devaney1989] and is also maximally non-injective; the tent map is therefore chaotic on $\mathcal{D}_{+}.$ In contrast, the examples of Secs. 1 and 2 are not chaotic as the maps are not topologically transitive, although the Liapunov exponents, as in the case of the tent map, are positive. Here the $(f_{n})$ are identified with the iterates of $f,$ and the “fixed point” as one through which graphs of all the functions on residual index subsets pass. When the set of points $\mathcal{D}_{+}$ is dense in $[0,1]$ and both $\mathcal{D}_{+}$ and $[0,1]-\mathcal{D}_{+}=[0,1]-\bigcup_{i
| 1,824
| 2,784
| 2,835
| 1,658
| 2,449
| 0.779621
|
github_plus_top10pct_by_avg
|
TR pCmdLine, int nCmdShow)
{
// Register the window class.
const wchar_t CLASS_NAME[] = L"Sample Window Class";
WNDCLASS wc = { };
wc.lpfnWndProc = WindowProc;
wc.hInstance = hInstance;
wc.lpszClassName = CLASS_NAME;
RegisterClass(&wc);
// Create the window.
HWND hwnd = CreateWindowEx(
0, // Optional window styles.
CLASS_NAME, // Window class
L"Learn to Program Windows", // Window text
WS_EX_APPWINDOW, // Window style
// Size and position
CW_USEDEFAULT, CW_USEDEFAULT, CW_USEDEFAULT, CW_USEDEFAULT,
NULL, // Parent window
NULL, // Menu
hInstance, // Instance handle
NULL // Additional application data
);
if (hwnd == NULL)
{
return 0;
}
ShowWindow(hwnd, nCmdShow);
// Run the message loop.
MSG msg = { };
while (GetMessage(&msg, NULL, 0, 0))
{
TranslateMessage(&msg);
DispatchMessage(&msg);
}
return 0;
}
LRESULT CALLBACK WindowProc(HWND hwnd, UINT uMsg, WPARAM wParam, LPARAM lParam)
{
switch (uMsg)
{
case WM_DESTROY:
PostQuitMessage(0);
return 0;
case WM_PAINT:
{
PAINTSTRUCT ps;
HDC hdc = BeginPaint(hwnd, &ps);
SelectObject(bmpdc, hBmp_red);
StretchBlt(memdc, 0, 0, 100, 100, bmpdc, 0, 0, 10, 10, SRCCOPY);
membmp = CreateCompatibleBitmap(hdc, 100, 100);
SelectObject(memdc, membmp);
BitBlt(hdc, 0, 0, 100, 100, memdc, 100, 100, SRCCOPY);
EndPaint(hwnd, &ps);
}
return 0;
}
return DefWindowProc(hwnd, uMsg, wParam, lParam);
}
A:
Пример из MSDN
HDC memDC = CreateCompatibleDC ( hDC );
HBITMAP memBM = CreateCompatibleBitmap ( hDC, nWidth, nHeight );
SelectObject ( memDC, memBM );
А рисовать нужно в обработчике сообщения WM_PAINT внутри Begin/EndPaint
В общем, логика т�
| 1,825
| 6,640
| 52
| 1,355
| 369
| 0.811851
|
github_plus_top10pct_by_avg
|
over the previous best result (47% $\rightarrow$ 53%) on the [*Success Rate weighted by Path Length*]{} metric.'
author:
- |
Xiujun Li^$\spadesuit\diamondsuit$^Chunyuan Li^$\diamondsuit$^Qiaolin Xia^$\clubsuit$^Yonatan Bisk^$\spadesuit\diamondsuit\heartsuit$^\
**[Asli Celikyilmaz]{}^$\diamondsuit$^**[Jianfeng Gao]{}^$\diamondsuit$^**[Noah A. Smith]{}^$\spadesuit\heartsuit$^**[Yejin Choi]{}^$\spadesuit\heartsuit$^\
^$\spadesuit$^Paul G. Allen School of Computer Science & Engineering, University of Washington\
^$\clubsuit$^Peking University^$\diamondsuit$^Microsoft Research AI^$\heartsuit$^Allen Institute for Artificial Intelligence\
[{xiujun,ybisk,nasmith,yejin}@cs.washington.edu]{}\
[xql@pku.edu.cn{xiul,chunyl,jfgao}@microsoft.com]{}********
bibliography:
- 'emnlp-ijcnlp-2019.bib'
title: Robust Navigation with Language Pretraining and Stochastic Sampling
---
---
abstract: 'The mass of the axion and its decay rate are known to depend only on the scale of Peccei-Quinn symmetry breaking, which is constrained by astrophysics and cosmology to be between $10^9$ and $10^{12}$ GeV. We propose a new mechanism such that this effective scale is preserved and yet the fundamental breaking scale of $U(1)_{PQ}$ is very small (a kind of inverse seesaw) in the context of large extra dimensions with an anomalous U(1) gauge symmetry in our brane. Unlike any other (invisible) axion model, there are now possible collider signatures in this scenario.'
---
plus 1pt ‘@=12 -0.5in 0.0in 0.0in 8.5in 6.5in
UCRHEP-T283\
July 2000
[**Low-Scale Axion from Large Extra Dimensions\
**]{}
Although CP violation has been observed in weak interactions [@cp1; @cp2] and it is required for an explanation of the baryon asymmetry of the universe [@asym], it becomes a problem in strong interactions. The reason is that the multiple vacua of quantum chromodynamics (QCD) connected by instantons [@insta] require the existence of the CP violating $\theta$ term [@theta] $${\cal L}_\theta = \theta_{QCD} {g_s^2 \over 32 \pi^2}
G
| 1,826
| 965
| 851
| 1,671
| null | null |
github_plus_top10pct_by_avg
|
distribution free framework. For notational convenience, in this section we let $\theta_{{\widehat{S}}}$ be any of the parameters of interest: $\beta_{{\widehat{S}}}$, $\gamma_{{\widehat{S}}}$, $\phi_{{\widehat{S}}}$ or $\rho_{{\widehat{S}}}$.
We will rely on sample splitting: assuming for notational convenience that the sample size is $2n$, we randomly split the data $\mathcal{D}_{2n}$ into two halves, $\mathcal{D}_{1,n}$ and $\mathcal{D}_{2,n}$. Next, we run the model selection and estimation procedure $w_{n}$ on $\mathcal{D}_{1,n}$, obtaining both ${\widehat{S}}$ and $\hat{\mu}_{{\widehat{S}}}$ (as remarked above, if we are concerned with the projection parameters, then we will only need ${\widehat{S}}$). We then use the second half of the sample $\mathcal{D}_{2,n}$ to construct an estimator $\hat \theta_{{\widehat{S}}}$ and a confidence hyper-rectangle $\hat{C}_{{\widehat{S}}}$ for $\theta_{{\widehat{S}}}$ satisfying the following properties:
$$\begin{aligned}
{\rm Concentration}: \phantom{xxxxxxxxx}&\ \displaystyle \limsup_{n
\rightarrow \infty} \sup_{w_{n}\in {\cal W}_{n}} \sup_{P\in {\cal Q}_n}
\mathbb{P}(||\hat\theta_{{\widehat{S}}}-\theta_{{\widehat{S}}}||_\infty > r_n) \to 0 \label{eq:concentration}\\
\vspace{.11pt} \nonumber\\
{\rm Coverage\ validity\ (honesty)}: &\
\displaystyle\liminf_{n\to\infty}\inf_{w_{n}\in {\cal W}_{n}} \inf_{P\in
{\cal Q}_{n}}
\mathbb{P}(\theta_{{\widehat{S}}}\in \hat{C}_{{\widehat{S}}})\geq 1-\alpha \label{eq::honest}\\
\vspace{.11pt} \nonumber \\
{\rm Accuracy}: \phantom{xxxxxxxx}&\
\displaystyle \limsup_{n
\rightarrow \infty} \sup_{w_{n}\in {\cal W}_{n}} \sup_{P\in {\cal
Q}_n}\mathbb{P}(\nu(\hat{C}_{{\widehat{S}}})> \epsilon_n)\to 0 \label{eq::accuracy}\end{aligned}$$
where $\alpha \in (0,1)$ is a pre-specified level of significance, $\mathcal{W}_n$ is the set of all the model selection and estimation procedures on samples of size $n$, $r_n$ and $\epsilon_n$ both vanish as $n \rightarrow \infty$ and $\nu$ is the size of the set (length of the sid
| 1,827
| 3,870
| 1,930
| 1,503
| null | null |
github_plus_top10pct_by_avg
|
ernabei:2013cfa; @Aalseth:2012if; @Angloher:2011uu; @Agnese:2013rvf; @Fan:2013faa] – begin to set tight limits (with some conflicting signal hints) on the standard WIMP scenario with a contact interaction to quarks. This makes it necessary to look for a more complete set of DM models which are theoretically motivated while giving unique experimental signatures.
![ The quark-level Feynman diagrams responsible for DM-nucleus scattering in *Dark Mediator Dark Matter* (dmDM). Left: the $2\rightarrow3$ process at tree-level. Right: the loop-induced $2\rightarrow2$ process. The arrows indicate flow of dark charge. []{data-label="f.feynmandiagram"}](dd_process_prl__1){width="8cm"}
Dark Mediator Dark Matter
=========================
Given its apparently long lifetime, most models of DM include some symmetry under which the DM candidate is charged to make it stable. An interesting possibility is that not only the DM candidate, but also the mediator connecting it to the visible sector is charged under this dark symmetry. Such a ‘dark mediator’ $\phi$ could only couple to the SM fields in pairs.
As a simple example, consider real or complex SM singlet scalars $\phi_i$ coupled to quarks, along with Yukawa couplings to a Dirac fermion DM candidate $\chi$. The terms in the effective Lagrangian relevant for direct detection are $$\mathcal{L}_\mathrm{DM} \supset
\displaystyle{\sum_{i,j}^{n_{\phi}}}\,\frac{1}{\Lambda_{ij}} \bar q\,q \,\phi_i \phi_j^* + \displaystyle{\sum_{i}^{n_{\phi}}}\left ( y^{\phi_i}_\chi \overline{\chi^c}\chi \phi_i + h.c. \right)+...
\label{eq:dmDM}$$ where $\ldots$ stands for $\phi, \chi$ mass terms, as well as the rest of the dark sector, which may be more complicated than this minimal setup. This interaction structure can be enforced by a $\mathbb{Z}_4$ symmetry. To emphasize the new features of this model for direct detection, we focus on the minimal case with a single mediator $n_\phi = 1$ (omitting the $i$-index). However, the actual number of dark mediators is important for interpreting
| 1,828
| 616
| 2,657
| 1,859
| 1,656
| 0.787126
|
github_plus_top10pct_by_avg
|
± 4.3 326.0 ± 3.8 284.5 ± 4.6 242.1 ± 5.6 280.2 ± 4.7 79.4 ± 0.5 66.4 ± 4.3 103.7 ± 3.5 96.2 ± 3.0
Sin 7 413.2 ± 2.8 427.6 ± 12.3 240.0 ± 8.1 302.8 ± 1.0 330.7 ± 11.5 306.9 ± 12.2 247.6 ± 7.8 291.8 ± 4.9 80.0 ± 2.8 71.8 ± 2.2 103.5 ± 4.5 96.4 ± 1.6
Sin 8 415.0 ± 11.1 418.1 ± 12.6 257.8 ± 6.4 296.9 ± 6.1 335.3 ± 11.9 298.8 ± 2.0 263.5 ± 4.3 313.1 ± 8.0 80.8 ± 1.2 71.7 ± 2.5 102.3 ± 0.9 105.7 ± 4.1
Sin 9 430.0 ± 9.1 426.2 ± 2.1 264.0 ± 4.3 307.3 ± 11.3 329.7 ± 3.8 293.2 ± 0.3 267.2 ± 5.3 309.4 ± 3.6 76.8 ± 2.3 68.8 ± 0.4 101.2 ± 0.9 101.1 ± 3.9
Sin 10 402.0 ± 12.6 432.3 ± 3.4 250.9 ± 2.1 298.6 ± 3.2 327.3 ± 10.3 286.4 ± 4.8 254.0 ± 1.6 316.8 ± 6.2 81.6 ± 3.0 66.3 ± 1.2 101.2 ± 0.2 106.1 ± 2.5
Hyb 1 450.6 ± 14.0 468.4 ± 13.4 284.3 ± 13.1 270.0 ± 9.3 317.3 ± 5.4 265.7 ± 4.0 269.7 ± 7.6 242.2 ± 23.6 70.5 ± 1.0 56.9 ± 2.1 95.4 ± 4.3 89.3 ± 7.0
Hyb 2 421.6 ± 7.8 472.6 ± 6.0 277.6 ± 5.4 272.8 ± 13.2 319.6 ± 4.5 268.3 ± 6.1 270.2 ± 5.2 245.0 ± 12.8 75.9 ± 1.5 56.8 ± 2.0 97.4 ± 2.1 90.3 ± 5.7
Hyb 3 419.5 ± 6.7 474.0 ± 14.6 255.7 ± 5.7 262.0 ± 5.3 316.6 ± 7.8 274.7 ± 8.1 250.2 ± 8.0 231.2 ± 4.2 75.5 ± 1.2 58.0 ± 1.5 97.8 ± 1.3 88.3 ± 1.8
Hyb 4 407.1 ± 6.1
| 1,829
| 5,177
| 879
| 998
| null | null |
github_plus_top10pct_by_avg
|
2}k_{j-2l+2, j-2l}'=0.$$ Then the sum of equations $$\sum_{0\leq l \leq m_j}\mathcal{Z}_{j-2l}'$$ is the same as $$\sum_{0\leq l \leq m_j}z_{j-2l}'=0$$ since $k_{j-2l+2, j-2l}'=k_{j-2l,j-2l+2}'$. Therefore, among Equations (\[ea19\]) for $j-2m_j \leq i \leq j$, only one of them is redundant. In conclusion, there are $$\#\{i:\textit{$i$ is even and $L_i$ is of type I}\}-\#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}$ is of type II}\}$$ independent linear equations of the form $z_i'=0$.\
We now combine all works done in this proof. Namely, we collect the above (i), (ii), (iii), (iv), (v), (vi) which are the interpretations of Equation (\[ea5\]), together with Equations (\[ea4\]) and (\[ea4’\]). To simplify notation, we say just in this paragraph that Equation (\[ea4’\]) is linear. Then there are exactly $$\sum_{i<j}n_in_j+\sum_{i:\mathrm{odd}}\frac{n_i^2+n_i}{2}-\#\{i:\textit{$i$ is odd and $L_i$ is free of type I}\}+
\sum_{i:\mathrm{even}}\frac{n_i^2-n_i}{2}+$$ $$\#\{i:\textit{$i$ is even and $L_i$ is of type I}\}-\#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}$ is of type II}\}$$ independent linear equations among the entries of $m\in \tilde{M}^1(R)$. Furthermore, all coefficients of these equations are in $\kappa$. Therefore, we consider $\tilde{G}^1$ as a subvariety of $\tilde{M}^1$ determined by these linear equations. Since $\tilde{M}^1$ is an affine space of dimension $n^2$, the underlying algebraic variety of $\tilde{G}^1$ over $\kappa$ is an affine space of dimension $$\sum_{i<j}n_in_j+\sum_{i:\mathrm{even}}\frac{n_i^2+n_i}{2}+\sum_{i:\mathrm{odd}}\frac{n_i^2-n_i}{2}
+\#\{i:\textit{$i$ is odd and $L_i$ is free of type I}\}$$ $$-\#\{i:\textit{$i$ is even and $L_i$ is of type I}\}
+ \#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}$ is of type II}\}.$$ This completes the proof by using a theorem of Lazard which is stated at the beginning of Appendix \[App:AppendixA\].
**
Let $R$ be a $\kappa$-algebra. We describe the functor of points of the scheme $\mathrm{Ker~}\
| 1,830
| 666
| 1,278
| 1,767
| 4,006
| 0.768673
|
github_plus_top10pct_by_avg
|
lde s)}2\tilde K}.\end{aligned}$$ Moreover, and imply that $$\tilde A\subset N\prod\{A_1^{(1)},\ldots,A_{r_1}^{(1)},\ldots,A_1^{(r_0)},\ldots,A_{r_{r_0}}^{(r_0)},X_1^{(1)},\ldots,X_{\ell_1}^{(1)},\ldots,X_1^{(r_0)},\ldots,X_{\ell_{r_0}}^{(r_0)},X\}$$ with the product taken in some order. We also have $$\begin{aligned}
(r_1+\ldots+r_{r_0})\,,\,(\ell_1+\ldots+\ell_{r_0}+1)&\le r_0e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s-1)}2\tilde K+1\\
&\le e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s)}2\tilde K+1\\
&\le e^{O(\tilde s^2)}\log^{O(\tilde s)}2\tilde K.\end{aligned}$$ Finally, since every $\langle\pi_i(A_j^{(i)})\rangle$ is abelian, every $\langle\pi(A_j^{(i)})\rangle$ certainly is, so the proof is complete.
\[prop:grp.in.normal\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$. Let $H\subset A^m$ be a subgroup of $G$. Then there exists a normal subgroup $N$ of $G$ such that $H\subset N\subset A^{K^{e^{O(s)}m}}$.
The bounds stated in [@nilp.frei Proposition 7.3] are less explicit than the ones claimed here; as usual, the bounds claimed here can be read out of the argument there, or alternatively found explicitly in [@book Corollary 6.5.2].
\[prop:prod.of.progs.and.small\] Let $s\in{\mathbb{N}}$ and $K\ge2$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist $k,\ell\le e^{O(s^2)}\log^{O(s)}2K$, ordered progressions $P_1,\ldots,P_k\subset A^{e^{O(s)}}$ of rank at most $e^{O(s)}\log^{O(1)}2K$, sets $X_1,\ldots,X_\ell\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$, and a subgroup $H<G$ normalised by $A$ satisfying $H\subset A^{K^{e^{O(s)}}}$ such that $$A\subset H\prod\{P_1,\ldots,P_k,X_1,\ldots,X_\ell\},$$ with the product taken in some order.
We may assume that $A$ generates $G$. Applying \[prop:tor-free.post.induc\] with $\tilde A=A$, we obtain integers $r,t\le e^{O(s^2)}\log^{O(s)}2K$; a normal subgroup $N\lhd G$ satisfying $N\subset A^{K^{e^{
| 1,831
| 760
| 1,584
| 1,726
| 3,179
| 0.774256
|
github_plus_top10pct_by_avg
|
e = d\^D x f\^(x) \_(x) f’\^(x) \[Killing-1\] up to an overall normalization constant factor. Here the measure $\sqrt{\hat{g}} = \det \hat{e}_{\mu}{}^a$ must be present to ensure that the integration is diffeomorphism-invariant.
The Killing form can be slightly simplified by a change of basis. Let us use the field-dependent basis $\{D_a\}$. The Killing form for two generators $T_f = f^a D_a$ and $T_{f'} = f'{}^b D_b$ is = d\^D x f\^[a]{}(x) \_[ab]{}(x) f’\^[b]{}(x), \[Killing-2\] where $f^a \equiv f^{\mu} \hat{e}_{\mu}{}^a$ and similarly for $f'{}^b$. (The factor of $\hat{g}_{\m\n}$ in (\[Killing-1\]) is replaced by $\eta_{ab}$.) In this basis, the structure constants are field-dependent: = F\_[ab]{}\^c D\_c, \[DDD\] and the Jacobi identity (the consistency of the Lie algebra) is equivalent to the Bianchi identity of the field strength.
YM as Gravity {#YM=GR}
=============
YM Action
---------
The YM action is given as the norm of the field strength: S\_[YM]{} = d\^D x F\^[abc]{} F\_[abc]{}. \[YM-action\] It is invariant under space-time diffeomorphism. However, it is not invariant under rotations of the local Lorentz frame: \_\^a(x) ’\_\^a(x) = \^a\_b(x) \_\^b(x). \[rotation\]
In the absence of the gauge symmetry of local Lorentz frame rotations, the variable $\hat{e}_{\m}{}^a$ contains more degrees of freedom than the genuine vielbein $e_{\m}{}^a$. (This is why we have used a hat to distinguish it from the vielbein.) The YM theory cannot be identified with pure GR.
To achieve a YM-like theory equivalent to Einstein’s theory, we should utilize the fact that the internal space index $a$ (for the basis $D_a$) can be contracted with the coordinate index $a$. It allows us to introduce quadratic terms in addition to (\[YM-action\]) in the action. The most general quadratic action is the superposition of three terms: S\_[YM-like]{} = d\^D x . \[general-action-0\] The action remains the same if we simultaneously scale $\kappa^2, \lambda, \alpha, \beta$ by the same factor. Up to this ambiguity, there is a u
| 1,832
| 1,054
| 2,371
| 1,804
| null | null |
github_plus_top10pct_by_avg
|
\in \Gamma_P\! \right\}\!.$$ Then, $$\dim H^1(Y,\mathcal{O}_Y(L^{(k)}))=\dim\operatorname{coker}\pi^{(k)}.$$ In particular, if $\cC$ is reduced, then $$\pi^{(k)}: H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w}\right) \right)
\longrightarrow \bigoplus_{P \in S}
\frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w}\right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}}$$ where $$\mathcal{M}_{\mathcal{C},P}^{(k)}:=
\left\{ g \in\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w}\right)
\vphantom{\frac{k m_{\v}}{d}}\right.
\left|\ \operatorname{mult}_{E_\v} \pi^* g >
\frac{k m_{\v}}{d} - \nu_\v, \ \forall \v \in \Gamma_P \right\}.$$
First note that by Theorem \[thm:h2Lk\] the kernel of $\pi^{(k)}$ is isomorphic to the dual of $H^2(Y,\cO_Y(L^{(k)}))$. Rewriting $\dim\ker\pi^{(k)}$ in terms of $\dim\operatorname{coker}\pi^{(k)}$ and using the Euler characteristic $\chi(Y,\cO_Y(L^{(k)})) = \sum_{q=0}^2 (-1)^q \dim H^q(Y,\cO_Y(L^{(k)}))$, one obtains $$\dim H^1(Y,\cO_Y(L^{(k)})) = \dim\operatorname{coker}\pi^{(k)} + A,$$ where $$\label{eq:A}
\begin{aligned}
A &= \dim H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)\right)
- \sum_{P \in S} \dim \frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}} \\
& \quad - \chi(Y,\cO_Y(L^{(k)})) + \dim H^0(Y,\cO_Y(L^{(k)})).
\end{aligned}$$
For $k=0$, $\dim H^0(Y,\cO_Y(L^{(k)})) = \chi (Y,\cO_Y(L^{(k)})) = 1$ and the rest of the terms appearing in $A$ are zero. Then $A = 0$ and the claim follows.
Assume from now on that $k\neq 0$, then $\dim H^0(Y,\cO_Y(L^{(k)})) = 0$. The first part of the proof is to rewrite $A$ in terms of local properties of the curve $\mathcal{C}$. To compute the zero cohomologies in we use the Riemann-Roch formula on normal surfaces (see Theorem \[thm:RR\]).
Recall from that $s_k - |w|$ is the degree of $kH+K_{\PP^2_w} - \mathcal{C}^{(k)}$. Then since $s_k>0$, one obtains $$\label{eq:H0skw}
\begin{aligned}
h^0(\PP^2_w, & \mathcal{O}_{\PP^2_w}(s_k-|w|))\\
&=h^0\
| 1,833
| 1,413
| 1,939
| 1,499
| null | null |
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the following compatibility condition: $${\label{eq:crossed}}
r \circ v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) =
r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2})\bigl)$$ holds for all $g_{1}, g_{2} \in G$. On the set $G$ we define a new multiplication $*$ and two new actions $\beta' : G\times H
\rightarrow G$, $\alpha' : G\times H \rightarrow H$ given by: $$\begin{aligned}
g_{1} * g_{2} &:=& v^{-1}\bigl((v(g_{1}) \lhd
r(g_{2}))v(g_{2})\bigl) {\label{eq:def1}}\\
g \lhd' h &:=& v^{-1}\bigl(v(g) \lhd \sigma(h)\bigl)
{\label{eq:def2}}\\
g \rhd' h &:=& \sigma^{-1}(r(g))\sigma^{-1}(v(g) \rhd \sigma(h))
\sigma^{-1}\bigl(r \circ v^{-1}(v(g) \lhd \sigma(h))^{-1}\bigl)
{\label{eq:def3}}\end{aligned}$$ for all $g_1$, $g_2$, $g \in G$ and $h\in H$. Then:
1. $(G,*)$ is a group structure on the set $G$ with $1_{G}$ as a unit;
2. $\bigl(H, (G,*), \alpha', \beta'\bigl)$ is a matched pair of groups and $$\psi : H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, (G,*)
\rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad
\psi(h,g) := \bigl(\sigma(h)r(g), v(g)\bigl)$$ is a $\sigma$-invariant isomorphism of groups.
3. Any $\sigma$-invariant isomorphism of groups $H\, {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G \cong H\, {}_{\alpha}\!\! \bowtie_{\beta} \,
G$ arises as above.
\(1) Let $g \in G$. Then $$g * 1_G = v^{-1}\bigl((v(g) \lhd
r(1_G))v(1)\bigl) = v^{-1}\bigl(v(g)\bigl) = g$$ and $$1_G * g =
v^{-1}\bigl((v(1_G) \lhd r(g))v(g)\bigl) = v^{-1}\bigl(v(g)\bigl)
= g$$ Hence $1_{G}$ is a unit for $*$. Let $g_{1}$, $g_{2}$, $g_{3} \in G$. Then: $$\begin{aligned}
v \bigl(\underline{(g_{1} * g_{2})} * g_{3}\bigl)
&\stackrel{{(\ref{eq:def1})}} {=}& v\bigl[\underline{
v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) * g_{3} }\bigl]\\
&\stackrel{{(\ref{eq:def1})}} {=}& \bigl[ \underline{\bigl((v(g_{1})
\lhd
r(g_{2}))v(g_{2}) \bigl) \lhd r(g_{3})}\bigl]v(g_{3})\\
&\stackrel{{(\ref{eq:3})}} {=}& \bigl[\bigl(\underline{v(g_{1}) \lhd
r(g_{2})\bigl) \lhd \bigl(v(g_{2}) \rhd r(g_{3})\bigl)\bigl]
\bigl(v(g_{2}) \lhd
r(g_{3}) \bigl)}
| 1,834
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| 1,737
| 1,713
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|
a$ by our choices; if $B> \frac{C-\lambda_0}2+1$ then the weight of the coefficient of $y^2$ exceeds $c$, so it does not survive the limiting process, and the limit is a line. If $B= \frac{C-\lambda_0}2+1$, the term in $y^2$ is dominant, and the limit is a conic. The explicit expressions given in the statement are obtained by reading the coefficients of the dominant terms.
We can now complete the proof of Proposition \[abc\]:
\[completion\] Assume $C>\lambda_0$. If $B>\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit.
We will show that the limit is necessarily a kernel star, which gives the statement by Lemma \[rank2lemma\].
As $B>1$, the coefficient $\gamma_{C-B+1}$ is determined by the truncation $f_{(C)}$, and in particular it is the same for all formal branches with that truncation. Since $B>\frac{C-\lambda_0}2+1$, by Lemma \[dominant\] the branches considered there contribute lines through the fixed point $(0:1:(C-B+1)\gamma_{C-B+1})$. We are done if we check that all other branches contribute a kernel line $x=0$: and this is implied by Lemma \[otherbranches\] for branches that are not tangent to $z=0$ (note $a<v(r)$ for the germs we are considering), and by Lemma \[nottrunc\] for formal branches $z=g(y)$ tangent to $z=0$ but whose truncation $g_{(C)}$ does not agree with $f_{(C)}$.
Quadritangent conics {#quadritangent}
--------------------
We are ready to complete the proof of Theorem \[mainmain\], by determining the limits of the last contributing germs. These have been reduced to the form listed as type V in §\[germlist\] (up to a coordinate change, and replacing $t$ by a root of $t$): $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0 \\
t^a & t^b & 0 \\
\underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c
\end{pmatrix}$$ for some branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ of ${{\mathscr C}}$ tangent to $z=0$ at $p=(0,0)$, and further satisfying $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$ for $B=\frac ba$, $C=\frac
ca$. Type V components of th
| 1,835
| 667
| 1,857
| 1,882
| 1,856
| 0.785042
|
github_plus_top10pct_by_avg
|
th the lowest $\sigma_\mathrm{{rms}}$ has $T_{\rmn{eff}}=12\,000$K, $M_*=0.585\,M_{\sun}$ and $M_\rmn{H}=10^{-5.0}\,M_*$, however, its $\sigma_\mathrm{{rms}}$ is relatively large ($6.8$s), which means that there are major differences between the observed and calculated periods. Table \[table:lp133params\] lists the stellar parameters and theoretical periods of the models mentioned above. For completeness, we included this last model solution, too.
We concluded, that our models predict at least $0.1\,M_{\sun}$ larger stellar mass for LP 133-144 than the spectroscopic value. Nevertheless, it is possible to find models with lower stellar masses, but in these cases not all the modes with triplet frequency structures has $l=1$ solutions and (or) the corresponding $\sigma_\mathrm{{rms}}$ values are larger than for the larger mass models. Considering the effective temperatures, the $T_{\rmn{eff}}=12\,000$K solutions are in agreement with the spectroscopic determination ($\sim12\,150$K) within its margin of error. As in the case of G 207-9, taking into account that the uncertainties for the grid parameters are of the order of the step sizes in the grid, the $T_{\rmn{eff}}=11\,800$K findings are still acceptable.
[llrrrr]{} $T_{\rmn{eff}}$ (K) & & & & & Reference\
\
12152$\pm$200 & 0.59$\pm$0.03 & & & & @2011ApJ...743..138G\
& & & & & @2013AA...559A.104T\
\
11700 & 0.520 & 2.0 & 5.0 & 209.2 (1,2), 305.7 (2,7), 327.3 (2,8) & @2009MNRAS.396.1709C\
12210 & 0.609 & 1.6 & $\sim6$ & 209.2 (1,2), 305.7 (2,8), 327.3 (2,9) & @2012MNRAS.420.1462R\
& & & & &\
& & &\
& & & & &\
11800 (0.46s) & 0.710 & 2.0 & 4.0 & 208.8 (1,3), 305.6 (2,11), 270.1 (1,5), &\
& & & & 327.2 (1,6), 140.6 (2,4), 180.4 (1,2) &\
11800 (1.46s) & 0.725 & 2.0 & 8.0 & 209.5 (1,2), 304.5 (1,4), 268.8 (1,3), &\
& & & & 328.3 (2,9), 138.5 (2,2), 181.0 (1,1) &\
12000 (2.89s) & 0.605 & 2.0 & 4.2 & 204.5 (1,2), 307.9 (1,4), 271.5 (1,3), &\
& & & & 326.2 (1,5), 138.4 (1,1), 183.7 (2,4) &\
12000 (6.83s) & 0.585 & 2.0 & 5.0 & 215.3 (1,2), 311.6 (1,4), 273.3 (1,3), &\
| 1,836
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| 2,921
| 2,016
| null | null |
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|
either ${\mathbf{f}} \in {\mathit{f}}$ or ${\mathbf{f}} \notin {\mathit{f}}$.
\[D:BASIC\_COVERING\] Let ${\mathbf{f}}$ be a frame and ${\mathit{f}}$ be a functionality. The functionality *covers* the frame if ${\mathbf{f}} \in {\mathit{f}}$ (that is, ${\mathbf{f}} = (\psi, \phi) = (\psi, {\mathit{f}}(\psi))$).
\[D:COVERING\_PROCEDURE\] Let $\lbrace {\mathbf{f}}_n \rbrace$ be a sequence of frames and $\lbrace {\mathit{f}}_n \rbrace$ be a procedure. The procedure *covers* the sequence of frames if ${\mathbf{f}}_i \in {\mathit{f}}_i$ for each $i \geq 1$ (that is, ${\mathbf{f}}_i = (\psi_i, \phi_i) = (\psi_i, {\mathit{f}}_i(\psi_i))$).
Any procedure covers some process.
\[T:PROCEDURE\_COVER\_PROCESS\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and catalog of functionality ${\mathscr{F}}$. Suppose $\lbrace {\mathit{f}}_n \rbrace \colon {\mathbb{N}}\to {\mathscr{F}}$ is a procedure. For each choice of persistent state $\varphi \in {\prod{\Phi}}$ and volatile excitation sequence $\lbrace \xi_n \rbrace \in {(\,{\prod{\Xi}})}^{{\mathbb{N}}}$ there is a process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ such that the procedure covers the process.
The persistent state $\varphi$ and volatile excitation $\lbrace \xi_n \rbrace$ are given. Inductively define sequence $\lbrace \phi_n \rbrace$ using base clause $\phi_1 = \varphi$ and recursive clause $\phi_{i+1} = {\mathit{f}}_i(\phi_i\xi_i)$ for $i \ge 1$ (dyadic notation, definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\] ff).
We first show that the sequence $\lbrace \phi_n \rbrace$ lies in ${\prod{\Phi}}$. Through $\varphi$, definition \[D:PERSISTENT\_VOLATILE\_COMPONENTS\] provides that $\phi_1 \in {\prod{\Phi}}$. For the iterative part, the hypothesis $\phi_i \in {\prod{\Phi}}$ implies that $\phi_i\xi_i \in {\prod{\Psi}}$ by Theorem \[T:DYADIC\_CHOICE\_PROD\]. Since ${\mathit{f}}_i$ maps ${\prod{\Psi}}$ to ${\prod{\Phi}}$, then ${\mathit{f}}_i(\phi_i\xi_i) = \phi_{i+1} \in {\pr
| 1,837
| 1,212
| 1,551
| 1,747
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|
ute**.
We may take $\Phi$ to be a class of categories instead of functors, in which case we identify a category $A$ with the unique functor $A\to\bbone$. In the next section we will show that left $\Phi$-stability coincides with right $\Phi\op$-stability.
A derivator is pointed if and only if it is left $\emptyset$-stable, if and only if it is right $\emptyset$-stable, and if and only if it is left stable for the class of cosieves, if and only if it is right stable for the class of sieves. Similarly, is stable if and only if it is left stable for the class of homotopy finite categories, if and only if it is right stable for the same class, if and only if it is left stable for the class of left homotopy finite functors, if and only if it is right stable for the class of right homotopy finite functors.
This notion of relatively stable derivators allows us to construct the following Galois correspondence.
Given a class $\Phi$ of functors between small categories, we write ${\mathsf{Stab}_L}(\Phi)$ for the collection of left $\Phi$-stable derivators. Dually, given a collection $\Upsilon$ of derivators, we write ${\mathsf{Abs}_L}(\Upsilon)$ for the class of left $\Upsilon$-absolute functors (i.e. functors that are left -absolute for all ${\sD}\in\Upsilon$). Then ${\mathsf{Stab}_L}$ and ${\mathsf{Abs}_L}$ are a Galois correspondence (a contravariant adjunction of partial orders) between the classes of functors and collections of derivators. In particular, we have $$\Phi \subseteq {\mathsf{Abs}_L}(\Upsilon) \iff \Upsilon \subseteq {\mathsf{Stab}_L}(\Phi)$$ and $$\Phi \subseteq {\mathsf{Abs}_L}({\mathsf{Stab}_L}(\Phi)) \qquad \Upsilon \subseteq {\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))$$ $${\mathsf{Stab}_L}(\Phi) = {\mathsf{Stab}_L}({\mathsf{Abs}_L}({\mathsf{Stab}_L}(\Phi))) \qquad {\mathsf{Abs}_L}(\Upsilon) = {\mathsf{Abs}_L}({\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))).$$ Dually, we have ${\mathsf{Stab}_R}$ and ${\mathsf{Abs}_R}$.
\[prop:ptd-comm\] can now be restated by saying that ${\mathsf{Stab}_L}(\{\em
| 1,838
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positionl\], we have $\lambda_2(-H(\theta)) \geq \frac{e^{2b}}{(1 + e^{2b})^2} \lambda_2(M)$, when $\lambda_{j,a} = 1/(\kappa_j-1)$ is substituted in the Hessian matrix $H(\theta)$, Equation . From Weyl’s inequality we have that $$\begin{aligned}
\label{eq:topl3}
\lambda_2(M) \;\; \geq \lambda_2(\E[M]) - {\|M - \E[M]\|}\,.\end{aligned}$$ We will show in that $\lambda_2(\E[M]) \geq e^{-2b}(\alpha/(d-1))\sum_{j = 1}^n \ell_j$ and in that ${\|M - \E[M]\|} \leq 32e^{b}\sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j}$. $$\begin{aligned}
\label{eq:topl4}
\lambda_2(M) \; \geq \; \frac{\alpha e^{-2b}}{d-1} \sum_{j=1}^n \ell_j \;-\; 32e^{b} \sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j} \;\geq \; \frac{\alpha e^{-2b}}{2(d-1)} \sum_{j=1}^n \ell_j\;, \end{aligned}$$ where the last inequality follows from the assumption that $\sum_{j=1}^n \ell_j \geq (2^{12}e^{6b}/\beta\alpha^2) d\log d$. This proves the desired claim.
To prove the lower bound on $\lambda_2(\E[M])$, notice that $$\begin{aligned}
\label{eq:topl5}
\E[M] &=& \sum_{j = 1}^n \frac{1}{\kappa_j -1} \sum_{i<\i \in S_j} \E\Bigg[ \sum_{a = 1}^{\ell_j} \I_{\{(i,\i) \in G_{j,a}\}} \Big| (i,\i \in S_j) \Bigg] (e_i - e_{\i})(e_i - e_{\i})^\top \;.\end{aligned}$$ Using the fact that $p_{j,a} = a$ for each $j \in [n]$, and the definition of rank-breaking graph $G_{j,a}$, we have that $$\begin{aligned}
\label{eq:topl6}
\E\Bigg[ \sum_{a = 1}^{\ell_j} \I_{\{(i,\i) \in G_{j,a}\}} \Big| (i,\i \in S_j) \Bigg] &=& \P\Big[\I_{\{\sigma_j^{-1}(i) \leq \ell_j\}} + \I_{\{\sigma_j^{-1}(\i) \leq \ell_j\}} \geq 1 \Big| (i,\i \in S_j) \Big] \nonumber\\
& \geq & \P\Big[(\sigma^{-1}(i) \leq \ell_j \Big| (i,\i \in S_j)\Big]\,.\end{aligned}$$ The following lemma provides a lower bound on $\P[(\sigma^{-1}(i) \leq \ell_j | (i,\i \in S_j)]$.
\[lem:prob\_toplbound\] Consider a ranking $\sigma$ over a set of items $S$ of size $\kappa$. For any item $i \in S$, $$\begin{aligned}
\label{eq:prob_toplbound_eq}
\P[(\sigma^{-1}(i) \leq \ell] \geq e^{-2b}\frac{\ell}{\kappa}\;.\end{aligned}$$
| 1,839
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| null | null |
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|
sf}{ord}}^n D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ for all $n\geq 0$. We therefore obtain an induced grading, again called the ${\mathbf{E}}$-grading, on the associated graded ring $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast {{W}}\cong
{\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^*]\ast W$. Clearly this is again given by $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}^*=1$ (which we define to mean that $\operatorname{{\mathbf{E}}\text{-deg}}(x)=1$ for every $0\not=x\in {\mathfrak{h}}^*$) while $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}=-1$ and $\operatorname{{\mathbf{E}}\text{-deg}}{{W}}=0$.
One should note that, in general, ${\mathbf{E}}\notin H_c$. However, there is a natural element in $H_c$ that has the same adjoint action. Indeed, let $$\label{hdefn} {\mathbf{h}}= {\mathbf{h}}_c =
\frac{1}{2} \sum_{i=1}^{n-1} (x_iy_i + y_ix_i) \in H_c.$$ \[boldh-defn\] This is independent of the choice of basis. By [@BEGqi (2.6)] we have $$\label{grading}
[{\mathbf{h}},x] = x, \quad [{\mathbf{h}}, y] = - y, \quad \text{and} \quad
[{\mathbf{h}}, w ] = 0 \qquad\mathrm{for \ all}\ x\in{\mathfrak{h}}^*,y\in {\mathfrak{h}}\ \mathrm{and}\ w\in
{{W}}.$$ Thus commutation with ${\mathbf{h}}$ also induces the Euler grading on $H_c$.
The spherical subalgebra {#shiftfunct}
------------------------
Let $e\in {\mathbb{C}}{{W}}$ be the trivial idempotent and $e_-\in {\mathbb{C}}{{W}}$\[e-defn\] be the sign idempotent; thus $e = |{{W}}|^{-1} \sum_{w\in {{W}}} w$ and $e_- = |{{W}}|^{-1} \sum_{w\in{{W}}} \text{sign}(w)w.$ The main algebra of study in this paper is not the Cherednik algebra itself, but its [*spherical subalgebra*]{} ${U}_c=eH_ce$\[spherical-defn\] and the related algebra ${U}^-_c=e_-H_ce_-$. We will use frequently and without comment that $\delta$ is a ${{W}}$-anti-invariant and so $e_-\delta =\delta e$. Also, as $\operatorname{{\mathbf{E}}\text{-deg}}{{W}}=0$, both $U_c$ and $U^-_c$ have an induced ${\mathbf{E}}$-graded structure.
Partitions {#dominance}
----------
| 1,840
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| 1,144
| 1,857
| 3,472
| 0.772089
|
github_plus_top10pct_by_avg
|
n is commutativity $\Theta\Phi = \Phi\Theta$, since $\Theta \cup \Phi = \Phi \cup \Theta$.
\[T:DYADIC\_PRODUCT\_IS\_ENSEMBLE\] Let $\Theta$ and $\Phi$ be disjoint ensembles. Their dyadic product $\Upsilon = \Theta\Phi$ is an ensemble with domain $({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$ and range $({{\operatorname{ran}{\Theta}}} \cup {{\operatorname{ran}{\Phi}}})$.
Suppose $i \in {{\operatorname{dom}{\Theta\Phi}}}$. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $\Theta\Phi = \Theta \cup \Phi$. From this it follows that $i \in {{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}}$. Therefore, $i \in {{\operatorname{dom}{\Theta}}}$, $i \in {{\operatorname{dom}{\Phi}}}$, or both. The stipulation that $\Theta$ and $\Phi$ be disjoint entails ${{\operatorname{dom}{\Theta}}} \cap {{\operatorname{dom}{\Phi}}} = \varnothing$ (definition \[D:DISJOINT\_ENSEMBLES\]). That stipulation eliminates the possibility of both memberships, leaving two feasible cases: either A) $i \in {{\operatorname{dom}{\Theta}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$, or B) $i \notin {{\operatorname{dom}{\Theta}}}$ and $i \in {{\operatorname{dom}{\Phi}}}$.
In case A, there exists $(i, \Theta_i) \in \Theta$, so $(i, \Theta_i) \in \Upsilon = \Theta \cup \Phi$. Furthermore, since $i \notin {{\operatorname{dom}{\Phi}}}$, $\Upsilon_i = \Theta_i$ is well-defined. Since by definition \[D:ENSEMBLE\] each member of ${{\operatorname{ran}{\Theta}}}$ is a non-empty set, then equivocally $\Upsilon_i = \Theta_i$ is a non-empty set.
Argumentation for case B, supporting that $\Upsilon_i = \Phi_i$ is well-defined and that $\Upsilon_i$ is a non-empty set, is obtained by interchanging the roles of $\Theta$ and $\Phi$ in case A.
Over all possibilities, $\Upsilon = \Theta \cup \Phi$ is well-defined as a mapping (family) and each element of its range is a non-empty set. Therefore the dyadic product of two disjoint ensembles is itself an ensemble.
The sub-case analysis establishes ${{\operatorname{dom}{\Theta\Phi}}} \sub
| 1,841
| 1,197
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| null | null |
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|
re $I_R$ and $I_M$ are the real and imaginary parts of the integral. Here, $\tilde{{\bf k}}$ is oriented along the $z$ direction in the ${\bf \xi}$- space and the angle $\Theta_0$ is measured from the $z$ direction. The real part can be evaluated by contour integration, with the result $$\begin{aligned}
I_R &=& \frac{1}{\pi} \int_{0}^\infty d\xi \frac{\xi^2\epsilon}{(\xi^2+\epsilon^2)^2} \cos(\xi |\tilde{{\bf k}}| \cos\Theta_0) \nonumber \\
&=& \frac{1}{4} \frac{\partial}{\partial \epsilon} \left( \epsilon e^ {- \epsilon |\tilde{{\bf k}}| |\cos{\Theta_0}| }\right) \nonumber \\
&=& \frac{1}{4} \left( 1-\epsilon |\tilde{{\bf k}}| |\cos\Theta_0| \right) e^{-\epsilon |\tilde{{\bf k}}| |\cos \Theta_0|}
\ .
\label{IR}\end{aligned}$$ The imaginary part can be reduced to the exponential integral $E_1$ [@dlmf], with the result $$\begin{aligned}
I_M = \frac{i \operatorname{sgn}(a)}{4\pi}
\Bigl[
(|a|+1) e^{|a|} E_1(|a|) + (|a|-1) e^{-|a|} \operatorname{Re}\bigl( E_1(-|a|) \bigr)
\Bigr]
\ , \end{aligned}$$ where $a = \epsilon |\tilde{{\bf k}}| \cos \Theta_0$. Note that the replacement $\Theta_0 \to \pi - \Theta_0$ leaves $I_R$ invariant but gives $I_M$ a minus sign.
To proceed further, we assume that the profile function is invariant under $\Theta_0 \rightarrow \pi - \Theta_0$, that is, under $\cos(\Theta_0) \rightarrow - \cos(\Theta_0)$. Since $I_M$ is an odd function of $\cos(\Theta_0)$, it does not contribute to $g_{\Theta_0} \left( {\bf k}, \tau \right)$ under such an invariance whereas $I_R$ being even in $\cos(\Theta_0)$ contributes. Physically, this would mean that the direction sensitive detector reads off the average of two transition rates from the $\Theta_0$ and $\pi - \Theta_0$ directions respectively. Eq.(\[gdef\]) then becomes $$g_{\Theta_0} \left( {\bf k}, \tau \right) = \frac{1}{4} \frac{\partial}{\partial \epsilon} \left( \epsilon \; e^ {- \epsilon |\tilde{{\bf k}}| |\cos{\Theta_0}| } \; e^{i k_b x^b(\tau)}\right)$$ Using the fact that $k_b$ is a null vector, it is straightforward to show that $|
| 1,842
| 3,900
| 1,860
| 1,826
| null | null |
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|
l{K-defn}
\tilde{H}_{\lambda}(\x;q,t)=
\sum_{\nu}\tilde{K}_{\nu\lambda}(q,t)s_{\nu}(\x).$$ These are $(q,t)$ generalizations of the $\tilde{K}_{\nu\lambda}(q)$ Kostka-Foulkes polynomial in Macdonald [@macdonald III, (7.11)], which are obtained as $q^{n(\lambda)}K_{\nu\lambda}(q^{-1})=
\tilde{K}_{\nu\lambda}(q)=\tilde{K}_{\nu\lambda}(0,q)$, i.e., by taking their pure part. In particular, $$\label{Hall-Littlewood}
\tilde{H}_{\lambda}(\x;q)=\sum_{\nu}\tilde{K}_{\nu\lambda}(q)s_{\nu}(\x).$$
For a partition $\lambda$, we denote by $\chi^\lambda$ the corresponding irreducible character of $S_{|\lambda|}$ as in Macdonald [@macdonald]. Under this parameterization, the character $\chi^{(1^n)}$ is the sign character of $S_{|\lambda|}$ and $\chi^{(n^1)}$ is the trivial character. Recall also that the decomposition into disjoint cycles provides a natural parameterization of the conjugacy classes of $S_n$ by the partitions of $n$. We then denote by $\chi^\lambda_\mu$ the value of $\chi^\lambda$ at the conjugacy class of $S_{|\lambda|}$ corresponding to $\mu$ (we use the convention that $\chi^\lambda_\mu=0$ if $|\lambda|\neq|\mu|$). The *Green polynomials* $\{Q_\lambda^\tau(q)\}_{\lambda,\tau\in\calP}$ are defined as
$$Q_\lambda^\tau(q)=\sum_\nu \chi^\nu_\lambda\tilde{K}_{\nu\tau}(q)$$
if $|\lambda|=|\tau|$ and $Q_\lambda^\tau=0$ otherwise.
### Exp and Log
Let $\Lambda(\x_1,\ldots,\x_k):=
\Lambda(\x_1)\otimes_\Z\cdots\otimes_\Z\Lambda(\x_k)$ be the ring of functions separately symmetric in each set $\x_1,\x_2,\ldots,\x_k$ of infinitely many variables. To ease the notation we will simply write $\Lambda_k$ for the ring $\Lambda(\x_1,\ldots,\x_k)\otimes_\Z \Q(q,t)$.
The power series ring $\Lambda_k[[T]]$ is endowed with a natural $\lambda$-ring structure in which the Adams operations are $$\psi_d(f(\x_1,\x_2,\dots,\x_k,q,t;T)):=f(\x_1^d,\x_2^d,\dots,\x_k^d,q^d,t^d;T^d).$$
Let $\Lambda_k[[T]]^+$ be the ideal $T\Lambda_k[[T]]$ of $\Lambda_k[[T]]$. Define $\Psi: \Lambda_k[[T]]^+\rightarrow\Lambda_k[[T]]^+$ by
$$\Psi(f):=\sum
| 1,843
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|
onary points.
In Fig. \[fig:Mrho0\] we illustrate the mass of the initial data as a function of bubble radius $\rho_0$ for $0\leq Q\leq Q_{max}$. Clearly even the perturbatively stable bubble at the local minimum is unstable against tunneling to larger radii. At $Q= Q_{max}$, the barrier disappears completely. On the other hand, for small $Q$, the barrier grows, and the rate for this transition may become extremely slow. In the next section we will show that a new, rapid instability arises in this limit.
In general the local minimum is a root of a cubic, equivalent to $b=0$ in Eq. (\[eq:b\]). For small $Q$ it simplifies to $$\begin{aligned}
\rho_0^2\approx Q/2.\end{aligned}$$ It is straightforward to verify that the local minimum is a static solution to the full Einstein equations with spacetime metric $$\begin{aligned}
ds^2=-h(\rho)dt^2+U(\rho)d\chi^2+\frac{d\rho^2}{U(\rho) h(\rho)}+\rho^2d\Omega_3\;
\label{eq:dsfull}\end{aligned}$$ with $U$ and $h$ given by Eqs. (\[eq:U\]), (\[eq:h\]) and $b=0$. The mass of this static bubble is $$\begin{aligned}
M_{min}=\pi^2 LM_6^4\left(\frac{3Q^2}{4\rho_0^2}+\rho_0^2\right),\end{aligned}$$ where $L$ is given by Eq. (\[eq:b\]) evaluated at $b=0$. The field strength (\[eq:cN\]) surrounding the static bubble simplifies to $$\begin{aligned}
C_{\rho t\chi}=\frac{Q_0}{2\pi^2 \rho^3}\;.
\label{eq:cN}\end{aligned}$$
The point in moduli space where the size $R$ of the KK circle vanishes lies an infinite proper distance $\int dR/R$ away from any point of finite circle size. $R=0$ is sampled locally on the wall of KK bubbles, since $V\rightarrow 0$ as $\rho\rightarrow\rho_0$, while $R=L$ at spatial infinity.
Typically, static neutral bubbles of nothing in asymptotically flat space have a single unstable mode, corresponding to perturbations of the bubble radius. The solution (\[eq:dsfull\]), lying at a local minimum of the energy, is perturbatively stabilized by the flux. It disappears for $Q=0$, leaving only the perturbatively unstable point corresponding to an ordinary neut
| 1,844
| 1,500
| 2,245
| 1,880
| 2,451
| 0.779615
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|
mega_M|=1 \mathrm{~and~} \int_{H'}|\omega_H|=1.$$
We choose another nonzero, translation-invariant forms $\omega^{\prime}_M$ and $\omega^{\prime}_H$ on $\mathrm{End}_EV$ and $H$, respectively, with normalization $$\int_{\underline{M}(A)}|\omega^{\prime}_M|=1 \mathrm{~and~} \int_{\underline{H}(A)}|\omega^{\prime}_H|=1$$ (cf. the second paragraph of page 499 in [@C2]). By Theorem \[t36\], we have an exact sequence of locally free sheaves on $\underline{M}^{\ast}$: $$0\longrightarrow \rho^{\ast}\Omega_{\underline{H}/A} \longrightarrow \Omega_{\underline{M}^{\ast}/A}
\longrightarrow \Omega_{\underline{M}^{\ast}/\underline{H}} \longrightarrow 0.$$ Put $\omega^{\mathrm{can}}=\omega^{\prime}_M/\rho^{\ast}\omega^{\prime}_H$. For a detailed explanation of what $\omega_M'/\rho^{\ast}\omega_H'$ means, we refer to Section 3.2 of [@GY]. It follows that $\omega^{\mathrm{can}}$ is a differential of top degree on $\underline{G}$, which is invariant under the generic fiber of $\underline{G}$, and which has nonzero reduction on the special fiber.
Recall that $2$ is a uniformizer of $A$.
\[l51\] We have: $$|\omega_M|=|2|^{N_M}|\omega_M^{\prime}|, \ \ \ \ N_M=\sum_{\textit{$L_i$:type I}}2n_i+\sum_{i<j}(j-i)\cdot n_i\cdot n_j-a,$$ $$|\omega_H|=|2|^{N_H}|\omega_H^{\prime}|, \ \ \ \ N_H=\sum_{L_i:\textit{type I}}n_i+\sum_{i<j}j\cdot n_i\cdot n_j+\sum_{\textit{i:even}} \frac{i+2}{2} \cdot n_i
+\sum_{\textit{i:odd}} \frac{i+3}{2} \cdot n_i+\sum_i d_i-a,$$ $$|\omega^{\mathrm{ld}}|=|2|^{N_M-N_H}|\omega^{\mathrm{can}}|.$$ Here,
- $a$ is the total number of $L_i$’s such that $i$ is odd and $L_i$ is *free of type I*.
- $d_i=i\cdot n_i\cdot (n_i-1)/2$.
The proof is similar to that of Lemma 5.1 in [@C2] and so we skip it.
Let $f$ be the cardinality of $\kappa$. The local density is defined as $$\beta_L= \frac{1}{[G:G^{\circ}]}\cdot \lim_{N\rightarrow \infty} f^{-N~dim G}\#\underline{G}'(A/\pi^N A).$$ Here, $\underline{G}'$ is the naive integral model described at the beginning of Section \[csm\] and $G$ is the generic fiber of
| 1,845
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|
}(x) = \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right]^{1/2} K_{i\omega /g} \left[ \frac{\sqrt{k_{\bot}^{2} + m^{2}}}{g e^{-g z}} \right] e^{ik_{\perp} \cdot x_{\perp} - i \omega t}
\label{modsol}$$ The smeared field operator defined in Eq.(\[smeared\]) can then be expressed as $$\phi(\tau) = \int d\omega \int d^2 k_{\perp} \left[ {\hat a}_{\omega, k_{\perp}} h_{\omega, k_{\perp}}(\tau) + {\hat a}^{\dagger}_{\omega, k_{\perp}} h^{\star}_{\omega, k_{\perp}}(\tau) \right]$$ with the corresponding smeared field modes to be $$\begin{aligned}
h_{\omega, k_{\perp}}(\tau) &=& \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right]^{1/2} e^{ - i \omega t} \; u_{\omega, k_{\perp}}\left(z_0(\tau), x_{\perp 0}(\tau) \right) \end{aligned}$$ and $$\begin{aligned}
u_{\omega, k_{\perp}}\left(z_0(\tau), x_{\perp 0}(\tau) \right) &=& \int dz \, d^2x_{\perp} e^{g z} f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right) \nonumber \\
&& \times K_{i\omega /g} \left[ \frac{\sqrt{k_{\bot}^{2} + m^{2}}}{g e^{-g z}} \right] e^{ik_{\perp} \cdot x_{\perp}} \end{aligned}$$ The pullback of the Wightman function given in Eq.(\[whitmannfunction\]) is then expressed in terms of the smeared field modes to become $$\begin{aligned}
W(\tau,\tau^\prime) &=& \int d\omega \int d^{2}k_{\perp} \left[ \left( \eta_{\omega}+1 \right) h_{\omega, k_{\perp}}(\tau) h^{\star}_{\omega, k_{\perp}}(\tau^\prime) + \eta_{\omega} \, h^{\star}_{\omega, k_{\perp}}(\tau) h_{\omega, k_{\perp}}(\tau^\prime) \right] \nonumber \\
&=& \int d\omega \int d^{2}k_{\perp} \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right] \bigg[ \left( \eta_{\omega}+1 \right) u_{\omega, k_{\perp}}(\tau) u^{\star}_{\omega, k_{\perp}}(\tau^\prime) e^{- i \omega (\tau - \tau^\prime)} \nonumber \\
\; \; \; & & + \; \eta_{\omega} \, u^{\star}_{\omega, k_{\perp}}(\tau) u_{\omega, k_{\perp}}(\tau^\prime) \, e^{ i \omega (\tau - \tau^\prime)} \bigg] \end{aligned}$$ where $\eta_{\omega} = 1/(\exp{(\beta \omega)} - 1)$ being the Planckian factor with the usual Unruh temperature. Since fo
| 1,846
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|
nguish these two SL(2,$\mathbb C$) representations, or equivalently the two Weyl spinors, the van der Waerden dotted and undotted index notation has been introduced. This notation proves particularly valuable for the construction of manifestly supersymmetric invariant Lagrangian densities.
The undotted indices $\alpha,\beta$, on the one hand, and dotted indices $\dot{\alpha},\dot{\beta}$, on the other hand, have the same meaning as the $\alpha,\beta$ indices for the SU(2) spinor representations. Consequently, Lorentz invariant quantities are readily constructed in terms of the Weyl spinors $\psi^\alpha$ and $\overline{\psi}_{\dot{\alpha}}$, through simple contraction of the indices using the invariant tensors available. Furthermore, given that we have $${x'}^\mu\sigma_\mu=X'=MXM^\dagger=M\left(x^\mu\sigma_\mu\right)M^\dagger\ ,$$ it follows that the SL(2,$\mathbb C$) or SO(1,3) Lorentz transformation properties of the matrices $\sigma_\mu$ are those characterised by the index structure, $$\sigma^\mu\ :\ \ \
\left(\sigma^\mu\right)_{\alpha\dot{\alpha}}\ \ ,\ \
\sigma_\mu=(\one,\sigma_i)\ \ ,\ \
\sigma^\mu=(\one,-\sigma_i)=(\one,\sigma^i)\ .$$ By raising the indices, one introduces the quantities $$\overline{\sigma}^\mu\ :\ \ \
\left(\overline{\sigma}^\mu\right)^{\dot{\alpha}\alpha}=
\epsilon^{\dot{\alpha}\dot{\beta}}\,
\epsilon^{\alpha\beta}\,\left(\sigma_\mu\right)_{\beta\dot{\beta}}\ \ ,\ \
\overline{\sigma}^\mu=(\one,\sigma_i)\ \ ,\ \
\overline{\sigma}_\mu=(\one,-\sigma_i)\ .$$ Note that these properties also justify why indeed a 4-vector $A_\mu$ is equivalent to the $(1/2,1/2)=(1/2,0)\oplus(0,1/2)$ Lorentz representation, $A^\mu\sigma_{\mu\alpha\dot{\alpha}}=A_{\alpha\dot{\alpha}}$, $A^\mu{\overline{\sigma}_{\mu}}^{\dot{\alpha}\alpha}=
\overline{A}^{\dot{\alpha}\alpha}$.
Let us now consider different Weyl spinors $\psi$, $\chi$, $\overline{\psi}$, $\overline{\chi}$, ... and the Lorentz invariant spinor bilinears that may constructed out of these quantities. For this purpose, it is important to realise
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rtunately, our gerbe example is not so well-behaved.
Second cautionary example {#sect:class3-caution2}
-------------------------
For completeness, we give here a second cautionary example, here involving a heterotic Spin$(32)/{\mathbb Z}_2$ compactification on a nontrivial toroidal orbifold. This will involve a rank 10 bundle over a ${\mathbb Z}_2$ gerbe on $[T^4/{\mathbb Z}_2]$, and although level matching holds, the spectrum is anomalous in six dimensions.
The ${\mathbb Z}_2$ gerbe is defined by $[T^4/{\mathbb Z}_4]$, where the ${\mathbb Z}_4$ acts on the $T^4$ by $$x \: \mapsto \: \exp\left( \frac{2\pi i (2k)k}{4} \right) x
\: = \: (-)^k x,$$ so that there is a trivially-acting ${\mathbb Z}_2$ subgroup. (This is the same ${\mathbb Z}_2$ gerbe discussed in a different context in section \[sect:class2-ex1\].) The gauge bundle is a rank 10 bundle, where the generator of ${\mathbb Z}_4$ acts on an ${\cal O}^{\oplus 2}$ factor by multiplication by $\exp(2 \pi i (2/4) ) = -1$, and on the ${\cal O}^{\oplus 8}$ factor by $\exp(2 \pi i /4)$. It is straightforward to check that this satisfies level-matching, in the sense of [@freedvafa], as well as the conditions in appendix \[app:spectra:fockconstraints\].
Let us now outline the massless spectrum.
In the untwisted sector, there are massless states in the (NS,NS) sector. It is straightforward to compute $E_{\rm left} = -1$, $E_{\rm right} = -1/2$, and one has ${\mathbb Z}_4$-invariant states of the form
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
State Count
----------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------
$\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{
| 1,848
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xe] is in progress.
---
abstract: 'Linear perturbation theory is a powerful toolkit for studying black hole spacetimes. However, the perturbation equations are hard to solve unless we can use separation of variables. In the Kerr spacetime, metric perturbations do not separate, but curvature perturbations do. The cost of curvature perturbations is a very complicated metric-reconstruction procedure. This procedure can be avoided using a symmetry-adapted choice of basis functions in highly symmetric spacetimes, such as near-horizon extremal Kerr. In this paper, we focus on this spacetime, and (i) construct the symmetry-adapted basis functions; (ii) show their orthogonality; and (iii) show that they lead to separation of variables of the scalar, Maxwell, and metric perturbation equations. This separation turns the system of partial differential equations into one of ordinary differential equations over a compact domain, the polar angle.'
author:
- Baoyi Chen
- 'Leo C. Stein'
bibliography:
- 'NHEK-and-stringy-corrections.bib'
title: 'Separating metric perturbations in near-horizon extremal Kerr'
---
Introduction {#sec:introduction}
============
Linear metric perturbation theory is widely used in studying weakly-coupled gravity [@Wald]. For example, it can be applied to investigating the stability of black holes, gravitational radiation produced by material sources moving in a curved background, and so on. In the context of linearized gravity, the equations that describe gravitational perturbations are the linearized Einstein equations (LEE). Although they are linear, the LEE are still difficult to solve unless we can separate variables. In the Kerr spacetime, while in Boyer-Lindquist (BL) coordinates $t$ and $\phi$ can be separated, $r$ and $\theta$ remain coupled due to lack of symmetry [@Teukolsky:2014vca].
A successful approach towards separating wave equations for perturbations of the Kerr black hole was first developed by Teukolsky [@Teukolsky:1972my; @Teukolsky:1973ha]. Instead of looking at metric pert
| 1,849
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l.com (match this)
I tried the following: (?!@)gmail\.com but this did not work. This is matching both the cases highlighted in the example above. Any suggestions?
A:
[^@\s]*(?<!@)\bgmail\.com\b
assuming you want to find strings in a longer text body, not validate entire strings.
Explanation:
[^@\s]* # match any number of non-@, non-space characters
(?<!@) # assert that the previous character isn't an @
\b # match a word boundary (so we don't match hogmail.com)
gmail\.com # match gmail.com
\b # match a word boundary
On a first glance, the (?<!@) lookbehind assertion appears unnecessary, but it isn't - otherwise the gmail.com part of abc@gmail.com would match.
A:
You want a negative lookbehind if your regex supports it, like (?<!@)gmail\.com and add \bs to avoid matching foogmail.comz, like: (?<!@)\bgmail\.com\b
Q:
Xubuntu 16.04 notification area and indicators glitch
Recently updated my Xubuntu 15.10 to 16.04 and noticed that my panel missed tray icons for ktorrent. After some research I found trouble:
notification area (системный лоток) has ktorrent tray icon, but it hides when I add indicators (индикаторы) and appears again after I remove indicators:
As you can see, the volume applet removes with removing indicators, it's not good.
I want to move the sound applet to the notification area and/or not have the notification area disappear when I add indicators. Is it mission possible?
A:
Delete Indicators plugin, install xfce4-pulseaudio-plugin with command
sudo apt install xfce4-pulseaudio-plugin
and add it to the panel. You will have your volume controls without messing with the buggy Indicators plugin.
There is also an alternative version of Indicators plugin which uses GTK2 and has a classic tray behavior, it might be free of bugs.
Q:
OpenGL VAO is pointing to address 0 for some reason
I am having some trouble with my VAO not binding properly (at least that's what I think is happening).
So, what I am doing is I have a class that is creati
| 1,850
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|
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|
the compatibility conditions of the system (\[eq:ms:12\]).
Example. Nonlinear interaction of waves and particles. {#sec:7}
======================================================
Let us consider the inhomogeneous system describing the propagation of shock waves intensity in nonlinear interaction of waves and particles [@Luneburg:1964]
\[eq:e1:1\] u\_x+\_y=2\^[1/2]{}a (u/2)(/2),u\_y-\_x=-2\^[1/2]{}a(u/2)(/2).
It should be noted that the compatibility condition of the mixed derivatives of $\phi$ corresponds to the Liouville equation
\[eq:e1:2\] u\_[xx]{}+u\_[yy]{}=a\^2.
Therefore, each solution of the system (\[eq:e1:1\]) also gives us a solution of the Liouville equation (\[eq:e1:2\]). We use the methods presented in Section \[sec:6\] to obtain the general solution of the system (\[eq:e1:1\]). First, write the system (\[eq:e1:1\]) in the matrix form
\[eq:e1:3\]
(
[c c]{} 1 & 0\
0 & -1
)
(
[c ]{} u\_x\
\_x
)
+
(
[c c]{} 0 & 1\
1 & 0
)
(
[c]{} u\_y\
\_y
)
=
(
[c]{} b\_1\
b\_2
)
,
where $$b_1=2^{1/2}a \exp(u/2)\sin(\phi/2),\qquad b_2=2^{1/2}a\exp{u/2}\cos(\phi/2).$$The matrices $\mathcal{A}^i$ introduced in equation (\[eq:ms:4\]) are given by $$\mathcal{A}^1={\left( \begin{array}{c c}
1 & 0\\
0 & -1
\end{array} \right)},\qquad \mathcal{A}^2={\left( \begin{array}{c c}
0 & 1\\
1 & 0
\end{array} \right)}.$$Condition (\[eq:ms:8\]) is then satisfied by the scalar function $\Omega$, the wave vector $\lambda$ and the rotation matrix $L$, defined by $$\Omega=12^{1/4}(1-\epsilon^i),\quad \lambda=(1,i),\quad L={\left( \begin{array}{c c} l_{11} & l_{12}\\
l_{21} & l_{22}\end{array} \right)},\quad \epsilon=\pm 1,$$together with their complex conjugates, where $$\begin{aligned}
l_{11}&=l_{22}=-108^{1/4}i{\left( \frac{3^{1/2}\epsilon (b_1+ib_2)^2+i(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)}\\
l_{12}&=-l_{21}=108^{1/4}i{\left( \frac{3^{1/2}\epsilon i (b_1+ib_2)^2+(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)}
\end{aligned}$$Since the wave vectors $\lambda$ and $\bar{\lambda}$ are constant,
| 1,851
| 795
| 323
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| null | null |
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|
M$, where $\alpha \geq 2$ will be determined shortly. This takes care of the all constraints except for $A \in PSD$. Note that since $M$ is regular, its eigenvectors are also eigenvectors of $A$. Moreover, if $M u = \lambda u$ for a non constant $u$, then $A u = \alpha d - n - \alpha \lambda$ (and $A \vec{1} = 0$). So take $\alpha = \frac {n} {d - \lambda_2}$, and by our assumption on $\lambda_2$, $\alpha \geq 2$.
Now $A$ gives an upper bound on (\[hs-norm\]): $$\begin{aligned}
{\frac 1 2}tr A =
{\frac 1 2}n(\alpha d - n + 1) = {\frac 1 2}n^2 \frac d {d-\lambda_2} - {\frac 1 2}n^2
+ {\frac 1 2}n= {\frac 1 2}n^2 \frac {\lambda_2} {d-\lambda_2} + {\frac 1 2}n.\end{aligned}$$ This, by (\[d-bound\]), shows that the dimension of the embedding is $\Omega\left(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))} \right)$.
A Quasi-random Graph of logarithmic Sphericity
----------------------------------------------
It is an intriguing problem to construct new examples of graphs of linear sphericity. Since random graphs have this property, it is natural to search among quasi-random graphs. There are several equivalent definitions for such graphs (see [@AlSp]). The one we adopt here is:
A family of graphs is called [*quasi-random*]{} if the graphs in the family are $(1+o(1))\frac n 2$-regular, and all their eigenvalues except the largest one are (in absolute value) $o(n)$ .
Counter-intuitively, perhaps, quasi-random graphs may have very small sphericity.
\[alex\] Let $\G$ be the family of graphs with vertex set $\{0,1\}^k$, and edges connecting vertices that are at Hamming distance at most $\frac k 2$. Then $\G$ is a family of quasi-random graphs of logarithmic sphericity.
The fact that the sphericity is logarithmic is obvious - simply map each vertex to the vector in $\{0,1\}^n$ associated with it. To show that all eigenvalues except the largest one are $o(2^k)$ we need the following facts about Krawtchouk polynomials (see [@vL]). Denote by $K_s^{(k)}(i) = \sum_{j=0}^s(-1)^j{i \choose j}{{k-i} \choose {s-j}}$ the Krawt
| 1,852
| 2,142
| 2,354
| 1,824
| 2,796
| 0.776869
|
github_plus_top10pct_by_avg
|
T}}_{m-1,r}(\lambda_{m}-s){\mathcal{T}}_{m,r}(\lambda_{m+1}-\lambda_{m})...{\mathcal{T}}_{l-2,r}(\lambda_{l-1}-\lambda_{l-2}){\mathcal{T}}_{l-1,r}(t-\lambda_{l-1})
\\&={\mathcal{T}}_{m-1}\Big((T-s)-(T-\lambda_{m})\Big){\mathcal{T}}_{m}\Big((T-\lambda_m)-(T-\lambda_{m+1})\Big)...{\mathcal{T}}_{l-1}\Big((T-\lambda_{l-1})-(T-t)\Big)\\&={{U}}_{\Lambda_T}(T-s,T-t)\end{aligned}$$ Finally, the desired equality follows by passing to the limit as $|\Lambda|=|\Lambda_T|\to 0.$
\[remark-rescaling\] The coerciveness assumption in may be replaced with $$\label{eq:Ellipticity-nonaut2}
{\operatorname{Re}}{\mathfrak{a}}(t,u,u) +\omega\Vert u\Vert_H^2\ge \alpha \|u\|^2_V \quad ( t\in [0,T], u\in V)$$ for some $\omega\in {\mathbb{R}}.$ In fact, ${\mathfrak{a}}$ satisfies (\[eq:Ellipticity-nonaut2\]) if and only if the form $a_\omega$ given by ${\mathfrak{a}}_\omega(t;\cdot,\cdot):={\mathfrak{a}}(t;\cdot,\cdot)+\omega (\cdot{\, \vert \,}\cdot)$ satisfies the second inequality in . Moreover, if $u\in MR(V,V')$ and $v:=e^{-w.}u,$ then $v\in MR(V,V')$ and $u$ satisfies (\[evolution equation u(s)=x\]) if and only if $v$ satisfies $$\dot{v}(t)+(\omega+\mathcal A(t))v(t)=0 \ \ \ t{\rm
-a.e.} \hbox{ on} \ [s,T],\
\ \ \ \ v(s)=x. $$
Norm continuous evolution family {#Sec2 Norm continuity}
================================
In this section we assume that the non-autonomous form ${\mathfrak{a}}$ satisfies (\[eq:continuity-nonaut\])-. Thus as mentioned in the introduction, under theses assumptions the Cauchy problem (\[evolution equation u(s)=x\]) has $L^2$-maximal regularity in $H$. Thus for each $x\in V,$ $$U(\cdot,s)x\in {\textit{MR}\,}(V,H):={\textit{MR}\,}(s,T;V,H):=L^2(s,T;V)\cap H^1(s,T;H).$$ Moreover, $ U(\cdot,s)x\in C[s,T];V)$ by [@Ar-Mo15 Theorem 4.2]. From [@LH17 Theorem 2.7] we known that the restriction of ${{U}}$ to $V$ defines an evolution family which norm continuous. The same is also true for the Cauchy problem (\[evolution equation u(s)=x returned\]) and the assocaited evolution family $U^*_r$ since the r
| 1,853
| 661
| 516
| 2,084
| null | null |
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|
skew-Hopf pairing ${\eta }$ of ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ such that for all $i,j\in I$ one has $$\begin{aligned}
{\eta }(E_i,F_j)=-\delta _{i,j},\quad
{\eta }(E_i,L_j)=0,\quad
{\eta }(K_i,F_j)=0,\quad
{\eta }(K_i,L_j)=q_{ij}.
\end{aligned}$$
\(ii) The skew-Hopf pairing ${\eta }$ satisfies the equations $$\begin{aligned}
{\eta }(EK,FL)={\eta }(E,F){\eta }(K,L)
\end{aligned}$$ for all $E\in {\mathcal{U}}^+(\chi )$, $F\in {\mathcal{U}}^-(\chi )$, $K\in {\mathcal{U}}^{+0}$, and $L\in {\mathcal{U}}^{-0}$.
\(iii) If $\beta ,\gamma \in {\mathbb{N}}_0^I$ with $\beta \not=\gamma $, $E\in {\mathcal{U}}^+(\chi )_\beta $, $F\in {\mathcal{U}}^-(\chi )_{-\gamma }$, then ${\eta }(E,F)=0$.
\(iv) The restriction of ${\eta }$ to ${\mathcal{U}}^+(\chi )\times {\mathcal{U}}^-(\chi )$ induces a non-degenerate pairing ${\eta }:U ^+(\chi )\times U^-(\chi )\to {\Bbbk }$.
\(i) and (ii) are [@p-Heck07b Prop.4.3]. (iii) follows from the definition of ${\eta }$ and since ${\varDelta }$ is a ${\mathbb{Z}}^I$-homogeneous map. (iv) was proven in [@p-Heck07b Thm.5.8].
By the general theory, see [@b-Joseph 3.2.2], the pairing ${\eta }$ in Prop. \[pr:sHpdef\] can be used to describe commutation rules in ${\mathcal{U}}(\chi )$ and $U(\chi )$. Namely, $$\begin{aligned}
\label{eq:Ucomm1}
yx=&{\eta }(x{_{(1)}},S(y{_{(1)}}))x{_{(2)}}y{_{(2)}}{\eta }(x{_{(3)}},y{_{(3)}}),\\
\label{eq:Ucomm2}
xy=&{\eta }(x{_{(1)}},y{_{(1)}})y{_{(2)}}x{_{(2)}}{\eta }(x{_{(3)}},S(y{_{(3)}}))\end{aligned}$$ for all $x\in {\mathcal{V}}^+(\chi )$ and $y\in {\mathcal{V}}^-(\chi )$. Note that the second formula follows from the first one and Eqs. –.
Later we will also need some other general facts about $U(\chi )$. Some of them are collected here. Let $$\begin{aligned}
\label{eq:kerderK}
U^+_{i,K}(\chi )=&\ker {\partial ^K}_i\subset U^+(\chi ),&
U^+_{i,L}(\chi )=&\ker {\partial ^L}_i\subset U^+(\chi ),\\
\label{eq:kerderL}
U^-_{i,K}(\chi )=&{\Omega }(U^+_{i,K}),&
U^-_{i,L}(\chi )=&{\Omega }(U^+_{
| 1,854
| 1,042
| 2,089
| 1,724
| null | null |
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|
X/R)^{cat}$ for affine schemes.
\[basic.quot.exmp\] Let $f:X\to Y$ be a finite and surjective morphism. Set $R:={\operatorname{red}}(X\times_YX)\subset X\times X$ and let $\sigma_i:R\to X$ denote the coordinate projections. Then the geometric quotient $X/R$ exists and $X/R\to Y$ is a finite and universal homeomorphism (\[univ.homeo.defn\]). Therefore, if $X$ is the normalization of $Y$, then $X/R$ is the weak normalization of $Y$. (See [@rc-book Sec.7.2] for basic results on semi-normal and weakly normal schemes.)
By taking the reduced structure of $X\times_YX$ above, we chose to focus on the set-theoretic properties of $Y$. However, as (\[sch.quot.exmp\]) shows, even if $X,Y$ and $X\times_YX$ are all reduced, $X/R\to Y$ need not be an isomorphism. Thus $X$ and $X\times_YX$ do not determine $Y$ uniquely.
In Section \[first.exmp.sec\] we give examples of finite, set theoretic equivalence relations $R\rightrightarrows X$ such that the categorical quotient $(X/R)^{cat}$ is non-Noetherian and there is no geometric quotient. This can happen even when $X$ is very nice, for instance a smooth variety over ${{\mathbb C}}$. Some elementary results about the existence of geometric quotients are discussed in Section \[basic.res.sec\].
An inductive plan to construct geometric quotients is outlined in Section \[induct.plan.sect\]. As an application, we prove in Section \[pos.char.sec\] the following:
\[quot.by.R.charp\] Let $S$ be a Noetherian ${{\mathbb F}}_p$-scheme and $X$ an algebraic space which is essentially of finite type over $S$. Let $R\rightrightarrows X$ be a finite, set theoretic equivalence relation. Then the geometric quotient $X/R$ exists.
There are many algebraic spaces which are not of finite type and such that the Frobenius map $F^q:X\to X^{(q)}$ is finite. By a result of Kunz (see [@mats-ca p.302]) such algebraic spaces are excellent. As the proof shows, (\[quot.by.R.charp\]) remains valid for algebraic spaces satisfying this property.
In the Appendix, C. Raicu constructs finite scheme theoretic equ
| 1,855
| 1,407
| 2,155
| 1,812
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ock notation (the “advanced-retarded” ordering of supermatrix elements), $\mathcal{F}_M$ reads $$\mathcal{F}_M = \prod_{k=1}^M \exp\left\{-\frac{1}{2} \mathrm{trg} \ln \left[ 1 + i
\left(\begin{array}{cc} \lambda_k & 0 \\ 0 & \lambda_k'^{*} \end{array}\right)
\sigma_G L \right]\right\},$$ where $\lambda_k=\lambda_k'$ for all channels save the varied one, $k\neq c$. By employing the “angular” parametrization of $\sigma_{G}$ in terms of the matrices $t_{12}$ and $t_{21}$, the subsequent evaluation of $\mathcal{F}_M$ goes in parallel with Sec. 7 of Ref. [@ver85a], with the final result being $$\label{eq:app1}
\mathcal{F}_M = \prod_{k=1}^M \exp \left[-\frac{1}{2} \mathrm{trg} \ln \left(
1 + T_k^{\mathrm{eff}} t_{12}t_{21} \right)\right]\,.$$ This is just a usual formula for the channel factor in the VWZ theory except for the effective transmission coefficient $T_c^{\mathrm{eff}}$ in the channel $c$ that is now given by expression (\[eq:Teff\]) (we note that $T_k^{\mathrm{eff}}=T_k$ if $k\neq c$) [@note2]. Performing finally the Fourier transform, the two-point correlation function in the time domain takes the form of Eqs. (\[eq:VWZ\])–(\[eq:I\]), with the above modification in the channel factor corresponding explicitly to the second line of Eq. (\[eq:I\]).
Although the subsequent evaluation cannot be made analytically and has to be done numerically, it is still useful to make some qualitative analysis. To this end we note that $P_{ab}(t)$ and $J_a(t)$ are quite similar in structure to the “norm leakage” decay function [@sav97] and the form factor of the Wigner time delays [@leh95b]. Following the analysis performed there (see also [@dit00]), one notices that the time dependence in question is mainly due to the channel factor \[Eq. (\[eq:app1\])\]. In the time domain, its typical behavior is $\sim\prod_{k=1}^M(1+\frac{2}{\beta}T_k t)^{-\beta/2}$, where $\beta=1$ is for the present case of time-invariant systems whereas $\beta=2$ is for the case of broken time invariance (GUE). The case of th
| 1,856
| 2,638
| 2,494
| 1,788
| 2,983
| 0.775613
|
github_plus_top10pct_by_avg
|
the (context-free or extended) Petri nets with place capacity.
Quite obviously, a context-free Petri net with place capacity regulates the defining grammar by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. A similar mechanism was discussed in [@gin:spa1] where the total number of nonterminals in each sentential form is bounded by a fixed integer. There it was shown that grammars regulated in this way generate the family of context-free languages of finite index, even if arbitrary nonterminal strings are allowed as left-hand sides. The main result of this paper is that, somewhat surprisingly, grammars with capacity bounds have a greater generative power.
This paper is organized as follows. Section \[sec:def\] contains some necessary definitions and notations from language and Petri net theory. The concepts of grammars with capacities and grammars controlled by Petri nets with place capacities are introduced in section \[sec:capacities\]. The generative power and closure properties of capacity-bounded grammars are investigated in sections \[sec:power-gs\] and \[sec:nb-cfg\]. Results on grammars controlled by Petri nets with place capacities are given in section \[sec:PNC\].
Preliminaries {#sec:def}
=============
Throughout the paper, we assume that the reader is familiar with basic concepts of formal language theory and Petri net theory; for details we refer to [@das:pau; @han; @rei:roz].
The set of natural numbers is denoted by ${\mathbb{N}}$, the power set of a set S by ${\mathcal{P}({S})}$. We use the symbol $\subseteq$ for inclusion and $\subset$ for proper inclusion. The *length* of a string $w \in X^*$ is denoted by $|w|$, the number of occurrences of a symbol $a$ in $w$ by $|w|_a$ and the number of occurrences of symbols from $Y\subseteq X$ in $w$ by $|w|_Y$. The *empty* string is denoted by ${\lambda}$.
A *phrase structure grammar* (due to Ginsburg and Spanier [@gin:spa1]) is a quadruple $G=(V, \Sigma, S, R)$ where $V$
| 1,857
| 4,452
| 2,020
| 1,901
| 1,651
| 0.787225
|
github_plus_top10pct_by_avg
|
ince each component of the inertia stack $I_{\mathfrak{X}}$ is isomorphic to the original stack $\mathfrak{X}$, the normal bundle $N_q$ vanishes, and each component of ${\rm ch}(d(\lambda_q))$ is $1$. Furthermore, as $\mathfrak{X}$ is essentially a $k$-fold quotient of ${\mathbb P}^n$, $$\int_{\mathfrak{X}} \: = \: \frac{1}{k} \int_{{\mathbb P}^n}.$$ Plugging into the index formula, $$\begin{aligned}
\int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal O}_X(m))
{\rm Td}(TI_{\mathfrak{X}})
& = & \sum_{\alpha} \int_{\mathfrak{X}} \alpha^{-m}
{\rm ch}({\cal O}_{\mathfrak{X}}(m)) {\rm Td}(T\mathfrak{X}),
\\
& = & \sum_{\alpha} \alpha^{-m} \int_{\mathfrak{X}}
\sum_i {\rm ch}_i({\cal O}_{\mathfrak{X}}(m))
{\rm Td}_{n-i}(T\mathfrak{X}).\end{aligned}$$ Now, since $\alpha$ is a $k$th root of unity, the sum $$\sum_{\alpha} \alpha^{-m}$$ will vanish unless $m$ is divisible by $k$. Thus, if $m$ is not divisible by $k$, we find that $\chi({\cal O}_{\mathfrak{X}}(m))$ vanishes. Next, suppose that $m=n k$ for some integer $n$. Then, $$\begin{aligned}
\int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal O}_{\mathfrak{X}}(m))
{\rm Td}(I_{\mathfrak{X}})
& = & \sum_{\alpha} \int_{\mathfrak{X}} \alpha^{-m}
{\rm ch}({\cal O}_{\mathfrak{X}}(m)) {\rm Td}(T\mathfrak{X}),
\\
& = & \sum_{\alpha} \int_{\mathfrak{X}} \pi^*
{\rm ch}({\cal O}_{{\mathbb P}^n}(n))
{\rm Td}(T {\mathbb P}^n), \\
& = & \sum_{\alpha} \frac{1}{k} \int_{ {\mathbb P}^n }
{\rm ch}({\cal O}_{{\mathbb P}^n}(n))
{\rm Td}(T {\mathbb P}^n), \\
& = &
\int_{ {\mathbb P}^n }
{\rm ch}({\cal O}_{{\mathbb P}^n}(n))
{\rm Td}(T {\mathbb P}^n), \\
& = & \chi\left({\mathbb P}^n, {\cal O}_{ {\mathbb P}^n }(n) \right).\end{aligned}$$
Now, let us compare to expectations. In the present case, if $m$ is not divisible by $k$, then all the sheaf cohomology groups of ${\cal O}_{\mathfrak{X}}(m)$ should vanish, so the Euler class $\chi({\cal O}_{\mathfrak{X}}(m))$ should vanish, exactly as we have computed. If $m$ is divisible by $k$, then $\chi({\cal O}_{\mathfrak{X}}(m)) =
\chi({\cal
| 1,858
| 1,377
| 2,070
| 1,618
| null | null |
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|
\Psi$ and $\Phi$, we write $\Phi \subseteq \Psi$ to express that $\Phi$ is contained in $\Psi$, following ordinary set theory that term $(i,P) \in \Phi$ implies $(i,P) \in \Psi$.
\[D:ENSEMBLE\_DIFFERENCE\] Let $\Psi$ and $\Phi$ be ensembles such that $\Phi \subseteq \Psi$. In classification of difference between $\Psi$ and $\Phi$, $\Psi$ is the *minuend*, $\Phi$ is the *subtrahend*, and the set difference[^11] $\Psi \setminus \Phi$ is the *remainder*.
\[L:DISJOINT\_AND\_COMPLEMENTARY\] Let $\Psi$ and $\Phi$ be ensembles such that $\Phi \subseteq \Psi$. The subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are disjoint and complementary with respect to $\Psi$.
Since $\Phi \subseteq \Psi$, definition \[D:ENSEMBLE\_DIFFERENCE\] applies. The minuend is $\Psi$, the subtrahend is $\Phi$, and the remainder is $\Psi \setminus \Phi$.
Definition \[D:DISJOINT\_ENSEMBLES\] asserts that two ensembles $\Theta$ and $\Upsilon$ are disjoint if ${{\operatorname{dom}{\Theta}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Upsilon}}} = \varnothing$. As hypothesis presume the lemma’s antithesis, namely that the subtrahend $\Phi$ and the remainder $\Psi \setminus \Phi$ are not disjoint. This implies that there exists some $i$ such that $i \in {{\operatorname{dom}{\Phi}}}\thickspace\cap\thickspace{{\operatorname{dom}{(\Psi \setminus \Phi)}}}$. For this $i$ to exist in the intersection of the domains of two mappings, there must be both a term $(i,P) \in \Phi$ and a term $(i,Q) \in (\Psi \setminus \Phi)$.
Because $(i,P) \in \Phi$ and the lemma’s premise states that $\Phi \subseteq \Psi$, then $(i,P) \in \Psi$. Since term $(i,Q) \in (\Psi \setminus \Phi)$, then $(i,Q) \in \Psi$ and $(i,Q) \notin \Phi$. Since $\Psi$ is a mapping, then $(i,P) \in \Psi$ and $(i,Q) \in \Psi$ together imply $P = Q$. With $P = Q$ and $(i,Q) \notin \Phi$, then also $(i,P) \notin \Phi$.
However, the immediately preceding conclusion that $(i,P) \notin \Phi$ contradicts the earlier inference that term $(i,P) \in \Phi$ must exist if $i$ is a member of
| 1,859
| 3,960
| 3,078
| 1,915
| null | null |
github_plus_top10pct_by_avg
|
ndence.
Let $g = (g_1,\ldots,g_s)^\top \colon \mathbb{R}^b \rightarrow \mathbb{R}^s$ be a twice-continuously differentiable vector-valued function defined over an open, convex subset $\mathcal{S}_n$ of $[-A,A]^b$ such that, for all $P \in
\mathcal{P}_n$, $\psi = \psi(P) = \mathbb{E}[W_1] \in \mathcal{S}_n$. Let $\widehat{\psi} = \widehat{\psi}(P) = \frac{1}{n} \sum_{i=1}^n W_i$ and assume that $\widehat{\psi} \in \mathcal{S}_n$ almost surely, for all $P \in
\mathcal{P}_n$. Finally, set $\theta = g(\psi)$ and $\widehat{\theta} = g(\widehat{\psi})$. For any point $\psi \in \mathcal{S}_n$ and $j\in \{ 1,\ldots,s\}$, we will write $G_j(\psi) \in \mathbb{R}^b$ and $H_j(\psi)\in \mathbb{R}^{b \times b}$ for the gradient and Hessian of $g_j$ at $\psi$, respectively. We will set $ G(\psi)$ to be the $s\times b$ Jacobian matrix whose $j^{\rm th}$ row is $G^\top_j(\psi)$.
[**Remark**]{} The assumption that $\hat{\psi}$ belongs to $\mathcal{S}_n$ almost surely can be relaxed to hold on an event of high probability, resulting in an additional error term in all our bounds.
To derive a high-dimensional Berry-Esseen bound on $g(\psi) - g(\hat{\psi})$ we will study its first order Taylor approximation. Towards that end, we will require a uniform control over the size of the gradient and Hessian of $g$. Thus we set $$\label{eq:H.and.B}
B = \sup_{P \in \mathcal{P}_n }\max_{j=1,\ldots,s} ||G_j(\psi(P))||
\quad \text{and} \quad \overline{H} =
\sup_{ \psi\in \mathcal{S}_n }\max_{j=1,\ldots,s}
\|H_j(\psi)\|_{\mathrm{op}}$$ where $\|H_j(\psi)\|_{\mathrm{op}}$ is the operator norm.
[**Remark.**]{} The quantity $\overline{H}$ can be defined differently, as a function of $\mathcal{P}_n$ and not $\mathcal{S}_n$. In fact, all that is required of $\overline{H}$ is that it satisfy the almost everywhere bound $$\label{eq:H.2}
\max_j \int_0^1 \left\| H_j \left( t\psi(P) - (1-t)\hat{\psi}(P) \right) \right\|_{\mathrm{op}} dt \leq \overline{H},$$ for each $P \in \mathcal{P}_n$ (see below). This allows us to establish a uniform
| 1,860
| 2,700
| 1,570
| 1,722
| 4,113
| 0.768023
|
github_plus_top10pct_by_avg
|
, $\cS(Y)=\emptyset$, $|\cS|=|\cS^*|+2$, and $|\cR|\ge 5+|Y|=5$. Using \[prop:rrstar\], equation (\[eq:general\]) becomes $$|\cR|\le 2+|\cS^*|-\frac{1}{2}|\cR^*|\le 2+3-0=5.$$ Thus $|\cR|=5$, $|\cS^*|=3$, $\cR^*=\emptyset$, $|\cS|=5$, and all inequalities hold with equality in inequality (\[eq:individual\]).\
We may assume, by relabeling, if necessary, that $\cS^*=\{\{1,2,4\},\{1,2,5\},\{1,3,4\}\}$. Then inequality (\[eq:individual\]) implies that, for $i\in\{2,4\}$, we have $\sum_{j\in C_i}|\cR(i,j)|=2$, and for $i\in\{3,5\}$ we have $\sum_{j\in C_i}|\cR(i,j)|=3$. This means that $|\cR(3,5)|-|\cR(2,4)|=1$ and, therefore, $|R(3,5)|>|R(2,4)|$. Hence (using \[prop:qij\]), we must have that $\cR(2,4)=\emptyset$ and $|\cR(3,5)|=1$. This means that $|\cR(2,5)|=|\cR(3,4)|=2$, which is a contradiction by \[prop:qij\].\
[**Case**]{} $|L|=1$\
Here we may assume that $L=\{2\}$. Then $Y=Y_2$, $|\cS|=2+|\cS^*|+|Y|$, $\{3,4,5\}\notin\cR^*$ (from \[prop:bi\], and from \[prop:ysmall\]) $|\cR(3,j)|\le 1$ for each $j\in \{4,5\}$. Using \[prop:rrstar\], equation (\[eq:general\]) becomes $$|\cR|\le 2+|\cS^*|+\frac{3}{2}|Y|-\frac{1}{2}|\cR^*|\le 5+\frac{3}{2}|Y|\ .$$\
1. $|Y|=|Y_2|=1$\
Since $5+|Y|=6\le |\cR|\le 6+\frac{1}{2}-\frac{1}{2}|\cR^*|$, we get $|\cR|=6$, $|\cS^*|=3$, and $|\cR^*|\le 1$. From \[prop:sstar\] we know that there are $i\in\{2,3\}$ and $j\in\{4,5\}$ such that, if $\{\ell,m\}\ne\{i,j\}$ then $\cR(\ell,m)=\emptyset$. Since $\cR\setminus\cR^*=\cR(i,j)$, this implies that $|\cR(i,j)|\ge 5$. Moreover, since $|\cR(3,k)|\le 1$ for each $k\in\{4,5\}$, we have $i=2$. Then, using inequality (\[eq:individual\]), we obtain $7\le 1+|Y_2|+|\cS_2^*|+\sum_{k\in C_2}|\cR(2,k)|+|\cR^*_2|\le |\cS|=6$, a contradiction.\
2. $|Y|=|Y_2|\ge 2$\
Here we have $\cR(3,4)=\cR(3,5)=\emptyset$, and so (since $\{3,4,5\}\notin\cR^*$) we get that $\sum_{j\in C_2}|\cR(2,j)|+|\cR^*_2|=|\cR|$. Using inequality (\[eq:individual\]) with $x=2$ yields $1+|Y|+|\cS_2^*|+|\cR|\le 2+|\cS^*|+|Y|$. In other words, $|\cR|\le 1+|\cS^*|-|\cS_2^*|<5$
| 1,861
| 645
| 1,035
| 1,921
| null | null |
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|
arameter $\eta$, defined through $D_\eta=4-\eta$, and on the renormalization scale $\mu$, for which we have taken $\mu=m_\pi$. In the following we use, $$\begin{aligned}
R=&-\frac{2}{\eta}-1+\gamma-\log(4\pi) \,,
\\
q_0''=&q_0'-q_0 \,.\end{aligned}$$ The integrals $A(m)$, $A(q_0,q_0')$ and $B(q_0,|{\vec{q}}|)$ appear, for example, in [@scherer02]. We have checked that both results coincide. It is important to maintain the $-i\epsilon$ prescription, otherwise the integrals may give a wrong result. We take it into account by replacing $q_0'\to q_0'-i\epsilon$ when evaluating the integrals.
### $A(m), A(q_0,q'_0)$ and $B(q_0,{\vec{q}})$
We have, $$A(m)=
-\frac{1}{8\pi^2}m^2\left(\frac12R+\log\left(\frac{m}{\mu}\right)\right) \,.$$ $$\begin{aligned}
A(q_0,q_0')\equiv
-\frac{q_0''}{8\pi^2}
\left[
\pi\frac{\sqrt{m^2-q_0''^2}}{q_0''}
+1-R-2\log\left(\frac{m}{\mu}\right)
\right. & \nonumber\\\left.
-\frac{2 \sqrt{{q_0''}^2(m^2-{q_0''}^2)} \tan
^{-1}\left(\frac{\sqrt{{q_0''}^2}}{\sqrt{m^2-{q_0''}^2
}}\right)}{{q_0''}^2}
\right] &\end{aligned}$$ $$\begin{aligned}
B(q_0,{\vec{q}})&=
-\frac{1}{16\pi^2}\left[-1+R+2\log\left(\frac{m}{\mu}\right)+2L(|q|)\right]\end{aligned}$$ with $$\begin{aligned}
L(|q|)\equiv &\frac{w}{|q|}\log\left(\frac{w+|q|}{2m}\right) \,,\end{aligned}$$ $w\equiv \sqrt{4m^2+|q|^2}$, $|q|\equiv \sqrt{{\vec{q}}^2-q_0^2}$, and $q^2\equiv q_0^2-{\vec{q}}^2\le0$.
### $C(q_0,q_0')$ and $D(q_0,q_0')$
$$\begin{aligned}
C(q_0,q_0')\equiv&
-\frac{1}{16\pi^2}{\int_0^1dx}{\int_{0}^1dy}\Bigg[
3y^{-\frac12}(1-y)
\\&
\left[
-\frac43
-\frac12(R-1+\log(4))
-\frac12\log\left(\frac{s_{xy}}{4\mu^2}\right)
\right]
\\&+y^{-\frac12}(1-y)(m^2+q_0''r_0')
s_{xy}^{-1}
\\&
-\pi(q_0''+r_0')s_x^{-\frac12}
\Bigg] \,,\end{aligned}$$ with $s_x=m^2-q_0^2+x(q_0^2-q_0''^2)$, $s_{xy}=m^2+(1-y)(-q_0^2+x(q_0^2-q_0''^2))$.
$$\begin{aligned}
D(q_0,q_0')=&
-C(q_0,q_0')+\frac{1}{q_0'}\frac{1}{4\pi}\sqrt{m^2-q_0^2} \,.\end{aligned}$$
### $I(q_0,|{\vec{q}}|,q_0')$
$$\begin{aligned}
I(q_0,q,q_0')&=
-\frac{1}{8\pi^2}\int_0^1dx\int_0^1dy
| 1,862
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| 1,145
| 2,021
| null | null |
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|
ow. The parameters in the potential of $\chi$ are thus guaranteed to be independent of the parameters of our brane.
The scalar potential in our brane excluding $V(\chi)$ is now given by $$\begin{aligned}
V &=& m_1^2 \Phi^\dagger \Phi + m_2^2 \sigma^\dagger \sigma + m_3^2
\eta^\dagger \eta + {1 \over 2} \lambda_1 (\Phi^\dagger \Phi)^2 + {1 \over 2}
\lambda_2 (\sigma^\dagger \sigma)^2 + {1 \over 2} \lambda_3 (\eta^\dagger
\eta)^2 \nonumber \\ && + \lambda_4 (\Phi^\dagger \Phi)(\sigma^\dagger \sigma)
+ \lambda_5 (\Phi^\dagger \Phi)(\eta^\dagger \eta) + \lambda_6
(\sigma^\dagger \sigma)(\eta^\dagger \eta) + (\mu z e^{i \varphi}
\sigma^\dagger \eta + h.c.),\end{aligned}$$ where $\mu$ has the dimension of mass and we assume that all mass parameters are of the same order of magnitude, i.e. 1 TeV.
The minimum of $V$ satisfies the following conditions: $$\begin{aligned}
m_1^2 + \lambda_1 v^2 + \lambda_4 u^2 + \lambda_5 w^2 &=& 0, \\
u(m_2^2 + \lambda_2 u^2 + \lambda_4 v^2 + \lambda_6 w^2) + \mu z w &=& 0, \\
w(m_3 ^2 + \lambda_3 w^2 + \lambda_5 v^2 + \lambda_6 u^2) + \mu z u &=& 0,\end{aligned}$$ where $\langle \phi^0 \rangle = v$. Hence $$\begin{aligned}
v^2 &\simeq& {-\lambda_2 m_1^2 + \lambda_4 m_2^2 \over \lambda_1 \lambda_2 -
\lambda_4^2}, \\ u^2 &\simeq& {-\lambda_1 m_2^2 + \lambda_4 m_1^2 \over
\lambda_1 \lambda_2 - \lambda_4^2},\end{aligned}$$ and $$w \simeq {- \mu z u \over m_3^2 + \lambda_5 v^2 + \lambda_6 u^2},$$ which is indeed of order $z$ as mentioned earlier.
Whereas Im$\phi^0$ becomes the longitudinal component of the usual $Z$ boson, $(u {\rm Im} \sigma + w {\rm Im} \eta)/\sqrt {u^2 + w^2}$ becomes that of the new $Z_A$ boson. Since the $3 \times 3$ mass matrix in the basis \[Im$\sigma$, Im$\eta$, $z\varphi$\] is given by $$\pmatrix{-\mu z w / u
& \mu z& \mu w \cr
\mu z & - \mu z u / w &
-\mu u \cr \mu w & -\mu u & - \mu u w / z },$$ the axion $a$ is identified as the following: $$\begin{aligned}
{a \over \sqrt 2} &=& {1 \over {N}} \left[ uw^2 {\rm Im} \sigma
- w u^2 {\rm Im} \eta + z (u^
| 1,863
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| 1,491
| null | null |
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|
other one composed of cross terms. The first one is given by $$\begin{aligned}
&& \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{1st}} =
\sum_{k, K} \sum_{l, L}
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times&
\biggl[
x^2 e^{- i ( h_{k} - h_{l} ) x}
- (ix) \frac{e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{K} - h_{k} ) }
+ (ix) \frac{e^{- i ( h_{k} - \Delta_{L} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{L} - h_{l} ) }
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\biggl\{
e^{- i ( \Delta_{K} - \Delta_{L} ) x} + e^{- i ( h_{k} - h_{l} ) x}
- e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - \Delta_{L} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&\times&
(UX)_{\alpha l}^* (UX)_{\beta l}
\left\{ (UX)^{\dagger} A W \right\}_{l L}
\left\{ W^{\dagger} A (UX) \right\}_{L l}
\nonumber \\
&+&
\sum_{k \neq m} \sum_{K} \sum_{l \neq n} \sum_{L}
\frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) }
\frac{ 1 }{ ( h_{n} - h_{l} ) (\Delta_{L} - h_{l}) (\Delta_{L} - h_{n}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{- i h_{m} x}
- \left( \Delta_{K} - h_{m} \right) e^{- i h_{k} x}
- ( h_{m} - h_{k} ) e^{- i \Delta_{K} x}
\biggr]
\nonumber \\
&\times&
\biggl[
\left( \Delta_{L} - h_{l} \right) e^{+ i h_{n} x}
- \left( \Delta_{L} - h_{n} \right) e^{+ i h_{l} x}
- ( h_{n} - h_{l} ) e^{+ i \Delta_{L} x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\nonumber \\
&\times&
(UX)_{\alpha l}^* (UX)_{\beta n}
\left\{ (UX)^{\dagger} A W \right\}_{n L}
\left\{ W^{\dagger} A (UX) \right\}_{L l}
\nonumber \\
&+&
\sum_{k, K} \sum_{l, L}
\frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\left( e^{- i \Delta_{K} x}
| 1,864
| 1,779
| 1,547
| 1,998
| null | null |
github_plus_top10pct_by_avg
|
) 0.7 Peritoneum Pancreas
10 F 80 Body, tail No No T3N1M0(III) 2.3 Pancreas
11 F 64 Tail No Gemcitabine/Cisplatin \#1 T4N1M1(IVB) 1.3 Liver Pancreas, Liver
12 M 59 Head No No T3N1M1(IVB) 1.4 Liver Pancreas
13 M 68 Body, tail No No T4N1M1(IVB) 0.6 Liver Pancreas, Liver
14 F 54 Neck, body No Gemcitabine/5 - FU \#2 T3N1M0(III) 5.5 Pancreas
15 M 57 Body Yes Gemcitabine, Cisplatin/5 FU \#6 T4N1M1(IVB) 5.2 Liver Pancreas
16 M 83 Head No No T4N1M0(IVA) 0.2 Pancreas
17 M 54 Head No No T4N1M0(IVA) 0.2 Pancreas
18 M 64 Yes Gemcitabine/xeloda \#9, Irinotecan \#2 M1(IVB) 63.3 Lung Lung
19 M 58 Head No No T4N1M0(IVA) 2.4
| 1,865
| 5,459
| 1,214
| 1,096
| null | null |
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|
*residuals* $\textrm{Res}(\mathbb{D})$ *in $\mathbb{D}$ given by* $$\textrm{Res}(\mathbb{D})=\{\mathbb{R}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for all }\beta\succeq\alpha\in\mathbb{D}\}\}.\label{Eqn: residual}$$
*The net* *adheres at* *$x\in X$*[^27] *if it is frequently in every neighbourhood of $x$, that is* $$((\forall N\in\mathcal{N}_{x})(\forall\mu\in\mathbb{D}))((\exists\nu\succeq\mu)\!:\chi(\nu)\in N).$$ *The point $x$ is known as the* *adherent* *of $\chi$ and the collection of all adherents of $\chi$ is the* *adherent set of the net, which* *may be expressed in terms of the* *cofinal subset* *of $\mathbb{D}$* $$\textrm{Cof}(\mathbb{D})=\{\mathbb{C}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{C}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for some }\beta\succeq\alpha\in\mathbb{D}\}\}\label{Eqn: cofinal}$$ (thus $\mathbb{D}_{\alpha}$ is cofinal in $\mathbb{D}$ iff it intersects every residual in $\mathbb{D}$), *as* $$\textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists\mathbb{C}_{\beta}\in\textrm{Cof}(\mathbb{D}))(\chi(\mathbb{C}_{\beta})\subseteq N)\}.\label{Eqn: adh net1}$$
*This recognizes, in keeping with the limit set, each subnet of a net to be a net in its own right, and is equivalent to* $${\textstyle \textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D}))(\chi(\mathbb{R}_{\alpha})\bigcap N\neq\emptyset)\}.\qquad\square}\label{Eqn: adh net2}$$
Intuitively, a sequence is eventually in a set $A$ if it is always in it after a finite number of terms (of course, the concept of a *finite number of terms* is unavailable for nets; in this case the situation may be described by saying that a net is eventually in $A$ if its *tail is in* $A$) and it is frequently in $A$ if it always returns to $A$ to leave it again. It can be shown that a net is eventually (resp. frequently) in a set iff it is not frequently (resp.eventually) in its complement.
The following examples illustrate graphical
| 1,866
| 728
| 2,006
| 1,773
| 2,875
| 0.776387
|
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|
s definition can be taken to suggest that their numerical plurality is a primitive fact. But this is just an artifact of the set-theoretical language. The elements of a set $\mathcal{M}$ *qua* set-theoretical objects have to be numerically distinct for $\mathcal{M}$ to be a well-defined set of $N$ objects. However, the referents of this formalism – the matter points *qua* physical objects – are individuated by the distance relations given by $\Delta$, so that these relations account for their numerical plurality.
To emphasize the indistinguishability of the matter points in the formalism, it is possible to make the above definition of $\Omega$ independent of the labelling by introducing the following equivalence relation:
Take $\Delta,\Delta'\in\Omega$, and consider $\mathbb S_N$ as the set of all possible permutations of elements of $\mathcal M$. We define $\Delta\simeq\Delta'$ if and only if there is a permutation $\sigma\in \mathbb S_N$ such that for all $(i,j)\in\mathcal E$ it is the case that $\Delta'_{ij}=\Delta_{\sigma(i)\sigma(j)}$.
The set $$\begin{aligned}
\widetilde \Omega = \Omega / {\simeq} := \left\{ [\Delta]_{\simeq} \, \big| \,
\Delta \in \Omega \right\},
\qquad
[\Delta]_{\simeq} = \{\Delta'\in\Omega\,|\,\Delta'\simeq\Delta\}\end{aligned}$$ then comprises all possible configurations of distance relations independently of a labelling of the matter points.
One way to envision an element of $[\Delta]_{\simeq}\in \widetilde \Omega$ is by a representative $\Delta\in\Omega$ that can be viewed as a coloured graph $G(\Delta)=(\mathcal M,\mathcal E,\Delta)$ in which $\mathcal M$ are the nodes, $\mathcal E$ are the edges, and to each edge $(i,j)\in\mathcal E$ the colour $\Delta_{ij}$ is attached. Also the graphs $G(\Delta)$ can be made label-independent by considering the equivalence classes $[G(\Delta)]_{\simeq}=\{G(\Delta')\,|\, \Delta'\simeq\Delta\}$ and treating $G(\Delta)$ only as the corresponding representative of the class.
This ontology follows Leibniz’ relationalism abou
| 1,867
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| 1,936
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|
t claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang’s Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.'
author:
- 'Alan Dow[$^1$]{} and Franklin D. Tall[$^2$]{}'
bibliography:
- 'normality.bib'
nocite: '[@*]'
title: Normality versus paracompactness in locally compact spaces
---
[^1]
[^2]
Introduction
============
The space of countable ordinals is locally compact, normal, but not paracompact. The question of what additional conditions make a locally compact normal space paracompact has a long history. At least 45 years ago, it was recognized that subparacompactness plus collectionwise Hausdorffness would do (see e.g. [@T1]), as would perfect normality plus metacompactness [@A]. Z. Balogh proved a variety of results under MA$_{\omega_1}$ [@B1] and **Axiom R** [@B2], and was the first to realize the importance of not including a perfect pre-image of $\omega_1$ (equivalently, the one-point compactification being countably tight [@B1]). However, he assumed collectionwise Hausdorffness in order to obtain paracompactness. A breakthrough came with S. Watson’s proof that:
$V = L$ implies locally compact normal spaces are collectionwise Hausdorff, and hence locally compact normal metacompact spaces are paracompact.
Watson’s proof crucially involved the idea of *character reduction*: if one wants to separate a closed discrete subspace of size $\kappa$, $\kappa$ regular, in a locally compact normal space, it suffices to separate $\kappa$ compact sets, each with an *outer base* of size $\leq \kappa$.
An [**outer base**]{} for a set $K \subseteq X$ is a collection ${\mathcal}{B}$ of open
| 1,868
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| 871
| 1,350
| null | null |
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|
ivalent to $\mathcal{C}$. If $K_{\mathbb{P}_w^2}$ denotes the canonical divisor of $\mathbb{P}_w^2$ and $$\mathcal{C}^{(k)} = \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \mathcal{C}_j, \qquad 0 \leq k < d,$$ then these dimensions are given as the cokernel of the evaluation linear maps $$\pi^{(k)}: H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right) \right)
\longrightarrow \bigoplus_{P \in S}
\frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}}$$ where $\mathcal{M}_{\mathcal{C},P}^{(k)}$ is defined as the following quasi-adjunction-type $\mathcal{O}_{\PP^2_w}$-module $$\mathcal{M}_{\mathcal{C},P}^{(k)}\!:=\!
\left\{ g \in\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)
\vphantom{\sum_{j=1}^r}\!\right.\!
\left|\ \operatorname{mult}_{E_\v} \pi^* g >
\sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} m_{\v j} -\! \nu_\v, \ \forall \v \in \Gamma_P\! \right\}\!.$$ The symbol $\{\cdot\}$ denotes the decimal part of a rational number and the multiplicities $m_{\v j}$ and $\nu_\v$ are provided by $\pi^{*} \mathcal{C}_j = \hat{\mathcal{C}}_j
+ \sum_{P \in \Si} \sum_{\v \in \Gamma_P} m_{\v j} E_\v$ and $K_{\pi} = \sum_{P \in S} \sum_{\v \in \Gamma_P}
(\nu_\v-1) E_\v$ for an embedded $\mathbb{Q}$-resolution $\pi$ of $\mathcal{C} \subset \mathbb{P}^2_w$, cf. Definition \[def:M\].
As a consequence, $$\label{eq:h1}
h^1(\tilde X, \CC)=2\sum_{k=0}^{d-1} \dim \operatorname{coker}\pi^{(k)}.$$ These formulas also reminisce the local and global interplay of conditions on linear systems on the base surface. Also, the local conditions can be obtained from a $\Q$-resolution of the singularities, which in particular allows for simpler theoretical and practical algorithms to calculate the irregularity. Moreover, in this paper the ramification along each irreducible component need not be the same, which translates into considering a non-reduced curve as a ramification locus. This allows for general formul
| 1,869
| 1,517
| 1,645
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| null | null |
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|
rators. Precisely, we assume the set of offerings $S_j$, the number of separators $\ell_j$, and their respective positions $\cP_j=\{p_{j,1},\ldots,p_{j,\ell_j}\}$ are predetermined. Each user draws the ranking of items from the PL model, and provides the partial ranking according to the separators of the form of $\{a>\{b,c,d\}>e>f\}$ in the example in the Figure \[fig:hasse\].
\[thm:main2\] Suppose there are $n$ users, $d$ items parametrized by $\theta^*\in\Omega_b$, each user $j$ is presented with a set of offerings $S_j\subseteq [d]$, and provides a partial ordering under the PL model. When the effective sample size $\sum_{j=1}^n \ell_j$ is large enough such that $$\begin{aligned}
\label{eq:main21}
\sum_{j=1}^n \, \ell_j \;\;\geq\;\; \frac{2^{11}e^{18b}\eta\log(\ell_{\max}+2)^2 }{\alpha^2\gamma^2\beta} d\log d\;,
\end{aligned}$$ where $b\equiv \max_{i}|\theta^*_i |$ is the dynamic range, $\ell_{\max} \equiv \max_{j\in[n]} \ell_j$, $\alpha$ is the (rescaled) spectral gap defined in , $\beta$ is the (rescaled) maximum degree defined in , $\gamma$ and $\eta$ are defined in Eqs. and , then the [*rank-breaking estimator*]{} in with the choice of $$\begin{aligned}
\lambda_{j,a} &=& \frac{1}{\kappa_j - p_{j,a}} \;,
\label{eq:deflambda}
\end{aligned}$$ for all $a\in[\ell_j]$ and $j\in[n]$ achieves $$\begin{aligned}
\label{eq:main22}
\frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq \;\; \frac{4\sqrt{2}e^{4b}(1+ e^{2b})^2}{\alpha\gamma} \sqrt{\frac{d\, \log d}{\sum_{j=1}^n \ell_j}} \;,
\end{aligned}$$ with probability at least $ 1- 3e^3d^{-3}$.
Consider an ideal case where the spectral gap is large such that $\alpha$ is a strictly positive constant and the dynamic range $b$ is finite and $\max_{j \in[n]}p_{j,\ell_j}/\kappa_j = C$ for some constant $C <1$ such that $\gamma$ is also a constant independent of the problem size $d$. Then the upper bound in implies that we need the effective sample size to scale as $O(d\log d)$, which is only a l
| 1,870
| 1,151
| 1,419
| 1,848
| 2,255
| 0.781386
|
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|
surface tension of the base solution without camphor and $\Gamma$ is a positive constant.
![\[fig:model\] Illustration of side view of a camphor boat.](model.eps){width="7cm"}
The time evolution on the camphor concentration $c$ is shown as $$\begin{aligned}
\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}-ac+f(x-X),
\label{eq:concentration}
\end{aligned}$$ where $a$ is the sum of sublimation rate and dissolution rate of the camphor molecules on solution surface, $D$ is the diffusion coefficient of the camphor molecule, and $f$ denotes the dissolution rate of the camphor molecules from the camphor disk to the aqueous solution surface. As for the term $f(z)$, we apply the following description, $$\begin{aligned}
f(z) = \begin{cases}
f_0, & ({-r < z <r}),\\
0, & \text({\rm otherwise}).
\end{cases}
\label{eq:provide2}
\end{aligned}$$ That is to say, the dissolution of camphor molecules from the disk occurs at $-r < z < r$. The above equation does not include Marangoni effect directly, although the flow has an influence on the camphor concentration. The previous paper [@Kitahata2] showed that Eq. (\[eq:concentration\]) was reasonable if $D$ was recognized as the spatially uniform effective diffusion coefficient of the camphor to include the transportation by the flow. In addition, this spatially-uniform effective diffusion coefficient is supported by the experimental results that the diffusion length is proportional to the square root of elapsed time [@Suematsu].
Theoretical analysis
====================
Our experimental results showed that the camphor boat moved with a constant velocity in time as shown in Fig. \[fig:velo\]. Thus, we should consider solutions for the motion of the camphor boat with a constant velocity $v$ in $x$-direction, i.e. $X = vt$. From this condition, Eq. (\[eq:motion\]) leads to $$\begin{aligned}
-hv+F=0.
\label{eq:motion1}
\end{aligned}$$ By setting $\xi=x-vt$ and $c=c(\xi)$,
| 1,871
| 3,066
| 3,147
| 2,108
| 3,881
| 0.769545
|
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|
\vert_{i \neq j}~(\text{single sum}) +
\hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{double sum})
\label{hatS-4th-order-ij-T-transf}\end{aligned}$$ where $$\begin{aligned}
&& \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{single sum})
\nonumber \\
&=&
\sum_{K}
\biggl[
(ix) e^{ - i h_{i} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{j} - h_{i} ) }
- (ix) e^{ - i \Delta_{K} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} ) }
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( h_{j} - h_{i} )^2 }
e^{ - i h_{j} x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( h_{j} - h_{i} )^2 }
\left( \Delta_{K} + 2 h_{j} - 3 h_{i} \right)
e^{ - i h_{i} x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{K} - h_{j} )^2 }
\left( h_{i} + 2 h_{j} - 3 \Delta_{K} \right)
e^{ - i \Delta_{K} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}
\nonumber \\
&+&
\sum_{K}
\biggl[
- (ix) e^{ - i h_{j} x}
\frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( h_{j} - h_{i} ) }
- (ix) e^{ - i \Delta_{K} x}
\frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{i} ) }
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} )^3 }
\left( h_{j} + 2 h_{i} - 3 \Delta_{K} \right)
e^{ - i \Delta_{K} x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{j} )^3 ( h_{j} - h_{i} )^2 }
\left( \Delta_{K} + 2 h_{i} - 3 h_{j} \right)
e^{ - i h_{j} x}
+ \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{j} - h_{i} )^2 }
e^{ - i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}
\left\{ (UX)^{\dagger} A W \right\}_{j K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}
\nonumber \\
&+&
\sum_{K} \sum_{k \neq i, j}
\biggl[
- (ix) e^{- i \Delta_{K} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{K} - h_{k} ) }
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} -
| 1,872
| 1,474
| 2,200
| 1,897
| null | null |
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|
&= \tau_D \times \frac{1}{16\pi m_D^3} \sqrt{
(m_D^2 - (m_{P_1} - m_{P_2})^2) (m_D^2 - ( m_{P_1} + m_{P_2})^2 )
}\,.\end{aligned}$$ The direct CP asymmetries are [@Golden:1989qx; @Pirtskhalava:2011va; @Nierste:2017cua] $$\begin{aligned}
a_{CP}^{\mathrm{dir}} &= \mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right) \mathrm{Im}\left(\frac{A_b}{A_{\Sigma}}\right)\,.\end{aligned}$$
Solving the complete U-spin System \[sec:solving\]
===================================================
We discuss how to extract the U-spin parameters of Eqs. (\[eq:decomp-1\])–(\[eq:decomp-4\]) from the observables. We are mainly interested in the ratios of parameters and less in their absolute sizes and therefore we consider only quantities normalized on $t_0$, that is $$\begin{aligned}
\label{eq:def-u-par}
&\tilde{t}_1 \equiv \frac{t_1}{t_0}\,, \qquad
\tilde{t}_2 \equiv \frac{t_2}{t_0}\,, \qquad
\tilde{s}_1 \equiv \frac{s_1}{t_0}\,, \qquad
\tilde{p}_0 \equiv \frac{p_0}{t_0}\,,\qquad
\tilde{p}_1 \equiv \frac{p_1}{t_0}\,.\end{aligned}$$ We choose, without loss of generality, the tree amplitude $t_0$ to be real. The relative phase between $\mathcal{A}(K\pi)$ and $\mathcal{A}(\pi K)$ is physical and can be extracted in experimental measurements. However, the relative phases between $\mathcal{A}(\pi\pi)$, $\mathcal{A}(KK)$ and $\mathcal{A}(K\pi)$ are unphysical, i.e. not observable on principal grounds. This corresponds to two additional phase choices that can be made in the U-spin parametrization. Consequently, without loss of generality, we can also choose the two parameters $\tilde{s}_1$ and $\tilde{t}_2$ to be real. Altogether, that makes eight real parameters, that we want to extract, not counting the normalization $t_0$. Of these, four parameters are in the CKM-leading part of the amplitudes and four in the CKM-suppressed one. In the CP limit $\mathrm{Im}\lambda_b \rightarrow 0$ we can absorb $\tilde{p}_0$ and $\tilde{p}_1$ into $\tilde{t}_2$ and $\tilde{s}_1$ respectively, which makes four real parameters in that limit.
The ei
| 1,873
| 2,303
| 2,505
| 1,737
| 2,067
| 0.783078
|
github_plus_top10pct_by_avg
|
mathit{f}}') \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\_PROJECTION\], $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathbf{f}}\,' = (\psi', \phi')$ and $\mho_{\mathscr{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathit{f}}'$
Definition \[D:ITERATIVE\_TRANSFORM\] evaluates ${\mathbf{f}}\,' = (\psi', \phi') = (\phi\xi, {\mathit{f}}'(\phi\xi))$ as the succeeding frame. This complex relation separates into simple conditions $\psi' = \phi\xi$ and $\phi' = {\mathit{f}}'(\phi\xi) = {\mathit{f}}'(\psi')$. The assertion ${\mathbf{f}}\,' = (\psi', \phi') = (\psi', {\mathit{f}}'(\psi'))$ bears the same meaning as ${\mathbf{f}}\,' = (\psi', \phi') \in {\mathit{f}}'$.
Definition \[D:CONSISTENT\_STEP\] states that step ${\mathfrak{A}}_\xi({\mathit{s}})$ is *consistent* if $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathbf{f}}\,' \in {\mathit{f}}' = \mho_{\mathscr{F}}({\mathfrak{A}}_\xi({\mathit{s}}))$, which is here satisfied.
\[T:AUTOMATON\_WALK\_PROCEDURE\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with consistent step ${\mathit{s}} \in {\mathbb{S}}$ and volatile excitation $\lbrace \xi_n \rbrace$. Suppose $\lbrace {\mathit{s}}_n \rbrace$ is the walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. Sequential procedure projection $\overline{\mho}_{\mathscr{F}}(\lbrace {\mathit{s}}_n \rbrace)$ covers sequential process projection $\overline{\mho}_{\mathbf{F}}(\lbrace {\mathit{s}}_n \rbrace)$.
We have the premises that step ${\mathit{s}} \in {\mathbb{S}}$ is consistent, that $\lbrace \xi_n \rbrace$ is an excitation, that ${\mathfrak{A}}$ is an automaton, and that $\lbrace {\mathit{s}}_n \rbrace$ is the walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. We temporarily suppress the repetitive lengthy expression ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$ through the abbreviation ${\mathfrak{A}}^{\mathbb{N}}$.
Induction demonstrates that each step of walk ${\mathfrak{A}}^{\mathbb{N}}$ is consistent. F
| 1,874
| 890
| 1,287
| 1,642
| null | null |
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|
e definition of $\Delta^2$ depends on the coordinates and the desired level of approximation. This section defines our computational domains and the finite-difference representations of $\Delta^2$, which are second-order accurate, in uniform Cartesian, uniform cylindrical, and logarithmic cylindrical coordinate systems.
Uniform Cartesian Grid {#s:uniform_cartesian_grid}
----------------------
In Cartesian coordinates, we discretize the computational domain $[x_{\rm min},x_{\rm max}]\times[y_{\rm min},y_{\rm max}]\times[z_{\rm min},z_{\rm max}]$ with size $L_x\times L_y\times L_z$ uniformly into $N_x\times N_y\times N_z$ cells. We define the face-centered coordinates as $x_{i+1/2} = x_{\rm min} + i \delta x$ where $\delta x \equiv L_x/N_x$ and similarly for $y_{j+1/2}$ and $z_{k+1/2}$. We also define the cell-centered coordinates as $x_i = (x_{i-1/2} + x_{i+1/2})/2$ with index $i$ running from $1$ to $N_x$, and similarly for $y_j$ with $j=1,2,\cdots, N_y$ and $z_k$ with $k=1,2,\cdots,N_z$.
We denote the cells inside the nominal index range given above as “active cells", because these are the places where other fluid variables are updated by a hydrodynamics solver. To the active cells, we add one extra layer of “ghost cells" to the boundaries of the computational domain, with their cell-centered coordinates are denoted, for example, by $i=0$ and $i=N_x+1$ in the $x$-direction. The boundary conditions for the Poisson equation and other equations of hydrodynamics are provided using these ghost cells. We similarly define the ghost cells in the other coordinate systems described below.
The second-order accurate, finite-difference approximation to Equation can be written as $$\label{eq:fd_uniform_cartesian}
\left(\Delta_x^2 + \Delta_y^2 + \Delta_z^2\right)\Phi_{i,j,k} = 4\pi G \rho_{i,j,k},$$ where $\Phi_{i,j,k}$ and $\rho_{i,j,k}$ are the cell-centered potential-density pair and the difference operators $\Delta_x^2$, $\Delta_y^2$, and $\Delta_z^2$ are defined by $$\begin{aligned}
\Delta_x^2\Phi_{i,j,k} &= \
| 1,875
| 4,105
| 2,870
| 1,906
| 2,712
| 0.777638
|
github_plus_top10pct_by_avg
|
tting $p=0$ and expanding these terms, B\_[LR\_i]{}\^x &=& - B\_0(0,[M\_[\_[i]{}\^]{}]{},m\_[\_x]{})\
&=& B\_0(0,[M\_[\_[i]{}\^]{}]{},m\_[\_x]{}) .
Neglecting terms proportional to $g_2$ and summing over the stops and charginos yields \_[i=1]{}\^2 \_[x=1]{}\^2 B\_[LR\_i]{}\^x &&\
&+& .\[Eq:chargino-B0\]
For $|\mu| > |M_2|$, one finds that $U^\dag_{12}V^\dag_{12}\simeq0$ and $ U^\dag_{22}V^\dag_{22}\simeq1$, whereas for $|\mu| < |M_2|$, one finds that $U^\dag_{12}V^\dag_{12}\simeq1$ and $U^\dag_{22}V^\dag_{22}\simeq0$. Furthermore, $\sin 2\theta_t = -2 \lam_t v_d \tanb(A_t - \frac{\mu}{\tanb})/(m^2_{\tilde{t}_2}-m^2_{\tilde{t}_1})$ so that[^4] (A\_t-) I(\^2,m\_[\_1]{}\^2,m\_[\_2]{}\^2) .\[Eq:chargino-app\] Among the two charginos, the dominant corrections are only from the Higgsino and are proportional to the Higgsino mass, $\mu$. Hence the chargino corrections tend be larger when $|\mu| > |M_2|$ (heavier Higgsino) and smaller when $|\mu| < |M_2|$ (lighter Higgsino) as shown in \[fig:ch-muM2\]. We refer to the term in \[Eq:fullchargino-1st\] containing the $B_{0(1)}$ Passarino-Veltman functions and its prefactor as the $``B_{0(1)}^{\widetilde{\chi}^\pm}"$ term.
![The plot shows that the dominant piece $B_0^{\widetilde{\chi}^\pm}$ (defined in the text) of the chargino corrections is small when $|\mu/M_2| < 1$ and can be large when $|\mu/M_2| >> 1$. The vertical dashed lines mark the crossover between these two regimes. Darker shades of blue represent increasing squark masses from 1 TeV to $\ge$4 TeV.[]{data-label="fig:ch-muM2"}](new_PLOTS/chargino_B0_vs_muM2.pdf){width="60.00000%"}
![The plot shows the exact, one-loop chargino-stop threshold correction to the bottom quark mass vs. the approximate form of this correction given in \[Eq:common-app\]. Darker shades of blue represent increasing squark masses from 1 TeV to $\ge$4 TeV. []{data-label="fig:ch-ex-app"}](new_PLOTS/chargino-stop.pdf){width="60.00000%"}
In \[fig:ch-ex-app\], the exact, one-loop chargino-stop threshold correction to the bottom quark mas
| 1,876
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| 2,023
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|
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|
E_0,\infty[$ where $E_0\geq 0$ but we omit this generalization here. We shall denote by $I^\circ$ the interior $]0,E_m[$ of $I$. The interval $I$ in equipped with the 1-dimensional Lebesgue measure ${\mathcal{L}}^1$, which we typically write as $dE$ in the sense that ${\mathcal{L}}^1(A)=\int_A dE$.
For $(x,\omega)\in G\times S$ the *escape time (in the direction $\omega$)* $t(x,\omega)=t_-(x,\omega)$ is defined by t(x,)&={s>0 | x-sG}\
&={T>0 | x-sG [for all]{} 0<s<T}, for $(x,\omega)\in G\times S$.
The escape time function $t(\cdot,\cdot)$ is known to be lower semicontinuous in general, and continuous if $G$ is convex, see e.g. [@tervo14].
We define $$\Gamma:={}&(\partial G)\times S\times I,$$ and $$\Gamma_0 :={}&\{(y,\omega,E)\in \Gamma\ |\ \omega\cdot\nu(y)=0\} \\[2mm]
\Gamma_{-}:={}&\{(y,\omega,E)\in \Gamma\ |\ \omega\cdot\nu(y)<0\} \\[2mm]
={}&\{(y,\omega)\in \partial G\times S\ |\ \omega\cdot\nu(y)<0\}\times I \\[2mm]
\Gamma_{+}:={}&\{(y,\omega,E)\in \Gamma\ |\ \omega\cdot\nu(y)>0\} \\[2mm]
={}&\{(y,\omega)\in \partial G\times S\ |\ \omega\cdot\nu(y)>0\}\times I.$$ Let $\mu_\Gamma = \sigma\otimes \mu_S\otimes {\mathcal{L}}^1$, written typically as ${d}\sigma{d}\omega{d}E$ in the same sense as discussed above. It follows that $\mu_{\Gamma}(\Gamma_0)=0$ and $$\Gamma=\Gamma_0\cup \Gamma_-\cup \Gamma_+.$$ Let \[N0\] N\_0:={(x,,E)GSI | (x-t(x,))=0}, and \[D\] D:=(GSI)N\_0. Recall that $N_{0}$ has a measure zero in $ G\times S\times I$ ([@tervo14 Theorem 3.8]).
Define *escape-time mappings $\tau_{\pm}(y,\omega)$ from boundary to boundary in the direction $\omega$* as follows $$\begin{aligned}
{3}
&
\tau_-(y,\omega):=
\inf\{s>0\ |\ y+s\omega\not\in G\},\quad && (y,\omega)\in \partial G\times S, \\
&
\tau_+(y,\omega):=
\inf\{s>0\ |\ y-s\omega\not\in G\},\quad && (y,\omega)\in \partial G\times S.\end{aligned}$$
Note that for $(y,\omega,E)\in\Gamma_-$ the vector $(y_+,\omega,E)\in\Gamma_+$ where $y_+:=y+
\tau_-(y,\omega)\omega\in\Gamma_+$ and $$\tau_-(y,\omega)=\tau_+(y_+,\omega).$$
\[re:S\_
| 1,877
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nts were found to be between $\sim2$ (close to the resolution limit) and $14\,\mu$Hz, but we mark these findings uncertain because of the effect of the 1d$^{-1}$ aliasing. The 2nd–6th panels of Fig. \[fig:g207FTa\] shows the FTs of the weekly datasets. We found only slight amplitude variations from one week to another. The amplitude of the dominant frequency varied between 8.6 and 10.5mmag.
![G 207-9: amplitude spectra of one night’s observation (*top panel*) and the weekly datasets (*lower panels*).[]{data-label="fig:g207FTa"}](g207FTa.eps){width="\columnwidth"}
The standard pre-whitening of the whole dataset resulted 26 frequencies above the 4S/N limit. Most of them are clustering around the frequencies already known by the analyses of the daily and weekly datasets. Generally, amplitude and (or) phase variations during the observations can be responsible for the emergence of such closely spaced peaks. In such cases, these features are just artefacts in the FT, as we fit the light curve with fixed amplitudes and frequencies during the standard pre-whitening process. Another possibility is that some of the closely spaced peaks are rotationally split frequencies. We can resolve such frequencies if the time base of the observations is long enough. The Rayleigh frequency resolution ($1/\Delta T$) of the whole dataset is $0.08\,\mu$Hz. We also have to consider the 1d$^{-1}$ alias problem of single-site observations, which results uncertainties in the frequency determination.
---------- -------------------- ------- ------ ------ -------
$f_1$ $3426.303\pm0.001$ 291.9 10.1 111.5
$f_2$ $1678.633\pm0.003$ 595.7 2.0 15.4
$f_1^-?$ $3414.639\pm0.004$ 292.9 11.7 1.6 18.0
$f_3$ $5098.861\pm0.003$ 196.1 1.2 12.5
$f_2^-?$ $1667.328\pm0.005$ 599.8 11.3
| 1,878
| 2,752
| 2,484
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is conformally invariant [@bida], and the conformally rescaled components of the fermion obey the flat space equation (\[fse\]) with Neumann boundary conditions. Thus, the spectrum (\[flateigenvalues\]) is also valid for massless fermions.
Flat Spacetime
---------------
Let us now consider the Casimir energy density in the conformally related flat space problem. We shall first look at the effective potential per unit area on the brane, ${\cal A}$. For bosons, this is given $$V^b_0 = {1\over 2 {\cal A}} {\rm Tr}\ {\rm\ln} (-\bar\Box^{(0)}/\mu^2).$$ Here $\mu$ is an arbitrary renormalization scale. Using zeta function regularization (see e.g. [@ramond]), it is straightforward to show that $$V^b_0 (L)= {(-1)^{\eta-1} \over (4\pi)^{\eta} \eta!}
\left({\pi\over L}\right)^{D-1} \zeta'_R(1-D).
\label{vboson}$$ Here $\eta=(D-1)/2$, and $\zeta_R$ is the standard Riemann’s zeta function. The contribution of a massless fermion is given by the same expression but with opposite sign: $$V^{f}_0(L) = - V_0^b(L).
\label{vfermion}$$ The expectation value of the energy momentum tensor is traceless in flat space for conformally invariant fields. Moreover, because of the symmetries of our background, it must have the form [@bida] $$\langle T^z_{\ z}\rangle_{flat}= (D-1) \rho_0(z),\quad
\langle T^i_{\ j}\rangle_{flat}= -{\rho_0(z)}\ \delta^i_{\ j}.$$ By the conservation of energy-momentum, $\rho_0$ must be a constant, given by $$\rho_0^{b,f} = {V_0^{b,f} \over 2 L} = \mp {A \over 2 L^D},$$ where the minus and plus signs refer to bosons and fermions respectively and we have introduced $$A\equiv{(-1)^{\eta} \over (4\pi)^{\eta} \eta!}
\pi^{D-1} \zeta'_R(1-D) > 0.$$ This result [@adpq; @dpq], which is a simple generalization to codimension-1 branes embedded in higher dimensional spacetimes of the usual Casimir energy calculation, and it reproduces the same kind of behaviour: the effective potential depends on the interbrane distance monotonously. So, depending on $D$ and the field’s spin, it induces an atractive or repulsiv
| 1,879
| 983
| 2,177
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|
pinorial $\bf{16}$ and $\bf{\overline{16}}$ representations of $SO(10)$, under the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group is given as follows: $$\begin{aligned}
\textbf{16} &= &\left({\textbf{4}},{\textbf{2}},
0\right) + \left(\overline{{\textbf{4}}},{\textbf{1}},
-1\right) + \left(\overline{{\textbf{4}}},{\textbf{1}},
+1\right),\nonumber\\
\overline{\textbf{16}} &= &\left(\overline{{\textbf{4}}},{\textbf{2}},
0\right) + \left({\textbf{4}},{\textbf{1}},
-1\right) + \left({\textbf{4}},{\textbf{1}},
+1\right).\nonumber\end{aligned}$$ Here to break the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group to the standard model group, we require the heavy higgs pair. This pair is given by $$\begin{aligned}
\left(\overline{{\textbf{4}}},{\textbf{1}},
-1\right) + \left({\textbf{4}},{\textbf{1}},
-1\right).\nonumber\end{aligned}$$ Similarly, the vectorial representation $\bf{10}$ of $SO(10)$ decomposed under the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group is given as follows $$\begin{aligned}
\textbf{10} &= &\left({\textbf{6}},{\textbf{1}},
0\right) + \left({\textbf{1}},{\textbf{2}},
-1\right) + \left({\textbf{1}},{\textbf{2}},
+1\right)\nonumber,\end{aligned}$$ Furthermore, we take the following normalizations of the hypercharge and electromagnetic charge $$\begin{aligned}
Y &=& \frac{1}{3} (Q_1 + Q_2 + Q_3) + \frac{1}{2} (Q_4 + Q_5), \nonumber\\
Q_{em} &=& Y + \frac{1}{2} (Q_4 - Q_5). \nonumber\end{aligned}$$ where the $Q_{i}$ charges of a state arise due to $\psi^{i}$ for $i =1,...,5$. The following table summaries the charges of the colour $SU(3)$ and electroweak $SU(2) \times U(1)$ Cartan generators of the states which form the $SU(4) \times SU(2)_L \times U(1)_L$ matter representations:
Representation $\overline{\psi}^{1,2,3}$ $\overline{\psi}^{4,5}$ $Y$ $Q_{em}$
---------------------------------------------------------------------- --------------------------- ------------------------- ------ ----------
| 1,880
| 2,480
| 2,955
| 1,980
| null | null |
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|
t(\sum_{\lambda}f_{\lambda}(v^{-1})
f_{\lambda}(1)\right) f_{\mu^t}(1) f_{\mu^t}(v^{-1}) v^N
v^{-d(n(\mu^t) - n(\mu))}.$$ The standard formula $\sum \dim
{\mathbb{C}}[{\mathfrak{h}}]^{\text{co}{{W}}}_iv^{-i}=[n]_{v^{-1}}!$ shows that the fake degrees satisfy the identity $$\sum_{\lambda} f_{\lambda}(v^{-1}) f_{\lambda}(1)=
\frac{\prod_{i=1}^n (1-v^{-i})}{(1-v^{-1})^n} = [n]_{v^{-1}}!.$$ Applying this and to we find that $$\label{weredone} p(\overline{J^d}, v) = \frac{\sum_{\mu}
f_{\mu^t}(1) f_{\mu^t}(v^{-1}) v^{-d(n(\mu^t) - n(\mu))}
v^N[n]_{v^{-1}}!}{\prod_{i=2}^n (1-v^{-i})}.$$ After changing the order of summation from $\mu$ to $\mu^t$ and using the equality $$v^N [n]_{v^{-1}} = v^N\frac{\prod_{i=1}^n
(1-v^{-i})}{(1-v^{-1})^n} = \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n}
= [n]_v!,$$ becomes the required equality , and so the corollary is proved.
${\mathbb{Z}}$–algebras {#zalg}
=======================
{#Z-alg-defn}
Typically in noncommutative algebra—and certainly in our case—one cannot apply the Rees ring construction since one is working with just right modules or homomorphism groups rather than bimodules. One way round this is to use ${\mathbb{Z}}$-algebras and in this section we describe the basic properties that we need from this theory. The reader is referred to [@BP] or [@SV Section 11] for the more general theory and to [@bgs Section 3] for applications of ${\mathbb{Z}}$-algebras to Koszul duality.
Throughout this paper a [*${\mathbb{Z}}$-algebra*]{} will mean a [*lower triangular ${\mathbb{Z}}$-algebra*]{}. By definition, this is a (non-unital) algebra $B=\bigoplus_{i\geq j\geq 0} B_{ij}$, where multiplication is defined in matrix fashion: $B_{ij}B_{jk}\subseteq B_{ik}$ for $i\geq j\geq k\geq 0$ but $B_{ij}B_{\ell k}=0$ if $j\not=\ell$. Although $B$ cannot have a unit element, we do require that each subalgebra $B_{ii}$ has a unit element $1_i$ such that $1_ib_{ij}=b_{ij}=b_{ij}1_j$, for all $b_{ij}\in B_{ij}$.
{#Z-alg-defn2}
Let $B$ be a ${\mathbb{Z}}$-algebra. We define the category
| 1,881
| 1,099
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| 1,801
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| 0.772956
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|
\pi^{*} ((h') + {\left \lfloor D \right \rfloor}) + \pi^{*} {\left \{ D \right \}} + \sum_{i} m_i E_i \geq 0.$$ Let $F$ be the $w$-homogeneous polynomial defined by the effective $(h')+{\left \lfloor D \right \rfloor}$. Using the isomorphism $H^0(X,\cO_X(D)) \cong \CC[x,y,z]_{w,d}$ described in Lemma \[lemma:h0-weighted\], the inequality simply becomes $\operatorname{mult}_{E_i} \pi^{*} F + m'_i \geq 0$, $\forall i=1,\ldots,s$, as claimed.
The second cohomology group {#sec:h2}
---------------------------
Let $\rho_X: \tilde{X} \to \mathbb{P}^2_w =:X$ be a cyclic branched covering ramifying on a (non-necessarily) reduced curve $\mathcal{C} = \sum_{j=1}^r n_j \mathcal{C}_j$ with $d:= \deg_w \mathcal{C}$ sheets, denote by $d_j = \deg_w \mathcal{C}_j$ so that $\sum_{j=1}^r n_j d_j = d$. Assume $\mathcal{C} \sim dH$ where $H$ is a divisor of $w$-degree one –it is neither effective nor reduced in general. In order to use the power of section \[sc:Esnault\], one needs to deal with a $\QQ$-normal crossing divisor. Hence, let $\pi: Y \to X$ be an embedded $\Q$-resolution of $\mathcal{C}$ and consider the maps $\tilde{\pi}: \tilde{Y} \to \tilde{X}$ and $\rho_Y: \tilde{Y} \to Y$ completing the following commutative diagram. $$\begin{tikzcd}
{ \arrow[mysymbol]{dr}[description]{\#}}\tilde{X}\ar[d,"\rho_X" left] &\ar[l,"\tilde{\pi}" above]\tilde{Y}\ar[d,"\rho_Y"]\\
X&\ar[l,"\pi"] Y
\end{tikzcd}$$ Denote by $S$ the points of $\mathbb{P}^2_w$ that have been blown up in the resolution $\pi: Y \to X$ and $\Gamma_P$ the dual graph associated with the resolution of $P \in \Si$. Then the total transform of $\mathcal{C}$ and $H$ and the relative canonical divisor can be written as $$\label{eq:notationEi}
\begin{aligned}
\pi^{*} \mathcal{C} &= \hat{\mathcal{C}} + \sum_{P \in \Si} \sum_{\v \in \Gamma_P} m_\v E_\v,
& \quad \pi^{*} \mathcal{C}_j &= \hat{\mathcal{C}}_j + \sum_{P \in \Si} \sum_{\v \in \Gamma_P}
m_{\v j} E_\v, \\
\pi^{*} H &= \hat{H} + \sum_{P \in \Si} \sum_{\v\in\Gamma_P} \b_\v E_\v,
& K_\pi &= \sum_{P \in S} \sum_
| 1,882
| 1,656
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| 1,931
| null | null |
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|
We use the ordering on ${{\textsf}{Irrep}({{W}})}$ arising from the dominance ordering; thus, as in [@MacD Example 1, p.116], the [trivial representation]{} $\operatorname{{\textsf}{triv}}$\[triv-defn\] is labelled by $(n)$ while the [sign representation]{} $\operatorname{{\textsf}{sign}}$\[sign-defn\] is parametrised by $(1^n)$ and so $\operatorname{{\textsf}{triv}}>\operatorname{{\textsf}{sign}}$. Note that the operation on ${{\textsf}{Irrep}({{W}})}$ given by tensoring by $\operatorname{{\textsf}{sign}}$ corresponds to the transposition of partitions.
Category $\mathcal{O}_c$ {#subsec-3.7}
------------------------
(See [@GGOR] and [@BEGqi Definition 2.4].) Let $\mathcal{O}_c$\[cat-O-defn\] be the abelian category of finitely-generated $H_c$-modules $M$ which are locally nilpotent for the subalgebra $\mathbb{C}[{\mathfrak{h}}^*]\subset H_c$. By [@guay Theorem 3] $\mathcal{O}_c$ is a highest weight category.
Given $\mu \in {{\textsf}{Irrep}({{W}})}$, we define $\Delta_c(\mu)$,\[standard-defn\] an object of $\mathcal{O}_c$ called the *standard module*, to be the induced module $\Delta_c(\mu) = H_c\otimes_{\mathbb{C}[{\mathfrak{h}}^*]\ast {{W}}} \mu,$ where $\mathbb{C}[{\mathfrak{h}}^*]\ast {{W}}$ acts on $\mu$ by $pw\cdot
m = p(0) (w\cdot m)$ for $p\in \mathbb{C}[{\mathfrak{h}}^*]$, $w\in {{W}}$ and $m\in \mu$. It is shown in [@BEGqi Section 2] that each $\Delta_c(\mu)$ has a unique simple quotient $L_c(\mu)$,\[L-defn\] that the set $\{ L_c(\mu) : \mu \in
{{\textsf}{Irrep}(W)}\}$ provides a complete list of non-isomorphic simple objects in $\mathcal{O}_c$ and that every object in $\mathcal{O}_c$ has finite length. Note that it follows from the PBW Theorem \[PBW\] that the standard module $\Delta_c(\mu)$ is a free left $\mathbb{C}[{\mathfrak{h}}]$-module of rank $\dim(\mu)$.
The ${{\textsf}{KZ}}$ functor {#subsec-3.11}
-----------------------------
Let $M\in \mathcal{O}_c$. Then its localisation ${M^{\text{reg}}}=M[\delta^{-1}]$ at the powers of $\delta$ is a $W$-equivariant $D$-module on ${\mathfrak{h}^
| 1,883
| 1,039
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| 1,909
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| 0.788564
|
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|
Of course this means that $s_{\beta_l}$ forces that $\dot{U}(\eta,0)$ meets $\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})$ for cofinally many $\xi<\delta$ such that $s_{\beta_l}\upharpoonright\gamma_\alpha\Vdash
\dot{f}_{\gamma_\alpha}(\xi)=n_l$. But $\bar{s}$ has already decided the value of $\dot{f}_{\gamma_\alpha}\upharpoonright\delta$, and $\bar{s}$ already forces $\dot{U}(\eta,0)\cap\dot{Z}(\xi,
C_{\zeta(\gamma_\alpha)})\neq\emptyset$ whenever $s_{\gamma_\beta}$ does. In particular then, $\bar{s}$ forces there is a $\xi$ with $\dot{f}_{\gamma_\alpha}(\xi)=n_l$ (and so $\dot{Z}(\xi,
C_{\zeta(\gamma_\alpha)})\subseteq\dot{W}(\gamma_\alpha,n_l)$) and $\dot{U}(\eta,0)\cap\dot{Z}(\xi,
C_\zeta(\gamma_\alpha))\neq\emptyset$.
For the record, let us state what we have accomplished:
$(S)[S]$ implies **LCN**$(\aleph_1)$.
There is a model of $(S)[S]$ in which **LCN** holds, *i.e. every locally compact normal space is collectionwise Hausdorff.*
Large Cardinals and the MOP
===========================
In [@DT2] we showed that large cardinals are not required to obtain the consistency of every *locally compact perfectly normal space is paracompact*. It is interesting to see which other PFA$(S)[S]$ results can be obtained without large cardinals. The standard method used was pioneered by Todorcevic in [@To3] and given several applications in [@D], all in the context of PFA results. In the context of PFA$(S)[S]$, it is referred to in [@To] and actually carried out in [@DT1] for a version of [P-ideal Dichotomy]{} and for ****. It is routine to get additionally that such models are of form MA$_{\omega_1}(S)[S]$ by interleaving additional forcing. In [@DT2] we pointed out that such methods can give models in which in addition the following holds:
**${\mathbf{\mathop{\pmb{\sum}}}}^{\bm{-}}$(sequential)**
In a compact sequential space, each locally countable subspace of size $\aleph_1$ is $\sigma$-discrete.
A modification of such a proof produces a model in which the following proposition (see [@FTT]) holds:
**${\mathbf{\mathop
| 1,884
| 1,018
| 1,554
| 1,735
| 1,849
| 0.785091
|
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|
rintf. That's where the Undefined Behaviour1 happens. No such thing in the C++ example.
1Do note that the C example is not guaranteed to SEGFAULT. Undefined behaviour is undefined.
Q:
Selectively record commands and output from terminal to MySQL table
In some situations I would like to store complete commands and output from terminal sessions in a MySQL database.
Ideally I would just copy the command and output from the terminal to my clipboard then just paste into a simple bash script that would update the relevant field.
I have experimented with using the read function in bash, see below. However when pasting multiple lines into the read it does not parse the line breaks correctly.
#!/bin/bash
read -e -p "name: " name
read -e -p "output: " ouput
mysql -u example@localhost << EOF
use database;
insert into table (name, output) values('$title', '$output');
EOF
I know this is not elegant, but would really useful reference for me in the future.
Perhaps somebody could shed some light on a better way to get this done...
Thanks in advance,
A:
To paste multilines in a variable, I would do :
#!/bin/bash
echo -en 'Paste your multiline content and ^D to submit >>> '
content="$(cat)"
echo "$content"
Q:
Methods to Make Site Content Editable
I've been doing some preliminary research into making the content on the internal, customer support website I maintain editable by the managers in the department, to make it easier to update the site's content when I'm not available to do so.
The design concept is that the manager would access a login-secured Edit mode on the site, and click a button next to the section of content they wish to change. Once they've completed their changes, upon exiting out of Edit mode, their changes would be written back to the site, where they would be permanently recorded for the customer support agents who use the site to see the updated content.
The easiest way to do this, if I understand right, would require have a SQL database that stores the actual content-- when the page lo
| 1,885
| 3,316
| 982
| 1,722
| 2,479
| 0.779454
|
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|
\frac{2k}{\alpha}\right)}.$$
Let $\hat{E}_{{\widehat{S}}} = \bigotimes_{j\in S} E(j)$.
------------------------------------------------------------------------
[^1]: For simplicity, we assume that the data are split into two parts of equal size. The problem of determining the optimal size of the split is not considered in this paper. Some results on this issue are contained in [@shao1993linear].
---
abstract: 'In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation formulas for double $L$-values.'
address: 'Department of Mathematics Faculty of Science and Technology Tokyo University of Science Noda, CHIBA 278-8510 JAPAN'
author:
- Takashi Nakamura
title: Simple proof of the functional relation for the Lerch type Tornheim double zeta function
---
Introduction and main results
=============================
We define the Lerch type Tornheim double zeta function by $$T (s,t,u \,; x,y) := \lim_{R \to \infty} \sum_{m,n=1}^{m+n=R} \frac{e^{2 \pi i mx} e^{ 2 \pi i ny}}{m^s n^t (m+n)^u},
\label{eq:defT}$$ where $0 \le x,y \le 1$, $\Re (s+t) >1$, $\Re (t+u) > 1$ and $\Re (s+t+u) > 2$. This function is continued meromorphically by [@NakamuraA Theorem 2.1]. Let $k \in {\mathbb{N}} \cup \{ 0 \}$. The function $T (s, t, u \,; x,y)$ can be continued meromorphically to ${\mathbb{C}}^3$, and all of its singularities are located on the subsets of ${\mathbb{C}}^3$ defined by the following equations; $$\begin{split}
t = 1 - k \qquad &{\rm{if}} \quad x \not \equiv 1, \,\, y \equiv 1 \mod 1, \\
s = 1 - k \qquad &{\rm{if}} \quad x \equiv 1, \,\, y \not \equiv 1 \mod 1, \\
{\mbox{no singularity}} \qquad &{\rm{if}} \quad x \not \equiv 1, \,\, y \not \equiv 1 \mod 1.
\end{split}$$
We write $T(s, t, u) := T (s, t, u \,; 1,1)$ and call this function the Tornheim double zeta function. The values $T(a,b,c)$ for $a,b,c \in{\mathbb{N}}$ were investigated by Tornheim in 1950 and later by Mordell in 1958, and some e
| 1,886
| 866
| 873
| 2,032
| 2,133
| 0.782497
|
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|
ac{d}{dt} \hat{\chi}(X,t)&=&
\frac{i}{\hbar}
\left[\hat{H},\hat{\chi}(X,t)\right]_{\mbox{\tiny\boldmath$\cal B$}}
-\frac{1}{2}\left\{\hat{H},\hat{\chi}(X,t)\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\nonumber\\
&+&\frac{1}{2}\left\{\hat{\chi}(X,t),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
%\nonumber \\
=\left(\hat{H},\hat{\chi}(X,t)\right)\;,
\label{eq:qcbracket}\end{aligned}$$ where $$\begin{aligned}
\left[\hat{H} , \hat{\chi}\right]_{\mbox{\tiny\boldmath$\cal B$}}
&=&
\left[\begin{array}{cc} \hat{H} & \hat{\chi}\end{array}\right]
\cdot\mbox{\boldmath$\cal B$}\cdot
\left[\begin{array}{c} \hat{H} \\
\hat{\chi} \end{array} \right]
\label{eq:qlm}\end{aligned}$$ is the commutator and $$\begin{aligned}
\{\hat{H},\hat{\chi}\}_{\mbox{\tiny\boldmath$\cal B$}}
&=&
\sum_{i,j=1}^{2N}
\frac{\partial \hat{H}}{\partial X_i}{\cal B}_{i j}
\frac{\partial \hat{\chi}}{\partial X_j}
\label{Lambda}\end{aligned}$$ is the Poisson bracket [@goldstein]. Both the commutator and the Poisson bracket are defined in terms of the antisymmetric matrix $$\mbox{\boldmath$\cal B$}=\left[\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right]\;.
\label{B}$$ The last equality in Eq. (\[eq:qcbracket\]) defines the quantum-classical bracket. Following Refs. [@b3; @bsilurante; @sergi], the quantum-classical law of motion can be easily casted in matrix form as $$\begin{aligned}
\frac{d}{dt} \hat{\chi}&=&\frac{i}{\hbar}
\left[\begin{array}{cc} \hat{H} & \hat{\chi} \end{array}\right]
\cdot\mbox{\boldmath$\cal D$}\cdot
\left[\begin{array}{c} \hat{H} \\ \hat{\chi} \end{array}\right]
\nonumber\\
&=&\frac{i}{\hbar}[\hat{H},\hat{\chi}]_{\mbox{\tiny\boldmath$\cal D$}}\;,
\label{qclm}\end{aligned}$$ where $$\mbox{\boldmath$\cal D$}=\left[\begin{array}{cc} 0&
1-\frac{\hbar}{2i}
\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\\
-1+\frac{\hbar}{2i}
\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
& 0\end{array}\right]\;.
\label{D}$$ The structure of Eq. (\[qclm\]) is that of a non-Hamiltonian commutator, which will be defined below i
| 1,887
| 1,104
| 2,100
| 1,858
| null | null |
github_plus_top10pct_by_avg
|
y^2\dot x \} \\
= \{ y^2x\dot x y\}.
\end{gathered}$$
The last two equalities imply $\alpha _2 =1$ and other coefficients are equal to zero. Therefore, $$\begin{gathered}
\phi (x,y,z) = \{y\dot xxz\}-2\delta_3\{yxx\dot z\}
-2\delta_5\{xxy\dot z\}-2\delta_4\{y\dot zxx\} \\
+2(\delta_3+\delta_5)\{\dot zyyy\}+2\delta_4\{y\dot zyy\}
+2\delta_3\{\dot zzy\}+2\delta_4\{z\dot zy\}+2\delta_5\{\dot zyz\}.\end{gathered}$$
By assumption this dipolynomial is Jordan. When we expand Jordan products then the central letter is preserved, hence the dipolynomials consisting of dimonomials from $\phi (x,y,z)$ with the fixed central letter must be Jordan. In particular, if we choose the central letter $x$ then the dipolynomial $\{y\dot xxz\}$ must be Jordan, but this is not true by the proof of Theorem \[thm:CohnForDialgebra\].
The contradiction obtained proves that $f\notin I$.
S-IDENTITIES
============
In this section ${\mathop{\mathrm{char}}\nolimits}\Bbbk=0$, so we can perform the process of full linearization of identities and varieties of algebras are always defined by multilinear identities.
Equality of varieties ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ and ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$
-------------------------------------------------------------------------------------------------------------
Consider a class of special Jordan dialgebras ${\mathrm{SJ}}$. The class ${\mathrm{SJ}}$ is not a variety because it is not close relative the taking of homomorphic images. Consider the operator ${\mathcal{H}}$ acting on classes of algebraic systems $${\mathcal{H}}(K)=\{A\mid A=\phi(B)\text{ for }B\in K,\phi\colon B\to A
\text{ is an epimorphism}\}.$$
It is well-known that ${\mathcal{H}}({\mathrm{SJ}})$ is a variety of algebras which we denote ${\mathcal{H}}{\mathrm{SJ}}$.
We recall (see Section \[subsec:DefDialg\]) that if $D\in{\mathrm{Di}}{\mathrm{Alg}}0$ then $D$ can be endowed with left and right actions of the algebra $\bar D$ by the rules $\bar xy=x{\mathbin\vdash}y$, $y\bar x=y{\mathbin\dashv}x$, where
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ion \[section::improving\].
[**Remark.**]{} Our results concern the bootstrap distribution and assume the ability to determine the quantities $\hat{t}^*_\alpha$ and $(\tilde{t}^*_j, j \in {\widehat{S}})$ in Equation . Of course, they can be approximated to an arbitrary level of precision by drawing a large enough number $B$ of bootstrap samples and then by computing the appropriate empirical quantiles from those samples. This will result in an additional approximation error, which can be easily quantified using the DKW inequality (and, for the set $\tilde{C}^*_{{\widehat{S}}}$, also the union bound) and which is, for large $B$, negligible compared to the size of the error bounds obtained above. For simplicity, we do not provide these details. Similar considerations apply to all subsequent bootstrap results.
### The Sparse Case {#the-sparse-case .unnumbered}
Now we briefly discuss the case of sparse fitting where $k = O(1)$ so that the size of the selected model is not allowed to increase with $n$. In this case, things simplify considerably. The standard central limit theorem shows that $$\sqrt{n}(\hat\beta - \beta)\rightsquigarrow N(0,\Gamma)$$ where $\Gamma = \Sigma^{-1} \mathbb{E}[(Y-\beta^\top X)^2] \Sigma^{-1}$. Furthermore, $\Gamma$ can be consistently estimated by the sandwich estimator $\hat\Gamma = \hat\Sigma^{-1} A \hat\Sigma^{-1}$ where $A = n^{-1}\mathbb{X}^\top R \mathbb{X}$, $\mathbb{X}_{ij} = X_i(j)$, $R$ is the $k\times k$ diagonal matrix with $R_{ii} = (Y_i - X_i^\top \hat\beta)^2$. By Slutsky’s theorem, valid asymptotic confidence sets can be based on the Normal distribution with $\hat\Gamma$ in place of $\Gamma$ ([@buja2015models]).
However, if $k$ is non-trivial relative to $n$, then fixed $k$ asymptotics may be misleading. In this case, the results of the previous section may be more appropriate. In particular, replacing $\Gamma$ with an estimate then has a non-trivial effect on the coverage accuracy. Furthermore, the accuracy depends on $1/u$ where $u = \lambda_{\rm min}(\Sigma)$. But
| 1,889
| 1,745
| 2,901
| 1,905
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| 0.772357
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\_x,\_x v\_[L\^2(GSI)]{} and \[fs5\] ,v\_[W\^2\_1(GSI)]{}=,v\_[L\^2(GSI)]{}+ \_x,\_x v\_[L\^2(GSI)]{} +,\_[L\^2(GSI)]{}.
These spaces are Hilbert spaces. Let C\^1(GSI):={\_[| GSI]{} | C\_0\^1(\^3S)}. We have (cf. [@friedrichs], [@bardos]. The proof can also be shown by the similar considerations as in [@friedman pp. 11-19])
\[denseth\] The space $C^1(\ol G\times S\times I)$ is a dense subspace of $W^2(G\times S\times I)$ and of $W^2_1(G\times S\times I)$.
For $\Gamma_-=\{(y,\omega,E)\in (\partial G)\times S\times I)\|\ \omega\cdot\nu(y)<0\}$ we define the space of $L^2$-functions with respect to the measure $|\omega\cdot\nu|\ d\sigma d\omega dE$ which is denoted by $T^2(\Gamma_-)$ that is, $T^2(\Gamma_-)=L^2(\Gamma_-,|\omega\cdot\nu|\ d\sigma d\omega dE)$. $T^2(\Gamma_-)$ is a Hilbert space and its inner product is (in this paper all functions are real-valued) \[fs8\] [h\_1]{},h\_2\_[T\^2(\_-)]{}=\_[\_-]{}h\_1(y,,E)h\_2(y,,E) || dddE. The space $T^2(\Gamma_+)$ (and its inner product) of $L^2$-functions on $\Gamma_+=\{(y,\omega,E)\in (\partial G)\times S\times I)|\ \omega\cdot\nu(y)>0\}$ with respect to the measure $|\omega\cdot\nu|\ d\sigma d\omega dE$ is similarly defined. We denote by $T^2(\Gamma)$ the space of $L^2$-functions with respect to the measure $|\omega\cdot\nu|\ d\sigma d\omega dE $ that is, $
T^2(\Gamma)=L^2(\Gamma,|\omega\cdot\nu|\ d\sigma d\omega dE).
$ The inner product in $T^2(\Gamma)$ is \[fs10\] [h\_1]{},h\_2\_[T\^2()]{}= \_h\_1(y,,E)h\_2(y,,E)|| dddE.
One can show ([@dautraylionsv6 pp. 230-231], [@tervo14]) that for any compact set $K\subset \Gamma_-$ \[ttha\] \_K|(y,,E)|\^2|| dddEC\_K\_[W\^2(GSI)]{}\^2 C\^1(GSI). Hence any element $\psi\in W^2(G\times S\times I)$ has well defined trace $\psi_{|\Gamma_-}$ in $L^2_{\rm loc}(\Gamma_-,|\omega\cdot\nu|\ d\sigma d\omega dE)$ defined by \[tthb\] \_[|K]{}:=\_[j]{} [\_j]{}\_[|K]{} [for any compact subset ]{} K\_-, where $\{\psi_j\}\subset C^1(\ol G\times S\times I)$ is a sequence such that $\lim_{j\to\infty}{\left\Vert \psi_j-\psi\rig
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n ${{\operatorname{edge}{{\mathcal{C}}}}}$, whereas software correctness examines only the structure within cone ${\mathcal{C}}$. The operational profile asserts the importance of relative excitational intensity to safety analysis. An accident that occurs more frequently is worse than an accident that happens less frequently, given that they are of comparable severity. This safety factor is ignored under software correctness alone.
Modeling accidents
------------------
Accidents are diverse in effect and mechanism, including injury, death, or damage either to equipment or environment. Since the causality of accidents is temporarily unknown, they manifest an apparent nature of unpredictability or randomness. However, under emulation as a stochastic process, the exact timing of accidents *is* truly a random phenomenon rather than causal. Nevertheless, it has proven useful to compare well-understood summary statistics of stochastic processes with those of deterministic but unknown physical processes.
### Compound Poisson process {#S:CPP}
Today’s prevalent safety model for the occurrence of accidents is the compound Poisson[^8] process. This model captures accidents’ two dominant attributes: rate of occurrence (intensity) and scalar measure of loss (severity). With some exceptions, neither the timing nor severity of one software accident affects another. The compound Poisson process (CPP) is appropriate to model accidents of this nature.
As stochastic processes are models rather than mechanisms, deriving their properties involves somewhat out-of-scope mathematics. The interested reader can immediately find greater detail in Wikipedia online articles: [@wW_Poisson_distribution], [@wW_Poisson_process], [@wW_total_expectation], [@wW_Compound_Poisson_process], and [@wW_Cumulant]. Relevant theorems will be documented here simply as facts.
### Poisson processes {#S:POISSON_PROCESSES}
We will consider three variants of basic stochastic process: the ordinary Poisson process, the compound Poisson process, and the
| 1,891
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| 1,081
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| 0.775834
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ATGACGAAGAGGATTAAGTATCTCGTGTAGGCTGGAGCTGCTTC
Rev cassette generation GAATCGTTAAAAAAGCGCGGCCAGAGGCGTTCTGACCGCATGCTTTGCTACATATGAATATCCTCCTTAG
Upstream CATCTGCGACCGCTAACTT
Downstream TTTATCCACCGAGGGTTATTCG
*fliZ* Fwd cassette generation CGAAAAGTGCCGCACAACGTATAGACTACCAGGAGTTCTCGTGTAGGCTGGAGCTGCTTC
Rev cassette generation CACGTTTCACCAACACGACTCTGCTACATCTTATGCTTTTCATATGAATATCCTCCTTAG
Upstream CATCGAACTGGTGACTGAAGA
Downstream CTACAGCCATTACTCCCATCAG
*flhDC* Fwd cassette generation GTGCGGCTACGTCGCACAAAAATAAAGTTGGTTATTCTGGGTGTAGGCTGGAGCTGCTTC
Rev cassette generation ATGACTTACCGCTGCTGGAGTGTTTGTCCACACCGTTTCGCATATGAATATCCTCCTTAG
Upstream CGAGTAGAGTTGCGTCGAATTA
Downstream ATCCTTCCGCTGTTGACTATG
*hilD* Fwd cassette generation CCAGTAAGGAACATTAAAATAACATCAACAAAGG
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|
to be normalized. We employed the ‘range’ method where each component of the data vector is normalized to lie in the intravel \[0,1\].
3. [**SOM Training:**]{} Select an input vector $x$ from the data set randomly. A best matching unit (BMU) for this input vector, is found in the map by the following metric $$\left\| x- m_c \right\| =
\min_i \left\{ \left\| x- m_i \right\| \right\}$$ where $m_i$ is the reference vector associated with the unit $i$.
4. [**Updating Step:**]{} The reference vectors of BMU and its neighbourhood are updated according to the following rule $$m_i(t+1)=\left\{
\begin{alignedat}{2}
&m_i(t) + \alpha(t)\cdot h_{ci}(t)\cdot[x(t)-m_i(t)], & & \qquad i \in N_c(t)
\\
&m_i(t), & & \qquad i \notin N_c(t)
\end{alignedat}
\right.$$ where\
$h_{ci}(t) $ is the kernel neighbourhood around the winner unit $c$.\
\
$t$ is the time constant.\
\
$x(t)$ is an input vector randomly drawn from the input data set at time $t$.\
\
$\alpha(t)$ is the learning rate at time $t$.\
\
$N_c(t)$ is the neighbourhood set for the winner unit $c$.\
The above equation make BMU and its neighbourhood move closer to the input vector. This adaptation to input vector forms the basis for the group formation in the map.
5. [**Data groups visualisation:**]{} steps 3 and 4 are repeated for selected number of trials or epochs. After the trails are completed the map unfolds itself to the distribution of the data set finding the number of natural groups exist in the data set. The output of the SOM is the set of reference vectors associated with the map units. This set is termed as a codebook. To view the groups and the outliers discovered by the SOM we have to visualize the codebook. U-Matrix is the technique typically used for this purpose.
The ON events data set has directly used with the SOM. No prior training is required. The unsupervised behavior of SOM had discovered the groups in the data set in an automatic way. We worked with two kernel
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finterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. This means that $hsea=htea$. Then $hse{\mathbf r}(a)=hte{\mathbf r}(a)$ and also $$hse{\mathbf r}(a)\cdot y =hte{\mathbf r}(a)\cdot y = z,$$ which proves that $(X,\mu)$ is free. Applying Proposition \[prop:trans\] (noting that filtered functors preserve pullbacks) and Proposition \[prop:torsors\], we conclude that $(X,\mu)$ is an $S$-torsor.
Principal bundles over inverse semigroups {#sec:bundles}
=========================================
In this section, we obtain an equivalence between the category of universal representations of an inverse semigroup on étale spaces over a topological space $X$ and the category of principal $L(S)$-bundles over $X$. This extends the well known result for groups [@MM VIII.1, VIII.2], and also is an analogue of Proposition \[prop:ff\], if in the latter one replaces the topos of sets by the topos ${\mathsf{Sh}}(X)$.
The following definition is taken from [@M]. Let $X$ be a topological space and ${\mathcal C}$ a small category. A functor $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is called a ${\mathcal C}$-[*bundle*]{}. If $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is a ${\mathcal C}$-[*bundle*]{}, $\alpha\colon c\to d$ is an arrow in ${\mathcal C}$ and $y\in E(c)$, we put $$\alpha\cdot y=E(\alpha)(y)\in E(d).$$
A ${\mathcal C}$-bundle $E$ is called [*principal*]{}, if for each point $x\in X$ the following axioms are satisfied by the stalks $E(C)_x$:
1. (non-empty) There is an object $c$ of $C$ such that $E(c)_x\neq\varnothing$;
2. (transitive) For any $y\in E(c)_x$ and $z\in E(d)_x$, there are arrows $\alpha\colon b\to c$ and $\beta\colon b\to d$ for some object $b$ of $C$, and a point $w\in E(b)_x$, so that $\alpha\cdot w=y$ and $\beta\cdot w=z$.
3. (free) For any two parallel arrows $\alpha,\beta\colon c\rightrightarrows d$ and any $y\in E(c)_x$, for which $\alp
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ge symmetry of general coordinate transformations.
The teleparallel gravity action (\[action-TP\]) is equivalent to the action (\[general-action-0\]) for the choice of parameters $\lam = 1/2, \alpha = 1, \beta = 1$. It is S\_[TP]{} = d\^D x . \[action-TP-2\] The first term is the YM action (\[YM-action\]). The rest of the terms provide the unique combination so that the action is invariant under rotations of local Lorentz frames (\[rotation\]). The field $\hat{e}_{\m}{}^{a}$ can now be identified with the vielbein $e_{\m}{}^a$ in gravity, and the modified YM theory is equivalent to GR.
Metric, $B$-field and Dilaton
-----------------------------
While the action (\[action-TP-2\]) is equivalent to pure GR, we investigate the most general quadratic action (\[general-action-0\]), which can be equivalently put in the form S = d\^D x , \[general-action\] assuming that the coefficient of the YM term is non-zero. It is invariant under general coordinate transformations for arbitrary constants $\a, \b$. (Compared with (\[general-action-0\]), $\lam = 1 - \a/2$.) The case of teleparallel gravity (\[action-TP\]) corresponds to the choice $\a = \b = 1$.
For generic values of $\a, \b$, local rotations of Lorentz frames are no longer gauge symmetries. With fewer gauge symmetries, there are more physical degrees of freedom in the theory. In $D$-dimensional space-time, the fundamental field $\hat{e}_a{}^{\m}$ has $D^2$ components. When the rotation of local Lorentz frames is a gauge symmetry, local Lorentz transformations identify $D(D-1)/2$ components of $\hat{e}_a{}^{\m}$ as gauge artifacts, with the remaining $D(D+1)/2$ components of $\hat{e}_a{}^{\m}$ to be matched with the $D(D+1)/2$ independent components of the metric.
Tuning the values of $\a, \b$ slightly away from $1$, we have $D(D-1)/2$ of the components that can no longer be gauged away. The theory with generic values of $\a, \b$ is expected to contain more physical fields in addition to the metric. For coefficients $\a, \b$ with values not too different from $1$
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Let us describe the transformations of $\R^4$ given by each generator. Consider a new base $(e_1, e_2,
e_3, e_4)$, given by $e_1=f_1+f_2$, $e_2=f_1-f_2$, $e_3=f_3+f_4$, $e_4=f_3-f_4$. The generator $r$ of order 4 is represented by the rotation in the plane $(e_2,e_4)$ through the angle $\frac{\pi}{2}$ and the reflection in the plane $(e_1,e_3)$ with respect to the line $e_1+e_3$. The generator $l$ of order 2 is represented by the central symmetry in the plane $(e_1, e_3)$.
Obviously, the described representation of $\I_4$ admits invariant (1,1,2)-dimensional subspaces. We will denote subspaces by $\lambda_1, \lambda_2, \tau$.
The lines $\lambda_1, \lambda_2$ are generated by the vectors $e_1+e_3$, $e_1-e_3$ correspondingly. The subspace $\tau$ is generated by the vectors $e_2, e_4$. The generator $r$ acts by the reflection in $\lambda_2$ and by the rotation in $\tau$ throught the angle $\frac{\pi}{2}$. The generator $l$ acts by reflections in the subspaces $\lambda_1$, $\lambda_2$.
In particular, if the structure group $\Z/2 \int \D_4$ of a 4-dimensional bundle $\zeta: E(\zeta) \to L$ admits a reduction to the subgroup $\I_4$, then the bundle is decomposed into the direct sum $\zeta = \lambda_1 \oplus \lambda_2 \oplus \tau$ of $1,1,2$–dimensional subbundles.
### Definition 6 {#definition-6 .unnumbered}
Let $(g: N^{n-2k} \looparrowright \R^n, \Xi_N, \eta)$ be an arbitrary $\D_4$-framed immersion. We shall say that this immersion is an $\I_b$–immersion (or a cyclic immersion), if the structure group $\Z/2 \int \D_4$ of the normal bundle over the double points manifold $L^{n-4k}$ of this immersion admits a reduction to the subgroup $\I_4 \subset \Z/2 \int \D_4$. In this definition we assume that the pairs $(f_1, f_2)$, $(f_3,f_4)$ are the vectors of the framing for the two sheets of the self-intersection manifold at a point in the double point manifold $L^{n-4k}$. $$$$
In particular, for a cyclic $\Z/2 \int \D_4$-framed immersion there exists the mappings $\kappa_a: L^{n-4k} \to K(\Z/2,1)$, $\mu_a: L^{n-4
| 1,896
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, as a direct application of the one-dimensional Berry-Esseen theorem, that:
Let $\hat{C}_{{\widehat{S}}}$ be the splitting-based confidence set. Then, $$\label{eq:lem12a}
\inf_{P\in {\cal P}_{n}}\mathbb{P}(\beta_{\hat{S}}\in \hat{C}_{{\widehat{S}}}) = 1-\alpha - \frac{c}{\sqrt{n}}$$ for some $c$. Also, $$\label{eq:lem12b}
\sup_{P\in {\cal P}_{n}}\mathbb{E}[\nu(\hat{C}_{{\widehat{S}}})] \preceq n^{-1/2}$$ where $\nu$ is Lebesgue measure. More generally, $$\inf_{w\in\mathcal{W}_n}\inf_{P\in {\cal
P}_{n}}\mathbb{P}(\beta_{\hat{S}}\in \hat{C}_{{\widehat{S}}}) = 1-\alpha - \frac{c}{\sqrt{n}}$$ for some $c$, and $$\sup_{w\in\mathcal{W}_n}\sup_{P\in {\cal P}_{n}}\mathbb{E}[\nu(\hat{C}_{{\widehat{S}}})] \preceq n^{-1/2}$$
Let $\hat{C}_{{\widehat{S}}}$ be the uniform confidence set. Then, $$\inf_{P\in {\cal P}_{n}} \mathbb{P}(\beta_{\hat{S}}\in \hat{C}_{{\widehat{S}}}) = 1-\alpha - \left(\frac{ c (\log D)^7 }{n}\right)^{1/6}$$ for some $c$. Also, $$\sup_{P\in {\cal P}_{2n}}\mathbb{E}[\nu(\hat{C}_{{\widehat{S}}})] \succeq \sqrt{\frac{\log D}{n}}.$$
The proof is a straightforward application of results in [@cherno1; @cherno2]. We thus see that the splitting method has better coverage and narrower intervals, although we remind the reader that the two methods may be estimating different parameters.
[**Can We Estimate the Law of $\hat\beta(\hat{S})$?**]{} An alternative non-splitting method to uniform inference is to estimate the law $F_{2n}$ of $\sqrt{2n}(\hat\beta_{{\widehat{S}}} - \beta_{{\widehat{S}}})$. But we show that the law of $\sqrt{2n}(\hat\beta_{{\widehat{S}}}-\beta_{{\widehat{S}}})$ cannot be consistently estimated even if we assume that the data are Normally distributed and even if $D$ is fixed (not growing with $n$). This was shown for fixed population parameters in [@leeb2008can]. We adapt their proof to the random parameter case in the following lemma.
\[lemma::contiguity\] Suppose that $Y_1,\ldots,Y_{2n} \sim N(\beta,I)$. Let $\psi_n(\beta) = \mathbb{P}(\sqrt{2n}(\hat\beta_{{\widehat{S}}} -
\b
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(X,\mathcal{U})\rightarrow(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))$ for a subspace $h(X)$ of $Y_{1}$, the converse need not be true unless — entirely like open functions again — either $h(X)$ is an open set of $Y_{1}$ or $i\!:(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))\rightarrow(X,\textrm{FT}\{\mathcal{U};h\})$ is an open map. Since an open preimage continuous map is image continuous, this makes $i\!:h(X)\rightarrow Y_{1}$ an ininal function and hence all the three legs of the commutative diagram image continuous.
Like preimage continuity, *an image continuous function $q\!:(X,\mathcal{U})\rightarrow Y$ need not be open.* However, although *the restriction of an image continuous function to the saturated open sets of its domain is an open function*, $q$ is unrestrictedly open iff the saturation of every open set of $X$ is also open in $X$. Infact it can be verified without much effort that a continuous, open surjection is image continuous.
Combining Eqs. (\[Eqn: IT’\]) and (\[Eqn: FT\]) gives the following criterion for ininality $$U\textrm{ and }V\in\textrm{IFT}\{\mathcal{U}_{\textrm{sat}};f;\mathcal{V}\}\Longleftrightarrow(\{ f(U)\}_{U\in\mathcal{U}_{\textrm{sat}}}=\mathcal{V})(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}}),\label{Eqn: INI}$$
which reduces to the following for a homeomorphism $f$ that satisfies both $\textrm{sat}(A)=A$ for $A\subseteq X$ and $_{f}B=B$ for $B\subseteq Y$ $$U\textrm{ and }V\in\textrm{HOM}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\mathcal{U}=\{ f^{-1}(V)\}_{V\in\mathcal{V}})(\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V})\label{Eqn: HOM}$$
and compares with $$\begin{gathered}
U\textrm{ and }V\in\textrm{OC}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\textrm{sat}(U)\in\mathcal{U}\!:\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge\\
\wedge(\textrm{comp}(V)\in\mathcal{V}\!:\{ f^{-}(V)\}_{V\in\mathcal{V}}=\mathcal{U}_{\textrm{sat}})\label{Eqn: OC}\end{gathered}$$
for an open-continuous $f$.
The following is a slightly mo
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|
(UX) \right\}_{K k}
\nonumber \\
&+&
2 \mbox{Re}
\biggl\{
\sum_{k, K} \sum_{l, L}
\left[
- (ix) e^{+ i h_{k} x} + \frac{e^{+ i \Delta_{K} x} - e^{+ i h_{k} x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\frac{e^{- i \Delta_{L} x} - e^{- i h_{l} x} }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times&
(UX)_{\alpha k}^* (UX)_{\beta k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&\times&
\biggl[
(UX)_{\alpha l} W^*_{\beta L}
\left\{ (UX)^{\dagger} A W \right\}_{l L}
+
W_{\alpha L} (UX)^*_{\beta l}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\biggr]
\biggr\}
\nonumber \\
&+&
2 \mbox{Re}
\biggl\{
\sum_{k, K} \sum_{L}
\left[
- (ix) e^{- i ( \Delta_{L} - h_{k} ) x} + \frac{ e^{- i ( \Delta_{L} - \Delta_{K} ) x} - e^{- i ( \Delta_{L} - h_{k} ) x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\frac{ 1 }{ \Delta_{K} - h_{k} }
\nonumber \\
&\times&
(UX)_{\alpha k}^* (UX)_{\beta k}
W_{\alpha L} W^*_{\beta L}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\biggr\}
\nonumber \\
&-&
2 \mbox{Re}
\biggl\{
\sum_{k \neq m} \sum_{K} \sum_{l, L}
\frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{+ i h_{m} x}
- \left( \Delta_{K} - h_{m} \right) e^{+ i h_{k} x}
- ( h_{m} - h_{k} ) e^{+ i \Delta_{K} x}
\biggr]
\frac{e^{- i \Delta_{L} x} - e^{- i h_{l} x} }{ ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times&
(UX)^*_{\alpha k} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&\times&
\biggl[
(UX)_{\alpha l} W^*_{\beta L}
\left\{ (UX)^{\dagger} A W \right\}_{l L}
+
W_{\alpha L} (UX)^*_{\beta l}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\biggr]
\nonumber \\
&-&
2 \mbox{Re}
\biggl\{
\sum_{k \neq m} \sum_{K} \sum_{L}
\frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right)
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| 1,908
| null | null |
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|
^{\prime\prime}$: $$\bar{N} \delta t = (\bar{g}^{1/2} \alpha) \delta t
= (\psi^6 g^{1/2}) \alpha \delta t = \psi^6 (N \delta t) \; .$$
The final relationships between the two physical Riemannian metrics $\bar{g}_{i j}$ and $\bar{g}^\prime_{i j} = \bar{g}_{i j} +
\dot{\bar{g}}_{i j} \delta t$ and the given data $g_{i j}$ and $g^\prime_{i j} = g_{i j} + u_{i j} \delta t$ are quite interesting. Of course, $\bar{g}_{i j} = \psi^4 g_{i j}$ is clear. But we have to calculate the relationship between $\bar{g}_{i j}$ and $\bar{g}^\prime_{i j}$ as $\bar{g}^\prime_{i j} = \bar{g}_{i j} + \dot{\bar{g}}_{i j} \delta t$, where, as in (\[Eq:gdot\]), $$\begin{aligned}
\dot{\bar{g}}_{i j} &=& \partial_t \left( \psi^4 g_{i j} \right) \nonumber\\
&=& -2 \bar{N} \left( \bar{A}_{i j} + \frac{1}{3} \bar{g}_{i j} K \right)
+ \left( \bar{\nabla}_i \bar{\beta}_j + \bar{\nabla}_j
\bar{\beta}_i \right) \; .
\label{Eq:gbardot}\end{aligned}$$ Working out (\[Eq:gbardot\]) gives a key result, namely, $$\begin{aligned}
\dot{\bar{g}}_{i j} &=& \psi^4 \left[ u_{i j} +
g_{i j} \partial_t \left(4 \log \psi\right) \right] \nonumber\\
&=& \bar{u}_{i j} + \bar{g}_{i j} \partial_t \left(4 \log \psi\right) \; ,\end{aligned}$$ where $$\begin{aligned}
\partial_t \left( 4 \log \psi \right) &=& \frac{2}{3} \left( \nabla_k \beta^k +
6 \beta^k \partial_k \log \psi - N K \psi^6 \right) \nonumber\\
&=& \partial_t \left( \bar{g}/g \right)^{1/3} = \frac{2}{3}
\left( \bar{\nabla}_k \bar{\beta}^k - \bar{N} \bar{K} \right) \; .
\label{Eq:DotLogPsi}\end{aligned}$$ Therefore, $$\begin{aligned}
\dot{\bar{g}}_{i j} &=& \psi^4 \left[ u_{i j} + \frac{2}{3} g_{i j}
\left( \nabla_k \beta^k + 6 \beta^k \partial_k \log \psi
- N K \psi^6 \right) \right] \nonumber\\
&=& \bar{u}_{i j} + \frac{1}{3} \bar{g}_{i j}
\left( 2 \bar{\nabla}_k \bar{\beta}^k - 2 \bar{N} K \right) \; .
\label{Eq:Bargdot}\end{aligned}$$ We see that $\dot{\psi}$ and $\dot{\bar{g}}_{i j}$ are fully determined by the constraints and, in the last e
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