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N}}$. The following construction generalizes the Poincaré-Birkhoff-Witt basis of quantized enveloping algebras given by Lusztig.
Let $i_1,i_2,\dots ,i_n\in I$ such that $\ell (1_{\chi }{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_n})=n$. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered}
\label{eq:betak}
\beta _\nu ^\chi
=1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_{\mu -1}}({\alpha }_{i_\nu }).\end{gathered}$$ Then the elements $\beta _\nu ^\chi $, $1\le \nu \le n$, are pairwise different and $$\begin{aligned}
R^\chi _+=\{\beta _\nu ^\chi \,|\,1\le \nu \le n\}
\label{eq:proots}\end{aligned}$$ by [@p-CH08 Prop.2.12]. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered}
\label{eq:Ebetak}
E_{\beta _\nu }= E_{\beta _\nu }^\chi
={T}_{i_1}\dots {T}_{i_{\nu -1}}(E_{i_\nu }),\quad
F_{\beta _\nu }= F_{\beta _\nu }^\chi
={T}_{i_1}\dots {T}_{i_{\nu -1}}(F_{i_\nu }),\\
\label{eq:Ebarbetak}
{\bar{E}}_{\beta _\nu }= {\bar{E}}_{\beta _\nu }^\chi
={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(E_{i_\nu }),\quad
{\bar{F}}_{\beta _\nu }= {\bar{F}}_{\beta _\nu }^\chi
={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(F_{i_\nu }),\end{gathered}$$ where $E_{i_\nu },F_{i_\nu }\in U(r_{i_{\nu -1}}\dots r_{i_2}r_{i_1}(\chi ))$. Then $$\begin{aligned}
E_{\beta _\nu },{\bar{E}}_{\beta _\nu }\in U^+(\chi )_{\beta _\nu },\qquad
F_{\beta _\nu },{\bar{F}}_{\beta _\nu }\in U^-(\chi )_{-\beta _\nu }
\label{eq:EbetainU+}\end{aligned}$$ for all $\nu \in \{1,\dots ,n\}$ by [@p-Heck07b Thm.6.19], Thm. \[th:Liso\](iii) and Prop. \[pr:LTdeg\].
\[le:rvrel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $\nu \in \{1,2,\dots ,n\}$, and assume that ${b^{\chi}} (\beta _\nu )<\infty $. Then $E_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}=
F_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}=0$ in $U(\chi )$.
By Eq. and since $T_i$ is an isomorphism for each $i\in I$, it suffices to prove that $E_i^{\bfun{\chi '}({\alpha }_i)}=F_i^{\bfun{\chi '}({\alpha }_i)}=0$ in $U(\chi ')$ for all $\chi '\in {\mathcal{X}}$ and $i\in I$
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x{\boldmath$\psi$}$ and $\mbox{\boldmath$\phi$}$ can be used as the weight for sampling the coordinates $X$ entering the classical integral in Eq. (\[eq:sigma-sb\]). Then, for each initial value $X$, Eqs. (\[eq:matrixSigma\]) must be integrated in time so that averages can be calculated. It is worth to note that in such a wave scheme the Eulerian point of view of quantum-classical dynamics [@kapral; @qc-sb] is preserved. This is different from what happens in the original operator approach [@kapral; @qc-sb], where in order to devise an effective time integration scheme by means of the Dyson expansion, one is forced to change from the Eulerian point of view (according to which the phase space point is fixed and the quantum degrees of freedom evolve in time *at* this fixed phase space point) to the Lagrangian point of view, where phase space trajectories are generated. Moreover, it must be noted that the numerical integration of Eqs. (\[eq:matrixSigma\]) provides directly the nonadiabatic dynamics without the need to introduce surface-hopping approximations.
In order to be able of comparing the results with those presented in Ref. [@qc-sb], the numerical values of the parameters specifying the spin-boson system have been chosen to be $\beta=0.3$, $\Omega=1/3$, $\omega_{\rm max}=3$, $\xi_K=0.007$, and $N=200$. Figure \[fig:fig1\] shows the results in the adiabatic case, obtained by setting $d_{12}=0$ in Eqs. (\[eq:matrixSigma\]). One can see that, in spite of the simple approximation of the form of the density matrix made in the equations of motion, the wave theory provides results which are in good agreement with those obtained with the operator approach of Ref. [@qc-sb]. Instead, Fig. \[fig:fig2\] shows the results of the nonadiabatic calculation. This is to be compared with the results of the operator theory [@qc-sb] (which are identical with the exact” ones of Ref. [@makri]). Of course, since different ways of dealing with the nonadiabatic effects are used in the two approaches the results do not
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sidering the effect of the angular momentum.
If the accretion disc is spatially resolved, we expect further mass loss happens due to, e.g., the disc winds [e.g., @Blandford:1999aa; @Zahra-Zeraatgari:2016aa; @Begelman:2016aa] and/or jets from a close vicinity of the BH [e.g., @Ohsuga:2005aa; @Jiang:2014aa; @Yuan:2015aa; @Sc-adowski:2016aa]. Outflows from the sink caused by such phenomena may change the outer gas dynamics on the scale of the Bondi radius. This should be studied in future work.
Parameter dependence {#sec:dependence}
--------------------
Here, we study how the flow structure changes with variation of the simulation parameters: the shadow opening angle $\theta_{\mathrm{shadow}}$ (in Sec. \[sec:sdep\]), BH mass $M_{\mathrm{BH}}$ (in Sec. \[sec:Mdep\]), and ambient density $n_\infty$ (in Sec. \[sec:ndep\]). In Sec. \[sec:comp\_prev\], we compare our results with previous 1D calculations.
### Dependence on shadow size {#sec:sdep}
run $\theta_{\mathrm{shadow}}$$^{a}$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$$^{b}$ $\dot{M}/\dot{M}_{\mathrm{B}}$$^{c}$
------------ ---------------------------------- -------------------------------------------------- --------------------------------------
s100 (Dds) $45^\circ$ $40^\circ$ $59\%$
s075 $37.75^\circ$ $29^\circ$ $42\%$
s050 $25^\circ$ $19^\circ$ $25\%$
s025 $11.25^\circ$ $9^\circ$ $6.5\%$
: Summary of the $\theta_{\mathrm{shadow}}$ dependence.[]{data-label="tab:s-model"}
\
NOTES. $^{a}$shadow opening angle of our subgrid model (equation \[eq:10\]); $^{b}$opening angle of equatorial neutral inflow region at $r_{\mathrm{B}}$; $^{c}$accretion rate normalized by Bondi one.
Considering uncert
| 1,203
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beta} \left\{ \sum_{i = 1}^{n} L(r_{i}(\beta)) \right\}. \end{aligned}$$ The case of $L(r_{i}(\beta)) = r_{i}(\beta)^{2}$ is well-known (see, e.g., Refs. [@doi:10.1111/j.1751-5823.1998.tb00406.x], [@legendre1805nouvelles], and [@stigler1981]). In the case of an asymmetric loss function, we refer the reader to, e.g., Refs. [@10.2307/2336317], [@10.2307/24303995], [@10.2307/1913643], and [@10.2307/2289234]. These studies estimate the parameter $\hat{\beta}$. In this paper, however, we do not make such the estimation, but instead give a solution to the minimization problems by correcting any predictions so that the prediction error follows a general normal distribution. In our method, we can not only minimize the expected value of the asymmetric loss, but also lower the variance of the loss.
Let $y$ be an observation value, and let $\hat{y}$ be a predicted value of $y$. We derive the optimized predicted value $y^{*} = \hat{y} + C$ minimizing the expected value of the loss under the assumption:
1. The prediction error $z := \hat{y} - y$ is the realized value of a random variable $Z$, whose density function is a generalized Gaussian distribution function (see, e.g., Refs. [@Dytso2018], [@doi:10.1080/02664760500079464], and [@Sub23]) with mean zero $$\begin{aligned}
f_{Z}(z) := \frac{1}{2 a b \G(a)} \exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)}, \end{aligned}$$ where $\G(a)$ is the gamma function and $a$, $b \in \mathbb{R}_{> 0}$.
2. Let $k_{1}$, $k_{2} \in \mathbb{R}_{> 0}$. If there is a mismatch between $y$ and $\hat{y}$, then we suffer a loss, $$\begin{aligned}
\Pe(z) :=
\begin{cases}
k_{1} z, & z \geq 0, \\
- k_{2} z, & z < 0.
\end{cases}\end{aligned}$$
That is, the solution to the minimization problem is $$\begin{aligned}
C = \arg\min_{c} \left\{ \operatorname{{E}}\left[ \Pe(Z + c) \right] \right\}. \end{aligned}$$
The motivation of our research is as follows: (1) Predictions usually cause prediction errors. Therefore, it is necessary to use predic
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t,0){\right\rangle}_{L^2(G\times S)}
+
{\left\langle}\psi(\cdot,\cdot,E_m),v(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}.\end{aligned}$$ The elements of $H_1$ are of the form $\tilde\psi=(\psi,q,p_0,p_m)\in
L^2(G\times S\times I)\times T^2(\Gamma)\times L^2(G\times S)^2$. More precisely, the elements of $H_1$ are the elements of the closure (in $L^2(G\times S\times I)\times T^2(\Gamma)\times L^2(G\times S)^2$) of the graph of the (trace) operator $C^1(\ol G\times S\times I)\to C^1(\partial G\times S\times I)\times C^1(\ol G\times S)^2$ defined by $\psi\mapsto (\gamma(\psi),\psi(\cdot,\cdot,0),\psi(\cdot,\cdot,E_m))$. The inner product in $H_1$ is $${\left\langle}\tilde\psi,\tilde \psi'{\right\rangle}_{H_1}={}&{\left\langle}\psi, \psi'{\right\rangle}_{L^2(G\times S\times I)}
+{\left\langle}q,q'{\right\rangle}_{T^2(\Gamma)}\nonumber\\
&
+
{\left\langle}p_0,p_0'{\right\rangle}_{L^2(G\times S)}+{\left\langle}p_{\rm m},p_{\rm m}'{\right\rangle}_{L^2(G\times S)},$$ for $\tilde\psi=(\psi,q,p_0,p_{\rm m})$, $\tilde\psi=(\psi',q',p_0',p_{\rm m}')\in H_1$.
Finally, let $H_2$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product \[inph2\] ,v\_[H\_2]{} :=&,v\_[W\^2(GSI)]{}+,[E]{} \_[L\^2(GSI)]{}\
=&,v\_[L\^2(GSI)]{}+ \_x ,\_x v\_[L\^2(GSI)]{}\
&+ (),(v)\_[T\^2()]{} + ,[E]{} \_[L\^2(GSI)]{}. Obviously $H_2\subset \tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I)$ and the inner product in $H_2$ is given by (\[inph2\]).
Some Details on (Inflow) Trace Theory {#trath}
-------------------------------------
We still bring up a refinement of the above explained trace theory. Let us define $$T^2_{\tau_{\pm}}(\Gamma_{\pm}):=L^2(\Gamma_{\pm},\tau_{\pm}(\cdot,\cdot)|\omega\cdot\nu| d\sigma d\omega dE),$$ equipped with the inner product \[fs8a\] [h\_1]{},h\_2\_[T\^2\_[\_]{}(\_)]{}=\_[\_]{}h\_1(y,,E)h\_2(y,,E)\_(y,) || dddE.
Suppose that $g\in C(\Gamma_-)$ such that ${{\frac{\partial g}{\partial \tilde y_i}}}\in C(\Gamma_-), i=1,2$, where ${\partial\over{\partial \tilde y_i}}$ denotes any
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s (g(N^{n-2k}) \cap (U_P))$ by $N^{n-2k}_{int}$, and the complement $N^{n-2k} \setminus
N^{n-2k}_{int}$ by $N^{n-2k}_{ext}$. The manifolds $N^{n-2k}_{ext}$, $N^{n-2k}_{int}$ are submanifolds in $N^{n-2k}$ of codimension 0 with the common boundary, this boundary is denoted by $N_Q^{n-2k-1}$. The self-intersection manifold of $g$ is denoted by $L^{n-4k}$. By the dimensional reason ($n-4k=q<<n$) $L^{n-4k}$ is a submanifold in $\R^n$, parameterized by an embedding $h$, equipped by the $\Z/2 \int \D_4$-framing of the normal bundle denoted by $(\Psi, \zeta)$. The triple $(h,\Psi,\zeta)$ determines an element in the cobordism group $Imm^{\Z/2 \int \D_4}(n-4k,4k)$.
### Definition 4 {#definition-4 .unnumbered}
We say that the $\D_4$–framed immersion $g$ is an $\I_b$–controlled immersion if the following conditions hold:
–1. The structure group of the $\D_4$–framing $\Xi_N$ restricted to the submanifold (with boundary) $g(N^{n-2k}_{ext})$ is reduced to the subgroup $\I_b \subset \D_4$ and the cohomology classes $\tau_{U_Q,1}, \tau_{U_Q,2} \in H^1(U_Q;\Z/2)$ are mapped to the generators $\tau_1, \tau_2 \in H^1(N_Q^{n-2k-1};\Z/2)$ of the cohomology of the structure group of this $\I_b$-framing by the immersion $g \vert_{N^{n-2k-1}_Q} : N^{n-2k-1}_Q
\looparrowright
\partial(U_Q) \subset U_Q$.
–2. The restriction of the immersion $g$ to the submanifold $N_Q^{n-2k-1} \subset N^{n-2k}$ is an embedding $g \vert_{N_Q^{n-2k-1}} : N_Q^{n-2k-1} \subset \partial U_Q$, and the decomposition $L^{n-4k} =L^{n-4k}_{int} \cup L^{n-4k}_{ext}
\subset (U_P \cup \R^n \setminus U_P)$ of the self-intersection manifold of $g$ into two (probably, non-connected) $\Z/2 \int
\D_4$-framed components is well-defined. The manifold $L^{n-4k}_{int}$ is a submanifold in $U_P$ and the triple $(L^{n-4k}_{int},\Psi_{int},\zeta_{int})$ represents an element in $Imm^{Ker \omega}(n-4k,4k)$ in the image of the homomorphism $\omega^!: Imm^{\Z/2 \int \D_4}(n-4k,4k) \to Imm^{Ker
\omega}(n-4k,4k)$. $$$$
### Definition 5 {#definition-5 .unnumbered}
Let
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|
akli], with the result [@li] that they are effectively proportional to the coupling $g_{37}^2$ in (\[coupl-N\]). The amplitudes depend on kinematical invariants expressible in terms of the Mandelstam variables: $s=-(k_1 + k_2)^2$, $t=-(k_2 + k_3)^2$ and $u=-(k_1 + k_3)^2$, which satisfy $s + t + u =0$ for massless particles. The ordered four-point amplitude $\mathcal{A}(1,2,3,4)$, describing *partly* the splitting and capture of a photon state by the D-particle, is given by $$\begin{aligned}
\mathcal{A} (1_{j_1 I_1},2_{j_2 I_2},3_{j_3 I_3},4_{j_4 I_4}) = &&
- { g_{37}^2} l_s^2 \int_0^1 dx \, \, x^{-1 -s\, l_s^2}\, \, \,
(1-x)^{-1 -t\, l_s^2} \, \, \, \frac {1}{ [F (x)]^2 } \, \times \nonumber \\
&& \left[ {\bar u}^{(1)} \gamma_{\mu} u^{(2)}
{\bar u}^{(4)} \gamma^{\mu} u^{(3)} (1-x) + {\bar u}^{(1)} \gamma_{\mu}
u^{(4)} {\bar u}^{(2)} \gamma^{\mu} u^{(3)} x \right ] \, \nonumber \\
&& \times \{ \eta
\delta_{I_1,{\bar I_2}} \delta_{I_3,{\bar I_4}}
\delta_{{\bar j_1}, j_4} \delta_{j_2,{\bar j_3}}
\sum_{m\in {\bf Z}} \, \,
e^{ - {\pi} {\tau}\,
m ^2 \, \ell_s^2 /R^{\prime 2} }
\nonumber \\
&& + \delta_{j_1,{\bar j_2}}
\delta_{j_3,{\bar j_4}}
\delta_{{\bar I_1}, I_4} \delta_{I_2,{\bar I_3}}
\sum_{n\in {\bf Z}} e^{- {\pi \tau} n^2
\, R^{\prime 2} / \ell_s^{2} } \}~,~\,
\label{4ampl}\end{aligned}$$ where $F(x)\equiv F(1/2; 1/2; 1; x)$ is the hypergeometric function, $\tau (x) = F(1-x)/F(x)$, $j_i$ and $I_i$ with $i=1, ~2, ~3, ~4$ are indices on the D7-branes and D3-branes, respectively, and $\eta=(1.55\ell_s)^4/(V_{A3} R')$ in the notation of [@benakli], $u$ is a fermion polarization spinor, and the dependence on the appropriate Chan-Paton factors has been made explicit [^6]. In the above we considered for concreteness the case where the photon state splits into two fermion excitations, represented by open strings. The results are qualitatively identical in the case of boson excitations. In fact, the spin of the incident open strings does not affect the delays or the relevant discussion on the qual
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a_\varphi$ is the largest relevant solution of the eigenvalue equation $$\label{csawlamdafi}
\mbox{det}\left| \left({\partial X^{(r+1)}_i\over \partial X^{(r)}_j}
\right)^{*}-
\lambda_\varphi\,\delta_{ij} \right|=0\>,$$ where the asterisk means that the derivatives should be taken at the tricritical fixed point. From here follows $\langle M^{(r)}\rangle\sim \lambda_\varphi^r$, which together with $\langle N_3^{(r)}\rangle\sim \lambda_{\nu_3}^r$ (where $\lambda_{\nu_3}$, as before, is the largest eigenvalue of the linearized RG equations for the bulk parameters $A$ and $B$), and (\[ficsaw\]), gives $$\label{fie2}
\varphi=\frac{\ln\lambda_{\varphi}}
{\ln\lambda_{\nu_3}}\>.$$
- For larger values of the interaction parameter $w>w_c(u,t)$, the RG parameters flow towards the fixed point $$\label{fp5}
(0,0,0,A_1^*=C^*,0,0,0,0,0)\>,$$ which describes the entangled phase of the two polymers, in which $P_3$ chain is completely attracted to $P_2$ chain.
[lllllllllll]{} $b$ & $A^*$&$B^*$&$C^*$ & $A_1^*$ & $A_2^*$ &$A_3^*$&$A_4^*$ &$B_1^*$ & $B_2^*$&$\varphi$\
\
2 & 0.4294&0.0499&0.6180&0.2654 &0.2654&0.2654&0.2654&0.0308 &0.0308&0.5428\
3 & 0.3420&0.0239&0.5511& 0.1884& 0.1884& 0.1884& 0.1884&0.0131&0.0131&0.4973\
\
2& 1/3 & 1/3&0.6180& 0.0510&0&0&0.0613 &0.2365&0.2362&0.6714\
3& 0.2071 & 0.4307 &0.5511&0.0810&0.0310&0.0250&0.0270 &0.3130&0.3150&0.6226\
\
2 & 0 & 0.3569 & 0.6180&0&0&0&0&0.2206& 0.2206&0.6261\
3 & 0 & $\infty$ &0.5511& 0&0&0&0&$\infty$&$\infty$&0.6073\
Critical self-attraction of the 3D chain
----------------------------------------
For $u=u_\theta$ the solitary 3D chain is in the state of the $\theta$-chain, for which $(A^*,B^*)=(A_\theta,B_\theta)$, whereas the two-polymer system can be in the following phases:
- For $w<w_c(u_\theta,t)$ the corresponding fixed point is of the form $$\label{fpt1}
(A_\theta,B_\theta,C^*,0,0,0,A_4^*,0,0)\>.$$ This is the case when the 3D $\theta$-chain is segregated from the 2D chain chain. Fixed point values of $A_4^*$ are: 0.061
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|
+2b} z_j^{\ast} \end{pmatrix}
&0 \\ 0& 0 &id \end{pmatrix}.$$ Here, the $(2,2)$-block of the above formal matrix corresponds to $B(-2be_1+e_2)\oplus$ $B(-ae_1+e_3)\oplus$ $B(e_2+e_3)$ with the Gram matrix $A(2b(2b-1), a(a+1), a(2b-1))\oplus (a+2b)$ and $id$ in the $(1,1)$-block corresponds to the direct summand $(\oplus_{\lambda}H_{\lambda})$ of $M_0$. The above formal matrix can be simplified by observing a formal matrix description of an element of $\underline{M}(R)$ for a $\kappa$-algebra $R$, explained in Section \[m\]. Since $A(2b(2b-1), a(a+1), a(2b-1))\oplus (a+2b)$ is *of type $I^o$*, the $(2,2)$-block of the above formal matrix turns to be $$\label{e4.4}
\begin{pmatrix}1&0 & \frac{-2a}{a+2b}z_j^{\ast}\\
0&1&0\\0&\frac{-2a}{a+2b}z_j^{\ast}&1+\frac{2a}{a+2b} z_j^{\ast} \end{pmatrix}.$$
Then the direct summand $M_0'$ of $C(L^j)=\oplus_{i\geq 0}M_i'$ is $B(e_2+e_3)$ of rank 1. Recall that $$M_0^{\prime}= B(e_2+e_3), ~~~M_1^{\prime}=M_1, ~~~ M_2^{\prime}=\left( \left(\pi B(-2be_1+e_2)\oplus \pi B(-ae_1+e_3)\right)\oplus(\oplus_{\lambda}\pi H_{\lambda})\right)\oplus M_2,$$ $$\mathrm{and}~ M_k^{\prime}=M_k \mathrm{~if~} k\geq 3.$$ The image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $C(L^j)$ is then $$\begin{pmatrix}
\begin{pmatrix}1+\frac{2a}{a+2b} z_j^{\ast}&0 & \frac{-2\pi a}{a+2b}z_j^{\ast}\\
\frac{-\pi a}{a+2b}z_j^{\ast}&1&0\\0&0&1 \end{pmatrix} &0 \\
0& id\end{pmatrix}.$$ Here, the $(1,1)$-block corresponds to $B(e_2+e_3)\oplus \left(\pi B(-2be_1+e_2)\oplus \pi B(-ae_1+e_3)\right)$.
We now describe the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $M_0'\oplus C(L^j)=(M_0^{\prime}\oplus M_0^{\prime})\oplus (\bigoplus_{i \geq 1} M_i^{\prime})$. If $(e_1', e_2')$ is a basis for $(M_0^{\prime}\oplus M_0^{\prime})$, then we choose another basis $(e_1', e_1'+e_2')$ for $(M_0^{\prime}\oplus M_0^{\prime})$. For this basis, based on the description of the morphism from the smooth i
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| 0.778197
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|
obi symbol properties
------------------------
The Jacobi symbol $(a | b)$ is defined for $b$ odd and positive, and arbitrary $a$. We work primarily with non-negative $a$, and make use of the following properties of the Jacobi symbol.
Assume that $a$ is positive and that $b$ is odd and positive. Then
(i) \[it:zero\] $(0 | b) = [b = 1]$.
(ii) \[it:negation\] $(a | b) = (-1)^{(b-1)/2} (-a | b)$
(iii) \[it:reciprocity\] If both $a$ and $b$ are odd, then $$(a | b) = (-1)^{(a-1)(b-1)/4} (b | a)$$
(iv) \[it:odd-reduction\] $(a | b) = (a - m b | b)$ for any $m$.
(v) \[it:even-reduction-4\] If $a = 0 \pmod 4$ and $1 \leq m \leq {\lfloor b/a \rfloor}$, then $$(a | b) = (a | b - ma)$$
(vi) \[it:even-reduction-2\] If $a = 2 \pmod 4$ and $1 \leq m \leq {\lfloor b/a \rfloor}$, then $$(a | b) = (-1)^{m(b-1)/2 + m(m-1)/2} (a | b - ma)$$
For to we refer to standard textbooks. The final two are not so well-known, and their use for Jacobi computation is suggested by Schönhage [@schoenhage-brent-communication]. To prove them, assume that $a$ is even and $a < b$. Then $$\begin{aligned}
(a | b) &= (a - b | b) && \text{By \eqref{it:odd-reduction}}\\
&= (-1)^{(b-1)/2} (b - a | b) && \text{By \eqref{it:negation}} \\
&= (-1)^{(b-1)/2 + (b-1)(b-a-1)/4} (b | b-a) &&\text{By \eqref{it:reciprocity}}\\
&= (-1)^{(b-1)/2 + (b-1)(b-a-1)/4}(a | b-a) && \text{By \eqref{it:odd-reduction}}
\end{aligned}$$ Since $b^2 - 1$ is divisible by 8 for any odd $b$, we get a resulting exponent, modulo two, of $$(b-1)/2 + (b-1)(b-a-1)/4 = a (b-1)/4$$ If $a = 0 \pmod 4$, this exponent is even and hence there is no sign change. And this continues to hold if the subtraction is repeated, which proves . Next, consider the case $a
= 2 \pmod 4$. Then $a/2 = 1 \pmod 2$, and repeating the subtraction $m$ times gives the exponent $$\begin{gathered}
a \{(b-1) + (b-a-1) + \cdots + (b - (m-1) a - 1)\} / 4 \\
= m(b-1)/2 + m(m-1)/2 \pmod 2
\end{gathered}$$ which proves .
Finally, note that in these formulas, all the signs are de
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|
m_j}\mathcal{F}_{_{j-2l}}(\tilde{m})$ is naturally identified with $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(m)$.
As for Equation (\[ea20\]) of Step (1), we need to express $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(m)$ precisely. Each entry $\tilde{x}_i$ of $(\tilde{m_{i,j}}, \tilde{s_i} \cdots \tilde{w_i})$ is expressed as $\tilde{x}_i=(\tilde{x}_i)_1+\pi (\tilde{x}_i)_2$. Then we have $$\begin{gathered}
\label{ea30}
\mathcal{F}_i(\tilde{m}) = \left\{\begin{array}{l l}
\left(1/2\cdot{}^t(\tilde{y}_i)_1a_i(\tilde{y}_i)_1+\bar{\gamma}_i(\tilde{z}_i)_1^2\right) & \quad \textit{if $L_i$ is of type $I^o$};\\
\left(1/2\cdot{}^t(\tilde{t}_i)_1a_i(\tilde{t}_i)_1+\bar{\gamma}_i(\tilde{w}_i)_1^2+(\tilde{z}_i)_1(\tilde{w}_i)_1\right)
& \quad \textit{if $L_i$ is of type $I^e$} \end{array}\right.\\
+(\tilde{z}_i)_2+(\tilde{z}_i)_2^2+(\tilde{z}_i)_1(\delta_{i-2}(\tilde{k}_{i-2, i})_1+\delta_{i+2}(\tilde{k}_{i+2,i})_1)+
\delta_{i-2}\delta_{i+2}(\tilde{k}_{i-2, i})_1(\tilde{k}_{i+2, i})_1+\\
\left({}^t(\tilde{m}_{i-1, i}')_1\cdot h_{i-1}\cdot (\tilde{m}_{i-1, i}')_2+{}^t(\tilde{m}_{i+1, i}')_1\cdot h_{i+1}\cdot (\tilde{m}_{i+1, i}')_2\right)+\\
\frac{1}{2}\left({}^t(\tilde{m}_{i-2, i})_1'\cdot a_{i-2}\cdot (\tilde{m}_{i-2, i})_1'+{}^t(\tilde{m}_{i+2, i})_1'\cdot a_{i+2}\cdot (\tilde{m}_{i+2, i})_1'\right)\\
+\left\{\begin{array}{l l}
\delta_{i-2}\left(\bar{\gamma}_{i-2}(\tilde{k}_{i-2, i})_1^2+(\tilde{k}_{i-2, i})_2^2\right) & \quad \textit{if $L_{i-2}$ is of type $I^o$}; \\
\delta_{i-2}\left(\bar{\gamma}_{i-2}(\tilde{k}_{i-2, i})_1'^2+ (\tilde{k}_{i-2, i})_2^2+ (\tilde{k}_{i-2, i})_1\cdot (\tilde{k}_{i-2, i})_1'\right)
& \quad \textit{if $L_{i-2}$ is of type $I^e$} \end{array}\right.\\
+ \left\{\begin{array}{l l}
\delta_{i+2}\left(\bar{\gamma}_{i+2}(\tilde{k}_{i+2, i})_1^2+(\tilde{k}_{i+2, i})_2^2\right) & \quad \textit{if $L_{i+2}$ is of type $I^o$};\\
\delta_{i+2}\left(\bar{\gamma}_{i+2}((\tilde{k}_{i+2, i})_1')^2+ (\tild
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\end{aligned}$$ and for the bundle ${\cal E}$, $$c_1^{\rm rep}({\cal E})|_{\alpha} \: = \:
\sum_a \frac{n_a}{k} J \alpha^{-n_a} \: - \: \frac{m}{k} J \alpha^{-m},$$ $$\begin{aligned}
{\rm ch}_2^{\rm rep}({\cal E})|_{\alpha} & = &
{\rm ch}_2^{\rm rep}( \oplus_a {\cal O}(n_a) )|_{\alpha} \: - \:
{\rm ch}_2^{\rm rep}( {\cal O}(m) )|_{\alpha}, \\
& = &
\frac{1}{2} \sum_a \left( \frac{n_a}{k} J \right)^2 \alpha^{-n_a}
\: - \:
\frac{1}{2} \left( \frac{m}{k} J \right)^2 \alpha^{-m}.\end{aligned}$$
By contrast, anomaly cancellation in the GLSM is merely the statement that $$\sum_a n_a^2 \: - \: m^2 \: = \: (n+1) k^2 \: - \: d^2,$$ a much weaker statement than demanding ${\rm ch}_2^{\rm rep}({\cal E}) =
{\rm ch}_2^{\rm rep}(T \mathfrak{X})$ in each sector $\alpha$. Anomaly cancellation in the GLSM is well-understood – in the present case, this is just the gauge anomaly in a $U(1)$ gauge theory, which is under extremely good control. Demanding matching ${\rm ch}^{\rm rep}$’s gives a stronger condition – some theories that would satisfy GLSM anomaly cancellation, would not satisfy the constraint of matching ${\rm ch}^{\rm rep}$’s.
For this reason, we do not believe that one should demand matching ${\rm ch}_2^{\rm rep}$’s. This is a somewhat puzzling conclusion, as these are not only the most natural notion of Chern classes on stacks, but they are also vital in index theory, which ordinarily would be a route to [*deriving*]{} their utility. (On the other hand, we briefly remark on a possible application of $c_1^{\rm rep}$ in appendix \[app:spectra:fockconstraints\].)
Combinations {#sect:combos}
============
So far we have discussed three fundamental classes of examples of heterotic string compactifications on gerbes.
Those three classes do not exhaust all possibilities; rather, one should think of them as ‘building blocks’ that can be used to assemble more complicated possibilities.
For one example, consider a string on a ${\mathbb Z}_4$ gerbe, of which a ${\mathbb Z}_2$ subgroup acts on a rank 8 bundle
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1273.001){#fig1}
{#fig2}
{#fig3}
{#fig4}
{#fig5}
{#fig6}
{#fig7}
######
Clinical features of case 2 and case 3.
------------------------------------------------------------------------------------
Case 2 Case 3
--------------------------- ------------------------------- ------------------------
Age 7 years 2 years
Gender Male Female
Gene ZNF469 ZNF469
Presenting visual acuity OD: counting fingers\ OD: 20/70 cc\
OS: 20/70 cc OS: 20/60 cc
IOP OD: 16 mmHg\ OD: 16 mmHg\
OS: 15 mmHg OS: 14 mmHg
Minimal corneal thickness OD: 366 *μ*m\ OD: 238 *μ*m\
OS: 232 *μ*m OS: 254 *μ*m
Re
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\le x_2$, $y_1 \le y \le y_2$, and $z_1 \le z \le z_2$ in a Cartesian grid. In a cylindrical grid, we consider a rectangular torus with density $\rho=1$, occupying the regions with $R_1 \le R \le R_2$, $\phi_1 \le \phi \le \phi_2$, and $z_1 \le z \le z_2$. Table \[tb:convergence\_test\] lists the parameters of the solid figures that we adopt: these values ensure that the mass distribution does not change with resolution.
We use our Poisson solver to calculate the gravitational potentials of the solid figures by varying resolution from $N_z=16$ to $512$, while keeping the domain sizes the same as in Section \[s:staticPot\]. As the reference potential, we take Equation (20) of @katz16 for the gravitational potential of a uniform cube. There is no algebraic expression for the potential of a rectangular torus, but @hure14 provided a closed-form expression, in terms of line integrals with smooth integrands, in their Equation (29). We use the Romberg’s method with a relative tolerance $10^{-10}$ to ensure that the numerical integrations are accurate enough to serve as a reference solution. The squares in Figure \[fig:convergence\] plot the resulting mean relative errors $\left<{\epsilon}\right>$ for the cube and rectangular torus. Note that $\left<{\epsilon}\right>$ against $N_z$ follows almost a straight line with slope of $-1.993$, $-2.003$, and $-2.003$ in the Cartesian, uniform cylindrical, and logarithmic cylindrical grid, respectively, confirming that our implementation of the Poisson solver retains a second-order accuracy.
[ccccccc]{} Cartesian & 0.0625 & 0.4375 & -0.375 & 0 & -0.125 & 0.25\
uniform cylindrical & 0.625 & 0.90625 & 0 & 1.570796327 & -0.0625 & 0.1875\
logarithmic cylindrical & 0.1 & 0.7498942093 & 0 & 1.570796327 & -0.0625 & 0.1875
Performance Test
----------------
To check the parallel performance of our implementation, we conduct a weak scaling test of our Poisson solver on the [TigerCPU]{} linux cluster at Princeton University[^4]. The [TigerCPU]{} cluster consists of 408 nodes, with each
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ptions, and , the integrals $\int_{S\times I}\sigma_{jk}(x,\omega',\omega,E',E)d\omega' dE'$ (resp. $\int_{S\times I}\sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'$) over $S\times I$ are to be replaced with $\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)d\omega'$ (resp. $\int_{S}\sigma_{jk}(x,\omega,\omega',E)d\omega'$) over $S$.
For the coupled BTE system (\[intro1\]), (\[intro2\]) we formulate the following result which is a slight modification of results given in [@tervo14]. Note that it is valid only for Schur collision operators and hence it does not govern completely the particle transport including charged particles, such as applications in radiation therapy.
\[origbte\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda3\]), (\[sda4\]) are valid, and that $c$ is strictly positive. Then for every $f\in L^2(G\times S\times I)^3$ and $g\in T^2_{\tau_-}(\Gamma_-)^3$ the following assertions hold.
1. The boundary value problem $$\begin{gathered}
\omega\cdot\nabla_x\psi_j+\Sigma_j\psi_j-K_j\psi=f_j, \nonumber\\
{\psi_j}_{|\Gamma_-}=g_j, \label{origbte1}\end{gathered}$$ for $j=1,2,3$, has a unique solution $\psi\in W^2(G\times S\times I)^3$.
2. There exists a constant $C>0$ such that \[origbte2\] \_[W\^2(GSI)\^3]{}C([f]{}\_[L\^2(GSI)\^3]{}+[g]{}\_[T\^2\_[\_-]{}(\_-)\^3]{}).
3. If $f\geq 0$ and $g\geq 0$, then $\psi\geq 0$, i.e. the solution $\psi$ is non-negative for non-negative data $f,g$.
The assertions follow from the considerations expressed in [@tervo14] noting that in Lemma 5.8 (see its proof) of [@tervo14] we actually have $${\left\Vert Lg\right\Vert}_{L^2(G\times S\times I)}\leq {\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)},\ {\rm for}\ g\in T^2_{\tau_-}(\Gamma_-).$$ We omit details here.
The corresponding result for time-dependent coupled system of BTEs has been proven in [@tervo14] as well.
Existence of Solutions Based on Variational Formulation {#LLM}
-------------------------------------------------------
As before, we perform a change of unknown functions, by set
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$q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1)$ is a character of $(\mathfrak{g},+)$.
\[charab\]
Absolutely indecomposable representations
=========================================
Generalities on quiver representations {#genquiv}
--------------------------------------
Let $\Gamma$ be a finite quiver, $I$ be the set of its vertices and let $\Omega$ be the set of its arrows. For $\gamma\in\Omega$, we denote by $h(\gamma),t(\gamma)\in I$ the head and the tail of $\gamma$. A *dimension vector* of $\Gamma$ is a collection of non-negative integers $\v=\{v_i\}_{i\in I}$ and a *representation* $\varphi$ of $\Gamma$ of dimension $\v$ over a field $\K$ is a collection of $\K$-linear maps $\varphihat=\{\varphi_\gamma:V_{t(\gamma)}\rightarrow
V_{h(\gamma)}\}_{\gamma\in\Omega}$ with ${\rm dim}\, V_i=v_i$. Let ${\rm Rep}_{\Gamma,\v}(\K)$ be the $\K$-vector space of all representations of $\Gamma$ of dimension $\v$ over $\K$. If $\varphihat\in {\rm Rep}_{\Gamma,\v}(\K)$, $\varphihat'\in {\rm
Rep}_{\Gamma,\v'}(\K)$, then a morphism $f:\varphihat\rightarrow\varphihat'$ is a collection of $\K$-linear maps $f_i: V_i\rightarrow V_i'$, $i\in I$ such that for all $\gamma\in\Omega$, we have $f_{h(\gamma)}\circ
\varphi_\gamma=\varphi'_\gamma\circ f_{t(\gamma)}$.
We define in the obvious way direct sums $\varphihat\,\oplus\,\varphihat'\in {\rm Rep}_\K(\Gamma,\v+\v')$ of representations. A representation of $\Gamma$ is said to be *indecomposable* over $\K$ if it is not isomorphic to a direct sum of two non-zero representations of $\Gamma$. If an indecomposable representation of $\Gamma$ remains indecomposable over any finite extension of $\K$, we say that it is *absolutely indecomposable*. Denote by ${\rm M}_{\Gamma,\v}(\K)$ be the set of isomorphism classes of ${\rm Rep}_{\Gamma,\v}(\K)$ and by ${\rm A}_{\Gamma,\v}(\K)$ the subset of absolutely indecomposable representations of ${\rm
Rep}_{\Gamma,\v}(\K)$.
By a theorem of Kac there exists a polynomial $A_{\Gamma,\v}(T)\in \Z[T]$ such that for
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e on-shell physical states defined by these asymptotic string fields, and thus the physical S-matrix. Thus the supersymmetry algebra is realized on the physical S-matrix, and we can identify the transformation (\[complete transformation\]) with space-time supersymmetry.
Extra unphysical symmetries {#extra symm}
---------------------------
We have shown that the supersymmetry algebra is realized on the physical S-matrix but this is not the end of the story. The extra transformation $\delta_{\tilde{p}}$ produces another extra transformation if we consider the nested commutator $[\delta_{{\mathcal{S}}_1},[\delta_{{\mathcal{S}}_2},\delta_{{\mathcal{S}}_3}]]$. The extra contribution comes from the commutator $[\delta_{\mathcal{S}},\delta_{\tilde{p}}]$ which is non-trivial because the first-quantized charges ${\mathcal{S}}$ and $\tilde{p}$ are not commutative: $[{\mathcal{S}},\tilde{p}]\ne0$. In fact, we can show that the algebra $$[\delta_{\mathcal{S}},\, \delta_{\tilde{p}}]\ \cong\
\delta_g + \delta_{[{\mathcal{S}},\tilde{p}]}\,,
\label{alg sp}$$ holds with the gauge parameters,
$$\begin{aligned}
\Lambda_{{\mathcal{S}}\tilde{p}}\ =&\
f\xi_0\big(D_{\tilde{p}}f\xi_0D_{\mathcal{S}}- D_{\mathcal{S}}F\Xi D_{\tilde{p}}\big) F\Psi
- [F\Psi, F\Xi D_{\tilde{p}}F\Xi A_{\mathcal{S}}]
\nonumber\\
&\
- [F\Xi A_{\mathcal{S}}, F\Xi D_{\tilde{p}}F\Psi]
- D_{\tilde{p}}f\xi_0\{F\Psi, F\Xi A_{\mathcal{S}}\}\,,\\
\lambda_{{\mathcal{S}}\tilde{p}}\ =&\
X\eta F\Xi D_\eta D_{\tilde{p}} F\Xi A_{\mathcal{S}}\,,\end{aligned}$$
and $\Omega_{{\mathcal{S}}\tilde{p}}$ in (\[Omega IJ\]). The new transformation $\delta_{[{\mathcal{S}},\tilde{p}]}$ is defined by
\[tf sp\] $$\begin{aligned}
A_{\delta_{[{\mathcal{S}},\tilde{p}]}}\ =&\
f\xi_0\Big(Qf\xi_0D_{[{\mathcal{S}},\tilde{p}]}F\Psi
+
[F\Psi,\,F\Xi\big(QA_{[{\mathcal{S}},\tilde{p}]}+[F\Psi,\,f\xi_0D_{[{\mathcal{S}},\tilde{p}]}F\Psi]\big)]\Big)\,,
\label{tf sp ns}\\
\delta_{[{\mathcal{S}},\tilde{p}]}\Psi\ =&\
X\eta F\Xi\big(
QA_{[{\mathcal{S}},\tilde{p}]}+[F\Psi,\,f\xi_0D_{[{\mathcal{S}},\ti
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estimator
===================
In this section we obtain the asymptotic size of the uniform deviation of the ideal estimator (\[ideal0\]) from the density $f$, that is, we will consider the a.s. asymptotic size of $$\sup_{t\in D_r} |\bar f(t;h_n)-f(t)|:=\|\bar f(t;h_n)-f(t)\|_{D_r}$$ As usual this quantity is divided into the bias part, $\|E\bar f(t;h_n)-f(t)\|_{D_r}$, and the stochastic part or variance part $\|\bar f(t;h_n)-E\bar f(t;h_n)\|_{D_r}$. Each is studied in a different subsection. There is no problem with extending the supremum for the variance part over the whole of $\mathbb R$; the problem is, as mentioned above, with the bias.
We will use the shorthand notations $$\bar f_n(t;h)=\bar f_n(t)=\bar f(t;h_n)$$ so that we display only either $h_n$ or $n$ but not both; the first expression is used in this section and the second in the next.
Stochastic part of the ‘ideal’ estimator
----------------------------------------
In this subsection we assume:
\[ass1\] The sequence $h_n$ will satisfy the following classical conditions: $$\label{band}
h_{n}\searrow0, \;\; \frac{nh_{n}}{|\log h_{n}|}\rightarrow\infty, \;\; \frac{|\log h_{n}|}{\log\log{n}}
\rightarrow\infty,\;\; and\:\: nh_n\nearrow,$$ as $n\to\infty$. The kernel $K$ will be a non-negative left or right continuous function, bounded, with support contained in $[-T,T]$ for some $T<\infty$, and of bounded variation. $f$ is a bounded density.
The proof of the following proposition is patterned after the proof of a similar theorem in Giné and Guillou (2002), and it consists of blocking and application of Talagrand’s inequality (\[tal\]). It extends to the variable bandwidth estimator a well known uniform rate for the usual kernel estimator (Silverman (1978), formula (9)).
\[varid\] Under the hypotheses in Assumptions \[ass1\], $$||\bar{f}_n-E\bar{f}_n||_{\infty}=O_{\rm a.s.}\left(\sqrt{\frac{\log h_{n}^{-1}}{n h_{n}}}\right)$$ uniformly over all densities $f$ such that $\|f\|_\infty\le C$, for any $0<C<\infty$.
We block the terms between
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ance which is equal to a lattice constant (red bonds, weighted with $v$). On the other hand, in the case of CSAWs model (b), the polymers $P_3$ and $P_2$ are cross-linked at the two sites, so that each contact contributes the weight factor $w$, while the red bonds (marked by $t$) correspond to the interactions between those monomers which are nearest neighbors to the cross-linked points. The two depicted examples for ASAWs (a) and CSAWs models (b), contribute the weights $x_3^{5}x_2^{3}u^4v^{12}$ and $x_3^{4}x_2^{3}w^2t^3$, respectively.[]{data-label="fig:interakcije"}](figure1.eps)
To describe exactly all possible configurations of the two-chain polymer system within the adopted model, we need four restricted partition functions $A^{(r)}$, $B^{(r)}$, $C^{(r)}$ and $D^{(r)}$, which are defined as $$\begin{aligned}
\fl A^{(r)}=\sum_{N_3,L} {\mathcal A}^{(r)}(N_3,L) x_3^{N_3} u^L,\quad && B^{(r)}=\sum_{N_3,L} {\mathcal B}^{(r)}(N_3,L) x_3^{N_3} u^L,\quad \nonumber\\
\fl C^{(r)}=\sum_{N_2} {\mathcal C}^{(r)}(N_2) x_2^{N_2},\quad && D^{(r)}= \sum_{N_2,N_3,L,M}{\mathcal D}^{(r)}(N_2,N_3,L,M) x_2^{N_2}x_3^{N_3}u^Lv^M, %\nonumber\end{aligned}$$ where ${\mathcal A}^{(r)}$, ${\mathcal B}^{(r)}$, ${\mathcal C}^{(r)}$, and ${\mathcal D}^{(r)}$ represent the numbers of particular configurations, consisting of one or two SAW strands on the $r$-th fractal structure (see figure \[fig:RGparametri\]). For instance, ${\mathcal D}^{(r)}(N_2,N_3,L,M)$ is the number of configurations consisting of $N_3$-step $P_3$ chain with $L$ pairs of non-consecutive nearest-neighbor monomers, and $N_2$-step $P_2$ chain, such that there are $M$ contacts between these two chains. The recursive nature of the fractal construction implies the following recursion relations for restricted partition functions $$\begin{aligned}
A'&=&\sum_{N_{A},N_{B}} a(N_{A},N_{B})\, A^{N_{A}}
B^{N_{B}}\,,
\label{eq:RGA}\\
B'&=&\sum_{N_{A},N_{B}} b(N_{A},N_{B})\, A^{N_{A}}
B^{N_{B}}\,,
\label{eq:RGB}\\
C'&=&\sum_{N_C} c(N_C)\, C^{N_C}\,,
\label{eq:RGC}\\
D'&=&
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al dimension of such operators as a function of the two parameters $(k,f)$ of the supergroup sigma-model. At the WZW point these operators are descendants in the highest-weight representations of the left affine Lie algebra.
Operators of the form $:j_L \phi :$ {#operators-of-the-form-j_l-phi .unnumbered}
-----------------------------------
Let us consider the operator $:j_{L,z}^a \phi:$ defined as the regular term in the OPE between the operators $j_{L,z}^a$ and $\phi$. To compute the holomorphic dimension of this operator we compute its OPE with the stress-tensor, and look at the second order pole. The computation is done following the method described in appendix \[compositeOPEs\]. The fact that the stress-tensor is holomorphic simplifies the calculation. We find: $$\begin{aligned}
T(z) :j^a_{L,z}\phi:(w)
& = \lim_{:x \to w:} T(z) j^a_{L,z}(x) \phi(w) \cr
%
& = \lim_{:x \to w:} \left \{
\left( \frac{j^a_{L,z}(x)}{(z-x)^2} + \frac{\p j^a_{L,z}(x)}{z-x} \right) \phi(w) %\right. \cr
%& \qquad \left.
+ j^a_{L,z}(x) \left( \frac{\Delta_{\phi} \phi(w)}{(z-w)^2} + \frac{\p \phi(w)}{z-w} \right)
\right\} \cr
%
&= \lim_{:x \to w:} \left \{
\frac{1}{(z-x)^2}\left(
-\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{x-w} + :j^a_{L,z} \phi:(w) \right.\right. \cr
&\qquad \qquad \left. \left.
+ {A^a}_c \log|x-w|^2 :j^c_{L,z} \phi:(w) + {B^a}_c \frac{\bar x - \bar w}{x-w} :j^c_{L,\bar z} \phi:(w) + ...
\right) \right. \cr
& \quad + \frac{1}{z-x}\left(
\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{(x-w)^2} + :\p j^a_{L,z} \phi:(w) \right. \cr
&\qquad \qquad \left. \left.
+ {A^a}_c \frac{1}{x-w} :j^c_{L,z} \phi:(w) - {B^a}_c \frac{\bar x - \bar w}{(x-w)^2} :j^c_{L,\bar z} \phi:(w) + ...
\right) \right. \cr
& \quad \left.
+ \frac{\Delta_{\phi}:j^a_{L,z} \phi:(w)}{(z-w)^2} + \frac{:j^a_{L,z}\p \phi:(w)}{z-w}
\right\} \cr
%
&=
-\frac{2}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:j^a_{L,z} \phi:(w)}{(z-w)^2} \cr
&\qquad + \frac{1}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:\p j^a_{L,z} \phi:(w)}{z-w} + {A^a}_c \frac{1}{(z-
| 1,220
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size specified by plot title. For each baseline SGD run, we have a corresponding large-noise SGD run, denoted by $\diamond$ with the same color. As mentioned, these $\diamond$ runs are designed to match the noise covariance of SGD with larger step size or smaller batch size. In addition to $\times$ and $\diamond$, we also plot using a small teal marker all the other runs from Section \[ss:acc\_rel\_var\]. This helps highlight the linear trend between the logarithm of noise covariance and test accuracy that we observed in Section \[ss:acc\_rel\_var\].
As can be seen, the (noise variance, test accuracy) values for the $\diamond$ runs fall close to the linear trend. More specifically, a run of large-noise SGD produces similar test accuracy to vanilla SGD runs with the same noise variance. We highlight two potential implications: First, just like in Section \[ss:acc\_rel\_var\], we observe that the test accuracy strongly correlates with relative variance, even for noise of the form , which can have rather different higher moments than $\zeta$ (standard SGD noise); Second, since the $\diamond$ points fall close to the linear trend, we hypothesize that the constant-noise limit SDE should also have similar test error. If true, then this implies that we only need to study the potential $U(x)$ and noise covariance $M(x)$ to explain the generalization properties of SGD.
Appendix {#appendix .unnumbered}
========
[Proofs for Convergence under Gaussian Noise (Theorem \[t:main\_gaussian\])]{}
[Proof Overview]{} The main proof of Theorem \[t:main\_gaussian\] is contained in Appendix \[ss:proof:t:main\_gaussian\].
Here, we outline the steps of our proof:
1. In Appendix \[ss:coupling\_construction\], we construct a coupling between $\eqref{e:exact-sde}$ and $\eqref{e:discrete-langevin}$ over a single step (i.e. for $t\in[k\delta, (k+1)\delta]$, for some $k$ and $\delta$).
2. Appendix \[ss:step\_gaussian\], we prove Lemma \[l:gaussian\_contraction\], which shows that under the coupling constructed in Step 1, a Lyapunov fu
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|
.54 92.71
36 months 47.71 81.45 68.13 79.76 66.71 84.34 62.36 77.73 66.18 73.17
*p* \<0.01 0.34 0.86 0.60 0.15
Adjusted for infant sex, gestational age, breastfeeding, period of breastfeeding, household income, and dietary supplementation.
Not estimated using least-squares means because of a missing value at 18 months.
Discussion {#S0004}
==========
To the best of our knowledge, this is the first study to show an association between levels of antioxidant vitamins and oxidative stress in mothers and infant growth during the first 3 years of life. Our findings indicate that maternal vitamin A and C increase the head circumference and weight of infants for 3 years within a normal growth range even after considering other relevant factors associated with infant growth. Maternal oxidative stress does not have an effect on infant growth in the first 3 years of life.
Although there are various methods for measuring maternal nutrition levels, including measuring the supplementation effect of a specific nutrient, the most precise method is to measure a nutrient directly in maternal serum. A few studies have used maternal serum samples to examine the effects of specific maternal nutrients on infant growth, but the results have been inconsistent.
Most studies that have examined the effects of maternal nutrition on infant growth have focused only on birth outcome, not on continuous growth after birth. In particular, some studies observed only weight ([@CIT0012]), and only a few studies used other anthropometric measurements such as infant height ([@CIT0013], [@CIT0014]), head circumference ([@CIT0013], [@CIT0015]), or the ponderal index (birth weight/height^3^) ([@CIT0016]). Along with weight and height, head circumference is an important body measurement for motor and cognitive development. There have been reports on an ass
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| 1,387
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|
that, by this definition, $r_i$ for $i\ge1$ equals $((d-2)\wedge(i+3))-{\epsilon}$ and increases until it reaches $d-2-{\epsilon}$. We prove below by induction that $\sum_x|x|^{r_i}|\Pi(x)|$ is finite for all $i\ge0$. This is sufficient for the finiteness of $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$, since $$\begin{aligned}
\lim_{i\uparrow\infty}r_i=d-2-{\epsilon}\ge d-2-\tfrac{2}3(d-4)
=\tfrac{d+2}3.\end{aligned}$$
Note that, by [(\[eq:Pi-bdNN\])]{}, $\sum_x|x|^{r_0}|\Pi(x)|<\infty$. Suppose $\sum_x|x|^{r_i}|\Pi(x)|<\infty$ for some $i\ge0$. Then, by Proposition \[prp:exp-bootstrap\](ii), $\bar W^{{\scriptscriptstyle}(t)}$ is finite for $t\in(0,\lfloor r_i\rfloor]\cap(0,d-4)$. Since $\lfloor r_0\rfloor=2$ and $\lfloor
r_i\rfloor=(d-3)\wedge(i+2)$ for $i\ge1$, $\bar W^{{\scriptscriptstyle}(T)}$ with $T=(i+2)\wedge(d-4-{\epsilon})$ is finite. Then, by Proposition \[prp:exp-bootstrap\](iii), $\sum_x|x|^{T+2}|\Pi(x)|$ is finite. Since $$\begin{aligned}
T+2=(i+4)\wedge(d-2-{\epsilon})\ge\big((d-2)\wedge(i+4)\big)-{\epsilon}=r_{i+1},\end{aligned}$$ we obtain that $\sum_x|x|^{r_{i+1}}|\Pi(x)|<\infty$. This completes the induction and the proof of [(\[eq:IR-xbd-nn\])]{}. The proof of Proposition \[prp:Pij-Rj-bd\] is now completed.
Diagrammatic bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ {#s:bounds}
=================================================================
In this section, we prove diagrammatic bounds on the expansion coefficients. In Section \[ss:diagram\], we construct diagrams in terms of two-point functions and state the bounds. In Section \[ss:pi0bd\], we prove a key lemma for the diagrammatic bounds and show how to apply this lemma to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$. In Section \[ss:pijbd\], we prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
Construction of diagrams {#ss:diagram}
------------------------
To state bounds on the expansion coefficients (as in Proposition \[prp:diagram-bd\] below), we first define diagra
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| 0.79358
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|
_{i,i}')a_i(\pi m_{i,i}')$ since its nondiagonal entries contain $\pi^2$ as a factor and its diagonal entries contain $\pi^4$ as a factor. Thus the above equation equals $$a_i'=a_i+\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'.$$ By letting $a_i'=a_i$, we have the following equation $$\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'=0.$$
Based on (2) of the description of an element of $\underline{H}(R)$ for a $\kappa$-algebra $R$, which is explained in Section \[h\], in order to investigate this equation, we need to consider the nondiagonal entries of $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'$ as elements of $B\otimes_AR$ and the diagonal entries of $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'$ as of the form $2 x_i$ with $x_i\in R$. Recall from Remark \[r33\].(2) that $$a_i=\begin{pmatrix} \begin{pmatrix} 0&1\\1&0\end{pmatrix}& & & \\ &\ddots & & \\ & &\begin{pmatrix} 0&1\\1&0\end{pmatrix}& \\ & & & \begin{pmatrix} 2\cdot 1&1\\1&2\cdot\bar{\gamma}_i\end{pmatrix} \end{pmatrix}.$$ Note that ${}^ta_i=a_i$ and $\sigma(\pi)=-\pi$ so that $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'=\pi\left( a_i m_{i,i}'-{}^t(a_i m_{i,i}')\right)$.
Then we can see that there is no contribution coming from diagonal entries and each nondiagonal (upper triangular) entry produces a linear equation. Thus there are exactly $(n_i^2-n_i)/2$ independent linear equations and $(n_i^2+n_i)/2$ entries of $m_{i,i}'$ determine all entries of $m_{i,i}'$.
For example, let $m_{i,i}'=\begin{pmatrix} x&y\\z&w\end{pmatrix}$ and $a_i=\begin{pmatrix} 2&1\\1&2\bar{\gamma}_i\end{pmatrix}$. Then $$\pi\left( a_i m_{i,i}'-{}^t(a_i m_{i,i}')\right)=\pi\begin{pmatrix} 0&w-x+2y-2\bar{\gamma}_iz\\w-x+2y-2\bar{\gamma}_iz&0\end{pmatrix}.$$ Thus there is only one linear equation $w-x=0$ and $x, y, z$ determine all entries of $m_{i,i}'$.\
4. Assume that $i$ is even and that $L_i$ is *of type $I^o$*. Then $\pi^ih_i=\xi^{i/2} \begin{pmatrix} a_i&\pi b_i\\ \sigma(\pi\cdot {}^t b_i) &1 +2\bar{\gamma}
| 1,224
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| 1,166
| null | null |
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|
1,j} \right)
\ = \ \operatorname{{\textsf}{ogr}}B_{ij}.$$
\(3) When $k=0$, the assertion $eJ^k\delta^{k}
= \operatorname{{\textsf}{ogr}}N(k)$ is just the statement that $e{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}= e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast {{W}}) $. When $k>0$, part (i) and Lemma \[abstract-products\] give $eJ^k\delta^k = eA^k\delta^ke{\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]\ast {{W}}\subseteq \operatorname{{\textsf}{ogr}}B_{k0}\operatorname{{\textsf}{ogr}}(eH_c)\subseteq \operatorname{{\textsf}{ogr}}N(k).$
{#h-defn}
The next several results will be aimed at getting a more detailed understanding of the bimodule structure of $N(k)$ and its factors. For the most part we are interested in their graded structure for which the actions of the elements ${\mathbf{h}}_{c+t}\in H_{c+t}$ from are particularly useful. Given an $({U}_{c+s},{U}_{c+t})$-bimodule $M$, define $${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}m = {\mathbf{h}}_{c+s} m - m{\mathbf{h}}_{c+t} \text{ for any $m\in M$}.$$ When $s=t$ this is just the adjoint action of ${\mathbf{h}}_{c+s}$ on $M$.
\[diaggrad\] [(1)]{} $ e{\mathbf{h}}_{c+t-1} e =
\delta^{-1}e_-{\mathbf{h}}_{c+t} e_-\delta$.
[(2)]{} The action of ${\mathbf{h}}$ is diagonalisable on the modules $N(i)$, $B_{ij}$ and $M(i)=H_{c+i}eB_{i0}$, for any $i\geq j\geq 0$.
\(1) Use the first paragraph of the proof of [@gordc Theorem 4.10].
\(2) We start with the $B_{ij}$. If $b_1\in B_{i\ell}$ and $b_2\in B_{\ell j}$, then ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}(b_1b_2) = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_1)b_2 + b_1({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_2)$. Thus, by induction, it suffices to prove the result for each $B_{t,t-1}
= e H_{c+t}\delta e$. Clearly $e{\mathbf{h}}={\mathbf{h}}e$. Thus, by part (1), for any $m\in H_{c+t}$ we have $$\label{diaggrad1}
{\mathbf{h}}{ \,{}_{^{^{\bullet}}}}e m \delta e \ =\
{\mathbf{h}}_{c+t}e m \delta e - e m \delta e {\mathbf{h}}_{c+t-1} \ =
\ e {\mathbf{h}}_{c+t} m \delta e - e m \delta (\delta^{-1}e_-{
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ss Type
0 __text 00000043 0000000100000f30 TEXT
[...]
You can also do the same trick to get the size of whole segments (as opposed to sections) by using the syntax segment$start$__TEXT / segment$end$__TEXT.
Q:
Why does new allocate 1040 extra bytes the first time?
I was creating this simple test program to demonstrate the way alignment works when allocating memory using standard new...
#include <iostream>
#include <iomanip>
#include <cstdint>
//
// Print a reserved block: its asked size, its start address
// and the size of the previous reserved block
//
void print(uint16_t num, uint16_t size_asked, uint8_t* p) {
static uint8_t* last = nullptr;
std::cout << "BLOCK " << num << ": ";
std::cout << std::setfill('0') << std::setw(2) << size_asked << "b, ";
std::cout << std::hex << (void*)p;
if (last != nullptr) {
std::cout << ", " << std::dec << (uint32_t)(p - last) << "b";
}
std::cout << "\n";
last = p;
}
int main(void) {
// Sizes of the blocks to be reserved and pointers
uint16_t s[8] = { 8, 8, 16, 16, 4, 4, 6, 6 };
uint8_t* p[8];
// Reserve some consecutive memory blocks and print
// pointers (start) and their real sizes
// std::cout << " size start prev.size\n";
// std::cout << "-----------------------------------\n";
for(uint16_t i = 0; i < 8; ++i) {
p[i] = new uint8_t[s[i]];
print(i, s[i], p[i]);
}
return 0;
}
But when I executed the program I found this odd behaviour:
[memtest]$ g++ -O2 mem.cpp -o mem
[memtest]$ ./mem
BLOCK 0: 08b, 0xa0ec20
BLOCK 1: 08b, 0xa0f050, 1072b
BLOCK 2: 16b, 0xa0f070, 32b
BLOCK 3: 16b, 0xa0f090, 32b
BLOCK 4: 04b, 0xa0f0b0, 32b
BLOCK 5: 04b, 0xa0f0d0, 32b
BLOCK 6: 06b, 0xa0f0f0, 32b
BLOCK 7: 06b, 0xa0f110, 32b
As you can see, the second block allocated by new is not at the next 32b memory alligned address, but far away (1040 bytes away). If this is not odd enough, uncommenting the 2 std::cout lines that print out the header of the table, yields
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| 11
| 0.841901
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80 (25.5) 1.26 (0.86, 1.85) A allele 88 (27.8) 16 (21.6) 1.40 (0.76, 2.56)
G allele 151 (70.0) 234 (74.5) 0.2416 1 G allele 228 (72.2) 58 (78.4) 0.2756 1
**rs6603797** **rs6603797**
C allele 193 (89.4) 270 (86.0) 1.37 (0.80, 2.34) C allele 285 (90.2) 67 (90.5) 0.96 (0.41, 2.28)
T allele 23 (10.6) 44 (14.0) 0.2521 1 T allele 31 (9.8) 7 (9.5) 0.9270 1
**rs4648727**^**a**^ **rs4648727**^**a**^
A allele 70 (32.4) 107 (34.3) 0.92 (0.64, 1.33) A allele 111 (35.1) 17 (23.6) 1.75 (0.97, 3.16)
C allele 146 (67.6) 205 (65.7) 0.6515 1 C allele 205 (64.9) 55 (76.4) 0.0607 1
**rs12126768**^**a**^ **rs12126768**
G allele 45 (20.8) 86 (27.6) 0.69 (0.46, 1.04) G allele 75 (23.7) 16 (21.6) 1.13 (0.61, 2.08)
T allele 171 (79.2) 226 (72.4) 0.0783 1 T allele 241 (76.3) 58 (78.4) 0.6989 1
^a^: Contains 1 missing data point in the RVR (−) group.
Abbreviations: SNP, single nucleotide polymorphism; RVR, rapid virological response; OR, odds ratio; CI, confidence interval.
Allele frequencies were determined by *χ*^2^ test using 2 × 2 tables. Odds ratios and 95% CI per allele were estimated by unconditional logistic regression. *P* values less than 0.05 were considered statistically significant.
######
**Genotype frequencies of*GNB1*single nuc
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|
potent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$ and a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $e^{O(s^2)}K\log^{O(s)}2K$ such that $$A\subset HP_{\text{\textup{nil}}}(x;L)\subset H\overline P(x;L)\subset HA^{e^{O(s^3)}K^{s+1}\log^{O(s^2)}2K}.$$
In particular, $|H\overline P(x;L)|\le\exp(K^{e^{O(s)}})|A|$, or $|P(x;L)|\le\exp(e^{O(s^3)}K^{s+1}\log^{O(s^2)}2K)|A|$ in the torsion-free setting.
We deduce these corollaries in \[sec:covering\].
<span style="font-variant:small-caps;">Applications to other groups.</span> A theorem of Breuillard, Green and Tao [@bgt] states, in one form, that an arbitrary finite $K$-approximate group $A$ is contained in a union of at most $O_K(1)$ translates of a coset nilprogression of rank and step $O(K^2\log K)$ and size at most $K^{11}|A|$. This result is powerful enough to have some quite general applications, such as those contained in [@bgt §11] and [@bt; @tao.growth; @tt], but its usefulness is slightly lessened by the fact that it does not give an explicit bound on the number of translates needed to contain $A$. Partly for this reason, various papers by several different authors have given explicit versions of this theorem for certain specific classes of groups.
The approach taken in these results is generally first to reduce to the nilpotent case, and then to apply \[thm:old\] (or an earlier result of Breuillard and Green [@bg] valid only in the torsion-free setting) to obtain the nilprogression. Unsurprisingly, using \[thm:new.gen\] or one of its corollaries in place of \[thm:old\] in these arguments leads to better bounds in a number of cases. In \[sec:non-nilp\] we present such better bounds for linear groups over ${\mathbb{F}}_p$ or fields of characteristic zero, and in residually nilpotent groups.
<span style="font-variant:small-caps;">Acknowledgement.</span> I was prompted to revisit the bounds in \[thm:old\] by a question from Harald Helfgot
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s is highly relevant for geometric complexity theory, since it was recently shown in [@kumar10b] and [@burgisserikenmeyer11] that the studied varieties (the orbit closures of the determinant and permanent on the one hand, and of the matrix multiplication tensor and the unit tensor on the other hand) are in fact never normal except in trivial situations.
The subgroup restriction problem for a rational group homomorphism $f \colon H \rightarrow G$ can be realized in the above setup: Indeed, for any highest weight $\lambda \in \Lambda^*_{G,+}$ consider $X = \mathcal O_{G,\lambda}$, the coadjoint orbit through $\lambda$, with the induced action of $H$. This variety can be canonically embedded into projective space as the orbit of the highest weight vector in ${\mathbb P}(V_{G,\lambda})$, and it is a consequence of the Borel–Weil theorem that ${\mathbb C}[\mathcal O_{G,\lambda}] = \bigoplus_{k=0}^\infty V_{G,k\lambda}^*$. By comparing with it follows that $m_{H,\mathcal O_{G,\lambda},k}(\mu) = m^{k \lambda}_{\mu}$. In particular, the above definition of the stretching function, $k \mapsto m_{H,\mathcal O_{G,\lambda},k}(k\mu)$, coincides with our previous usage, $k \mapsto m^{k\lambda}_{k\mu}$.
By the Hilbert–Serre theorem, the function $k \mapsto \dim {\mathbb C}[X]_k$ is a polynomial of degree $\dim X$ for large $k$ [@hartshorne77 Theorem I.7.5]. Hence there exists a constant $A > 0$ such that $$\begin{aligned}
&A \, k^{\dim X}
\sim \dim {\mathbb C}[X]_k \\
~=~ &\sum_{\mathclap{\mu \in \Lambda^*_{H,+}}} m_{H,X,k}(\mu) \, \dim V_\mu
~=~ \sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu},
\end{aligned}$$ where for the last equality we have used the definition of the moment polytope $\Delta_X$. By the Weyl dimension formula, we have $$\begin{aligned}
&\dim V_{k\mu} ~=~
\prod_{\mathclap{\alpha \in R_{H,+}}} \frac {\langle \alpha, k \mu + \rho \rangle} {\langle \alpha, \rho \rangle} \\
~=~ &\left( ~~~
\prod_{\mathclap{\substack{\al
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\[fig2\] and Fig. \[fig3\], Fig. \[fig4\] depicts the mean momentum $p(t)$ of the atoms for two different values of the occupation number $\bar{n}=N/L$ (number of atoms per lattice cite) – $\bar{n}=1$ (upper panel) and $\bar{n}=2/7$ (lower panel). As to be expected, the dynamics of the system depends on the value of $\bar{n}$, and for a larger occupation number the deviations of many-particle BO from the non-interacting result $p(t)=NJ\sin(2\pi Ft)$ becomes larger.
![Normalised mean momentum ($p/NJ\rightarrow p$) as a function of time, for two different values of the occupation number $\bar{n}=N/L$: $N=L=7$ (upper panel), and $N=4$, $L=14$ (lower panel). The dashed lines show the analytical result for the envelope function in the thermodynamic limit.[]{data-label="fig4"}](fig4.eps){width="8cm"}
Our explanation for the numerical results is the following. It is convenient to treat the atom-atom interaction as a perturbation. Let us denote by $U_0(t)$ the evolution operator of the system for $W=0$, by $U(t)$ the evolution operator for $W\ne0$, and by $U_W(t)$ the evolution operator defined by the decomposition $U(t)=U_W(t)U_0(t)$. Since $U_0(T_B)$ is the identity matrix, one has to find $U_W(T_B)$ to reproduce the result of Fig. \[fig3\]. In the interaction representation, the formal solution for $U_W(T_B)$ has the form $$\label{5}
U_W(T_B)=\widehat{\exp}\left(-i\frac{W}{2}
\int_0^{T_B} dt U_0^\dag(t)
\sum_{l=1}^L \hat{n}_l(\hat{n_l}-1)U_0(t)\right) \;,$$ where the hat over the exponential denotes time ordering. Now we make use of the above strong-field condition $dF>J$. Under this premise the Wannier states are the eigenstates of the atom in the optical lattice \[see Eq. (\[1a\])\]. In the multi-particle case this means that the Fock states $|{\bf n}\rangle$ are the eigenstates of the system (\[2\]) and, thus, that the operator $U_0(t)$ is diagonal in the Fock state basis. Then the integral in Eq. (\[5\]) can be calculated explicitely, which yields $$\label{6}
\langle {\bf n}|U_W(T_B)|{\bf n}\rangle=
\ex
| 1,230
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| 878
| 1,324
| 2,810
| 0.77681
|
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|
times of measurement to record random variations in detected x-ray intensity are acquired. However, in this work, the collected datasets are not supported by the aforementioned factors and they fall outside the scope of this paper. The results presented here are focusing on the implementation of a new algorithm to limited-data CT reconstruction and are reported as a preliminary study.
Reconstruction using a conventional method is computed as well with the built-in <span style="font-variant:small-caps;">Matlab</span> function [iradon]{}, which uses the FBP to invert the Radon transform. It reconstructs a two-dimensional slice of the sample from the corresponding projections. The angles for which the projections are available are given as an argument to the function. Linear interpolation is applied during the backprojection and a Ram–Lak or ramp filter is used. The FBP reconstruction of the chest phantom is shown in Figure \[fig:ChestPhantomRec\](b). For comparison, FBP reconstructions computed using some other filters are seen in Figure \[fig:FBPs\].
------------ ----------------- -------------- ---------------
Covariance $\sigma_f$ (SD) $l$ (SD) $\sigma$ (SD)
function
SE 0.12 (0.04) 5.03 (0.03) 0.60 (0.02)
Matérn 0.12 (0.07) 10.14 (0.08) 0.34 (0.03)
Laplacian 0.05 (0.10) 4.49 (0.02) 0.39 (0.03)
Tikhonov 0.64 (0.02) - 0.35 (0.03)
------------ ----------------- -------------- ---------------
: The GP parameter estimates for the chest phantom. The estimates are calculated using the conditional mean, and the standard deviation (SD) values are also reported in parentheses.[]{data-label="GP parameters"}
(100,600) (52,415)[![Histogram of the 1-d marginal distribution of the GP parameters. Left, middle and right columns are the marginal distribution for parameter $\sigma_f$, $l$ and $\sigma$ with corresponding covariance functions indicated in the vertical text in the le
| 1,231
| 4,280
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| 1,073
| null | null |
github_plus_top10pct_by_avg
|
y order in a semi-classical expansion. We will show that the knowledge of the poles in these OPEs is enough to fix all the subleading terms. The idea driving the bootstrap is to ask for the compatibility of the elementary OPEs with both current conservation and the Maurer-Cartan equation.
Current-current OPEs {#current-current-opes .unnumbered}
--------------------
First let us consider the current-current OPEs. Current conservation gives the first constraints: j\^a\_[L,z]{}(z) = 0 j\^a\_[L,|z]{}(z) = 0. The first line implies a one-to-one correspondence between the terms in the $j^a_{L,z} j^b_{L,z}$ and $j^a_{L,z} j^b_{L,\bar z}$ OPEs. The second line then links the $j^a_{L,\bar z}
j^b_{L,\bar z}$ and the $j^a_{L,z} j^b_{L,\bar z}$ OPEs. These OPEs are expected to vanish up to contact terms. Indeed the same OPEs code the Ward identity for the global symmetry $G_L$. It follows that the contact terms in these OPEs are given by the transformation properties of the left current under the left action of the group on itself [^4].
The second constraint comes from the Maurer-Cartan equation : j\^a\_[L,z]{}(z) = 0. Contact terms in this OPE should vanish. Using current conservation and the fact that $c_++c_-=-f^{-2}$ we rewrite this constraint as : \[jmodMC\] j\^a\_[L,z]{}(z) |j\^b\_[L,z]{}(w) = f\^2 j\^a\_[L,z]{}(z) i [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w). Thanks to the factor of $f^2$ on the right-hand side of the previous equation, it becomes manifest that the knowledge of the current algebra at a given order in $f^2$ will also determine the current algebra at the next order. The discussion of appendix \[XXOPEs\] shows that the terms appearing in the current-current OPEs at order $f^{2n}$ are composites of at most $n+1$ currents. This allows us to make an ansatz for the current-current OPE at higher-order. Then equation fixes the coefficients in this ansatz. This method is illustrated in appendix \[jMCOPE\] where we compute the current algebra up to order $f^2$.
Current-primary OPEs {#current-primary-o
| 1,232
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nd the periodicity of the KK circle at infinity $\chi\sim\chi+L$ will be determined by smoothness of the metric at $\rho=\rho_0$.
We now add electric 3-form flux to the bubbles, $C=\frac{Q_0}{2\pi^2}(\star \epsilon_3)$, where $\epsilon_3$ is the volume element of the spatial $S^3$. Concretely, the field strength is $$\begin{aligned}
C_{\rho t\chi}=\frac{NQ_0}{2\pi^2 \sqrt{h}\rho^3}\;,
\label{eq:cN}\end{aligned}$$ where $N$ is the lapse function. The Hamiltonian constraint is then[^3] $$\begin{aligned}
{}^5R=\frac{Q^2}{\rho^6}\;,\end{aligned}$$ where $Q\equiv Q_0/(2\pi^2M_6^2)$ and $M_6=(8\pi G)^{-1/4}$ is the 6D Planck scale. We find $$\begin{aligned}
h(\rho)\equiv1+\frac{b}{3\rho^2-2\rho_0^2}-\frac{Q^2}{4\rho_0^2\rho^2}
\label{eq:h}\end{aligned}$$ where $b$ is an arbitrary constant.
To make the geometry smooth everywhere, we impose periodicity $\chi\sim\chi+L$ on the KK circle, where $$\begin{aligned}
L=\frac{2\pi \rho_0}{\left(1+\frac{b}{\rho_0^2}-\frac{Q^2}{4\rho_0^4}\right)^{1/2}}\;.
\label{eq:b}\end{aligned}$$ With this periodicity, space ends on a smooth cap at $\rho=\rho_0$.
![ The energy (Eq. (\[eq:Mbubble\])) of a one-parameter family of initial data labeled by bubble radius $\rho_0$. From bottom to top, curves correspond to $Q=0,\dots,Q=Q_{max}$. The most stable bubbles lie at small $Q$; for $Q>Q_{\max}$, no stable solution exists. []{data-label="fig:Mrho0"}](Mrho0.pdf){width="0.5\linewidth"}
Thus far we have a family of charged bubble initial data. For a given charge $Q$ and asymptotic circle size $L$, it is a one-parameter family of bubbles labeled by the radius $\rho_0$. The energy of the family is $$\begin{aligned}
M=\pi^2 LM_6^4\left(\frac{Q^2}{2\rho_0^2}+2\rho_0^2-\frac{4\pi^2\rho_0^4}{L^2}\right)\;.
\label{eq:Mbubble}\end{aligned}$$ There is a stable local minimum of the energy $M(\rho_0)$ for all $Q$, $L$ such that $$\begin{aligned}
Q<Q_{max}=\frac{L^2}{3\pi^2\sqrt{3}}\;.
\label{eq:stab}\end{aligned}$$ There is also an unstable maximum at larger $\rho_0$. For $Q>Q_{max}$, there are no stati
| 1,233
| 751
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| 1,419
| 3,583
| 0.77139
|
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u_i&1+\pi w_i \end{pmatrix}\in \mathrm{GL}_{n_i}(B\otimes_AR),$$ where $s_i$ is an $(n_i-2) \times (n_i-2)-$matrix, etc.\
3. Let $i=2m$ be even. Then $g$ stabilizes $Z_i$ and induces the identity on $W_i/(X_i\cap Z_i)$. For the proof, it is easy to show that $g$ stabilizes $Z_i$. To prove the latter, we choose an element $w$ in $W_i$. Since $gw-w\in X_i$ by the above step (2), it suffices to show that $gw-w\in Z_i$. Recall that $Z_i$ is the sublattice of $A_i$ such that $Z_i/\pi B_i$ is the radical of the quadratic form $\frac{1}{2^{m+1}}q$ mod $2$ on $B_i/\pi B_i$, where $\frac{1}{2^{m+1}}q(a)=\frac{1}{2^{m+1}}h(a,a)$ for $a\in B_i$. The lattice $Z_i$ can also be described as the sublattice of $B_i$ such that $Z_i/\pi B_i$ is the kernel of the additive polynomial $\frac{1}{2^{m+1}}q$ mod $2$ on $Y_i/\pi B_i$. Now it is also easy to show that $gw-w\in Y_i$. Thus our claim that $gw-w\in Z_i$ follows from the following computation:
$$\frac{1}{2}\cdot \frac{1}{2^m} q(gw-w)=\frac{1}{2}(2\cdot \frac{1}{2^m}q(w)- \frac{1}{2^m}(h(gw,w)+h(w,gw)))=$$ $$\frac{1}{2^m}(q(w)-\frac{1}{2}(h(w+x,w)+h(w,w+x))) =-\frac{1}{2^m}\cdot\frac{1}{2}(h(x,w)+h(w,x))=0 \textit{ mod } 2,$$ where $gw=w+x$ for some $x\in X_i$, since $\frac{1}{2^m}h(x,w)\in \pi B$ and $\pi + \sigma(\pi) \in 4A$. Recall that $\frac{1}{2^m}q(w)=\frac{1}{2^m}h(w,w)$.
We interpret this in terms of matrices.
- If $L_i$ is *free of type II*, then $W_i=X_i=Z_i$ and so there is no contribution.
- If $L_i$ is *bound of type II*, then $W_i=X_i$ and $Z_i$ is as explained in Remark 2.11 of [@C2]. Matrix interpretation in this case is covered by matrix interpretation of Steps (2) and (4) below. Thus there is no new contribution.
- If $L_i$ is *of type I*, then $$z_i+\delta_{i-2}k_{i-2, i}+\delta_{i+2}k_{i+2, i} \in (\pi).$$ Here,
- $z_i$ is an entry of $g_{i,i}$ as described in *Step (2)*.
- $k_{i-2, i}$ (resp. $k_{i+2, i}$) is the $(n_{i-2}, n_i)^{th}$-entry (resp. $(n_{i+2}, n_i)^{th}$-entry) of the matrix $g_{i-2, i}
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u i}_\alpha(u) \right)}$ are $m\times q$ matrix functions of $u$ and we denote the partial derivatives by $u^\alpha_i={\partial}u^\alpha/{\partial}x^i$. The matrix $L_i^\alpha$ satisfying the conditions[@Burnat:1972]
\[eq:intelem\] u\_i\^[{ L\_i\^:\_\^[i]{}L\_i\^=0, =1,…,q }]{}
at some open given point $u_0\in U$ is called an integral element of the system (\[eq:2.1\]). This matrix $L={\left( {{\partial}u^\alpha}/{{\partial}x^i} \right)}$ is a matrix of the tangent mapping $du:X{\rightarrow}T_uU$ given by the formula $$X\ni (\delta x^i){\rightarrow}(\delta u^\alpha)\in T_uU,\quad \text{where }\delta u^\alpha=u_i^\alpha \delta x^i.$$The tangent mapping $du(x)$ determines an element of linear space $L(X,T_uU)$, which can be identified with the tensor product $T_uU\otimes X^\ast$, (where $X^\ast$ is the dual space of $X$, [*i.e.* ]{}the space of linear forms). It is well known [@Burnat:1972; @Grundland:1974; @Perad:1985] that each element of this tensor product can be represented as a finite sum of simple tensors of the form $$L=\gamma\otimes \lambda,$$where $\lambda\in X^\ast$ is a covector and $\gamma=\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\in
T_uU$ is a tangent vector at the point $u\in U$. Hence, the integral element $L_i^\alpha$ is called a simple element if ${\operatorname{rank}}(L_i^\alpha)=1.$ To determine a simple integral element $L_i^\alpha$ we have to find a vector field $\gamma\in T_uU$ and a covector $\lambda\in X^\ast$ satisfying the so-called wave relation
\[eq:2.2\] [( \_i\^[i]{}\_(u) )]{}\^=0,=1,…,m.
The necessary and sufficient condition for the existence of a nonzero solution $\gamma$ for the equation (\[eq:2.2\]) is
\[eq:2.3\] [( \_i\^i(u) )]{}<(m,q).
This relation is known as the dispersion relation. If the covector $\lambda=\lambda_i(u)dx^i$ satisfies the dispersion relation (\[eq:2.3\]) then there exists a polarization vector $\gamma\in
T_uU$ satisfying the wave relation (\[eq:2.2\]). The algebraic approach which has been used in [@Burnat:1972; @Grundland:1974; @Pera
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d B.-J. Schaefer for discussions. We thank O. Jahn for discussions and collaboration at an early state of this project. FM acknowledges financial support from the state of Baden-Württemberg and the Heidelberg Graduate School of Fundamental Physics.
Faddeev-Popov determinant {#app:FPdet}
=========================
From the gauge fixing functionals and we can compute the Faddeev-Popov determinant given by $$\begin{aligned}
\Delta_{FP}[A] = \mathrm{det}\left[\frac{
\delta F^a(A^\omega) }{ \delta \omega^b } \right]\,,\end{aligned}$$ where $A^\omega$ is the gauge transformed gauge field $A$. For infinitesimal gauge transformations it is given by $$\begin{aligned}
A_\mu^\omega &=& A_\mu - (\partial_\mu \sigma^a
+ i g A_\mu^b [\sigma^a, \sigma^b ]) \omega^a\,.\end{aligned}$$ In the following we use the representation $\omega^a \sigma^a =
\omega^+ \sigma^- + \omega^- \sigma^+ + \omega^3 \sigma^3 $, and the related derivatives w.r.t. $\omega^{\pm},\omega^3$. The matrix elements related to $\omega$-derivatives of $F^+$ read $$\begin{aligned}
\nonumber
\frac{\delta F^+ (A^\omega)}{ \delta \omega^+ }
&=& - {\mathrm{Tr}}\,\sigma^+ \left(
\partial_0 \sigma^- + i A_0^3 [\sigma^-, \sigma^3]
\right) \,,\\\nonumber
\frac{\delta F^+ (A^\omega)}{ \delta \omega^- }
&=&0\,,\\
\frac{\delta F^+ (A^\omega)}{ \delta \omega^3 }
&=& -{\mathrm{Tr}}\,\sigma^+ \left( \partial_0
\sigma^3 + i A_0^+ [\sigma^3, \sigma^-] \right)\,. \hspace{.5cm}
\label{eq:coef+}\end{aligned}$$ Analogously we get for the $\omega$-derivatives of $F^-$ $$\begin{aligned}
\nonumber
\frac{\delta F^- (A^\omega)}{ \delta \omega^+ }
&=&0\,,\\\nonumber
\frac{\delta F^- (A^\omega)}{ \delta \omega^- }
&=& -{\mathrm{Tr}}\,\sigma^- \left( \partial_0 \sigma^+
+ i A_0^3 [\sigma^+, \sigma^3] \right)\,,\\
\frac{\delta F^- (A^\omega)}{ \delta \omega^3 }
&=& -{\mathrm{Tr}}\,\sigma^- \left( \partial_0
\sigma^3 + i A_0^- [\sigma^3, \sigma^+] \right)\,.
\label{eq:coef-}\end{aligned}$$ The $\omega$-derivatives of $F^3$ yi
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ssion. The application of an integer constructor labels the free variable as an integer variable. An integer condition, e.g. $\geq/2$, is applicable if both arguments are integer expressions. Since integer variables denote unknown integers, integer expressions are allowed to contain integer variables. Applications of integer constructors and integer conditions in the moded SLD-tree are denoted by derivation steps $N_i:G_i\Longrightarrow_{cons} N_{i+1}:G_{i+1}$ and $N_i:G_i\Longrightarrow_{cond} N_{i+1}:G_{i+1}$, respectively.
\[example:count\_to\] The following program, $count\_to$, is a faulty implementation of a predicate generating the list starting from 0 up to a given number. The considered class of queries is represented by the moded query $\leftarrow count\_to(\underline{N},L)$ with $\underline{N}$ an integer variable.
count_to(N,L):- count(0,N,L). count(N,N,[N]).
count(M,N,[M|L]):- M > N, M1 is M+1, count(M1,N,L).
In the last clause, the integer condition should be `M < N` instead of `M > N`. Due to this error, the program:
- fails for the queries for which $\underline{N}>0$ holds,
- succeeds for $\leftarrow count\_to(0,L)$,
- loops for the queries for which $\underline{N} < 0$ holds.
![Moded SLD-tree $count\_to$[]{data-label="fig:count_to"}](figs/count_to.pdf){width="60ex"}
Figure \[fig:count\_to\] shows the moded SLD-tree for the considered query, constructed using LP-check. LP-check cuts clause 3 at node $N_9$. $\hfill \square$
Note that by ignoring the possible values for the integer variables when constructing the tree, some derivations in it may not be applicable to any considered query. For example the refutations at nodes $N_6$ and $N_{10}$ in the previous example cannot be reached by the considered queries.
Adapting the non-termination condition
--------------------------------------
In [@DBLP:conf/iclp/VoetsS09], programs are shown to be non-terminating for a moded query, by proving that a path in the moded SLD-tree can be repeated infinitely often. Such a path, fr
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}_{1}}+{{\theta}_{2}}), \\
& {{J}_{12}}=-{{L}_{1}}\cos {{\theta}_{1}}-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{13}}={{J}_{14}}=0, \\
& {{J}_{21}}=-{{L}_{2}}\sin ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{22}}=-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{23}}={{J}_{24}}=0, \\
& {{J}_{31}}={{J}_{32}}=0, \\
& {{J}_{33}}={{L}_{3}}\sin {{\theta}_{3}}+{{L}_{4}}\sin ({{\theta}_{3}}+{{\theta}_{4}}), \\
& {{J}_{34}}=-{{L}_{3}}\cos {{\theta}_{3}}-{{L}_{4}}\cos ({{\theta}_{3}}+{{\theta}_{4}}), \\
& {{J}_{41}}={{J}_{42}}=0, \\
& {{J}_{43}}={{L}_{4}}\sin ({{\theta}_{3}}+{{\theta}_{4}}), \\
& {{J}_{44}}=-{{L}_{4}}\cos ({{\theta}_{3}}+{{\theta}_{4}}).
\end{array}$$
Control Approach {#sec_3}
================
In order to design a control law to efficiently and automatically adjust the robot manipulators, we first discuss a control scheme based on the dynamic surface control method. It is noted that due to the system uncertainties including parameter variations, actuator nonlinearities and external disturbances, system parameters in the designed controller are practically uncertain and unknown; then we introduce a radial basis function network based technique to adaptively estimate those uncertain and unknown dynamics.
Generally speaking, the dynamic model of the dual arm robot (DAR) (\[eq12\]) can be represented as follows, $$\label{eq14}
\begin{array}{r@{}l@{\qquad}l}
& {{{{\dot{x}}}}_{1}}={{{{x}}}_{2}} \\
& {{{{\dot{x}}}}_{2}}={{M}^{-1}}(\theta )u+{{M}^{-1}}(\theta )[{{J}^{T}}\left( \theta \right)F(\theta ,\dot{\theta },\ddot{\theta })-T_d-\beta-C(\theta ,\dot{\theta })],
\end{array}$$ where ${{{x}}_{1}}={{({{\theta }_{1}},{{\theta }_{2}},{{\theta }_{3}},{{\theta }_{4}})}^{T}}$ and ${{{x}}_{2}}={{({{\dot{\theta }}_{1}},{{\dot{\theta }}_{2}},{{\dot{\theta }}_{3}},{{\dot{\theta }}_{4}})}^{T}}$. Let $K(\theta,\dot{\theta},\ddot{\theta})={{J}^{T}}(\theta)F(\theta,\dot{\theta},\ddot{\theta})-C(\theta,\dot{\theta})-G(\theta)-\beta -{{
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{\ast}=\underline{M}$. As a matrix, each element of $\underline{M}^{\ast}(R)$ for a flat $A$-algebra $R$ can be written as $\begin{pmatrix} 1+2 z \end{pmatrix}$.
We define another functor from the category of commutative flat $A$-algebras to the category of sets as follows. For any commutative flat $A$-algebra $R$, let $\underline{H}(R)$ be the set of hermitian forms $f$ on $L\otimes_{A}R$ (with values in $B\otimes_AR$) such that $f(a,a)$ mod 4 = $h(a, a)$ mod 4, where $a \in L \otimes_{A}R$. As a matrix, each element of $\underline{H}(R)$ for a flat $A$-algebra $R$ is $\begin{pmatrix} 1+4 c \end{pmatrix}$.
Then for any flat $A$-algebra $R$, the group $\underline{M}^{\ast}(R)$ acts on the right of $\underline{H}(R)$ by $f\circ m = \sigma({}^tm)\cdot f\cdot m$ and this action is represented by an action morphism (Theorem \[t34\]) $$\underline{H} \times \underline{M}^{\ast} \longrightarrow \underline{H} .$$ Let $\rho$ be the morphism $\underline{M}^{\ast} \rightarrow \underline{H}$ defined by $\rho(m)=h \circ m$, which is obtained from the above action morphism. As a matrix, for a flat $A$-algebra $R$, $$\rho(m)=\rho(\begin{pmatrix} 1+2 z \end{pmatrix})
=\begin{pmatrix} 1+2 z+\sigma(2 z)+4\cdot z\sigma(z) \end{pmatrix}.$$
Then $\rho$ is smooth of relative dimension 1 (Theorem \[t36\]). Let $\underline{G}$ be the stabilizer of $h$ in $\underline{M}^{\ast}$. The group scheme $\underline{G}$ is smooth, and $\underline{G}(R)=\mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$ for any étale $A$-algebra $R$ (Theorem \[t38\]).\
We now describe the structure of the special fiber $\tilde{G}$ of $\underline{G}$. For a $\kappa$-algebra $R$, each element of $\underline{M}(R)$ (resp. $\underline{H}(R)$) can be written as a formal matrix $m=\begin{pmatrix} 1+2 z \end{pmatrix}$ (resp. $f=\begin{pmatrix} 1+4 c \end{pmatrix}$). Firstly, it is obvious that the morphism $\varphi$ in Section \[red\] is trivial since the dimension of $B(L)/Z(L)=\pi L/2L$ is $1$ as a $\kappa$-vector space so that the associated reduced ortho
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imal categories in the Pascal VOC Part dataset. For the ILSVRC 2013 DET Animal-Part dataset and the CUB200-2011 dataset, we learned an AOG for the head part[^3] of each category. It is because all categories in the two datasets contain the head part. We did not train human annotators. Shape differences between two part templates were often very vague, so that an annotator could assign a part to either part template.
{width="\linewidth"}
{width="\linewidth"}
![Image patches corresponding to different latent patterns.[]{data-label="fig:patches"}](patches.pdf){width="0.8\linewidth"}
Table \[tab:stat\] shows how the AOG grew when people annotated more parts during the QA process. Given AOGs learned for the PASCAL VOC Part dataset, we computed the average number of children for each node in different AOG layers. The AOG mainly grew by adding new branches to represent new part templates. The refinement of an existing AOG branch did not significantly change the node number of the AOG.
Fig. \[fig:energyCurve\] analyzes activation states of latent patterns in AOGs that were learned with different numbers of part annotations. Given a testing image $I$ for part parsing, we only focused on the inferred latent patterns and neural units, *i.e.* latent patterns and their inferred neural units under the selected part template. Let ${\bf V}$ and ${\bf V'}\subset{\bf V}$ denote all units in a specific conv-layer and the inferred units, respectively. $a_{v}$ denotes the activation score of $v\in{\bf V}$ after the ReLU operation. $a_{v}$ is also normalized by the average activation level of $v$’s corresponding feature maps *w.r.t.* different images. Thus, in Fig. \[fig:energyCurve\](left), we computed the ratio of the inferred activation energy as $\frac{\sum_{v\in{\bf V'}}a_{v}}{\sum_{v\in{\bf V}}a_{v}}$. For each inferred latent pattern $u$, $a_{u}$ denotes the activation score of its selected neural unit[^4]. Fig. \[fig:energyCurve\](middle) measures the relative magni
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phic to $\PP^2_w$ and the isomorphism is induced by the branched covering $$\label{covering}
\PP^2\ni [X_0:X_1:X_2] \overset{\phi}{\longmapsto} [X_0^{w_0}:X_1^{w_1}:X_2^{w_2}]_w \in\PP^2_w.$$
Note that this branched covering is unramified over $$\PP^2_{w} \setminus \{ [X_0,X_1,X_2]_{w} \mid X_0\cdot X_1\cdot X_2 = 0 \}$$ and has $\bar{w}=w_0w_1w_2$ sheets. Moreover, one can decompose $\PP^2_w = U_0 \cup U_1 \cup U_2$, where $U_i$ is the open set consisting of all elements $[X_0:X_1:X_2]_w$ with $X_i\neq 0$. The map $$\widetilde{\psi}_2: \CC^2 \longrightarrow U_2,\quad
\widetilde{\psi}_2(x_0,x_1):= [x_0:x_1:1]_w$$ induces an isomorphism $\psi_2$ replacing $\CC^2$ by $\frac{1}{w_2}(\w_0,\w_1)$. Analogously, $\frac{1}{w_0}(\w_1,\w_2) \cong U_0$ and $\frac{1}{w_1}(w_2,\w_0) \cong U_1$ under the obvious analytic maps. This shows that $\PP^2_w$ is a cyclic $V$-manifold.
The following result justifies why the weights can be assumed to be pairwise coprime.
\[propPw\] Let $d_0 := \gcd (w_1,w_2)$, $d_1 := \gcd (w_0,w_2)$ , $d_2 := \gcd (w_0,w_1)$, $e_0:= d_1\cdot d_2$, $e_1:= d_0\cdot d_1$, $e_2:= d_0\cdot d_1$ and $p_i:=\frac{w_i}{e_i}$. The following map is an isomorphism: $$\begin{array}{rcl}
\PP^2_{(w_0,w_1,w_2)} & \longrightarrow & \PP^2_{(p_0,p_1,p_2)}, \\[0.15cm]
\,[X_0:X_1:X_2] & \mapsto &
\big[\,X_0^{d_0}:X_1^{d_1}:X_2^{d_2}\,\big].
\end{array}$$
Note that, due to the preceding proposition, one can always assume the weight vector satisfies that $(w_0,w_1,w_2)$ are pairwise relatively prime numbers. Note that following a similar argument, $\PP^1_{(w_0,w_1)} \cong \PP^1$.
Embedded Q-resolutions {#sec:Qres}
----------------------
Classically an embedded resolution of $\{f=0\} \subset (\CC^n,0)$ is a proper map $\pi: X \to (\CC^n,0)$ from a smooth variety $X$ satisfying, among other conditions, that $\pi^{-1}(\{f=0\})$ is a normal crossing divisor. For a weaker notion of resolution one can relax the condition on the preimage of the singularity by allowing the ambient space $X$ to be a projective $V$-manifold
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\[r31\]. Thus $\tilde{T}$ is an element of $\underline{M}_j^{\ast}(R)$, where $\underline{M}_j^{\ast}$ is the group scheme in Section \[m\] associated to $M_0^{\prime}\oplus C(L^j)$ so that $\underline{G}_j'$ is defined as the closed subgroup scheme of $\underline{M}_j^{\ast}$ stabilizing the hermitian form on $M_0^{\prime}\oplus C(L^j)$.
In conclusion, $\tilde{T}$ is an element of $\underline{M}_j^{\ast}(R)$ preserving the hermitian form on $M_0^{\prime}\oplus C(L^j)$. Therefore, it is an element of $\underline{G}_j'(R)$.
To summarize, if $R$ is a nonflat $A$-algebra, then we can write an element of $\underline{G}_j(R)$ formally as $m= \begin{pmatrix} 1+2 z_0^{\ast} &m_1\\m_2&m_3 \end{pmatrix}$. Then the image of $m$ in $\underline{G}_j'(R)$ is $\tilde{T}$ with respect to a basis as explained above.
**
\(3) We choose any odd integer $j$ such that $L_{j}$ is *of type I* and $L_{j-1}, L_{j+1}, L_{j+2}, L_{j+3}$ are *of type II* (possibly zero, by our convention). Note that the lattice $L_{j}$ is then *free of type I*. Consider the Jordan splitting $$L^{j-1}=\bigoplus_{i \geq 0} M_i,$$ where $$M_0=\pi^{(j-1)/2}L_0\oplus\pi^{(j-1)/2-1}L_2\oplus \cdots \oplus \pi L_{j-3}\oplus L_{j-1},$$ $$M_1=\pi^{(j-1)/2}L_1\oplus\pi^{(j-1)/2-1}L_3\oplus \cdots \oplus \pi L_{j-2}\oplus L_{j}$$ $$\mathrm{and}~ M_k=L_{j-1+k} \mathrm{~if~} k\geq 2.$$ We stress that $$\left\{
\begin{array}{l }
\textit{$M_1$ is nonzero and \textit{of type I}, since it contains $L_{j}$ (of type I) as a direct summand};\\
\textit{All of $M_2 (=L_{j+1}), M_3 (=L_{j+2}), M_4 (=L_{j+3}$) are \textit{of type II}.}
\end{array} \right.$$
We now follow the arguments of the above two cases.
1. If $M_0$ is *of type $I^e$*, then we follow the argument (1) with $j-1$. We briefly summarize it below. Since $M_0$ is *of type $I$* and $n(M_1)=(2)$, we choose another basis for $M_0\oplus M_1$ whose associate Jordan splitting is $M_0'\oplus M_1'$ with $n(M_1')=(4)$. Consider the lattice $Y(C(L^{j-1}))=\bigoplus_{i \geq 0} M_i^{\prime\prime}$. The
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\hat{b}^{il}_{-}
( u - \frac{1}{2} (k+l) \hbar )
- \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-}
( u- \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\
& & ~~~~- :\mbox{exp} \left( (b+c)^{i,i+1}
( u + \frac{1}{2} (k+i) \hbar ) \right) \nonumber\\
& & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_+ (u)
+ \sum_{l=i+1}^N \hat{b}^{il}_{+}
( u + \frac{1}{2} (k+l) \hbar )
- \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-}
( u+ \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\
& &~~~~- \sum_{m=i+2}^N :\mbox{exp} \left( (b+c)^{im}
( u + \frac{1}{2} (k+m-1) \hbar ) \right) \nonumber\\
& & ~~~~~~~~\times \left[
\mbox{exp} \left( \hat{b}^{i+1,m}_+ ( u + \frac{1}{2} (k+m-1) \hbar )
-(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m) \hbar ) \right) \right. \nonumber\\
& &~~~~~~~~~~~~ - \left. \mbox{exp} \left( \hat{b}^{i+1,m}_-
( u + \frac{1}{2} (k+m-1) \hbar )
-(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m-2) \hbar ) \right) \right] \nonumber\\
& &~~~~~~~~\times \left.
\mbox{exp} \left( \sum_{l=m}^{N} \left[
\hat{b}^{il}_+ ( u + \frac{1}{2} (k+l) \hbar )
-\hat{b}^{i+1,l}_+ ( u + \frac{1}{2} (k+l-1) \hbar ) \right]
\right) : \right\}~. \label{en}\end{aligned}$$
The following proposition is the main result of this paper:
\[prop1\] The fields $H^{\pm}_i(u)$, $E^{\pm}_i(u)$ defined in equations (\[hpn\]), (\[ep\]) and (\[en\]) are well-defined on the Fock space ${\cal F}(l_a,l_b,l_c)$ and satisfy equations (\[y1\]-\[y4\]) with $c=k$ and
$$\begin{aligned}
& & E^{\pm}_i(u) E^{\pm}_j(v) \simeq
E^{\pm}_j(v) E^{\pm}_i(u) \sim reg. ~~\mbox{for}~B_{ij}=0,\\
& & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v) \simeq
(u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u)
\sim reg. ~~\mbox{for}~B_{ij} \neq 0,\\
& & E^{+}_i(u) E^{-}_j(v) - E^{-}_j(v) E^{+}_i(u) \\
& &~~~~ \sim reg. + \frac{1}{\hbar} \left(
\delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right),\end{aligned}$$
where $reg.$ means some regular expressions and $\simeq$ and $\sim$ imply “equals up to” such expressions.
[*Proof*]{}: The proposition follow by straightforward but tedious calculations. Actuall
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=3$ system, i.e., equation (\[eq:Bog\]). Other systems with similar behaviour have been presented in [@BW].
Acknowledgements {#acknowledgements .unnumbered}
----------------
PX acknowledges support from the EPSRC grant [*Structure of partial difference equations with continuous symmetries and conservation laws*]{}, EP/I038675/1.
[99]{} =-1pt
Bogoyavlensky O.I., Integrable discretizations of the [K]{}d[V]{} equation, [*Phys. Lett. A*](https://doi.org/10.1016/0375-9601(88)90542-7) **134** (1988), 34–38.
Fordy A.P., Xenitidis P., [$\mathbb{Z}_N$]{} graded discrete [Lax]{} pairs and discrete integrable systems, [arXiv:1411.6059](https://arxiv.org/abs/1411.6059).
Fordy A.P., Xenitidis P., [${\mathbb Z}_N$]{} graded discrete [L]{}ax pairs and integrable difference equations, [*J. Phys. A: Math. Theor.*](https://doi.org/10.1088/1751-8121/aa639a) **50** (2017), 165205, 30 pages.
Marì Beffa G., Wang J.P., Hamiltonian evolutions of twisted polygons in [${\mathbb{RP}}^n$]{}, [*Nonlinearity*](https://doi.org/10.1088/0951-7715/26/9/2515) **26** (2013), 2515–2551, [arXiv:1207.6524](https://arxiv.org/abs/1207.6524).
Mikhailov A.V., Xenitidis P., Second order integrability conditions for difference equations: an integrable equation, [*Lett. Math. Phys.*](https://doi.org/10.1007/s11005-013-0668-8) **104** (2014), 431–450, [arXiv:1305.4347](https://arxiv.org/abs/1305.4347).
Yamilov R., Symmetries as integrability criteria for differential difference equations, [*J. Phys. A: Math. Gen.*](https://doi.org/10.1088/0305-4470/39/45/R01) **39** (2006), R541–R623.
---
abstract: 'Population transfer between two identical, communicating defects in a one-dimensional tight-binding lattice can be systematically controlled by external time-periodic forcing. Employing a force with slowly changing amplitude, the time it takes to transfer a particle from one defect to the other can be altered over several orders of magnitude. An analytical expression is derived which shows how the forcing effectively changes the energy
| 1,244
| 50
| 2,211
| 1,437
| 2,455
| 0.779594
|
github_plus_top10pct_by_avg
|
P51884 Lumican 26.62 35.50 38.4 6.61
P01876 Immunoglobulin heavy constant alpha 1 25.67 29.18 37.6 6.51
Q08380 Galectin-3-binding protein 25.15 17.78 65.3 5.27
P67936 Tropomyosin alpha-4 chain 25.00 44.35 28.5 4.69
P01859 Immunoglobulin heavy constant gamma 2 17.12 25.77 35.9 7.59
P07225 Vitamin K-dependent protein S 16.75 9.62 75.1 5.67
P35443 Thrombospondin-4 16.66 5.52 105.8 4.68
P12259 Coagulation factor V 10.18 1.98 251.5 6.05
P04070 Vitamin K-dependent protein C 7.36 5.86 52.0 6.28
P16070 CD44 antigen 5.64 2.70 81.5 5.33
P13591 Neural cell adhesion molecule 1 5.50 1.75 94.5 4.87
P22105 Tenascin-X 4.42 2.33 458.1 5.17
Q99436 Proteasome subunit beta type-7 4.03 8.30 29.9 7.68
P12814 Alpha-actinin-1 3.68 2.35 103.0 5.41
P27348 14-3-3 protein theta 3.53 6.53 27.7 4.78
P07900 Heat shock protein HSP 90-alpha 1.73 3.14 84.6 5.02
biomedicines-08-00069-t0A5_Table A5
######
LCMS analysis of FIX-PCC permeate following two-step filtration with 11 μm/33 μm filters.
Accession Description Score Coverage MW \[kDa\] calc. pI
----------- ---------------------------------------------- --------- ---------- ------------ ----------
P0C0L5 Complement C4-B 2266.18 76.26 192.6 7.27
| 1,245
| 4,474
| 659
| 767
| null | null |
github_plus_top10pct_by_avg
|
a trial‐specific adjustment term for the baseline outcome value (here, centered at the mean for each trial ( ${\bar{Y}}_{\mathit{Bi}}$) to aid interpretation of the trial‐specific intercepts). For example, when there are *K* = 10 trials, there would be 10 *β* ~*i*~ terms and 10 *λ* ~*i*~ terms. Of main interest is an estimate of the model parameter θ, as this denotes the summary (average) treatment effect. The random effect, , indicates that the true treatment effects in each trial are assumed to arise from a distribution of true effects with mean and between‐trial variance *τ* ^*2*^. This assumption could be constrained if considered appropriate, with a common (fixed) treatment effect (ie, constrain *τ* ^*2*^ = 0). Lastly, $\sigma_{i}^{2}$ denotes a distinct residual variance per trial.
The flexibility of the one‐stage IPD approach allows us to make further modifications by considering, for example, a common baseline adjustment term (ie, *λ*) across trials, or common residual variances (ie, $\sigma_{i}^{2} =$ *σ* ^*2*^) if necessary([5](#sim7930-bib-0005){ref-type="ref"}, [10](#sim7930-bib-0010){ref-type="ref"}, [11](#sim7930-bib-0011){ref-type="ref"}); however, this should be justified (eg, based on computational reasons or estimation problems), and sensitivity analysis to the choice of assumptions is often sensible.
2.2. Model (2): random intercept {#sim7930-sec-0004}
--------------------------------
When there are a large number of trials to be synthesized, a stratified intercept approach to clustering can be computationally intensive (as Equation [(1)](#sim7930-disp-0001){ref-type="disp-formula"} requires estimation of 3* K + *2 parameters).[4](#sim7930-bib-0004){ref-type="ref"} An alternative approach for dealing with clustering, which is preferred by some researchers,[12](#sim7930-bib-0012){ref-type="ref"} is to use a random intercept term. $$\begin{matrix}
Y_{\mathit{Fij}} & {= \left( {\beta + u_{1i}} \right) + \lambda_{i}\left( {Y_{\mathit{Bij}} - {\bar{Y}}_{\mathit{Bi}}} \ri
| 1,246
| 766
| 2,329
| 1,231
| 344
| 0.812502
|
github_plus_top10pct_by_avg
|
+ \int_0^t -\nabla U(w_0) ds + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^t} dV_s + \int_0^T N(x_s) dW_s
\end{aligned}$$
Where $\gamma_t := \frac{x_t - y_t}{\|x_t-y_t\|_2} \cdot \ind{\|x_t-y_t\|_2 \in [2\epsilon, \Rq)}$. The coupling $(x_t,y_t)$ defined in and is identical to the coupling in (with $y_0 = w_0$).
4. We now define a process $v_{k\delta}$ for $k=0...n$: $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_v}
v_{k\delta} =& w_0 + \sum_{i=0}^{k-1} - \delta \nabla U(w_0) + {\sqrt{\delta}} \sum_{i=0}^{k-1} \xi(w_0,\eta_i)
\end{aligned}$$ where marginally, the variables $\lrp{\eta_0...\eta_{n-1}}$ are drawn $i.i.d$ from the same distribution as in .
Notice that $y_T - w_0 - T \nabla U(w_0) = \int_0^T \cm dB_t + \int_0^T N(w_0) dW_t$, so that $\Law(y_T - w_0 - T \nabla U(w_0)) = \N(0, T M(w_0)^2)$. Notice also that $v_T - w_0 - T\nabla U(w_0) = \sqrt{\delta} \sum_{i=0}^{n-1} \xi(w_0, \eta_i)$. By Corollary \[c:clt\_sum\], $W_2(y_T - w_0 - T \nabla U(w_0), v_T - w_0 - T \nabla U(w_0)) = 6 \sqrt{d\delta}\beta \sqrt{\log n}$. Let the joint distribution between and be the one induced by the optimal coupling between $y_T - w_0 - T \nabla U(w_0)$ and $v_T - w_0 - T \nabla U(w_0)$, so that $$\begin{aligned}
& \sqrt{\E{\lrn{y_T - v_T}_2^2}} \\
=& \sqrt{\E{\lrn{y_T - T \nabla U(w_0) - v_T + T \nabla U(w_0)}_2^2}} \\
=& W_2(y_T - w_0 - T\nabla U(w_0), v_T - w_0 - T\nabla U(w_0)) \\
\leq& 6 \sqrt{d\delta}\beta \sqrt{\log n}
\numberthis \label{e:yt-vt}
\end{aligned}$$ where the last inequality is by Corollary \[c:clt\_sum\].
5. Given the sequence $\lrp{\eta_0...\eta_{n-1}}$ from , we can define $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_w}
w_{k\delta} =& w_0 + \sum_{i=0}^{k-1} -\delta\nabla U(w_{i\delta}) + {\sqrt{\delta}} \sum_{i=0}^{k-1} \xi(w_{i\delta},\eta_i)
\end{aligned}$$
| 1,247
| 1,977
| 1,354
| 1,199
| 3,892
| 0.769479
|
github_plus_top10pct_by_avg
|
cO_{X}(D))$ is isomorphic to $\CC[x,y,z]_{w,d}$, the $w$-homogeneous polynomials of degree $d := \deg_w({\left \lfloor D \right \rfloor})$.
It is a well-known result for integral Weil divisors. The general rational case follows from the fact that by definition $\cO_X(D) = \cO_X({\left \lfloor D \right \rfloor})$. The isomorphism converts $(h)$ into the $w$-homogeneous polynomial defined by $(h) + {\left \lfloor D \right \rfloor} \geq 0$.
\[prop:H0YD\] For any $D \in \operatorname{Weil}_{\QQ}(X)$, $m_i \in \QQ$, and $D' \in \operatorname{Weil}_{\QQ}(Y)$, one has:
1. \[item1-lemma-h0\] Under the previous isomorphism $H^0(Y,\cO_Y(\pi^{*}D + \sum_{i=1}^{s} m_i E_i))$ can be identified with the subspace $\{ F \in \CC[x,y,z]_{w,d} \mid \operatorname{mult}_{E_i} \pi^{*} F + m'_i \geq 0, \forall i \}$, where $m'_i := m_i + \operatorname{mult}_{E_i} \pi^{*} {\left \{ D \right \}}$, $i=1,\ldots,s$.
2. \[item2-lemma-h0\] The cohomology $H^0(Y,\cO_Y(D'))$ consists of all $w$-homogeneous polynomials $F$ of degree $\deg_w {\left \lfloor \pi_{*} D' \right \rfloor}$ such that $\operatorname{mult}_{E_i} \pi^{*} F \geq \operatorname{mult}_{E_i} ( \pi^{*} {\left \lfloor \pi_{*} D' \right \rfloor} - D' )$ for all $i=1,\ldots,s$.
The statement \[item2-lemma-h0\] is a direct consequence of \[item1-lemma-h0\] after writing $D'$ in the form $\pi^{*}D + \sum_{i=1}^{s} m_i E_i$ for $D=\pi_{*} D'$ and $m_i = \operatorname{mult}_{E_i}(D'-\pi^{*} D)\in\QQ$.
In order to prove \[item1-lemma-h0\], let us consider $h \in K(Y)$. According to Proposition \[prop:H0D\], its associated divisor $(h)$ belongs to $H^0(Y,\cO_Y(\pi^{*}D + \sum_{i=1}^{s} m_i E_i))$ if and only if $$\label{eq:condition-h}
(h) + \pi^{*}D + \sum_{i} m_i E_i \geq 0.$$ Since $K(Y)$ and $K(X)$ are isomorphic under $\pi$, the global section $h$ can be written as $h = \pi^{*} h'$ for some $h' \in K(X)$. The condition from in terms of $h'$ is $(h') + D \geq 0$, or equivalently $h \in H^0(X,\cO_X(D))$, and $$\label{eq:condition-h'}
\pi^{*} ((h')+D) + \sum_i m_i E_i =
| 1,248
| 980
| 1,221
| 1,158
| 2,709
| 0.777697
|
github_plus_top10pct_by_avg
|
36 (32.7) 40 (27.0) 1.31 (0.77, 2.25) A allele 43 (28.7) 6 (23.1) 1.34 (0.50, 3.56)
G allele 74 (67.3) 108 (73.0) 0.3206 1 G allele 107 (71.3) 20 (76.9) 0.5572 1
**rs6603797** **rs6603797**
C allele 99 (90.0) 124 (83.8) 1.74 (0.81, 3.73) C allele 135 (90.0) 25 (96.2) 0.36 (0.05, 2.85)
T allele 11 (10.0) 24 (16.2) 0.1493 1 T allele 15 (10.0) 1 (3.8) 0.3136 1
**rs4648727** **rs4648727**^**a**^
A allele 39 (35.5) 49 (33.1) 1.11 (0.66, 1.87) A allele 57 (38.0) 5 (20.8) 2.33 (0.82, 6.58)
C allele 71 (64.5) 99 (66.9) 0.6942 1 C allele 93 (62.0) 19 (79.2) 0.1030 1
**rs12126768** **rs12126768**
G allele 23 (20.9) 40 (27.0) 0.71 (0.40, 1.28) G allele 35 (23.3) 4 (15.4) 1.67 (0.54, 5.18)
T allele 87 (79.1) 108 (73.0) 0.2580 1 T allele 115 (76.7) 22 (84.6) 0.3676 1
**Females**
**rs10907185** **rs10907185**
A allele 29 (27.4) 40 (24.1) 1.19 (0.68, 2.07) A allele 45 (27.1) 10 (20.8) 1.41 (0.65, 3.07)
| 1,249
| 4,800
| 308
| 669
| null | null |
github_plus_top10pct_by_avg
|
r to the literature for general results.
We begin by fixing a field $\bbf$; all our modules will be modules for the group algebra $\bbf{\mathfrak{S}_}n$. We assume familiarity with James’s book [@j2]; in particular, we refer the reader there for the definitions of partitions, the dominance order, the permutation modules $M^\la$, the Specht modules $S^\la$ and the simple modules $D^\la$. We shall also briefly use the Nakayama Conjecture [@j2 Theorem 21.11] which describes the block structure of the symmetric group.
We also need the following two results; recall that if $\la$ is a partition then $\la'$ denotes the conjugate partition.
\[isospecht\] Suppose ${\operatorname{char}}(\bbf)=2$ and $\la$ is a partition such that $S^\la$ is irreducible. Then $S^\la\cong S^{\la'}$.
By [@j2 Theorem 8.15] we have $S^\la\cong(S^{\la'})^\ast$, since the sign representation is trivial in characteristic $2$. But by [@j2 Theorem 11.5], all simple modules for the symmetric group are self-dual.
\[815hom\] If $\la,\mu$ are partitions of $n$, then $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^\mu)=\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^{\mu'},S^{\la'}).$$
This also follows from [@j2 Theorem 8.15].
Regularisation
--------------
We recall here a useful lemma which we shall use later; this is due to James, although it does not appear in the book [@j2]. We concentrate on the special case where $\bbf$ has characteristic $2$, referring to [@j1] for the full result.
For any $l\gs1$, the $l$th *ladder* in $\bbn^2$ is $$\call_l=\lset{(i,j)}{i+j=l+1}.$$ If $\la$ is a partition, the *$2$-regularisation* of $\la$ is the partition $\la{^{\operatorname{reg}}}$ whose Young diagram is obtained by moving the nodes in $[\la]$ as high as possible within their ladders. For example, $(8,3,1^6){^{\operatorname{reg}}}=(8,7,2)$, as we see from the following Young diagrams, in which nodes are labelled according to the ladders in which they lie. 0 $$\young(12345678,234,3,4,5,6,7,8)\qquad\qquad
\young(12345678,234567
| 1,250
| 982
| 1,031
| 1,207
| 1,657
| 0.787113
|
github_plus_top10pct_by_avg
|
on_{\star}=-2/mr_{\star}^2$. After resonance when $\delta' \to +\infty$, the CI bound state energy behaves as $\epsilon_b'\simeq -1/ma^2$, which translates into $\epsilon_b'/\epsilon_{\star}'\simeq (\delta'+A)^2/2\simeq
\delta^{'2}/2$ in terms of the parameter $\delta'$ introduced above. Similarly, in the BFRM, the bound state energy decreases monotonically with the behavior $\epsilon_b/\epsilon_{\star} \simeq
\sqrt{2}\delta$ as a function of the dimensionless detuning $\delta$.
![Confinement induced bound state energy $\epsilon_b'$ \[in units of the bound state energy on resonance $|\epsilon_{\star}'|$\] as a function of the dimensionless parameter $\delta'$ (full line). The dashed line corresponds to the asymptotic behavior $\epsilon_b' \simeq -mg_1^{'2}/4$ and the dotted line to the asymptotic behavior $\epsilon_b' \simeq -1/ma^2$.](BMOboundstate.eps "fig:"){height="6cm"} \[BMOboundstate\]
Many-body problem
=================
The grand partition function at temperature $T\equiv 1/\beta$ and chemical potential $\mu$ can be written as a path integral $$\begin{aligned}
Z=\int \mathcal{D}(\bar{\psi}_{\sigma},\psi_{\sigma})
\mathcal{D}(\bar{\psi}_B,\psi_B) e^{-S}
\label{Z}\end{aligned}$$ over Grassmann fields $\bar{\psi}_{\sigma}$, $\psi_{\sigma}$ with $\sigma=\uparrow,\downarrow$ and complex fields $\bar{\psi}_B$, $\psi_B$ [@AS]. The action corresponding to the Hamiltonian (\[BFM\]) is: $$\begin{aligned}
&S&=\int_0^{\beta}d\tau \int dx \bigg(
\sum_{\sigma={\uparrow,\downarrow}} \bar{\psi}_{\sigma}
\Big[\partial_{\tau}-\frac{\partial_x^2}{2m}-\mu\Big]\psi_{\sigma}
\nonumber \\
&+&\bar{\psi}_{B}\Big[\partial_{\tau}-\frac{\partial_x^2}{4m}-2\mu+\nu\Big]
\psi_{B} + g \Big( \bar{\psi}_{B}\psi_{\uparrow}
\psi_{\downarrow}+c.c.\Big)\bigg).\nonumber\\ \left. \right.
\label{originalaction}\end{aligned}$$ The average total number of atoms is obtained from $$\begin{aligned}
\langle N \rangle = \frac{\partial F}{\partial \mu}\end{aligned}$$ where $F\equiv-T \ln Z$ is the grand potential and we are interested in the $T\to 0$
| 1,251
| 417
| 1,277
| 1,331
| null | null |
github_plus_top10pct_by_avg
|
any solution $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ of the problem , , that further satisfies \[asscl\] \_[|\_+]{}T\^2(\_+)(,,0)L\^2(GS), is unique and obeys the estimate \[csda40aa\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}), where $c'$ is given in .
The proof is based on “variations” and it is quite standard.
\(i) We apply Theorem \[glm\] with $X=H_{1}$, $Y=H_{2}$, and with $B(\cdot,\cdot)=\tilde B(\cdot,\cdot)$ and $F$ given by and , respectively. As mentioned above $\tilde B(\cdot,\cdot)$ satisfies (\[csda38\]) and (\[csda37a\]), while $F$ is a bounded linear functional, hence Theorem \[glm\] guarantees the existence of a solution $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ such that (\[csda40a\]) holds.
We verify that $\phi\in L^2(G\times S\times I)$ is a weak solution of the equation (\[csda3A\]). Let $I^\circ:=]0,E_m[$. From (\[csda40a\]) it follows that \[csda41\] B(,v)=F(v),vC\_0\^(GSI\^). Since for $v\in C_0^\infty(G\times S\times I^\circ)$ we have $v(\cdot,\cdot,0)=v(\cdot,\cdot,E_m)=0$ and $v_{|\Gamma}=0$, we see from (\[coex\]) that \[csda42\] (,v) =&,S\_0[E]{}\_[L\^2(GSI)]{} -, \_x v\_[L\^2(GSI)]{}\
&+CS\_0,v\_[L\^2(GSI)]{}+(-K\_C),v\_[L\^2(GSI)]{}\
=&F(v) =[**f**]{},v\_[L\^2(GSI)]{}, for all $v\in C_0^\infty(G\times S\times I^\circ)$, which means that holds in the weak sense.
Since $\phi\in L^2(G\times S\times I)$, and by the above $${\left\langle}\phi,P'(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}={\left\langle}-(CS_0+(\Sigma-K_C))\phi + {\bf f},v{\right\rangle}_{L^2(G\times S\times I)},$$ we see that $\phi\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$.
\(ii) Suppose that the assumption ${\bf TC}$ holds and that $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ satisfies (\[csda40a\]). Then for all $v\in H_2$ \[coexpr\] &,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{}\
&+C,S\_0v\_[L\^2(GSI)]{}+,(\^\*-K\_C\^\*) v\_[L\^2(GSI)]{}\
&+q,\_+(v)\_[T\^2(\_+)]{}+p\_0,S\_0(,0) v(,,0)\_[L\^2(GS)]{}\
& = B(,v)= [**f**]{},v\_[L\^2(GSI)]{}+[**g**]{}, \_-(v)\_[T\^2(\_-)]{}.
Recall that $
| 1,252
| 528
| 1,763
| 1,319
| null | null |
github_plus_top10pct_by_avg
|
k$ vertices, there are at most $\binom{k}{2}$ edges whose addition may produce a cycle. [This includes edges already present in the component, as an edge with multiplicity 2 (double edge) forms a 2-cycle.]{} Thus, the $p$ edges can be chosen in $\binom{k}{2}^p<k^{2p}$ ways. Let $\{e_1,e_2,\ldots, e_p\}$ denote a set of $p$ cycle-producing edges (some of these may create 2-cycles). Also let $(t_1,\ldots, t_p)$ denote a sequence of rounds, where $t_i\in\{1,\ldots, n\}$ is the round in which the $t_i$-th ball picks edge $e_i$. For each round $t=1,2,\ldots, n$ and $i=1,\ldots, p$, let us define ${\mathbb{I}}_t(e_i)$ as follows: $${\mathbb{I}}_t(e_i)= \begin{cases}
1 & \text{ if $e_i\in E_t$,}\\
0 & \text{ otherwise.}
\end{cases}$$ It is easy to see that $${\ensuremath{\operatorname{\mathbf{Pr}}\left[\text{ball $t$ picks edge $e_i $ of $G^{(t)}$}\right]}}=\frac{{\mathbb{I}}_t(e_i)}{|E_t|}.$$ Now ${\ensuremath{\operatorname{\mathtt{vis}}(e_i)}}=\sum_{t=1}^n {\mathbb{I}}_t(e_i)$ for $i=1,\ldots, p$. Using this, and the fact that $|E_t|{\geqslant}n/2$ for each $t$ (since $G^{(t)}$ is regular with degree at most 1), the probability that $e_1,e_2,\ldots, e_p$ are chosen is at most $$\begin{aligned}
\label{in:cyc}
\sum_{(t_1,\ldots,t_p)}\left\{\prod_{i=1}^p
\frac{{\mathbb{I}}_{t_i}(e_i)}{E_{t_i}}\right\}
{\leqslant}\prod_{i=1}^p\left\{\sum_{t=1}^{n}\frac{{\mathbb{I}}_{t}(e_i)}{E_{t}}\right\}
{\leqslant}\prod_{i=1}^p\frac{{4{\ensuremath{\operatorname{\mathtt{vis}}(e_i)}}}}{n}{\leqslant}\left(\frac{{4n^{1-\varepsilon}}}{n}\right)^p=\left(\frac{{4}}{ n^{\varepsilon}}\right)^p.
\end{aligned}$$ Moreover, applying Lemma \[lem:tree\] shows that [the probability that ${\mathcal{C}}_n$ contains]{} a $c$-loaded $k$-vertex tree [is at most]{} $$\label{new}
n\cdot 8^k\cdot \left(\frac{2{\mathrm{e}}}{c}\right)^{ck}{\leqslant}n\cdot 2^{-k},$$ as $c {\geqslant}4{\mathrm{e}}$. So, with high probability, [${\mathcal{C}}_n$ does not contain]{} any $c$-loaded tree with at least $(\log n
| 1,253
| 1,423
| 1,281
| 1,213
| 2,961
| 0.775804
|
github_plus_top10pct_by_avg
|
s. Here's the code.
int main (void)
{
const char *alp = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
char *ptr = &alp[3];
printf("%s\n", ptr);
return 0;
}
Edit- Sorry for not mentioning the errors. The thing is I get tons of different errors depending on where I put different asterisks and ampersands. There is no one particular error. One of the more frequent ones I get says
"incompatible integer to pointer conversion assigning to 'char *'
from 'const char';"
In the end I just want "ptr" to be equal to a pointer pointing to "D" in the array "alp".
A:
If you only want one character to print, change the %s to %c and dereference the pointer
printf("%c\n", *ptr);
It's true that you had a character pointer but %s tells printf to print from that pointer until it reads a null character. So we switch to %c which will print one character but it expects a value rather than a pointer to a value.
Q:
How to generate javascript code with javascript if posible
Im quite new at javascript so dont expect me to know much else than whats written :)
So i have a javascript file with a lot of javascript codes that almost looks the same except a number is changing with each element.
Most of my code are inside a function just not written since it's a cut out.
The relevant part of my code looks something like:
//Upgrade 1
if (CREDITS >= UpgradePrice1 && Upgrade1 === 0){
document.getElementById("Upgrade1").style.display = "block";}
else{document.getElementById("Upgrade1").style.display = "none";}
//Upgrade 2
if (CREDITS >= UpgradePrice2 && Upgrade2 === 0){
document.getElementById("Upgrade2").style.display = "block";}
else{document.getElementById("Upgrade2").style.display = "none";}
//Upgrade 3
if (CREDITS >= UpgradePrice3 && Upgrade3 === 0){
document.getElementById("Upgrade3").style.display = "block";}
else{document.getElementById("Upgrade3").style.display = "none";
And so on...
The values are assigned in a file named StandardValues.js:
var Up
| 1,254
| 6,702
| 225
| 196
| 42
| 0.832662
|
github_plus_top10pct_by_avg
|
.
\[thm:DefDiVar\] Let $D\in{\mathrm{Di}}{\mathrm{Alg}}0$. Then the following conditions are equivalent:
1. $D\in{\mathrm{Di}}{\mathrm{Var}}$
2. $\widehat D=\bar D\oplus D\in{\mathrm{Var}}$ the definition in the sense of Eilenberg
3. $D\vDash \Psi^{x_i}_{\mathrm{Alg}}\,f$ for every $f\in T_0({\mathrm{Var}})$, $\deg f=n$, $i=1,\,\ldots,\,n$ the definition in the sense of
We prove the following
\[prop:PsiDiVarVar\] Let $f=f(x_1,\ldots,x_n)\in {\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be multilinear, $f=\Psi^{x_j}_{\mathrm{Alg}}\,\bar f$ for some $j$. Then $$f\in T_0({\mathrm{Di}}{\mathrm{Var}})\Leftrightarrow \bar f\in T_0({\mathrm{Var}}).$$
Since evidently ${\mathrm{Var}}\subseteq{\mathrm{Di}}{\mathrm{Var}}$, the statement “$\Rightarrow$” is trivial.
To prove “$\Leftarrow$” consider an identity $\bar f\in T_0({\mathrm{Var}})$. By Theorem \[thm:DefDiVar\] for arbitrary $D\in{\mathrm{Di}}{\mathrm{Var}}$ we have $D\vDash \Psi^{x_i}_{\mathrm{Alg}}\,\bar{f}$ for all $i=1,\,\ldots,\,n$, but $\Psi^{x_j}_{\mathrm{Alg}}\,\bar{f}=f$ and so $f\in T_0({\mathrm{Di}}{\mathrm{Var}})$.
\[prop:f1fnDiVarDiVar\] Let $f=f(x_1,\ldots,x_n)\in{\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be multilinear, $f=f_1+\ldots+f_n$ where $f_i$ consists of all dimonomials in $f$ with a central letter $x_i$. Then $$f\in T_0({\mathrm{Di}}{\mathrm{Var}})\Leftrightarrow f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})\text{ for all }
i=1,\,\ldots,\,n.$$
“$\Leftarrow$” is evident.
We prove “$\Rightarrow$”. Let $f\in T_0({\mathrm{Di}}{\mathrm{Var}})$, consider an arbitrary algebra $A\in{\mathrm{Var}}$. Then by Proposition \[prop:VarCur0DiVar\] we obtain ${({\mathop{\mathrm{Cur}}\nolimits}A)}^{(0)}\in
{\mathrm{Di}}{\mathrm{Var}}$, hence ${({\mathop{\mathrm{Cur}}\nolimits}A)}^{(0)}\vDash f$, where $f=f(x_1,\ldots,x_n)$. Fix $i\in\{1,\,\ldots,\,n\}$ and assign the following values to variables: $x_i:=Ta_i$, $a_i\in A$, $x_j:=a_j$ for all $j\not=i$, $a_j\in A$. The properties of a conformal product imply $$0=f(a_1,\ldots,Ta_i,\l
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|
e cannot be sure that system (\[eq:SW:15\]) for $f(r)$ is well-defined in the sense that it represents a system for $f(r)$ express in terms of $r$ only. To ensure this, we begin by introducing vector fields orthogonal to the wave vector $\lambda$, that is, vector fields of the form
\[eq:SW:16\] X\_a=\^i\_a(u)\_[x\^i]{},a=1,…,p-1,i=1,…,p,
where
\[eq:SW:17\] \_a\^i\_i=0,a=1,…,p-1,(\_a\^i)=p-1.
Consequently, the wave vector $\lambda$ together with the vectors $\xi_a$, $a=1,\ldots,p-1$ form a basis for ${\mathbb{R}}^p$. Moreover, let us note that the vector fields $X_a$ form an Abelian Lie algebra of dimension $p-1$. Application of the vector field (\[eq:SW:16\]) to equation (\[eq:SW:15\]) cancels out the left side since we suppose that $f=f(r)$, so it must be the same on the right side. Therefore, the conditions for the system of ODEs (\[eq:SW:15\]) to be well-defined in terms of $r$ are
\[eq:SW:18\] X\_a=0,a=1,…,p-1.
####
In summary, system (\[eq:SW:1\]) admits a simple wave solution if the following conditions are satisfied:
- there exist a scalar function $\Omega(x,u)$, a wave vector $\lambda(u)$ and a rotation matrix $L(x,u)$ satisfying the wave equation (\[eq:SW:13\]);
- there exist $p-1$ vector fields (\[eq:SW:16\]) which satisfy the orthogonality relation (\[eq:SW:17\]);
- the right-hand side of equation (\[eq:SW:15\]) is annihilated by the vector fields (\[eq:SW:16\]), [*i.e.* ]{}conditions (\[eq:SW:18\]) are satisfied;
- $\det \Phi\neq 0$, where $\Phi$ is given by (\[eq:SW:4\]);
- $1+\Omega \frac{{\partial}\lambda_i}{{\partial}u}Lbx^i\neq 0$, $\lambda\in C^1$.
These conditions are sufficient, but not necessary, since the vanishing of $\det\Phi$ does not imply that solutions of form (\[eq:SW:2\]) do not exist. We finish the present analysis of the simple wave solutions of system (\[eq:SW:1\]) by several remarks:
- In general, there are more parameters (arbitrary functions) defining $\Omega$ and the rotation matrix $L$ than the minimum number required to satisfy condition (
| 1,256
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|
t $D^{(k)} = (d_{ij}^{(k)}-\delta_{i,1}\cdot \delta_{kj})_{i,j=1,\dots,n}$. Expressing $\lambda$ as linear combination of the dual basis then leads to the system of linear equations
$D^{(k)} x = 0$ $(k=1,...,n)$. As pointed out by Mariano it has a unique solution (up to scalars). Again, this is no closed formula, but it can be used in practise to compute integrals.
Q:
Como inserir dados XML dentro de um banco MYSQL?
Gerando o arquivo XML com PHP recebo o seguinte resultado:
Eu gostaria de mandar isso agora para um banco de dados MYSQL inserindo primeiro as colunas na tabela Imovel e depois preenchendo estas colunas com os dados que estão sendo puxados dentro de cada field.
Alguém poderia me ajudar a fazer um script para inserir dados do XML dentro do banco de dados MYSQL?
A:
Recomendo você arrumar esse XML para uma forma mais pratica e evitar repetir os nodes. Repare que até seu CDATA contém espaços indevidos, para remove-los use a função TRIM
Talvez a melhor opção seja de XML para seu caso seja:
<CODIGO>CL501</CODIGO>
<DATA>2012-03-16</DATA>
Com o XML que você passou, o melhor que se pode fazer é da forma abaixo:
XML
$string = '<Imovel>
<field name="CODIGO"><![CDATA[ CL501 ]]></field>
<field name="DATA"><![CDATA[ 2012-03-16 ]]></field>
<field name="ENDERECO"><![CDATA[ CASEMIRO DE ABREU ]]></field>
</Imovel>';
Montando a SQL
$root = simplexml_load_string( $string );
foreach( $root as $element )
{
foreach( $element-> attributes() as $field )
{
$clear = trim( $element );
$fields[] = "'$field'";
$values[] = "'$clear'";
}
}
SQL
INSERT INTO `TABLE` (" . implode( ', ' , $fields ) . ")
VALUES (" . implode( ', ' , $values ) . ")
OUTPUT
INSERT INTO `TABLE` ('CODIGO', 'DATA', 'ENDERECO')
VALUES ('CL501', '2012-03-16', 'CASEMIRO DE ABREU')
Q:
Prompt confirm before Leaving edited html form
If user tries to leave unsaved edited form, a message box pop-up
"This page is asking you to confirm that you want to leave - data you
| 1,257
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|
$, one has $\bar{a}\bar{b} = [a+F^{n-1}R][b+F^{m-1}R]
\subseteq [ab+F^{n+m-1}R]$. Since $\bar{a}\bar{b}\not=0$, $ab\in F^{n+m}R\smallsetminus F^{n+m-1}R$, whence $\bar{a}\bar{b}=\sigma(ab)$ is the image of $ab$ in $\operatorname{gr}_F(AB)$.
\(2) Define a map $\rho: \operatorname{gr}_FA\times \operatorname{gr}_FB \to \operatorname{gr}_F(A\otimes_RB)$ by $\rho(\bar{a}, \bar{b} ) = [a\otimes b + F^{n+m-1}(A\otimes B)]$, for $\bar{a}\in \operatorname{gr}_F^nA$, $\bar{b}\in \operatorname{gr}_F^mB$ and where the rest of the notation is the same as for part (1). This clearly defines a ${\mathbb{C}}$-bilinear map that is $\operatorname{gr}_F R$-balanced in the sense that $\rho(\bar{a}\bar{r}, \bar{b}) = \rho(\bar{a}, \bar{r} \bar{b})$ for $\bar{r}\in \operatorname{gr}^s_FR$. By universality, $\rho$ therefore induces a map $ \operatorname{gr}_FA\otimes_{\operatorname{gr}R}\operatorname{gr}_FB\to \operatorname{gr}_F(A\otimes_RB)$. It is surjective since $F^{n+m}(A\otimes B)/ F^{n+m-1}(A\otimes B)$ is spanned by elements of the given form $[a\otimes b + F^{n+m-1}(A\otimes B)].$
Lemma {#grade-elements}
-----
Let $R=\bigcup_{i\geq 0}F^iR$ be a filtered ring, pick $r\in R$ and let $ I$ be a subset of $R$. Under the induced filtrations, $\operatorname{gr}_F(rI) = \sigma(r)\operatorname{gr}_F(I)$ in the following cases:
1. $\sigma(r)$ is regular in $\operatorname{gr}_FR$;
2. $ r=r^2 \in F^0(R)$ and $rI\subseteq I$.
Assume that $r\in F^sR\smallsetminus F^{s-1}R$. We claim that, in both cases, it suffices to prove that $F^n(rI) = rF^{n-s}I$ for all $n\geq s$. Indeed, if this is true then the identity $F^m(rI) = rI\cap F^mR$ implies that the $n^{\mathrm{th}}$ summand of $\operatorname{gr}(rI)$ equals $$\frac{F^n(rI)}{F^{n-1}(rI)} \ = \ \frac{F^n(rI)}{F^n(rI)\cap F^{n-1}R}
\ \cong\ \frac{ F^n(rI) + F^{n-1}R}{F^{n-1}R}
\ = \ \frac{rF^{n-s}I + F^{n-1}R}{F^{n-1}R},$$ which is the $n^{\mathrm{th}}$ summand of $\sigma(r)gr(I)$.
\(1) In this case, $rt\in F^n(rI) = rI\cap F^n(R) \Leftrightarrow
t\in I\ \mathrm
| 1,258
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|
restate and prove Theorem \[bicirc\].
\[bicirctech\] There is a function $f_{\ref{bicirctech}}\colon \bZ \to \bZ$ so that, for every integer $s \ge 2$, if $M$ is a matroid without a $U_{s,2s}$-minor and with a $B^+(K_{r(M)})$-restriction framed by $B$, then there is a set $\wh{B} \subseteq B$ and a $\wh{B}$-clique $\wh{M}$ such that $\dist(M,\wh{M}) \le f_{\ref{bicirctech}}(s)$.
For each $s\ge 2$ set $f_{\ref{bicirctech}}(s) = n = 7f_{\ref{threenonsingular}}(s,f_{\ref{bicircpg}}(s))$. Let $M$ be a matroid with a $B^+(K_{r(M)})$-restriction framed by $B$ and with no $U_{s,2s}$-minor. By Theorem \[maintech\], either $M$ has a rank-$f_{\ref{bicircpg}}(s)$ projective geometry minor $N$, or there is a set $\wh{B} \subseteq B$ and a $\wh{B}$-clique $\wh{M}$ with $\dist(M,\wh{M}) \le n$. In the second case the theorem is immediate. In the first case, let $C$ be such that $N$ is a spanning restriction of $M \con C$. Now $M \con C$ also has a $B^+(K_{r(M \con C)})$-restriction, and the result follows from Lemma \[bicircpg\].
References {#references .unnumbered}
==========
[\[\]]{}
\[highlyconnected\] J. Geelen, B. Gerards, G. Whittle, The highly-connected matroids in minor-closed classes, Ann. Comb. 19 (2015), 107-123.
\[gkep\] J. Geelen, K. Kabell, The [E]{}rd[ő]{}s-[P]{}ósa property for matroid circuits, J. Combin. Theory Ser. B 99 (2009), 407–419.
\[covering1\] J. Geelen, P. Nelson, Projective geometries in exponentially dense matroids. I, J. Combin. Theory Ser. B 113 (2015), 185–207.
\[gr\] C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001.
\[covering2\] P. Nelson, Projective geometries in exponentially dense matroids. II, J. Combin. Theory Ser. B 113 (2015), 208–219.
\[margulis\] G. A. Margulis, Explicit constructions of graphs without short cycles and low density codes, Combinatorica 2 (1982), 71–78.
\[oxley\] J. G. Oxley, Matroid Theory, Oxford University Press, New York (2011).
[^1]: This research is partly supported by a Discovery Grant \[203110-2011\] from the Natural Scien
| 1,259
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ring sites
*wR*(*F*^2^) = 0.089 H-atom parameters constrained
*S* = 1.14 *w* = 1/\[σ^2^(*F*~o~^2^) + (0.0341*P*)^2^ + 13.8363*P*\] where *P* = (*F*~o~^2^ + 2*F*~c~^2^)/3
19520 reflections (Δ/σ)~max~ = 0.009
950 parameters Δρ~max~ = 1.20 e Å^−3^
0 restraints Δρ~min~ = −1.36 e Å^−3^
------------------------------------- --------------------------------------------------------------------------------------------------
Special details {#specialdetails}
===============
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.
Refinement. Refinement of *F*^2^ against ALL reflections. The weighted *R*-factor *wR* and goodness of fit *S* are based on *F*^2^, conventional *R*-factors *R* are based on *F*, with *F* set to zero for negative *F*^2^. The threshold expression of *F*^2^ \> σ(*F*^2^) is used only for calculating *R*-factors(gt) *etc*. and is not relevant to the choice of reflections for refinement. *R*-factors based on *F*^2^ are statistically about twice as
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required to be large enough, but still independent of the mesh size $h$, in order to guarantee the well-posedness of the discontinuous Galerkin formulation. Details will be given later.
It is clear that the exact solution $u$ to Equation (\[eq:ellipticeq\]) satisfies $$\label{eq:dg-exactsol}
A(u,v) = (f,v)\qquad\textrm{for all } v\in V_h,$$ as $[u]$ vanishes on all $e\in\mathcal{E}_h$. Hence the following interior penalty discontinuous Galerkin formulation is consistent with Equation (\[eq:ellipticeq\]): find $u_h \in V_h$ satisfying $$\label{eq:dg}
A(u_h,v) = (f,v)\qquad\textrm{for all } v\in V_h.$$
Finally, we would like to point out that the formulation (\[eq:dg\]) is computable, as long as each finite dimensional space $V_K$ has a clearly defined and computable basis.
Abstract theory
===============
Define a norm $\3bar\cdot\3bar$ on $V(h)$ as following: $$\begin{aligned}
\3bar v\3bar^2=\sum_{K\in\mathcal{T}_h}\|\nabla v\|_K^2+\sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2+\alpha\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v]\|_e^2.\end{aligned}$$ By the Poincaré inequality, $\3bar\cdot\3bar$ is obviously a well-posed norm on $V(h)$.
Next, we give a set of assumptions, which form the minimum requirements guaranteeing the well-posedness and the approximation properties of the interior penalty discontinuous Galerkin method.
- (The trace inequality) There exists a positive constant $C_T$ such that for all $K\in\mathcal{T}_h$ and $\theta\in H^1(K)$, we have $$\label{eq:TraceIn}
\|\theta\|_{\partial K}^2\le C_{T}(h_K^{-1}\|\theta\|_K^2+h_K\|\nabla\theta\|_K^2).$$
- (The inverse inequality) There exists a positive constant $C_I$ such that for all $K\in\mathcal{T}_h$, $\phi\in V_K$ and $\phi\in \frac{\partial}{\partial x_i}V_K$ where $i=1,\ldots, d$, we have $$\label{eq:InverseIn}
\|\nabla\phi\|_K\le C_I\, h_K^{-1}\|\phi\|_K.$$
- (The approximability) There exist positive constants $s$ and $C_A$ such that for all $v\in H^{s+1}(\Omega)$, we have $$\label{eq:approximability}
\inf_{\
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|
t \int_{S\times I}\sigma(\cdot,\omega',\cdot,E',\cdot)d\omega' dE'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{S\times I}\sigma(x,\omega',\omega,E',E)d\omega' dE', \nonumber\\
& {\left\Vert \int_{S\times I}\sigma(\cdot,\cdot,\omega',\cdot,E')d\omega' dE'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{S\times I}\sigma(x,\omega,\omega',E,E')d\omega' dE'.\end{aligned}$$
In what follows, we assume that the stopping power $S_0:\ol G\times I\to{\mathbb{R}}$ satisfies the following assumptions: & S\_0L\^(GI), \[csda9\]\
& L\^(GI), \[csda9aa\]\
& :=\_[(x,E)GI]{}S\_0(x,E)>0,\[csda9a\]\
& \_x S\_0L\^(GI). \[csda9b\] We remark that the assumption (\[csda9b\]) will be needed only in the context of the theory of evolution operators in section \[evcsd\].
\[rese1a\] Assume that $\tilde{\sigma}:G\times S^2\times I\to{\mathbb{R}}$ is a measurable non-negative function, and that the cross-section $\sigma$ is (formally) of the form $$\sigma(x,\omega',\omega,E',E)=\tilde\sigma(x,\omega',\omega,E)\delta(E'-E),$$ in the sense that the collision operator $K$ is given by \[collb\] (K)(x,,E)=\_S(x,’,,E)(x,’,E)d’,L\^2(GSI). In this case the assumptions (\[ass2\]) and (\[ass3\]) for any $C$ mean that \[ass2a\] &\_S(x,’,,E)d’ M\_1,\
&\_S(x,,’,E)d’M\_2, for a.e. $(x,\omega)\in G\times S\times I$, and \[ass2b\] &(x,,E)-\_S(x,,’,E) d’ c,\
&(x,,E)-\_S(x,’,,E) d’ c, for a.e. $(x,\omega,E)\in G\times S\times I$. All results proved in this paper are valid for these (simplified) collision operators. The estimate (\[k-norma\]) in this case is \[k-normb\] & [K]{} [\_S(,’,,)d’]{}\_[L\^(GSI)]{}\^[1/2]{} [\_S(,,’,)d’]{}\_[L\^(GSI)]{}\^[1/2]{} \^[1/2]{}[M\_2]{}\^[1/2]{}, where ${\left\Vert K\right\Vert}$ is the norm of $K$ as an operator in $L^2(G\times S\times I)$, and $$&{\left\Vert \int_S\tilde\sigma(\cdot,\omega',\cdot,\cdot)d\omega'\right\Vert}_{L^\infty(G\times S\
| 1,262
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\bar{b}_{ki}-f_j\bar{h}_i=0 \label{nobraneE84} \\
& \bar{g}_{ki}\bar{b}_{jj}-\bar{f}_ib_{ij}-\bar{g}_{jj}\bar{b}_{ki}+g_{ij}\bar{h}_i=0 \nonumber \\
& -g_{ji}b_{kj}+\bar{g}_{ii}h_j+g_{kj}b_{ji}-f_j\bar{b}_{ii}=0 \nonumber \\
& g_{ji}\bar{b}_{jj}-\bar{g}_{ii}b_{ij}-\bar{g}_{jj}b_{ji}+g_{ij}\bar{b}_{ii}=0 \quad .\nonumber\end{aligned}$$
For the $E_{9,2,1}$ potential, the generalised Chern-Simons term has the form $$\frac{1}{2}\int E_{9,2,1} \wedge (P_1^2 \cdot Q)^{2,1}_1 \quad ,$$ with $(P_1^2\cdot Q)^{ab,c}_d= \frac{1}{3} (-P^{be}_d Q^{ac}_e-2P^{ce}_dQ^{ab}_e-P^{ae}_dQ^{cb}_e+Q^{ae}_d P^{cb}_e+Q^{be}_dP^{ac}_e+2Q^{ce}_dP^{ab}_e)$. The exotic branes are the $6_3^{1,1}$-branes, and denoting with $//abc$ the internal directions wrapped by the branes, and with $\bigcirc d ,\bigcirc e$ the isometries corresponding to the index $d$ repeated twice and the index $e$ repeated three times, the constraints are $$\begin{aligned}
& N_{6_3^{1,1}}(//x^jy^jy^i, \bigcirc x^k ,\bigcirc x^i)-\tfrac{1}{2}[f_kb_{jj}-g_{jk}h_j-h_kg_{jj}+b_{jk}f_j]=0\nonumber \\
& N_{6_3^{1,1}}(// x^ix^jy^j,\bigcirc x^k, \bigcirc y^i)-\tfrac{1}{2}[g_{ik}\bar{b}_{kj}-\bar{g}_{kk}b_{ij}-b_{ik}\bar{g}_{kj}+\bar{b}_{kk}g_{ij}]=0 \nonumber \\
& N_{6_3^{1,1}}(//y^ix^jy^j, \bigcirc y^k, \bigcirc x^i)+\tfrac{1}{2}[-g_{kk}\bar{b}_{ij}+\bar{g}_{ik}b_{kj}+b_{kk}\bar{g}_{ij}-\bar{b}_{ik}g_{kj}]=0 \nonumber \\
& N_{6_3^{1,1}}(// x^ix^jy^j,\bigcirc y^k, \bigcirc y^i)+\tfrac{1}{2}[-\bar{g}_{jk}\bar{h}_j+\bar{f}_k\bar{b}_{jj}+\bar{b}_{jk}\bar{f}_j-\bar{h}_k\bar{g}_{jj}]=0 \quad .\end{aligned}$$ As in the previous case, we must also consider the quadratic constraints that do not correspond to branes. These are $$\begin{aligned}
& g_{kk}\bar{b}_{kj}-\bar{g}_{ik}b_{ij}-\bar{g}_{jk}b_{jj}+\bar{f}_kh_j-b_{kk}\bar{g}_{kj}+\bar{b}_{ik}g_{ij}+\bar{b}_{jk}g_{jj}-\bar{h}_kf_j=0\nonumber \\
& -f_k\bar{h}_j+g_{jk}\bar{b}_{jj}+g_{ik}\bar{b}_{ij}-\bar{g}_{kk}b_{kj}+h_k\bar{f}_j-b_{jk}\bar{g}_{jj}-b_{ik}\bar{g}_{ij}+\bar{b}_{kk}g_{kj}=0\nonumber
\\
& -g_{kk}\bar{h}_j+\bar{g}_{ik}\
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|
finition \[D:DISJOINT\_ENSEMBLES\], disjointness entails that ${{\operatorname{dom}{\Psi}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Phi}}} = \varnothing$. Thus, if $i \in {{\operatorname{dom}{\Upsilon}}}$, exactly one of two cases hold: either A: $i \in {{\operatorname{dom}{\Psi}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$, or B: $i \in {{\operatorname{dom}{\Phi}}}$ and $i \notin {{\operatorname{dom}{\Psi}}}$.
Assume case A, that $i \in {{\operatorname{dom}{\Psi}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$. With $\Upsilon = \Psi \cup \Phi$, it follows from the definition of set union that for any $i \in {{\operatorname{dom}{\Psi}}}$, $(i,P) \in \Psi$ implies $(i,P) \in \Upsilon$ – that is, $\Psi \subseteq \Upsilon$.
For case B, similar argument leads to $\Phi \subseteq \Upsilon$.
\[C:ENSEMBLE\_PROD\_MEMBERS\] If $\Upsilon$, $\Psi$, and $\Phi$ are ensembles such that $\Upsilon = \Psi\Phi$, then $\Psi(i) = \Upsilon(i)$ for $i \in {{\operatorname{dom}{\Psi}}}$, and $\Phi(j) = \Upsilon(j)$ for $j \in {{\operatorname{dom}{\Phi}}}$.
Under identical premises, lemma \[L:ENSEMBLE\_PROD\_SUBSETS\] provides $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$. Suppose $i \in {{\operatorname{dom}{\Psi}}}$. If $(i,P) \in \Psi$, then $(i,P) \in \Upsilon$ since $\Psi \subseteq \Upsilon$. The notation $\Psi(i) = \Upsilon(i)$ (both equaling $P$) is equivalent. A similar argument demonstrates $\Phi(j) = \Upsilon(j)$ for $j \in {{\operatorname{dom}{\Phi}}}$.
\[T:ENSEMBLE\_PROD\_CHOICE\_PROD\] Let $\Upsilon$, $\Psi$, and $\Phi$ be ensembles such that $\Upsilon = \Psi\Phi$. For each $\upsilon \in {\prod{\Upsilon}}$, there exist unique $\psi \in {\prod{\Psi}}$ and $\phi \in {\prod{\Psi}}$ such that $\upsilon = \psi\phi$.
Suppose $\upsilon \in {\prod{\Upsilon}}$. Since any choice has the same domain as its generating ensemble, ${{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{\upsilon}}}$. Theorem \[T:DYADIC\_PRODUCT\_IS\_ENSEMBLE\] states that ${{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{\Ps
| 1,264
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|
Big(\PP^2_w, \mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right)\Big) =
\chi \left( \PP^2_w,\mathcal{O}_{\PP^2_w} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right) \right) \\
& =1 + \frac{1}{2} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right)
\cdot \left( kH - \mathcal{C}^{(k)} \right)
+ R_{\PP^2_w} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right).
\end{aligned}$$ After applying Serre’s duality to $R_{\PP^2_w}$ the last term can be replaced: $$\label{eq:RXlocal}
R_{\PP^2_w} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right)=
R_{\PP^2_w} \left( -kH + \mathcal{C}^{(k)} \right) =
\sum_{P \in S} R_{{\PP^2_w},P} \left( -kH + \mathcal{C}^{(k)} \right).$$
Now we are interested in studying the Euler characteristic of $\cO_Y(L^{(k)})$. From the proof of Theorem \[thm:h2Lk\], see , it follows that $$L^{(k)} = \pi^{*} \left( -kH + \mathcal{C}^{(k)} \right)
+ \sum_{P \in S} \sum_{\v\in\Gamma_P} \left( {\left \lfloor \frac{k(m_\v-d\b_\v)}{d} \right \rfloor} + k \b_\v
- e_{\v k} \right) E_\v.$$ Let us denote by $\alpha_{\cC,P}^{(k)}$ the rational number
$$\label{eq:def-alpha}
\begin{aligned}
& \alpha_{\cC,P}^{(k)} =\\
&\quad \frac{1}{2} \sum_{\v,\vv \in \Gamma_P}
\left( {\left \lfloor \frac{k(m_\v-d\b_\v)}{d} \right \rfloor} + k \b_\v - e_{\v k} \right)
\left( {\left \lfloor \frac{k(m_\vv-d\b_\vv)}{d} \right \rfloor} + k \b_\vv - e_{\vv k} - (\nu_\vv - 1) \right)
E_\v \cdot E_\vv.
\end{aligned}$$
Applying the Riemann-Roch formula (Theorem \[thm:RR\]) to the $V$-surface $Y$ one has $$\label{eq:chiLk}
\begin{aligned}
& \chi(Y,\cO_Y(L^{(k)})) = 1 + \frac{L^{(k)} \cdot (L^{(k)}-K_Y)}{2} + R_Y(L^{(k)}) \\
&= 1 + \frac{1}{2} \left( -kH + \mathcal{C}^{(k)} \right)
\cdot \left( -kH + \mathcal{C}^{(k)} - K_{\PP^2_w} \right)
+ \sum_{P \in S} \alpha_{\cC,P}^{(k)} + R_Y(L^{(k)}).
\end{aligned}$$ Rearranging the local contributions to the correction term $R_Y(L^{(k)})$ one obtains $$\label{eq:RYlocal}
R_Y(L^{(k)}) = \sum_{Q \in Y} R_{Y,Q} (L^{(k)}) = \sum_{P \in S} \sum_{Q \in \pi^{-1}(P)} R_{Y,Q}(L^{(k)}).$$
P
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|
\begin{aligned}
\hat{u}_{ {\bf k} }(\tau) &=& \hat{a}_{{\bf k} } f(k,\tau)+
\hat{a}_{-{\bf k} }^{\dagger} f^{*}(k,\tau), \label{sol11} \\
\hat{\pi}_{{\bf k} }(\tau) &=& \hat{a}_{{\bf k} } g(k,\tau)+
\hat{a}_{-{\bf k} }^{\dagger} g^{*}(k,\tau). \label{sol22}\end{aligned}$$ where $f(k,\tau)'=g(k,\tau)$. When we insert these solutions to the Fourier decompositions (\[decomp1\]) and (\[decomp2\]) we simply obtain $$\begin{aligned}
\hat{u}(\tau,{\bf x} ) &=& \frac{1}{(2\pi)^{3/2}} \int d^3{\bf k } \left[ f(k,\tau) \hat{a}_{{\bf k}}
e^{i{\bf k}\cdot {\bf x}} +
f^*(k,\tau) \hat{a}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \label{decomp11} \ , \\
\hat{\pi}(\tau,{\bf x} ) &=& \frac{1}{(2\pi)^{3/2}} \int d^3{\bf k } \left[ g(k,\tau)
\hat{a}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} + g^*(k,\tau) \hat{a}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right].
\label{decomp22} \end{aligned}$$ The mode functions fulfils the so called Wronskian condition $$f^*(k,\tau) g(k,\tau) - f(k,\tau) g^*(k,\tau)=-i \label{Wronskian}$$ as a result of relations of commutation (\[com1\]-\[com4\]). These relations is important to normalise properly the mode functions.
The Hamilton equations (\[Ham11\]) and (\[Ham22\]) together with (\[sol11\]) give us the equation for the mode function $$\frac{d^2}{d\tau^2}f(k,\tau) + \left[ D k^2 +m^2_{\text{eff}} \right] f(k,\tau) = 0. \label{modeeq}$$ This equations has two regimes. The first one called adiabatic corresponds to the situation when $D k^2 +m^2_{\text{eff}} \equiv \Gamma \gg 0 $. The second one leads to the super-adiabatic amplification and corresponds to the situation when $\Gamma \ll 0$. The creation of the gravitational waves corresponds to the case of the super-adiabatic amplification.
We can now investigate which modes are amplified. The condition $\Gamma \ll 0$ corresponds to $D k^2 \ll -m^2_{\text{eff}}$. In Fig. \[pumpfield\] we see the evolution of $-m^2_{\text{eff}}$. As we see the condition for the creation of gravitational waves is
| 1,266
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ion parameters.
Similarly to what we did in , we derive two approximate confidence sets: one is an $L_\infty$ ball and the other is a hyper-rectangle whose $j^{\mathrm{th}}$ side length is proportional to the standard deviation of the $j^{\mathrm{th}}$ coordinate of $\hat{\gamma}_{{\widehat{S}}}$. Both sets are centered at $\hat{\gamma}_{{\widehat{S}}}$.
Below, we let $\alpha \in (0,1)$ be fixed and let $$\label{eq:Sigma.loco}
\hat \Sigma_{{\widehat{S}}} = \frac{1}{n} \sum_{i=1}^n \left( \delta_i - \hat{\gamma}_{{\widehat{S}}}
\right)\left( \delta_i - \hat{\gamma}_{{\widehat{S}}} \right)^\top,$$ be the empirical covariance matrix of the $\delta_i$’s. The first confidence set is the $L_\infty$ ball $$\label{eq::gamma.conf-rectangle}
\hat{D}_{{\widehat{S}}} = \Big\{ \gamma \in \mathbb{R}^k \colon \|\gamma -
\hat{\gamma}_{{\widehat{S}}} \|_\infty \leq \hat{t}_\alpha \Big\},$$ where $\hat{t}_\alpha$ is such that $$\mathbb{P}\left( \| Z_n \|_\infty \leq \hat{t}_\alpha \right) = 1 -
\alpha,$$ with $Z_n \sim N(0,\hat{\Sigma}_{{\widehat{S}}}$). The second confidence set we construct is instead the hyper-rectangle $$\label{eq:gamma.hyper:CI}
\tilde{D}_{{\widehat{S}}} = \bigotimes_{j \in {\widehat{S}}} \hat{D}(j),$$ where, for any $j \in {\widehat{S}}$, $\tilde{D}(j) = \left[ \hat{\gamma}_{{\widehat{S}}}(j) -\hat{t}_{j,\alpha}, \hat{\gamma}_{{\widehat{S}}}(j) +\hat{t}_{j,\alpha}
\right]$, with $ \hat{t}_{j,\alpha} = z_{\alpha/2k} \sqrt{
\frac{\hat\Sigma_{{\widehat{S}}}(j,j)}{n} }.$
The above confidence sets have the same form as the confidence sets for the projection parameters . The key difference is that for the projection parameters we use the estimated covariance of the linear approximation to $\hat{\beta}_{{\widehat{S}}}$, while for the LOCO parameter $\hat{\gamma}_{{\widehat{S}}}$ we rely on the empirical covariance , which is a much simpler estimator to compute.
In the next result we derive coverage rates for both confidence sets.
\[thm::CLT2\] There exists a universal const
| 1,267
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ion $\varphi^a=G_{ret}^{ab}q_{c}F^c_b$ we get
\^a\^bG\_s\^[ab]{}=G\_[ret]{}\^[ac]{}F\^d\_cG\_[ret]{}\^[be]{}F\^f\_eQ\_[df]{}=G\_[ret]{}\^[ac]{}G\_[ret]{}\^[be]{}N\_[ce]{} \[ne21\]
From effective actions to Langevin equations
---------------------------------------------
Let us now investigate a scalar field theory described by the Heisenberg operators $\Phi_H^a$. The fields have one and two-particle expectation values
\_H\^a=\^a
\_H\^a\_H\^b=\^a\^b+G\^[ab]{} These expectation values cannot be derived either from the euclidean generating functional or its analytic continuation to Minkowski space, which generate IN-OUT matrix elements instead [@CH08]. To find a suitable generating functional, we must consider two external sources $J^A=\left(J^{1a},J^{2a}\right)$ and introduce the Schwinger - Keldysh or closed time-path (CTP) 1 particle-irreducible (1PI) generating functional $W_{1PI}\left[J_A\right]$ as
Z\_[1PI]{}=e\^[iW\_[1PI]{}]{}=()(T)\[ne22\] Where $T$ ($\tilde{T}$) means (anti) temporal ordering. We may now derive the expectation value
\^a=.|\_[J\^1=J\^2=0]{} \[ne23\] More generally, we may consider the expectation value of the Heisenberg operator driven by an external source $J_a$. This is
\^a=.|\_[J\^1=J\^2=J]{} \[ne24\] Even more generally, we may consider this as the theory of a field doublet $\Phi^A=\left(\Phi^{1a},\Phi^{2a}\right)$ coupled to sources $J_A=\left(J_{1a}=J^{1a},J_{2a}=-J^{2a}\right)$ (observe that lowering or raising a $2$ index involves a sign change). In this theory we have two backgrounds fields
\^A=W\_[1PI]{}\^[,A]{} \[ne25\] but the physical situation is when
\_-\^a=\^[1a]{}-\^[2a]{}=0 \[ne26\] When this obtains, then
\_+\^a=12(\^[1a]{}+\^[2a]{})=\^a \[ne27\] is the physical expectation value
The 1PI effective action is the Legendre transform
\_[1PI]{}=W\_[1PI]{}-J\_A\^A Therefore
\_[1PI,A]{}==-J\_A \[mean\]
The 1PI CTP EA can be written generically as [@CH08]
$$\Gamma \left[ \phi_{-},\phi_{+}\right] =\phi_{-}^a
\mathbf{D}_a\left[ \phi_{+}\right] +\frac{i}{2}\phi_{-}^a
| 1,268
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LB($n^{0.8}$) 0.000 0.002 0.002 0.000 0.002 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.070 0.074 0.050 0.082 0.062 0.058 0.066
: Empirical sizes comparison for Example \[example1\].
\[table2\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.013 0.013 0.013 0.013 0.013 0.013 0.013
K=100 0.013 0.013 0.013 0.013 0.013 0.013 0.013
K=150 0.013 0.013 0.013 0.013 0.013 0.013 0.013
BLB($n^{0.6}$) 0.106 0.107 0.111 0.110 0.109 0.109 0.106
BLB($n^{0.8}$) 0.034 0.035 0.034 0.035 0.035 0.034 0.034
SDB($n^{0.6}$) 0.127 0.129 0.129 0.129 0.129 0.129 0.127
SDB($n^{0.8}$) 0.042 0.042 0.043 0.042 0.042 0.042 0.042
TB 0.012 0.012 0.012 0.012 0.012 0.012 0.012
2 K=50 0.014 0.014 0.014 0.014 0.014 0.014 0.014
K=100 0.014 0.015 0.015 0.015 0.015 0.014 0.014
K=150 0.014 0.015 0.014 0.0
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make the following approximation:
- When assimilating a non-baseline observation, ${\bar{\mathcal{A}}}$ is kept fixed at its previous value, and for updating ${\mathcal{A}}$ it is assumed that ${\bar{\nu}}^{-1}=0$.
Approximation B2 implies that for ${\mathbf{x}}_t\neq{\mathbf{0}}$, $$\begin{aligned}
Y_t|~ {\bar{\nu}},~ {\boldsymbol{\mu}},~ \nu,~ {\bar{\mathcal{A}}}_{t-1},~ {\mathcal{A}}_{t-1} & \stackrel{apx.}{\sim}
\mathrm{N} \left(~{\bar{m}}_{t-1} + {\mathbf{x}}_t^\top{\boldsymbol{\mu}},~ \nu^{-1}{\mathbf{x}}_t^\top{\mathbf{1}}~ \right), $$ which, given distributions at time $t-1$ as specified in , leads to the desired tractable updates for the sufficient statistics for as follows: ${\bar{\mathcal{A}}}_t = {\bar{\mathcal{A}}}_{t-1}$ and $$\begin{aligned}
{\mathbf{m}}_t &= {\mathbf{m}}_{t-1} + q_t C_{t-1}{\mathbf{x}}_t(y_t - {\bar{m}}_{t} - {\mathbf{x}}_t^\top{\mathbf{m}}_{t-1}) &
C_t &= C_{t-1} - q_t C_{t-1} {\mathbf{x}}_t {\mathbf{x}}_t^\top C_{t-1} \nonumber\\
a_t &= a_{t-1} + \frac{1}{2} &
b_t &= b_{t-1} + \frac{q_t}{2}(y_t - {\bar{m}}_{t} - {\mathbf{x}}_t^\top{\mathbf{m}}_{t-1})^2, $$ where $q_t=({\mathbf{x}}_t^\top{\mathbf{1}}+ {\mathbf{x}}_t^\top C_{t-1} {\mathbf{x}}_t)^{-1}$. In essence, the approximate observation process decouples the learning about the observational parameters: when no MU fires then $({\bar{\mu}},{\bar{\nu}})$ is updated, else $(\mu,\nu)$ is updated.
After assimilating the baseline observations $y_1,\dots,y_{\tau-1}$, both ${\bar{\nu}}^{-1}$ and ${\bar{\mu}}$ are known (and known to be small) with considerable certainty. Thus, approximating ${\bar{\nu}}^{-1}$ as $0$ and considering ${\bar{\mu}}$ to be a point mass at ${\bar{m}}$ is reasonable. Furthermore, the prior for $\nu$ does not need to be set until just before the observation $y_{\tau}$ is assimilated. Given the tight posterior for ${\bar{\nu}}$ at this juncture it is, therefore, possible to incorporate the knowledge that ${\bar{\nu}}>>\nu$ into the vague prior for $\nu$ (which is conceptually equivale
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, where $\sigma_n$ is the time it takes for the walk-on-spheres to exit the $n$th sphere. Thus $\sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq\kappa \sum_{n=0}^{N-1} \sigma_n = \kappa\,\sigma_D$. We thus have that $$\mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right)^2\right]\leq \kappa^2 \mathbb{E}_x[\sigma_D^2].$$ However, the latter expectation can be bounded by $\mathbb{E}_x[\sigma_{B^*}^2]$, where $B^* = B(x, R)$ for some suitably large $R$ such that $D$ is compactly embedded in $B^*$. Moreover, appealing to [@GET], we know that $\mathbb{E}_x[\sigma_{B^*}^2]$ is bounded.
Numerical experiments {#numerics}
=====================
In the following section, all of the routines associated with the simulations are publicly available at the following repository:
<https://bitbucket.org/wos_paper/wos_repo>
For the Monte Carlo procedure, independent copies of the walk-on-spheres $(\rho_n, n\leq N)$ need to be simulated whereby, by the Markov property, every new point in the sequence can be expressed as $\rho_{n+1}=\rho_n+X'_{\sigma'_{B(0,{r_n})} }$, where $X'$ is an independent version of $X$ and $$\sigma'_{B(0,{r_n})} = \inf\{t>0\colon X'_t \not\in B(0,{r_n})\}.$$ In other words, $\rho_{n+1}$ is an exit point from a ball $B(0,{r_n})$ under $\mathbb{P}_0$ translated by $\rho_n$. A consequence of Lemma \[BGR\] is that the exit distribution of $X'_t$ from $B(0,r_n)$, $r_n>0$, can be, via a change of variable $y = \tilde{y}/r_n$, written as $$\mathbb{P}_0(X_{\sigma_{B(0,r_n)}}\in {\rm d}\tilde{y}) = \pi^{-(d/2+1)}\Gamma(d/2)\sin(\pi\alpha/2)\left|r_n^2-|\tilde{y}|^2\right|^{-\alpha/2}|\tilde{y}|^{-d} r_n^{\alpha}\,{\rm d}\tilde{y}, \qquad |\tilde{y}|>r_n.
\label{rdy}$$ For $d=2$, it is more convenient to work with polar coordinates $(r,\theta)$ in order to separate variables in . Indeed, recalling that ${\rm d}\tilde{y}=r\,{\rm d}r\,{\rm d}\theta$, we have $$\begin{aligned}
\mathbb{P}_0(X_{\sigma_{B(0,r_n)}}\in {\rm d}\tilde{y}) & = \
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|
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|
$$
The Poincaré series of $N(k)$ is now easy to compute. First, in the [*canonical grading*]{}, shows that $$p(\Delta_d(\mu), v, W) = v^{D(d,\mu)}\frac{\sum_{\lambda}
f_{\lambda}(v) [\lambda\otimes \mu]}{\prod_{i=2}^n(1-v^i)}
\qquad\mathrm{and\ so}\qquad p(e\Delta_d(\mu), v)=
v^{D(d,\mu)} \frac{f_{\mu}(v)}{\prod_{i=2}^n (1-v^i)}$$ for any $d\in {\mathbb{C}}$. Therefore, implies that $p(e\widetilde{\Delta}_d(\mu), v)=
f_{\mu}(v) \prod_{i=2}^n (1-v^i)^{-1}
$ in the graded category $\widetilde{{\mathcal{O}}}_{d}$. Combined with this shows that $$\label{wrongsideformulaA}
p(\underline{N(k)},v) \ = \
\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) -
n(\mu^t))}}{\prod_{i=2}^n
(1-v^{i})}.$$
Finally, we calculate the Poincaré series of $\overline{N(k)}$. By Lemma \[Bbar-freeA\](2,3), an ${\mathbf{h}}$-homogeneous basis for this module is given by lifting a homogeneous ${\mathbb{C}}$-basis from $\overline{N(k)}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]}
{\mathbb{C}}= {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}} {\underline{N(k)}}.$ Thus, combining with the formulæ $p({\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}, v) = \prod_{i=2}^n (1-v^i)^{-1}$ and $ p({\mathbb{C}}[{\mathfrak{h}}^*], v) = (1-v^{-1})^{n-1}$ gives $$\label{wrongsideformula2A}
p(\overline{N(k)}, v) =
\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) -
n(\mu^t))}}{(1-v^{-1})^{n-1}}.$$ This needs to be adjusted to yield . Set $N=n(n-1)/2$. Then Lemma \[babyverma\](1) and combine to show that $$\begin{aligned}
\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) - n(\mu^t))} &=&
\sum_{\mu} f_{\mu^t}(1)f_{\mu^t}(v^{-1}) v^{k(n(\mu) - n(\mu^t))}\\
&=&
v^N\sum_{\lambda} f_{\lambda}(1)f_{\lambda}(v^{-1})
v^{k(n(\lambda^t) - n(\lambda))}.\end{aligned}$$ Moreover, rearranging gives $$[n]_v!
\ = \ \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n} \ =\
v^N\frac{\prod_{i=1}^n (1-v^{-i})}{(1-v^{-1})^n}.$$ Combining these formulæ with gives .
{#poincare-S2A}
Recall the Euler gradation $\operatorname{{\mathbf{E}}\text{-deg}}$ on $D({{\mathfrak{h}}^{\text{reg}}})\as
| 1,272
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| 0.768366
|
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|
ivalence relations $R$ on $X={{\mathbb A}}^2$ (in any characteristic) such that the geometric quotient $X/R$ exists yet $R$ is strictly smaller than the fiber product $X\times_{X/R}X$. Closely related examples are in [@venken; @philippe].
In characteristic zero, this leaves open the following:
\[sch.th.quot.quest\] Let $R\subset X\times X$ be a scheme theoretic equivalence relation such that the coordinate projections $R\rightrightarrows X$ are finite.
Is there a geometric quotient $X/R$?
A special case of the quotient problem, called gluing or pinching, is discussed in Section \[glue.sec\]. This follows the works of [@artin70], [@ferrand] (which is based on an unpublished manuscript from ’70) and [@raoult].
First examples {#first.exmp.sec}
==============
The next examples show that in many cases, the categorical quotient of a very nice scheme $X$ can be non-Noetherian. We start with a nonreduced example and then we build it up to smooth ones.
\[first.exmp\] Let $k$ be a field and consider $g_i:k[x,\epsilon]\to k[x,\epsilon]$ where $$g_1\bigl(a(x)+\epsilon b(x)\bigr)=a(x)+\epsilon b(x)
{\quad\mbox{and}\quad}
g_2\bigl(a(x)+\epsilon b(x)\bigr)=a(x)+\epsilon \bigl(b(x)+a'(x)\bigr).$$
If ${\operatorname{char}}k=0$ then the coequalizer is the spectrum of $$\ker\Bigl[k[x,\epsilon]\stackrel{g_1^*-g_2^*}{\longrightarrow}
k[x,\epsilon]\Bigr]=k+\epsilon k[x].$$ Note that $k+\epsilon k[x]$ is not Noetherian and its only prime ideal is $\epsilon k[x]$.
If ${\operatorname{char}}k=p$ then the coequalizer is the spectrum of the finitely generated $k$-algebra $$\ker\Bigl[k[x,\epsilon]\stackrel{g_1^*-g_2^*}{\longrightarrow}
k[x,\epsilon]\Bigr]=k[x^p]+\epsilon k[x].$$
It is not surprising that set theoretic equivalence relations behave badly on nonreduced schemes. However, the above example is easy to realize on reduced and even on smooth schemes.
(cf. [@holmann p.342]) Let $p_i:Z\to Y_i$ be finite morphisms for $i=1,2$. We can construct out of them an equivalence relation on $Y_1\amalg Y_2$ where $R$ is the union
| 1,273
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| 1,328
| 0.790923
|
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|
I have no idea how to read what address is currently stored in whatever OpenGL has currently bound as the vertex attrib array, so I cannot test whether or not it is pointing to the vertex data. I'm pretty sure that it's pointing to address 0 for some reason though.
Edit:
It turns out it was not the hard-coded 0 that was a problem. It removed the errors that Visual Studio and OpenGL were giving me, but the actual error was somewhere else. I realized that I was passing in the vaoID as the attributeNumber in some of the code above, when I should have been passing in a hard-coded 0. I edited my code here:
RawModel* Loader::loadToVao(float* positions, int sizeOfPositions) {
unsigned int vaoID = this->createVao();
this->storeDataInAttributeList(0, positions, sizeOfPositions);
this->unbindVao();
return new RawModel(vaoID, sizeOfPositions / 3);
}
I changed the line this->storeDataInAttributeList(vaoID, positions, sizeOfPositions); to what you see above, with a hard-coded 0. So, it turns out I wasn't even binding the array to the correct location in the vbo. But, after changing that it worked fine.
A:
You should be using your vertex attribute index with glVertexAttribPointer, glEnableVertexAttribArray and glDisableVertexAttribArray but what you've got is:
VAO id used with glVertexAttribPointer
hard coded 0 used with glEnableVertexAttribArray and glDisableVertexAttribArray (this isn't necessarily a bug though if you're sure about the value)
If you are not sure about the index value (e.g. if you don't specify the layout in your shader) then you can get it with a glGetAttribLocation call:
// The code assumes `program` is created with glCreateProgram
// and `position` is the attribute name in your vertex shader
const auto index = glGetAttribLocation(program, "position");
Then you can use the index with the calls mentioned above.
Q:
Randomly generate integers with a distribution that prefers low ones
I have an list ordered by some quality function from which I'd like to take elements, preferring t
| 1,274
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| 1,195
| 676
| 0.802583
|
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|
rmation up to an additive term. $$\label{transform}
G_2 \left( \frac{a\tau +b}{c\tau +d}\right)
= (c\tau +d)^2 G_2 (\tau) - \frac{c}{4\pi i} (c\tau +d).$$ The ring $\Q [G_2,\ G_4,\ G_6]$ is called the ring of *quasi-modular* forms (see [@kaneko-zagier]).
We have $$1+\sum_{n\geq
1}\H_{(n-1,1)}\left(e^{u/2},e^{-u/2}\right)T^n
=\frac{1}{u}\left(e^{u/2}-e^{-u/2}\right)\exp\left(2\sum_{k\geq
2}G_k(T)\frac{u^k}{k!}\right).$$ In particular, the coefficient of any power of $u$ on the left hand side is in the ring of quasi-modular forms.
The relation between the $E$-polynomial of the Hilbert scheme of points on a surface and theta functions goes back to Göttsche [@Gottsche].
Consider the classical theta function $$\label{classical-theta}
\theta (w) = (1-w) \prod_{n\ge1}
\frac{(1-q^nw)(1-q^nw^{-1})}{(1-q^n)^2},$$ with simple zeros at $q^n$, $n\in \Z$ and functional equations $$\label{functional-equation}
\begin{split}
\text{i)}\qquad& \theta (qw) = - w^{-1}\theta (w)\\
\text{ii)}\qquad& \theta (w^{-1}) = - w^{-1}\theta (w)
\end{split}$$ We have the following expansion $$\label{expansion}
\frac1{\theta (w)} = \frac1{1-w} +
\sum_{\substack{n,m >0 \\ n\not\equiv m\mod 2}}
(-1)^n\ q^{\frac{nm}2}\ w^{\frac{m-n-1}2}$$ This is classical but not that well known. For a proof see, for example, [@jordan Chap.VI, p. 453], where it is deduced from a more general expansion due to Kronecker. Namely, $$\frac{\theta(uv)}{\theta(u)\theta(v)}=\sum_{m,n\geq
0}q^{mn}u^mv^n-\sum_{m,n\geq 1}q^{mn}u^{-m}v^{-n}.$$ (To see this set $v=u^{-\tfrac12}$ and use the functional equation to get $$\frac1{\theta (w)} = \frac1{1-w} + \sum_{m,n\geq 1}q^{mn}
(w^{m-\tfrac12(n+1)}-w^{m+\tfrac12(n-1)}),$$ which is equivalent to .) It is not hard, as was shown to us by J. Tate, to give a direct proof using .
From we deduce, switching $q$ to $T$ and $w$ to $q$, that $$\label{expansion2}
\prod_{n\ge1} \frac{(1-T^n)^2}{(1-qT^n)(1-q^{-1}T^n)}
= 1 + \sum_{\substack{r,s>0\\ r\not\equiv s \mod 2}}
(-1)^r T^{\frac{rs}2} \left( q^{\frac{s-r-1}2
| 1,275
| 3,715
| 1,769
| 1,082
| 2,887
| 0.776306
|
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|
entity \otimes \identity + p(\mathbf{P})]\enspace.\end{aligned}$$ The solution is $$\label{superop-soln}
S_t = \exp\left([i(\identity \otimes H - H \otimes \identity) - p \identity \otimes \identity + p(\mathbf{P})]t\right)\enspace.$$
We now define the decoherence operator $\mathbf P$. This operator will correspond to choosing a coordinate uniformly at random and measuring it by projecting to the computational basis $\{|0\rangle, |1\rangle\}.$ Let $\Pi_0$ and $\Pi_1$ be the single qubit projectors onto $\vert 0 \rangle$ and $\vert 1
\rangle$, respectively. We define $$\mathbf P = \frac{1}{n} \sum_{1 \leq i \leq n} [\Pi^i_0 \otimes \Pi^i_0 + \Pi^i_1 \otimes \Pi^i_1]$$ where $\Pi^i_0 = \identity \otimes \cdots \otimes \identity \otimes \Pi_0 \otimes
\identity \otimes \cdots \otimes \identity$ with the nonidentity projector appearing in the $i$th place. We define $\Pi^i_1$ similarly, so that $\Pi^i_j$ ignores all the qubits except the $i$th one, and projects it onto $\vert j \rangle$ where $j \in \{0,1\}$. Note that $$\Pi^i_j \otimes \Pi^i_j = [\identity \otimes \identity]
\otimes \cdots \otimes [\Pi_j \otimes \Pi_j]
\otimes \cdots \otimes [\identity \otimes \identity]$$ for $j \in \{0,1\}$.
The superoperator as an $n$-fold tensor product
-----------------------------------------------
The pure continuous quantum walk on the $n$-dimensional hypercube is easy to analyze, in part, because it is equivalent to a system of $n$ non-interacting qubits. We now show that, with the model of decoherence described above, each dimension still behaves independently. In particular, the superoperator that dictates the behavior of the walk is decomposable into an $n$-fold tensor product.
Recall the product formulation of the non-decohering Hamiltonian $$H = \sum_{j=1}^n \identity \otimes \cdots \otimes \sigma_x
\otimes \cdots \otimes \identity$$ where $$\sigma_x = \left(\begin{matrix} 0 & k/n \\ k/n & 0 \end{matrix} \right)$$ with $\sigma_x$ appearing in the $j$th place in the tensor product. We have given each sing
| 1,276
| 2,948
| 1,530
| 1,174
| 3,655
| 0.770882
|
github_plus_top10pct_by_avg
|
ary morphisms since all morphisms preserve left Kan extensions along left adjoint functors (see [@groth:can-can Prop. 5.7] and [@groth:can-can Rmk. 6.11]). As for the second statement, there is an adjoint triple $0\dashv\pi_{[1]}\dashv 1$ and hence an induced adjoint triple $1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast$. This yields canonical isomorphisms $1_\ast\cong\pi_{[1]}^\ast$ and $0_!\cong\pi_{[1]}^\ast$.
We refer to $\pi_{[1]}^\ast\colon{\sD}\to{\sD}^{[1]}$ as the **constant morphism morphism**.
In every derivator there is an adjoint $5$-tuple $$\label{eq:5tuple}
1_!\dashv 1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast\dashv 0_\ast.$$
This is immediate from .
A derivator is pointed if and only if the adjoint $5$-tuple extends to an adjoint $7$-tuple, which is then given by $$\label{eq:7tuple}
C\dashv 1_!\dashv 1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast\dashv 0_\ast\dashv F.$$
This is immediate from and .
While in any pointed derivator there is by [@groth:ptstab Prop. 3.20] an adjunction $$({\mathsf{cof}},{\mathsf{fib}})\colon{\sD}^{[1]}\rightleftarrows{\sD}^{[1]},$$ in pointed derivators the morphism $C$ is the sixth left adjoint of $F$.
\[thm:stable-fun\] The following are equivalent for a pointed derivator ${\sD}$.
1. The derivator ${\sD}$ is stable.\[item:sf1\]
2. The cone morphism $C\colon{\sD}^{[1]}\to{\sD}$ is a right adjoint.\[item:sf2\]
3. For any homotopy finite category $A$, the colimit morphism $\colim : {\sD}^A \to {\sD}$ is a right adjoint.\[item:sf3\]
4. For any left homotopy finite functor $u:A\to B$, the left Kan extension morphism $u_!: {\sD}^A \to {\sD}^B$ is a right adjoint.\[item:sf4\]
5. The fiber morphism $F\colon{\sD}^{[1]}\to{\sD}$ is a left adjoint.\[item:sf2a\]
6. For any homotopy finite category $A$, the limit morphism $\lim : {\sD}^A \to {\sD}$ is a left adjoint.\[item:sf3a\]
7. For any right homotopy finite functor $u:A\to B$, the right Kan extension morphism $u_*: {\sD}^A \to {\sD}^B$ is a left adjoint.\[item:sf4a\]
8. The adjoint $7$-tuple extends to a doubly-i
| 1,277
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f({\mathbf x}) &\sim {\GP \left(m({\mathbf x}),\,\, k({\mathbf x},{\mathbf x}') \right)}, \\
y_i &= \mathcal{H}_{{\mathbf x},i} f({\mathbf x})+\varepsilon_i.
\end{aligned}$$
As discussed, for example, in [@Sarkka:2011; @SolinSarkka2015] the GP regression equations can be extended to this kind of models, which in this case leads to the following:
\[eq:GPreg\] $$\begin{aligned}
\mathbb{E}[f({\mathbf x}_*)|{\mathbf y}] &=
{\mathbf q}_*^{\mathsf{T}}(K +\sigma^2I)^{-1}{\mathbf y}, \\
\mathbb{V}[f({\mathbf x}_*)|{\mathbf y}] &=
k({\mathbf x}_*,{\mathbf x}_*)-
{\mathbf q}_*^{\mathsf{T}}(Q +\sigma^2I)^{-1}{\mathbf q}_*,
\end{aligned}$$
where $\mathbf{y} = \begin{bmatrix} y_1 & \cdots & y_n \end{bmatrix}^{\mathsf{T}}$ and
$$\begin{aligned}
\label{eq:crosscov}
({\mathbf q}_*)_i &= \int_{-R}^R k({\mathbf x}_i^0+s\hat{{\mathbf u}}_i,{\mathbf x}_*) ds, \\
\label{eq:Gram}
Q_{ij} &= \int_{-R}^R\int_{-R}^R k({\mathbf x}_i^0+s\hat{{\mathbf u}}_i,{\mathbf x}_j^0+s'\hat{{\mathbf u}}_j) ds ds'.\end{aligned}$$
In general we can not expect closed form solutions to – and numerical computations are then required. However, even with efficient numerical methods, the process of selecting the hyperparameters is tedious since the hyperparameters are in general not decoupled from the integrand and the integrals need to be computed repeatedly in several iterations. In this paper, we avoid this by using the basis function expansion that will be described in Section \[sec:approx\].
Squared exponential and Matérn covariance functions {#sec:cov_func}
---------------------------------------------------
An important modeling parameter in Gaussian process regression is the covariance function $k({\mathbf x},{\mathbf x}')$ which can be selected in various ways. Because the basis function expansion described in Section \[sec:approx\] requires the covariance function to be *stationary*, we here limit our discussion to covariance functions of this form. *Stationarity* means that $k({\ma
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| 0.771947
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------------------------------------------------------
The zeroth and first order $\hat{S}$ matrix elements can be calculated as follows: $$\begin{aligned}
\hat{S}_{ii}^{(0+1)} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{i k} (\Omega_{k i}) +
\left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K i}) =
e^{-i h_{i} x} (\Omega_{i i}) = e^{-i h_{i} x},
\nonumber \\
\hat{S}_{i j}^{(1)} \vert_{i \neq j} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{i k} (\Omega_{k j}) +
\left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K j}) =
e^{-i h_{i} x} (\Omega_{i j}) = 0,
\nonumber \\
\hat{S}_{i J}^{(1)} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{i j} (\Omega_{j J}) +
\left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K J}) =
e^{- i h_{i} x} (\Omega_{i J}) =
\frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\left\{ (UX)^{\dagger} A W \right\}_{i J},
\nonumber \\
\hat{S}_{J i}^{(1)} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{J k} (\Omega_{k i}) +
\left( e^{-i \hat{H}_{0} x} \right)_{J K} (\Omega_{K i}) =
\frac{e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\left\{ W ^{\dagger} A (UX) \right\}_{J i},
\nonumber \\
\hat{S}_{J K}^{(1)} \vert_{J \neq K} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{J i} (\Omega_{i K}) +
\sum_{I} \left( e^{-i \hat{H}_{0} x} \right)_{J I} (\Omega_{I K})
= \frac{e^{- i \Delta_{K} x } - e^{- i \Delta_{J} x} }{ ( \Delta_{K} - \Delta_{J} ) }
\left( W ^{\dagger} A W \right)_{J K},
\nonumber \\
\hat{S}_{J J}^{(0+1)} &=&
\sum_{i} \left( e^{-i \hat{H}_{0} x} \right)_{J i} (\Omega_{i J}) +
\sum_{I} \left( e^{-i \hat{H}_{0} x} \right)_{J I} (\Omega_{I J}) =
e^{-i \Delta_{J} x}
\left[
1 - (i x) \left( W ^{\dagger} A W \right)_{J J}
\right].
\label{hat-S-elements-1st}\end{aligned}$$
Contribution to $\hat{S}$ matrix elements from second order in $H_{1}$ {#sec:hatS-2nd}
----------------------------------------------------------------------
Likewise, $\hat{S}$ matrix elements can be calculated in second order in $\hat{H}_{1}$ by using the formula for $\Omega$ in (\[
| 1,279
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| 1,900
| 1,380
| null | null |
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2,2)$-position (because $T$ is not $d$-bad). So we can repeat the above argument and show that that there is a tableau $T'\domby T$ such that ${\hat\Theta_{T'}}$ occurs in $\theta$; contradiction.
We now know that every semistandard homomorphism occurring in $\theta$ has at least two $2$s in the second row. This means in particular that the entries $3,\dots,b+2$ lie in different columns. So we can repeat the argument from Proposition \[cdhomdim1\] and Proposition \[cd2homdim1\] to show that $\theta$ must be a linear combination of $\tau_0$ and $\tau_1$, where $\tau_i$ is the sum of all homomorphisms labelled by semistandard tableau with $i$ $2$s in the first row. If $u=a$, then $\tau_1=0$, and so the space of homomorphisms $S^\mu\to S^\la$ has dimension at most $1$. if $u>a$, then ${\psi_{2,1}}\circ\tau_0={\psi_{2,1}}\circ\tau_1\neq0$, so $\theta$ must be a scalar multiple of $\tau_0+\tau_1$ and again the homomorphism space has dimension at most $1$.
Composing the homomorphisms
---------------------------
We have constructed homomorphisms $S^\mu\stackrel{{\hat\Theta_{C}}}\longrightarrow S^\la\stackrel\sigma\longrightarrow S^{\mu'}$, and shown that these homomorphisms are unique up to scaling. To complete this section, we just need to compute the composition of these homomorphisms.
Let $E$ be the $\mu$-tableau $${\text{\footnotesize$\gyoungx(1.2,;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;{v\!\!+\!\!1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v,;1;2)$}}$$ of type $\mu'$. Then we have the following.
\[comp2\] For $T\in{\calu}$, we have ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}\neq0$, and therefore we have $\sigma\circ{\hat\Theta_{C}}\neq0$ if and only if $\mbinom{u-v}{a-v}$ is odd.
It is easy to express ${\hat\Theta_{E}}$ as a linear com
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| 1,189
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| 0.80236
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|
by stacking the $n-k+j-1$ eigenvectors associated to the smallest eigenvalues. Computation of constraints of the $l_1$ minimization problem Compute $T=V_{2,n}^t$ Compute $W=T^{-Index[j]}$ and $w=T^{Index[j]}$(where $T^{Index[j]}$ is the column $Index[j]$ of $T$ and $V^{-I_j}$ the matrix $T$ wihtout the column $Index[j]$) Compute the solution $v^j$ of the following problem
$\underset{Wv=-w}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $.
Recovery of the $j_{th}$ cluster:
$\tilde{v}_j=[v^j_{1} \ v^j_{2} \ \dots \ v^j_{Index[j] -1} \ 1 \ v^j_{Index[j]+1} \ \dots \ v^j_{n}] $ $F$ concatenation of the $j_{th}$ clusters and deflation of $A=A - \tilde{v}_j ^t \tilde{v}_j$ to recover the other indicator vectors $F((F>0.5))=1$ $F((F \leq 0.5))=0$ $k$ column vectors $F$.
$\ell_1$-spectral clustering applications {#section6}
==========================================
Spectral clustering and $\ell_1$-spectral clustering on simulated dataset
-------------------------------------------------------------------------
### Performances
In Section \[section5\], we introduced a new algorithm (called $\ell_1$-spectral clustering) that aims to detect cluster structures in complex graphs. To illustrate graphically the performances of this method, we simulated a perturbed version of a graph with an exact group structure. The associated adjacency matrix is composed of $k \in [5,10]$ blocks of size $c_{n-k+1},\dots, c_n \in [10,20]$. Let $p$ be the level of Bernoulli noise applied on the adjacency matrix.
Once the matrix is disturbed by a strictly positive coefficient, we no longer have exact block structures. To recover it, we applied the traditional spectral clustering algorithm and the new $\ell_1$-spectral clustering algorithm. Figure \[recovery\] gives the performances of both algorithm with a perturbation coefficient of $p=2$.
0.2in
![Input Adjacency matrix: Adjacency matrix with exact community structure. Perturbed Adjacency matrix $p=0.2,0.3$: Adjacency matrix after perturbation. L1 Adjacency matrix: Adj
| 1,281
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x_k\tilde x_l\biggr)+
\biggl(\sum_m \tilde
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_k
x_l\biggr)\leq\sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
k}}\tilde x_m+\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr).$$ The right hand side equals $$\begin{aligned}
\sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
k}}\tilde x_m+\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr)&= \sum_k
x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde x_m\biggr)+
\sum_{\substack{{k,l}\\l<k}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
l}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
l}}\tilde x_m\biggr)\\
&= \sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde x_m\biggr)+
\sum_{\substack{{k,l}\\k\neq
l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr)\\
&=\sum_{k,l}x_kx_l\biggl(\sum_{\substack{m\\m\neq l}}\tilde
x_m\biggr)=\biggl(\sum_k
x_k\biggr)\biggl(\sum_{\substack{{m,l}\\m\neq l}}x_l\tilde
x_m\biggr),\end{aligned}$$ hence the previous inequality is the same as $$\biggl(\sum_m x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}\tilde x_k\tilde x_l\biggr)+
\biggl(\sum_m \tilde
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_k x_l\biggr)\leq
\biggl(\sum_k x_k\biggr)\biggl(\sum_{\substack{{m,l}\\m\neq
l}}x_l\tilde x_m\biggr).$$ The first factors are equal and positive, hence after renaming $m,l$ to $k,l$ when $m<l$ and to $l,k$ when $m>l$ on the right hand side we are left with proving $$\sum_{\substack{{k,l}\\k<l}}(\tilde x_k\tilde x_l+x_k x_l)\leq
\sum_{\substack{{k,l}\\k<l}}(\tilde x_k x_l+x_k\tilde x_l).$$ This can be written in the elegant form $$\sum_{\substack{{k,l}\\k<l}}(\tilde x_k-x_k)(\tilde x_l-x_l)\leq
0.$$ However, $$0=\biggl(\sum_k (\tilde x_k-x_k)\biggr)^2=\sum_{k,l}(\tilde x_k-x_k)(\tilde x_l-x_l)
=\sum_k(\tilde x_k-x_k)^2 +2\sum_{\substack{{k,l}\\k<l}}(\tilde
x_k-x_k)(\tilde x_l-x_l),$$ so that $$\sum_{\substack{{k,l}\\k<l}}(\
| 1,282
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| 1,491
| 1,332
| null | null |
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|
and $33$% for $A_1$ units ($p < 0.0005$). For $R_1$ units, the median %LS decrease was $2$% ($p=0.18$). Indeed, the average $R_1$ response did not exhibit much length suppression (see Supplement), though, there were particular examples with a strong effect.
#### Sequence Learning Effects in Visual Cortex
Predictions are often informed by recent experience, and violations of these predictions can be highly salient. Meyer and Olson [@Meyer_2011] provided a striking example of this in visual cortex via image sequence learning. The authors exposed monkeys to image pairs in a fixed order for over $800$ trials for each pair. The left panel of Fig. \[image\_pairing\] shows the mean response of $81$ IT neurons in a subsequent testing period, for predicted and unpredicted pairs. When the second image differs from expectations, the response is much stronger than when the expected image is presented.
![Predicted vs. unpredicted image transitions. Left: Mean of $81$ neurons recorded in macaque (IT). Reproduced with permission from [@Meyer_2011]. Right: Mean ($\pm$ SE) across PredNet $E_3$ units.[]{data-label="image_pairing"}](image_pairing-relu.pdf){width="90.00000%"}
![Learned image transitions. Top: Predictions of a KITTI-trained PredNet model on an example sequence. Middle: PredNet predictions after repeated “training” on the sequence. Bottom: PredNet predictions for an unpredicted image transition.[]{data-label="image_pairing_examples"}](image_pairing_examples-relu_v2.pdf){width="90.00000%"}
The right panel of Fig. \[image\_pairing\] demonstrates a similar effect in the PredNet after an analogous experiment. The model was trained on five image pairs for $800$ epochs. Fig. \[image\_pairing\_examples\] contains an example sequence and the corresponding next-frame predictions before and after the training. The model, prior to exposure to the images in this experiment (trained only on KITTI), settles into a noisy, copy-last-frame prediction mode. After exposure, the model is able to successfully make predictions
| 1,283
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| 886
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|
github_plus_top10pct_by_avg
|
_{c_{\k\sigma} ,d^\dagger_\sigma}(z)
&=& V_k G_{d_\sigma,d^\dagger_\sigma}(z) \\
&& \nonumber
+\lambda_c \frac{V_k}{V_0} N_\sigma(z)
.\end{aligned}$$ The off-diagonal composite correlation function $$\begin{aligned}
N_\sigma(z) &=& G_{\hat X_0 c_{0\sigma} ,d^\dagger_\sigma}(z)\end{aligned}$$ accounts for the correlations between hybridization process and the vibrational displacement $\hat X_0$. We have seen its explicit importance for the renormalization of bare hybridization via Ref. . Defining $$\begin{aligned}
\Delta_\sigma(z)&=& \sum_k \frac{V_k^2}{z- \e_{\k\sigma}}\end{aligned}$$ and using the standard parametrization of the GF in terms of self-energy corrections $\Sigma_\sigma$, $$\begin{aligned}
G_{d_\sigma,d^\dagger_\sigma}(z) &=& \frac{1}{z-\e_d -\Delta_\sigma(z) - \Sigma_\sigma(z)}\end{aligned}$$ the self-energy can be expressed [@BullaHewsonPruschke98] as $$\begin{aligned}
\label{eq:APP-self-energy}
\Sigma_\sigma(z) &=&
\frac{U F_\sigma(z) +\lambda_d M_\sigma(z) +
\frac{ \lambda_c}{V_0}\Delta(z) N_\sigma(z) }{G_{d_\sigma,d^\dagger_\sigma}(z)}\end{aligned}$$ Since the NRG can calculate each individual Green function $F_\sigma(z), M_\sigma(z),
N_\sigma(z)$ and $G_{d_\sigma,d^\dagger_\sigma}(z)$, Bulla et al. [@BullaHewsonPruschke98] have shown that replacing the GFs on the right side of by the NRG results yields a self-energy that becomes almost independent of the NRG discretization parameters and, therefore, is an accurate representation of the true self-energy for the continuum model. Eq. is analytically exact and is also used in the main text to present an better analytical understanding of the numerical finding.
[117]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1103/PhysRevLett.17.1139) [****, ()](\doibase 10.1103/PhysRev.165.821) @noop [
| 1,284
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| 1,553
| 1,229
| 2,677
| 0.777926
|
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|
of the diagonal with two copies of $Z$, one of which maps as $$(p_1,p_2):Z\to Y_1\times Y_2\subset \bigl( Y_1\amalg Y_2\bigr)
\times
\bigl( Y_1\amalg Y_2\bigr),$$ the other its symmetric pair. The categorical quotient $\bigl(\bigl( Y_1\amalg Y_2\bigr)/R\bigr)^{cat}$ is also the universal push-out of $Y_1\stackrel{p_1}{\leftarrow} Z
\stackrel{p_2}{\to}Y_2$. If $Z$ and the $Y_i$ are affine over $S$, then it is the spectrum of the $S$-algebra $$\ker\Bigl[{{\mathcal O}}_{Y_1}+{{\mathcal O}}_{Y_2}\stackrel{p_1^*-p_2^*}{\longrightarrow} {{\mathcal O}}_Z\Bigr].$$
For the first example let $Y_1\cong Y_2:={\operatorname{Spec}}k[x,y^2,y^3]$ and $Z:={\operatorname{Spec}}k[u,v]$ with $p_i$ given by $$p_1^*: (x,y^2,y^3)\mapsto (u,v^2,v^3)
{\quad\mbox{and}\quad}
p_2^*:(x,y^2,y^3)\mapsto (u+v,v^2,v^3).$$ Since the $p_i^*$ are injective, the categorical quotient is the spectrum of the $k$-algebra $k[u,v^2,v^3]\cap k[u+v,v^2,v^3]$. Note that $$\begin{array}{lll}
k[u,v^2,v^3] & = & \bigl\{f_0(u)+
\sum_{i\geq 2} v^i f_i(u)\ :\ f_i\in k[u]\bigr\}{\quad\mbox{and}\quad}\\
k[u+v,v^2,v^3] & = & \bigl\{f_0(u)+vf'_0(u)+\sum_{i\geq 2} v^i f_i(u)
\ :\ f_i\in k[u]\bigr\}.
\end{array}$$ As in (\[first.exmp\]), if ${\operatorname{char}}k=0$ then the categorical quotient is the spectrum of the non-Noetherian algebra $k+\sum_{n\geq 2}v^nk[u]$. If ${\operatorname{char}}k=p$ then the geometric quotient is given by the finitely generated $k$-algebra $$k[u^p]+\sum_{n\geq 2}v^nk[u].$$
This example can be embedded into a set theoretic equivalence relation on a smooth variety.
Let $Y_1\cong Y_2:={{\mathbb A}}^3_{xyz}$, $Z:={{\mathbb A}}^2_{uv}$ and $$p_1^*: (x_1,y_1,z_1)\mapsto (u,v^2,v^3)
{\quad\mbox{and}\quad}
p_2^*(x_2,y_2,z_2)\mapsto (u+v,v^2,v^3).$$ By the previous computations, in characteristic zero the categorical quotient is given by $$k+(y_1,z_1)+(y_2, z_2)\subset k[x_1,y_1,z_1]+k[x_2,y_2,z_2],$$ where $(y_i,z_i)$ denotes the ideal $(y_i,z_i)\subset k[x_i,y_i,z_i]$. A minimal generating set is given by $$y_1x_1^m, z_1x_1^m, y_2x_2^m, z_
| 1,285
| 1,374
| 1,310
| 1,215
| 2,491
| 0.779352
|
github_plus_top10pct_by_avg
|
ae} j_{\bar z}^d (w)
\nonumber \\
& = - c_- {f^{ac}}_g (c_2-g) j_{\bar z}^g (w)+ c_+ {f^{ac}}_g c_4 j_{\bar z}^g(w)
-i (-1)^a (-1)^a {f^{ac}}_{g} \tilde{c} j_{\bar z}^g (w)\end{aligned}$$ which also vanishes thanks to the relation : $$\begin{aligned}
-c_- (c_2-g) + c_+ c_4 - i \tilde{c} &=& 0.\end{aligned}$$
3\. There are terms proportional to $1/(\bar z - \bar w)^2$ with coefficients: $$\begin{aligned}
c_- (c_4-g) {f^{ac}}_g j^g_z + c_+ (c_4-g) {f^{ac}}_g j_z^g
-i (-1)^{ea} {f^c}_{de} c_3 \kappa^{ad} j_z^e
-i {f^c}_{de} {f^{ae}}_g (c_4-g)^2 {f^{gd}}_h j^h_z
\nonumber\end{aligned}$$ where the last term arises from expanding $1/(z-x)$ and taking into account the further contraction in Term 4. This last term vanishes thanks to the super-Jacobi identity combined with the vanishing of the Killing form. Note that this implies that the second line in Term 4 does not contribute when the contraction between $j^g_z$ and $j^d_{\bar z}$ gives rise to either a metric or structure constant. Thus, it can potentially contribute starting at order zero in the separation only. The coefficient of the terms under consideration then vanishes since the coefficient satisfies the relation : $$\begin{aligned}
(c_-+c_+)(c_4-g) +i c_3
&=& 0 .\end{aligned}$$
4\. We now turn to the calculation which is new compared to [@Ashok:2009xx]. In the operator product expansion the simple pole in $1/(\bar z - \bar w)$ comes with the coefficient : $$\begin{aligned}
c_-& {f^{ac}}_g (c_4-g) \partial_{\bar z} j_z^g(w)
-c_+ {f^{ac}}_g c_4 \partial_{z} j_{\bar z}^g (w)
\nonumber \\ &
-c_- \frac{g}{4} {f^{ac}}_g (\partial_z j_{\bar z}^g-
\partial_{\bar z} j_{ z}^g)
+c_- {B^{ac}}_{gh} : j_z^{g} j^{h}_{\bar z}:(w)
\nonumber \\ &
+c_+ \frac{g}{4} {f^{ac}}_g (\partial_z j_{\bar z}^g-
\partial_{\bar z} j_{ z}^g)
+c_+ {B^{ac}}_{gh} : j_z^{g} j^{h}_{\bar z}:(w)
\nonumber \\ &
-i (-1)^{ea} {f^{c}}_{de} {f^{ad}}_g c_4 : j_z^e j_{\bar z}^g:(w)
\nonumber \\ &
-i {f^{c}}_{de} {f^{ae}}_g (c_4-g) :j_z^g j^d_{\bar z}
\nonumber \\ &
- i {f^{c}}_{de} {f^{ae}
| 1,286
| 1,965
| 1,499
| 1,244
| null | null |
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|
isms to graded $R_{\mathbb{Z}}$-module homomorphisms, $\Phi$ is a functor.
Conversely, suppose that $\widetilde{N}\in R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$ and pick a preimage $N\in
R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Then $N$ is generated by $\bigoplus_{i=0}^a N_i$, for some $a$, and so $N_j=R_{ja}N_a$, for all $j\geq a$. For $j\geq i\geq a$ we have natural maps of $R_a$-modules $$\theta_{ji}: R^*_{ia}\otimes N_i \cong
R^*_{ia}\otimes R^*_{ji}\otimes R_{ji}\otimes N_i \cong
R^*_{ja}\otimes (R_{ji}\otimes N_i )
\twoheadrightarrow R^*_{ja}\otimes N_j,$$ where the tensor products are over the appropriate $R_k$. By the associativity of tensor products, $\theta_{ki}=\theta_{kj}\theta_{ji}$, for all $k\geq j\geq
i\geq a$. Since each $N_i$ is a noetherian $R_i$-module, each $R^*_{ia}\otimes N_i$ is a noetherian $R_a$-module and so $\theta_{ji}$ is an isomorphism for all $j\geq i\gg 0$. Equivalently, $N_j\cong R_{ji}\otimes N_i$ for all such $j\geq i$.
Set $\Theta(\widetilde{N})
=R^*_{j0}\otimes N_j\in R_0{\text{-}{\textsf}{mod}}$ for some $j\gg 0$. Since any two preimages of $\widetilde{N}$ in $R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$ agree in high degree, $\Theta(\widetilde{N})$ is independent of the choice of $N$. Moreover, as $R^*_{j0}=R^*_{k0}R_{kj}$, $$\phi(\Theta(\widetilde{N}))_{\geq j}
\cong \bigoplus_{k\geq j} R_{k0}\otimes R^*_{j0}\otimes N_j
\cong \bigoplus_{k\geq j} R_{kj}\otimes N_j = \bigoplus_{k\geq j}N_k,$$ and so $\Phi\Theta(\widetilde{N}) = \widetilde{N}.$ Checking that $\Theta$ and $\Phi$ are inverse equivalences is now routine.
{#section-1}
We remark that many of the standard techniques and results concerned with associated graded modules for unital algebras extend routinely to ${\mathbb{Z}}$-algebras. These only appear in peripheral ways in this paper and so we refer the reader to [@GS2] for a discussion of these results.
The main theorem {#sect-filt}
================
{#sect601}
In this section we prove the main theorem of the paper by proving Theorem \[mainthm-intro\] fr
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eq{\vbx'-z{|\!|\!|}}\vee{\vbz-y'{|\!|\!|}}$. Suppose that ${\vbx-z{|\!|\!|}}\leq{\vbz-y{|\!|\!|}}$ and ${\vbx'-z{|\!|\!|}}\leq{\vbz-y'{|\!|\!|}}$. Then, by [(\[eq:conv\])]{} with $a=b=q$, the contribution from this case is bounded by $$\begin{aligned}
\frac{2^{2q}}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}\sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\,
\frac1{{\vbx'-z{|\!|\!|}}^q}\leq\frac{2^{2q}c{\vbx-x'{|\!|\!|}}^{d-2q}}{{\vbx-y{|\!|\!|}}^q
{\vbx'-y'{|\!|\!|}}^q},\end{aligned}$$ for some $c<\infty$, where we note that ${\vbx-x'{|\!|\!|}}^{d-2q}\leq1$ because of $\frac12d<q$. The other three possible cases can be estimated similarly (see Figure \[fig:star\](a)). This completes the proof of Proposition \[prp:conv-star\].
$$\begin{aligned}
\begin{array}{cc}
\text{(a)}&{\displaystyle}\sum_z~~\raisebox{-1.4pc}{\includegraphics[scale=0.2]
{star1}}~~~~\lesssim~~~~\raisebox{-1.4pc}{\includegraphics[scale=0.2]
{star2}}\\[2pc]
\text{(b)}&\qquad{\displaystyle}\sum_{u_j,v_j}~~\raisebox{-21pt}{\includegraphics
[scale=0.2]{fish1}}~~~~\lesssim~~~~\sum_{v_j}~~\raisebox{-14pt}{
\includegraphics[scale=0.2]{fish2}}~~~~\lesssim~~~\raisebox{-14pt}{
\includegraphics[scale=0.2]{fish3}}
\end{array}\end{aligned}$$
Before going into the proof of Proposition \[prp:GimpliesPix\], we summarize prerequisites. Recall that [(\[eq:Q’-def\])]{}–[(\[eq:Q”-def\])]{} involve $\tilde G_\Lambda$, and note that, by [(\[eq:G-delta-bd\])]{}, $$\begin{aligned}
{\label{eq:pi0-1stbd}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\leq\delta_{o,x}+\tilde G_\Lambda(o,x)^3.\end{aligned}$$ We first show that $$\begin{aligned}
{\label{eq:tildeG-bd}}
\tilde G_\Lambda(o,x)\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q},&&
\sum_{b:{\underline{b}}=o}\tau_b\big(\delta_{{\overline{b}},x}+\tilde G_\Lambda({\overline{b}},x)\big)
\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}\end{aligned}$$ hold assuming the bounds in [(\[eq:IR-xbd\])]{}.
By the assumed bound $\tau\leq2$ in [(\[eq:IR-xbd\])]{}, we have $$\begin{aligned}
{\label{eq:tildeG-1stbd}}
\tilde G_\Lambda(o,x)=\t
| 1,288
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mm} \put(20,40){\line (1,0){20}}
\put(80,40){\line(1,0){20}} \qbezier(40,40)(40,10)(70,10)
\qbezier(70,10)(100,10)(100,40)
\put(160,40){\line (1,0){20}} \put(220,40){\line(1,0){20}}
\qbezier(180,40)(180,70)(210,70) \qbezier(210,70)(240,70)(240,40)
\end{picture}$$ with order two group of automorphisms each. Thus, they generate $3+3=6$ tree-rooted cubic maps.
So, we have $$30+18+8+2+6+6=70$$ tree-rooted cubic maps, as expected.
[99]{} Albenque M. and Poulalhon D., *Generic method for bijections between blossoming trees and planar maps*, arXiv: 1305.1312. Bernardi O., *Bijective counting of tree-rooted maps and shuffles of parenthesis systems*, arXiv:math/0601684. Goulden I.P. and Jackson D.M., *The KP hierarchy, branched covers, and triangulations*, arXiv: 0803.3980. Lando S. and Zvonkin A., *Graphs on surfaces and their applications. Encyclopedia of mathematical sciences* **141**, Springer-Verlag, Berlin, 2004. Mullin R.C., *On the enumeration of tree-rooted maps*, Canad. J. Math., 1967, 19, 174-183. Poulalhon D. and Schaeffer G., *Optimal coding and sampling of triangulations*, Algorithmica, 2006, 46(3), 505-527. Stanley R.P., *Enumerative combinatorics, volume 2*, Wadsworth & Brooks, 1999. Tutte W.T., *A census of planar triangulations*, Canad. J. Math., 1962, 14, 21-38.
---
abstract: 'We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.'
address:
- 'Jürgen Herzog, Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany'
- 'Dorin Popescu, Institute of Mathematics “Simion Stoilow”, University of Bucharest, P.O.Box 1-764, Bucharest 014700, Romania'
author:
- Jürgen Herzog and Dorin Popescu
title: Finite filtrations of modules and shellable multicomplexes
---
[^1]
Introduction {#introduction .unnumbered}
============
Let $R$ be a Noetherian ring, and $M$ a finitely generate
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in by giving a general context for all three results.
{#tens-defn-sect}
For fixed $i\geq j\geq 0$ we are interested in the following tensor product decompositions $$\label{tensor-1}
B_{ij}\cong Q_{c+i-1}^{c+i}\otimes Q_{c+i-2}^{c+i-1}\otimes\cdots \otimes
Q_{c+j}^{c+j+1},$$ $$\label{tensor-101}
N(i)\cong Q_{c+i-1}^{c+i}\otimes \cdots \otimes
Q_{c}^{c+1}\otimes eH_c\qquad\mathrm{or}\qquad
N(i) \cong B_{i0}\otimes eH_c$$ and $$\label{app-c-cor}
M(i)\cong H_{c+i}\delta e\otimes_{U_{c+i-1}} B_{i-1,i-2}\otimes
\cdots\otimes_{U_{c+1}} B_{10}
\qquad\mathrm{or}\qquad
M(i) \cong H_{c+i}\delta e\otimes_{U_{c+i-1}}B_{i-1,0}$$ where the tensor products are over the appropriate rings $U_k$. Corresponding to these decompositions we have the [*tensor product filtration*]{} $\operatorname{{\textsf}{ten}}$ defined by the following convention: Given a module $C=C_1\otimes\cdots \otimes C_r$, where each $C_j$ is filtered by the $\operatorname{{\textsf}{ord}}$ filtration, define $$\label{tensor-2}
\operatorname{{\textsf}{ten}}^n(C) = \Big\{\sum c_{1}\otimes \cdots\otimes c_r,
\ \mathrm{where}\ c_m\in \operatorname{{\textsf}{ord}}^{\ell(m)}(C_m) \ \mathrm{with}\
\sum_{m=1}^{r} \ell(m)\leq n\Big\}.$$ As usual, we will write the associated graded module as $\operatorname{{\textsf}{tgr}}C = \bigoplus
\operatorname{{\textsf}{ten}}^n C/\operatorname{{\textsf}{ten}}^{n-1} C$.
\[ord-tens\] Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\]. Let $C$ denote one of the objects $B_{ij}$, $N(i)$ or $M(i)$ and consider the tensor product filtrations induced from one of the tensor product decompositions (\[tensor-1\]–\[app-c-cor\]). Then $\operatorname{{\textsf}{ord}}^mC=\operatorname{{\textsf}{ten}}^mC$, for all $m\geq 0$.
We will prove the result for the decomposition and the first decomposition in each of and . The proof in the remaining cases is left to the reader as it uses essentially the same argument, although one needs to use the conclusion of the lemma for .
In each of the three cases we are given a decompositi
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e{R}}$.
From ${{\Psi}\negmedspace\mid\negmedspace{R}} \subseteq \Phi$ and $\Phi \subseteq {{\Psi}\negmedspace\mid\negmedspace{R}}$ we infer $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$.
\[L:SUBSET\_SUBSPACE\] Let $\Psi$ and $\Phi$ be ensembles. If $\Phi \subseteq \Psi$, then ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$.
Set $R = {{\operatorname{dom}{\Phi}}}$. Since $\Phi \subseteq \Psi$ by hypothesis, then by applying lemma \[L:SUBSET\_RESTRICTION\] we infer $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$. With this equality and Theorem \[T:SPACE\_UNIQ\_ENSEMBLE\] (invertibility of the Cartesian product), we have ${\prod{\Phi}} = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\] asserts that the restriction of the choice space equals the choice space of the restriction: $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Transitivity of equality implies ${\prod{\Phi}} = (\thinspace\prod\Psi) \mid R$. This last equality is exactly the premise of definition \[D:SUBSPACE\]: ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$.
\[T:SUBSET\_IFF\_SUBSPACE\] Let $\Psi$ and $\Phi$ be ensembles. ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$ if and only if $\Phi \subseteq \Psi$.
Lemma \[L:SUBSPACE\_SUBSET\] asserts that if ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$, then $\Phi \subseteq \Psi$. Lemma \[L:SUBSET\_SUBSPACE\] asserts that if $\Phi \subseteq \Psi$, then ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$. This pair of converse implications establishes the biconditional.
\[L:ENSEMBLE\_PROD\_SUBSETS\] If $\Upsilon$, $\Psi$, and $\Phi$ are ensembles such that $\Upsilon = \Psi\Phi$, then $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$.
Since $\Upsilon$ is the dyadic product of $\Psi$ and $\Phi$, then by definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $\Psi$ and $\Phi$ are disjoint ensembles and $\Upsilon = \Psi \cup \Phi$.
Suppose $i \in {{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{(\Psi \cup \Phi)}}}$. Through de
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tcome, two died without requiring dialysis, and their uNGAL levels decreased within 48 h (details in Table [3](#Tab3){ref-type="table"} and Supplemental Table [7](#MOESM2){ref-type="media"}).Table 3Association between tracheotomy and discharge modality with regard to patients suffering from AKI with and without required dialysis.Discharge modality depending on tracheotomy and AKI req. dialysisAKI req. dialysis andNo AKI andTracheotomyTracheotomyYesNoTotalYesNoTotaldischarge mortalityfavorable13403939adverse707202Total831123941p (Fisher Test)P = 0.0242\*P = 0.0012\*Data listed in each of the combination cell corresponds to absolute frequency. \**p* \< 0.05.
Repeated measures analysis of uNGAL {#Sec10}
-----------------------------------
In a longitudinal analysis, a significant impact of the baseline value of uNGAL on the post-surgical measured levels of uNGAL within the first 48 h after admission to ICU (Table [4](#Tab4){ref-type="table"}; base: *p* = 0.0011) was observed. Overall, uNGAL did not change significantly over time (base: *p* = 0.6935). The repeated factors urea, urine, and serum creatinine were also identified as significantly related to uNGAL (Table [4](#Tab4){ref-type="table"}; urea: *p* = 0.0462; urine: *p* = 0.0044; serum creatinine: *p* = 0.0018). Nevertheless, the slope estimate of urine was close to zero.Table 4Repeated measures analysis of uNGAL in a single parameter and Multivariable analysis of the longitudinal model.Linear mixed model for log(uNGAL)CovariablesEstimateSEM (Estimate)DF resp Num DF/Den DFt- resp. F-valuep-valueBase model Intercept0.96060.222871.54.31\<0.0001\* Baseline (uNGAL)0.10310.029750.13.470.0011\* Time Point (overall)3/1130.480.6935 ICU (Reference)0\.... 12 h after ICU0.11540.14871270.780.4390 24 h after ICU−0.01520.1496118−0.100.9192 48 h after ICU−0.06080.1568122−0.390.6989**Single parameter analysis of the longitudinal model**Continuous covariables Age0.015830.0143548.71.1
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at\beta(\hat{S})=\overline{Y}(\hat{S})$ for $2n$ (non-splitting) and $n$ (splitting). In this example we see that indeed, the splitting estimator suffers a larger risk. In this example, $D=1,000$, $n=50$, and $\beta = (a,0,\ldots, 0)$. The horizontal axis is $a$ which is the gap between the largest and second largest mean.
![*Horizontal axis: the gap $\beta_{(1)} - \beta_{(2)}$. Blue risk: risk of splitting estimator. Black line: risk of non-splitting estimator.*[]{data-label="fig::price"}](PriceOfSplitting)
To summarize: splitting gives more precise estimates and coverage for the selected parameter than non-splitting (uniform) inference. But the two approaches can be estimating different parameters. This manifests itself by the fact that splitting can lead to less precise estimates of the population parameter $\theta$. In the regression setting, this would correspond to the fact that splitting the data can lead to selecting models with poorer prediction accuracy.
Comments on Non-Splitting Methods {#section::comments}
=================================
There are several methods for constructing confidence intervals in high-dimensional regression. Some approaches are based on debiasing the lasso estimator [e.g., @zhang2014confidence; @vandegeer2014asymptotically; @javanmard2014confidence; @nickl2013confidence See Section \[sec:related\]]. These approaches tend to require that the linear model to be correct as well as assumptions on the design, and tend to target the true $\beta$ which is well-defined in this setting. Some partial exceptions exist: [@peter.sarah.2015] relaxes the requirement of a correctly-specified linear model, while [@meinshausen2015group] removes the design assumptions. In general, these debiasing approaches do not provide uniform, assumption-free guarantees.
[@lockhart2014significance; @lee2016exact; @taylor2014exact] do not require the linear model to be correct nor do they require design conditions. However, their results only hold for parametric models. Their method works by inverting a
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E}\left[\left.{g}(X_{\sigma_D}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s
+ \int_{\sigma_{B(x')}}^{\sigma_D} {f}(X_s)\,{\rm d}s\right|\mathcal{F}_{\sigma_{B(x')}}\right]\right] \notag\\
& = \mathbb{E}_x\left[\upsilon (X_{\sigma_{B(x')}}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s\right], \qquad x\in D,
\label{localisedv}
\end{aligned}$$ where $(\mathcal{F}_t, t\geq 0)$ is the natural filtration generated by $X$. Thanks to the fact that Theorem \[hasacorr\] is valid on balls, we see immediately that the right-hand side of is the unique solution to $$\begin{gathered}
\begin{aligned}
-(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in B(x'), \\
u(x) & = \upsilon(x), & x & \in B(x')^{\rm c}.
\end{aligned}
\label{upsilondirichlet}
\end{gathered}$$ That is to say, $\upsilon$ solves . Note that it is at this point in the argument that we are using the condition ${f}\in C^{\alpha +\varepsilon}(\overline{D}) $. Since the solution to is defined on $B(x')$ and $x'$ is chosen arbitrarily in $D$, we conclude that $\upsilon$ solves $$\begin{gathered}
\begin{aligned}
-(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in D, \\
u(x) & = \upsilon(x), & x & \in D^{\rm c}.
\end{aligned}
\label{upsilondirichletonD}
\end{gathered}$$ On account of the fact that $\mathbb{P}_x(\sigma_D =0) = 1$ for all $x\in D^{\rm c}$, it follows that $\upsilon = {g}$ on $D^{\rm c}$ and hence is identical to .
[*Uniqueness:*]{} Suppose that $\hat{u}$ solves , then, in particular, for any $x'\in D$, it must solve $$\begin{gathered}
\begin{aligned}
-(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in B(x'), \\
u(x) & = \hat{u}(x), & x & \in B(x')^{\rm c}.
\end{aligned}
\end{gathered}$$ As we k
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instantaneous capacity of a MIMO system whose channel matrix has correlated zero-mean complex Gaussian entries can be approximated by a Gaussian variable [@Moustakas_03_Mctccitpocian; @Martin_03_aedacfcucf]. Based on the discussion above, the distribution of $R_{\psi}$ is approximated by a Gaussian distribution, and the accuracy of the approximation will also be numerically shown later in Section \[sec:numersim\]. It is worth mentioning that practical mmWave channels may have relatively small numbers of clusters and paths. Different from the results in [@Gustafson_14_ommcacm; @Maltsev_10_cmf60gwsmodl] for 60 GHz, one may observe only 3-6 clusters at 28 GHz [@Raghavan8255763; @Raghavan8053813]. Although the distribution of ${\widetilde{\mathbf{H}}_{\psi,V}}(i,j)$ may deviate from Gaussian when the number of distinct paths associated with it becomes small, we find that $R_{\psi}$ is still approximately Gaussian distributed. Note that Gaussian approximated distribution for the rate of MIMO systems has been shown many times in the literature under various assumptions [@Telatar_99_CmGaucs; @Moustakas_03_Mctccitpocian; @Smith_04_Aappcdfms; @Martin_03_aedacfcucf], and the Gaussianity of $R_{\psi}$ with randomness in the angular profiles of the clusters is reasonable.
Denoting the approximated Gaussian distribution of $R_{\psi}$ as $\mathcal{N}({\bar{R}_{\psi}},{\sigma^2_{R_\psi}})$, where ${\bar{R}_{\psi}}$ and ${\sigma^2_{R_\psi}}$ denote the mean and the variance of $R_\psi$, respectively, we have the following proposition giving the approximated average throughput gain of employing the reconfigurable antennas.
\[Prop:1\] The average throughput gain of employing the reconfigurable antennas with $\Psi$ distinct reconfiguration states is approximated by $$\label{eq:th_gain_close}
G_{\bar{R}} \approx\int_0^\infty \frac{1}{{\bar{R}_{\psi}}}-\frac{1}{2^\Psi{\bar{R}_{\psi}}}\left(1+\mathrm{erf}\left(\frac{x-{\bar{R}_{\psi}}}{\sqrt{2{\sigma^2_{R_\psi}}}}\right)\right)^\Psi\mathrm{d}x. $$
See Appendix \[App:proofav
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] Let $G^{\ddag}$ be the subfunctor of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ consisting of those $m$ satisfying Equations (\[ea20\]), (\[ea22\]), (\[24\]), (\[24’\]), (\[ea25\]), (\[ea27\]), and (\[ea32\]). Note that such $m$ also satisfies Equation (\[32’\]). Then $G^{\ddag}$ is represented by a smooth closed subscheme of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ and is isomorphic to $ \mathbb{A}^{l^{\prime}}\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$ as a $\kappa$-variety, where $\mathbb{A}^{l^{\prime}}$ is an affine space of dimension $l^{\prime}$. Here, $$\begin{aligned}
l^{\prime}=\sum_{i<j}n_in_j +\sum_{i:\mathrm{even~and~} L_i:\textit{of type }I^e}(n_i-1) +
\sum_{i:\mathrm{odd~and~}L_i:\textit{free of type I}}(2n_i-2) \notag \\
- \sum_{i:\mathrm{even~and~} L_i:\textit{bound of type II}}n_i +\#\{i:\textit{$i$ is even and $L_i$ is of type I}\} ~~~~~~~~~~~~~~~\notag \\ -\#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}$ is of type II}\}. ~~~~~~~~~~~~~~\end{aligned}$$
Let $\mathcal{B}$ be the set of integers $j$ such that $L_j$ is *of type I* and $L_{j+2}, L_{j+3}, L_{j+4}$ (resp. $L_{j-1}, L_{j+1},$ $L_{j+2}, L_{j+3}$) are *of type II* if $j$ is even (resp. odd). Equation (\[32’\]) implies that $G^{\ddag}$ is disconnected with at least $2^\beta$ connected components (Exercise 2.19 of [@H]). Here $\beta=\# \mathcal{B}$.
Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be a pair of two (possibly empty) subsets of $\mathcal{B}$ such that $\mathcal{B}$ is the disjoint union of $\mathcal{B}_1$ and $\mathcal{B}_2$. Let $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ be the subfunctor of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ consisting of those $m$ satisfying Equations(\[ea20\]), (\[ea22\]), (\[24\]), (\[24’\]), (\[ea25\]), (\[ea27\]), and (\[ea32\]), the equations $\sum_{l=0}^{k_j}z_{j-l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=0$ for any $j\in \mathcal{B}_1$, and the equations $\sum_{l=0}^{k_j}z_{j-l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=1$ for
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{{\widehat{S}}}$:
Get $\hat\beta_{{\widehat{S}}}$ from ${\cal D}_{2,n}$ by least squares.
Output $\hat{C}_{{\widehat{S}}} = \bigotimes_{j\in {\widehat{S}}} C(j)$ where $C(j) = \hat\beta_{{\widehat{S}}}(j) \pm z_{\alpha/(2k)} \sqrt{\hat\Gamma_n(j,j)}$ where $\hat\Gamma$ is given by (\[eq::Ga\]).
For $\gamma_{{\widehat{S}}}$:
Get $\hat\beta_{{\widehat{S}}}$ from ${\cal D}_{1,n}$. This can be any estimator. For $j\in {\widehat{S}}$ let $\hat\gamma_{{\widehat{S}}}(j) = \frac{1}{n}\sum_{i=1}^n r_i$ where $r_i = (\delta_i(j) + \epsilon \xi_i(j))$, $\delta_i(j) = \left| Y_i - t_{\tau} \left( \hat\beta_{{\widehat{S}},j}^\top X_i \right) \right| - \left|Y_i - t_{\tau} \left( \hat\beta_{{\widehat{S}}}^\top X_i \right)\right|$ and $\xi_i(j)\sim {\rm Unif}(-1,1)$. Let $\hat\gamma_{{\widehat{S}}} = (\hat\gamma_{{\widehat{S}}}(j):\ j\in {\widehat{S}})$.
Output $\hat{D}_{{\widehat{S}}} = \bigotimes_{j\in {\widehat{S}}} D(j)$ where $D(j) = \hat\gamma_{{\widehat{S}}}(j) \pm z_{\alpha/(2k)}\hat{\Sigma}(j,j)$, with $\hat{\Sigma}(j,j)$ given by .
------------------------------------------------------------------------
------------------------------------------------------------------------
[Median-Split]{}
[Input]{}: Data ${\cal D} = \{(X_1,Y_1),\ldots, (X_{2n},Y_{2n})\}$. Confidence parameter $\alpha$.\
[Output]{}: Confidence set $\hat{E}_{{\widehat{S}}}$.
Randomly split the data into two halves ${\cal D}_{1,n}$ and ${\cal D}_{2,n}$.
Use ${\cal D}_{1,n}$ to select a subset of variables ${\widehat{S}}$. This can be forward stepwise, the lasso, or any other method. Let $k= |{\widehat{S}}|$.
Write ${\cal D}_{2,n}=\{(X_1,Y_1),\ldots, (X_n,Y_n)\}$. For $(X_i,Y_i)\in {\cal D}_{2,n}$ let $$W_i(j) =
|Y_i-\hat\beta_{{\widehat{S}},j}^\top X_i| - |Y_i-\hat\beta_{{\widehat{S}}}^\top X_i|,$$
Let $W_{(1)}(j) \leq \cdots \leq W_{(n)}(j)$ be the order statistics and let $E(j) = [W_{(n-k_2)}(j),W_{(n-k_1+1)}(j)]$ where $$k_1 = \frac{n}{2} + \sqrt{n \log\left( \frac{2k}{\alpha}\right)},\ \ \
k_2 = \frac{n}{2} - \sqrt{n \log\left(
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\^[i]{}\_[( \^[-1]{} )]{}\^\_\_i=b\^.
Note that the expression $\Phi^{-1}\frac{{\partial}f}{{\partial}r}\in {\mathbb{R}}^q$ is a contravariant vector as well as $b\in{\mathbb{R}}^q$. Hence there exists a nonzero scalar function $\Omega=\Omega(x,u)$, a rotation matrix $L=L(x,u)\in SO(q)$ and a vector $\tau=\tau(x,u)\in{\mathbb{R}}^q$ such that
\[eq:SW:8\] \^[-1]{}=L b+,\^i\_i=0.
It should be noted that we assume appropriate levels of differentiability of the functions $\Omega$, $\lambda$, $L$, $\tau$ and $b$, as necessary in order to justify all the following steps. Using relation (\[eq:SW:8\]), we eliminate the vector $\Phi^{-1}\frac{{\partial}f}{{\partial}r}\in {\mathbb{R}}^q$ from equation (\[eq:SW:7\]), which allows us to factor out the vector $b$ on the right after regrouping all terms on the left of the resulting equation. Therefore, we obtain the condition
\[eq:SW:13\] [( \^[i]{}\_i L-I\_q )]{}b=0,
on the scalar function $\Omega$, the wave vector $\lambda$ and the rotation matrix $L$. This implies that we have the following dispersion relation
\[eq:SW:14\] [( \^[i]{}\_i L-I\_q )]{}=0.
Once a scalar function $\Omega$, a wave vector $\lambda$ and a matrix $L$ satisfying (\[eq:SW:13\]) have been obtained, equation (\[eq:SW:8\]) must be used in order to determine the function $f$. Replacing the expression (\[eq:SW:4\]) for the matrix $\Phi$ into equation (\[eq:SW:8\]) and solving for the vector $\frac{{\partial}f}{{\partial}r}$ and taking into account the relation ${\partial}r/{\partial}u=({\partial}\lambda_i/{\partial}u) x^i$, we find that
\[eq:SW:15\] =,
which cannot admit the gradient catastrophe. Indeed, if we suppose that $1+\Omega({\partial}\lambda_i/{\partial}u) x^i L b=0$ when we proposed from equation (\[eq:SW:8\]) to equation (\[eq:SW:15\]), we easily conclude that $\Omega L b=0$. Consequently, since the matrix $\Phi$ is invertible, we conclude from (\[eq:SW:8\]) that the solution for $f(r)$ is constant, so it cannot admit the gradient catastrophe if $\det \Phi\neq 0$.
####
Up till now, w
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,0}$ have degree 0, 2 or 4 in $x$. Since $f$ contains summands of only 2-nd and 4-th degree in $x$, we have $\phi_{1,1}+\phi_{1,3}+\phi_{2,1}=0$.
Therefore, $\phi=\phi_{1,0}+\phi_{1,2}+\phi_{2,0}$.
Since $x$ is the central letter of the dipolynomial $f$, central letters of dimonomials from $\phi$ can be variables $x$ and $z$. Every dipolynomial from ${\mathrm{Di}}{\mathrm{H}}\,\langle x,y,z \rangle$ with this property is equal to a linear combination of the next dipolynomials: $$\begin{gathered}
\{ \dot xyxz \}, \ \{ xy\dot xz \}, \ \{ xyx\dot z \},\quad
\{ y\dot xxz \}, \ \{ yx \dot xz \}, \ \{ yxx\dot z \},\\
\{ \dot xxyz \}, \ \{ x\dot xyz \}, \ \{ xxy\dot z \},\quad
\{ \dot xyzx \}, \ \{ xyz \dot x \}, \ \{ xy\dot zx \},\\
\{ yz\dot xx \}, \ \{ yzx\dot x \}, \ \{ y\dot zxx \} ,\quad
\{ y\dot xzx \}, \ \{ yxz\dot x \}, \ \{ yx\dot zx \},\\
\{\dot zyyy\},\ \{y\dot zyy\},\quad \{\dot zzy\},\ \{z\dot zy\},
\{\dot zyz\}.
\end{gathered}$$
Consequently $\phi (x,y,z)$ has the form $$\begin{gathered}
\alpha_1 \{ \dot xyxz \}
+\alpha_2 \{ y\dot xxz \} +\alpha_3 \{ \dot xxyz \} +\alpha_4 \{
\dot xyzx \} +\alpha_5 \{ yz\dot xx \}
+\alpha_6 \{ y\dot xzx \} \\
+\beta_1 \{ xy\dot xz \} +\beta_2 \{ yx \dot xz \} +\beta_3 \{ x\dot
xyz \} +\beta_4 \{ xyz \dot x \} +\beta_5 \{ yzx\dot x \}
+\beta_6 \{ yxz\dot x \} \\
+2\gamma_1 \{ xyx\dot z \} +2\gamma_2 \{ yxx\dot z \} +2\gamma_3 \{
xxy\dot z \} +2\gamma_4 \{ xy\dot zx \} +2\gamma_5 \{ y\dot zxx \}
+2\gamma_6 \{ yx\dot zx \} \\
+2\delta_1\{\dot zyyy\}+2\delta_2\{y\dot zyy\}+2\delta_3\{\dot
zzy\}+2\delta_4\{z\dot zy\}+2\delta_5\{\dot zyz\}.
\end{gathered}$$
Substituting $z=k$ and using the equalities $$\begin{gathered}
2\dot zz=(\dot xx+x\dot x-\dot yy-y\dot y)\mathbin\dashv(xx-yy) \\
=\dot xx^3+x\dot xx^2-\dot yyx^2-y\dot yx^2-\dot xxy^2-x\dot
xy^2+\dot y y^3+y\dot yy^2, \\
2z\dot z=(xx-yy)\mathbin\vdash(\dot xx+x\dot x-\dot yy-y\dot y) \\
=x^2\dot xx+x^3\dot x-x^2\dot yy-x^2y\dot y-y^2\dot xx-y^2x\dot x
+y^2\dot yy+y^3\dot y,\end{gathered}$$ we obtain $\phi(x,y,k)$ is e
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| 1,285
| null | null |
github_plus_top10pct_by_avg
|
s a normal subgroup of $G=N{{\operatorname}{C}_{G}(y)}$. Since $N$ is a minimal normal subgroup of $G$, we deduce that either $N={{\operatorname}{C}_{N}(y)}$ or ${{\operatorname}{C}_{N}(y)}=1$. The first case yields to the contradiction $G={{\operatorname}{C}_{G}(y)}$. So we may assume ${{\operatorname}{C}_{N}(y)}=1$. If we take $1 \neq x\in N\cap A$ of prime power order (which is a $p$-regular element) then, by our hypotheses, there exists some $n\in N$ such that $\langle y\rangle^n\leq {{\operatorname}{C}_{G}(x)}$, and so $x \in {{\operatorname}{C}_{N}(\langle y\rangle^n)}=({{\operatorname}{C}_{N}(\langle y\rangle)}^n$, a contradiction.
\[xi\] ${\ensuremath{\left| N \right|}}{\ensuremath{\left| \langle y\rangle \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}.$
Recall that $p\in\pi(AN)\smallsetminus\pi(BN)$ and $G=AN=BPN$ for any $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Hence $G=AN=B\langle y\rangle N$. Now it is enough to make some computations having in mind that ${\ensuremath{\left| B\cap \langle y\rangle N \right|}}={\ensuremath{\left| B\cap N \right|}}$ (recall that $G$ is $p$-separable).
\[7\] The Sylow $p$-subgroups of $G$ are cyclic of order $p$, i.e. $\langle y\rangle \cong C_p$.
Take $x\in \langle y\rangle$ of order $p$, and set $H:=BN\langle x\rangle$; this is a subgroup of $G$ since $BN={{\operatorname}{O}_{p'}(G)}\unlhd G$. Assume that $H=(H \cap A) B<G$. Now if $h\in (H\cap A)\cup B$ is $p$-regular of prime power order, by the hypotheses there exists $n\in N$ with $\langle x\rangle ^n \leq \langle y\rangle^n \leq {{\operatorname}{C}_{G}(h)}$, so $h\in {{\operatorname}{C}_{H}(\langle x\rangle^n)}$, where $\langle x\rangle^n\in{{\operatorname}{Syl}_{p}\left(H\right)}$. By minimality, $\langle x\rangle={{\operatorname}{O}_{p}(H)}$, so (recall that $G=BPN$, by Lemma \[1\], with $P=\langle y \rangle$) $1\neq {{\operatorname}{O}_{p}(H)}^G={{\operatorname}{O}_{p}(H)}^{BNP}={{\
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github_plus_top10pct_by_avg
|
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