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d $\sigma_i^2(\Delta t)$ within each group. The obtained scaling plots are shown in Fig. \[fig:scavg\].
![The normalized variance $\frac{1}{2}\log \sigma_i^2(\Delta t)-\frac{1}{2}\log \Delta t$ for the six groups of companies, with average traded values $\ev{f}\in[0,10^4)$, $\ev{f}\in[10^4,10^5)$, …, $\ev{f}\in[10^8,\dots)$ USD/min, increasing from bottom to top. A horizontal line would mean the absence of autocorrelations in the data. Instead, one observes a crossover phenomenon in the regime $\Delta t = 60-390$ mins, indicated by darker background. Below the crossover all stocks show very weakly correlated behavior, $H^-\approx 0.5$. Above the crossover, the strength of correlations, and thus the slope corresponding to $H^+-\frac{1}{2}$, increases with the liquidity of the stock. The asymptotic values of $H^{\pm}$ are indicated in the plot.[]{data-label="fig:scavg"}](EislerFig2){height="200pt"}
![Value of the Hurst exponents of traded value for the time period $1994-1995$. For short time windows (O, $\Delta t < 60$ min) all signals are nearly uncorrelated, $H^-\approx 0.51 - 0.53$. The fitted slope is $\gamma^-=0.00\pm 0.01$. For larger time windows ($\blacksquare$, $\Delta t > 390$ min) the strength of correlations depends logarithmically on the mean trading activity of the stock, $\gamma^+=0.053\pm 0.01$ for $1994-1995$. Shuffled data ($\bigstar$) display no correlations, thus $H_{\mathrm{shuff}} = 0.5$, which also implies $\gamma_\mathrm{shuff} = 0$. [*Note*]{}: Groups of stocks were binned, and their Hurst exponents were averaged. The error bars correspond to the standard deviations in the bins.[]{data-label="fig:hurst"}](EislerFig3){height="205pt"}
All stocks display a crossover around window sizes of $\Delta t = 60-390$ min, and there are two sets of Hurst exponents: $H^-_i$ valid below, and $H^+_i$ above the crossover. These characterize the strength of intraday and long time correlations, respectively. The behavior on these two time scales is very different.
1. For intraday fluctuations, regardle
| 1,101
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15 0.014 0.015 0.014
BLB($n^{0.6}$) 0.120 0.122 0.122 0.121 0.122 0.123 0.118
BLB($n^{0.8}$) 0.037 0.039 0.038 0.038 0.038 0.038 0.037
SDB($n^{0.6}$) 0.142 0.144 0.144 0.144 0.144 0.145 0.141
SDB($n^{0.8}$) 0.046 0.047 0.047 0.047 0.048 0.048 0.046
TB 0.014 0.014 0.014 0.014 0.014 0.014 0.014
3 K=50 0.018 0.018 0.018 0.018 0.018 0.018 0.018
K=100 0.018 0.018 0.018 0.018 0.018 0.018 0.018
K=150 0.018 0.018 0.018 0.018 0.018 0.018 0.018
BLB($n^{0.6}$) 0.150 0.155 0.155 0.154 0.155 0.153 0.150
BLB($n^{0.8}$) 0.047 0.048 0.049 0.049 0.049 0.048 0.047
SDB($n^{0.6}$) 0.179 0.182 0.183 0.182 0.182 0.182 0.178
SDB($n^{0.8}$) 0.059 0.060 0.060 0.060 0.060 0.060 0.059
TB 0.017 0.018 0.017 0.018 0.017 0.017 0.017
: Lengths of confidence interval comparison for Example \[example1\].
\[table3\]
\[example2\] In this example, we consider a $p$-dimensional multiple Logistic regression model. Given covariates $Z_{i}\in {\mathbb{R}}^{p}$, $${\mathbb{P}}(Y_{i}=1|Z_{i})=\frac{\exp(Z_{i}^\top \beta)}{1+ \exp(Z_{i}^\top \beta)}, \quad i=1,\dots, n,$$ where $Y_{i}\in\{0,1\}$ is the response and $\beta$ is a $p$-dimensional unknown parameter. The interesting problem is
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sublinear expectation, As $\mathcal{M}_1\subset \mathcal{P}$, it is clear that On the other hand, for $\lambda_1, \lambda_2{\geqslant}0$ such that $\lambda_1+\lambda_2=1$ and $\mu_1, \mu_2\in \overline{\mathcal{M}}_1$, the closure of $\mathcal{M}_1$, Therefore, and by the choice of $\mu_k$, we have This, together with [claim1-1]{}, proves Claim \[claim1\].
Here, $\mathcal{P}$ is a bounded convex polytope with $m$ vertices. Denote the set of vertices by $\mathcal{V}$. For each vertex $v\in \mathcal{V}$, define It is clear that $\mathcal{P}=\cap_{v\in \mathcal{V}} T_v$ where the intersection is over all the $m$ vertices. (Just to clarify the definitions, consider, for example, $d=1$ and $\mathcal{P}=[\underline{\mu},\overline{\mu}]$. It has two vertices $\mathcal{V}=\{\underline{\mu},\overline{\mu}\}$. Thus, we have $T_{\underline{\mu}}=[\underline{\mu},\infty)$, $T_{\overline{\mu}}=(-\infty, \overline{\mu}]$ and $\mathcal{P}=T_{\underline{\mu}} \cap T_{\overline{\mu}}$.) We will prove that and hence To prove [c3]{}, we take the function $\varphi$ in Theorem \[t4\] to be where $T_v-v=\{u-v: u\in T_v\}$. We will prove the following lemma.
\[l4\] For this $\varphi$, we have that $\varphi$ is differentiable, $D\varphi: \mathbb{R}^d\to \mathbb{R}^d$ is a Lipschitz function, and
On the basis of this lemma, we can take $\mu_i=v$ for all $i$ in Theorem \[t4\]. This implies the following: The left-hand side is precisely $d_{T_v}^2(\frac{\sum_{i=1}^n X_i}{n})$; hence, we obtain [c3]{}.
We now prove Lemma \[l4\].
Without loss of generality, we assume that $v=0$; hence, $T_v-v=T_v=T_0$. For each $x$ such that $d(x,T_0)>0$, define Because of the convexity of $T_0$, $x_0$ is unique for each $x$, and moreover, $x_0$ as a function of $x$ is continuous. Based on this definition, Let $\mathcal{E}$ and $\mathcal{S}$ denote the set of “edges" and “surfaces" of $T_0$, respectively. The $d$-dimensional set $\mathcal{R}=\{x: d(x, T_0)>0\}$ can be divided into a finite number of disjoint parts as where and For each $x\in \mathc
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3723 1.77 (0.50, 6.25)
**Females**^**a**^ **Females**
G/G 3 (5.7) 6 (7.3) 0.62 (0.14, 2.65) G/G 4 (4.8) 4 (16.7) 0.34 (0.08, 1.52)
G/T 16 (30.2) 34 (41.5) 0.58 (0.28, 1.23) G/T 32 (38.6) 4 (16.7) 2.72 (0.83, 8.90)
T/T 34 (64.1) 42 (51.2) 0.3339 1 T/T 47 (56.6) 16 (66.6) 0.0350\* 1
G/G + G/T 19 (35.8) 40 (48.8) 0.1391 0.59 (0.29, 1.19) G/G + G/T 36 (43.4) 8 (33.4) 0.3786 1.53 (0.59, 3.97)
^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.
Genotype frequencies were determined by *χ*^2^ test using 2 × 3 or 2 × 2 tables as appropriate. Odds ratios and 95% CI per genotype were estimated by unconditional logistic regression. *P* values less than 0.05 were considered statistically significant, and are denoted with an asterisk.
######
**Allele frequencies of*GNB1*single nucleotide polymorphisms stratified by gender in HCV-1- and HCV-2-infected patients receiving PEG-IFNα-RBV therapy with and without RVR in a Chinese population in Taiwan**
**HCV-1** **HCV-2**
----------------------- ---------------------- ------------ -------- ------------------- ---------- ------------ ----------- -------- -------------------
**Males**
**rs10907185** **rs10907185**
A allele
| 1,104
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|
^ -01.40 \[-5.50, 2.71\] 0.51
Family situation (single parent vs. stepfamily) 6.68 \[-6.21, 19.57\] 0.31 10.16 \[-17.43, 37.74\] 0.48
Family situation (divorced vs. stepfamily) 4.81 \[-8.17, 17.80\] 0.47 -16.72 \[-40.24, 6.79\] 0.18
Family situation (married vs. stepfamily) 1.24 \[-4.90, 7.38\] 0.69 -3.23 \[-22.05, 15.58\] 0.74
Age (of ill child at diagnosis) 0.08 \[-0.16, 0.32\] 0.51 1.76 \[0.47, 3.04\] 0.01^\*^
Sex (female vs. male)^3^ 5.04 \[-1.09, 11.16\] 0.12
1
Note that only 48 children could be included in the analyses, due to the age restrictions of some of the questionnaires (FES and PSS).
2
Obtained by fitting a second model, including the subscales of the FES, instead of the FRI and FSI.
3
Note that
sex
was redundant and was thus left out of the model assessing quality of life for mothers and fathers, since the variable
Family member
(father vs. mother) was identical in this case.
∗
p
\< 0.05,
∗∗
p
\> 0.001.
### Mothers and Fathers
The interaction effects between *fa
| 1,105
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|
bf x}}^{\prime} \right]}{\left(-T^2+X^2 \right)^2} \nonumber \\
&& - \frac{32 \pi \epsilon^{\prime} \epsilon^{\prime\prime} |\cos\Theta_{0}| \left[ -T \dot{t}^\prime + X \dot{{\bf x}}^{\prime} \right] \left[ -T \dot{t}+X\dot{{\bf x}} \right] }{\left( -T^2+X^2 \right)^3} \nonumber \\
&& + \frac{8 \pi \epsilon^{\prime} \epsilon^{\prime\prime} |\cos\Theta_{0}|^{2} \left[- \dot{t} \dot{t}^{\prime} + \dot{{\bf x}}\dot{{\bf x}}^{\prime} \right] }{ \left( -T^2+X^2 \right)^2} \; \bigg\}_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{Wfinal}\end{aligned}$$ The first term is of the familiar form one gets for the total transition rate, however one must note that both the functions $T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ and $X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ are dependent on the angle $\Theta$.
Another feature of Eq.(\[Wfinal\]), is that, $\epsilon$ and $|\cos\Theta_0|$ always appear as a product in the expression. Given that $|\cos\Theta_0|$ is always non-negative, one can formally absorb it in the definition of $\epsilon$ itself. Then, in the point-like limit of the detector, that is, when taking the $\epsilon \rightarrow 0$, one will arrive at an expression which is independent of the angular direction. Thus to have a direction dependence in the transition rate, one needs to have the spatial extension of the detector modelled using a finite positive $\epsilon$ parameter in the present model.
We have thus finished our construction of the direction dependent spatially extended detector. Substituting Eq.(\[Wfinal\]) in Eq.(\[angtransitionrate\]) gives us the angular transition rate of the detector. The expression is general and will hold for any accelerating trajectory in a flat spacetime.
Rindler trajectory
------------------
We shall now analyse the direction dependent transition rate for the special case of the Rindler trajectory.
Substituting for the trajectory $t(\tau) = (1/g)\sinh (g\tau) $, $x(\tau) =(1
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talks are invariant) For any $s\in S$ and $e\in E$, if $s\cdot e$ is defined then $\pi(s\cdot e)=\pi(e)$.
4. (universal on stalks) For any $s\in S$ and $x\in X$, $(E_x,\mu|_{S\times E_x})$ (which is well-defined by (U3)) is a universal $S$-set.
5. (continuous) The partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous ($S$ is considered as a discrete space and $S\times E$ as a product space).
It is easy to see that (U4) implies (U1), so that (U1) may be omitted from the above list.
Let $\pi \colon E\to X$, $\pi'\colon E'\to X$ be étale spaces and $(E,\mu)$, $(s,e)\mapsto s\cdot x$, if defined, $(E',\nu)$, $(s,e)\mapsto s\circ x$, if defined, be universal $S$-sets in the topos ${\mathsf{Sh}}(X)$. A morphism $$f\colon (E,\mu) \to (E',\nu)$$ is defined as a morphism $f\colon E\to E'$ of étale spaces (that is, a continuous map such that $\pi=\pi'f$, cf. [@MM]) which is simultaneously a morphism of $S$-sets (that is, if $s\cdot e$ is defined then $s\circ f(e)$ is defined and $f(s\cdot x)=s\circ f(x)$). We denote the category of universal $S$-sets in the topos ${\mathsf{Sh}}(X)$ by ${\mathsf{Univ}}(S,X)$.
\[th:sheaves\] There is an equivalence of categories $${\mathsf{Prin}}(S, X) \,\,
{\mathrel{
\settowidth{\@tempdima}{$\scriptstyle\tau$}
\settowidth{\@tempdimb}{$\scriptstyle\rho$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}} \,\, {\mathsf{Univ}}(S,X).$$
We begin with the construction of the functor $$\tau\colon {\mathsf{Prin}}(S, X)\to {\mathsf{Univ}}(S,X).$$ Let $E\colon L(S) \to {\mathsf{Sh}}(X)$ be a principal bundle over $X$ and $x\in X$. We first describe the colimit sheaf $\widetilde{E}\in {\mathsf{Sh}}(X)$. By definition, for each $x\in X$ we have a filtered functor $f_x\colon L(S)\to {\mathsf{Sets}}$ obtained by restricting $E$ to the stalks over $x$. We now apply the functor $\Psi$
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{n>0} l_{a^i} \frac{1}{k+g} (B^{-1})^{ji} a^j_n
- \sum_{1 \leq i < j \leq N} l_{b^{ij}} q_{b^{ij}}
+ \sum_{1 \leq i < j \leq N} l_{c^{ij}} q_{c^{ij}} \right) | 0 \rangle,\end{aligned}$$
it can be shown that the following equations hold,
$$\begin{aligned}
& & a^i_n| l_a,l_b,l_c \rangle = b^{ij}_n| l_a,l_b,l_c \rangle
= c^{ij}_n| l_a,l_b,l_c \rangle =0 ~( n > 0),\\
& & p_{a^i} | l_a,l_b,l_c \rangle = l_{a^i} | l_a,l_b,l_c \rangle,\\
& & p_{b^{ij}} | l_a,l_b,l_c \rangle = l_{b^{ij}} | l_a,l_b,l_c \rangle,\\
& & p_{c^{ij}} | l_a,l_b,l_c \rangle = l_{c^{ij}} | l_a,l_b,l_c \rangle.\end{aligned}$$
The Fock space ${\cal F}(l_a,l_b,l_c)$ is then generated by the actions of the negative modes of $a^i,~b^{ij},~c^{ij}$. We shall see later that this Fock space actually forms a (Wakimoto-like [@wakimoto; @fff]) module for the Yangian double $DY_\hbar(sl_N)$ with level $k$.
For $X= a^i,~b^{ij},~c^{ij}$, let us now define
$$\begin{aligned}
& & X(u;A,B)=\sum_{n>0} \frac{X_{-n}}{n} (u+A \hbar)^n -
\sum_{n>0} \frac{X_{n}}{n} (u+B \hbar)^{-n} + \mbox{log}(u+B \hbar) p_X
+q_X, \\
& & X_+(u;B)= - \sum_{n>0} \frac{X_{n}}{n} (u+B \hbar)^{-n}
+ \mbox{log}(u+B \hbar) p_X,\\
& & X_-(u;A)=\sum_{n>0} \frac{X_{-n}}{n} (u+A \hbar)^n
+q_X,\\
& & X(u;A) = X(u;A,A),~~X(u)=X(u,0).\end{aligned}$$
Then we have
$$: \mbox{exp} \left( X(u;A,B) \right) :
= \mbox{exp} \left( X_-(u;A) \right) \mbox{exp} \left( X_+(u;B) \right).$$
Following the standard quantum field theory we have
$$X^{\alpha}(u;A,B) X^{\beta}(v;C,D) =
\langle X^{\alpha}(u;A,B) X^{\beta}(v;C,D) \rangle
+ :X^{\alpha}(u;A,B) X^{\beta}(v;C,D):, \label{contract}$$
where [^1]
$$\begin{aligned}
& & \langle a^{i}(u;A,B) a^{j}(v;C,D) \rangle
=(k+g) B_{ij} \mbox{log}(u-v+(B-C) \hbar),\\
& & \langle b^{ij}(u;A,B) b^{i'j'}(v;C,D) \rangle
=- \delta^{ii'} \delta^{jj'} \mbox{log}(u-v+(B-C) \hbar),\\
& & \langle c^{ij}(u;A,B) c^{i'j'}(v;C,D) \rangle
= \delta^{ii'} \delta^{jj'} \mbox{log}(u-v+(B-C) \hbar),\end{aligned}$$
and all other contractions vanish. From eq.(\[contract\]) it is easy
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a way that $S_j$ are assumed to be constant in subdomains.
In this example, we write out Eq. in special case of constant $S_0\geq 0$ and $\Sigma\geq 0$. Moreover, we consider a single particle CSDA transport equation only. Let $${}&P(x,\omega,E,D)u:=-{\frac{\partial (S_0u)}{\partial E}}+\omega\cdot\nabla_x u+\Sigma u,$$ In this case, $R(E)=\int_0^E \frac{1}{S_0}d\tau=\frac{1}{S_0}E$, and Eq. gives, when $S_0>0$, using the notation $\eta(E):=(E_m-E)/S_0$ and noticing that $r_m:=R(E_m)=\frac{E_m}{S_0}$, $$(\widetilde {P}_0^{-1}h)(x,\omega,E)=\int_0^{\min\{\eta(E),t(x,\omega)\}} e^{-\Sigma s} h\big(x-s\omega,\omega,E+S_0s\big)ds,$$ for $h\in L^2(G\times S\times I)$. It is clear that this last formula gives the correct (explicit) expression for $\widetilde{P}_0^{-1}$ also in the case where $S_0=0$, if we make the convention that $\eta(E)=+\infty$ for all $E\in I$ when $S_0=0$.
An Approximative Solution Based on the Theory of Evolution Equations {#appevo-o}
--------------------------------------------------------------------
In this section we for simplicity restrict ourselves to a single particle CSDA-equation $$\begin{gathered}
-{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+
\Sigma\psi
- K\psi=f, \label{mb1} \\
\psi_{|\Gamma_-}=g,\quad
\psi(\cdot,\cdot,E_m)=0. \label{mb2}\end{gathered}$$ Suppose that the assumptions of Theorem \[coth3-dd\] are valid.
Another method to compute approximately the solution of the problem (\[mb1\]), (\[mb2\]) which avoids the explicit inversions of matrices, can be (formally) described as follows. Note that we will be using throughout this section the notations of Section \[possol\]. After the change of unknown $\phi=e^{CE}\psi$ the problem is \[ps1\] T\_[C]{}=[**f**]{},\_[\_-]{}=[**g**]{},(,,E\_[m]{})=0, where $$T_C\phi:=-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi-K_C\phi.$$ Assume that $g\in H^1(I,T^2(\Gamma'_-))$ and that $g(\cdot,\cdot,E_{\rm m})=0$ on $G\times S$. Let $u:=\phi-L({\bf g})$. Then $u$ satisfies \[ps0\
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, T., [Ellis]{}, R. S., [Liao]{}, T. X., & [van Dokkum]{}, P. G. 2005, , 622, L5
, T., [Ellis]{}, R. S., [Liao]{}, T. X., [van Dokkum]{}, P. G., [Tozzi]{}, P., [Coil]{}, A., [Newman]{}, J., [Cooper]{}, M. C., & [Davis]{}, M. 2005, , 633, 174
, I., [Conselice]{}, C. J., [Bundy]{}, K., [Cooper]{}, M. C., [Eisenhardt]{}, P., & [Ellis]{}, R. S. 2007, , 382, 109
, I., [Ferreras]{}, I., & [de La Rosa]{}, I. G. 2011, , 415, 3903
, I. [et al.]{} 2006, , 650, 18
, T., [Poggianti]{}, B. M., [Saglia]{}, R. P., [Arag[ó]{}n-Salamanca]{}, A., [Simard]{}, L., [S[á]{}nchez-Bl[á]{}zquez]{}, P., [D’onofrio]{}, M., [Cava]{}, A., [Couch]{}, W. J., [Fritz]{}, J., [Moretti]{}, A., & [Vulcani]{}, B. 2010, , 721, L19
, A., [Bell]{}, E. F., [van den Bosch]{}, F. C., [Gallazzi]{}, A., & [Rix]{}, H.-W. 2009, , 698, 1232
, A. [et al.]{} 2007, , 670, 206
, P. G., [Franx]{}, M., [Kelson]{}, D. D., & [Illingworth]{}, G. D. 2001, , 553, L39
, P. G. [et al.]{} 2008, , 677, L5
—. 2010, , 709, 1018
, A., [Best]{}, P. N., [Kauffmann]{}, G., & [White]{}, S. D. M. 2007, , 379, 867
, J. V. & [Jenkins]{}, C. R. 2003, [Practical Statistics for Astronomers]{} (Princeton Series in Astrophysics)
, B. J. [et al.]{} 2009, , 692, 187
, S. M., [Kauffmann]{}, G., [van den Bosch]{}, F. C., [Pasquali]{}, A., [McIntosh]{}, D. H., [Mo]{}, H., [Yang]{}, X., & [Guo]{}, Y. 2009, , 394, 1213
, A. R., [Schulz]{}, A. E., [Holz]{}, D. E., & [Warren]{}, M. S. 2008, , 683, 1
, S. D. M. [et al.]{} 2005, , 444, 365
, R. J., [Quadri]{}, R. F., [Franx]{}, M., [van Dokkum]{}, P., [Toft]{}, S., [Kriek]{}, M., & [Labb[é]{}]{}, I. 2010, , 713, 738
, C. N. A. [et al.]{} 2006, , 647, 853
, R. & [Blanton]{}, M. R. 2011, ArXiv e-prints
, R., [Newman]{}, J. A., [Faber]{}, S. M., [Konidaris]{}, N., [Koo]{}, D., & [Davis]{}, M. 2006, , 648, 281
, R. [et al.]{} 2011, , 728, 38
, D. G. [et al.]{} 2000, , 120, 1579
, H. 2002, , 336, 159
, A. W. [et al.]{} 2007, , 656, 66
[^1]: Note that the sampling density
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t users and robots or the employed workers, the fake transaction records will always cause a bias or distortion of the original transaction distribution. To better observe the problem, we downloaded a real world data set containing Taobao online sellers’ transaction records and emulated the circumstances if it had been click farmed (see section \[sec:exp-methodology\]). Fig. \[fig:example-ecdf\] shows the difference between normal and click farmed distributions of one day in the data set. Thus, if we can measure the similarity between different transaction distributions, there is still a chance for us to detect dishonest sellers.
Preliminaries {#sec:preliminaries}
=============
Statistical divergence, also called statistical distance, measures the similarity between two or more distributions. Mathematically, statistical divergence is a function which describes the “distance” of one probability distribution to the other on a statistical manifold. Let $\mathbb{S}$ be a space of probability distributions, then a divergence is a function from $\mathbb{S}$ to non-negative real numbers: $$D(\cdot || \cdot): \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{R^+} \cup \{0\}$$
Divergence between two distributions $P$ and $Q$, written as $D(P||Q)$, satisfies:
1. $D(P||Q) \ge 0, \forall P, Q \in \mathbb{S}$
2. $D(P||Q) = 0$, if and only if $P=Q$
For our purposes, we do not require the function $D$ to have the property: $D(P||Q) = D(Q||P)$. But we do need it to be true that if $Q$ is more similar with $P$ than $U$, then $D(Q||P) < D(U||P)$. There are ways to calculate divergence, several frequently used divergence metrics are as follows:
Kullback-Leibler Divergence
---------------------------
Let $P,Q$ be discrete probability distributions, $Q(x)=0$ implies $P(x)=0$ for $\forall x$, the *Kullback-Leibler Divergence* from $Q$ to $P$ is defined to be:
$$KLD(P||Q) = \sum_{Q(x)\ne 0} P(x)log\Big(\frac{P(x)}{Q(x)}\Big)$$
For $P,Q$ being continuous distributions:
$$KLD(P||Q) = \int_{q(x) \ne 0} p(x)log\frac{p(x
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|
\[Cor:aveGst5\] {#App:proofaveGast5}
==================================
We rewrite $G_{\bar{R}}$ as $$\label{eq:th_gain_close_alt}
G_{\bar{R}}(\Psi=i) \approx 1+\frac{\sqrt{{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}E_{i}, $$ where $E_{i}$ denotes the mean of the maximum of $i$ independent standard normal random variables. Denoting the cdf of standard normal distribution by $\Phi(\cdot)$, we have $$\begin{aligned}
\label{eq:proofeiniaj}
E_i&=\int_{-\infty}^\infty x\frac{\mathrm{d}\left(\Phi(x)\right)^i}{\mathrm{d}x}\mathrm{d}x \notag\\
&=i\left(i-1\right)\int_{-\infty}^\infty \frac{\exp\left(-x^2\right)}{2\pi}\left(\Phi(x)\right)^{i-2}\mathrm{d}x
\notag\\
&=\frac{i\left(i-1\right)}{2\pi}\sum_{j=0}^{\lfloor \frac{i}{2}-1\rfloor}\left(\frac{1}{2}\right)^{i-2-2j}\binom{i-2}{2j}A_j,\end{aligned}$$ where $A_j=\int_{-\infty}^\infty \exp\left(-x^2\right)\left(\Phi(x)-\frac{1}{2}\right)^{2j}\mathrm{d}x $. We note that $A_0$ and $A_1$ can be derived[^8], which are given by $
A_0=\sqrt{\pi}
$ and $
A_1=\frac{1}{2\sqrt{\pi}\tan^{-1}\left(\frac{\sqrt{2}}{4}\right)},
$ respectively. Substituting $A_0$ and $A_1$ into , we can obtain $E_1=0$, $E_2=\pi^{-\frac{1}{2}}$, $E_3=\frac{3}{2}\pi^{-\frac{1}{2}}$, $E_4=3\pi^{-\frac{3}{2}}\arccos\left(-\frac{1}{3}\right)$, and $E_5=\frac{5}{2}\pi^{-\frac{3}{2}}\arccos\left(-\frac{23}{27}\right)$. Finally, substituting the expressions for $E_1, \cdots, E_5$ into completes the proof.
Proof of Corollary \[Cor:aveGlsn\] {#App:proofaveGlsn}
==================================
When $\Psi$ is large, we can adopt the Fisher-–Tippett theorem to approximate the distribution of the maximum of $\Psi$ independent standard normal random variables as a Gumbel distribution, whose cumulative distribution function (cdf) is given by $$\label{}
F_{E_\Psi}(x)=\exp\left(-\exp\left(-\frac{x-\Phi^{-1}\left(1-\frac{1}{\Psi}\right)}{\Phi^{-1}\left(1-\frac{1}{e\Psi}\right)-\Phi^{-1}\left(1-\frac{1}{\Psi}\right)}\right)\right),$$ where $\Phi^{-1}(\cdot)$ denotes the inverse cdf of the standard normal dist
| 1,112
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| 0.772712
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|
}( \Delta_{m-n+1}+ \Delta_{n-m+1}-\Delta_{1-m-n})
+{1\over 2}( - \Delta_{m-n+2}- \Delta_{m-n-2}+\Delta_{2-m-n})$$ $$+{\alpha\over 4}( - \Delta_{m-n+3}- \Delta_{m-n-3}+\Delta_{3-m-n})
) \big]. \eqno(A14)$$
The matrix elements connecting different parities are
$$\bigg < \bar{K}^+ \bar{\nu} \bigg | -i {\tau_1 \alpha^3 \over F}
{\rm sin\theta cos\phi} {\partial \over \partial \phi} \bigg | K^-
\nu \bigg
> = {\pi \tau_1\alpha^3 \over 4}\sum_{m=0}^{\bar{K}}\sum_{n=1}^{K}
c_{\bar{K}m} d_{Kn}(-\nu)
(\Delta_{\bar{\nu}-\nu+1}+\Delta_{\bar{\nu}-\nu-1})$$ $$\big[\Delta_{m+n+1}+ \Delta_{n-m+1}
-\Delta_{m-n+1}-\Delta_{m+n1-1}\big] \eqno(A15)$$
$$\bigg < \bar{K}^+ \bar{\nu} \bigg |\ i \alpha \tau_1 {\rm
sin\phi}
({\alpha +\rm cos\theta }) {\partial \over \partial \theta}
\bigg | K^- \nu \bigg
> = \tau_1\alpha^3 \pi \sum_{m=0}^{\bar{K}}\sum_{n=1}^{K}
c_{\bar{K}m} d_{Kn}(n)
(\Delta_{\nu-\bar{\nu}+1}-\Delta_{\nu-\bar{\nu}-1})$$ $$\big[{3\alpha \over 4}(\Delta_{m+n}+ \Delta_{m-n})+
{(1+\alpha^2)\over
4}(\Delta_{m+n+1}+\Delta_{m-n+1}+\Delta_{m+n-1}+\Delta_{m-n-1})$$ $$+ {\alpha \over 4}
(\Delta_{m+n+2}+\Delta_{m-n-2}+\Delta_{m+n-2}+\Delta_{m-n-2})\big]
\eqno(A16)$$
$$\bigg < \bar{K}^+ \bar{\nu} \bigg |{{\tau_0 \tau_1 \alpha^3}
\over 2} \ {\rm sin\theta \rm cos\phi}
F
\bigg | K^- \nu \bigg
> = {\tau_0 \tau_1 \alpha^3 \pi \over 4} \sum_{m=0}^{\bar{K}}\sum_{n=1}^{K}
n \ c_{\bar{K}m} d_{Kn}
(\Delta_{\nu-\bar{\nu}+1}-\Delta_{\nu-\bar{\nu}-1})$$ $$\big[{1\over 2}(1+{\alpha^2 \over 2})(\Delta_{m+n-1}-
\Delta_{m+n+1}+ \Delta_{m-n-1}- \Delta_{m-n+1} )+$$ $${\alpha \over 2}(\Delta_{m+n-2}- \Delta_{m+n+2}+ \Delta_{m-n-2}-
\Delta_{m-n+2} )$$ $$+ {\alpha^2 \over 8}
(\Delta_{n+m+1}-\Delta_{m+n-1}+\Delta_{m-n+1}-\Delta_{m-n-1}$$ $$+\Delta_{n+m-3}-\Delta_{m+n+3}+\Delta_{n-m-3}-\Delta_{n-m+3})\big].
\eqno(A17)$$ The negative to positive elements are obtained by interchanging all indices, or equivalently, by taking their transpose.
[**Figure Captions**]{}
Fig. 1: $\varepsilon$ as a function of $\tau_0$ for five low-lying states. D
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|
Bbb{C}^1 \ar[d]^\psi \\
Ver \ar[r]^{\iota \gamma} & Ver
}$$ Now let $\tau\in\PSL(3,\Bbb{C})$ be an element which leaves $Ver$ invariant. Define $$\begin{array}{l}
\widetilde{\tau}:\Bbb{P}_\Bbb{C}^1\rightarrow \Bbb{P}_\Bbb{C}^1 \quad.\\
\widetilde{\tau}(z)=\psi^{-1}(\tau(\psi(z))).
\end{array}$$ Clearly $\widetilde{\tau}$ is well defined and biholomorphic, thus $\widetilde{\tau}\in \PSL(2,\Bbb{C})$ and the following diagram commutes. $$\xymatrix{
\Bbb{P}_\Bbb{C}^1 \ar[r]^{\widetilde \tau}\ar[d]^\psi & \Bbb{P}_\Bbb{C}^1 \ar[d]^\psi \\
Ver \ar[r]^{\tau} & Ver
}$$ From diagram \[e:aut\], we conclude that $\tau\mid_{Ver}=\iota\widetilde\tau\mid_{Ver}$. Since the Veronese curve has four points in general position, we conclude $\tau=\iota\widetilde \tau$ in $\Bbb{P}_\Bbb{C}^2$, which concludes the proof.
\[l:ltanver\] Given $[1,k]\in \Bbb{P}^1_\Bbb{C}$, the tangent line to $Ver$ at $\psi[1,k]$, denoted $T_{\psi[1,k]}Ver$, is given by $$T_{\psi[x,y]}Ver=\{[x,y,z]\in \Bbb{P}^2_\Bbb{C}\vert z=ky-k^2x\}.$$
Let us consider the chart $(W_1=\{[x,y,z]\in\Bbb{P}^2_\Bbb{C}\vert x\neq 0\},\phi_1:W_1\rightarrow\Bbb{C}^2) $ of $\Bbb{P}^2_\Bbb{C}$ where $\phi_1[x,y,z]=(yx^{-1},zx^{-1})$ and $(W_2=\{[x,y]\in\Bbb{P}^1_\Bbb{C}\vert x\neq 0\},\phi_2:W_2\rightarrow\Bbb{C}^1)$ of $\Bbb{P}^1_\Bbb{C}$ where $\phi_1[x,y]=yx^{-1}$. Let us define $$\begin{array}{l}
\phi:\Bbb{C}\rightarrow \Bbb{C}^2\\
\phi(z)=\phi_1(\psi(\phi_2^{-1}( z)))
\end{array}.$$
A straightforward calculation shows that $\phi(z)=(2z,z^2)$, thus the tangent space to the curve $\phi$ at $\phi(k)$ is $\Bbb{C}(1, k)+(2k,k^2)$. Therefore the tangent line to $Ver$ at $[1,2k,k^2]$ is $\overleftrightarrow{[1,2k,k^2], [1,2k+1,k+k^2]}$. A simple verification shows
$$T_{\psi[x,y]}Ver=\{[x,y,z]\in \Bbb{P}^1_\Bbb{C} \vert z=ky-k^2x\}.$$
\[l:3gen\] Let $\Gamma\subset\PSL(2,\Bbb{C})$ be a non-elementary subgroup and $x,y,z\in \Lambda(\Gamma)$ be distinct points, then the lines $T_{\psi(x)}Ver,T_{\psi(y)}Ver,T_{\psi(z)}Ver$ are in general position.
Let us assume that $[1
| 1,114
| 736
| 1,036
| 1,042
| 3,867
| 0.769629
|
github_plus_top10pct_by_avg
|
M(C^\lambda)$, since both $\cO_{X,\zeta^{\s_{\lambda,k}}}=\cO_{X}$ and the conditions on $g \in \cO_{X}$ in are independent of $k$. Moreover, if $0\leq \lambda<\frac{1}{\max\{n_i\}}$, then $\cM(C^\lambda)$ coincides with the multiplier ideal associated with the principal ideal generated by an equation of $C$ (for a definition of multiplier ideals see for instance [@Blickle-Lazarsfeld-informal]). This follows from the fact that $\left\{\lambda n_i\right\}=\lambda n_i$ for all $i=1,...,r$.
\[rem:qaideals\] Ideals of quasi-adjunction $A(j_1,\dots,j_r|m_1,\dots,m_r)$ ($0\leq j_i<m_i$) for $X=\CC^2$ were originally defined by A.Libgober in [@Libgober-characteristic section 2.3]. Given a local divisor $C=\sum_{i=1}^r n_iC_i$ in $(\CC^2,0)$ and $\lambda=\frac{k}{d}$ such that $0\leq \lambda<\frac{1}{\max\{n_i\}}$, one can write $\cM(C^\lambda)$ as a quasi-adjunction ideal choosing $m_i=d$, and $j_i+1=d(1-\lambda n_i)=d-kn_i$, that is, $$\cM(C^\lambda)=A(j_1,\dots,j_r|d,\dots,d).$$
\[rem:independenciak\] In general, note that $\s_{\lambda,k}$ is independent of $\lambda$ as long as $0\leq \lambda<\frac{1}{\max\{n_i\}}$. Moreover, in this case the expected monotonicity property holds $$\cO_{X,\zeta^{\s_{\lambda,k}}}=
\cO_{X,\zeta^{k-|w|}}\supseteq \cM(C^{\lambda_1},k)\supseteq \cM(C^{\lambda_2},k), \quad \text{ for }
0\leq \lambda_1\leq \lambda_2<\frac{1}{\max\{n_i\}}.$$ Example \[ex:non-reduced\] exhibits that this poperty does not need to hold for $\lambda\geq \frac{1}{\max\{n_i\}}$.
Consider $X=\frac{1}{5}(2,3)$ and the irreducible Weil divisor $C$ defined as the zero set of $x^3-y^2$. By Remark \[rem:independenciak\], if $\lambda\in [0,1)$ one has a stratification for $\cM(C^\lambda,k)$ for the different $k=0,...,4$ as shown in Table \[tab:MX\].
$[0,\frac{1}{6})$ $[\frac{1}{6},\frac{1}{3})$ $[\frac{1}{3},\frac{1}{2})$ $[\frac{1}{2},\frac{2}{3})$ $[\frac{2}{3},\frac{5}{6})$ $[\frac{5}{6},1)$
--- ------------------- ------
| 1,115
| 844
| 1,467
| 1,121
| 2,739
| 0.77737
|
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|
)}H_\lambda(q)\right)^k$ is $1$ (see ). Also, the coefficient of the lowest power of $q$ in each $m_{\lambda}(\y)$ is always $1$; hence so is the coefficient of the lowest power of $q$ in $C_{\nu\mu}(\y)$.
In the course of the proof of Proposition \[maxima\] we found that when $v(\lambda)$ is minimal, and $\rho^1,\ldots,\rho^r$ achieve the minimum in the right hand side of , then $\lambda=\rho^1\cup\cdots\cup\rho^r$. Hence by Lemma \[v-fmla-lemma\], the coefficient of the lowest power of $q$ in $\langle
h_{\mu}(\x),s_\lambda(\x\y)\rangle=\sum_{\nu\unlhd\lambda}
K_{\lambda\nu}C_{\nu\mu}(\y)$ equals the coefficient of the lowest power of $q$ in $K_{\lambda\lambda}C_{\lambda\mu}(\y)=C_{\lambda\mu}(\y)$ which we just saw is $1$. This completes the proof.
### Leading terms of $\Log\,\Omega$ {#step-2}
We now proceed to the second step in the proof of connectedness where we analyze the smallest power of $q$ in the coefficients of $\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$. Write $$\Omega\left(\sqrt{q},1/\sqrt{q}\right)=\sum_\muhat
P_\muhat(q)\,m_\muhat
\label{pmu}$$ with $P_\muhat(q):=\sum_\lambda\calA_{\lambda\muhat}$ and $\calA_{\lambda \muhat}$ as in .
Then by Lemma \[Log-w\] we have $$\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)
=\sum_\omhat C_\omhat^0 P_\omhat(q)\,m_\omhat(q)$$ where $\omhat$ runs over [*multi-types*]{} $(d_1,\omhat^1)\cdots(d_s,\omhat^s)$ with $\omhat^p\in(\calP_{n_p})^k$ and $P_\omhat(q):=\prod_pP_{\omhat^p}(q^{d_p}),
m_\omhat(\x):=\prod_pm_{\omhat^p}(\x^{d_p})$..
Now if we let $\gamma_{\muhat\omhat}:=\langle
m_\omhat,h_\muhat\rangle$ then we have $$\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)
=\frac{(q-1)^2}{q}\left(\sum_{\omhat\in\mathbf{T}^k}
C_{\omhat}^0P_\omhat(q)\gamma_{\muhat\omhat}\right).$$ By Theorem \[step1\], $v_q\left(P_\omhat(q)\right)=-d\sum_{p=1}^s\Delta(\omhat^p)$ for a multi-type $\omhat=(d,\omhat^1)\cdots(d,\omhat^s)$.
\[perm-sum-lemma\] Let $\nu^1,\ldots,\nu^s$ be partitions. Then $$\langle m_{\nu^1}\cdots m_{\nu^s}, h_\mu\rangle \neq 0$
| 1,116
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ws the occupation probabilities $p_n = |a_n|^2$ for a defect located at the site $n = 0$ in the absence of the periodic force; the discrete values have been connected by lines to guide the eye. The lower panel shows the state under the influence of a periodic force with high frequency $\hbar\omega/W = 7.5$ and scaled amplitude $eFd/(\hbar\omega) = j_{0,1} \simeq 2.4048$, equal to the first zero of ${\rm J}_0$: As expected, the state now is confined almost entirely to the defect site. Hence, the extension of the defect state is governed by the driving force’s amplitude. This effect will be exploited in Sec. \[sec:twodefs\] to control the population transfer between two communicating defects.
Energy splitting and quasienergy splitting for communicating defects {#sec:split}
====================================================================
In the following, the real amplitudes $a_\ell$ for the state bound by a single defect placed at a site $\gamma > 0$ are normalized such that $$\label{eq:normdef}
\sum_{\ell=1}^{\infty} \left|a_{\ell}\right|^2 = 1 \; ,$$ which implies that the normalization constant ${\cal N}$ in Eq. (\[eq:al\]) is now given by $$\begin{aligned}
{\cal N} &=& \left({{\sum_{i=1}^{\gamma}x_-^{2(\gamma-i)}+
\sum_{i=\gamma+1}^{\infty}x_-^{2(i-\gamma)}}}\right)^{-1/2}
\nonumber \\
&=& \sqrt{\frac{1-x_-^2}{1+x_-^2-x_-^{2\gamma}}} \; .
\label{eq:normerg} \end{aligned}$$ This defect state obeys the Schrödinger equation $$\label{eq:E0}
\left( \hat{H}_0 + \hat{V}_{\rm r} \right)
\left|\psi_0\right> = E_0\left|\psi_0\right> \; .$$ Next, a second, identical defect is introduced into the left half of the lattice at the site labeled $-\gamma$, as described by $$\label{eq:V_l}
\hat{V}_{\rm l} = {{\left|-\gamma\right.\rangle}}\nu{{\langle \left.-\gamma\right|}} \;,\quad \gamma>0\; .$$ The two defects then carry two localized states $\left|\psi_{1}\right>$ and $\left|\psi_{2}\right>$, obeying the eigenvalue equations $$\begin{aligned}
\label{eq:E1}
\left(\hat{H}_0 + \hat{V}_{\rm r
| 1,117
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| 0.771203
|
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|
r the Rindler trajectory, we have $z_0(\tau) =0 = x_{\perp 0}(\tau) $, hence the $u_{\omega, k_{\perp}}$ are just constants. Then $W(\tau, \tau^\prime) = W(\tau - \tau^\prime) = W(s)$ as expected for a Killing trajectory. The transition rate is straightforward to obtain and we get, $$\begin{aligned}
{\dot {\cal F}}(E)
&=& \int d\omega \int d^{2}k_{\perp} \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right] \bigg[ \Theta(\omega + E) \left( \eta_{\omega}+1 \right) u_{\omega, k_{\perp}} u^{\star}_{\omega, k_{\perp}} \nonumber \\
\; \; \; & & + \; \Theta(\omega - E)\; \eta_{\omega} \, u^{\star}_{\omega, k_{\perp}} u_{\omega, k_{\perp}} \bigg] \end{aligned}$$ Thus, ${\dot {\cal F}}(E) $, satisfies the KMS condition for an arbitrary profile function, $$\frac{{\dot {\cal F}}(E)}{{\dot {\cal F}}(-E)} = \frac{\eta_{E}}{\eta_{E}+1} = e^{- \beta E}$$ with the usual Unruh temperature.
The above result regarding the thermality is quite general and holds for any arbitrary smooth profile function which falls off to Rindler’s spatial infinity. One could even have included a direction dependent angle as in the case of Eq.(\[angprof\]). However, the result would still be the same, since the spatial part does not contribute to the $\tau$ integral in the co-moving frame.
Discussion {#discsection}
==========
We have analysed two models for a spatially extended detectors having direction dependence on the Rindler trajectory. The first model is based on Schlicht type construction with a direction-sensitive Lorentz-function profile for the smeared field operator with a characteristic length $\epsilon$ and defined in the Fermi co-ordinates attached to the uniformly accelerated trajectory. Whereas the second model has a very general direction sensitive profile for the smeared field operator but defined in the Rindler wedge corresponding to the trajectory. The transition rate for the two models were found to differ significantly when evaluated on the Rindler trajectory. In the first model, the spectrum was obtained to be anisotr
| 1,118
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|
eq:3.39\] v=f(z\^1,…, z\^k, |[z]{}\^1,…, |[z]{}\^k)+c.c.
for an arbitrary function $f:X{\rightarrow}U$. The expression (\[eq:3.39\]) is the general solution of the invariance conditions
\[eq:3.40\] v\_[z\^[2k+1]{}]{},…, v\_[z\^p]{}=0.
In general, the initial system (\[eq:3.1\]) described in the new coordinates $(x,v)\in X\times U$ is a nonlinear system of first-order PDEs, of the form
\[eq:3.42\]
&\^[l]{}\_(v)=0,&&i=1,…, k,\
&\^[l]{}\_(v)=0,&&i=k+1,…, 2k,\
&\^[i]{}\_(v)=0, &&i=2k+1,…,p.
We obtain the following Jacobi matrix in the coordinates $(z,\bar{z},v)$
\[eq:3.43\] =[( \^[-1]{} )]{}\^j\_l\^l\_i\^[pp]{},=\_s\^i-x\^l,
whenever the invariance conditions (\[eq:3.40\]) are satisfied. Appending to the system (\[eq:3.42\]) the invariance condition (\[eq:3.40\]), we obtain the quasilinear reduced system of PDEs
\[eq:3.44\] [( \^(v)[( I\_[q]{}- )]{}\^[-1]{})]{}=0,,…,=0,=1,…,m,
or
\[eq:3.45\] [( \^(v)\^[-1]{})]{}=0,,…,=0,=1,…,m.
####
We now provide some basic definitions that we require for the use of the conditional symmetry method in order to encompass the use of Riemann invariants.
####
A vector field $X_a$ is called a conditional symmetry of the original system (\[eq:3.1\]) if $X_a$ is tangent to the manifold ${\mathbb{S}}={\mathbb{S}}_\Delta\cap {\mathbb{S}}_Q$, [*i.e.* ]{}
\[eq:3.46\] .\^[(1)]{}X\_a|\_T\_[(x,u\^[(1)]{})]{},
where the first prolongation of $X_a$ is given by
\[eq:3.47\] \^[(1)]{}X\_a=X\_a-\^i\_[a,u\^]{}u\_j\^u\_i\^,a=1,…,p-2k
and the submanifolds of the solution spaces are given by
\[eq:3.48\] \_=[{ (x,u\^[(1)]{}):\_\^[i]{}u\_i\^=0,=1,…,m }]{},
and
\[eq:3.49\] \_Q=[{ (x,u\^[(1)]{}):\_a\^i(u)u\_i\^=0,=1,…, q, a=1,…,p-2k }]{}.
Consequently, an Abelian Lie algebra $\mathcal{L}$ generated by the vector fields $X_1,\ldots, X_{p-2k}$ is called a conditional symmetry algebra of the original system (\[eq:3.1\]) if the conditions
\[eq:3.50\] .\^[(1)]{}X\_a[( \^iu\_i )]{}|\_=0,a=1,…, p-2k,
are satisfied.
####
Supposing that $\mathcal{L}$, spanned
| 1,119
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1.4 1.5
sensors-20-02119-t003_Table 3
######
*RMSE* and *MAE* of I--V curve obtained by equivalent circuit method.
Working Conditions 1 2 3 4
----------------------------- ------- ------- ------- -------
**Irradiance *G* (W/m^2^)** 153.7 328.7 537.9 726.9
***MAE*** 0.443 0.975 1.236 0.998
***RMSE*** 0.335 0.766 0.895 0.864
sensors-20-02119-t004_Table 4
######
Comparison between artificial intelligence and equivalent circuit method.
Evaluation Method Equivalent Circuit Method CNN Model MLP Model
------------------- --------------------------- ----------- -----------
***MAE*** 0.864 0.0307 0.0671
***RMSE*** 0.445 0.0322 0.0816
Pincus, R., et al. (2015), Radiative flux and forcing parameterization error in aerosol‐free clear skies, Geophys. Res. Lett., 42, 5485--5492, doi:[10.1002/2015GL064291](10.1002/2015GL064291).26937058
This article was corrected on 14 SEP 2015. See the end of the full text for details.
1. Assessing the Accuracy of Radiation Parameterizations in Climate Models {#grl53099-sec-0001}
==========================================================================
Radiative transfer is unique among parameterization problems for global atmospheric models because the governing equations are deeply grounded in fundamental physics, the approximations (e.g., of one‐dimensional radiative transfer) applicable across many relevant scales, and the result entirely deterministic. In aerosol‐free clear skies, where scattering is small relative to absorption and emission, the problem is defined by the profile of extinction of the gaseous atmosphere, which is itself determined by the profiles of temperature, pressure, and the concentrations of radiatively active gases. Fluxes of longwave or terrestrial radiation also depend on how the extinction profile is relate
| 1,120
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ection lines, which can also be added/removed through the web interface.Fig. 3Study template creation
When completed, the resulting workflow must be saved to serve as a template for workflow executions, i.e., conducting a clinical study, coordinating teamwork, or similar processes. Each execution of the template is an independent process, having a particular set of participants and deadlines. This means that the same template can be reused to manage several studies over time.
Users and Studies {#Sec7}
-----------------
The web client can be described from two different user perspectives: the study manager's perspective and the task assignee's perspective. Figure [4](#Fig4){ref-type="fig"} presents the main phases of each study and the role of each user in this pipeline. The components displayed in grey refer to the actions taken by the assignees, and the remainder are the responsibility of the SM.Fig. 4Users workflow - for the study manager (in white boxes), and for the assignee (in gray)
### Study manager {#Sec8}
The SM role is assumed automatically by any user that creates a study template and decides to execute the workflow within a team of other users (acting here as assignees). Besides defining and coordinating the study pipeline, the SM is responsible for task assignment, scheduling management, and results compilation at the end of the study.
Each user may create study templates, which work as a model that can be used to initiate a study. This template is kept private, in the user workspace, unless they decide to share it with other platform users. The latter may then clone or reuse, but not change the original template. To start a study, the user may create a new template, or select an existing one from the available list. Then, the user, now in an SM role, needs to choose the users that will be involved in the study. In this phase, the SM can also activate some general reminders that will be sent to the task assignees, before and after each task deadline.
The next step is to assign users to tasks.
| 1,121
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|
predecessor walks into conventional sequences.
\[D:TEST\] Let ${\mathit{w}}$ be a localized predecessor walk of $n = {\lvert{{\mathit{w}}}\rvert}$ steps, indexed from $0$ down to $-(n-1)$. Define the *test* function $\test({\mathit{w}}) = \widetilde{{\mathit{w}}}$ according to formula $\widetilde{{\mathit{w}}}_i = {\mathit{w}}_{i - n}$ for $i = 1, 2, \cdots , n$.
Assuming that a localized predecessor walk is indexed from $0$ down to $-(n-1)$, its corresponding test will be indexed from $1$ to $n$. In sense of direction, the localized predecessor walk traverses steps from ${\mathit{s}}_{\text{crux}}$ to ${\mathit{s}}_{\text{edge}}$, while the corresponding test traverses steps from ${\mathit{s}}_{\text{edge}}$ to ${\mathit{s}}_{\text{crux}}$.
Suppose ${\mathcal{C}}$ is an acyclic cone and ${\mathbb{W}} \subset {\mathcal{C}}$ is a (unique) set of localized predecessor walks. If $\widetilde{{\mathbb{W}}} = \test({\mathbb{W}})$ is its converted set of reversed and re-indexed tests, then ${\mathbb{W}}$ and $\widetilde{{\mathbb{W}}}$ are in one-to-one correspondence.
By virtue of construction, $\test$ is already a mapping. Remaining to show is that $\test$ is additionally a bijection. Let ${\mathit{u}}$ and ${\mathit{v}}$ be localized predecessor walks and ${\mathit{x}}$ be a finite walk. As hypothesis set ${\mathit{x}} = \test({\mathit{u}}) = \test({\mathit{v}})$. These sequences cannot be equal unless they possess the same number of terms, $n = {\lvert{{\mathit{x}}}\rvert} = {\lvert{\test({\mathit{u}})}\rvert} = {\lvert{\test({\mathit{v}})}\rvert}$. Since transformation $\test$ preserves the number of steps (from Definition \[D:TEST\] ${\lvert{{\mathit{w}}}\rvert} = {\lvert{\test({\mathit{w}})}\rvert}$), then $n = {\lvert{{\mathit{x}}}\rvert} = {\lvert{{\mathit{u}}}\rvert} = {\lvert{{\mathit{v}}}\rvert}$.
Again invoking Definition \[D:TEST\] on the first part of the hypothesis, we write ${\mathit{x}}_i = {\mathit{u}}_{i - n}$. The second part similarly yields ${\mathit{x}}_i = {\mathit{v}}_{i - n}$. By equatin
| 1,122
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|
m\] we deduce that ${\ensuremath{\left| A_s \right|}}$ divides ${\ensuremath{\left| {\operatorname}{Aut}(M) \right|}}$, and so $s^{(\delta+n)m}\cdot s^{\frac{m-1}{s-1}}$.
On the other hand, if ${\ensuremath{\left| G/N \right|}}_s=s^{\gamma}$, then ${\ensuremath{\left| G \right|}}_s={\ensuremath{\left| G/N \right|}}_s {\ensuremath{\left| N \right|}}_s=s^{\gamma+nr}$. Further, ${\ensuremath{\left| B_s \right|}}={\ensuremath{\left| G/N \right|}}_s {\ensuremath{\left| B_s\cap N \right|}}$ divides $s^{\gamma+nm}$. Since ${\ensuremath{\left| G \right|}}_s$ divides ${\ensuremath{\left| A_s \right|}} {\ensuremath{\left| B_s \right|}}$, so $s^{\gamma+nr}$ divides $s^{\frac{m-1}{s-1}}\cdot s^{\gamma+nm} s^{(\delta + n)m}$. This fact, after some straightforward computations, leads to the desired conclusion, having in mind that $r=pm$.
$N$ is not a $p'$-group.
We take a prime $s\in \pi(N_1)\smallsetminus \pi({\operatorname}{Out}(N_1))$ (such prime always exists, see for instance [@KMP3 Lemma 5]). Note that $s \neq p$, since $N$ is a $p'$-group. Applying the previous Lemma for such prime we obtain that $n(p-2)<\delta +1$, but $\delta=0$ so necessarily $p=2$. This cannot happen, as it would imply that $N$ is a $2'$-group, so soluble, which is a contradiction by Lemma \[5\].
The almost simple case
======================
Let $N$ be a non-abelian simple group with $p \in \pi(N)$, and let $N \unlhd G \leq {{\operatorname}{\textup{Aut}}({N})}$ such that $G=AB=AN=BN$. Assume that $G$ satisfies the hypotheses of our main theorem, i.e. $p$ does not divide $i_G(x) $ for every $p$-regular element of prime power order $x \in A \cup B$.
We will carry out a case-by-case analysis of the simple group $N$ occuring as the socle of $G$ to prove that there is no a counterexample to our Main Theorem. Our strategy will apply the following lemma and the results in Section 3.
\[cent\] For any $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ $$\pi(G)=\pi(C_G(P)) =\pi(G/N) \cup \pi(C_N(P)).$$ In particular, $p \in {{\operatorname}{\mathcal{Z}}
| 1,123
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| 1,102
| null | null |
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|
as caused to age restrictions of the questionnaires, we assumed that the data are missing completely at random (MCAR). Therefore, listwise deletion was used. The ANOVA table was inspected to check for significant main and interaction effects and specific hypotheses were tested. Satterthwaite's approximation was used to obtain the degrees of freedom ([@B44]). Model assumptions of linearity, independence, normality and homogeneity of variance were checked. Significance was evaluated at the 5% significance level. To get insight into the magnitude of the effects, 95% confidence intervals (CI) are reported.
Results
=======
[Table 2](#T2){ref-type="table"} shows the means, standard deviations, and observed range for the variables in our study.
######
Descriptive statistics of the study variables.
Patient Mother Father Sibling
------------------------- ------------------------ --------- -------- -------- --------- ------ -------- ------- ------ -------- ------- ------- --------
Cancer appraisal 18.81 5.31 8--28 21.03 6.55 9--39 17.97 6.28 5--32 20.82 6.19 10--36
Family functioning Family relation index 56.22 7.91 37--68 53.76 7.99 28--68 52.66 7.78 26--68 54.82 8.04 37--68
Family structure index 54.09 7.73 39--68 49.68 7.55 20--64 49.34 8.41 18--64 51.06 8.34 35--65
Cancer-related emotions Loneliness 5.91 3.63 1--14 7.82 6.81 0--30 5.34 5.13 0--22 5.49 4.70 0--18
Uncertainty 5.65 3.78 0--15 8.88 4.26 0--18 7.40 3.82 0--15 7.29 5.56 0--24
Helplessness 12.87 4.70 1--23 13.36 4.67 3--21 11.23 4.51 1--21 13.37 5.14 1--21
| 1,124
| 1,031
| 1,948
| 1,336
| null | null |
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only if one of the following occurs.
1. $\mu$ or $\mu'$ equals $(u,v)$, where $v\equiv3\ppmod4$ and $\mbinom{u-v}{a-v}$ is odd.
2. $\mu$ or $\mu'$ equals $(u,v,2)$, where $\mbinom{u-v}{a-v}$ is odd.
Using this result, we can show that most of the Specht modules under consideration are decomposable. Specifically, we have the following result.
\[maincor\] Suppose $a,b$ are positive even integers with $a\gs4$, and let $\la=(a,3,1^b)$. Then $S^\la$ has a summand isomorphic to an irreducible Specht module if and only if at least one of the following occurs:
- $a+b\equiv0$ or $2\pmod 8$, $a\gs6$ and $b\gs4$;
- $a+b\equiv2\ppmod4$ and $\mbinom{a+b-3}{a-3}$ is odd;
- $a+b\equiv0\ppmod4$ and $\mbinom{a+b-9}{a-5}$ is odd.
Computing the space of homomorphisms between two Specht modules {#homsec}
===============================================================
In this section, we explain the set-up for computing the space of homomorphisms between two Specht modules. We begin with a revision of some material from [@j2], before citing some results of the second author and Martin.
Homomorphisms from Specht modules to permutation modules
--------------------------------------------------------
Suppose $\mu$ and $\la$ are partitions of $n$. Since $S^\la\ls M^\la$, any homomorphism from $S^\mu$ to $S^\la$ can be regarded as a homomorphism from $S^\mu$ to $M^\la$. This is very useful, because if $\mu$ is $2$-regular (or if ${\operatorname{char}}(\bbf)\neq2$), then the space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,M^\la)$ can be described explicitly. Furthermore, using the Kernel Intersection Theorem below, one can check whether the image of a homomorphism $\theta:S^\mu\to M^\la$ lies in $S^\la$.
We now make some more precise definitions. We take $\la,\mu$ as above, but we now allow $\la$ to be any composition of $n$, not necessarily a partition. A *$\mu$-tableau of type $\la$* is a function $T$ from the Young diagram $[\mu]$ to $\bbn$ with the property that for each $i\in\bbn$ there are exactly $\la_i$
| 1,125
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| 782
| 1,110
| 1,421
| 0.789802
|
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|
res \[fig:fg\](a) and \[fig:fg\](b) we observe that for a short period of time $f(\tau)$ exhibits a very similar growth to the upper bound $g(\tau)$, but then this growth slows down and $f(\tau)$ eventually starts to decrease short of ever approaching the limit $1/\E_0$.
We further characterize the time evolution by showing the maximum enstrophy increase $\delta\E := \mathop{\max}_{t \geq 0} \, \{ \E(t) -
\E(0) \}$ and the time when the maximum is achieved $T_{\max} :=
\mathop{\arg\max}_{t \geq 0}\, \E(t)$ as functions of $\E_0$ in figures \[fig:Emax\_vsE0\_fixE\](a) and \[fig:Emax\_vsE0\_fixE\](b), respectively. In both cases approximate power laws in the form $$\delta\E \sim \E^{\alpha_1}_0, \quad \alpha_1 = 0.95 \pm 0.06 \qquad\mbox{and}\qquad
T_{\max} \sim \E^{\alpha_2}_0, \quad \alpha_2 = -2.03 \pm 0.02$$ are detected in the limit $\E_0 \rightarrow \infty$ (as regards the second result, we remark that $T_{\max}$ is not equivalent to the time until which the enstrophy grows at the sustained rate proportional to $\E_0^3$, cf. figure \[fig:fg\]). To complete presentation of the results, the dependence of the quantities $$\mathop{\max}_{t \geq 0} \, \left\{\frac{1}{\E_0} - \frac{1}{\E(t)}\right\} \qquad\mbox{and}\qquad [\K(0) - \K(T_{\max})]$$ on the initial enstrophy $\E_0$ is shown in figures \[fig:Emax\_vsE0\_fixE\](c) and \[fig:Emax\_vsE0\_fixE\](d), respectively. It is observed that both quantities approximately exhibit a power-law behaviour of the form $\E^{-1}_0$. Discussion of these results in the context of the estimates recalled in §\[sec:intro\] is presented in the next section.
\
Discussion {#sec:discuss}
==========
In this section we provide some comments about the results reported in §§\[sec:3D\_InstOpt\_E0to0\], \[sec:3D\_InstOpt\_E\] and \[sec:timeEvolution\]. First, we need to mention that our gradient-based approach to the solution of optimization problem \[pb:maxdEdt\_E\] can only yield local maximizers and, due to nonconvexity of the problem, it is not possible to guarantee a pr
| 1,126
| 1,216
| 1,521
| 1,212
| 2,953
| 0.775857
|
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|
)) \leq D$, $d(x_i, x_{i+1}) \leq K$, and $d(g\circ f(x_i), g\circ f(x_{i+1})) \leq L$, the unioned sequence is an $L$-sequence. Further, because the two sequences ${x_n}$ and ${g\circ f(x_n)}$ are visited in order, we can say that ${x_n}$ and ${g\circ f(x_n)}$ are both subsequences of this union. Thus, the diagram commutes.
Since $z_L\circ\phi_{KL}$ is a bijection, $f_K$ must be one-to-one.
Symmetrically we can see that the following diagram commutes where $S$ is chosen so that ${\text{d}}(y,f\circ g(y))\leq S$ for all $y\in Y$ and ${\text{d}}(f(x),f(y))\leq S$ whenever ${\text{d}}(x,y)\leq L$.
& & & & \_S(Y, fgf(x\_0))\
& & & (4,4)\^[f\_L]{} & \^[z\_M]{}\
& & & & \_S(Y, f(x\_0))\
& & & & \^[\_[MS]{}]{}\
\_L(X, gf(x\_0) & & \_[g\_M]{} & & \_M(Y, f(x\_0))\
Thus $g_M$ must be one-to-one which forces $f_K$ to be onto. Then we have that $f_K$ is a bijection.
Some examples
=============
We begin with the standard example of a coarse equivalence.
\[integers\][@Roe] Consider $\mathbb R$ and $\mathbb Z$ as metric spaces under the usual metric. Let $f:\mathbb R\to \mathbb Z$ be the floor function, $x\mapsto \lfloor x\rfloor$. Let $g:\mathbb Z\to \mathbb R$ be the inclusion, $n\mapsto n$. It is easy to see that $f$ and $g$ are coarse and that $g\circ f$ and $f\circ g$ are close to the identities ($g\circ f$ is the identity). Corollary 3.7 in [@MMS] says that $\sigma(\mathbb R)=2$. Since $\mathbb Z$ is coarsely equivalent to $\mathbb R$ we must have $\sigma(\mathbb Z)=2$ also. Of course we can see these two sequences in $\mathbb Z$.
Next we give another way to calculate $\sigma(V)$ where $V$ is the vase from [@MMS Example 1.3]. We first give a basic lemma.
Suppose $f:X\to Y$ is any function and $g:Y\to X$ is bornologous. Suppose that $g\circ f$ is close to the identity on $X$. Then $f$ is proper.
Suppose $A\subset Y$ is bounded, say ${\text{d}}(x,y)\leq N$ for all $x,y\in A$. Suppose $x,y\in f^{-1}(A)$. Then $f(x),f(y)\in A$ so ${\text{d}}(f(x),f(y))\leq N$. Since $g$ is bornologous there is an $M>0$ so that ${\t
| 1,127
| 2,198
| 2,540
| 1,184
| 4,090
| 0.768179
|
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|
`
`SRR022865_71442` `3` `ctgagtaacga`
`cttaataaaca`
1252956 NONSYN T:9 G:125 G:37 `ttaaagaataa` DNA topoisomerase I
`gi 87162241 ref` `319` `SMDNVVTVGSTD`
`SM NVVTVGSTD`
`SMYNVVTVGSTD` lantibiotic epidermin leader
`SRR022865_97728` `2` tataggaggtag
1948255 NONSYN A:5 C:169 C:37 ctaattctgcca peptide processing serine
`tgctctaaatat` protease EpiP
1309034 TRUNC C:115 G:5 C:37 `Query: 35 ASAGKKSSI*N` `3 b` DNA mismatch r
| 1,128
| 4,367
| 2,898
| 1,013
| null | null |
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|
on $A$ and $U$, and since $U \leq A$, we can reduce the dependence of such constant on $A$ only.
$\Box$
Appendix 6: Anti-concentration and comparison bounds for maxima of Gaussian random vectors and Berry-Esseen bounds for polyhedral sets {#app:high.dim.clt}
======================================================================================================================================
Now we collect some results that can be are derived from [@chernozhukov2015comparison], [@cherno2] and [@nazarov1807maximal]. However, our statement of the results is slightly different than in the original papers. The reason for this is that we need to keep track of some constants in the proofs that affect our rates.
The following anti-concentration result for the maxima of Gaussian vectors follows from Lemma A.1 in [@cherno2] and relies on a deep result in [@nazarov1807maximal].
\[thm:anti.concentration\] Let $(X_1\ldots,X_p)$ be a centered Gaussian vector in $\mathbb{R}^p$ with $\sigma_j^2 = \mathbb{E}[X_j^2] > 0$ for all $j=1,\ldots,p$. Moreover, let $\underline{\sigma} = \min_{1 \leq j \leq p} \sigma_j$. Then, for any $y = (y_1,\ldots,y_p) \in \mathbb{R}^p$ and $a > 0$ $$\mathbb{P}( X_j \leq y_j + a, \forall j) - \mathbb{P}( X_j
\leq y_j, \forall j) \leq \frac{a}{\underline{\sigma}} \left( \sqrt{2 \log p} + 2 \right).$$
The previous result implies that, for any $a > 0$ and $y = (y_1,\ldots,y_p) \in
\mathbb{R}^p_+$, $$\mathbb{P}( |X_j| \leq y_j + a, \forall j) - \mathbb{P}( |X_j|
\leq y_j, \forall j) \leq \frac{a}{\underline{\sigma}} \left(
\sqrt{2 \log 2p} + 2 \right)$$ and that, for any $y > 0$, $$\mathbb{P}( \max_j |X_j | \leq y + a) - \mathbb{P}(\max_j |X_j|
\leq y) \leq \frac{a}{\underline{\sigma}} \left( \sqrt{2 \log 2p} + 2 \right).$$
The following high-dimensional central limit theorem follows from Proposition 2.1 in [@cherno2] and . Notice that we have kept the dependence on the minimal variance explicit.
\[thm:high.dim.clt\] Let $X_1,\ldots,X_n$ be independent centered
| 1,129
| 609
| 1,339
| 1,085
| 3,639
| 0.771027
|
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|
a continuous measure, coded (1) not at all, (2) once or twice a month, (3) once a week, (4) a few times a week, and (5) everyday (mean is 3.43). Results of the regression analysis are presented in [Table 4](#table4-0192513X17710773){ref-type="table"}. This analysis reveals that especially poor functioning transnational child-raising arrangements are associated with job instability. That is, Angolan transnational parents who have limited contact with their children have changed their jobs in the Netherlands more often. One unit decrease in contact leads to 0.41 more job changes. Next to happiness (β = .35) the amount of contact with the child is also one of the most important predictors of job change (β = .28) in this regression model.
######
Amount of Contact With Child on Job Instability of Transnational Parents.
Job instability (*n* = 82)
-------------------------------------------------------------------------------- ----------------------------------------------------------------------- -------
Happiness^[a](#table-fn9-0192513X17710773){ref-type="table-fn"}^ −1.39 (0.44)[\*\*](#table-fn11-0192513X17710773){ref-type="table-fn"} −0.35
Family-to-work conflict^[b](#table-fn9-0192513X17710773){ref-type="table-fn"}^ 0.33 (0.64) 0.07
Age −0.01 (0.05) −0.03
Sex^[c](#table-fn9-0192513X17710773){ref-type="table-fn"}^ 0.64 (0.46) 0.15
Marital status^[d](#table-fn9-0192513X17710773){ref-type="table-fn"}^ 0.35 (0.56) 0.07
Education
| 1,130
| 275
| 1,670
| 1,369
| null | null |
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|
omplete action (\[complete action\]). If this is in fact space-time supersymmetry, the commutator of two transformations should satisfy the supersymmetry algebra $$[\delta_{{\mathcal{S}}_1},\,\delta_{{\mathcal{S}}_2}]\ =^{\hspace{-2mm} ?}\ \delta_{p(v_{12})}\,,
\label{susy alg}$$ up to the equations of motion (\[equations of motion\]) and gauge transformation (\[full gauge\]) generated by some field-dependent parameters, where $\delta_{p(v_{12})}$ is the space-time translation defined by $$\delta_{p(v)}A_\eta\ =\ - p(v)A_\eta\,,
\qquad \delta_{p(v)}\Psi\ =\ - p(v)\Psi\,,
\label{translation}$$ with the parameter $v_{12}$ in (\[1st quantized alg\]). In this section, we show that the algebra (\[susy alg\]) is slightly modified, but still the transformation (\[complete transformation\]) can be identified with space-time supersymmetry.
Preparation
-----------
As preparation, note that the relations
\[large small\] $$\begin{aligned}
\delta A_\eta\ =&\ D_\eta A_\delta\,,
\label{delta eta}\\
A_\delta\
=&\ f\xi_0 \delta A_\eta + D_\eta \Omega_\delta\,,
\label{A delta}\end{aligned}$$
hold with $\Omega_\delta=f\xi_0A_\delta$, for general variation of the NS string field $A_\delta$. The former, (\[delta eta\]), is the case of $(\mathcal{O}_1,\mathcal{O}_2)=(\delta,\eta)$ in (\[MC\]), and the latter, (\[A delta\]), is obtained by decomposing $A_\delta$ by the projection operators (\[proj ns\]) and using (\[delta eta\]). These relations (\[large small\]) show that two variations $A_\delta$ and $\delta A_\eta$ are in one-to-one correspondence up to the $\Omega$-gauge transformation. Since any transformation of the string field is a special case of the general variation, (\[large small\]) holds for any symmetry transformation $\delta_I$,
\[delta I\] $$\begin{aligned}
\delta_I A_\eta\ =&\ D_\eta A_{\delta_I}\,,
\label{delta I 1}\\
A_{\delta_I}\ =&\ f\xi_0\delta_IA_\eta + D_\eta\Omega_I\,.
\label{delta I 2}\end{aligned}$$
This is the case even for the commutator of the two transformations $[\delta_I, \delta_J]$,
\[delt
| 1,131
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| 2,066
| 1,237
| null | null |
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|
de of is defined via Theorem \[thm:intersection\]. It is easily seen that and match with Sakai’s definition ([@Sakai84]) of intersection multiplicity assuming $\pi$ is a resolution, not a $\Q$-resolution.
\[ex:QNC:pullback\] Following Example \[ex:QNC:resolution\] note that $\pi_1^*D_1=\hat D_1+mE$ and the equation $E\cdot \pi_1^*D_1=E\cdot \hat D_1+mE^2=1-3m=0$ matches the solution $m=\frac{1}{3}$ given by Theorem \[thm:intersection\].
Given $D\in \operatorname{Weil}(X)$, the sheaf $\cO_X(D)$ is defined as $i_*\cO_{\operatorname{Reg}(X)}(D|_{\operatorname{Reg}(X)})$ via the inclusion $i:\operatorname{Reg}(X)\to X$ of the regular part the $X$. In the general case $D\in \operatorname{Weil}_\QQ(X)$ one defines $$\label{eq:OD}
\cO_X(D):=\cO_X(\lfloor D \rfloor).$$ As a word of caution, note that this notation differs from the one used in [@Nemethi-Poincare]. Using this definition the following projection formula is obtained.
\[thm:projection\] If $D\in \operatorname{Weil}_\QQ(X)$ and $\pi:Y\to X$ is a resolution of $X$, then $$\pi_*(\cO_Y(\pi^*D))=\cO_X(D).$$
This shows how $\cO_X(D)$ can be interpreted via the divisor $D$.
\[prop:H0D\] Let $X$ be a normal surface and $D \in \operatorname{Weil}_{\QQ}(X)$. Then, the cohomology $H^ 0(X, \cO_X(D))$ can be identified with $\{ h \in K(X) \mid (h) + D \geq 0 \}$.
The statement of this result is true for an integral divisor on a smooth variety. Let $\pi: \tilde{X} \to X$ be a resolution of $X$ consisting in a composition of blowing-ups. Then, by [@Sakai84 Theorem 1.2], $\cO_X(D) = \pi_{*} \cO_{\tilde{X}}(\pi^{*} D) = \pi_{*} \cO_{\tilde{X}}({\left \lfloor \pi^{*} D \right \rfloor})$ and thus $H^0 (X,\cO_X(D)) = H^0(\tilde{X},\cO_{\tilde{X}}({\left \lfloor \pi^{*} D \right \rfloor}))$. Since the latter is an integral divisor in a smooth variety, $$H^0(\tilde{X},\cO_{\tilde{X}}({\left \lfloor \pi^{*} D \right \rfloor})) = \{ \tilde h \in K(\tilde{X}) \mid (\tilde h) + {\left \lfloor \pi^* D \right \rfloor} \geq 0 \}.$$ The condition $(\tilde h) + {\left \lfl
| 1,132
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| 1,096
| null | null |
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|
EA~ clade dissemination in east Africa.\
We provide snapshots of the dispersal pattern for the years 1960, 1965, 1970, 1975, 1980, 1985, 1990 and 2000. Lines between locations represent branches in the Bayesian MCC tree along which location transition occurs. Location circle diameters are proportional to square root of the number of Bayesian MCC branches maintaining a particular location state at each time-point. The white-green color gradient informs the relative age of the transitions (older-recent). The maps are based on satellite pictures made available in Google^™^ Earth (<http://earth.google.com>).](pone.0041904.g004){#pone-0041904-g004}
10.1371/journal.pone.0041904.t002
###### Estimated number of migration events of HIV-1 C~EA~ clade among east African countries.
{#pone-0041904-t002-2}
From/To Burundi Ethiopia Kenya Tanzania Uganda
---------- --------- ---------- ------- ---------- --------
Burundi \- 4 5 8 8
Ethiopia 0 \- 0 0 1
Kenya 0 0 \- 1 1
Tanzania 0 0 3 \- 7
Uganda 0 0 0 0 \-
The Bayesian analysis also supports an important phylogeographic subdivision within the C~EA~ lineage. Consistent with the ML topology ([Figure S2](#pone.0041904.s002){ref-type="supplementary-material"}), most subtype C sequences from Ethiopia, Kenya, Tanzania and Uganda branched in country-specific monophyletic sub-clusters that most probably (*PP*≥0.93) had a Burundian origin ([Fig. 3](#pone-0041904-g003){ref-type="fig"}). The C~ET1~ and C~ET2~ lineages, that correspond to the so called Ethiopian-C clade, comprise 44% of all Ethiopian sequences here included and were almost exclusively composed by sequences from this country. The C~KE~ and C~UG~ lineages comprise 33% and 37% of all sequences from Kenya and Uganda, respectively, and their circulation seems to be mainly restricted to thos
| 1,133
| 1,229
| 2,718
| 1,367
| null | null |
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|
frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(\delta x)^2}, \\
\Delta_y^2\Phi_{i,j,k} &= \frac{\Phi_{i,j-1,k}-2\Phi_{i,j,k}+\Phi_{i,j+1,k}}{(\delta y)^2}, \\
\Delta_z^2\Phi_{i,j,k} &= \frac{\Phi_{i,j,k-1}-2\Phi_{i,j,k}+\Phi_{i,j,k+1}}{(\delta z)^2}.
\label{eq:delz2}\end{aligned}$$
Uniform Cylindrical Grid
------------------------
In uniform cylindrical coordinates, we discretize the computational domain $[R_{\rm min},R_{\rm max}]\times[\phi_{\rm min},\phi_{\rm max}]\times[z_{\rm min},z_{\rm max}]$ with size $L_R\times L_\phi\times L_z$ uniformly into $N_R\times N_\phi\times N_z$ cells. We require that $L_\phi=\phi_{\rm max}-\phi_{\rm min}$ should be an integer fraction of 2$\pi$ to impose periodic boundary condition along the azimuthal direction. We define the face-centered radial and azimuthal coordinates as $R_{i+1/2} = R_{\rm min} + i \delta R$ and $\phi_{j+1/2} = \phi_{\rm min} + j \delta \phi$, where $\delta R \equiv L_R/N_R$ and $\delta \phi \equiv L_\phi/N_\phi$. Unlike in Cartesian coordinates, the definition of the radial cell-center is ambiguous in cylindrical coordinates because the geometric center does not coincide with the volumetric center. When the radial grid is uniform, finite difference of quantities defined at the geometric centers can retain second-order accuracy. We thus define the cell-centered coordinates as $R_{i} = (R_{i-1/2} + R_{i+1/2})/2$ with $i=1,2,\cdots,N_R$ and $\phi_{j} = (\phi_{j-1/2} + \phi_{j+1/2})/2$ with $j=1,2,\cdots,N_\phi$. Discretization in the vertical direction is the same as in the uniform Cartesian coordinates.
The second-order finite-difference approximation to Equation can be written as $$\label{eq:fd_uniform_cylindrical}
\left( \Delta_R^2 + \Delta_\phi^2 + \Delta_z^2 \right)\Phi_{i,j,k} = 4\pi G \rho_{i,j,k},$$ where the difference operators $\Delta_R^2$ and $\Delta_\phi^2$ are defined by $$\begin{aligned}
\Delta_R^2\Phi_{i,j,k} =& \frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(\delta R)^2}
+ \frac{\Phi_{i+1,j,k}-\Phi_{i-1,j,k}}{2R_i\de
| 1,134
| 4,035
| 1,528
| 996
| null | null |
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|
ibution of . From Lemma \[l:energy\_x\], we know that $\bx_0$ satisfies the required initial conditions in this Lemma. Continuing from , $$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \bw_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \E{\lrn{\bx_0}_2^2 + \lrn{\bw_0}_2^2} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + \epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \frac{16}{\lambda} \exp\lrp{2\frac{7\aq\Rq^2}{3}}\lrp{L + \LN^2} \epsilon\\
=& 4\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \hat{\epsilon}
\end{aligned}$$ By our assumption that $i\geq \frac{1}{ \delta} \cdot 3\aq\Rq^2 \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, the first term is also bounded by $\hat{\epsilon}$, and this proves our second claim.
[Coupling Properties]{} \[s:coupling\_properties\]
\[l:marginal\_of\_coupling\] Consider the coupled $(x_t,y_t)$ in . Let $p_t$ denote the distribution of $x_t$, and $q_t$ denote the distribution of $y_t$. Let $p_t'$ and $q_t'$ denote the distributions of and .
If $p_0 = p_0'$ and $q_0 = q_0'$, then $p_t = p_t'$ and $q_t=q_t'$ for all $t$.
Consider the coupling in , reproduced below for ease of reference: $$\begin{aligned}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s\\
y_t =& y_0 + \int_0^t -\nabla U(y_0) dt + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s
\end{aligned}$$
Let us define the stochastic process $A_t := \int_0^t M(x_s)^{-1} \cm dV_s + \int_0^t M(x_s)^{-1} N(x_s) dW_s$. We can verify using Levy’s characterization that $A_t$ is a standard Brownian motion: first, since $V_t$ and $W_t$ are Brownian motions, and $N(x)$ is differentiable with bounded derivatives, we know that $A_t$ has continuous sample paths. We now verify that $A_t^i A_t^j - \ind{i=j} t$ is a martingale.
Notice that $d A_t = \cm dV_t + M(x_s)^{-1} N(x_s) dW_s$. T
| 1,135
| 1,408
| 687
| 1,063
| null | null |
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|
File file = new File(name);
if (file.exists()) {
this.keyStore = KeyStore.getInstance(KeyStore.getDefaultType());
try (FileInputStream fileInputStream = new FileInputStream(name)) {
this.keyStore.load(fileInputStream, password.toCharArray());
}
} else {
this.keyStore = KeyStore.getInstance(KeyStore.getDefaultType());
this.keyStore.load(null, null);
}
} catch (KeyStoreException | IOException | NoSuchAlgorithmException | CertificateException ex) {
}
}
Q:
Can't display more than 1 sprites - PyGame
Problem
My problem is that I have a game that generates 8 obstacles at the start of the game. The issue is that when I loop through the obstacles list, and update the sprites group, it only generates 1 sprite.
What I Want To Happen
When the game loads, I want 8 squares to fly down from the top of the window at random speeds, and starting at random positions.
What Is Currently Happening
Currently, when the game loads, only one square is falling from the screen.
PYthon Code
OBSTICLES_AMOUNT = 8
class Obstacle(pygame.sprite.Sprite):
def __init__(self):
pygame.sprite.Sprite.__init__(self)
self.image = pygame.Surface((30, 30))
self.image.fill(BLUE)
self.rect = self.image.get_rect()
self.rect.x = random.randrange(0, WIDTH - self.rect.width)
self.rect.y = random.randrange(-100, -40)
self.velY = 6
def animate(self):
self.rect.y += self.velY
class Game(pygame.sprite.Sprite):
def __init__(self):
pygame.sprite.Sprite.__init__(self)
pygame.init()
pygame.mixer.init()
self.screen = pygame.display.set_mode((WIDTH, HEIGHT))
pygame.display.set_caption(TITLE)
self.running = True
self.clock = pygame.time.Clock()
self.obstaclesList = []
self.allSprites = pygame.sprite.Group()
self.obstacles = pygame.sprite.Group()
def new(self):
# create a new game
# add obstacles to list
for i in range(OBS
| 1,136
| 6,289
| 97
| 805
| 218
| 0.817801
|
github_plus_top10pct_by_avg
|
$.
Moreover, depending on the values of $K_{\pm}$ bosonization opens the possibility of two consecutive phase transitions with increasing $U$ starting from the CSF phase [@Supplementary], which we have confirmed with our DMRG calculations (Fig. \[fig:2\](a)). First a KT transition occurs from CSF to chiral-Mott (CMI), a narrow Mott phase with finite chirality. Then an Ising transition is produced from CMI to non-chiral MI. At both KT transition lines in Fig. \[fig:2\](a) (SF-MI and CSF-CMI), up to a logarithmic prefactor, $G_{ij}\sim (-1)^{i-j}e^{-i \kappa(i-j)}{|i-j|^{-1/4}}$, where in CSF $ \kappa\neq 0$.
#### Constrained bosons.
As mentioned above, sufficiently large three-body losses may result in a three-body constraint $(b_i^\dag)^3=0$ ($U_3=\infty$) [@Daley2009]. In that case, Model may be mapped to a large extent onto a frustrated spin-$1$ chain model [@commentSpin1], which, presents the possibility of a gapped Haldane phase, characterized by a non-local string order. Hence, interestingly, constrained bosons in a zig-zag lattice may be expected to allow for the observation of the HI phase in the absence of polar interactions.
Indeed, a model with $U=0$ and finite $U_3$ shows that at the Lifshitz point, $j=1/4$, a HI phase is stabilized for arbitrarily weak $U_3$ (Fig. \[fig:2\](b)). The effective theory describing the HI is again the sine-Gordon model (\[sine-Gordon\]) with $K<2$. However, now ${\cal M}<0$, which selects a hidden string order ${\cal O}^2_S\equiv\lim_{|i-j|\rightarrow\infty}\langle \delta n_i \exp [i\pi\sum_{i<l<j}\delta n_l]\delta n_j\rangle\sim \langle \sin \sqrt{\pi}\theta\rangle^2$ [@Berg2008]. Resembling the case of Fig. \[fig:2\](a), SF, HI, chiral-HI (CHI) and CSF phases occur (Fig. \[fig:2\](b)). These phases are expected for $U_3=\infty$ from known results in frustrated spin-$1$ chains [@Kolezhuk; @Hikihara2000; @Hikihara2002]. Our DMRG simulations suggest that all these phases meet at $j=1/4$ for $U_3\to 0$.
Figure \[fig:3\] shows the phase diagram for co
| 1,137
| 194
| 1,951
| 1,276
| 2,211
| 0.781834
|
github_plus_top10pct_by_avg
|
verline{g(a)},$$ and $\ell^2(\widehat{G})$ is defined analogously. The Fourier transform of $f \in \ell^2(G)$ is the function $\widehat{f} \in
\ell^2(\widehat{G})$ given by $$\widehat{f}(\chi) = {\left\langle f, \overline{\chi} \right\rangle}
= \sum_{a \in G} f(a) \chi(a).$$ This includes as special cases both the classical discrete Fourier transform (when $G$ is cyclic) and the Walsh–Hadamard transform (when $G$ is a product of cyclic groups of order $2$). The following lemma summarizes the most important fundamental facts about the Fourier transform for our purposes.
\[T:FT-isometry\] Let $G$ be a finite abelian group with ${\left\vert G \right\vert}$ elements.
1. \[I:onb\] The functions $\bigl\{
\frac{1}{\sqrt{{\left\vert G \right\vert}}}\chi \mid \chi \in \widehat{G}\bigr\}$ form an orthonormal basis of $\ell^2(G)$.
2. \[I:isometry\] The map $f \mapsto
\frac{1}{\sqrt{{\left\vert G \right\vert}}}\widehat{f}$ is a linear isometry of $\ell^2(G)$ onto $\ell^2(\widehat{G})$.
3. \[I:convolution\] If $f, g \in \ell^2(G)$, then for each $\chi \in \widehat{G}$, $\widehat{f*g}(\chi) = \widehat{f}(\chi)
\widehat{g}(\chi)$ (where the convolution $f*g$ is defined in .
<!-- -->
1. See Theorem 6 on [@Serre p. 19].
2. This follows easily from Proposition 7 on [@Serre p. 20] (which is a consequence of part (\[I:onb\])).
3. This follows directly from the definitions by a straightforward computation.
Observe that contained in Lemma \[T:FT-isometry\](\[I:onb\]) is the fact that ${\left\vert G \right\vert} = \bigl\vert\widehat{G}\bigr\vert$.
We will need two additional facts about characters of finite abelian groups which are not as easily located in standard references.
\[T:p2\] The number of elements $a \in G$ such that $a^2 = 1$ is equal to the number of characters $\chi \in \widehat{G}$ such that $\chi =
\overline{\chi}$.
For $a \in G$, define $\delta_a : G \to {\mathbb{C}}$ by $\delta_a(b) =
\delta_{a,b}$, where the latter is the Kronecker delta function, and observe that $\{
| 1,138
| 4,415
| 1,302
| 842
| 3,942
| 0.769088
|
github_plus_top10pct_by_avg
|
chi (\Lambda )$. Then Eq. follows from Thm. \[th:PBWtau\].
Assume now that $\nu \in \{2,3,\dots ,n\}$ and that the lemma holds for $\nu -1$. Let $\chi _\mu =r_{i_{\mu -1}}\cdots
r_{i_2}r_{i_1}(\chi )$ and $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}
{t}_{i_1}^\chi (\Lambda )$ for all $\mu \in \{1,2,\dots ,\nu \}$. By Lemma \[le:VTinv\], the assumptions on $\Lambda $ are equivalent to the relations $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi _\mu }}({\alpha }_{i_\mu })-1}
(\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1})
-{\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^{m-1}) \not=0$$ and $\Lambda _\nu (K_{i_\nu }L_{i_\nu }^{-1})=
{\rho ^{\chi _\nu }}({\alpha }_{i_\nu })^{t-1}$. Let $$\beta '_\nu ={\sigma }_{i_1}^\chi (\beta _\nu )=1_{\chi _2}{\sigma }_{i_2}\s
_{i_3}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }).$$ By induction hypothesis there exists a $U(\chi _2)$-submodule $V'$ of $M^{\chi _2}(\Lambda _2)$ with $$\begin{aligned}
\fch{V'}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}.
\end{aligned}$$ Moreover, $\Lambda (K_{i_1}L_{i_1}^{-1})\not=\rho ^\chi ({\alpha }_{i_1})^{m-1}$ for all $m\in
\{1,2,\dots ,{b^{\chi}} ({\alpha }_{i_1})-1\}$, and hence ${\hat{T}}_{i_1}:M^{\chi _2}(\Lambda _2)\to M^\chi (\Lambda )$ is an isomorphism. Let $V={\hat{T}}_{i_1}(V')$. By Lemmata \[le:MLmap\] and \[le:Tpfch\], $V$ is a $U(\chi )$-submodule of $M^\chi (\Lambda )$ and $$\begin{aligned}
{\mathrm{ch}\,V}={\dot{\sigma }}_{i_1}^{\chi _2}(\fch{V'})=\sum _{{\alpha }\in {\mathbb{N}}_0^I}
{\dot{\sigma }}_{i_1}^{\chi _2}({P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}).
\end{aligned}$$ Thus, by Lemma \[le:P1\], $$\begin{aligned}
{\mathrm{ch}\,V}=&e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big(
\frac {e^{-t\beta '_\nu}-e^{-\bfun{\chi _2}(\beta '_\nu )\beta '_\nu }}
{1-e^{-\beta '_\nu }}
\prod _{\beta \in R_+^{\chi _2}\setminus \{\beta '_\nu \}}
\frac {1-e^{-\bfun{\chi _2}(\beta )\beta }}
{1-e^
| 1,139
| 845
| 926
| 1,143
| null | null |
github_plus_top10pct_by_avg
|
cause the tableaux involved are not semistandard.
\[sigmanz\] With the notation above, $\sigma\neq0$.
The version of this paper published in the Journal of Algebra includes a fallacious proof of Proposition \[sigmanz\]; the proof below replaces it. The authors are grateful to Sinéad Lyle for pointing out the error.
We’ll use Lemma \[semidom\]. Consider the semistandard tableau $$S={\text{\footnotesize$\gyoungx(1.2,;1;1;2_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;{b\!\!+\!\!3};{b\!\!+\!\!4},;3,;4,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}.$$ We’ll show that when $\sigma$ is expressed as a linear combination of semistandard homomorphisms, ${\hat\Theta_{S}}$ occurs with non-zero coefficient, and hence $\sigma\neq0$. Given $T\in{\calu}$, consider expressing ${\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms. By Lemma \[semidom\], ${\hat\Theta_{S}}$ can only occur if $S\dom T$; so we can ignore all $T\in{\calu}$ for which $S\ndom T$. In particular, we need only consider those tableaux in ${\calu}$ which have $b+5,\dots,u$ in the first row and $b+3,b+4$ in the top two rows. If we assume for the moment that $v<b+3$, then the tableaux $T\in{\calu}$ that we need to consider are those of the following forms: [$$\begin{aligned}
T[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\dra
| 1,140
| 777
| 1,060
| 1,127
| 375
| 0.811673
|
github_plus_top10pct_by_avg
|
$ is defined in . To prove the lower bound on $\lambda_2(\E[\lM])$, notice that $$\begin{aligned}
\label{eq:bottoml_hess2}
\E\big[\lM\big] &=& \sum_{j = 1}^n \sum_{i<\i \in [\ld]} \E\Bigg[ \sum_{a = 1}^{\ell} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \Big| (i,\i \in S_j) \Bigg] \P\Big[i,\i \in S_j\Big] (\le_i - \le_{\i})(\le_i - \le_{\i})^\top \;. \end{aligned}$$ Since the sets $S_j$ are chosen uniformly at random, $\P[i,\i \in S_j] = \kappa(\kappa -1)/d(d-1)$. Using the fact that $p_{j,a} = \kappa - \ell +a$ for each $j \in [n]$, and the definition of rank breaking graph $G_{j,a}$, we have that $$\begin{aligned}
\label{eq:bottoml_hess3}
\E\Bigg[ \sum_{a = 1}^{\ell} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \Big| (i,\i \in S_j) \Bigg] = \P\Big[\big(\sigma_j^{-1}(i), \sigma_j^{-1}(\i) > \kappa - \ell\big) \Big| (i,\i \in S_j) \Big]\;.\end{aligned}$$ The following lemma provides a lower bound on $\P[(\sigma_j^{-1}(i),\sigma_j^{-1}(\i)) > \kappa - \ell | (i,\i \in S_j)]$.
\[lem:prob\_bottomlbound\] Under the hypotheses of Theorem \[thm:bottoml\_upperbound\_general\], for any two items $i,\i \in [\ld]$, the following holds: $$\begin{aligned}
\label{eq:prob_bottomlbound_eq}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;\Big|\; i,\i \in S\Big] \;\geq\;
\frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{2}\frac{\ell^2}{\kappa^2}\;,\end{aligned}$$ where $\gamma_{\beta_1} \equiv \ld (\kappa-2)/(\lfloor \ell\beta_1 \rfloor+1 )(d-2) $ and $\eta_{\beta_1} \equiv (\lfloor \ell \beta_1 \rfloor +1)^2/2(\kappa-2)$.
Therefore, using Equations , and we have, $$\begin{aligned}
\label{eq:bottoml_expec}
\E\big[\lM\big] &\succeq& \frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{2} \frac{\ell^2}{\kappa^2} \frac{\kappa(\kappa-1)}{d(d-1)} \sum_{j = 1}^n \sum_{i<\i \in [\ld]} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top\;.\end{aligned}$$ Define $\lL = \sum_{j = 1}^n \sum_{i<\i \in [\ld]} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top$. We have, $\lambda_1(\lL
| 1,141
| 821
| 1,433
| 1,030
| null | null |
github_plus_top10pct_by_avg
|
complex integral elements. {#sec:2}
============================================================
The methodological approach assumed in this section is based on the generalized method of characteristics which has been extensively developed (*e.g. in* [@Burnat:1972; @DoyleGrundland:1996; @Grundland:1974; @GrundlandTafel:1996; @Perad:1985] and references therein) for multidimensional homogeneous and inhomogeneous systems of first-order PDEs. A specific feature of that approach is an algebraic and geometric point of view. An algebraization of systems of PDEs was made possible by representing the general integral elements as linear combinations of some special elements associated with those vector fields which generate characteristic curves in the spaces of independent variables $X$ and dependent variables $U$, respectively (see [@GrundlandZelazny:1983; @Perad:1985]). The introduction of these elements (called simple integral elements) proved to be very useful for constructing certain classes of rank-$k$ solutions in closed form. These integral elements proved to correspond to Riemann wave solutions in the case of nonelliptic systems and serve to construct multiple waves ($k$-waves) as a superposition of several single Riemann waves.
####
The generalized method of characteristics for solving quasilinear hyperbolic first-order systems can be extended to the case of complex characteristic elements (see *e.g.* [@Perad:1985; @Sobolev:1934]). These elements were introduced not only for elliptic systems but also for hyperbolic systems by allowing the wave vectors to be the complex solutions of the dispersion relation associated with the initial system of equations. The starting point is to make an algebraization, according to [@Burnat:1972; @Grundland:1974; @Perad:1985], of a first-order system of PDEs (\[eq:1\]) in $p$ independent and $q$ dependent variables written in its matrix form
\[eq:2.1\] &\^i(u)u\_i=0,i=1,…,p,\
&x=(x\^1,…,x\^p)X\^p, u=(u\^1,…,u\^q)U\^q,
where $\mathcal{A}^i(u)={\left( \mathcal{A}^{\m
| 1,142
| 214
| 2,261
| 1,255
| null | null |
github_plus_top10pct_by_avg
|
eratorname{ss}}} \rightarrow \Lambda^*_H$ and $C_z \colon \Lambda^*_{Z(G)} \rightarrow \Lambda^*_H$.
Let us now choose group homomorphisms $A \colon {\mathbb Z}^s \rightarrow {\mathbb Z}^t$ and $B = B_1 \oplus B_2 \colon \Lambda^*_{G_{\operatorname{ss}}} \oplus \Lambda^*_{G_{\operatorname{ss}}} \rightarrow {\mathbb Z}^t$ such that holds for the weight multiplicities for $G_{\operatorname{ss}}$. For this, $s$ and $t$ can be taken of order $O(r_G^2)$ [@bliem08 Proposition 19]. Then, $$\begin{aligned}
\label{constructive}
&\sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}} m_{T_{G_{\operatorname{ss}}},V_{G_{\operatorname{ss}},\lambda_{\operatorname{ss}}}}(\beta_{\operatorname{ss}}) \\
=~&\sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}}
\# \{ x \in {\mathbb Z}^s_{\geq 0} : A x = B {\left(\begin{smallmatrix}\lambda_{\operatorname{ss}} \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} \} \\
=~&\# \{ (x,\beta_{\operatorname{ss}})
:
{\left(\begin{smallmatrix}
A & - B_2\\
0 & C_{\operatorname{ss}}
\end{smallmatrix}\right)}
{\left(\begin{smallmatrix}x \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} =
{\left(\begin{smallmatrix}B_1 \lambda_{\operatorname{ss}} \\ - C_z \lambda_z + \delta\end{smallmatrix}\right)} \}\\
=~&\# \{ (x,\beta_{\operatorname{ss}})
:
\underbrace{{\left(\begin{smallmatrix}
A & - B_2\\
0 & C_{\operatorname{ss}}
\end{smallmatrix}\right)}}_{=: \mathcal A}
{\left(\begin{smallmatrix}x \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} =
\underbrace{{\left(\begin{smallmatrix}
B_1 & 0 & 0\\
0 & -C_z & {\mathbf 1}\end{smallmatrix}\right)}}_{=: \mathcal B}
{\left(\begin{smallmatrix}\lambda_{\operatorname{ss}} \\ \la
| 1,143
| 3,130
| 1,337
| 1,136
| null | null |
github_plus_top10pct_by_avg
|
in S$*,* $\pi _{S}(\delta _{1})=J$* implies* $\pi
_{S}(\delta _{2})=J$*.*
The binary relation $\prec $ can then be characterized as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.*
The relation $\prec $ is obviously a pre-order relation on $\psi _{A}^{Q}$, hence it induces canonically an equivalence relation $\approx $ on $\psi
_{A}^{Q}$, defined as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $\delta _{1}\prec $* *$\delta _{2}$* and* $\delta _{2}\prec $* *$\delta _{1}$*.*
The equivalence relation $\approx $ can then be characterized as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $S_{\delta _{1}}=S_{\delta _{2}}$*.*
Decidability versus justifiability in $\mathcal{L}_{Q}^{P}$
-----------------------------------------------------------
We have commented rather extensively in Sec. 3.3 on the notion of justification formalized in $\mathcal{L}_{Q}^{P}$, for every $S\in \mathcal{S}$, by the pragmatic evaluation function $\pi _{S}$. It must still be noted, however, that the definition of $\pi _{S}$ on all afs in $\psi _{A}^{Q}$ does not grant that an empirical procedure of proof exists which allows one to establish, for every $S\in \mathcal{S}$, the justification value of every af of $\mathcal{L}_{Q}^{P}$. In order to understand how this may occur, note that the notion of empirical proof is defined by A$_{5}$ for atomic rfs of $\mathcal{L}_{Q}^{P}$ and makes explicit reference, for every $E(x)\in \psi
_{R}^{Q}$, to the closed subset $\mathcal{S}_{E}\in \mathcal{L(S)}$ associated with $E$ by the function $\rho $ introduced in Sec. 2.2. Basing on this notion, the justification value $\pi _{S}(\vdash E(x))$ of an elementary af $\vdash E(x)\in \psi _{A}^{Q}$ can be determined by means of the same empirical procedure, making reference to the closed subset $\ma
| 1,144
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| 1,654
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| 4,093
| 0.768157
|
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|
& \quad +P{\left(a_7=2,b_8=2\right)}+P{\left(a_8=0,b_4=2\right)}+P{\left(a_8=1,b_5=1\right)}+P{\left(a_8=2,b_7=2\right)}+\\
& \quad +P{\left(a_1=0,b_6=1\right)}+P{\left(a_1=1,b_3=0\right)}+P{\left(a_1=2,b_2=2\right)}
+P{\left(a_2=0,b_6=0\right)}+\\
& \quad +P{\left(a_2=1,b_1=0\right)}+P{\left(a_2=2,b_3=1\right)}+P{\left(a_3=0,b_6=2\right)}+P{\left(a_3=1,b_1=2\right)}+\\
& \quad +P{\left(a_3=2,b_2=0\right)}+P{\left(a_4=0,b_5=0\right)}+P{\left(a_4=1,b_8=1\right)}+P{\left(a_4=2,b_7=2\right)}+\\
& \quad +P{\left(a_5=0,b_8=2\right)}+P{\left(a_5=1,b_4=2\right)}+P{\left(a_5=2,b_7=1\right)}+P{\left(a_6=0,b_3=2\right)}+\\
& \quad +P{\left(a_6=1,b_2=1\right)}+P{\left(a_6=2,b_1=1\right)}+P{\left(a_7=0,b_4=1\right)}+P{\left(a_7=1,b_8=0\right)}+\\
& \quad +P{\left(a_7=2,b_5=1\right)}+P{\left(a_8=0,b_5=2\right)}+P{\left(a_8=1,b_7=0\right)}+P{\left(a_8=2,b_4=0\right)}+\\
& \quad +P{\left(a_1=0,b_1=0\right)}+P{\left(a_1=1,b_1=1\right)}+P{\left(a_1=2,b_1=2\right)}
+P{\left(a_2=0,b_2=0\right)}+\\
& \quad +P{\left(a_2=1,b_2=1\right)}+P{\left(a_2=2,b_2=2\right)}+P{\left(a_3=0,b_3=0\right)}+P{\left(a_3=1,b_3=1\right)}+\\
& \quad +P{\left(a_3=2,b_3=2\right)}+P{\left(a_4=0,b_4=0\right)}+P{\left(a_4=1,b_4=1\right)}+P{\left(a_4=2,b_4=2\right)}+\\
& \quad +P{\left(a_5=0,b_5=0\right)}+P{\left(a_5=1,b_5=1\right)}+P{\left(a_5=2,b_5=2\right)}+P{\left(a_6=0,b_6=0\right)}+\\
& \quad +P{\left(a_6=1,b_6=1\right)}+P{\left(a_6=2,b_6=2\right)}+P{\left(a_7=0,b_7=0\right)}+P{\left(a_7=1,b_7=1\right)}+\\
& \quad +P{\left(a_7=2,b_7=2\right)}+P{\left(a_8=0,b_8=0\right)}+P{\left(a_8=1,b_8=1\right)}+P{\left(a_8=2,b_8=2\right)}
\end{split}$$
Example III:
$$\begin{split}
& S_3\equiv P{\left(a_1=0,b_5=2\right)}+P{\left(a_1=1,b_7=0\right)}+P{\left(a_1=2,b_4=0\right)}
+P{\left(a_2=0,b_8=0\right)}+\\
& \quad +P{\left(a_2=1,b_5=1\right)}+P{\left(a_2=2,b_4=1\right)}+P{\left(a_3=0,b_7=1\right)}+P{\left(a_3=1,b_4=2\right)}+\\
& \quad +P{\left(a_3=2,b_8=2\right)}+P{\left(a_4=0,b_2=1\right)}+P{\left(a_4=1,b_3=2\right)}+P{\left(a_4=2,b_1=1\right)}+\\
& \quad +P{\left
| 1,145
| 1,083
| 1,358
| 1,290
| null | null |
github_plus_top10pct_by_avg
|
use this method to detect the pair for $(a,b) = (2,2)$, since $H^1(\tilde{X}_\lambda,\CC) = 0$ for $\lambda = 1,\zeta$.
Assume from now on that $(a,b) = (1,3)$. The curve $\mathcal{C}_{\lambda}$ has only three singular points at the origins of the projective plane, $P_1 = [1:0:0]$, $P_2=[0:1:0]$, $P_3=[0:0:1]$ and note that $(S,P_i)$ is regular at $i=1,2$ whereas $S_3=(S,P_3)=\frac{1}{3}(1,1)$. From Theorem \[thm:conucleo\_singular\] one needs to study the cokernel of the map $$\label{eq:pik-ex}
\pi^{(k)}: H^0(S,\mathcal{O}_{S}(k-5)) \longrightarrow
\frac{\cO_{\CC^2,0}}{\cM_{\cC_\lambda,P_1}^{(k)}}
\oplus
\frac{\cO_{\CC^2,0}}{\cM_{\cC_\lambda,P_2}^{(k)}}
\oplus
\frac{\cO_{S_3,\zeta^{(k-5)}}}{\mathcal{M}_{\cC_\lambda,P_3}^{(k)}}$$ for $k=0,\ldots,11$. To understand the three vector spaces on the right-hand side one resolves the singularities of the curve as shown in Figure \[fig:res\]. The local type of the surface singularities and the self-intersections are shown for convenience.
\(A) at (0,2); (B) at (2,0); (C) at (-2,0.2);
at (A) [$\frac{1}{3}(1,1)$]{}; at (A) [$P_3$]{}; (A) circle \[radius=.1cm\];
at (C) [$P_1$]{}; at (C) [$(\mathbb C^2,0)$]{}; (C) circle \[radius=.1cm\];
at (B) [$P_2$]{}; at (B) [$(\mathbb C^2,0)$]{}; (B) circle \[radius=.1cm\];
\(A) to\[out=-90, in=150\] (B) to\[out=150,in=0\] ($(B)+(C)-.25*(A)$) to\[out=180,in=9\] (C) to\[out=0,in=-90\] (A);
at ($.5*(C)+.5*(B)$) [$48$]{};
\(A) at (-1,4); (B) at (2,.5); (C) at (-1,-1); (D) at (1,-1); (E) at (1,2); (F) at (-2,.5);
($.5*(A)+.5*(E)+(0,-1)$) – ($.5*(A)+.5*(E)+(0,1)$); ($.5*(A)+.5*(E)+(0,.75)$) circle \[radius=.1cm\]; at ($.5*(A)+.5*(E)+(0,.75)$) [$\frac{1}{3}(1,2)$]{}; ($.5*(A)+.5*(E)+(0,-.2)$) circle \[radius=.1cm\]; at ($.5*(A)+.5*(E)+(0,-.2)$) [$\frac{1}{6}(1,1)$]{}; at ($.5*(A)+.5*(E)+(-.2,1.2)$) [$-\frac{1}{6}$]{}; at ($.5*(A)+.5*(E)+(-.2,-1)$) [$E_3$]{};
($.5*(B)+.5*(D)+(0,-1)$) node\[below=3\] [$-\frac{1}{12}$]{} – ($.5*(B)+.5*(D)+(0,1.5)$); ($.5*(B)+.5*(D)+(0,.75)$) circle \[radius=.1cm\]; at ($.5*(D)+.5*(B)+(-.25,.5)$)
| 1,146
| 597
| 599
| 1,141
| 2,883
| 0.776338
|
github_plus_top10pct_by_avg
|
x_\mu )\in U^+(r_{i_1}(\chi ))$ for all $\mu \in \{0,1,\dots
,m\}$ by [@p-Heck07b Prop.5.19,Lemma6.7(d)]. Since ${T}^-_{i_1}(E_{\beta _\nu })\in U^+(r_{i_1}(\chi ))$, triangular decomposition of $U(r_{i_1}(\chi ))$ implies that $x_\mu =0$ for all $\mu >0$. Hence $E_{\beta _\nu }=x_0\in \ker {\partial ^K}_{i_1}$. Then the claim of the lemma follows from the inclusions $$\langle E_{\beta _\kappa }\,|\,\kappa \in \{2,3,\dots ,n\}\rangle
\subset \ker {\partial ^K}_{i_1} \subset
\mathop{\oplus }_{ {(m_2,m_3,\dots ,m_n) \atop
0\le m_\kappa <{b^{\chi}} (\beta _\kappa )\,\text{for all $\kappa $}}}
{\Bbbk }E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n},$$ where the second inclusion is obtained from Thm. \[th:PBW\] and the formula $${\partial ^K}_{i_1}( E_{\beta _1}^{m_1}E_{\beta _2}^{m_2}\cdots
E_{\beta _n}^{m_n})= \qnum{m_1}{q_{i_1i_1}}
E_{\beta _1}^{m_1-1}K_{i_1}{\boldsymbol{\cdot}}(E_{\beta _2}^{m_2}\cdots
E_{\beta _n}^{m_n}).$$
The analogous version of Lemma \[le:kerderK\] for $\ker {\partial ^L}_{i_1}$ is obtained by replacing $E_{\beta _\nu }$ by ${\bar{E}}_{\beta _\nu }$ for all $\nu \in \{2,3,\dots ,n\}$.
\[th:EErel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Then $$\begin{aligned}
E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu )
E_{\beta _\nu }E_{\beta _\mu }
\in \, & \langle E_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
\subset U^+(\chi ),\\
{\bar{E}}_{\beta _\mu }{\bar{E}}_{\beta _\nu }
-\chi ^{-1}(\beta _\nu ,\beta _\mu ){\bar{E}}_{\beta _\nu }{\bar{E}}_{\beta _\mu }
\in \, & \langle {\bar{E}}_{\beta _\kappa }
\,|\,\mu <\kappa <\nu \rangle \subset U^+(\chi ),\\
F_{\beta _\mu }F_{\beta _\nu }-\chi (\beta _\nu ,\beta _\mu )
F_{\beta _\nu }F_{\beta _\mu }
\in \, & \langle F_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
\subset U^-(\chi ),\\
{\bar{F}}_{\beta _\mu }{\bar{F}}_{\beta _\nu }-\chi ^{-1}(\beta _\mu ,\beta _\nu )
{\bar{F}}_{\beta _\nu }{\bar{F}}_{\beta _\mu }
\in \, & \langle {\b
| 1,147
| 2,701
| 996
| 1,028
| null | null |
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|
\max_{j \in [n]} \ell_j$ and $\kappa_{\max} = \max_{j \in [n]} \kappa_j$. The second inequality follows from the Jensen’s inequality.
Consider a case when the comparison graph is an expander such that $\alpha$ is a strictly positive constant, and $b=O(1)$ is also finite. Then, the Cramér-Rao lower bound show that the upper bound in is optimal up to a logarithmic factor.
Optimality of the Choice of the Weights
---------------------------------------
We propose the optimal choice of the weights $\lambda_{j,a}$’s in Theorem \[thm:main2\]. In this section, we show numerical simulations results comparing the proposed approach to other naive choices of the weights under various scenarios. We fix $d = 1024$ items and the underlying preference vector $\theta^*$ is uniformly distributed over $[-b,b]$ for $b = 2$. We generate $n$ rankings over sets $S_j$ of size $\kappa$ for $j \in [n]$ according to the PL model with parameter $\theta^*$. The comparison sets $S_j$’s are chosen independently and uniformly at random from $[d]$.
![Data-driven rank-breaking is consistent, while a random rank-breaking results in inconsistency.[]{data-label="fig:consistent"}](Plot7-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-92,-5) (-180,50)
Figure \[fig:consistent\] illustrates that a naive choice of rank-breakings can result in inconsistency. We create partial orderings data set by fixing $\kappa = 128$ and select $\ell=8$ random positions in $\{1,\ldots,127\}$. Each data set consists of partial orderings with separators at those $8$ random positions, over $128$ randomly chosen subset of items. We vary the sample size $n$ and plot the resulting mean squared error for the two approaches. The data-driven rank-breaking, which uses the optimal choice of the weights, achieves error scaling as $1/n$ as predicted by Theorem \[thm:main2\], which implies consistency. For fair comparisons, we feed the same number of pairwise orderings to a naive rank-breaking estimator. This estimator uses randomly chosen pairwise orderings with uni
| 1,148
| 605
| 205
| 1,216
| null | null |
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|
}}^I$.
By Eq. , the claim is equivalent to the equation $$\begin{aligned}
{\zeta ^{\chi}} \Big(-\sum _{k=1}^{m-1}({b^{\chi}} (\beta _k)-1)\beta _k\Big )
(K_{\alpha }L_{\alpha }^{-1})=
\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}.
\label{eq:Fhweight}
\end{aligned}$$ For each $k\in \{1,\dots ,n\}$ let $\chi _k=r_{i_{k-1}}\cdots r_{i_2}r_{i_1}(\chi )$, $w_k={\sigma }_{i_{k-1}}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Since $$\begin{aligned}
\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}=
\prod _{k=1}^{m-1}\frac{{\rho ^{\chi _k}}(w_k({\alpha }))}{
{\rho ^{\chi _{k+1}}}(w_{k+1}({\alpha }))},
\label{eq:rhochiprod}
\end{aligned}$$ Lemma \[le:rho\] gives that $$\begin{aligned}
\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}=
\prod _{k=1}^{m-1} \chi _k({\alpha }_{i_k},w_k({\alpha }))
^{1-\bfun{\chi _k}({\alpha }_{i_k})}
\prod _{k=1}^{m-1} \chi _k (w_k({\alpha }),{\alpha }_{i_k})
^{1-\bfun{\chi _k}({\alpha }_{i_k})}.
\end{aligned}$$ Using Eqs. and , this implies that $$\begin{aligned}
\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha })))}=
\prod _{k=1}^{m-1} \chi
(\beta _k,{\alpha })^{1-{b^{\chi}} (\beta _k)}
\chi ({\alpha },\beta _k)^{1-{b^{\chi}} (\beta _k)}.
\end{aligned}$$ Thus Eq. holds, and the lemma is proven.
Let ${\alpha }\in {\mathbb{Z}}^I$ and $\Lambda '={t}_{i_m}\cdots {t}_{i_2}\VT
_{i_1}^\chi (\Lambda )$. Then $$\begin{aligned}
&{\rho ^{\chi '}}(w({\alpha }))K_{\alpha }L_{\alpha }^{-1}
F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\
&={\hat{T}}_{i_1}\big(
{\rho ^{\chi '}}(w({\alpha }))K_{{\sigma }_{i_1}^{\chi }({\alpha })}
L_{{\sigma }_{i_1}^{\chi }({\alpha })}^{-1}
{T}_{i_1}^-(F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1})v_{{t}_{i_1}^\chi (\Lambda )}
\big)\\
&={\hat{T}}_{i_1}{\hat{T}}
| 1,149
| 2,482
| 1,121
| 1,125
| null | null |
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|
ow \pi^+\pi^-)}{A_{\Sigma}(D^0\rightarrow \pi^+\pi^-)} \right) &=
2\, \mathrm{Re}(2 \tilde{p}_0 \tilde{s}_1 + \tilde{p}_1 ) {\nonumber}\\
&= 2 \left[ 2\, \mathrm{Re}(\tilde{p}_0) \tilde{s}_1 + \mathrm{Re}(\tilde{p}_1)\right] \,. \label{eq:retildep1}\end{aligned}$$ As $\tilde{s}_1$ is already in principle determined from the other observables, this gives us then the full information on $\tilde{p}_0$ and $\tilde{p}_1$.
As the observables $\delta_{KK}$ and $\delta_{\pi\pi}$ are the hardest to measure, we are not providing here the explicit relation of Eq. (\[eq:retildep0\]) and Eq. (\[eq:retildep1\]) to these observables, acknowledging just that the corresponding parameter combinations can be determined from these in a straight forward way.
Taking everything into account, we conclude that the above system of eight observables for eight parameters can completely be solved. This is done where the values of the CKM elements are used as inputs. We emphasize that in principle with correlated double-tag measurements at a future charm-tau factory [@Gronau:2001nr; @Goldhaber:1976fp; @Bigi:1986dp; @Xing:1994mn; @Xing:1995vj; @Xing:1995vn; @Xing:1996pn; @Xing:1999yw; @Asner:2005wf; @Asner:2008ft; @Asner:2012xb; @Xing:2019uzz] we could even overconstrain the system.
Numerical Results \[sec:numerics\]
===================================
We use the formalism introduced in Sec. \[sec:solving\] now with the currently available measurements. As not all of the observables have yet been measured, we cannot determine all of the U-spin parameters. Yet, we use the ones that we do have data on to get useful information on some of them.
- Using Gaussian error propagation without taking into account correlations, from the branching ratio measurements [@Tanabashi:2018oca] $$\begin{aligned}
\mathcal{BR}(D^0\rightarrow K^+K^-) &= (3.97\pm 0.07) \cdot 10^{-3}\,, \\
\mathcal{BR}(D^0\rightarrow \pi^+\pi^-) &= (1.407\pm 0.025)\cdot 10^{-3}\,, \\
\mathcal{BR}(D^0\rightarrow K^+\pi^- ) &= (1.366 \pm 0.028) \cdot 10^{-4
| 1,150
| 305
| 1,479
| 1,226
| 287
| 0.81473
|
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|
geometrical/combinatorial explanation.
In [@Mu] a nice formula was proposed for the number tree-rooted planar maps, i.e. edge-rooted planar maps with distinguished spanning tree: the number of such maps with $n$ edges is $C_n\cdot
C_{n+1}$, where $C_k$ is $k$-th Catalan number. An elegant proof of this formula see in [@Be].
There are four planar maps with two edges: $$\begin{picture}(215,40) \multiput(0,25)(15,0){3}{\circle*{3}}
\put(0,25){\line(1,0){30}} \put(13,2){\small 1}
\put(70,25){\oval(20,20)} \put(80,25){\circle*{3}}
\put(95,25){\circle*{3}} \put(80,25){\line(1,0){15}}
\put(68,2){\small 2}
\put(125,25){\circle*{3}} \put(145,25){\circle*{3}}
\put(135,25){\oval(20,20)} \put(134,2){\small 3}
\put(195,25){\circle*{3}} \put(185,25){\oval(20,20)}
\put(205,25){\oval(20,20)} \put(194,2){\small 4}
\end{picture}$$
- There is one way to choose a spanning tree in the first map and two ways to choose a directed edge.
- There is one way to choose a spanning tree in the second map and four ways to choose a directed edge.
- There is one way to choose a spanning tree in the third map and two ways to choose a directed edge.
- There is no spanning trees in the forth map and two ways to choose a directed edge.
Thus we have $10=C_2\cdot C_3$ tree rooted planar maps with two edges.
We will study tree-rooted cubic maps with additional property: a root edge *does not* belong to the spanning tree.
**Theorem.** *The number of such tree-rooted cubic maps with $2n$ vertices is $C_{2n}\cdot
C_{n+1}$, where $C_k$ is $k$-th Catalan number.*
The main construction: from map to curve
========================================
By tree-rooted plane cubic map we will understand a cubic graph imbedded into plane (sphere) with
- marked spanning tree;
- marked directed edge that *does not* belong to the spanning tree.
Let $G$ be a tree-rooted pane cubic map with $2n$ vertices. We draw triangles, one triangle for each vertex, in such way that:
- triangles are disjoint;
- each vertex is inside the corresponding t
| 1,151
| 3,981
| 1,057
| 909
| 1,873
| 0.784818
|
github_plus_top10pct_by_avg
|
Consider the following two bootstrap confidence sets: $$\label{eq:ci.boot.loco}
\hat{D}^*_{{\widehat{S}}} = \left\{ \gamma \in \mathbb{R}^{{\widehat{S}}} \colon \|
\gamma - \hat{\gamma}_{{\widehat{S}}}
\|_\infty \leq \frac{ \hat{t}^*_{\alpha}}{\sqrt{n}} \right\} \quad
\text{and} \quad
\tilde{D}^*_{{\widehat{S}}} = \left\{ \gamma \in \mathbb{R}^{{\widehat{S}}} \colon |
\gamma_j - \hat{\gamma}_{{\widehat{S}}}
| \leq \frac{ \tilde{t}^*_{j}}{\sqrt{n}}, \forall j \right\}.$$
\[thm:boot.loco\] Using the same notation as in , assume that $n$ is large enough so that $\epsilon_n = \sqrt{ \epsilon^2 - N_n }$ is positive. Then there exists a universal constant $C>0$ such that the coverage of both confidence sets in is at least $$1 - \alpha
- C\left( \mathrm{E}^*_{1,n} +
\mathrm{E}_{2,n} + \frac{1}{n} \right),$$ where $$\mathrm{E}^*_{1,n} = \frac{2(A+\tau) + \epsilon_n }{\epsilon_n} \left(\frac{ (\log n
k)^7}{n}\right)^{1/6}.$$
Median LOCO parameters
----------------------
For the median loco parameters $(\phi_{{\widehat{S}}}(j), j \in {\widehat{S}})$ given in finite sample inference is relatively straightforward using standard confidence intervals for the median based on order statistics. In detail, for each $j \in {\widehat{S}}$ and $i \in \mathcal{I}_{2,n}$, recall the definition of $\delta_i(j)$ in and let $\delta_{(1)}(j) \leq
\ldots \leq \delta_{(n)}(j)$ be the corresponding order statistics. We will not impose any restrictions on the data generating distribution. In particular, for each $j \in {\widehat{S}}$, the median of $\delta_i(j)$ needs not be unique. Consider the interval $$E_j = [ \delta_{(l)}(j), \delta_{(u)}(j)]$$ where $$\label{eq:lu}
l = \Big\lceil \frac{n}{2} - \sqrt{\textcolor{black}{\frac{n}{2}} \log\left( \frac{2k}{\alpha}\right)} \Big\rceil
\quad \text{and} \quad
u = \Big\lfloor \frac{n}{2} + \sqrt{\textcolor{black}{\frac{n}{2}} \log\left( \frac{2k}{\alpha}\right)}
\Big\rfloor$$ and construct the hyper-cube $$\hat{E
| 1,152
| 3,330
| 1,278
| 1,016
| null | null |
github_plus_top10pct_by_avg
|
nd analysis of all examined cases.
**Clinicopathological features** **OS**
---------------------------------- ------------ -------------------- --------- --------- ---------- -----------
**N (%)** **Median, months** **UVA** **MVA**
***P*** ***P*** ***HR*** **95%CI**
Gender 0.79 0.34 1.2 0.36--3.8
Male 100 (55.6) 54
Female 80 (44.4) 51.5
Age 0.13 0.12 1.7 0.85--3.6
\>60 98 (54.4) 52.5
≤60 82 (45.6) 56.5
Size 0.9 0.32 1.0 0.52--2.1
\>5 55 (30.6) 53
≤5 125 (69.4) 56
T 0.19 0.92 1.6 0.78--3.4
T1 36 (20) 66
T2 102 (56.7) 54
T3 32 (17.8) 49.5
T4 10 (6) 14
N 0.04
| 1,153
| 5,599
| 476
| 372
| null | null |
github_plus_top10pct_by_avg
|
tive curves.
In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partially resolved and need not be reduced, and finally global and local conditions are given to describe the irregularity of cyclic branched coverings of the weighted projective plane.
The techniques required for these results are conceptually different and provide simpler proofs for the classical results. For instance, the local contribution comes from certain modules that have the flavor of quasi-adjunction and multiplier ideals on singular surfaces.
As an application, a Zariski pair of curves on a singular surface is described. In particular, we prove the existence of two cuspidal curves of degree 12 in the weighted projective plane $\PP^2_{(1,1,3)}$ with the same singularities but non-homeomorphic embeddings. This is shown by proving that the cyclic covers of $\PP^2_{(1,1,3)}$ of order 12 ramified along the curves have different irregularity. In the process, only a partial resolution of singularities is required.
address:
- |
Departamento de Matemáticas, IUMA\
Universidad de Zaragoza\
C. Pedro Cerbuna 12, 50009, Zaragoza, Spain
- |
Departamento de Matemáticas, IUMA\
Universidad de Zaragoza\
C. Pedro Cerbuna 12\
50009 Zaragoza, Spain
- |
Centro Universitario de la Defensa, IUMA\
Academia General Militar\
Ctra. de Huesca s/n.\
50090, Zaragoza, Spain
author:
- Enrique Artal Bartolo
- 'Jos[é]{} Ignacio Cogolludo-Agust[í]{}n'
- 'Jorge Martín-Morales'
title: Cyclic branched coverings of surfaces with abelian quotient singularities
---
Introduction {#introduction .unnumbered}
============
Motivated by the Riemann’s Existence Theorem and the classification of Riemann surfaces by their projection onto the projective line, Zariski ([@Zariski-problem]) started the classification of surfaces via a projection onto the projective plane, the study of the fundamental group
| 1,154
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| 1,371
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| null | null |
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|
egion between 0.1 and 1.0[ $(\text{GeV}\! / c)^2$]{} was selected for the analysis (see [[Fig. \[fig:tPrime\]]{}]{}).
![[${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected data sample for $t' \in [0.1, 1.0]{~\ensuremath{(\text{GeV}\! / c)^2}}$.[]{data-label="fig:threePiMass"}](tprime_zoom){width="\textwidth"}
![[${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected data sample for $t' \in [0.1, 1.0]{~\ensuremath{(\text{GeV}\! / c)^2}}$.[]{data-label="fig:threePiMass"}](Invariant_Mass_of_2008_data){width="\textwidth"}
[[Figure \[fig:threePiMass\]]{}]{} shows the [${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected 2008 data sample. It exhibits clear structures in the mass regions of the well-known resonances ${\ensuremath{a_1}}(1260)$, ${\ensuremath{a_2}}(1320)$, and ${\ensuremath{\pi_2}}(1670)$. In order to find and disentangle the various resonances in the data, a PWA was performed, in which the total cross section was assumed to factorize into a resonance and a recoil vertex. The isobar model[@isobar] is used to decompose the decay $X^- \to {\ensuremath{{\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}}}$ into a chain of successive two-body decays: The $X^-$ with quantum numbers [$J^{PC}$]{} and spin projection $M^\epsilon$ decays into a di-pion resonance, the so-called isobar, and a bachelor pion. The isobar has spin $S$ and a relative orbital angular momentum $L$ with respect to ${\ensuremath{\pi^-}}_\text{bachelor}$. A partial wave is thus defined by ${\ensuremath{J^{PC}}}M^\epsilon[\text{isobar}]L$, where $\epsilon = \pm 1$ is the reflectivity[@reflectivity]. The production amplitudes are determined by extended maximum likelihood fits performed in 40[ $\text{MeV}\! / c^2$]{} wide bins of the three-pion invariant mass $m_X$. In these fits no assumption is made on the produced resonances $X^-$ other then that their
| 1,155
| 562
| 1,349
| 1,294
| 1,026
| 0.795479
|
github_plus_top10pct_by_avg
|
(solid lines) and theoretical results for the experimental parameter $\lambda_W$ (dotted line) and the fit parameter $\lambda_{\rm fit}$ (dashed line) for two openings $d=\rm 6.5\,mm$ with $\lambda_W=0.05$ and $\lambda_{\rm fit}=-0.18i$ (black), and $d=\rm 11.2\,mm$ with $\lambda_W=0.52$ and $\lambda_{\rm fit}=-0.55i$ (orange, light gray). The frequency window of the Fourier transform of the measured $S_{\rm aa}(\nu)$ was 13 to $\rm14\,GHz$ and the transmission coefficient for antenna $a$ was $T_a=0.95$.](fig5){width=".98\columnwidth"}
In Fig. \[fig:05\] the coupling fidelity decay is shown for two different perturbation strengths. The solid lines show the experimental results. With the formulas derived in Sec. \[sec:theory\], we calculated the expected theoretical fidelity decay assuming that the channel is totally open, i.e. the coupling is purely imaginary (dotted lines). There is obviously no agreement. This shows that something is wrong in the argumentation. For a further check we removed the absorbing end and the antenna in the channel and replaced it by a reflecting end thus closing the system. We did not find any noticeable difference to the case with the absorbing end and the antenna in the channel experimentally. So there is only one explanation: by far the major part of the wave is reflected directly at the slit, and only a minor part really penetrates into the channel. This means that the coupling is not imaginary but mainly real (up to perhaps a minor imaginary contribution), and we should use Eq. (\[eq:s14\]) instead of Eq. (\[eq:s13\]) for the interpretation of our results. The dashed lines in Fig. \[fig:05\] show the resulting theoretical curves with $\lambda_{\rm fit}$ as a free parameter according to the definition preceding Eq. (\[eq:s14\]). Now a perfect agreement between experiment and theory is found.
As a resume we can state that the variable slit works essentially as a scattering center leading to partial masking of the change of coupling by the change of scattering properties in
| 1,156
| 706
| 2,593
| 1,308
| null | null |
github_plus_top10pct_by_avg
|
eta complaining about low quality first user questions. And there's always suggestions about how to tweak the question wizard... Have you thought that maybe you're inviting bad post by being a little more difficult to use than you should?
A:
Tour
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Q:
How to disable future date picker in android
I'm new to android
How to do disable future date in date picker, I tried many codes in stack over flow but it couldn't help.can anybody help me pleas.
Have a look Code
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.daty);
click=(ImageView)findViewById(R.id.click);
hdate=(TextView)findViewById(R.id.hdate);
timeStamp = new SimpleDateFormat("dd/MM/yyyy").format(new Date(System.currentTimeMillis()));
hdate.setText(timeStamp);
final Calendar myCalendar = Calendar.getInstance();
// listener for date picker
final DatePickerDialog.OnDateSetListener date = new DatePickerDialog.OnDateSetListener() {
@Override
public void onDateSet(DatePicker view, int year, int monthOfYear,
int dayOfMonth) {
| 1,157
| 1,653
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| 744
| 251
| 0.816016
|
github_plus_top10pct_by_avg
|
note that the low-frequency part of the all spectral functions in Fig. \[NRG\_fig2\] is governed by the same energy scale $T_{K}$ that is reduced with increasing $\lambda_c$.
![Contributions to the inelastic spectrum due to tunneling into the molecular orbital $d_{0\sigma}$, the local surface orbital $c_{0\sigma}$ and a coupling $\lambda^\text{tip}_{\mu\nu}$ of the vibrational mode $\omega_0$ to the STM tip, leading to the constituents (a) $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}}(\w)$, (b) $\rho_{\hat{X}_0d_{0\sigma},d^\dagger_{0\sigma}}(\w)$ and (c) $\rho_{\hat{X}_0d_{0\sigma},c^\dagger_{0\sigma}}(\w)$. The spectral functions have been calculated with NRG. We set $U/\Gamma_0=10$, $\omega_0/\Gamma_0=0.1$ and different colors indicate various Holstein couplings $\lambda_c$. []{data-label="NRG_fig3"}](fig18-Inelastic-Lc){width="50.00000%"}
The constituents of the inelastic spectrum $$\begin{aligned}
\label{eq:transmission_inelastic-two_channel_model}
\tau^{(1)}_\sigma(\omega)&+&\tau^{(2)}_\sigma(\omega)= \lambda^{\rm tip}
\Big\{\lambda^{\rm tip}t^2_{d}
\rho_{\hat{X}_0 d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}} (\w)\nonumber \\
&+&t^2_{d}
\big[
\rho_{ \hat{X}_0d_{0\sigma} ,d^\dagger_{0\sigma}}(\w)
+ \rho_{d_{0\sigma},\hat{X}_0 d^\dagger_{0\sigma}}(\w)
\big]
\\
&+&t_{d} t_{c}\big[
\rho_{ \hat{X}_0 d_{0\sigma} ,c_{0\sigma}^\dagger}(\w) + \rho_{c_{0\sigma},\hat{X}_0 d^\dagger_{0\sigma}}(\w) \big]
\Big\}
\nonumber\end{aligned}$$ are shown in Fig. \[NRG\_fig3\]. Here we have assumed that $\lambda^{\rm tip}_{\mu\nu}=0$ except for $\mu=0$ and $\nu=0$, i.e. only the molecular orbital $d_0$ (but not the local effective substrate orbital $c_0$) is coupled through the vibration $\hat X_0$ to the STM tip, with coupling constant $\lambda^{\rm tip}$.
We note that $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}}(\w)$, displayed in panel (a) of Fig. \[NRG\_fig3\], can in principle also be obtained from Eq. in the limit $\lambda_c=0$. It consists of two peaks at $\pm\omega_0$ that indicate th
| 1,158
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| 1,472
| 1,305
| null | null |
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|
ft(A^{18}\cap\pi^{-1}(H)\right)\prod_{i=1}^r\left(A^{24}\cap\pi^{-1}(\langle x_i\rangle)\right)\right|\ge\frac{|A|}{\exp(\log^{O(1)}2K)}.$$
We prove \[prop:pre-chang.tor.free\] shortly.
\[lem:x\[G,G\]\] Let $s\ge2$. Let $G$ be an $s$-step nilpotent group, write $\pi:G\to G/[G,G]$ for the quotient homomorphism, and let $x\in G/[G,G]$. Then the group $\pi^{-1}(\langle x\rangle)$ has step at most $s-1$.
This is implicitly shown in the proofs of [@nilp.frei Propositions 4.2 & 4.3]; it is also proved explicitly in [@book Lemma 6.1.6 (i)].
\[prop:lower.step.in.quotient\] Let $m>0$ and $s\ge\tilde s\ge2$ be integers, and let $K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde G$ be an $\tilde s$-step nilpotent subgroup of $G$. Write $\pi:\tilde G\to\tilde G/[\tilde G,\tilde G]$ for the quotient homomorphism, and suppose that $H\subset\pi(A^m\cap\tilde G)$ is a finite group. Then there is a normal subgroup $N\lhd G$ such that $[\,\pi^{-1}(H),\ldots,\pi^{-1}(H)\,]_{\tilde s}\subset N\subset A^{K^{e^{O(s)}m}}$.
The bounds stated here are more precise than those stated in [@nilp.frei Proposition 7.1], but the bounds claimed here can be read out of the argument there. Alternatively, \[prop:lower.step.in.quotient\] is proved exactly as stated here in [@book Proposition 6.6.2].
Combine Proposition \[prop:pre-chang.tor.free\] with Lemmas \[lem:slicing\] and \[lem:x\[G,G\]\] and \[prop:lower.step.in.quotient\].
Before we prove \[prop:pre-chang.tor.free\] we isolate the following lemma, which is inspired by a lemma of Tao [@tao.product.set Lemma 7.7].
\[lem:splitting\] Let $G$ be a group, let $N\lhd G$ be a normal subgroup, and let $\pi:G\to G/N$ be the quotient homomorphism. Let $A$ be a symmetric subset of $G$, and define a map $\varphi:\pi(A)\to A$ by choosing, for each element $x\in\pi(A)$, an element $\varphi(x)\in A$ such that $\pi(\varphi(x))=x$. Then
1. \[eq:split.inv\] for every $a\in A$ we have $a\in\left(A^2\cap N\right)\varphi(\pi(a))$; and
2. \[
| 1,159
| 810
| 1,307
| 1,137
| 2,421
| 0.779882
|
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|
defect to the other, or to create superposition states with well-defined weights. This might open up new perspectives for the design of quantum logical devices.
I would like to thank M. Holthaus for his continuous support and insightful discussions.
\[sec:eigen\]Eigenfunction for a single defect
==============================================
In order to demonstrate that ${{\left|\psi_0\right.\rangle}}$ as defined by Eqs. (\[eq:psi0\]) and (\[eq:al\]) indeed is an eigenfunction of the single-defect Hamiltonian (\[eq:h1\]) with the energy eigenvalue (\[eq:E\_0\]), it is helpful to invoke the definition (\[eq:defp\]), and to write $\nu = W\frac{\nu}W = W p\left|\frac{\nu}W\right|$. One then obtains $$\begin{aligned}
\hat{H}{{\left|\psi_0\right.\rangle}}=&&
{{\left|\gamma\right.\rangle}}pW\left|\frac{\nu}{W}\right|{{\langle \left.\gamma\right|}}{\psi_0}\rangle
\\
&-&\frac{W}4 \sum_{\ell = -\infty}^{\infty}
\left\{{{\left|\ell\right.\rangle}}{{\langle \left.\ell+1\right|}} + {{\left|\ell\right.\rangle}}{{\langle \left.\ell-1\right|}}\right\}{{\left|\psi_0\right.\rangle}}
\nonumber\\
=&& pW\left|\frac{\nu}{W}\right|a_{\gamma}{{\left|\gamma\right.\rangle}}
-\frac{W}4 \sum_{\ell = -\infty}^{\infty}
\left(a_{\ell-1}+a_{\ell+1}\right){{\left|\ell\right.\rangle}}
\nonumber\;.\nonumber\end{aligned}$$ Singling out the defect site, one has $$\begin{aligned}
\hat{H}{{\left|\psi_0\right.\rangle}}=&&
\left(pW\left|\frac{\nu}{W}\right|a_{\gamma}
-\frac{W}4\left(a_{\gamma-1}+a_{\gamma+1}\right)\right){{\left|\gamma\right.\rangle}}
\nonumber\\&
-&\frac{W}4 \sum_{\genfrac{}{}{0pt}{1}{\ell =
-\infty}{\ell\ne\gamma}}^{\infty}\left(a_{\ell-1}
+a_{\ell+1}\right){{\left|\ell\right.\rangle}}\;.\end{aligned}$$ Utilizing the explicit expression (\[eq:al\]) for the amplitudes $a_{\ell}$, this leads to $$\begin{aligned}
\hat{H}{{\left|\psi_0\right.\rangle}}=&&
\left(pW\left|\frac{\nu}{W}\right|+p\frac{W}2x_-\right)a_{\gamma}{{\left|\gamma\right.\rangle}}
\nonumber\\
&-&\frac{W}4\s
| 1,160
| 4,626
| 340
| 875
| null | null |
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|
|
| | 25‐250 for mRNA | | | |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Process scale | 0.5‐50 | L | Demand, scale‐up optimization | [46](#amp210060-bib-0046){ref-type="ref"}, [49](#amp210060-bib-0049){ref-type="ref"}, [55](#amp210060-bib-0055){ref-type="ref"}, [60](#amp210060-bib-0060){ref-type="ref"} |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Production titres | 1.5‐7 | g/L | Reaction optimization, process development | [46](#amp210060-bib-0046){ref-type="ref"}, [55](#amp210060-bib-0055){ref-type="ref"} |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 1,161
| 204
| 1,539
| 1,225
| null | null |
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|
0.084 0.084 0.083 0.083 0.084 0.083
mVC 0.103 0.103 0.103 0.103 0.103 0.103 0.103
mMSE 0.096 0.095 0.096 0.096 0.096 0.096 0.095
BLB($n^{0.6}$) 0.707 0.702 0.697 0.696 0.686 0.701 0.690
BLB($n^{0.8}$) 0.218 0.219 0.217 0.217 0.219 0.214 0.216
SDB($n^{0.6}$) 0.827 0.826 0.827 0.824 0.825 0.824 0.825
SDB($n^{0.8}$) 0.270 0.268 0.268 0.269 0.268 0.268 0.269
TB 0.078 0.078 0.078 0.078 0.078 0.078 0.078
: Lengths of confidence interval for Cases 1-3 in Example \[example2\]
\[table8\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50 0.207 0.207 0.208 0.207 0.206 0.206 0.206
K=100 0.213 0.213 0.213 0.214 0.213 0.213 0.212
K=150 0.219 0.218 0.218 0.218 0.218 0.218 0.218
mVC 0.250 0.249 0.250 0.250 0.250 0.250 0.250
mMSE 0.240 0.240 0.241 0.240 0.241 0.241 0.240
BLB($n^{0.6}$) 1.793 1.790 1.810 1.785 1.800 1.772 1.792
BLB($n^{0.8}$) 0.531 0.541 0.539 0.534
| 1,162
| 5,112
| 420
| 786
| null | null |
github_plus_top10pct_by_avg
|
^!})(\vec X;\varnothing)$ with “dashed” first and last inputs, can uniquely be written in the form $\varphi *_\sigma\psi$ or $\varphi\#_\sigma\psi$ for some $\varphi,\psi$ and $\sigma$.
For each $r\geq 1$, we show that the $(-r)$th homology of $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)$ vanishes by decomposing every element in $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)^{-r}$ as a sum of terms of the form $\varphi *_\sigma\psi$ or $\varphi\#_\sigma\psi$, and then performing and induction on the degree of $\psi$. More precisely, let us define the order of $\varphi*_\sigma \psi$ or $\varphi\#_\sigma \psi$ to be the degree of $\psi$ in $\textbf{D}(\mathcal O^!)$. Then for $s\in \mathbb N$, we claim the following statement:
- Let $\chi\in \textbf{D}(\widehat{\mathcal O^!})
(\vec X;\varnothing)^{-r}$ be a closed element ${\partial}(\chi)=0$. Then $\chi$ is homologous to a sum $\sum_i \sum_\sigma \varphi_i *_\sigma
\psi_i+\sum_j \sum_\sigma \varphi'_j \#_\sigma\psi'_j$, where the order of each term is less or equal to $(-s)$.
Rather intuitively this means, that for smaller $(-s)$, $\chi$ is homologous to decorated trees whose total degree is more and more concentrated on the right branch of the tree.
It is easy to the that the above is true for $s=1$. As for the inductive step, let $\chi=\sum_i \sum_\sigma \varphi_i *_\sigma \psi_i+\sum_j\sum_\sigma \varphi'_j
\#_\sigma\psi'_j$, where we assume an expansion so that $\{\varphi_i\}_i$ are linear independent, $\{\varphi'_j \}_j$ are linear independent, but the $\psi_i$ and the $\psi'_j$ are allowed to be linear combinations in $\textbf{D}(\mathcal O^!)$. We claim that those elements $\psi_i$ and $\psi'_j$, which are of degree $-s$, are closed in $\textbf{D} (\mathcal O^!)$. This follows from ${\partial}(\chi)=0$ and the inductive hypothesis, because the only terms of the boundary ${\partial}(\chi)$, which are of order $-s+1$, are terms of the form $\varphi_i *_\sigma {\partial}(\psi_i)$ and $\varphi'_j \#_\sigma
{\partial}(\psi'
| 1,163
| 4,076
| 760
| 733
| 2,891
| 0.776269
|
github_plus_top10pct_by_avg
|
le qubit energy $k/n$, resulting in a system with energy $k$. This choice will allow us to precisely describe the behavior of the walk in terms of the relationship between the energy of the system and the rate of decoherence.
We can write each of the terms in the exponent of the superoperator from (\[superop-soln\]) as follows: $$\begin{aligned}
\identity \otimes H &= \sum_{j=1}^n [\identity \otimes \identity] \otimes \cdots \otimes [\identity \otimes \sigma_x] \otimes \cdots \otimes [\identity \otimes \identity]\enspace, \\
H \otimes \identity &= \sum_{j=1}^n [\identity \otimes \identity] \otimes \cdots \otimes [\sigma_x \otimes \identity ] \otimes \cdots \otimes [\identity \otimes \identity]\enspace.\end{aligned}$$ Our decoherence operator can also be written in this form: $$\begin{aligned}
{\mathbf P} &=& \frac{1}{n} \sum_{j=1}^n [\Pi^i_0 \otimes \Pi^i_0 + \Pi^i_1 \otimes \Pi^i_1] \\
&=& \frac{1}{n} \sum_{j=1}^n ([{\identity} \otimes {\identity}] \otimes \cdots \otimes [\Pi_0 \otimes \Pi_0] \otimes \cdots \otimes [{\identity} \otimes {\identity}]\\
&&+ [{\identity} \otimes {\identity}] \otimes \cdots \otimes [\Pi_1 \otimes \Pi_1] \otimes \cdots \otimes [{\identity} \otimes {\identity}])\enspace.\end{aligned}$$ The identity operator has a consistent decomposition: $\identity \otimes
\identity = \frac{1}{n}\sum_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes
[{\identity} \otimes {\identity}].$ We can now put these pieces together to form the superoperator: $$\begin{aligned}
S_t
& = \exp\left(it(\identity \otimes H) - it(H \otimes \identity) - pt \identity \otimes \identity + pt\mathbf{P}\right) \\
& = \exp\left(\sum_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes \mathbf{A} \otimes \cdots [{\identity} \otimes {\identity}]\right) \\
& = \prod_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes e^\mathbf{A} \otimes \cdots [{\identity} \otimes {\identity}]\\
& = \left[e^\mathbf{A}\right]^{\otimes n}\end{aligned}$$
| 1,164
| 2,690
| 1,383
| 1,147
| null | null |
github_plus_top10pct_by_avg
|
'
- 'Primož Škraba, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, SLOVENIA '
author:
- Ganna Kudryavtseva
- Primož Škraba
title: The principal bundles over an inverse semigroup
---
Introduction
============
The classifying topos ${\mathcal{B}}(S)$ of an inverse semigroup $S$ has recently begun to be investigated [@F; @FH; @FLS; @FS; @KL; @St]. This topos is by definition the presheaf topos over Loganathan’s category $L(S)$ of $S$. There are several equivalent characterizations of this topos, cf. [@F; @FS; @KL]. An immediate question one can ask about ${\mathcal{B}}(S)$ is “What does ${\mathcal{B}}(S)$ classify?” A direct application of well-known results [@MM Theorems VII.7.2, VII.9.1] of topos theory, provides the answer: for an arbitrary Grothendieck topos ${\mathcal E}$, the presheaf topos ${\mathcal{B}}(S)$ classifies filtered functors $L(S)\to {\mathcal E}$.
The category of geometric morphisms from ${\mathcal E}$ to ${\mathcal{B}}(S)$ is equivalent to the category of filtered functors $L(S)\to {\mathcal E}$.
The construction of the functors establishing the correspondence in the above theorem can be found in [@MM]. In particular, if $\gamma^*\colon {\mathcal{B}}(S)\to {\mathcal E}$ is the inverse image functor of a geometeric morphism, its composition with the Yoneda embedding $L(S)\to {\mathcal{B}}(S)$ is a filtered functor, and any such a functor is obtained this way.
This answer, however, is not quite satisfactory. We expect more structure given that for groups the category of filtered functors $G\to {\mathcal E}$ is equivalent to the category of $G$-[*torsors*]{}. The latter are just objects of ${\mathcal E}$ with a particular type of internal action of the group object obtained by applying the canonical constant sheaf functor $\Delta$ to $G$ [@MM VIII.2]. This naturally raises a question of how to define $S$-torsors, where $S$ is an inverse semigroup. The latter question was raised by Funk and Hofstra in [@FH], where in [@FH Theorem 3.9] they show that, for the topos of sets, $S$-
| 1,165
| 226
| 1,707
| 800
| null | null |
github_plus_top10pct_by_avg
|
\end{pmatrix}\quad.$$ By (\[refMPI\]) in Lemma \[MPI\], $\ell(t)$ has entries in ${{\mathbb{C}}}[[t]]$ and is invertible; in fact, $L=\ell(0)$ is lower triangular, with 1’s on the diagonal. Therefore Lemma \[MPI\] gives $$h_1(t)\cdot
\begin{pmatrix}
t^a & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}=
\begin{pmatrix}
1 & 0 & 0 \\
q & 1 & 0 \\
r & s & 1
\end{pmatrix}\cdot
\begin{pmatrix}
t^a & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}\cdot \ell(t)\quad,$$ from which the statement follows.
The gist of this result is that, up to equivalence, matrices ‘to the left of a 1-PS’ and centered at the identity may be assumed to be lower triangular, and to have polynomial entries, with controlled degrees.
###
We denote by $v$ the order of vanishing at $0$ of a polynomial or power series; we define $v(0)$ to be $+\infty$. The following statement is a refined version of Proposition \[keyreduction\].
\[faber\] Let $\alpha(t)$ be a $\/3\times 3$ matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $\alpha(0)\ne 0$ and $\det \alpha(t)\not\equiv 0$. Then there exist constant invertible matrices $H$, $M$ such that $\alpha(t)$ is equivalent to $$\beta(t)=H\cdot \begin{pmatrix}
1 & 0 & 0 \\
q & 1 & 0 \\
r & s & 1
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix} \cdot M\quad,$$ with
- $b\le c$ nonnegative integers, $q,r,s$ polynomials;
- $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$;
- $q(0)=r(0)=s(0)=0$.
If, further, $b=c$ and $q$, $r$ are not both zero, then we may assume that $v(q)<v(r)$.
Finally, if $q(t)\not\equiv 0$ then we may choose $q(t)=t^a$, with $a=v(q)<b$ (and thus $a<v(r)$ if $b=c$).
By standard diagonalization of matrices over Euclidean domains, every $\alpha(t)$ as in the statement can be written as a product $$h_0(t)\cdot
\begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}
\cdot k(t)\quad,$$ where $b\le c$ are nonnegative integers, and $h_0(t)$, $k(t)$ are invertible (over ${{\mathbb{C}}}[[t]]$). Letting $H=h_0(0)$, $h_1(t)
| 1,166
| 427
| 916
| 1,221
| 4,141
| 0.76782
|
github_plus_top10pct_by_avg
|
S {#sec-lambda-0}
In order to make the connection to the literature and also to point out the major difference of our theory in comparison with earlier ones, we consider the limit of vanishing electron-phonon coupling in the system S, but maintain a small but non-zero $\lambda^{\rm tip}_{\mu\nu}$. Then, $\langle\hat X_\nu \rangle=0$. As argued in the previous section, as a consequence $G^{(1)}_{d\sigma}(z)=0$, $\tau^{(1)}_{\sigma}(\w)=0$ and $I^{(1)}_{\rm inel}=0$ hold, while $I^{(2)}_{\rm inel}$ is reduced to a simplified result [@MolecularVibrationTunnel1968], because the correlation function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t)$ in Eq. factorizes in the time domain into the product of the electronic Green’s function and the phonon propagator $G_{\hat X_\nu, \hat X_\nu}(t)$, $$\begin{aligned}
\label{eqn:26}
G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t) &=&
G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(t) G_{\hat X_\nu, \hat X_\nu}(t) \delta_{\nu\nu'}.\end{aligned}$$ Therefore, the spectral function is given by a convolution in the frequency domain. Summing over all free vibrational modes $\nu$ on a molecule with frequency $\w_{\nu}$, we obtain [@Mahan81]
$$\begin{aligned}
\label{eqn:27}
\tau^{(2)}_{\sigma}(\w)
&=& \sum_{\mu\mu'\nu}
t_{\mu \sigma}t_{\mu' \sigma}
\lambda^{\rm tip}_{\mu \nu}
\lambda^{\rm tip}_{\mu' \nu}
\left[
\rho_{d_{\mu \sigma},d^\dagger_{\mu'\sigma}}(\w-\w_\nu)(g(\w_\nu) +f_{S}(\w_\nu-\w))
+
\rho_{d_{\mu \sigma},d^\dagger_{\mu'\sigma}}(\w+\w_\nu)(g(\w_\nu) +f_{S}(\w_\nu+\w))
\right]
\nonumber \\
\label{eqn:freemode_inelastic}\end{aligned}$$
where $g(\w)$ denotes the Bose function. Using the approximation in Eq. yields the identical inelastic current contribution as derived in Refs. [@Caroli72].
We briefly discuss two extreme cases for the electronic spectral function. For simplicity, we restrict ourselves to $M=1$ and a single vibrational mode with frequency $\w_0$. This excludes the possibility of a Fano res
| 1,167
| 2,462
| 1,470
| 1,118
| 4,050
| 0.768449
|
github_plus_top10pct_by_avg
|
6]), so provision was made for the elimination of data recorded on days when conditions were especially adverse (e.g., intense rain or cold). However, no data were required to be removed on these grounds.
In addition to calculating the average number of steps taken by participants, the analysis also grouped results according to the index categories established by [@B103]: "sedentary" (\<5,000 steps per day), "low active" (5,000--7,499 steps), "somewhat active" (7,500--9,999), "active" (10,000--12,499), and "highly active" (\>12,500). In view of the low number of participants in the final two ranges, these scores were classed together as "active or highly active."
### Number of Physical Activities Carried Out {#S2.SS2.SSS4}
Participants were asked to state the different physical activities performed over the course of the week. Spaces were provided in the questionnaire for two unstructured activities and two organized activities on weekdays, and two unstructured activities and two organized activities on the weekend. The range of values for each type of activity was 0--4, with a maximum total value of 8 as the sum of the week's activities.
### Level of Autonomy in Physical Activities Carried Out {#S2.SS2.SSS5}
Level of autonomy was calculated based on the number of activities carried out. Participants who practiced just one activity were classified as "unstructured" or "organized" depending on the type of activity performed. Where the difference between activity types was more than 1, participants were classified according to the higher scoring activity. Where the difference between activity types was less than or equal to 1, participants were classified as "mixed" (both unstructured and organized).
Procedure {#S2.SS3}
---------
The principals of the schools selected for the study were contacted and informed of the objectives and procedure of the study. Once the schools' agreement had been obtained, permission for students to participate in the study was requested from their families using an informed conse
| 1,168
| 433
| 2,711
| 1,070
| null | null |
github_plus_top10pct_by_avg
|
O(s)}}}$; finite $K^{e^{O(s)}}$-approximate groups $A_1,\ldots,A_r\subset A^{e^{O(s)}}$ such that, writing $\pi:G\to G/N$ for the quotient homomorphism, each group $\langle\pi(A_i)\rangle$ is abelian; and sets $X_1,\ldots,X_t\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$ such that $$A\subset N\prod\{A_1,\ldots,A_r,X_1,\ldots,X_t\}$$ with the product taken in some order.
For each $i=1,\ldots,r$, apply \[cor:sanders\] to the set $\pi(A_i)$ to obtain a subgroup $H_i\subset A_i^8N\subset A^{e^{O(s)}}N$ containing $N$, an ordered progression $P_i\subset A_i^8\subset A^{e^{O(s)}}$ of rank at most $e^{O(s)}\log^{O(1)}2K$, and a set $Y_i\subset A_i\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$, such that $A_i\subset Y_iH_iP_i$. Since $G/N$ is gererated by the $K$-approximate group $\pi(A)$, applying Proposition \[prop:grp.in.normal\] in $G/N$ implies that for each $i$ there is a normal subgroup $N_i\lhd G$ such that $H_i\subset N_i\subset A^{K^{e^{O(s)}}}N$. The subgroup $H=N_1\cdots N_r$ is then normal in $G$, and satisfies $$\begin{aligned}
H&\subset A^{rK^{e^{O(s)}}}N\\
&\subset A^{K^{e^{O(s)}}}\end{aligned}$$ and $$A\subset H\prod\{P_1,\ldots,P_r,Y_1,\ldots,Y_r,X_1,\ldots,X_t\},$$ with the product taken in some order. This completes the proof.
Note that if $K<2$ then $A$ is a finite subgroup of $G$, and so the theorem holds with $A=H$. We may therefore assume that $K\ge2$. Let $k,\ell\le e^{O(s^2)}\log^{O(s)}2K$, $$\label{eq:P.S.contained}
P_1,\ldots,P_k,X_1,\ldots,X_\ell\subset A^{e^{O(s)}},$$ and $H\subset A^{K^{e^{O(s)}}}$ be as coming from \[prop:prod.of.progs.and.small\], noting in particular that $$\label{eq:prod.order}
AH\subset H\prod\{P_1,\ldots,P_k,X_1,\ldots,X_\ell\}.$$ The pigeonhole principle therefore implies that there exist elements $u_1,\ldots,u_\ell$ with $u_i\in X_i$ such that the product $\prod\{P_1,\ldots,P_k,u_1,\ldots,u_\ell\}$, taken in the same order as the product in , satisfies $$\begin{aligned}
\left|H\prod\{P_1,\ldots,P_k,u_1,\ldots,u_\ell\}\
| 1,169
| 640
| 1,239
| 1,099
| 2,928
| 0.776015
|
github_plus_top10pct_by_avg
|
<!-- <thead class="thead-dark"> -->
<tr>
<th>Item</th>
<th>Quantity</th>
<th>Rate</th>
<th>Subtotal</th>
<th> </th>
</tr>
<!-- </thead> -->
<tr v-for="item in cart" :key="{{item.id}}">
<td><a href="#">{{ item.name }}</a></td>
<td><button class="btn btn-flat btn-xs btn-info p-1 mx-1" @click="inc(item.id)">+</button>[[ item.quantity ]]<button class="btn btn-flat p-1 mx-1 btn-xs btn-info" @click="dec(item.id)">-</button></td>
<td><span class="text-muted">{{ item.sell_price }}</span> </td>
<td>Rs {{ item.sell_price * item.quantity }}</td>
<td><button class="btn btn-xs btn-outline-primary" @click="removeFromCart(item)"><i class="fas fa-trash-alt"></i></button></td>
</tr>
</table>
</div>
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| 1,170
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omega_2$ the stable one. Because $b$ depends on $\omega$ through Eq. and the eigenmode structure $\phi(z)$ depends on $b$ through Eq. , the two solutions $\omega_j$ correspond to two different eigenmodes $\phi_j(z)$. We identify $b_j$ and $\phi_j$ as the $b$ and $\phi$ corresponding to $\omega_j$. The eigenmodes are then $$\label{solved phi}
\phi_j(k,z) =
\begin{cases}
\left(e^{2|k|}+b_j\right)e^{-|k|z} & z > 1 \\
e^{|k|z}+b_je^{-|k|z} & -1 < z < 1 \\
\left(1+b_je^{2|k|}\right)e^{|k|z} & z < -1,
\end{cases}$$ where $$\label{bj}
b_j = e^{2|k|}\frac{2|k|(\omega_j+k)-k}{k}$$ satisfies $b_1(k) = b_2(-k) = b_2^*(k)$ for $|k|<k_c$, and $b_j(k) = b_j(-k) = b_j^*(k)$ for $|k|>k_c$.
![Equilibrium (left column) compared with velocity profiles of the unstable $\phi_1$ (middle column) and the stable $\phi_2$ (right column) at wavenumbers $k = 0.4$ (top row) and $k=1$ (bottom row) plotted over one wavelength in $x$ and from $z=-2$ to $z=2$. Streamlines are plotted with color indicating flow speed. The first row is in the unstable range, where $\phi_1$ grows exponentially while $\phi_2$ decays exponentially. The second row is a marginally stable wavenumber, where both $\phi_1$ and $\phi_2$ oscillate without any growth \[see Fig. \[dispersion\_plot\]\].[]{data-label="eigenmode_plot"}](Fig2.eps){width="16cm"}
For $\omega^2<0$, the eigenmodes are nearly identical but satisfy $\phi_1(k,z) = \phi^*_2(k,z)$. Figure \[eigenmode\_plot\] shows the flows corresponding to these eigenmodes for four wavenumbers sampling the unstable and stable ranges. Previous work has shown that the physical mechanisms for instability of $\phi_1$ and stability of $\phi_2$ can be understood in terms of resonant vorticity waves in both the hydrodynamic[@Baines] and MHD[@Heifetz] systems.
In standard descriptions of turbulence and quasilinear transport calculations, it is common practice to neglect stable modes given their exponential decay from a small initial value. In this paper we account for the nonlinear drive of the stable mode by the unstable
| 1,171
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| 1,377
| 3,715
| 0.770528
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\Q$-resolution as defined in section \[sec:h2\]. Recall that $$H^1(\tilde X,\CC)=H^1(\tilde Y,\CC)=H^1(Y,\cO_Y)\oplus H^0(Y,\Omega^1_Y)=H^1(Y,\cO_Y)\oplus \overline{H^1(Y,\cO_Y)}.$$ In this section the dimension of the $e^{\frac{2\pi ik}{d}}$-eigenspace of $H^1(Y,\mathcal{O}_Y)$ will be computed using the tools developed in section \[sc:Esnault\]. Hence the classical formula (see Theorem \[thm:conucleo\_liso\]) will be generalized in three directions: first, it applies to a surface with quotient singularities (the weighted projective plane); second, the ramification locus need not be completely resolved (a partial resolution is enough); and finally, the result also applies to non-reduced ramification divisors. For this latter purpose we define the following divisor $\mathcal{C}^{(k)}$ which is trivial when the ramification divisor is reduced: $$\mathcal{C}^{(k)} := \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \mathcal{C}_j.$$ The divisor $\cC^{(k)}$ appeared implicitly in . Recall there is a natural isomorphism between $H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right) \right)$ and the vector space of $w$-homogeneous polynomials of total degree $s_k - |w|$ (see and statement of Theorem \[thm:h2Lk\]).
\[thm:conucleo\_singular\] For $0\leq k<d$, let $$\pi^{(k)}: H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right) \right)
\longrightarrow \bigoplus_{P \in S}
\frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}}$$ be the evaluation map where $\mathcal{M}_{\mathcal{C},P}^{(k)}:=\mathcal{M}_P(\cC^{1/d},k)$ is defined as the following quasi-adjunction $\mathcal{O}_{\PP^2_w,P}$-module $$\mathcal{M}_{\mathcal{C},P}^{(k)}\!=\!
\left\{ g \in\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)}\right)
\vphantom{\sum_{j=1}^r}\!\right.\!
\left|\ \operatorname{mult}_{E_\v} \pi^* g >
\sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} m_{\v j} -\! \nu_\v, \ \forall \v
| 1,172
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| 1,030
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| null | null |
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g Theorem \[theorem:aLinear\].
5. \[item:outline:Quantize\] Quantize $\{ {{\bar{\a}}^\dagger}_k \}$ to integer-valued vectors $\{ {{\bar{\a}}^\diamond}_k \}$ with Algorithm \[agorithm:SuccessiveQuantization\].
6. \[item:outline:Select\] Select a vector from $\{ {{\bar{\a}}^\diamond}_k \}$ to be a suboptimal coefficient vector ${{\bar{\a}}^\diamond}$ for ${\bar{\h}}$ using .
7. \[item:outline:Recover\] Recover a suboptimal coefficient vector ${\a^\diamond}$ for $\h$ from ${{\bar{\a}}^\diamond}$ according to Remark \[remark:Transformation\].
$\bst\leftarrow$ $({\bar{\h}},\p)\leftarrow$ $b\leftarrow 1 + P\norm{\h}^2$
$\u\leftarrow (P/b)^{1/2}{\bar{\h}}$ $\bsr\leftarrow \frac{\u(L)}{1-\norm{\u(1:L-1)}^2}\u(1\!:\!L-1)$ ${{\bar{\a}}^\dagger}_1(1\!:\!L-1)\leftarrow \bsr$ ${{\bar{\a}}^\dagger}_1(L)\leftarrow 1$
()[ $K_l\leftarrow 1$ $K\leftarrow K_l$ ]{}
${{\bar{\a}}^\diamond}(1\!:\!L-1)\leftarrow \0$ ${{\bar{\a}}^\diamond}(L)\leftarrow 1$ $f_{\min}\leftarrow \norm{{{\bar{\a}}^\diamond}}^2 - \big(({{\bar{\a}}^\diamond})^T\u\big)^2$
Complexity Analysis
-------------------
Here we analyze the complexity of our algorithm, in terms of the number of flops required. Referring to the outline in Algorithm \[agrm:QPRApproachOutline\], the processing of $\h$ in step \[item:outline:Preprocess\] involves recording the signs of the elements and sorting the elements, and takes $O(L\log(L))$ flops. Calculating ${{\bar{\a}}^\dagger}_1$ in step \[item:outline:CalculateaBarDagger1\] has a complexity of $O(L)$. For the bi-section search applied to determine $K$ in step \[item:outline:DetermineK\], the maximum number of loops required to execute is $O(\log(K_u))$, the number of flops in each loop is $O(L)$, and thus the maximum cost is $O(\log(K_u)L)$. Step \[item:outline:CalculateaBarDaggerk\] takes $O(KL)$ flops. By introducing appropriate temporary variables $b$ and $d$ as shown in Algorithm \[algorithm:QPRApproachCode\], the successive quantization of a real-valued approximation ${{\bar{\a}}^\dagger}_k$ can be imple
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\
[ letellier.emmanuel@math.unicaen.fr]{}
- |
Fernando Rodriguez-Villegas\
[*University of Texas at Austin*]{}\
[ villegas@math.utexas.edu]{}\
\
[with an appendix by Gergely Harcos]{}
title: |
Arithmetic harmonic analysis on\
character and quiver varieties II
---
Introduction
============
Character varieties {#char}
-------------------
Given a non-negative integer $g$ and a $k$-tuple $\muhat=(\mu^1,\mu^2,\dots,\mu^k)$ of partitions of $n$, we define the generic character variety $\M_\muhat$ of type $\muhat$ as follows (see [@hausel-letellier-villegas] for more details). Choose a *generic* tuple $(\calC_1,\dots,\calC_k)$ of semisimple conjugacy classes of $\GL_n(\C)$ such that for each $i=1,2,\dots,k$ the multiplicities of the eigenvalues of $\calC_i$ are given by the parts of $\mu^i$.
Define $\cal{Z}_\muhat$ as $$\calZ_\muhat:= \left\{(a_1,b_1,\dots,a_g,b_g,x_1,\dots,x_k)\in
(\GL_n)^{2g} \times\calC_1\times\cdots\times\calC_k\,\left|\,
\prod_{j=1}^g(a_i,b_i)\prod_{i=1}^kx_i=1\right.\right\},$$ where $(a,b)=aba^{-1}b^{-1}$. The group $\GL_n$ acts diagonally by conjugation on $\calZ_\muhat$ and we define $\M_\muhat$ as the affine GIT quotient $$\M_\muhat:=\calZ_\muhat/\!/\GL_n:={\rm
Spec}\,\left(\C[\calZ_\muhat]^{\GL_n}\right).$$ We prove in [@hausel-letellier-villegas] that, if non-empty, $\M_\muhat$ is nonsingular of pure dimension $$d_\muhat:=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2.$$ We also defined an [*a priori*]{} rational function $\H_\muhat(z,w)\in\Q(z,w)$ in terms of Macdonald symmetric functions (see § \[Cauchy\] for a precise definition) and we conjecture that the compactly supported mixed Hodge numbers $\{h_c^{i,j;k}(\M_\muhat)\}_{i,j,k}$ satisfies $h_c^{i,j;k}(\M_\muhat)=0$ unless $i=j$ and $$H_c(\M_\muhat;q,t){\stackrel{?}{=}}(t\sqrt{q})^{d_\muhat}\H_\muhat
\left(-t\sqrt{q},\frac{1}{\sqrt{q}}\right),
\label{mainconj}$$ where $H_c(\M_\muhat;q,t):=\sum_{i,j}h_c^{i,i;j}(\M_\muhat)q^it^j$ is the compactly supported mixed Hodge polynomial.
In particular, $\H_\m
| 1,174
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riangleleft {(b\triangleright a^2)}) (b\triangleleft {a^2})
\stackrel{ {(\ref{eq:2.4.750})} } =( b\triangleleft a)( b\triangleleft
{a^2}) = b^{2(l + t + 1 )}$$ $$\beta (b^3, a^2) = (bb^2)\triangleleft {a^2} \stackrel{{(\ref{eq:3})}
}= (b\triangleleft {(b^2\triangleright a^2)}) (b^2 \triangleleft
{a^2}) \stackrel{ {(\ref{eq:2.4.750})} } = (b\triangleleft {a^2})
(b^2\triangleleft {a^2}) = b^{2l + 4t + 3}$$ Using the induction we can easily prove that $${\label{eq:4.445'}}
\beta (b^{2k}, a^2) = b^{2k(l+t+1)} , \qquad \beta (b^{2k+1},
a^2) = b^{ 2kl + (2k+2)t + 2k + 1}$$ for any $k = 0, 1, \cdots$. Moreover, keeping in mind [(\[eq:4.445\])]{} we find that $$\beta (b^{2k}, a) = \beta (b^{2k}, a^2) = b^{2k(l+t+1)}$$ On the other hand $\beta$ is a right action and $a^3 = 1$. Hence: $$b^{2k} = \beta (b^{2k}, 1) = (b^{2k}\triangleleft
{a^2})\triangleleft a = (b^{2k}\triangleleft {a})\triangleleft a =
b^{2k}\triangleleft {a^2} = b^{2k(l+t+1)}$$ As the order of $b$ is $m$ we obtain a second compatibility condition between $l$ and $t$: $m | 2k(l+t)$ for any $k = 0, 1,
\cdots$ which is equivalent to: $${\label{eq:c5}}
m | 2(l+t)$$ From this condition and [(\[eq:c1\])]{} we obtain $${\label{eq:c1'}}
m | 2(2l - t)$$ Let now $m = 2r$. We have to find $l$, $t \in \{1, 2, \cdots, r-1
\}$ such that $$m | 2(l+t) \quad {\rm and} \quad m | 2(2l -t)$$ Equivalently, we have to solve in $\ZZ_r$ the system of equations $${\label{eq:4.600}}
\left \{\begin{array}{rcl}
\hat{l} + \hat{t} = \hat{0} \\
\hat{2}\hat{l} - \hat{t} = \hat{0}
\end{array} \right.$$ The equation $\hat{3} \hat{l} = \hat{0}$ has $(3, r)$ solutions in $\ZZ_r$. If $3$ does not divide $m$ then the unique solution of the system is $\hat{l} = \hat{t} = \hat{0}$ and therefore $\beta$ is the trivial action. If $3$ divides $r$ let $u$ be such that $r
= 3u$. Then the system [(\[eq:4.600\])]{} has three solutions $$\hat{l_1} =
\hat{t}_1 = \hat{0}; \qquad \hat{l}_2 = \hat{u}, \, \hat{t}_2 =
\hat{2}\hat{u}, \qquad \hat{l_3} = \hat{2}\hat{u}, \, \hat{t_3} =
\hat{4}\hat{u}$$ The fi
| 1,175
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re also linearly independent. Then (\[eq:linfq2\]) implies that their coefficients are zero, that is, the probability (\[eq:p\_q\]) of the outcome $q$ is independent of the input state $|\Phi\rangle_A$. Reversely, if (\[eq:p\_q\]) is independent of $|\Phi\rangle_A$, then $LL_q^\dag$ is injective and $f_q$ is linear. We can conclude that the condition that “the probability of the measurement outcome $q$ does not depend on the input state” (that is, Alice learns nothing about the input state due to the measurement) is equivalent to that the teleportation channel is linear. Moreover, it can be proven in a way not detailed here that the linearity of the channel is equivalent to its unitarity—therefore its unitary reversibility.
Entanglement matching {#sec:matching}
=====================
In this section we answer the question what measurements provide fidelity 1 teleportation for an arbitrarily given (even partially) entangled shared state. Suppose that the shared state $|\sigma\rangle$ is described by an invertible antilinear operator $L$. If Bob applies a unitary transformation $U_q\colon {{\mathcal H}}_C \to {{\mathcal H}}_C$ which may depend on the result $q$ of Alice’s measurement, then the final state of system $C$ reads $|\text{out}\rangle_C =
p_q^{-1/2} U_q LL_q^\dag |\Phi\rangle_A$. Let $i_{AC}$ be a unitary isomorphism between ${{\mathcal H}}_A$ and ${{\mathcal H}}_C$ so that we can compare the states of systems $A$ and $C$. The teleportation condition is $$\frac1{\sqrt{p_q}} U_q LL_q^\dag = i_{AC}$$ which also guarantees that $p_q(|\Phi\rangle_A)$ is independent of $|\Phi\rangle_A$. From this we conclude that a measurement with an outcome described by the antilinear operator $$L_q = \sqrt{p_q} i_{AC}^{-1} U_q L^{-1}\strut^\dag
\label{eq:L_q}$$ supports fidelity 1 conditional teleportation. The appropriate recovering unitary transformation applied by Bob is to be $U_q$.
Although $p_q$ in (\[eq:p\_q\]) depends on $L_q$, this can be resolved by the fact that $L_q$ has a norm ${\mathrm{tr}}_B (L_q
| 1,176
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\;\leq\; \frac{e^{6b}\ell_j}{\max\{ \ell_j, \kappa_j - p_{j,\ell_j}\}} \leq \frac{e^{6b}\eta \ell_j}{ \kappa_j}\,,\end{aligned}$$ where we used $\eta$ defined in Equation . Define a diagonal matrix $D^{(j)} \in \cS^{d}$ and a matrix $A^{(j)} \in \cS^d$, $$\begin{aligned}
A^{(j)}_{i\i} &\equiv & \I_{\big\{i,\i \in S_j \big\}} \,\sum_{a=1}^{\ell_j} \lambda_{j,a} \I_{\big\{(i,\i) \in G_{j,a}\big\}}\;,\; \text{for all} \;\; i,\i \in [d] \,, \label{eq:hess_posl_8}\end{aligned}$$ and $D^{(j)}_{ii} = \sum_{i'\neq i} A^{(j)}_{ii'}$. Observe that $M^{(j)} = D^{(j)} - A^{(j)}$. For all $i \in [d]$, we have, $$\begin{aligned}
D^{(j)}_{ii} &=& \I_{\big\{i \in S_j \big\}} \sum_{\i = 1}^{\kappa_j} \I_{\big\{\sigma_j^{-1}(i) = \i \big\}} \sum_{a=1}^{\ell_j} \lambda_{j,a} \deg_{G_{j,a}}(\sigma_j^{-1}(\i)) \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}}\Bigg\{ \I_{\big\{\sigma^{-1}_j(i) \in \cP_j\big\}}\Bigg( \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a} \Bigg) + \I_{\big\{\sigma^{-1}_j(i) \notin \cP_j\big\}} \Bigg( \sum_{a = 1}^{\ell_j} \lambda_{j,a} \Bigg)\Bigg\} \nonumber\\
&=& \I_{\big\{i \in S_j \big\}}\bigg\{ \I_{\big\{\sigma^{-1}_j(i) \in \cP_j\big\}}\delta_{j,1} \; + \; \I_{\big\{\sigma^{-1}_j(i) \notin \cP_j\big\}} \delta_{j,2}\bigg\}, \label{eq:hess_posl_9}\end{aligned}$$ where the last equality follows from the definition of $\delta_{j,1}$ and $\delta_{j,2}$ in Equation . Note that $\max_{i \in [d]} \{D_{ii}\} = \delta_{j,1}$. Using and , we have, $$\begin{aligned}
\label{eq:hess_posl_14}
\E\Big[D^{(j)}_{ii}\Big] &\leq & \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j} \bigg(\delta_{j,1} + \frac{\delta_{j,2}\kappa_j}{\eta\ell_j} \bigg) \Bigg\} \,.\end{aligned}$$ Similarly we have, $$\begin{aligned}
\label{eq:hess_posl_10}
\E\Big[\big(D^{(j)}_{ii}\big)^2\Big] &\leq & \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{ e^{6b}\eta\ell_j}{\kappa_j} \bigg( \delta_{j,1}^2 + \frac{\delta_{j,2}^2\kappa_j}{\eta\ell_j} \bigg) \Bigg\} \end{aligned
| 1,177
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_i+4c_i \end{pmatrix}$ as explained in Section \[h\] and thus we have $$\label{ea12}
\begin{pmatrix} a_i'&\pi b_i'\\ \sigma(\pi\cdot {}^t b_i') &1 +2\bar{\gamma}_i+4c_i' \end{pmatrix}=
\sigma(1+\pi\cdot {}^tm_{i,i}')\cdot\begin{pmatrix} a_i&\pi b_i\\ \sigma(\pi\cdot {}^t b_i) &1 +2\bar{\gamma}_i+4c_i \end{pmatrix}\cdot(1+\pi m_{i,i}')+\pi^3(\ast).$$ Here, the nondiagonal entries of $a_i'$ as well as the entries of $b_i'$ are considered in $B\otimes_AR$, each diagonal entry of $a_i'$ is of the form $2 x_i$ with $x_i\in R$, and $c_i'$ is in $R$. In addition, $b_i=0, c_i=0$ as explained in Remark \[r33\].(2) and $a_i$ is the diagonal matrix with $\begin{pmatrix} 0&1\\1&0\end{pmatrix}$ on the diagonal.
Note that in this case, $m_{i,i}'=\begin{pmatrix} s_i'&\pi y_i'\\ \pi v_i' &\pi z_i' \end{pmatrix}$. Compute $\sigma(\pi\cdot {}^tm_{i,i}')\cdot\begin{pmatrix} a_i&0\\ 0 &1 +2\bar{\gamma}_i \end{pmatrix}\cdot (\pi m_{i,i}')$ formally and this equals $\sigma(\pi)\pi\begin{pmatrix} {}^ts_i'a_is_i'+\pi^2X_i &\pi Y_i\\ \sigma(\pi\cdot {}^tY_i) &-\pi^2(z_i')^2+\pi^4Z_i \end{pmatrix}$ for certain matrices $X_i, Y_i, Z_i$ with suitable sizes.
Thus we can ignore the $(1,1)$ and $(1,2)$-blocks of the term $\sigma(\pi\cdot {}^tm_{i,i}')\begin{pmatrix} a_i&0\\ 0 &1 +2\bar{\gamma}_i \end{pmatrix}(\pi m_{i,i}')$ in Equation (\[ea12\]). On the other hand, we should consider the $(2,2)$-block of this term because of the appearance of $\pi^4(z_i')^2$. By the same reason, we can ignore the $(1,1)$ and $(1,2)$-blocks of the term $\pi^3(\ast)$, whereas the $(2,2)$-block of this term should be considered.
We interpret each block of Equation (\[ea12\]) below:
1. Firstly, we consider the $(1,1)$-block. The computation associated to this block is similar to that for the above case (iii). Hence there are exactly $((n_i-1)^2-(n_i-1))/2$ independent linear equations and $((n_i-1)^2+(n_i-1))/2$ entries of $s_i'$ determine all entries of $s_i'$.
2. Secondly, we consider the $(1, 2)$-block. Then it equals $$\la
| 1,178
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her partitioned into smaller regions, corresponding to various phases of the system (see figure \[fig:fdCSAWs\]).
![Phase diagrams in the space of interaction parameters for CSAWs model in the case of $b=2$ and $b=3$ SG fractal, for $t=0.5$. In both cases the critical line $w=w_c(u,t)$ separates the $u-w$ plane into the area $w>w_c(u,t)$ of entangled phase and area $w<w_c(u,t)$, in which the two chains are segregated. The latter area is divided by vertical line $u=u_\theta$ into regions, corresponding to three segregated phases: (i) 2D chain (always extended) and extended 3D chain ($u<u_\theta$), (ii) 2D chain and 3D $\theta$-chain ($u=u_\theta$), and (iii) 2D chain and 3D globule ($u>u_\theta$). One should observe that there appears the multi-critical point (full red circle) at the crossing of the $\theta$–line and the critical line $w=w_c(u,t)$. For other values of $t$ ($0<t<1$), the critical line $w_c(u,t)$ also monotonically decreases, for both $b=2$ and $b=3$ fractals.[]{data-label="fig:fdCSAWs"}](figure5.eps)
To each of these area different fixed point of the general type $$\label{fpgen}
(A^*,B^*,C^*,A_1^*,A_2^*,A_3^*,A_4^*,B_1^*,B_2^*)\>,$$ pertains. We describe general features of the fixed points and the corresponding phases in the three following subsections.
Weak self-attraction of the 3D chain
------------------------------------
For each value of $0<t<1$, and small values of the interaction parameter $1\leq u<u_\theta$, there is some critical value $w=w_c(u,t)$ such that
- For $w<w_c(u,t)$ the fixed point of the form $$\label{fp1}
(A_E,B_E,C^*,0,0,0,A_{4}^*,0,0)\>,$$ is reached. This point corresponds to the phase in which 2D chain and extended 3D chain are segregated, since as it is approached, after some number $r\gg1$ of RG iterations, the average number of contacts between the two chains, quickly becomes constant. Values of $A_E$ and $B_E$ are fixed values of the RG parameters for the solitary extended chain on 3D SG fractal, and they are presented in table \[tab:CSAWs\], toge
| 1,179
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p\times q$ matrix functions of $u$ ([*i.e.* ]{}$\mathcal{A}^\mu=(\mathcal{A}^{\mu i}_\alpha(u))\in
{\mathbb{R}}^{p\times q}, \mu=1,\ldots, m$). For the given initial system of equations (\[eq:3.1\]), the matrices $\mathcal{A}^\mu$ are known functions of $u$ and the trace conditions (\[eq:3.22\]) or (\[eq:3.23\]) are conditions on the functions $f$, $\bar{f}$, $\lambda$, $\bar{\lambda}$ (or on $\xi$ due to the orthogonality conditions (\[eq:3.14\])). From the computational point of view, it is useful to split $x^i$ into $x^{i_A}$ and $x^{i_a}$ and to choose a basis for the wave vector $\lambda^A$ and $\bar{\lambda}^A$ such that
\[eq:3.24\] \^A=dx\^[i\_A]{}+\_[i\_a]{}\^Adx\^[i\_a]{},|\^A=dx\^[i\_A]{}+|\_[i\_a]{}\^Adx\^[i\_a]{}, A=1,…, k,
where $(i_A,i_a)$ is a permutation of $(1,\ldots, p)$. So, the expressions (\[eq:3.9\]) become
\[eq:3.25\] =x\^[i\_a]{},=x\^[i\_a]{}.
Substituting (\[eq:3.25\]) into (\[eq:3.22\]) (or (\[eq:3.23\])) yields
\[eq:3.26\] [( \^\^[-1]{}[( + )]{})]{}=0,=1…,m,
or
\[eq:3.27\] [( \^\^[-1]{})]{}=0,=1…,m,
where $R=(r^1,\ldots, r^k,\bar{r}^1,\ldots,\bar{r}^k)^T$ and
\[eq:3.28\] Q\_a=[( + )]{}+[( + )]{}=[( + )]{} \_a\^[qq]{},\
K\_a=
(
[cc]{} &\
&
)
=\_a[( + )]{}\^[2k2k]{},
and for simplicity of notation, we note
\[eq:3.28b\] =
(
&\
&|
)
\^[2kp]{},\_a=
(
&\
&
)
\^[2kq]{},=[( , )]{}\^[q2k]{},
for $i_A$ fixed and $i_a=1,\ldots, p-1$. In (\[eq:3.28\]) the $2k\times 2k$ matrix $K_a$ is defined in terms of the $k\times k$ subblocks of the form ${{\partial}\lambda_{i_a}}/{{\partial}u}{\left( {{\partial}f}/{{\partial}r}+{{\partial}\bar{f}}/{{\partial}r} \right)}$, where $\eta_a$ is a matrix form of the block ${\partial}\lambda_{a}/{\partial}u$ over the block ${\partial}\bar{\lambda}_a/{\partial}u$. The notation ${\left( {{\partial}f}/{{\partial}r}, {{\partial}f}/{{\partial}\bar{r}} \right)}$ represents the matrix formed of the left block ${\partial}f/{\partial}r$ and the right block ${\partial}f/{\partial}\bar{r}$. Note that the functions $r^A$, $\bar{r}^A$ and $x^{i_a}$
| 1,180
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oreover, one sees that every strong solution with homogeneous inflow boundary conditions is a weak solution of $P(x,\omega,E,D)\phi=f$, that is $$\begin{aligned}
\label{eq:P_0_in_P-dot-star}
\tilde P_{0}\subset P'^*.\end{aligned}$$
Since $\tilde P_{0}$ is a closed operator, the space \[eq:H\_tilde-P\_0\_is\_D-tilde-P\_0\] [H]{}\_[P\_[0]{}]{}(GSI\^):= D(P\_[0]{}) is a Hilbert space when equipped with the inner product $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}_{P_0}(G\times S\times I^\circ)}:={\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}\tilde{P}_0\phi,\tilde{P}_0v{\right\rangle}_{L^2(G\times S\times I)}.$$
When $\phi\in {{{\mathcal{}}}H}_{P_{0}}(G\times S\times I^\circ)$ we say that the (homogeneous) initial and boundary conditions $$\phi_{|\Gamma_-}=0,\quad \phi(\cdot,\cdot,E_{\rm m})=0$$ are valid in the strong sense.
We show the $m$-dissipativity of $\tilde{P}_0$ using the theory of evolution operators presented in section \[evcsd\] below.
\[md-evoth\] Suppose that $$\begin{aligned}
{}& S_0\in C^2(I,L^\infty(G)),\label{evo16} \\[2mm]
{}& \kappa:=\inf_{(x,E)\in \ol{G}\times I}S_0(x,E)>0, \label{evo8-a} \\[2mm]
{}& \nabla_xS_0\in L^\infty(G\times I), \label{evo9-a} \\[2mm]
{}& -{{\frac{\partial S_0}{\partial E}}}+2CS_0\geq 0. \label{flp1-ab}\end{aligned}$$ Then \[inf4-a\] R(I+P\_0)=L\^2(GSI) and \[inf5-a\] P\_0,\_[L\_2(GSI)]{}0,D(P\_0).
We apply Theorem \[evoth1\] (see below) with $K=0,\ \Sigma=CS_0+1,\ g=0$. Let $f\in C_0^\infty(G\times S\times I^\circ)$. Then it follows that the problem -[E]{}+\_x+(CS\_0+1)= (I + P(x,,E,D))=f,(,,E\_[m]{})=0, has a unique solution $\phi\in C(I,\tilde{W}^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$. We find that $$\big\{\psi\in C(I,\tilde W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))\ \big|\ \psi(\cdot,\cdot,E_m)=0\big\}\subset D(P_0),$$ and so for any $f\in C_0^\infty(G\times S\times I)$ the equation $(I+ P_0)\phi=f$ has a solution. Since $C_0^\infty(G\times S\times I^\circ)$ is dense in $L^2(G\times S\times I)$ we find tha
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$$\partial_t\hat\psi({\boldsymbol{k}},t) = \hat\omega({\boldsymbol{k}})\hat\psi({\boldsymbol{k}}, t) + \hat N({\boldsymbol{k}} ,t),
\label{eq:DecompositionGPE}$$ which can be solved by an operator-splitting and exponential-time differentiating method [@cox2002exponential]. It means that we exploit the fact that the linear part of Eq. (\[eq:DecompositionGPE\]) can be solved exactly by multiplying with the integrating factor $e^{-\hat\omega({\boldsymbol{k}})t}$. This leads to $$\partial_t \left(\hat\psi({\boldsymbol{k}},t) e^{-\hat\omega({\boldsymbol{k}})t}\right) = e^{-\hat\omega({\boldsymbol{k}})t}\hat N({\boldsymbol{k}}, t).
\label{eq:ETD}$$ The nonlinear term $\hat N({\boldsymbol{k}},t)$ is linearly approximated in time for a small time-interval $(t,t+\Delta t)$, i.e $$\hat N({\boldsymbol{k}}, t+\tau) = N_0 + \frac{N_1}{\Delta t}\tau$$ where $N_0 = \hat N(t)$ and $N_1 = \hat N(t+\Delta t) -N_0$. Inserting this into Eq. (\[eq:ETD\]) and integrating from $t$ to $t+\Delta t$ we get $$\begin{aligned}
&\hat{\psi}({\boldsymbol{k}},t+\Delta t) = \hat{\psi}({\boldsymbol{k}},t) e^{\hat{\omega}({\boldsymbol{k}})\Delta t} + \frac{N_0}{\hat\omega({\boldsymbol{k}})}\left(e^{\hat \omega({\boldsymbol{k}}) \Delta t} -1\right) \nonumber\\
&+ \frac{N_1}{\hat\omega({\boldsymbol{k}})} \left[\frac{1}{\hat\omega({\boldsymbol{k}}) \Delta t}(e^{\hat\omega({\boldsymbol{k}}) \Delta t} -1) -1 \right].\end{aligned}$$ Since computing the value of $N_1$ requires knowledge of the state at $t+\Delta t$ before we have computed it, we start by setting it to zero and find a value for the state at $t +\Delta
t$ given that $\hat N(t)$ is constant in the interval. We then use this state to calculate $N_1$, and add corrections to the value we got when assuming $N_1=0$.
---
abstract: 'The performance of millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems is limited by the sparse nature of propagation channels and the restricted number of radio frequency chains at transceivers. The introduction of reconfigurable antennas
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ntegral model associated to $C(L^j)$ to the smooth integral model associated to $M_0'\oplus C(L^j)$ explained in Remark \[r410\], the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $M_0'\oplus C(L^j)$ is $$\begin{pmatrix}\begin{pmatrix}1&\frac{-2a}{a+2b} z_j^{\ast} &0 & \frac{2\pi a}{a+2b}z_j^{\ast}\\
0&1+\frac{2a}{a+2b} z_j^{\ast}&0 & \frac{-2\pi a}{a+2b}z_j^{\ast}\\
0&\frac{-\pi a}{a+2b}z_j^{\ast}&1&0\\0&0&0&1 \end{pmatrix}&0
\\ 0&id\end{pmatrix}.$$ Here, the $(1,1)$-block corresponds to $(M_0^{\prime}\oplus M_0^{\prime})\oplus \left(\pi B(-2be_1+e_2)\oplus \pi B(-ae_1+e_3)\right)$.
We now follow Step (2) with $M_0'\oplus C(L^j)$ such that the above formal matrix corresponds to the formal matrix (\[e4.3\]). Recall that a Jordan splitting of $M_0'\oplus C(L^j)$ is $$M_0'\oplus C(L^j)=(M_0^{\prime}\oplus M_0^{\prime})\oplus (\bigoplus_{i \geq 1} M_i^{\prime}),$$ where, $M_0^{\prime}\oplus M_0^{\prime}$ with a basis $(e_1', e_1'+e_2')$ is $\pi^0$-modular, $M_i'$ is $\pi^i$-modular, and $M_2'=\left(\pi B(-2be_1+e_2)\oplus \pi B(-ae_1+e_3)\right)\oplus (\oplus_{\lambda}\pi H_{\lambda})\oplus M_2$. We consider a Jordan splitting $Y(C(M_0'\oplus C(L^j)))=\bigoplus_{i \geq 0} M_i''$. Then $M_0''$, which is the only lattice needed in the desired orthogonal group, can be described by using $M_0^{\prime}\oplus M_0^{\prime}$, $M_1^{\prime}$, and $M_2'$. The only difference between the above formal matrix and the formal matrix (\[e4.3\]) of Step (2) is the appearance of $\frac{2\pi a}{a+2b}z_j^{\ast}$ in the $(1, 4)$ and $(2, 4)$-entries, and $\frac{-\pi a}{a+2b}z_j^{\ast}$ in the $(3, 2)$-entry of the above formal matrix. However, these entries will be zero after reduction to the orthogonal group associated to $M_0''$ since $M_0''$ is *free of type II* so that the diagonal block associated to $M_0''$ has no congruence condition. Thus in the corresponding orthogonal group, all entries having $\pi$ as a factor become zero.
Then by using the result of St
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ries) was proved by Bose and Mitra in [@BoMi].
We will not attempt to deal thoroughly with the question of when the convergence in probability in the results above can be strengthened to almost sure convergence. However, the following result gives some sufficient conditions. Each of the conditions stated automatically implies the Lindeberg-type condition ; for the first part this follows from exponential tail decay which is implied by a Poincaré inequality (see [@Ledoux Corollary 3.2]), and for the other parts it is elementary.
\[T:as-convergence\] In the setting of Theorem \[T:circular-law-correlated\], \[T:circular-law-uncorrelated\], \[T:semicircle-law-general\], or \[T:semicircle-law-special\], suppose in addition that ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$ for some ${\varepsilon}> 0$ and that one of the following conditions holds:
1. There is a constant $K > 0$ such that for every $n$ and every $a \in G^{(n)}$, $Y_a^{(n)}$ satisfies a Poincaré inequality with constant $K$. That is, $$\operatorname{Var}f\bigl(Y_a^{(n)}\bigr)
\le K {\mathbb{E}}\bigl\vert \nabla f\bigl(Y_a^{(n)}\bigr)\bigr\vert^2$$ for every smooth $f : {\mathbb{R}}^2 \to {\mathbb{R}}$.
2. There is a constant $K > 0$ such that $\bigl\vert
Y_a^{(n)}\bigr\vert \le K$ a.s. for every $n$ and every $a \in
G^{(n)}$.
3. For some $\delta \in (0,1]$, $\sup_{n \in {\mathbb{N}}} \max_{a \in G^{(n)}} {\mathbb{E}}\bigl\vert Y_a^{(n)} \bigr\vert^{2 + \delta} < \infty$, and $\sum_{n = 1}^\infty
{\left\vert G^{(n)} \right\vert}^{-\delta/2} < \infty$.
4. For some $\delta \in (0,1]$, $\sup_{n \in {\mathbb{N}}} \max_{a \in
G^{(n)}} {\mathbb{E}}\bigl\vert Y_a^{(n)}\bigr\vert^{2 + \delta} <
\infty$, and $p_2^{(n)} \to p > 0$.
Then $\mu^{(n)}$ converges to the stated limit almost surely.
We now turn to the proofs of our main results. Unsurprisingly, generalizing the results of the last section to non-Gaussian matrix entries is achieved by using an appropriate version of the cen
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\delta_t}, {\boldsymbol \omega} \rangle$, $t\geq 0$.
Another useful corollary of Theorem \[charthm\] concerns integration of a family of generalized functions, see [@PS91; @HKPS93; @KLPSW96].
\[intcor\] Let $(\Lambda, {\mathcal{A}}, \nu)$ be a measure space and $\Lambda \ni\lambda \mapsto \Phi(\lambda) \in (S)'$ a mapping. We assume that its $T$–transform $T \Phi$ satisfies the following conditions:
1. The mapping $\Lambda \ni \lambda \mapsto T(\Phi(\lambda))({\bf f})\in {\mathbb{C}}$ is measurable for all ${\bf f} \in S_d({\mathbb{R}})$.
2. There exists a $p \in {\mathbb{N}}_0$ and functions $D \in L^{\infty}(\Lambda, \nu)$ and $C \in L^1(\Lambda, \nu)$ such that $${\left|T(\Phi(\lambda))(z{\bf f})\right|} \leq C(\lambda)\exp(D(\lambda) {\left|z\right|}^2 {\left\|{\bf f}\right\|}^2),$$ for a.e. $ \lambda \in \Lambda$ and for all ${\bf f} \in S_d({\mathbb{R}})$, $z\in {\mathbb{C}}$.
Then, in the sense of Bochner integration in $H_{-q} \subset (S)'$ for a suitable $q\in {\mathbb{N}}_0$, the integral of the family of Hida distributions is itself a Hida distribution, i.e. $\!\displaystyle \int_{\Lambda} \Phi(\lambda) \, d\nu(\lambda) \in (S)'$ and the $T$–transform interchanges with integration, i.e. $$T\left( \int_{\Lambda} \Phi(\lambda) \, d\nu(\lambda) \right)(\mathbf{f}) =
\int_{\Lambda} T(\Phi(\lambda))(\mathbf{f}) \, d\nu(\lambda), \quad \mathbf{f} \in S_d({\mathbb{R}}).$$
Based on the above theorem, we introduce the following Hida distribution.
\[D:Donsker\] We define Donsker’s delta at $x \in {\mathbb{R}}$ corresponding to $0 \neq {\boldsymbol\eta} \in L_{d}^2({\mathbb{R}})$ by $$\delta_0(\langle {\boldsymbol\eta},\cdot \rangle-x) :=
\frac{1}{2\pi} \int_{{\mathbb{R}}} \exp(i \lambda (\langle {\boldsymbol\eta},\cdot \rangle -x)) \, d \lambda$$ in the sense of Bochner integration, see e.g. [@HKPS93; @LLSW94; @W95]. Its $T$–transform in ${\bf f} \in S_d({\mathbb{R}})$ is given by $$T(\delta_0(\langle {\boldsymbol\eta},\cdot \rangle-x)({\bf f})
= \frac{1}{\sqrt{2\pi \langle {
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uses . Substituting these observations into Corollary \[gr\] gives the second equality in .
Proof of proposition \[app-c-prop\] {#subsec-6.21C}
-----------------------------------
We first show that the map $\theta:J^{k-1}\delta^ke \to M(k)$ is an isomorphism for all $k\geq 1$. This is analogue of Proposition \[grsameA\]. In that case, a purely formal argument showed that Proposition \[grsameA\] followed from . The same argument can be used, essentially without change, to show that the bijectivity of $\theta$ follows from .
Combined with Lemma \[filter-injC\](ii) this says that $\operatorname{{\textsf}{ogr}}M(k) = \operatorname{{\textsf}{ogr}}\theta(J^{k-1}\delta^{k}e) =
J^{k-1}\delta^ke$, as required.
Index of Notation {#index .unnumbered}
=================
[2]{}
${\mathbb{A}}^1$, $A^1$, alternating polynomials,
${\mathbb{A}}=\bigoplus {\mathbb{A}}^i$, $A=\bigoplus A^i$, ,
$\widehat{A} = \bigoplus_{i\geq j \geq 0} A^{i-j}$,
${\mathcal{B}}_1$, the tautological rank $n$ bundle,
$B=\bigoplus B_{ij}$ for $B_{ij}= \prod_{a=u}^{v-1} Q_{a}^{a+1}$,
canonical grading $W_\alpha $,
$d(\mu)= \{ (i,j)\in {\mathbb{N}}\times {\mathbb{N}}: j <
\mu_{i+1}\}$,
$\delta=\prod_{s\in \mathcal{S}} \alpha_s$,
$\Delta_c(\mu) $, the standard module,
$\widehat{\Delta}_c(\mu) $, the graded standard module,
dominance ordering on ${{\textsf}{Irrep}({{W}})}$,
Dunkl-Cherednik representation $\theta_c$,
${\mathbf{E}}=\sum x_i\delta_i$, the Euler operator,
$\operatorname{{\mathbf{E}}\text{-deg}}$, the Euler grading,
$e, e_-$, trivial and sign idempotents,
fake degrees $f_\mu$,
$H_c$, the rational Cherednik algebra, ${\mathfrak{h}},{\mathfrak{h}}^*$,
${\mathfrak{h}^{\text{reg}}}$,
$\mathbf{h} = \mathbf{h}_c=
\frac{1}{2} \sum_{i=1}^{n-1} x_iy_i + y_ix_i \in H_c$,
$\operatorname{{\mathbf{h}}\text{-deg}}$, the ${\mathbf{h}}$-grading,
Hecke algebra ${\mathcal{H}_{q}}$,
Hilbert schemes $\operatorname{Hilb^n{\mathbb{C}}^2}$, $\operatorname{Hilb(n)}$, ,
$I_\mu$, monomial ideal for a partition $\mu$,
${\mathbb{J}}^1= {
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s no longer appropriate to model the channel for each reconfiguration state as a full-rank matrix with i.i.d. entries due to the sparse nature of mmWave channels.[^4] In the following, we present the channel model of mmWave MIMO systems with reconfigurable antennas.
### Physical Channel Representation
The physical channel representation is also often known as Saleh-Valenzuela (S-V) geometric model. The mmWave MIMO channel can be characterized by physical multipath models. In particular, the clustered channel representation is usually adopted as a practical model for mmWave channels [@akdeniz2014millimeter; @Gustafson_14_ommcacm; @Health_16_OverviewSPTmmMIMO]. The channel matrix for reconfiguration state $\psi$ is contributed by $N_{\psi,{\mathrm{cl}}}$ scattering clusters, and each cluster contains $N_{\psi,{\mathrm{ry}}}$ propagation paths. The 2D physical multipath model for the channel matrix ${\mathbf{H}}_{\psi}$ is given by $$\label{eq:H_PhyscialModeling2D}
\mathbf{H_\psi}=
\sum^{N_{\psi,{\mathrm{cl}}}}_{i=1}\sum^{N_{\psi,{\mathrm{ry}}}}_{l=1}
\alpha_{\psi,i,l} \mathbf{a}_{R}\left(\theta^r_{\psi,i,l}\right)
\mathbf{a}_{T}^H\left(\theta^t_{\psi,i,l}\right),$$ where $\alpha_{\psi,i,l}$ denotes the path gain, $\theta^r_{\psi,i,l}$ and $\theta^t_{\psi,i,l}$ denote the angle of arrival (AOA) and the angle of departure (AOD), respectively, $\mathbf{a}_{R}\left(\theta^r_{\psi,i,l}\right)$ and $\mathbf{a}_{T}^H\left(\theta^t_{\psi,i,l}\right)$ denote the steering vectors of the receive antenna array and the transmit antenna array, respectively. In this work, we consider the 1D uniform linear array (ULA) at both the transmitter and the receiver. Then, the steering vectors are given by $$\label{}
\mathbf{a}_{R}\left(\theta^r_{\psi,i,l}\right)=\left[1,e^{-j2\pi\vartheta^r_{\psi,i,l}},\cdots,e^{-j2\pi\vartheta^r_{\psi,i,l}(N_r-1)}\right]^T,$$ and $$\label{}
\mathbf{a}_{T}\left(\theta^t_{\psi,i,l}\right)=\left[1,e^{-j2\pi\vartheta^t_{\psi,i,l}},\cdots,e^{-j2\pi\vartheta^t_{\psi,i,l}(N_t-1)}\right]^T,$$
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,l}) =
E^{(1)}_{k,l} (A_{1} \chi^{(1)}_{k,l}).
\label{eq.2.3.2}$$ We see, that the function $f(r)=A_{1} \chi^{(1)}_{k,l}(r)$ is the eigen-function of the operator $\hat{H}_{2}$ with quantum number $l$ to a constant factor, i. e. it represents the wave function $\chi^{(2)}_{k^{\prime},l}(r)$ of the hamiltonian $H_{2}$. The energy level $E^{(1)}_{k,l}$ must be the eigen-value of this operator exactly, i. e. it represents the energy level $E^{(2)}_{k^{\prime},l}$ of this hamiltonian. Here, new wave function and energy level have the same index ${k^{\prime}}$. One can write: $$\begin{array}{lcr}
A_{1} \chi^{(1)}_{k,l} (r) =
N_{2} \chi^{(2)}_{k^{\prime},l} (r), &
E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, &
N_{2} = const.
\end{array}
\label{eq.2.3.3}$$
Taking into account (\[eq.2.3.1\]), one can write: $$H_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} A_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} (A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} (E^{(2)}_{k^{\prime},l^{\prime}}
\chi^{(2)}_{k^{\prime},l^{\prime}}) =
E^{(2)}_{k^{\prime},l^{\prime}}
(A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}).
\label{eq.2.3.4}$$ and obtain: $$\begin{array}{lcr}
A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r) =
N_{1} \chi^{(1)}_{k,l^{\prime}} (r), &
E^{(2)}_{k^{\prime},l^{\prime}} = E^{(1)}_{k,l^{\prime}}, &
N_{1} = const.
\end{array}
\label{eq.2.3.5}$$
Thus, we obtain the following interdependences between the wave functions and the levels of the continuous energy spectra of two systems SUSY-partners in the radial problem: $$\begin{array}{cccc}
\chi^{(1)}_{k,l^{\prime}} (r) =
\displaystyle\frac{1}{N_{1}}
A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r), &
\chi^{(2)}_{k^{\prime},l} (r) =
\displaystyle\frac{1}{N_{2}}
A_{1} \chi^{(1)}_{k,l} (r), &
E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, &
E^{(1)}_{k,l^{\prime}} = E^{(2)}_{k^{\prime},l^{\prime}}.
\end{array}
\label{eq.2.3.6}$$ Darboux transformations establish the interdepend
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\widehat{S}}} = \mathbb{E}[X({\widehat{S}}) X({\widehat{S}})^\top].$$ We will be studying the ordinary least squares estimator $\hat{\beta}_{{\widehat{S}}}$ of $\beta_{{\widehat{S}}}$ computed using the sub-sample $\mathcal{D}_{2,n}$ and restricted to the coordinates ${\widehat{S}}$. That is, $$\label{eq:least.squares}
\hat{\beta}_{{\widehat{S}}} = \widehat{\Sigma}_{{\widehat{S}}}^{-1} \widehat{\alpha}_{{\widehat{S}}}$$ where, for any non-empty subset $S$ of $\{1,\ldots,d\}$, $$\label{eq:alpha.beta.hat}
\widehat{\alpha}_{S} = \frac{1}{n} \sum_{i \in \mathcal{I}_{2,n} } Y_i X_i(S)
\quad \text{and} \quad
\widehat{\Sigma}_{S} = \frac{1}{n} \sum_{i \in \mathcal{I}_{2,n}} X_i({\widehat{S}}) X_i(S)^\top.$$ Since each $P \in \mathcal{P}_n^{\mathrm{OLS}}$ has a Lebesgue density, $\hat{\Sigma}_{{\widehat{S}}}$ is invertible almost surely as long as $n \geq k \geq |{\widehat{S}}|$. Notice that $\hat{\beta}_{{\widehat{S}}}$ is not an unbiased estimator of $\beta_{{\widehat{S}}}$ , conditionally or unconditionally on $\mathcal{D}_{2,n}$.
In order to relate $\hat{\beta}_{{\widehat{S}}}$ to $\beta_{{\widehat{S}}}$, it will first be convenient to condition on ${\widehat{S}}$ and thus regard $\beta_{{\widehat{S}}}$ as a $k$-dimensional deterministic vector of parameters (recall that, for simplicity, we assume that $|{\widehat{S}}| \leq k$), which depends on some unknown $P \in \mathcal{P}_n^{\mathrm{OLS}}$. Then, $\hat{\beta}_{{\widehat{S}}}$ is an estimator of a fixed parameter $\beta_{{\widehat{S}}} =
\beta_{{\widehat{S}}}(P)$ computed using an i.i.d. sample $\mathcal{D}_{2,n}$ from the same distribution $P \in
\mathcal{P}_n^{\mathrm{OLS}}$. Since all our bounds depend on ${\widehat{S}}$ only through its size $k$, those bounds will hold also unconditionally.
For each $P \in \mathcal{P}_n^{\mathrm{OLS}}$, we can represent the parameters $\Sigma_{{\widehat{S}}} = \Sigma_{{\widehat{S}}}(P)$ and $\alpha_{{\widehat{S}}} = \alpha_{{\widehat{S}}}(P)$ in in vectorized form as $$\label{eq:psi.beta}
\psi = \psi_{{\widehat{S}}} =
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rements, say $X = (X_t, t\geq 0)$ with probabilities $(\mathbb{P}_x, x\in\mathbb{R}^d)$, whose semi-group is represented by the Fourier transform
$$\mathbb{E}_0\bp{{\rm e}^{{\rm i}\langle\theta ,X_t\rangle}} = {\rm e}^{-|\theta|^\alpha t}, \qquad \theta\in \mathbb{R}^d, \;t\geq 0,$$ where $\langle \cdot,\cdot \rangle$ represents the usual Euclidian inner product. Stable processes enjoy an isotropy in the following sense: if $U$ is any orthogonal matrix in $\mathbb{R}^{d\times d}$, then $(UX_t, t\geq 0)$ under $\mathbb{P}_0$ has the same law as $(X, \mathbb{P}_0)$. Moreover, we have the following important scaling property: for all $c>0$, $$\label{levyscaling}
((cX_{c^{-\alpha} t}, t\geq 0), \mathbb{P}_0) \text{ is equal in law to }((X_t, t\geq 0), \mathbb{P}_0).$$
In dimension two or greater, the operator $-(-\Delta)^{\alpha/2}$ can be expressed in the form $$-(-\Delta)^{\alpha/2} u(x) =-\frac{2^\alpha \,\Gamma((d+\alpha)/2)}{\pi^{d/2}\,\Gamma(-\alpha/2)} \lim_{\varepsilon\downarrow0}\int_{\mathbb{R}^d\backslash B(0,\varepsilon)}\frac{[u(y)- u(x)]}{|y-x|^{d+\alpha}}\,{\rm d}y,\qquad x\in \mathbb{R}^d,$$ where $B(0,\varepsilon) = \{x\in\mathbb{R}^d: |x|<\varepsilon\}$ and $u$ is smooth enough for the limit to make sense.
Noting that $-(-\Delta)^{\alpha/2}$ is no longer a local operator, the analogous formulation of (\[Dirichlet\]) needs a little more care. In particular, the boundary condition on the domain $D$ is no longer stated on $\partial D$, but must now be stated on the complement of $D$, written $D^{\rm c}$. To avoid pathological cases, we must assume throughout that $D^{\rm c}$ has positive $d$-dimensional Lebesgue measure. The Dirichlet problem for $-(-\Delta)^{\alpha/2}$ requires one to find $u\colon D \to\mathbb{R}$ such that $$\begin{gathered}
\begin{aligned}
-(-\Delta )^{\alpha/2}u(x) & = 0, & \qquad x & \in D, & \\
u(x) & = {g}(x), & x & \in D^{\rm c},
\end{aligned}
\label{aDirichlet}\end{gathered}$$ where ${g}$ is a suitabl
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s crucial to distinguish between low-energy and high-energy unitarity violation, is of order $W^4$. The other normalization term, the second term in (\[P-beta-alpha-ave-vac\]), also deviates from the one in unitary case by a quantity of order $W^4$ in the appearance channels, but in an implicit way. The resulting expressions of $\hat{S}$ matrix elements to order $W^4$ are summarized in appendix \[sec:hatS-elements\].
There exists important consistency check in the calculation. That is, the identity relation between $\hat{S}$ matrix elements that follows from generalized T invariance:[^10] $$\begin{aligned}
\hat{S}_{i j} (U, W, X, A) &=& \hat{S}_{j i} (U^*, W^*, X^*, A^*),
\nonumber \\
\hat{S}_{i J} (U, W, X, A) &=& \hat{S}_{J i} (U^*, W^*, X^*, A^*),
\nonumber \\\hat{S}_{I J} (U, W, X, A) &=& \hat{S}_{J I} (U^*, W^*, X^*, A^*),
\label{T-invariance}\end{aligned}$$ where $\hat{S}_{J i}$ is obtained by performing the exchange $h_{i} \leftrightarrow \Delta_{J}$ in $\hat{S}_{i J}$. The generalized T invariance relation will be explicitly verified by the computed results of $\hat{S}$ matrix elements in appendix \[sec:hatS-elements\].[^11]
To carry out a complete consistency check, we have obtained all the $\hat{S}$ matrix elements (including $\hat{S}_{I J}$) to fourth order in $W$, and verified their generalized T invariance. However, only the ones which are required to obtain $S$ matrix elements to fourth order are exhibited in appendix \[sec:hatS-elements\].
### Computation of $S$ matrix elements {#sec:S-matrix}
Given the results of $\hat{S}$ matrix elements it is straightforward to calculate $S$ matrix elements by using the formulas in eq. (\[flavor-hat-relation2\]). The active neutrino space $S$ matrix elements can be written in perturbative forms, $S_{\alpha \beta} = S_{\alpha \beta}^{(0)} + S_{\alpha \beta}^{(2)} + S_{\alpha \beta}^{(4)}$, where $$\begin{aligned}
S_{\alpha \beta}^{(0)} &=&
\sum_{k l} (UX)_{\alpha k} (UX)^*_{\beta l}
\hat{S}_{kl}^{(0)},
\nonumber \\
S_{\alpha \beta}^{(2)} &=&
\sum_{k l
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|
17.99 37.74 11.3 7.24
Q08380 Galectin-3-binding protein 17.44 15.73 65.3 5.27
P01876 Immunoglobulin heavy constant alpha 1 17.06 25.78 37.6 6.51
P35443 Thrombospondin-4 9.76 4.79 105.8 4.68
P12259 Coagulation factor V 8.36 2.11 251.5 6.05
P04070 Vitamin K-dependent protein C 7.60 8.89 52.0 6.28
P61981 14-3-3 protein gamma 5.93 10.53 28.3 4.89
P16070 CD44 antigen 4.96 2.70 81.5 5.33
Q99436 Proteasome subunit beta type-7 4.05 8.30 29.9 7.68
Q14515 SPARC-like protein 1 4.05 3.77 75.2 4.81
P07900 Heat shock protein HSP 90-alpha 2.46 2.60 84.6 5.02
P01023 Alpha-2-macroglobulin 1.97 1.49 163.2 6.46
P22105 Tenascin-X OS=Homo sapiens 1.66 0.66 458.1 5.17
P13591 Neural cell adhesion molecule 1 0.00 1.75 94.5 4.87
Q5UIP0 Telomere-associated protein RIF1\] 0.00 1.62 274.3 5.52
{#biomedicines-08-00069-f001}
{#biomedicines-08-00069-f002}
{#biomedicines-08-00069-f003}
![Typical cryoporometry DSC curves (**A**) and boxplot of pore width modes (**B**) for 6, 11, and 33 μm filter pape
| 1,192
| 2,710
| 1,793
| 1,219
| null | null |
github_plus_top10pct_by_avg
|
-bottom: as shown in Fig. \[fig:sdntn\_prop2\], an attacker launches a DoS attack against an OpenFlow switch, which is connected to a primary SDN-controller. In order to increase the resilience of the system, the switch has been assigned a secondary SDN-controller. The switch needs to contact the SDN-controller for every packet. The SDN-controller is not able to handle all the new flow requests and fails. Then, the switch redirects its traffic to the secondary SDN-controller, where the same situation can happen. If the failing SDN-controller is the only controller available for other OpenFlow switches, then the switches lose their connection with the controller and can be considered as failed.
- *Horizontal DP propagation*: as observed in Fig. \[fig:sdntn\_prop1\], an attacker injects huge quantities of traffic to an edge OpenFlow switch (e.g., a DoS attack). The destination of the injected traffic is outside the SDNTN, and to reach it, the traffic has to be routed through core switches and exit via another edge switch. Since SDN allows heterogeneous network scenarios (e.g., hardware from different vendors), we assume that edge OpenFlow switches have high-performance capabilities, while core OpenFlow switches are commodity devices. Therefore, the huge amount of injected traffic can eventually overflow the buffer of core switches, causing successive failures in the DP plane.
Second, it is possible to observe the same scenario shown in Fig. \[fig:sdntn\_prop1\] caused by a fault. In such case, instead of being caused by a DoS attack, the switch failures would occur due to, for instance, a software bug in the routing protocol of the SDN-controller. Lastly, it is worth noting that the proposed horizontal DP propagation, as well as the vertical propagation, are closer to cascading failures. However, in the case of a distributed CP with several SDN-controllers, an epidemic-like spreading could happen if, for instance, the SDN-controllers where infected by masterful worms such as Tuxera[^4].
Transport Network Pr
| 1,193
| 2,902
| 2,038
| 1,203
| null | null |
github_plus_top10pct_by_avg
|
tive to LINQ you can use operations over the array with Array.FindLastIndex + AsSpan + ToArray (note that all solutions here forward assume that it's possible for your array to be all nulls):
string[] strings = { "Hello", null, "World", null, null, null, null, null, null };
var idx = Array.FindLastIndex(strings, c => c != null);
var result = idx >= 0 ? strings.AsSpan(0, idx + 1).ToArray()
: Array.Empty<string>();
If Span<T> is not available to you, we could combine with LINQ's Take:
var idx = Array.FindLastIndex(strings, c => c != null);
var result = idx >= 0 ? strings.Take(idx + 1).ToArray()
: Array.Empty<string>();
Or without LINQ:
var idx = Array.FindLastIndex(strings, c => c != null);
var result = Array.Empty<string>();
if (idx >= 0)
{
result = new string[idx +1 ];
Array.Copy(strings, 0, result, 0, idx + 1);
}
If you have later C# features, we can use System.Range/System.Index and create a sub-array of the existing array:
string[] strings = { "Hello", null, "World", null, null, null, null, null, null };
var idx = Array.FindLastIndex(strings, c => c != null);
var result = idx >= 0 ? strings[..(idx + 1)]
: Array.Empty<string>();
Or as a (kind of hard to read) one-liner if that's what you are after:
var result = Array.FindLastIndex(strings, c => c != null) is int idx && idx >= 0 ? strings[..(idx + 1)] : Array.Empty<string>();
Removing Elements from Array
You're title asks to remove elements. While I'm fairly sure you're looking to create a new collection (given your mention of LINQ), I figured I'd point out that you can't really remove elements from an array--they are fixed in size. However you can make use of Array.Resize to "crop" your array:
var idx = Array.FindLastIndex(strings, c => c != null);
if (idx >= 0)
Array.Resize(ref strings, idx + 1);
Q:
Dynamic organizational chart
I'm in a bit of a pickle and don't really know if can solve my problem with SharePoint.
My company wants to build an organizational chart (eas
| 1,194
| 6,328
| 35
| 688
| 90
| 0.826928
|
github_plus_top10pct_by_avg
|
sidered in this work.
Heating and cooling rates {#sec:heat_cool}
-------------------------
[lllc]{} No. & Process & Rate \[${\mathrm{erg\, cm^{-3}\, s^{-1}}}$\] & Ref.\
\
1 & ${\mathrm{H}}$ photoionization & $\Gamma_1$ (see text)&\
2 & ${\mathrm{He}}$ photoionization & $\Gamma_2$ (see text)&\
3 & ${\mathrm{He^+}}$ photoionization & $\Gamma_3$ (see text)&\
\
$1^{a}$ & ${\mathrm{H^+}}$ recombination & $\Lambda_1=$ & 1\
$2^{b}$ & ${\mathrm{He^+}}$ recombination & & 2,3\
$3^{c}$ & ${\mathrm{He^{2+}}}$ recombination & $\Lambda_3= 1.38\times10^{-16}T_{\mathrm{K}} \left(0.684-0.0416\,\ln(T_{\mathrm{K}}/40000)\right)
k_6\,n({\mathrm{e}})\,n({\mathrm{He^{2+}}})$ & 4\
4 & ${\mathrm{H}}$ excitation & $\Lambda_4=7.50\times10^{-19} \left(1 + (T_{\mathrm{K}}/100000)^{1/2}\right)^{-1} \exp[-118348/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{H}})$ & 5\
$5^{d}$ & ${\mathrm{He}}$ excitation & $\Lambda_5=1.1\times10^{-19} T_{\mathrm{K}}^{0.082} \exp[-230000/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{He}})$ & 6\
6 & ${\mathrm{He^+}}$ excitation & $\Lambda_6=5.54\times10^{-17} T_{\mathrm{K}}^{-0.397} \left(1 + (T_{\mathrm{K}}/100000)^{1/2}\right)^{-1} \exp[-473638/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{He^{+}}})$ & 5\
7 & ${\mathrm{H}}$ ionization & $\Lambda_7=2.18\times10^{-11}k_1\,n({\mathrm{e}})\,n({\mathrm{H}})$ & 7\
$8^{c}$ & ${\mathrm{He}}$ ionization & $\Lambda_8=3.94\times10^{-11}k_2\,n({\mathrm{e}})\,n({\mathrm{He}})$ & 7\
9 & ${\mathrm{He^+}}$ ionization & $\Lambda_9=8.72\times10^{-11}k_3\,n({\mathrm{e}})\,n({\mathrm{He^{+}}})$ & 7\
10 & Free-free & & 8\
$11^{e}$ & Compton & $\Lambda_{11}=1.017\times10^{-37} T_{\mathrm{CMB}}^4 (T_{\mathrm{K}} - T_{\mathrm{CMB}}) \,n({\mathrm{e}})$ & 5\
\
NOTES. $^{a}$Case B, our fit to [@Ferland:1992aa]; $^{b}$radiative [Case B; singlet; our fit to @Hummer:1998aa] and dielectric [@Black:1981aa] recombination cooling; $^{c}$Case B [@Draine:2011aa with typo about the charge dependence corrected]; $^{d}$singlet; $^{e}$$T_{\mathrm{CMB}}=2.73(1 + z)$ with $z=15$.\
REFERE
| 1,195
| 1,847
| 2,455
| 1,312
| null | null |
github_plus_top10pct_by_avg
|
he different aspects of the optical processes raised by the selection of distinct sidebands.
We proceed to apply the NL in Eq. (\[nl1\]) to reveal the irreversibility of the work protocol. Just like what happened with the full Hamiltonian (\[hf\]), the first moment of the work distribution or simply the average work is again null, i.e., $\langle W \rangle = 0$. For this reason, the NL in Eq. (\[nl1\]) for the sudden quench of the sideband Hamiltonian in Eq. (\[hs\]) reads $$\label{nlfinal}
\mathcal L = {\rm ln}\, \frac{\mathcal Z(\lambda_f)}{\mathcal Z(\lambda_i)},$$ with $$\label{zi}
\mathcal Z(\lambda_i) = 2(\bar n + 1) \cosh{ \tfrac{\beta \hbar\omega_0}{2}},$$ obtained using Eq. (\[h0\]), and $$\label{zf}
\!\!\mathcal Z_{\pm}(\lambda_f)\! = \!
\sum_{n = 0}^{\infty} \left[ {\rm e}^{-\beta \mu_{\pm}^{ (n,m) } } \!\! + \!
{\rm e}^{-\beta \gamma_{\pm}^{(n,m)}}\right] +
\sum_{n = 0}^{m-1} {\rm e}^{-\beta \zeta_{\pm}^{(n,m)}} \!\!
,$$ obtained with Eq. (\[hs\]). The functions $\mu_{\pm}$, $\gamma_{\pm}$, and $\zeta_{\pm}$ are the eigenvalues of the Hamiltonians in (\[hs\]), and their expressions can be found in Eqs. (\[eigval1+\]), (\[eigval+\]), (\[eigval1-\]), and (\[eigval-\]), which allows one to get $$\begin{aligned}
\label{partf1}
\!\!\!\!\!\!\!\mathcal Z_{\pm}(\lambda_f) &=& 2
\sum_{n=0}^{\infty} \!{\rm e}^{-\beta \hbar \nu( n + \frac{m}{2})}
\! \cosh\!\!\left[ \tfrac{\beta \hbar}{2}
\sqrt{{\omega_L}^{2} \! +\! \Omega^{2}\left|f_n^m\right|^{2}} \right] \nonumber\\
&& + \,
(\bar n + 1 ) (1- {\rm e}^{- \beta \hbar m \nu} )
{\rm e}^{\pm \tfrac{1}{2}\beta\hbar \omega_0 } , \end{aligned}$$ with $\omega_L = \omega_{0} \mp m\nu$, and $$\begin{aligned}
\label{auxf1}
\!\!\!\!\!\!f_n^m\!:=\! \frac{2}{\Omega} \!
\left\langle n\right|\!\hat{\Omega}_{m}^+\!\left|n+m\right\rangle
| 1,196
| 3,832
| 1,168
| 1,026
| null | null |
github_plus_top10pct_by_avg
|
{\bigcup\{\dot{Z}(\xi,C_{\zeta(\xi)}):\xi\in f_\gamma^{-1}(n)\}}
\subseteq\dot{W}(\gamma,n).$$ Then we can get a $\zeta(\gamma)$ that works for all $n$.
By recursion on $\gamma\in\omega_2$, we can choose $\zeta(\gamma)\geq\gamma$ as above, so that the sequence $\{\zeta(\gamma):\gamma\in\omega_2\}$ is strictly increasing. For each $\gamma$, we have the $S$-name $\dot A_{\zeta(\gamma)}$ as above. It is immediate that $A_\gamma = \{ \delta :
(\exists s\in S) s\Vdash \delta\in \dot A_{\zeta(\gamma)}\}$ is a stationary set. In other words, $\delta \in A_\gamma$ implies there is some $s\in S$ and $\eta \in [\delta,\beta(\delta)]$ such that $s\Vdash \eta \in
\langle\dot{Z}(\xi,
C_{\zeta(\gamma)}):\xi\in\delta\rangle'$.
By **SCC** and \[larson\] we may assume there is an elementary submodel $M$ of some $\langle H(\theta),\{\langle
\gamma,\zeta(\gamma), A_\gamma\rangle:\gamma\in\omega_2 \}\rangle$, with $M\cap\omega_1=\delta<\omega_1$, $|M\cap\omega_2|=\aleph_1$, and an uncountable $\{\gamma_\alpha:\alpha\in\omega_1\}\subseteq
M\cap\omega_2$, so that $\delta\in A_{\gamma_\alpha}$ for all $\alpha\in\omega_1$.
For each $\alpha\in\omega_1$ choose $s_{\alpha}\in S$, $\eta_\alpha\in[\delta,\beta(\delta)]$ such that $s_\alpha\Vdash
\eta_\alpha\in\langle\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)}):
\xi\in\delta\rangle'$. We may assume $s_\alpha$ is on a level at least as high as $\delta^+(C_{\gamma_\alpha})$. We may also assume that if $\alpha<\beta\in\omega_1$, then $\gamma_\alpha<\gamma_\beta$. We may also assume that the height of $s_\alpha$ is less than the height of $s_\beta$, for $\alpha<\beta$, so that $\{s_\alpha:\alpha\in\omega_1\}$ is an uncountable subset of $S$. Therefore there is an $\eta\in [\delta,\beta(\delta)]$ such that $L = \{ \alpha : \eta_\alpha = \eta\}$ is uncountable. Also, as is well-known for Souslin trees, there is an $\bar{s}\in S$, such that $\{s_\alpha :\alpha\in L\}$ includes a dense subset of $\{s\in S:\bar{s}<s\}$. By passing to an uncountable subset, we may assume that $\bar{s} <s_\alpha$ for all
| 1,197
| 1,115
| 950
| 1,152
| null | null |
github_plus_top10pct_by_avg
|
-(\[csda3\]), and related operators. Write &T\_1:=\_x\_1+\_1\_1-K\_1,\
&T\_j:=-[E]{}+\_x\_j+ \_j\_j - K\_j,j=2,3,and define a (densely defined) linear operator $T:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by D(T):=&{L\^2(GSI)\^3 | T\_jL\^2(GSI), j=1,2,3},\
T:=&(T\_1,T\_2,T\_3). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. The problem (\[csda1a\])-(\[csda3\]) can be expressed equivalently as the problem \[adjoint1\] T=f,\_[|\_-]{}=g,\_j(,,E\_m)=0,j=2,3.
As in section \[esols\], an application of integration by parts and the Green’s formula (\[green\]) implies \[adjoint2\] T,v\_[L\^2(GSI)\^3]{}=,T\^\*v\_[L\^2(GSI)\^3]{}vC\_0\^1(GSI\^), where $T^*v=(T_1^*v,T_2^*v,T_3^*v)$, and \[adjoint3a\] T\_1\^\*v:=&-\_x v\_1+\_1\^\*v\_1-K\_1\^\*v,\
T\_j\^\*v:=&S\_j[E]{}-\_x v\_j +\_j\^\*v\_j-K\_j\^\*v,j=2,3. Moreover, we have $\Sigma_j^*=\Sigma_j$ and for $v\in L^2(G\times S\times I)^3$, $j=1,2,3$, $$(K_j^*v)(x,\omega,E)
=\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')\psi_k(x,\omega',E')d\omega' dE'.$$
Let $f^*\in L^2(G\times S\times I)^3$ and $g^*\in T^2(\Gamma_+)^3$. The *adjoint problem* of (\[adjoint1\]) (or equivalently (\[desol10\])-(\[desol12\])) is defined by (cf. [@agoshkov pp. 24-28]) \[adjoint4\] T\^\*\^\*=f\^\*,\_[|\_+]{}=g\^\*,\_j\^\*(,,0)=0, j=2,3, or more explicitly -\_x\_1\^\*+\_1\^\*\_1\^\*-K\_1\^\*\^\*=& f\_1\^\*, \[adesol10:1\]\
S\_j[E]{}-\_x\_j\^\* +\_j\^\*\_j\^\* - K\_j\^\*\^\*=& f\_j\^\*,j=2,3,\[adesol10:2\] holding a.e. on $G\times S\times I$, together with the *outflow* boundary and initial values $$\begin{aligned}
{3}
\psi^*_{|\Gamma_+}={}&g^* && \quad {\rm a.e.\ on}\ \Gamma_+, \label{adesol11} \\[2mm]
\psi_j^*(\cdot,\cdot,0)={}&0\ && \quad {\rm a.e.\ on}\ G\times S,\quad j=2,3. \label{adesol12}\end{aligned}$$ The adjoint problem has various kind of applications both in the existence theory of solutions and in computations. At the end of this section we give an example which is related to the dose calculation in radiation therapy. We refer also to co
| 1,198
| 201
| 1,162
| 1,218
| null | null |
github_plus_top10pct_by_avg
|
he Standard Model parameters, we use the measured values of the top quark, W, Z, and Higgs masses. Note that we use $m_h=125.3$ GeV for all points when calculating threshold corrections. After running through , the points that survive all have a Higgs mass within 3 GeV of this value. This is at most a $\sim$2% difference. Furthermore, the Higgs mass only occurs in the calculation of the neutral Higgs contribution. The error in this approximation is therefore negligible and the results remain unaffected. For the bottom quark mass, we use the package [@Chetyrkin:2000yt] to run $m_b(m_b)$ to $m_b(M_Z)$.
----------------- --------------------------------------------- ---------------
$g_1$ = 0.46 $g_2$ = 0.64 $g_3$ = 1.2
$M_t$ = 173.36 $m_b$ = 2.69 $V_{tb}$ = 1
$M_Z$ = 91.1876 $M_W$ = 80.385 $m_h$ = 125.3
$v$ = 246 $\tanb$ = 50
$1000 < \{m_{Q_3},m_{u_3},m_{d_3}\} < 5000$
$~100 < \{m_{L_3},m_{e_3}\} < 5000$
$-15000 < \{A_t,A_b\} < 15000~~~$
$-1000 < \{M_1,M_2\} < 1000~~$
$~500 < M_3 < 2000$
$~1000 < M_A < 2000$
$-2000 < \mu < 2000~~$
----------------- --------------------------------------------- ---------------
: Parameter values and ranges at $M_Z$.\
All masses in GeV.[]{data-label="tab:params"}
Exact vs. Approximation \[Sec:ex-app\]
======================================
The complete set of one loop corrections to the bottom quark mass is given by [@Pierce:1996zz] m\_b (M\_Z) = m\_b\^ + m\_b\^[\^]{} + m\_b\^[\^0]{} + m\_b\^[H\^]{} + m\_b\^[A]{} + m\_b\^[h]{} + m\_b\^[W]{} + m\_b\^[Z]{} , with the tree level mass given by $\lambda_b (M_Z) \frac{v}{\sqrt{2}} {\rm cos} \beta$.
![The plot sho
| 1,199
| 3,233
| 2,137
| 1,296
| null | null |
github_plus_top10pct_by_avg
|
vation by Idea program \[VIREPAP 19104\]. EIT Health is supported by the European Institute of Innovation and Technology (EIT), a body of the European Union that receives support from the European Union' s Horizon 2020 Research and innovation program.
The corresponding author (A.M. (Albert Mihranyan)) is the inventor behind the IP pertaining to virus removal filter paper.
biomedicines-08-00069-t0A1_Table A1
######
LCMS analysis of FIX-PCC feed solution.
Accession Description Score Coverage MW \[kDa\] calc. pI
----------- ---------------------------------------------- --------- ---------- ------------ ----------
P0C0L5 Complement C4-B 1920.89 76.38 192.6 7.27
P0C0L4 Complement C4-A 1898.90 76.03 192.7 7.08
P00734 Prothrombin 1767.89 75.56 70.0 5.90
P19823 Inter-alpha-trypsin inhibitor heavy chain H2 917.12 53.59 106.4 6.86
P19827 Inter-alpha-trypsin inhibitor heavy chain H1 685.72 55.76 101.3 6.79
Q06033 Inter-alpha-trypsin inhibitor heavy chain H3 257.18 40.22 99.8 5.74
P02760 Protein AMBP 199.11 27.56 39.0 6.25
P00740 Coagulation factor IX 134.15 47.29 51.7 5.47
P00742 Coagulation factor X 59.86 38.52 54.7 5.94
P02768 Serum albumin 48.49 28.41 69.3 6.28
P49747 Cartilage oligomeric matrix protein 41.93 14.27 82.8 4.60
P67936 Tropomyosin alpha-4 chain 32.67 39.11 28.5 4.69
P01857 Immunoglobulin heavy constant gamma 1 32.57 35.15 36.1 8.19
P01834 Immunoglobulin kappa constant 30.33 39.
| 1,200
| 4,953
| 823
| 591
| null | null |
github_plus_top10pct_by_avg
|
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