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w you would display, page, and retrieve the documents.
I would prefer embedding and updating multiple schema when there's a change, as opposed to doing a ref, for multiple reasons.
Get would be fast and easy and filter is not a problem (like you've said)
Retrieve operations usually happen a lot more often than updates and with proper indexing, you wouldn't really have to bother about performance.
It leverages on NoSQL's schema-less nature and you'll be less prone restructuring due to requirement changes (new sorting, new filters, etc)
Paging would be a lot less of a hassle, and UI would not be restricted with it's design with paging and limit.
Joining could become expensive. Redundant data might be a hassle to update but it's always better than not being able to display a data in a particular way because your schema is normalized and joining is difficult.
I'd say that the rule of thumb is that only split them when you do not need to display them together. It is not impossible to join them back if you do, but definitely more troublesome.
Q:
Pass a splatted param list to a nested function
I have a set of nested functions that each take an arbitrary list of arguments:
def foo *args
bar args
end
def bar *args
baz args
end
def baz *args
end
Calling foo with a set of args like :a => :foo, :b => :bar gives us a single element array after the splat:
[{:a => :foo, :b => :bar}]
And then passing that along to the nested function, and again through a splat, makes for this:
[[{:a => :foo, :b => :bar}]]
Is it appropriate to pass args[0] along to the nested function, or is there some kind of reverse splat that I should be using instead?
A:
If you want to relay splatted arguments to another function, just splat them again (the operator behaves the opposite way when used in a method call (vs. method definition))
def foo(*args)
bar *args
end
Q:
Putting labels on a bar chart in R with ggplot2
I am having a little bit of trouble inserting labels on a bar chart in ggplot2.
So far, I have been able to create
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|
49
VIII. Experiences of Others 58
IX. The American Lakes 66
X. The Happy Life 74
XI. Boy Desperadoes 82
XII. American and English Beggars 89
XIII. Beggars' Slang 97
XIV. Bony's Wits 105
XV. Favouritism 114
XVI. A Law to Suppress Vagrancy 122
XVII. Stubborn Invalids 130
XVIII. The Earnings of Beggars 138
XIX. Charity in Strange Quarters 146
XX. Enemies of Beggars 154
XXI. The Lowest State of Man 161
XXII. The Lodger Lover 169
XXIII. The Handy Man 176
XXIV. On Books 183
XXV. Narks 191
XXVI. The Scribe in a Lodging-house 199
XXVII. Licensed Beggars 207
XXVIII. Navvies and Frauds 213
XXIX. A First Night in a Lodging-house 222
XXX. Gentleman Bill 230
XXXI. Fallacies Concerning Beggars 238
XXXII. Lady Tramps 247
XXXIII. Meeting Old Friends 256
XXXIV. The Comparison 263
XXXV. The Supper 270
XXXVI. The Literary Life 278
XXXVII. The Sport of Fame 285
XXXVIII. Beggars in the Making 293
BEGGARS
I
The Nationalities as Beggars
There is no question but that the American beggar is the finest in
his country; but in that land of many nationalities he has a number
of old-country beggars to contend with. Perhaps it would interest--it
certainly should--a number of people to know how well or ill their
own nation is represented by beggars in t
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|
is similar to that of the proof of Theorem 3.4 in [@C2] and we may skip it.
\[r35\] Let $R$ be a $\kappa$-algebra. We explain the above action morphism in terms of $R$-points. Choose an element $(m_{i,j}, s_i\cdots w_i)$ in $ \underline{M}^{\ast}(R) $ as explained in Section \[m\] and express this element formally as a matrix $m=\begin{pmatrix}\pi^{max\{0,j-i\}}m_{i,j}\end{pmatrix}$ with $z_i^{\ast}, m_{i, i}^{\ast}, m_{i, i}^{\ast\ast}$. We also choose an element $(f_{i,j}, a_i \cdots f_i)$ of $\underline{H}(R)$ and express this element formally as a matrix $f=\begin{pmatrix}\pi^{max\{i,j\}}f_{i,j}\end{pmatrix}$ with $f_{i, i}^{\ast}$ as explained in Section \[h\].
We then compute the formal matrix product $\sigma({}^tm)\cdot f\cdot m$ and denote it by the formal matrix $\begin{pmatrix}\pi^{max\{i,j\}}\tilde{f}_{i,j}'\end{pmatrix}$ with $(\tilde{f}_{i,j}', \tilde{a}_i' \cdots \tilde{f}_i')$. Here, the description of the formal matrix $\begin{pmatrix}\pi^{max\{i,j\}}\tilde{f}_{i,j}'\end{pmatrix}$ with $(\tilde{f}_{i,j}', \tilde{a}_i' \cdots \tilde{f}_i')$ is as explained in Section \[h\]. We emphasize that in the above formal computation, we distinguish $-1$ from $1$ formally. If $L_i$ is *bound of type I* with $i$ odd, then let $$\pi (\tilde{f}_{i, i}^{\ast})'=\delta_{i-1}(0,\cdots, 0, 1)\cdot \tilde{f}_{i-1,i}'+\delta_{i+1}(0,\cdots, 0, 1)\cdot \tilde{f}_{i+1,i}'$$ formally. We formally compute the right hand side and then it is of the form $\pi X$. Here, $X$ involves $m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}$. Then $(\tilde{f}_{i, i}^{\ast})'$ is defined as $X$.
We now let $\pi^2$ be zero in each entry of the formal matrices $$(\tilde{f_{i,j}'})_{i< j}, (\tilde{b_i'})_{\textit{$L_i$ of type $I^o$}},
(\tilde{b_i'}, \tilde{d_i'}, \tilde{e_i'})_{\textit{$L_i$ of type $I^e$ or free of type I with i odd}},
$$(b\_i, d\_i, e\_i)\_[*$L_i$ free of type $I$ with $i$ odd*]{}, ()\_[*$L_i$ bound of type $I$ with $i$ odd*]{} $$ and in each nondiagonal entry of the formal matrix $(\tilde{a}_i')$. Then these entries are elemen
| 2,503
| 1,823
| 2,582
| 2,289
| 2,089
| 0.782838
|
github_plus_top10pct_by_avg
|
stem of wave functions of the form $$\left|\psi(t)\right> = {{\rm e}}^{-{{\rm i}}\varepsilon t/\hbar}\left|u(t)\right>\;,$$ where $\left|u(t)\right>$ is a $T$-periodic function. The quantity $\varepsilon$ is called “quasienergy”, in analogy to the quasimomentum in solid state physics.
If there is no defect, the quasienergies for a periodically driven particle in a tight-binding lattice with nearest-neighbor coupling read [@Holthaus92; @HolthausHone93] $$\varepsilon(k) = -\frac{W}{2}
{\rm J}_0\left(\frac{eFd}{\hbar\omega}\right)\cos(kd)
\quad \bmod \hbar\omega \; ,$$ so that the bandwidth is effectively quenched according to the zero-order Bessel function ${\rm J}_0$: $$\label{eq:qeband}
W_{\rm eff} = W{\rm J}_0\left(\frac{eFd}{\hbar\omega}\right)\;.$$ At the zeros of ${\rm J}_0$, the band “collapses”. This band collapse manifests itself as dynamical localization of the driven particle [@DunlapKenkre86], an effect which should be observable in far-infrared driven semiconductor superlattices [@MeierEtAl95].
Without periodic forcing, the strength of a defect of the form (\[eq:def\]) is determined not by the on-site energy $\nu$ alone, but rather by the ratio $\nu/W$: The larger $|\nu/W|$, the shorter is the localization length of the defect state, as witnessed by Eq. (\[eq:loc\]). This leads to the conjecture that in the presence of periodic forcing the defect Floquet state again is described by Eq. (\[eq:loc\]), but with $\nu/W$ replaced by $\nu/W_{\rm eff}$, so that the localization length of the defect state should become strongly dependent on the amplitude of the forcing. In particular, when $eFd/(\hbar\omega)$ approaches a zero of ${\rm J}_0$, the state should be confined entirely to the defect site, if only $\nu \neq 0$.
As shown in Ref. , this conjecture indeed is correct in the high-frequency regime, where $\hbar\omega$ is significantly larger than the bare band width $|W|$. This is illustrated in Fig. \[fig:defect\] for a defect with strength $\nu/W = 0.1$: The upper panel sho
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| 3,073
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|
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|
$G$-equivariantizable structure, which means that for every $g \in G$, we need a lift $\tilde{g}: {\cal E} \rightarrow {\cal E}$ such that $$\xymatrix{
{\cal E} \ar[r]^{\tilde{g}} \ar[d] & {\cal E} \ar[d] \\
X \ar[r]^{g} & X
}$$ and also such that the lifts obey the group law: $\tilde{g} \circ
\tilde{h} = \widetilde{gh}$.
We need such an equivariant structure on the bundle ${\cal E}$ for the following two reasons:
- In the worldsheet theory, such an equivariant structure enables us to define a group action on the worldsheet fermions/bosons describing the bundle, such that summing over twisted sectors in the orbifold yields an honest projection operator onto $G$-invariant states.
- In the low-energy supergravity, if ${\cal E}$ does not have an equivariant structure, then even if $G$ acts freely, on the quotient $X/G$ the bundle ${\cal E}$ will descend to a ‘twisted’ bundle, not an honest bundle, whose transition functions $g_{\alpha \beta}$ obey $$g_{\alpha \beta} g_{\beta \gamma} g_{\gamma \alpha} \: = \: h_{\alpha
\beta \gamma} I$$ on triple overlaps, and whose gauge field $A$ obeys $$A_{\beta} \: = \: g_{\alpha \beta} A_{\alpha} g_{\alpha \beta}^{-1}
\: + \: g_{\alpha \beta}^{-1} d g_{\alpha \beta} \: + \:
\Lambda_{\alpha \beta} I$$ across intersections, for some affine translation $\Lambda_{\alpha \beta}$. As ten-dimensional super-Yang-Mills only describes honest bundles and ordinary gauge transformations, the structure above cannot be used to define a consistent string compactification.
However, there is a workaround. If the bundle ${\cal E}$ is invariant (meaning, its characteristic classes are invariant under the group action), but not equivariant, then we can find a larger group $\tilde{G}$, an extension of $G$ by a trivially-acting subgroup, such that ${\cal E}$ does admit a $\tilde{G}$-equivariant structure, and then take a $G'$ orbifold. This is precisely an example of a heterotic string on a gerbe, in this case a gerbe over $[X/G]$.
First, let us review some generalities on th
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| 3,907
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|
d by the ratio (w/w) of epididymal fat to liver decreased (*P*= 0.044) by 16% compared to the corresponding control group. None of the above mentioned effects were observed in the non-fasting group. Liver weight at the end of the study was 11% smaller (*P*= 0.039) among non-fasting animals in the control group, as compared to non-fasting animals in the PS-HOSO group. This difference was abolished in the fasting groups. Analysis of liver fat weight to liver weight ratio didn\'t reveal any significant difference between treatment groups among non-fasting animals, indicating that the liver weight difference didn\'t result from liver fat accumulation. Interestingly, among fasting animals, this ratio was 27% lower (*P*= 0.04) in the PS-HOSO group in comparison with that of the control group.
######
PS-HOSO effect on body weight, liver weight and fat accumulation of phase C gerbils^1^.
**Control** **PS-HOSO**
---------------------------------------- --------------- ----------------- --------------- ---------------- ----------------- ------------------
**Total** **Non-fasting** **Fasting** **Total** **Non-fasting** **Fasting**
Bodyweight at endpoint (gr) 219.05 ± 16.2 214.60 ± 15.8 222.75 ± 16.2 211.58 ± 19.8 216.63 ± 23.0 \* 207.91 ± 17.3
Liver weight (gr) 8.23 ± 0.77 8.36 ± 0.80 8.12 ± 0.77 8.68 ± 1.19 \* 9.40 ± 1.17 8.15 ± 0.93
Liver fat weight to liver weight ratio 0.11 ±
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|
,1\}$ let $N_i = N'|(A_i \cup B_i)$. Now $$\begin{aligned}
\lambda_{N'}(A_1 \cup B_1) &= r_{N'}(A_1 \cup B_1) + r_{M'}(A_0 \cup B_0 \cup X \cup Y) - r(M')\\
&= r_{N'}(A_1) + r_{M'}(A_0 \cup X) - r(M')\\
&\le |A_1| + |A_0 \cup X| - r(M') = 0,
\end{aligned}$$ Therefore $N' = N_0 \oplus N_1$. Moreover, since $\sqcap_{M'}(A_1,X \cup Y) = 0$ and $B_1 \subseteq \cl_{M' \con Y}(A_1)$ we have $0 = \sqcap_{M' \con Y}(A_1,X) = \sqcap_{M' \con Y}(A_1 \cup B_1,X)$ and so $N_1 = (M' \con Y \del X)/A_0\del B_0$; by Lemma \[staytriangular\] it follows that $(\bar{A},\bar{B}) \cap E(N_1)$ is an upper-triangular pair in $N_1$. Since $r(N_0) = |B_1| \ge (s4^s)^{t-1}$, the inductive hypothesis gives a $(t-1)U_{1,2}$-minor $\hat{N}$ of $N_1$ in which $E(\hat{N}) \cap \bar{A}$ and $E(\hat{N}) \cap \bar{B}$ are bases. Since $\sqcap_{M'}(A_0,X) = 0$ and $\sqcap_{M'}(A_0,B_0) = s = r_{M'}(A_0)$, we have $\sqcap_{M' \con X}(A_0,B_0) = s$ and therefore $\sqcap_{N'}(A_0,B_0) \ge s - |Y| > 0$. Therefore $N_0$ contains a circuit intersecting both $A_0$ and $B_0$, and so $N_0$ has a $U_{1,2}$-minor $N_0$ intersecting $A_0$ and $B_0$. It follows that $N' = N_0 \oplus N_1$ has the required $tU_{1,2}$-minor.
\[selfdual\] Let $s \ge 2$ and $t \ge 0$. If $A$ and $B$ are disjoint bases of a matroid $M$ with $r(M) \ge 4^{s(s4^s)^t}$, then either
- $M$ has a $U_{s,2s}$-minor $U$ in which $E(U) \cap A$ and $E(U) \cap B$ are bases, or
- $M$ has a $tU_{1,2}$-minor $N$ in which $E(N) \cap A$ and $E(N) \cap B$ are bases.
Let $t' = (s4^s)^t$. Assume that the first outcome does not hold. Note that $A$ and $B$ are disjoint bases of $M_1 = (M|(A \cup B))^*$. By Lemma \[triangularone\] applied to $M_1$, we see that $M_1$ has a rank-$t'$-minor $M_2$ having a lower-triangular pair $(\bar{A},\bar{B}) \in A^{t'} \times B^{t'}$. Now $M_2^*$ is a rank-$t'$ minor of $M$ and $(\bar{A},\bar{B})$ is an upper-triangular pair of $M_2^*$; the result now follows from Lemma \[triangulartwo\].
We can now prove Theorem \[unavoidable
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|
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|
400 1
13 7-back 19 1 2500 1 1-back 25 5 300 5
14 7-back 21 1 2500 1 1-back 25 5 300 1
15 8-back 21 1 2500 1 1-back 25 5 250 5
16 8-back 23 1 2500 1 1-back 25 5 250 1
17 9-back 23 1 2500 1 1-back 25 5 230 5
18 9-back 25 1 2500 1 1-back 25 5 230 1
19 10-back 25 1 2500 1 1-back 25 5 220 5
20 10-back 27 1 2500 1 1-back 25 5 220 1
Task, the task participants should complete in this level; Spatial, the number of possible locations of the stimuli; Block(s), the number of blocks participants should complete in this level; Time, maximum response time of each trial in this level (ms); Errors, maximum error times allowed to pass this level
.
There were 15 training sessions for both groups, and participants would complete one session per day. After each training session, participants were required to answer 3 manipulation check questions which were (a) how concentrated do you think you were in this training session (i.e., perceived attention level); (b) how difficult do you think the task was in this training session (i.e., perceived difficulty level); and (c) how attractive do you think the task was in this training session (i.e., perceived attraction level). All these questions were 7-points Likert evaluation. Participants could get "points" after they achieved a higher level, and they could get extra monetary reward based on how many points they have at the end of all training sessions.
### Measurement of WMC, Measurement of Attentional Control, and the SA Condition
All were the same with Experiment 1.
### Procedure
Participants were invited to take part in Experiment 2 and matched into WM training group and control group based on pre
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|
& = & \E_0,\end{aligned}$$ which, for $\alpha \neq 0$, forces $\E({\mathbf{u}}_0) = 0$. Hence, ${\mathbf{u}}_0
\equiv 0$, $\alpha = 1/2$ and $\E({\mathbf{u}}_1) = 1$. The systems at orders $\E_0^{1/2}$ and $\E_0^1$ are given by:
\[eq:maxdEdt\_Asympt\_1\] $$\begin{aligned}
\E^{1/2}_0:\quad\qquad\qquad\qquad\qquad\qquad 2\nu{\Delta}^2{\mathbf{u}}_1 + \lambda_0{\Delta}{\mathbf{u}}_1 +
\nabla q_1 & = 0 \quad\mbox{in}\,\,\Omega , \label{eq:maxdEdt_Asympt_1_PDE}\\
\nabla\cdot{\mathbf{u}}_1 & = 0 \quad\mbox{in}\,\,\Omega , \label{eq:maxdEdt_Asympt_1_Div0Constr} \\
\E({\mathbf{u}}_1) & = 1, \label{eq:maxdEdt_Asympt_1_E0Constr} \end{aligned}$$
\[eq:maxdEdt\_Asympt\_2\] $$\begin{aligned}
\E_0:\qquad 2\nu{\Delta}^2{\mathbf{u}}_2 + \lambda_0{\Delta}{\mathbf{u}}_2 + \nabla q_2 - {\mathcal{B}}({\mathbf{u}}_1,{\mathbf{u}}_1) +
\lambda_1{\Delta}{\mathbf{u}}_1 & = 0 \quad\mbox{in}\,\,\Omega, \\
\nabla\cdot{\mathbf{u}}_2 & = 0 \quad\mbox{in}\,\,\Omega , \\
\langle {\Delta}{\mathbf{u}}_1, {\mathbf{u}}_2 \rangle_{L_2} & = 0, \end{aligned}$$
where the fact that ${\mathcal{B}}({\mathbf{u}}_0,{\mathbf{u}}_j) = 0$ for all $j$ has been used. While continuing this process to larger values of $m$ may lead to some interesting insights, for the purpose of this investigation it is sufficient to truncate expansions at the order $\O(\E_0)$. The corresponding approximation of the objective functional then becomes $$\label{eq:R03D}
\R({\widetilde{\mathbf{u}}}) = - \nu\E_0\int_{\Omega} \left| {\Delta}{\mathbf{u}}_1 \right|^2 \, d{\mathbf{x}}+ \O(\E_0^{3/2}).$$ It is worth noting that, in the light of relation , the maximum rate of growth of enstrophy in the limit of small $\E_0$ is in fact negative, meaning that, for sufficiently small $\E_0$, the enstrophy itself is a decreasing function for all times. This observation is consistent with the small-data regularity result discussed in Introduction.
As regards problem defining the triplet $\{{\mathbf{u}}_1, q_1,\lambda_0\}$, taking the divergence of equation and using the condition $\
| 2,509
| 2,062
| 2,568
| 2,469
| null | null |
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|
\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1}) q^{2(b-1)(\al
,{\alpha }_p)}\\
=&q^{2({\sigma }_p^\chi ({\alpha }),\lambda )}q^{-2({\alpha },{\alpha }_p)}
=q^{2({\alpha },{\sigma }_p^\chi (\lambda )-{\alpha }_p)},
\end{aligned}$$ which recovers the dot action of the Weyl group on the weight lattice.
If we consider a composition ${t}_i^{\chi '}{t}_j^{\chi ''}$, where $i,j\in I$, $\chi ',\chi ''\in {\mathcal{X}}$, then we will always assume that $$\chi '=r_j(\chi '').$$ For simplicity, we will omit the upper index $\chi '$ if it is uniquely determined by another bicharacter in the expression.
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then ${t}_p {t}_p^\chi (\Lambda )
=\Lambda $. \[le:VTMrel1\]
By Eqs. and , and since $r_p^2(\chi )=\chi $, $$\begin{aligned}
&{t}_p^{r_p(\chi )}{t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )=
{t}_p^\chi (\Lambda )(K_{{\sigma }_p^\chi ({\alpha })}L_{{\sigma }_p^\chi (\beta )})
\frac{\chi ({\alpha },{\alpha }_p)^{{b}-1}}{\chi ({\alpha }_p,\beta )^{{b}-1}}
=\Lambda (K_{\alpha }L_\beta )
\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. This proves the lemma.
\[le:MLmap\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. There exist unique ${\mathbb{K}}$-linear maps $$\begin{gathered}
{\hat{T}}_p={\hat{T}}^\chi _{p,\Lambda },\,
{\hat{T}}^-_p={\hat{T}}^{\chi ,-}_{p,\Lambda }: \,\,
M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^{\chi }(\Lambda ),\\
\intertext{such that for all $u\in U(\chi )$,}
{\hat{T}}_p(uv_{{t}_p^\chi (\Lambda )})=
{T}_p(u)F_p^{{b}-1}v_{\Lambda },\quad
{\hat{T}}^-_p(uv_{{t}_p^\chi (\Lambda )})=
{T}^-_p(u)F_p^{{b}-1}v_{\Lambda }.
\end{gathered}$$ If $V\subset M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$ is a $U(r_p(\chi ))$-submodule, then ${\hat{T}}_p(V), {\hat{T}}^-_p(V)$ are $U(\chi )$-submodules of $M^{\chi }(\Lambda )$.
The uniqueness of the maps ${\hat{T}}_p$, ${\hat{T}}^-_p$ is clear. We prove that ${\hat{T}}_p$ is well-defined. The
| 2,510
| 1,972
| 2,014
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| null | null |
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|
U^+(\chi ).$$ The vector spaces ${\mathcal{F}}^{{\underline{m}}}U^+(\chi )$, where ${\underline{m}}\in N$, are finite-dimensional, since the degrees of their elements are bounded. Moreover, $${\mathcal{F}}^0 U^+(\chi )={\Bbbk }1,\qquad
{\mathcal{F}}^{{\underline{m}}}U^+(\chi ) {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\subset
{\mathcal{F}}^{{\underline{m}}+{\underline{m}}'}U^+(\chi )$$ for all ${\underline{m}},{\underline{m}}'\in N$ by Thm. \[th:EErel\] and since $U^+(\chi )$ is ${\mathbb{Z}}^I$-graded. Thus ${\mathcal{F}}$ defines a filtration of $U^+(\chi )$ by the monoid $N$, and the corresponding graded algebra $$\mathop{\oplus }_{{\underline{m}}\in N} \Big({\mathcal{F}}^{{\underline{m}}}U^+(\chi )
/ \sum _{{\underline{m}}'<{\underline{m}},\ulm
'\not={\underline{m}}} {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\Big)$$ is a skew-polynomial ring in $n$ variables by Thm. \[th:EErel\]. By a standard conclusion we obtain that the first set in the claim of the theorem is indeed a basis of $U^+(\chi )$.
For later purpose we define additional elements in $U(\chi )$.
For each $\nu \in \{n+1,n+2, \dots ,2n\}$ there exists a unique element $i_\nu \in I$ such that $$\begin{aligned}
\ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
{\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_\nu })=n.
\label{eq:ellsss}
\end{aligned}$$ For this $i_\nu $ we get $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
{\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu })
={\alpha }_{i_{\nu -n}}$. \[le:longlongw\]
We proceed by induction on $\nu $. First, $$\ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
{\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}})=n-1$$ by induction hypothesis (if $\nu >n+1$) or by the choice of $i_1,\dots ,i_n$ (for $\nu =n+1$). Thus, by [@a-HeckYam08 Lemma8(iii)] there exists a unique positive root ${\alpha }$ such that $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
{\sigma }_{i_{\nu +1-n}}
| 2,511
| 2,422
| 2,213
| 2,357
| null | null |
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|
vered by the equatorial neutral region. Note that, in our simulations, we neglect possible photoevaporation outflow coming out from the sink. We discuss it later in Sec. \[sec:mass\_loss\_inner\]. In the next section, we investigate the structure of the flow in more detail.
### Analysis of flow structure in case with shadowing effect {#sec:anl_modeling}
In this section, we develop an analytical model and compare it with our result to examine the inflow-outflow structure presented in Fig. \[fig:snap\_Ds\]. The overall structure of our model is schematically depicted in Fig. \[fig:anl\_model\] and can be summarized as follows: in the equatorial neutral region where ionizing photons cannot penetrate, the gas inflows in a Bondi accretion fashion; in the bipolar [H[ii]{} ]{}regions where ionizing photons heat up the gas via photoionization, the outflows are launched due to the thermal and radiation pressure; through their boundaries, the photoevaporating gas is lost from the neutral region and supplied into the [H[ii]{} ]{}regions.
We begin with considering the density profiles of the inflow and outflow. The radial density profile in the equatorial neutral region $n_{\mathrm{inflow}}(r)$ is well approximated by that of the Bondi solution, which we further simplify as $$\begin{aligned}
n_{\mathrm{inflow}}(r) &=
\begin{cases}
{\displaystyle}n_\infty\left(\frac{r}{r_{\mathrm{B}}}\right)^{-3/2}&r < r_{\mathrm{B}}\\[0.4cm]
n_\infty&r > r_{\mathrm{B}}
\end{cases}\label{eq:9} \,.
\end{aligned}$$ This expression slightly over- and underestimates the obtained density profile at $r\ll r_{\mathrm{B}}$ and $r\sim r_{\mathrm{B}}$. As for the bipolar ionized outflows, the density profile $n_{\mathrm{outflow}}(r)$ can be estimated by assuming the pressure equilibrium at the conical boundaries between the neutral and ionized gas, $$\begin{aligned}
n_{\mathrm{outflow}}(r)&=n_{\mathrm{inflow}}(r)\left(\frac{T_{\mathrm{HI}}}{2T_{\mathrm{HII}}}\right)\,,
\label{eq:7}
\end{aligned}$$ where $T_{\mathrm{HI}} \simeq 10^4{\,\mathrm{
| 2,512
| 2,535
| 3,200
| 2,550
| 3,395
| 0.772703
|
github_plus_top10pct_by_avg
|
ix and $\bar{A}=(\bar{a}_{nk})$ the associated matrix defined by (2.9). Then, by combining Lemmas 2.2, 2.3 and 3.1, we have the following result:
Let $1<p<\infty$ and $q=p/(p-1)$. Then we have:
\(a) If $A\in(\ell_{p}(\widehat{F}),\ell_{\infty})$, then$$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q} \tag{3.1}$$ and $$L_{A}\text{ is compact if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q}=0. \tag{3.2}$$
\(b) If $A\in(\ell_{p}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q} \tag{3.3}$$ and $$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q}=0. \tag{3.4}$$
\(c) If $A\in(\ell_{p}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}
\tag{3.5}$$ and $$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}=0,
\tag{3.6}$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
It is obvious that (3.2), (3.4) and (3.6) are respectively obtained from (3.1), (3.3) and (3.5) by using (1.5). Thus, we have to proof (3.1), (3.3) and (3.5).
Let $A\in(\ell_{p}(\widehat{F}),\ell_{\infty})$ or $A\in(\ell_{p}(\widehat
{F}),c_{0}).$ Since $A_{n}\in \{ \ell_{p}(\widehat{F})\}^{\beta}$ for all $n\in\mathbb{N}
$, we have from Lemma 2.2(c)
| 2,513
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| 1,119
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of $\mathcal{L}^* \mathcal{L}$ and then inverting it to form the spectral density: $$S({\boldsymbol{\omega}}) = \frac{\sigma_f^2}{\mathcal{F}[\mathcal{L}^* \mathcal{L}]}.$$ In particular, the minimum norm or (classical) Tikhonov regularization can be recovered by using a white noise prior which is given by the constant spectral density $$\label{eq:STikhonov}
S_{\textrm{Tikhonov}}({\boldsymbol{\omega}}) = \sigma_f^2,$$ where $\sigma_f$ is a scaling parameter. Another interesting case is the Laplacian operator based regularization which corresponds to $$\label{eq:SLaplacian}
S_{\textrm{Laplacian}}({\boldsymbol{\omega}}) = \frac{\sigma_f^2}{\| {\boldsymbol{\omega}} \|^4_2}.$$ It is useful to note that the latter spectral density corresponds to a $l \to \infty$ limit of the Matérn covariance function with $\nu + d/2 = 2$ and the white noise to $l \to 0$ in either the SE or the Matérn covariance functions. The covariance functions corresponding to the above spectral densities would be degenerate, but this does not prevent us from using the spectral densities in the basis function expansion method described in Section \[sec:approx\] as the method only requires the availability of the spectral density.
Basis function expansion {#sec:approx}
------------------------
To overcome the computational hazard described in Section \[GP Xray\], we consider the approximation method proposed in [@SolinSarkka2015], which relies on the following truncated basis function expansion $$\begin{aligned}
\label{eq:BFE}
k({\mathbf x},{\mathbf x}')\approx \sum_{i=1}^{m}S({\sqrt{\lambda}_i})\phi_i({\mathbf x})\phi_i({\mathbf x}'), \end{aligned}$$ where $S$ denotes the spectral density of the covariance function, and $m$ is the truncation number. The basis functions $\phi_i({\mathbf x})$ and eigenvalues ${\lambda}_i$ are obtained from the solution to the Laplace eigenvalue problem on the domain $\Omega$ $$\begin{aligned}
\label{eq:lapl_eig}
\begin{cases}
\hspace{-4mm}
\begin{split}
-\Delta\phi_i({\mathbf x})& ={\lambda}_i\p
| 2,514
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| 2,657
| 1,264
| 0.791906
|
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|
the matrix $\mathbf{M}$. The governing nonlinear PDE, Eq. has been rewritten as a system of nonlinear ODEs, Eq. . The linearized system of ODEs (Eq. with $N_{\pm} \to 0$) can be diagonalized: substituting $\hat{\Delta}_{\pm}$ for $\beta_j$ via Eq. and multiplying by $\mathbf{M}^{-1}$ on the left gives $$\label{Leigenmode}
\begin{pmatrix}\dot{\beta}_1\\\dot{\beta}_2\end{pmatrix} = \mathbf{M}^{-1}\mathcal{D}\mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix},$$ where the matrix $\mathbf{M}^{-1}\mathcal{D}\mathbf{M}$ is diagonal with entries $i\omega_j$.
The nonlinear interactions between the eigenmodes can now be investigated. Applying the steps that led to Eq. to the full, nonlinear Eq. yields $$\label{NLeigenmode}
\begin{pmatrix}\dot{\beta}_1\\\dot{\beta}_2\end{pmatrix} = \mathbf{M}^{-1}\mathcal{D}\mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix} + \mathbf{M}^{-1}\begin{pmatrix}N_+\\N_-\end{pmatrix},$$ where, again, $N_{\pm}$ are the nonlinearities in Eq. . Using Eq. and the forms for $\phi_j$ given by Eq. , $N_{\pm}$ can be written in terms of products of the form $\beta_i \beta_j$ with $i,j$ each taking values $1,2$. Equation then becomes $$\label{unstablemode}
\begin{split}
\dot{\beta}_1(k) = i\omega_1(k)\beta_1(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\bigg[ &C_1(k,k') \beta_1(k')\beta_1(k'') + C_2(k,k') \beta_1(k')\beta_2(k'')\\
+ &C_3(k,k') \beta_1(k'')\beta_2(k') + C_4(k,k')\beta_2(k')\beta_2(k'')\bigg],
\end{split}$$ $$\label{stablemode}
\begin{split}
\dot{\beta}_2(k) = i\omega_2(k)\beta_2(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\bigg[ &D_1(k,k')\beta_1(k')\beta_1(k'') + D_2(k,k')\beta_1(k')\beta_2(k'')\\
+ &D_3(k,k')\beta_1(k'')\beta_2(k') + D_4(k,k')\beta_2(k')\beta_2(k'')\bigg].
\end{split}$$ The coefficients $C_j,D_j$ arise from writing the nonlinearities $N_{\pm}$ in the basis of the linear eigenmodes, so their functional forms include information about both the linear properties of the system and the nonlinearities $N_{\pm}$. The exact expressions for
| 2,515
| 1,698
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| 2,396
| null | null |
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|
%
\fl A'_1&=&A_1^2 + A_1^3 + A{A_2^2} + 2A^2A_2A_3 + A{A_3^2} +
2AA_1{A_3^2} + 2AA_2A_3B_1 + 2AA_2A_3B_2 \nonumber\\
\fl &+& 4A^2A_1B_2 + 4A^2B_1B_2 +
4AA_1B_1B_2 + 2A^2{B_2^2}+ 2AA_1{B_2^2} +
{A_2^2}C + A{A_3^2}C\>, \label{eq:b2jednacinaA1}\\
%
\fl A'_2&=&AA_1A_2 + A_2^3 + A^2A_1A_3 + AA_2A_3^2 +
A^2A_3A_4 + AA_2A_4^2 + 2AA_2A_4B+ AA_3A_4C\nonumber\\
\fl &+& A^2A_3B_1+
AA_1A_3B_1 + 2AA_2BB_1 +
AA_2B_1^2 + A^2A_3B_2 +4AA_2BB_2 +A^2A_3C
\nonumber\\
\fl &+& 2AA_2B_1B_2 + 3AA_2B_2^2 + AA_2C +
A_1A_2C + AA_1A_3B_2 +
2AA_2A_4B_2 \, ,\\
\fl A'_3&=&A^2A_1A_2 + AA_1A_3 + AA_1^2\, A_3 +
AA_2^2A_3 + A^2A_2A_4 + 2A_3^3B+
4AA_3BB_2+ AA_3C^2
\nonumber\\ \fl&+& 2AA_3A_4B+
A^2A_2B_1 + AA_1A_2B_1 + 2AA_3BB_1 +
2A_3A_4BB_1+AA_2A_4C \nonumber\\
\fl&+& A^2A_2B_2+4A_3A_4BB_2+A^2A_2C + AA_3C +
AA_1A_3C + AA_1A_2B_2 \, ,\\
\fl A'_4&=&2A^2A_2A_3 + 2AA_2^2A_4 + 2AA_2^2B + 2AA_3^2\,B +
2A_4^3B + 6A_4^2B^2 + 2A_3^2BB_1
\nonumber\\ \fl&+& 2AA_2^2B_2 +
4A_3^2BB_2 + 2AA_2A_3C + 2A^2A_4C + A^2C^2 + AA_4C^2\, , \label{eq:b2jednacinaa4}
\\
\fl B'_1&=&A^2A_1^2 + AA_2^2A_4 + AA_2^2B + AA_3^2B + A_3^2A_4B +
2A^2A_1B_1 + AA_1^2B_1+ 8BB_2^3\nonumber\\ \fl&+& 6B^2B_1^2+
2BB_1^3 + 2AA_2^2B_2 + 8B^2B_1B_2 +
4BB_1^2B_2+ 8B^2B_2^2 + 8BB_1B_2^2 \, ,
\\
\fl B'_2&=&A^2A_2A_3 + AA_1A_2A_3 + AA_2^2B + AA_3^2B + A_3^2 A_4B
+ AA_2^2B_1+10BB_1B_2^2 + 6BB_2^3\nonumber\\ \fl&+& 2A^2A_1B_2+
AA_1^2B_2 + AA_2^2B_2 + 12B^2B_1B_2 + 6BB_1^2B_2 +
10B^2B_2^2 \, .
\label{eq:b2jednacinaB2}\end{aligned}$$ One can check, by inserting $A_1=A_2=A_3=B_1=B_2=0$ and $A_4=D$, into equation (\[eq:b2jednacinaa4\]) for the function $A_4$, that RG equation (\[eq:Db2\]) for the function $D$ in the case of the ASAWs model is recovered. This is not surprising, since it follows from the definitions of $A_4$ and $D$, and it is certainly also correct for the $b=3$ fractal equations. However, here we do not quote the $b=3$ RG equations because they are extremely intricate. For instance, equa
| 2,516
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|
g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & h_{--}\ r^{1-\mu l}+f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric mu Neg}%$$ where $f_{\mu\nu}$ and $h_{\mu\nu}$ depend only on $x^{+}$ and $x^{-}$ and not on $r$. We use the convention that the $f$-terms are the standard deviations from AdS already encountered in (\[Standard-Asympt\]), while the $h$-terms represent the relaxed terms that need to be included in order to accommodate the solutions of the topologically massive theory with slower fall-off. We see that only the negative chirality $h$-terms $h_{r-}$ and $h_{--}$ are present, hence the terminology.
*Positive chirality.* The boundary conditions are in that case $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+\cdot\cdot\cdot\\
\Delta g_{r+} & = & h_{r+}\ r^{-2+\mu l}+f_{r+}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{r-} & = & f_{r-}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{++} & = & h_{++}\ r^{1+\mu l}+f_{++}+\cdot\cdot\cdot\\
\Delta g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric mu Pos}%$$ with only the positive chirality $h$-terms $h_{r+}$ and $h_{++}$.
Although the known solutions [@DS; @OST] are of a given chirality and hence completely covered by the above boundary conditions, one might try to be more general and include both chiralities simultaneously. This cannot be done, however, in a manner that is compatible with the other consistency requirements as it will be explained below.
Asymptotic symmetry
-------------------
One easily verifies that both sets of asymptotic conditions are invariant under diffeomorphisms that behave at infinity as $$\begin{aligned}
\eta^{+} & =T^{+}+\frac{l^{2}}{2r^{2}}\partial_{-}^{2}T^{-}+\cdot\cdot
\cdot\nonumber\\
\eta^{-} & =T^{-}+\frac{l^{2}}{2r^{2}}\partial_{+}^{2}T^{+}+\cdot\cdot
\cdot\label{Asympt KV}\\
\eta^{r} & =-\frac{r}{2}\left( \partial_{+}T^{+}+\partial_{-}T^{-}\right)
+\cdot\cdot\cdot\nonumber\end{aligned}$$ where $T^{\pm}=T^{\pm}(x^{\pm})$. The $\
| 2,517
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| 2,812
| null | null |
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|
)=g''_1(q_1)$ and $g''_1(p_1)=g''_1(q_0)$ and then $Y''_1$ consists of a connected curve with 2 nodes and 2 irreducible components.
Both of these are étale double covers of $Y_2$.
As in (\[pf.of.glue.thm.asp\]), the next lemma will be used to reduce quasi projective gluing to the affine case.
\[affine.red.lem\] Let $X$ be an $A$-scheme, $Z\subset X$ a closed subscheme and $g:Z\to V$ a finite surjection.
Let $P\subset V$ be a finite subset and assume that there are open affine subsets $P\subset V_1\subset V$ and $g^{-1}(P)\subset X_1\subset X$.
Then there are open affine subsets $P\subset V_P\subset V_1$ and $g^{-1}(P)\subset X_P\subset X_1$ such that $g$ restricts to a finite morphism $g: Z\cap X_P\to V_P$.
Proof. There is an affine subset $g^{-1}(P)\subset X_2\subset X_1$ such that $g^{-1}(V\setminus V_1)$ is disjoint from $X_2$. Thus $g$ maps $Z\cap X_2$ to $V_1$. The problem is that $(Z\cap X_2)\to V_1$ is only quasi finite in general. The set $Z\setminus X_2$ is closed in $X$ and so $g( Z\setminus X_2)$ is closed in $V$. Since $V_1$ is affine, there is a function $f_P$ on $V_1$ which vanishes on $g( Z\setminus X_2)\cap V_1$ but does not vanish on $P$. Then $f_P\circ g$ is a function on $g^{-1}(V_1)$ which vanishes on $(Z\setminus X_2)\cap g^{-1}(V_1)$ but does not vanish at any point of $g^{-1}(P)$. Since $Z\cap X_1$ is affine, $f_P\circ g$ can be extended to a regular function $F_p$ on $X_2$.
Set $V_P:=V_1\setminus (f_P=0)$ and $X_P:=X_2\setminus (F_P=0)$. The restriction $(Z\cap X_P)\to V_P$ is finite since, by construction, $X_P\cap Z$ is the preimage of $V_P$.
We say that an algebraic space $X$ has the [*Chevalley-Kleiman property*]{} if $X$ is separated and every finite subscheme is contained in an open affine subscheme. In particular, $X$ is necessarily a scheme.
These methods give the following interesting corollary.
\[CK.prop.desc\] Let $\pi:X\to Y$ be a finite and surjective morphism of separated, excellent algebraic spaces. Then $X$ has the Chevalley-Kleiman property iff $Y$ has.
Proof.
| 2,518
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github_plus_top10pct_by_avg
|
ts in $B\otimes_AR$.
We also let $\pi^2$ be zero in $(\tilde{x}_i^j)', (\tilde{c}_i')_{\textit{$L_i$ of type $I^o$}},$ $(\tilde{f}_i', \tilde{c}_i')_{\textit{$L_i$ of type $I^e$ or free of type $I$ with $i$ odd}}$. Note that $(\tilde{x}_i^j)'$ is a diagonal entry of a formal matrix $\tilde{a}_i'$. Then these entries are elements in $R$.
Let $(f_{i,j}', a_i' \cdots f_i')$ be the reduction of $(\tilde{f}_{i,j}', \tilde{a}_i' \cdots \tilde{f}_i')$ as explained above, i.e. by letting $\pi^2$ be zero in the entries of formal matrices as described above. Then $(f_{i,j}', a_i' \cdots f_i')$ is an element of $\underline{H}(R)$ and the composition $(f_{i,j}, a_i \cdots f_i)\circ (m_{i,j}, s_i\cdots w_i)$ is $(f_{i,j}', a_i' \cdots f_i')$.
We can also write $(f_{i,j}', a_i' \cdots f_i')$ explicitly in terms of $(f_{i,j}, a_i \cdots f_i)$ and $(m_{i,j}, s_i\cdots w_i)$ like the product of $(m_{i,j}, s_i\cdots w_i)$ and $(m_{i,j}', s_i'\cdots w_i')$ explained in Section \[m\]. However, this is complicated and we do not use it in this generality. On the other hand, we explicitly calculate $(f_{i,j}, a_i \cdots f_i)\circ (m_{i,j}, s_i\cdots w_i)$ when $(f_{i,j}, a_i \cdots f_i)$ is the given hermitian form $h$ and $(m_{i,j}, s_i\cdots w_i)$ satisfies certain conditions on each block. This explicit calculation will be done in Appendix \[App:AppendixA\].
\[t36\] Let $\rho$ be the morphism $\underline{M}^{\ast} \rightarrow \underline{H}$ defined by $\rho(m)=h \circ m$, which is induced by the action morphism of Theorem \[t34\]. Then $\rho$ is smooth of relative dimension dim $\mathrm{U}(V, h)$.
The theorem follows from Theorem 5.5 in [@GY] and the following lemma.
\[l37\] The morphism $\rho \otimes \kappa : \underline{M}^{\ast}\otimes \kappa \rightarrow \underline{H}\otimes \kappa$ is smooth of relative dimension $\mathrm{dim~} \mathrm{U}(V, h)$.
The proof is based on Lemma 5.5.2 in [@GY] and is parallel to Lemma 3.7 of [@C2]. It is enough to check the statement over the algebraic closure $\bar{\kappa}$ of $\kappa$. By [@H
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ther with the values of $C^*$, corresponding to 2D chain, which can exist only in extended state. RG fixed point value $A_4^*$ is equal to $0.1165$ and $0.0779$, for $b=2$ and 3 respectively, and they coincide with the values of $D^*$ for $v<v_c(u<u_\theta)$ case in the ASAWs model (see table \[tab:avoiding\]).
- For $w=w_c(u,t)$ one obtains the symmetrical fixed point $$\label{fp2}
(A_E,B_E,C^*,A_EC^*,A_EC^*,A_EC^*,A_EC^*,B_EC^*,B_EC^*)\>,$$ which appears to be a tricritical fixed point. As one approaches this fixed point, the average number of contacts ${\langle M^{(r)}\rangle}$ becomes infinitely large (although large parts of $P_2$ and $P_3$ are not in contact), and it turns out that it scales with the average length ${\langle {N_3}^{(r)}\rangle}$ of the 3D chain, according to the power law $$\label{ficsaw}
{\langle M^{(r)}\rangle}\sim \langle
{N_3}^{(r)}\rangle^{\varphi}\>.$$ To calculate the contact critical exponent $\varphi$, within the CSAWs model, we find the average number of contacts between two chains at the $r$th stage of fractal construction, through the formula $$\begin{aligned}
\langle M^{(r)}\rangle&=& {\sum_{N_2,N_3,L,M,K}M\left(\sum_{i=1}^4{\mathcal A}_i^{(r)}+{\sum_{j=1}^2{\mathcal B}_i^{(r)}}\right) x_2^{N_2}x_3^{N_3}u^L w^M t^K\over \sum_{i=1}^4 A_i^{(r)}+\sum_{j=1}^2 B_j^{(r)}}\nonumber\\
&=& {w\over \sum_{i=1}^4 A_i^{(r)}+\sum_{j=1}^2 B_j^{(r)}}
\left(\sum_{i=1}^{4}\frac{\partial A_i^{(r)}}{\partial w}+
\sum_{j=1}^{2}\frac{\partial B_j^{(r)}}{\partial w}\right)\nonumber\\
&=&
{w\over \sum_{i=1}^6 X_i^{(r)}}
\sum_{i=1}^{6}\frac{\partial X_i^{(r)}}{\partial w}
\>,\end{aligned}$$ where $X_i=A_i$ ($i=1,2,3,4$), $X_5=B_1$, and $X_6=B_2$. Since $$\label{nov1}
\frac{\partial X_i^{(r+1)}}{\partial w}=\sum_{j=1}^6 \frac{\partial X_i^{(r+1)}}{\partial X_j^{(r)}}\frac{\partial X_j^{(r)}}{\partial w}\>,\quad i=1,\ldots,6\>,$$ one expects, for large $r$, that $\frac{\partial X_i^{(r)}}{\partial w}$ behaves as $\lambda_\varphi^r$, where $\lambd
| 2,520
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| 2,301
| 3,055
| 0.775141
|
github_plus_top10pct_by_avg
|
0.026 (0.054) 0.001 (0.002)
Long-term Unemployment -0.032 (0.035) -0.022 (0.023) -0.116 (0.077) -0.015 (0.023)
**Intentional Homicides** **-0.383**[\*\*](#t003fn003){ref-type="table-fn"} **(0.121)** **-0.025 (0.088)** \- \-
**Property Crimes** \- \- **-0.156**[\*](#t003fn002){ref-type="table-fn"} **(0.073)** **-0.001**[\*\*](#t003fn003){ref-type="table-fn"} **(0.0004)**
**Institutional Trust** \- **0.979**[\*\*\*](#t003fn004){ref-type="table-fn"} **(0.119)** \- **0.905**[\*\*\*](#t003fn004){ref-type="table-fn"} **(0.098)**
**Intentional Homicides Indirect Effect** \- **-0.354**[\*\*](#t003fn003){ref-type="table-fn"} **(0.114)** \- \-
**% of Total Effect** \- **93%** \- \-
**Property Crimes Indirect Effect** \- \- \- **-0.0002 (0.0004)**
**% of Total Effect** \-
| 2,521
| 4,935
| 2,022
| 1,360
| 1,019
| 0.795543
|
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|
mes S\times I)}$) \[trpr15\] [L\_[e,-]{}g\_[e]{}]{}\_[W\^2(G\_eSI)]{} =&\_[W\^2(G\_eSI)]{}= \_[L\^2(G\_eSI)]{}\
=& [g\_[e]{}]{}\_[T\^2(\_[e,-]{})]{}, since (cf. Remark \[changevar\]) $${\left\Vert \Psi\right\Vert}^2_{L^2(G_e\times S\times I)}
={}&\int_{\Gamma_{e,-}} \int_0^{\tau_{e,-}(x,\omega)} \big(e^{-\lambda s}g_{e}(y,\omega,E)\big)^2 |\omega\cdot\nu(y)| ds d\sigma(y)d\omega dE \\
={}&\frac{1}{2\lambda}\int_{\Gamma_{e,-}} g_{e}(y,\omega,E)^2 |\omega\cdot\nu(y)| d\sigma(y)d\omega dE \\
={}&\frac{1}{2\lambda}{\left\Vert g_e\right\Vert}_{T^2(\Gamma_{e,-})}^2,$$ where $\tau_{e,-}$ is $\tau_-$ for the domain $G_e\times S\times I$. We omit further details.
\[wla\] A. By (\[exle2\]) one can show that $$L_{e,-}(g_{e,-})_{|\Gamma_{e,+}}=0.$$
B. The lift $L_{e,+}:T^2(\Gamma_{e,+})\to W^2(G_e\times S\times I)$ is given by $$(L_{e,+})(x,\omega,E)=\begin{cases}e^{-\lambda t_e(x,-\omega)}g_{e}(x+t_e(x,-\omega)\omega,\omega,E),\ &{\rm when}\ t_e(x,-\omega)\ {\rm is\ finite},\\ 0,\ &{\rm otherwise}.\end{cases}$$ We find that $$L_{e,+}(g_{e,+})_{|\Gamma_{e,-}}
=0.$$
C. Let $g_e\in T^2(\Gamma_e)$ and let $g_{e,\pm}:={g_e}_{|\Gamma_{e,\pm}}$. Then for $$\tilde{\Psi}:=L_{e,-}(g_{e,-})+L_{e,+}(g_{e,+})\in W^2(G_e\times S\times I),$$ we find that $\tilde{\Psi}\in \tilde{W}^2(G_e\times S\times I)$ and $$\tilde\Psi_{|\Gamma_{e,\pm}}=g_{e,\pm}.$$ (See Remark \[wl\], Part E.)
As a corollary we show the following extension result. In the proof we explicitly construct the [*extension of*]{} $\psi$ (cf. [@dautraylionsv6 p. 415, proof of Lemma 2]).
\[exle\] Suppose that $G\subset{\mathbb{R}}^3$ is as above and that it is convex. Then for any $\psi\in \tilde W^2(G\times S\times I)$ there exists an extension ${{{\mathcal{}}}E}\psi\in W^2({\mathbb{R}}^3\times S\times I)$ of $\psi$ that is, ${{{\mathcal{}}}E}\psi_{|G\times S\times I}=\psi$. In addition, the linear operator ${{{\mathcal{}}}E}:\tilde W^2(G\times S\times I)\to W^2({\mathbb{R}}^3\times S\times I)$ is bounded.
Suppose that $\psi\in\tilde W^2(G\times S\times I)$. Denote $g:=\
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|
t$, then clearly $g(c)\preceq t_{\leftarrow}$. Else, $t\preceq x$ for *each* $t\in C_{g}$ shows that $t_{\leftarrow}\preceq c$ because $t_{\leftarrow}$ is the smallest of all the upper bounds $c$ of $C_{g}$. Hence $t_{\leftarrow}\in C_{g}$.
Property (ST3) for $C_{g}$ follows from a small yet significant modification of the above arguments in which the immediate successors $g(t)$ of $t\in C_{g}$ formally replaces the supremum $t_{\leftarrow}$ of $C_{g}$. Thus given a $c\in C$, if there is *some* $t\in C_{g}$ for which $g(c)\preceq t$ then $g(c)\prec g(t)$; this combined with $(c=t)\Rightarrow(g(c)=g(t))$ yields $g(c)\preceq g(t)$. On the other hand, $t\prec c$ for *every* $t\in C_{g}$ requires $g(t)\preceq c$ as otherwise $(t\prec c)\Rightarrow(c\prec g(t))$ would, from the resulting consequence $t\prec c\prec g(t)$, contradict the assumed hypothesis that $g(t)$ is the immediate successor of $t$. Hence, $C_{g}$ is a $g$-tower in $X$.
To complete the proof that $g(c)\in C_{\textrm{T}}$, and thereby the argument that $C_{\textrm{T}}$ is a tower, we first note that as $_{\rightarrow}T$ is the smallest tower and $C_{g}$ is built from it, $C_{g}=\,_{\rightarrow}T$ must infact be $_{\rightarrow}T$ itself. From Eq. (\[Eqn: chain\_g\]) therefore, for every $t\in\,_{\rightarrow}T$ either $t\preceq g(c)$ or $g(c)\preceq t$, so that $g(c)\in C_{\textrm{T}}$ whenever $c\in C_{\textrm{T}}$. This concludes the proof that $C_{\textrm{T}}$ is actually the tower $_{\rightarrow}T$ in $X$.
From (ST2), the implication of the chain $C_{\textrm{T}}$ $$C_{\textrm{T}}=\,_{\rightarrow}T=C_{g}\label{Eqn: ChainedTower}$$ being the minimal tower $_{\rightarrow}T$ is that the supremum $t_{\leftarrow}$ of the totally ordered $_{\rightarrow}T$ *in its own tower* (as distinct from in the tower $X$: recall that $_{\rightarrow}T$ is a subset of $X$) must be contained in itself, that is $$\sup_{C_{\textrm{T}}}(C_{\textrm{T}})=t_{\leftarrow}\in\,_{\rightarrow}T\subseteq X.\label{Eqn: sup chain}$$ This however leads to the contradiction from (ST3
| 2,523
| 1,901
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| 2,527
| 2,014
| 0.783512
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github_plus_top10pct_by_avg
|
im$12%. By considering the forms of the effective couplings of the Higgses to the bottom quark, we can determine if this enhancement translates to a nonnegligible correction. The effective couplings are given by [@Carena:1999py] \_b\^h &=& (1 - )\
\_b\^H &=& (1 + )\
\_b\^A &=& (1 - ) , where $g^{h,H,A}_b$ are the tree level couplings. In the decoupling limit, $\tana {\ensuremath{\rightarrow}}-\cotb$ and we obtain \_b\^h &=& g\^h\_b\
\_b\^H &=& (1 - )\
\_b\^A &=& (1 - ) . We therefore only need to determine the extent to which $\Delta_1$ affects the size of the factor $(1+\Delta_b)^{-1}$. From \[Eq:deltas\], the factor may be written as = \_[12]{} . Let us define $\delta_2\equiv (1+\Delta_2\tanb)^{-1}$ and $\delta_\Phi$ to be the relative change between ignoring $\Delta_1$ and including it, \_ . By setting $\Delta_1=0.12$, $\delta_\Phi$ can be plotted as a function of $\Delta_2\tanb$ as shown in \[fig:deltas\]. For positive values of $\Delta_2\tanb$, the relative change is never more than 6%. Unless $\Delta_2\tanb$ is $\mathcal{O}(1)$, the relative correction to the heavy Higgs couplings is only a few percent. The effect of including $\Delta_1$ can be more drastic if $\Delta_2\tanb$ is negative. As $\Delta_2\tanb$ approaches $-\mathcal{O}(1)$, the relative change increases quickly to the nearly the same magnitude. Such large, negative values of $\Delta_2\tanb$ may be a more extreme case however. For most values of $\Delta_2\tanb$ obtained in the parameter scan $(<40\%)$, the relative change is again only a few percent. Thus unless the magnitude of the $\tanb$-enhanced corrections to the bottom quark mass are $\mathcal{O}(1)$ it is safe to neglect the $\Delta_1$ correction to the couplings of the bottom quark with the heavy Higgses.[^5] Note that in calculating the $B_1^{\widetilde{g}}$ contribution to $\Delta_1$ we take $Q=M_Z$. If the scale is chosen to be higher, then $B_1^{\widetilde{g}}$ would be smaller and the relative change, $\delta_\Phi$, would be more suppressed.
![The plot shows the relative size of
| 2,524
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��j-bd\])]{} with $j\ge2$, we first note that, by applying [(\[eq:IR-xbd\])]{} and [(\[eq:psi-bd\])]{} to the definition [(\[eq:Pj-def\])]{} of $P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)$, we have $$\begin{gathered}
P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)\leq\sum_{\substack{v_2,\dots,v_j\\ v'_1,\dots,
v'_{j-1}}}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbv_2-y{|\!|\!|}}^q{\vbv'_1
-v_2{|\!|\!|}}^q}\prod_{i=2}^{j-1}\frac{O(\theta_0^2)}{{\vbv'_i-v_i{|\!|\!|}}^{2q}{{|\!|\!|}v_{i+1}-v'_{i-1}{|\!|\!|}}^q{\vbv'_i-v_{i+1}{|\!|\!|}}^q}{\nonumber}\\
\times\frac{O(\theta_0^2)}{{\vbx-v_j{|\!|\!|}}^{2q}{\vbx-v'_{j-1}{|\!|\!|}}^q}.{\label{eq:Pj-bd}}\end{gathered}$$ By definition, the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(j)}(y,x)$ is obtained by “embedding $u$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ (not ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$) and then summing over all these $2j-1$ choices. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ is replaced by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ is bounded, similarly to [(\[eq:piNgeq2-bd\])]{}, by $$\begin{aligned}
&\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbv_2
-u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j-1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx
-v_2{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{{|\!|\!|}x-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^{2q}{\vbu
-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}.\end{aligned}$$ The other $2j-2$ contributions can be estimated in a similar way, with the same form of the bound. This completes the proof of [(\[eq:P’j-bd\])]{}.
By [(\[eq:psipsi-bd\])]{}, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(j)}(y,x)$ is also obtained by “embedding $u$ and $v$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ and one of the $j$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])
| 2,525
| 1,692
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| 2,467
| null | null |
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|
ac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}.
\end{aligned}$$
If we assume that $\bx_0 = \bw_0$, then there exists a coupling between $\bx_t$ and $\bw_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{\bx_{k\delta} - \bw_{k\delta}}_2} \leq \hat{\epsilon}.
\end{aligned}$$
Alternatively, if we assume that $k \geq \frac{3\aq\Rq^2}{ \delta} \cdot \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^w_{k\delta}} \leq 2\hat{\epsilon},
\end{aligned}$$ where $p^w_t := \Law(\bw_t)$.
Let $f$ be defined as in Lemma \[l:fproperties\] with parameter $\epsilon$. $$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \bw_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\E{f(\bx_{i\delta} - \bw_{i\delta})} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{16\lrp{L+\LN^2}}{\lambda}\exp\lrp{\frac{7\aq\Rq^2}{3}} \cdot \epsilon
\numberthis \label{e:t:asdkjas:1}\\
=& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \hat{\epsilon}
\end{aligned}$$ where the first inequality is by item 4 of Lemma \[l:fproperties\], the second inequality is by Corollary \[c:main\_nongaussian:1\] (notice that $\delta$ satisfies the requirement on $T$ in Theorem \[t:main\_gaussian\], for the given $\epsilon$). The third inequality uses the fact that $1\leq L/m \leq \frac{\lrp{L+\LN^2}}{\lambda}$.
The first claim follows from substituting $\bx_0 = \bw_0$ into , so that the first term is $0$, and using the definition of $\epsilon$, so that the second term is $0$.
For the second claim, let $\bx_0 \sim p^*$, the invariant distr
| 2,526
| 3,269
| 1,923
| 2,261
| null | null |
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|
,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(
\prod_{i\ne l}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}}
\bigg)\sum_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\}$}}}.\end{aligned}$$ Note that the last sum of the indicators is the only difference from [(\[eq:Theta’-2ndindbd3\])]{}.
When $j=1$, the second line of [(\[eq:Theta”-2ndindbd1\])]{} equals $$\begin{aligned}
{\label{eq:Theta''-2ndindbd1:j=1}}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ As described in [(\[eq:Theta’-2ndindbd3:j=1\])]{}–[(\[eq:Theta’-2ndindbd3:j=1bd\])]{}, we can bound [(\[eq:Theta”-2ndindbd1:j=1\])]{} without ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ by a chain of bubbles $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. If ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}=1$, then, by the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, one of the bubbles has an extra vertex $v'$ that is further connected to $v$ with another chain of bu
| 2,527
| 1,333
| 2,555
| 2,568
| null | null |
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|
a _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
\end{aligned}$$ form vector space bases of $U ^+(\chi )$, and the sets $$\begin{aligned}
\big\{ F_{\beta _{\tau (1)}}^{m_{\tau (1)}} F_{\beta _{\tau (2)}}^{m_{\tau
(2)}}\cdots F_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
\big\{ {\bar{F}}_{\beta _{\tau (1)}}^{m_{\tau (1)}}
{\bar{F}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots
{\bar{F}}_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
\end{aligned}$$ form vector space bases of $U ^-(\chi )$.
It suffices to prove that the first set is a basis of $U^+(\chi )$. Indeed, the proof for the second set can be obtained by using the maps ${T}^-_i$, where $i\in I$, instead of ${T}_i$. The second part of the claim follows from the first part by applying the algebra antiautomorphism ${\Omega }$ and using Eq. .
For any ${\underline{m}}=(m_1,\ldots ,m_n)\in {\mathbb{N}}_0^n$ let $|{\underline{m}}|=\sum _{\mu =1}^n
m_\mu |\beta _\mu |$, where $|{\alpha }|=\sum _{j\in I}a_j$ for all ${\alpha }=\sum _{j\in I}a_j{\alpha }_j \in {\mathbb{N}}_0^I$. Let $N$ be the (additive) monoid ${\mathbb{N}}_0^n$ equipped with the following ordering: $${\underline{m}}'<{\underline{m}}\quad \Leftrightarrow \quad |{\underline{m}}'|<|{\underline{m}}| \quad
\text{or} \quad |{\underline{m}}'|=|{\underline{m}}|,\,{\underline{m}}'<_{\mathrm{lex}}{\underline{m}},$$ where $<_{\mathrm{lex}}$ means lexicographical ordering. We use the convention ${\underline{m}}<_{\mathrm{lex}}{\underline{m}}$. The ordering $<$ is a total ordering. For all ${\underline{m}}\in {\mathbb{N}}_0^n$ define $${\mathcal{F}}^{{\underline{m}}}U^+(\chi )=\mathop{\oplus }_{{\underline{m}}'\in N,{\underline{m}}'<\ulm}
{\Bbbk }E_{\beta _1}^{m'_1}
E_{\beta _2}^{m'_2}\cdots E_{\beta _n}^{m'_n} \subset
| 2,528
| 4,152
| 1,994
| 2,184
| null | null |
github_plus_top10pct_by_avg
|
$Here also we will work with weighted Sobolev norms. And a similar hypothesis is supposed to hold for the adjoint $P_{t}^{\ast ,n}$ (see Assumption [A2A\*2]{} for a precise statement).
Finally we assume the following regularity property: for every $t\in (0,1]$, $P_{t}^{n}(x,dy)=p_{t}^{n}(x,y)dy$ with $p_{t}^{n}\in C^{\infty }({\mathbb{R}%
}^{d}\times {\mathbb{R}}^{d})$ and for every $\kappa \geq 0$, $t\in (0,1]$, $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}^{n}(x,y)\right\vert \leq \frac{C}{(\lambda _{n}t)^{\theta
_{0}(\left\vert \alpha \right\vert +\left\vert \beta \right\vert +\theta
_{1})}}\times \frac{(1+\left\vert x\right\vert ^{2})^{\pi (\kappa )}}{%
(1+\left\vert x-y\right\vert ^{2})^{\kappa }}. \label{i3}$$Here, $\alpha ,\beta $ are multi-indexes and $\partial _{x}^{\alpha
},\partial _{y}^{\beta }$ are the corresponding differential operators. Moreover, $\pi (\kappa )$, $\theta _{0}$ and $\theta _{1}$ are suitable parameters and $\lambda _{n}\rightarrow 0$ as $n\rightarrow \infty $ (we refer to Assumption \[H3\]).
By (\[i1\])–(\[i3\]), the rate of convergence is controlled by $%
\varepsilon _{n}\rightarrow 0$ and the blow up of $p_{t}^{n}$ is controlled by $\lambda _{n}^{-\theta _{0}}\rightarrow \infty $. So the regularity property may be lost as $n\rightarrow \infty $. However, if there is a good equilibrium between $\varepsilon _{n}\rightarrow 0$ and $\lambda
_{n}^{-\theta _{0}}\rightarrow \infty $ and $\Lambda _{n}\rightarrow \infty $ then the regularity is saved: we ask that for some $\delta >0$ $$\overline{\lim_{n}}\frac{\varepsilon _{n}\Lambda _{n}}{\lambda _{n}^{\theta
_{0}(a+b+\delta )}}<\infty , \label{i4}$$the parameters $a$, $b$ and $\theta _{0}$ being given in (\[i1\]), ([i2]{}) and (\[i3\]) respectively. Then $P_{t}(x,dy)=p_{t}(x,y)dy$ with $%
p_{t}\in C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d})$ and the following upper bound holds: for every $\varepsilon >0$ and $\kappa \in \N$ one may find some constants $C,\pi(\kappa)>0$ such that for every $(x,y)\in
\R^d\
| 2,529
| 1,028
| 1,457
| 2,607
| null | null |
github_plus_top10pct_by_avg
|
ered.
Moore and Russell [@MR01] analyzed both the discrete and the continuous quantum walk on a hypercube. Kendon and Tregenna [@KT03] performed a numerical analysis of the effect of decoherence in the discrete case. In this article, we extend the continuous case with the model of decoherence described above. In particular, we show that up to a certain rate of decoherence, both linear instantaneous mixing times and linear instantaneous hitting times still occur. Beyond the threshold, however, the walk behaves like the classical walk on the hypercube, exhibiting $\Theta(n \log n)$ mixing times. As the rate of decoherence grows, mixing is retarded by the quantum Zeno effect.
Results
-------
Consider the continuous quantum walk on the $n$-dimensional hypercube with energy $k$ and decoherence rate $p$, starting from the initial wave function $\Psi_0 = \vert 0 \rangle ^{\otimes n}$, corresponding to the corner with Hamming weight zero. We prove the following theorems about this walk.
When $p < 4k$, the walk has instantaneous mixing times at $$t_{mix} = \frac {n (2\pi c - \arccos(p^2/8k^2-1))}{\sqrt{16k^2 - p^2}}$$ for all $c \in \mathbb{Z}$, $c > 0$. At these times, the total variation distance between the walk distribution and the uniform distribution is zero.
This result is an extension of the results in [@MR01], and an improvement over the classical random walk mixing time of $\Theta(n \log n)$. Note that the mixing times decay with $p$ and disappear altogether when $p \geq 4k$. Further, for large $p$, we will see that the walk is retarded by the quantum Zeno effect.
When $p < 4k$, the walk has approximate instantaneous hitting times to the opposite corner $(1, \dots , 1)$ at times $$t_{hit} = \frac{2 \pi n (2c + 1)}{\sqrt{16k^2 - p^2}}$$ for all $c \in \mathbb{Z}$, $c \geq 0$. However, the probability of measuring an exact hit decays exponentially in $c$; the probability is $$P_{hit} = \left[\frac{1}{2} + \frac{1}{2}e^{-\frac{p \pi (2c + 1)}
{\sqrt{16k^2 - p^2}}}\right]^n\enspace.$$ In particular, w
| 2,530
| 4,686
| 2,809
| 1,957
| 2,788
| 0.776938
|
github_plus_top10pct_by_avg
|
,\mathbf{C},\mathbf{D}\in\mathbb{C}^3$ suitably chosen so that $\mathbf{A}\cdot[1,1,1] = 0$, $\mathbf{B}\cdot[-1,1,1]
= 0$, $\mathbf{C}\cdot[1,-1,1] = 0$ and $\mathbf{D}\cdot[1,1,-1] = 0$, which ensures that incompressibility condition is satisfied, and that $\E({\mathbf{u}}_1) = 1$; in this case, $|{\mathbf{k}}|^2 =3$, $\forall\,{\mathbf{k}}\in{\mathcal{W}}_3$, and the optimal asymptotic value of $\R$ is $$\label{eq:R0_kvec_3D_k3}
\R({\widetilde{\mathbf{u}}}) \approx - 24\pi^2\nu\E_0.$$ \[c3\]
The three constructions of the extremal field ${\mathbf{u}}_1$ given in , and are all defined up to arbitrary shifts in all three directions, reflections with respect to different planes and rotations by angles which are multiples of $\pi / 2$ about the different axes. As a result of this nonuniqueness, there is some freedom in choosing the constants $\mathbf{A},\ldots,\mathbf{F}$. Given that the optimal asymptotic value of $\R$ depends exclusively on the wavevector magnitude $|{\mathbf{k}}|$, cf. , any combination of constants $\mathbf{A},\ldots,\mathbf{F}$ will produce the [*same*]{} optimal rate of growth of enstrophy. Thus, to fix attention, in our analysis we will set $\mathbf{A}=\mathbf{B}=\mathbf{C}$ in case (i), $\mathbf{A}=\mathbf{B}=\ldots=\mathbf{F}$ in case (\[c2\]) and $\mathbf{A}=\ldots=\mathbf{D}$ in case (\[c3\]). With these choices, the contribution from each component of the field ${\mathbf{u}}_1$ to the total enstrophy is the same. The maximum (i.e., least negative) value of $\R$ can be thus obtained by choosing the smallest possible $|{\mathbf{k}}|^2$. This maximum is achieved in case (\[c1\]) with the wavevectors ${\mathbf{k}}_1 = [1,0,0]$, ${\mathbf{k}}_2 = [0,1,0]$, ${\mathbf{k}}_3 =
[0,0,1]$, and $-{\mathbf{k}}_1$, $-{\mathbf{k}}_2$ and $-{\mathbf{k}}_3$, for which $|{\mathbf{k}}|^2 = 1$. Because of this maximization property, this is the field we will focus on in our analysis in §\[sec:3D\_InstOpt\_E\] and §\[sec:timeEvolution\].
The three fields constructed in , and are visualized in
| 2,531
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| 0.771992
|
github_plus_top10pct_by_avg
|
sing restrictions on an adversary it was shown by Lo in [@Lo97] and and Buhrman et al. in [@Buhrman12] that these constructions are impossible, even in a quantum setting. As a consequence, constructions for generic unrestricted adversaries in the quantum setting are doomed to failure.
All in all, the necessity for authentication in QKD has led to many authors considering approaches which are strictly quantum in nature, such as those in [@Penghao16; @Zeng00; @Huang11] which are based off entanglement or more recently [@Zawadzki19; @Hong17] which do not rely on entanglement. These are known as *quantum identity authentication* (QIA) protocols. For protocols such as BB84 that do not rely on entanglement it would be more appealing to not rely on entanglement for entity authentication purposes.
[*Our Contribution.*]{} Recently, an original work about authentication without entanglement by Hong et. al. in [@Hong17] was improved by Zawadzki using tools from classical cryptography in [@Zawadzki19]. We start this contribution by summarizing in section \[sec:impossibility\] the impossibility results from Lo [@Lo97] and Buhrman et al. [@Buhrman12], concerning generic quantum two party protocols. Further, we present and discuss the Zawadzki protocol in section \[sec:zawadzki\_protocol\] and show how it succumbs under a simple attack, which we outline in section \[sec:attack\]. Our attack evidences the practical implications of the proven impossibility of identification schemes as conceived in Zawadki’s design, and thus we stress that fundamental changes in the original proposal, beyond preventing our attack, would be needed in order to derive a secure identification scheme.
Quantum Equality Tests are Impossible {#sec:impossibility}
=====================================
A *one sided equality test* is a cryptographic protocol in which one party, Alice, convinces another, Bob, that they share a common key by revealing nothing to either party except equality (or inequality) to Bob. Formally we define a key space $K$ an
| 2,532
| 1,825
| 3,274
| 2,232
| null | null |
github_plus_top10pct_by_avg
|
hbf{e}}+ \left(1-\frac 1 m + \frac t m \right) {\mathbf{1}}\in \Gamma_+.$$
Hence by (the homogeneity of $\Gamma_+$ and) Remark \[hypid\], the maximal zero is at most $$\inf \left\{ \frac {\epsilon t+ \left(1-\frac 1 m\right)\frac t {t-1}} {1-\frac 1 m + \frac t m } : t >1\right\}.$$ It is a simple exercise to deduce that the infimum is exactly what is displayed in the statement of the theorem.
Proof of Theorem \[t1\]
=======================
To prove Theorem \[t1\] we use the following theorem which for $h=\det$ appears in [@MSS1; @MSS2]:
\[hypprob\] Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}$. Let ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ be independent random vectors in $\Lambda_+$ of rank at most one and with finite supports such that $$\label{hypeta2}
{\mathbb{E}}\sum_{i=1}^m {\mathsf{X}}_i ={\mathbf{e}},$$ and $$\label{hyptr}
\tr({\mathbb{E}}{\mathsf{X}}_i) \leq \epsilon \mbox{ for all } 1\leq i \leq m.$$ Then $$\label{hypbig}
{\mathbb{P}}\left[ {\lambda_{\rm max}}\left(\sum_{i=1}^m {\mathsf{X}}_i \right) \leq \delta(\epsilon,m) \right] >0.$$
Let $V_i$ be the support of ${\mathsf{X}}_i$, for each $1 \leq i \leq m$. By Theorem \[mixedchar\], the family $$\{h(t{\mathbf{e}}- {\mathbf{v}}_1-\cdots-{\mathbf{v}}_m)\}_{{\mathbf{v}}_i \in V_i}$$ is compatible. By Theorem \[expfam\] there are vectors ${\mathbf{v}}_i \in V_i$, $1\leq i \leq m$, such that the largest zero of $h(t{\mathbf{e}}- {\mathbf{v}}_1-\ldots-{\mathbf{v}}_m)$ is smaller or equal to the largest zero of $${\mathbb{E}}h(t{\mathbf{e}}- {\mathsf{X}}_1-\cdots-{\mathsf{X}}_m)= {\mathbb{E}}h[{\mathsf{X}}_1,\ldots, {\mathsf{X}}_m](t{\mathbf{e}}+{\mathbf{1}})= h[{\mathbb{E}}{\mathsf{X}}_1,\ldots, {\mathbb{E}}{\mathsf{X}}_m](t{\mathbf{e}}+{\mathbf{1}}).$$ The theorem now follows from Theorem \[mainbound\].
For $1\leq i \leq k$, let ${\mathbf{x}}^i=(x_{i1},\ldots,x_{in})$ where ${\mathbf{y}}=\{x_{ij} : 1\leq i \leq k, 1\leq j \leq k\}$ are independent variables. Consider the polynomial $$g({\mathbf{y}}) = h({\mathbf{x}}^1)h({\mathbf{x}}^2) \
| 2,533
| 1,372
| 2,067
| 2,277
| null | null |
github_plus_top10pct_by_avg
|
------ -------
\(1\) Weekly -- 7428.08 3197.22 1.13 1.49
\(2\) Weekday 0.96\*\*\* -- 8295.71 3585.12 1.30 2.44
\(3\) Weekend 0.71\*\*\* 0.49\*\* -- 5187.05 3375.77 0.61 −0.22
\*\*p \< 0.01, \*\*\*p \< 0.001; M = mean; SD = standard deviation; As = asymmetry; K = kurtosis.
Based on the indexing criteria for weekly step counts set out in the methodology section, the distribution of subjects by step index category is as follows: 23.1% sedentary, 37.6% low active, 23.1% somewhat active, and 16.2% active or very active.
Regarding the number of activities performed, the average for unstructured activities was 2.57 (*SD* = 1.31) out of a possible 4, indicating that participation in this type of PA extends to both weekdays and weekends. The average number of organized activities was 1.46 (*SD* = 1.30). The Wilcoxon signed-rank test shows a highly significant difference between participation in each type of activity (*Z* = −6.05, *p* \< 0.001). The adolescents surveyed in this study thus present a clear preference for unstructured activities over institutionalized activities. The overall average number of activities (both structured and unstructured) was 3.99 (*SD* = 1.95) out of a possible 8.
A similar distribution of values was obtained among subjects at the more autonomous end of the scale, with 48.8% classed as unstructured only and 43.9% as mixed. Only 7.3% of the sample reported participating in organized PA only.
The comparison of PA levels according to sex ([Table 2](#T2){ref-type="table"}) reveals a largely homogeneous group, with little significant difference between average step counts for male and female subjects, either on weekdays or at the weekend. It is important to note, however, the high dispersion of the dataset, as denoted by the very high standard deviation across all parameters.
######
Physical activity (PA) comparison between male and female subjects.
PA (steps) Sex *N* *M* (*SD*) *T* *
| 2,534
| 1,033
| 2,916
| 2,710
| null | null |
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|
s n_i}(B\otimes_AR)$. This equation should be interpreted as follows. We formally compute the right hand side then it is of the form $\pi\cdot X$, where $X$ involves $\tilde{m}_{i,i}^{\ast}$ and $\tilde{m}_{i,i}^{\ast\ast}$. The left hand side $\mathcal{X}_{i,i}^{\ast}(\tilde{m})$ is then defined to be $X$. Then by using an argument similar to the paragraph just before Equation (\[ea20\]) of Step (1), the image of $\mathcal{X}_{i,i}^{\ast}(\tilde{m})$ in $M_{1\times n_i}(B\otimes_AR)/(\pi\otimes 1)M_{1\times n_i}(B\otimes_AR)$ is independent of the choice of the lift $\tilde{m}$ of $m$. Therefore, we may denote this image by $\mathcal{X}_{i,i}^{\ast}(m)$. As for Equation (\[ea20\]) of Step (1), we need to express $\mathcal{X}_{i,i}^{\ast}(m)$ as matrices. By using conditions (5) and (6) of the description of an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ given in Remark \[ra5\], we have that $$\mathcal{X}_{i,i}^{\ast}(m)=m_{i,i}^{\ast}+ m_{i,i}^{\ast\ast}\cdot \bar{h}_i+\mathcal{P}^{\ast}_i.$$ Here, $\mathcal{P}^{\ast}_i$ is a polynomial with variables in the entries of $m$ not including $m_{i,i}^{\ast}$ and $m_{i,i}^{\ast\ast}$. Since $m$ actually belongs to $\mathrm{Ker~}\varphi(R)/\tilde{G}^1(R)$, we have the following equation by the argument made at the beginning of this paragraph: $$\label{ea22}
\mathcal{X}_{i,i}^{\ast}(m)=m_{i,i}^{\ast}+ m_{i,i}^{\ast\ast}\cdot \bar{h}_i+\mathcal{P}^{\ast}_i=\bar{f}_{i,i}^{\ast}=0.$$ Thus we get polynomials $\mathcal{X}_{i,i}^{\ast}(m)$ on $\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$, vanishing on the subscheme $\mathrm{Ker~}\varphi/\tilde{G}^1$.\
3. Assume that $i$ is odd and that $L_i$ is *free of type I*. By Equations (\[ea7\]), (\[ea8\]), (\[ea9\]), and (\[ea10\]) which involve an element of $\tilde{M}^1(R)$, each entry of $b_i', e_i', d_i'$ has $\pi$ as a factor and $f_i-f_i'$ has $\pi^2$ as a factor so that $b_i'\equiv b_i=0, e_i'\equiv e_i=0, d_i'\equiv d_i=0$ mod $(\pi\otimes 1)(B\otimes_AR)$, and $f_i'= f_i=\bar{\gamma}_i$. Let $\tilde{m}\in
| 2,535
| 1,235
| 1,384
| 2,644
| 2,495
| 0.779339
|
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|
eta_{{\widehat{S}}})\leq t)$. There is no uniformly consistent estimator of $\psi_n(\beta)$.
[**Prediction Accuracy.**]{} Now we discuss prediction accuracy which is where splitting pays a price. The idea is to identify a population quantity $\theta$ that model selection is implicitly targeting and compare splitting versus non-splitting in terms of how well they estimate $\theta$. The purpose of model selection in regression is to choose a model with low prediction error. So, in regression, we might take $\theta$ to be the prediction risk of the best linear model with $k$ terms. In our many-means model, a natural analog of this is the parameter $\theta = \max_j \beta(j)$.
We have the following lower bound, which applies over all estimators both splitting and non-splitting. For the purposes of this lemma, we use Normality. Of course, the lower bound is even larger if we drop Normality.
\[lemma::many-means-bound\] Let $Y_1,\ldots, Y_n \sim P$ where $P=N(\beta,I)$, $Y_i\in\mathbb{R}^D$, and $\beta \in \mathbb{R}^D$. Let $\theta = \max_j \beta(j)$. Then $$\inf_{\hat\theta}\sup_{\beta}E[ (\hat\theta - \theta)^2] \geq \frac{2\log D}{n}.$$
To understand the implications of this result, let us write $$\hat\beta(S) - \theta =
\underbrace{\hat\beta(S) - \beta(S)}_{L_1} +
\underbrace{\beta(S) - \theta}_{L_2}.$$ The first term, $L_1$, is the focus of most research on post-selection inference. We have seen it is small for splitting and large for non-splitting. The second term takes into account the variability due to model selection which is often ignored. Because $L_1$ is of order $n^{-1/2}$ for splitting, and the because the sum is of order $\sqrt{\log D/n}$ it follows that splitting must, at least in some cases, pay a price by have $L_2$ large. In regression, this would correspond to the fact that, in some cases, splitting leads to models with lower predictive accuracy.
Of course, these are just lower bounds. To get more insight, we consider a numerical example. Figure (\[fig::price\]) shows a plot of the risk of $\h
| 2,536
| 3,348
| 2,604
| 2,225
| null | null |
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|
neered the research in this field by presenting a new epidemic model called *Susceptible-Infected-Disabled* (SID), which relates each state with a specific functionality of a node in the network [@calle2010multiple]. The state diagram of the SID model (*Susceptible$\leftrightarrows$Infected$\rightarrow$Disabled$\rightarrow$Susceptible*), as seen from a single node, is shown in Fig. \[fig:sid\]. Each node, at each moment of time, can be either susceptible (S), infected (I) or disabled (D). A susceptible node can be infected with probability $\beta$ by receiving the infection from a neighbor (e.g., a bug in the routing or signaling protocol). An infected node can be repaired with probability $\delta_1$ (e.g., the network operator might manually reboot the CP). Finally, the disabled state takes into account the fact that the CP failure eventually affects the DP of the node with probability $\tau$ (e.g., the forwarding tables of the DP become unaccessible). After that, the model states that a repairing time, such as the mean time to repair (MTTR) of the node, determines when it becomes susceptible again ($\gamma$) (e.g., the required time to replace the node).
No operations can be performed during control plane node failures. However, as long as the data plane of the node does not fail, established connections should not fail or be re-routed as a result of control plane node failures. The routing protocol is assumed to be capable of detecting the failure of the control plane and informing all other nodes. Once the routing protocol has converged with this new information, a new connection will not be routed through this node. This same behavior is taken into account by the SID epidemic model.
![State diagram of the SID model and its relationship with the STN and DTN planes.[]{data-label="fig:sid"}](sidstates.pdf)
\
According to the SID model, Fig. \[fig:failurepropagation\] illustrates how a failure can propagate in a BTN. From Fig. \[fig:prop1\], the network operates properly and thus, all nodes are in the sus
| 2,537
| 1,201
| 3,747
| 2,445
| 3,000
| 0.775513
|
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|
\sum_{N_{A},N_{B},N_{C},N_{D}}
d(N_{A},N_{B},N_{C},N_{D})\,
A^{N_A}B^{N_B}C^{N_C}
D^{N_{D}}
\>,
\label{eq:RGA4}\end{aligned}$$ where we have used the prime symbol as a superscripts for $(r+1)$-th restricted partition functions and no indices for the $r$-th order partition functions. These relations can be considered as the RG equations for the problem under study, with the initial conditions $$\label{initalmodel2}
A^{(0)}=x_3\,,\quad B^{(0)}=x_3^2u^4\,,\quad C^{(0)}=x_2\,,\quad
\,D^{(0)}=x_3x_2v^4\,,$$ which correspond to the unit tetrahedron[^1].
![Schematic depiction of restricted generating functions used in the description of all possible two-SAW configurations, within the $r$-th stage of the 3D SG fractal structure, for ASAWs model. The 3D floating chain is depicted by green line, while the 2D surface-adhered chain is depicted by the yellow one. The interior details of the $r$-th stage fractal structure, as well as details of the chains, are not shown (for the chains, they are manifested by the wiggles of the SAW paths). The functions $A^{(r)}$, $B^{(r)}$, and $C^{(r)}$, describe one-polymer configurations (they are the same for both ASAWs and CSAWs models), while the function $D^{(r)}$ depicts the inter-chain configurations of ASAWs model.[]{data-label="fig:RGparametri"}](figure2.eps)
Equation (\[eq:RGC\]), alone, describes a single SAW on 2D SG fractal, whereas (\[eq:RGA\]) and (\[eq:RGB\]) are RG equations for a single SAW on 3D SG fractal. Critical properties of the SAW, based on the analysis of these equations, have been well established previously, and here we recall their basic properties relevant for the present work.
First, we describe the behavior of a single 2D SG chain. The RG equation (\[eq:RGC\]), for any $b$, has only one non-trivial fixed point $C^*$, corresponding to the extended polymer phase [@dhar78; @EKM], that is, the 2D SG chain is always swollen, and it cannot be in the compact phase. The corresponding eigenvalue $\lambda_{\nu_2}$ of (\[eq:RGC\]) is larger than 1, and deter
| 2,538
| 173
| 1,834
| 2,869
| 2,206
| 0.781847
|
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|
rametric, of order $\frac{1}{\sqrt{n}}$.
Prediction/Accuracy Tradeoff: Comparing Splitting to Uniform Inference {#section::splitornot}
======================================================================
There is a price to pay for sample splitting: the selected model may be less accurate because only part of the data are used to select the model. Thus, splitting creates gains in accuracy and robustness for inference but with a possible loss of prediction accuracy. We call this the [*inference-prediction tradeoff*]{}. In this section we study this phenomenon by comparing splitting with uniform inference (defined below). We use uniform inference for the comparison since this is the any other method we know of that achieves (\[eq::honest\]). We study this use with a simple model where it is feasible to compare splitting with uniform inference. We will focus on the [*many means problem*]{} which is similar to regression with a balanced, orthogonal design. The data are $Y_1,\ldots, Y_{2n} \sim P$ where $Y_i\in\mathbb{R}^D$. Let $\beta = (\beta(1),\ldots, \beta(D))$ where $\beta(j) = \mathbb{E}[Y_i(j)]$. In this section, the model ${\cal P}_{n}$ is the set of probability distributions on $\mathbb{R}^D$ such that $\max_j \mathbb{E}|Y(j)|^3 < C$ and $\min_j {\rm Var}(Y(j)) > c$ for some positive $C$ and $c$, which do not change with $n$ or $D$ (these assumptions could of course be easily relaxed). Below, we will only track the dependence on $D$ and $n$ and will use the notation $\preceq$ to denote inequality up to constants.
To mimic forward stepwise regression — where we would choose a covariate to maximize correlation with the outcome — we consider choosing $j$ to maximize the mean. Specifically, we take $$\label{eq::J}
{\widehat{S}}\equiv w(Y_1,\ldots, Y_{2n}) =\operatorname*{argmax}_j \overline{Y}(j)$$ where $\overline{Y}(j) = (1/2n)\sum_{i=1}^{2n} Y_i(j)$. Our goal is to infer the random parameter $\beta_{{\widehat{S}}}$. The number of models is $D$. In forward stepwise regression with $k$ steps and $d$
| 2,539
| 913
| 2,546
| 2,299
| 1,007
| 0.795862
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|
{Var}}$.
The simplest example of a conformal algebra can be constructed as follows. Let $A$ be an ordinary algebra, then a conformal product is uniquely defined on $\Bbbk[T]\otimes A$ by the following formulas for $a,b\in
A$: $$a{\mathbin{{}_{(n)}}}b=\begin{cases}ab, & n=0,\\ 0, & n>0.\end{cases}$$
The conformal algebra obtained is denoted by ${\mathop{\mathrm{Cur}}\nolimits}A$ and is called a current conformal algebra. If an algebra $A \in {\mathrm{Var}}$, then ${\mathop{\mathrm{Cur}}\nolimits}A \in {\mathrm{Var}}_\mathrm{C}$. In the language of category theory, we can say that ${\mathop{\mathrm{Cur}}\nolimits}$ is a functor from the category of algebras to the category of conformal algebras. If $\phi\colon A\to B$ is a homomorphism of algebras, then the mapping ${\mathop{\mathrm{Cur}}\nolimits}\phi\colon{\mathop{\mathrm{Cur}}\nolimits}A\to{\mathop{\mathrm{Cur}}\nolimits}B$ acting by the rule ${\mathop{\mathrm{Cur}}\nolimits}\phi(f(T)\otimes a) = f(T)\otimes \phi(a)$ is a morphism of conformal algebras.
In [@GubKol:09] it was proved that an arbitrary dialgebra $D$ is embedded into the dialgebra $({\mathop{\mathrm{Cur}}\nolimits}\widehat D)^{(0)}$.
Notation for varieties of algebras and dialgebras
-------------------------------------------------
An arbitrary variety of ordinary algebras we denote ${\mathrm{Var}}$, the free algebra in this variety generated by a set $X$ is denoted by ${\mathrm{Var}}\,\langle X\rangle$. The corresponding variety of dialgebras is denoted by ${\mathrm{Di}}{\mathrm{Var}}$, the free dialgebra is denoted by ${\mathrm{Di}}{\mathrm{Var}}\,\langle X\rangle$. The denotation for concrete varieties is analogous, for example, ${\mathrm{Jord}}$ is the variety of Jordan algebras, ${\mathrm{Di}}{\mathrm{Jord}}\,\langle X\rangle$ is the free Jordan dialgebra.
SPECIAL JORDAN DIALGEBRAS
=========================
In this section ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2$. This is necessary to define the Jordan product correctly.
Special and exceptional Jordan dialgebras
-----------------
| 2,540
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se from ten dimensions have been classified. Typically, from the point of view of the ten-dimensional theory these solutions are locally well-defined only in the presence of isometries, and are dubbed ‘exotic branes’ in the literature [@Elitzur:1997zn]. Denoting with $p+1$ the world-volume directions and with $n$ the number of isometries, such branes are commonly dubbed $p_{-\alpha}^n$-branes. So the $\alpha=-2$ brane with two isometries of [@deboer] is a $5^2_2$-brane.
In the brane classification of [@Bergshoeff:2010xc; @stringsolitons; @bergshoeffriccionimarrani], the non-geometric nature of a particular brane corresponds to the fact that the brane is charged with respect to a mixed-symmetry potential.[^5] In particular, denoting with $A_{p,q,r,..}$ a ten-dimensional mixed-symmetry potential in a representation such that $p,q,r, ...$ (with $p\ \geq q \geq r ...$) denote the length of each column of its Young Tableau, this corresponds to a brane if some of the indices $p$ are isometries and contain all the indices $q$, which themselves contain all the indices $r$ and so on. In particular, the exotic defect branes discussed in [@LozanoTellechea:2000mc] correspond to mixed-symmetry potentials with $p=8$ [@defectbranes], but the analysis of [@Bergshoeff:2010xc; @stringsolitons; @bergshoeffriccionimarrani] is more general because it includes domain walls and space-filling branes by also including mixed-symmetry potentials with $p=9$ and $p=10$.
The D-branes, [*i.e.*]{} the $\alpha=-1$ branes, are special because they always arise from D-branes of the ten-dimensional theory. This means that the corresponding potentials are forms, which are indeed the RR forms of the ten-dimensional theory and their duals, together with the 10-form $C_{10}$ associated to the D9-brane in IIB and the 9-form $C_9$ associated to the D8-brane in IIA. In total one gets $$\begin{aligned}
& C_2 \quad C_4 \quad C_6 \quad C_8 \quad C_{10} \quad ({\rm IIB}) \nonumber \\
& C_1 \quad C_3 \quad C_5 \quad C_7 \quad C_9 \quad \ ({\rm IIA}) \q
| 2,541
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on_{n,j}^{\pm}$ on $\tilde{A_{n}}$ such that $\varepsilon_{n,j}^{\pm}(x_i)=\tau_j^{\pm}(x_i)$, $$\varepsilon_{n,j}^{\pm}(\partial_j)=\mp x_j^{2}\partial_j \, \mbox{ and }
\, \varepsilon_{n,j}(\partial_i)=\partial_i \, \mbox{ if } i\neq j \, ,$$ $i,j=1, \ldots, n$. We show it for $n=1$. Suppose we have a group action $G\times \mathbb{T}^1\rightarrow \mathbb{T}^1$ on one dimensional torus. Then the induced action on $\tilde{A}_{1}$ is given as $$x \stackrel{g}\longmapsto x^{g}, \partial\stackrel{g}\longmapsto \partial^{g}= g\partial g^{-1}.$$ Hence, if $\tau^{-}$ sends $x$ to $-x^{-1}$ then we have
$$\varepsilon^{-}(\partial)(x^r)=\partial^{\tau^{-}}(x^r)= \tau^{-}\partial \tau^{-}(x^r)=(-1)^r \tau^{-}\partial(x^{-r})=(-1)^{r+1} r \tau^{-}(x^{-(r+1)})=$$ $=rx^{r+1},$ and $\varepsilon^{-}(x)(x^r)=(-1)^r \tau^{-}(x^{-r+1})=-x^{r-1}.$\
We obtain $\varepsilon^{-}(\partial)=x^{2}\partial$, $\varepsilon^{-}(x)=-x^{-1}$. This is easily generalized to $n$-dimensional torus $\mathbb{T}^n$ and $\tau_i^{\pm}$, $i=1, \ldots, n$.
We will consider the action of the reflection group of type $B_n$ ($n\geq 2$) and the reflection group of type $D_n$ ($n\geq 4$) on $X$. We recall, the group $B_n$ is the semi-direct product of the symmetric group $S_{n}$ and $(\mathbb{Z}/2\mathbb{Z})^n$. There is a natural action of $B_n$ on $\D(X)=\tilde{A_{1}}^{\otimes n}$, where $S_n$ acts by permutations and $(\mathbb{Z}/2\mathbb{Z})^n$ acts by $\varepsilon_{n,i}^{-}$, $i=1, \ldots, n$.
We have
\[proposition-action-in-odd-case\] (i) The subalgebra of $B_n$-invariants of $\D(X)$ is a polynomial algebra in $$s_{i}=e_{i}(x_{1}-x_{1}^{-1},\dots,x_{n}-x_{n}^{-1}), \, i=1, \dots, n,$$ where $e_i$ is the $i$-th elementary symmetric polynomial. In particular, $X/B_n$ is $n$-dimensional affine space.
\(ii) Let $Z\subset \mathbb{T}^{n}$ be the subvariety defined by the following equation $$\prod_{1\leq i\leq j\leq n} (x_{i}^{2}-\frac{1}{x_{j}^{2}})\prod_{1\leq i<j\leq n}(x_{i}^{2}-x_{j}^{2})=0$$ and $U=\mathbb{T}^{n}\setminus Z$. Then $U$ is an aff
| 2,542
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| null | null |
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|
au^2 +(\delta_{ij}+h_{ij} )dx^i dx^j \right]
\label{metric1}$$ where $|h_{ij}|\ll 1$. Using constraints $h^i_i=\nabla_ih^i_j=0 $ we can see that tensor $h_{ij}$ have only two independent components $h^1_1=-h^2_2=h_+$ and $h^2_1=h^1_2=h_{\times}$. These components correspond to two different polarisations of gravitational waves. Inserting the perturbed metric (\[metric1\]) to the Hilbert-Einstein action $S_{\text{H-E}}=(1/16 \pi G) \int d^4x \sqrt{-g}R$ gives the series $S_{\text{H-E}}=S^{(0)}+S^{(1)}+S^{(2)}+\dots$, where the second order term have a form $$S^{(2)}_t=\frac{1}{64\pi G} \int d^4x a^3 \left[
\partial_t h^i_j\partial_t h^j_i-\frac{1}{a^2}\nabla_k h^i_j\nabla_k h^j_i \right]
= \frac{1}{32\pi G} \int d^4x a^3 \left[ \dot{h}_{\times}^2+\dot{h}_{+}^2-
\frac{1}{a^2}\left(\vec{\nabla} h_{\times} \right)^2-\frac{1}{a^2}\left(\vec{\nabla} h_{+}\right)^2 \right]
\label{action1}$$ and give us the action for the gravitational waves. The two kinds of polarisations are not coupled and can be treated separately. To normalise the action and simplify the notation it us useful to introduce the variable $$h=\frac{h_{+}}{\sqrt{16\pi G}}=\frac{h_{\times}}{\sqrt{16\pi G}},$$ what leads to the expression for the action in the form $$S_t = \frac{1}{2} \int d^4x a^3 \left[ \dot{h}^2-\frac{1}{a^2} \left( \vec{\nabla} h \right)^2 \right].
\label{actionh}$$ This action is the same like the action for an inhomogeneous scalar field without the potential. Inverse volume corrections can be therefore introduced in the same way like in the case of the scalar field. As we mentioned in Introduction there are also holonomy corrections to this action. Here we consider however the influence from the better examined inverse volume corrections.
As it was shown by Mulryne and Nunes [@Mulryne:2006cz], in the context of scalar field perturbations, it is useful to introduce the variable $u=a D^{-1/2} h $ and rewrite the action (\[actionh\]) with quantum corrections to the form $$S_{\text{t}} = \frac{1}{2}\int d \tau d^3 {\bf x} [ u^{'2
| 2,543
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|
0\\
t^2 & 0 & 0\\
0 & 0 & 1
\end{pmatrix}\quad{\rm and}\quad
{\rm V}:
\begin{pmatrix}
1 & 0 & 0\\
t^4 & t^5 & 0\\
t^8 & 2t^9 & t^{10}
\end{pmatrix}$$ and the corresponding limits of ${{\mathscr C}}_2$ are given by $$z(y^2z-x^3)\quad{\rm and}\quad(y^2-xz+x^2)(y^2-xz-x^2),$$ respectively: a cuspidal cubic with its inflectional tangent and a pair of quadritangent conics. The connected component of the stabilizer of the latter limit is the additive group. The germ with entries $1$, $t$, and $t^2$ on the diagonal and zeroes elsewhere leads to the limit $(y^2-xz)^2$, a double conic; its orbit is too small to produce an additional component of type IV.
Proof of the main theorem: key reductions and components of type I–IV {#setth}
=====================================================================
Outline {#preamble}
-------
In this section we show that, for a given curve ${{\mathscr C}}$, any germ $\alpha(t)$ contributing to the PNC is ‘equivalent’ (up to a coordinate and parameter change, if necessary) to a marker germ as listed in §\[germlist\]. As follows from §\[rough\] and Lemma \[PNCtolimits\], we may assume that $\det\alpha(t)\not\equiv 0$ and that the image of $\alpha(0)$ is contained in ${{\mathscr C}}$.
Observe that if the center $\alpha(0)$ has rank 2 and is a point of ${{\mathscr S}}$, then $\alpha(t)$ is already of the form given in §\[germlist\], Type I; it is easy to verify that the limit is then as stated there. This determines completely the components of type I. Thus, we will assume in most of what follows that $\alpha(0)$ has rank 1, and its image is a point of ${{\mathscr C}}$.
### Equivalence of germs
\[equivgermsnew\] Two germs $\alpha(t)$, $\beta(t)$ are [*equivalent*]{} if $\beta(t\nu(t))\equiv \alpha(t)\circ m(t)$, with $\nu(t)$ a unit in ${{\mathbb{C}}}[[t]]$, and $m(t)$ a germ such that $m(0)=I$ (the identity).
For example: if $n(t)$ is a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm PGL}}(3)$, then $\alpha(t)\circ n(t)$ is equivalent to $\alpha(t)\circ n(0)$. W
| 2,544
| 1,034
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| 0.788036
|
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|
ad
O^{R,L}_{\Delta S=2} =\bar s \gamma_\mu (1\pm\gamma_5) d \,
\bar s \gamma^\mu (1\pm\gamma_5) d \ .$$ The $W$-boson diagrams yield a purely real Wilson coefficient $C^L_{\Delta S=2}(\mu)$; CP violation in kaon matrix elements is due solely to the operator $O^R_{\Delta S=2}$ rather than $O^L_{\Delta
S=2}$, in contrast to the KM model. The complex coefficient $C^R_{\Delta S=2}(\mu)$ is generated by the charged Higgs through a box diagram with vertices given by Eq.(\[eq:QqH\]). At the scale $\mu=
M_Q$, we have $$C^R_{\Delta S=2}(M_Q) =
2 \xi_{d1} \xi_{s1}^* \xi_{d2} \xi_{s2}^{*}
\frac{2m_W^2}{M_Q^2} \frac{f(x_2)-f(x_1)}{x_2-x_1}
+\sum_{i=1,2} (\xi_{di}\xi_{si}^{*})^2 \frac{2m_W^2}{M_Q^2}\,
{df \over dx}(x_i)
\ ,$$ with $x_{1,2}=m_{H_{1,2}}^2/M_Q^2$, $f(x)= (1-x+ x^2\log x )/(1-x)^2$, and $df/dx\-(1) = 1/3$. Clearly, $C^R_{\Delta S=2}(M_Q)$ is real when $m_2 = m_1$ as it should be. For illustration, we shall take $m_2 \gg m_1$ and $m_1 = M_Q$, in which case the first term is negligible and $C^R_{\Delta S=2}(M_Q) =
{2\over3}(\xi_{d1}\xi_{s1}^{*})^2m_W^2/M_Q^2$.
Following Ref.[@renormgroup] for the renormalization group evolution and numerical evaluation of hadronic matrix elements to leading order, we obtain $$\begin{aligned}
C^R_{\Delta S=2}(\mu \le m_c) &=&
{\left[ \frac{\alpha_s(m_c)}{\alpha_s(\mu)} \right]}^{6/27}
{\left[ \frac{\alpha_s(m_b)}{\alpha_s(m_c)} \right]}^{6/25}
{\left[ \frac{\alpha_s(m_t)}{\alpha_s(m_b)} \right]}^{6/23}
{\left[ \frac{\alpha_s(M_Q)}{\alpha_s(m_t)} \right]}^{6/21}
C^R_{\Delta S=2}(M_Q) \nonumber \\
&\approx& 0.59 \,\alpha_s^{-2/9}(\mu) \,C^R_{\Delta S=2}(M_Q) \ .\end{aligned}$$ We will assume that the $W$-boson contributions dominate the real part of all relevant matrix elements; analysis of $\epsilon^{'}/\epsilon$ below shows this to be consistent. We will thus take, [*e.g*]{}., $\hbox{Re}M_{12} = {1\over2}\Delta m_K$ from experiment, and have no need of the explicit value of $W$-boson contributions to, e.g., $C^L_{\Delta S=2}$. Let $M_{12}^R$ be the contribution of $O^R_{\De
| 2,545
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|
, $$\begin{aligned}
& \ddt \E{\lrn{x_t}_2^2} \\
=& 2\E{\lin{\nabla U(x_t), x_t - x_0}} + \E{ \tr\lrp{M(x_t)^2}}\\
\leq& 2L \E{\lrn{x_t}_2 \lrn{x_t - x_0}_2} + \beta^2\\
\leq& 2L \E{\lrn{x_t - x_0}_2^2} + 2L\E{\lrn{x_0}_2\lrn{x_t - x_0}_2} + \beta^2\\
\leq& 2L \E{\lrn{x_t - x_0}_2^2} + L^2 T\E{\lrn{x_0}_2^2} + \frac{1}{T} \E{\lrn{x_t - x_0}_2^2} + \beta^2\\
\leq& \frac{2}{T} \E{\lrn{x_t - x_0}_2^2} + \lrp{L^2 T \E{\lrn{x_0}_2^2} + \beta^2}
\end{aligned}$$ where the first inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\], the second inequality is by triangle inequality, the third inequality is by Young’s inequality, the last inequality is by our assumption on $T$.
Applying Gronwall’s inequality for $t\in[0,T]$, $$\begin{aligned}
& \lrp{\E{\lrn{x_t - x_0}_2^2} + L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2} \\
\leq& e^{2}\lrp{\E{\lrn{x_0 - x_0}} + L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2}\\
\leq& 8 L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2
\end{aligned}$$ This concludes our proof.
\[l:divergence\_yt\] Let $y_t$ be as defined in (or equivalently or ), initialized at $y_0$. Then for any $T$, $$\begin{aligned}
\E{\lrn{y_T - y_0}_2^2} \leq T^2 L^2 \E{\lrn{y_0}_2^2} + T\beta^2
\end{aligned}$$ If we additionally assume that $\E{\lrn{y_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{y_T - y_0}_2^2} \leq 2 T\beta^2
\end{aligned}$$
Notice from the definition in that $y_T - y_0\sim \N \lrp{-T \nabla U(y_0), T M(y_0)^2}$, the conclusion immediately follows from where the inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\], and the fact that $$\begin{aligned}
\tr\lrp{M(x)^2} = \tr\lrp{\E{\xi(x,\eta) \xi(x,\eta)^T}} = \E{\lrn{\xi(x,\eta)}_2^2}
\end{aligned}$$
\[l:diverg
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}\,, \\
\mathcal{BR}(D^0\rightarrow K^- \pi^+ ) &= (3.89\pm 0.04)\cdot 10^{-2}\,,\end{aligned}$$ we obtain the normalized combinations $$\begin{aligned}
R_{K\pi} &= -0.11 \pm 0.01\,, \\
R_{KK,\pi\pi} &= 0.534 \pm 0.009\,, \\
R_{KK,\pi\pi,K\pi} &= 0.071 \pm 0.009\,.\end{aligned}$$
- The strong phase between DCS and CF mode for the scenario of no CP violation in the DCS mode is [@Amhis:2016xyh] $$\begin{aligned}
\delta_{K\pi} &= \left(8.6^{+9.1}_{-9.7}\right) ^{\circ}\,.\end{aligned}$$
- The world average of $\Delta a_{CP}^{\mathrm{dir}}$ is given in Eq. (\[eq:HFLAVav\]).
- The sum of CP asymmetries $\Sigma a_{CP}^{\mathrm{dir}}$ in which CP violation has not yet been observed. In order to get an estimate we use the HFLAV averages for the single measurements of the CP asymmetries [@Amhis:2016xyh; @Aaij:2014gsa; @Aaltonen:2011se; @Aubert:2007if; @Staric:2008rx; @Csorna:2001ww; @Link:2000aw; @Aitala:1997ff] $$\begin{aligned}
A_{CP}(D^0\rightarrow \pi^+\pi^-) &= 0.0000 \pm 0.0015\,, \\
A_{CP}(D^0\rightarrow K^+K^-) &= -0.0016 \pm 0.0012\,, \end{aligned}$$ and subtract the contribution from indirect charm CP violation $a_{CP}^{\mathrm{ind}} = (0.028 \pm 0.026)\%$ [@Carbone:2019]. We obtain $$\begin{aligned}
\Sigma a_{CP}^{\mathrm{dir}} &= A_{CP}(D^0\rightarrow K^+K^-) + A_{CP}(D^0\rightarrow \pi^+\pi^-) - 2 a_{CP}^{\mathrm{ind}} {\nonumber}\\
&= -0.002\pm 0.002\,, \end{aligned}$$ where we do not take into account correlations, which may be sizable.
- The phases $\delta_{KK}$ and $\delta_{\pi\pi}$ have not yet been measured, and we cannot get any indirect information about them.
From Eqs. (\[eq:Ret1tilde\])–(\[eq:strongphase\]) it follows that $$\begin{aligned}
\mathrm{Re}( \tilde{t}_1 ) &= 0.109 \pm 0.011\,, \label{eq:result-ret1tilde}\\
\mathrm{Im}( \tilde{t}_1 ) &= -0.15^{+0.16}_{-0.17}\,, \label{eq:result-imt1tilde} \\
\tilde{s}_1 &= -0.2668 \pm 0.0045\,, \label{eq:result-res1tilde} \\
-\frac{1}{4} \left(\mathrm{Im}
| 2,547
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|
0}\right)$ matrix is positive definite. In particular the DE for the first order perturbation is $$\frac{\partial\Ket{\Psi_{1}}}{\partial\tau}=-\left(\hat{H}_{0}-E_{0}\right)\Ket{\Psi_{1}}-V\Ket{\Psi_{0}}.\label{eq:psi1diffeq}$$
This equation is similar to the one used for $\Ket{\Psi_{0}}$ , with the addition of a source term due to the second term of the right hand side. In the next section we will show how the simultaneous solving of Eq.\[eq:imtimeSEH0\] and Eq.\[eq:psi1diffeq\] has been implemented in the NECI program.
Implementation\[sec:Implementation\]
====================================
To simultaneously sample the zeroth and first order wavefunctions, we use the multi-replica technique[@overy_unbiased_2014]. A first replica, labeled 0, is sampling the 0 order wavefunctions by propagating the ITDSE of Eq.\[eq:imtimeSEH0\] while another one, labeled 1, is sampling the first order perturbation. We first start with a small amount of walkers on a reference determinant, typically the Hartree-Fock one, on replica 0 and no walkers on replica 1.
In replica 0 new walkers are spawned by applying one and two electrons operators that belong to $\hat{H}_{0}$, thus only determinants that belong to the CAS space are generated. Because the sampling of $\Ket{\Psi_{0}}$ is equivalent to a standard FCIQMC sampling in a CASCI, we can use all the optimizations and approximations that have been introduced in previous publications such as initiator approximation[@cleland_communications:_2010; @cleland_taming_2012] or the semi-stochastic approximation[@blunt_semi-stochastic_2015].
Once the population on replica 0 is equilibrated we start to sample $\Ket{\Psi_{1}}$. This equilibration of the zeroth order wavefunction can be monitored by looking at the variational energy for this replica and checking that is correspond to the CASCI energy. At this point we attempt multiple spawning from each walkers of replica 0. In addition to the excitation that belong to $\hat{H}_{0}$ we also generate excitation belonging to $\hat{V}$. Thi
| 2,548
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| 0.793706
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|
teq S=F_D \cup F \cup \Lambda_{I,p,d}\cup \Sigma_{p,d,P}$. Then, $d|(n-m)$. For, as $0\in P$, $a^pb^{p+d}\in S$ and we have that $a^pb^{p+d}a^mb^n=a^pb^{n-m+p+d} \in \Sigma_{p,d,P}$, so that $d|(m-n-d)$, that is, $m-n=(t+1)d$ for some $t\in \mathbb{N}^0$. Hence for any $a^ib^j \in S$ we have that $d|i-j$. By Lemma \[dimpact\], the first part of the following lemma is clear.
\[dequal1\] If a two-sided subsemigroup $S=F_{D}\cup F\cup \Lambda_{I,p,d} \cup \Sigma_{p,d,P}$ of $\mathcal{B}$ is a left I-order in $\mathcal{B}$, then $d=1$ and $q=0$. Consequently, $R_{1} \subseteq S$.
Let $a^0b^h\in \mathcal{B}$ where $h\in \mathbb{N}$. Then, $$a^0b^h=(a^ib^j)^{-1}(a^mb^n)=a^{j-i+t}b^{n-m+t}$$ where $t=$max$\{m,i\}$, so that $0=j-i+t$. Hence we deduce that $j=0$. If $a^ib^j \in \Sigma_{p,d,P}$, then as $\Sigma_{p,d,P}$ is an inverse subsemigoup of $\mathcal{B}$ we have that $a^0b^h \in S$. In the case where $a^ib^j \notin \Sigma_{p,d,P}$ we must have that $a^ib^j\in F_{D}\cup F\cup \Lambda_{I,p,d}$. Hence $j\geq i$ so that $i=j=0$. It follows that $a^0b^h=a^mb^n\in S$. Hence $q=0$.
Since $F_D\subseteq \{1\}$ we have that $F_D=\{1\}$ or $F_D=\emptyset$. In either case, $S\cap L_1=\{1\}$. Then the following corollaries are clear.
If a two-sided subsemigroup $S=F_{D}\cup F\cup \Lambda_{I,p,d} \cup \Sigma_{p,d,P}$ of $\mathcal{B}$ is a left I-order in $\mathcal{B}$, then $F_D=\{1\}$ or $F_D=\emptyset$.
\[twosided\] A two-sided subsemigroup $S=F_{D}\cup F\cup \Lambda_{I,p,d} \cup \Sigma_{p,d,P}$ of $\mathcal{B}$ is a left I-order in $\mathcal{B}$ iff $R_{1} \subseteq S$.
\[twostraightr\] If $S=F_{D}\cup F\cup\Lambda_{I,p,d}\cup \Sigma_{p,d,P}$ is a left I-order in $\mathcal{B}$, then it is straight.
Now, we start studying the second form which has the form (2).($ii$) in Proposition \[subbicyclic\].
Let $a^mb^n \in \widehat{F} \subseteq S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$. Then, $d|n-m$. For, since $a^pb^{p+d} \in \Sigma_{p,d,P}$, it follows that $a^mb^na^pb^{p+d}=a^{m-n+p}b^{p+
| 2,549
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|
l m)\leq T$. This is because the numerator in the middle expression in (\[eqpoi2\]) is a finite sum of polynomials. However, there is no universal lower bound.
3. For any $g_0$, the number of $a_{g\ell m}$ with $g=g_0$ equals the number of $b_{gu}$ with $g=g_0$. This is simply because $\sum v^{a(g\ell m)} = \sum v^{b(gu)}$ and the numbers are finite by ($\dagger$1).
We aim to adjust the basis $\{b_{gu}\}$ to be equal to the basis $\{a_{g\ell m}\}$, and we achieve this by a downwards induction on $g$. The induction starts since, by ($\dagger$3), there are no basis elements $b_{gu}$ with $g>T$.
Let $-\infty < G\leq T$ and, by induction, suppose that $\{b_{gu} : u\in {\mathbb{Z}}\}= \{a_{g\ell m} : \ell,m \in {\mathbb{Z}}\}$ for all $g>G$. Suppose that there exists a basis element $b_{Gw} \not\in \{a_{G\ell m}\}$. By Lemma \[step-1\](2), $\theta({\mathcal{J}})[\delta^{-2}]={\mathcal{N}}[\delta^{-2}]$ and so there exists a homogeneous element $\mathbf{x}^m\in {\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ of ${\mathbf{E}}$-degree $m$ such that $\mathbf{x}^mb_{Gw}\in \theta({\mathcal{J}})$. Thus we have the ${\mathbf{E}}$-homogeneous equation $$\label{bigger-kk}
\mathbf{x}^mb_{Gw} = \sum_{g<G} c_{gfh}a_{gfh}
+\sum c_{Gfh}a_{Gfh}+
\sum_{g>G} c'_{gz}b_{gz},$$ where $c_{gfh}, c'_{gz}\in \mathbb C[{\mathfrak{h}}]^W$ and summation over $f,h,z$ is suppressed. Since $\theta({\mathcal{J}})\subseteq {\mathcal{N}}$, we may write each $a_{gfh}$ as an ${\mathbf{E}}$-homogeneous sum $a_{gfh} =\sum d_{\bullet}b_{uz}$ for some $d_\bullet = d_{fghuz} \in {\mathbb{C}}[{\mathfrak{h}}]^W$ and obtain $$\label{bigger-k}
\mathbf{x}^mb_{Gw} = \sum_{g<G} c_{gfh}d_{\bullet}b_{uz}
+\sum c_{Gfh}d_{\bullet}b_{uz} +
\sum_{g>G} c'_{gz}b_{gz}.$$ Both the last two displayed equations are ${\mathbf{E}}$-homogeneous of ${\mathbf{E}}$-degree $G+m$ and so, by , each element $c_{gfh}$ must have ${\mathbf{E}}$-degree $\geq m$. Thus the $b_{uz} $ appearing in the first two terms on the right hand side of must have ${\mathbf{E}}$-degree $\leq G$. Thus the only
| 2,550
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|
rtial_y d_y+d h_y,\ee
\be
[\bar{H},b_x]=\partial_y b_x,\hs{2ex}[\bar{H},b_y]=\partial_yb_y,$$ $$[D,h_x]=(c x\partial_x+d y\partial_y)h_x+2c h_x,\hs{2ex}[D,h_y]=(c x\partial_x+d y\partial_y)h_y+(c+d) h_y,$$ $$[D,\bar{h}_x]=(c x\partial_x+d y\partial_y)\bar{h}_x+(c+d) \bar{h}_x,\hs{2ex}[D,\bar{h}_y]=(c x\partial_x+d y\partial_y)\bar{h}_y+2d \bar{h}_y,$$ $$[D,d_x]=(c x\partial_x+d y\partial_y)d_x+c d_x,\hs{2ex}[D,d_y]=(c x\partial_x+d y\partial_y)d_y+d d_y,$$ $$[D,b_x]=(c x\partial_x+d y\partial_y)b_x+d b_x,\hs{2ex}[D,b_y]=(c x\partial_x+d y\partial_y)b_y+(2d-c) b_y,$$ $$[B,h_x]=x\partial_yh_x- h_x,\hs{2ex}[B,h_y]=x\partial_yh_y,\ee
\be
[B,\bar{h}_x]=x\partial_y\bar{h}_x,\hs{2ex}[B,\bar{h}_y]=x\partial_y\bar{h}_y+\bar{h}_y,$$ $$[B,d_x]=x\partial_yd_x- d_x,\hs{2ex}[B,d_y]=x\partial_yd_y,\ee
\be
[B,b_x]=x\partial_yb_x,\hs{2ex}[B,b_y]=x\partial_yb_y+b_y.$$ We choose the above commutation relations by the following two requirements. One is that the differential operators must act on the field properly, while the other is that we must recover the commutators of the generators.
It is remarkable that there are ambiguities in defining the Noether currents. One can shift the currents by some local operators to get the same commutation relations of the generators and still have the conservation laws. One may organize the currents with respect to the canonical commutation relations to define the local operators. The above canonical commutation relations imply that the dilation and boost currents can be expressed by the translation current up to some local operators $$d_x=c xh_x+d y\bar{h}_x+s_x,\hs{2ex}d_y=c xh_y+d y\bar{h}_y+s_y$$ $$b_x=x\bar{h}_x+w_x,\hs{2ex}b_y=x\bar{h}_y+w_y,$$ where $s_{x},s_y$ and $w_{x},w_y$ are local operators. In the following we will study the shifts of the currents that do not change the canonical commutation relations.
Enhanced symmetries
-------------------
Let us first study the boost symmetry and the boost current. The boost currents are related to the translation currents $$b_x=x\bar{h}_x+w_
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|
r{\kappa}, \bar C>0$ such that $P_t\psi_\kappa(x)\leq \bar C\psi_{\bar \kappa}(x)$, for all $x\in \R^d$ and $t>0$. Then $%
P_{t}(x,y)=p_{t}(x,y)$ with $p_{t}\in C^{\infty }(\R^{d}\times \R^{d})$ and for every $\kappa \in \N$, $\varepsilon >0$ and for every multi-indexes $\alpha $ and $\beta $ there exists $%
C=C(\kappa ,\varepsilon ,\delta,\alpha ,\beta )$ such that for every $t>0$ and $x,y\in \R^{d}$ $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq C\times t^{-\theta _{0}(1+\frac{a+b}{\delta }%
)(\left\vert \alpha \right\vert +\left\vert \beta \right\vert
+2d +\varepsilon)}\times \frac{\psi_{\eta+\kappa}(x)}{\psi _{\kappa }(x-y)} \label{TR7e-new}$$ with $\theta _{0}$ from (\[TR5\]).
We give now a result which goes in another direction (but the techniques used to prove it are the same): we assume that the semigroup $%
P_{t}:C_{b}^{\infty }({\mathbb{R}}^{d})\rightarrow C_{b}^{\infty }({\mathbb{R%
}}^{d})$ verifies hypothesis of type $(A_{2})$ (see (\[TR2\]) and ([TR2’]{})) and $(A_{3})$ (see (\[TR5\])), we perturb it by a compound Poisson process, and we prove that the new semigroup still verifies a regularity property of type $(A_{3})$. This result will be used in [@BCW] in order to cancel the big jumps.
Let us give our hypotheses.
\[H2H\*2-P\] For every $q\in {\mathbb{N}},\kappa \geq 0$ and $p\geq 1$ there exist $C_{q,\kappa ,p}(P),C_{q,\kappa ,\infty }(P)\geq 1$ such that $$\begin{aligned}
(H_{2})& \qquad \left\Vert P_{t}f\right\Vert _{q,-\kappa ,\infty }\leq
C_{q,\kappa ,\infty }(P)\left\Vert f\right\Vert _{q,-\kappa ,\infty },
\label{J6a} \\
(H_{2}^{\ast })& \qquad \left\Vert P_{t}^{\ast }f\right\Vert _{q,\kappa
,p}\leq C_{q,\kappa ,p}(P)\left\Vert f\right\Vert _{q,\kappa ,p} \label{J6b}\end{aligned}$$
This means that the hypotheses $(A_{2})$ and$(A_{2}^{\ast })$ (see (\[TR2\]) and (\[TR2’\])) from Section \[sect:3.2\] hold for $P_{t}$ (instead of $P_{t}^{n})$ with $\Lambda _{n}$ replaced by $C_{q,\kappa ,\infty }(P)\vee
C_{q,\kappa ,p}
| 2,552
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| 2,046
| 2,438
| 4,160
| 0.767674
|
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|
2.67 ± 0.52 X10^6^/µl - p-value 0.833) between the two groups. The mean post-transplant hemoglobin (8.39 ± 0.91 gm/dl vs 8.37 ± 0.85 gm/dl - p-value 0.85), hematocrit (26.54 ± 2.96 % vs 26.34 ± 2.93 % - p-value 1), and RBC count (3.24 ± 0.41 X10^6^/µl vs 3.09 ± 0.54 X10^6^/µl - p-value 0.571) were comparable in group A vs B, respectively. The median days to expiration of the PRBC unit was four in group A as compared to 16 in group B. There was no statistically significant difference in the mean time lapse between transfusion and rechecking hemoglobin between the two groups, i.e., nine hours (range: 3-17) in group A vs 8.47 hours (range:4-23) in group B. Patients who received new blood had a higher mean red blood cell distribution width (RDW) at the time of transfusion as compared to the group that received old blood (19.82 ± 5.32 % vs 18.36 ± 4.05 % - p-value 0.034). There was no difference in the prevalence of coronary artery disease (22 vs 18 - p-value 0.479), chronic kidney disease (23 vs 16 - p-value 0.211), cancer (36 vs 48 - p-value 0.084), infections (5 vs 6 - p-value 0.756) in the group that received old blood vs. new blood, respectively. Group A received a mean of 2.06 ± 3.95 blood transfusions in the previous six months as compared to 3.33 ± 4.5 in group B with no statistically significant difference.
The mean rise in hemoglobin after transfusing old blood was 1.01 gm/dl with no statistically significant difference from the mean rise in hemoglobin of patients who received new blood, i.e., 1.08 gm/dl - p-value 0.298. The mean rise in hematocrit and RBC count were also similar in both groups (3.37 % vs 3.61 % - p-value 0.249) and (0.42 X10^6^/µl vs 0.44 X10^6^/µl - p-value 0.097), respectively (Table [2](#TAB2){ref-type="table"}).
###### Post-transfusion rise in hemoglobin, hematocrit, and red blood cell count
----------------------------------- ------------ ------------ ---------
Variable Old blood New blood
| 2,553
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el{450}$$ $${\check{V}}^{crv}_{1} = {{[{\bar{{\alpha}^{2}}} {\sin}^{2} (\theta)]}^{1
\over 2} \over {N}^{-{1 \over 2}}[{r}^{av} -{\cos}^{2}
(\theta){R}^{crv}_{N}] }.
\label{451}$$
The results just derived demonstrate the powerful volatility reduction effect of diversification coupled with short sales for the market-orthogonal portfolio ${{\bf e}^{crv}}^{1}$. To see this, let us assume a typical value for $\tan (\theta)$ of the order of unity \[for reasonably large $N$; cf. Eq. (\[447\])\]. We then find from the above results $${\lim}_{N \rightarrow \infty} \; {{V}^{crv}_{N}}={[{{\bm{\beta}}
\cdot {\bm{\beta}} \over N {\cos}^{2} (\theta)}
{\bar{{\rho}^{2}}}_{mkt}]}^{1 \over 2}, \; {\lim}_{N \rightarrow
\infty}\;{{V}^{crv}_{1}}=0. \label{452}$$ Note that the quantity ${\bm{\beta}} \cdot {\bm{\beta}}$ in general grows in proportion to $N$, and therefore that ${\bm{\beta}} \cdot
{\bm{\beta}} / N {\cos}^{2}(\theta)$ is typically of the order of unity for large $N$. Thus the variance of the market-aligned portfolio will be of the order of ${\bar{{\rho}^{2}}}_{mkt}$ for large $N$, as would be expected. The variance of the market-orthogonal portfolio, on the other hand, vanishes altogether in proportion to ${N}^{-1}$ in the same limit of large $N$. These conclusions echo our results in §2, Eq. (\[442\]) et seq.
Note that the vanishing of the market risk for the market-orthogonal portfolio, which is in addition to the vanishing of the “diversifiable” (or specific) risk expected for large $N$ (Elton and Gruber 1991), is a specific result of leveraging coupled with hedging (or diversification). Similarly, the infinite volatility and expected return levels of the $N-2$ remaining portfolios of this model underscore the dramatic levels of volatility as well as return that can be expected of highly leveraged portfolios.
We are now in a position to determine the composition of the efficient frontier for the constant residual variance case. As stated above, we find from the allocation rule of the efficient frontier that
| 2,554
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|
al A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})}
{Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_i={\varnothing}~
{{}^\forall}i=1,\dots,4\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}\prod_b\frac{N_b!}{n_b!\prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!},\end{aligned}$$ where we have used the notation $m_b^{{\scriptscriptstyle}(i)}={{\bf m}}_i|_b$. (iii) The sum over ${{\bf n}},{{\bf m}}_1,\dots,{{\bf m}}_4$ in [(\[eq:Theta”-bd1stprebd\])]{} is bounded by the cardinality of ${\mathfrak{S}}$ in Lemma \[lmm:GHS-BK\] with $k=4$, ${{\cal V}}=\{y,x\}$, $\{z_1,z'_1\}=\{y,u\}$, $\{z_2,z'_2\}=\{u,x\}$, $\{z_3,z'_3\}=\{y,v\}$ and $\{z_4,z'_4\}=\{v,x\}$. Bounding the cardinality of ${\mathfrak{S}}'$ in Lemma \[lmm:GHS-BK\] for this setting, we obtain $$\begin{aligned}
{\label{eq:Theta''-bd1stbd1}}
{(\ref{eq:Theta''-bd1stprebd})}&\leq\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_1=y{\vartriangle}u,~{\partial}{{\bf m}}_2=u{\vartriangle}x\\ {\partial}{{\bf m}}_3=y{\vartriangle}v,~{\partial}{{\bf m}}_4=v{\vartriangle}x\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}\prod_b\frac{N_b!}{n_b!
\prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!}{\nonumber}\\
&\leq\sum_{u\in{{\cal A}}}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_v \rangle}}_\Lambda{{\lan
| 2,555
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|
or each }N\in\mathcal{N}_{x}\}\label{Eqn: TBx}$$ *determines the full neighbourhood system* $$\mathcal{N}_{x}=\{ N\subseteq X\!:x\in B\subseteq N\textrm{ for some }B\textrm{ }\in\,\mathcal{B}_{x}\}\label{Eqn: TBx_nbd}$$
*reciprocally as all supersets of the basic elements.$\qquad\square$*
The entire neighbourhood system $\mathcal{N}_{x}$, which is recovered from the base by forming all supersets of the basic neighbourhoods, **is trivially a local base at $x$; non-trivial examples are given below.
The second example of a base, consisting as usual of a subcollection of a given collection, is the topological base $_{\textrm{T}}\mathcal{B}$ that allows the specification of the topology on a set $X$ in terms of a smaller collection of open sets.
**Definition A1.2.** *A base* $_{\textrm{T}}\mathcal{B}$ *in a topological space $(X,\mathcal{U})$ is a subcollection of the topology $\mathcal{U}$ having the property that each $U\in\mathcal{U}$ contains some member of* $_{\textrm{T}}\mathcal{B}$*.* *Thus* $$_{\textrm{T}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{U}\!:B\subseteq U\textrm{ for each }U\in\mathcal{U}\}\label{Eqn: TB}$$ *determines reciprocally the topology $\mathcal{U}$ as* $$\mathcal{U}=\left\{ U\subseteq X\!:U=\bigcup_{B\in\,\!_{\textrm{T}}\mathcal{B}\,}B\right\} \qquad\square\label{Eqn: TB_topo}$$
This means that the topology on $X$ can be reconstructed form the base by taking all possible unions of members of the base, and a collection of subsets of a set $X$ is a topological base iff Eq. (\[Eqn: TB\_topo\]) of arbitrary unions of elements of $_{\textrm{T}}\mathcal{B}$ generates a topology on $X$. This topology, which is the coarsest (that is the smallest) that contains $_{\textrm{T}}\mathcal{B}$, is obviously closed under finite intersections. Since the open set $\textrm{Int}(N)$ is a neighbourhood of $x$ whenever $N$ is, Eq. (\[Eqn: TBx\_nbd\]) and the definition Eq. (\[Eqn: Def: nbd system\]) of $\mathcal{N}_{x}$ implies that *the open neighbourhood system of any point in a topological space i
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, ... _I ever_ ... _ever I_.
... ... 97, ... _wage-pasty_ ... _way-pasty_.[100]
... ... 99, ... _he_ ... _ye_.
... ... _ib._ ... _ield_ ... _yelde_.
... ... 105, ... _to please_ ... _it please_.
... ... 108, ... _a master_ ... _an M_.
... ... 117, ... _as much_ ... _so much_.
... ... 118, ... _make a_ ... _make me a_.
... ... 121, ... _another ... _another but_.
than_
... ... _ib._ ... _readiness_ ... _a readiness_.
... ... 122, ... _other's_ ... _others'_.
... ... _ib._ ... _point ... _point
whereof_ wherefore_.
... ... 125, ... _draw ye_ ... _draw we_.
... ... 128, ... _thou goose_ ... _you goose_.
... ... 139, ... _Not if all ... _Nor if all the_.
the_
... ... 140, ... _where or ... _where nor how_.
how_
... ... 158, ... _all men_ ... _of all men_.
... ... 178, ... _halse-aker_ ... _half-acre_.[101]
VOL. V. ... 115, ... _Alvearic_ ... _Alvearie_.
... ... 285, ... _Got_ ... _Get_.
VoL. IX. ... 98, ... _collection_ ... _collation_.
... ... _ib._} ... _moldash_ ... _molash_.
... ... 332,} ... _moldash_ ... _molash_.
... ... 205, ... _Amoretta_ ... _Amoretto_.
VOL. X. ... 274, ... _Foresaw_ ... _Foreseen_.
VOL. XI. ... 436, ... _Sir Thomas_ ... _St. Thomas_.
FOOTNOTES:
[98] See Nares. ed. 1859, _v._ Nott. We still have the vulgarism _nut_
for the head; but it more properly means a head with the h
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| null | null |
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ollowing relation: $$\Lambda = - \Lambda_5 .$$ We will find the solution by construction. Assume the ansatz in Eqn. (7) for the warp factor $f(z)$. Then its derivatives are given by Eqns. (8,9). $$\begin{aligned}
f(z) = {\alpha}
{\displaystyle Sinh[g(z)]^{-1} }
& & \\
f'( z) = -{\alpha} \displaystyle \frac {Cosh [g(z)]}{ Sinh[g(z)]^{2} } g'(z)
& & \\
f''(z) = -{\alpha } (\displaystyle \frac{ g'(z)^2 }{ Sinh[g(z)] } - - \frac { 2 Cosh[g(z)]^{2} }{ Sinh[g(z)]^{3} } {g'}^{2} +
\frac { Cosh[g] }{ Sinh[g]^2 } g'')\end{aligned}$$ The function $g(z)$ \[1,3\], describing even branes (domain walls) of positive and negative energy positioned along $S^1$ is of the following form $$\begin{aligned}
f(z) & =\alpha \displaystyle{Sinh[g(z]^{-1}} \\
g(z) &=\left(\displaystyle{\sum^{n-1}_{i=1}}(-1)^{i+1}|z-L_i|+L\right)(-\beta)\\
g''(z)&=2\left(\displaystyle{\sum^{n-1}_{i=1}}(-1)^{i+1}
\delta |z - L_i| \right) (-\beta) \end{aligned}$$ The even $(n-1)$ branes are located at $L_i (i = 2, \dots n-1)$. One always ends up with even branes as it is clear from topological consideration. Each brane is a pointlike gravitational source in the fifth dimension, i.e. there are flux lines extending from one brane to the others (mutual interaction). Those flux lines should close in order to preserve the stability of the 5th dimension (see \[4\]). Thus each positive energy brane in $S^1$ is alternating with its counterpart (negative tension brane) so that the number of positive energy branes equals the number of negative energy branes.\
Replacing the above Eqns. (7-12) for $f(z)$ in the Einstein Eqns.(6) we get the following relation $$\begin{aligned}
\nonumber (H^2 - {g'}^2)+ \frac {\Lambda /3 - 2{g'}^2}{ \displaystyle Sinh[g]^2 }
&= & \frac{( \sum_i - \kappa_{5} ({\cal L}_i + V_i) - g''
{\displaystyle Cosh[g(L_i)]}) \delta(z - L_i)}{Sinh[g]} \\
\nonumber (H^2 - {g'}^2) & =& - \displaystyle \frac{\Lambda \alpha^2}{6 Sinh[g]^2} +
\frac { {g'}^2 }{ Sinh[g]^2 } \end{aligned}$$ There
| 2,558
| 4,088
| 2,508
| 2,318
| 3,047
| 0.775227
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|
. For a massless representation, one has $\hat{P}^2=0$ and $\hat{W}^2=0$. Such representations are characterised by the helicity $s$ of the state, namely a specific representation of the helicity group SO(2), the rotation subgroup of the Wigner little group for a massless particle.[^9] For instance, for a light-like energy-momentum 4-vector $P^\mu=E(1,0,0,1)$, one has $W^\mu=M_{12}P^\mu$, so that $M_{12}$ takes the possible eigenvalues $\pm s$.
In the case of the scalar field, it is then straightforward to identify the particle content of its Hilbert space. A 1-particle state $|\vec{k}\rangle=a^\dagger(\vec{k}\,)|0\rangle$ is characterised by the eigenvalues $$\hat{P}^0|\vec{k}\rangle=\omega(\vec{k}\,)|\vec{k}\rangle\ \ ,\ \
\hat{\vec{P}}|\vec{k}\rangle=\vec{k}\,|\vec{k}\rangle\ \ ,\ \
\hat{W}^2|\vec{k}\rangle=0\ ,$$ thus showing that indeed, the quanta of such a quantum field may be identified with particles of definite energy-momentum and mass $m$, carrying a vanishing spin (in the massive case) or helicity (in the massless case). Relativistic quantum field theories are thus the natural framework in which to describe all the relativistic quantum properties, including the processes of their annihilation and creation in interactions, of relativistic quantum point-particles. It is the Poincaré invariance properties, namely the relativistic covariance of such systems, that also justifies, on account of Noether’s theorem, this physical interpretation.
One has to learn how to extend the above description to more general field theories whose quanta are particles of nonvanishing spin or helicity. Clearly, one then has to consider collections of fields whose components also mix under Lorentz transformations, namely nontrivial representations[^10] of the Lorentz group.
Spinor Representations of the Lorentz Group\
and Spin 1/2 Particles {#Sec3}
============================================
The Lorentz Group and Its Covering Algebra {#Sec3.1}
------------------------------------------
Let us now consider the possi
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gR \left( \h, \a \right) = \bigR \left( \S\h, \S\a \right)$ always holds. Therefore, $\S{\a^\star}$ is optimal for $\S\h$ with the same power constraint $P$.
(Nonnegative Ordered Vector) A vector $\h$ is said to be nonnegative ordered if its elements are nonnegative and in nondecreasing order according to their indices.
\[lemma:h2hBar\] For any vector $\h$, there exists a signed permutation matrix $\S$ such that $\S\h$ is nonnegative ordered.
To find such an $\S$ in Lemma \[lemma:h2hBar\], we can simply choose $\S=\P\T$, where $\T$ is a signature matrix that converts all the elements in $\h$ to nonnegative, and $\P$ is a permutation matrix that sorts the elements in $\T\h$ in nondecreasing order.
With Theorem \[theorem:ProblemTransformation\] and Lemma \[lemma:h2hBar\], for any channel vector $\h$, we can first find a signed permuation matrix $\S$ and transform $\h$ to the nonnegative ordered ${\bar{\h}}=\S\h$, then obtain the optimal coefficient vector ${{\bar{\a}}^\star}$ for ${\bar{\h}}$, and finally recover the desired optimal coefficient vector ${\a^\star}=\S^{-1}{{\bar{\a}}^\star}$ for $\h$. In this way, it suffices to focus on solving the problem in for nonnegative ordered channel vectors ${\bar{\h}}$.
\[remark:Transformation\] In implementation, there is no need to use the signed permutation matrix $\S$. It is merely necessary to: 1) record the sign of the elements in $\h$ with a vector $\bst=\fsign{\h}$, and 2) sort $\boldsymbol{\hbar}=\fabs{\h}$ in ascending order as ${\bar{\h}}$ and record the original indices of the elements with a vector $\p$ such that ${\bar{\h}}(\ell)=\boldsymbol{\hbar}(\p(\ell))$, $\ell=1,2,\cdots,L$. After ${{\bar{\a}}^\star}$ for ${\bar{\h}}$ is obtained, ${\a^\star}$ for $\h$ can be recovered with ${\a^\star}(\p(\ell))=\bst(\p(\ell)){{\bar{\a}}^\star}(\ell)$, $\ell=1,2,\cdots,L$.
Given a channel vector as $\h=[-1.9,0.1,1.1]^T$, then $\bst=[-1,1,1]^T$, $\fabs{\h}=[1.9,0.1,1.1]^T$, ${\bar{\h}}=[0.1,1.1,1.9]^T$, and $\p=[2,3,1]^T$. If for certain power $P$, ${{\bar{\a}}^\
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| 0.77404
|
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|
onnected* *if it has no separation, that is if it cannot be partitioned into two open or two closed nonempty subsets. $X$ is* *separated (disconnected)* *if it is not connected.$\qquad\square$*
It follows from the definition, that for a disconnected space $X$ the following are equivalent statements.
\(a) There exist a pair of disjoint nonempty open subsets of $X$ that cover $X$.
\(b) There exist a pair of disjoint nonempty closed subsets of $X$ that cover $X.$
\(c) There exist a pair of disjoint nonempty clopen subsets of $X$ that cover $X.$
\(d) There exists a nonempty, proper, clopen subset of $X$.
By a *connected subset* is meant a subset of $X$ that is connected *when provided with its relative topology making it a subspace of $X$.* Thus any connected subset of a topological space must necessarily be contained in any clopen set that might intersect it: if $C$ and $H$ are respectively connected and clopen subsets of $X$ such that $C\bigcap H\neq\emptyset$, then $C\subset H$ because $C\bigcap H$ is a nonempty clopen set in $C$ which must contain $C$ because $C$ is connected.
For testing whether a subset of a topological space is connected, the following relativized form of (a)$-$(d) is often useful.
**Lemma A3.1.** *A subset $A$ of $X$ is disconnected iff there are disjoint open sets $U$ and $V$ of $X$ satisfying* $${\textstyle U\bigcap A\neq\emptyset\neq V\bigcap A\textrm{ such that }A\subseteq U\bigcup V,\;\textrm{with }U\bigcap V\bigcap A=\emptyset}\label{Eqn: SubDisconnect1}$$ *or there are disjoint closed sets $E$ and $F$ of $X$ satisfying* $${\textstyle E\bigcap A\neq\emptyset\neq F\bigcap A\textrm{ such that }A\subseteq E\bigcup F,\;\textrm{with }E\bigcap F\bigcap A=\emptyset.}\label{Eqn: SybDisconnect2}$$ *Thus $A$ is disconnected iff there are disjoint clopen subsets in the relative topology of $A$ that cover $A$.$\qquad\square$*
**Lemma A3.2.** *If $A$ is a subspace of $X$, a* *separation of* *$A$ is a pair of disjoint nonempty subsets $H_{1}$ and $H_{2}$ of $A$ whose union is $A$ neither of
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| 3,346
| 2,284
| 2,525
| 0.77918
|
github_plus_top10pct_by_avg
|
lta_{ij}L
\right)_{,j}=0,$$ which amounts to the momentum conservation law $$m_{i,t}+\left(\lambda^2u_{k,i}u_{k,j}-u_jm_i-\delta_{ij}\left(\frac{1}{2}u_ku_k+\frac{\lambda^2}{2}u_{k,l}u_{k,l}\right)\right)_{,j}.$$
Conservation of vorticity
-------------------------
Next, consider the coefficient of each ${\mathrm{d}}x_r\wedge{\mathrm{d}}x_s$ in the pull-back of the structural (two-form) conservation law (\[epsympcl\]). This is $$\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right)_{,t}
+\left(\lambda^2\left(W_{ij,r}u_{i,s}-W_{ij,s}u_{i,r}\right)
+u_j\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right)
+\pi_k\left(u_{j,r}l_{k,s}-u_{j,s}l_{k,r}\right)\right)_{,j}=0,$$ which amounts to $$\left(m_{r,s}-m_{s,r}\right)_{,t}+\left(\lambda^2\left(u_{i,s}u_{i,jr}-u_{i,r}u_{i,js}\right)+(u_jm_r)_{,s}-(u_jm_s)_{,r}\right)_{,j}=0.$$ One can regard this as a vorticity conservation law for EPDiff($H^1$); it is a differential consequence of the momentum conservation law.
Particle relabelling symmetry
-----------------------------
As we discussed in Section \[inverse map sec\], fluid equations in general, and EPDiff in particular, are invariant under relabelling of particles. In the context of the inverse map variables, relabelling is accomplished by the action of the diffeomorphism group $\operatorname{Diff}(\Omega)$ defined by $$\MM{l}\mapsto
\eta\circ\MM{l}\equiv\eta(\MM{l}), \qquad \eta\in\operatorname{Diff}(\Omega).$$ The corresponding infinitesimal action of the vector fields $\mathfrak{X}(\Omega)$ is then $$\MM{l} \mapsto \MM{\xi}\circ\MM{l}\equiv\MM{\xi}(\MM{l}), \qquad
\MM{\xi}\in\mathfrak{X}(\Omega),$$ and the cotangent lift of this action is $$(\MM{\pi},\MM{l}) \mapsto
\left(-(\nabla\MM{\xi}(\MM{l}))^T\cdot\MM{\pi}
,\MM{\xi}(\MM{l})\right).$$ To obtain the symmetry generator (\[X\]), we extend the above action to first derivatives as follows: $$\begin{aligned}
X &=& \xi_k(\MM{l}){\frac{\partial }{\partial l_k}} + (\xi_k(\MM{l}))_{,t}{\frac{\partial }{\partial l_{k,t}}}
+ (\xi_k(\MM{l}))_{,i}{\frac{\partial }{\partial l_{k,i}}}
| 2,562
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|
that the determinant of the matrix $M$ (Eq. \[determinant\]) doesn’t vanish under this homomorphism.
For example, we can work over the ring $\Z_6$ and use the element $-1$ as a substitute for $\gamma$. Since $(-1)^6 = 1$ all of the calculations we did with $\gamma$ carry through. In addition, the resulting determinant of $M$ is non zero when setting $\gamma= -1$ and so we can complete the recovery process. More formally, define the homomorphism $\tau:\Z_6[\gamma]/(\gamma^6 - 1)\mapsto \Z_6$ by extending the identity homomorphsim on $\Z_6$ using $\tau(\gamma)=-1$. Observe that the determinant of the matrix $M$ in Eq. (\[determinant\]) doesn’t vanish under this homomorphism, $\tau(\det(M))=-4=2$.
A more interesting example is the ring of integers modulo $3$, which we denote by $\F_3$ to highlight that it is also a field. We can use the homomorphsim $\phi: \Z_6[\gamma]/(\gamma^6-1)\mapsto \F_3$ by extending the natural homomorphsim from $\Z_6$ to $\F_3$ (given by reducing each element modulo $3$) using $\phi(\gamma)=-1$. Again the determinant in Eq. (\[determinant\]) doesn’t vanish. This also shows that our scheme can be made to be [*bilinear*]{}, as defined in [@RazborovY06], since the answers of each server become linear combinations of database entries over a field.
An Alternative Construction
---------------------------
In the construction above we used the special properties of Grolmusz’s construction, namely that the non-zero inner products are in the special set $S = {\{1,3,4\}}$. Here we show how to make the construction work with any matching vector family (over $\Z_6$). This construction also introduces higher order differential operators, which could be of use if one is to generalize this work further.
Suppose we run our protocol (with $\cR = \cR_{6,6}$) using a matching vector family with $S=\Z_6\setminus{\{0\}}$. Then, we cannot claim that $c_2=c_5=0$, but we still have $c_0=a_\tau\gamma^{{\langle \bu_\tau,\bz \rangle}}$. We can proceed by asking for the ‘second order’ derivative of $F(
| 2,563
| 4,334
| 3,348
| 2,376
| 3,380
| 0.772757
|
github_plus_top10pct_by_avg
|
thbb{Z}})$ such that $m'_{1i}=m_i/m_0$ for all $i\in \{1,2,\dots ,k\}$. Then $(X_1^{(M')})^{m_0}-q\in J$, and hence $J$ is the intersection of the (finite number of) ideals $J+(X^{(M')}_1-q')$, where $q'\in {\bar{{\Bbbk }}}$, ${q'}^{m_0}=q$. By assumption, $J+(X^{(M')}_1-q')$ is generated by $X^{(M')}_1-q'$ and by elements of the form $q''-\prod _{i=2}^k (X^{(M')}_i)^{m'_i}$, where $m'_2,\dots ,m'_k\in {\mathbb{Z}}$ and $q''\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then the claim follows by the induction hypothesis.
The ideals in Eq. are prime ideals of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$, since the quotient ring ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J
\simeq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k-l]$ is an integral domain.
Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by polynomials of the form $$\big(q_1-\prod _{i=1}^k x_i^{m_{1i}}\big)
\big(q_2-\prod _{i=1}^k x_i^{m_{2i}}\big)\cdots
\big(q_l-\prod _{i=1}^k x_i^{m_{li}}\big),$$ where $l\in {\mathbb{N}}$, $m_{j1},\dots ,m_{jk}\in {\mathbb{Z}}$ and $q_j\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$ for all $j\in \{1,2,\dots ,l\}$. Let $V\subset {\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$ be an open subset. Then the union of the subsets $$V_{n_1,\dots ,n_k}=\{p\in V\,|\,p_1^{n_1}=1,\dots ,p_k^{n_k}=1\},
\quad n_1,\dots ,n_k\in {\mathbb{N}},$$ is dense in $V$ with respect to the Zariski topology. \[le:gendensity\]
We can assume that $V={\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$. Moreover, it suffices to prove the lemma for the irreducible components of $V$. Thus, as a first reduction, $J$ can be assumed to be as in the assumptions of Lemma \[le:torusideal\]. Then by Lemma \[le:torusideal\] we may assume that $J$ is an ideal as in Eq. , where $M\in \mathrm{GL}(k,{\mathbb{Z}})$, $l\in \{0,1,\dots ,k\}$, and $q_1,\dots ,q_l\in {{\bar{{\Bbbk }}}^\times }$ are roots of $1$. Then $V\simeq ({{\bar{{\Bb
| 2,564
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|
m{on}\quad\mathcal{D}_{-}=\{0\}=\mathcal{D}_{+},\qquad G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$
The graphical limit is $(0,[0,1])$.$\qquad\blacksquare$
[1.4]{} In these examples that we consider to be the prototypes of graphical convergence of functions to multifunctions, $G(y)=0$ on $\mathcal{R}_{-}$ because $g_{n}(y)\rightarrow0$ for all $y\in\mathcal{R}_{-}$. Compare the graphical multifunctional limits with the corresponding usual pointwise functional limits characterized by discontinuity at $x=0$. Two more examples from @Sengupta2000 that illustrate this new convergence principle tailored specifically to capture one-to-many relations are shown in Fig. \[Fig: Example2\_1\] which also provides an example in Fig. \[Fig: Example2\_1\](c) of a function whose iterates do not converge graphically because in this case both the sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$are empty. The power of graphical convergence in capturing multifunctional limits is further demonstrated by the example of the sequence $(\sin n\pi x)_{n=1}^{\infty}$ that converges to $0$ both $1$-integrally and test-functionally, Eqs. (\[Eqn: intsin\]) and (\[Eqn: testsin\]).
[$\quad$(b) $F(x)=1$ on $\mathcal{D}_{-}=\{0\}$ and $G(y)=0$ on $\mathcal{R}_{-}=\{1\}$. Also $F(x)=-1/2,\textrm{ }0,\textrm{ }1,\textrm{ }3/2$ respectively on $\mathcal{D}_{+}=(0,3],\textrm{ }\{2\},\textrm{ }\{0\},\textrm{ }(0,2)$ and $G(y)=0,\textrm{ }0,\textrm{ }2,\textrm{ }3$ respectively on $\mathcal{R}_{+}=(-1/2,1],\textrm{ }[1,3/2),\textrm{ }[0,3/2),\textrm{ }[-1/2,0)$. ]{}
[$\quad$(c) For $f(x)=-0.05+x-x^{2}$, no graphical limit as $\mathcal{D}_{-}=\emptyset=\mathcal{R}_{-}$.]{}
[$\quad$(d) For $f(x)=0.7+x-x^{2}$, $F(x)=\alpha$ on $\mathcal{D}_{-}=[a,c]$, $G_{1}(y)=a$ and $G_{2}(y)=c$ on $\mathcal{R}_{-}=(-\infty,\alpha]$. Notice how the two fixed points and their equivalent images define the converged limit rectangular multi. As in example (1) one has $\mathcal{D}_{-}=\mathcal{D}_{+}$; also $\mathcal{R}_{-}=\mathcal{R}_{+}$.]{}
It is
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| 0.781736
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a_{3k}\}\setminus\{a_{n-3k}\}$ (size $k-1$).
The reasoning behind this lemma is that there exist sets $X \subset \mathbb{Z}_{k+1}^{2} \times \mathbb{Z}$ that are missing exactly $k+1$ points in every $\mathbb{Z}_{k+1}^{2}$ layer and can be tiled with strings. If we take $d = 2$ in Lemma \[otherlemma\], we would like to choose such a set $X$ and a set $A \subset \mathbb{Z}_{k+1}^{2}$ (abusing notation slightly, as $\mathbb{Z}_{k+1}^{2}$ is not actually a subset of $\mathbb{Z}^2$) such that the resulting $B$ in Lemma \[otherlemma\] is disjoint from $X$. Then $(\mathbb{Z}_{k+1}^{2} \times \mathbb{Z})\setminus(B \cup X)$ contains either 2 or 0 points in each $\mathbb{Z}_{k+1}^{2}$ layer, which is what we wanted.
In order for this construction to work, we need the set $B \cap (A \times \{n\})$ to be a hole whenever it has size $k+1$, and to be a subset of a hole of size $k+1$ whenever it has size $k-1$, so that we actually can tile the required points with strings. By observing the forms of the sets $B \cap (A \times \{n\})$ in the proof of Lemma \[otherlemma\], we see that it is sufficient to choose the $a_n$ such that for all $n$, $\{a_n, \ldots, a_{n+k}\}$ is a hole. Here we regard the indices $n$ of the points $a_n$ of $A$ as integers mod $3k$, so $a_{3k+1} = a_1$ and so on. The following proposition says that we can do this.
\[anprop\] There exists a set $A = \{a_1, \ldots, a_{3k}\} \subset \mathbb{Z}_{k+1}^{2}$ such that for all $n$, $\{a_n, \ldots, a_{n+k}\}$ contains either one point in every row or one point in every column. Here the indices are regarded as integers *mod* $3k$.
For $n = 1, \ldots, k+1$, let $a_n = (n-1,n-1)$.\
For $n = k+2, \ldots, 2k-1$, let $a_n = (n-k-2,n-k-1)$.\
For $n = 2k, 2k+1, 2k+2$, let $a_n = (n-k-2,n-2k)$.\
For $n = 2k+3, \ldots, 3k$, let $a_n = (n-2k-3,n-2k)$.\
Note that all the $a_n$ are distinct. Let us regard the first coordinate as horizontal and the second as vertical.\
Then, for $n = 1, \ldots, 2k$, $\{a_n, \ldots, a_{n+k}\}$ contains one point in every column.\
For $n
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| 0.782494
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github_plus_top10pct_by_avg
|
\sim \Sigma (1,0) - \frac{\lambda_b}{2} (0,0)\,,\end{aligned}$$ where $(i,j) = \mathcal{O}^{\Delta U=i}_{\Delta U_3=j}$, and the appearing combination of CKM matrix elements are $$\begin{aligned}
\Sigma &\equiv \frac{V_{cs}^* V_{us} - V_{cd}^* V_{ud}}{2}\,, \qquad
-\frac{\lambda_b}{2} \equiv -\frac{V_{cb}^* V_{ub}}{2} = \frac{V_{cs}^* V_{us} + V_{cd}^* V_{ud} }{2}\,, \end{aligned}$$ where numerically, $|\Sigma| \gg |\lambda_b|$. The corresponding amplitudes have the structure $$\begin{aligned}
\mathcal{A} = \Sigma ( A_{\Sigma}^s - A_{\Sigma}^d ) - \frac{\lambda_b}{2} A_b\,,\end{aligned}$$ where $A_{\Sigma}^s$, $A_{\Sigma}^d$ and $A_b$ contain only strong phases and we write also $A_{\Sigma}\equiv A_{\Sigma}^s - A_{\Sigma}^d$.
For the amplitudes we use the notation $$\begin{aligned}
\mathcal{A}(K\pi) &\equiv \mathcal{A}(\overline{D}^0 \rightarrow K^+\pi^-)\,, \\
\mathcal{A}(\pi\pi) &\equiv \mathcal{A}(\overline{D}^0 \rightarrow \pi^+\pi^-)\,, \\
\mathcal{A}(KK) &\equiv \mathcal{A}(\overline{D}^0 \rightarrow K^+K^-)\,, \\
\mathcal{A}(\pi K) &\equiv \mathcal{A}(\overline{D}^0 \rightarrow \pi^+ K^-)\,.\end{aligned}$$ The U-spin related quartet of charm meson decays into charged final states can then be written as [@Brod:2012ud; @Muller:2015lua; @Muller:2015rna] $$\begin{aligned}
{\mathcal{A}}(K\pi) &= V_{cs} V_{ud}^* \left(t_0 - \frac{1}{2} t_1 \right)\,, \label{eq:decomp-1}\\
{\mathcal{A}}(\pi\pi) &= -\Sigma^*\left(t_0 + s_1 + \frac{1}{2} t_2 \right)
-\lambda_b^*\left(p_0 - \frac{1}{2} p_1 \right)\,, \label{eq:decomp-2}\\
{\mathcal{A}}(KK) &= \Sigma^*\left(t_0 - s_1 + \frac{1}{2} t_2 \right)
-\lambda_b^* \left(p_0 + \frac{1}{2} p_1 \right)\,, \label{eq:decomp-3}\\
{\mathcal{A}}(\pi K) &= V_{cd} V_{us}^* \left(t_0 + \frac{1}{2} t_1 \right)\,. \label{eq:decomp-4}\end{aligned}$$ The subscript of the parameters denotes the level of U-spin breaking at which they enter. We write $A(K\pi)$ and $A(\pi K)$ for the Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) amplitud
| 2,567
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|
gamma^{\mu}\ell)+B^{'}q_{\mu}(\bar{\ell}\gamma^{\mu}\ell)
+C^{'}P_{\mu}(\bar{\ell}\gamma^{\mu}\gamma_5\ell)\nonumber\\&+&
D^{'}q_{\mu}(\bar{\ell}\gamma^{\mu}\gamma_5\ell)+A(\bar{\ell}\ell)
+B(\bar{\ell}\gamma_5\ell)+iC(P_{\mu}q_{\nu}-P_{\nu}q_{\mu})(\bar{\ell}\sigma^{\mu\nu}\ell)
\nonumber\\&+&D(P_{\mu}q_{\nu}-P_{\nu}q_{\mu})(\epsilon^{\mu\nu\alpha\beta}
\bar{\ell}\sigma_{\alpha\beta}\ell)\Bigg]\,,\end{aligned}$$ where $$\begin{aligned}
A^{'}&=&(2C^{eff}_9+C_{LL}+C_{LR}+C_{RL}+C_{RR})f_{+}-4\hat{m}_bC^{eff}_7
\frac{f_T}{1+\hat{m}_\pi}\,, \nonumber \\
B^{'}&=&(2C^{eff}_9+C_{LL}+C_{LR} +C_{RL}+C_{RR})f_{-}
+4\hat{m}_bC^{eff}_7\frac{1-\hat{m}_\pi}{\hat{s}}f_T\,, \nonumber \\
C^{'}&=&\Big[2C_{10}+C_{LR}+C_{RR}-(C_{LL}+C_{RL})\Big]f_{+}\,, \nonumber \\
D^{'}&=&\Big[2C_{10}+C_{LR}+C_{RR}-(C_{LL}+C_{RL})\Big]f_{-}\,, \nonumber \\
A&=&(C_{LRLR}+C_{LRRL}+C_{RLLR}+C_{RLRL})\frac{m_B(1-\hat{m}^2_\pi)}{\hat{m}_b}f_0\,, \nonumber \\
B&=&\Big[C_{LRLR}+C_{RLLR}-(C_{LRRL}+C_{RLRL})
\Bigg]\frac{m_B(1-\hat{m}^2_\pi)}{\hat{m}_b}f_0\,, \nonumber \\
C&=&-4C_T\frac{f_T}{m_B+m_\pi}\,, \nonumber \\
D&=&4C_{TE}\frac{f_T}{m_B+m_\pi}\,.\nonumber\end{aligned}$$ with $$\begin{aligned}
f_-=\frac{1-\hat{m}^2_\pi}{\hat{s}}(f_0-f_+)\,. \nonumber\end{aligned}$$ We would like to note that in calculating the double-decay width we take into account of the massless case. However, the calculation for the massive case is given in the Appendix. Using the matrix element in Eq.(9), the double differential decay width can be calculated as: $$\begin{aligned}
\frac{d^2\Gamma}{d\hat{s}dcos\theta}&=&\,\frac{G^2_F\alpha^2}{2^{13}\pi^5}\,
|V_{tb}V^*_{td}|^2m^3_B\lambda^{1/2}(1,\hat{m}^2_{\pi},\hat{s})\nonumber\\
&\times&\Bigg[\Big[-\Big(\left|A^{'}\right|^2+\left|C^{'}\right|^2\Big)\lambda+4\lambda
\hat{s}m^2_B\Big(\left|C\right|^2+4\left|D\right|^2\Big)\Big]cos^2\theta\nonumber\\
&-&\Big[16Re\Big(D^{'}D^{*}\Big)+Re(AC^{*})+8Re(BD^{*})\Big]\lambda^{1/2}\hat{s}cos\theta
\nonumber\\&+&\Big(\left|A^{'}\right|^2+\left|C^{'}\right|^2\Big)\lambda+
\Big(\left|A\right|^
| 2,568
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|
sure the number of unbounded subtrees of different colors is exactly $m$, an additional precaution must be taken. Precisely, a fourth condition is added to the event $\hat{N}\in B_{\varepsilon}$:
- For any $k\in\{1,\dots,m\}$, the argument of the point $Y_{k}$ of $\hat{N}\cap B(r e^{\i 2k\pi/m},\varepsilon)$ belongs to $(2k\pi/m-\varepsilon,2k\pi/m)$.
Thanks to $(\spadesuit)$, each sector of the ball $B(O,r+1)$ with angle $2\pi/m$ contains (at least) one of the points $Y_1,\dots Y_m$. Assume $N\in A_{3\varepsilon}\cap B_{\varepsilon}$ which still occurs with positive probability. By construction, the origin $O$ has exactly $m$ children in the ball $B(O,R)$. Let us consider a point $X\in N\setminus \{O,Y_1,\dots,Y_m\}$ such that $|X|\geq R\geq r+1$. Then $B(O,|X|)\cap B(X,|X|)$ contains a sector of the ball $B(O,r+1)$ with angle $2\pi/3$ and so one of the $Y_1,\dots Y_m$. The origin $O$ cannot be the ancestor of $X$. This proves that $O$ is exactly of degree $m$ and ends the proof.\
This latter argument no longer works when $m$ is equal to $1$ or $2$.
Following the construction for $m=5$, there exists $r_1 $ and $r_2>0$ such that there exist with positive probability two semi-infinite paths $\gamma_1$ and $\gamma_2$ included in the cones $C_{0,\varepsilon,r_1}$ and $C_{\pi,\varepsilon,r_2}$. The following event has a positive probability:
- For a given increasing subsequence $(\theta_j)_{j\in {{\mathbb N}}}$ of $[0,\pi)$ with a sufficiently small step, and for a sufficiently small $\varepsilon>0$: $$N\Big(B\big((r_1\wedge r_2)(1+\cos(\theta_j))e^{\i \theta_j},\varepsilon\big)\Big)=1,\qquad N\Big(B\big(-(r_1\wedge r_2)(1+\cos(\theta_j))e^{\i \theta_j},\varepsilon\big)\Big)=1,$$
- For all integers $n$ and $m$ such that $0\leq n\leq (r_2 -r_1)/2\varepsilon$ and $0\leq n\leq (r_1 -r_2)/2\varepsilon$, if they exist: $$N\Big(B\big((r_1+2n\varepsilon,0),\varepsilon\big)\Big)=1,\qquad N\Big(B\big((0,r_2-2n\varepsilon),\varepsilon\big)\Big)=1.$$
- The rest of $B(O,r_1\vee r_2)$ is empty.
The idea is that in
| 2,569
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| 0.778782
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|
s to prove the statements resumed in Section \[sect:results\]: in Section \[sect:3.1\] we give an abstract regularity criterion, in Section \[sect:3.2\] we prove a regularity result for iterated integrals.
A regularity criterion based on interpolation {#sect:3.1}
---------------------------------------------
Let us first recall some results obtained in [@[BC]] concerning the regularity of a measure $\mu $ on ${\mathbb{R}}^{d}$ (with the Borel $\sigma$-field). For two signed finite measures $\mu ,\nu $ and for $k\in {\mathbb{N}%
}$ we define the distance$$d_{k}(\mu ,\nu )=\sup \Big\{\Big\vert \int fd\mu -\int fd\nu \Big\vert %
:\Vert f\Vert _{k,\infty }\leq 1\Big\}. \label{reg2}$$If $\mu $ and $\nu $ are probability measures, $d_0$ is the total variation distance and $d_1$ is the Fortét Mourier distance. In this paper we will work with an arbitrary $k\in {\mathbb{N}}$. Notice also that $d_{k}(\mu ,\nu
)=\left\Vert \mu -\nu \right\Vert _{W_{\ast }^{k,\infty }}$ where $W_{\ast
}^{k,\infty }$ is the dual of $W^{k,\infty }.$
We fix now $k,q, h\in {\mathbb{N}}$, with $h\geq 1$, and $p>1$. Hereafter, we denote by $p_{\ast }=p/(p-1)$ the conjugate of $p.$ Then, for a signed finite measure $\mu $ and for a sequence of absolutely continuous signed finite measures $\mu _{n}(dx)=f_{n}(x)dx$ with $f_{n}\in C^{2h+q}({\mathbb{R}%
}^{d}),$ we define$$\pi _{k,q,h,p}(\mu ,(\mu _{n})_{n})=\sum_{n=0}^{\infty }2^{n(k+q+d/p_{\ast
})}d_{k}(\mu ,\mu _{n})+\sum_{n=0}^{\infty }\frac{1}{2^{2nh}}\left\Vert
f_{n}\right\Vert _{2h+q,2h,p}. \label{reg3}$$
Notice that $\pi _{k,q,h,p}$ is a particular case of $\pi _{k,q,h,\mathbf{e}%
} $ treated in [@[BC]]: just choose the Young function $\mathbf{e}%
(x)\equiv \mathbf{e}_{p}(x)=|x|^{p}$, giving $\beta _{\mathbf{e}%
_{p}}(t)=t^{1/p_{\ast }}$ (see Example 1 in [@[BC]]). Moreover, $\pi
_{k,q,h,p}$ is strongly related to interpolation spaces. More precisely, let $$\overline{\pi }_{k,q,h,p}(\mu )=\inf \{\pi _{k,q,h,p}(\mu ,(\mu
_{n})_{n}):\mu _{n}(dx)=f_{n}(x)dx,\quad f_{n}\in C^{2h+q}({\mathbb
| 2,570
| 1,062
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| 3,255
| 0.773746
|
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|
)_R$ model[^1] was discussed in ref. [@su421]. The SU421 class of heterotic–string models differs from the HSPSM models in the breaking of $SU(2)_R\rightarrow U(1)_R$ directly at the string level. Similar to the HSPSM, the SU421 heterotic–string models admit the $SO(10)$ embedding and the chiral states are obtained from the spinorial [**16**]{} representations of $SO(10)$ which decomposes in the following way: F\_L\^[i]{} &=& ( 4 ,2, 0) = (3 ,2, [13]{}, 0) + (1,2, -[1]{}, 0) = [ud]{}\^i+[e]{}\^i,\[SU421fl\]\
U\_R\^[i]{} &=& ([4]{},1,-[12]{}) = ([3]{},1,-[13]{},-[12]{}) + (1,1,+[1]{},-[12]{}) = [u\^[c]{}]{}\^i+[N\^[c]{}]{}\^i,\[SU421ur\]\
D\_R\^[i]{} &=& ([4]{},1,+[12]{}) = ([3]{},1,-[13]{},+[12]{}) + (1,2,+[1]{},+[12]{}) = [d\^[c]{}]{}\^i+[e\^[c]{}]{}\^i. \[SU421dr\] The first and second equalities show the decomposition under $SU(4)_C\times SU(2)_L\times U(1)_R$ and $SU(3)_C\times SU(2)_L\times U(1)_{B-L}\times U(1)_R$, respectively. The electroweak $U(1)_Y$ current is given by U(1)\_Y=[12]{}U(1)\_[B-L]{}+U(1)\_R. \[ewu1current\] From eq. (\[SU421fl\]) we note that $F_L$ produces the quarks and leptons weak doublets, and that $U_R$ and $D_R$ produces the right–handed weak singlets. The two Higgs multiplets of the Minimal Supersymmetric Standard Model, $h^u$ and $h^d$, are given by, h\^d &=& ( 1 ,2,-[12]{}),\
h\^u &=& ( 1 ,2,+[12]{}). \[SU421mssmhigss\] The heavy Higgs states that are responsible for breaking $SU(4)_C\times U(1)_{R}$ gauge symmetry to the Standard Model groups $SU(3)\times U(1)_Y$ are given by the fields &=& ([4]{},1,-[12]{})\
[H]{} &=& ([ 4]{},1,+[12]{}) \[SU421Higgs\] The SU421 heterotic–string models may also contain states that transform as $$(6,1,0)= ({3},1,{1\over3},0)+ ({\overline 3},1,-{1\over3},0).$$ These multiplets arise from the vectorial [**10**]{} representation of $SO(10)$. These coloured states generate proton decay from dimension five operators, and therefore must be sufficiently heavy to be in agreement with the proton lifet
| 2,571
| 1,015
| 3,609
| 2,694
| null | null |
github_plus_top10pct_by_avg
|
tinuing from above, and using item 2 and 3 of Lemma \[l:N\_is\_regular\], $$\begin{aligned}
\circled{5}
\leq& q'(g(z_t)) \cdot \lrp{\frac{8\beta^2 \LN}{\cm} + \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}\\
\leq& q'(g(z_t)) \cdot \lrp{\frac{m}{2} \|z_t\|_2}
+ q'(g(z_t)) \cdot \lrp{ \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}
\end{aligned}$$ Where the second inequality is by our definition of $\Rq$ in the Lemma statement, which ensures that $\frac{8\beta^2 \LN}{\cm} \leq \frac{m}{2} \Rq \leq \frac{m}{2} \|z_t\|_2 $.
Thus $$\begin{aligned}
& \circled{1} + \circled{2} + \circled{4} + \circled{5}\\
\leq& -m q'(g(z_t)) \|z_t\|_2 + \LR \lrn{y_t - y_0}_2 + \frac{m}{2} q'(g(z_t)) \|z_t\|_2 + q'(g(z_t)) \cdot \lrp{ \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}\\
\leq& -\frac{m}{2} q'(g(z_t)) \|z_t\|_2 + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2\\
\leq& - \lambda f(z_t) + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2
\end{aligned}$$
where the second inequality uses $q'\leq 1$ from item 3 of Lemma \[l:qproperties\], the third inequality uses our definition of $\lambda$ in .
Combining the three cases, can be upper bounded with probability 1: $$\begin{aligned}
&d f(z_t)
\leq - \lambda f(z_t) + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2 + \lin{\nabla f(z_t), 2 \cm \gamma_t \gamma_t^T dV_t + \lrp{N_t + N(y_t) - N(y_0)} dW_t }
\end{aligned}$$
To simplify notation, let us define $G_t \in \Re^{1\times 2d}$ as $G_t:= \lrb{\nabla f(z_t)^T 2\cm \gamma_t \gamma_t^T, \nabla f(z_t)^T \lrp{N_t + N(y_t) - N(y_0)}}$, and let $A_t$ be a $2d$-dimensional Brownian motion from concatenating $A_t = \cvec{V_t}{W_t}$. Thus $$\begin{aligned}
d f(z_t) \leq -\lambda f(z_t) dt + \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2} + G_t dA_t.
\end{aligned}$$ We will study the Lyapunov function $$\begin{aligned}
\mathcal{L}_t:= f(z_t) - \int_0^t e^{
| 2,572
| 2,941
| 1,934
| 2,295
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|
a certain tree can be found as a subgraph of ${\mathcal{C}}_m$.
\[lem:col\] [Let $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$ be a dynamic $s$-uniform hypergraph which satisfies the $\beta$-balanced, $\varepsilon$-visibility and $c_0$-size properties.]{} Suppose that $c>0$ is an arbitrary constant and $k=C\log n$ for some constant $C{\geqslant}1$. There exists $\Theta(n){\leqslant}m{\leqslant}n$ such that the probability that ${\mathcal{C}}_{m}$ contains a $c$-loaded $k$-vertex tree [with $r$ red vertices in its blue-red colouring]{} is at most $$n^{c_0+3}\, \exp\{4k\log({2\beta d})-r\varepsilon \log(n)/2-c(d-1)(k-r-1)\}.$$ Moreover, with high probability, $r={\mathcal{O}}(1/\varepsilon)$.
The proof, presented in Appendix \[sub:exist\], involves an extension of the witness tree technique. This method might be of independent interest in the study of random hypergraphs.
We now explain how to recursively build a witness graph if there exists a bin whose load is higher than a certain threshold. The [*minimum load*]{} of $D_t$ is the number of balls in the least-loaded bin in $D_t$ (the set of $d$ choices of $D_t$). Clearly, if ball $t$ is placed at height $h$ then $D_t$ has minimum load at least $h$.
#### Construction of the Witness Graph {#construction-of-the-witness-graph .unnumbered}
Suppose that there exists a bin with load $\ell+c+1$. Let $R$ be the $d$-choice corresponding to the ball at height $\ell+c$ in this bin. Then the minimum load of $R$ is $\ell+c$. We start building the witness tree in ${\mathcal{C}}_m$ whose root is $R$. For every bin $i\in R$, consider the $\ell$ balls in bin $i$ at height $\ell + c - j$, for $j=1,\ldots, \ell$, and let $D^{i}_{t_j}$ be the $d$-choice corresponding to the ball in bin $i$ with height $\ell+c-j$. These $\ell$ balls exist as the minimum load of $R$ is $\ell + c$. We refer to set $\{D^i_{t_j} \mid i\in R,\,\, 1{\leqslant}j{\leqslant}\ell\}$ as the set of *children* of $R$, where the minimum load of $D^i_{t_j}$ is $\ell+c-j-1$. All children of $R$ are conne
| 2,573
| 655
| 1,737
| 2,546
| 1,112
| 0.794264
|
github_plus_top10pct_by_avg
|
\; \Bigg( \frac{1}{16 \pi}\bigg[ R +\nu
\sqrt{
-5 R^4 -9 \Big( 8 R^{\mu\nu}R^{\alpha\beta}R^{\sigma\rho}_{\;\,\;\,\,\mu\alpha}R_{\sigma\rho\nu\beta} - 32 R\,R^{\mu\nu\alpha\beta}R_{\mu\;\,\alpha}^{\;\,\sigma\;\,\rho}R_{\nu\sigma\beta\rho}
} \nonumber\\
& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \overline{ ~ -7 \big( R^{\mu\nu\alpha\beta} R_{\mu\nu\alpha\beta} \big)^2 ~\Big)
} \;\bigg] + \curv{L}_m \Bigg). \nonumber\\\end{aligned}$$
#### Correction to the Friedmann equation
\
From this action, we get the acceleration equation : $$\begin{aligned}
3 H(t)^2+2 \dot{H}(t)-6 \nu \sqrt{66} H(t)^2 \left(3 H(t)^2+4 \dot{H}(t)\right) = -8\pi p .
\end{aligned}$$ In analogy with equations (12) to (14), this last modification leads to an $H^4$ modification of the Friedmann equation. Now we are going to see that this unique correction is very interesting regarding the problem of the Big Bang singularity as it was shown in Ref [@16]. It is convenient to introduce the dimensional quantity $\rho_c$, critical energy density of the universe, and the constant parameter $ \epsilon $, such that $$\begin{aligned}
\frac{\epsilon}{8 \pi \rho_c}=- 2 \nu \sqrt{66}\,.\end{aligned}$$ Thus, the modified Friedmann equation induced by the action $S_4$ is : $$\begin{aligned}
3 H(t)^2+ \epsilon \; \frac{9 H(t)^4}{8 \pi \rho_c}=8 \pi
\rho.\end{aligned}$$ The solution $H(t)^2$ that reduces to the standard Friedmann equation in the limit when $ \rho_c $ goes to infinity is : $$\begin{aligned}
3 H(t)^2 = \, \frac{4 \pi \rho_c }{ \epsilon }\left(-1+\sqrt{\frac{4 \epsilon \rho }{\rho_c}+1} \, \right).\end{aligned}$$ Furthermore, if the energy density of the universe is small compared to the critical one, $\rho \ll \rho_c$, this equation becomes : $$\begin{aligned}
H(t)^2 = \, \frac{8 \pi \rho }{3} \Big( 1 - \epsilon \, \frac{\rho}{\rho_c} \Big) + O(\rho^3).\end{aligned}$$ Choosing $\epsilon=1$, gives the loop quantum cosmology correction to the Friedmann equation [@19], and choosing $\epsilon=-\frac{1}{2}$, giv
| 2,574
| 4,995
| 745
| 2,019
| null | null |
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|
proof of Theorem \[sequentially\], and induction on the length of the pretty clean filtration it follows easily that $$\Hilb(\Ext_S^i(M,\omega_S))=\sum_{j\atop \dim S/P_j=n-i}\Hilb(\omega_{S/P_j})t^{-a_j} \quad\text{for}\quad i=0,\ldots, \dim M.$$
In particular we have
\[hilbert\] With the assumptions and notation of [*\[monomial\]*]{}, one has $$\Hilb(\Ext_S^i(S/I,S(-n)))=(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}= (-1)^{n-i}H_{n-i}(t^{-1}).$$
The first equality follows from the fact that $\omega_{S/P_j}=S/P_j(-(n-i))$ if $\dim S/P_j=n-i$, so that $\Hilb(\omega_{S/P_j})=t^{n-i}/(1-t)^{n-i}$. To obtain the second equality, we divide numerator and denominator of $(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}$ by $t^{n-i}$ and get $$(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}=(\sum_k h_{k,n-i}t^{-k})/(t^{-1}-1)^{n-i}=(-1)^{n-i}H_{n-i}(t^{-1}).$$
Let $S=K[x_1,\ldots, x_n]$ and $M$ be a graded $S$-module. We set $$b_j=\min\{k\: \Ext_S^j(M, S(-n))_k\neq 0\}.$$ Then the regularity of $M$ is given by $$\reg(M)=\max\{n-j-b_j\: j=0,\ldots, \},$$ cf. [@Ei Section 20.5].
\[regularity\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. Assume that $S/I$ is a graded pretty clean ring with filtration as in [*\[monomial\]*]{}. Then
1. $\reg(S/I)=\max\{k\: h_{ki}\neq 0 \ \ \mbox{for some i}\}=\max\{a_j\: j=0,\ldots r\}$;
2. $\Hilb(D_i(S/I)/D_{i-1}(S/I))=H_i(t)$ for all $i$.
\(a) the first equality follows immediately from Proposition \[hilbert\] and the definition of $\reg(S/I)$. The second equality results from the definition of the numbers $h_{ki}$.
\(b) By Proposition \[exti\] we have $$D_i(S/I)/D_{i-1}(S/I)\iso\Ext_S^{n-i}(\Ext_S^{n-i}(S/I,\omega_S),\omega_S).$$ Thus the assertion follows from Proposition \[hilbert\] and [@BH Theorem 4.4.5(a)].
We denote by $e(M)$ the multiplicity of a graded module.
\[independent\] Let $i$ and $k$ be integers. Then the number of factors $S/P(-k)$ in a graded pretty clean filtration of $S/I$ satisfying $\dim S/P=i$ is independent of the chosen filtration.
| 2,575
| 1,464
| 1,982
| 2,379
| 3,293
| 0.773395
|
github_plus_top10pct_by_avg
|
the coefficients $\gamma_{\lambda_0}$, $\gamma_{\frac {\lambda_0+C}2}$ for these $S$ branches agree. Let $\gamma_C^{(1)},\dots,\gamma_C^{(S)}$ be the coefficients of $y^C$ in these branches (so that at least two of these numbers are distinct, by the choice of $C$). Then the limit is defined by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2
\gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2
\gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad.$$ This is a union of quadritangent conics with (possibly) a multiple of the distinguished tangent, which must be supported on the kernel line.
The main theorem, and the structure of its proof {#proof}
------------------------------------------------
Simple dimension counts show that, for each type as listed in §\[germlist\], the union of the orbits of the marker centers is a set of dimension $7$ in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$; hence it is a dense set in a component of the PNC. In fact, marker centers of type I, III, IV, and V have 7-dimensional orbit, so the corresponding components of the PNC are the orbit closures of these points.
Type II marker centers are points $(\alpha, {{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, where $\alpha$ is a rank-1 matrix whose image is a general point of a nonlinear component of ${{\mathscr C}}$. The support of ${{\mathscr X}}$ contains a conic tangent to the kernel line; this gives a 1-parameter family of 6-dimensional orbits in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, accounting for a component of the PNC.
We can now formulate a more precise version of Theorem \[main\]:
\[mainmain\] Let ${{\mathscr C}}\subset {{\mathbb{P}}}^2_{{\mathbb{C}}}$ be an arbitrary plane curve. The marker germs listed in §\[germlist\] determine components of the PNC for ${{\mathscr C}}$, as explained above. Conversely, all components of the PNC are determined by the marker germs of type I–V listed in §\[germlist\].
By the considerations in §\[ident\], thi
| 2,576
| 514
| 1,736
| 2,629
| null | null |
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|
Normal 0.8-1.4 ml EDTA & ACT tubes 2-8°C To confirm the viral infection RT-PCR
RDT
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RT-PCR: Reverse Transcription-Polymerase Chain Reaction; RDT: Rapid Diagnostic Test; ELISA: Enzyme-linked immunosorbent assay.
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------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 2,577
| 5,421
| 548
| 1,753
| null | null |
github_plus_top10pct_by_avg
|
$& 21.9& 1.6\
05498&01& 22.91& 25.5& 25.2$\pm0.2$ & 23.1& 7.1& 22.28& 25.0?& 24.4 $\pm1.2$& 22.5& 6.0\
51835&55& 23.01& 26.0& 25.7$\pm0.3$ & 23.1& 11.2& 22.41& 23.8 & 23.4 $\pm0.2$& 23.0& 1.6\
00784&05& 23.28& 25.0& 24.7$\pm0.2$ & 23.6& 2.8& 22.80& 24.0 & 23.7 $\pm0.2$& 23.4& 1.2\
05696&02& 22.73& 24.7& 24.4$\pm0.2$ & 23.0& 3.6& 22.32& 23.8 & 23.5 $\pm0.2$& 22.8& 1.9\
05696&03& 23.14& 24.2& 23.9$\pm0.15$& 23.9& 0.9& 22.68& 24.0 & 23.7 $\pm0.2$& 23.2& 1.6\
07671&07& 22.35& 25.6& 25.2$\pm0.2$ & 22.4& 13.2& 22.24& 25.0?& 24.4 $\pm1.2$& 22.4& 6.3\
07671&15& 22.36& 25.5& 25.1$\pm0.2$ & 22.5& 11.9& 22.22& 24.6 & 24.3 $\pm0.3$& 22.4& 5.5\
11922&11& 22.65& 24.9& 24.6$\pm0.2$ & 22.9& 4.8& 21.93& 25.2?& 24.5 $\pm1.5$& 22.0& 9.3\
stack& & 21.47& 24.9& 24.3$\pm0.05$& 21.6& 12.3& 21.60& &&\
[ccccc]{} ID&Tile&$z$&$(\mathrm{{F606W}} - \mathrm{{F850LP}})$&SFR [F606W]{}\
& & & &\[$\mathrm{M}_\odot/\mathrm{year}$\]\
19965& 23 &1.90&–0.2$\pm0.2$& 11\
30792& 82 &1.929& 0.5$\pm0.3$& 4\
18324& 19 &1.990& 0.6$\pm0.2$& 15\
05498& 01 &2.075& 0.8$\pm1.2$& 2\
51835& 55 &2.179& 2.3$\pm0.4$& 1.5\
00784& 05 &2.282& 1.0$\pm0.3$& 4\
05696& 02 &2.386& 0.9$\pm0.3$& 6\
05696& 03 &2.386& 0.1$\pm0.3$& 9\
07671& 07 &2.436& 0.8$\pm1.2$& 3\
07671& 15 &2.436& 0.9$\pm0.4$& 3\
11922& 11 &2.539& 0.1$\pm1.5$& 5\
stack& &2.3 & $<0.0$ & 6\
![\[fig:SSP\_colours\_AB\] Observed colours ($\mathrm{{F606W}} - \mathrm{{F850LP}}$) of the sample from PSF peak subtraction (circles), the open symbol marks the upper limit for the ‘stacked’ AGN. Overplotted are two single burst models from [solar metallicity @bruz03] (solid lines). The upper curve is for a passively evolving burst at $z=5$, the lower for burst of 100 Myr age, relative to each redshift. The dot-dashed lines are mixtures between the two, with a (from top) 0.1%, 1% and 10% fraction of mass of the 100 Myr population on top of 99.9%, 99% and 90% of the $z=5$ population. ](f6.eps){width="8.5cm"}
![\[fig:nuc\_host\_col\] Nuclear vs. host galaxy colours for the objects in the sample with available col
| 2,578
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|
a\|}_p^p + C(M,p),$$ which immediately concludes the lemma with ${\|\theta\|}_p^p \leq \max({\|\theta_0\|}_p^p, C(M,p)\delta^{-1})$.
We now turn to proving that implies something analogous to Lemma \[lem:finite\_p\_bounded\]. Let $u(t)$ be as in *(ii)* of Theorem \[thm:Decay\]. One can verify that is equivalent to $$\lim_{k \rightarrow \infty} \sup_{\tau \in [0, \infty)} {\|\left(\theta(\tau) - k\right)_+\|}_1 = 0, \label{def:theta_equi}$$ which is precisely the condition of uniform equi-integrability which plays a key role in [@CalvezCarrillo06; @Blanchet09; @BRB10]. We may refine Lemma \[lem:finite\_p\_bounded\] in the following fashion, adapting the techniques in [@CalvezCarrillo06; @Blanchet09; @BRB10] to this setting.
\[lem:finite\_p\_bounded\_unifint\] If $\theta(\tau)$ satisfies then we have ${\|\theta(\tau)\|}_{p} \in L_\tau^\infty({\mathbb R}^+)$ for all $p < \infty$.
We proceed similar to the proof of Lemma \[lem:finite\_p\_bounded\], but now slightly refined to take advantage of . Since similar arguments have appeared in several locations (for example [@CalvezCarrillo06; @Blanchet09; @BRB10]) we sketch a proof and highlight mainly the differences that appear due to the rescaling in . Define $\theta_k(\tau,\eta) := (\theta(\tau,\eta) - k)_+$ and $$\mathcal{I} = \int \theta_k^{m-1}{\left\vert{\nabla}\theta_k^{p/2}\right\vert}^2 dx.$$ The $L^p$ norms of $\theta$ and $\theta_k$ are related through the following inequality for $1 \leq p < \infty$, $${\|\theta\|}_p^p \lesssim_p {\|\theta_k\|}_p^p + k^{p-1}{\|\theta\|}_1. \label{ineq:slicing}$$ It is important to note that the implicit constant in does not depend on $k$. Estimating the time evolution of $\theta_k$ as in Lemma \[lem:finite\_p\_bounded\], using Lemma \[lem:CZ\_rescale\] and implies, $$\begin{aligned}
\frac{d}{d\tau}{\|\theta_k\|}_p^p & = -C(p)\mathcal{I} - \int \left((p-1)\theta_k^{p} + kp\theta^{p-1}_k \right) {\nabla}\cdot \left(e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot) \ast \theta_k\right)d\eta \\
& \leq -C(p)\mathcal{I} + C(p){\|\th
| 2,579
| 1,974
| 501
| 2,685
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|
mathbb{Z}_{k+1}^2$ have size $(k+1)^2-mk$ for some $m$, and this is always odd, so we cannot use Lemma \[biglemma\]. The same is true if we replace 2 with a larger dimension, or if, as in [@gltan16], we use strings in which every $(2k+1)$th point, rather than every $(k+1)$th point, is removed. We will therefore need a new idea.
Instead of using strings in $d-1$ out of $d$ directions, we could only use them in $d-2$ directions and fill the gaps with copies of $T$ in the 2 remaining directions. We will show that this approach works in the case $d = 2$, giving a tiling of $\mathbb{Z}^4$. The strategy will be to produce a partial tiling of each $\mathbb{Z}^3$ slice and use the construction from Lemma \[biglemma\] to fill the gaps with tiles in the fourth direction.
We will again build partial tilings of $\mathbb{Z}^{2}$, and therefore of higher dimensions, from partial tilings of the discrete torus $\mathbb{Z}_{k+1}^{2}$. The following result is a special case of one proved in [@gltan16]:
\[onepoint\] If $x \in \mathbb{Z}_{k+1}^{2}$, then $\mathbb{Z}_{k+1}^{2}\setminus\{x\}$ can be tiled with strings.
Let $x = (x_1,x_2)$, where the first coordinate is horizontal and the second vertical. Since a string is a row or column minus one point, we can place a string $(\{n\} \times \mathbb{Z}_{k+1})\setminus\{(n,x_2)\}$ in each column, leaving only the row $\mathbb{Z}_{k+1} \times \{x_2\}$ empty. Placing the string $(\mathbb{Z}_{k+1} \times \{x_2\})\setminus \{x\}$ in this row completes the tiling of $\mathbb{Z}_{k+1}^{2}\setminus\{x\}$.
The sets $S$ of size 3 that we will use in Lemma \[biglemma\] will have 2 points, say $x_1$ and $x_2$, in one $\mathbb{Z}_{k+1}^{2}$ layer and one point, say $x_3$, in another layer. Every layer will contain points from exactly one such set $S$. Let $Y$ be the set constructed from $S$ in the proof of Lemma \[biglemma\]. In a given slice $\mathbb{Z}^3 \times \{n\}$, there are therefore two cases:
1. $Y \cap (S \times \{n\}) = \{x_1, x_3\} \times \{n\}$ or $\{x_2, x_3\} \times \{n\}$.
2
| 2,580
| 4,656
| 2,784
| 2,170
| 1,586
| 0.787937
|
github_plus_top10pct_by_avg
|
; @penas; @shukla1; @andriot; @shukla2]. By S-duality, the $Q$ flux is mapped to $P_a^{bc}$, and in [@Aldazabal:2006up] it was indeed shown that the second constraint in eq. is modified by the addition of the term $-P^{ae}_{[b}F_{cd]e}$, while the fourth constraint, which is the only other one that is relevant in the case of the O3 orbifold, is mapped to an equivalent quadratic constraint for $P_a^{bc}$. We now determine how the full set of quadratic constraints in eq. is modified by the inclusion of all the $P$ fluxes in the general case, and then we will analyse the particular case of the IIB/O3 and IIA/O6 models.
Our method is as follows: we first write down all the possible terms of the form $F \cdot P$ to the first equation in . These terms can only be $F_3 \wedge P_1$ and $P_1^2 \cdot F_5$ in IIB, and $P_1^1 \cdot F_4$ in IIA. Then we consider particular components of these constraints and we act on them with all possible T-dualities using eqs. , and . Requiring closure under T-duality fixes all the coefficients of all the possible terms of the form $F \cdot P$ that can be added to all the NS-NS quadratic constraints. Finally, we write the resulting expressions in covariant notation. The final result is that the Bianchi identities become $$\begin{aligned}
& 6f^{e}_{[ab}H_{cd]e}+4F_{[abc}P_{d]}+2P^{ef}_{[a}F_{bcd]ef}=0 \nonumber\\
& 3 Q^{ae}_{[b}H_{cd]e}+3f^{e}_{[bc}f^{a}_{d]e}-3P^{ae}_{[b}F_{cd]e}-P^{a,ae}F_{bcdae} +\tfrac{1}{2} P^{aefg}_{[b} F_{cd]efg}=0 \nonumber \\
& -Q^{ab}_e f^e_{cd}-4Q^{[a|e|}_{[c}f^{b]}_{d]e}-R^{abe}H_{cde}
+2{F}_{[c}P^{ab}_{d]}+P^{a,ab}{F}_{cda}+P^{b,ab}{F}_{cdb}\nonumber \\
& \qquad \qquad +P_{[c}^{abef}{F}_{d]ef}+\tfrac{1}{2}P^{a,baef}{F}_{cdaef}-\tfrac{1}{2}P^{b,abef}{F}_{cdbef}=0 \label{NSNSBianchiIIBwithP} \\
& 3R^{[ab|e|}f^{c]}_{de}+3Q^{[ab}_eQ^{c]e}_d+P^{abce}_d {F}_e-P^{a,abce}{F}_{ade}-P^{b,abce}{F}_{bde}-P^{c,abce}{F}_{cde} \nonumber \\
& \qquad \qquad + \tfrac{1}{6} P^{a,abcefg} F_{adefg} + \tfrac{1}{6} P^{b,abcefg} F_{bdefg} + \tfrac{1}{6} P^{c,abcefg} F_{cdefg} =0 \no
| 2,581
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|
xplicit formulas for them were obtained. Subbarao and Sitaramachandrarao, Huard, Williams and Zhang, and Tsumura researched the explicit formulas for $T(a,b,c)$ for $a,b,c \in{\mathbb{N}}$. The value $T (0, a, b \,; x,y)$ and their multiple sum versions have been already defined in Arakawa and Kaneko [@AraKa] for the case $x,y \in {\mathbb{Q}}$ as special cases of their multiple $L$-values.
As a three-variable function, Matsumoto continued $T(s, t, u)$ meromorphically to the whole ${\mathbb{C}}^3$ plane in [@Ma Theorem 1]. Tsumura [@Tsumurap2 Theorem 4.5], afterwards Nakamura [@Na Theorem 1.2] found functional relations for the Tornheim double zeta function. Moreover, generalizations of the functional relations are proved by Matsumoto and Tsumura [@MaTsu], and Nakamura [@NakamuraA].
In this paper, we show the following functional relation. This functional relation is essentially the same as [@NakamuraT Theorem 3.1]. Therefore we can obtain the all results in [@NakamuraT] by this formula. Zhou gave a simple proof of [@Na Theorem 1.2] in [@Zhou2]. Recently, Li gave the proof similar to Zhou’s one in [@Li], independently. By modifying their methods, we can prove the following theorem.
For $0 < x \ne y < 1$, $a,b \in {\mathbb{N}}$ and $s \in {\mathbb{C}}$, we have $$\begin{split}
&T (a,b,s \,; x,y) + (-1)^b T(b,s,a \,; x-y,x) + (-1)^a T(s,a,b \,; y, y-x) \\
&=\sum_{j=1}^{a} \binom{a+b-j-1}{a-j} \zeta (a+b+s-j \, ; y ) \bigl( \zeta (j \, ; x-y) + (-1)^j \zeta (j \, ; y-x) \bigr) \\
&\quad + \sum_{j=1}^{b} \binom{a+b-j-1}{b-j} \zeta (a+b+s-j \, ; x ) \bigl( \zeta (j \, ; y-x) + (-1)^j \zeta (j \, ; x-y )\bigr) \\
&\quad - \binom{a+b-1}{a} \zeta (a+b+s \, ; y) - \binom{a+b-1}{b} \zeta (a+b+s \, ; x) .
\label{eq:th1}
\end{split}$$ \[th:1\]
Taking $x \to y$ in the above formula, we have [@NakamuraT (3.1)] since $$\lim_{x \to y} \bigl( \zeta (a+b+s-1 \, ; y ) - \zeta (a+b+s-1 \, ; x ) \bigr)
\bigl( \zeta (1 \, ; x-y) - \zeta (1 \, ; y-x) \bigr) = 0 .$$ Define $K (a,b \,; x,y)$ by the right-hand side of (\[eq:th1
| 2,582
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|
{-1} j)^{-1} (j^{T}R^{-1} r(X) - 1)^2 ) \label{eqn:varC}\end{aligned}$$ where $\sigma_{\mathcal{C}}^2$ is the variance of the costs $\mathcal{C}$, and we define the constant $\beta \equiv (j^{T}R^{-1} j)^{-1} j^{T}R^{-1} Y$, the $N \times 1$ vector $r(X)$ such that $\{r(X)\}_{1,i} = K(X,X_i,H)$, the $N \times 1$ vector $\gamma \equiv R^{-1}(Y - j \beta)$, the $N \times 1$ vector $Y$ of the costs defined by $\{Y\}_{1,i} = C_i$, the $N \times 1$ vector $\{j\}_{1,i} = 1$, the $N\times N$ matrix $R$ defined as $\{R\}_{i,j} = K(X_i,X_j,H) + \delta_{i,j} U_i^2$, and where $\delta_{i,j}$ is the Kronecker delta function. $\{\cdot\}_{i,j}$ is our notation for the $i$th row and $j$th column of a matrix or vector.
When finding the most likely hyperparameters we maximize the likelihood function. The likelihood $L(H | \mathcal{O})$ is defined as the probability of the costs given the parameters, uncertainties and hyperparameters: $P(\mathcal{C}| \mathcal{X},\mathcal{U},H)$, the log of which is: $$\begin{aligned}
\log P(\mathcal{C}| \mathcal{X},\mathcal{U},H) = & \frac{1}{2}( - \log |R| - \log j^{T}R^{-1} j {\nonumber}\\
& - (N - 1) \log 2\pi {\nonumber}- Y^{T}(R^{-1} \\
& - (j^{T}R^{-1} j)^{-1} R^{-1} j j^{T}R^{-1}) Y) \end{aligned}$$
*Parameterizations of evaporation ramps:* The simple parameterization of the evaporation ramps is $$\begin{aligned}
\mathcal{R}_{s}(y_{i},y_{f},t_{f}) = y_{i}+\left(y_{f}-y_{i}\right)\frac{t}{t_{f}}\end{aligned}$$ where $y_i$ and $y_f$ specify the start and end amplitudes of the ramps and $t_f$ specifies the length in time.
The complex parameterization an extension of the simple form: $$\begin{aligned}
\mathcal{R}_{c}&(y_{i},y_{f},A_{1},A_{2},A_{3},t_{f}) = {\nonumber}\\
& y_{i}+\left(y_{f}-y_{i}\right)\frac{t}{t_{f}}+A_{2}t\left(t-t_{f}\right) {\nonumber}\\
& +A_{3}t\left(t-t_{f}\right)\left(t+\frac{1}{2}t_{f}\right) {\nonumber}\\
& +A_{4}t\left(t-t_{f}\right)\left(t+\frac{2}{3}t_{f}\right)\left(t+\frac{1}{3}t_{f}\right)\end{aligned}$$ where $A_1$, $A_2$ and $A_3$ correspond to the
| 2,583
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| 2,119
| 1,985
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States of
the far West:
PLACE. RECEIPTS. EXPENSES. BALANCE.
California Red Cross State $22,119.74 $10,472.63 $11,647.11
Association, Cal.
Red Cross Society, San Francisco, Cal. 55,408.83 33,434.18 21,974.65
" " " San Jose, Cal. 2,274.66 1,465.03 809.63
" " " Lompoc, Cal. 234.70 124.35 110.35
" " " Palo Alto, Cal. 222.90 153.15 69.75
" " " Ventura, Cal. 193.40 179.95 13.45
" " " San Leandro, Cal. 73.50 69.65 3.85
" " " Centerville, Cal. 165.90 133.55 2.35
" " " Suisun, Cal. 405.80 154.65 251.15
" " " Tulare, Cal. 55.70 53.45 2.25
" " " Sacramento, Cal. 6,373.43 2,749.75 3,623.68
" " " Mendocino, Cal. 105.10 102.29 2.81
" " " Grass Valley, Cal. 787.10 571.09 216.01
" " " Berkeley, Cal. 1,092.91 485.37 607.54
" " " Sausalito, Cal. 612.30 322.20 290.10
" " " Redwood City, Cal. 335.55 222.63 112.92
" " " Galt, Cal. 67.75 59.04 8.71
" " " Auburn, Cal. 257.67 200.77 56.90
" " " Santa Cruz, Cal. 493.45 393.60 99.85
" " " San Diego, Cal. 410.25 257.39 152.86
" " " Fresno, Cal. 326.00 292.30 33.70
" " " Los Angeles, Cal. 2,586.28 1,397.92 1,188.36
" " " Walnut Creek, Cal. 171.75 142.28 29.47
" " " Belvedere, Cal. 310.00 192.35 117.65
" " " Martinez, Cal. 233.31 199.80 33.51
" " " Monterey, Cal. 312.38 177.95 134.43
" "
| 2,584
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|
-hand side is the normalized outer potential. An analytic solution to this equation is given by $$\label{sol}
r_{{\rm c}}=\lambda_{{\rm D}} W\left(\frac{\alpha}{4\pi \Lambda}\frac{q}{e}\right),$$ where $W\left(x\right)$ is the inverse function of $x=W\exp\left(W\right)$, which is also known as the Lambert W-function. The potential at $r=r_{{\rm c}}$ may be approximated by $\phi\left(r_{{\rm c}}\right)=\alpha q \exp{\left(-r_{{\rm c}}/\lambda\right)}/r_{{\rm c}}$ and should be bounded by $q \exp{\left(-a/\lambda\right)}/a$ and the bare Coulomb potential $q/a$, leading to the inequality $1\leq\alpha\leq \exp{\left(r_{{\rm c}}/\lambda\right)}$. Using $\Lambda$ and $q/e$, we can rewrite this inequality as $$\label{ine1}
1\leq\alpha\leq \exp{\left(\frac{1}{4\pi \Lambda} \frac{q}{e}\right)}.$$ This estimate must be modified when $\Lambda$ is much larger than the critical value $\Lambda_{{\rm c}}$ for which the condition $a=r_{{\rm c}}$ is satisfied. When $a\gg r_{{\rm c}}$, $\phi\left(a\right)$ rather than $\phi\left(r_{{\rm c}}\right)$ must be used for a similar comparison, yielding $$\label{ine2}
1\leq\alpha\leq \exp{\left(\sqrt[3]{\frac{3}{4\pi\Lambda}}\right)}.$$ The condition $r_{{\rm c}}=a$ leads to $\Lambda_{{\rm c}}\sim\left(q/e\right)^{3/2}$, which can also be expressed as $kT_{{\rm c}}\sim eq/a$ with a critical temperature $T_{{\rm c}}$. From this, it is clear that when the temperature is above the critical value, the plasma is weakly coupled even with dust grains having relatively large charge. This indicates that the OML correction in this regime is only a minor modification, and essentially, the Yukawa-type potential in the far zone directly connects to the bare Coulomb potential.
In our simulations, since we used large dust charges with relatively small numbers of particles, the plasma parameter is smaller than the critical value. Note that the plasma parameter of dusty plasmas in space is usually huge, and so is almost always above the critical value. Our choice of dust charge was mo
| 2,585
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eta)}{\partial\theta_i \partial \theta_{\i}}$ is given by $$\begin{aligned}
H(\theta) = - \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \sum_{m = 1}^{\ell_j} \frac{\exp(\theta_i+ \theta_{\i})\I_{\{\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m\}}}{[\exp(\theta_{\sigma_j(m)})+\exp(\theta_{\sigma_j(m+1)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})]^2}.\end{aligned}$$ It follows from the definition that $-H(\theta)$ is positive semi-definite for any $\theta \in \reals^n$.
The Fisher information matrix is defined as $I(\theta) = -\E_\theta[H(\theta)]$ and given by $$\begin{aligned}
I(\theta) = \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \sum_{m = 1}^{\ell_j} \E\Bigg[ \frac{\I_{\{\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m\}}}{[\exp(\theta_{\sigma_j(m)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})]^2}\Bigg]\exp(\theta_i+ \theta_{\i}).\end{aligned}$$ Since $-H(\theta)$ is positive semi-definite, it follows that $I(\theta)$ is positive semi-definite. Moreover, $\lambda_1(I(\theta))$ is zero and the corresponding eigenvector is the all-ones vector. Fix any unbiased estimator $\widehat{\theta}$ of $\theta \in \Omega_b$. Since, $\widehat{\theta} \in \mathcal{U}$, $\widehat{\theta} - \theta$ is orthogonal to ${\boldsymbol{1}}$. The Cramér-Rao lower bound then implies that ${\E[{\|\widehat{\theta} - \theta^*\|}^2] \geq \sum_{i = 2}^d \frac{1}{\lambda_i(I(\theta))}}$. Taking the supremum over both sides gives $$\begin{aligned}
\sup_{\theta}\E[{\|\widehat{\theta} - \theta\|}^2] \geq \sup_{\theta} \sum_{i=2}^d \frac{1}{\lambda_i(I(\theta))} \geq \sum_{i = 2}^d \frac{1}{\lambda_i(I({\boldsymbol{0}}))}\;.\end{aligned}$$ If $\theta$ equals the all-zero vector, then $$\begin{aligned}
\P_\theta[\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m] = \frac{{\kappa_j-m+1 \choose 2}}{{\kappa_j \choose 2}} = \frac{(\kappa_j - m +1)(\kappa_j - m)}{\kappa_j(\kappa_j - 1)}.\end{aligned}$$ It follows from the definition that $$\begin{aligned}
I(0) &=& \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i
| 2,586
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for decay estimates. Indeed, is simply the requirement that the solution of the rescaled system remain uniformly equi-integrable. The proofs of Theorems \[thm:Decay\],\[thm:IA\] and \[thm:IA2\] are outlined in more detail in §\[sec:outline\]. Remarks on the limitations and possible extensions are made after the statements.
\[thm:Decay\] Let $d \geq 2$, $m \in [1,2-2/d]$ and ${\mathcal{K}}$ admissible. Let $u_0 \in L^1_+({\mathbb R}^d;(1+{\left\vertx\right\vert}^2)dx)\cap L^\infty({\mathbb R}^d)$.
- There exists an $\epsilon_0 > 0$ (independent of $u_0$) such that if ${\|u_0\|}_1 + {\|u_0\|}_{(2-m)d/2} < \epsilon_0$, then the weak solution $u(t)$ to which satisfies $u(0) = u_0$ is global and satisfies the decay estimate $${\|u(t)\|}_\infty \lesssim (1+t)^{-d\beta}. \label{ineq:LinftyDecay}$$
- If $m = 2-2/d$ and $u(t)$ is a global weak solution to which satisfies $$\lim_{k \rightarrow \infty}\sup_{t \in [0,\infty)}\int \left(u(t,x) - k\left(\frac{t}{\beta}+1\right)^{d\beta}\right)_+ dx = 0, \label{def:EquiInTheta}$$ then $u(t)$ satisfies the decay estimate .
Once the decay estimate has been established, entropy-entropy dissipation methods can be adapted to deduce the following intermediate asymptotics theorems, as the decay estimate provides sufficient control of the nonlocal terms.
\[thm:IA\] Let $d \geq 2$, $m \in [1,2 - 2/d]$ and ${\mathcal{K}}\in W^{1,1}$ be admissible. Suppose $u(t)$ is a global weak solution of which satisfies the decay estimate . If $m < 2-2/d$, then $u(t)$ satisfies $$t^{d\beta\left(1 - \frac{1}{p}\right)}{\|u(t) - \mathcal{U}(t;M)\|}_p \lesssim (1+t)^{-\frac{\beta}{p}}, \; \forall p, \, 1 \leq p < \infty, \label{ineq:conv_supercrit_IA}$$ and if $m = 2-2/d$, then for all $\delta > 0$, $u(t)$ satisfies $$t^{d\beta\left(1 - \frac{1}{p}\right)}{\|u(t) - \mathcal{U}(t;M)\|}_p \lesssim_{\delta} (1+t)^{-\frac{\beta}{p}(1-\delta)}, \;\; \forall p, \, 1 \leq p < \infty. \label{ineq:conv_IA}$$ Here $\beta$ is defined in and $\mathcal{U}(x,t;M)$ is the self-similar solution to with mass
| 2,587
| 1,968
| 690
| 2,618
| null | null |
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|
12.1 ± 2.0 0.001
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
jcdd-05-00035-t003_Table 3
######
Logistic regression analysis for the prediction of systolic cardiac dysfunction using traditional and specific risk factors for hemodiafiltration patients.
Characteristic *p*-Value Odds Ratio Confidence Interval
------------------- ----------- ------------ ---------------------
age 0.01 1.06 1.01--1.11
gender 0.8 1.1 0.3--3.8
Diabetes mellitus 0.6 1.7 0.2--11.3
hypertension 0.1 0.4 0.1--1.4
smoking 0.4 1.9 0.4--8.6
spKt/V for urea 0.9 0.9 0.2--4.5
nPCR 0.2 0.5 0.2--1.5
Vitamin D therapy 0.02 0.2 0.06--0.8
Ox-LDL/LDL-C 0.004 6.2 1.8--21.2
[^1]: Running title: ox-LDL and cardiovascular disease in hemodialysis
1. Introduction {#sec1-sensors-17-00869}
===============
There are an estimated 2 million hand amputees in the United States and approximately the same in Europe. More than 200,000 new amputation surgeries are performed each year and approximately 10,000 children receive amputations resulting in a lifelong disability \[[@B1-sensors-17-00869]\]. Amputee patients are supported by a long standing research and development of prosthetic devices, which can be divided in passive and active ones. Passive prostheses have only a cosmetic purpose and do not support any of the hand functionalities. Modern active prostheses are externally powered and feature advanced grasping and control functionalities.
The first active prostheses were composed by a split hook able to perform a power grasp, restoring just one Degree of Freedom (DoF). During the last years, the advancemen
| 2,588
| 4,703
| 2,215
| 2,173
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|
ngle layer case, RPA is never accurate for dipolar interactions, since it neglects exchange correlations [@babadi2011; @sieberer2011] which are important even in the long-wavelength limit [@parish2012].
A straightforward and physically motivated way of incorporating correlations beyond RPA is by means of local field factors $G_{ij} ({{\mathbf q}})$ (for an introduction to this method see, e.g., Ref. [@vignale_book]). Here, the (inverse) response function now reads: $${\chi^{-1}}_{ij} ({{\mathbf q}},\omega) = \frac{\delta_{ij}}{\Pi
(q,\omega)} - v_{ij}({{\mathbf q}}) \left[1 - G_{ij}({{\mathbf q}}) \right]
\; .
\label{eq:response}$$ Note that we clearly recover both RPA and the non-interacting case if we take, respectively, $G_{ij}=0$ or $G_{ij}=1$. This response function can be related to the “layer-resolved” static structure factor $S_{ij}({{\mathbf q}})$ by the fluctuation-dissipation theorem: $$S_{ij}({{\mathbf q}}) = -\frac{\hbar}{\pi n} \int_0^{\infty} d\omega
\chi_{ij}({{\mathbf q}},i\omega)\; .
\label{eq:struc}$$ In turn, we can approximate the local field factors using the STLS scheme [@STLSpaper]: $$G_{ij}({{\mathbf q}}) = \frac{1}{n} \int \frac{d{{\mathbf k}}}{(2\pi)^2}
\frac{{{\mathbf q}} \cdot {{\mathbf k}}}{q^2} \frac{v_{ij} ({{\mathbf k}})}{v_{ij}
({{\mathbf q}})} \left[ \delta_{ij} - S_{ij} ({{\mathbf q}}-{{\mathbf k}}) \right] \; .
\label{eq:local}$$ The response function $\chi_{ij}$ (and associated structure factor $S_{ij}$) can now be determined by solving Eqs. - self-consistently. The STLS scheme has been heavily utilized for Coulomb interactions and it has proven to be very successful for describing the dielectric function of several strongly-correlated electron systems (see [@vignale_book] and references therein). Following Ref. [@parish2012], we consider an improved version of the STLS scheme that has been adapted to the dipolar system. In essence, it ensures that our results are insensitive to $\Lambda$ and $V_0$, by requiring that the intralayer correlations be
| 2,589
| 3,136
| 3,326
| 2,436
| 1,126
| 0.794036
|
github_plus_top10pct_by_avg
|
mathrm{e}} \rightarrow {\mathrm{He^{2+}}} + 2{\mathrm{e}}$ $k_3=$ 2
$4^{a}$ ${\mathrm{H^+}}+{\mathrm{e}} \rightarrow {\mathrm{H}} + h\nu$ $k_4=2.753\times10^{-14} (T_{\mathrm{K}}/315614)^{-3/2} (1+(T_{\mathrm{K}}/115188)^{-0.407})^{-2.242}$ 3
$5^{b}$ ${\mathrm{He^+}}+{\mathrm{e}} \rightarrow {\mathrm{He}} + h\nu$ $k_5=k_{\mathrm{5rr}}+k_{\mathrm{5di}}$
$k_{\mathrm{5rr}}=$ 4
$k_{\mathrm{5di}}=1.9\times10^{-3} T_{\mathrm{K}}^{-3/2} \exp[-473421/T_{\mathrm{K}}] (1 + 0.3\exp[-94684/T_{\mathrm{K}}])$ 5
$6^{c}$ ${\mathrm{He^{2+}}}+{\mathrm{e}} \rightarrow {\mathrm{He^+}} + h\nu$ $k_6=5.08\times10^{-13} (T_{\mathrm{K}}/40000)^{-0.8163-0.0208\log_{10}(T_{\mathrm{K}}/40000)}$ 6
7 ${\mathrm{He^+}}+{\mathrm{H}} \rightarrow {\mathrm{He}} + {\mathrm{H^+}}$ $k_7=1.25\times10^{-15}(T_{\mathrm{K}}/300)^{0.25}$ 7
8 ${\mathrm{H^+}}+{\mathrm{He}} \rightarrow {\mathrm{H}} + {\mathrm{He^+}}$ $k_8=$ 8
9 $2{\mathrm{H}} \rightarrow {\mathrm{H^+}} + {\mathrm{H}} + {\mathrm{e}}$ $k_9=1.7\times10^{-4}k_1$ 9
\
NOTES. The $T_{\mathrm{K}}$ and $T_{\mathrm{eV}}$ are the gas
| 2,590
| 4,798
| 803
| 2,203
| null | null |
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|
heta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-1}}\exp(\ltheta_{k})} \Bigg)\Bigg)\Bigg) \nonumber\\
&\geq& \frac{e^{-4b} \exp(\ltheta_i)}{\widetilde{W}} \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\ltheta_{j_1})}{\widetilde{W}-\exp(\ltheta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\ltheta_{j_2})}{\widetilde{W}-\exp(\ltheta_{j_1})-\exp(\ltheta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-1}}\exp(\ltheta_{k})} \Bigg)\Bigg)\Bigg)\nonumber\\
&=& \big(e^{-4b}\big) \P_{\ltheta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \;.\end{aligned}$$ The second inequality uses $\frac{\exp(\theta_i)}{W} \geq e^{-2b}/\kappa$ and $\frac{\exp(\ltheta_i)}{\widetilde{W}} \leq e^{2b}/\kappa$. Observe that $\exp(\ltheta_j) = 1$ for all $j \neq i,\i$ and $\exp(\ltheta_i) + \exp(\ltheta_{\i}) = \widetilde{\alpha}_{i,i',\ell,\theta} \leq {\left \lceil{\widetilde{\alpha}_{i,i',\ell,\theta}} \right \rceil} = \alpha_{i,i',\ell,\theta} \geq 1$. Therefore, we have $$\begin{aligned}
&&\P_{\ltheta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&=& {\kappa-2 \choose \ell-1} \frac{(\widetilde{\alpha}_{i,i',\ell,\theta}/2)(\ell-1)!}{(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta})(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta} - 1)\cdots(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta} - (\ell-1))} \nonumber\\
&\geq& \frac{(\kappa-2)!}{(\kappa-\ell-1)!} \frac{e^{-2b}}{(\kappa + \alpha_{i,i',\ell,\theta}-2)(\kappa + \alpha_{i,i',\ell,\theta} - 3)\cdots(\kappa + \alpha_{i,i',\ell,\theta} - (\ell+1))} \label{eq:posl_5} \\
& =& \frac{e^{-2b}(\kappa-\ell + \alpha_{i,i',\ell,\theta}-2)(\kappa -\ell +\alpha_{i,i',\ell,\theta}-3)\cdots (\kappa -\ell)}{(\kappa+\alpha_{i,i',\ell,\theta}-
| 2,591
| 1,524
| 1,868
| 2,365
| null | null |
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|
etilde{\sigma}_9(v_i,x_i) \; \dot{H}(t) H^{(3)}(t) \Bigg).
\end{split}\end{aligned}$$
To find squares, we use the following general procedure that is usefull for FLRW space-time, and necessary for spherical symmetry : Take the higher order perfect square $\ddot{H}(t)^2$. In the expansion of our square, each term will be multiply by $\ddot{H}(t)$, so the only terms that can enter inside are those $K_i(t)$ for which $K_i(t) \ddot{H}(t)$ and $K_i(t)K_j(t)$ exist in the expansion of order 6 scalars. Because of this, we need to impose the conditions : $ \widetilde{\sigma}_9\, =\, 0 \;$, $\widetilde{\sigma}_8\, =\, 0 \, \;$ and $\, \widetilde{\sigma}_4\, =\, 0$, what give $x_1=x_3=0$ and $v_7=-v_4-5 v_6/12$. Therefore, $\widetilde{\sigma}_5\, =\, 0$ and there are only two possible forms of squares made of order 6 scalars : $$\begin{aligned}
\sum\limits_{i,j} \big( v_i \mathcal{L}_i + x_j \curv{L}_j \big) = \Big( \delta H(t) \dot{H}(t) + \gamma H(t)^3 \Big)^2,\end{aligned}$$ $$\begin{aligned}
\text{And :} \quad \quad \quad \sum\limits_{i,j} \big( v_i \mathcal{L}_i + x_j \curv{L}_j \big) = \Big( \xi \ddot{H}(t) + \delta H(t) \dot{H}(t) \Big)^2.\end{aligned}$$
It means that all their square-roots can be decomposed in the basis $\big( H(t)^3, \, H(t) \dot{H}(t) , \, \ddot{H}(t) \big)$. Moreover, the general Lagrangian density : $$\begin{aligned}
\sqrt{\sum\limits_{i,j} \big( v_i \mathcal{L}_i + x_j \curv{L}_j \big) }= \xi \ddot{H}(t) + \delta H(t) \dot{H}(t) + \gamma H(t)^3 \, ,\end{aligned}$$ leads to second order differential equations for all $\big( \xi, \delta, \gamma \big) $, so we do not need to impose additional conditions on these coefficients and there are then only 3 independent second order corrections that we can find in this way.
Now let us see what are the actual perfect squares that one can find. Solving the natural conditions for respectively the first and the second kind of perfect squares, $\, \widetilde{\sigma}_2^2\, =\, 4 \, \widetilde{\sigma}_3\, \widetilde{\sigma}_1 \, \;$, $\widetild
| 2,592
| 3,990
| 1,972
| 2,139
| 4,070
| 0.768289
|
github_plus_top10pct_by_avg
|
B''$.
{#app-to-main}
Analogues of Theorem \[main\] also hold for certain important $U_{c+k}$-modules and we will derive the theorem from one of these. The module in question is the $(U_{c+k},H_c)$-bimodule $N(k)=B_{k0}eH_c$ with the induced $\operatorname{{\textsf}{ord}}$ filtration coming from the inclusion $N(k)\subset D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}$. Recall the definition of $J^d$ from .
\[pre-cohh\] Assume that $c\in{\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and let $k\in {\mathbb{N}}$. Then $\operatorname{{\textsf}{ogr}}N(k)=e J^k\delta^k$ as submodules of $\operatorname{{\textsf}{ogr}}D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$.
Outline of the proof of the theorem and proposition {#surjstrat}
---------------------------------------------------
[*For the rest of the section, we will assume that $c\in
{\mathbb{C}}$ satisfies Hypothesis \[main-hyp\].*]{} Thus the notation from and is available and, by Corollary \[morrat-cor\], $N(k)\cong B_{k0}\otimes_{U_c} eH_c$ is a progenerative $(U_{c+k},\,H_c)$-bimodule. As will be shown in , Theorem \[main\] follows easily from Proposition \[pre-cohh\], so we need only discuss the proof of the latter result. This is nontrivial and will take most of the section but, in outline, is as follows.
It is easy to see that $eJ^k\delta^k\subseteq \operatorname{{\textsf}{ogr}}N(k)$ (see Lemma \[thetainjA\]). The other inclusion is considerably harder. The philosophy behind the proof is to note that we can grade both $J^{k}\delta^{k} $ and $ N(k)$ by the ${\mathbf{E}}$-gradation. This is not immediately useful since the graded pieces of the two sides are infinite dimensional but both sides have factor modules for which the graded pieces are finite dimensional. For $eJ^{k}\delta^{k}\cong J^k\delta^k$ the factor is the module $\overline{J^{k}}\delta^{k}$ described by Corollary \[gr\], while the analogous factor $\overline{N(k)}$ of $\operatorname{{\textsf}{ogr}}N(k)$ is described in and Corol
| 2,593
| 1,072
| 932
| 2,606
| 1,732
| 0.786284
|
github_plus_top10pct_by_avg
|
b P}}^1\cong C_1\cong C_2\subset X$ such that $C_1+C_2$ is homologous to $0$. Let $g:C_1\cong C_2$ be an isomorphism and $R$ the corresponding equivalence relation.
We claim that there is a no quasi projective open subset $U\subset X$ which intersects both $C_1$ and $C_2$. Assume to the contrary that $U$ is such. Then there is an ample divisor $H_U\subset U$ which intersects both curves but does not contain either. Its closure $H\subset X$ is a Cartier divisor which intersects both curves but does not contain either. Thus $H\cdot (C_1+C_2)>0$, a contradiction.
This shows that if $p\in X/R$ is on the image of $C_i$ then $p$ does not have any affine open neighborhood since the preimage of an affine set by a finite morphism is again affine. Thus $X/R$ is not a scheme.
[@lip-lip]\[sch.quot.exmp\] Fix a field $k$ and let $a_1,\dots, a_n\in k$ be different elements. Set $A:=k[x,y]/\bigl( \prod_i (x-a_iy)\bigr)$. Then $Y:={\operatorname{Spec}}R$ is $n$ lines through the origin. Let $f:X\to Y$ its normalization. Thus $X=\amalg_i {\operatorname{Spec}}k[x,y]/(x-a_iy)$. Note that $$k[x,y]/(x-a_iy)\otimes_A k[x,y]/(x-a_jy)=
\left\{
\begin{array}{l}
k[x,y]/(x-a_iy){\quad\mbox{if $a_i=a_j$, and}\quad}\\
k \qquad \qquad\qquad\ \ {\quad\mbox{if $a_i\neq a_j$.}\quad}
\end{array}
\right.$$ Thus $X\times_YX$ is reduced. It is the union of the diagonal $\Delta_X$ and of $f^{-1}(0,0)\times f^{-1}(0,0)$. Thus $X/\bigl(X\times_YX\bigr)$ is a seminormal scheme which is isomorphic to the $n$ coordinate axes in ${{\mathbb A}}^n$. For $n\geq 3$, it is not isomorphic to $Y$.
One can also get similar examples where $Y$ is integral. Indeed, let $Y\subset {{\mathbb A}}^2$ be any plane curve whose only singularities are ordinary multiple points and let $f:X\to Y$ be its normalization. By the above computations, $X\times_YX$ is reduced and $X/\bigl(X\times_YX\bigr)$ is the seminormalization of $Y$.
If $Y$ is a reduced scheme with normalization $\bar Y\to Y$. Then, as we see in (\[quot.X/S.finite.lem\]), the geometric quotient $\bar Y/\b
| 2,594
| 1,359
| 1,499
| 2,377
| 2,586
| 0.77867
|
github_plus_top10pct_by_avg
|
}$ and $\E{\lrn{y_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$, we can verify that $$\begin{aligned}
& \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} ds \leq \frac{1}{4} TL_N^2 \epsilon + TL_N^2 \epsilon\\
& L \E{\lrn{y_s - y_0}_2} \leq \frac{1}{2} TL\epsilon + TL\epsilon
\end{aligned}$$ Combining the above gives $$\begin{aligned}
\E{f(z_T)} \leq e^{-\lambda T} \E{f(z_0)} + 3 T\lrp{L+L_N^2} \epsilon
\end{aligned}$$
\[c:main\_gaussian:1\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$.
Let $\delta\leq \min\lrbb{\frac{\epsilon^2}{\beta^2}, \frac{\epsilon}{8 L\sqrt{R^2 + \beta^2/m}}}$, and let $\bx_t$ and $\by_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\by_0}_2^2}\leq R^2 + \beta^2/m$. Then there exists a coupling between $\bx_t$ and $\by_t$ such that $$\begin{aligned}
\E{f(\bx_{i\delta} - \by_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
From Lemma \[l:energy\_x\] and \[l:energy\_y\], our initial conditions imply that for all $t$, $\E{\|\bx_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}$ and $\E{\|\by_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}$.
Consider an arbitrary $k$, and for $t\in[k\delta,(k+1)\delta)$, define $$\begin{aligned}
x_t := \bx_{k\delta+t} \quad \text{and} \quad
y_t := \by_{k\delta+t}
\end{aligned}$$ Under this definition, $x_t$ and $y_t$ have dynamics described in and . Thus the coupling in , which describes a coupling between $x_t$ and $y_t$, equivalently describes a coupling between $\bx_t$ and $\by_t$ over $t\in[k\delta, (k+1)\delta)$.
We now apply Lemma \[l:gaussian\_contraction\]. Given our assumed bound on $\delta$ and our proven bounds on $\E{\lrn{\bx_t}_2^2}$ and $\E{\lrn{\by_t}_2^2}$, $$\begin{aligned}
& \E{f(\b
| 2,595
| 3,131
| 2,500
| 2,300
| null | null |
github_plus_top10pct_by_avg
|
GP17 CA 67 7 (3 + 4) T3A 7.4 \+ 7 (3 + 4) 95 80
######
Characteristics of the analyzed prostate tumor and matched normal blood whole genomes.
Characteristics of the analyzed prostate tumor and matched normal blood whole genomes
--------------------------------------------------------------------------------------- ------- ---- -------- ------- -------- -------- ------- ----- ------- ------ ------ -----
GP02 AA \- 114.7 37.8 112.2 95.6 2331 25 0.82 528 284 31
GP04 AA \- 116.7 38.3 111.8 95.6 2800 25 0.98 701 429 40
GP10 AA \- 118.5 39.2 111.6 95.7 2570 23 0.90 602 242 5
GP18 AA \- 116.2 38.6 102.4 95.7 1976 20 0.69 431 251 7
GP12 AA \+ 112.6 37.2 117.7 95.6 2069 8 0.72 452 286 15
GP13 AA \+ 107.6 34.9 113.4 95.3 1934 13 0.68 435 164 4
GP15 AA \+ 114.2 38.1 108.1 95.6 2167 12 0.76 455 399 6
GP06 CA \- 123.5 41.0 113.3 95.8 3635 38 1.27 667 148 36
GP11
| 2,596
| 5,430
| 1,031
| 1,653
| null | null |
github_plus_top10pct_by_avg
|
)}{\ell_j}\label{eq:tau_def}\\
\delta_{j,1} & \equiv & \bigg\{ \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a} \bigg\} \;\;, \;\text{and}\;\;\;\;\;\; \delta_{j,2} \equiv \sum_{a = 1}^{\ell_j} \lambda_{j,a} \label{eq:delta12_def} \\
\delta & \equiv & \max_{j \in [n]} \bigg\{ 4 \delta_{j,1}^2 + \frac{2\big(\delta_{j,1}\delta_{j,2} + \delta_{j,2}^2\big)\kappa_j}{\eta_{j}\ell_j} \bigg\} \;\;\,. \label{eq:delta_def} \end{aligned}$$ Note that $\delta \geq \delta_{j,1}^2 \geq \max_a \lambda_{j,a}^2 (\kappa_j-p_{j,a})^2 \geq \tau^2$, and for the choice of $\lambda_{j,a}=1/(\kappa_j-p_{j,a})$ it simplifies as $\tau=\tau_j=1$. We next define a comparison graph $\H$ for general $\lambda_{j,a}$, which recovers the proposed comparison graph for the optimal choice of $\lambda_{j,a}$’s
\[def:comparison\_graph2\] (Comparison graph $\H$). Each item $i \in [d]$ corresponds to a vertex $i$. For any pair of vertices $i,\i$, there is a weighted edge between them if there exists a set $S_j$ such that $i, \i \in S_j$; the weight equals $\sum_{j: i,\i \in S_j} \frac{\tau_{j}\ell_j}{\kappa_j(\kappa_j-1)}$.
Let $A$ denote the weighted adjacency matrix, and let $D = {\rm diag}(A {\boldsymbol{1}})$. Define, $$\begin{aligned}
\label{eq:posl_Dlmax}
D_{\max} \;\;\equiv \;\; \max_{i \in [d]} D_{ii} \;= \;\max_{i \in [d]} \bigg\{ \sum_{j: i \in S_j} \frac{\tau_{j}\ell_j}{\kappa_j} \bigg\} \;\;\geq\;\; \tau_{\rm min} \max_{i \in [d]} \bigg\{ \sum_{j: i \in S_j} \frac{\ell_j}{\kappa_j} \bigg\} \,. \end{aligned}$$ Define graph Laplacian $L $ as $L = D - A$, i.e., $$\begin{aligned}
\label{eq:comparison2_L}
L \; =\; \sum_{j = 1}^n \frac{\tau_{j}\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top.\end{aligned}$$ Let $ 0 = \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_d(L)$ denote the sorted eigenvalues of $L$. Note that $\Tr(L ) = \sum_{i =1}^d \sum_{j: i \in S_j}\tau_{j}\ell_j/\kappa_j = \sum_{j = 1}^n \tau_{j}\ell_j$. Define
| 2,597
| 3,890
| 2,383
| 2,285
| null | null |
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|
((\tau+2)/3)} - \frac{27\eta'(3\tau)}{\eta(3\tau)} \right), \nonumber \\
y_{2}(\tau) &=& \frac{-i}{\pi}\left( \frac{\eta'(\tau/3)}{\eta(\tau/3)} +\omega^2\frac{\eta'((\tau +1)/3)}{\eta((\tau+1)/3)}
+\omega \frac{\eta'((\tau +2)/3)}{\eta((\tau+2)/3)} \right) , \label{eq:Yi} \\
y_{3}(\tau) &=& \frac{-i}{\pi}\left( \frac{\eta'(\tau/3)}{\eta(\tau/3)} +\omega\frac{\eta'((\tau +1)/3)}{\eta((\tau+1)/3)}
+\omega^2 \frac{\eta'((\tau +2)/3)}{\eta((\tau+2)/3)} \right)\,.
\nonumber\end{aligned}$$ Then, any couplings of higher weights are constructed from the multiplication rules of $A_4$. One finds the following expressions: $$\begin{aligned}
&Y^{(4)}_{\bf1}=y^2_1+2y_2y_3,\quad
Y^{(6)}_{\bf3}=
\left[\begin{array}{c}
y^3_1+2y_1y_2y_3 \\
y^2_1y_2+2y_2^2y_3 \\
y^2_1y_3+2y_3^2y_2 \\
\end{array}\right],\quad
Y'^{(6)}_{\bf3}=
\left[\begin{array}{c}
y^3_3 + 2y_1y_2y_3 \\
y^2_3y_1+2y_1^2y_2 \\
y^2_3y_2+2y_2^2y_1 \\
\end{array}\right].\end{aligned}$$ To construct a nonzero neutrino mass matrix, we also need a quartic term in the Higgs potential, [*i.e.*]{} $(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)$, which can be realized as follows: $$\begin{aligned}
a_0\frac{Y^{(4)}_{\bf1}}{i(\tau^*-\tau)}(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)+h.c.
\equiv \lambda_0(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)+h.c.,\end{aligned}$$ where $a_0$ is an arbitrary complex number. Although it mixes the neutral complex Higgs bosons, we assume that the mixing angle is very small and the mass eigenstates are almost the flavor eigenstates, which we denote as $\eta_{1,2}$.
Neutrino mass matrix
--------------------
Due to the modular symmetry, the heavy Dirac neutrino mass matrix is diagonal with the eigenvalue, $M_D$: $$\begin{aligned}
{\cal M}&= {M_D}
\left[\begin{array}{ccc}
1 &0 & 0 \\
0 & 1 &0 \\
0& 0 & 1 \\
\end{array}\right] . \end{aligned}$$
We write down the relevant interactions for the generation of the neutrino mass matrix as $$\begin{aligned}
-{\cal L}_\nu&=\bar\nu_L y_{N_R} N_R \eta_1^*
+\bar N_Ly_{N_L} \nu^C_L \eta_2^*+\frac{\lambda_0v_H^2}{2}\et
| 2,598
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| 2,312
| 2,343
| null | null |
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|
,0],[0,1]\notin \Lambda(\Gamma)$. Then there are $k,r,s\in \Bbb{C}$ such that
$$\begin{array}{l}
\psi(x)=[1,2k,k^2] \\
\psi(y)=[1,2r,r^2] \\
\psi(z)=[1,2s,s^2].
\end{array}$$ From Lemma \[l:ltanver\] we know $$\begin{array}{l}
T_{\psi(x)} Ver=\{
[x,y,z]
\in \Bbb{P}^1_\Bbb{C} \vert z=ky-k^2x\} \\
T_{\psi(y)} Ver=\{
[x,y,z]
\in \Bbb{P}^1_\Bbb{C} \vert z=ry-r^2x\} \\
T_{\psi(z)} Ver =\{
[x,y,z]
\in \Bbb{P}^1_\Bbb{C} \vert z=sy-s^2x\}.
\end{array}$$ Since $$\left \vert
\begin{array}{lll}
k^2&-k&1\\
r^2&-r&1\\
s^2&-s&1\\
\end{array}\right\vert=(s-r)(k-s)(k-r)\neq 0$$ we conclude the proof.
\[l:pseudo\] Let $(\gamma_n)\subset \PSL(2,\Bbb{C})$ be a sequence of distinct elements such that $\gamma_n\xymatrix{\ar[r]_{\rightarrow\infty}&}x$ uniformly on compact sets of $\Bbb{P}^1_\Bbb{C}\setminus\{y\}$. Then $\iota\gamma_n\xymatrix{\ar[r]_{\rightarrow\infty}&}\psi(x)$ uniformly on compact sets of $\Bbb{P}^2_\Bbb{C}\setminus T_{\psi(y)}Ver$.
Let us assume that $\gamma_n=\left [\left [a_{ij}^{(n)}\right ]\right ]$. Note that we can assume $a_{ij}^{(n)}\xymatrix{\ar[r]_{n\rightarrow\infty}&}a_{ij}$ and $\sum_{i,j=1}^2\mid a_{ij} \mid\neq 0$. Then $\gamma_n\xymatrix{\ar[r]_{n\rightarrow\infty}&}\gamma=\left [\left [a_{ij}\right ]\right ]$ uniformly on compact sets of $\Bbb{P}^1_{\Bbb{C}}\setminus Ker(\gamma)$, thus $Ker(\gamma)=\{y\}$ and $Im(\gamma)=\{x\}$. Therefore there is a $k\in \Bbb{C}^*$ such that $x= [1,k]$, thus $a_{11}=-ka_{12}$ and $a_{21}=-ka_{22}$. In consequence $$\iota\gamma_n
\xymatrix{ \ar[r]_{n \rightarrow\infty}&}
B=
\left [ \left [
\begin{array}{lll}
k^2a_{12}^2&-ka_{12}^2& a_{12}^2\\
2k^2a_{12}a_{22}&-2ka_{12}a_{22}&2a_{12}a_{22}\\
k^2a_{22}^{2}&-ka_{22}^2&a_{22}^2\\
\end{array}
\right ]\right ].$$
A simple calculation shows that $Ker(B)$ is the line $\ell=\overleftrightarrow{ [e_1-k^2e_3],[e_2+ke_3]}$. Also it is not hard to check that $$\ell=\{[x,y,z]\vert k^2x-ky+z=0 \},$$ which concludes the proof.
Let $\Gamma\subset\PSL(2,\Bbb{C})$ be a non-elementary group. Then $\iota(\Gamma)$ does not hav
| 2,599
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| 1,077
| 2,656
| null | null |
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|
\wedge^{\ell_n} T_n^{\alpha}
\otimes \wedge^{k_n} T_n^{\alpha *} \right) \otimes
{\cal F}
\right) ,$$ where $${\cal F}^{\alpha} \: = \: \sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 }
\otimes_{n>0} \left( \left( \det {\cal E}^{\alpha}_n \right) \left(
\det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[
- \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ (reflecting the Fock vacuum). Strictly speaking, not all states need be of the form above – for example, one might also be able to multiply in bosonic $\partial \phi$ modes. As their inclusion is standard and their treatment should now be clear, for reasons of brevity we shall move on.
For example, if $I_{\mathfrak{X}}|_{\alpha} = [ {\rm point}/{\mathbb Z}_2 ]$, then this becomes $$H^{k_0}\left( {\rm point},
\left( \wedge^{m_0} {\cal E}_0^{\alpha *} \right)
\otimes_{n>0}\left( \wedge^{m_n} {\cal E}_n^{\alpha *}
\otimes \wedge^{p_n} {\cal E}_n^{\alpha *}
\otimes \wedge^{\ell_n} T_n^{\alpha}
\otimes \wedge^{k_n} T_n^{\alpha *} \right) \otimes
{\cal F}^{\alpha}
\right)^{\mathbb{Z}_2}.$$ (Taking group invariants is encoded implicitly in taking sheaf cohomology on the quotient stack.) This group vanishes if $k_0 \neq 0$, and when $k_0 = 0$, is the dimension of the ${\mathbb Z}_2$-invariant part of the vector space fibers.
Finally, in a physical computation, one must impose the left- and right- GSO projections. For states of the form above, this will amount to a chirality constraint on $k_0$ and $m_0$. As the procedure is standard, we will say no more.
One of the central observations of the heterotic spectrum computation on smooth manifolds in [@dist-greene] is that it is closed under Serre duality. The same is true here. First, for any component of the inertia stack indexed by an automorphism $\alpha$, there is another (not necessarily distinct) component indexed by $\alpha^{-1}$, which is isomorphic: $$I_{\mathfrak{X}}|_{\alpha} \:
\cong \: I_{\mathfrak{X}} |_{\alpha^{-1}}.$$ Eigenbundle decompositions are closely related: $$\
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