prompt stringlengths 0 512 | answer stringlengths 1 8.58k |
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how the English language works.
You will also want a grammar reference. Our favorite grammar reference is the Handbook of Grammar and Composition published by A Beka. We purchased it on a recommendation in The Well-Trained Mind and haven’t looked back.
Now | s Friday Farmers Market to the big Saturday markets in Winter Garden and Winter Park – you are hard pressed to find a locale without a marketplace of fresh produce. Dogs at these events have become so prominent that many vendors now cater exclusively to do |
tay ahead of your child’s work. Of course, at some point, if you feel you are losing ground there are many qualified and reputable online writing tutors av | ert, D. T., Wilson, T. D., Sievers, B., Amodio, D. M., & Wheatley, T. （2018）。 Prevalence-induced concept change in human judgment. Science, 360（6396）， 1465-1467. |
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ing.
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h mentioning!
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[Coroner] Have you seen any of these articles since? - Yes; I saw the black overcoat in a room in the court on Friday afternoon.
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Inspector Beck, H Division, depo |
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\section{Introduction}
Let $G$ be a simple undirected graph with the \textit{vertex set} $V(G)$ and the \textit{edge set} $E(G)$. A vertex with degree one is called a \textit{pendant vertex}. The distance between the vertices $u$ and $v$ in graph $G$ is de | p) x^3 + (2p + c_p^2 - c_p -1)x^2 + p(1 - 2c_p)x + p^2.\]
Once more we claim that $f_p$ is the Weil polynomial of a simple ordinary isogeny
class ${\mathcal C}_p$ over~${\mathbf F}_p$, and again we prove this by showing that the algebra
$K = {\mathbf Q}[x] |
S\rangle$. A path that starts in $v$ and ends in $u$ is denoted by $\stackrel\frown{v u}$.
A \textit{traceable} graph is a graph that possesses a Hamiltonian path.
In a graph $G$, we say that a cycle $C$ is \textit{formed by the path} $Q$ if $ | E(C) \set | ll} Ideal: & Macbeath,\cite{Macbeath}: \\
& $ x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,$ \\
& $ x_0^2+\zeta_{7} x_1^2+\zeta_{7}^2 x_2^2+\zeta_{7}^3 x_3^2+\zeta_{7}^4 x_4^2+\zeta_{7}^5 x_5^2+\zeta_{7}^6 x_6^2,$ \\
& $x_0^2+\zeta_{7}^{-1} x_1^2+\zeta_{7} |
hing and a family of cycles.
\end{conjecture}
Conjecture \theconjecture$\,$ also appears in Problem 516 \cite{cameron}. There are a few partial results known for Conjecture \theconjecture. Kostochka \cite{kostocha} noticed that the Petersen graph, the pri | eta_2, \xi)}\partial_{\eta_1} e^{it\Phi(\eta_1,\eta_2, \xi)} \hat h_{j_1}(\eta_1)\hat h_{j_2}(\eta_2) \hat h_{j_3}(\xi-\eta_1-\eta_2) \,\mathrm{d}{\eta_1} \,\mathrm{d}{\eta_2}\\
=~& -W_1-W_2-W_3,
\end{align*}
where
\begin{align*}
W_1(t, \xi) & = \iint_{\ |
onjecture \theconjecture$\,$ is true for every 3-connected cubic graph embedded in torus or Klein-bottle. Akbari, Jensen and Siggers \cite[Theorem 9]{akbari} showed that Conjecture \theconjecture$\,$ is true for Hamiltonian cubic graphs.
In this paper, we | \caption{Average anomalous publications throughout the years.}
\label{fig:year}
\end{figure}
Next, we plot the trends with respect to the size of the anomalous journals, defined in terms of the number of papers published in it. Figure \ref{fig:siz |
tree $T$ and finitely many cycles such that there is no edge between any pair of cycles (not necessarily distinct cycles), and every pendant vertex of $T$ is adjacent to at least one vertex of a cycle. Then, Conjecture \theconjecture$\,$ holds for $G$.
\en |
\begin{equation}\label{eq:kolmepsilon}
K(G)\leq (1+\epsilon)nd\log(n)
\end{equation}
Let $X$ be the set of all graphs $G$ such that
\begin{equation}\label{eq:kolmcondlog}
K(G|n) \geq \frac{n(n-1)}{2} - \log(n)
\end{equation}
Using Lemma \ref{lem:kolc |
up Q$ is a spanning tree of $G$. We call it $T'$. Note that every edge between a pendant vertex of $T$ and the union of cycles in the partition is also contained in $T'$. Thus, every pendant vertex of $T'$ is contained in a cycle of the partition. Now, con | version) in 180 products including WLAN routers, smart cameras, and solar panels, from well-known manufacturers' official releases and third-party providers such as DD-WRT~\cite{ddwrt}, as shown in Table~\ref{tab:firmware}.
We collect firmware images with |
y there exists a chordless cycle formed by $Q$.
Now, we are in a position to prove the main result.
\begin{theorem}
Conjecture \theconjecture$\,$ holds for traceable cubic graphs.
\end{theorem}
\begin{proof}
Let $G$ be a traceable cubic graph and $P : v_ | dence on
a control sequence.
In this paper we extend the above to the case where the drift function $h$ in recursion \eqref{recmark1}, is a set-valued map. The recursion now
takes the form,
\begin{equation*}
X_{n+1}-X_n-a(n)M_{n+1}\in a(n)H(X_n,S_n),
\ |
1 < i < i'$. Two cases can occur:
\begin{enumerate}[leftmargin=0pt,label=]
\item
\textbf{Case 1.}
Assume that $i < j$. Consider the following graph in Figure \ref{fig:overlapping} in which the thick edges denote the path $P$. Call the three paths between $ | :2019ewn}, but when considering that the interpolations between experimental data
have also horizontal and vertical, type A and type B errors, the signicance of the Odderon signal is further reduced to
a 6.26 $\sigma$ effect~\cite{Csorgo:2020msw,Csorgo:202 |
\caption{Paths $P_1$, $P_2$ and $P_3$}
\label{fig:overlapping}
\end{center}
\end{figure}
If $P_2$ has order $2$, then $G$ is Hamiltonian and so by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds. Thus we can assume that $P_1$, $P_2$ | process took place in wells of capacity $C_w=500 \,\mu l$ of such RPMI. This was carried out in the presence of isobutylmethylxanthine, otherwise known as IBMX, a broad
spectrum inhibitor of cAMP phosphodiesterase (PDE) \cite{Biondi:2010}. This was perfor |
_i$ has degree 3 for every $i$, by \hyperref[remark:1]{Remark 1}$\,$ there are two chordless cycles $C_1$ and $C_2$ formed by $P_1$ and $P_2$, respectively.
Define a tree $T$ with the edge set
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \r | thy brain tissue. Thus, surgical damage is often unavoidable. Understanding the relationship between the damaged cerebellar regions and postoperative neurological and cognitive outcomes has the potential to guide surgery and significantly affect quality of |
o loss of generality, assume that $r=1$ and $s=2$. Suppose that $ab \in E(G)$, where $a \in V(P_1)$, $b \in V(P_2)$ and $d_{P_1}(v_j, a) + d_{P_2}(v_j, b)$ is minimum.
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/ab.pdf}
| propose to employ this stabilization term selectively only at the solid-liquid and liquid-gas interface
to avoid oscillations originating from the phase transition solid-liquid and from the high liquid-gas interface forces typical for metal AM melt pool h |
rdless cycle $\stackrel\frown{v_j a}\stackrel\frown{ b v_j}$. Define $T$ with the edge set
$$ E\Big(\langle V(G) \setminus V(C)\rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C\}$. | olling robotic arms to assist human hands in reaching tasks~\cite{maimon2017towards} , and Shaft et al. used eye movement controlling robotic arms to perform tasks\cite{shafti2019gaze}. Other studies have adopted unnatural methods to recognize the grasping |
\setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \; \bigcap \; E\Big(\bigcup_{i=1}^3 P_i\Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
(c) With no loss of generality, there exists a chordless cycle formed by the path $\stackrel\frown{v_j a}$ and the | 19UnderstandingSimulations}.
$M_z^\mathrm{eq}$ is the extrapolated equilibrium dipole reached at late times.
Comparing the individual contributions to the total electrolytic dipole reveals that $\tau_1$ describes relatively fast water dipole relaxations an |
the edges with one end in $C_t$ and other endpoint in $V(G) \setminus V(P_1)$ by $e_t$. Let $v_j=w_0, w_1, \dots, w_l=a$ be all vertices of the path $\stackrel\frown{v_j a}$ in $P_1$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < | tion (A2) (note that $2k-2\geq k+1$ for $k\geq3$) gives}
\begin{align}
\frac{1}{p}\sum_{i=1}^p h\left(\vct{h}_i,\sqrt{p}\betab^\star_i,\bSi_{i,i}\right) \stackrel{{P}}{\longrightarrow} \operatorname{\mathbb{E}}_{\mathcal{N}(0,1)\otimes \mu}\left[h(H,B,\Lam |
contained in $\stackrel\frown{v_j a}$ remain. Note that there exists a path of order $2$ in $C'$ which by adding this path to $Q_i$ we find a cycle $C_{t_i}$, for some $i$. Hence there exists an edge $e_{t_i}$ connecting $Q_i$ to $V(G) \setminus V(P_1)$. | ffects include depletion of metals in the diffuse interstellar gas due to their incorporation into grains; anomalous interstellar abundance ratios, especially between isotopes of the same element, caused by chemical fractionation\footnote{This is a case of |
lemma:1]{Lemma 1} $\,$for the partition $\{T,C'\}$.\\
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/deltaCycle.pdf}
\caption{The cycle $C'$ and the tree $T$}
\label{fig:deltaCycle}
\end{center}
\end{figure}
Ne | ll respond with
some~$y_j$ such that $a_i=b_j$ and $(x_i,y_j)\in Z$.
\end{expl}
\section{Infinite-depth behavioural
equivalence}\label{sec:infinte-depth}
\sloppypar
\noindent We have seen in \cref{sec:determinization} that in case
\mbox{$M_01=1$}, $ |
can be considered:\\
\begin{enumerate}[leftmargin=5pt,label=(\roman*)]
\item There exists a cycle $C_2$ formed by $P_2$ or $P_3$. Define the partition $C_1$, $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V( | the LOGO algorithm
as was done for the SOO algorithm in a previous study \cite{val13}. To
consider categorical and/or discrete parameters in addition to continuous
parameters, a possibility could be to use the LOGO algorithm as a subroutine to deal with t |
$a \in V(P_2)$ and $b \in V(P_3)$ and moreover $d_{P_2}(v_j, a) + d_{P_3}(v_j,b)$ is minimum. Notice that the cycle $\stackrel\frown{v_j a} \stackrel\frown{b v_j}$ is chordless. Let us call this cycle by $C_2$. Now, define the partition $C_2$ and a tree wi | dashed and magenta dotted lines), which are calculated with the K20 yields (red solid and green short-dashed lines) or the yields from \citet{lim18} with the rotational velocity of 300 km s$^{-1}$ (blue long-dashed and magenta dotted lines).
The solar sym |
en $V(P_2)$ and $V(P_3)$. Let $C_2$ be the cycle consisting of two paths $P_2$ and $P_3$. Define the partition $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
an | and conceptual difficulties. However, these no-go theorems are avoided if one allows for non-deterministic coupling as in \cite{blanchard1995event,diosi1995quantum,diosi2011gravity,alicki2003completely,poulinKITP2,oppenheim_post-quantum_2018,unrav,UCLcons |
ee cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=]
\item \textbf{Subcase 1.} There is no $s < j'$ and $t > i'$ such that $v_s v_t \in E(G)$. By \hyperref[remark:1]{Remark 1} $\,$ there are two chordless cycles $C_1$ and $C_2$, respecti | of the Volterra integral representation. However, it will be instructive for the reader to think of the expression
\begin{equation}\label{vp}
X^{\tau}(t):=X(\tau,t)=\int_{0}^{t}K(\tau,r)dW(r),
\end{equation}
where we have chosen to let a Volterra process |
ig( E(P) \cup \{v_{i'}v_n, v_{j'}v_1\} \Big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\item \textbf{Subcase 2.}
\label{subcase:22} There are at least four indices $s, s' < j$ and $t, t' > i$ such that $v_s v_t, | lty{t}{\mbox{\lstinline+$pre+}}{d}\mbox{,}\\
t\in\set{\uintmodty{\mbox{\lstinline+N+}},\boolmodty{\mbox{\lstinline+N+}}}\mbox{,}\quad s=\mbox{\lstinline+$post+}\mbox{.}
\end{array}
\]
Furthermore,
\[
\begin{array}{l}
\semc{e_1}\gamma\phi\nu=\uopl{pure}((\m |
edges $v_h v_e$ and $v_g v_f$}
\label{fig:non-overlapping}
\end{center}
\end{figure}
Three cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=(\alph*)]
\item There is no chordless cycle formed by $\stackrel\frown{v_g v_h}$ and by $\ | nicability centrality cannot produce unlocalized results, except in the case of regular networks as discussed in Sec.~\ref{sec:reg}, or in the case of nearly-regular networks such as $\mathcal{B}(L=15)$.
The pattern is reversed with the ground-current ce |
v_n\} \Big),$$
apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C\}$.\\
\item With no loss of generality, there exists a chordless cycle formed by $\stackrel\frown{v_e v_f}$ and there is no chordless cycle formed by the path $\stackrel\frown{ | chastic process $\underline{\boldsymbol{\omega}}$. Let $\{\underline{\mathbf{x}}^k\}_{k\in\mathbb{N}_+}$ be the
sequence generated by the algorithm, given
${\mathbf{x}}^0\in {\cal K}$. Suppose that Assumptions A',B-D hold and that
$
\gamm
|
\stackrel\frown{v_1 v_j}$. By assumption there is no edge between $V(C_1)$ and $V(C_2)$. Now, define a tree $T$ with the edge set,
$$\quad\quad\quad\quad E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v | 1}{X_2}}{\Tens{Y_1}{Y_2}}}}
\end{align*}
by
$(\Tens{s_1}{s_2})_{((a_1,a_2),(b_1,b_2))}=(s_1)_{(a_1,b_1)}(s_2)_{(a_2,b_2)}$
and one must check that
\[
\Tens{s_1}{s_2}\in\PCOH({\Tens{X_1}{X_2},\Tens{Y_1}{Y_2}})\,.
\]
This follows directly from Lemma~\ref{ |
v_j\}$ such that $x_r y_r \in E(G)$. Let $v_e=w_0, w_1, \dots, w_l=v_f$ be all vertices of the path $\stackrel\frown{v_e v_f}$ in $P$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$. Consider the cycle $w_0 w | y in the central phase of the evolution. The measurement-based trajectories shown in Figure \ref{fig:results} are the average obtained from 1000 realizations. The true trajectories would resemble the dashed ones in Figure \ref{fig:case_detail}. These resu |
ath to $Q_i$ we find a cycle $C_{r_i}$, for some $i$. Hence there exists an edge $x_{r_i} y_{r_i}$ connecting $Q_i$ to $V(G) \setminus V(\stackrel\frown{v_e v_f})$. We define a tree $T$ whose edge set is the edges,
$$\quad\quad\quad\quad\quad\quad E\Big(\l | especially for movement prediction. It takes as input the embedded positions of the pedestrian over the last five time steps, and outputs the predicted positions over the next five time steps.
These embedded positions are called a displacement volume and |
r}
\includegraphics[width=90mm]{engImages/deltaNonOverlapping.pdf}
\caption{The tree $T$ and the shortest path $w_0 w_{i_1}\dots w_l$}
\label{fig:delta-non-overlapping}
\end{center}
\end{figure}
\item There are at least two chordless cycles, | {F}'$ yields an infinite subsequence of $\{G_n\}$ which contains arbitrarily large $t$-blowups of $F_i$. The densities of these blowups all tend to at least $c$. Therefore, for any positive integer $m$, there exists an $m$-set of each $\{G_n\}$ for suffici |
set,
$$\quad\quad\quad\quad E\Big( \langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_1 v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\end{enumerate}
| 19.833 & 19.833 & 19.833 & 19.832 & 19.829 & 19.829 \\
9.4 & 20.033 & 20.033 & 20.033 & 20.033 & 20.033 & 20.033 & 20.033 & 20.033 & 20.033 & 20.029 & 20.029 \\
9.6 & 20.233 & 20.233 & 20.233 & 20.233 & 20.23 |
ed by $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$, to see this by symmetry consider a cycle $C$ formed by $\stackrel\frown{v_{s+1} v_j}$. By \hyperref[remark:1]{Remark 1} $\,$ there exist chordless cycles $C_1$ formed by $\stackrel\fr | htt{A}$ are mostly dissolved before reaching the receiver. To overcome this, the transmitter may release molecules of a different type $\mathtt{B}$, which would also react with the enzyme $\mathtt{C}$ and thereby reduce the concentration of $\mathtt{C}$ in |
\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.\\
$\quad$Furthermore, we can also assume that | ref{fig:denprof} shows the radial surface-density profile for $S_1$.
Error bars are estimated from Poisson noise based on the counts in each annulus. In order to see details, we used high-resolution and logarithmic spacing in radius. We also request a mini |
suppose that $s \neq j'-1$. Let $v_k$ be the vertex adjacent to $v_{j'-1}$, and $k \notin \{j'-2, j'\}$. It can be shown that $k > j'-1$, since otherwise by considering the Hamiltonian path $P': \; \stackrel\frown{ v_{k+1} v_{j'-1}}\stackrel\frown{v_k v_1 | constraint that the policy must lie on the manifold of policies with the chosen parameterization. Let $\Pi_\Theta$ be the set of all possible policies $\pi_\theta$ for a given parameterization, where we overload notation to also let $\Pi_\Theta$ denote the |
form a cycle contradicts the case that $j' < k \le j$. So $j < k < i$. Consider two cycles $C_1$ and $C_2$, respectively with the vertices $v_1 \stackrel\frown{v_{j'} v_{j}} v_1$ and $v_n \stackrel\frown{v_{i'} v_{i}} v_n$. The cycles $C_1$ and $C_2$ are | mmary of methods for measuring the SP-induced ISHE.
It is written which effect can be separated (\Checkmark ) or cannot
be separated ($\times$) by using combination of the corresponding
material for the spin injector (F) layer and the measurement. $-$
deno |
(P) \cup \{v_s v_t, v_k v_{j'-1}\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.
\end{enumerate}
\end{enumerate}
\end{proof}
\noindent\textbf{Re | \end{array} \right),
\text{ and }
B_r^{(iii)} = \left( \begin{array}{cc} S_{r-1}\\
S_{r-1} F_{r-1}
\end{array} \right).
$$
If $c_r$ and $\rho_r$ denote the number of columns and rows of |
\section{Principle of nano strain-amplifier}
\begin{figure*}[t!]
\centering
\includegraphics[width=5.4in]{Fig1}
\vspace{-0.5em}
\caption{Schematic sketches of nanowire strain sensors. (a)(b) Conventional non-released and released NW structure; | ection of $n$ convex integral polytopes in $(\ensuremath{\mathbb{R}}_{\geq 0})^n$. Let $\mathscr{S}$ be a collection of subsets of $[n]$. Define
\begin{align}
\scrT_{\mscrS,\mscrP} &:= \{I \subseteq [n]: I \not \in \overbar \mathscr{S} \cup \scrE_\mscrP,\ |
stive elements are either released from, or kept on, the substrate. The sensitivity ($S$) of the sensors is defined based on the ratio of the relative resistance change ($\Delta R/R$) of the sensing element and the strain applied to the substrate ($\vareps | ubseteq [n]|\\|S|=|g|\\S_j=i}} \sum_{\substack{f\in\mathcal{A}^n|\\f_i=a}} \mathbbm 1_{f_S=g}. \numberthis \label{eq:lemmasmapsum}
\end{align*}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.25]{lemma1.pdf}
\vspace*{-2mm}
\caption{Figure illust |
tive elements: $\Delta R/R = GF \varepsilon_{ind}$, where $\varepsilon_{ind}$ is the strain induced into the piezoresistor. In most of the conventional strain gauges as shown in Fig. \ref{fig:fig1} (a,b), the thickness of the sensing layer is typically bel | al equations of motion and by extension also their discretizations). Both couplings are now described by,
\begin{align}
\lambda = \lambda_0 a^{2(1-s)}
&&
e = e_0 a^{(1-s)}
\end{align}
such that the value of $s$ can be used to either fix the comovin |
to improve the sensitivity of strain sensors (e.g. enlarging $\Delta R/R$), electrical approaches which can enlarge the gauge factor ($GF$) are required. Nevertheless, as aforementioned, the existence of the large gauge factor in nanowires due to quantum c | 2mm]
&\text{under the constraints}\nonumber\\[2mm]
&\qquad \theta_i >0, \label{eq:c0}\\[1mm]
&
\qquad \frac{q_i-q_{i-1}}{\tau_i}=\frac{p_i}{m},\label{eq:c1}\\
&\qquad c\frac{\theta_i-\theta_{i-1}}{\tau_i}=-\frac{\kappa q_i p_i}{m}
-\frac{p_i}{m}\frac{p |
resistive element. Figure \ref{fig:fig1}(c) shows our proposed nano strain-amplifier structure, in which the piezoresistive nanowires are locally fabricated at the centre of a released bridge. The key idea of this structure is that, under a certain strain | 17 \\
\textsc{Bert-Ext}\cite{bae2019summary} & 42.29 & 19.38 & 38.63 \\
\textsc{Bert-Ext} + RL & 42.76 & 19.87 & 39.11 \\
{BertSum}~\cite{liu2019text} & 42.57 & 19.96 & 39.04 \\
{BertSum}-large & \underline{43.85} & 20.34 & 39.90 \\
\midrule
\textsc{BartE |
le strength of materials \cite{dogbone1,dogbone2}. That is, when a stress is applied to the dogbone-shape of a certain material, a crack, if generated, will occur at the middle part of the dogbone. The large strain concentrated at the narrow area located a | \caption{NSC 1}
\label{Fig:Interconnection1}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.23\textwidth}
\centering
\captionsetup{justification=centering}
\includegraphics[width=1.4in]{Figures/Interconnectio |
wire (single wire or array) of the nano strain-amplifier can be considered as solid springs, Fig. \ref{fig:fig1}(d). The stiffness of these springs are proportional to their width ($w$) and inversely proportional to their length (l): $K \propto w/l$. Conse | ensity on the interface between two media most of them are limited to planar electrodes.\cite{Nguyen2019}
Those approaches faithfully reproduce the behavior of electrolytes near electrodes, particularly the correlation between thermal fluctuations in the e |
-known in classical physics that, for serially connected springs, a larger strain will be concentrated in the low--stiffness string, while a smaller strain will be induced in the high--stiffness string \cite{Springbook}. The following analysis quantitative | dashed] (bmv11) to (b1v1k);
\draw [thick, loosely dashed] (bmv11) to (b2v11);
\draw [thick, loosely dashed] (bmv11) to (b2v12);
\draw [thick, loosely dashed] (bmv11) to (b2v1k);
\draw [thick, loosely dashed] (bmv12) to (b1v11);
\draw [thick, loosely |
el{fig:fig2}
\end{figure}
When a tensile mechanical strain ($\varepsilon_{sub}$) is applied to the substrate, the released structure will also be elongated. Since the stiffness of the released frame is much smaller than that of the substrate, it is safe | et}). But instead of tracking the persistence of features, which underpins most published applications of topological data analysis, we will take a counting process approach based on the birth levels of new features in the filtration. As well as being co |
ent}
\end{equation}
where $L_m$, $L_n$ are the length; $\Delta L_m$, $\Delta L_n$ are the displacement; and $\varepsilon_m$, $\varepsilon_n$ are the strains induced into the micro spring and nano spring, respectively. The subscripts m and n stand for th | }_{\cat{Q}_{1}}$ with the following action on objects:
\[
\mathcal{S}_{\cat{Q}_{1}}\left(\vec{E}(Q),\vec{\epsilon}_{Q},\vec{V}(Q)\right)
=
\left(\Ran\left(\vec{\epsilon}_{Q}\right),m_{\vec{\epsilon}_{Q}},\vec{V}(Q)\right),
\]
where $\Ran\left(\vec{\epsilon |
f the released micro frames and nanowires, respectively. Consequently the relationship between the displacement of the micro frame (higher stiffness) and nanowires (lower stiffness) is:
\begin{equation}
\frac{\Delta L_m}{\Delta L_n}=\frac{K_n}{K_m}=\frac | al}_\nu X^N.
\label{OSp-current-M}
\end{eqnarray}
The above proof makes sense only under the assumption that the metric $\tilde g^{\mu\nu} \phi^2$
and the other terms except for the last term in (\ref{Choral-Lag}) are invariant under the $OSp$ rotation |
w_m-w_n}{w_m}\frac{L_m}{L}}\varepsilon_{sub}
\label{eq:strainamp}
\end{equation}
Equation \ref{eq:strainamp} indicates that increasing the ratio of $w_m/w_n$ and $L_m/L_n$ significantly amplifies the strain induced into the nanowire from the strain a | cal{H}}$-template.
\end{proof}
\begin{lemma}\label{boundinglem}
Suppose $k\geq r$, $W$ is a finite set, $\mathcal{H}$ is a nonempty collection of finite $\mathcal{L}$-structures, and $\sigma \subseteq S_r(C_W,\mathcal{H})$ is complete. Suppose $\mathcal{ |
we compare the strain induced into (i) non released nanowires, (ii) the conventionally released nanowires, and (iii) our nano strain-amplifier structure, using COMSOL Multiphysics \texttrademark. In our nano strain amplifying structure, the width of the r | nction of concentration index $CI$. The contours indicate the 90th percentiles for each LINER sub-sample selected by using the S06, K01, and KW lines (black, purple, and blue lines, respectively). The 90th-percentile contours in cyan and green mark the loc |
uent experimental demonstration easier. The ratio between the length of nanowires and micro bridge was set to be 1: 20. With this geometrical dimensions, strain induced into nanowires array $\varepsilon_n$ was numerically calculated to be approximately 6 t | c{w}{h_\mathit{max}(n)+w}\biggl(\frac{\gamma^{-w}-1}{\gamma^{-1}-1}\biggr)^{\!-1}(\gamma^{-wd/D}-1)\gamma^{2wd/D}\Biggr)^{\!-1/d}.
\]
From Theorem~\ref{the1} and Assumption~\ref{assA3},
\[
r_n\le\max\Biggl(\biggl(\frac{n}C\frac{w}{h_\mathit{max}(n)+w}
\big |
o conventional released and non-released structures. Furthermore, the ratio of the strain induced in to the locally fabricated nanowires was estimated to be 5.9 times larger than that of the substrate, Fig. \ref{fig:fig2}. These results are in solid agreem | e.
In the other case, $\mathbf{j}_{{\rm rf}}$ is generated by the induction
around a sample forming a pick-up coil with wires. In both the cases,
rf currents only appear along the electrode direction.
As we discussed above, there are several mechanisms fo |
igh sensitivity of SiC nanowire strain sensors using the nano strain-amplifier. A thin 3C-SiC film with its thickness of 300 nm was epitaxially grown on a 150 mm diameter Si wafer using low pressure chemical vapour deposition \cite{SiC_growth}. The film wa | enario, the i-vector approach outperforms the x-vector one and paralinguistic features. For disorder symptoms that are mainly manifested in phonological errors, extraction of i-vector using posterior-based features can improve detection performance over co |
te{Phan_JMC}. Subsequently, I-shape p-type SiC resistors with aluminum electrodes deposited on the surface were patterned using inductive coupled plasma (ICP) etching. As the piezoresistance of p-type 3C-SiC depends on crystallographic orientation, all SiC | ich is then passed to the query $q_2$. In turn,
$q_2$ then processes its \texttt{FROM} clause, and appends the schema
of $R_2$ to the context ($\node{\sigma_{R_1}}{\sigma_{R_2}}$), and
this context is used to evaluate the path expression $q_2$'s predicate
|
he centre of the released bridge using focused ion beam (FIB). Two types of nanowire array were fabricated with three nanowires for each array. The average width of each nanowire in each type were 380 nm and 470 nm, respectively. Figure \ref{fig:fig3} show | are the lines of the initial plane.
Let $\Theta=(A_1,A_3, B_3, B_1; A_2, B_2)$ be a convex marked box in ${\mathbb {RP}}^2$. Its dual, $\Theta^*$, is a marked box in the dual projective plane whose points are the lines
$$
(A_2B_1,A_2B_3,A_1B_2,A_3B_2; a,b |
nsors. (a) Released SiC micro bridge used for the subsequent fabrication of the nano strain-amplifier; (b) SEM of a micro SiC resistor where the SiC nanowires array were formed using FIB; (c) SEM of non-released SiC nanowires; (d) SEM of locally fabricated | satisfies the test in the 66'th iteration.
\begin{figure}[tbh!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{merged_averaged_remaining-cars_vs_iterations}
\caption{}
\label{fig:remaining-tests-1-car |
linear relationship between the applied voltage and measured current, indicated that Al made a good Ohmic contact with the highly doped SiC resistance, Fig. \ref{fig:IV}. Additionally, the electrical conductivity of both nanowires and micro frame estimated | plant}, \eqref{eq:quadtrig}} is $f^* = \max_{\sigma\in\mathcal{X}^{{\mathrm{S'}}}}(|\sigma|/(h{\sum_{k_i\in\sigma}k_i}))$. \end{prop}
\begin{proof} In the worst case, $\mathcal{S}_{{\mathrm{S'}}}$ generates $\sigma^* \coloneqq \arg\!\max_{\sigma\in\math |
h=3in]{Fig4}
\vspace{-1.5em}
\caption{Current voltage curves of the fabricated SiC resistors.}
\label{fig:IV}
\end{figure}
The bending experiment was used to characterize the piezoresistive effect in micro size SiC resistors and locally fabrica | fer simply to {\bf $F$-matroids}.
\end{remark}
The following is part of Theorem 2.24 in~\cite{BB17}.
\begin{thm} Let $\mathcal{C}$ be the $F$-circuit set of a strong $F$-matroid. Then $\mathcal{C}^*:=\mathrm{Minsupp}(\mathcal{C}^\perp-\{\mathbf 0\})$ is a |
nd of the Si cantilever was approximately 45 mm. The strain induced into the Si substrate is $\varepsilon_\text{sub} = Mt/2EI$, where $M$ is the applied bending moment; and $t$, $E$ and $I$ are the thickness, Young's modulus and the moment of inertia of th | -order theory. At one-loop order, these features lead to $\log^2$ divergences, rather than the more standard single-logarithm divergences found in conventional field theories. Re-exponentiating these corrections, we predict that the universal properties o |
on{Experimental results. (a) A comparision between the relative resistance change in the nano strain-amplifiers, non released nanowires and released micro frames; (b) The repeatability of the SiC nanowires strain sensors utilizing the proposed structure.}
| RQ2:} Can \textsc{DIME}\ help researchers gain additional insight in whether unimodal contributions or multimodal interactions are the dominant factors behind a model's prediction?}
\label{rq2}
\
Disentangling the model into UC and MI and generating vis |
d samples, the relative resistance change shows a good linear relationship with the applied strain ($\varepsilon_{sub}$). In addition, with the same applied strain to the Si substrate, the resistance change of the SiC nanowires using the nano strain-amplif | case,
the following result holds in the punctured case.
\begin{theorem}[Topology of relative representation spaces of punctured surfaces in $\PSL_2(\RR)$]\label{thm:rep-punctured}
Assume $n>0$ and let $\bCo$ and $\bm{\{r\}}$ be as above.
\begin{it |
n the nano strain-amplifier as well as conventional structures were then quantitatively evaluated based on the effective gauge factor ($GF_{eff}$), which is defined as the ratio of the relative resistance change to the applied strain to the substrate: $GF_ | h}) + \frac{\tilde\epsilon^2}{2C},~\text{provided that}~ \|\vct{w}-\vct{w}_n({\vct{g}},\vct{h})\|_2 \geq \tilde\epsilon.
\end{align}
Now, we argue that $\boldsymbol{\beta}\in\mathcal{S}$ implies that $\|\vct{w}-\vct{w}_n({\vct{g}},\vct{h})\|_2\geq \tilde\v |
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