set stringclasses 1
value | id stringlengths 5 9 | chunk_text stringlengths 1 115k | chunk_num_tokens int64 1 106k | document_num_tokens int64 58 521k | document_language stringclasses 2
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|---|---|---|---|---|---|
train | 0.0.0 | \begin{document}
\begin{abstract} We prove the Liv\v{s}ic Theorem for H\"{o}lder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${\ensuremath{\mathbf{\sigma}}:X\to X}$ that satisfies the Anosov Closing Lemma, and a H\"{o}lder continuous map ${a:X\to B^\times}$ from a compa... | 3,893 | 10,708 | en |
train | 0.0.1 | \section{Proof of the Theorem \ref{t3}}
The following result proven in \cite[Proposition 4.2]{MK} will be used.
\begin{lemman}[A. Karlsson, G. A. Margulis]\ensuremath{\lambda}bel{MK} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a measurable map, $\mu$ an ergodic measure, $s(n,x)$ a subadditive cocycle. For any $\ensu... | 3,863 | 10,708 | en |
train | 0.0.2 | \begin{lemma}\ensuremath{\lambda}bel{l7} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a homeomorphism. For any $\ensuremath{\varepsilon},\delta>0$ let $P_{\ensuremath{\epsilon},\delta}$ be the set of points $x$ in $X$ for which there is an integer number $N=N(x,\ensuremath{\epsilon},\delta)$ such that if $n>N$ then ... | 1,867 | 10,708 | en |
train | 0.0.3 | \section{Proof of the Main Theorem }
After Theorem \ref{t3} is established we can use Corollary \ref{c3} to show that the growth of $\|a(n,x)\|$ is sub-exponential. It allows to use the idea of the original Liv\v{s}ic proof for cocycles with values in Banach rings.
H.Bercovici and V.Nitica in \cite{BN} (Theorem 3.... | 1,085 | 10,708 | en |
train | 0.1.0 | \begin{document}
\pagenumbering{gobble}
\begin{titlepage}
\title{Sensitivity Oracles for All-Pairs Mincuts}
\author{
Surender Baswana\thanks{Department of Computer Science \& Engineering, IIT Kanpur, Kanpur -- 208016, India, sbaswana@cse.iitk.ac.in}
\and
Abhyuday Pandey\thanks{Department of Computer Science \&... | 1,068 | 1,068 | en |
train | 0.2.0 | \begin{document}
\mainmatter
\title{Learning with a Drifting Target Concept}
\titlerunning{Learning with a Drifting Target Concept}
\author{Steve Hanneke \and Varun Kanade \and Liu Yang}
\authorrunning{Steve Hanneke, Varun Kanade, and Liu Yang}
\institute{Princeton, NJ USA.\\
\email{steve.hanneke@gmail.com}
\an... | 2,164 | 33,201 | en |
train | 0.2.1 | \section{Definitions and Notation}
\label{sec:definitions}
Formally, in this setting, there is a fixed distribution $\mathcal{P}$ over the instance space $\mathcal X$,
and there is a sequence of independent $\mathcal{P}$-distributed unlabeled data $X_{1},X_{2},\ldots$.
There is also a concept space $\mathbb C$, and a s... | 3,547 | 33,201 | en |
train | 0.2.2 | \section{Adapting to Arbitrarily Varying Drift Rates}
\label{sec:general}
This section presents a general bound on the error rate at each time,
expressed as a function of the rates of drift, which are allowed to be \emph{arbitrary}.
Most-importantly, in contrast to the methods from the literature discussed above,
the m... | 205 | 33,201 | en |
train | 0.2.3 | \subsection{Adapting to a Changing Drift Rate}
\label{sec:adaptive-varying-rate}
Recall that the method yielding \eqref{eqn:hl94} (based on the work of \cite{helmbold:94})
required access to the sequence $\mathcal Deltaseq$ of changes to achieve the stated guarantee
on the expected number of mistakes. That method is ... | 3,466 | 33,201 | en |
train | 0.2.4 | Let us denote
\begin{equation*}
\tilde{m}_{T} = \mathop{\rm argmin}_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m}.
\end{equation*}
Note that, for any $m^{\prime} \in \{1,\ldots,T-1\}$ and $\delta \in (0,1)$,
if $\tilde{m}_... | 2,792 | 33,201 | en |
train | 0.2.5 | \end{proof} | 5 | 33,201 | en |
train | 0.2.6 | \subsection{Conditions Guaranteeing a Sublinear Number of Mistakes}
\label{sec:sublinear}
\input{tex-files/sublinear.tex} | 38 | 33,201 | en |
train | 0.2.7 | \section{Polynomial-Time Algorithms for Linear Separators}
\label{sec:halfspaces}
In this section, we suppose $\mathcal Delta_{t} = \mathcal Delta$ for every $t \in \mathbb{N}$, for a fixed constant $\mathcal Delta > 0$,
and we consider the special case of learning homogeneous linear separators in $\mathbb{R}^{k}$ und... | 709 | 33,201 | en |
train | 0.2.8 | \subsection{An Improved Guarantee for a Polynomial-Time Algorithm}
\label{sec:efficient-linsep}
We have the following result.
\begin{theorem}
\label{thm:linsep-uniform}
When $\mathbb C$ is the space of homogeneous linear separators (with $d \geq 4$)
and $\mathcal{P}$ is the uniform distribution on the surface of
the... | 3,010 | 33,201 | en |
train | 0.2.9 | Next, we consider the execution of ${\rm ABL}(t,\tilde{h})$, and let the sets $W_{k}$ be as in that execution.
We will denote by $w^{*}$ the weight vector with $\|w^{*}\|=1$ such that $h_{t+m_{0}+1}^{*} = h_{w^{*}}$.
Also denote by $M_{1} = M-m_{0}$.
The proof relies on a few results proven in the work of \cite{awasth... | 3,642 | 33,201 | en |
train | 0.2.10 | Next, note that because $h_{w_{k}}(x) \neq y \Rightarrow \ell_{\tau_{k}}(y (v_{k} \cdot x)) \geq 1$,
and because (as proven above) $\|w^{*} - w_{k-1}\| \leq r_{k}$,
\begin{equation*}
|W_{k}| {\rm er}_{W_{k}}( h_{w_{k}} )
\leq \sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (v_{k} \cdot x))
\leq \sum_{(x,y) \in W_{k}} \ell_{... | 3,040 | 33,201 | en |
train | 0.2.11 | Lemma~\ref{lem:vc-ratio} (applied under the conditional distribution given $|W_{k}|$)
and the law of total probability imply that with probability at least $1-\delta_{k}/3$,
\begin{align*}
|W_{k}| &\mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \Big| |w_{k-1} \cdot x| \leq b_{k-1}\right)
\\ & \leq \sum_{(x,y) \in... | 2,565 | 33,201 | en |
train | 0.2.12 | \end{proof} | 5 | 33,201 | en |
train | 0.2.13 | \begin{proof}[Proof of Theorem~\ref{thm:linsep-uniform}]
We begin with the bound on the error rate.
If $\mathcal Delta > \frac{\pi^{2}}{400 \cdot 2^{27} (d+\ln(4/\delta))}$, the result trivially holds, since then $1 \leq \frac{400 \cdot 2^{27}}{\pi^{2}} \sqrt{\mathcal Delta (d+\ln(4/\delta))}$.
Otherwise, suppose $\mat... | 2,626 | 33,201 | en |
train | 0.2.14 | \section{General Results for Active Learning}
\label{sec:general-active}
As mentioned, the above results on linear separators also provide results
for the number of queries in \emph{active learning}. One can also state
quite general results on the expected number of queries and mistakes
achievable by an active learn... | 3,794 | 33,201 | en |
train | 0.2.15 | We are now ready for the proof of Theorem~\ref{thm:general-active}.
\begin{proof}[Proof of Theorem~\ref{thm:general-active}]
Fix any $i \in \mathbb{N}$, and consider running ${\rm Active}(M(i-1))$.
Since $h^{*}_{M(i-1)+1} \in \mathbb C$,
by Lemma~\ref{lem:active-subroutine}, a union bound, and induction,
with probabi... | 1,593 | 33,201 | en |
train | 0.3.0 | \begin{equation}gin{document}
\date{}
\title{ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE
\footnote{2000 Mathematics Subject Classification. Primary 20NO5 ;
Secondary 08A05.}
\thanks{{\bf Keywords and Phrases :} Smarandache quasigroups, Smarandache loops, universality, $f,g$-principal isotopes}}
\... | 3,655 | 25,145 | en |
train | 0.3.1 | \section{Main Results}
\subsection*{Universality of Smarandache Loops}
\begin{equation}gin{myth}\leftarrowbel{1:4}
A Smarandache quasigroup is universal if all its $f,g$-principal
isotopes are Smarandache $f,g$-principal isotopes.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a Smarandache quasigroup with a S-subquasigr... | 3,528 | 25,145 | en |
train | 0.3.2 | The proof of the converse is as follows. If a SRBL(SLBL) $(G,\oplus
)$ is universal then every isotope $(H,\otimes)$ is an SRBL(SLBL)
i.e there exists an S-RB(LB)-subloop $(S,\otimes )$ in $(H,\otimes
)$. Let $(G,\circ )$ be the $f,g$-principal isotope of $(G,\oplus)$,
then by Corollary~\ref{1:2}, $(G,\circ)$ is an SRB... | 4,067 | 25,145 | en |
train | 0.3.3 | Again, for an SM-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
(x\circ y)\circ (z\circ x)=x\circ [(y\circ z)\circ x]
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
(xR_g^{-1... | 3,885 | 25,145 | en |
train | 0.3.4 | The proof of the converse is as follows. If a SEL $(G,\oplus )$ is
universal then every isotope $(H,\otimes)$ is an SEL i.e there
exists an SE-subloop $(S,\otimes )$ in $(H,\otimes )$. Let $(G,\circ
)$ be the $f,g$-principal isotope of $(G,\oplus)$, then by
Corollary~\ref{1:2}, $(G,\circ)$ is an SEL with say an SE-subl... | 3,970 | 25,145 | en |
train | 0.3.5 | Conversely, if $(G,\oplus)$ is SLBL, then there exists a
SLB-subloop $(S,\oplus )$ in $(G,\oplus)$. If $(G,\circ )$ is an
arbitrary $f,g$-principal isotope of $(G,\oplus)$, then by
Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of $(G,\circ)$ if
$(S,\circ )$ is a Smarandache $f,g$-principal isotope of $(S,\oplus
)$. Let ... | 3,108 | 25,145 | en |
train | 0.3.6 | Conversely, if $(G,\oplus)$ is SML, then there exists a SM-subloop
$(S,\oplus )$ in $(G,\oplus)$. If $(G,\circ )$ is an arbitrary
$f,g$-principal isotope of $(G,\oplus)$, then by Lemma~\ref{1:3},
$(S,\circ )$ is a subloop of $(G,\circ)$ if $(S,\circ )$ is a
Smarandache $f,g$-principal isotope of $(S,\oplus )$. Let us ... | 2,932 | 25,145 | en |
train | 0.4.0 | \begin{document}
\newcounter{algnum}
\newcounter{step}
\newtheorem{alg}{Algorithm}
\newenvironment{algorithm}{\begin{alg}\mathcal End{alg}}
\mathcal Title[Joint spectral radius, Sturmian measures, finiteness conjecture]
{Joint spectral radius, Sturmian measures, and the finiteness conjecture}
\author{O.~Jenkinson \... | 732 | 61,706 | en |
train | 0.4.1 | \section{Introduction}\mathcal Label{generalsection}
\subsection{Problem and setting}\mathcal Label{problemsetting}
For a square matrix $A$ with real entries, its \mathcal Emph{spectral radius} $r(A)$, defined as the maximum modulus of its eigenvalues, satisfies \mathcal Emph{Gelfand's formula}
$$
r(A)= \mathcal Lim_... | 2,976 | 61,706 | en |
train | 0.4.2 | \subsection{Statement of results}\mathcal Label{statementsubsection}
We use $M_2(\mathcal Mathbb{R})$ to denote the set of real $2 \mathcal Times 2$ matrices,
and focus attention on certain of its open subsets:
\begin{notation}
$M_2(\mathcal Mathbb{R}^+)$ will denote the set of \mathcal Emph{positive matrices}, i.e... | 4,068 | 61,706 | en |
train | 0.4.3 | \begin{theorem}\mathcal Label{maxtheoremintro}
If $\mathcal Mathcal A\in\mathcal Mm$ then
\begin{equation}\mathcal Label{maxtheoremintroeq}
\mathcal Log r(\mathcal Mathcal A) =
\mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu \,.
\mathcal End{equation}
\mathcal End{t... | 1,872 | 61,706 | en |
train | 0.4.4 | \subsection{Relation with previous results}\mathcal Label{relatsubsection}
The methods of this paper can also be used to give an alternative proof of some of the results mentioned above,
namely establishing the analogue of Theorem \mathcal Ref{maintheorem} in certain cases treated by
Bousch \& Mairesse \cite{bouschmai... | 1,174 | 61,706 | en |
train | 0.4.5 | \section{Preliminaries}\mathcal Label{prelimsection}
\subsection{The induced map for a positive matrix}\mathcal Label{inducproj}
\begin{notation}
Throughout we use the notation $X=[0,1]$.
\mathcal End{notation}
A positive matrix $A\in M_2(\mathcal Mathbb{R}^+)$ gives a self-map
$v \mathcal Mapsto Av$
of $(\mathcal ... | 3,513 | 61,706 | en |
train | 0.4.6 | \subsection{Perron-Frobenius theory and the joint spectral radius}
\begin{lemma}\mathcal Label{pflemma}
The dominant (Perron-Frobenius) eigenvalue $\mathcal Lambda_A>0$ of the matrix
$A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$
is given by
\begin{equation*}
\mathcal Label{lambdaad... | 2,150 | 61,706 | en |
train | 0.4.7 | \section{Projective convexity and projective concavity}\mathcal Label{projconvprojconc}
\begin{remark}\mathcal Label{derivativesremark}
\item[\, (a)]
For $A\in\mathbb M$ and $x\in X$, the derivative formula
\begin{equation}\mathcal Label{taderiv}
T_A'(x)
=
\det A\mathcal Left( \alpha_A x+b+d\mathcal Right)^{-2}
\mathc... | 3,953 | 61,706 | en |
train | 0.4.8 | We can now prove the following result mentioned in \S \mathcal Ref{statementsubsection}
(note, however, that there is no constraint on the sign of $\sigma_{A_1}$ when $(A_0,A_1)\in\mathcal Mm$):
\begin{cor}\mathcal Label{rhoposlessminusone}
If $(A_0,A_1)\in \mathcal Mm$ then
$\sigma_{A_0}<0<\mathcal Rho_{A_0}$ and $\... | 2,432 | 61,706 | en |
train | 0.4.9 | \section{The induced dynamical system for a concave-convex matrix pair}\mathcal Label{induceddynsyssection}
\subsection{The induced dynamical system and joint spectral radius}
\begin{defn}\mathcal Label{induceddefn}
For a matrix pair $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$,
define the
\mathcal Emph{induced space... | 3,892 | 61,706 | en |
train | 0.4.10 | \subsection{The finiteness property and periodic orbits}
In view of Theorem \mathcal Ref{maxtheorem}, we shall be interested in those measures $m\in\mathcal M_\mathcal Mathcal A$ which
are $f_\mathcal Mathcal A$-maximizing, in the sense of Definition \mathcal Ref{eomax},
i.e.~$m$ attains the maximum in (\mathcal Ref{... | 2,264 | 61,706 | en |
train | 0.4.11 | \subsection{Monotonicity properties and formulae}
The following simple lemma records that for $\mathcal Mathcal A\in\mathcal Mm$,
the induced dynamical system $T_{\mathcal Mathcal A(t)}$
is independent of $t$, and that the induced function $f_{\mathcal Mathcal A(t)}$ differs from $f_\mathcal Mathcal A$
only by the ad... | 2,909 | 61,706 | en |
train | 0.4.12 | \section{Sturmian measures associated to a concave-convex matrix pair}\mathcal Label{sturmasection}
For $\mathcal Mathcal A\in\mathcal Mm$, the induced space $X_\mathcal Mathcal A$ becomes an ordered set when equipped
with the usual order on $X=[0,1]$. In particular,
by a \mathcal Emph{sub-interval} of $X_\mathcal Ma... | 2,447 | 61,706 | en |
train | 0.4.13 | \section{The Sturmian transfer function}\mathcal Label{transfersection}
In order to
show that the maximizing measure for $f_{\mathcal Mathcal A(t)}$ is supported in some $\mathcal Mathcal A$-Sturmian interval
$\Gamma\in\mathcal I_\mathcal Mathcal A$, our strategy will be to add a coboundary $\varphi_\Gamma - \var... | 2,681 | 61,706 | en |
train | 0.4.14 | \section{The extremal Sturmian intervals}\mathcal Label{extsturmsection}
\subsection{Formulae involving extremal intervals}
As noted in Remark \mathcal Ref{neglectsingleton}, an $\mathcal Mathcal A$-Sturmian interval is the disjoint union of two closed intervals when viewed as a subset of $X=[0,1]$. However, the two ... | 2,790 | 61,706 | en |
train | 0.4.15 | \section{Associating $\mathcal Mathcal A$-Sturmian intervals to parameter values}\mathcal Label{particularsection}
\begin{notation}
For a Sturmian interval $\Gamma\in\mathcal I_\mathcal Mathcal A$,
let $s_\Gamma\in\mathcal S_\mathcal Mathcal A$
denote the $\mathcal Mathcal A$-Sturmian measure supported by $\Gamma$, i.... | 3,636 | 61,706 | en |
train | 0.4.16 | A consequence is the following property:
\begin{cor}
For $\mathcal Mathcal A\in \mathcal Mm$, $t\in\mathcal Mathbb{R}^+$, and $i\in\{0,1\}$,
\begin{equation}\mathcal Label{tti}
t_i( \mathcal Mathcal A(t)) = \frac{t_i(\mathcal Mathcal A)}{t}\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{proof}
This follows easily... | 2,955 | 61,706 | en |
train | 0.4.17 | \section{The case when one matrix dominates}\mathcal Label{dominatessec}
It will be useful to record the value of the induced function
$f_\mathcal Mathcal A$ at the two fixed points of $T_\mathcal Mathcal A$:
\begin{lemma}\mathcal Label{fapai}
For $\mathcal Mathcal A\in\mathcal Mm$ and $i\in\{0,1\}$,
\begin{equation*... | 3,645 | 61,706 | en |
train | 0.4.18 | As a consequence of Theorem \mathcal Ref{t0larger1theorem} we obtain:
\begin{cor}\mathcal Label{t0largerttheorem}
If $\mathcal Mathcal A\in\mathcal Mm$ and $t\in \mathcal Mathbb{R}^+$ are such that
\begin{equation}\mathcal Label{t0largert}
t \mathcal Le t_0(\mathcal Mathcal A) \,,
\mathcal End{equation}
then the Dirac... | 2,441 | 61,706 | en |
train | 0.4.19 | \section{Sturmian maximizing measures}\mathcal Label{technicalsection}
It is at this point that we make the extra hypothesis
that the matrix pair $\mathcal Mathcal A$ lies in the class $\mathfrak D \subset \mathcal Mm$.
By Lemma \mathcal Ref{posnegf}(ii) we know that if $\mathcal Mathcal A\in\mathcal Mm$ then $f_\mat... | 3,385 | 61,706 | en |
train | 0.4.20 | \section{The parameter map is a devil's staircase}\mathcal Label{devilsection}
As noted in Remark \mathcal Ref{conjugacymeasures}, if $\mathcal Mathcal A\in\mathcal Mm$ then there is a topological conjugacy
$h_\mathcal Mathcal A:\Omega\mathcal To Y_\mathcal Mathcal A$ between the the shift map $\sigma:\Omega\mathcal T... | 4,014 | 61,706 | en |
train | 0.4.21 | In view of (\mathcal Ref{rot0}) and (\mathcal Ref{rot1}), it suffices to
establish the required properties of $\mathcal Mathcal{P}_\mathcal Mathcal A$ on the sub-interval $\mathcal T_\mathcal Mathcal A=(t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A))$.
Using
the factorisation
(\mathcal Ref{rfactor}),
we see that this... | 1,777 | 61,706 | en |
train | 0.5.0 | \begin{document}
\begin{center}
{A heuristic for the non-unicost set covering problem using local branching} ~\\
~\\
{J.E. Beasley} ~\\
~\\
~\\
Mathematics, Brunel University, Uxbridge UB8 3PH, UK
~\\
~\\
john.beasley@brunel.ac.uk
~\\
{http://people.brunel.ac.uk/$\sim$mastjjb/jeb/jeb.html}
~\\
~\\
April 2023... | 3,486 | 12,865 | en |
train | 0.5.1 | \section{The set covering problem}
\label{app}
\begin{comment}It is clear that to investigate the worth of the
approach given above we need to apply it to an example optimisation problem.
For this purpose we choose to use a classical zero-one optimisation problem, the set covering problem.
\end{comment}
The set c... | 2,588 | 12,865 | en |
train | 0.5.2 | \section{Computational results}
\label{results}
In this section we first discuss the non-unicost set covering test problems which we used. We then give computational results for our
approach when applied to these test problems. We also give a comparison between the results from our approach and eight other approache... | 947 | 12,865 | en |
train | 0.5.3 | \subsection{Results}
Table~\ref{table2} shows the results obtained by our optimisation approach.
In that table we show the optimal/best-known solution value (OBK) for each problem, as taken from Lan et al~\cite{lan07}. We also show the solution value as obtained by our approach,
the final value of $K$ at terminati... | 3,820 | 12,865 | en |
train | 0.5.4 | \hline
\end{tabular}
\caption{Computational results}
\label{table2}
\end{table}
\normalsize
In order to compare our results with previous results in the literature we have taken the detailed results given by various authors and computed percentage deviation from the \\
optimal/best-known value, OBK, as shown in Table... | 1,670 | 12,865 | en |
train | 0.5.5 | \section{Conclusions}
\label{conc}
In this paper we have presented
a heuristic for the non-unicost set covering problem using local branching.
Local branching eliminates the need to define a problem specific search neighbourhood for any particular (zero-one) optimisation problem. It does this by incorporating a gene... | 354 | 12,865 | en |
train | 0.6.0 | \betaegin{document}
\title{Differential systems with reflection\ and matrix invariants}
\footnotetext{\footnotemark Département de Physique Théorique et Section de Mathématiques. Université de Genève, Genève, CH-1211 Switzerland. \href{mailto:santiago.codesido@unige.ch}{santiago.codesido@unige.ch}}
\footnotetext{C... | 1,876 | 12,170 | en |
train | 0.6.1 | \section{The $\mathbf Y$ matrix}
For $X(t)$ the fundamental matrix of the problem, define
\betaegin{equation}
Y(t) := X(t)^{-1} X'(t).
\lambdaabel{Ydef}
\varepsilonnd{equation}
We have that $X = S_1-M_+ S_2$ where $S_1$ and $S_2$ s are power series in $E$ which we can formally give, by using $\Omega = \sqrt{E}$, as $... | 3,841 | 12,170 | en |
train | 0.6.2 | \section{Generalized matrix invariants}
In the following section we use the concept of crossed or generalized matrix invariants which can be found in \gammaite{simon} and \gammaite{cgm} among others.
\subsection{Definition and basic properties}
Let $X_1,\deltaots,X_N \in GL\lambdaeft(n\right)$. Define
\betaegin{equat... | 3,674 | 12,170 | en |
train | 0.6.3 | \subsection{Small-$\mathbf\varepsilonpsilon$ expansion}
By using expression \varepsilonqref{z} we can easily derive distributivity properties, which can be applied to calculate
\betaegin{align*}
& Z_{l,m_1,\deltaots,m_N}\lambdaeft(A_1+\varepsilonpsilon A_2+O\lambdaeft(\varepsilonpsilon^2\right),B_1,\deltaots,B_N\right)... | 2,779 | 12,170 | en |
train | 0.7.0 | \begin{document}
\title{Discriminating between L\"uders and von Neumann measuring devices: \\ An NMR investigation }
\author{C. S. Sudheer Kumar, Abhishek Shukla, and T. S. Mahesh
}
\email{mahesh.ts@iiserpune.ac.in}
\affiliation{Department of Physics and NMR Research Center,\\
Indian Institute of Science Education a... | 1,523 | 7,694 | en |
train | 0.7.1 | \section{Theory}
For the sake of clarity, and also to match the experimental details described in the next section, we consider a system of two qubits.
Since the system is to be measured projectively, dimension of the pointer basis should be greater than or equal to that of the system, and hence we need at least two a... | 3,622 | 7,694 | en |
train | 0.7.2 | \section{Experiment}
We utilize the four spin-1/2 nuclei of 1,2-dibromo-3,5-difluorobenzene (DBDF) as our quantum register.
About 12 mg of DBDF was partially oriented in 600 $\upmu$l of liquid crystal MBBA. The molecular structure of DBDF and its NMR Hamiltonian parameters are shown in Fig. \ref{mol_H}.
The expe... | 2,549 | 7,694 | en |
train | 0.8.0 | \begin{document}
\begin{center} {Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment}
\end{center}
\begin{center}
{\small \textsc{Erika Rivero-Esquivel}\footnote{email: erika.rivero@correo.uady.mx}, \textsc{Eric \'Avila-Vales}\footnote{email: avila@uady.mx}, \texts... | 3,744 | 38,693 | en |
train | 0.8.1 | \section{Existence and positivity of equilibria}
Assume that system (\ref{ruanmod}) has a constant solution $(S_{0},I_{0})$, then:
\begin{align}
-\dfrac{\beta S_{0}I_{0}}{1+\alpha I_{0}}-bS_{0}+bm(1-I_{0})+p\delta I_{0} &
=0\label{eqec1}\\
\dfrac{\beta S_{0}I_{0}}{1+\alpha I_{0}}-p\delta I_{0}-\gamma I_{0}
-\dfrac{\... | 1,990 | 38,693 | en |
train | 0.8.2 | \begin{theorem}
Let $0<\mathcal{R}_{0}\leq1$. For system (\ref{ruanmod}), if $ \mathcal{R}_{0} ^{*} \leq 1 $ then there are no positive endemic equilibria. Otherwise, if
$ \mathcal{R}_{0} ^{*}>1 $ the following propositions hold:
\begin{enumerate}
\item If $\mathcal{R}_{0}=1$ and $(\beta_{2},\alpha_{2})\in A_{3}$ the ... | 3,797 | 38,693 | en |
train | 0.8.3 | moreover $R_{0}^{-}<R_{0}^{+}$ making $\Delta$ positive for $\mathcal{R}
_{0}>R_{0}^{+}$ or $\mathcal{R}_{0}<R_{0}^{-}$. Nevertheless $$R_{0}
^{-}=1+\frac{1}{b\alpha(p\delta+\gamma+\beta_{2})}(\beta(\gamma+\beta
_{2}+bm-bm\alpha_{2}+bm \alpha))-\epsilon,$$ while $$P_{1}=1+\frac
{1}{b\alpha(p\delta+\gamma+\beta_{2})}(\b... | 1,466 | 38,693 | en |
train | 0.8.4 | \begin{corollary}
If $\mathcal{R}_{0}=1$, $ \mathcal{R}_{0} ^{*}>1 $ and $(\beta,\alpha_{2})\in A_{3}$, system
(\ref{ruanmod}) has a backward bifurcation of the disease-free equilibrium $E$.
\end{corollary}
\begin{proof}
First we note that if $(\beta_{2},\alpha_{2})\in A_{3}$ then $R_{0}^{+}$ is
real less than one and... | 269 | 38,693 | en |
train | 0.8.5 | \section{Characteristic Equation and Stability}
The characteristic equation of the linearization of system (\ref{ruanmod}) in
the equilibrium $(S_{0},I_{0})$ is given by:
\begin{equation}
\det(DF-\lambda I),
\end{equation}
where
\begin{equation}
DF=\left(
\begin{matrix}
\frac{\partial f_{1}}{\partial S} & \frac{\parti... | 3,515 | 38,693 | en |
train | 0.8.6 | \subsection{Stability of endemic equilibria}
The general form of the Jacobian matrix is
\begin{equation}
DF=\left(
\begin{matrix}
-\dfrac{\beta I}{1+\alpha I}-b & & -\dfrac{\beta S}{(1+\alpha I)^{2}
}-bm+p\delta\\
\dfrac{\beta I}{1+\alpha I} & & \dfrac{\beta S}{(1+\alpha I)^{2}}
-p\delta-\gamma-\dfrac{\beta_{2}}{(1+... | 3,171 | 38,693 | en |
train | 0.8.7 | \section{Hopf bifurcation}
By previous section we know that the system \eqref{ruanmod} has two positive endemic equilibria under the conditions of theorem \eqref{teo4} . Equilibrium $E_1$ is always a saddle, so its stability does not change and there is no possibility of a Hopf bifurcation in it. So let us analyse th... | 2,613 | 38,693 | en |
train | 0.8.8 | \section{Discussion}
As we said in the introduction, traditional epidemic models have always stability results in terms of $\mathcal{R}_{0} $, such that we need only reduce $\mathcal{R}_{0} <1$ to eradicate the disease. However, including the treatment function brings new epidemic equilibria that make the dynamics of ... | 2,585 | 38,693 | en |
train | 0.8.9 | \section{Computing center manifold}
The Jacobian matrix of system \eqref{carac6} is
\begin{equation}
DF(m,0)=\left(
\begin{matrix}
-b & - \beta m - bm + p \delta\\
0 & 0
\end{matrix}
\right) . \label{eqst1}
\end{equation}
With eigenvalues $\lambda_{1}=-b$ and $\lambda_{2}=0$ and
respective eigenvectors $v_{1}=(1,0)... | 1,916 | 38,693 | en |
train | 0.8.10 | Substituting $S=u-{\frac{\left( \gamma+\beta_2 +bm\right) v}{b}},I=v$
and $\beta m=p\delta+\gamma+\beta_{2}$ we obtain:
\begin{align}
\frac{dv}{dt} & =0u+f(v,u)\nonumber\\
\frac{du}{dt} & =-bu+g(v,u), \label{eqst2}
\end{align}
where
\begin{align}
& f(u,v)=-{\frac{v\left( -\beta\,b-\beta\,b\mathit{\alpha_{2}}\,v... | 4,046 | 38,693 | en |
train | 0.8.11 | \section{Hopf bifurcation}
To analyze the behaviour of the solutions of \eqref{ruanmod} when $ s=0 $ we make a change of coordinates $x=S-S_2$, $y=I-I_2$, to obtain a new equivalent system to \eqref{ruanmod} with an equilibrium in $(0,0)$ in the $x-y$ plane. Under this change the system becomes in:
\begin{align}
\fra... | 3,638 | 38,693 | en |
train | 0.8.12 | Where
\begin{align}
G_1 &= \dfrac{1}{(1 + \alpha_1 y+ \alpha_1 I_2)(1 + \alpha_1 I_2)} \{[(1 + \alpha_1 I_2)c_1-a_{11} \alpha_1] xy + [ c_2 ( 1 + \alpha_1 I_2 ) - \alpha_1 a_{12} ] y^{2} \} \\
G_2 &= \dfrac{1}{(1 + \alpha_1 y+ \alpha_1 I_2)(1 + \alpha_2 y+ \alpha_2 I_2)(1 + \alpha_1 I_2)(1 + \alpha_2 I_2)} \{ [c_3 (1 ... | 4,079 | 38,693 | en |
train | 0.8.13 | Let
\begin{align}
\bar{a}_2 &= \dfrac{1}{16} [ (H_1)_{uuu} + (H_1)_{uvv} + (H_{2})_{uuv} + (H_2)_{vvv} ] + \dfrac{1}{16( - \Lambda)} [(H_1)_{uv} ((H_1)_{uu} + (H_1)_{vv}) \nonumber \\
& - (H_{2})_{uv} ( (H_{2})_{uu} + (H_2)_{vv} ) - (H_1)_{uu}(H_2)_{uu} + (H_1)_{vv} (H_2)_{vv}].
\end{align}
Then
\begin{align}
\bar{a... | 1,864 | 38,693 | en |
train | 0.9.0 | \begin{document}
\title{Temporal profile of biphotons generated from a hot atomic vapor and spectrum of electromagnetically induced transparency}
\author{
Shih-Si Hsiao,$^{1,}$\footnote{Electronic address: {\tt shihsi.hsiao@gmail.com}}
Wei-Kai Huang,$^1$
Yi-Min Lin,$^1$
Jia-Mou Chen,$^1$
Chia-Yu Hsu,$^1$ and
Ite A. ... | 2,421 | 12,811 | en |
train | 0.9.1 | \section{Theoretical Predictions}
Biphotons or a pair of single photons are produced from a hot atomic vapor with the spontaneous four-wave mixing process (SFWM). Figure~\ref{fig:transition_diagram} shows the relevant energy levels and transitions of SFWM process for the generation of biphotons. All population is place... | 4,065 | 12,811 | en |
train | 0.9.2 | \FigTwo
We compare the biphoton's EIT spectrum calculated from the numerical integral of Eq.~(\ref{eq:T_exact}) with that calculated from the analytical formula of Eq.~(\ref{eq:T_approximation}). Figure~\ref{fig:eit_simulation}(a) shows the comparisons at different values of the coupling Rabi frequency, $\Omega_c$. As... | 2,883 | 12,811 | en |
train | 0.9.3 | \section{Experimental Setup}
We experimentally studied the temporal width and spectral linewidth of SFWM biphotons generated from a hot vapor of $^{87}$Rb atoms. The SFWM transition scheme is shown in Fig.~\ref{fig:transition_diagram}, and the actual energy levels in the experiment are specified in the caption. The ti... | 3,442 | 12,811 | en |
train | 0.10.0 | \begin{document}
\title{Galloping in fast-growth natural merge sorts}
\begin{abstract}
We study the impact of sub-array merging routines
on merge-based sorting algorithms.
More precisely, we focus on the
\emph{galloping} sub-routine that
TimSort\xspace uses to merge monotonic (non-decreasing)
sub-arrays, hereafter ca... | 430 | 49,320 | en |
train | 0.10.1 | \section{Introduction}\label{sec:intro}
In 2002, Tim Peters, a software engineer, created a new sorting algorithm,
which was called TimSort\xspace~\cite{Peters2015} and was built
on ideas from McIlroy~\cite{McIlroy1993}. This algorithm immediately
demonstrated its efficiency for sorting actual data, and was
adopted a... | 3,948 | 49,320 | en |
train | 0.10.2 | \paragraph*{Contributions}
We study the time complexity of various natural
merge sort algorithms in a context where arrays are not just
parametrised by their lengths.
More precisely, we focus on a
decomposition of input arrays that is
dual to the decomposition of arrays into
monotonic runs,
and that was proposed by Mc... | 1,698 | 49,320 | en |
train | 0.10.3 | \section{The galloping sub-routine for merging runs}
\label{sec:description}
Here, we describe the galloping sub-routine that
the algorithm TimSort\xspace uses to merge adjacent non-decreasing runs.
This sub-routine is a blend between a naïve
merging algorithm, which requires~$a+b-1$
comparisons to merge runs~$A$ and~... | 3,426 | 49,320 | en |
train | 0.10.4 | The above discussion immediately provides us with
a cost model for
the number of comparisons performed when merging two runs.
\begin{proposition}\label{pro:2}
Let~$A$ and~$B$ be two non-decreasing runs of
lengths~$a$ and~$b$, with values in~$\{1,2,\ldots,\sigma\}$.
For each integer~$i \leqslant \sigma$, let~$a_{\right... | 2,995 | 49,320 | en |
train | 0.10.5 | \section{Fast-growth and (tight) middle-growth properties}
\label{sec:fast-growth}
In this section, we focus on two novel properties of
stable natural merge sorts, which we call \emph{fast-growth}
and \emph{middle-growth}, and on a variant of
the latter property, which we call
\emph{tight middle-growth}.
These propert... | 2,728 | 49,320 | en |
train | 0.10.6 | Similar, weaker results also holds for
algorithms with the (tight)
middle-growth property.
\begin{theorem}
\label{thm:middle-few-naïve}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the middle-growth property.
If~$\mathcal{A}$ uses either
the galloping or the naïve
sub-routine for merging runs, it
... | 2,605 | 49,320 | en |
train | 0.10.7 | \section{A few algorithms with the fast- and (tight) middle-growth properties}
\label{sec:pos-fast-growth}
In this section, we briefly present the algorithms
mentioned in Section~\ref{sec:intro} and prove
that each of them enjoys the fast-growth property
and/or the (tight) middle-growth property.
Before treating these... | 897 | 49,320 | en |
train | 0.10.8 | \subsection{Algorithms with the fast-growth property}
\label{subsec:other-fast}
\subsubsection{PowerSort\xspace}
\label{subsubsec:PoS}
The algorithm PowerSort\xspace is best defined by introducing
the notion of \emph{power}
of a run endpoint or of a run, and then
characterising the merge trees that PowerSort\xspace i... | 3,367 | 49,320 | en |
train | 0.10.9 | \subsubsection{PeekSort\xspace}
\label{subsubsec:PeS}
Like its sibling PowerSort\xspace, the algorithm PeekSort\xspace is best defined
by characterizing the merge trees it induces.
\begin{definition}\label{def:tree:PeS}
Let~$\mathcal{T}$ be the merge tree induced by PeekSort\xspace on an array~$A$.
The children of ea... | 2,598 | 49,320 | en |
train | 0.10.10 | \subsubsection{TimSort\xspace}
\label{subsubsec:TS}
The algorithm TimSort\xspace is presented in
Algorithm~\ref{alg:TS}.
\begin{algorithm}[h]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on ... | 3,286 | 49,320 | en |
train | 0.10.11 | \subsubsection{\textalpha-MergeSort\xspace}
\label{subsubsec:aMS}
The algorithm \textalpha-MergeSort\xspace is
parametrised by a real number~$\alpha > 1$ and
is presented in Algorithm~\ref{alg:aMS}.
\begin{algorithm}[h]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort, pa... | 2,893 | 49,320 | en |
train | 0.10.12 | \subsection{Algorithms with the tight middle-growth property}
\label{subsec:other-tight}
\subsubsection{PowerSort\xspace}
\label{subsubsec:PoS:2}
\begin{proposition}
\label{pro:POS:tight}
The algorithm PowerSort\xspace has the tight middle-growth
property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge... | 3,203 | 49,320 | en |
train | 0.10.13 | \subsection{Algorithms with the middle-growth property}
\label{subsec:other-middle}
\subsubsection{\textalpha-StackSort\xspace}
\label{subsubsec:aSS}
The algorithm \textalpha-StackSort\xspace, which
predated and inspired its variant
\textalpha-MergeSort\xspace, is presented in
Algorithm~\ref{alg:aSS}.
\begin{algorit... | 2,116 | 49,320 | en |
train | 0.10.14 | \section{Refined complexity bounds}
\label{sec:precise-bounds-PoS}
One weakness of Theorem~\ref{thm:middle-few} is that it cannot
help us to distinguish the complexity upper bounds of those
algorithms that have the middle-growth property, although
the constants hidden in the~$\mathcal{O}$ symbol could be dramatically
... | 1,976 | 49,320 | en |
train | 0.10.15 | \subsection{Parameter t with logarithmic growth and tight middle-growth property}
\label{subsec:variable-t}
Letting the parameter~$\mathbf{t}$ vary,
we minimise the upper bound provided by Theorem~\ref{thm:PoS-constant}
by choosing~$\mathbf{t} = \mathcal{T}heta(\mathcal{H}^\ast+1)$, in which case
this upper bound simp... | 2,747 | 49,320 | en |
train | 0.10.16 | \subsection{Refined upper bounds for adaptive ShiversSort\xspace}
\label{subsec:cASS:precise}
\begin{proposition}
\label{pro:cASS:not-tight}
The algorithm adaptive ShiversSort\xspace does not have the
tight middle-growth property.
\end{proposition}
\begin{proof}
Let~$A_k$ be an array whose run decomposition
consists ... | 1,605 | 49,320 | en |
train | 0.10.17 | \subsection{Refined upper bounds for PeekSort\xspace}
\label{subsec:PeS:precise}
\begin{proposition}
\label{pro:PeS:not-tight}
The algorithm PeekSort\xspace does not have the
tight middle-growth property.
\end{proposition}
\begin{proof}
For all~$k \geqslant 0$,
let~$\mathcal{S}_k$ be the integer-valued
sequence defin... | 3,954 | 49,320 | en |
train | 0.10.18 | \begin{theorem}
\label{thm:PeS-log}
Theorem~\ref{thm:PoS-log} remains valid if
we consider the algorithm
PeekSort\xspace instead of
an algorithm with the tight middle-growth property.
\end{theorem}
\begin{proof}
Let us reuse the notations we introduced while
proving Theorem~\ref{thm:PeS-constant}.
With these notations... | 2,224 | 49,320 | en |
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