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
values |
|---|---|---|---|---|---|
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 |
End of preview. Expand in Data Studio
MathPile ArXiv (subset)
Description
This dataset consists of a toy subset of 8834 (5000 training + 3834 testing) TeX files found in the arXiv subset of MathPile, used for testing. You should not use this dataset. Training and testing sets are already split
Source
The data was obtained from the training + validation portion of the arXiv subset of MathPile.
Format
- Given as JSONL files of JSON dicts each containing the single key: "text"
Usage
- LaTeX stuff idk
License
The original data is subject to the licensing terms of the arXiv. Users should refer to the arXiv's terms of use for details on permissible usage.
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