Model: Master-thesis-NAP/ModernBERT-DAPT-Embed-DAPT-Math
Author: Master-thesis-NAP
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, modernbert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:79876, loss:TripletLoss, en, arxiv:1908.10084, arxiv:1703.07737, base_model:Master-thesis-NAP/ModernBert-DAPT-math, base_model:finetune:Master-thesis-NAP/ModernBert-DAPT-math, license:apache-2.0, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: Master-thesis-NAP/ModernBert-DAPT-math
Card metadata: language: en
license: apache-2.0
tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:79876, loss:TripletLoss
base_model: Master-thesis-NAP/ModernBert-DAPT-math
widget: source_sentence: What is the error estimate for the difference between the exact solution and the local oscillation decomposition (LOD) solution in terms of the $L_0$ norm?
sentences: \label{RL1}
The system \eqref{R3} has the following positive fixed points if $0 <\alpha\leq1$ and $b>d$
$$E^*=\left(\dfrac{d}{b}, \dfrac{(b-d) r}{b^2}\right)$$, \label{theo1d}
With the assumptions and setting is this section,  the finite difference solution  computed using the improved harmonic average method applied to \eqn{eq1d} or \eqn{eq1dB}  has second order convergence in the infinity norm, that is,
\eqm
  \|\mathbf{E} \|_{\infty}\le C h^2,
\enm
assuming that the true solution of \eqn{eq1d} is piecewise $C^4$ excluding the interface $\alf$, that is, 
$u(x) \in C^4(0,\alf)  \cup C^4(\alf,1)$. 
%where $C$ is a generic error constant., \label{Corollary}
     Let Assumptions~\ref{assum_1} and~\ref{assump2} be satisfied. Let $u$ be the solution of~\eqref{WeakForm} and let $u_{H,k}$ be the LOD solution of~\eqref{local_probelm }. Then we have 
     \begin{equation}\label{L2Estimate}
         \|u-I_Hu_{H,k}\|_0\lesssim  \|u-I_Hu\|_0+\|u-u_{H,k}\|_0 +H|u-u_{H,k}|_1.
     \end{equation}
     %\[\|u-I_Hu_{H,k}\|_0\lesssim H |u|_1 +|u-u_{H,k}|_1.\], source_sentence: What is the expected value of the number of individuals in a Markov branching process with non-homogeneous Poisson immigration (MBPNPI) at time $t=0$, given that the immigration rate is $\lambda$?
sentences: \label{lemma-sampling}
Fix an integer~$n\geq 1$.
Consider the initial configuration with one active particle on each
site of~$V_n$ and let the system evolve, with particles being killed
when they jump out of~$V_n$, until no active particle remains
in~$V_n$.
Then the distribution of the resulting stable configuration is exactly
the stationary distribution of the driven-dissipative Markov chain
on~$V_n$.
In particular, the number of sleeping particles remaining in~$V_n$ is
distributed as~$S_n$., The process $Y(t)$, $t\geq 0,$ is called Markov branching process with
non-homogeneous Poisson immigration (MBPNPI)., For any $\lambda \in(0,1)$ and $s \in\mathbb N$,
  \begin{equation*}
\sum_{k=s}^{\infty}\binom {k}{s}
(1-\lambda)^{k-s}=    \lambda^{-s-1}.
\end{equation*}, source_sentence: Does the theorem imply that the rate of convergence of the sequence $T_{m,j}(E)$ to $T_{m+k_n,j+k_n}(E)$ is exponential in the distance between $m$ and $j$, and that this rate is bounded by a constant $C$ times an exponential decay factor involving the parameter $\gamma$?
sentences: \label{lem1}
		For all $m,j\in\Z$,  we have
		\begin{equation*}
		|| T_{m,j} (E)-T_{m+k_n,j+k_n}(E)||\leq C e^{-\gamma  k_n}  e^{(\mathcal L(E)+\varepsilon) |m-j|}. 
		\end{equation*}, [Divergence Theorem or Gauss-Green Theorem for Surfaces in $\R^3$]
	\label{thm:surface_int}
	        Let $\Sigma \subset \Omega\subseteq\R^3$ be a bounded smooth surface.
	        Further, $\bb a:\Sigma\to\R^3$ is a continuously differentiable vector field that is either defined on the
					boundary $\partial\Sigma$ or has a bounded continuous extension to this boundary.
	        Like in \eqref{eq:decomp} it may be decomposed into tangential and normal components
					as follows $\bb a = \bb a^\shortparallel + a_\nu\bs\nu_\Sigma$. By $\dd l$ we denote the line element on 
					the curve $\partial \Sigma$. We assume that the curve is continuous and consists of finitely many
					smooth pieces.
	        Then the following divergence formula for surface integrals holds
	        %
	        \begin{align}
	            %
	            \int\limits_\Sigma \left[\nabla_\Sigma\cdot\bb a^\shortparallel\right](\x)\;\dd S
							= \int\limits_{\partial\Sigma} \left[\bb a\cdot\bs\nu_{\partial\Sigma}\right](\x)\,\dd l .
	            \label{eq:surface_div}
	            %
	        \end{align}
					%
					From this we obtain the formula
					%
	        \begin{align}
	            %
	            \int\limits_\Sigma \left[\nabla_\Sigma\cdot\bb a\right](\x)\;\dd S
							= \int\limits_{\partial\Sigma} \left[\bb a\cdot\bs\nu_{\partial\Sigma}\right](\x)\,\dd l 
							-\int\limits_\Sigma\left[ 2\kappa_Ma_\nu\right](\x)\;\dd S.
	            \label{eq:surface_div_2}
	            %
	        \end{align}
	    %, \label{theo:helper3}
Assume that $\{\PP_N\}_{N\ge 1}$ is a sequence of probability measures that is HT-appropriate in the sense of \cref{def:appropriate} and satisfies the LLN in the sense of \cref{def:LLN}.
Let $(\kappa_n)_{n\ge 1}$ and $(m_n)_{n\ge 1}$ be the sequences that arise from these definitions.
Moreover, assume that there exists a constant $C>0$ such that $|\kappa_n|\leq C^n$, for all $n \geq 1$.
Then $(m_n)_{n\ge 1}$ is the sequence of moments of a unique probability measure on $\R$., source_sentence: What is the error estimate for the eigenfunction approximation in terms of the weak eigenvalue and the norm of the difference between the exact and approximate eigenfunctions?
sentences: Consider dynamics \eqref{avg} and define the corresponding average dynamics as $\label{T-avg}
\mathring{\chi} = \epsilon h_{av}(\chi)$, with the average function defined as
\begin{equation*} 
h_{av}(\chi):=\lim_{T \to \infty} \frac{1}{T}\int_{t}^{t+T} h(\mu, \chi, 0) d \mu, \ T>0,
\end{equation*}
both \eqref{avg} and \eqref{T-avg} twice differentiable and bounded in every compact set of the $\chi$-domain $\mathcal{D} \subset \mathbb{R}^{3}$. 
%
Let $\chi(\tau,\epsilon)$ and $\chi_{av}(\epsilon\tau)$ denote the solutions of \eqref{avg} and \eqref{T-avg}, respectively. If $\chi_{av}(\epsilon\tau)\in \mathcal{D}$ for all $\tau\in[0,\zeta/\epsilon]$, $\zeta\geq 0$, and $\chi(0,\epsilon) - \chi_{av}(0)=\mathcal{O}(\nu(\epsilon))$, then there exists an $\epsilon^{*}>0$ such that for all $0<\epsilon<\epsilon^{*}$, $\chi(\tau,\epsilon)$ is well defined and
$$
\chi(\tau,\epsilon) - \chi_{av}(\epsilon\tau) = \mathcal{O}(\nu(\epsilon)) \ \textnormal{on} \ \tau \in [0, \zeta/\epsilon],
$$
for some function $\nu\in \mathcal{K}$., (\cite{DangWangXieZhou})\label{Theorem_Error_Estimate_k}
Let us define the spectral projection $F_{k,h}^{(\ell)}: V\mapsto {\rm span}\{u_{1,h}^{(\ell)}, \cdots, u_{k,h}^{(\ell)}\}$ for any integer $\ell \geq 1$ as follows:
\begin{eqnarray*}
a(F_{k,h}^{(\ell)}w, u_{i,h}^{(\ell)}) = a(w, u_{i,h}^{(\ell)}), \ \ \ i=1, \cdots, k\ \ {\rm for}\ w\in V.
\end{eqnarray*}
Then the exact eigenfunctions $\bar u_{1,h},\cdots, \bar u_{k,h}$ of (\ref{Weak_Eigenvalue_Discrete}) and the eigenfunction approximations $u_{1,h}^{(\ell+1)}$, $\cdots$,  $u_{k,h}^{(\ell+1)}$ from Algorithm \ref{Algorithm_k} with the integer $\ell > 1$ have the following error estimate:
\begin{eqnarray*}\label{Error_Estimate_Inverse}
 \left\|\bar u_{i,h} - F_{k,h}^{(\ell+1)}\bar u_{i,h} \right\|_a \leq
 \bar\lambda_{i,h} \sqrt{1+\frac{\eta_a^2(V_H)}{\bar\lambda_{1,h}\big(\delta_{k,i,h}^{(\ell+1)}\big)^2}}
\left(1+\frac{\bar\mu_{1,h}}{\delta_{k,i,h}^{(\ell)}}\right)\eta_a^2(V_H)\left\|\bar u_{i,h} - F_{k,h}^{(\ell)}\bar u_{i,h} \right\|_a,
\end{eqnarray*}
where $\delta_{k,i,h}^{(\ell)} $ is defined as follows:
\begin{eqnarray*}
\delta_{k,i,h}^{(\ell)} = \min_{j\not\in \{1, \cdots, k\}}\left|\frac{1}{\lambda_{j,h}^{(\ell)}}-\frac{1}{\bar\lambda_{i,h}}\right|,\ \ \ i=1, \cdots, k.
\end{eqnarray*}
Furthermore, the following $\left\|\cdot\right\|_b$-norm error estimate holds:
\begin{eqnarray*}
\left\|\bar u_{i,h} -F_{k,h}^{(\ell+1)}\bar u_{i,h} \right\|_b\leq 
\left(1+\frac{\bar\mu_{1,h}}{\delta_{k,i,h}^{(\ell+1)}}\right)\eta_a(V_H) \left\|\bar u_{i,h} -F_{k,h}^{(\ell+1)}\bar u_{i,h}\right\|_a.
\end{eqnarray*}, \big[{\bf Condition $SD1(h)$}\big]\label{DefnSD1(h)}

In \cite{MDL} an approximation order $O(h^s)$, as $h\to 0$, is proved, where $h$ is the sampling distance. The achievable order $s$ is of course limited by the smoothness order of the boundaries of $Graph(F)$. Then, the order $s$ depends upon the degree of the polynomials used to approximate the boundary near the neighborhood of points of topology change and upon the degree of splines used at regular regions. 

For example, let us view Step C of the approximation algorithm described in Section 5.2 of \cite{MDL}. 
It is assumed that the boundary curves are $C^{2k}$ smooth, and it is implicitly assumed that $h$ is small enough so that there are $2k$ sample points close to the point of topology change, for computing the polynomial $p_{2k-1}$ therein.
This condition is related to the more general condition $SD(h)$ and it can serve as a practical way of checking it for the case $d=1$. That is, near a point of topology change, we check whether there are enough sample points for applying the approximation algorithm in \cite{MDL}. We denote this condition as the $SD1(h)$ condition., source_sentence: Does Werner-Young's inequality imply that the convolution of two $L^p$ spaces is always $L^r$ for $1 < r < \infty$?
sentences: $\cE^{(0)}_{p,\alpha}$ satisfies the second Beurling-Deny criterion.  If $1 < p_- \leq p_+ < \infty$, it is reflexive and satisfies the $\Delta_2$-condition.  
 %, A \emph{bond system} is a tuple $(B,C,s,t,1,\cdot)$, where $B$ is a set of \emph{bonds}, $C$ is a set of \emph{content} relations, and $s,t:C\to B$ are \emph{source} and \emph{target} functions. For $c\in C$ with $s(c)=x$ and $t(c)=y$, we write $x\xrightarrow{c}y$ or $c:x\to y$, indicating that $x$ \emph{contains} $y$. Each bond $x\in B$ has an \emph{identity} containment $1_x:x\to x$, meaning every bond trivially contains itself. For $c:x\to y$ and $c':y\to z$, their composition is $cc':x\to z$. These data must satisfy:
    \begin{enumerate}
        \item Identity laws: For each $c:x\to y$, $1_x c= c=c1_y$
        \item Associativity: For $c:x\to y$, $c':y\to z$, $c'':z\to w$, $c(c'c'')=(cc')c''$
        \item Anti-symmetry: For $c:x\to y$ and $c':y\to x$, $x=y$
        \item Left cancellation: For $c,c':x\to y$ and $c'':y\to z$, if $cc''=c'c''$, then $c=c'$
    \end{enumerate}, [Werner-Young's inequality]\label{Young op-op}
Suppose $S\in \cS^p$ and $T\in \cS^q$ with $1+r^{-1}=p^{-1}+q^{-1}$.
Then $S\star T\in L^r(\R^{2d})$ and
\begin{align*}
    \|S\star T\|_{L^{r}}\leq \|S\|_{\cS^p}\|T\|_{\cS^q}.
\end{align*}
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics: cosine_accuracy@1, cosine_accuracy@3, cosine_accuracy@5, cosine_accuracy@10, cosine_precision@1, cosine_precision@3, cosine_precision@5, cosine_precision@10, cosine_recall@1, cosine_recall@3, cosine_recall@5, cosine_recall@10, cosine_ndcg@10, cosine_mrr@10, cosine_map@100
model-index: name: ModernBERT DAPT Embed DAPT Math
results: task: type: information-retrieval
name: Information Retrieval
dataset: name: TESTING
type: TESTING
metrics: type: cosine_accuracy@1
value: 0.5679510844485464
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.6324411628980157
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.6586294416243654
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.6938163359483156
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.5679510844485464
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.36494385479157054
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.27741116751269035
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.18192201199815417
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.026541702012005317
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.048742014322369596
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.0598887341486898
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.07516536747041261
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.25320633940615317
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.6070309695944213
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.07416668442975916
name: Cosine Map@100
verified: no
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Created: 2025-05-27 11:19:56