Model: Master-thesis-NAP/ModernBERT-DAPT-Embed-DAPT-Math-v2
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:MultipleNegativesRankingLoss, en, arxiv:1908.10084, arxiv:1705.00652, 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:MultipleNegativesRankingLoss
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{thm1}
Suppose $\kappa$ and $\bar a$ are as above. Then $|\Pcut(\bar a)| \leq 2^\kappa$. Indeed if
$2^\kappa=\aleph_\alpha,$ then $|\Pcut(\bar a)| \leq |\alpha+1|^2$., \cite{kyushu}
    For every discrete group $\G$ and every 2-dimensional  representation $\varrho$ of $\G$, $\varrho-$equivariant functions for $\G$ always exist., \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: 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{thm:weibull}
Suppose random variable $X$ follows Weibull distribution, and $E(X^i)$ denotes the $i$-th moment of $X$. Then the random variable $X$ satisfy the following inequality: 
\begin{equation}\label{eq:moments}
    E(X^n)^{\frac{1}{n}} \geq E(X^m)^{\frac{1}{m}},
\end{equation}
where $n > m$., \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*}, If the problem \eqref{eq:Model-based_Program} is convex, then under the primal-dual dynamics \eqref{eq:PDD}-\eqref{eq:AlgebraicConstruction}, the system \eqref{eq:Input-OutputMap} asymptotically converges to a steady state that is the optimal solution of \eqref{eq:Model-based_Program}., source_sentence: What is the rate of convergence for the total error in the given problem, assuming the conditions in Theorem~\ref{convergence-rates} are met?
sentences: \label{convergence-rates}
Under the assumptions of Theorem~\ref{well-posedness}. Given $(\bu,{p},\bzeta,\varphi)\in (\bH^{s_1+1}(\Omega)\cap \bV_1)\times (\text{H}^{s_1}(\Omega)\cap Q_{b_1}) \times (\bH^{s_2}\cap \bV_2) \times (\text{H}^{s_2}\cap Q_{b_2})$, $(\bu_h,{p}_h,\bzeta_h,\varphi_h)\in \bV_1^{h,k_1}\times Q_1^{h,k_1}\times \bV_2^{h,k_2}\times Q_2^{h,k_2}$ be the respective solutions of the continuous and discrete problems, with the data satisfying $\fb\in \bH^{s_1-1}\cap \bQ_{b_1}$ and $g\in H^{s_2}(\Omega)\cap Q_{b_2}$. If $\overline{C}_1 \sqrt{M} L_\ell + \overline{C}_2^2 \sqrt{M^3} L_\bbM\sqrt{2\mu}   (\norm{\varphi_D}_{1/2,\Gamma_D} + \norm{g}_{0,\Omega}) < 1/2.$ Then, the total error $\overline{\textnormal{e}}_h:=\norm{(\bu-\bu_h,{p}-{p}_h, \bzeta-\bzeta_h,\varphi-\varphi_h)}_{\bV_1\times Q_{1} \times \bV_2\times Q_2}$ decays with the following rate for $s:= \min \left\{s_1,s_2\right\}$
    \begin{align*}\label{convergence-rate}
         \overline{\textnormal{e}}_h &\lesssim h^{ s} (|\fb|_{s_1-1,\bQ_{b_1}} + |\bu|_{s_1+1,\bV_1} + |{p}|_{s_1,Q_{b_1}} + |g|_{s_2,Q_{b_2}} + |\bzeta|_{s_2,\bV_2}+|\varphi|_{s_2,Q_{b_2}}).
    \end{align*}, \label{thm}
For vector linear secure aggregation defined above, the optimal total key rate is 
\begin{eqnarray}
     R_{Z_{\Sigma}}^* %= \left\{R_{Z_{\Sigma}}: R_{Z_{\Sigma}} \geq 
    = \mbox{rank} \left( \left[ \mathbf{F} ; \mathbf{G} \right] \right)
     - \mbox{rank} \left( \mathbf{F} \right) = \mbox{rank}({\bf G} | {\bf F}).
     %\right\}.
%     \\ \mbox{rank}
\end{eqnarray}, The process $Y(t)$, $t\geq 0,$ is called Markov branching process with
non-homogeneous Poisson immigration (MBPNPI)., source_sentence: Is the local time of the horizontal component of the Peano curve ever greater than 1?
sentences: [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}
	    %, There exists  local time of the horizontal component $x$ of the Peano curve. Moreover, this local time attains values no greater than $1$., [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*}, source_sentence: What is the meaning of the identity containment $1_x:x\to x$ in the context of the bond system?
sentences: \label{lem:opt_lin}
Consider the optimization problem
\begin{equation}\label{eq:max_tr_lem}
\begin{aligned}
    \max_{\bs{U}}&\;\; \Re\{\mrm{tr}(\bs{U}^\mrm{H}\bs{B}) \}\\
    \mrm{s.t. \;\;}& \bs{U}\in \mathcal{U}(N),
\end{aligned}
\end{equation}
where $\bs{B}$ may be an arbitrary $N\times N$ matrix with singular value decomposition (SVD) $\bs{B}=\bs{U}_{\bs{B}}\bs{S}_{\bs{B}}\bs{V}_{\bs{B}}^\mrm{H}$. The solution to \eqref{eq:max_tr_lem} is given by
\begin{equation}\label{eq:sol_max}
    \bs{U}_\mrm{opt} = \bs{U}_{\bs{B}}^\mrm{H}\bs{V}_{\bs{B}}.
\end{equation}
\begin{skproof}
    A formal proof, which may be included in the extended version, can be obtained by defining the Riemannian gradient over the unitary group and finding the stationary point where it vanishes. However, an intuitive argument is that the solution to \eqref{eq:max_tr_lem} is obtained by positively combining the singular values of $\bs{B}$, leading to \eqref{eq:sol_max}.
\end{skproof}, \label{AM_BA_lem1}
Let $$\Omega =\left\{a={{\left(k_1x_1+k_2,\dots,k_1x_n+k_2\right)}}\mid k_1, k_2\in \mathbb{R}\right\} .$$ Then ${\displaystyle\underset{a\in \Omega}{\operatorname{argmin}} {J_{\alpha }}(a)=\overline{a}\ },$ where $\overline{a}=\left(\overline{a}_1,\dots,\overline{a}_n\right)$, $$\overline{a}_i=\frac{1}{n}\sum^n_{j =1}{y_j},\quad\forall i=1,\dots,n.$$ In other words, on the class of lines $J_{\alpha }\left(a\right)$ reaches a minimum on a straight line parallel to the $Ox$ axis. So, this is the average line for the ordinates of all points of set $X$., 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}
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: 0.868020304568528
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.9183202584217812
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.9325103830179973
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9495846792801107
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.868020304568528
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.6118674050146131
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.49353945546838945
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.34758883248730965
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.04186710795480722
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.08315252408701693
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.1073909448198794
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.14207392775097807
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.4493273991613623
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.8963655316764447
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.16376932233660765
name: Cosine Map@100
verified: no
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Created: 2025-05-27 18:33:20