pair_id stringlengths 23 50 | family stringclasses 16
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|---|---|---|---|---|
GPT-5.1::2008.05796::prose5 | GPT-5.1 | 1ai | prose | In this work we therefore combine three complementary approaches in order to obtain a coherent bulk and element resolved picture of the magnetism in selected Cantor-Wu alloys. Magnetic Compton scattering provides a truly bulk-sensitive probe of the spin-resolved electron momentum density and yields the spin component o... |
Llama-4-Scout::2012.06637::prose0 | human | 0human | prose | The Coleman-Weinberg mechanism \cite{Coleman:1973jx} is a natural realization of spontaneous symmetry breaking as a consequence of radiative quantum corrections.
While the mechanism is not directly relevant within the electroweak symmetry breaking of the Standard Model, it can be, in principle, responsible for symmetry... |
GLM-4.5-Air::1903.08278::thm5 | human | 0human | proof | 1.) (Setting up the equations and solving them formally)
Let $M\in \R^{3 \times 12}$ be the coordinate matrix. If the first
column is zero, then so is the twelfth, since $d$ interchanges 1 and
12. But we are dealing with the case that all twelve vertices are
pairwise different. Hence the first column is not zero. ... |
Llama-3.3-70B-Instruct::1908.06409::prose0 | human | 0human | prose | A finite $p$-group $G$ is called \emph{special $p$-group of rank $k$} if
$G'=\Z(G)$ is an elementary abelian $p$-group of order $p^k$ and $G/G'$ is elementary abelian.
A group $G$ is called \emph{capable group} if there exists a group $H$ such that $G \cong H/\Z(H)$.
The \emph{epicenter} of a group $G$ is denoted by $... |
GPT-4.1::2009.12529::thm1 | human | 0human | proof | Taking the inner product of \eqref{1} with $u$ and noticing \eqref{2}--\eqref{3}, we have
\begin{align*}
(v,u)
=\;&\left(\delta_{x}^{2}u-\frac{h^{2}}{12}\delta_{x}^{2}v+S,u\right)\nonumber\\
=\;&(\delta_{x}^{2}u,u)-\frac{h^{2}}{12}(\delta_{x}^{2}v,u)+(S,u)\nonumber\\
=\;&-|u|_{1}^{2}-\frac{h^{2}}{12}(v,\delta_{x}^{2}u)... |
Claude-Sonnet-5::2009.08548::prose5 | human | 0human | prose | While the regularity of the 1-skeleton $X$ is obtained from the geometry of $\cP_{W,M}$, proving that the quotient graphs are expander graphs requires arguments of a completely different nature.
To this end, we use the fact that Coxeter groups are linear, and when $(W,S)$ is indefinite, that they are not virtually sol... |
Phi-4::2002.04444::thm0 | Phi-4 | 1ai | proof | To show that $T_{\vec{\alpha}}$ is uniquely ergodic if and only if it is ergodic, we use Weyl's criterion. For $T_{\vec{\alpha}}$ to be ergodic, the sequence $\{n\vec{\alpha}\}_{n\in\N}$ must be uniformly distributed in $X_{\cQ}^d$. By Lemma \ref{lem:Weyl}, this is equivalent to
\[
\lim_{N\rar\infty}\frac{1}{N}\sum_{n=... |
Llama-4-Scout::2012.07129::thm5 | human | 0human | proof | \sloppypar For $\gamma\neq 0$ the result of (i) follows immediately from \cref{cont}, since $f_\mu\to f_\gamma$ as $\mu\to\gamma$ and there is at least one $\mu$-minimal matching. For $\gamma=0$, note that applying an increasing affine transformation to a function $f$ does not change the notion of $f(\cdot)$-minimalit... |
GPT-4.1::2109.02732::prose4 | GPT-4.1 | 1ai | prose | This construction provides a convenient framework to analyze the distinguishing advantage between distributions generated by even and odd subsets of $k$ independent Brownian motions, each stopped at their respective times. By considering all possible subsets $S \subseteq [k]$, we can systematically compute expectations... |
Phi-4::1907.02974::prose7 | Phi-4 | 1ai | prose | In Ricci cosmology, the interplay between curvature and matter fundamentally alters the dynamics of the universe's expansion. The presence of curvature-matter coupling terms introduces modifications to the effective pressure, potentially leading to early-time inflation without the need for exotic fields. This model dem... |
DeepSeek-R1-Distill-Llama-70B::1905.07124::thm2 | DeepSeek-R1-Distill-Llama-70B | 1ai | proof | The number of type-1 $cLRC$s is $O(m^2)$ as each corresponds to a maximal empty rectangle (MER) on the top face of $\cal A$, which is known to be $O(m^2)$ for $m$ points. To compute the maximum $cLRC$, we project $P_b$ onto the top face, compute all MERs, and for each, count red points in the corresponding cuboid using... |
Qwen3-32B::1910.05976::prose5 | Qwen3-32B | 1ai | prose | The key idea is to generate the phase GHZ state $|GHZ\rangle_p$, which encodes the modulo zero-sum condition in its structure. When each player performs measurements in the computational basis, the resulting outcomes $X_1, \ldots, X_m$ satisfy $\sum_{i=1}^m X_i=0$, fulfilling the modulo zero condition. Moreover, the in... |
gpt-oss-120b::1905.05272::prose9 | human | 0human | prose | The ventral stream, the \emph{``what''} pathway, is considered to
encode, among other things, information related to object identity, that
is, both category and specific object identity. The word stream
indicates a feed-forward visual processing network. The predominant idea
has been that the network creates a hierarch... |
DeepSeek-R1-Distill-Llama-70B::2005.10174::prose2 | human | 0human | prose | In Algorithm~\ref{algorithm:: Cheby} we present an efficient algorithm for approximating $\normP{\A}$ using the Chebyshev-Monte Carlo method, based on the discussion in~\cite{han2017approximating,trefethen2008gauss}. The method combines the Chebyshev polynomial approximation for $x^{p/2}$ in $[\lambda_\text{min},\lambd... |
GLM-4.5-Air::1909.01266::prose3 | GLM-4.5-Air | 1ai | prose | In the frequency domain, functional connectivity is typically quantified through the cross-power spectrum, which captures the linear dependencies between different signals as a function of frequency. This approach is particularly well-suited for M/EEG data analysis, as it allows researchers to study brain interactions ... |
Qwen3-32B::1910.03614::prose13 | Qwen3-32B | 1ai | prose | The transition off the plateau is further supported by the spectral evolution observed in SN 2017erp, where the grism spectrum at 605 days reveals a marked shift in the dominant emission features. This suggests that the ejecta have cooled sufficiently to allow the formation of neutral iron species, consistent with a ch... |
gpt-oss-120b::1902.04074::prose30 | gpt-oss-120b | 1ai | prose | The grain‑size distribution therefore carries a fossil record of the preceding dynamical history, because the growth and evaporation time scales are comparable to the characteristic lifetimes of the shock‑induced density enhancements. In regions where a shock has recently compressed the gas, the local density may incre... |
Qwen3-235B-A22B-Instruct-2507::2002.02900::thm7 | human | 0human | proof | Let us assume firstly that $m_{i}\in\mathbb{N}$ for any $i.$ Consider the multiplication
\begin{equation}\label{u*v}
u\ast v= \sum_{\alpha \in BC_{N}(r,s,q)}c_{\alpha} (\alpha,u)(\alpha,v)\coth (\alpha,x)\alpha
\end{equation}
on the tangent space $T_{x}W$ for $x \in W.$ By Theorem \ref{assoiativity&WDVV} the multipl... |
Llama-3.3-70B-Instruct::2007.01972::prose3 | human | 0human | prose | We also propose ACbag which is defined as bagging on an ensemble of wSigDirect classifiers. Motivated by the approach proposed by Breiman in \cite{breiman1996bagging}, we use an ensemble model of wSigDirect classifiers trained in parallel over different training datasets, and perform a majority voting over the ensemble... |
Qwen2.5-72B-Instruct::1901.08230::thm2 | human | 0human | proof | It is obvious that $x=1$ is a solution of (\ref{eqnplus}). Suppose that $\theta\in\gf(3^m)\setminus \gf(3)$ is a solution of (\ref{eqnplus}). Through a straight calculation, we have
$$\theta^{3^h}(\theta^4-\theta^3+\theta^2+1)=\theta^4+\theta^2-\theta+1.$$
First, we assert that $\theta^4-\theta^3+\theta^2+1\neq0$. Othe... |
Qwen3-235B-A22B-Instruct-2507::2005.11847::prose1 | Qwen3-235B-A22B-Instruct-2507 | 1ai | prose | This work builds upon these foundations by focusing on data placement strategies within a heterogeneous IoT healthcare scenario, where diverse devices generate varying workloads. The selected algorithms—Cloud-only, Mapping, and Edge-ward—are evaluated not only in terms of latency and energy but also considering their a... |
DeepSeek-R1-Distill-Qwen-32B::1902.06548::prose13 | human | 0human | prose | Latent Dirichlet allocation (LDA) is a three-level hierarchical Bayesian model, in which each document is represented as a finite mixture, according to a Dirichlet distribution, over an underlying fixed set of topics. In turn, each topic is modeled by a probability distribution over words \cite{ref:19}.
Word2Vec is a ... |
GLM-4.5-Air::1901.01165::thm0 | human | 0human | proof | Let us prove first that a minimizer exists. In fact, let
$$
\mathcal{K}=\Big\{v\in W^{1,p(\cdot)}(\Omega)\colon v - \phi\in W_0^{1,p(\cdot)}(\Omega)\Big\}.
$$
In order to prove that $J$ is bounded from below in $\mathcal{K}$, we observe that if $v\in\mathcal{K}$, then
$$
J(v)\ge\frac{1}{p_{\max}}\int_\Omega |\nabla v|... |
Qwen3-32B::1903.01153::prose11 | human | 0human | prose | To formalize our compilation we first define a set of classical planning instances $P_t=\tup{F,\emptyset,I_t,G_t}$ that belong to the same planning frame (i.e. same fluents and actions but different initial states and goals). Fluents $F$ are built instantiating the predicates in $\Psi$ with the objects of the input lab... |
GLM-4.5-Air::1901.09175::thm0 | GLM-4.5-Air | 1ai | proof | Let $\Fcal$ be a graph property and let $E$ be the event that $\GPo(n;\alpha,\nu) \in \Fcal$. We have
\begin{equation*}
\Pee(E) = \Pee(E_n) \cdot \Pee(|\PP|=n) + \Pee(E \mid |\PP| \neq n) \cdot \Pee(|\PP| \neq n),
\end{equation*}
where $E_n$ is the event that $G(n;\alpha,\nu) \in \Fcal$. Since $\Pee(E \mid |\PP| \neq n... |
gpt-oss-120b::1902.00406::prose7 | gpt-oss-120b | 1ai | prose | The simultaneous analysis of the three quantities shown in Fig.~\ref{fig:pars}(d) reveals a clear correlation between the spin‑filter efficiency and the suppression of the Josephson coupling: as the barrier thickness grows, the increase of the spin‑mixing angle $\Theta$ (see Table~\ref{tab:tab1}) drives a rapid reducti... |
GPT-4.1::2009.13435::thm2 | GPT-4.1 | 1ai | proof | Let us denote by $D^s$ a generic spatial differential operator of order $s$, and apply $D^s$ to system \eqref{system-MHD}. Taking the $L^2$ inner product with $(D^s u, D^s b)$ and integrating over $\Omega$, we obtain
\begin{equation}
\label{Hs-energy}
\begin{split}
&\frac{1}{2} \frac{d}{dt} \left( \|u(t)\|_{H^s}^2 + \|... |
Phi-4::1906.02177::prose0 | Phi-4 | 1ai | prose | \textcolor{black}{
The $p$-Laplacian operator $\Delta_p$ is a nonlinear generalization of the classical Laplace operator, defined by $\Delta_p u = \text{div}(|Du|^{p-2}Du)$. This operator introduces significant complexity into the problem due to its nonlinearity, particularly affecting the existence and uniqueness of s... |
GLM-4.5-Air::1911.04091::prose4 | human | 0human | prose | This transition represents the set of monomophic types $\{\T{a} \to t \to \T{a} \mid t \in Type\}$
and \emph{over-approximates} the set of instances of \T{f}
where the first argument unifies with \T{a} and the second argument unifies with \T{_t}
(which in this case is a singleton set $\{\T{a -> M a -> a}\}$). |
GPT-5.1::2008.01148::prose1 | human | 0human | prose | To fluently and fluidly collaborate with people, CRS needs to recognize the activities performed by their human teammates robustly \cite{tiqbal_joint_action,Iqbal2015TAC,mit_ucsd}. Although modern robots are equipped with various sensors, robust human activity recognition (HAR) remains a fundamental problem for CRS \ci... |
gpt-oss-120b::1908.02373::prose44 | gpt-oss-120b | 1ai | prose | The eight horizontal‑branch variables identified here therefore expand the known phenomenology of M 4’s HB population beyond the classic RR Lyrae pulsators, revealing a continuum of low‑amplitude, multi‑harmonic oscillations that bridge the gap between the well‑studied instability strip and the more quiescent HB stars.... |
Phi-4::2001.05583::thm1 | human | 0human | proof | ({\em Only if direction.}) First, we show that if $\aperm$ is an automorphism of
$\agraph$ then there is an annotated tree decomposition $\annotatedtreedecomposition$ such that
$\aperm = \eaut(\etd)$. Let $\td$ be a width-$\treewidth$ tree decomposition of
$\agraph$ and suppose $\aperm$ is an automorphism. Then, we ... |
Llama-4-Scout::2011.02346::thm3 | Llama-4-Scout | 1ai | proof | For $\bs{v}\in \Omega$ we have $\bs{v}\prec \bs{w}$, which implies that there exists $i$ such that $\bs{v}_i<\bs{w}_i$ and $\bs{v}_j=\bs{w}_j$ for all $j>i$.
Denote by $\mathcal{H}_{\bs{w},i}$ the hyperplane
\[ \mathcal{H}_{\bs{w},i} := \{ \bs{x}\in \V \; | \; \bs{x}_i=\bs{w}_i \}. \]
It is clear that $\mathcal{H}_... |
DeepSeek-R1-Distill-Qwen-32B::2003.02125::thm4 | human | 0human | proof | By Lemma \ref{lee}, $D$ is bipartite (Eulerian) if and only if both $M/A$ and $(M\backslash A^{c})^{*}$ are bipartite (Eulerian) if and only if both $(M/A)^{*}$ and $M\backslash A^{c}$ are Eulerian (bipartite) by reason that any minor of a binary matroid is binary and Lemma \ref{led}. So if and only if $M\setminus A^{c... |
Llama-3.3-70B-Instruct::1901.06001::thm8 | Llama-3.3-70B-Instruct | 1ai | proof | Assume that $|A|\geq \omega I(t_1)$, then by the conservation of the angular momentum, $|A(t)|\geq \omega I(t)$ for all $t$. By the definition of $K_\omega$, $K_\omega(\bf x(t))=\omega^2I(t)+\alpha U(t)\geq \alpha U(t)+\omega^2\frac{|A(t)|}{\omega}\geq \alpha U(t)+\omega |A(t)|$. By the conservation of the energy and t... |
Llama-3.3-70B-Instruct::2001.07265::prose2 | human | 0human | prose | In this work, we perform {\it ab initio} path integral molecular dynamics (AI-PIMD) simulations, which provide a quantum mechanical description of both the electrons and nuclei, of liquid water and aqueous solutions containing the monatomic Na$^+$ and Cl$^-$ ions and the polyatomic HPO$_4^{2-}$ ion. Using these simulat... |
Phi-4::1901.00730::thm6 | Phi-4 | 1ai | proof | By the definition of the action of $\g$ on $\CS(G, E)$, we have for $X \in \g$ and $f \in \CS(G, E)$, the action is given by
\[
(X.f)(g) = -\left.\frac{d}{dt}\right|_{t=0} f(g \exp(-tX)) = -\sum_{k=0}^\infty \frac{t^k}{k!} \left.\frac{d^k}{dt^k} f(g \exp(-tX))\right|_{t=0}.
\]
This implies
\[
(X.f)(g) = -\sum_{k=1}^\in... |
Qwen3-32B::2001.01404::thm1 | Qwen3-32B | 1ai | proof | By Lemma~\ref{lem:pw}, we know that for each $\x \in \cX$, the sequence of probabilities $ q^* \left(\prod_{i=1}^{m}\In_{(-\infty,0]}(g_i(\x,\vxi)) |\nX \right) $ converges pointwise $P_0$-almost surely to $ \prod_{i=1}^{m}\In_{(-\infty,0]}(g_i(\x,\vxi_0)) $. To show hypo-convergence, we need to verify the two properti... |
Qwen3-32B::1905.04179::thm3 | human | 0human | proof | For any $p\in P$, we have $r_{A^2-A^2}(p)\gg \frac{|A|}{K}$, so
\[T_w\cdot \frac{|A|}{K}\cdot |A|^2\ll |\{(x, y, z, t, u, v)\in (A-A)^2\times A^2\times P_w^2\colon (x+z)^2+(y+t)^2=u-v\}|.\]
By repeating the argument of the proof of Lemma \ref{e4} with $P_w$ in the place of $X_k$, we have the number of such tuples $(x,... |
Qwen3-235B-A22B-Instruct-2507::1905.09343::thm7 | Qwen3-235B-A22B-Instruct-2507 | 1ai | proof | Let $a,b\in P$. \\
(i) Assume $[a]\Theta\leq'[b]\Theta$. Then $[a*b]\Theta=[1]\Theta$, i.e.\ $a*b\mathrel{\Theta}1$. Since $(P,\leq)$ is strongly sectionally pseudocomplemented, we have $a\leq(a*b)*b$. Put $d:=(a*b)*b$. Then $a\leq d$ and $d\mathrel{\Theta}1*b=b$, i.e.\ $d\in[b]\Theta$. \\
(ii) If $a\leq b$ then $a*b=1... |
DeepSeek-R1-Distill-Qwen-32B::2007.01897::prose0 | human | 0human | prose | Articulation points and bridges of an undirected graph can be identified in linear time \cite{T72,T74,JS13}. In $2010$, Georgiadis \cite{G10} gave a linear time algorithm to test whether a directed graph is $2$-vertex-connected. Italiano et al. \cite{ILS12} gave linear time algorithms for calculating strong articulati... |
GLM-4.5-Air::2001.09075::thm9 | GLM-4.5-Air | 1ai | proof | The isomorphism is induced by the natural transformation $\varepsilon: E \to \id$. Specifically, for $f: \bM \to E(\bP)$, we get $\varepsilon_P \circ f: \bM \to \bP$. Conversely, for $g: \bM \to \bP$, we define $f: \bM \to E(\bP)$ by $f(x)(h) = h \cdot g(h^{-1} \cdot x)$. This is well-defined and makes the diagram comm... |
gpt-oss-120b::1901.03630::prose6 | gpt-oss-120b | 1ai | prose | The overall architecture consists of three mechanically isolated sections linked by the inertial pendulum: a vertical segment (I) housing the pulse‑tube cold head, a horizontal pendulum segment (II) that provides the low‑pass filtering of vibrations, and a second vertical segment (III) delivering the cooled payload to ... |
Llama-4-Scout::1907.08934::thm0 | Llama-4-Scout | 1ai | proof | We start by expressing the total number of released molecules in terms of the increments $\{\Delta_i\}_{i=1}^{J}$.
\begin{align}
\sum_{j=1}^{J}\Delta_j &\leq \beta(T-T_M) \label{positive_sum}
\end{align}
From (\ref{min_ai s}), we have
\begin{align}
a_i - a_{i+1} &= \frac{1}{2^{i}}\frac{M}{\lambda} \label{ai-ai+1}
\end{... |
Qwen3-32B::2001.05087::prose2 | human | 0human | prose | Iterative Deepening Alpha-Beta associated to a heuristic evaluation
function and a transposition table is the standard algorithm for
playing games such as Chess and Checkers. Iterative Deepening
Alpha-Beta has also been used to solve games such as small board Go
\cite{van2009solving}, Renju \cite{WagnerRenju01}, Lines ... |
gpt-oss-120b::2007.03309::prose1 | gpt-oss-120b | 1ai | prose | In what follows we develop a systematic method to compute both spectra and spectral measures for the families of Schreier graphs introduced above. After recalling the necessary background on spinal groups, we derive explicit recurrences for the finite level spectra, pass to the limit to obtain the infinite Schreier spe... |
Llama-4-Scout::2011.14138::thm4 | Llama-4-Scout | 1ai | proof | To prove that any singular flat surface $S$ has an ideal triangulation, we proceed by induction on the number of boundary components, genus, and the number of singular points of $S$.
First, consider the case of a flat disk. If the label set contains three points, then the disk is a triangle and has an obvious ideal t... |
GLM-4.5-Air::1906.06468::prose1 | GLM-4.5-Air | 1ai | prose | Our extracted collinear gluon density $xg(x,Q^2)$ from KC-BFKL evolution shows reasonable agreement with global data fits NNPDF3.1sx and CT14 across the studied kinematic range. At small $x$ values ($x \leq 10^{-4}$), our prediction lies between the two global PDF sets, with NNPDF3.1sx exhibiting slightly higher values... |
gpt-oss-120b::1905.07677::thm0 | human | 0human | proof | Since $ f_{1},...,f_{l} $ are linearly dependent over $\Bbbk((t_{1},t_{2})),$ we can obtain a nonzero linear relation of the form,
\begin{equation}\label{equ2}
\begin{array}{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}
&\varphi_{1}f_{1}+\cdots +\varphi_{l}f_{l}=0,
\end{array}
\end{equation}
where $ \varphi_{... |
Qwen3-32B::2003.06957::prose2 | Qwen3-32B | 1ai | prose | We sample multiple subsets for training and evaluate the performance on the base and novel classes to compute the average and confidence intervals. On the LVIS dataset, we follow the same approach, sampling subsets for training while maintaining the class imbalance characteristics of the original dataset. This revised ... |
gpt-oss-120b::2007.01121::prose1 | gpt-oss-120b | 1ai | prose | Thus, once a definite Hermitian determinantal representation is fixed, the associated spectrahedron furnishes a concrete convex body in \(\R\P^{3}\) whose boundary is precisely the real part of the quartic surface \(\V(f)\). In particular, every essential singularity of \(\V(f)\) lies either on the interior of this sp... |
DeepSeek-R1-Distill-Llama-70B::1910.00937::prose1 | human | 0human | prose | The construction of the
moduli space of stable pairs given in \cite[Sec.4.9]{k-modbook} relies on
a suitable moduli theory of divisors. In characteristic 0, all
restrictions on bases schemes come from the moduli theory of divisors. In
\cite[Sec.4.7]{k-modbook} we used Cayley-Chow theory, which was at that time
worked ... |
Qwen3-32B::1909.05305::prose3 | Qwen3-32B | 1ai | prose | The proposed two-stage framework leverages edge information to guide the super-resolution process. In the first stage, we generate a high-resolution edge map from the LR input using a generator-discriminator pair trained with adversarial and feature matching losses. In the second stage, this edge map is combined with a... |
Phi-4::2007.07880::thm0 | Phi-4 | 1ai | proof | We construct a hierarchical decomposition of the rectangle family $\famR$ by iteratively removing rectangles according to specific criteria. Initially, let $\famR_0 = \famR$. For $i \geq 0$, define $\famR_{i+1}$ as follows: remove all rectangles $R \in \famR_i$ such that there exists a rectangle $R' \in \famV(R)$ with ... |
Llama-3.3-70B-Instruct::2003.08830::thm4 | Llama-3.3-70B-Instruct | 1ai | proof | The characteristic root $s_0$ of $G_{cl}$ has multiplicity more than one at $h=h_0$ if and only if $f(s_0,h_0)=0$ and $\frac{\partial f}{\partial s}(s_0,h_0)=0$. By (\ref{eq:complex}) and (\ref{eq:RT}), we have
\bea
f(s_0,h_0)&=&1+G(s_0)e^{-h_0s_0}=0, \\
\frac{\partial f}{\partial s}(s_0,h_0)&=&\left(G'(s_0)-h_0G(s_0)\... |
GPT-5.1::2008.10434::prose0 | GPT-5.1 | 1ai | prose | Classical analyses based on Darcy's law and a constant capillary pressure jump predict a one-parameter family of steadily translating bubbles of identical velocity but different shapes, so that the observed bubble profile cannot be selected uniquely from the theory alone. In particular, in the Taylor–Saffman framework ... |
Qwen3-32B::2001.11971::thm0 | Qwen3-32B | 1ai | proof | To prove the theorem, we consider the optimization problem in \eqref{E:V}, where the optimal control input $U_k^*$ is to be determined as a function of $\I^c_k$. Since the problem is linear-quadratic in nature, we expect the optimal control policy to be linear in the estimate of the state $\tilde{X}_k = \E[X_k|\I^c_k]$... |
gpt-oss-120b::2005.12360::thm2 | gpt-oss-120b | 1ai | proof | Let
\[
\lambda_j(Q):=\frac{e^{\beta q_j}}{\sum_{i=1}^{n}e^{\beta q_i}},\qquad
\lambda(Q):=(\lambda_1(Q),\dots ,\lambda_n(Q))\in\Delta_{n},
\]
so that \(f_{\alpha}(Q)=\sum_{j=1}^{n}\lambda_j(Q)\,\alpha_j\).
For any constant \(c\in\mathbb{R}\),
\[
f_{\alpha}(Q)-f_{\alpha}(\widetilde Q)
=\sum_{j=1}^{n}\bigl(\la... |
Llama-3.3-70B-Instruct::1905.02238::thm9 | Llama-3.3-70B-Instruct | 1ai | proof | The space complexity of the Approx-FSMI-RLE algorithm is determined by the storage required for the look-up tables of $\alpha$, $\beta$, $\theta$, and $\gamma$. The look-up tables for $\alpha$ and $\beta$ each require $O(|\mathcal{X}|n^2)$ space, where $|\mathcal{X}|$ is the number of quantized occupancy probability va... |
Phi-4::2007.10657::prose0 | Phi-4 | 1ai | prose | In the study of prolongations of convenient Lie algebroids, we extend the classical notions of prolongations to the setting of convenient vector bundles. A key aspect is the definition of the prolongation of a convenient Lie algebroid over a fibred manifold, which adapts the finite-dimensional framework to the infinite... |
Qwen3-32B::1903.08841::thm2 | human | 0human | proof | Let $\cA(x)$ denote the indicator function of the set $\mathcal{A}$ and let $\widehat \cA(y)$ denote the Fourier coefficients of $\cA(x)$, so that
\begin{align}
\label{eq:afc}
|\widehat \cA(y)|\ll \frac{1}{q}\prod_{i=1}^{d}\left(H,\frac{1}{\|\alpha_i y/q\|}\right) \quad \text{and} \quad \widehat \cA(0)=\frac{|\cA|}{q}.... |
Phi-4::1904.02921::prose8 | Phi-4 | 1ai | prose | Once \(\theta\) and \((\zb_i)_{1 \leq i \leq n}\) are learnt on the estimation set, we can simulate an arbitrary number of patients and visits per individual that will be used as the training set. To have similar characteristics in the training and test set, the time between two simulated visits is kept similar to the ... |
GLM-4.5-Air::1903.08968::thm6 | human | 0human | proof | \begin{displaymath}
\begin{split}
Q_{D_{m,n}} =& \big[ l_1 , (l_2 , \ldots , l_n) , (l_{n+1} , \ldots , l_{m+n-1})\big] \cdot \big[ l_2 , (l_3 , \ldots , l_n) \big] \cdots \big[ l_{n-2} , (l_{n-1} , l_n) \big] \cdot \\ \cdot& \big[ l_{n-1} , l_n \big] \cdot \big[ l_{n+1}, (l_{n+2} , \ldots , l_{m+n-1}) \big] \cdots \bi... |
DeepSeek-R1-Distill-Qwen-32B::1903.02847::thm0 | DeepSeek-R1-Distill-Qwen-32B | 1ai | proof | To prove the theorem, we consider both directions:
\begin{itemize}
\item \textbf{Forward Direction:} Assume $R/aR$ is a Kasch ring. Let $M$ be a maximal ideal in $R$. Then $M/aR$ is an ideal in $R/aR$. Since $R/aR$ is Kasch, the annihilator of $M/aR$ is non-zero. Thus, there exists $x \in R$ such that $xM \subseteq aR... |
Qwen2.5-72B-Instruct::1901.08037::thm4 | human | 0human | proof | Apostolov \cite[Proposition 3.2 + Remark 3.3.(1)]{Apostolov14} shows that this moduli space has only one
connected component.
To see that this consists of a single irreducible componen, we use the period map for marked irreducible
symplectic varieties $\dP \colon \Def(X) \to Q_\Lambda$ as introduced in \cite[Sec... |
Phi-4::1911.03041::prose11 | human | 0human | prose | A scale recurrence lemma holds for the renormalization operators $\tilde{T}_{\ul{b}^1,...,\ul{b}^n}$ and for sets $F(\ul{a}^1,...,\ul{a}^n)$ in $\tilde{J}_r$, its statement is very similar to theorem A. The only difference is that we do not need the hypothesis that all the $n$ Cantor sets are not essentially affine and... |
DeepSeek-R1-Distill-Qwen-32B::1902.00099::prose7 | human | 0human | prose | This extension is based on a task introduced in the Ziv-Zakai inequality
where the ideal observer is trying to detect a change in a parameter
and the prior probabilities for the two hypotheses are determined
by the prior PDF on the parameter. In this case the AUC for the ideal-observer
is no longer relevant and it is t... |
GPT-5.1::2010.06941::thm0 | GPT-5.1 | 1ai | proof | Using Definition \ref{deffuncaodiscr} and (\ref{eqScore}), (\ref{eqScoreMedia}), the score quantile function for unit $i$ in subinterval $\ell$ can be written as
\[
\Psi_{S(i)}^{-1}(t)=\alpha_{i\ell}+
\beta_{i\ell}\left(\frac{2(t-w_{\ell-1})}{w_{\ell}-w_{\ell-1}}-1\right),
\qquad t\in[w_{\ell-1},w_{\ell}],
\]
where
\[
... |
gpt-oss-120b::2001.10555::prose2 | gpt-oss-120b | 1ai | prose | The probabilistic viewpoint emphasized in the preceding paragraph suggests that the heat kernel on a finite Cayley graph can be interpreted as the transition matrix of a time‑homogeneous Markov chain whose one‑step distribution is supported on the neighbours of each vertex. In this setting the kernel evolves by discre... |
DeepSeek-R1-Distill-Llama-70B::2001.02703::thm4 | DeepSeek-R1-Distill-Llama-70B | 1ai | proof | For the forward direction, assume that \( A \) is prime. By Proposition \ref{prop:prime}, \( X \) is essentially full and both the linking algebra \( \mathcal{L}(X) \) and the extended linking algebra \( \mathcal{M}(X) \) are prime. Using the canonical isometry \( \iota_X \) and the properties of the Haagerup tensor pr... |
Llama-3.3-70B-Instruct::1901.07156::thm4 | human | 0human | proof | We have that $F/k$ is abelian. Let $F\subseteq {_n\cicl N{}_m}$ and
$E=FM\cap \cicl N{}$. Then $\g EM=\g FM=FM=EM$
(see \cite{BaMoReRzVi2018}) and therefore $\g E=E$.
Since $e_{\infty}(\ge E|\g E)|q-1$ and $e_{\infty}(M|k)=q^n$, it follows
that $e_{\infty}(\ge EM/\g EM)=e_{\infty}(\ge E|\g E)=[\ge E:\g E]$.
\[
\xymatr... |
Qwen3-32B::1903.11521::thm0 | Qwen3-32B | 1ai | proof | Assume that $\hat{\Theta}^{k}$ are EQSPPs w.r.t. \eqref{eq:gentensorop}. Then, for any $i \in \mathbb{S}(U)$, the value $U_i = \sum_{j \in R_U(i)} Z_j$, where $Z_j = T^{1}_{\tp{j}} \cdot \ldots \cdot T^{n}_{\tp{j}}$. Since the sparsity patterns $\hat{\Theta}^{k}$ are such that $U = \hat{U}$, it follows that $\hat{Z}_j ... |
Llama-3.3-70B-Instruct::1903.03994::thm0 | Llama-3.3-70B-Instruct | 1ai | proof | We prove the two directions of the equivalence separately. Suppose that $(V,L,R,\phi,\psi)$ is a bimodule of a BiHom-alternative algebra $( A,\mu,\a,\b)$. Then, for any $x,y,z \in A$ and $v \in V$, we have
\begin{align*}
&\quad as_{\a+\phi,\b+\psi}((\b+\psi)(x+v),(\a+\phi)(y+w),z+u) \\
&\quad + as_{\a+\phi,\b+\psi}((\b... |
DeepSeek-R1-Distill-Llama-70B::2001.00056::prose3 | DeepSeek-R1-Distill-Llama-70B | 1ai | prose | The decoder is a feed-forward neural network that computes a score for each sentence in the paragraph, enabling efficient parallel processing. This approach eliminates the need for complex sequential decoding methods, such as beam search, which are often required in pointer network-based decoders. By assigning a releva... |
Llama-3.3-70B-Instruct::2001.06134::thm8 | Llama-3.3-70B-Instruct | 1ai | proof | Suppose $L$ is simple. Then, as $L\in \mathbf{M}_n$, it is also subdirectly irreducible. So, its dual space $P$ has exactly one order component $Q$ or exactly two order components $Q$ and $\zeta(Q)$, with $Q\cap\zeta(Q)=\emptyset$. Since $P$ is compact and $Q$ is closed, by Theorem \ref{X4}, $Q$ is also compact. Suppos... |
GPT-5.1::2008.07246::prose3 | human | 0human | prose | \clearpage
So far, a majority of studies \citep{eigen2014depth,li2015depth,laina2016deeper} have focused on a supervised training of \glspl*{CNN} for the task of monocular depth estimation.
They show that, with appropriate training data, state-of-the-art networks can even outperform conventional methods that rely on i... |
DeepSeek-R1-Distill-Qwen-32B::2001.10009::prose0 | human | 0human | prose | In this paper, I examine the nature of physical reality in the context of unavoidable unobservables in physics \cite{Unni-Beyond,Unni-Bili}, and discuss some examples. For the purpose of this discussion I will work with a definition of an unobservable (in physics) as a quantitative entity, created and mathematically r... |
Qwen3-235B-A22B-Instruct-2507::1902.07141::prose10 | human | 0human | prose | We will consider two different types of translation-invariant Hamiltonian. They are defined either by (a) nearest-neighbor interactions and arbitrary $D$, or (b) arbitrary finite-range interactions and $D=3$. We work with periodic boundary conditions for simplicity; the techniques from \cite{LM} can be used to extend t... |
Phi-4::1909.02129::prose0 | Phi-4 | 1ai | prose | \section*{Section Heading}
The use of the \texttt{chapter.tex} template, combined with the appropriate Springer document class (SVMono or SVMult), ensures consistent styling of chapter elements in line with Springer's format. This approach not only aids in maintaining a uniform appearance but also streamlines the type... |
DeepSeek-R1-Distill-Llama-70B::2001.05550::prose0 | DeepSeek-R1-Distill-Llama-70B | 1ai | prose | The variance of linear statistics of random zero sets of sections of $L^k$ is studied here, sharpening the leading-order asymptotics of Shiffman--Zelditch. We consider a compact Kähler manifold $M$ with a positive line bundle $L$, and analyze the variance of integrals of smooth forms over the zero sets of random sectio... |
DeepSeek-R1-Distill-Qwen-32B::1905.06117::prose8 | DeepSeek-R1-Distill-Qwen-32B | 1ai | prose | Null curves in $\bbC^3$ correspond, via the Klein correspondence, to contact curves in $\bbP^3$. A null curve $\gamma:S\to\bbC^3$ with $d$ simple poles extends uniquely to a null curve $g:S\to\bbQ^3$ of degree $d$ as a map to $\bbP^4$. The ramification of $\gamma$ at its poles translates to the ramification of $g$ at t... |
Llama-3.3-70B-Instruct::1902.05511::prose21 | human | 0human | prose | The problem thus described is a complex one, but the similarities with
the lattice gas models of condensed matter physics provide guidelines
for how to proceed. These prescriptions, however, are aimed at the
treatment of a periodic crystalline lattice, and, although the DNA
sites exhibit helical symmetry, they do not ... |
Phi-4::2001.08825::thm4 | Phi-4 | 1ai | proof | The proof follows a similar strategy to that of Proposition \ref{c:corolCHSH}. We express $\CC_{2n}(x)$ and $\SC_{2n+1}(x)$ in terms of $\mu_n(x)$ using their definitions:
$$
\CC_{2n}(x) = \Re(\mu_n(x)), \quad \SC_{2n+1}(x) = \Im(\mu_{n+1}(x)).
$$
From Euler's formula and \eqref{e:munpm}, we have for $n=0$, $1$, $\ldot... |
DeepSeek-R1-Distill-Llama-70B::2007.06852::prose3 | DeepSeek-R1-Distill-Llama-70B | 1ai | prose | Our analysis focuses on the stochastic heavy ball method, where the dynamics can be described by a nonlinear kinetic Fokker-Planck equation. We introduce a Lyapunov functional that combines the objective function, kinetic energy, and entropy, which allows us to characterize stationary points and prove global convergenc... |
o4-mini::2012.05229::prose0 | human | 0human | prose | DHQM is a generalization of CQM that applies to closed systems like a spatially closed universe. It is an extension, and to some extent a completion, of the ideas pioneered by Everertt \cite{Eve57}. Given theories of the closed system's quantum state and dynamics, DHQM predicts probabilities for the individual memb... |
gpt-oss-120b::1903.05863::thm9 | gpt-oss-120b | 1ai | proof | Let \(\{v_{s}\}_{s\geq0}\) be the Haar basis of \(L^{2}([0,T])\) and \(\{e_{k}\}_{k\geq1}\) the orthonormal basis of \(\Hcal\). Define the linear operator \(C:L^{2}([0,T])\otimes \ell^{2}\to L^{2}([0,T])\otimes \ell^{2}\) by
\[
C\bigl(v_{s}\otimes e_{k}\bigr)=\mu_{s,k}\,v_{s}\otimes e_{k},
\qquad
\mu_{s,k}=2^{-i\alp... |
Llama-3.3-70B-Instruct::2001.11082::thm1 | human | 0human | proof | The strategy of this proof is to use \cref{prop:IC}.
Let $\Phi=\left\{F_{-1}, \dots, F_{n} \right\} $ be the base flag of $\cP$.
Let $F'_{n-1}$ be the facet of $\Phi^{n-1}$.
Observe that the groups $H_{n-1}=\left\langle \rho_{0}, \dots, \rho_{n-2} \right\rangle $ and $H_{n-2}=\left\langle \rho_{0},\dots, \rho_{n-... |
Gemini-3.1-Pro::2009.12316::prose1 | Gemini 3.1 Pro | 1ai | prose | Our proposed end-to-end ML-based visualization recommendation system takes as input a large corpus of datasets and visualizations, and learns a model based on this data. Then, given a new unseen dataset from an arbitrary user, the model automatically generates visualizations for that new dataset, derives scores for the... |
Llama-3.3-70B-Instruct::1901.07162::thm1 | human | 0human | proof | If $ a^{(q-1)/(p^d+1)} \neq (-1)^s $, by Lemma \ref{lem:fun}, we know the equation
$ f_a(X) =a^{p^k}X^{p^{2k}} + a X $ is a permutation polynomial over $ \mathbb{F}_q $ and
$ f_a(X) =-b^{p^k} $ has a unique solution $ \gamma $ in
$ \mathbb{F}_{q} $. Thus $ y^{-1}z\gamma $ is the unique solution for
$ f_{ay}(X) =-... |
Qwen2.5-72B-Instruct::2007.08608::thm2 | Qwen2.5-72B-Instruct | 1ai | proof | \begin{enumerate}[label=(\alph*)]
\item We first show that $\hat{s}_n$ can be obtained by considering only the cycles of length up to $a_n$. Suppose there exists a cycle length $a > a_n$ such that $\hat{s}_n$ is the minimizer of $L_{na}(y) + v_{n+a} \leq v_n$. Then, from the definition of $a_n$, we have $v_n - v_{n+a} ... |
gpt-oss-120b::1905.07753::thm8 | gpt-oss-120b | 1ai | proof | Let $G^{\bullet}h_{0}^{\et}(X)_{R}$ be a filtration satisfying the axioms of Conjecture~\ref{conj: ABV}.
By Proposition~\ref{prop: A} and Proposition~\ref{prop: B} the filtration $F^{\bullet}h_{0}^{\et}(X)_{R}$ constructed in Definition~\ref{defn: filtration} also satisfies (A) and (B).
Moreover, Corollary~\ref{cor... |
Llama-4-Scout::2012.07590::prose0 | Llama-4-Scout | 1ai | prose | The proposed experiment with a very long baseline between the International Space Station (ISS) and the Lunar Gateway (LG) may provide a unique opportunity to test fundamental issues in quantum foundations and relativistic quantum information. With the ISS at an altitude of $\sim 400$ km and the LG in an orbit around t... |
Phi-4::2003.10670::prose2 | human | 0human | prose | The proposal generation firstly involves a pruning step for the removal of points estimated to be part of the ground. Because it is assumed that the OOIs are not part of the ground, the segmentation process can ignore all those points assumed to belong to that surface. The majority of the approaches used for estimating... |
DeepSeek-R1-Distill-Qwen-32B::1903.03687::thm2 | DeepSeek-R1-Distill-Qwen-32B | 1ai | proof | To prove the proposition, suppose $x^u \notin D$, $x^u \succ x^v$, and $\calA u = \calA v$. We must show $u = v$.
Since $x^u \notin D$, by Lemma~\ref{lemxunotinD}, $x^u$ is either of the form
\[
u = (0 \ldots 0 \, a_1 | 0 \ldots 0 \, a_2 | \ldots | 0 \ldots 0 \, a_{i-1} | 0 \ldots 0 \, a_{i,\ell} \, a_{i,\ell+1} \, 0... |
Claude-Sonnet-5::2009.06574::prose0 | Claude Sonnet 5 | 1ai | prose | Since the mesh quality of the resulting elements is critical for the accuracy and stability of subsequent simulations, hex-mesh generation has received considerable attention over the last decades. In many applications, the elements need to conform to a curved boundary surface while at the same time keeping the deviati... |
DeepSeek-R1-Distill-Qwen-32B::1901.06191::thm8 | human | 0human | proof | (1) $\iff$ (2) According to Proposition \ref{l,rho,isomorphism}, the following properties are equivalent:
$L(A)$ is a Boolean algebra iff $L(A)=B(L(A))$ iff for any $c\in K(A)$, $\l(c)\in B(L(A))$ iff for any $c\in K(A)$, there exists $e\in B(A)$ such that $\l(c)=\l(e)$ iff for any $c\in K(A)$, there exists $e\in B(A)$... |
Qwen2.5-72B-Instruct::2003.06041::prose1 | human | 0human | prose | Focusing on learning-based approaches, \ac{STL} is of special interest; unlike most other temporal logics, it allows definitions of robustness metrics associated to the degree of task satisfaction \cite{donze2010robust}. These metrics quantify how much a task is satisfied or violated, and serve as more descriptive rewa... |
Qwen2.5-72B-Instruct::1902.06686::prose21 | human | 0human | prose | Figure \ref{Propagation and coalescence of the three pre-existing flaws in a square rock sample}(a)-(f) shows the propagation and coalescence of the three pre-existing flaws. We also calculate the reaction force on the upper boundary of the rock sample, and present the load-displacement curves in Fig. \ref{Load-displac... |
Qwen3-32B::1907.02253::prose0 | human | 0human | prose | There have been several important attempts in this direction by focusing on synthesizing parts of the face (around the mouth)~\cite{Suwajanakorn17,Kumar17}. However, as instructors' emotional states are communicated not only with facial expressions, but also through body posture, movement, and gestures~\cite{Dael12,Lho... |
Qwen3-32B::2003.13086::thm3 | Qwen3-32B | 1ai | proof | To prove the theorem, we analyze the cones of effective cycles in dimension 2 and nef cycles in codimension 2 on the Hilbert scheme $\mathbb{P}^{2[3]}$. The key strategy is to exploit the orbit structure of the $\mathrm{PGL}(3)$-action on $\mathbb{P}^{2[3]}$, as well as the duality between nef and effective cones under... |
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