Model: Stergios-Konstantinidis/MNLP_M2_document_encoder
Author: Stergios-Konstantinidis
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:21000, loss:ContrastiveTensionLoss, arxiv:1908.10084, base_model:sentence-transformers/all-MiniLM-L6-v2, base_model:finetune:sentence-transformers/all-MiniLM-L6-v2, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: sentence-transformers/all-MiniLM-L6-v2
Card metadata: tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:21000, loss:ContrastiveTensionLoss
base_model: sentence-transformers/all-MiniLM-L6-v2
widget: source_sentence:         "The lemma follows by invoking Lemma 4.1 and Lemma A.1.\n\u220e",
sentences:         "To better address non-stationarity with changing uncertainty, we introduce Location-Scale Noise Model (LSNM) into DDPMs, which relaxes the traditional Additive Noise Model (ANM) by incorporating a contextually changing variance: \ud835\udc18=f\u2062(\ud835\udc17)+g\u2062(\ud835\udc17)\u2062\u03f5\ud835\udc18\ud835\udc53\ud835\udc17\ud835\udc54\ud835\udc17bold-italic-\u03f5\\mathbf{Y}=f(\\mathbf{X})+\\sqrt{g(\\mathbf{X})}\\boldsymbol{\\epsilon}bold_Y = italic_f ( bold_X ) + square-root start_ARG italic_g ( bold_X ) end_ARG bold_italic_\u03f5, where g\u2062(\ud835\udc17)\ud835\udc54\ud835\udc17g(\\mathbf{X})italic_g ( bold_X ) is an \ud835\udc17\ud835\udc17\\mathbf{X}bold_X-dependent variance model. LSNM is capable of modeling both the contextual mean through f\u2062(\ud835\udc17)\ud835\udc53\ud835\udc17f(\\mathbf{X})italic_f ( bold_X ) and the contextual uncertainty through g\u2062(\ud835\udc17)\ud835\udc54\ud835\udc17\\sqrt{g(\\mathbf{X})}square-root start_ARG italic_g ( bold_X ) end_ARG. In the special case where g\u2062(\ud835\udc17)\u22611\ud835\udc54\ud835\udc171g(\\mathbf{X})\\equiv 1italic_g ( bold_X ) \u2261 1, this simplifies to the standard ANM. Building upon this more flexible and expressive assumption, we propose the Non-stationary Diffusion Model (NsDiff) framework, which provides an uncertainty-aware noise schedule for both forward and reverse diffusion processes. In summary, our contributions are as:\n\n\n\u2022\n\nWe observe that the ANM is inadequate for capturing the varying uncertainty and propose a novel framework that integrates LSNM to allow for explict uncertainty modeling. This work is the first attempt to introduce LSNM into probabilistic time series forecasting.\n\n\n\n\u2022\n\nTo fundamentally elevate the noise modeling capabilities of DDPM, we seamlessly integrate time-varying variances into the core diffusion process through an uncertainty-aware noise schedule that dynamically adapts the noise variance at each step.\n\n\n\n\n\u2022\n\nExperimental results indicate that NsDiff achieves superior performance in capturing uncertainty. Specifically, in comparison to the second-best recent baseline TMDM, NsDiff improves up to 66.3% on real-world datasets and 88.3% on synthetic datasets.",,         "The deep neural network representation of the Bifrost simulations is highly compressed compared to the original Bifrost data: the deep neural network has 44,261 floating point values whereas the Bifrost simulation cube has 96\u22c596\u22c564\u22c520=11,796,480\u22c5969664201179648096\\cdot 96\\cdot 64\\cdot 20=11,796,48096 \u22c5 96 \u22c5 64 \u22c5 20 = 11 , 796 , 480 floating point values. This corresponds to a compression by a factor of 267; this compression factor may be different for other numerical simulations and depends on their smoothness. In addition, the deep neural network can be evaluated at any point in space and time covered by the simulations, therefore enabling a trivial way to interpolate between grid points; furthermore, gradients are calculate with high efficiency with automatic differentiation. As such, it might be worth considering releasing deep-neural-network representations of (magneto)hydrodynamic simulations.",,         "\u03f5y\u2062(\u03bc)={1nt\u2062\u2211i=nkntey\u2062(ti,\u03bc)=1nt\u2062\u2211i=nknt|y~\u2062(ti,\u03bc)\u2212y\u2062(ti,\u03bc)|if\u00a0\u20621nt\u2062\u2211i=nknt|y\u2062(ti,\u03bc)|\u22641,1nt\u2062\u2211i=nkntey,r\u2062e\u2062l\u2062(ti,\u03bc)=1nt\u2062\u2211i=nknt|y~\u2062(ti,\u03bc)\u2212y\u2062(ti,\u03bc)|/|y\u2062(ti,\u03bc)|if\u00a0\u20621nt\u2062\u2211i=nknt|y\u2062(ti,\u03bc)|>1.subscriptitalic-\u03f5\ud835\udc66\ud835\udf07cases1subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61subscript\ud835\udc52\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf071subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61~\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07if\u00a01subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf0711subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61subscript\ud835\udc52\ud835\udc66\ud835\udc5f\ud835\udc52\ud835\udc59subscript\ud835\udc61\ud835\udc56\ud835\udf071subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61~\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07if\u00a01subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf071\\centering\\epsilon_{y}(\\mu)=\\begin{cases}\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^%\n{n_{t}}e_{y}(t_{i},\\mu)=\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}|\\tilde{y}%\n(t_{i},\\mu)-y(t_{i},\\mu)|&\\text{if }\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t%\n}}|y(t_{i},\\mu)|\\leq 1,\\\\\n\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}e_{y,rel}(t_{i},\\mu)=\\frac{1}{n_{t%\n}}\\sum\\limits_{i=n_{k}}^{n_{t}}|\\tilde{y}(t_{i},\\mu)-y(t_{i},\\mu)|/|y(t_{i},%\n\\mu)|&\\text{if }\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}|y(t_{i},\\mu)|>1.%\n\\end{cases}\\@add@centeringitalic_\u03f5 start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_\u03bc ) = { start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | over~ start_ARG italic_y end_ARG ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) - italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | end_CELL start_CELL if divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | \u2264 1 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_y , italic_r italic_e italic_l end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | over~ start_ARG italic_y end_ARG ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) - italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | / | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | end_CELL start_CELL if divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | > 1 . end_CELL end_ROW\n\n(12)",, source_sentence:         "While significant research addresses design tolerance optimisation in manufacturing, there is very little focus on production inspection machines such as AOIs for manufactured products. For AOIs inspecting PCBs, each component may demand a distinct tolerance for each type of inspection, leading to thousands of possible scenarios. Consequently, a general paradigm is needed that accommodates inspection of all components, including new introductions that the system has not previously encountered.",
sentences:         "Indeed, for any e\u2208D0\ud835\udc52subscript\ud835\udc370e\\in D_{0}italic_e \u2208 italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, there exists a \u03b4\ud835\udeff\\deltaitalic_\u03b4-tube Te\u03b4\u2062(ae)subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52T^{\\delta}_{e}(a_{e})italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) centred at some ae\u2208Asubscript\ud835\udc4e\ud835\udc52\ud835\udc34a_{e}\\in Aitalic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT \u2208 italic_A such that\n\n\n\n1|Te\u03b4\u2062(ae)|n\u2062|E\u2229Te\u03b4\u2062(ae)|n>\u03bb.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscript\ud835\udc38subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b\ud835\udf06\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|E\\cap T^{\\delta}_{e}(a_{%\ne})\\right|_{n}>\\lambda.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | italic_E \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_\u03bb .\n\n\n\nSince Emsubscript\ud835\udc38\ud835\udc5aE_{m}italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT and E\u00af\u00af\ud835\udc38\\overline{E}over\u00af start_ARG italic_E end_ARG form a partition of E\ud835\udc38Eitalic_E, we obtain\n\n\n\n1|Te\u03b4\u2062(ae)|n\u2062|Em\u2229Te\u03b4\u2062(ae)|n+1|Te\u03b4\u2062(ae)|n\u2062|E\u00af\u2229Te\u03b4\u2062(ae)|n>\u03bb.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscriptsubscript\ud835\udc38\ud835\udc5asubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscript\u00af\ud835\udc38subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b\ud835\udf06\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|E_{m}\\cap T^{\\delta}_{e}%\n(a_{e})\\right|_{n}+\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|%\n\\overline{E}\\cap T^{\\delta}_{e}(a_{e})\\right|_{n}>\\lambda.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | over\u00af start_ARG italic_E end_ARG \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_\u03bb .\n\n\n\nThus, at least one of the terms on the left-hand side must be greater than \u03bb2\ud835\udf062\\frac{\\lambda}{2}divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG, implying e\u2208Dm\u222aD\u00af\ud835\udc52subscript\ud835\udc37\ud835\udc5a\u00af\ud835\udc37e\\in D_{m}\\cup\\overline{D}italic_e \u2208 italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT \u222a over\u00af start_ARG italic_D end_ARG from the definition (3.14) and (3.19). Since\n\n\n\n|Dm|n\u22121+|D\u00af|n\u22121\u2a7e|D0|n\u22121=\u03b50subscriptsubscript\ud835\udc37\ud835\udc5a\ud835\udc5b1subscript\u00af\ud835\udc37\ud835\udc5b1subscriptsubscript\ud835\udc370\ud835\udc5b1subscript\ud835\udf000\\left|D_{m}\\right|_{n-1}+\\left|\\overline{D}\\right|_{n-1}\\geqslant\\left|D_{0}%\n\\right|_{n-1}=\\varepsilon_{0}| italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT + | over\u00af start_ARG italic_D end_ARG | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT \u2a7e | italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT = italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT\n\n\n\nand the stopping condition ensures\n\n\n\n|Dm|n\u22121<14\u2062\u03b50,subscriptsubscript\ud835\udc37\ud835\udc5a\ud835\udc5b114subscript\ud835\udf000\\left|D_{m}\\right|_{n-1}<\\frac{1}{4}\\varepsilon_{0},| italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT < divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,\n\n\n\nit follows that\n\n\n(3.20)\n\n|D\u00af|n\u22121\u2a7e14\u2062\u03b50.subscript\u00af\ud835\udc37\ud835\udc5b114subscript\ud835\udf000\\left|\\overline{D}\\right|_{n-1}\\geqslant\\frac{1}{4}\\varepsilon_{0}.| over\u00af start_ARG italic_D end_ARG | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT \u2a7e divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .\n\n\n\nFor any \u03be\u2208D\u00af\ud835\udf09\u00af\ud835\udc37\\xi\\in\\overline{D}italic_\u03be \u2208 over\u00af start_ARG italic_D end_ARG, there exists a \u03b4\ud835\udeff\\deltaitalic_\u03b4-tube T\u03be\u03b4\u2062(a\u03be)subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09T^{\\delta}_{\\xi}(a_{\\xi})italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) centred at a\u03be\u2208Asubscript\ud835\udc4e\ud835\udf09\ud835\udc34a_{\\xi}\\in Aitalic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT \u2208 italic_A such that\n\n\n\n1|T\u03be\u03b4\u2062(a\u03be)|n\u2062|\u22c3i=0m\u22121(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n>\u03bb2.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsuperscriptsubscript\ud835\udc560\ud835\udc5a1\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5b\ud835\udf062\\frac{1}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}\\left|\\bigcup_{i=0}^{m-1}(%\nE\\cap\\mathcal{B}_{i})\\cap T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}>\\frac{\\lambda}{%\n2}.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | \u22c3 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG .\n\n\n\nThis implies\n\n\n(3.21)\n\n\u2211i=0m\u22121|\u212ci\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n\u2a7e\u2211i=0m\u22121|(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n\u2a7e|\u22c3i=0m\u22121(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n>\u03bb2superscriptsubscript\ud835\udc560\ud835\udc5a1subscriptsubscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsuperscriptsubscript\ud835\udc560\ud835\udc5a1subscript\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsuperscriptsubscript\ud835\udc560\ud835\udc5a1\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5b\ud835\udf062\\begin{split}\\frac{\\sum_{i=0}^{m-1}\\left|\\mathcal{B}_{i}\\cap T^{\\delta}_{\\xi}(%\na_{\\xi})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}&\\geqslant%\n\\frac{\\sum_{i=0}^{m-1}\\left|(E\\cap\\mathcal{B}_{i})\\cap T^{\\delta}_{\\xi}(a_{\\xi%\n})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}\\\\\n&\\geqslant\\frac{\\left|\\bigcup_{i=0}^{m-1}(E\\cap\\mathcal{B}_{i})\\cap T^{\\delta}%\n_{\\xi}(a_{\\xi})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}>\\frac{%\n\\lambda}{2}\\end{split}start_ROW start_CELL divide start_ARG \u2211 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT | caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG end_CELL start_CELL \u2a7e divide start_ARG \u2211 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT | ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL \u2a7e divide start_ARG | \u22c3 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG > divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG end_CELL end_ROW",,         "In [kipvar], the authors first add and subtract terms to\nexplicitly express\nIn\u2062(f,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5I_{n}(f,\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) in terms of\nDynkin martingale and then pass to\nthe limit \u03bb\u21930\u2193\ud835\udf060\\lambda\\downarrow 0italic_\u03bb \u2193 0, before\nanalyzing that result in a second limit as\nn\u2192\u221e\u2192\ud835\udc5bn\\to\\inftyitalic_n \u2192 \u221e. This is the approach of\n[varadhan95, liggett99, landim] as well.\nThe essential idea of the present proof is to first note that\nfor f\u2208\ud835\udc9f(\u2212A^)\u221212\u2283\u211bA^\ud835\udc53subscript\ud835\udc9fsuperscript^\ud835\udc3412superset-ofsubscript\u211b^\ud835\udc34f\\in\\mathscr{D}_{(-\\hat{A})^{-\\frac{1}{2}}}\\supset\\mathscr{R}_{\\hat{A}}italic_f \u2208 script_D start_POSTSUBSCRIPT ( - over^ start_ARG italic_A end_ARG ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_POSTSUBSCRIPT \u2283 script_R start_POSTSUBSCRIPT over^ start_ARG italic_A end_ARG end_POSTSUBSCRIPT, the sequence\n\u039bn\u2062(f,\u03bbn,\u22c5)subscript\u039b\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5\\Lambda_{n}(f,\\lambda_{n},\\cdot)roman_\u039b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) converges to zero\nin probability as n\u2192\u221e\u2192\ud835\udc5bn\\to\\inftyitalic_n \u2192 \u221e for a choice of the sequence \u03bbnsubscript\ud835\udf06\ud835\udc5b\\lambda_{n}italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT\ntending to zero. From this it follows that\nIn\u2062(f,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5I_{n}(f,\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) and In\u2062(f,\u22c5)\u2212\u039bn\u2062(f,\u03bbn,\u22c5)\u2261An\u2062(f,\u03bbn,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5subscript\u039b\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5subscript\ud835\udc34\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5I_{n}(f,\\cdot)-\\Lambda_{n}(f,\\lambda_{n},\\cdot)\\equiv A_{n}(f,\\lambda_{n},\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) - roman_\u039b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) \u2261 italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) have the same limit distribution, provided that the limit exists.\nThe proof is then completed by showing that the latter\nlimit exists and can be obtained by an\nargument using Theorem 1 in which n\ud835\udc5bnitalic_n tends to infinity for a\nfixed small, but positive\n\u03bb\u2113subscript\ud835\udf06\u2113\\lambda_{\\ell}italic_\u03bb start_POSTSUBSCRIPT roman_\u2113 end_POSTSUBSCRIPT, to be determined. Thus, this new\nproof exhibits the asymptotic distribution of\n1n\u2062\u222b0n\u2062tf\u2062(X\u2062(s))\u2062\ud835\udc51s,t\u226501\ud835\udc5bsuperscriptsubscript0\ud835\udc5b\ud835\udc61\ud835\udc53\ud835\udc4b\ud835\udc60differential-d\ud835\udc60\ud835\udc610\\frac{1}{\\sqrt{n}}\\int_{0}^{nt}f(X(s))ds,t\\geq 0divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_n end_ARG end_ARG \u222b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_t end_POSTSUPERSCRIPT italic_f ( italic_X ( italic_s ) ) italic_d italic_s , italic_t \u2265 0,\nf\u2208\ud835\udc9f(\u2212A^)\u221212\ud835\udc53subscript\ud835\udc9fsuperscript^\ud835\udc3412f\\in\\mathscr{D}_{(-\\hat{A})^{-\\frac{1}{2}}}italic_f \u2208 script_D start_POSTSUBSCRIPT ( - over^ start_ARG italic_A end_ARG ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, explicitly\nas the limit of\n1n\u2062\u222b0n\u2062tA^\u2062R\u03bbn\u2062f\u2062(X\u2062(s)),t\u226501\ud835\udc5bsuperscriptsubscript0\ud835\udc5b\ud835\udc61^\ud835\udc34subscript\ud835\udc45subscript\ud835\udf06\ud835\udc5b\ud835\udc53\ud835\udc4b\ud835\udc60\ud835\udc610\\frac{1}{\\sqrt{n}}\\int_{0}^{nt}\\hat{A}R_{\\lambda_{n}}f(X(s)),t\\geq 0divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_n end_ARG end_ARG \u222b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_A end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f ( italic_X ( italic_s ) ) , italic_t \u2265 0,\nA^\u2062R\u03bbn\u2062f\u2208\u211bA^^\ud835\udc34subscript\ud835\udc45subscript\ud835\udf06\ud835\udc5b\ud835\udc53subscript\u211b^\ud835\udc34\\hat{A}R_{\\lambda_{n}}f\\in\\mathscr{R}_{\\hat{A}}over^ start_ARG italic_A end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f \u2208 script_R start_POSTSUBSCRIPT over^ start_ARG italic_A end_ARG end_POSTSUBSCRIPT,\nfor a sequence of positive \u201ctuning\u201dparameters \u03bbnsubscript\ud835\udf06\ud835\udc5b\\lambda_{n}italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.\nSo this new approach\nmay have added value in\ncomputational and further theoretical refinements of the fclt.",,         "Few-shot Voice Cloning: This follows the central concept of speaker adaptation. However, the difference is the amount of data required. Thus, the reference audio can range from a few seconds to a maximum of 5 minutes, which is decided based on previous work, and anything more is challenging to obtain in real-life scenarios.",, source_sentence:         "For any \u03b3\u2208(0,2\u2062d)\ud835\udefe02\ud835\udc51\\gamma\\in(0,\\sqrt{2d})italic_\u03b3 \u2208 ( 0 , square-root start_ARG 2 italic_d end_ARG ), define a stochastic process\n{P\u03b3(\u03bb)\u2062(\ud835\udc2d):\ud835\udc2d\u2208[0,1]d}conditional-setsuperscriptsubscript\ud835\udc43\ud835\udefe\ud835\udf06\ud835\udc2d\ud835\udc2dsuperscript01\ud835\udc51\\{P_{\\gamma}^{(\\lambda)}(\\mathbf{t}):\\mathbf{t}\\in[0,1]^{d}\\}{ italic_P start_POSTSUBSCRIPT italic_\u03b3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_\u03bb ) end_POSTSUPERSCRIPT ( bold_t ) : bold_t \u2208 [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT } by\n\n\n(3.52)\n\nP\u03b3(\u03bb)\u2062(\ud835\udc2d):=exp\u2061(\u03b3\u2062Z\u03bb\u2062(\ud835\udc2d)\u2212\u03b322\u2062\ud835\udd3c\u2062[Z\u03bb\u2062(\ud835\udc2d)2])=exp\u2061(\u03b3\u2062Z\u03bb\u2062(\ud835\udc2d)\u2212\u03b322\u2062R\u03bb\u2062(\ud835\udc2d,\ud835\udc2d)).assignsuperscriptsubscript\ud835\udc43\ud835\udefe\ud835\udf06\ud835\udc2d\ud835\udefesubscript\ud835\udc4d\ud835\udf06\ud835\udc2dsuperscript\ud835\udefe22\ud835\udd3cdelimited-[]subscript\ud835\udc4d\ud835\udf06superscript\ud835\udc2d2\ud835\udefesubscript\ud835\udc4d\ud835\udf06\ud835\udc2dsuperscript\ud835\udefe22subscript\ud835\udc45\ud835\udf06\ud835\udc2d\ud835\udc2d\\displaystyle P_{\\gamma}^{(\\lambda)}(\\mathbf{t}):=\\exp\\Big{(}\\gamma Z_{\\lambda%\n}(\\mathbf{t})-\\frac{\\gamma^{2}}{2}\\mathbb{E}[Z_{\\lambda}(\\mathbf{t})^{2}]\\Big{%\n)}=\\exp\\Big{(}\\gamma Z_{\\lambda}(\\mathbf{t})-\\frac{\\gamma^{2}}{2}R_{\\lambda}(%\n\\mathbf{t},\\mathbf{t})\\Big{)}.italic_P start_POSTSUBSCRIPT italic_\u03b3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_\u03bb ) end_POSTSUPERSCRIPT ( bold_t ) := roman_exp ( italic_\u03b3 italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) - divide start_ARG italic_\u03b3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG blackboard_E [ italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ) = roman_exp ( italic_\u03b3 italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) - divide start_ARG italic_\u03b3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t , bold_t ) ) .",
sentences:         "In this section, we highlight open challenges and future directions in network-level ISAC design and the practical implementation of distributed ISAC systems.",, },         "Warning: As before, we need to restrict ourselves to a smaller class of perturbation data (i.e. sufficiently small Hamiltonian perturbations) to ensure that the element on the right is in \u039b0subscript\u039b0\\Lambda_{0}roman_\u039b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, in other words such that for any quilted strip u\u00af\u00af\ud835\udc62\\underline{u}under\u00af start_ARG italic_u end_ARG we have \u03c9\u2062(u\u00af)=0\ud835\udf14\u00af\ud835\udc620\\omega(\\underline{u})=0italic_\u03c9 ( under\u00af start_ARG italic_u end_ARG ) = 0 if and only if [u\u00af]=0delimited-[]\u00af\ud835\udc620[\\underline{u}]=0[ under\u00af start_ARG italic_u end_ARG ] = 0.",, source_sentence:         "For the regular planar lattice graphs, \ud835\udca2\u25b3,\ud835\udca2\u25a1,\ud835\udca2\u2394subscript\ud835\udca2\u25b3subscript\ud835\udca2\u25a1subscript\ud835\udca2\u2394\\mathcal{G}_{\\triangle},\\,\\mathcal{G}_{\\square},\\,\\mathcal{G}_{\\hexagon}caligraphic_G start_POSTSUBSCRIPT \u25b3 end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT \u25a1 end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT \u2394 end_POSTSUBSCRIPT,\n\n\n\nvol\u27c2\u2062(G)=vol\u2062(G)=vol\u25c6\u2062(G)+vol\u25c6\u2062(G\u2217)=2\u2062\u03c0\u2062m\u2062(p\u2062(z,w))=2\u2062\u03c0\u2062zGfd.superscriptvolperpendicular-to\ud835\udc3avol\ud835\udc3asuperscriptvol\u25c6\ud835\udc3asuperscriptvol\u25c6superscript\ud835\udc3a2\ud835\udf0bm\ud835\udc5d\ud835\udc67\ud835\udc642\ud835\udf0bsubscriptsuperscript\ud835\udc67fd\ud835\udc3a{\\rm vol}^{\\perp}(G)={\\rm vol}(G)={\\rm vol}^{\\lozenge}(G)+{\\rm vol}^{\\lozenge}%\n(G^{*})=2\\pi\\,\\mathrm{m}(p(z,w))=2\\pi\\,z^{\\rm fd}_{G}.roman_vol start_POSTSUPERSCRIPT \u27c2 end_POSTSUPERSCRIPT ( italic_G ) = roman_vol ( italic_G ) = roman_vol start_POSTSUPERSCRIPT \u25c6 end_POSTSUPERSCRIPT ( italic_G ) + roman_vol start_POSTSUPERSCRIPT \u25c6 end_POSTSUPERSCRIPT ( italic_G start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT ) = 2 italic_\u03c0 roman_m ( italic_p ( italic_z , italic_w ) ) = 2 italic_\u03c0 italic_z start_POSTSUPERSCRIPT roman_fd end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT .\n\n\n\nThus, the lower bound in Conjecture\u00a01 holds with equality.",
sentences:         "Let F\ud835\udc39Fitalic_F denote a target model, which will now be trained on a modified dataset Dp\u2062o\u2062i\u2062s\u2062o\u2062n\u2062e\u2062d=D\u2217subscript\ud835\udc37\ud835\udc5d\ud835\udc5c\ud835\udc56\ud835\udc60\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc51superscript\ud835\udc37D_{poisoned}=D^{*}italic_D start_POSTSUBSCRIPT italic_p italic_o italic_i italic_s italic_o italic_n italic_e italic_d end_POSTSUBSCRIPT = italic_D start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT, where D\u2217superscript\ud835\udc37D^{*}italic_D start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT is a surreptitiously modified version of the clean training dataset D\ud835\udc37Ditalic_D. The aim of data poisoning F\ud835\udc39Fitalic_F is creating a poisoned model F\u2217superscript\ud835\udc39F^{*}italic_F start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT that makes incorrect predictions, often without an observable degradation in its overall accuracy. Data poisoning compromises the model integrity by introducing systematic biases that serve the attacker\u2019s objectives while evading detection during model training.",,         "Figure 2 illustrates a comparison between the observed low-medium resolution and the high-resolution spectral profiles of the oxygen A band, depicting observations of (telluric) molecular oxygen. The upper panel of Figure 2 shows low to medium resolution telluric oxygen features. These were obtained from the ESO Science Archive Facility using X-shooter[141] observations during February and March 2024 by the UVES team, as part of Program ID: 60.A-9022(c), OB ID:2024672, 2024624 and 2024822, at various resolutions with short exposures (12 seconds). The results indicate that higher resolution enables the observation of more detailed features within the molecular oxygen spectrum, revealing the signal more distinctly within each spectral line. The lower panel of Figure 2 shows performance tests for future HRS instrumentation by observing the Sun through the Earth\u2019s atmosphere. These profiles demonstrate the measurement outcomes obtained using two types of interferometers: Michelson-based and FPI-based. Firstly, the FTS from the National Solar Observatory at Kitt Peak [126] reported R=700,000 in the oxygen A-band. Secondly, the FIOS-demo[133] showcases spectral profiles based on a chained FPI array with a spectral resolution of R=250,000. This resolution can potentially increase up to R=350,000 with the addition of each array. The throughput of each additional unit, however, decreases by 10-15% [50]. One benefit of achieving this level of resolution is the increase in signal-to-noise ratio and the sampling frequency for each spectral line, which may reduce the required observing time, as predicted in [46, 93].",,         "At this point, we can reconcile what we observe with the evidence from the last paragraphs on TFP in Figure 5. We argue that a critical mass is needed in either case to record a significant impact of the exporting activity. At lower levels of exporting activity, the company starts to benefit from economies of scale but also needs to invest in productive capacity. To keep up with the technological frontier is costly, and it often requires an upgrade of obsolete tangible assets. We argue that the combined evidence of rising operational capacity (sales and costs) and investment in fixed assets explains why we observe a negative albeit small productivity loss in an intermediate range of export intensity. It is only when the company operates abroad at a larger scale that positive albeit small TFP gains come as a consequence of exporting. In this case, we argue, economies of scale become evident and the capital adjustment unveils its impact on firms\u2019 performance.",, source_sentence:         "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",
sentences:     "title": "Always skip attention",,         "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",,         "Assuming an adequately sized Bloom filter, the proportion of false positives is small, ensuring that XAcomsuperscriptsubscript\ud835\udc4b\ud835\udc34comX_{A}^{\\text{com}}italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT com end_POSTSUPERSCRIPT and XBcomsuperscriptsubscript\ud835\udc4b\ud835\udc35comX_{B}^{\\text{com}}italic_X start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT com end_POSTSUPERSCRIPT are highly similar. This minimizes the occurrence of similar but non-identical buckets, thereby mitigating the redundancy issue inherent in bucketing. Furthermore, the use of bucketing not only detects false positives but also ensures convergence, addressing the limitation of Bloom filters alone. This combined approach is analogous to the RSync protocol, where Bloom filters act as the weak checksum and bucketing serves as the strong checksum.",
pipeline_tag: sentence-similarity
library_name: sentence-transformers
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Created: 2025-05-27 18:20:42