To prove Theorem 1, we need to consider the two directions of the iff conditions. If we are given h(c1,X1) = h(c2,X2), we are able to prove that the conditions mentioned in the theorem are necessary by showing contradictions occur when they are not satisfied. As Eq. (4) equals Eq. (6), we have: X Obviously, the above equation does not hold as the terms in the summation operator are positive. We may now assume S1 = S2 = S. Eliminating the irrational terms in Eq. (4), we have: X Eq. (9) can be simplified and rewritten as: µ1(x) µ2(x) = However, the RHS of Eq. (10) can be an irrational number. It is obvious that the above equality does not hold as the RHS is an irrational number, while LHS is a rational number.

In this section, we provide more information on the application backgrounds, including the detailed structures of the RAS and VAS, the structures of the simulated advertising system. We also discuss the importance and universality of the IBOO problem in auto-bidding, which acts as the motivation of this work.

The dataset and the baseline code will be made publicly available in a dedicated GitHub repository upon acceptance. License TempEL is distributed under Creative Commons Attribution-ShareAlike 4.0 International license (CCBY-SA 4.0).1 Maintenance The maintenance and extension to further temporal snapshots of TempEL will be carried out by the authors of the paper. Additionally, we will make the code public to create potential new variations and extensions of TempEL using a number of hyperparameters (see Sections A.4 and A.5 for further details). A.2 Datasheet for TempEL In this section we provide a more detailed documentation of the dataset with the intended uses. We base ourselves on the datasheet proposed by [1]. A.2.1 Motivation For what purpose was the dataset created? The TempEL dataset was created to evaluate how the temporal change of anchor mentions and that of target Knowledge Base (KB; i.e., modification or creation of new entities) affects the entity linking (EL) task. This contrasts with the currently existing datasets [9, 7, 8, 6], which are associated with a single version of the target KB such as the Wikipedia 2010 for the widely adopted CoNLL-AIDA[2] dataset. We expect that TempEL will encourage research in devising new models and architectures that are robust to temporal changes both in mentions as well as in the target KBs. Who created the dataset and on behalf of which entity?

We present the Manuscripts of Handwritten Arabic (Muharaf) dataset, which is a machine learning dataset consisting of more than 1,600 historic handwritten page images transcribed by experts in archival Arabic. Each document image is accompanied by spatial polygonal coordinates of its text lines as well as basic page elements. This dataset was compiled to advance the state of the art in handwritten text recognition (HTR), not only for Arabic manuscripts but also for cursive text in general. The Muharaf dataset includes diverse handwriting styles and a wide range of document types, including personal letters, diaries, notes, poems, church records, and legal correspondences. In this paper, we describe the data acquisition pipeline, notable dataset features, and statistics. We also provide a preliminary baseline result achieved by training convolutional neural networks using this data.

This section from Archimedes Palimpsest has a mixture of ancient geometry and Byzantine prayers. The missing page still has traces of geometric diagrams based on Greek mathematician Archimedes of Syracuse's work. Breakthroughs, discoveries, and DIY tips sent six days a week. The Archimedes Palimpsest is a Byzantine prayerbook written in 1229, but the artifact holds more than what immediately meets the eye. The original writing on its pages was erased and replaced--making it a palimpsest--a common practice during the medieval period for expensive writing materials made from animal-skin like parchment.

This paper introduces a distillation framework for an ensemble of entropy-optimal Sparse Probabilistic Approximation (eSPA) models, trained exclusively on satellite-era observational and reanalysis data to predict ENSO phase up to 24 months in advance. While eSPA ensembles yield state-of-the-art forecast skill, they are harder to interpret than individual eSPA models. We show how to compress the ensemble into a compact set of "distilled" models by aggregating the structure of only those ensemble members that make correct predictions. This process yields a single, diagnostically tractable model for each forecast lead time that preserves forecast performance while also enabling diagnostics that are impractical to implement on the full ensemble. An analysis of the regime persistence of the distilled model "superclusters", as well as cross-lead clustering consistency, shows that the discretised system accurately captures the spatiotemporal dynamics of ENSO. By considering the effective dimension of the feature importance vectors, the complexity of the input space required for correct ENSO phase prediction is shown to peak when forecasts must cross the boreal spring predictability barrier. Spatial importance maps derived from the feature importance vectors are introduced to identify where predictive information resides in each field and are shown to include known physical precursors at certain lead times. Case studies of key events are also presented, showing how fields reconstructed from distilled model centroids trace the evolution from extratropical and inter-basin precursors to the mature ENSO state. Overall, the distillation framework enables a rigorous investigation of long-range ENSO predictability that complements real-time data-driven operational forecasts.