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In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer $n$ and an asymptotic formula for this quantity.

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $\pi$ into distinct parts of any integer $n$, namely $s_k(\pi)$. Using $s_k(\pi)$, we generalize the above result for the $k$th smallest parts of partitions for any positive integer $k$ and show its connection with divisor functions for general $k$ and derive interesting special cases. We also study weighted partitions involving $s_k(\pi)$ with another parameter $z$, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving $s_k(\pi)$.

Let $p_{\textrm{dsd}} (n)$ be the number of partitions of $n$ into distinct squarefree divisors of $n$. In this note, we find a lower bound for $p_{\textrm{dsd}} (n)$, as well as a sequence of $n$ for which $p_{\textrm{dsd}} (n)$ is unusually large., Comment: 8 pages

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $\lambda(n+1)=\lambda(n+2)=\cdots=\lambda(n+k)$ holds for some $n\leq x$, where $\lambda$ is the Carmichael $\lambda$ function, is bounded above by $O\left((\log x\log\log x)^2\right)$. Their method involved bounding the value of $\lambda(n + i)$ from below using the prime factorization of $n + i$ for each $i \leq k$. They then used the fact that every $\lambda(n + i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime.

Recommending a diversity of product types (PTs) is important for a good shopping experience when customers are looking for products around their high-level shopping interests (SIs) such as hiking. However, the SI-PT connection is typically absent in e-commerce product catalogs and expensive to construct manually due to the volume of potential SIs, which prevents us from establishing a recommender with easily accessible knowledge systems. To establish such connections, we propose to extract PTs from the Web pages containing hand-crafted PT recommendations for SIs. The extraction task is formulated as binary HTML node classification given the general observation that an HTML node in our target Web pages can present one and only one PT phrase. Accordingly, we introduce TrENC, which stands for Tree-Transformer Encoders for Node Classification. It improves the inter-node dependency modeling with modified attention mechanisms that preserve the long-term sibling and ancestor-descendant relations. TrENC also injects SI into node features for better semantic representation. Trained on pages regarding limited SIs, TrEnc is ready to be applied to other unobserved interests. Experiments on our manually constructed dataset, WebPT, show that TrENC outperforms the best baseline model by 2.37 F1 points in the zero-shot setup. The performance indicates the feasibility of constructing SI-PT relations and using them to power downstream applications such as search and recommendation.

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary., Comment: 2 pages

Extracting structured information from HTML documents is a long-studied problem with a broad range of applications, including knowledge base construction, faceted search, and personalized recommendation. Prior works rely on a few human-labeled web pages from each target website or thousands of human-labeled web pages from some seed websites to train a transferable extraction model that generalizes on unseen target websites. Noisy content, low site-level consistency, and lack of inter-annotator agreement make labeling web pages a time-consuming and expensive ordeal. We develop LEAST -- a Label-Efficient Self-Training method for Semi-Structured Web Documents to overcome these limitations. LEAST utilizes a few human-labeled pages to pseudo-annotate a large number of unlabeled web pages from the target vertical. It trains a transferable web-extraction model on both human-labeled and pseudo-labeled samples using self-training. To mitigate error propagation due to noisy training samples, LEAST re-weights each training sample based on its estimated label accuracy and incorporates it in training. To the best of our knowledge, this is the first work to propose end-to-end training for transferable web extraction models utilizing only a few human-labeled pages. Experiments on a large-scale public dataset show that using less than ten human-labeled pages from each seed website for training, a LEAST-trained model outperforms previous state-of-the-art by more than 26 average F1 points on unseen websites, reducing the number of human-labeled pages to achieve similar performance by more than 10x.

Recently, neural models have been leveraged to significantly improve the performance of information extraction from semi-structured websites. However, a barrier for continued progress is the small number of datasets large enough to train these models. In this work, we introduce the PLAtE (Pages of Lists Attribute Extraction) benchmark dataset as a challenging new web extraction task. PLAtE focuses on shopping data, specifically extractions from product review pages with multiple items encompassing the tasks of: (1) finding product-list segmentation boundaries and (2) extracting attributes for each product. PLAtE is composed of 52, 898 items collected from 6, 694 pages and 156, 014 attributes, making it the first largescale list page web extraction dataset. We use a multi-stage approach to collect and annotate the dataset and adapt three state-of-the-art web extraction models to the two tasks comparing their strengths and weaknesses both quantitatively and qualitatively., Comment: Accepted to ACL Industry Track 2023

HTML documents are an important medium for disseminating information on the Web for human consumption. An HTML document presents information in multiple text formats including unstructured text, structured key-value pairs, and tables. Effective representation of these documents is essential for machine understanding to enable a wide range of applications, such as Question Answering, Web Search, and Personalization. Existing work has either represented these documents using visual features extracted by rendering them in a browser, which is typically computationally expensive, or has simply treated them as plain text documents, thereby failing to capture useful information presented in their HTML structure. We argue that the text and HTML structure together convey important semantics of the content and therefore warrant a special treatment for their representation learning. In this paper, we introduce a novel representation learning approach for web pages, dubbed DOM-LM, which addresses the limitations of existing approaches by encoding both text and DOM tree structure with a transformer-based encoder and learning generalizable representations for HTML documents via self-supervised pre-training. We evaluate DOM-LM on a variety of webpage understanding tasks, including Attribute Extraction, Open Information Extraction, and Question Answering. Our extensive experiments show that DOM-LM consistently outperforms all baselines designed for these tasks. In particular, DOM-LM demonstrates better generalization performance both in few-shot and zero-shot settings, making it attractive for making it suitable for real-world application settings with limited labeled data.

We derive a continuum-level plasticity model for polycrystalline materials in the high energy density regime, based on a single dislocation density and single mobility mechanism, with an evolution model for the dislocation density. The model is formulated explicitly in terms of quantities connected closely with equation of state (EOS) theory, in particular the shear modulus and Einstein temperature, which reduces the number of unconstrained parameters while increasing the range of applicability. The least constrained component is the Peierls barrier $E_P$, which is however accessible by atomistic simulations. We demonstrate an efficient method to estimate the variation of $E_P$ with compression, constrained to fit a single flow stress datum. The formulation for dislocation mobility accounts for some or possibly all of the stiffening at high strain rates usually attributed to phonon drag. The configurational energy of the dislocations is accounted for explicitly, giving a self-consistent calculation of the conversion of plastic work to heat. The configurational energy is predicted to contribute to the mean pressure, and may reach several percent in the terapascal range, which may be significant when inferring scalar EOS data from dynamic loading experiments. The bulk elastic strain energy also contributes to the pressure, but appears to be much smaller. Although inherently describing the plastic relaxation of elastic strain, the model can be manipulated to estimate the flow stress as a function of mass density, temperature, and strain rate, which is convenient to compare with other models and inferences from experiment. The deduced flow stress reproduces systematic trends observed in elastic waves and instability growth experiments, and makes testable predictions of trends versus material and crystal type over a wide range of pressure and strain rate., Comment: Various corrections; updates and additions to references; response to referee comments

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$., Comment: 24 pages

Atom-in-jellium calculations of the Einstein frequency in condensed matter and of the equation of state were used to predict the variation of shear modulus from zero pressure to ~$10^7$ g/cm$^3$, for several elements relevant to white dwarf (WD) stars and other self-gravitating systems. This is by far the widest range reported electronic structure calculation of shear modulus, spanning from ambient through the one-component plasma to extreme relativistic conditions. The predictions were based on a relationship between Debye temperature and shear modulus which we assess to be accurate at the o(10%) level, and is the first known use of atom-in-jellium theory to calculate a shear modulus. We assessed the overall accuracy of the method by comparing with experimental measurements and more detailed electronic structure calculations at lower pressures.

Atom-in-jellium calculations of the electron states, and perturbative calculations of the Einstein frequency, were used to construct equations of state (EOS) from around $10^{-5}$ to $10^7$g/cm$^3$ and $10^{-4}$ to $10^{6}$eV for elements relevant to white dwarf (WD) stars. This is the widest range reported for self-consistent electronic shell structure calculations. Elements of the same ratio of atomic weight to atomic number were predicted to asymptote to the same $T=0$ isotherm, suggesting that, contrary to recent studies of the crystallization of WDs, the amount of gravitational energy that could be released by separation of oxygen and carbon is small. A generalized Lindemann criterion based on the amplitude of the ion-thermal oscillations calculated using atom-in-jellium theory, previously used to extrapolate melt curves for metals, was found to reproduce previous thermodynamic studies of the melt curve of the one component plasma with a choice of vibration amplitude consistent with low pressure results. For elements for which low pressure melting satisfies the same amplitude criterion, such as Al, this melt model thus gives a likely estimate of the melt curve over the full range of normal electronic matter; for the other elements, it provides a useful constraint on the melt locus.

Information extraction from semi-structured webpages provides valuable long-tailed facts for augmenting knowledge graph. Relational Web tables are a critical component containing additional entities and attributes of rich and diverse knowledge. However, extracting knowledge from relational tables is challenging because of sparse contextual information. Existing work linearize table cells and heavily rely on modifying deep language models such as BERT which only captures related cells information in the same table. In this work, we propose a novel relational table representation learning approach considering both the intra- and inter-table contextual information. On one hand, the proposed Table Convolutional Network model employs the attention mechanism to adaptively focus on the most informative intra-table cells of the same row or column; and, on the other hand, it aggregates inter-table contextual information from various types of implicit connections between cells across different tables. Specifically, we propose three novel aggregation modules for (i) cells of the same value, (ii) cells of the same schema position, and (iii) cells linked to the same page topic. We further devise a supervised multi-task training objective for jointly predicting column type and pairwise column relation, as well as a table cell recovery objective for pre-training. Experiments on real Web table datasets demonstrate our method can outperform competitive baselines by +4.8% of F1 for column type prediction and by +4.1% of F1 for pairwise column relation prediction.

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^{\beta}$ and $G(n)^{\beta}$ up to $x$ for all real $\beta$ and the asymptotic bounds for $f(n)^{\beta}$ and $g(n)^{\beta}$ for all negative $\beta$., Comment: 26 pages

The formation and evolution of stars depends on various physical aspects of stellar matter, including the equation of state (EOS) and transport properties. Although often dismissed as `ideal gas-like' and therefore simple, states occurring in stellar matter are dense plasmas, and the EOS has not been established precisely. EOS constructed using multi-physics approaches found necessary for laboratory studies of warm dense matter give significant variations in stellar regimes, and vary from the EOS commonly used in simulations of the formation and evolution of stars. We have investigated the sensitivity of such simulations to variations in the EOS, for sun-like and low-mass stars, and found a strong sensitivity of the lifetime of the Sun and of the lower luminosity limit for red dwarfs. We also found a significant sensitivity in the lower mass limit for red dwarfs. Simulations of this type are also used for other purposes in astrophysics, including the interpretation of absolute magnitude as mass, the conversion of inferred mass distribution to the initial mass function using predicted lifetimes, simulations of star formation from nebulae, simulations of galactic evolution, and the baryon census used to bound the exotic contribution to dark matter. Although many of the sensitivities of stellar physics to the EOS are large, some of the inferred astrophysical quantities are also constrained by independent measurements, although the constraints may be indirect and non-trivial. However, it may be possible to use such measurements to constrain the EOS more than presently possible by established plasma theory.

In many documents, such as semi-structured webpages, textual semantics are augmented with additional information conveyed using visual elements including layout, font size, and color. Prior work on information extraction from semi-structured websites has required learning an extraction model specific to a given template via either manually labeled or distantly supervised data from that template. In this work, we propose a solution for "zero-shot" open-domain relation extraction from webpages with a previously unseen template, including from websites with little overlap with existing sources of knowledge for distant supervision and websites in entirely new subject verticals. Our model uses a graph neural network-based approach to build a rich representation of text fields on a webpage and the relationships between them, enabling generalization to new templates. Experiments show this approach provides a 31% F1 gain over a baseline for zero-shot extraction in a new subject vertical., Comment: Accepted to ACL 2020

Ru and Rh are interesting cases for comparing equations of state (EOS), because most general purpose EOS are semi-empirical, relying heavily on shock data, and none has been reported for Ru. EOS were calculated for both elements using all-electron atom-in-jellium theory, and cold compression curves were calculated for the common crystal types using the multi-ion pseudopotential approach. Previous EOS constructed for these elements used Thomas-Fermi (TF) theory for the electronic behavior at high temperatures, which neglects electronic shell structure; the atom-in-jellium EOS exhibited pronounced features from the excitation of successive electron shells. Otherwise, the EOS matched surprisingly well, especially considering the lack of experimental data for Ru. The TF-based EOS for Ru may however be inaccurate in the multi-terapascal range needed for some high energy density experiments. The multi-ion calculations predicted that the hexagonal close-packed phase of Ru remains stable to at least 2.5 TPa and possibly 10 TPa, and that its c/a should gradually increase to the ideal value. A method was devised to estimate the variation in Debye temperature from the cold curve, and thus estimate the ion-thermal EOS without requiring relatively expensive dynamical force calculations, in a form convenient for adjusting EOS or phase boundaries. The Debye temperature estimated in this way was similar to the result from atom-in-jellium calculations. We also predict the high-pressure melt loci of both elements.

We have used an electron beam ion trap to measure electron-density-diagnostic line-intensity ratios for extreme ultraviolet lines from F XII, XIII, and XIV at wavelengths of 185-205 255-276 Angstroms. These ratios can be used as density diagnostics for astrophysical spectra and are especially relevant to solar physics. We found that density diagnostics using the Fe XIII 196.53/202.04 and the Fe XIV 264.79/274.21 and 270.52A/274.21 line ratios are reliable using the atomic data calculated with the Flexible Atomic Code. On the other hand, we found a large discrepancy between the FAC theory and experiment for the commonly used Fe XII (186.85 + 186.88)/195.12 line ratio. These FAC theory calculations give similar results to the data tabulated in CHIANTI, which are commonly used to analyze solar observations. Our results suggest that the discrepancies seen between solar coronal density measurements using the Fe XII (186.85 + 186.88)/195.12 and Fe XIII 196.54/202.04 line ratios are likely due to issues with the atomic calculations for Fe XII.

Equations of state (EOS) calculated from a computationally efficient atom-in-jellium treatment of the electronic structure have recently been shown to be consistent with more rigorous path integral Monte Carlo (PIMC) and quantum molecular dynamics (QMD) simulations of metals in the warm dense matter regime. Here we apply the atom-in-jellium model to predict wide-ranging EOS for the cryogenic liquid elements nitrogen, oxygen, and fluorine. The principal Hugoniots for these substances were surprisingly consistent with available shock data and Thomas-Fermi (TF) EOS for very high pressures, and exhibited systematic variations from TF associated with shell ionization effects, in good agreement with PIMC, though deviating from QMD and experiment in the molecular regime. The new EOS are accurate much higher in pressure than previous widely-used models for nitrogen and oxygen in particular, and should allow much more accurate predictions for oxides and nitrides in the liquid, vapor, and plasma regime, where these have previously been constructed as mixtures containing the older EOS.