Search

In this article, we are interested in whether a product of three consecutive integers $a (a+1) (a+2)$ divides another such product $b (b+1) (b+2)$. If this happens, we prove that there is some gaps between them, namely $b \gg \frac{a \log a)^{1/6}}{\log \log a)^{1/3}}$. We also consider other polynomial sequences such as $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$ for some fixed integer $l$. Our method is based on effective Liouville-Baker-Feldman theorem., Comment: 9 pages

In this article, we derive a few more facts concerning powered numbers in short intervals both unconditionally and conditionally on the $abc$-conjecture. We make use of sieve method, a polynomial identity, and a recent breakthrough result on density of sets with no $k$-term arithmetic progression., Comment: 7 pages

Attributed bipartite graphs (ABGs) are an expressive data model for describing the interactions between two sets of heterogeneous nodes that are associated with rich attributes, such as customer-product purchase networks and author-paper authorship graphs. Partitioning the target node set in such graphs into k disjoint clusters (referred to as k-ABGC) finds widespread use in various domains, including social network analysis, recommendation systems, information retrieval, and bioinformatics. However, the majority of existing solutions towards k-ABGC either overlook attribute information or fail to capture bipartite graph structures accurately, engendering severely compromised result quality. The severity of these issues is accentuated in real ABGs, which often encompass millions of nodes and a sheer volume of attribute data, rendering effective k-ABGC over such graphs highly challenging. In this paper, we propose TPO, an effective and efficient approach to k-ABGC that achieves superb clustering performance on multiple real datasets. TPO obtains high clustering quality through two major contributions: (i) a novel formulation and transformation of the k-ABGC problem based on multi-scale attribute affinity specialized for capturing attribute affinities between nodes with the consideration of their multi-hop connections in ABGs, and (ii) a highly efficient solver that includes a suite of carefully-crafted optimizations for sidestepping explicit affinity matrix construction and facilitating faster convergence. Extensive experiments, comparing TPO against 19 baselines over 5 real ABGs, showcase the superior clustering quality of TPO measured against ground-truth labels. Moreover, compared to the state of the arts, TPO is often more than 40x faster over both small and large ABGs., Comment: The technical report for the paper was accepted to KDD 2024. 14 pages

In the domain of chip design, Hardware Description Languages (HDLs) play a pivotal role. However, due to the complex syntax of HDLs and the limited availability of online resources, debugging HDL codes remains a difficult and time-intensive task, even for seasoned engineers. Consequently, there is a pressing need to develop automated HDL code debugging models, which can alleviate the burden on hardware engineers. Despite the strong capabilities of Large Language Models (LLMs) in generating, completing, and debugging software code, their utilization in the specialized field of HDL debugging has been limited and, to date, has not yielded satisfactory results. In this paper, we propose an LLM-assisted HDL debugging framework, namely HDLdebugger, which consists of HDL debugging data generation via a reverse engineering approach, a search engine for retrieval-augmented generation, and a retrieval-augmented LLM fine-tuning approach. Through the integration of these components, HDLdebugger can automate and streamline HDL debugging for chip design. Our comprehensive experiments, conducted on an HDL code dataset sourced from Huawei, reveal that HDLdebugger outperforms 13 cutting-edge LLM baselines, displaying exceptional effectiveness in HDL code debugging., Comment: 13 pages,5 figures

Boolean Satisfiability problems are vital components in Electronic Design Automation, particularly within the Logic Equivalence Checking process. Currently, SAT solvers are employed for these problems and neural network is tried as assistance to solvers. However, as SAT problems in the LEC context are distinctive due to their predominantly unsatisfiability nature and a substantial proportion of UNSAT-core variables, existing neural network assistance has proven unsuccessful in this specialized domain. To tackle this challenge, we propose IB-Net, an innovative framework utilizing graph neural networks and novel graph encoding techniques to model unsatisfiable problems and interact with state-of-the-art solvers. Extensive evaluations across solvers and datasets demonstrate IB-Net's acceleration, achieving an average runtime speedup of 5.0% on industrial data and 8.3% on SAT competition data empirically. This breakthrough advances efficient solving in LEC workflows., Comment: 7 pages, 12 figures

In this paper, we continue the study on variance of the number of squarefull numbers in short intervals $(x, x + 2 \sqrt{x} H + H^2]$ with $X \le x \le 2X$. We obtain the expected asymptotic for this variance over the range $X^\epsilon \le H \le X^{0.180688...}$ unconditionally and over the optimal range $X^\epsilon \le H \le X^{0.25 - \epsilon}$ conditionally on the Riemann Hypothesis or the Lindel\"{o}f Hypothesis., Comment: 17 pages, welcome any comments. arXiv admin note: text overlap with arXiv:2205.12108

Analogy-making between narratives is crucial for human reasoning. In this paper, we evaluate the ability to identify and generate analogies by constructing a first-of-its-kind large-scale story-level analogy corpus, \textsc{StoryAnalogy}, which contains 24K story pairs from diverse domains with human annotations on two similarities from the extended Structure-Mapping Theory. We design a set of tests on \textsc{StoryAnalogy}, presenting the first evaluation of story-level analogy identification and generation. Interestingly, we find that the analogy identification tasks are incredibly difficult not only for sentence embedding models but also for the recent large language models (LLMs) such as ChatGPT and LLaMa. ChatGPT, for example, only achieved around 30% accuracy in multiple-choice questions (compared to over 85% accuracy for humans). Furthermore, we observe that the data in \textsc{StoryAnalogy} can improve the quality of analogy generation in LLMs, where a fine-tuned FlanT5-xxl model achieves comparable performance to zero-shot ChatGPT., Comment: Accepted by EMNLP 2023 main conference

Let $s_1, s_2, s_3, \cdots$ be the set of squarefree numbers in ascending order. In this paper, we prove that the following asymptotic on moments of gaps between squarefree numbers \[ \sum_{s_{k+1} \le x} (s_{k+1} - s_k)^\gamma \sim B(\gamma) x \; \; \mbox{ with some constant} \; \; B(\gamma) > 0 \] is true for $0 \le \gamma < 3.75$. This improves the previous best range $0 \le \gamma < 3.6875$., Comment: 10 pages

Abduction has long been seen as crucial for narrative comprehension and reasoning about everyday situations. The abductive natural language inference ($\alpha$NLI) task has been proposed, and this narrative text-based task aims to infer the most plausible hypothesis from the candidates given two observations. However, the inter-sentential coherence and the model consistency have not been well exploited in the previous works on this task. In this work, we propose a prompt tuning model $\alpha$-PACE, which takes self-consistency and inter-sentential coherence into consideration. Besides, we propose a general self-consistent framework that considers various narrative sequences (e.g., linear narrative and reverse chronology) for guiding the pre-trained language model in understanding the narrative context of input. We conduct extensive experiments and thorough ablation studies to illustrate the necessity and effectiveness of $\alpha$-PACE. The performance of our method shows significant improvement against extensive competitive baselines., Comment: Accepted at IJCNLP-AACL 2023 main track

Centrifuges serve as essential instruments in modern experimental sciences, facilitating a wide range of routine sample processing tasks that necessitate material sedimentation. However, the study for real time observation of the dynamical process during centrifugation has remained elusive. In this study, we developed an innovative Lab_in_a_Tube imaging spectrophotometer that incorporates capabilities of real time image analysis and programmable interruption. This portable LIAT device costs less than 30 US dollars. Based on our knowledge, it is the first Wi Fi camera built_in in common lab centrifuges with active closed_loop control. We tested our LIAT imaging spectrophotometer with solute solvent interaction investigation obtained from lab centrifuges with quantitative data plotting in a real time manner. Single re circulating flow was real time observed, forming the ring shaped pattern during centrifugation. To the best of our knowledge, this is the very first observation of similar phenomena. We developed theoretical simulations for the single particle in a rotating reference frame, which correlated well with experimental results. We also demonstrated the first demonstration to visualize the blood sedimentation process in clinical lab centrifuges. This remarkable cost effectiveness opens up exciting opportunities for centrifugation microbiology research and paves the way for the creation of a network of computational imaging spectrometers at an affordable price for large scale and continuous monitoring of centrifugal processes in general., Comment: 21 Pages, 6 Figures

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact K\"ahler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-\varphi_F}$. Via the use of the adjoint ideal sheaves (constructed from $\varphi_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact K\"ahler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular K\"ahler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Koll\'ar-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact K\"ahler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and P\u{a}un., Comment: 30 pages

This note reviews the authors' approach to Fujino's conjecture, i.e.~the injectivity theorem for lc pairs on compact K\"ahler manifolds, via the use of adjoint ideal sheaves coupled with the associated residue computations in their previous work. Using only the techniques and results obtained from that study and under a slightly stronger positivity assumption, a ``qualitative'' extension result is obtained. Such extension result guarantees the existence of a global holomorphic extension $F$ of any holomorphic section $f$ on some $\sigma$-lc centres of the given lc pair. The extension $F$ can be shown to take values in the corresponding adjoint ideal sheaf, even though it comes without any $L^2$ estimate of $F$ in terms of $f$. Moreover, the proof invokes and implies no vanishing theorem in general., Comment: 13 pages

Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerful numbers. We prove that its real logarithmic spectrum takes values from $- \sqrt{2} / (4 + \sqrt{2}) = -0.26160...$ to $1$ while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from $0$ to $1$. In the course of this study, we correct a proof of Granville and Soundararajan concerning contribution of small primes in the study of mean value of multiplicative functions., Comment: 13 pages

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterl\'e.

The "qualitative" extension theorem of Demailly guarantees existence of holomorphic extensions of holomorphic sections on some subvariety under certain positive-curvature assumption, but that comes without any estimate of the extensions, especially when the singular locus of the subvariety is non-empty and the holomorphic section to be extended does not vanish identically there. Residue functions are analytic functions which connect the $L^2$ norms on the subvarieties (or their singular loci) to $L^2$ norms with specific weights on the ambient space. Motivated by the conjectural "dlt extension", this note discusses the possibility of retrieving the $L^2$ estimates for the extensions in the general situation via the use of the residue functions. It is also shown in this note that the $1$-lc-measure defined via the residue function of index $1$ is indeed equal to the Ohsawa measure in the Ohsawa--Takegoshi $L^2$ extension theorem., Comment: 10 pages; v2: minor changes and updates to some citations and references

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d \ll N^{1/2}$ unconditionally. We also prove some partial results when $k \ge 4$ and pose some open questions., Comment: 7 pages, welcome any comments

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{\log^{1+\epsilon} y})$ for squarefull numbers and the bound $O(y^{(2 + \epsilon)/k})$ for $k$-full numbers when $k \ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numbers., Comment: 7 pages, welcome any comments

In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0 < \theta < 1/2$, almost all short intervals $(x, x + x^{1/2 + \theta}]$ contain about $\frac{\zeta(3/2)}{2 \zeta(3)} x^\theta$ squarefull numbers., Comment: Corrected and improved arguments. Accepted by Illinois Journal of Mathematics

As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact K\"ahler manifolds for plt pairs by Matsumura is improved in this article to allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic singularities in the coefficients of the relevant cohomology groups. With the use of a refined version of analytic adjoint ideal sheaves, a plan towards a solution to the generalised version of Fujino's conjecture (i.e. an injectivity theorem on compact K\"ahler manifolds for lc pairs with multiplier ideal sheaves) is laid down and, in addition to the result for plt pairs, a proof for lc pairs in dimension 2, which is also an improvement of Matsumura's result, is given., Comment: 43 pages; v2: presentations in Remark 2.4.3 and proofs of Propositions 3.3.2 and 3.3.4 are expanded; accepted for publication in the Trans. Amer. Math. Soc

A new definition of analytic adjoint ideal sheaves for quasi-plurisubharmonic (quasi-psh) functions with only neat analytic singularities is studied and shown to admit some residue short exact sequences which are obtained by restricting sections of the newly defined adjoint ideal sheaves to some unions of $\sigma$-log-canonical ($\sigma$-lc) centres. The newly defined adjoint ideal sheaves induce naturally some residue $L^2$ norms on the unions of $\sigma$-lc centres which are invariant under log-resolutions. They can also describe unions of $\sigma$-lc centres without the need of log-resolutions even if the quasi-psh functions in question are not in a simple-normal-crossing configuration. This is hinting their potential use in discussing the $\sigma$-lc centres even when the quasi-psh functions in question have more general singularities. Furthermore, their relations between the algebraic adjoint ideal sheaves of Ein--Lazarsfeld as well as those of Hacon--McKernan are described in order to illustrate their role as a (potentially finer) measurement of singularities in the minimal model program. In the course of the study, a local $L^2$ extension theorem is proven, which shows that holomorphic sections on any unions of $\sigma$-lc centres can be extended holomorphically to some neighbourhood of the unions of $\sigma$-lc centres with some $L^2$ estimates. The proof does not rely on the techniques in the Ohsawa--Takegoshi-type $L^2$ extension theorems., Comment: 56 pages; v6: typos fixed; published in J. Geom. Anal., see https://rdcu.be/deUDt ; v5: a mistake concerning the direct image of the residue sheaf via a log-resolution in the original proof of Theorem 5.2.1 is fixed; Examples 2.3.4, 5.2.2 and 5.2.3, Propositions 5.2.6 and 5.2.7, and Theorem 5.2.8 are added or revised from the previous version; v4: reference to an erratum by Guenancia added

In this paper, we show that, for some constant $C > 0$, the interval $(x, x + C x^{5/26}]$ always contains a squarefree number when $x$ is sufficiently large (in terms of $C$). Our improvement comes from establishing asymptotic relations between the shifts $a$ and $b$ when $m n^2 \approx (m - a) (n + b)^2$ We apply them to study quadruples $(m + a_1) (n - b_1)^2 \approx m n^2 \approx (m - a_2)(n + b_2)^2 \approx (m - a_2 - a_3)(n + b_2 + b_3)^2$ and generalize Roth differencing and Filaseta-Trifonov differencing by allowing $b_1$ to be different from $b_3$. We also introduce a new differencing and exploit the interplay among these three differencings., Comment: There is an error in the article. The recursive argument for Proposition 4 does not work if there is only one element in a "short interval"

High Dynamic Range (HDR) imaging via multi-exposure fusion is an important task for most modern imaging platforms. In spite of recent developments in both hardware and algorithm innovations, challenges remain over content association ambiguities caused by saturation, motion, and various artifacts introduced during multi-exposure fusion such as ghosting, noise, and blur. In this work, we propose an Attention-guided Progressive Neural Texture Fusion (APNT-Fusion) HDR restoration model which aims to address these issues within one framework. An efficient two-stream structure is proposed which separately focuses on texture feature transfer over saturated regions and multi-exposure tonal and texture feature fusion. A neural feature transfer mechanism is proposed which establishes spatial correspondence between different exposures based on multi-scale VGG features in the masked saturated HDR domain for discriminative contextual clues over the ambiguous image areas. A progressive texture blending module is designed to blend the encoded two-stream features in a multi-scale and progressive manner. In addition, we introduce several novel attention mechanisms, i.e., the motion attention module detects and suppresses the content discrepancies among the reference images; the saturation attention module facilitates differentiating the misalignment caused by saturation from those caused by motion; and the scale attention module ensures texture blending consistency between different coder/decoder scales. We carry out comprehensive qualitative and quantitative evaluations and ablation studies, which validate that these novel modules work coherently under the same framework and outperform state-of-the-art methods.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. One may similarly consider the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that $\tau_2 < 0.8$, which directly implies the Erd\H{o}s conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that $\tau_k$ tends to zero as $k\to\infty$., Comment: 14 pages

With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with $L^2$ estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the $L^2$ estimates on compact K\"ahler manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is introduced which is shown to "deform holomorphically" to an $L^2$ norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a "non-universal" $L^2$ estimate on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation are presented to verify the expected $L^2$ estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general., Comment: 22 pages; v3: minor typos fixed; published online in Math. Z., view-only version via https://rdcu.be/cFDPA

A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erd\H{o}s question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements., Comment: 13 pages

Range aggregate queries find frequent application in data analytics. In some use cases, approximate results are preferred over accurate results if they can be computed rapidly and satisfy approximation guarantees. Inspired by a recent indexing approach, we provide means of representing a discrete point data set by continuous functions that can then serve as compact index structures. More specifically, we develop a polynomial-based indexing approach, called PolyFit, for processing approximate range aggregate queries. PolyFit is capable of supporting multiple types of range aggregate queries, including COUNT, SUM, MIN and MAX aggregates, with guaranteed absolute and relative error bounds. Experiment results show that PolyFit is faster and more accurate and compact than existing learned index structures., Comment: 13 pages

With a view to proving the conjecture of "dlt extension" related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the "lc-measures", which hopefully could provide the $L^2$ estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of $L^2$ extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where the classical Ohsawa-Takegoshi $L^2$ extension works, with some improvements on the assumptions on the metrics involved. The second half of the paper provides a simplified proof of the result of Demailly-Hacon-Paun on the "plt extension" with the superfluous assumption "$\operatorname{supp} D \subset \operatorname{supp}(S+B)$" in their result removed. Most arguments in the proof are readily adopted to the "dlt extension" once the $L^2$ estimates with respect to the lc-measures of holomorphic extensions of sections on subvarieties with singular locus are ready., Comment: 44 pages; dedicated to Prof. Fabrizio Catanese on the occasion of his 71st birthday; to appear in Math. Ann. arXiv admin note: text overlap with arXiv:1811.02204