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Paul Johnson

Publications and source records attributed to Paul Johnson.

At least 19 recordsLinked to original sources

A cohomology theory of supercommutative algebras and grading-restricted vertex superalgebras

This paper constructs the cohomology theory for grading-restricted vertex superalgebras, generalizing Yi-Zhi Huang's cohomology theory of grading-restricted vertex algebras. To simplify the discussion, motivate the construction, and make it easier for the reader to understand the technical points, we also include the construction of the cohomology theory of supercommutative associative algebras, a generalization of the Harrison cohomology theory of a commutative algebra that has not been explicitly written down. The paper will serve as the foundation for many subsequent studies, especially, the deformation theory of vertex superalgebras.

math.QA

(In)stability in the Dynamics of the Cross-Country Distribution of Income Per Capita

Using a panel of 102 countries from PWT 10.0 covering 1970-2019, we examine the veracity of the assumption that a time-homogeneous, first-order process describes the evolution of the cross-country distribution of per capita output, an assumption often made in studies of the convergence hypothesis employing the distribution dynamics approach pioneered by Quah (1993). To test homogeneity, we compare transition kernels estimated for different time periods and, for those periods exhibiting evidence of homogeneity, we test the first-order assumption using an implication of such a process's Chapman-Kolmogorov equations. Both tests require measurement of the distance between probability distributions which we do with several different metrics, employing bootstrap methods to assess the statistical significance of the observed distances. We find that the process was time-homogeneous and first-order in the 1970-1995 period during which the distribution dynamics imply a bimodal long-run distribution, consistent with convergence clubs. Following the apparent break in the process in the late 1990s, the 2000-2010 distribution dynamics imply a unimodal long-run distribution suggestive of a single convergence club, consistent with recent claims of short-term beta-convergence from the late 1990s and beyond made by Patel et al. (2021) and Kremer et al (2022). After 2010, there is some evidence of a return to non-convergent dynamics similar to those of the 1970-1995 period.

econ.GN

MTrainS: Improving DLRM training efficiency using heterogeneous memories

Recommendation models are very large, requiring terabytes (TB) of memory during training. In pursuit of better quality, the model size and complexity grow over time, which requires additional training data to avoid overfitting. This model growth demands a large number of resources in data centers. Hence, training efficiency is becoming considerably more important to keep the data center power demand manageable. In Deep Learning Recommendation Models (DLRM), sparse features capturing categorical inputs through embedding tables are the major contributors to model size and require high memory bandwidth. In this paper, we study the bandwidth requirement and locality of embedding tables in real-world deployed models. We observe that the bandwidth requirement is not uniform across different tables and that embedding tables show high temporal locality. We then design MTrainS, which leverages heterogeneous memory, including byte and block addressable Storage Class Memory for DLRM hierarchically. MTrainS allows for higher memory capacity per node and increases training efficiency by lowering the need to scale out to multiple hosts in memory capacity bound use cases. By optimizing the platform memory hierarchy, we reduce the number of nodes for training by 4-8X, saving power and cost of training while meeting our target training performance.

cs.IR

Damage Detection in a laboratory-scale wellbore applying Time Reverse Nonlinear Elastic Wave Spectroscopy (TR NEWS)

Time Reverse Nonlinear Elastic Wave Spectroscopy (TR-NEWS) has been used to focus acoustic energy, and make measurements correlated with damage in a variety of industrial materials. Most studies using TR-NEWS in Earth Science have focused on relatively small objects and may have multiple acoustic sources. In Earth, within energy extraction settings, the structure and scale of wellbores makes acoustic focusing challenging. This paper explores the feasibility of applying TR NEWS for damage detection in wellbores by constructing a laboratory-scale wellbore, and using TR to focus and make dynamic linear and nonlinear elastic measurements. After successive cycles of induced, localized mechanical damage the sample, the hysteretic nonlinear elastic parameter {\alpha}, increased with damage cycle indicating progressive mechanical damage. In addition to these strain-dependent changes, TR peak width and changes to peak amplitude near the damage sites was also observed. To deploy TR in a wellbore, it will be necessary to choose sensors that are suitable for the environment, and that can be distributed along the wellbore. Thus, this paper demonstrates that acoustic TR can be conducted using both an intrinsic Fabry Perot interferometer fiber optic strain sensor, and an intrinsic Michaelson interferometer fiber optic strain sensor, as a first step towards deployable sensing for TR in a wellbore.

physics.app-ph

Tremor Waveform Denoising and Automatic Location with Neural Network Interpretation

Active faults release tectonic stress imposed by plate motion through a spectrum of slip modes, from slow, aseismic slip, to dynamic, seismic events. Slow earthquakes are often associated with tectonic tremor, non-impulsive signals that can easily be buried in seismic noise and go undetected. We present a new methodology aimed at improving the detection and location of tremors hidden within seismic noise. After identifying tremors with a classic convolutional neural network, we rely on neural network attribution to extract core tremor signatures and denoise input waveforms. We then use these cleaned waveforms to locate tremors with standard array-based techniques. We apply this method to the Cascadia subduction zone, where we identify tremor patches consistent with existing catalogs. In particular, we show that the cleaned signals resulting from the neural network attribution analysis correspond to a waveform traveling in the Earth's crust and mantle at wavespeeds consistent with local estimates. This approach allows us to extract small signals hidden within the noise, and therefore to locate more tremors than in existing catalogs.

physics.geo-ph

Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

We study the stationary descendant Gromov-Witten theory of toric surfaces by combining and extending a range of techniques - tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block-Göttche, Cooper -Pandharipande, and Block-Gathmann-Markwig.

math.AG

Attention network forecasts time-to-failure in laboratory shear experiments

Rocks under stress deform by creep mechanisms that include formation and slip on small-scale internal cracks. Intragranular cracks and slip along grain contacts release energy as elastic waves termed acoustic emissions (AE). AEs are thought to contain predictive information that can be used for fault failure forecasting. Here we present a method using unsupervised classification and an attention network to forecast labquakes using AE waveform features. Our data were generated in a laboratory setting using a biaxial shearing device with granular fault gouge intended to mimic the conditions of tectonic faults. Here we analyzed the temporal evolution of AEs generated throughout several hundred laboratory earthquake cycles. We used a Conscience Self-Organizing Map (CSOM) to perform topologically ordered vector quantization based on waveform properties. The resulting map was used to interactively cluster AEs. We examined the clusters over time to identify those with predictive ability. Finally, we used a variety of LSTM and attention-based networks to test the predictive power of the AE clusters. By tracking cumulative waveform features over the seismic cycle, the network is able to forecast the time-to-failure (TTF) of lab earthquakes. Our results show that analyzing the data to isolate predictive signals and using a more sophisticated network architecture are key to robustly forecasting labquakes. In the future, this method could be applied on tectonic faults monitor earthquakes and augment current early warning systems.

physics.geo-ph

Earthquake Detection in 1-D Time Series Data with Feature Selection and Dictionary Learning

Earthquakes can be detected by matching spatial patterns or phase properties from 1-D seismic waves. Current earthquake detection methods, such as waveform correlation and template matching, have difficulty detecting anomalous earthquakes that are not similar to other earthquakes. In recent years, machine-learning techniques for earthquake detection have been emerging as a new active research direction. In this paper, we develop a novel earthquake detection method based on dictionary learning. Our detection method first generates rich features via signal processing and statistical methods and further employs feature selection techniques to choose features that carry the most significant information. Based on these selected features, we build a dictionary for classifying earthquake events from non-earthquake events. To evaluate the performance of our dictionary-based detection methods, we test our method on a labquake dataset from Penn State University, which contains 3,357,566 time series data points with a 400 MHz sampling rate. 1,000 earthquake events are manually labeled in total, and the length of these earthquake events varies from 74 to 7151 data points. Through comparison to other detection methods, we show that our feature selection and dictionary learning incorporated earthquake detection method achieves an 80.1% prediction accuracy and outperforms the baseline methods in earthquake detection, including Template Matching (TM) and Support Vector Machine (SVM).

physics.geo-ph

Simultaneous cores with restrictions and a question of Zaleski and Zeilberger

IMPORTANT NOTE: This paper is much rougher than I'd usually submit, and not entirely complete, though the main theorems and proofs should not be hard to follow. Given the ongoing strike at UK Universities it may be some time before I get to complete it to my satisfaction, and in the meantime people I've shared the preliminary draft with would like to be able to reference it. Hence I'm uploading it in its current form, and will update it later. The main new result of this paper is to count the number of (n,n+1)-core partitions with odd parts, answering a question of Zaleski and Zeilberger with bounty a charitable contribution to the OEIS. Along the way, we prove a general theorem giving a recurrence for (n,n+1)-core parts whose smallest part and consecutive part differences are restricted to lie in an arbitrary set M. This theorem unifies many known results about (n,n+1)-core partitions with restrictions. We end with discussions of extensions of the general theorem that keep track of the largest part, number of parts, and size of the partition, and about a few cases where the same methods work on more general simultaneous cores.

math.CO

Cascaded Region-based Densely Connected Network for Event Detection: A Seismic Application

Automatic event detection from time series signals has wide applications, such as abnormal event detection in video surveillance and event detection in geophysical data. Traditional detection methods detect events primarily by the use of similarity and correlation in data. Those methods can be inefficient and yield low accuracy. In recent years, because of the significantly increased computational power, machine learning techniques have revolutionized many science and engineering domains. In this study, we apply a deep-learning-based method to the detection of events from time series seismic signals. However, a direct adaptation of the similar ideas from 2D object detection to our problem faces two challenges. The first challenge is that the duration of earthquake event varies significantly; The other is that the proposals generated are temporally correlated. To address these challenges, we propose a novel cascaded region-based convolutional neural network to capture earthquake events in different sizes, while incorporating contextual information to enrich features for each individual proposal. To achieve a better generalization performance, we use densely connected blocks as the backbone of our network. Because of the fact that some positive events are not correctly annotated, we further formulate the detection problem as a learning-from-noise problem. To verify the performance of our detection methods, we employ our methods to seismic data generated from a bi-axial "earthquake machine" located at Rock Mechanics Laboratory, and we acquire labels with the help of experts. Through our numerical tests, we show that our novel detection techniques yield high accuracy. Therefore, our novel deep-learning-based detection methods can potentially be powerful tools for locating events from time series data in various applications.

cs.LG

A graphical interface for the Gromov--Witten theory of curves

We explore the explicit relationship between the descendant Gromov--Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov--Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient "graphical user interface" for Okounkov and Pandharipande's celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.

math.AG

Lattice points and simultaneous core partitions

We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate (a,b)-cores. We conjecture a unimodality result for q rational Catalan numbers, and make preliminary investigations in applying these methods to the (q,t)-symmetry and specialization conjectures. We prove these conjectures for low degree terms and when a=3, connecting them to the Catalan hyperplane arrangement and quadratic permutation statistics.

math.CO

A Naive Bayes machine learning approach to risk prediction using censored, time-to-event data

Predicting an individual's risk of experiencing a future clinical outcome is a statistical task with important consequences for both practicing clinicians and public health experts. Modern observational databases such as electronic health records (EHRs) provide an alternative to the longitudinal cohort studies traditionally used to construct risk models, bringing with them both opportunities and challenges. Large sample sizes and detailed covariate histories enable the use of sophisticated machine learning techniques to uncover complex associations and interactions, but observational databases are often ``messy,'' with high levels of missing data and incomplete patient follow-up. In this paper, we propose an adaptation of the well-known Naive Bayes (NB) machine learning approach for classification to time-to-event outcomes subject to censoring. We compare the predictive performance of our method to the Cox proportional hazards model which is commonly used for risk prediction in healthcare populations, and illustrate its application to prediction of cardiovascular risk using an EHR dataset from a large Midwest integrated healthcare system.

stat.ML

Hurwitz numbers, ribbon graphs, and tropicalization

Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain collections of permutations. More recently, double Hurwitz numbers have been expressed as a count of certain ribbon graphs, or as a weighted count of certain labeled graphs. This note is an expository account of the equivalences between these definitions, with a few novelties. In particular, we give a simple combinatorial algorithm to pass directly between the permutation and ribbon graph definitions. The two graph theoretic points of view have been used to give proofs that double Hurwitz numbers are piecewise polynomial. We use our algorithm to compare these two proofs.

math.AG

Double Hurwitz numbers via the infinite wedge

We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Goulden, Jackson and Vakil's Strong Piecewise Polynomiality conjecture. The method is a straightforward application of Okounkov's expression for double Hurwitz numbers in terms of operators on the infinite wedge. We begin with a introduction to the infinite wedge tailored to our use.

math.CO

Tropical Hurwitz Numbers

Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers.

math.AG

Chamber Structure of Double Hurwitz numbers

Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0. This paper gives a unified approach to these results and strengthens them in several ways --- the most important being the extension of the results of Shapiro, Shadrin and Vainshtein to arbitrary genus. The main tool is the authors' previous work expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which are well known to give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko, and could have broader applications.

math.AG