arXiv subjects
Publications and source records attributed to Joseph McDonald.
Partial orthomodular lattices are an intermediate class of algebras between ortholattices and orthomodular lattices. In this short note, we obtain a duality for partial orthomodular lattices via a subcategory of the category of spectral spaces, independently of the Axiom of Choice.
The purpose of this study is to investigate the bitopological duality theory of De Morgan algebras equipped with a closure operator, known as S4 De Morgan algebras. We first introduce certain expansions of pairwise Stone spaces, which we call pairwise S4 De Morgan Stone spaces (henceforth, PS4D-spaces). These consist of a pairwise Stone space $X$ equipped with a twist continuous involution $g\colon X\to X$, as well as a binary relation $R\subseteq X\times X$ that is reflexive and transitive. We first demonstrate that the bitopological spectrum $S_0(A)$ of prime filters of an S4 De Morgan algebra $A$ gives rise to a PS4D-space. A topological representation is then obtained by exhibiting an isomorphism from $A$ to the S4 De Morgan algebra $A_0(S_0(A))$ of $(\tau_1,\delta_2)$-biclopen subsets of $S_0(A)$ whose operation of De Morgan involution is defined through $g$ and whose closure operator is defined through $R$. We then provide an algebraic realization theorem by showing that every PS4D-space $X$ is bihomeomorphic and relationally isomorphic to the bitopological spectrum $S_0(A_0(X))$ of prime filters of $A_0(X)$. With the introduction of suitable bicontinuous frame morphisms, we show that the category $\mathbf{S4D}$ of S4 De Morgan algebras is dually equivalent to the category $\mathbf{PStone_{S4D}}$ of PS4D-spaces. As an application, we provide bitopological characterizations of filters and ideals in general De Morgan algebras under our established duality as well as bitopological soundness and completeness results for an S4-type modal extension of the calculus FDE of first-degree entailment.
Stone demonstrated that the category $\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $\mathbf{S4DM}$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimb\'o's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.
Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.
In this note, we study the operation of Sasaki hook within the setting of quantum cylindric algebras by introducing cylindric quasi-implication algebras. It is first demonstrated that every quantum cylindric algebra can be converted into a cylindric quasi-implication algebra and conversely that every cylindric quasi-implication algebra gives rise to a quantum cylindric algebra. These constructions are then shown to induce an isomorphism between the category $\mathbf{CQIA}$ of cylindric quasi-implication algebras and the category $\mathbf{QCA}$ of quantum cylindric algebras. We then give two alternative constructions of a cylindric orthoframe $X_A$ from a cylindric quasi-implication algebra $A$. The first construction of $X_A$ arises via the non-zero elements of $A$ and generalizes the construction given by Harding in the setting of cylindric ortholattices from the perspective of MacNeille completions. The second construction of $X_A$ arises via the proper filters of $A$ and generalizes the construction given by McDonald in the setting of cylindric ortholattices from the perspective of canonical completions.
In this note, we investigate the algebraic and topological representation theory of cylindric ortholattices and cylindric Boolean algebras. The first contribution demonstrates that cylindric ortholattices are closed under canonical completions. By equipping a spectral topology to the dual space associated with the canonical completion, we then establish a dual equivalence between the category of cylindric ortholattices and a certain subcategory of the category of spectral spaces. This work builds on the completion and duality results obtained by Harding, McDonald, and Peinado in the setting of monadic ortholattices combined with the duality results obtained by McDonald and Yamamoto in the setting of general ortholattices. By working with the duality theory for Boolean algebras established by Bezhanishvili and Holliday, we then obtain completion and duality results for cylindric Boolean algebras. A key aspect of our duality results is that they are constructive in the sense that they obtain in Zermelo-Fraenkel set theory independently of the Axiom of Choice.
In this note, we study various relational and algebraic aspects of the bounded quasi-implication algebras introduced by Hardegree. By generalizing the constructions given by MacLaren and Goldblatt within the setting of ortholattices, we construct various orthogonality relations from bounded quasi-implication algebras. We then introduce certain bounded quasi-implication algebras with an additional operator, which we call monadic quasi-implication algebras, and study them within the setting of quantum monadic algebras. A quantum monadic algebra is an orthomodular lattice equipped with a closure operator, known as a quantifier, whose closed elements form an orthomodular sub-lattice. It is shown that every quantum monadic algebra can be converted into a monadic quasi-implication algebra with the underlying magma structure being determined by the operation of Sasaki implication on the underlying orthomodular lattice. It is then conversely demonstrated that every monadic quasi-implication algebra can be converted into a quantum monadic algebra. These constructions are shown to induce an isomorphism between the category of quantum monadic algebras and the category of monadic quasi-implication algebras. Finally, by generalizing the constructions given by Harding as well as Harding, McDonald, and Peinado in the setting of monadic ortholattices, we construct various monadic orthoframes from monadic quasi-implication algebras.
In this paper, we show that every monadic ortholattice is isomorphic to a functional one, thereby resolving a recent question posed by Harding. We then study certain substitution-free reducts of the polyadic ortholattices, which we call locally finite $\sigma$-free polyadic ortholattices, and provide an analogous functional representation result.
IIn this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors.
We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of $L$ is obtained by forming an associated dual space $X$ that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, $X$ is formed from the non-zero elements of $L$, and for the canonical completion, $X$ is formed from the proper filters of $L$. The corresponding completion of $L$ is then obtained as the ortholattice of bi-orthogonally closed subsets of $X$ with an additional operation defined through the binary relation of $X$. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimbó, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.
As research and deployment of AI grows, the computational burden to support and sustain its progress inevitably does too. To train or fine-tune state-of-the-art models in NLP, computer vision, etc., some form of AI hardware acceleration is virtually a requirement. Recent large language models require considerable resources to train and deploy, resulting in significant energy usage, potential carbon emissions, and massive demand for GPUs and other hardware accelerators. However, this surge carries large implications for energy sustainability at the HPC/datacenter level. In this paper, we study the aggregate effect of power-capping GPUs on GPU temperature and power draw at a research supercomputing center. With the right amount of power-capping, we show significant decreases in both temperature and power draw, reducing power consumption and potentially improving hardware life-span with minimal impact on job performance. While power-capping reduces power draw by design, the aggregate system-wide effect on overall energy consumption is less clear; for instance, if users notice job performance degradation from GPU power-caps, they may request additional GPU-jobs to compensate, negating any energy savings or even worsening energy consumption. To our knowledge, our work is the first to conduct and make available a detailed analysis of the effects of GPU power-capping at the supercomputing scale. We hope our work will inspire HPCs/datacenters to further explore, evaluate, and communicate the impact of power-capping AI hardware accelerators for more sustainable AI.
Weather radar is the primary tool used by forecasters to detect and warn for tornadoes in near-real time. In order to assist forecasters in warning the public, several algorithms have been developed to automatically detect tornadic signatures in weather radar observations. Recently, Machine Learning (ML) algorithms, which learn directly from large amounts of labeled data, have been shown to be highly effective for this purpose. Since tornadoes are extremely rare events within the corpus of all available radar observations, the selection and design of training datasets for ML applications is critical for the performance, robustness, and ultimate acceptance of ML algorithms. This study introduces a new benchmark dataset, TorNet to support development of ML algorithms in tornado detection and prediction. TorNet contains full-resolution, polarimetric, Level-II WSR-88D data sampled from 10 years of reported storm events. A number of ML baselines for tornado detection are developed and compared, including a novel deep learning (DL) architecture capable of processing raw radar imagery without the need for manual feature extraction required for existing ML algorithms. Despite not benefiting from manual feature engineering or other preprocessing, the DL model shows increased detection performance compared to non-DL and operational baselines. The TorNet dataset, as well as source code and model weights of the DL baseline trained in this work, are made freely available.
Large language models (LLMs) have exploded in popularity due to their new generative capabilities that go far beyond prior state-of-the-art. These technologies are increasingly being leveraged in various domains such as law, finance, and medicine. However, these models carry significant computational challenges, especially the compute and energy costs required for inference. Inference energy costs already receive less attention than the energy costs of training LLMs -- despite how often these large models are called on to conduct inference in reality (e.g., ChatGPT). As these state-of-the-art LLMs see increasing usage and deployment in various domains, a better understanding of their resource utilization is crucial for cost-savings, scaling performance, efficient hardware usage, and optimal inference strategies. In this paper, we describe experiments conducted to study the computational and energy utilization of inference with LLMs. We benchmark and conduct a preliminary analysis of the inference performance and inference energy costs of different sizes of LLaMA -- a recent state-of-the-art LLM -- developed by Meta AI on two generations of popular GPUs (NVIDIA V100 \& A100) and two datasets (Alpaca and GSM8K) to reflect the diverse set of tasks/benchmarks for LLMs in research and practice. We present the results of multi-node, multi-GPU inference using model sharding across up to 32 GPUs. To our knowledge, our work is the one of the first to study LLM inference performance from the perspective of computational and energy resources at this scale.
A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak p-morphisms. It is well-known that orthomodular lattices provide an algebraic semantics for the quantum logic Q. Hence, as an application of our duality, we develop a topological semantics for Q using orthomodular spaces and prove soundness and completeness.
Efficient structural repair of wind turbine blades is essential to limiting global warming and reducing the Levelized Cost of Energy (LCOE). Repairs carried out up-tower are sensitive to environmental conditions whose effect on the material properties during processing needs to be accounted for to accurately predict the repair outcome. This study investigates the effect of moisture content from environmental exposure on the cure kinetics of an infusion resin system used in wind turbine blade manufacturing and repair and provides an experimentally validated finite element tool for the analysis of cure cycle repairs as a function of repair geometry and moisture content. Moisture absorption tests on the two-part infusion system reported up to 12% moisture uptake by the curing agent under high temperature and relative humidity conditions. Differential scanning calorimetric measurements of resin in the presence of moisture revealed an accelerated cure behavior. Numerical predictions of a repair model agreed very well with the corresponding lab-scale repair and revealed a substantial temperature lag within the repair patch which resulted in thermal gradients and spatial distribution of the degree of cure. It was shown that the repair geometry and the accelerated-cure kinetics greatly influenced the temperature and cure distribution within the repair. The proposed approach can be used to reduce turbine downtime by minimizing the curing time.
Radial velocity estimates provided by Doppler weather radar are critical measurements used by operational forecasters for the detection and monitoring of life-impacting storms. The sampling methods used to produce these measurements are inherently susceptible to aliasing, which produces ambiguous velocity values in regions with high winds, and needs to be corrected using a velocity dealiasing algorithm (VDA). In the US, the Weather Surveillance Radar-1988 Doppler (WSR-88D) Open Radar Product Generator (ORPG) is a processing environment that provides a world-class VDA; however, this algorithm is complex and can be difficult to port to other radar systems outside of the WSR-88D network. In this work, a Deep Neural Network (DNN) is used to emulate the 2-dimensional WSR-88D ORPG dealiasing algorithm. It is shown that a DNN, specifically a customized U-Net, is highly effective for building VDAs that are accurate, fast, and portable to multiple radar types. To train the DNN model, a large dataset is generated containing aligned samples of folded and dealiased velocity pairs. This dataset contains samples collected from WSR-88D Level-II and Level-III archives, and uses the ORPG dealiasing algorithm output as a source of truth. Using this dataset, a U-Net is trained to produce the number of folds at each point of a velocity image. Several performance metrics are presented using WSR-88D data. The algorithm is also applied to other non-WSR-88D radar systems to demonstrate portability to other hardware/software interfaces. A discussion of the broad applicability of this method is presented, including how other Level-III algorithms may benefit from this approach.
As research and practice in artificial intelligence (A.I.) grow in leaps and bounds, the resources necessary to sustain and support their operations also grow at an increasing pace. While innovations and applications from A.I. have brought significant advances, from applications to vision and natural language to improvements to fields like medical imaging and materials engineering, their costs should not be neglected. As we embrace a world with ever-increasing amounts of data as well as research and development of A.I. applications, we are sure to face an ever-mounting energy footprint to sustain these computational budgets, data storage needs, and more. But, is this sustainable and, more importantly, what kind of setting is best positioned to nurture such sustainable A.I. in both research and practice? In this paper, we outline our outlook for Green A.I. -- a more sustainable, energy-efficient and energy-aware ecosystem for developing A.I. across the research, computing, and practitioner communities alike -- and the steps required to arrive there. We present a bird's eye view of various areas for potential changes and improvements from the ground floor of AI's operational and hardware optimizations for datacenters/HPCs to the current incentive structures in the world of A.I. research and practice, and more. We hope these points will spur further discussion, and action, on some of these issues and their potential solutions.
In this paper we address the application of pre-processing techniques to multi-channel time series data with varying lengths, which we refer to as the alignment problem, for downstream machine learning. The misalignment of multi-channel time series data may occur for a variety of reasons, such as missing data, varying sampling rates, or inconsistent collection times. We consider multi-channel time series data collected from the MIT SuperCloud High Performance Computing (HPC) center, where different job start times and varying run times of HPC jobs result in misaligned data. This misalignment makes it challenging to build AI/ML approaches for tasks such as compute workload classification. Building on previous supervised classification work with the MIT SuperCloud Dataset, we address the alignment problem via three broad, low overhead approaches: sampling a fixed subset from a full time series, performing summary statistics on a full time series, and sampling a subset of coefficients from time series mapped to the frequency domain. Our best performing models achieve a classification accuracy greater than 95%, outperforming previous approaches to multi-channel time series classification with the MIT SuperCloud Dataset by 5%. These results indicate our low overhead approaches to solving the alignment problem, in conjunction with standard machine learning techniques, are able to achieve high levels of classification accuracy, and serve as a baseline for future approaches to addressing the alignment problem, such as kernel methods.