Algebraic Framework Research Articles

Network effects in multi-species fisheries.

Managing sustainable marine fisheries is one of the greatest challenges for humanity. The complexity of the issue calls for the development of socio-ecological models and the integration of our knowledge from several disciplines. Here we focus on the ecological aspects of sustainability: how can we increase the catch and, at the same time, possibly decrease negative effects on the marine ecosystem. Coexisting species live in richly interconnected interaction networks. This means that changes in their biomass are caused and may cause various direct and indirect effects on all other coexisting species in the food web. The assessment of maximum sustainable yield values is typically based on single-species analyses, poorly considering this multi-species context. If several fish species are exploited in particular combinations, their effects may not be additive and non-additivity may mean dampening. In these cases, the community response to fishing species A and B together may be smaller than the sum of fishing species A and B separately. We report on some preliminary results on how to develop a network algebra framework for better understanding food web simulation results for pairwise perturbations and their counter-intuitive effects.

Matroids over skew tracts

Matroids over tracts provide an algebraic framework simultaneously generalizing the notions of matroids, oriented matroids, and valuated matroids, presented by Baker and Bowler. Pendavingh partially extended this theory to skew hyperfields and presented a new axiom system in terms of quasi-Plücker coordinates. We present a theory of matroids over skew tracts, which generalizes both the theory of matroids over tracts and the theory of weak matroids over skew hyperfields developed by Pendavingh. We give several cryptomorphic axiom systems for such matroids in terms of circuits, quasi-Plücker coordinates and dual pairs.

Design, analysis, and experiment of a new parallel manipulator with two rotational and one translational motion

• A new overconstrained 2R1T parallel manipulator with fixed actuators. • Only 13 single-DOF joints, simple inverse kinematics and few singularities . • Great performances in the levels of kinematics, stiffness and dynamics. • Suitable parallel module of five-axis hybrid machines through machining experiment. • Addition and multiplication in the geometric algebra avoid complex calculations. Five-axis hybrid machine tools are important equipment for highly efficient precision machining of complex workpieces, in which a key part is parallel manipulators (PMs) with two rotations and one translation (2R1T). In view of many single-degree-of-freedom (DOF) joints and virtual output rotational axes in most existing 2R1T PMs, a new 2R1T 2PRU-PSR PM with two certain rotational axes is proposed, which is composed of 13 single-DOF joints and only one S joint and is fully actuated with stationary actuators. Using geometric algebra as a mathematical tool, the mobility and kinematic performance evaluation considering the motion/force transmissibility are presented. With the motion/force transmissibility taken as the optimal index, the dimensional parameters of the 2PRU-PSR PM are optimized to obtain an improved transmission workspace without singular configurations. Based on the optimized parameters, stiffness and dynamic analyses of the 2PRU-PSR PM are presented in the framework of geometric algebra. The results reveal that the 2PRU-PSR PM exhibits excellent performance in the levels of kinematics, stiffness, and dynamics. Finally, a five-axis hybrid prototype is built based on the optimized 2PRU-PSR PM and two serial guides. Machining experiments show that the 2PRU-PSR PM is suitable as a parallel module for five-axis hybrid machines.

Computational Differential Algebraic Equation Framework and Multi Spatial and Time Discretizations Preserving Consistent Second-Order Accuracy: Nonlinear Dynamics

Abstract We propose the novel design, development, and implementation of the well-known generalized single step single solve (GS4) family of algorithms into the differential algebraic equation framework which not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains but also additionally allows for the selection of different space discretized methods such as the finite element method and particle methods, and other spatial methods as well in a single analysis unlike existing state-of-the-art. For the first time, the user has the flexibility and robustness to embed different algorithms for time integration and different spatial methods for space discretization in a single analysis. In addition, the GS4 family enables a wide variety of choices of time integration methods in a single analysis and also ensures the second-order accuracy in time of all primary variables and Lagrange multipliers. This is not possible to date. However, the present framework provides the fusion of a wide variety of choices of time discretized methods and spatial methods and has the bandwidth and depth to engage in various types of research investigations as well and features for fine tuning of numerical simulations. It provides generality/versatility of the computational framework incorporating subdomains with different spatial and time integration algorithms with improved accuracy. The robustness and accuracy of the present work is not feasible in the current state of technology. Various numerical examples illustrate the significant capabilities and generality and effectiveness for general nonlinear dynamics.

LLT Cumulants and Graph Coloring

The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the Macdonald cumulant has an explicit positive expansion in terms of LLT cumulants of ribbon shapes, generalizing the classical decomposition of Macdonald polynomials. We also find a natural combinatorial interpretation of the LLT cumulant of a given directed graph as a weighted generating function of colorings of its subgraphs.We use this graph theoretical framework to prove various positivity results. This includes monomial positivity, positivity in fundamental quasisymmetric functions and related positivity of the coefficients of Schur polynomials indexed by hook shapes. We also prove $e$-positivity for vertical-shape LLT cumulants, after the shift of variable $q \to q+1$, which refines a recent result of Alexandersson and Sulzgruber. All these results give evidence towards Schur-positivity of LLT cumulants, which we conjecture here. We prove that this conjecture implies Schur-positivity of Macdonald cumulants, and we give more evidence by proving the conjecture for LLT cumulants of melting lollipops that refines a recent result of Huh, Nam and Yoo.

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

Algebraic reasoning in years 5 and 6: classifying its emergence and progression using reverse fraction tasks

This paper builds on our previous research and investigates how students’ fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This included three reverse fraction tasks which required students to find an unknown whole when presented with a quantity representing a fraction of that whole. Seventeen students from one participating primary school undertook a semi-structured interview which included reverse fraction tasks, similar to those on the written test, but with progressive levels of abstraction, starting with particular instances and becoming more generalised. Two important products of the study are the Classification Framework for Reverse Fraction Tasks and the Emerging Algebraic Reasoning Framework. The interview results highlight two critical transition points for the emergence of students’ algebraic reasoning. The first is the ability to transition from additive strategies to multiplicative strategies to solve reverse fraction problems. Students reliant on diagrams and additive strategies struggled to solve more generalised tasks that required multiplicative rather than additive strategies. The second transition is the shift from multiplicative thinking to algebraic reasoning where students could generalise their multiplicative knowledge to deal with any quantity represented in a reverse fraction task.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

Codebase release 3.4 for NetKet

We introduce version 3 of NetKet, the machine learning toolbox for many-body quantum physics. NetKet is built around neural quantum states and provides efficient algorithms for their evaluation and optimization. This new version is built on top of JAX, a differentiable programming and accelerated linear algebra framework for the Python programming language. The most significant new feature is the possibility to define arbitrary neural network ansätze in pure Python code using the concise notation of machine-learning frameworks, which allows for just-in-time compilation as well as the implicit generation of gradients thanks to automatic differentiation. NetKet 3 also comes with support for GPU and TPU accelerators, advanced support for discrete symmetry groups, chunking to scale up to thousands of degrees of freedom, drivers for quantum dynamics applications, and improved modularity, allowing users to use only parts of the toolbox as a foundation for their own code.

On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain h-adic quantum vertex algebra $${\mathcal {V}}_c (R)$$ via the Etingof–Kazhdan construction, while, in the braided case, they produce ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $${\mathcal {V}}_c (R)$$ , as well as the invariants of such ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $${\mathcal {V}}_c (R)$$ .

NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems

We introduce version 3 of NetKet, the machine learning toolbox for many-body quantum physics. NetKet is built around neural quantum states and provides efficient algorithms for their evaluation and optimization. This new version is built on top of JAX, a differentiable programming and accelerated linear algebra framework for the Python programming language. The most significant new feature is the possibility to define arbitrary neural network ansätze in pure Python code using the concise notation of machine-learning frameworks, which allows for just-in-time compilation as well as the implicit generation of gradients thanks to automatic differentiation. NetKet 3 also comes with support for GPU and TPU accelerators, advanced support for discrete symmetry groups, chunking to scale up to thousands of degrees of freedom, drivers for quantum dynamics applications, and improved modularity, allowing users to use only parts of the toolbox as a foundation for their own code.

An algebraic framework for structured epidemic modelling.

Pandemic management requires that scientists rapidly formulate and analyse epidemiological models in order to forecast the spread of disease and the effects of mitigation strategies. Scientists must modify existing models and create novel ones in light of new biological data and policy changes such as social distancing and vaccination. Traditional scientific modelling workflows detach the structure of a model—its submodels and their interactions—from its implementation in software. Consequently, incorporating local changes to model components may require global edits to the code base through a manual, time-intensive and error-prone process. We propose a compositional modelling framework that uses high-level algebraic structures to capture domain-specific scientific knowledge and bridge the gap between how scientists think about models and the code that implements them. These algebraic structures, grounded in applied category theory, simplify and expedite modelling tasks such as model specification, stratification, analysis and calibration. With their structure made explicit, models also become easier to communicate, criticize and refine in light of stakeholder feedback.This article is part of the theme issue ‘Technical challenges of modelling real-life epidemics and examples of overcoming these’.

Processes against tests: On defining contextual equivalences

In this paper, we would like to offer and defend a template to study equivalences between programs—in the particular framework of process algebras for concurrent computation. We believe that our layered model of development will clarify the distinction that is too often left implicit between the tasks and duties of the programmer and of the tester. It will also enlighten pre-existing issues that have been running across process algebras such as the calculus of communicating systems, the π-calculus—also in its distributed version—or mobile ambients. Our distinction starts by subdividing the notion of process in three conceptually separated entities, that we call process terms, (completed) processes and tests, and by stressing the importance of formalizing the completion of process terms and the instrumentation that results from placing a (completed) process into a test. While the role of what can be observed and the subtleties in the definitions of congruences have been intensively studied, the fact that not every term can be tested, and that the tester should have access to a different set of tools than the programmer is curiously left out, or at least not often formally discussed–in this respect, the theory of monitor is a counter-example that we discuss and compare to our approach. We argue that this blind spot comes from the under-specification of contexts—environments in which comparisons occur—that play multiple distinct roles but are generally—at least, on the surface of it—given only one definition that fails to capture all of their aspects.

Fractionalization of subsystem symmetries in two dimensions

The fractionalization of global symmetry charges is a striking hallmark of topological quantum order. Here, we discuss the fractionalization of subsystem symmetries in two-dimensional topological phases. In line with previous no-go arguments, we show that subsystem symmetry fractionalization is not possible in many cases due to the additional rigid geometric structure of the symmetries. However, we identify a new mechanism that allows fractionalization, involving global relations between macroscopically many symmetry generators. We find that anyons can fractionalize such relations, meaning that the total charge carried under all generators involved in the global relation is non-trivial, despite the fact that these generators multiply to the identity. We first discuss the general algebraic framework needed to characterize this new type of fractionalization, and then explore this framework using a number of exactly solvable models with $\mathbb{Z}_2$ topological order, including models having line and fractal symmetries. These models all showcase another necessary property of subsystem symmetry fractionalization: fractionalized anyons must have restricted mobility when the symmetry is enforced, such that they are confined to a single line or point in the case of line and fractal symmetries, respectively. Looking forward, we expect that our identification of the importance of global relations in fractionalization will hold significance for the classification of phases with subsystem symmetries in all dimensions.

An Algebraic Evaluation Framework for a Class of Car-Following Models

Car-following models describe how a driver follows the leading vehicle in the same lane. They serve as the cornerstone of microscopic traffic-flow simulations and play an essential role in analyzing human factors in traffic casualty, congestion, efficiency, and emissions. An extensive and continuously growing number of car-following models in the literature raises the requirement to evaluate and compare different models objectively. Generally, a car-following model is evaluated after model parameter calibration: the optimal residual between the calibrated model output and the measured counterpart is used as a metric to assess a car-following model’s performance. However, model parameter calibration, usually formed as a numerical optimization problem, suffers from several issues, such as local optimality and heavy computational burden. More importantly, different formulations of the cost function can lead to distinct calibration outcomes and contradictory conclusions of the model evaluation results. This paper proposes instead a purely algebraic framework for evaluating a class of car-following models whose parameters can be linearly identified. Car-following models with nonlinear relationships among parameters, e.g., the behavioral car-following models, are out of the scope of analysis in this paper. Algebraic manipulations performed on a model finally produce a system error index, which is a uniform metric for evaluating and comparing different car-following models. During the whole process, no cost function needs to be designed a priori, and no computationally expensive numerical optimization is involved. Three car-following models are evaluated and compared under the proposed algebraic framework.

Graph Spectral Regularized Tensor Completion for Traffic Data Imputation

In intelligent transportation systems (ITS), incomplete traffic data due to sensor malfunctions and communication faults, seriously restricts the related applications of ITS. Recovering missing data from incomplete traffic data becomes an important issue for ITS. Existing works on traffic data imputation cannot achieve satisfactory accuracy due to inefficiently exploiting the underlying topological structure of the traffic data. In this paper, we model the topology of the road network as a graph and introduce graph Fourier transform (GFT) to process the traffic data. Then we utilize an algebraic framework termed as graph-tensor singular value decompositions (GT-SVD) to extract the hidden spatial information of traffic data. Furthermore, we propose a novel graph spectral regularized tensor completion algorithm based on GT-SVD and construct temporal regularized constraints to improve the recovery accuracy. The extensive experimental results on real traffic datasets demonstrate that the proposed algorithm outperforms the state-of-the-art methods under different missing patterns.

Aggregation and definition of an algebraic framework for fuzzy time series: An application in the supply-demand domain

This paper proposes a method for aggregating the information contained in sets of Time Series (TS) into a Fuzzy Time Series (FTS). First, an aggregation technique is defined, which is based on the algorithm known as Kernel Density Estimation (KDE) which reconstructs the probabilistic density function of a set of points, in this case a TS. Second, to operate with FTS, an algebraic framework is created based on Zadeh's extension principle and as a result of operating with FTS, a new FTS is obtained, which allows obtaining richer information and operating under conditions of uncertainty. Finally, the operations needed to compute the membership function of any TS in the aggregated FTS are introduced too. As a use case, it is proposed to work with sets of TS in the supply and demand domain, in such a way that information regarding the satisfaction of demand over time can be extracted. The specific application domain chosen will be that of the electricity market, analysing the consumption of buildings and their self-generation of energy to obtain information about the dependence on the electricity grid.

A Unifying Algebraic Framework for Discontinuous Galerkin and Flux Reconstruction Methods Based on the Summation-by-Parts Property

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136–155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.

New interpretations of the higher Stasheff–Tamari orders

In 1996, Edelman and Reiner defined the two higher Stasheff–Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type A, denoted And. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of And, namely as equivalence classes of d-maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of 2d-dimensional triangulations as sets of non-intersecting d-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of d-simplices produces a (2d+1)-dimensional triangulation.

A variety of algebras closely related to subordination algebras

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.