Relational Algebra Research Articles

Stabilizing Decomposition of Multiparameter Persistence Modules

Abstract While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We introduce new tools, in particular $$\epsilon $$ ϵ -refinements and $$\epsilon $$ ϵ -erosion neighborhoods, to start building such a theory. We then define the $$\epsilon $$ ϵ -pruning of a module, which is a new invariant acting like a “refined barcode” that shows great promise to extract features from a module by approximately decomposing it. Our main theorem can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of 2r, where r is the maximal pointwise dimension of one of the modules. Furthermore, we show that the factor 2r is close to optimal. Finally, we discuss the possibility of strengthening the stability theorem for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the computational complexity of approximating the interleaving distance, and recent applications of relative homological algebra to multipersistence.

Bialgebras and related algebras for Novikov-Poisson algebras

In this paper, we develop the bialgebra theory for Novikov-Poisson algebras and establish the equivalence between matched pairs, Manin triples and Novikov-Poisson bialgebras. We study coboundary Novikov-Poisson bialgebras which leads to the introduction of the Novikov-Poisson Yang-Baxter equation (NPYBE). A skew-symmetric solution of the NPYBE naturally gives a coboundary Novikov-Poisson bialgebra. Moreover, O -operators and related algebras are considered.

The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

Functional L1-Lp inequalities in the CAR algebra

In the present paper, we use the semigroup method to investigate various functional inequalities invoking L1 and Lp norms in the framework of canonical anti-commuting relations algebra (CAR algebra for short). As the main results, we obtain the Poincaré inequality for Talagrand type sum, Eldan-Gross inequality for projections, and the Talagrand influence inequality along with its strengthening form in the CAR algebra. All our results strengthen the noncommutative Poincaré inequality of Efraim and Lust-Piquard at several points. We conclude the paper with two applications of our inequalities. In the first application, we apply the noncommutative Eldan-Gross inequality to derive two KKL-type inequalities in the CAR algebra, which are closely related to the quantum KKL conjecture of Montanaro and Osborne. The second application is the CAR algebra counterpart of the superconcentration phenomenon derived from the noncommutative Talagrand influence inequality.

Extreme learning with projection relational algebraic secured data transmission for big cloud data

SummaryCloud Computing (CC) and big data are growing technology in the business. Big data is demonstrated in terms of volume, variety, and velocity. CC is employed for storing, processing, and accessing data. Many cryptographic techniques have been developed to enhance big data security in cloud computing. However, security and privacy are the primary concerns in protecting data, as it is highly sensitive. Yet, it faces the major problems of inefficient performance, increased time consumption, and lack of data confidentiality and integrity. To address this issue, proposed Extreme Learning with Projection Relational Algebraic Secured Data Transmission (ELPRA‐SDT) is introduced to secure data transactions from cloud users to cloud servers with enhanced data confidentiality and reduced time consumption for big cloud data. The proposed ELPRA‐SDT consists of two major processes namely registration and key generation. At first, the user's IP address is registered employing a transitive advanced set relation theory graph model in a cloud server (CS) for retrieving the numerous services. The CS generates private and public keys for each registered user's IP address using the Transitive Operational and Time Synchronized Random Winternitz Key generation model. After, the user sends a request to the CS for acquiring data. The CS validates the requested user based on security policy attributes. Second, the Projection Relational Algebraic Signcryption and Unsigncryption algorithm performs signature verification to ensure secure data access for protecting the data. Results of experiments carried out by using Coburg Intrusion Detection Data Sets‐001 dataset in Java. ELPRA‐SDT method is more efficient and more suitable for providing security and privacy to network traces in the Cloud. The result shows maximum performance with data confidentiality by 10% and data integrity by 13%. In addition, delay is reduced by 32%, and data delivery time and communication complexity is decreased by 28% and 24% to other existing methods.

The Duck’s Brain

Although database systems perform well in data access and manipulation, their relational model hinders data scientists from formulating machine learning algorithms in SQL. Nevertheless, we argue that modern database systems perform well for machine learning algorithms expressed in relational algebra. To overcome the barrier of the relational model, this paper shows how to transform data into a coordinate relational representation for training neural networks in SQL: We first describe building blocks for data transformation, model training and inference in SQL-92 and their counterparts using an extended array data type. Then, we compare the implementation for model training and inference using array data types to the one using a coordinate relational representation in SQL-92 only. The evaluation in terms of runtime and memory consumption proves the suitability of modern database systems for matrix algebra, although specialised array data types perform better than matrices in coordinate relational representation.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

From free fields to interacting SCFTs via representation theory

We ask when it is possible to construct arbitrary unitary multiplets of the superconformal algebra with eight Poincaré supercharges that are compatible with locality from (continuous deformations of) representations in free field theories. We answer this question in two, three, and five dimensions. In four dimensions, we resort to an intricate but self-consistent web of conjectures. If correct, these conjectures imply various new non-perturbative constraints on short multiplets in any local unitary 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal field theory and on an unusual set of related vertex algebras. Throughout, we connect our results with properties of deformations in the space of theories.

Relation-Algebraic Verification of Disjoint-Set Forests

This paper studies how to use relation algebras, which are useful for high-level specification and verification, for proving the correctness of lower-level array-based implementations of algorithms. We give a simple relation-algebraic semantics of read and write operations on associative arrays. The array operations seamlessly integrate with assignments in computation models supporting while-programs. As a result, relation algebras can be used for verifying programs with associative arrays. We verify the correctness of an array-based implementation of disjoint-set forests using the union-by-rank strategy and find operations with path compression, path splitting and path halving. All results are formally proved in Isabelle/HOL. This paper is an extended version of [1].

Gaussian Markov Random Fields over graphs of paths and high relative accuracy

The present paper presents some results that allow us to perform with High Relative Accuracy linear algebra operations with correlation and covariance matrices of Gaussian Markov Random Fields over graphs of paths. Some numerical experiments are carried out showing the computational benefits of this approach.

Matching relative Rota-Baxter algebras, matching dendriform algebras and their cohomologies

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo, and Zhang motivated by the study of algebraic renormalization of regularity structures. In this paper, we introduce matching relative Rota-Baxter algebras that are closely related to matching dendriform algebras. We define the cohomology of a matching relative Rota-Baxter algebra using the classical Hochschild cohomology and a new cohomology induced by the matching operators. As an application, we show that our cohomology governs the formal deformation theory of the matching relative Rota-Baxter algebra. Finally, we define the cohomology of a matching dendriform algebra and study homotopy matching dendriform algebras.

Nearly associative algebras

We present a comprehensive study of nearly associative algebras. Our research shows that these algebras are power associative Lie-admissible for which the related Lie algebra is solvable, and are also Jordan-admissible.Furthermore, we describe the features of finite-dimensional nearly associative algebras that are nilpotent. In addition, we define the concepts of radical and semisimplicity for this type of algebras. We give a characterization of semisimple nearly associative algebras and prove the Wedderburn Principal Theorem for them.Lastly, we deal with quadratic nearly associative algebras. We provide a characterization and then an inductive description of them through the process of double extension, enriching our understanding of the structural intricacies of this class of algebras.

Koszul Complexes and Relative Homological Algebra of Functors Over Posets

AbstractUnder certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

3-pre-Leibniz Algebras, Deformations and Cohomologies of Relative Rota-Baxter Operators on 3-Leibniz Algebras

In this paper, first we introduce the notions of 3-pre-Leibniz algebras and relative Rota-Baxter operators on 3-Leibniz algebras. We show that a 3-pre-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the identity map is a relative Rota-Baxter operator. Conversely, a relative Rota-Baxter operator naturally induces a 3-pre-Leibniz algebra. Then we construct a Lie 3-algebra, and characterize relative Rota-Baxter operators as its Maurer-Cartan elements. Consequently, we obtain the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra that controls deformations of relative Rota-Baxter operators on 3-Leibniz algebras. Next we define the cohomology of relative Rota-Baxter operators on 3-Leibniz algebras and show that infinitesimal deformations of a relative Rota-Baxter operator are classified by the second cohomology group. Finally, we construct an L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra whose Maurer-Cartan elements are relative Rota-Baxter 3-Leibniz algebra structures, and define the cohomology of relative Rota-Baxter 3-Leibniz algebras.

Modal algebra of multirelations

Abstract We formalize the modal operators from the concurrent dynamic logics of Peleg, Nerode and Wijesekera in a multirelational algebraic language based on relation algebras and power allegories, using relational approximation operators on multirelations developed in a companion article. We relate Nerode and Wijesekera’s box operator with a relational approximation operator for multirelations and two related operators that approximate multirelations by different kinds of deterministic multirelations. We provide an algebraic soundness proof of Goldblatt’s axioms for concurrent dynamic logic and one for a multirelational Hoare logic based on Nerode and Wijesekera’s box as applications.

TypeQL: A Type-Theoretic & Polymorphic Query Language

Relational data modeling can often be restrictive as it provides no direct facility for modeling polymorphic types, reified relations, multi-valued attributes, and other common high-level structures in data. This creates many challenges in data modeling and engineering tasks, and has led to the rise of more flexible NoSQL databases, such as graph and document databases. In the absence of structured schemas, however, we can neither express nor validate the intention of data models, making long-term maintenance of databases substantially more difficult. To resolve this dilemma, we argue that, parallel to the role of classical predicate logic for relational algebra, contemporary foundations of mathematics rooted in type theory can guide us in the development of powerful new high-level data models and query languages. To this end, we introduce a new polymorphic entity-relation-attribute (PERA) data model, grounded in type-theoretic principles and accessible through classical conceptual modeling, with a near-natural query language: TypeQL. We illustrate the syntax of TypeQL as well as its denotation in the PERA model, formalize our model as an algebraic theory with dependent types, and describe its stratified semantics.

Determinism of multirelations

Binary multirelations allow modelling alternating nondeterminism, for instance, in games or nondeterministically evolving systems interacting with an environment. Such systems can show partial or total functional behaviour at both levels of alternation, so that nondeterministic behaviour may occur only at one level or both levels, or not at all. We study classes of inner and outer partial and total functional multirelations in a multirelational language based on relation algebra and power allegories. While it is known that general multirelations do not form a category, we show in the multirelational language that the classes of deterministic multirelations mentioned form categories with respect to Peleg composition from concurrent dynamic logic, and sometimes quantaloids. Some of these categories are isomorphic to the category of binary relations. We also introduce determinisation maps that approximate multirelations either by binary relations or by deterministic multirelations. Such maps are useful for defining modal operators on multirelations.

Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ, which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L∞[1]-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ, the corresponding twisted L∞[1]-algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version.

TorchQL: A Programming Framework for Integrity Constraints in Machine Learning

Finding errors in machine learning applications requires a thorough exploration of their behavior over data. Existing approaches used by practitioners are often ad-hoc and lack the abstractions needed to scale this process. We present TorchQL, a programming framework to evaluate and improve the correctness of machine learning applications. TorchQL allows users to write queries to specify and check integrity constraints over machine learning models and datasets. It seamlessly integrates relational algebra with functional programming to allow for highly expressive queries using only eight intuitive operators. We evaluate TorchQL on diverse use-cases including finding critical temporal inconsistencies in objects detected across video frames in autonomous driving, finding data imputation errors in time-series medical records, finding data labeling errors in real-world images, and evaluating biases and constraining outputs of language models. Our experiments show that TorchQL enables up to 13x faster query executions than baselines like Pandas and MongoDB, and up to 40% shorter queries than native Python. We also conduct a user study and find that TorchQL is natural enough for developers familiar with Python to specify complex integrity constraints.

QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.