Algebraic Framework Research Articles

Termination of Graph Transformation Systems Using Weighted Subgraph Counting

We introduce a termination method for the algebraic graph transformation framework PBPO+, in which we weigh objects by summing a class of weighted morphisms targeting them. The method is well-defined in rm-adhesive quasitoposes (which include toposes and therefore many graph categories of interest), and is applicable to non-linear rules. The method is also defined for other frameworks, including SqPO and left-linear DPO, because we have previously shown that they are naturally encodable into PBPO+ in the quasitopos setting. We have implemented our method, and the implementation includes a REPL that can be used for guiding relative termination proofs.

Conditionally monotone cumulants via shuffle algebra

In this work, we study conditional monotone cumulants and additive convolution in the shuffle-algebraic approach to non-commutative probability. We describe c-monotone cumulants as an infinitesimal character and identify the c-monotone additive convolution as an associative operation in the set of pairs of characters in the dual of a double tensor Hopf algebra. In this algebraic framework, we understand previous results on c-monotone cumulants and prove a combinatorial formula that relates c-free and c-monotone cumulants. We also identify the notion of [Formula: see text]-Boolean cumulants in the shuffle-algebraic approach and introduce the corresponding notion of [Formula: see text]-monotone cumulants as a particular case of c-monotone cumulants.

Correction: Path Enumeration Solution for Evaluating Reliability, Based on the Path Algebra Framework

Correction: Path Enumeration Solution for Evaluating Reliability, Based on the Path Algebra Framework

Braided Scalar Quantum Electrodynamics

AbstractWe formulate scalar electrodynamics in the braided ‐algebra formalism and study its perturbative expansion in the algebraic framework of Batalin–Vilkovisky quantization. We also confirm that UV/IR mixing is absent at one‐loop order in this noncommutative field theory, and that the non‐anomalous Ward‐Takahashi identities for the braided gauge symmetry are satisfied.

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.

Volichenko-type metasymmetry of braided Majorana qubits

Abstract This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, "mixed-bracket" Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the Volichenko algebras defined in 1990 by Leites-Serganova as metasymmetries which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators.
In a previous paper braided Z2-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at 
roots of unity and realizes, for a given integer s=2,3,4, ..., an interpolation between ordinary Majorana fermions (recovered at s=2) and bosons (recovered in the s →∞ limit); it implements a parastatistics where at most s-1 indistinguishable particles are accommodated in a multi-particle sector.
The structures discussed in this work are:
- the quantum group interpretation of the roots of unity truncations recovered from a (superselected) set of reps of the quantum superalgebra Uq(osp(1|2));
- the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products
(in a minimal representation, the N-particle sector of the braided Majorana qubits is described by 2N x 2N;
- the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras;
- the s→∞ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators;
- (meta)symmetries of ordinary differential equations given by matrix Schrödinger equations in 0+1 dimension induced by the braided creation/annihilation operators;
- in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system
as a ternary algebra.

On compact extensions of tracial $W^{*}$-dynamical systems

We establish several classification results for compact extensions of tracial W^{*} -dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg–Zimmer type dichotomies on the L^{2} -level. Although in general an adequate generalization of the Furstenberg–Zimmer tower of intermediate compact extensions does not seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host–Kra–Ziegler tower for any ergodic action of a countable abelian group.

Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

Analyzing cyclobutane: A vibrational hamiltonian approach using dynamical U(2) Lie Algebras

This study uses a one-dimensional Lie algebraic framework, precisely adapted to the symmetry of molecules, to investigate the vibrational frequencies of cyclobutane. A vibrational Hamiltonian is also constructed using one-dimensional Morse oscillators, preserving the D2d point group symmetry for each molecule. The analysis compares the calculated fundamental vibrational frequencies with the observed experimental frequencies, resulting in a root mean square deviation of 0.886 cm-1. The result highlights the exceptional precision of the U(2) Lie algebraic Hamiltonian in accurately predicting the vibrational frequencies and their combination bands at the sub-cm-1 level of precision. Specifically, this computational approach has the potential to achieve these results with lower computational costs compared to traditional theoretical approaches. The broader implications of our study suggest that the U(2) Lie algebraic framework can be effectively applied to a wide range of molecular systems, offering significant advantages in fields such as material science, drug design, and environmental monitoring by providing precise and efficient vibrational spectral analyses. KEY WORDS: Vibrational Hamiltonian, Vibrational frequencies, Morse oscillator, Cyclobutane, Lie algebraic method Bull. Chem. Soc. Ethiop. 2024, 38(6), 1887-1896. DOI: https://dx.doi.org/10.4314/bcse.v38i6.29

Vibrational Frequencies of Tetrachloroethylene using Lie Algebraic Framework

This study employs a sophisticated computational approach to predict tetrachloroethylene's higher overtone vibrational frequencies (C2Cl4) up to the fourth overtone. We utilize a Lie algebraic framework within the context of the vibrational Hamiltonian. The method involves replacing tetrachloroethylene's carbon-chlorine (C-Cl) bonds with unitary Lie algebras. The resulting Hamiltonian is written in terms of Casimir and Majorana's invariant operators and parameters. The derived Hamiltonian operator effectively characterizes the stretching vibrations inherent to the molecular structure. This approach enhances our understanding of the vibrational dynamics of tetrachloroethylene at higher overtones, offering valuable insights for applications in diverse scientific and technological fields.

A fine-grained RDF graph model for fuzzy spatiotemporal data

The uncertainty and spatiotemporal dynamics of information necessitate the urgent modeling of fuzzy spatiotemporal knowledge across various applications, with the Resource Description Framework (RDF) serving as a widely recognized data representation model. Existing models suffer from incomplete semantic representation and poor robustness in modeling fuzzy spatiotemporal data, e.g., lack of fuzziness in spatiotemporal semantics; lack of altitude in spatial semantics. Meanwhile, the algebraic framework regarding the model has not been investigated. Thus, in this paper, we first propose a new fine-grained fuzzy spatiotemporal RDF model. This model can represent fine-grained uncertain spatiotemporal semantics that may be associated with any element of a spatiotemporal RDF. We further define its graph algebraic operations. Note that we demonstrate the use of the algebraic operations for fuzzy spatiotemporal RDF querying. Finally, we establish a set of transformation rules for SPARQL query syntax to algebraic operations in fuzzy spatiotemporal RDF. In addition, we used experiments to evaluate the validity and rationality of our model.

Yang Model Revisited

A long time ago, C.N. Yang proposed a generalization of the Snyder model to the case of a curved background spacetime, based on an algebra isomorphic to o(1, 5) which includes, as subalgebras both the Snyder and the de Sitter algebras. His proposal can, therefore, be interpreted as a model of noncommutative curved spacetime, and could be useful for relating physics on very small and very large scales. We review this model and some recent progress concerning its generalizations and its interpretation in the framework of Hopf algebras. We also report some possibilities to relate it to more phenomenological aspects.

Path Enumeration Solution for Evaluating Reliability, Based on the Path Algebra Framework

Path Enumeration Solution for Evaluating Reliability, Based on the Path Algebra Framework

Polynomial solutions to the first order difference equations in the bivariate difference field

The bivariate difference field provides an algebraic framework to study sequences satisfying a recurrence of order two, and it can be used to transform summations involving such sequences into first order difference equations over the bivariate difference field. Based on this, we present an algorithm for finding a series of polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree of any possible polynomial solutions, which in turn is sufficient to compute all polynomial solutions by using the method of undetermined coefficients.

Analysis of Complex Entities in Algebra B

We present an exploration of algebra B, a recently published unital and non-associative algebra. Unique complex entities emerged from this algebra, distinct from both quaternion and complex number systems, which we termed treons. We defined the treonic number system and established an isomorphism between this system and the real vector space. Our findings revealed that the treonic representation maintained structural integrity under the defined operations. Based on this foundation, we proceed to determine Euler identity for algebra B within the treonic system. By presenting the fundamental definitions and properties of algebra B, we derived a generalized version of Euler identity applicable within this algebra. This formula revealed the emergence of hyperbolic trigonometric entities, extending the applicability of Euler identity beyond traditional complex numbers. Our results provide a theoretical foundation for a deeper understanding of the properties and behaviors of complex entities within this expanded algebraic framework, thus enabling new theoretical developments and practical applications in the realms of advanced mathematics and theoretical physics.

Efficient analysis of belief properties in process algebra

Protocols are typically specified in an operational manner by specifying the communication patterns among the different involved principals. However, many properties are of epistemic nature, e.g., what each principal believes after having seen a run of the protocol. We elaborate on a unified algebraic framework suitable for epistemic reasoning about operational protocols. This reasoning framework is based on a logic of beliefs and allows for the operational specification of untruthful communications. The information recorded in the semantic models to support reasoning about the interaction between the operational and epistemic aspects intensifies the state-space explosion. We propose an efficient on-the-fly reduction for such a unifying framework by providing a set of operational rules. These operational rules automatically generate efficient reduced semantics for a class of epistemic properties, specified in a rich extension of modal μ-calculus with past and belief modality, and can potentially reduce an infinite state space into a finite one. We reformulate and prove criteria that guarantee belief consistency for credulous agents, i.e., agents that are ready to believe what is told unless it is logically inconsistent. We adjust our reduction so that the belief consistency of an original model is preserved. We prove the soundness and completeness result for the specified class of properties.

A Formal Concept Analysis approach to hierarchical description of malware threats

The problem of intelligent malware detection has become increasingly relevant in the industry, as there has been an explosion in the diversity of threats and attacks that affect not only small users, but also large organisations and governments. One of the problems in this field is the lack of homogenisation or standardisation in the nomenclature used by different antivirus programs for different malware threats. The lack of a clear definition of what a category is and how it relates to individual threats makes it difficult to share data and extract common information from multiple antivirus programs. Therefore, efforts to create a common naming convention and hierarchy for malware are important to improve collaboration and information sharing in this field.Our approach uses as a tool the methods of Formal Concept Analysis (FCA) to model and attempt to solve this problem. FCA is an algebraic framework able to discover useful knowledge in the form of a concept lattice and implications relating to the detection and diagnosis of suspicious files and threats. The knowledge extracted using this mathematical tool illustrates how formal methods can help prevent new threats and attacks. We will show the results of applying the proposed methodology to the identification of hierarchical relationships between malware.

Elastostatics within multi-layer metamaterial structures and an algebraic framework for polariton resonances

Multi-layer structures are ubiquitous in constructing metamaterial devices to realise various frontier applications including super-resolution imaging and invisibility cloaking. In this paper, we develop a general mathematical framework for studying elastostatics within multi-layer material structures in Rd, d = 2, 3. The multi-layer structure is formed by concentric balls and each layer is filled by either a regular elastic material or an elastic metamaterial. The number of layers can be arbitrary and the material parameters in each layer may be different from one another. In practice, the multi-layer structure can serve as the building block for various material devices. Considering the impingement of an incident field on the multi-layer structure, we first derive the exact perturbed field in terms of an elastic momentum matrix, whose dimension is the same as the number of layers. By highly intricate and delicate analysis, we derive a comprehensive study of the spectral properties of the elastic momentum matrix. This enables us to establish a handy algebraic framework for studying polariton resonances associated with multi-layer metamaterial structures, which forms the fundamental basis for many metamaterial applications.

Volumetric nonwoven structures: An algebraic framework for systematic design of infinite polyhedral frames using nonwoven fabric patterns

In this paper, we present an algebraic framework that can be used to construct a large class of 3D shapes and structures that can potentially provide unusual material properties. We formalized this framework as a 3D generalization of planar nonwoven textile structures that are used to mimic the woven structures. Our extension is based on the fact that it is straightforward to extend planar nonwoven textile structures into volumetric nonwoven textile structures, which we also call nonwoven volumetric fabrics. This property is essential because such an extension is impossible with planar woven structures. In other words, using this approach, it can be possible to easily produce volumetric structures that mimic the fabric behavior as if they were planar nonwoven textile structures, which is impossible to produce. These volumetric structures also correspond to regular & semiregular frame structures and are capable of representing previously unknown infinite regular polyhedra and flexible wood structures.