Algebraic Relations Research Articles

Categorical Bernstein operators and the Boson-Fermion correspondence

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov’s Heisenberg category $${\mathcal {H}}$$ and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of $${\mathcal {H}}$$ is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of $${\mathcal {H}}$$ by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of $${\mathcal {H}}$$ .

Topological Basis Realization Associated with Spin-1 Non-Hermitian XXZ Model

We investigate the spin-1 topological basis realization of a non-Hermitian Heisenberg XXZ chain model. For the non-Hermitian case, the Hamiltonian and the topological ground states of the system are constructed by Birman-Murakami-Wenzl algebra. The results show that all the spin singlet states belong to the topological ground states. The topological subspace and the spin singlet subspace are equivalent. The topological basis can reduce Birman-Murakami-Wenzl algebra generators, and the reduced matrices satisfy reduced Birman-Murakami-Wenzl algebra relations.

Topological Phase Transition in the Non-Hermitian Coupled Resonator Array.

In a two-dimensional non-Hermitian topological photonic system, the physics of topological states is complicated, which brings great challenges for clarifying the topological phase transitions and achieving precise active control. Here, we prove the topological phase transition exists in a two-dimensional parity-time-symmetric coupled-resonator optical waveguide system. We reveal the inherent condition of the appearance of topological phase transition, which is described by the analytical algebraic relation of coupling strength and the quantity of gain-loss. In this framework, the system can be switched between the topological and trivial states by pumping the site rings. This work provides a new degree of freedom to control topological states and offers a scheme for studying non-Hermitian topological photonics.

Unifying semantic foundations for automated verification tools in Isabelle/UTP

The growing complexity and diversity of models used in the engineering of dependable systems implies that a variety of formal methods, across differing abstractions, paradigms, and presentations, must be integrated. Such an integration relies on unified semantic foundations for the various notations, and co-ordination of a variety of automated verification tools. The contribution of this paper is Isabelle/UTP, an implementation of Hoare and He's Unifying Theories of Programming, a framework for unification of formal semantics. Isabelle/UTP permits the mechanisation of computational theories for diverse paradigms, and their use in constructing formalised semantic models. These can be further applied in the development of verification tools, harnessing Isabelle's proof automation facilities. Several layers of mathematical foundations are developed, including lenses to model variables and state spaces as algebraic objects, alphabetised predicates and relations to model programs, including algebraic and axiomatic semantics, proof tools for Hoare logic and refinement calculus, and UTP theories to encode computational paradigms.

Maximum Varma Entropy Distribution with Conditional Value at Risk Constraints.

It is well known that Markowitz’s mean-variance model is the pioneer portfolio selection model. The mean-variance model assumes that the probability density distribution of returns is normal. However, empirical observations on financial markets show that the tails of the distribution decay slower than the log-normal distribution. The distribution shows a power law at tail. The variance of a portfolio may also be a random variable. In recent years, the maximum entropy method has been widely used to investigate the distribution of return of portfolios. However, the mean and variance constraints were still used to obtain Lagrangian multipliers. In this paper, we use Conditional Value at Risk constraints instead of the variance constraint to maximize the entropy of portfolios. Value at Risk is a financial metric that estimates the risk of an investment. Value at Risk measures the level of financial risk within a portfolio. The metric is most commonly used by investment bank to determine the extent and occurrence ratio of potential losses in portfolios. Value at Risk is a single number that indicates the extent of risk in a given portfolio. This makes the risk management relatively simple. The Value at Risk is widely used in investment bank and commercial bank. It has already become an accepted standard in buying and selling assets. We show that the maximum entropy distribution with Conditional Value at Risk constraints is a power law. Algebraic relations between the Lagrangian multipliers and Value at Risk constraints are presented explicitly. The Lagrangian multipliers can be fixed exactly by the Conditional Value at Risk constraints.

New Results on Stability of Delayed Cohen–Grossberg Neural Networks of Neutral Type

This research work conducts an investigation of the stability issues of neutral-type Cohen–Grossberg neural network models possessing discrete time delays in states and discrete neutral delays in time derivatives of neuron states. By setting a new generalized appropriate Lyapunov functional candidate, some novel sufficient conditions are proposed for global asymptotic stability for the considered neural networks of neutral type. This paper exploits some basic properties of matrices in the derivation of the results that establish a set of algebraic mathematical relationships between network parameters of this neural system. A key feature of the obtained stability criteria is to be independent from time and neutral delays. Therefore, the derived results can be easily tested. Moreover, a constructive numerical example is studied to check the verification of presented global stability conditions.

New set of fractional-order generalized Laguerre moment invariants for pattern recognition

This article presents a new series of invariant moments, called Fractional-order Generalized Laguerre Moment Invariants (FGLMI), based on Fractional-order Generalized Laguerre polynomials (FGLPs). To begin, we provide the relations and the properties necessary to define the fractional-order generalized Laguerre moments. Then, we present the theoretical framework to derive invariants from fractional-order moments with respect to the change in orientation, size and position based on the algebraic relationships between FGLM and fractional-order geometric moments. In addition, a fast and precise algorithm has been proposed for the calculation of FGLM in order to speed up the calculation time and ensure the numerical stability of the invariant moments. Numerical experiments are carried out to demonstrate the efficiency of FGLM and their proposed invariants compared to existing methods, with regard to the reconstruction of 2D and 3D images, the computation time, the global entity extraction capacity and image localization, invariability property and 2D / 3D image classification performance on different 2D and 3D image databases. The theoretical and experimental results presented clearly show the efficiency of the descriptors proposed for the representation and classification of 2D and 3D images by other types of orthogonal moments.

Equivariance and algebraic relations for curves

We generalise a classical argument for deducing algebraic models of Riemann surfaces from the Riemann–Roch theorem. The method involves counting arguments based on equivariant resolutions.

Generic triangular solutions of the reflection equation: case

We consider intertwining relations of the triangular q-Onsager algebra, and obtain generic triangular boundary K-operators in terms of the Borel subalgebras of U q (sl 2). These K-operators solve the reflection equation.

Simple transport models for karst systems

Tracer tests are useful field methods with a wide range of applications in karst aquifers. The measured tracer breakthrough curve at a karst spring allows the estimation of the important transport properties of the karst system. The breakthrough curve is typically characterized by a long tailing and a low tracer recovery that is not captured by the classical one-dimensional advection dispersion model. Partitioning models are often used and require a large number of parameters. In the present work, simplified transport models are proposed to simulate the typical karst breakthrough curve. The model setup consists of a karst conduit that is interacting laterally with the surrounding matrix. The solute transport in both the conduit and matrix are governed by the one-dimensional advection-dispersion equation and the resulting mathematical model consists of a coupled system of differential equations. The Laplace transform method is used to derive the transport models for various flow and boundary conditions. The models are in terms of four physically meaningful parameters that represent the advection and dispersion properties of the flow in the conduit and matrix. Algebraic relationships to estimate the model parameters are also derived in terms of the key features of the breakthrough curves. The main advantage of the proposed models are their simple formulation and ability to capture the long tailing of the breakthrough curve in terms of just a few parameters. Application of the models to two real karst aquifers demonstrate their effectiveness in simulating the observed breakthrough curves and estimating the key transport parameters.

A finite Hopfield neural network model for the oxygenation of hemoglobin

A model of the oxygenation of hemoglobin is developed based in a finite Hopfield neural network. The model is based in two transition probabilities, and between the states of each neuron. The oxygen partial pressure, is introduced by an external field and the cooperativity phenomenon appears as a field this includes the interaction between the four binding sites of the hemoglobin. The results of oxygen saturation as a function of present a sigmoidal curve, similar to oxygen-hemoglobin saturation curves (ODCs) from Severinghaus. These simulated ODCs present properties that are compatible with those from ODCs that are studied in medical practice and scientific research. The curves obtained shift to the left if cooperativity increases, and shift to the right if cooperativity decreases. Other parameters that are obtained are: the Hill coefficient, the at half saturation, the maximum saturation when reaches 100 torr; and the equilibrium constant The Hill curve is satisfactorily fitted with the simulation data. Algebraic relation between and the cooperativity parameter are also found; as well as a relation between and We establish a relation between the network temperature, and the temperature of the medium of the hemoglobin, measured in Celsius. It is found that when increases, and also increase. The change in is found to be directly proportional to this means that alkaline mediums favor cooperative behavior. We also calculate the change in Gibbs free energy, and notice that the values taken are negative and decrease as increases; this indicates that the oxygenation of hemoglobin is a spontaneous reaction. The change in enthalpy, is also calculated for each value of

Gauge theories on \u03ba-Minkowski spaces: twist and modular operators

We discuss the construction of κ-Poincaré invariant actions for gauge theories on κ-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the κ-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the κ-Poincaré invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the κ-Poincaré invariance still holds.

Single‐valued neutrosophic relations and their application to factors affecting oil prices

The recent boom of economic activities has escalated the demand for oil that eventually will affect its prices. However, oil prices are difficult to predict because most of the factors affecting oil prices are vague and intangible. The challenges in predicting oil prices urgently require a novel approach where issues related to multiple-factors, uncertainty, and periodicity can be addressed. The authors propose the single-valued neutrosophic relations based decision-making method inspired by the three memberships of neutrosophic sets, and amplitude and periodicity of complex numbers. The proposed method is applied to the case of oil prices where six factors affecting oil prices and six benchmarks measuring oil prices are employed. This new method combines with complex neutrosophic numbers and algebraic relations, and can suggest the most influential factor that affects oil prices. Considering the periodicity of 24 months, computation results verify the ‘global economic rate’ as the most influential factor that affects the prices of oil. The main contribution of this study is the development of a neutrosophic relations-based decision-making method to suggest the most influential factor that affects oil prices. The result provides evidence on the feasibility of the proposed method in suggesting the influential factors that affect oil prices.

Exact Nonequilibrium Steady State of Open XXZ/XYZ Spin-1/2 Chain with Dirichlet Boundary Conditions.

We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by the Lindblad master equation. The nonequilibrium steady state is expressed in terms of a matrix product ansatz using novel site-dependent Lax operators. The components of Lax operators satisfy a simple set of linear recurrence equations that generalize the defining algebraic relations of the quantum group U_{q}(sl_{2}). We reveal connection between the nonequilibrium steady state of the nonunitary dynamics and the respective integrable model with edge magnetic fields, described by coherent unitary dynamics.

Exponents Associated with Y-Systems and their Relationship with q-Series

Let $X_r$ be a finite type Dynkin diagram, and $\ell$ be a positive integer greater than or equal to two. The $Y$-system of type $X_r$ with level $\ell$ is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair $(X_r, \ell)$, we define an integer sequence called exponents using formulation of the $Y$-system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type $X_r$, and prove this conjecture for $(A_1,\ell)$ and $(A_r, 2)$ cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from $q$-series identities for this relationship.

Elliptic Multizetas and the Elliptic Double Shuffle Relations

Abstract We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in $\mathcal{O}({{\mathfrak{H}}})$, the ring of functions on the Poincaré upper half-plane ${{\mathfrak{H}}}$. The elliptic multizetas generate a ${{\mathbb{Q}}}$-algebra ${{\mathcal{E}}}$, which is an elliptic analogue of the algebra of multiple zeta values. Working modulo $2\pi i$, we show that the algebra ${{\mathcal{E}}}$ decomposes into a geometric and an arithmetic part and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (1) the classical double shuffle relations give all algebraic relations among multiple zeta values and (2) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck–Teichmüller Lie algebra.

Exact discretization of non-commutative space-time

Non-commutative space-time is discussed for discrete case. An exact discretization of the non-commutative space-time is proposed. New discrete operators, which satisfy the same algebraic relations as standard derivatives of integer orders, are used. Exact discretization of the non-commutative space-time is suggested by using generalization of the star-product (the Moyal–Weyl product) based on the recently proposed exact finite differences. The suggested discrete non-commutative space-time corresponds to the standard “continuous” non-commutative space-time without any approximations. The suggested discrete non-commutative space-time can be interpreted as a special physical (crystal) lattice with long-range interaction.

Volume of the set of LOCC-convertible quantum states

The class of quantum operations known as local operations and classical communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipartitions. In the latter case the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most states there exist local strategies that succeed in achieving the conversion with a probability arbitrarily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.

Обробка розвідувальних відомостей з використанням автоматизованих інформаційних систем

У статті розглядається підхід до використання автоматизованих інформаційних систем в якості інструменту оперативної обробки отримуваних розвідувальних відомостей. Отримувані розвідувальні відомості класифікуються за напрямками проведеного аналізу можливих дій противника, виходячи із реального стану його сил і засобів. Ці відомості і оцінки експертів за можливими сценаріями дій противника формалізуються з використанням апарату реляційної алгебри. Такий підхід дозволяє автоматизувати процес аналізу отримуваних розвідувальних відомостей, і дати кількісну оцінку поточної оперативної обстановки, що підвищує обґрунтованість прийнятих рішень у стислі терміни часу.

A New Mathematical Model for the Prediction of Internal Recirculation in Impinging Streams Reactors

A mathematical model for the prediction of internal recirculation of complex impinging stream reactors has been presented. The model constitutes a repetition of a series of ideal plug flow reactors and CSTR reactors with recirculation. The simplicity of the repeating motif allows for the derivation of an algebraic relation of the whole system using the Laplace transform. An impinging streams reactor system with one axial and two tangential inlet fluid streams was constructed and considered as a case study. The model predicts satisfactorily the complex and flow rate dependent experimental residence time distribution functions obtained employing a pulse tracer method for different total flow rates of the incoming feed. The variation of the controlling parameters with changing the total inlet flow rate is discussed. The presented model can predict complex internal recirculation streams within the impinging streams reactor system.