Algebraic Expansion Research Articles

A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions

We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.

Extended kinematical 3D gravity theories

In this work, we classify all extended and generalized kinematical Lie algebras that can be obtained by expanding the so\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{so} $$\\end{document} (2, 2) algebra. We show that the Lie algebra expansion method based on semigroups reproduces not only the original kinematical algebras but also a family of non- and ultra-relativistic algebras. Remarkably, the extended kinematical algebras obtained as sequential expansions of the AdS algebra are characterized by a non-degenerate bilinear invariant form, ensuring the construction of a well-defined Chern-Simons gravity action in three spacetime dimensions. Contrary to the contraction process, the degeneracy of the non-Lorentzian theories is avoided without extending the relativistic algebra but considering a bigger semigroup. Using the properties of the expansion procedure, we show that our construction also applies at the level of the Chern-Simons action.

Resurgent aspects of applied exponential asymptotics

AbstractIn many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.

Bäcklund transformations of multi-component Boussinesq and Degasperis–Procesi equations

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Bäcklund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Bäcklund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Bäcklund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis–Procesi (DP) equation, and construct a Lax pair of it. In addition, the Bäcklund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a [Formula: see text]-DP equation. Considering the superposition principle, we obtain the solution of an associated [Formula: see text]-DP equation by using Bäcklund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena.

Counterfactuals as modal conditionals, and their probability

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.

HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method

Abstract A derivative-based uncertainty quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. Hypercomplex automatic differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities are used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices using algebraic expansions. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

Nonrelativistic spin-3 symmetries in 2+1 dimensions from expanded and extended Nappi-Witten algebras

We show that infinite families of Galilean spin-3 symmetries in $2+1$ dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, nonrelativistic Maxwell, and nonrelativistic AdS-Lorentz algebras, can be obtained as Lie algebra expansions of two different spin-3 extensions of the Nappi-Witten symmetry. These higher-spin Nappi-Witten algebras, in turn, are obtained by means of In\"on\"u-Wigner contractions applied to suitable direct product extensions of $\mathfrak{s}\mathfrak{l}(3,\mathbb{R})$. Conversely, we show that the same result can be obtained by considering contractions of expanded $\mathfrak{s}\mathfrak{l}(3,\mathbb{R})$ algebras. The method can be used to define nonrelativistic higher-spin Chern-Simon gravity theories in $2+1$ dimensions in a systematic way.

Three-dimensional Newtonian gravity with cosmological constant and torsion

In this paper we present an alternative cosmological extension of the three-dimensional extended Newtonian Chern–Simons gravity by switching on the torsion. The theory is obtained as a non-relativistic limit of an enhancement and U(1)-enlargement of the so-called teleparallel algebra and can be seen as the teleparallel analogue of the Newtonian gravity theory. The infinite-dimensional extension of our result is also explored through the Lie algebra expansion method. An infinite-dimensional torsional Galilean gravity model is presented which in the vanishing cosmological constant limit reproduces the infinite-dimensional extension of the Galilean gravity theory.

Development of explicit algebraic models of Reynolds stresses for flows in channels using gene expression programming

To build a data-driven approximation for the Reynolds-stress anisotropy (RSA), the symbolic regression method of gene expression programming (GEP) is applied. Two tensor-basis terms from the algebraic expansion for the RSA tensor are used in the GEP algorithm. A new RANS-GEP model is tested in several flows in channels without/with bumps at different physical and geometrical parameters, where DNS data of high fidelity involved as a target for RSA are available. The results of RANS-DNS runs are also obtained where the RSA values in the mean momentum equation are taken directly from DNS to show ability to improve the model performance versus the conventional linear eddy viscosity model (LEVM). Next, the training with carefully selected input features is performed to get an explicit non-linear algebraic model for RSA. The results of RANS-GEP study show potentials of the new tool to improve predictions.

A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.

Extending the nonrelativistic string AdS coset

Inspired by Lie algebra expansion, we consider an extension of the algebra introduced in A. Fontanella and S. J. van Tongeren [Coset space actions for nonrelativistic strings, J. High Energy Phys. 06 (2022) 80.] for the nonrelativistic string coset action in ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$. We show that the extended algebra admits a nondegenerate inner product which is adjoint invariant under the full extended algebra. Furthermore, we provide a finite-dimensional representation of the extended algebra.

Lewis–Riesenfeld invariants for PT-symmetrically coupled oscillators from two-dimensional point transformations and Lie algebraic expansions

We construct Lewis–Riesenfeld invariants from two-dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic sp(4) Lie algebra. This allows for an alternative systematic approach to find Lewis–Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time-dependent Dyson maps from their respective time-independent counterparts in the reference system.

Companions of fields of rational and real algebraic numbers

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories

In this paper, we present novel and known non-relativistic and ultra-relativistic spin-3 algebras, by considering the Lie algebra expansion method. We start by applying the expansion procedure using different semigroups to the spin-3 extension of the AdS algebra, leading to spin-3 extensions of known non-relativistic and ultra-relativistic algebras. We then generalize the procedure considering an infinite-dimensional semigroup, which allows to obtain a spin-3 extension of two new infinite families of the Newton-Hooke type and AdS Carroll type. We also present the construction of the gravity theories based on the aforementioned algebras. In particular, the expansion method based on semigroups also allows to derive the (non-degenerate) invariant bilinear forms, ensuring the proper construction of the Chern-Simons gravity actions. Interestingly, in the vanishing cosmological constant limit we recover the spin-3 extensions of the infinite-dimensional Galilean and infinite-dimensional Carroll gravity theories.

Minimizing Binary Decision Diagrams for Systems of Incompletely Defined Boolean Functions Using Algebraic Cofactor Expansions

Minimizing Binary Decision Diagrams for Systems of Incompletely Defined Boolean Functions Using Algebraic Cofactor Expansions

Highly dispersive optical solitons in birefringent fibers for complex-Ginzburg–Landau equation with parabolic law of nonlinearity using two integration techniques

Object:This paper aims to deduce optical soliton solutions of the highly dispersive optical solitons with perturbed complex-Ginzburg–Landau equation having parabolic law nonlinear refractive index in birefringent fibers. Methods:The addendum Kudryashov’s method and algebraic sub-equation expansion method are used for solving the given system. Results:Combo soliton solutions, bright soliton solutions and singular soliton solutions as well as rational and periodic wave solutions have been deduced.

ALGEBRAIC EXPANSIONS OF LOGICS

AbstractAn algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$ . For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of abelian $\ell $ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

Asymptotics of the Mittag-Leffler function $$E_a(z)$$ on the negative real axis when $$a \rightarrow 1$$

We consider the asymptotic expansion of the one-parameter Mittag-Leffler function \(E_a(-x)\) for \(x\rightarrow +\infty \) as the parameter \(a\rightarrow 1\). The dominant expansion when \(0<a<1\) consists of an algebraic expansion of \(O(x^{-1})\) (which vanishes when \(a=1\)), together with an exponentially small contribution that approaches \(e^{-x}\) as \(a\rightarrow 1\). Here we concentrate on the form of this exponentially small expansion when a approaches the value 1. Numerical examples are presented to illustrate the accuracy of the expansion so obtained.

Three-dimensional non-relativistic supergravity and torsion

In this paper we present a torsional non-relativi-stic Chern–Simons (super)gravity theory in three spacetime dimensions. We start by developing the non-relativistic limit of the purely bosonic relativistic teleparallel Chern–Simons formulation of gravity. On-shell the latter yields a non-Riemannian setup with non-vanishing torsion, which, at non-relativistic level, translates into a non-vanishing spatial torsion sourced by the cosmological constant. Then we consider the three-dimensional relativistic {mathcal {N}}=2 teleparallel Chern–Simons supergravity theory and obtain its non-relativistic counterpart by exploiting a Lie algebra expansion method. The non-relativistic supergravity theory is characterized, on-shell, by a non-vanishing spatial super-torsion, again sourced by the cosmological constant.

Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion

The extension from the planar case to three dimensions of the clothoid curve (Euler spiral) is herein presented, that is, a curve parametrised by arc length, whose curvature and torsion are linear (affine) functions of the arc length. The problem is modelled as a linear time variant system and its stability is studied with Lyapunov techniques. Closed form solutions in terms of the standard Fresnel integrals are for the first time presented and they are valid for a wide family of clothoids; for the remaining cases, numerical methods of order four, based on Lie algebra, Magnus Expansions and Commutator-Free Expansions are provided. These geometric integrators have been optimised to be easily implemented with few lines of code and require only matrix-vector products (avoiding explicit matrix exponentials at each iteration). The achieved computational efficiency has a dramatic impact for several real-time applications, especially for designing trajectories for flying vehicles, as previous approaches were fully numeric and could not take advantage of the present closed form solutions, which render the computation of the 3D clothoid as computationally expensive as its planar companion.