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Wavelets and multiwavelets have lately been adopted in quantum chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few), they provide a framework that narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for multiwavelet calculations (algebra and integral and differential operator application) and exposes the required functionality to write a simple Python code to solve, among others, the Hartree–Fock equations, the generalized Poisson equation, the Dirac equation, and the time-dependent Schrödinger equation up to any predefined precision. In this study, we will outline the main features of multiresolution analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms. [ABSTRACT FROM AUTHOR]

We work within the framework of the variational bicomplex: we define A ∞ -algebra structures on horizontal and vertical cohomologies of (formally integrable) partial differential equations with the help of Merkulov's theorem. Since higher order A ∞ -algebra operations are related to Massey products, our observation implies the existence of invariants for differential equations that go beyond conservation laws. We also propose notions of formality for PDEs, and we present examples of formal equations. [ABSTRACT FROM AUTHOR]

We generalise two facts about finite‐dimensional algebras to finite‐dimensional differential graded algebras. The first is the Nakayama lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality, respectively. As an application, we prove a corepresentability result for finite‐dimensional differential graded algebras. [ABSTRACT FROM AUTHOR]

A Gelfand–Dorfman algebra is called special if it can be embedded into a differential Poisson algebra. We find a new basis of the free Novikov algebra. With its help, we construct the monomial basis of the free special Gelfand–Dorfman algebra. [ABSTRACT FROM AUTHOR]

We look at classes of semiassociative algebras, with an emphasis on those that canonically generalize associative (generalized) cyclic algebras, and at their behaviour in the semiassociative Brauer monoid defined by Blachar, Haile, Matzri, Rein, and Vishne. A possible way to generalize this monoid in characteristic p that includes nonassociative differential algebras is briefly considered. [ABSTRACT FROM AUTHOR]

We introduce a cohomology theory that classifies differential objects that arise from Picard-Vessiot theory, using differential Hopf-Galois descent. To do this, we provide an explicit description of Picard-Vessiot theory in terms of differential torsors. We then use this cohomology to give a bijective correspondence between differential objects and differential torsors. As an application, we prove a universal bound for the differential splitting degree of differential central simple algebras. [ABSTRACT FROM AUTHOR]

This study examines the interactions between healthy target cells, infected target cells, virus particles, and immune cells within an HIV model. The model exhibits two equilibrium points: an infection-free equilibrium and an infection equilibrium. Stability analysis shows that the infection-free equilibrium is locally asymptotically stable when R 0 < 1 . Further, it is unstable when R 0 > 1 . The infection equilibrium is locally asymptotically stable when R 0 > 1 . The structural and practical identifiabilities of the within-host model for HIV infection dynamics were investigated using differential algebra techniques and Monte Carlo simulations. The HIV model was structurally identifiable by observing the total uninfected and infected target cells, immune cells, and viral load. Monte Carlo simulations assessed the practical identifiability of parameters. The production rate of target cells (λ) , the death rate of healthy target cells (d) , the death rate of infected target cells (δ) , and the viral production rate by infected cells (π) were practically identifiable. The rate of infection of target cells by the virus (β) , the death rate of infected cells by immune cells (Ψ) , and antigen-driven proliferation rate of immune cells (b) were not practically identifiable. Practical identifiability was constrained by the noise and sparsity of the data. Analysis shows that increasing the frequency of data collection can significantly improve the identifiability of all parameters. This highlights the importance of optimal data sampling in HIV clinical studies, as it determines the best time points, frequency, and the number of sample points required to accurately capture the dynamics of the HIV infection within a host. [ABSTRACT FROM AUTHOR]

By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout. [ABSTRACT FROM AUTHOR]

We prove that if a system of first-order partial differential equations in one variable has a nonzero Kolchin dimension polynomial, then its leading coefficient is equal to 1. The notion of typical differential dimension plays an important role in differential algebra. Some of its estimations were proved by J. Ritt and E. Kolchin; they also advanced several conjectures that were later refuted. There are bounds for the typical differential dimension in codimension 1 (E. Kolchin) and in the case of linear differential equations (D. Grigoriev). Note that in codimension 2 for systems of linear differential equations in one indeterminate the well-known B´ezout theorem holds, and in the case of several variables, we have earlier proved its analogue, which is not satisfied in higher codimensions. For nonlinear systems, in the general case, there are no exponential bounds yet (although it is known that the growth of the typical dimension is bounded by the Ackermann function). [ABSTRACT FROM AUTHOR]

In this paper, we establish a bialgebraic structure on a Coder pair of any weight λ (i.e., a coassociative coalgebra with a coderivation of weight λ ), which is consistent with the differential antisymmetric infinitesimal bialgebra in (SIGMA 19:018, 2023) when the weight λ = 0 . [ABSTRACT FROM AUTHOR]

AbstractLet K be a differential field with algebraically closed field of constants CK. Let KPV∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X(KPV∞) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X(KPV∞)=X(Kdiff) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K. [ABSTRACT FROM AUTHOR]

In pharmacometric modeling, it is often important to know whether the data is sufficiently rich to identify the parameters of a proposed model. While it may be possible to assess this based on the results of a model fit, it is often difficult to disentangle identifiability issues from other model fitting and numerical problems. Furthermore, it can be of value to ascertain identifiability beforehand. This paper compares four methods for parameter identifiability, namely Differential Algebra for Identifiability of SYstems (DAISY), the sensitivity matrix method (SMM), Aliasing, and the Fisher information matrix method (FIMM). We discuss the characteristics of the methods and apply them to a set of applications, consisting of frequently used PK model structures, with suitable dosing regimens and sampling times. These applications were selected to validate the methods and demonstrate their usefulness. While traditional identifiability analysis provides a categorical result [PLoS One, 6, 2011, e27755; CPT Pharmacometrics Syst Pharmacol, 8, 2019, 259; Bioinformatics, 30, 2014, 1440], we argue that in practice a continuous scale better reflects the limitations on the data and is more informative. The methods were generally consistent in their evaluation of the applications. The Fisher information matrix method seemed to provide the most consistent answers. All methods provided information on the parameters that were unidentifiable. Some of the results were unexpected, indicating identifiability issues where none were foreseen, but could be explained upon further analysis. This illustrated the usefulness of identifiability assessment. [ABSTRACT FROM AUTHOR]

This paper considers a coupled hybrid thermostat system driven by the Ã-Caputo fractional derivative in a Banach algebra. We employ a version of Darbo’s fixed-point theorem combined with the measure of noncompactness (MNC) technique to establish certain existence results. Additionally, an example is provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A , m ¯) such that the maximal ideal m ¯ ⊆ H 0 (A) can be generated by an A -regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. [ABSTRACT FROM AUTHOR]

We study rank functions on a triangulated category 풞 via its abelianisation mod ⁡ C . We prove that every rank function on 풞 can be interpreted as an additive function on mod ⁡ C . As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod ⁡ C . We study the connection between rank functions and functors from 풞 to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T c for a compactly generated triangulated category 풯, this connection becomes particularly nice, providing a link between rank functions on 풞 and smashing localisations of 풯. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in 풯. Finally, if C = per ⁡ (A) for a differential graded algebra 퐴, we classify homological epimorphisms A → B with per ⁡ (B) locally finite via special rank functions which we call idempotent. [ABSTRACT FROM AUTHOR]

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov [ABSTRACT FROM AUTHOR]

Let |$(M,I,J,K,\Omega)$| be a compact HKT manifold, and let us denote with |$\partial$| the conjugate Dolbeault operator with respect to I , |$\partial_J:=J^{-1}\overline\partial J$|⁠ , |$\partial^\Lambda:=[\partial,\Lambda]$|⁠ , where Λ is the adjoint of |$L:=\Omega\wedge-$|⁠. Under suitable assumptions, we study Hodge theory for the complexes |$(A^{\bullet,0},\partial,\partial_J)$| and |$(A^{\bullet,0},\partial,\partial^\Lambda)$| showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT |$\mathrm{SL}(n,\mathbb{H})$| -manifold, the differential graded algebra |$(A^{\bullet,0},\partial)$| is formal and this will lead to an obstruction for the existence of an HKT |$\mathrm{SL}(n,\mathbb{H})$| structure |$(I,J,K,\Omega)$| on a compact complex manifold (M ,  I). Finally, balanced HKT structures on solvmanifolds are studied. [ABSTRACT FROM AUTHOR]

Let be a nonassociative finite-dimensional simple Novikov algebra over an algebraically closed field of characteristic . Then the right multiplication algebra is a differential simple algebra with respect to some derivation . The algebra is isomorphic to a Novikov algebra for some operator of right multiplication by and multiplication is given by . Moreover, the algebra is a truncated polynomial algebra. [ABSTRACT FROM AUTHOR]

We prove that dg manifolds of finite positive amplitude, that is, bundles of positively graded curved |$L_{\infty }[1]$| -algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved |$L_{\infty }[1]$| -algebras. As an application, we study the derived intersections of manifolds. [ABSTRACT FROM AUTHOR]

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space’s algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. [ABSTRACT FROM AUTHOR]

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem L y = λ y , for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ , defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L − λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. [ABSTRACT FROM AUTHOR]

The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models. [ABSTRACT FROM AUTHOR]

We prove that there exists a version of Weil descent, or Weil restriction, in the category of D -algebras. The objects of this category are k -algebras R equipped with a homomorphism e : R → R ⊗ k D for some fixed field k and finite-dimensional k -algebra D. We do this under a mild assumption on the so-called associated endomorphisms. In particular, this yields the existence of the Weil descent functor in the category of difference algebras, which, to our knowledge, does not appear elsewhere. [ABSTRACT FROM AUTHOR]

We prove that after inverting the Planck constant $h$ , the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$. [ABSTRACT FROM AUTHOR]

Let be a compact Kähler Lagrangian in a holomorphic symplectic variety X / C . We use deformation quantisation to show that the endomorphism differential graded algebra RHom (i ∗ K L 1 / 2 , i ∗ K L 1 / 2 ) is formal. We prove a generalisation to pairs of Lagrangians, along with auxiliary results on the behaviour of formality in families of A ∞ -modules. [ABSTRACT FROM AUTHOR]

Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking a uniform approach, this paper presents an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly stronger restraints among the replicated copies. The linear compatibility is in Koszul duality to the total compatibility, while the matching compatibility is self dual. Further, each compatibility condition can be expressed in terms of either one or both of the two Manin square products. Finally it is shown that the operads defined by these compatibility conditions from the associative algebra and differential algebra are Koszul utilizing rewriting systems. [ABSTRACT FROM AUTHOR]

Procedural flexibility is an important skill for algebra. Although prior work has focused on measuring students' procedural flexibility using arithmetic problems, word problems may also capture students' flexibility because of their open-ended nature. To date, no published study has examined the use of word problems as another measure of procedural flexibility. The present study aims to establish flexibility on fraction word problems as a predictor of algebra learning and to determine whether flexibility demonstrated on fraction arithmetic versus fraction word problems differentially predicts two types of algebra learning (e.g., algebraic feature knowledge and algebra equation-solving). Middle-school students (N = 350) completed fraction arithmetic problems, fraction word problems, and algebra measures at the start and end of the school year. We coded fraction arithmetic and word problems for demonstrated procedural flexibility. Path models showed that overall flexibility significantly predicted both algebraic feature knowledge and equation-solving. When examining flexibility by type, procedural flexibility on arithmetic problems significantly predicted end-of-year algebraic feature knowledge and algebra equation-solving, whereas procedural flexibility on word problems significantly predicted algebraic feature knowledge. Procedural flexibility on arithmetic problems was still predictive of performance when accounting for fraction magnitude skill. These findings build on prior literature that establishes procedural flexibility as a crucial predictor of algebra learning and extends measures of procedural flexibility to include word problems. Educational Impact and Implications Statement: We found that students' ability to appropriately choose between problem-solving strategies, or their procedural flexibility, was predictive of their knowledge of algebraic features when measured by flexibility on both fraction arithmetic and word problems. Furthermore, students' overall procedural flexibility was predictive of their algebraic equation-solving skills. These results suggest that procedural flexibility with fractions is an important predictor of algebra learning and can be measured both through arithmetic and word problem measures. Practically, results suggest that procedural flexibility is a skill that should be given more consideration in classroom instruction, an area that may be explored in future research. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept and representations of modified λ-differential Hom-Lie algebras. We then develop the cohomology of modified λ-differential Hom-Lie algebras with coefficients in a suitable representation. As applications, abelian extensions and skeletal modified λ-differential Hom-Lie 2-algebras are characterized in terms of cohomology groups. [ABSTRACT FROM AUTHOR]

Frequency security is the premise of realizing the net‐zero transition. A novel assessment scheme is proposed to quantify the frequency security levels of transmission systems under different inverter‐based resources (IBRs) and their control parameter combinations. A novel system frequency dynamic model is proposed as a parametric optimization method, where the saturation of generators, energy storage systems, and renewable energy sources is incorporated as differential algebra equations. This problem is further reformulated as a mixed‐integer linear programming problem to generate a sufficient amount of data under different IBR integration and control parameters. The frequency security assessment problem is formulated as a data‐driven multivariate classification problem, which is solved by the optimal classification tree (OCT) algorithm with better interptretionability. Simulations are conducted on a transmission system. Numerical results indicate that the proposed system frequency dynamic model can capture the frequency dynamic under different IBR reinforcement plans and the OCT can realize accurate classification of the synthetic data regarding frequency security. [ABSTRACT FROM AUTHOR]

We propose a novel discretization procedure for the classical Euler equation based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define algorithmically a new discrete model that inherits from the continuous Euler equation a class of exact solutions. [ABSTRACT FROM AUTHOR]

The development of construction theory of generalized and ordinary Hilbert series for Sibirsky graded algebras of comitants and invariants of autonomous polynomial differential systems is examined. The mentioned series are of particular importance for some problems of qualitative theory of differential systems. These researches have their beginnings in the works of Professor M. N. Popa and his disciples. But the construction of these series for more complicated differential systems encounter insurmountable computational difficulties, especially, for the generalized Hilbert series, from which easily are obtained the ordinary Hilbert series. In this paper it is shown how using the adaptation of the Residue Theorem from the paper [3] to overshoot to the mentioned problem, an ordinary Hilbert series were obtained for Sibirsky graded algebras of comitants and invariants for some rather complicated differential systems. [ABSTRACT FROM AUTHOR]

Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra |${{\mathcal {V}}}$|⁠ , we have a surjective homomorphism of differential algebras |$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$|⁠ ; equivalently, the singular support of |${{\mathcal {V}}}$| is a closed subscheme of the arc space of the associated scheme |$X_{{{\mathcal {V}}}}$|⁠. We give many new examples of classically free vertex algebras (i.e. this map is an isomorphism), including |$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$| for all positive integers |$n$| and |$k$|⁠. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular |${{\mathcal {W}}}$| -algebra of |${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$| at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for the preprocessing of observation data aimed at improving the robustness of orbit determination tools. Two objectives are fulfilled: obtain a refined solution to the initial orbit determination problem and detect possible outliers in the processed measurements. The uncertainty on the initial estimate is propagated forward in time and progressively reduced by exploiting sensor data available in said propagation window. Differential algebra techniques and a novel automatic domain splitting algorithm for second-order Taylor expansions are used to efficiently propagate uncertainties over time. A multifidelity approach is employed to minimize the computational effort while retaining the accuracy of the propagated estimate. At each observation epoch, a polynomial map is obtained by projecting the propagated states onto the observable space. Domains that do no overlap with the actual measurement are pruned thus reducing the uncertainty to be further propagated. Measurement outliers are also detected in this step. The refined estimate and retained observations are then used to improve the robustness of batch orbit determination tools. The effectiveness of the algorithm is demonstrated for a geostationary transfer orbit object using synthetic and real observation data from the TAROT network. • Uncertainty on the initial orbit determination is reduced using other sensor data. • Measurement outliers are detected and isolated. • Multifidelity methods are used for fast and accurate uncertainty propagation. • Robustness of batch orbit determination tools is improved. [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of a transposed Poisson superalgebra.We prove that a transposed Poisson superalgebra can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Making use of this, we construct two examples of the transposed Poisson superalgebra. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum Yang--Mills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra. [ABSTRACT FROM AUTHOR]

Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the description of generalized gauge symmetries using two specific examples. The first example deals with the non-commutative gauge symmetry obtained using Drinfel'd twist of the symmetry Hopf algebra. The homotopy Lie algebra compatible with the twisted gauge symmetry turns out to be the recently proposed braided L ∞ -algebra. In the second example, we focus on the generalized gauge symmetry of the double field theory. The symmetry includes both diffeomorphisms and gauge transformation and can consistently be defined using a curved L ∞ -algebra. [ABSTRACT FROM AUTHOR]

The paper provides an explicit solution to the well-known problem of differentiation of hyperelliptic functions with respect to parameters of the corresponding hyperelliptic curve. [ABSTRACT FROM AUTHOR]

In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and C2-algebras. We also introduce the corresponding notions of modules, and investigate the relations between the different module categories. [ABSTRACT FROM AUTHOR]

The Molien-Weyl integral formula and the Hilbert-Poincaré series have proven to be powerful mathematical tools in relation to gauge theories, allowing to count the number of gauge invariant operators. In this paper we show that these methods can also be employed to construct Free Differential Algebras and, therefore, reproduce the associated pure supergravity spectrum and nonperturbative objects. Indeed, given a set of fields, the Hilbert-Poincaré series allows to compute all possible invariants and consequently derive the cohomology structure. [ABSTRACT FROM AUTHOR]

The successful application of epidemic models hinges on our ability to estimate model parameters from limited observations reliably. An often-overlooked step before estimating model parameters consists of ensuring that the model parameters are structurally identifiable from the observed states of the system. In this tutorial-based primer, intended for a diverse audience, including students training in dynamic systems, we review and provide detailed guidance for conducting structural identifiability analysis of differential equation epidemic models based on a differential algebra approach using differential algebra for identifiability of systems (DAISY) and Mathematica (Wolfram Research). This approach aims to uncover any existing parameter correlations that preclude their estimation from the observed variables. We demonstrate this approach through examples, including tutorial videos of compartmental epidemic models previously employed to study transmission dynamics and control. We show that the lack of structural identifiability may be remedied by incorporating additional observations from different model states, assuming that the system’s initial conditions are known, using prior information to fix some parameters involved in parameter correlations, or modifying the model based on existing parameter correlations. We also underscore how the results of structural identifiability analysis can help enrich compartmental diagrams of differential-equation models by indicating the observed state variables and the results of the structural identifiability analysis. [ABSTRACT FROM AUTHOR]

It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. In [4] a counterpart of this result was studied in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of differential matrix algebras divide their degrees. [ABSTRACT FROM AUTHOR]

Let A be a proper non-positive dg algebra over a field k. For a simple-minded collection of the finite-dimensional derived category D f d (A) , we construct a "dual" silting object of the perfect derived category per (A) by using the Koszul duality for dg algebras. This induces a one-to-one correspondence between the equivalence classes of silting objects in per (A) and algebraic t-structures of D f d (A) . [ABSTRACT FROM AUTHOR]

Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace–Beltrami operator has no L 2 -eigenvalues ≥ 1 / 4 . In this article we prove a generalization of this result for the joint L 2 -eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces Γ \ G / K of higher rank. We derive dynamical assumptions on the Γ -action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups. [ABSTRACT FROM AUTHOR]

We show that free algebras of the variety of algebras generated by the Witt algebra W n , the left-symmetric Witt algebra L n , and the symplectic Poisson algebra P n can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that W n , L n , P n , and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian. [ABSTRACT FROM AUTHOR]

We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field k is always split by a quadratic extension of k. However, a differential quaternion algebra need not be split over any algebraic extension of k. We use solutions of certain Riccati equations to provide bounds on the transcendence degree of splitting fields of a differential quaternion algebra. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right. [ABSTRACT FROM AUTHOR]