Polynomial Space Research Articles

Bianchi period polynomials: Hecke action and congruences

Let Γ=PSL2(O)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma ={{\\,\\mathrm{\ extrm{PSL}}\\,}}_2(\\mathcal {O})$$\\end{document} be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is “dual” to the space of weight k modular symbols for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, reflecting the duality between the first and second cohomology groups of the arithmetic group Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. Using this result, we describe the action of Hecke operators on the space of period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

Finite element simulations of double diffusion in a staggered cavity filled with a power-law fluid

Double diffusion refers to a phenomenon where two different components of a fluid (such as heat and mass) exhibit distinct diffusive behaviors. In this study, we employ finite element-based numerical simulations to investigate the phenomenon of double diffusion in a non-Newtonian fluid within a staggered cavity. Mathematically, this system can be understood by coupling the two-dimensional continuity, momentum, energy, and concentration equations. Since the governing equations have been written in a dimensionless form, Galerkin's finite element method is used to find a solution. The velocity profile and temperature are calculated in a function space of quadratic polynomials (P2), while the pressure is calculated in a linear (P1) finite element function space. Discrete systems of nonlinear algebraic equations are resolved through the implementation of Newton's method with appropriate damping and PARDISO solver in the inner loops for solving the sparse linear systems. In this work, the data are presented graphically in the form of streamlines, isotherms, iso-concentration, average Nusselt numbers, average Sherwood numbers, and kinetic energy distribution. Code validation and grid independence study are also provided. Moreover, convective mass transfer is significantly correlated with the Lewis number, as demonstrated by the results. As the power-law index increases, convection also exhibits enhanced as a means of transmitting heat and mass.

The R-matrix presentation for the rational form of a quantized enveloping algebra

Let Uq(g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g, and let V be the corresponding type 1 finite-dimensional irreducible representation of Uq(g). Starting from this data, the R-matrix formalism for quantum groups outputs a Hopf algebra URλ(g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of Uq(g) by adding a single invertible quantum Cartan element. We simultaneously establish that URλ(g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining URλ(g), with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End(V).

A space–time DG method for the Schrödinger equation with variable potential

We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

Hermite interpolation of type total degree associated with certain spaces of polynomials

Abstract We study Hermite interpolation for the space of polynomials of total degree in ℝ N and the space of homogeneous polynomials in ℝ N+1. We investigate the relations between the two types of Hermite interpolation. We show that they have the same regularity and continuity property.

A family of [formula omitted] four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties

The four-point interpolatory scheme and the cubic B-spline are examples of the most well-known stationary subdivision procedures. They are based on the space of cubic polynomials and have their respective strengths and weaknesses. In this regard, the purpose of this study is to introduce a new type of subdivision scheme that integrates the advantages of both the four-point and the cubic B-spline schemes. The proposed scheme achieves approximation order ‘four’ and an improved smoothness C2, while keeping the same support of the basic limit function as the four-point scheme. Moreover, under some mild conditions, the new scheme has the properties of monotonicity and convexity preservation. Whereas most high-order shape-preserving schemes are non-linear and rather computationally complicated, the proposed scheme is linear and stationary. Several numerical examples are provided to illustrate the efficiency of our subdivision algorithm.

Minimal Roman Dominating Functions: Extensions and Enumeration

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman dominating functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document} and a function f:V→{0,1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:V\\rightarrow \\{0,1,2\\}$$\\end{document}, is there a minimal Roman dominating function f~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ilde{f}$$\\end{document} with f≤f~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\le \ ilde{f}$$\\end{document}? Here, ≤\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le $$\\end{document} lifts 0<1<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< 1< 2$$\\end{document} pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of O(1.9332n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(1.9332^n)$$\\end{document} for graphs of order n; this is complemented by a lower bound example of Ω(1.7441n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (1.7441^n)$$\\end{document}.

Numerical approximation of the boundary control for the wave equation in a square domain with a spectral collocation method

We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document} in space. We prove that we can choose a sequence of controls fN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f^N$$\\end{document} associated with the approximate control problem in such a way that they converge, as N→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\rightarrow \\infty $$\\end{document}, to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods.

On the convergence of 2D P4+$$ {P}_{4+} $$ triangular and 3D P6+$$ {P}_{6+} $$ tetrahedral divergence‐free finite elements

AbstractWe show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the ‐ mixed finite element method for on 2D triangular grids or on tetrahedral grids, even in the case the inf‐sup condition fails. By a simple ‐projection of the discrete pressure to the space of continuous polynomials, we show this post‐processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.

Generalised local polynomial estimators of smooth functionals of a distribution function with nonnegative support

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for x ∈ [ 0 , ∞ ) , and for a functional, Φ ( x , F ) of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space L 2 ( q ) , where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.

Geometric properties of the lattice of polynomials with integer coefficients

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let \(r\), \(n\) be positive integers with \(n \ge 6r\). Let \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) be the space of polynomials of degree at most \(n\) on \([0,1]\) with integer coefficients such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\) and let \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) from \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) in \(L_2(0,1)\). We give rather precise quantitative estimations for successive minima of \(\boldsymbol{P}_n^\mathbb{Z}\) in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\).

On Polynomial Space Curves with Flc-frame

The first and second derivatives of a curve provide us fundamental&#x0D; information in the study of the behavior of curve near a point. However,&#x0D; if a curve is a polynomial space curve of degree n, we don’t know what&#x0D; is the geometric meaning of the n-th derivative of the curve? There is no&#x0D; doubt that the Frenet frame is not suitable for this purpose because it is&#x0D; constructed by using first and second derivatives of a curve. On the other&#x0D; hand, in this paper by using a new frame called as Flc-frame we are able&#x0D; to give the geometric meaning of the n-th derivative of a curve. Moreover,&#x0D; we explore some basic concepts regarding polynomial space curves from&#x0D; point of view of Flc-frame in three dimensional Euclidean space.

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree [Formula: see text] and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces [Formula: see text] of real monic univariate polynomials of degree [Formula: see text] whose real divisors avoid sequences of root multiplicities, taken from a given poset [Formula: see text] of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of [Formula: see text] and of some related topological spaces. We find explicit presentations for the groups [Formula: see text] in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in [Formula: see text]. We also show that [Formula: see text] stabilizes for [Formula: see text] large. The mechanism that generates [Formula: see text] has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree [Formula: see text] polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups [Formula: see text] admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder [Formula: see text], whose images avoid the tangency patterns from [Formula: see text] with respect to the generators of the cylinder.

Partitions, multiple zeta values and the q-bracket

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as q→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\rightarrow 1$$\\end{document}, we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be ‘lifted’ to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular.

Exact algorithms for restricted subset feedback vertex set in chordal and split graphs

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The task is to determine whether there exists a subset S⊆V∖T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP-complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal and split graphs can be solved in O(1.1550|V|) time and exponential space or in O(1.1605|V|) time and polynomial space, significantly improving all previous results. As a by-product, we show that the Maximum Independent Set problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to Vertex Cover, we obtain an O⁎(1.2738k)-time parameterized algorithm and a tight O(k2)-kernel for R-SFVS in chordal and split graphs.

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [ 7 ], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [ 25 ] and Kedlaya &amp; Umans [ 16 ] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most \(d^{o(1)}\) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e., \(n \notin d^{o(1)}\) ) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field \(\mathbb {F}_{q}\) of characteristic p , which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time \(\begin{equation*} \left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, n, p)\right), \end{equation*}\) provided that p is at most d o (1) , and q is at most (exp (exp (exp (...(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: — We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [ 21 ] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. — We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [ 26 ]. More precisely, for every fixed prime p , we show that for every constant ɛ &gt; 0, and large enough n , the rank of any \(n \times n\) Vandermonde matrix V over the field \(\mathbb {F}_{p^a}\) can be reduced to ( n /exp (Ω (poly(ɛ)log 0.53 n ))) by changing at most n Θ (ɛ) entries in every row of V , provided a ≤ poly(log n ). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [ 9 ].

A 𝐶1-𝑃7 Bell Finite Element on Triangle

Abstract We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in L 2 L^{2} -norm. By eliminating all degrees of freedom on edges of C 1 C^{1} - P 7 P_{7} Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27 ⁢ V 27V to 12 ⁢ V 12V asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C 1 C^{1} - P 6 P_{6} Argyris finite element is 19 ⁢ V 19V , the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

In this paper, we are interested in algorithms that take in input an arbitrary graph G, and that enumerate in output all the (inclusion-wise) maximal “subgraphs” of G which fulfil a given property Π. All over this paper, we study several different properties Π, and the notion of subgraph under consideration (induced or not) will vary from a result to another.More precisely, we present efficient algorithms to list all maximal split subgraphs, maximal induced cographs and maximal threshold graphs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal stable sets of an auxiliary graph. For cographs and threshold graphs, the algorithms rely on a framework recently introduced by Conte & Uno (2019) called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every non-trivial hereditary property Π. We extend the hardness result to some specific edge version of the extension problem.

Probabilistic temporal logic with countably additive semantics

This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.

ℤ-gradings on the Grassmann algebra over infinite fields: Graded identities and central polynomials

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of E equipped with its [Formula: see text] and 3-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.