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This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results. [ABSTRACT FROM AUTHOR]

Disaster management requires detailed data from past disasters for policy planning as well as for the generation of disaster exercises and simulations. Post-analysis of disasters is often distributed in official reports, which provide detailed analysis of the events that have happened. However, some information in such reports is often only given implicitly as part of natural language and thus not accessible to classical natural language processing-based text mining. To address this problem, in this paper, we propose to consider the information extraction tasks related to post-disaster report analysis as algebra stories, that can be treated with computer algebra systems together with natural language processing. We applied our enhanced information extraction approach in preliminary experiments to the report of a bushfire in 2009 in Victoria, Australia and used four different tools for solving fire-specific algebra story problems (ASP). Our evaluation shows that these tools have difficulty handling the occurring ASPs. [ABSTRACT FROM AUTHOR]

The paper considers the possible relationship between the local integrability of an autonomous two-dimensional ODE system with polynomial right hand sides and its global integrability. A hypothesis is put forward that local integrability in a neighborhood of each point of some region of the phase space is necessary for the existence of the first integral in this region. By integrability in some domain of the phase space it is meant the existence there of a differentiable function which is constant along the orbit of the system. Using the example of a polynomial resonance case of the Liénard-type equation with parameters, we have written out the conditions for local integrability near stationary points and found restrictions on the parameters under which these conditions are satisfied. The resulting constraint is written as a system of algebraic equations for the ODE parameters. It is shown that for parameter values that are solutions of such an algebraic system, the ODE turns out to be integrable. In this way we have found several cases of integrability. We propose a heuristic method that allows one to a priori determine the cases of integrability of an autonomous ODE with a polynomial right-hand side. The paper has an experimental character. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

Railway interlocking systems are crucial safety components in rail transportation, designed to prevent train collisions by regulating switch positions and signal indications. These systems delineate potential train movements within a railway station by connecting sections into routes, which are further divided into blocks. To ensure safety, the system prohibits the simultaneous allocation of the same block or intersecting routes to multiple trains. In this study, we characterize the 'interlocking problem' as a safety verification task for a single real-time station configuration, rather than a 'command and control' function. This is a matter of verification, not solution, typically managed by an interlocking system that receives movement authority requests. Over the years, we have developed various algebraic models to address this issue, suggesting the potential use of computer algebra systems in implementing interlocking systems. However, some of these models exhibit limitations. In this paper, we propose a novel algebraic model for decision-making in railway interlocking systems that overcomes the limitations of previous approaches, making it suitable for large railway stations. Our primary objective is to offer a mathematical solution to interlocking problems in linear time, which our approach accomplishes. [ABSTRACT FROM AUTHOR]

Decision-making in a railway station regarding the compatibility of the positions of the switches of the turnouts and the indications (proceed/stop) of the railway colour light signals is a safety-critical issue that is considered very labor-intensive. Different authors have proposed alternative solutions to automate its supervision, which is performed by the so-called railway interlocking systems. The classic railway interlocking systems are route-based and their compatibility is predetermined (usually by human experts): only some chosen routes are simultaneously allowed. Some modern railway interlocking systems are geographical and make decisions on the fly, but are unsuitable if the station is very large and the number of trains is high. In this paper, we present a completely new algebraic model for decision-making in railway interlocking systems, based on other computer algebra techniques, that bypasses the disadvantages of the approaches mentioned above (its performance does not depend on the number of trains in the railway station and can be used in large railway stations). The main goal of this work is to provide a mathematical solution to the interlocking problems. We prove that our approach solves it in linear time. Although our approach is interesting from a theoretical perspective, it has a significant limitation: it can hardly be adopted in an actual interlocking implementation, mainly due to the heavy certification requirements for this kind of safety-critical application. Nevertheless, the results may be useful for simulations that do not require certification credit. [ABSTRACT FROM AUTHOR]

In this paper, we describe an educational experience in the context of the Master's degree that is compulsory in Spain to become a secondary education mathematics teacher. Master's students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didactic and mathematical issues related to linkage mechanisms, such as those arising in the 18th and 19th centuries during the development of the steam engine. Afterwards, participants were asked to provide three different kinds of feedback: (i) working on an assigned group task, (ii) individually answering a questionnaire, and (iii) proposing some classroom activity, imagining it would be addressed to their prospective pupils. All three issues focused on the specific topic of the attended lectures. In the framework of Mason's reflective discourse analysis, the information supplied by the participants has been analyzed. The objective was to explore what they have learned from the experience and what their perception is of the potential interest in linkages as a methodological instrument for their future professional activity as teachers. This analysis is then the basis upon which to reflect on the opportunities (and problems) that this particular bar-joint linkages methodological approach could bring towards providing future mathematics teachers with attractive tools that would contribute to enhancing a STEAM-oriented education. Finally, the students' answers allow us to conclude that the experience was beneficial for these pre-service teachers, both in improving their knowledge on linkages history, mathematics, industrial, technological and artistic applications, and in enhancing the use in the classroom of this very suitable STEAM context. [ABSTRACT FROM AUTHOR]

Tensor formalism (and its special case—vector formalism) is a mathematical technique that is widely used in physical and engineering problems. Even though this formalism is fairy universal and suitable for describing many spaces, the application of other special mathematical techniques is sometimes required. For example, the problem of rotation in a 3D space is not very well described in tensor representation, and it is reasonable to use the formalism of Clifford algebra, in particular, quaternions and geometric algebra representations for its solution. In this paper, computer algebra is used to demonstrate the solution of the problem of rotation in a 3D space using both the quaternion and geometric algebra formalisms. It is shown that although these formalisms are fundamentally similar, the latter one seems to be clearer both for computations and interpretation of results. [ABSTRACT FROM AUTHOR]

We presented a new and complete algorithm for detecting isometries and symmetries of implicit algebraic surfaces. First, our method reduced the problem to the case of isometries fixing the origin. Second, using tools from elimination theory and polynomial factoring, we determined the desired isometries between the surfaces. We have implemented the algorithm in Maple to provide evidences of the efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as { 2 , 5 } -inverses) of a matrix, which commute with it. This class of { 2 , 5 } -inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Inspired by a recent paper by Wares, we discuss an extension of another well-known geometric problem involving arcs which, since the solution is computer-assisted, could be a source of a mini-project for an introductory course on a computer algebra system. [ABSTRACT FROM AUTHOR]

This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear Schrödinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. As you can see, we give all the classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation. The obtained results substantially improve or complement the corresponding conditions in the literature (Esen and Sulaiman in Optik 167:150-156, 2018), (Eslami in Appl. Math. Comput. 258:141-148, 2016), (Han et al. in Phys. Lett. 395:127217, 2021). Finally, in order to further explain the propagation of the fractional coupled nonlinear Schrödinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn. [ABSTRACT FROM AUTHOR]

This paper discusses some of the philosophical and historical underpinnings of the talk "The Mathieu Functions: Computational and Historical Perspectives" given at the Maple Conference 2022. I also comment on the role Problem Solving Environments (PSEs) play in curating the computational knowledge of the 19th century, which is so necessary for us to think of special functions as answers rather than questions. [ABSTRACT FROM AUTHOR]

Fidelity is one of the most widely used quantities in quantum information that measures the distance of two quantum states through a noisy channel, a kind of quantum operations. In this paper, we consider the model of quantum Markov chain (QMC), in which transitions are weighted by super-operators to characterize quantum operations and the initial quantum state is left parametric. A quantum analogy of probabilistic computation tree logic, called QCTL, is introduced to take into account fidelity, instead of probability measure, over QMC. The key to the model checking problem lies in computing the fidelity of the super-operator valued measure specified by a path formula in QCTL. It is minimized over all initial quantum states, which is intended for analyzing the system performance in the worst case. We achieve it by a reduction to quantifier elimination in the existential theory of the reals. The method is absolutely exact, so that model checking QCTL formulas against QMCs is proved to be decidable in exponential time. [ABSTRACT FROM AUTHOR]

In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson's method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method. [ABSTRACT FROM AUTHOR]

Railway interlocking systems are essential safety components in rail transportation, designed to prevent train collisions. They regulate the transitions between sections of a railway station using rail traffic control elements. An interlocking system can assess whether the configuration of these control elements poses a collision risk. Over the years, researchers have developed various algebraic models to tackle this issue, highlighting the potential use of computer algebra systems in implementing interlocking systems. In this work, we aim to enhance these systems' capabilities. Not only will they indicate whether a situation is dangerous, but if it is, they will also provide guidance on how to configure certain rail traffic control elements to ensure safety. In this paper, we introduce an algebraic model that represents the railway station through polynomials. This approach transforms the task of identifying dangerous situations into calculating the residue of a polynomial over a set of polynomials. The monomials contained in this residue polynomial encode all possible configurations that would render the situation safe.

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group. [ABSTRACT FROM AUTHOR]

Considered are 80 sets of layer groups, each set consisting of four groups: ordinary single and double, and grey single and double layer groups. The structural properties of layer groups (factorization into cyclic subgroups and the existence of grading according to the sequence of halving subgroups) enable efficient symbolic computation (by the POLSym code) of the relevant properties, real and complex irreducible and allowed (half‐)integer (co‐)representations in particular. This task includes, as the first step, classification of the irreducible domains based on the group action in the Brillouin zone combined with torus topology. Also, the band (co‐)representations induced from the irreducible (co‐)representations of Wyckoff‐position stabilizers (site‐symmetry groups) are decomposed into the irreducible components. These, and other layer group symmetry related theoretical data relevant for physics, layered materials in particular, are tabulated and made available through the web site https://nanolab.group/layer/. [ABSTRACT FROM AUTHOR]

Computer algebra methods were used to find the roots of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies, connected by a spherical hinge, that moves along a circular orbit under the action of gravitational torque. To determine the equilibrium orientations of two connected bodies the system of 12 algebraic equations was decomposed using algorithms for Gröbner basis construction. The number of equilibria was found by analyzing the real roots of the algebraic equations from the calculated Gröbner basis. Evolution of the conditions for equilibria existence in the dependence of the parameter of the problem was investigated. The effectiveness of the algorithms for Gröbner basis construction was analyzed depending on the number of parameters for the problem under consideration. [ABSTRACT FROM AUTHOR]

Railway interlocking systems are crucial safety components in rail transportation, designed to prevent train collisions by regulating switch positions and signal indications. These systems delineate potential train movements within a railway station by connecting sections into routes, which are further divided into blocks. To ensure safety, the system prohibits the simultaneous allocation of the same block or intersecting routes to multiple trains. In this study, we characterize the 'interlocking problem' as a safety verification task for a single real-time station configuration, rather than a 'command and control' function. This is a matter of verification, not solution, typically managed by an interlocking system that receives movement authority requests. Over the years, we have developed various algebraic models to address this issue, suggesting the potential use of computer algebra systems in implementing interlocking systems. However, some of these models exhibit limitations. In this paper, we propose a novel algebraic model for decision-making in railway interlocking systems that overcomes the limitations of previous approaches, making it suitable for large railway stations. Our primary objective is to offer a mathematical solution to interlocking problems in linear time, which our approach accomplishes.

We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation. [ABSTRACT FROM AUTHOR]

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (ℂ k ∖ Z) / / G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂ k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G -orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of X Σ . In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound. [ABSTRACT FROM AUTHOR]

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results.

A novel scheme based on the generalized differential operator and computer algebra was used to construct solitary solutions to a system of Riccati differential equations with diffusive coupling. The presented approach yields necessary and sufficient existence conditions of solitary solutions with respect to the system parameters. The proposed stepwise approach enabled the derivation of the explicit analytic solution, which could not be derived using direct balancing techniques due to the complexity of algebraic relationships. Computational experiments were used to demonstrate the efficacy of proposed scheme. [ABSTRACT FROM AUTHOR]

Computer algebra methods are used to investigate the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque in the plane of the orbit. An algebraic method based on the resultant approach is applied to reduce the satellite stationary motion system of algebraic equations to a single algebraic equation in one variable, which determines the equilibrium configurations of the two-body system in the orbital plane. Classification of domains with equal numbers of equilibrium solutions is carried out using algebraic methods for constructing discriminant hypersurfaces. Discriminant curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria for the two-body system are obtained symbolically. Using the proposed approach it is shown that the satellite-stabilizer system can have up to 24 equilibrium orientations in the plane of a circular orbit. Some simple cases of the problem were studied in detail. [ABSTRACT FROM AUTHOR]

Decision-making in a railway station regarding the compatibility of the positions of the switches of the turnouts and the indications (proceed/stop) of the railway colour light signals is a safety-critical issue that is considered very labor-intensive. Different authors have proposed alternative solutions to automate its supervision, which is performed by the so-called railway interlocking systems. The classic railway interlocking systems are route-based and their compatibility is predetermined (usually by human experts): only some chosen routes are simultaneously allowed. Some modern railway interlocking systems are geographical and make decisions on the fly, but are unsuitable if the station is very large and the number of trains is high. In this paper, we present a completely new algebraic model for decision-making in railway interlocking systems, based on other computer algebra techniques, that bypasses the disadvantages of the approaches mentioned above (its performance does not depend on the number of trains in the railway station and can be used in large railway stations). The main goal of this work is to provide a mathematical solution to the interlocking problems. We prove that our approach solves it in linear time. Although our approach is interesting from a theoretical perspective, it has a significant limitation: it can hardly be adopted in an actual interlocking implementation, mainly due to the heavy certification requirements for this kind of safety-critical application. Nevertheless, the results may be useful for simulations that do not require certification credit.

We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal enveloping Poisson algebra. Finally, we briefly discuss operadic aspects of Poisson triple systems. [ABSTRACT FROM AUTHOR]

Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants. arXiv preprint. arXiv:1907.00205, 2019), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued fractions, and generalize them to evaluate infinite families (naturally generalizing theirs). Our work complements their beautiful approach, since we use symbolic rather than numeric computations, and we instruct the computer to not only discover such evaluations, but at the same time to prove them rigorously. [ABSTRACT FROM AUTHOR]

Verifying arithmetic circuits and most prominently multiplier circuits is an important problem which in practice is still considered to be challenging. One of the currently most successful verification techniques relies on algebraic reasoning. In this article, we present AMulet2, a fully automatic tool for verification of integer multipliers combining SAT solving and computer algebra. Our tool models multipliers given as and-inverter graphs as a set of polynomials and applies preprocessing techniques based on elimination theory of Gröbner bases. Finally, it uses a polynomial reduction algorithm to verify the correctness of the given circuit. AMulet2 is a re-factorization and improved re-implementation of our previous verification tool AMulet1 and cannot only be used as a stand-alone tool but also serves as a polynomial reasoning framework. We present a novel XOR-based slicing approach and discuss improvements on the data structures including monomial sharing. [ABSTRACT FROM AUTHOR]

In this study, we manufacture an exact solution for a set of 2D thermochemical mantle convection problems. The derivation begins with the specification of a stream function corresponding to a non‐stationary velocity field. The method of characteristics is then applied to determine an expression for composition consistent with the velocity field. The stream function formulation of the Stokes equation is then applied to solve for temperature. The derivation concludes with the application of the advection‐diffusion equation for temperature to solve for the internal heating rate consistent with the velocity, composition, and temperature solutions. Due to the large number of terms, the internal heating rate is computed using Maple™, and code is also made available in Fortran and Python. Using the method of characteristics allows the compositional transport equation to be solved without the addition of diffusion or source terms. As a result, compositional interfaces remain sharp throughout time and space in the exact solution. The exact solution presented allows for precision testing of thermochemical convection codes for correctness and accuracy. Plain Language Summary: We manufacture an exact solution for a set of 2D thermochemical mantle convection problems, for which both thermal and compositional gradients impact buoyancy. Such problems must typically be solved approximately via computer models and are notoriously difficult to solve accurately. Our derivation uses a mathematical technique known as the method of characteristics that allows us to solve for composition and temperature variables without adding artificial terms to the model equations. Accordingly, our solution is able to feature sharp compositional gradients, which are difficult to model numerically. The exact solution facilitates the testing of thermochemical convection codes for both correctness and accuracy. Key Points: An exact solution is manufactured for a realistic thermochemical mantle convection flow in two dimensionsAn exact solution is highly useful for software verification and validation for problems featuring sharply varying thermochemical flowsMaple™ is used to assist with symbolic computations, and the resulting formulas are also provided in Fortran and Python [ABSTRACT FROM AUTHOR]

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups. [ABSTRACT FROM AUTHOR]