Case studies (CS) attempt to help students increase critical thinking skills and engagement while working through a real-life scenario in various disciplines, including medicine, law, and business. However, the CS method has not been heavily utilized in biological sciences. The present study investigated the effect of the CS method on undergraduate biology students’ conceptual understanding, academic outcomes, and perspectives. A case study was applied in a one-semester undergraduate biology course, which was compared to ten semesters of standard sections. Participants completed course pre- and post-tests, pre- and post-case tests, and an online survey to assess their conceptual understanding and engagement. The initial lowest quartiles were determined from the individual course pre-test scores, which were lower than class averages. Results suggested that the CS method helped students in learning outcomes, critical thinking, and conceptual understanding toward biology. In post-test learning gains, the CS group did 20% better than the non-CS group, with the largest benefit seen in the initially lowest pre-test quartile of the class. Moreover, post-case learning gains were 55% improved in the case test. Survey results indicated that students had positive attitudes toward CS for their engagement in plant biology content. Overall, the distribution of A grades improved by 2.6-fold from standard to CS groups. We conclude that the use of CS may address course content engagement and have the potential to effectively boost academic performance, especially for the initially lowest quartile in undergraduate plant biological sciences courses.
We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and $π$, such as Chebyshev polynomials, $\left(\sin^2\left(n\,π/4\right)\cdot\cos\left(n\,π/6\right)\right)_n$, and compositions like $\left(\sin\left(\cos(nπ/3)π\right)\right)_n$. We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic $n\text{th}$ term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions.
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec et al (2017) on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation.
We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on Bour-Mayer method to determine compatibility conditions via Jacobi-Mayer brackets. We solve compatible systems recursively by imitating what one would do with pen and paper: Solve one equation, substitute the solution into the remaining equations and iterate the process until the equations of the system are exhausted. The method we employ for assessing the consistency of the underlying system differs from the traditional use of differential Gröbner bases yet seems more efficient and straightforward to implement. We are not aware of a computer algebra system that adopts the procedure we advocate in this work.
We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems. Ordinary points of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.
A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us several properties of input ideals and the output of our algorithm completely describes the multiplicity structure of input ideals. An efficient algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output "reduced" standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.
The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary $\partial$-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper $\partial$-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.
We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,τ)$, $d$ resp., $τ$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.
We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.
Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of matrices of integer numbers: not only it computes the determinants wrongly, but also it produces different results if one evaluates the same determinant twice.
The foundational theory of differentiation was developed as part of the original release of ACL2(r). In work reported at the last ACL2 Workshop, we presented theorems justifying the usual differentiation rules, including the chain rule and the derivative of inverse functions. However, the process of applying these theorems to formalize the derivative of a particular function is completely manual. More recently, we developed a macro and supporting functions that can automate this process. This macro uses the ACL2 table facility to keep track of functions and their derivatives, and it also interacts with the macro that introduces inverse functions in ACL2(r), so that their derivatives can also be automated. In this paper, we present the implementation of this macro and related functions.
We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results.
In this paper, we consider an extended concept of invariant for polynomial dynamical system (PDS) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAI) for such PDSs. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one using this template.
The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular methods and fast polynomial arithmetic. Our strategies take also advantage of the context in which these operations are performed. We report on extensive experimentation, comparing our code to pre-existing Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.
The Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of formal power series and its principal ideal of non units, equipped with a product that extends the multiplication of the Riordan group. The later is naturally embedded as a subgroup of units into the former. In this paper, we prove the existence of a formal calculus on the Riordan algebra. This formal calculus plays a role similar to those of holomorphic calculi in the Banach or Fréchet algebras setting, but without the constraint of a radius of convergence. Using this calculus, we define \emph{en passant} a notion of generalized powers in the Riordan group.
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra Groebner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a polynomial over the ground field by a generic linear map. Then this polynomial is factorized over the ground field. From these factors, the factorization of the polynomial over the extension field is obtained. The new algorithm has been implemented and computer experiments indicate that the new algorithm is very efficient, particularly in complicated examples.
These lectures give a brief introduction to the Computer Algebra systems Reduce and Maple. The aim is to provide a systematic survey of most important commands and concepts. In particular, this includes a discussion of simplification schemes and the handling of simplification and substitution rules (e.g., a Lie Algebra is implemented in Reduce by means of simplification rules). Another emphasis is on the different implementations of tensor calculi and the exterior calculus by Reduce and Maple and their application in Gravitation theory and Differential Geometry. I held the lectures at the Universidad Autonoma Metropolitana-Iztapalapa, Departamento de Fisica, Mexico, in November 1999.
The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev-Petviashvili, Karney-Sen-Chu-Verheest, Boussinesq, Tzetzeica, Benney.
This paper developed a systematic strategy establishing RBF on the wavelet analysis, which includes continuous and discrete RBF orthonormal wavelet transforms respectively in terms of singular fundamental solutions and nonsingular general solutions of differential operators. In particular, the harmonic Bessel RBF transforms were presented for high-dimensional data processing. It was also found that the kernel functions of convection-diffusion operator are feasible to construct some stable ridgelet-like RBF transforms. We presented time-space RBF transforms based on non-singular solution and fundamental solution of time-dependent differential operators. The present methodology was further extended to analysis of some known RBFs such as the MQ, Gaussian and pre-wavelet kernel RBFs.
The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the utility of the parameterized categorical approach by deriving a multivariate polynomial category (over various coefficient domains), which is used by our Mathematica implementation of Buchberger's algorithms for determining the Groebner basis. These implementations are designed according to domain and category parameterization principles underlining their advantages: operation protection, inheritance, generality, easy extendibility. In particular, such an extension of Mathematica, a widely used symbolic computation system, with a new type system has a certain practical importance. The approach we propose for Mathematica is inspired from D. Gruntz and M. Monagan's work in Gauss, for Maple.