Computer Algebra Approach Research Articles

Solving the Master Equation on river networks: A computer algebra approach

Solving the Master Equation on river networks: A computer algebra approach

An Algorithm for Detecting Sentence Validity

In this paper, we introduce an algorithm for determining the grammatical validity of a sentence. We take a similar approach as in (Preller, 2007) and (Lambek, Type Grammar Revisited, 1999) (Lambek, From word to sentence: a computational algebraic approach to grammar, 2008) by encoding the English words based on word type which we call components. A sentence can be described both algebraically and geometrically. Our algorithm generates the geometric portion called underlinks from the generalized reductions of the algebraic portion. Underlinks uniquely determine the reduction of the components leading to the empty string. This is the mathematical basis for determining if a sentence is valid. We also provide a proof for the algorithm’s time complexity of O(n2) along with a Python implementation. This paper is part of a bigger project based on (Coecke, Mehrnoosh, Clarky, 2010) where we explore the combination of a sentence’s grammar and meaning. This is done by combining two compact closed categories; pregroups represent the grammar of a sentence, and finite dimensional vector spaces describe the meaning of a sentence. Together one compact closed category is created, representing both aspects of the sentence.

Implementation of analytic projective geometry for computer graphics

In their research, the authors actively exploit different branches of geometry. For geometric constructions, computer algebra approaches and systems are used. Currently, we are interested in computer geometry, more specifically, the implementation of computer graphics. The use of the projective space and homogeneous coordinates has actually become a standard in modern computer graphics. In other words, the problem is reduced to the application of analytic projective geometry. The authors failed to find a computer algebra system that could implement projective geometry in its entirety. Therefore, it was decided to partially implement computer algebra for visualization of algebraic relations. For this purpose, the Asymptote system was employed.

Computer-algebraic approach to first differential approximation: Van der Pol oscillator

First differential approximation has been used to analyze various numerical methods for solving systems of ordinary differential equations. This has made it possible to estimate the stiffness of the ODE system that models the oscillations of the Van der Pol oscillator and the error of the method as well as to propose simple criteria for choosing a calculation step. The presented methods allow one to perform efficient calculations using computer algebra systems.

Computer-Algebraic Approach to First Differential Approximations: Van der Pol Oscillator

Computer-Algebraic Approach to First Differential Approximations: Van der Pol Oscillator

A computational approach to analyze the Hadamard quasigroup product

<abstract><p>Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular.</p></abstract>

Frataxin Deficit Leads to Reduced Dynamics of Growth Cones in Dorsal Root Ganglia Neurons of Friedreich’s Ataxia YG8sR Model: A Multilinear Algebra Approach

Computational techniques for analyzing biological images offer a great potential to enhance our knowledge of the biological processes underlying disorders of the nervous system. Friedreich’s Ataxia (FRDA) is a rare progressive neurodegenerative inherited disorder caused by the low expression of frataxin, which is a small mitochondrial protein. In FRDA cells, the lack of frataxin promotes primarily mitochondrial dysfunction, an alteration of calcium (Ca2+) homeostasis and the destabilization of the actin cytoskeleton in the neurites and growth cones of sensory neurons. In this paper, a computational multilinear algebra approach was used to analyze the dynamics of the growth cone and its function in control and FRDA neurons. Computational approach, which includes principal component analysis and a multilinear algebra method, is used to quantify the dynamics of the growth cone (GC) morphology of sensory neurons from the dorsal root ganglia (DRG) of the YG8sR humanized murine model for FRDA. It was confirmed that the dynamics and patterns of turning were aberrant in the FRDA growth cones. In addition, our data suggest that other cellular processes dependent on functional GCs such as axonal regeneration might also be affected. Semiautomated computational approaches are presented to quantify differences in GC behaviors in neurodegenerative disease. In summary, the deficiency of frataxin has an adverse effect on the formation and, most importantly, the growth cones’ function in adult DRG neurons. As a result, frataxin deficient DRG neurons might lose the intrinsic capability to grow and regenerate axons properly due to the dysfunctional GCs they build.

Solution of nonlinear equations via Pade approximation. A Computer Algebra approach

"We generate automatically several high order numerical methods for the solution of nonlinear equations using Pad e approximation and Maple CAS."

Dynamic Simulation of Quantum Entanglement in Finite Quantum Mechanics: A Computer Algebra Approach

In this paper, we investigate the behavior of quantum entanglement in the process of unitary evolution in constructive models of multicomponent quantum systems. Symmetry groups of quantum systems that admit the occurrence of geometric structures associated with quantum entanglement are described. Algorithms for dynamic simulation of quantum entanglement are based on methods of computer algebra and computational group theory. Some examples of practical computations are presented.

On some configurations of oppositely charged trapped vortices in the plane

Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary symmetric polynomials, which can then be fully solved via computational algebra packages. We retrieve a number of previously identified configurations, including collinear ones and polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also retrieve a configuration with 4 positive charges and 2 negative ones, which is unprecedented, to the best of our knowledge, in BEC studies. We corroborate these predictions via numerical computations in the fully two-dimensional PDE system of the Gross-Pitaevskii type which characterizes the BEC at the mean-field level.

Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field $\mathbb {K}$ as the set of zeros of the binomial ideal $\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]$ . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.

Detecting truth, just on parts

We introduce, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true over some relevant collections of instances and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test within the dynamic geometry program GeoGebra. Our framework has some significant differences regarding some alternative, recent formulation of the “true on components” idea; differences and similarities between both approaches are discussed here.

Symbolic linearization and vibration analysis of constrained multibody systems

A computer algebraic approach for linearization of the equations of constrained multibody systems is discussed in this paper. Based on linearized differential equations, the Newmark method is applied to calculate steady-state periodic vibrations of the parametric vibration of constrained dynamical models. The numerical calculation is also demonstrated on a model of a mechanism with elastic connecting link.

An elementary solution of Gessel's walks in the quadrant

Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in {→,↗,←,↙} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is, steps in {−1,0,1}2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function (algebraic, D-finite or not) is now available, but Gessel's walks remained mysterious because they were the only model among the 23 D-finite ones that had not been given an elementary solution. Instead, Gessel's conjecture was first proved using an inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.

Measuring the robustness of a network using minimal vertex covers

We define two quantities associated to each of the vertices of a simple graph, based on the collection of minimal vertex covers of the graph. They are called covering degree and covering index. We use them to describe new strategies for measuring the robustness of a network. We study the correlation between the defined quantities and other quantities used in the context of network attacks. Using the attack strategies associated to these quantities we study their effect on the connectedness of several network models. We also consider the complexity of the computation of the defined quantities and use a computational commutative algebra approach for their actual computation.

Center conditions and cyclicity for a family of cubic systems: Computer algebra approach

Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. To that end we overcome the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we also determine the number of limit cycles bifurcating from each component of the center variety.

A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general chemistry. In order to test the impact of our method on student learning, we measured the performance of students who used the method. We designated a group of students in our course as the study group, followed their progress through the course, and compared them to other students learning with more traditional approaches. As discussed in the article, we see significant increases in the performance indicators.

SD-CAS: Spin Dynamics by Computer Algebra System

A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples.

Computer algebra for guaranteed accuracy. How does it help?

Today simulation technologies (based on numerical computation) are definitely vital in many fields of science and engineering. The accuracy of numerical simulations grows in importance as simulation technologies develop and prevail, and many researches have been carried out for establishing numerically stable algorithms. In recent years, combining computer algebra and other guaranteed accuracy approaches draws much attention as one of promising directions for developing guaranteed accuracy algorithms for a wider class of problems. This paper illustrates several typical usages of symbolic and algebraic methods for guaranteed accuracy computation, highlighting some of the recent applications in control problems.