This work is devoted to the symbolic computation of centralizers of ordinary differential operators (ODOs), in the ring of differential operators. Starting with an operator L of order n and the order 𝔪 of a non-trivial operator in its centralizer, which is not a multiple of n , a finite set of generators of a subalgebra of the centralizer is obtained, maximal of a certain rank R , the greatest common divisor of all orders of its elements. The true rank r of the centralizer is unknown to start unless (n,𝔪)=1 since 1≤ r≤ R≤ (n,𝔪) , and remains unknown unless our algorithm returns R=1 . Ours is a direct approach based on solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchies, which after substituting the coefficients of L become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of L belong to a computable differential field. In addition, by considering parametric coefficients, we develop an algorithm to generate families of ODOs with non-trivial centralizer, whose coefficients belong to a previously chosen differential field. Our algorithms are implemented in SageMath.
In this work, we address the problem of expressing sequences of d -orthogonal polynomials in terms of either orthogonal polynomials or the canonical basis. We derive a general recurrence relation that enables the symbolic recursive computation of these coefficients. Several results are established to investigate the impact of the Appell character, as well as the symmetry and d -symmetry of the polynomial sequences, on these coefficients. Additionally, we deduce some results concerning the zeros of d -orthogonal polynomials.
Motzkin paths represent many mathematical objects from different contexts with combinatorial flavor. In this paper, we exhibit an unexpected appearance of this family of paths in the network coding setting through a particular case of multishot codes called flag codes. A flag code is a set of flags, i.e., sequences of nested subspaces of a vector space over a finite field, with prescribed increasing dimensions. The flag distance is defined as the sum of the respective subspace distances and can be obtained in different ways, which considerably complicates the manipulation of these codes. Throughout this work we define the set of Motzkin paths associated with a flag code. This new invariant allows us to characterize some important families of flag codes and to extract relevant information about them. In particular, it gives us a direct way to compute the maximum possible number of combinations behind a prescribed value of the minimum distance.
Given a curve X in the complex plane ℂ^2 and a smooth point p∈ X , an osculating conic to X at p generalizes the notion of a tangent line, and of an osculating circle. Beyond degree two, one may consider more general osculating spaces of X at p , where the degree and order of tangency are prescribed. This paper lays out a computational framework for computing such spaces. We work in local coordinates y=f(x) for X near p and assume that p is the origin. Then, the computation of an osculating space amounts to solving a linear system of equations in the derivatives of f(x) at x=0 . We showcase our work by producing a formula for the osculating conic of a generic analytic curve and exhibit several specific examples with interesting properties.
We define the notion of approximate affine equivalence for polynomially parametrized curves and trigonometric curves, i.e., curves parametrized by truncated Fourier series, whose coefficients are floating point numbers, therefore known up to finite precision, and provide symbolic-numeric algorithms in any dimension to compute the approximate affine equivalences between such curves. In order to do this, we first develop the notion of approximate equivalence in norm, which leads to concrete algorithms, and then we analyze the relationship between closeness in norm, and closeness in terms of the Hausdorff distance. The algorithms have been implemented in the computer algebra system Maple; we report on experiments carried out with these algorithms to provide evidence of their efficiency.
We consider the topic of integrability, i.e. when an analytical solution to a system of differential equations can be found. It is not always possible to find such solutions, and one approach to simplify the problem is to search for specific parameter values at which the system is locally integrable. This leads to the hypothesis that we need local integrability at each point of the phase space to have integrability of the system overall. We explore this hypothesis experimentally for a two-dimensional polynomial autonomous ODE system. In the case when the system under study depends on parameters, then Bruno’s normal form method allows us to reduce the requirements of this hypothesis toa system of algebraic equations for the parameters of the system: by solving these equations, we find integrable cases. The method works in cases of resonance in the linear part of the system. We applied this method to study the Bautin system, a two-dimensional ODE with quadratic nonlinearity on the right-hand sides. For resonance cases 1:1and 1:2, a few dozen first integrals were found and are presented. We also report that the algebraic system of parameters obtained by combining systems of equations for resonances 1:1, 1:2 and 1:3 has 11 solutions, each of which corresponds to an integrable case of the Bautin system of a general (non-resonant) type. This demonstrates the possibility of using this method outside resonant regions.
Given a multivariate polynomial f, we consider an approximate decomposition in the Hamming distance, that is, f=g∘h + δ, where g is a univariate polynomial, h and δ are multivariate polynomials, and δ has few terms. We propose an algorithm for computing an approximate decomposition and introduce its application to multivariate Horner’s scheme.
We consider the first-order theory of order over real numbers extended with monadic uninterpreted predicates. It has long been established that this theory is decidable, by different approaches such as a reduction to linear-time temporal logic over the reals, expressing the problem as a particular case of deciding the first-order theory of all linear orderings, or reasoning about restrictions of the monadic second-order theory of order. The main contribution of this work is to provide a complete self-contained proof of decidability for this logic. Additionally, our proof develops a symbolic notation system for models of satisfiable formulas, which can be seen as a first step towards an actual implementation of a decision procedure for the theory of order over the reals.
A secret sharing scheme involves dividing a confidential piece of information among multiple users, and a specific number of these users can work together to recover the original secret. In this article, we propose an innovative secret sharing scheme that is dynamic and hierarchical, relying on elliptic curves and pairings. The primary motivation for incorporating bilinear pairings with elliptic curves is to maintain a high level of security while reducing the key size, as seen in various existing schemes. The proposed scheme is computationally secure, efficient and verifiable. The time computation for the proposed scheme has been determined. We have included explicit examples using SageMath.
Program verification using abstract interpretation involves the symbolic calculation of fixpoints over lattices. Integral to these fixpoint calculations is widening, an operation that trades precision for guarantees of termination. Abstract interpretation often works with lattices of convex sets of points in n-dimensional space, represented by sets of linear inequalities. When the form of these inequalities is restricted these are known as weakly relational domains. This paper addresses weakly relational domains, the detail of their representation and the way in which widening is applied, including study of how the closure operators used in weakly relational domains interact with widening, and considers how sequences of constraints might be widened. Satisfiability checking for numeric constraints is one tool used in this work.
A staggered linear basis (SLB) provides a particular linear basis for the ideal it generates and contains a Gröbner basis for this ideal. These properties are enhanced through the additional structure of sets of allowed and forbidden terms assigned to each polynomial within the SLB. In the first part of this paper, we develop the theory of SLBs by introducing and studying minimal staggered linear bases, and by presenting an algorithm for their computation. The second part of the paper explores several applications of SLBs, including the computation of Hilbert functions, Hilbert polynomials, and Hilbert series for polynomial ideals. Furthermore, by leveraging the combinatorial structure of SLBs, we introduce an algorithm for constructing irreducible complementary decompositions for a given monomial ideal. Finally, we present several algorithms that address various aspects of complementary decompositions of ideals. We conclude the paper by reporting the first implementation of SLBs in the CoCoALib. We compare its efficiency with that of the built-in function for Gröbner basis computations, showing that for some classes of examples and coefficient fields, our algorithm outperforms the built-in one.
A Hamiltonian path in the complete graph K_v whose vertices are labeled with the integers 0,1,… ,v-1 is a linear realization for the multiset L of the linear edge-lengths, given by |x-y| for the edge between vertices x and y , of the edges in the path. A linear realization is standard if an end-point is 0 and perfect if the end-points are 0 and v-1 . Linear realizations are useful in the study of the Buratti-Horak-Rosa Conjecture on the existence of cyclic realizations, where cyclic edge-lengths are given by distance modulo v , for given multisets. In this paper, we focus on multisets of the form {1^a, (y-k)^b, y^c} . Using core perfect linear realizations for supports of size 2 (which have the forms {x^y-1,y^x+1} whenever (x,y)=1 ), we construct standard linear realizations (with a=k-1 , b=j(y-k) , c=jy ) when k| y or k ≤ 4 . When k=2 , these allow us to show that there is a linear realization whenever a ≥ y . This is in line with the known results for the case of k=1 . We also supplement these results for k=1 by constructing linear realizations whenever b+c < y and a ≥ y - min (b,c) , from which the coprime version of the conjecture, which requires that v is coprime with each element of the multiset, follows for k=1 when y ≤ 16 . Our methods show promise for constructing linear realizations for arbitrary k , in the direction of a resolution of the conjecture for supports of size 3.
We propose three floating-point validated algorithms to compute respectively fast inversion, Euclidean division and Hensel lifting over ℂ[[x]][y] . This is the second step (after Bréhard, Poteaux and Soudant in ISSAC 2023) towards a validated numerical Newton–Puiseux algorithm, and will also be useful towards a validated OM-algorithm over ℂ[[x]][y] . Our strategy is simply to first compute a floating-point approximation using the classical algorithm, then to a posteriori validate the result using a Newton-like fixed-point operator. We also provide a prototype Julia implementation of these algorithms and several examples.
An Alphabet Reduction Pair of Arrays (ARPA) on a set Σ _q of q symbols consists of two arrays of q columns each. The first array contains a target word composed of all symbols from Σ _q . For some p< q , each row of the second array contains at most p distinct symbols from Σ _q . For some k≤ p , the two arrays, when restricted to any subset of k columns, yield the same multiset of rows. ARPAs were introduced in the study of Constraint Satisfaction Problems with bounded constraint arity ( k -CSPs) to transfer positive differential approximation results from instances over alphabets of size p≥ k to instances over alphabets of larger size q . In this context, the goal is to maximize the frequency of the target word in the first array. This work focuses on ARPAs that achieve this maximum frequency, which we refer to as optimal ARPAs. We show that the frequency of the target word in an optimal ARPA can be determined by solving a linear program with Θ (q) continuous variables and Θ (k) constraints. In addition, we prove the optimality of previously known ARPAs for the case p=k and provide new optimal constructions for the cases k=1 and k=2 . These results are obtained by relating ARPAs to simpler objects called Cover Pairs of Arrays (CPAs), which partially encode the structure of ARPAs in Boolean terms. This connection highlights the relevance of CPAs in the study of the approximability of k -CSPs.
Over the years, the development of computer algebra has derived enormous benefits for the qualitative theory of ordinary differential equations. In this paper, with the aid of algebraic manipulator-Mathematica, the integrability and linearizability of symmetric planar cubic differential systems are investigated thoroughly. Based on the coefficients and eigenvalues, four simple normal forms are obtained. Furthermore, the integrable and linearizable conditions are classified for each case.
Supine-dependent OSA is a well-recognised OSA phenotype and may relate to craniofacial structure. Our aim was to assess whether craniofacial photos are able to identify positional OSA, specifically supine-isolated OSA, in comparison to non-positional OSA. Frontal and profile craniofacial photographs of participants were acquired according to a standardised protocol. Photographs were analysed and compared between non-positional and supine-isolated OSA groups. A total of 156 OSA patients were included (54.5% supine-isolated OSA, 45.5% with non-positional OSA). The supine-isolated group had a longer upper face height and greater upper-to-lower face height ratio, smaller face width, reduced face width-to-height ratio, and smaller mandibular width, length, and size of the mandibular base. Differences in facial measurements were no longer significant after adjustment for body size and OSA severity. Our study demonstrates that supine-isolated OSA can be identified using facial photography. Larger studies in groups matched for BMI and OSA severity are needed to confirm whether this technique may also capture other features related to supine-isolated OSA (i.e., differences in underlying skeletal structure).
In commutative algebra, the theory of Gröbner bases enables one to compute in any finitely generated algebra over a given computable field. For non-finitely generated algebras however, other methods have to be pursued. For instance, it follows from the Cohen structure theorem that standard bases of formal power series ideals offer a similar prospect but for complete local equicharacteristic rings whose residue field is computable. Using the language of rewriting theory, one can characterise Gröbner bases in terms of confluence of the induced rewriting system. It has been shown, so far via purely algebraic tools, that an analogous characterisation holds for standard bases with a generalised notion of confluence. Subsequently, that result is utilised to prove that two generalised confluence properties, where one is actually in general strictly stronger than the other, are actually equivalent in the context of formal power series. In the present paper, we propose alternative proofs making use of tools purely from the new theory of topological rewriting to recover both the characterisation of standard bases and the equivalence between generalised confluence properties. The objective is to extend the analogy between Gröbner basis theory together with classical algebraic rewriting theory and standard basis theory with topological rewriting theory.
Lorenz-84 system was proposed about four decades ago, however, there are almost no analytical results on the equilibria and their local stability. The first objective of this paper is to fill this gap. We discuss the possibility of the existence of multiple equilibria and establish the conditions for a given number of equilibria to exist by using algebraic methods of resultant. Furthermore, we derive the stability conditions on the parameters of the system by using symbolic methods for solving semi-algebraic systems. The second objective is to investigate the zero-Hopf bifurcation of the Lorenz-84 system. By using the averaging method, we provide sufficient conditions for the existence of one limit cycle bifurcating from a zero-Hopf equilibrium of Lorenz-84 system. Several examples and numerical simulations are presented to verify the established results.
We prove that cocyclic Hadamard matrices of order 8p, with p > 3 prime, can be described using an 8 × 8 block template array. In this framework, underlying cocycles are translated into signs and actions related to the blocks and their columns, respectively. Our method uses the result that if a cocyclic Hadamard matrix has an indexing group G = K ⋉ N , where N is cyclic of odd order, then H^2(K, A) ≅ H^2(G, A) whenever A is a finite trivial G-module of order coprime to |N|. This result implies that cocyclic Hadamard matrices of order 8p exhibit no coboundary dependency, allowing for a more concise description of such matrices. Our findings generalise to certain complex cocyclic Hadamard matrices, and we also investigate cocyclic Hadamard matrices of orders 24 and 40.
