We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive bounds on their degree. Here we give bounds that apply under some technical condition about the defining differential equations.
We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.
Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.
Computers are good at evaluating finite sums in closed form, but there are finite sums which do not have closed forms. Summands which do not produce a closed form can often be ``fixed'' by multiplying them by a suitable polynomial. We provide an explicit description of a class of such polynomials for simple hypergeometric summands in terms of the Bell numbers.
We propose a functional implementation of \emph{Multivariate Tower Automatic Differentiation}. Our implementation is intended to be used in implementing $C^\infty$-structure computation of an arbitrary Weil algebra, which we discussed in the previous work.
AbstractA pivotal challenge posed by unconventional superconductors is to unravel how superconductivity emerges upon cooling from the generally complex normal state. Here, we use nonlinear magnetic response, a probe that is uniquely sensitive to the superconducting precursor, to uncover remarkable universal behaviour in three distinct classes of oxide superconductors: strontium titanate, strontium ruthenate, and the cuprate high-Tc materials. We find unusual exponential temperature dependence of the diamagnetic response above the transition temperature Tc, with a characteristic temperature scale that strongly varies with Tc. We correlate this scale with the sensitivity of Tc to local stress and show that it is influenced by intentionally-induced structural disorder. The universal behaviour is therefore caused by intrinsic, self-organized structural inhomogeneity, inherent to the oxides’ perovskite-based structure. The prevalence of such inhomogeneity has far-reaching implications for the interpretation of electronic properties of perovskite-related oxides in general.
The optimal calculation order of a computational graph can be represented by a set of algebraic expressions. Computational graph and algebraic expression both have close relations and significant differences, this paper looks into these relations and differences, making plain their interconvertibility. By revealing different types of multiplication relations in algebraic expressions and their elimination dependencies in line-graph, we establish a theoretical limit on the efficiency of face elimination.
This paper presents new fast algorithms for Hermite interpolation and evaluation over finite fields of characteristic two. The algorithms reduce the Hermite problems to instances of the standard multipoint interpolation and evaluation problems, which are then solved by existing fast algorithms. The reductions are simple to implement and free of multiplications, allowing low overall multiplicative complexities to be obtained. The algorithms are suitable for use in encoding and decoding algorithms for multiplicity codes.
We present an algorithm which, given a linear recurrence operator $L$ with polynomial coefficients, $m \in \mathbb{N}\setminus\{0\}$, $a_1,a_2,\ldots,a_m \in \mathbb{N}\setminus\{0\}$ and $b_1,b_2,\ldots,b_m \in \mathbb{K}$, returns a linear recurrence operator $L'$ with rational coefficients such that for every sequence $h$, \[ L\left(\sum_{k=0}^\infty \prod_{i=1}^m \binom{a_i n + b_i}{k} h_k\right) = 0 \] if and only if $L' h = 0$.
This volume contains the proceedings of the Fifteenth International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2-2018), a two-day workshop held in Austin, Texas, USA, on November 5-6, 2018, immediately after FMCAD'18. The proceedings of ACL2-2018 include eleven long papers and two extended abstracts.
Inspired by Faugère and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.
We write a procedure for constructing noncommutative Groebner bases. Reductions are done by particular linear projectors, called reduction operators. The operators enable us to use a lattice construction to reduce simultaneously each S-polynomial into a unique normal form. We write an implementation as well as an example to illustrate our procedure. Moreover, the lattice construction is done by Gaussian elimination, which relates our procedure to the F4 algorithm for constructing commutative Groebner bases.
Tensor expression simplification is an "ancient" topic in computer algebra, a representative of which is the canonicalization of Riemann tensor polynomials. Practically fast algorithms exist for monoterm canonicalization, but not for multiterm canonicalization. Targeting the multiterm difficulty, in this paper we establish the extension theory of graph algebra, and propose a canonicalization algorithm for Riemann tensor polynomials based on this theory.
It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree curve obtained from the usual linear algebra reasoning.
We show that a linear-time algorithm for computing Hong's bound for positive roots of a univariate polynomial, described by K. Mehlhorn and S. Ray in an article "Faster algorithms for computing Hong's bound on absolute positiveness", is incorrect. We present a corrected version.
In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of polynomials in one step using the theory of multivariate resultant. This seems to be the first differential triangular decomposition algorithm with elementary computation complexity.
In this paper, the author present reliable symbolic algorithms for solving a general bordered tridiagonal linear system. The first algorithm is based on the LU decomposition of the coefficient matrix and the computational cost of it is O(n). The second is based on The Sherman-Morrison-Woodbury formula. The algorithms are implementable to the Computer Algebra System (CAS) such as MAPLE, MATLAB and MATHEMATICA. Three examples are presented for the sake of illustration.
We present an algorithm for efficient computation of the constant term of a power of a multivariate Laurent polynomial. The algorithm is based on univariate interpolation, does not require the storage of intermediate data and can be easily parallelized. As an application we compute the power series expansion of the principal period of some toric Calabi-Yau varieties and find previously unknown differential operators of Calabi-Yau type.
AbstractNeu dargestellt wurden Cs2KGaF6 (a = 8,975), Cs2KInF6 (a = 9,219), Cs2KTlF6 (a = 9,36), Cs2KScF6 (a = 9,32), Cs2KYF6 (a = 9,443) sowie Cs2KFeF6 (a = 9,015 Å), die kubisch kristallisieren und mit K2NaCrF6 isotyp sind. Alle Verbindungen sind farblos; Cs2KTlF6 zersetzt sich an feuchter Luft sofort. Der Existenzbereich der Cs2KMF6‐Struktur als Funktion des Ionenradius von M3+ wird diskutiert.