What is a closed-form number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.Comment: 11 pages; submitted to Amer. Math. Monthl

Duality of Abuse and Care. Empathy in Sara Gruen’s Water for Elephants

In an era of Anthropocene, habitat loss and species extinction due to anthropogenic factors, and the upsurge in animal exploitation force us to reconsider the “animal question” and relationships between humans and animals. All forms of animal abuse violate the subjectivity of the animals by othering them as objects who are mercilessly exploited. Purportedly influenced by the social consciousness of the moral rights of animals and the animal advocacy movement, Sara Gruen’s novel “Water for Elephants” (2006), exposes the horrible reality of animals being mistreated for entertainment in the circus industry through a fictitious description of the events in the Benzini Brothers’ Shows. The framework of this research is based on two arguments: the crucial link between human insensitivity or empathy erosion and animal abuse; and the significance of empathy, in particular, “entangled empathy”, in acknowledging animals as moral subjects, taking care of them, and creating the harmonious human-animal relationship in the novel

Applications of transcendental number theory to decision problems for hypergeometric sequences

A rational-valued sequence is hypergeometric if it satisfies a first-order linear recurrence relation with polynomial coefficients. In this note we discuss two decision problems, the membership and threshold problems, for hypergeometric sequences. The former problem asks whether a chosen target is in the orbit of a given sequence, whilst the latter asks whether every term in a sequence is bounded from below by a given value. We establish decidability results for restricted variants of these two decision problems with an approach via transcendental number theory. Our contributions include the following: the membership and threshold problems are both decidable for the class of rational-valued hypergeometric sequences with Gaussian integer parameters

Nonnegativity Problems for Matrix Semigroups

The matrix semigroup membership problem asks, given square matrices $M,M_1,\ldots,M_k$ of the same dimension, whether $M$ lies in the semigroup generated by $M_1,\ldots,M_k$. It is classical that this problem is undecidable in general but decidable in case $M_1,\ldots,M_k$ commute. In this paper we consider the problem of whether, given $M_1,\ldots,M_k$, the semigroup generated by $M_1,\ldots,M_k$ contains a non-negative matrix. We show that in case $M_1,\ldots,M_k$ commute, this problem is decidable subject to Schanuel's Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability result is a procedure to determine, given a matrix $M$, whether the sequence of matrices $(M^n)_{n\geq 0}$ is ultimately nonnegative. This answers a problem posed by S. Akshay (arXiv:2205.09190). The latter result is in stark contrast to the notorious fact that it is not known how to determine effectively whether for any specific matrix index $(i,j)$ the sequence $(M^n)_{i,j}$ is ultimately nonnegative (which is a formulation of the Ultimate Positivity Problem for linear recurrence sequences)

Nonnegativity problems for matrix semigroups

The matrix semigroup membership problem asks, given square matrices M, M1, ..., Mk of the same dimension, whether M lies in the semigroup generated by M1, ..., Mk. It is classical that this problem is undecidable in general, but decidable in case M1, ..., Mk commute. In this paper we consider the problem of whether, given M1, ..., Mk, the semigroup generated by M1, ..., Mk contains a non-negative matrix. We show that in case M1, ..., Mk commute, this problem is decidable subject to Schanuel's Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mn)∞n=0 is ultimately nonnegative. This answers a problem posed by S. Akshay [1]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i, j), whether the sequence (Mn)i,j is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem

Recent progress in exact geometric computation

AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters