Related papers: Parameterized Telescoping Proves Algebraic Indepen…
Astronomical instruments make intensity measurements; any precise astronomical experiment ought to involve modeling those measurements. People make catalogues, but because a catalogue requires hard decisions about calibration and detection,…
Natural philosophy integrates scientific observation with abstract frameworks, often using a mathematical Ansatz to hypothesise about physical phenomena. Exploring the possibility of other universes, however, challenges assumptions that…
Our main result, Theorem 2.5, shows the existence of a vast infinity of subset sum problems solvable in polynomial time. The only proof we have of this result uses the ZFC independent Jump Free Theorem of Harvey Friedman, thus putting…
We show that the number of digits in the integers of a creative telescoping relation of expected minimal order for a bivariate proper hypergeometric term has essentially cubic growth with the problem size. For telescopers of higher order…
Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions.…
For a hypergeometric series $\sum_k f(k,a, b, ...,c)$ with parameters $a, b, >...,c$, Paule has found a variation of Zeilberger's algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general…
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…
Hou and Liu developed a telescoping method to prove the congruence of partial sums of P-recursive sequences. We release the requirement on the telescoper and utilize the congruence of the sequence. With this approach, we are able to confirm…
In difference algebra, summability arises as a basic problem upon which rests the effective solution of other more elaborate problems, such as creative telescoping problems and the computation of Galois groups of difference equations. In…
In this paper, we introduce the super telescoping formula, a natural generalization of well-known telescoping formula. We explore various aspects of the formula including its origin and the telescoping cancellations emerging from symmetric…
We study the properties of algebraic independence and pointwise algebraic independence in a class of continuous theories, the randomizations $T^R$ of complete first order theories $T$. If algebraic and definable closure coincide in $T$,…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…
Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…
This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalsch\"utz' summation formula in the…
The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results…
In this paper, we consider whether existence of a sums-of-squares formula depends on the base field. We reformulate the question of existence as a question in algebraic geometry. We show that, for large enough p, existence of…
This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…
Most major discoveries in astronomy are unplanned, and result from surveying the Universe in a new way, rather than by testing a hypothesis or conducting an investigation with planned outcomes. For example, of the 10 greatest discoveries…
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…