Related papers: A note on series with recursively defined terms
Testing convergence of infinite series is an important part of mathematics. A very basic test of convergence is to upper-bound a given series with a known series, term by term. In $19^{th}$ century, Kummer proposed a test of convergence for…
We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the…
The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero.…
The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification…
We introduce an operational rewriting-based semantics for strictly positive nested higher-order (co)inductive types. The semantics takes into account the "limits" of infinite reduction sequences. This may be seen as a refinement and…
Conceiving of premises as collected into sets or multisets, instead of sequences, may lead to triviality for classical and intuitionistic logic in general proof theory, where we investigate identity of deductions. Any two deductions with…
We show that any affine invariant function on the set of positive definite matrices must factor through the determinant function, as long as the restriction of the function to scalar matrices is surjective. A motivation from robust…
This is a reminder that an infinite series can be defined other than as the limit of a sequence of finite series. An example is provided in which a circuit element comprised of an infinite series of resistors has negative resistance.
We give extensional and intensional characterizations of functional programs with nondeterminism: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which…
A notion of a coring extension is defined and it is related to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable…
We describe techniques for synthesis and verification of recursive functional programs over unbounded domains. Our techniques build on top of an algorithm for satisfiability modulo recursive functions, a framework for deductive synthesis,…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of…
In this note we axiomatize the classes of rudimentary functions, primitive recursive functions, safe recursive set functions, and predicatively computable functions.
In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.
The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of…
In this note we provide a simple formula of general term of recurrent sequence.
We provide estimates for the convolution product of an arbitrary number of "resurgent functions", that is holomorphic germs at the origin of $C$ that admit analytic continuation outside a closed discrete subset of $C$ which is stable under…
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…
Strong typicality and the Markov lemma have been used in the proofs of several multiterminal source coding theorems. Since these two tools can be applied to finite alphabets only, the results proved by them are subject to the same…