Related papers: Extending the Algebraic Manipulability of Differen…
The goal of this thesis is threefold: first, to provide a general semantic setting for reasoning about incremental computation. Second, to establish and clarify the connection between derivatives in the incremental sense and derivatives in…
We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of practical rules allowing significant…
We explore when it is legal to differentiate a polynomial evaluated at a root of unity using modular arithmetic.
We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for…
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
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…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We observe algebraic derivations on an affine domain B defined over an algebraically closed field of characteristic 0, which are called locally finite derivations in commutative and non-commutative contexts in other references. We observe…
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.
Algebraic differentiators have attracted much interest in recent years. Their simple implementation as classical finite impulse response digital filters and systematic tuning guidelines may help to solve challenging problems, including, but…
The fractional calculus of variations is now a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. In this paper the fractional operators are defined…
The fractional integrals and fractional derivatives problem is tackled by using the operator approach. The definition domain E of operators is causal functions.Many properties of fractional integrals are given. Fractional derivatives…
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential…
The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this geometrical description is less trivial, it can be…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…