Related papers: Generalized improper integral definition for finit…
In this paper we study an index of a critical orbit, defined in terms of the degree for invariant strongly indefinite functionals. We establish a relationship of this index with the index of a critical point of the mapping restricted to the…
Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain…
We study generalized means whose domain may contain unbounded sets as well. We investigate usual properties of this type of means and also new attributes that regard for such means only. We examine how a mean defined on bounded sets can be…
We define the indefinite logarithm [log x] of a real number x>0 to be a mathematical object representing the abstract concept of the logarithm of x with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed number). The…
The basic formalism of a novel scale invarinat nonlinear analysis is presented. A few analytic number theoretic results are derived independent of standard approaches.
The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation.…
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…
A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order…
Many results on the convex order in the literature were stated for random variables with finite mean. For instance, a fundamental result in dependence modeling is that the sum of a pair of random random variables is upper bounded in convex…
If the non-zero finite floating-point numbers are interpreted as point intervals, then the effect of rounding can be interpreted as computing one of the bounds of the result according to interval arithmetic. We give an interval…
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
A fundamental problem in artificial intelligence is that nobody really knows what intelligence is. The problem is especially acute when we need to consider artificial systems which are significantly different to humans. In this paper we…
Many Engineering Problems could be mathematically described by Final Value Problem, which is the inverse problem of Initial Value Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the…
An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some…
We generalize nonequilibrium integral equalities to situations involving absolutely irreversible processes for which the forward-path probability vanishes and the entropy production diverges, rendering conventional integral fluctuation…
It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…
This paper introduces a notion of integrality that is suitable for non-commutative varieties. It is compatible with the usual notion of integrality for schemes. The function field and generic point of a non-commutative integral space are…
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…