In this paper we show that
it is possible to integrate functions with concepts and fundamentals of
Paraconsistent Logic (PL). The PL is a non-classical Logic that tolerates the
contradiction without trivializing its results. In several works the PL in his annotated
form, called Paraconsistent logic annotated with annotation of two values
(PAL2v), has presented good results in analysis of information signals. Geometric
interpretations based on PAL2v-Lattice associate were obtained forms of
Differential Calculus to a Paraconsistent Derivative of first and second-order
functions. Now, in this paper we extend the calculations for a form of
Paraconsistent Integral Calculus that can be viewed through the analysis in the
PAL2v-Lattice. Despite improvements that can develop calculations in complex
functions, it is verified that the use of Paraconsistent Mathematics in
differential and Integral Calculus opens a promising path in researches
developed for solving linear and nonlinear systems. Therefore the Paraconsistent
Integral Differential Calculus can be an important tool in systems by modeling
and solving problems related to Physical Sciences.

The Paraconsistent
Logic (PL) is a non-classical logic and its main property is to present
tolerance for contradiction in its fundamentals without the invalidation of the
conclusions. In this paper, we use the PL in its annotated form, denominated Paraconsistent
Annotated Logic with annotation of two values-PAL2v. This type of
paraconsistent logic has an associated lattice that allows the development of
a Paraconsistent Differential Calculus based on fundamentals and equations obtained
by geometric interpretations. In this paper (Part II), it is presented a
continuation of the first article (Part I) where the Paraconsistent
Differential Calculus is given emphasis on the second-order Paraconsistent
Derivative. We present some examples applying Paraconsistent Derivatives at
functions of first and second-order with the concepts of Paraconsistent
Mathematics.

Abstract:
In a series of publications in the early 1990s, L D Nel set up a study of non-normable topological vector spaces based on methods in category theory. One of the important results showed that the classical operations of derivative and integral in Calculus can in fact be obtained by a rather simple construction in categories. Here we present this result in a concise form. It is important to note that the respective differentiation does not lead to any so called generalized derivatives, for instance, in the sense of distributions, hyperfunctions, etc., but it simply corresponds to the classical one in Calculus. Based on that categorial construction, Nel set up an infinite dimensional calculus which can be applied to functions defined on non-convex domains with empty interior, a situation of great importance in the solution of partial differential equations

Abstract:
Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra $M_n({\mathbb{C}})$ and the algebra of matrix valued functions $C^\infty(M)\otimes M_n({\mathbb{C}})$. The UV/IR mixing problem of the resulting Yang-Mills-Higgs models is also discussed.

A type of Inconsistent Mathematics structured on Paraconsistent Logic
(PL) and that has, as the main purpose, the study of common mathematical
objects such as sets, numbers and functions, where some contradictions are
allowed, is called Paraconsistent Mathematics. The PL is a non-Classical logic
and its main property is to present tolerance for contradiction in its
fundamentals without the invalidation of the conclusions. In this paper (part
1), we use the PL in its annotated form, denominated Paraconsistent Annotated
Logic with annotation of two values—PAL2v for present a first-order Paraconsistent
Derivative. The PAL2v has, in its representation, an associated lattice FOUR
based on Hasse Diagram. This PAL2v-Lattice
allows development of a Para-consistent Differential Calculus based on
fundamentals and equations obtained by geometric interpretations. In this first
article it is presented some examples applying derivatives of first-order with
the concepts of Paraconsistent Mathematics. In the second part of this work we
will show the Paraconsistent Derivative of second-order with application
examples.

Abstract:
We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products.

Abstract:
We show that bicovariant bimodules as defined by Woronowicz are in one to one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)--dimensional representation of the double. An example of bicovariant differential calculus on the non quasitriangular quantum group E_q(2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.

Abstract:
Given a spectral triple $(\mathcal{A},\mathcal{H},D)\,$ Connes associated a canonical differential graded algebra $\,\Omega_D^\bullet(\mathcal{A})$. However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes' calculus for its quantum double suspension. This allows one to compute $\,\Omega_D^\bullet$ for spectral triples obtained by iterated quatum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes' calculus for a large family of spectral triples.

Abstract:
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.

Abstract:
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework generalizes the usual (lattice) discretization.