Related papers: Natural discrete differential calculus in physics
Differential linear logic (DiLL) provides a fine analysis of resource consumption in cut-elimination. We investigate the subsystem of DiLL without promotion in a deep inference formalism, where cuts are at an atomic level. In our system…
Automatic differentiation plays a prominent role in scientific computing and in modern machine learning, often in the context of powerful programming systems. The relation of the various embodiments of automatic differentiation to the…
Almost all theories of physics have expressed physical laws by means of differential equations. One can ask: why differential equations? What is special about them? This article addresses these questions and is presented as an inquiry-based…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
Adjoint logic is a general approach to combining multiple logics with different structural properties, including linear, affine, strict, and (ordinary) intuitionistic logics, where each proposition has an intrinsic mode of truth. It has…
Short introduction to exotic differential structures on manifolds is given. The possible physical context of this mathematical curiosity is discussed. The topic is very interesting although speculative.
We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra…
We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The…
In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via…
This text can be considered as a non-technical and arithmetically motivated introduction to the definition of the limiting mixed Hodge structure. We state several assertions in terms natural to the classical theory of ordinary differential…
For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. However, we observe that if $ f\left( x\right) $ is a sum of two functions $f\left( x\right) =f_{1}\left( x\right)…
We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four…
Making a linguistic theory is like making a programming language: one typically devises a type system to delineate the acceptable utterances and a denotational semantics to explain observations on their behavior. Via this connection, the…
In this letter we briefly investigate the mathematical structure of space-time in the framework of discretization. It is shown that the discreteness of space-time may result in a new mechanical system which differ from the usual quantum…
Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a '{\it cellular network}'…
The Lambek calculus provides a foundation for categorial grammar in the form of a logic of concatenation. But natural language is characterized by dependencies which may also be discontinuous. In this paper we introduce the displacement…
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on…
The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of…
A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a…