相关论文: Virtual Calculus - Part I

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…

Differential Linear Logic enriches Linear Logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for Differential Linear Logic…

We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of…

We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually…

This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular…

We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit…

The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinite-dimensional Lie groups and dynamical systems.

In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational…

We study first-order expansions of the reals which do not define the set of natural numbers. We also show that several stronger notions of tameness are equivalent to each others.

Vop\v{e}nka's Alternative Set Theory has been considered as a framework for modelling vague notions. This paper takes feasibility, pertaining to numbers as per some of Yessenin-Volpin's work, and tries to assess how this notion could be…

We obtain complete classification of in-equivalent realizations of the Virasoro algebra by Lie vector fields over the three-dimensional field of real numbers. As an application we construct new classes of nonlinear second-order partial…

Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…

The apparent impossibility of extending non-relativistic quantum mechanics to a relativistic quantum theory is shown to be due to the insufficient structural richness of the field of complex numbers over which quantum mechanics is built. A…

Calculations on atomistic scale are necessary for understanding of physical phenomena occurring during advanced processing of liquids, slurries, and nano-ceramics composite materials. This paper describes some new ideas for using the…

Category theory provides an alternative to Hilbert's Formal Axiomatic method and goes beyond Mathematical Structuralism

Distributed representations (such as those based on embeddings) and discrete representations (such as those based on logic) have complementary strengths. We explore one possible approach to combining these two kinds of representations. We…

We analyse the family of $C^1$-Virtual Elements introduced in \cite{Brezzi:Marini:plates} for fourth-order problems and prove optimal estimates in $L^2$ and in $H^1$ via classical duality arguments.

This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…

This note is intended to be a friendly introduction to virtual classes. We review virtual classes and we give a number of properties and applications. We also include a new virtual push-forward theorem and many computations of virtual…