Search results for "DIFFERENTIAL CALCULUS"

A non-local model of fractional heat conduction in rigid bodies

2011

In recent years several applications of fractional differential calculus have been proposed in physics, chemistry as well as in engineering fields. Fractional order integrals and derivatives extend the well-known definitions of integer-order primitives and derivatives of the ordinary differential calculus to real-order operators. Engineering applications of fractional operators spread from viscoelastic models, stochastic dynamics as well as with thermoelasticity. In this latter field one of the main actractives of fractional operators is their capability to interpolate between the heat flux and its time-rate of change, that is related to the well-known second sound effect. In other recent s…

Stochastic dynamics of nonlinear systems driven by non-normal delta-correlated processes

1993

In this paper, nonlinear systems subjected to external and parametric non-normal delta-correlated stochastic excitations are treated. A new interpretation of the stochastic differential calculus allows first a full explanation of the presence of the Wong-Zakai or Stratonovich correction terms in the Itoˆ’s differential rule. Then this rule is extended to take into account the non-normality of the input. The validity of this formulation is confirmed by experimental results obtained by Monte Carlo simulations.

Direct evaluation of jumps for nonlinear systems under external and multiplicative impulses

2015

In this paper the problem of the response evaluation of nonlinear systems under multiplicative impulsive input is treated. Such systems exhibit a jump at each impulse occurrence, whose value cannot be predicted through the classical differential calculus. In this context here the correct jump evaluation of nonlinear systems is obtained in closed form for two general classes of nonlinear multiplicative functions. Analysis has been performed to show the different typical behaviors of the response, which in some cases could diverge or converge to zero instantaneously, depending on the amplitude of the Dirac's delta.

Stochastic response of linear and non-linear systems to α-stable Lévy white noises

2005

Abstract The stochastic response of linear and non-linear systems to external α -stable Levy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed. Approximate CF solutions to the …

Heat Flow on Metric Measure Spaces

2020

In order to develop a second-order differential calculus on spaces with curvature bounds we need to make use of the regularising effects of the heat flow, to which this chapter is dedicated.

Differential calculus on 'non-standard' (h-deformed) Minkowski spaces

1997

Rectifiability of RCD(K,N) spaces via δ-splitting maps

2021

In this note we give simplified proofs of rectifiability of RCD(K,N) spaces as metric measure spaces and lower semicontinuity of the essential dimension, via -splitting maps. The arguments are inspired by the Cheeger-Colding theory for Ricci limits and rely on the second order differential calculus developed by Gigli and on the convergence and stability results by Ambrosio-Honda. peerReviewed

Path Integrals in Noncommutative Geometry

2006

Quasi-Continuous Vector Fields on RCD Spaces

2021

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined $\mathfrak {m}$ -a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.

Can (noncommutative) geometry accommodate leptoquarks?

1997

We investigate the geometric interpretation of the Standard Model based on noncommutative geometry. Neglecting the $S_0$-reality symmetry one may introduce leptoquarks into the model. We give a detailed discussion of the consequences (both for the Connes-Lott and the spectral action) and compare the results with physical bounds. Our result is that in either case one contradicts the experimental results.