Vassiliev measures of complexity of open and closed curves in 3-space

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

Solution of structural mechanic’s problems by machine learning

This article proposes analysis procedure of structural mechanic’s problem as integral formulation. The analysis procedure was proposed as stressed-based analysis procedure as before plying the procedure, it is required to define stress distribution within the structural body by proper modelling and structural idealization assumptions. The methodology can suitable be applied for finding the solution of engineering applications with required accuracy. The methodology exploits the unfolded part of the structural analysis problems which were not so easy to solve such as geometric and material nonlinearity together with simple integration technique [11]. It has already unfolded the misery of physically exploiting plastic behaviour structures before the start of fracture of elastic materials [13]. The formulation is integral formulation rather than differential formulation in which whole stress –strain behaviour is utilised in the analysis procedure by using neural network as regression tool. In this article, one dimensional problem of uniaxial bar and plane strain axis symmetric problem of cylinder subjected to internal pressure is solved. The results are compared with the classical differential formulation or linear theory.

Techniques for adaptive metamodelling of propeller arrays far-field noise

The fast development of Urban-Air-Mobility as well as the constant growth of the air transport have made the acoustic pollution abatement a crucial requirement for the aviation industries in order to comply with the increasingly demanding constraints for the community acceptance. The aeroacoustic characterization of arrays of electrically-powered propellers is one of the most challenging issues. The vast majority of the UAM concepts under development adopt propulsion systems based on multiple propellers, for which reliable and cost-efficient aeroacoustic models are still lacking. The present paper proposes the development of surrogate models for the description of acoustic emission of multi-propeller configurations. The numerical investigation focuses on surrogate models able to take into account the effects of the propeller blade geometry (e.g., chord and twist distributions) and global propeller-array geometric parameters (e.g., propellers clearance) on acoustic performances of the whole system. An innovative Artificial Neural Network adaptive metamodelling technique is applied on a numerical database obtained through a boundary integral formulation for the solution of incompressible potential flows around lifting/thrusting bodies, followed by the application of the Farassat 1A boundary integral formulation for the noise field evaluation.

Path-integral formulation of spreading processes in complex networks

We introduce a path-integral formulation of network-based measures that generalize the concept of geodesic distance and that provides fundamental insights into the dynamics of disease transmission as well as an efficient numerical estimation of the infection arrival time.