Free nilpotent and HH-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices

NUMERICAL APPLICATIONS OF THE METHOD OF HURWITZ-RADON MATRICES

Computer sciences need suitable methods for numerical calculations of interpolation, extrapolation, quadrature, derivative and solution of nonlinear equation. Classical methods, based on polynomial interpolation, have some negative features: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomial considerably, also the Runge‘s phenomenon cannot be forgotten. To deal with numerical interpolation, extrapolation, integration and differentiation dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skewsymmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz- Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical differentiation. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the function point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes, interpolation of L points of the curve is connected with the computational cost of rank O(L), MHR interpolation is not a linear interpolation

NUMERICAL DIFFERENTIATION VIA THE INTERPOLATION METHOD OF HURWITZ-RADON MATRICES

Mathematics and computer science need suitable method for numerical calculation of derivative. Classical methods, based on polynomial interpolation, have some negative features: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomial considerably, also the Runge’s phenomenon cannot be forgotten. To deal with numerical interpolation and differentiation dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical differentiation. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the function point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes, interpolation of L points of the curve is connected with the computational cost of rank O(L), MHR interpolation is not a linear interpolation

Machine Learning

Machine Learning can be defined in various ways related to a scientific domain concerned with the design and development of theoretical and implementation tools that allow building systems with some Human Like intelligent behavior. Machine learning addresses more specifically the ability to improve automatically through experience

Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability

Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies

Aerial Drone-based System for Wildfire Monitoring and Suppression

Wildfire, also known as forest fire or bushfire, being an uncontrolled fire crossing an area of combustible vegetation, has become an inherent natural feature of the landscape in many regions of the world. From local to global scales, wildfire has caused substantial social, economic and environmental consequences. Given the hazardous nature of wildfire, developing automated and safe means to monitor and fight the wildfire is of special interest. Unmanned aerial vehicles (UAVs), equipped with appropriate sensors and fire retardants, are available to remotely monitor and fight the area undergoing wildfires, thus helping fire brigades in mitigating the influence of wildfires. This thesis is dedicated to utilizing UAVs to provide automated surveillance, tracking and fire suppression services on an active wildfire event. Considering the requirement of collecting the latest information of a region prone to wildfires, we presented a strategy to deploy the estimated minimum number of UAVs over the target space with nonuniform importance, such that they can persistently monitor the target space to provide a complete area coverage whilst keeping a desired frequency of visits to areas of interest within a predefined time period. Considering the existence of occlusions on partial segments of the sensed wildfire boundary, we processed both contour and flame surface features of wildfires with a proposed numerical algorithm to quickly estimate the occluded wildfire boundary. To provide real-time situational awareness of the propagated wildfire boundary, according to the prior knowledge of the whole wildfire boundary is available or not, we used the principle of vector field to design a model-based guidance law and a model-free guidance law. The former is derived from the radial basis function approximated wildfire boundary while the later is based on the distance between the UAV and the sensed wildfire boundary. Both vector field based guidance laws can drive the UAV to converge to and patrol along the dynamic wildfire boundary. To effectively mitigate the impacts of wildfires, we analyzed the advancement based activeness of the wildfire boundary with a signal prominence based algorithm, and designed a preferential firefighting strategy to guide the UAV to suppress fires along the highly active segments of the wildfire boundary

Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.Comment: 88 pages, 8 figure