Related papers: Beyond Linear Algebra
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
In practice, most of the optimization problems are non-linear requiring certain interactive solutions and approaches to model. In 5G Advanced and Beyond network slicing, mathematically modeling the users, type of service distributions and…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm…
Decision making algorithms are used in a multitude of different applications. Conventional approaches for designing decision algorithms employ principled and simplified modelling, based on which one can determine decisions via tractable…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…
Countless applications cast their computational core in terms of dense linear algebra operations. These operations can usually be implemented by combining the routines offered by standard linear algebra libraries such as BLAS and LAPACK,…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from…
Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing…
The paper presents the aspect of use of modern graphics accelerators supporting CUDA technology for high-performance computing in the field of linear algebra. Fully programmable graphic cards have been available for several years for both…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
In these lectures we briefly cover some of the main lines of research in particle physics beyond the Standard Model.
Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
In this note we aim at putting more emphasis on the fact that trying to solve non-convex optimization problems with coordinate-descent iterative linear matrix inequality algorithms leads to suboptimal solutions, and put forward other…
The principal innovative idea in this paper is to transform the original complex nonlinear modeling problem into a combination of linear problem and very simple nonlinear problems. The key step is the generalized linearization of nonlinear…
Bases, mappings, projections and metrics, natural for Neural network training, are introduced. Graph-theoretical interpretation is offered. Non-Gaussianity naturally emerges, even in relatively simple datasets. Training statistics,…