Related papers: Compendium of vector analysis with applications to…
The paper discusses the relationships between electrical quantities, such as voltages, currents, and frequency, and geometrical ones, namely curvature and torsion. The proposed approach is based on the Frenet frame utilized in differential…
We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are…
We expose the Schr\"odinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schr\"odinger, Pauli and Dirac equations, as well as…
We review the concept of a graded bundle, which is a generalisation of a vector bundle, its linearisation, and a double structure of this kind. We then present applications of these structures in geometric mechanics including systems with…
We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical…
In this note, we provide a important considerations of a familiar topic: the gradient of a vector field. The gradient of a vector field is a common quantity represented in continuum mechanics. However, even for Cartesian coordinate systems,…
Different approaches to the problem of a gauge-invariant operator definition of transverse-momentum dependent parton densities (TMDs) are reviewed and compared from the point of view of their compatibility with the operator definition of…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by…
Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure…
Phase transitions with spontaneous symmetry breaking and vector order parameter are considered in multidimensional theory of general relativity. Covariant equations, describing the gravitational properties of topological defects, are…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems.…
Basic notions of continuous media mechanics are introduced for spaces with affine connections and metrics. The physical interpretation of the notion of relative velocity is discussed. The notions of deformation velocity tensor, shear…
A continuum model to study the influence of dislocations on the electronic properties of condensed matter systems is described and analyzed. The model is based on a geometrical formalism that associates a density of dislocations with the…
This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and…
A new approach to quantum mechanics based on independence of the Continuum Hypothesis is proposed. In one-dimensional case, it is shown that the properties of the set of intermediate cardinality coincide with quantum phenomenology.
Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and…
The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed).…
In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…